# Properties

 Label 1200.3.l.a.401.1 Level $1200$ Weight $3$ Character 1200.401 Self dual yes Analytic conductor $32.698$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(401,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.l (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 300) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 401.1 Character $$\chi$$ $$=$$ 1200.4

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -13.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -13.0000 q^{7} +9.00000 q^{9} -23.0000 q^{13} -11.0000 q^{19} +39.0000 q^{21} -27.0000 q^{27} -59.0000 q^{31} -26.0000 q^{37} +69.0000 q^{39} +83.0000 q^{43} +120.000 q^{49} +33.0000 q^{57} -121.000 q^{61} -117.000 q^{63} -13.0000 q^{67} +46.0000 q^{73} +142.000 q^{79} +81.0000 q^{81} +299.000 q^{91} +177.000 q^{93} -167.000 q^{97} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −1.00000
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −13.0000 −1.85714 −0.928571 0.371154i $$-0.878962\pi$$
−0.928571 + 0.371154i $$0.878962\pi$$
$$8$$ 0 0
$$9$$ 9.00000 1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ −23.0000 −1.76923 −0.884615 0.466321i $$-0.845579\pi$$
−0.884615 + 0.466321i $$0.845579\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ −11.0000 −0.578947 −0.289474 0.957186i $$-0.593480\pi$$
−0.289474 + 0.957186i $$0.593480\pi$$
$$20$$ 0 0
$$21$$ 39.0000 1.85714
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −27.0000 −1.00000
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −59.0000 −1.90323 −0.951613 0.307299i $$-0.900575\pi$$
−0.951613 + 0.307299i $$0.900575\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −26.0000 −0.702703 −0.351351 0.936244i $$-0.614278\pi$$
−0.351351 + 0.936244i $$0.614278\pi$$
$$38$$ 0 0
$$39$$ 69.0000 1.76923
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 83.0000 1.93023 0.965116 0.261822i $$-0.0843232\pi$$
0.965116 + 0.261822i $$0.0843232\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 120.000 2.44898
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 33.0000 0.578947
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ −121.000 −1.98361 −0.991803 0.127774i $$-0.959217\pi$$
−0.991803 + 0.127774i $$0.959217\pi$$
$$62$$ 0 0
$$63$$ −117.000 −1.85714
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −13.0000 −0.194030 −0.0970149 0.995283i $$-0.530929\pi$$
−0.0970149 + 0.995283i $$0.530929\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 46.0000 0.630137 0.315068 0.949069i $$-0.397973\pi$$
0.315068 + 0.949069i $$0.397973\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 142.000 1.79747 0.898734 0.438494i $$-0.144488\pi$$
0.898734 + 0.438494i $$0.144488\pi$$
$$80$$ 0 0
$$81$$ 81.0000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 299.000 3.28571
$$92$$ 0 0
$$93$$ 177.000 1.90323
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −167.000 −1.72165 −0.860825 0.508902i $$-0.830052\pi$$
−0.860825 + 0.508902i $$0.830052\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 194.000 1.88350 0.941748 0.336321i $$-0.109183\pi$$
0.941748 + 0.336321i $$0.109183\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 71.0000 0.651376 0.325688 0.945477i $$-0.394404\pi$$
0.325688 + 0.945477i $$0.394404\pi$$
$$110$$ 0 0
$$111$$ 78.0000 0.702703
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −207.000 −1.76923
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 121.000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 146.000 1.14961 0.574803 0.818292i $$-0.305079\pi$$
0.574803 + 0.818292i $$0.305079\pi$$
$$128$$ 0 0
$$129$$ −249.000 −1.93023
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 143.000 1.07519
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 22.0000 0.158273 0.0791367 0.996864i $$-0.474784\pi$$
0.0791367 + 0.996864i $$0.474784\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −360.000 −2.44898
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ −59.0000 −0.390728 −0.195364 0.980731i $$-0.562589\pi$$
−0.195364 + 0.980731i $$0.562589\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 193.000 1.22930 0.614650 0.788800i $$-0.289297\pi$$
0.614650 + 0.788800i $$0.289297\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −37.0000 −0.226994 −0.113497 0.993538i $$-0.536205\pi$$
−0.113497 + 0.993538i $$0.536205\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 360.000 2.13018
$$170$$ 0 0
$$171$$ −99.0000 −0.578947
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ −1.00000 −0.00552486 −0.00276243 0.999996i $$-0.500879\pi$$
−0.00276243 + 0.999996i $$0.500879\pi$$
$$182$$ 0 0
$$183$$ 363.000 1.98361
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 351.000 1.85714
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ −143.000 −0.740933 −0.370466 0.928846i $$-0.620802\pi$$
