# Properties

 Label 1170.2.b.d.181.4 Level $1170$ Weight $2$ Character 1170.181 Analytic conductor $9.342$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(181,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1170.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.b (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: $$9.34249703649$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 181.4 Root $$2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 1170.181 Dual form 1170.2.b.d.181.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +4.60555i q^{7} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +4.60555i q^{7} -1.00000i q^{8} -1.00000 q^{10} +3.60555 q^{13} -4.60555 q^{14} +1.00000 q^{16} -4.60555 q^{17} +4.60555i q^{19} -1.00000i q^{20} +1.39445 q^{23} -1.00000 q^{25} +3.60555i q^{26} -4.60555i q^{28} -4.60555 q^{29} -6.00000i q^{31} +1.00000i q^{32} -4.60555i q^{34} -4.60555 q^{35} +9.21110i q^{37} -4.60555 q^{38} +1.00000 q^{40} +3.21110i q^{41} +8.00000 q^{43} +1.39445i q^{46} -9.21110i q^{47} -14.2111 q^{49} -1.00000i q^{50} -3.60555 q^{52} -6.00000 q^{53} +4.60555 q^{56} -4.60555i q^{58} -9.21110i q^{59} -11.2111 q^{61} +6.00000 q^{62} -1.00000 q^{64} +3.60555i q^{65} +3.21110i q^{67} +4.60555 q^{68} -4.60555i q^{70} +9.21110i q^{71} +1.39445i q^{73} -9.21110 q^{74} -4.60555i q^{76} -14.4222 q^{79} +1.00000i q^{80} -3.21110 q^{82} +2.78890i q^{83} -4.60555i q^{85} +8.00000i q^{86} +15.2111i q^{89} +16.6056i q^{91} -1.39445 q^{92} +9.21110 q^{94} -4.60555 q^{95} -1.39445i q^{97} -14.2111i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 4 q^{10} - 4 q^{14} + 4 q^{16} - 4 q^{17} + 20 q^{23} - 4 q^{25} - 4 q^{29} - 4 q^{35} - 4 q^{38} + 4 q^{40} + 32 q^{43} - 28 q^{49} - 24 q^{53} + 4 q^{56} - 16 q^{61} + 24 q^{62} - 4 q^{64} + 4 q^{68} - 8 q^{74} + 16 q^{82} - 20 q^{92} + 8 q^{94} - 4 q^{95}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^10 - 4 * q^14 + 4 * q^16 - 4 * q^17 + 20 * q^23 - 4 * q^25 - 4 * q^29 - 4 * q^35 - 4 * q^38 + 4 * q^40 + 32 * q^43 - 28 * q^49 - 24 * q^53 + 4 * q^56 - 16 * q^61 + 24 * q^62 - 4 * q^64 + 4 * q^68 - 8 * q^74 + 16 * q^82 - 20 * q^92 + 8 * q^94 - 4 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$$.

 $$n$$ $$911$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ 4.60555i 1.74073i 0.492403 + 0.870367i $$0.336119\pi$$
−0.492403 + 0.870367i $$0.663881\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 3.60555 1.00000
$$14$$ −4.60555 −1.23089
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.60555 −1.11701 −0.558505 0.829501i $$-0.688625\pi$$
−0.558505 + 0.829501i $$0.688625\pi$$
$$18$$ 0 0
$$19$$ 4.60555i 1.05659i 0.849062 + 0.528293i $$0.177168\pi$$
−0.849062 + 0.528293i $$0.822832\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.39445 0.290763 0.145381 0.989376i $$-0.453559\pi$$
0.145381 + 0.989376i $$0.453559\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 3.60555i 0.707107i
$$27$$ 0 0
$$28$$ − 4.60555i − 0.870367i
$$29$$ −4.60555 −0.855229 −0.427615 0.903961i $$-0.640646\pi$$
−0.427615 + 0.903961i $$0.640646\pi$$
$$30$$ 0 0
$$31$$ − 6.00000i − 1.07763i −0.842424 0.538816i $$-0.818872\pi$$
0.842424 0.538816i $$-0.181128\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ − 4.60555i − 0.789846i
$$35$$ −4.60555 −0.778480
$$36$$ 0 0
$$37$$ 9.21110i 1.51430i 0.653243 + 0.757148i $$0.273408\pi$$
−0.653243 + 0.757148i $$0.726592\pi$$
$$38$$ −4.60555 −0.747119
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 3.21110i 0.501490i 0.968053 + 0.250745i $$0.0806756\pi$$
−0.968053 + 0.250745i $$0.919324\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 1.39445i 0.205600i
$$47$$ − 9.21110i − 1.34358i −0.740743 0.671789i $$-0.765526\pi$$
0.740743 0.671789i $$-0.234474\pi$$
$$48$$ 0 0
$$49$$ −14.2111 −2.03016
$$50$$ − 1.00000i − 0.141421i
$$51$$ 0 0
$$52$$ −3.60555 −0.500000
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.60555 0.615443
$$57$$ 0 0
$$58$$ − 4.60555i − 0.604739i
$$59$$ − 9.21110i − 1.19918i −0.800306 0.599592i $$-0.795330\pi$$
0.800306 0.599592i $$-0.204670\pi$$
$$60$$ 0 0
$$61$$ −11.2111 −1.43543 −0.717717 0.696335i $$-0.754813\pi$$
−0.717717 + 0.696335i $$0.754813\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 3.60555i 0.447214i
$$66$$ 0 0
$$67$$ 3.21110i 0.392299i 0.980574 + 0.196149i $$0.0628437\pi$$
−0.980574 + 0.196149i $$0.937156\pi$$
$$68$$ 4.60555 0.558505
$$69$$ 0 0
$$70$$ − 4.60555i − 0.550469i
$$71$$ 9.21110i 1.09316i 0.837408 + 0.546578i $$0.184070\pi$$
−0.837408 + 0.546578i $$0.815930\pi$$
$$72$$ 0 0
