# Properties

 Label 7938.2.a.bt.1.2 Level $7938$ Weight $2$ Character 7938.1 Self dual yes Analytic conductor $63.385$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7938 = 2 \cdot 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7938.a (trivial)

## 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: $$63.3852491245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1134) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 7938.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +3.73205 q^{5} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +3.73205 q^{5} +1.00000 q^{8} +3.73205 q^{10} +4.19615 q^{11} -0.464102 q^{13} +1.00000 q^{16} +7.00000 q^{17} +2.73205 q^{19} +3.73205 q^{20} +4.19615 q^{22} -6.19615 q^{23} +8.92820 q^{25} -0.464102 q^{26} -8.46410 q^{29} +2.19615 q^{31} +1.00000 q^{32} +7.00000 q^{34} -6.66025 q^{37} +2.73205 q^{38} +3.73205 q^{40} +9.46410 q^{41} +5.46410 q^{43} +4.19615 q^{44} -6.19615 q^{46} +1.26795 q^{47} +8.92820 q^{50} -0.464102 q^{52} +2.53590 q^{53} +15.6603 q^{55} -8.46410 q^{58} -6.19615 q^{59} +9.92820 q^{61} +2.19615 q^{62} +1.00000 q^{64} -1.73205 q^{65} -3.26795 q^{67} +7.00000 q^{68} -13.4641 q^{71} -11.7321 q^{73} -6.66025 q^{74} +2.73205 q^{76} +15.1244 q^{79} +3.73205 q^{80} +9.46410 q^{82} -14.5885 q^{83} +26.1244 q^{85} +5.46410 q^{86} +4.19615 q^{88} -3.92820 q^{89} -6.19615 q^{92} +1.26795 q^{94} +10.1962 q^{95} +2.92820 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{8} + 4 q^{10} - 2 q^{11} + 6 q^{13} + 2 q^{16} + 14 q^{17} + 2 q^{19} + 4 q^{20} - 2 q^{22} - 2 q^{23} + 4 q^{25} + 6 q^{26} - 10 q^{29} - 6 q^{31} + 2 q^{32} + 14 q^{34} + 4 q^{37} + 2 q^{38} + 4 q^{40} + 12 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{46} + 6 q^{47} + 4 q^{50} + 6 q^{52} + 12 q^{53} + 14 q^{55} - 10 q^{58} - 2 q^{59} + 6 q^{61} - 6 q^{62} + 2 q^{64} - 10 q^{67} + 14 q^{68} - 20 q^{71} - 20 q^{73} + 4 q^{74} + 2 q^{76} + 6 q^{79} + 4 q^{80} + 12 q^{82} + 2 q^{83} + 28 q^{85} + 4 q^{86} - 2 q^{88} + 6 q^{89} - 2 q^{92} + 6 q^{94} + 10 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^8 + 4 * q^10 - 2 * q^11 + 6 * q^13 + 2 * q^16 + 14 * q^17 + 2 * q^19 + 4 * q^20 - 2 * q^22 - 2 * q^23 + 4 * q^25 + 6 * q^26 - 10 * q^29 - 6 * q^31 + 2 * q^32 + 14 * q^34 + 4 * q^37 + 2 * q^38 + 4 * q^40 + 12 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^46 + 6 * q^47 + 4 * q^50 + 6 * q^52 + 12 * q^53 + 14 * q^55 - 10 * q^58 - 2 * q^59 + 6 * q^61 - 6 * q^62 + 2 * q^64 - 10 * q^67 + 14 * q^68 - 20 * q^71 - 20 * q^73 + 4 * q^74 + 2 * q^76 + 6 * q^79 + 4 * q^80 + 12 * q^82 + 2 * q^83 + 28 * q^85 + 4 * q^86 - 2 * q^88 + 6 * q^89 - 2 * q^92 + 6 * q^94 + 10 * q^95 - 8 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 3.73205 1.66902 0.834512 0.550990i $$-0.185750\pi$$
0.834512 + 0.550990i $$0.185750\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 3.73205 1.18018
$$11$$ 4.19615 1.26519 0.632594 0.774484i $$-0.281990\pi$$
0.632594 + 0.774484i $$0.281990\pi$$
$$12$$ 0 0
$$13$$ −0.464102 −0.128719 −0.0643593 0.997927i $$-0.520500\pi$$
−0.0643593 + 0.997927i $$0.520500\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 0 0
$$19$$ 2.73205 0.626775 0.313388 0.949625i $$-0.398536\pi$$
0.313388 + 0.949625i $$0.398536\pi$$
$$20$$ 3.73205 0.834512
$$21$$ 0 0
$$22$$ 4.19615 0.894623
$$23$$ −6.19615 −1.29199 −0.645994 0.763343i $$-0.723557\pi$$
−0.645994 + 0.763343i $$0.723557\pi$$
$$24$$ 0 0