−0.370466 + 0.928846i $$0.620802\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 109.000 0.547739 0.273869 0.961767i $$-0.411696\pi$$
0.273869 + 0.961767i $$0.411696\pi$$
$$200$$ 0 0
$$201$$ 39.0000 0.194030
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −419.000 −1.98578 −0.992891 0.119027i $$-0.962022\pi$$
−0.992891 + 0.119027i $$0.962022\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 767.000 3.53456
$$218$$ 0 0
$$219$$ −138.000 −0.630137
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 83.0000 0.372197 0.186099 0.982531i $$-0.440416\pi$$
0.186099 + 0.982531i $$0.440416\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ −409.000 −1.78603 −0.893013 0.450031i $$-0.851413\pi$$
−0.893013 + 0.450031i $$0.851413\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −426.000 −1.79747
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 479.000 1.98755 0.993776 0.111397i $$-0.0355327\pi$$
0.993776 + 0.111397i $$0.0355327\pi$$
$$242$$ 0 0
$$243$$ −243.000 −1.00000
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 253.000 1.02429
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 338.000 1.30502
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ −242.000 −0.892989 −0.446494 0.894786i $$-0.647328\pi$$
−0.446494 + 0.894786i $$0.647328\pi$$
$$272$$ 0 0
$$273$$ −897.000 −3.28571
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −407.000 −1.46931 −0.734657 0.678439i $$-0.762657\pi$$
−0.734657 + 0.678439i $$0.762657\pi$$
$$278$$ 0 0
$$279$$ −531.000 −1.90323
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ −517.000 −1.82686 −0.913428 0.407001i $$-0.866574\pi$$
−0.913428 + 0.407001i $$0.866574\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 289.000 1.00000
$$290$$ 0 0
$$291$$ 501.000 1.72165
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1079.00 −3.58472
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −253.000 −0.824104 −0.412052 0.911160i $$-0.635188\pi$$
−0.412052 + 0.911160i $$0.635188\pi$$
$$308$$ 0 0
$$309$$ −582.000 −1.88350
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 457.000 1.46006 0.730032 0.683413i $$-0.239505\pi$$
0.730032 + 0.683413i $$0.239505\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −213.000 −0.651376
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −362.000 −1.09366 −0.546828 0.837245i $$-0.684165\pi$$
−0.546828 + 0.837245i $$0.684165\pi$$
$$332$$ 0 0
$$333$$ −234.000 −0.702703
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −167.000 −0.495549 −0.247774 0.968818i $$-0.579699\pi$$
−0.247774 + 0.968818i $$0.579699\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −923.000 −2.69096
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ −502.000 −1.43840 −0.719198 0.694805i $$-0.755490\pi$$
−0.719198 + 0.694805i $$0.755490\pi$$
$$350$$ 0 0
$$351$$ 621.000 1.76923
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ −240.000 −0.664820
$$362$$ 0 0
$$363$$ −363.000 −1.00000
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 227.000 0.618529 0.309264 0.950976i $$-0.399917\pi$$
0.309264 + 0.950976i $$0.399917\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 577.000 1.54692 0.773458 0.633847i $$-0.218525\pi$$
0.773458 + 0.633847i $$0.218525\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −611.000 −1.61214 −0.806069 0.591822i $$-0.798409\pi$$
−0.806069 + 0.591822i $$0.798409\pi$$
$$380$$ 0 0
$$381$$ −438.000 −1.14961
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 747.000 1.93023
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 793.000 1.99748 0.998741 0.0501728i $$-0.0159772\pi$$
0.998741 + 0.0501728i $$0.0159772\pi$$
$$398$$ 0 0
$$399$$ −429.000 −1.07519
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 1357.00 3.36725
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −769.000 −1.88020 −0.940098 0.340905i $$-0.889267\pi$$
−0.940098 + 0.340905i $$0.889267\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −66.0000 −0.158273
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ −358.000 −0.850356 −0.425178 0.905110i $$-0.639789\pi$$
−0.425178 + 0.905110i $$0.639789\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1573.00 3.68384
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ −503.000 −1.16166 −0.580831 0.814024i $$-0.697272\pi$$
−0.580831 + 0.814024i $$0.697272\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 709.000 1.61503 0.807517 0.589844i $$-0.200811\pi$$
0.807517 + 0.589844i $$0.200811\pi$$
$$440$$ 0 0
$$441$$ 1080.00 2.44898
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 177.000 0.390728
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 814.000 1.78118 0.890591 0.454805i $$-0.150291\pi$$
0.890591 + 0.454805i $$0.150291\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ −526.000 −1.13607 −0.568035 0.823005i $$-0.692296\pi$$