$$73$$ 1.39445i 0.163208i 0.996665 + 0.0816039i $$0.0260043\pi$$
−0.996665 + 0.0816039i $$0.973996\pi$$
$$74$$ −9.21110 −1.07077
$$75$$ 0 0
$$76$$ − 4.60555i − 0.528293i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −14.4222 −1.62262 −0.811312 0.584613i $$-0.801246\pi$$
−0.811312 + 0.584613i $$0.801246\pi$$
$$80$$ 1.00000i 0.111803i
$$81$$ 0 0
$$82$$ −3.21110 −0.354607
$$83$$ 2.78890i 0.306121i 0.988217 + 0.153061i $$0.0489130\pi$$
−0.988217 + 0.153061i $$0.951087\pi$$
$$84$$ 0 0
$$85$$ − 4.60555i − 0.499542i
$$86$$ 8.00000i 0.862662i
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 15.2111i 1.61237i 0.591661 + 0.806187i $$0.298472\pi$$
−0.591661 + 0.806187i $$0.701528\pi$$
$$90$$ 0 0
$$91$$ 16.6056i 1.74073i
$$92$$ −1.39445 −0.145381
$$93$$ 0 0
$$94$$ 9.21110 0.950053
$$95$$ −4.60555 −0.472520
$$96$$ 0 0
$$97$$ − 1.39445i − 0.141585i −0.997491 0.0707924i $$-0.977447\pi$$
0.997491 0.0707924i $$-0.0225528\pi$$
$$98$$ − 14.2111i − 1.43554i
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 7.39445 0.735775 0.367888 0.929870i $$-0.380081\pi$$
0.367888 + 0.929870i $$0.380081\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ − 3.60555i − 0.353553i
$$105$$ 0 0
$$106$$ − 6.00000i − 0.582772i
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ − 1.39445i − 0.133564i −0.997768 0.0667820i $$-0.978727\pi$$
0.997768 0.0667820i $$-0.0212732\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.60555i 0.435184i
$$113$$ 13.8167 1.29976 0.649881 0.760036i $$-0.274819\pi$$
0.649881 + 0.760036i $$0.274819\pi$$
$$114$$ 0 0
$$115$$ 1.39445i 0.130033i
$$116$$ 4.60555 0.427615
$$117$$ 0 0
$$118$$ 9.21110 0.847951
$$119$$ − 21.2111i − 1.94442i
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ − 11.2111i − 1.01501i
$$123$$ 0 0
$$124$$ 6.00000i 0.538816i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ 1.21110 0.107468 0.0537340 0.998555i $$-0.482888\pi$$
0.0537340 + 0.998555i $$0.482888\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ −3.60555 −0.316228
$$131$$ −22.6056 −1.97506 −0.987528 0.157443i $$-0.949675\pi$$
−0.987528 + 0.157443i $$0.949675\pi$$
$$132$$ 0 0
$$133$$ −21.2111 −1.83924
$$134$$ −3.21110 −0.277397
$$135$$ 0 0
$$136$$ 4.60555i 0.394923i
$$137$$ 3.21110i 0.274343i 0.990547 + 0.137172i $$0.0438011\pi$$
−0.990547 + 0.137172i $$0.956199\pi$$
$$138$$ 0 0
$$139$$ 17.2111 1.45983 0.729913 0.683540i $$-0.239560\pi$$
0.729913 + 0.683540i $$0.239560\pi$$
$$140$$ 4.60555 0.389240
$$141$$ 0 0
$$142$$ −9.21110 −0.772979
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 4.60555i − 0.382470i
$$146$$ −1.39445 −0.115405
$$147$$ 0 0
$$148$$ − 9.21110i − 0.757148i
$$149$$ 15.2111i 1.24614i 0.782165 + 0.623071i $$0.214115\pi$$
−0.782165 + 0.623071i $$0.785885\pi$$
$$150$$ 0 0
$$151$$ − 6.00000i − 0.488273i −0.969741 0.244137i $$-0.921495\pi$$
0.969741 0.244137i $$-0.0785045\pi$$
$$152$$ 4.60555 0.373560
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 20.4222 1.62987 0.814935 0.579553i $$-0.196773\pi$$
0.814935 + 0.579553i $$0.196773\pi$$
$$158$$ − 14.4222i − 1.14737i
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 6.42221i 0.506141i
$$162$$ 0 0
$$163$$ − 24.4222i − 1.91289i −0.291905 0.956447i $$-0.594289\pi$$
0.291905 0.956447i $$-0.405711\pi$$
$$164$$ − 3.21110i − 0.250745i
$$165$$ 0 0
$$166$$ −2.78890 −0.216460
$$167$$ 9.21110i 0.712777i 0.934338 + 0.356388i $$0.115992\pi$$
−0.934338 + 0.356388i $$0.884008\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 4.60555 0.353230
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ 12.4222 0.944443 0.472221 0.881480i $$-0.343452\pi$$
0.472221 + 0.881480i $$0.343452\pi$$
$$174$$ 0 0
$$175$$ − 4.60555i − 0.348147i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −15.2111 −1.14012
$$179$$ 19.8167 1.48117 0.740583 0.671965i $$-0.234549\pi$$
0.740583 + 0.671965i $$0.234549\pi$$
$$180$$ 0 0
$$181$$ 8.42221 0.626018 0.313009 0.949750i $$-0.398663\pi$$
0.313009 + 0.949750i $$0.398663\pi$$
$$182$$ −16.6056 −1.23089
$$183$$ 0 0
$$184$$ − 1.39445i − 0.102800i
$$185$$ −9.21110 −0.677214
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 9.21110i 0.671789i
$$189$$ 0 0
$$190$$ − 4.60555i − 0.334122i
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ − 7.81665i − 0.562655i −0.959612 0.281328i $$-0.909225\pi$$
0.959612 0.281328i $$-0.0907747\pi$$
$$194$$ 1.39445 0.100116
$$195$$ 0 0
$$196$$ 14.2111 1.01508
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 22.4222 1.58947 0.794734 0.606958i $$-0.207610\pi$$
0.794734 + 0.606958i $$0.207610\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 0 0