$$25$$ 8.92820 1.78564
$$26$$ −0.464102 −0.0910178
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −8.46410 −1.57174 −0.785872 0.618389i $$-0.787786\pi$$
−0.785872 + 0.618389i $$0.787786\pi$$
$$30$$ 0 0
$$31$$ 2.19615 0.394441 0.197220 0.980359i $$-0.436809\pi$$
0.197220 + 0.980359i $$0.436809\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 7.00000 1.20049
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.66025 −1.09494 −0.547470 0.836826i $$-0.684409\pi$$
−0.547470 + 0.836826i $$0.684409\pi$$
$$38$$ 2.73205 0.443197
$$39$$ 0 0
$$40$$ 3.73205 0.590089
$$41$$ 9.46410 1.47804 0.739022 0.673681i $$-0.235288\pi$$
0.739022 + 0.673681i $$0.235288\pi$$
$$42$$ 0 0
$$43$$ 5.46410 0.833268 0.416634 0.909074i $$-0.363210\pi$$
0.416634 + 0.909074i $$0.363210\pi$$
$$44$$ 4.19615 0.632594
$$45$$ 0 0
$$46$$ −6.19615 −0.913573
$$47$$ 1.26795 0.184949 0.0924747 0.995715i $$-0.470522\pi$$
0.0924747 + 0.995715i $$0.470522\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 8.92820 1.26264
$$51$$ 0 0
$$52$$ −0.464102 −0.0643593
$$53$$ 2.53590 0.348332 0.174166 0.984716i $$-0.444277\pi$$
0.174166 + 0.984716i $$0.444277\pi$$
$$54$$ 0 0
$$55$$ 15.6603 2.11163
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −8.46410 −1.11139
$$59$$ −6.19615 −0.806670 −0.403335 0.915052i $$-0.632149\pi$$
−0.403335 + 0.915052i $$0.632149\pi$$
$$60$$ 0 0
$$61$$ 9.92820 1.27118 0.635588 0.772028i $$-0.280758\pi$$
0.635588 + 0.772028i $$0.280758\pi$$
$$62$$ 2.19615 0.278912
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.73205 −0.214834
$$66$$ 0 0
$$67$$ −3.26795 −0.399244 −0.199622 0.979873i $$-0.563971\pi$$
−0.199622 + 0.979873i $$0.563971\pi$$
$$68$$ 7.00000 0.848875
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −13.4641 −1.59789 −0.798947 0.601401i $$-0.794609\pi$$
−0.798947 + 0.601401i $$0.794609\pi$$
$$72$$ 0 0
$$73$$ −11.7321 −1.37313 −0.686566 0.727067i $$-0.740883\pi$$
−0.686566 + 0.727067i $$0.740883\pi$$
$$74$$ −6.66025 −0.774239
$$75$$ 0 0
$$76$$ 2.73205 0.313388
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 15.1244 1.70162 0.850811 0.525471i $$-0.176111\pi$$
0.850811 + 0.525471i $$0.176111\pi$$
$$80$$ 3.73205 0.417256
$$81$$ 0 0
$$82$$ 9.46410 1.04514
$$83$$ −14.5885 −1.60129 −0.800646 0.599138i $$-0.795510\pi$$
−0.800646 + 0.599138i $$0.795510\pi$$
$$84$$ 0 0
$$85$$ 26.1244 2.83358
$$86$$ 5.46410 0.589209
$$87$$ 0 0
$$88$$ 4.19615 0.447311
$$89$$ −3.92820 −0.416389 −0.208194 0.978087i $$-0.566759\pi$$
−0.208194 + 0.978087i $$0.566759\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −6.19615 −0.645994
$$93$$ 0 0
$$94$$ 1.26795 0.130779
$$95$$ 10.1962 1.04610
$$96$$ 0 0
$$97$$ 2.92820 0.297314 0.148657 0.988889i $$-0.452505\pi$$
0.148657 + 0.988889i $$0.452505\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 8.92820 0.892820
$$101$$ −4.92820 −0.490375 −0.245187 0.969476i $$-0.578849\pi$$
−0.245187 + 0.969476i $$0.578849\pi$$
$$102$$ 0 0
$$103$$ −12.3923 −1.22105 −0.610525 0.791997i $$-0.709042\pi$$
−0.610525 + 0.791997i $$0.709042\pi$$
$$104$$ −0.464102 −0.0455089
$$105$$ 0 0
$$106$$ 2.53590 0.246308
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −7.19615 −0.689266 −0.344633 0.938737i $$-0.611997\pi$$
−0.344633 + 0.938737i $$0.611997\pi$$
$$110$$ 15.6603 1.49315
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.26795 0.213351 0.106675 0.994294i $$-0.465979\pi$$
0.106675 + 0.994294i $$0.465979\pi$$
$$114$$ 0 0