−0.568035 + 0.823005i $$0.692296\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 169.000 0.360341
$$470$$ 0 0
$$471$$ −579.000 −1.22930
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 598.000 1.24324
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −613.000 −1.25873 −0.629363 0.777111i $$-0.716684\pi$$
−0.629363 + 0.777111i $$0.716684\pi$$
$$488$$ 0 0
$$489$$ 111.000 0.226994
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −851.000 −1.70541 −0.852705 0.522392i $$-0.825040\pi$$
−0.852705 + 0.522392i $$0.825040\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1080.00 −2.13018
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ −598.000 −1.17025
$$512$$ 0 0
$$513$$ 297.000 0.578947
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 803.000 1.53537 0.767686 0.640826i $$-0.221408\pi$$
0.767686 + 0.640826i $$0.221408\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 529.000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −241.000 −0.445471 −0.222736 0.974879i $$-0.571499\pi$$
−0.222736 + 0.974879i $$0.571499\pi$$
$$542$$ 0 0
$$543$$ 3.00000 0.00552486
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 506.000 0.925046 0.462523 0.886607i $$-0.346944\pi$$
0.462523 + 0.886607i $$0.346944\pi$$
$$548$$ 0 0
$$549$$ −1089.00 −1.98361
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −1846.00 −3.33816
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ −1909.00 −3.41503
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1053.00 −1.85714
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 181.000 0.316988 0.158494 0.987360i $$-0.449336\pi$$
0.158494 + 0.987360i $$0.449336\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1033.00 1.79029 0.895147 0.445770i $$-0.147070\pi$$
0.895147 + 0.445770i $$0.147070\pi$$
$$578$$ 0 0
$$579$$ 429.000 0.740933
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ 649.000 1.10187
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −327.000 −0.547739
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 1199.00 1.99501 0.997504 0.0706077i $$-0.0224939\pi$$
0.997504 + 0.0706077i $$0.0224939\pi$$
$$602$$ 0 0
$$603$$ −117.000 −0.194030
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −814.000 −1.34102 −0.670511 0.741900i $$-0.733925\pi$$
−0.670511 + 0.741900i $$0.733925\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 1126.00 1.83687 0.918434 0.395574i $$-0.129454\pi$$
0.918434 + 0.395574i $$0.129454\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 949.000 1.53312 0.766559 0.642174i $$-0.221967\pi$$
0.766559 + 0.642174i $$0.221967\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 1261.00 1.99842 0.999208 0.0398015i $$-0.0126726\pi$$
0.999208 + 0.0398015i $$0.0126726\pi$$
$$632$$ 0 0
$$633$$ 1257.00 1.98578
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2760.00 −4.33281
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 314.000 0.488336 0.244168 0.969733i $$-0.421485\pi$$
0.244168 + 0.969733i $$0.421485\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −2301.00 −3.53456
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 414.000 0.630137
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 122.000 0.184569 0.0922844 0.995733i $$-0.470583\pi$$
0.0922844 + 0.995733i $$0.470583\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −249.000 −0.372197
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1154.00 −1.71471 −0.857355 0.514725i $$-0.827894\pi$$
−0.857355 + 0.514725i $$0.827894\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 2171.00 3.19735
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 1227.00 1.78603
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1318.00 1.90738 0.953690 0.300790i $$-0.0972504\pi$$
0.953690 + 0.300790i $$0.0972504\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 286.000 0.406828
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1391.00 1.96192 0.980959 0.194214i $$-0.0622158\pi$$
0.980959 + 0.194214i $$0.0622158\pi$$
$$710$$ 0 0
$$711$$ 1278.00 1.79747
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ −2522.00 −3.49792
$$722$$ 0 0
$$723$$ −1437.00 −1.98755
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 947.000 1.30261 0.651307 0.758815i $$-0.274221\pi$$
0.651307 + 0.758815i $$0.274221\pi$$
$$728$$ 0 0
$$729$$ 729.000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −1034.00 −1.41064 −0.705321 0.708888i $$-0.749197\pi$$
−0.705321 + 0.708888i $$0.749197\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 1222.00 1.65359 0.826793 0.562506i $$-0.190163\pi$$
0.826793 + 0.562506i $$0.190163\pi$$
$$740$$ 0 0
$$741$$ −759.000 −1.02429
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1202.00 −1.60053 −0.800266 0.599645i $$-0.795309\pi$$
−0.800266 + 0.599645i $$0.795309\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 673.000 0.889036 0.444518 0.895770i $$-0.353375\pi$$