$$202$$ 7.39445i 0.520272i
$$203$$ − 21.2111i − 1.48873i
$$204$$ 0 0
$$205$$ −3.21110 −0.224273
$$206$$ − 4.00000i − 0.278693i
$$207$$ 0 0
$$208$$ 3.60555 0.250000
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −17.2111 −1.18486 −0.592431 0.805622i $$-0.701832\pi$$
−0.592431 + 0.805622i $$0.701832\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.00000i 0.545595i
$$216$$ 0 0
$$217$$ 27.6333 1.87587
$$218$$ 1.39445 0.0944440
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −16.6056 −1.11701
$$222$$ 0 0
$$223$$ − 1.81665i − 0.121652i −0.998148 0.0608261i $$-0.980627\pi$$
0.998148 0.0608261i $$-0.0193735\pi$$
$$224$$ −4.60555 −0.307721
$$225$$ 0 0
$$226$$ 13.8167i 0.919070i
$$227$$ 24.0000i 1.59294i 0.604681 + 0.796468i $$0.293301\pi$$
−0.604681 + 0.796468i $$0.706699\pi$$
$$228$$ 0 0
$$229$$ 19.8167i 1.30952i 0.755836 + 0.654761i $$0.227231\pi$$
−0.755836 + 0.654761i $$0.772769\pi$$
$$230$$ −1.39445 −0.0919472
$$231$$ 0 0
$$232$$ 4.60555i 0.302369i
$$233$$ 1.81665 0.119013 0.0595065 0.998228i $$-0.481047\pi$$
0.0595065 + 0.998228i $$0.481047\pi$$
$$234$$ 0 0
$$235$$ 9.21110 0.600866
$$236$$ 9.21110i 0.599592i
$$237$$ 0 0
$$238$$ 21.2111 1.37491
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 6.42221i 0.413691i 0.978374 + 0.206845i $$0.0663197\pi$$
−0.978374 + 0.206845i $$0.933680\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ 0 0
$$244$$ 11.2111 0.717717
$$245$$ − 14.2111i − 0.907914i
$$246$$ 0 0
$$247$$ 16.6056i 1.05659i
$$248$$ −6.00000 −0.381000
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 13.3944 0.845450 0.422725 0.906258i $$-0.361074\pi$$
0.422725 + 0.906258i $$0.361074\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 1.21110i 0.0759913i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 28.6056 1.78437 0.892183 0.451675i $$-0.149173\pi$$
0.892183 + 0.451675i $$0.149173\pi$$
$$258$$ 0 0
$$259$$ −42.4222 −2.63599
$$260$$ − 3.60555i − 0.223607i
$$261$$ 0 0
$$262$$ − 22.6056i − 1.39658i
$$263$$ 7.81665 0.481996 0.240998 0.970526i $$-0.422525\pi$$
0.240998 + 0.970526i $$0.422525\pi$$
$$264$$ 0 0
$$265$$ − 6.00000i − 0.368577i
$$266$$ − 21.2111i − 1.30054i
$$267$$ 0 0
$$268$$ − 3.21110i − 0.196149i
$$269$$ −25.8167 −1.57407 −0.787035 0.616909i $$-0.788385\pi$$
−0.787035 + 0.616909i $$0.788385\pi$$
$$270$$ 0 0
$$271$$ 0.422205i 0.0256471i 0.999918 + 0.0128236i $$0.00408198\pi$$
−0.999918 + 0.0128236i $$0.995918\pi$$
$$272$$ −4.60555 −0.279253
$$273$$ 0 0
$$274$$ −3.21110 −0.193990
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −16.4222 −0.986715 −0.493357 0.869827i $$-0.664230\pi$$
−0.493357 + 0.869827i $$0.664230\pi$$
$$278$$ 17.2111i 1.03225i
$$279$$ 0 0
$$280$$ 4.60555i 0.275234i
$$281$$ 27.2111i 1.62328i 0.584159 + 0.811639i $$0.301424\pi$$
−0.584159 + 0.811639i $$0.698576\pi$$
$$282$$ 0 0
$$283$$ 10.4222 0.619536 0.309768 0.950812i $$-0.399749\pi$$
0.309768 + 0.950812i $$0.399749\pi$$
$$284$$ − 9.21110i − 0.546578i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −14.7889 −0.872961
$$288$$ 0 0
$$289$$ 4.21110 0.247712
$$290$$ 4.60555 0.270447
$$291$$ 0 0
$$292$$ − 1.39445i − 0.0816039i
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 0 0
$$295$$ 9.21110 0.536291
$$296$$ 9.21110 0.535384
$$297$$ 0 0
$$298$$ −15.2111 −0.881156
$$299$$ 5.02776 0.290763
$$300$$ 0 0
$$301$$ 36.8444i 2.12368i
$$302$$ 6.00000 0.345261
$$303$$ 0 0
$$304$$ 4.60555i 0.264146i
$$305$$ − 11.2111i − 0.641946i
$$306$$ 0 0
$$307$$ 8.78890i 0.501609i 0.968038 + 0.250804i $$0.0806951\pi$$
−0.968038 + 0.250804i $$0.919305\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 6.00000i 0.340777i
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 3.57779 0.202229 0.101114 0.994875i $$-0.467759\pi$$
0.101114 + 0.994875i $$0.467759\pi$$
$$314$$ 20.4222i 1.15249i
$$315$$ 0 0
$$316$$ 14.4222 0.811312
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ − 1.00000i − 0.0559017i
$$321$$ 0 0
$$322$$ −6.42221 −0.357895
$$323$$ − 21.2111i − 1.18022i
$$324$$ 0 0
$$325$$ −3.60555 −0.200000
$$326$$ 24.4222 1.35262
$$327$$ 0 0
$$328$$ 3.21110 0.177303
$$329$$ 42.4222 2.33881
$$330$$ 0 0
$$331$$ 16.6056i 0.912724i 0.889794 + 0.456362i $$0.150848\pi$$
−0.889794 + 0.456362i $$0.849152\pi$$
$$332$$ − 2.78890i − 0.153061i
$$333$$ 0 0
$$334$$ −9.21110 −0.504009
$$335$$ −3.21110 −0.175441
$$336$$ 0 0
$$337$$ 13.6333 0.742654 0.371327 0.928502i $$-0.378903\pi$$
0.371327 + 0.928502i $$0.378903\pi$$
$$338$$ 13.0000i 0.707107i
$$339$$ 0 0
$$340$$ 4.60555i 0.249771i
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 33.2111i − 1.79323i