$$115$$ −23.1244 −2.15636
$$116$$ −8.46410 −0.785872
$$117$$ 0 0
$$118$$ −6.19615 −0.570402
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 6.60770 0.600700
$$122$$ 9.92820 0.898857
$$123$$ 0 0
$$124$$ 2.19615 0.197220
$$125$$ 14.6603 1.31125
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −1.73205 −0.151911
$$131$$ 17.4641 1.52585 0.762923 0.646490i $$-0.223764\pi$$
0.762923 + 0.646490i $$0.223764\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −3.26795 −0.282308
$$135$$ 0 0
$$136$$ 7.00000 0.600245
$$137$$ 11.7321 1.00234 0.501168 0.865350i $$-0.332904\pi$$
0.501168 + 0.865350i $$0.332904\pi$$
$$138$$ 0 0
$$139$$ 6.73205 0.571005 0.285503 0.958378i $$-0.407839\pi$$
0.285503 + 0.958378i $$0.407839\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −13.4641 −1.12988
$$143$$ −1.94744 −0.162853
$$144$$ 0 0
$$145$$ −31.5885 −2.62328
$$146$$ −11.7321 −0.970951
$$147$$ 0 0
$$148$$ −6.66025 −0.547470
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ −16.1962 −1.31802 −0.659012 0.752132i $$-0.729025\pi$$
−0.659012 + 0.752132i $$0.729025\pi$$
$$152$$ 2.73205 0.221599
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.19615 0.658331
$$156$$ 0 0
$$157$$ 1.00000 0.0798087 0.0399043 0.999204i $$-0.487295\pi$$
0.0399043 + 0.999204i $$0.487295\pi$$
$$158$$ 15.1244 1.20323
$$159$$ 0 0
$$160$$ 3.73205 0.295045
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −6.53590 −0.511931 −0.255966 0.966686i $$-0.582393\pi$$
−0.255966 + 0.966686i $$0.582393\pi$$
$$164$$ 9.46410 0.739022
$$165$$ 0 0
$$166$$ −14.5885 −1.13228
$$167$$ −12.1962 −0.943767 −0.471883 0.881661i $$-0.656426\pi$$
−0.471883 + 0.881661i $$0.656426\pi$$
$$168$$ 0 0
$$169$$ −12.7846 −0.983432
$$170$$ 26.1244 2.00365
$$171$$ 0 0
$$172$$ 5.46410 0.416634
$$173$$ 9.73205 0.739914 0.369957 0.929049i $$-0.379372\pi$$
0.369957 + 0.929049i $$0.379372\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.19615 0.316297
$$177$$ 0 0
$$178$$ −3.92820 −0.294431
$$179$$ −8.19615 −0.612609 −0.306305 0.951934i $$-0.599093\pi$$
−0.306305 + 0.951934i $$0.599093\pi$$
$$180$$ 0 0
$$181$$ −4.39230 −0.326477 −0.163239 0.986587i $$-0.552194\pi$$
−0.163239 + 0.986587i $$0.552194\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −6.19615 −0.456786
$$185$$ −24.8564 −1.82748
$$186$$ 0 0
$$187$$ 29.3731 2.14797
$$188$$ 1.26795 0.0924747
$$189$$ 0 0
$$190$$ 10.1962 0.739707
$$191$$ −11.6603 −0.843706 −0.421853 0.906664i $$-0.638620\pi$$
−0.421853 + 0.906664i $$0.638620\pi$$
$$192$$ 0 0
$$193$$ −8.85641 −0.637498 −0.318749 0.947839i $$-0.603263\pi$$
−0.318749 + 0.947839i $$0.603263\pi$$
$$194$$ 2.92820 0.210233
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 25.7846 1.83708 0.918539 0.395331i $$-0.129370\pi$$
0.918539 + 0.395331i $$0.129370\pi$$
$$198$$ 0 0
$$199$$ −5.12436 −0.363256 −0.181628 0.983367i $$-0.558137\pi$$
−0.181628 + 0.983367i $$0.558137\pi$$
$$200$$ 8.92820 0.631319
$$201$$ 0 0
$$202$$ −4.92820 −0.346747
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 35.3205 2.46689
$$206$$ −12.3923 −0.863413
$$207$$ 0 0
$$208$$ −0.464102 −0.0321797
$$209$$ 11.4641 0.792988
$$210$$ 0 0
$$211$$ −20.7321 −1.42725 −0.713627 0.700526i $$-0.752949\pi$$
−0.713627 + 0.700526i $$0.752949\pi$$
$$212$$ 2.53590 0.174166
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 20.3923 1.39074
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −7.19615 −0.487385
$$219$$ 0 0
$$220$$ 15.6603 1.05581