0.444518 + 0.895770i $$0.353375\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ −923.000 −1.20970
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 671.000 0.872562 0.436281 0.899811i $$-0.356295\pi$$
0.436281 + 0.899811i $$0.356295\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −1014.00 −1.30502
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −613.000 −0.778907 −0.389454 0.921046i $$-0.627336\pi$$
−0.389454 + 0.921046i $$0.627336\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 2783.00 3.50946
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 1261.00 1.55487 0.777435 0.628963i $$-0.216520\pi$$
0.777435 + 0.628963i $$0.216520\pi$$
$$812$$ 0 0
$$813$$ 726.000 0.892989
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −913.000 −1.11750
$$818$$ 0 0
$$819$$ 2691.00 3.28571
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ 563.000 0.684083 0.342041 0.939685i $$-0.388882\pi$$
0.342041 + 0.939685i $$0.388882\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ 458.000 0.552473 0.276236 0.961090i $$-0.410913\pi$$
0.276236 + 0.961090i $$0.410913\pi$$
$$830$$ 0 0
$$831$$ 1221.00 1.46931
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1593.00 1.90323
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 841.000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1573.00 −1.85714
$$848$$ 0 0
$$849$$ 1551.00 1.82686
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 1177.00 1.37984 0.689918 0.723888i $$-0.257647\pi$$
0.689918 + 0.723888i $$0.257647\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ −1418.00 −1.65076 −0.825378 0.564580i $$-0.809038\pi$$
−0.825378 + 0.564580i $$0.809038\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −867.000 −1.00000
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 299.000 0.343284
$$872$$ 0 0
$$873$$ −1503.00 −1.72165
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1727.00 −1.96921 −0.984607 0.174785i $$-0.944077\pi$$
−0.984607 + 0.174785i $$0.944077\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 443.000 0.501699 0.250849 0.968026i $$-0.419290\pi$$
0.250849 + 0.968026i $$0.419290\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ −1898.00 −2.13498
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 3237.00 3.58472
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −214.000 −0.235943 −0.117971 0.993017i $$-0.537639\pi$$
−0.117971 + 0.993017i $$0.537639\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −971.000 −1.05658 −0.528292 0.849063i $$-0.677167\pi$$
−0.528292 + 0.849063i $$0.677167\pi$$
$$920$$ 0 0
$$921$$ 759.000 0.824104
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1746.00 1.88350
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ −1320.00 −1.41783
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1847.00 −1.97118 −0.985592 0.169138i $$-0.945902\pi$$
−0.985592 + 0.169138i $$0.945902\pi$$
$$938$$ 0 0
$$939$$ −1371.00 −1.46006
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ −1058.00 −1.11486
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 2520.00 2.62227
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1534.00 −1.58635 −0.793175 0.608994i $$-0.791573\pi$$
−0.793175 + 0.608994i $$0.791573\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 0 0
$$973$$ −286.000 −0.293936
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 639.000 0.651376
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −1739.00 −1.75479 −0.877397 0.479766i $$-0.840722\pi$$
−0.877397 + 0.479766i $$0.840722\pi$$
$$992$$ 0 0
$$993$$ 1086.00 1.09366
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1894.00 1.89970 0.949850 0.312707i $$-0.101236\pi$$
0.949850 + 0.312707i $$0.101236\pi$$
$$998$$ 0 0
$$999$$ 702.000 0.702703
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1200.3.l.a.401.1 1
3.2 odd 2 CM 1200.3.l.a.401.1 1
4.3 odd 2 300.3.g.c.101.1 yes 1
5.2 odd 4 1200.3.c.b.449.2 2
5.3 odd 4 1200.3.c.b.449.1 2
5.4 even 2 1200.3.l.e.401.1 1
12.11 even 2 300.3.g.c.101.1 yes 1
15.2 even 4 1200.3.c.b.449.2 2
15.8 even 4 1200.3.c.b.449.1 2
15.14 odd 2 1200.3.l.e.401.1 1
20.3 even 4 300.3.b.b.149.2 2
20.7 even 4 300.3.b.b.149.1 2
20.19 odd 2 300.3.g.a.101.1 1
60.23 odd 4 300.3.b.b.149.2 2
60.47 odd 4 300.3.b.b.149.1 2
60.59 even 2 300.3.g.a.101.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
300.3.b.b.149.1 2 20.7 even 4
300.3.b.b.149.1 2 60.47 odd 4
300.3.b.b.149.2 2 20.3 even 4
300.3.b.b.149.2 2 60.23 odd 4
300.3.g.a.101.1 1 20.19 odd 2
300.3.g.a.101.1 1 60.59 even 2
300.3.g.c.101.1 yes 1 4.3 odd 2
300.3.g.c.101.1 yes 1 12.11 even 2
1200.3.c.b.449.1 2 5.3 odd 4
1200.3.c.b.449.1 2 15.8 even 4
1200.3.c.b.449.2 2 5.2 odd 4
1200.3.c.b.449.2 2 15.2 even 4
1200.3.l.a.401.1 1 1.1 even 1 trivial
1200.3.l.a.401.1 1 3.2 odd 2 CM
1200.3.l.e.401.1 1 5.4 even 2
1200.3.l.e.401.1 1 15.14 odd 2