$$344$$ − 8.00000i − 0.431331i
$$345$$ 0 0
$$346$$ 12.4222i 0.667822i
$$347$$ 27.6333 1.48343 0.741717 0.670713i $$-0.234012\pi$$
0.741717 + 0.670713i $$0.234012\pi$$
$$348$$ 0 0
$$349$$ 7.81665i 0.418416i 0.977871 + 0.209208i $$0.0670886\pi$$
−0.977871 + 0.209208i $$0.932911\pi$$
$$350$$ 4.60555 0.246177
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 8.78890i − 0.467786i −0.972262 0.233893i $$-0.924853\pi$$
0.972262 0.233893i $$-0.0751465\pi$$
$$354$$ 0 0
$$355$$ −9.21110 −0.488875
$$356$$ − 15.2111i − 0.806187i
$$357$$ 0 0
$$358$$ 19.8167i 1.04734i
$$359$$ 15.6333i 0.825094i 0.910936 + 0.412547i $$0.135361\pi$$
−0.910936 + 0.412547i $$0.864639\pi$$
$$360$$ 0 0
$$361$$ −2.21110 −0.116374
$$362$$ 8.42221i 0.442661i
$$363$$ 0 0
$$364$$ − 16.6056i − 0.870367i
$$365$$ −1.39445 −0.0729888
$$366$$ 0 0
$$367$$ −19.6333 −1.02485 −0.512425 0.858732i $$-0.671253\pi$$
−0.512425 + 0.858732i $$0.671253\pi$$
$$368$$ 1.39445 0.0726907
$$369$$ 0 0
$$370$$ − 9.21110i − 0.478862i
$$371$$ − 27.6333i − 1.43465i
$$372$$ 0 0
$$373$$ −20.4222 −1.05742 −0.528711 0.848802i $$-0.677324\pi$$
−0.528711 + 0.848802i $$0.677324\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −9.21110 −0.475026
$$377$$ −16.6056 −0.855229
$$378$$ 0 0
$$379$$ − 35.0278i − 1.79925i −0.436658 0.899627i $$-0.643838\pi$$
0.436658 0.899627i $$-0.356162\pi$$
$$380$$ 4.60555 0.236260
$$381$$ 0 0
$$382$$ 12.0000i 0.613973i
$$383$$ − 27.6333i − 1.41200i −0.708214 0.705998i $$-0.750499\pi$$
0.708214 0.705998i $$-0.249501\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 7.81665 0.397857
$$387$$ 0 0
$$388$$ 1.39445i 0.0707924i
$$389$$ −4.60555 −0.233511 −0.116755 0.993161i $$-0.537249\pi$$
−0.116755 + 0.993161i $$0.537249\pi$$
$$390$$ 0 0
$$391$$ −6.42221 −0.324785
$$392$$ 14.2111i 0.717769i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ − 14.4222i − 0.725660i
$$396$$ 0 0
$$397$$ 3.63331i 0.182350i 0.995835 + 0.0911752i $$0.0290623\pi$$
−0.995835 + 0.0911752i $$0.970938\pi$$
$$398$$ 22.4222i 1.12392i
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 8.78890i 0.438897i 0.975624 + 0.219448i $$0.0704257\pi$$
−0.975624 + 0.219448i $$0.929574\pi$$
$$402$$ 0 0
$$403$$ − 21.6333i − 1.07763i
$$404$$ −7.39445 −0.367888
$$405$$ 0 0
$$406$$ 21.2111 1.05269
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.7889i 0.731264i 0.930760 + 0.365632i $$0.119147\pi$$
−0.930760 + 0.365632i $$0.880853\pi$$
$$410$$ − 3.21110i − 0.158585i
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 42.4222 2.08746
$$414$$ 0 0
$$415$$ −2.78890 −0.136902
$$416$$ 3.60555i 0.176777i
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4.18335 −0.204370 −0.102185 0.994765i $$-0.532583\pi$$
−0.102185 + 0.994765i $$0.532583\pi$$
$$420$$ 0 0
$$421$$ 19.8167i 0.965805i 0.875674 + 0.482902i $$0.160417\pi$$
−0.875674 + 0.482902i $$0.839583\pi$$
$$422$$ − 17.2111i − 0.837823i
$$423$$ 0 0
$$424$$ 6.00000i 0.291386i
$$425$$ 4.60555 0.223402
$$426$$ 0 0
$$427$$ − 51.6333i − 2.49871i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ 0 0
$$433$$ 19.2111 0.923227 0.461613 0.887081i $$-0.347271\pi$$
0.461613 + 0.887081i $$0.347271\pi$$
$$434$$ 27.6333i 1.32644i
$$435$$ 0 0
$$436$$ 1.39445i 0.0667820i
$$437$$ 6.42221i 0.307216i
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 16.6056i − 0.789846i
$$443$$ −15.6333 −0.742761 −0.371380 0.928481i $$-0.621115\pi$$
−0.371380 + 0.928481i $$0.621115\pi$$
$$444$$ 0 0
$$445$$ −15.2111 −0.721075
$$446$$ 1.81665 0.0860211
$$447$$ 0 0
$$448$$ − 4.60555i − 0.217592i
$$449$$ 33.6333i 1.58725i 0.608405 + 0.793627i $$0.291810\pi$$
−0.608405 + 0.793627i $$0.708190\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −13.8167 −0.649881
$$453$$ 0 0
$$454$$ −24.0000 −1.12638
$$455$$ −16.6056 −0.778480
$$456$$ 0 0
$$457$$ − 38.2389i − 1.78874i −0.447330 0.894369i $$-0.647625\pi$$
0.447330 0.894369i $$-0.352375\pi$$
$$458$$ −19.8167 −0.925971
$$459$$ 0 0
$$460$$ − 1.39445i − 0.0650165i
$$461$$ − 33.6333i − 1.56646i −0.621733 0.783230i $$-0.713571\pi$$
0.621733 0.783230i $$-0.286429\pi$$
$$462$$ 0 0
$$463$$ 31.3944i 1.45902i 0.683968 + 0.729512i $$0.260253\pi$$
−0.683968 + 0.729512i $$0.739747\pi$$
$$464$$ −4.60555 −0.213807
$$465$$ 0 0
$$466$$ 1.81665i 0.0841549i
$$467$$ 30.4222 1.40777 0.703886 0.710313i $$-0.251447\pi$$
0.703886 + 0.710313i $$0.251447\pi$$
$$468$$ 0 0
$$469$$ −14.7889 −0.682888
$$470$$ 9.21110i 0.424876i
$$471$$ 0 0
$$472$$ −9.21110 −0.423975
$$473$$ 0 0
$$474$$ 0 0
$$475$$ − 4.60555i − 0.211317i
$$476$$ 21.2111i 0.972209i