$$221$$ −3.24871 −0.218532
$$222$$ 0 0
$$223$$ 18.5359 1.24126 0.620628 0.784105i $$-0.286878\pi$$
0.620628 + 0.784105i $$0.286878\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.26795 0.150862
$$227$$ −5.07180 −0.336627 −0.168313 0.985734i $$-0.553832\pi$$
−0.168313 + 0.985734i $$0.553832\pi$$
$$228$$ 0 0
$$229$$ −4.46410 −0.294996 −0.147498 0.989062i $$-0.547122\pi$$
−0.147498 + 0.989062i $$0.547122\pi$$
$$230$$ −23.1244 −1.52477
$$231$$ 0 0
$$232$$ −8.46410 −0.555695
$$233$$ 13.1962 0.864509 0.432254 0.901752i $$-0.357718\pi$$
0.432254 + 0.901752i $$0.357718\pi$$
$$234$$ 0 0
$$235$$ 4.73205 0.308685
$$236$$ −6.19615 −0.403335
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −28.0526 −1.81457 −0.907285 0.420517i $$-0.861849\pi$$
−0.907285 + 0.420517i $$0.861849\pi$$
$$240$$ 0 0
$$241$$ 17.7321 1.14222 0.571111 0.820873i $$-0.306513\pi$$
0.571111 + 0.820873i $$0.306513\pi$$
$$242$$ 6.60770 0.424759
$$243$$ 0 0
$$244$$ 9.92820 0.635588
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.26795 −0.0806777
$$248$$ 2.19615 0.139456
$$249$$ 0 0
$$250$$ 14.6603 0.927196
$$251$$ 16.0526 1.01323 0.506614 0.862173i $$-0.330897\pi$$
0.506614 + 0.862173i $$0.330897\pi$$
$$252$$ 0 0
$$253$$ −26.0000 −1.63461
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −0.464102 −0.0289499 −0.0144749 0.999895i $$-0.504608\pi$$
−0.0144749 + 0.999895i $$0.504608\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1.73205 −0.107417
$$261$$ 0 0
$$262$$ 17.4641 1.07894
$$263$$ 23.6603 1.45895 0.729477 0.684005i $$-0.239764\pi$$
0.729477 + 0.684005i $$0.239764\pi$$
$$264$$ 0 0
$$265$$ 9.46410 0.581375
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −3.26795 −0.199622
$$269$$ −25.5885 −1.56016 −0.780078 0.625682i $$-0.784821\pi$$
−0.780078 + 0.625682i $$0.784821\pi$$
$$270$$ 0 0
$$271$$ −25.5167 −1.55003 −0.775013 0.631945i $$-0.782257\pi$$
−0.775013 + 0.631945i $$0.782257\pi$$
$$272$$ 7.00000 0.424437
$$273$$ 0 0
$$274$$ 11.7321 0.708759
$$275$$ 37.4641 2.25917
$$276$$ 0 0
$$277$$ 22.7846 1.36899 0.684497 0.729015i $$-0.260022\pi$$
0.684497 + 0.729015i $$0.260022\pi$$
$$278$$ 6.73205 0.403762
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2.80385 0.167264 0.0836318 0.996497i $$-0.473348\pi$$
0.0836318 + 0.996497i $$0.473348\pi$$
$$282$$ 0 0
$$283$$ 19.3205 1.14848 0.574242 0.818685i $$-0.305297\pi$$
0.574242 + 0.818685i $$0.305297\pi$$
$$284$$ −13.4641 −0.798947
$$285$$ 0 0
$$286$$ −1.94744 −0.115155
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ −31.5885 −1.85494
$$291$$ 0 0
$$292$$ −11.7321 −0.686566
$$293$$ 20.6603 1.20698 0.603492 0.797369i $$-0.293775\pi$$
0.603492 + 0.797369i $$0.293775\pi$$
$$294$$ 0 0
$$295$$ −23.1244 −1.34635
$$296$$ −6.66025 −0.387119
$$297$$ 0 0
$$298$$ −9.00000 −0.521356
$$299$$ 2.87564 0.166303
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −16.1962 −0.931984
$$303$$ 0 0
$$304$$ 2.73205 0.156694
$$305$$ 37.0526 2.12162
$$306$$ 0 0
$$307$$ −5.85641 −0.334243 −0.167121 0.985936i $$-0.553447\pi$$
−0.167121 + 0.985936i $$0.553447\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 8.19615 0.465510
$$311$$ −0.196152 −0.0111228 −0.00556139 0.999985i $$-0.501770\pi$$
−0.00556139 + 0.999985i $$0.501770\pi$$
$$312$$ 0 0
$$313$$ 5.58846 0.315878 0.157939 0.987449i $$-0.449515\pi$$
0.157939 + 0.987449i $$0.449515\pi$$
$$314$$ 1.00000 0.0564333
$$315$$ 0 0
$$316$$ 15.1244 0.850811
$$317$$ 10.6077 0.595788 0.297894 0.954599i $$-0.403716\pi$$
0.297894 + 0.954599i $$0.403716\pi$$
$$318$$ 0 0
$$319$$ −35.5167 −1.98855
$$320$$ 3.73205 0.208628
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 19.1244 1.06411
$$324$$ 0 0
$$325$$ −4.14359 −0.229845
$$326$$ −6.53590 −0.361990
$$327$$ 0 0
$$328$$ 9.46410 0.522568
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −8.39230 −0.461283 −0.230641 0.973039i $$-0.574082\pi$$
−0.230641 + 0.973039i $$0.574082\pi$$
$$332$$ −14.5885 −0.800646
$$333$$ 0 0
$$334$$ −12.1962 −0.667344
$$335$$ −12.1962 −0.666347
$$336$$ 0 0
$$337$$ 4.39230 0.239264 0.119632 0.992818i $$-0.461829\pi$$
0.119632 + 0.992818i $$0.461829\pi$$
$$338$$ −12.7846 −0.695391
$$339$$ 0 0
$$340$$ 26.1244 1.41679
$$341$$ 9.21539 0.499041
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 5.46410 0.294605
$$345$$ 0 0
$$346$$ 9.73205 0.523198
$$347$$ −14.5359 −0.780328 −0.390164 0.920745i $$-0.627582\pi$$
−0.390164 + 0.920745i $$0.627582\pi$$
$$348$$ 0 0
$$349$$ 5.46410 0.292487 0.146243 0.989249i $$-0.453282\pi$$
0.146243 + 0.989249i $$0.453282\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.19615 0.223656
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ −50.2487 −2.66692
$$356$$ −3.92820 −0.208194
$$357$$ 0 0
$$358$$ −8.19615 −0.433180
$$359$$ −2.92820 −0.154545 −0.0772723 0.997010i $$-0.524621\pi$$
−0.0772723 + 0.997010i $$0.524621\pi$$
$$360$$ 0 0
$$361$$ −11.5359 −0.607153
$$362$$ −4.39230 −0.230854
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −43.7846 −2.29179
$$366$$ 0 0
$$367$$ 13.1244 0.685086 0.342543 0.939502i $$-0.388712\pi$$
0.342543 + 0.939502i $$0.388712\pi$$
$$368$$ −6.19615 −0.322997
$$369$$ 0 0
$$370$$ −24.8564 −1.29222
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −33.8564 −1.75302 −0.876509 0.481385i $$-0.840134\pi$$
−0.876509 + 0.481385i $$0.840134\pi$$
$$374$$ 29.3731 1.51885
$$375$$ 0 0
$$376$$ 1.26795 0.0653895
$$377$$ 3.92820 0.202313
$$378$$ 0 0
$$379$$ 17.5167 0.899770 0.449885 0.893086i $$-0.351465\pi$$
0.449885 + 0.893086i $$0.351465\pi$$
$$380$$ 10.1962 0.523052
$$381$$ 0 0
$$382$$ −11.6603 −0.596590
$$383$$ −35.7128 −1.82484 −0.912420 0.409256i $$-0.865788\pi$$
−0.912420 + 0.409256i $$0.865788\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −8.85641 −0.450779
$$387$$ 0 0
$$388$$ 2.92820 0.148657
$$389$$ 6.53590 0.331383 0.165692 0.986178i $$-0.447014\pi$$
0.165692 + 0.986178i $$0.447014\pi$$
$$390$$ 0 0
$$391$$ −43.3731 −2.19347
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 25.7846 1.29901
$$395$$ 56.4449 2.84005
$$396$$ 0 0
$$397$$ 21.0000 1.05396 0.526980 0.849878i $$-0.323324\pi$$
0.526980 + 0.849878i $$0.323324\pi$$
$$398$$ −5.12436 −0.256861
$$399$$ 0 0
$$400$$ 8.92820 0.446410
$$401$$ 34.5167 1.72368 0.861840 0.507180i $$-0.169312\pi$$
0.861840 + 0.507180i $$0.169312\pi$$
$$402$$ 0 0
$$403$$ −1.01924 −0.0507719
$$404$$ −4.92820 −0.245187
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −27.9474 −1.38530
$$408$$ 0 0
$$409$$ −34.6603 −1.71384 −0.856920 0.515450i $$-0.827625\pi$$
−0.856920 + 0.515450i $$0.827625\pi$$
$$410$$ 35.3205 1.74436
$$411$$ 0 0
$$412$$ −12.3923 −0.610525
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −54.4449 −2.67259
$$416$$ −0.464102 −0.0227545
$$417$$ 0 0
$$418$$ 11.4641 0.560728
$$419$$ −2.53590 −0.123887 −0.0619434 0.998080i $$-0.519730\pi$$
−0.0619434 + 0.998080i $$0.519730\pi$$
$$420$$ 0 0