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 5.57779i 0.254856i 0.991848 + 0.127428i $$0.0406722\pi$$
−0.991848 + 0.127428i $$0.959328\pi$$
$$480$$ 0 0
$$481$$ 33.2111i 1.51430i
$$482$$ −6.42221 −0.292523
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 1.39445 0.0633187
$$486$$ 0 0
$$487$$ 0.972244i 0.0440566i 0.999757 + 0.0220283i $$0.00701239\pi$$
−0.999757 + 0.0220283i $$0.992988\pi$$
$$488$$ 11.2111i 0.507503i
$$489$$ 0 0
$$490$$ 14.2111 0.641992
$$491$$ −7.81665 −0.352761 −0.176380 0.984322i $$-0.556439\pi$$
−0.176380 + 0.984322i $$0.556439\pi$$
$$492$$ 0 0
$$493$$ 21.2111 0.955300
$$494$$ −16.6056 −0.747119
$$495$$ 0 0
$$496$$ − 6.00000i − 0.269408i
$$497$$ −42.4222 −1.90290
$$498$$ 0 0
$$499$$ − 23.0278i − 1.03086i −0.856930 0.515432i $$-0.827631\pi$$
0.856930 0.515432i $$-0.172369\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ 0 0
$$502$$ 13.3944i 0.597824i
$$503$$ −23.4500 −1.04558 −0.522791 0.852461i $$-0.675109\pi$$
−0.522791 + 0.852461i $$0.675109\pi$$
$$504$$ 0 0
$$505$$ 7.39445i 0.329049i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −1.21110 −0.0537340
$$509$$ − 33.6333i − 1.49077i −0.666634 0.745385i $$-0.732266\pi$$
0.666634 0.745385i $$-0.267734\pi$$
$$510$$ 0 0
$$511$$ −6.42221 −0.284102
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 28.6056i 1.26174i
$$515$$ − 4.00000i − 0.176261i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ − 42.4222i − 1.86392i
$$519$$ 0 0
$$520$$ 3.60555 0.158114
$$521$$ −21.6333 −0.947772 −0.473886 0.880586i $$-0.657149\pi$$
−0.473886 + 0.880586i $$0.657149\pi$$
$$522$$ 0 0
$$523$$ 32.8444 1.43619 0.718093 0.695947i $$-0.245015\pi$$
0.718093 + 0.695947i $$0.245015\pi$$
$$524$$ 22.6056 0.987528
$$525$$ 0 0
$$526$$ 7.81665i 0.340822i
$$527$$ 27.6333i 1.20373i
$$528$$ 0 0
$$529$$ −21.0555 −0.915457
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ 21.2111 0.919618
$$533$$ 11.5778i 0.501490i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 3.21110 0.138699
$$537$$ 0 0
$$538$$ − 25.8167i − 1.11303i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 6.97224i − 0.299760i −0.988704 0.149880i $$-0.952111\pi$$
0.988704 0.149880i $$-0.0478888\pi$$
$$542$$ −0.422205 −0.0181353
$$543$$ 0 0
$$544$$ − 4.60555i − 0.197461i
$$545$$ 1.39445 0.0597316
$$546$$ 0 0
$$547$$ 14.4222 0.616649 0.308324 0.951281i $$-0.400232\pi$$
0.308324 + 0.951281i $$0.400232\pi$$
$$548$$ − 3.21110i − 0.137172i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 21.2111i − 0.903623i
$$552$$ 0 0
$$553$$ − 66.4222i − 2.82456i
$$554$$ − 16.4222i − 0.697713i
$$555$$ 0 0
$$556$$ −17.2111 −0.729913
$$557$$ 11.5778i 0.490567i 0.969451 + 0.245283i $$0.0788810\pi$$
−0.969451 + 0.245283i $$0.921119\pi$$
$$558$$ 0 0
$$559$$ 28.8444 1.21999
$$560$$ −4.60555 −0.194620
$$561$$ 0 0
$$562$$ −27.2111 −1.14783
$$563$$ −34.0555 −1.43527 −0.717634 0.696420i $$-0.754775\pi$$
−0.717634 + 0.696420i $$0.754775\pi$$
$$564$$ 0 0
$$565$$ 13.8167i 0.581271i
$$566$$ 10.4222i 0.438078i
$$567$$ 0 0
$$568$$ 9.21110 0.386489
$$569$$ −33.6333 −1.40998 −0.704991 0.709216i $$-0.749049\pi$$
−0.704991 + 0.709216i $$0.749049\pi$$
$$570$$ 0 0
$$571$$ −30.0555 −1.25778 −0.628892 0.777493i $$-0.716491\pi$$
−0.628892 + 0.777493i $$0.716491\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ − 14.7889i − 0.617277i
$$575$$ −1.39445 −0.0581525
$$576$$ 0 0
$$577$$ 37.3944i 1.55675i 0.627799 + 0.778376i $$0.283956\pi$$
−0.627799 + 0.778376i $$0.716044\pi$$
$$578$$ 4.21110i 0.175159i
$$579$$ 0 0
$$580$$ 4.60555i 0.191235i
$$581$$ −12.8444 −0.532876
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 1.39445 0.0577027
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ − 6.42221i − 0.265073i −0.991178 0.132536i $$-0.957688\pi$$
0.991178 0.132536i $$-0.0423121\pi$$
$$588$$ 0 0
$$589$$ 27.6333 1.13861
$$590$$ 9.21110i 0.379215i
$$591$$ 0 0
$$592$$ 9.21110i 0.378574i
$$593$$ − 24.4222i − 1.00290i −0.865187 0.501450i $$-0.832800\pi$$
0.865187 0.501450i $$-0.167200\pi$$
$$594$$ 0 0
$$595$$ 21.2111 0.869570
$$596$$ − 15.2111i − 0.623071i
$$597$$ 0 0
$$598$$ 5.02776i 0.205600i
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 1.63331 0.0666240 0.0333120 0.999445i $$-0.489394\pi$$
0.0333120 + 0.999445i $$0.489394\pi$$
$$602$$ −36.8444 −1.50167
$$603$$ 0 0
$$604$$ 6.00000i 0.244137i
$$605$$ 11.0000i 0.447214i
$$606$$ 0 0
$$607$$ 17.2111 0.698577 0.349289 0.937015i $$-0.386423\pi$$
0.349289 + 0.937015i $$0.386423\pi$$
$$608$$ −4.60555 −0.186780
$$609$$ 0 0
$$610$$ 11.2111 0.453924
$$611$$ − 33.2111i − 1.34358i
$$612$$ 0 0
$$613$$ 33.2111i 1.34138i 0.741736 + 0.670692i $$0.234003\pi$$