$$421$$ −24.1244 −1.17575 −0.587875 0.808952i $$-0.700035\pi$$
−0.587875 + 0.808952i $$0.700035\pi$$
$$422$$ −20.7321 −1.00922
$$423$$ 0 0
$$424$$ 2.53590 0.123154
$$425$$ 62.4974 3.03157
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 20.3923 0.983404
$$431$$ 21.4641 1.03389 0.516945 0.856019i $$-0.327069\pi$$
0.516945 + 0.856019i $$0.327069\pi$$
$$432$$ 0 0
$$433$$ −12.2679 −0.589560 −0.294780 0.955565i $$-0.595246\pi$$
−0.294780 + 0.955565i $$0.595246\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −7.19615 −0.344633
$$437$$ −16.9282 −0.809786
$$438$$ 0 0
$$439$$ 11.3205 0.540298 0.270149 0.962818i $$-0.412927\pi$$
0.270149 + 0.962818i $$0.412927\pi$$
$$440$$ 15.6603 0.746573
$$441$$ 0 0
$$442$$ −3.24871 −0.154525
$$443$$ 18.7321 0.889987 0.444993 0.895534i $$-0.353206\pi$$
0.444993 + 0.895534i $$0.353206\pi$$
$$444$$ 0 0
$$445$$ −14.6603 −0.694963
$$446$$ 18.5359 0.877700
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 11.8564 0.559538 0.279769 0.960067i $$-0.409742\pi$$
0.279769 + 0.960067i $$0.409742\pi$$
$$450$$ 0 0
$$451$$ 39.7128 1.87000
$$452$$ 2.26795 0.106675
$$453$$ 0 0
$$454$$ −5.07180 −0.238031
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 20.8564 0.975622 0.487811 0.872949i $$-0.337796\pi$$
0.487811 + 0.872949i $$0.337796\pi$$
$$458$$ −4.46410 −0.208594
$$459$$ 0 0
$$460$$ −23.1244 −1.07818
$$461$$ 34.7846 1.62008 0.810040 0.586374i $$-0.199445\pi$$
0.810040 + 0.586374i $$0.199445\pi$$
$$462$$ 0 0
$$463$$ −32.5885 −1.51451 −0.757257 0.653117i $$-0.773461\pi$$
−0.757257 + 0.653117i $$0.773461\pi$$
$$464$$ −8.46410 −0.392936
$$465$$ 0 0
$$466$$ 13.1962 0.611300
$$467$$ 14.5885 0.675073 0.337537 0.941312i $$-0.390406\pi$$
0.337537 + 0.941312i $$0.390406\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 4.73205 0.218273
$$471$$ 0 0
$$472$$ −6.19615 −0.285201
$$473$$ 22.9282 1.05424
$$474$$ 0 0
$$475$$ 24.3923 1.11920
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −28.0526 −1.28309
$$479$$ 23.5167 1.07450 0.537252 0.843422i $$-0.319462\pi$$
0.537252 + 0.843422i $$0.319462\pi$$
$$480$$ 0 0
$$481$$ 3.09103 0.140939
$$482$$ 17.7321 0.807673
$$483$$ 0 0
$$484$$ 6.60770 0.300350
$$485$$ 10.9282 0.496224
$$486$$ 0 0
$$487$$ −28.5885 −1.29547 −0.647733 0.761867i $$-0.724283\pi$$
−0.647733 + 0.761867i $$0.724283\pi$$
$$488$$ 9.92820 0.449429
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −33.4641 −1.51021 −0.755107 0.655602i $$-0.772415\pi$$
−0.755107 + 0.655602i $$0.772415\pi$$
$$492$$ 0 0
$$493$$ −59.2487 −2.66843
$$494$$ −1.26795 −0.0570477
$$495$$ 0 0
$$496$$ 2.19615 0.0986102
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 30.1962 1.35177 0.675883 0.737009i $$-0.263763\pi$$
0.675883 + 0.737009i $$0.263763\pi$$
$$500$$ 14.6603 0.655626
$$501$$ 0 0
$$502$$ 16.0526 0.716461
$$503$$ 1.94744 0.0868321 0.0434161 0.999057i $$-0.486176\pi$$
0.0434161 + 0.999057i $$0.486176\pi$$
$$504$$ 0 0
$$505$$ −18.3923 −0.818447
$$506$$ −26.0000 −1.15584
$$507$$ 0 0
$$508$$ 12.0000 0.532414
$$509$$ 4.14359 0.183662 0.0918308 0.995775i $$-0.470728\pi$$
0.0918308 + 0.995775i $$0.470728\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −0.464102 −0.0204706
$$515$$ −46.2487 −2.03796
$$516$$ 0 0
$$517$$ 5.32051 0.233996
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −1.73205 −0.0759555
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ 29.1769 1.27582 0.637909 0.770112i $$-0.279800\pi$$
0.637909 + 0.770112i $$0.279800\pi$$
$$524$$ 17.4641 0.762923
$$525$$ 0 0
$$526$$ 23.6603 1.03164
$$527$$ 15.3731 0.669661
$$528$$ 0 0
$$529$$ 15.3923 0.669231
$$530$$ 9.46410 0.411094
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −4.39230 −0.190252
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −3.26795 −0.141154
$$537$$ 0 0
$$538$$ −25.5885 −1.10320
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 20.6603 0.888254 0.444127 0.895964i $$-0.353514\pi$$
0.444127 + 0.895964i $$0.353514\pi$$
$$542$$ −25.5167 −1.09603
$$543$$ 0 0
$$544$$ 7.00000 0.300123
$$545$$ −26.8564 −1.15040
$$546$$ 0 0
$$547$$ −19.2679 −0.823838 −0.411919 0.911220i $$-0.635141\pi$$
−0.411919 + 0.911220i $$0.635141\pi$$
$$548$$ 11.7321 0.501168
$$549$$ 0 0
$$550$$ 37.4641 1.59747
$$551$$ −23.1244 −0.985131
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 22.7846 0.968025
$$555$$ 0 0
$$556$$ 6.73205 0.285503
$$557$$ −10.0718 −0.426756 −0.213378 0.976970i $$-0.568447\pi$$
−0.213378 + 0.976970i $$0.568447\pi$$
$$558$$ 0 0
$$559$$ −2.53590 −0.107257
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.80385 0.118273
$$563$$ −35.7128 −1.50512 −0.752558 0.658526i $$-0.771180\pi$$
−0.752558 + 0.658526i $$0.771180\pi$$
$$564$$ 0 0
$$565$$ 8.46410 0.356087
$$566$$ 19.3205 0.812102
$$567$$ 0 0
$$568$$ −13.4641 −0.564941
$$569$$ −12.8038 −0.536765 −0.268383 0.963312i $$-0.586489\pi$$
−0.268383 + 0.963312i $$0.586489\pi$$
$$570$$ 0 0
$$571$$ −19.2679 −0.806339 −0.403169 0.915125i $$-0.632091\pi$$
−0.403169 + 0.915125i $$0.632091\pi$$
$$572$$ −1.94744 −0.0814266
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −55.3205 −2.30702
$$576$$ 0 0
$$577$$ −7.33975 −0.305558 −0.152779 0.988260i $$-0.548822\pi$$
−0.152779 + 0.988260i $$0.548822\pi$$
$$578$$ 32.0000 1.33102
$$579$$ 0 0
$$580$$ −31.5885 −1.31164
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 10.6410 0.440706
$$584$$ −11.7321 −0.485476
$$585$$ 0 0
$$586$$ 20.6603 0.853467
$$587$$ 11.2679 0.465078 0.232539 0.972587i $$-0.425297\pi$$
0.232539 + 0.972587i $$0.425297\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ −23.1244 −0.952015
$$591$$ 0 0
$$592$$ −6.66025 −0.273735
$$593$$ 40.1769 1.64987 0.824934 0.565229i $$-0.191212\pi$$
0.824934 + 0.565229i $$0.191212\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −9.00000 −0.368654
$$597$$ 0 0
$$598$$ 2.87564 0.117594
$$599$$ −9.12436 −0.372811 −0.186406 0.982473i $$-0.559684\pi$$
−0.186406 + 0.982473i $$0.559684\pi$$
$$600$$ 0 0
$$601$$ −8.80385 −0.359116 −0.179558 0.983747i $$-0.557467\pi$$
−0.179558 + 0.983747i $$0.557467\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −16.1962 −0.659012
$$605$$ 24.6603 1.00258
$$606$$ 0 0
$$607$$ −6.58846 −0.267417 −0.133709 0.991021i $$-0.542689\pi$$
−0.133709 + 0.991021i $$0.542689\pi$$
$$608$$ 2.73205 0.110799
$$609$$ 0 0
$$610$$ 37.0526 1.50021
$$611$$ −0.588457 −0.0238064
$$612$$ 0 0
$$613$$ −14.7846 −0.597145 −0.298572 0.954387i $$-0.596510\pi$$
−0.298572 + 0.954387i $$0.596510\pi$$
$$614$$ −5.85641 −0.236345
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 39.9808 1.60956 0.804782 0.593570i $$-0.202282\pi$$
0.804782 + 0.593570i $$0.202282\pi$$
$$618$$ 0 0
$$619$$ −31.7128 −1.27465 −0.637323 0.770597i $$-0.719958\pi$$
−0.637323 + 0.770597i $$0.719958\pi$$
$$620$$ 8.19615 0.329165
$$621$$ 0 0
$$622$$ −0.196152 −0.00786500
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 10.0718 0.402872
$$626$$ 5.58846 0.223360