−0.741736 + 0.670692i $$0.765997\pi$$
$$614$$ −8.78890 −0.354691
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.4222i 0.500099i 0.968233 + 0.250050i $$0.0804469\pi$$
−0.968233 + 0.250050i $$0.919553\pi$$
$$618$$ 0 0
$$619$$ − 25.8167i − 1.03766i −0.854878 0.518829i $$-0.826368\pi$$
0.854878 0.518829i $$-0.173632\pi$$
$$620$$ −6.00000 −0.240966
$$621$$ 0 0
$$622$$ − 12.0000i − 0.481156i
$$623$$ −70.0555 −2.80671
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 3.57779i 0.142997i
$$627$$ 0 0
$$628$$ −20.4222 −0.814935
$$629$$ − 42.4222i − 1.69148i
$$630$$ 0 0
$$631$$ − 3.21110i − 0.127832i −0.997955 0.0639160i $$-0.979641\pi$$
0.997955 0.0639160i $$-0.0203590\pi$$
$$632$$ 14.4222i 0.573685i
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ 1.21110i 0.0480611i
$$636$$ 0 0
$$637$$ −51.2389 −2.03016
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −0.422205 −0.0166761 −0.00833805 0.999965i $$-0.502654\pi$$
−0.00833805 + 0.999965i $$0.502654\pi$$
$$642$$ 0 0
$$643$$ − 9.63331i − 0.379901i −0.981794 0.189950i $$-0.939167\pi$$
0.981794 0.189950i $$-0.0608327\pi$$
$$644$$ − 6.42221i − 0.253070i
$$645$$ 0 0
$$646$$ 21.2111 0.834540
$$647$$ 34.6056 1.36048 0.680242 0.732987i $$-0.261875\pi$$
0.680242 + 0.732987i $$0.261875\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ − 3.60555i − 0.141421i
$$651$$ 0 0
$$652$$ 24.4222i 0.956447i
$$653$$ −39.2111 −1.53445 −0.767225 0.641379i $$-0.778363\pi$$
−0.767225 + 0.641379i $$0.778363\pi$$
$$654$$ 0 0
$$655$$ − 22.6056i − 0.883272i
$$656$$ 3.21110i 0.125372i
$$657$$ 0 0
$$658$$ 42.4222i 1.65379i
$$659$$ 26.2389 1.02212 0.511060 0.859545i $$-0.329253\pi$$
0.511060 + 0.859545i $$0.329253\pi$$
$$660$$ 0 0
$$661$$ − 50.2389i − 1.95407i −0.213090 0.977033i $$-0.568353\pi$$
0.213090 0.977033i $$-0.431647\pi$$
$$662$$ −16.6056 −0.645393
$$663$$ 0 0
$$664$$ 2.78890 0.108230
$$665$$ − 21.2111i − 0.822531i
$$666$$ 0 0
$$667$$ −6.42221 −0.248669
$$668$$ − 9.21110i − 0.356388i
$$669$$ 0 0
$$670$$ − 3.21110i − 0.124056i
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −37.6333 −1.45066 −0.725329 0.688403i $$-0.758312\pi$$
−0.725329 + 0.688403i $$0.758312\pi$$
$$674$$ 13.6333i 0.525135i
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 28.0555 1.07826 0.539130 0.842222i $$-0.318753\pi$$
0.539130 + 0.842222i $$0.318753\pi$$
$$678$$ 0 0
$$679$$ 6.42221 0.246462
$$680$$ −4.60555 −0.176615
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 9.21110i 0.352453i 0.984350 + 0.176227i $$0.0563891\pi$$
−0.984350 + 0.176227i $$0.943611\pi$$
$$684$$ 0 0
$$685$$ −3.21110 −0.122690
$$686$$ 33.2111 1.26801
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ −21.6333 −0.824163
$$690$$ 0 0
$$691$$ − 20.2389i − 0.769922i −0.922933 0.384961i $$-0.874215\pi$$
0.922933 0.384961i $$-0.125785\pi$$
$$692$$ −12.4222 −0.472221
$$693$$ 0 0
$$694$$ 27.6333i 1.04895i
$$695$$ 17.2111i 0.652854i
$$696$$ 0 0
$$697$$ − 14.7889i − 0.560169i
$$698$$ −7.81665 −0.295865
$$699$$ 0 0
$$700$$ 4.60555i 0.174073i
$$701$$ −47.0278 −1.77621 −0.888107 0.459637i $$-0.847980\pi$$
−0.888107 + 0.459637i $$0.847980\pi$$
$$702$$ 0 0
$$703$$ −42.4222 −1.59998
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 8.78890 0.330775
$$707$$ 34.0555i 1.28079i
$$708$$ 0 0
$$709$$ 1.39445i 0.0523696i 0.999657 + 0.0261848i $$0.00833584\pi$$
−0.999657 + 0.0261848i $$0.991664\pi$$
$$710$$ − 9.21110i − 0.345687i
$$711$$ 0 0
$$712$$ 15.2111 0.570060
$$713$$ − 8.36669i − 0.313335i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −19.8167 −0.740583
$$717$$ 0 0
$$718$$ −15.6333 −0.583430
$$719$$ 51.6333 1.92560 0.962799 0.270220i $$-0.0870963\pi$$
0.962799 + 0.270220i $$0.0870963\pi$$
$$720$$ 0 0
$$721$$ − 18.4222i − 0.686079i
$$722$$ − 2.21110i − 0.0822887i
$$723$$ 0 0
$$724$$ −8.42221 −0.313009
$$725$$ 4.60555 0.171046
$$726$$ 0 0
$$727$$ −14.4222 −0.534890 −0.267445 0.963573i $$-0.586179\pi$$
−0.267445 + 0.963573i $$0.586179\pi$$
$$728$$ 16.6056 0.615443
$$729$$ 0 0
$$730$$ − 1.39445i − 0.0516109i
$$731$$ −36.8444 −1.36274
$$732$$ 0 0
$$733$$ 34.0555i 1.25787i 0.777458 + 0.628935i $$0.216509\pi$$
−0.777458 + 0.628935i $$0.783491\pi$$
$$734$$ − 19.6333i − 0.724679i
$$735$$ 0 0
$$736$$ 1.39445i 0.0514001i
$$737$$ 0 0
$$738$$ 0 0
$$739$$ − 20.2389i − 0.744498i −0.928133 0.372249i $$-0.878587\pi$$
0.928133 0.372249i $$-0.121413\pi$$
$$740$$ 9.21110 0.338607
$$741$$ 0 0
$$742$$ 27.6333 1.01445
$$743$$ 36.8444i 1.35169i 0.737044 + 0.675845i $$0.236221\pi$$
−0.737044 + 0.675845i $$0.763779\pi$$
$$744$$ 0 0
$$745$$ −15.2111 −0.557292
$$746$$ − 20.4222i − 0.747710i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.4222 −0.380312 −0.190156 0.981754i $$-0.560899\pi$$
−0.190156 + 0.981754i $$0.560899\pi$$
$$752$$ − 9.21110i − 0.335894i
$$753$$ 0 0
$$754$$ − 16.6056i − 0.604739i
$$755$$ 6.00000 0.218362
$$756$$ 0 0
$$757$$ 12.7889 0.464820 0.232410 0.972618i $$-0.425339\pi$$
0.232410 + 0.972618i $$0.425339\pi$$
$$758$$ 35.0278 1.27227
$$759$$ 0 0
$$760$$ 4.60555i 0.167061i
$$761$$ 33.6333i 1.21921i 0.792707 + 0.609603i $$0.208671\pi$$
−0.792707 + 0.609603i $$0.791329\pi$$
$$762$$ 0 0
$$763$$ 6.42221 0.232499
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 27.6333 0.998432
$$767$$ − 33.2111i − 1.19918i
$$768$$ 0 0
$$769$$ 12.8444i 0.463181i 0.972813 + 0.231591i $$0.0743930\pi$$
−0.972813 + 0.231591i $$0.925607\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 7.81665i 0.281328i
$$773$$ 30.0000i 1.07903i 0.841978 + 0.539513i $$0.181391\pi$$
−0.841978 + 0.539513i $$0.818609\pi$$
$$774$$ 0 0
$$775$$ 6.00000i 0.215526i
$$776$$ −1.39445 −0.0500578
$$777$$ 0 0
$$778$$ − 4.60555i − 0.165117i
$$779$$ −14.7889 −0.529867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 6.42221i − 0.229658i
$$783$$ 0 0
$$784$$ −14.2111 −0.507539
$$785$$ 20.4222i 0.728900i
$$786$$ 0 0
$$787$$ − 49.2666i − 1.75617i −0.478509 0.878083i $$-0.658823\pi$$
0.478509 0.878083i $$-0.341177\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 0 0
$$790$$ 14.4222 0.513119
$$791$$ 63.6333i 2.26254i
$$792$$ 0 0
$$793$$ −40.4222 −1.43543
$$794$$ −3.63331 −0.128941
$$795$$ 0 0
$$796$$ −22.4222 −0.794734
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ 42.4222i 1.50079i
$$800$$ − 1.00000i − 0.0353553i
$$801$$ 0 0
$$802$$ −8.78890 −0.310347
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −6.42221 −0.226353
$$806$$ 21.6333 0.762001
$$807$$ 0 0
$$808$$ − 7.39445i − 0.260136i
$$809$$ 6.84441 0.240637 0.120318 0.992735i $$-0.461608\pi$$
0.120318 + 0.992735i $$0.461608\pi$$
$$810$$ 0 0
$$811$$ − 32.2389i − 1.13206i −0.824385 0.566030i $$-0.808479\pi$$
0.824385 0.566030i $$-0.191521\pi$$
$$812$$ 21.2111i 0.744364i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 24.4222 0.855473
$$816$$ 0 0
$$817$$ 36.8444i 1.28902i
$$818$$ −14.7889 −0.517082
$$819$$ 0 0
$$820$$ 3.21110 0.112137
$$821$$ − 3.21110i − 0.112068i −0.998429 0.0560341i $$-0.982154\pi$$
0.998429 0.0560341i $$-0.0178456\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 4.00000i 0.139347i
$$825$$ 0 0
$$826$$ 42.4222i 1.47606i
$$827$$ − 27.6333i − 0.960904i −0.877021 0.480452i $$-0.840473\pi$$
0.877021 0.480452i $$-0.159527\pi$$
$$828$$ 0 0
$$829$$ 46.8444 1.62697 0.813487 0.581583i $$-0.197567\pi$$
0.813487 + 0.581583i $$0.197567\pi$$
$$830$$ − 2.78890i − 0.0968040i
$$831$$ 0 0
$$832$$ −3.60555 −0.125000
$$833$$ 65.4500 2.26771
$$834$$ 0 0
$$835$$ −9.21110 −0.318763
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 4.18335i − 0.144511i
$$839$$ − 18.4222i − 0.636005i −0.948090 0.318003i $$-0.896988\pi$$
0.948090 0.318003i $$-0.103012\pi$$
$$840$$ 0 0
$$841$$ −7.78890 −0.268583
$$842$$ −19.8167 −0.682927
$$843$$ 0 0
$$844$$ 17.2111 0.592431
$$845$$ 13.0000i 0.447214i
$$846$$ 0 0
$$847$$ 50.6611i 1.74073i
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ 4.60555i 0.157969i
$$851$$ 12.8444i 0.440301i
$$852$$ 0 0
$$853$$ 14.7889i 0.506362i 0.967419 + 0.253181i $$0.0814769\pi$$
−0.967419 + 0.253181i $$0.918523\pi$$
$$854$$ 51.6333 1.76686
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 23.0278 0.786613 0.393307 0.919407i $$-0.371331\pi$$
0.393307 + 0.919407i $$0.371331\pi$$
$$858$$ 0 0
$$859$$ 25.2111 0.860192 0.430096 0.902783i $$-0.358480\pi$$
0.430096 + 0.902783i $$0.358480\pi$$
$$860$$ − 8.00000i − 0.272798i
$$861$$ 0 0
$$862$$ −12.0000 −0.408722
$$863$$ 51.6333i 1.75762i 0.477173 + 0.878809i $$0.341661\pi$$
−0.477173 + 0.878809i $$0.658339\pi$$
$$864$$ 0 0
$$865$$ 12.4222i 0.422368i
$$866$$ 19.2111i 0.652820i
$$867$$ 0 0
$$868$$ −27.6333 −0.937936
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 11.5778i 0.392299i
$$872$$ −1.39445 −0.0472220
$$873$$ 0 0
$$874$$ −6.42221 −0.217234
$$875$$ 4.60555 0.155696
$$876$$ 0 0
$$877$$ 24.8444i 0.838936i 0.907770 + 0.419468i $$0.137783\pi$$
−0.907770 + 0.419468i $$0.862217\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −39.2111 −1.32106 −0.660528 0.750802i $$-0.729667\pi$$
−0.660528 + 0.750802i $$0.729667\pi$$
$$882$$ 0 0
$$883$$ 9.57779 0.322318 0.161159 0.986928i $$-0.448477\pi$$
0.161159 + 0.986928i $$0.448477\pi$$
$$884$$ 16.6056 0.558505
$$885$$ 0 0
$$886$$ − 15.6333i − 0.525211i
$$887$$ −6.97224 −0.234105 −0.117053 0.993126i $$-0.537345\pi$$
−0.117053 + 0.993126i $$0.537345\pi$$
$$888$$ 0 0
$$889$$ 5.57779i 0.187073i
$$890$$ − 15.2111i − 0.509877i
$$891$$ 0 0
$$892$$ 1.81665i 0.0608261i
$$893$$ 42.4222 1.41960
$$894$$ 0 0
$$895$$ 19.8167i 0.662398i
$$896$$ 4.60555 0.153861
$$897$$ 0 0
$$898$$ −33.6333 −1.12236
$$899$$ 27.6333i 0.921622i
$$900$$ 0 0
$$901$$ 27.6333 0.920599
$$902$$ 0 0
$$903$$ 0 0
$$904$$ − 13.8167i − 0.459535i
$$905$$ 8.42221i 0.279964i
$$906$$ 0 0
$$907$$ 21.5778 0.716479 0.358239 0.933630i $$-0.383377\pi$$
0.358239 + 0.933630i $$0.383377\pi$$
$$908$$ − 24.0000i − 0.796468i
$$909$$ 0 0
$$910$$ − 16.6056i − 0.550469i
$$911$$ 27.6333 0.915532 0.457766 0.889073i $$-0.348650\pi$$
0.457766 + 0.889073i $$0.348650\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 38.2389 1.26483
$$915$$ 0 0
$$916$$ − 19.8167i − 0.654761i
$$917$$ − 104.111i − 3.43805i
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 1.39445 0.0459736
$$921$$ 0 0
$$922$$ 33.6333 1.10765
$$923$$ 33.2111i 1.09316i
$$924$$ 0 0
$$925$$ − 9.21110i − 0.302859i
$$926$$ −31.3944 −1.03169
$$927$$ 0 0
$$928$$ − 4.60555i − 0.151185i
$$929$$ 39.2111i 1.28647i 0.765667 + 0.643237i $$0.222409\pi$$
−0.765667 + 0.643237i $$0.777591\pi$$
$$930$$ 0 0
$$931$$ − 65.4500i − 2.14504i
$$932$$ −1.81665 −0.0595065
$$933$$ 0 0
$$934$$ 30.4222i 0.995445i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.3667 −0.338665 −0.169333 0.985559i $$-0.554161\pi$$
−0.169333 + 0.985559i $$0.554161\pi$$
$$938$$ − 14.7889i − 0.482875i
$$939$$ 0 0
$$940$$ −9.21110 −0.300433
$$941$$ − 54.0000i − 1.76035i −0.474650 0.880175i $$-0.657425\pi$$
0.474650 0.880175i $$-0.342575\pi$$
$$942$$ 0 0
$$943$$ 4.47772i 0.145815i
$$944$$ − 9.21110i − 0.299796i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 15.6333i − 0.508014i −0.967202 0.254007i $$-0.918251\pi$$
0.967202 0.254007i $$-0.0817487\pi$$
$$948$$ 0 0
$$949$$ 5.02776i 0.163208i
$$950$$ 4.60555 0.149424
$$951$$ 0 0
$$952$$ −21.2111 −0.687456
$$953$$ 20.2389 0.655601 0.327800 0.944747i $$-0.393693\pi$$
0.327800 + 0.944747i $$0.393693\pi$$
$$954$$ 0 0
$$955$$ 12.0000i 0.388311i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −5.57779 −0.180210
$$959$$ −14.7889 −0.477558
$$960$$ 0 0
$$961$$ −5.00000 −0.161290
$$962$$ −33.2111 −1.07077
$$963$$ 0 0
$$964$$ − 6.42221i − 0.206845i
$$965$$ 7.81665 0.251627
$$966$$ 0 0
$$967$$ 8.23886i 0.264944i 0.991187 + 0.132472i $$0.0422914\pi$$
−0.991187 + 0.132472i $$0.957709\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ 0 0
$$970$$ 1.39445i 0.0447731i
$$971$$ 53.0278 1.70174 0.850871 0.525375i $$-0.176075\pi$$
0.850871 + 0.525375i $$0.176075\pi$$
$$972$$ 0 0
$$973$$ 79.2666i 2.54117i
$$974$$ −0.972244 −0.0311527
$$975$$ 0 0
$$976$$ −11.2111 −0.358859
$$977$$ 18.8444i 0.602886i 0.953484 + 0.301443i $$0.0974683\pi$$
−0.953484 + 0.301443i $$0.902532\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 14.2111i 0.453957i
$$981$$ 0 0
$$982$$ − 7.81665i − 0.249439i
$$983$$ − 42.4222i − 1.35306i −0.736416 0.676529i $$-0.763483\pi$$
0.736416 0.676529i $$-0.236517\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 21.2111i 0.675499i
$$987$$ 0 0
$$988$$ − 16.6056i − 0.528293i
$$989$$ 11.1556 0.354727
$$990$$ 0 0
$$991$$ −22.4222 −0.712265 −0.356132 0.934436i $$-0.615905\pi$$
−0.356132 + 0.934436i $$0.615905\pi$$
$$992$$ 6.00000 0.190500
$$993$$ 0 0
$$994$$ − 42.4222i − 1.34555i
$$995$$ 22.4222i 0.710832i
$$996$$ 0 0
$$997$$ −16.4222 −0.520096 −0.260048 0.965596i $$-0.583738\pi$$
−0.260048 + 0.965596i $$0.583738\pi$$
$$998$$ 23.0278 0.728931
$$999$$ 0 0
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 1170.2.b.d.181.4 4
3.2 odd 2 390.2.b.c.181.2 4
12.11 even 2 3120.2.g.q.961.1 4
13.12 even 2 inner 1170.2.b.d.181.1 4
15.2 even 4 1950.2.f.n.649.3 4
15.8 even 4 1950.2.f.m.649.2 4
15.14 odd 2 1950.2.b.k.1351.3 4
39.5 even 4 5070.2.a.z.1.1 2
39.8 even 4 5070.2.a.bf.1.2 2
39.38 odd 2 390.2.b.c.181.3 yes 4
156.155 even 2 3120.2.g.q.961.4 4
195.38 even 4 1950.2.f.n.649.1 4
195.77 even 4 1950.2.f.m.649.4 4
195.194 odd 2 1950.2.b.k.1351.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 3.2 odd 2
390.2.b.c.181.3 yes 4 39.38 odd 2
1170.2.b.d.181.1 4 13.12 even 2 inner
1170.2.b.d.181.4 4 1.1 even 1 trivial
1950.2.b.k.1351.2 4 195.194 odd 2
1950.2.b.k.1351.3 4 15.14 odd 2
1950.2.f.m.649.2 4 15.8 even 4
1950.2.f.m.649.4 4 195.77 even 4
1950.2.f.n.649.1 4 195.38 even 4
1950.2.f.n.649.3 4 15.2 even 4
3120.2.g.q.961.1 4 12.11 even 2
3120.2.g.q.961.4 4 156.155 even 2
5070.2.a.z.1.1 2 39.5 even 4
5070.2.a.bf.1.2 2 39.8 even 4