$$627$$ 0 0
$$628$$ 1.00000 0.0399043
$$629$$ −46.6218 −1.85893
$$630$$ 0 0
$$631$$ 13.6603 0.543806 0.271903 0.962325i $$-0.412347\pi$$
0.271903 + 0.962325i $$0.412347\pi$$
$$632$$ 15.1244 0.601615
$$633$$ 0 0
$$634$$ 10.6077 0.421285
$$635$$ 44.7846 1.77722
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −35.5167 −1.40612
$$639$$ 0 0
$$640$$ 3.73205 0.147522
$$641$$ 19.4449 0.768026 0.384013 0.923328i $$-0.374542\pi$$
0.384013 + 0.923328i $$0.374542\pi$$
$$642$$ 0 0
$$643$$ 40.5885 1.60065 0.800326 0.599565i $$-0.204660\pi$$
0.800326 + 0.599565i $$0.204660\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 19.1244 0.752438
$$647$$ 16.3923 0.644448 0.322224 0.946663i $$-0.395570\pi$$
0.322224 + 0.946663i $$0.395570\pi$$
$$648$$ 0 0
$$649$$ −26.0000 −1.02059
$$650$$ −4.14359 −0.162525
$$651$$ 0 0
$$652$$ −6.53590 −0.255966
$$653$$ −18.2487 −0.714127 −0.357064 0.934080i $$-0.616222\pi$$
−0.357064 + 0.934080i $$0.616222\pi$$
$$654$$ 0 0
$$655$$ 65.1769 2.54667
$$656$$ 9.46410 0.369511
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 15.6077 0.607989 0.303995 0.952674i $$-0.401679\pi$$
0.303995 + 0.952674i $$0.401679\pi$$
$$660$$ 0 0
$$661$$ −14.8564 −0.577847 −0.288924 0.957352i $$-0.593297\pi$$
−0.288924 + 0.957352i $$0.593297\pi$$
$$662$$ −8.39230 −0.326176
$$663$$ 0 0
$$664$$ −14.5885 −0.566142
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 52.4449 2.03067
$$668$$ −12.1962 −0.471883
$$669$$ 0 0
$$670$$ −12.1962 −0.471178
$$671$$ 41.6603 1.60828
$$672$$ 0 0
$$673$$ 16.3205 0.629109 0.314555 0.949239i $$-0.398145\pi$$
0.314555 + 0.949239i $$0.398145\pi$$
$$674$$ 4.39230 0.169185
$$675$$ 0 0
$$676$$ −12.7846 −0.491716
$$677$$ 36.0000 1.38359 0.691796 0.722093i $$-0.256820\pi$$
0.691796 + 0.722093i $$0.256820\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 26.1244 1.00182
$$681$$ 0 0
$$682$$ 9.21539 0.352876
$$683$$ −25.8564 −0.989368 −0.494684 0.869073i $$-0.664716\pi$$
−0.494684 + 0.869073i $$0.664716\pi$$
$$684$$ 0 0
$$685$$ 43.7846 1.67292
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 5.46410 0.208317
$$689$$ −1.17691 −0.0448369
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 9.73205 0.369957
$$693$$ 0 0
$$694$$ −14.5359 −0.551775
$$695$$ 25.1244 0.953021
$$696$$ 0 0
$$697$$ 66.2487 2.50935
$$698$$ 5.46410 0.206819
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −6.60770 −0.249569 −0.124785 0.992184i $$-0.539824\pi$$
−0.124785 + 0.992184i $$0.539824\pi$$
$$702$$ 0 0
$$703$$ −18.1962 −0.686281
$$704$$ 4.19615 0.158148
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −28.1244 −1.05623 −0.528116 0.849172i $$-0.677101\pi$$
−0.528116 + 0.849172i $$0.677101\pi$$
$$710$$ −50.2487 −1.88580
$$711$$ 0 0
$$712$$ −3.92820 −0.147216
$$713$$ −13.6077 −0.509612
$$714$$ 0 0
$$715$$ −7.26795 −0.271806
$$716$$ −8.19615 −0.306305
$$717$$ 0 0
$$718$$ −2.92820 −0.109280
$$719$$ −2.53590 −0.0945731 −0.0472865 0.998881i $$-0.515057\pi$$
−0.0472865 + 0.998881i $$0.515057\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −11.5359 −0.429322
$$723$$ 0 0
$$724$$ −4.39230 −0.163239
$$725$$ −75.5692 −2.80657
$$726$$ 0 0
$$727$$ −16.6795 −0.618608 −0.309304 0.950963i $$-0.600096\pi$$
−0.309304 + 0.950963i $$0.600096\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −43.7846 −1.62054
$$731$$ 38.2487 1.41468
$$732$$ 0 0
$$733$$ −12.6795 −0.468328 −0.234164 0.972197i $$-0.575235\pi$$
−0.234164 + 0.972197i $$0.575235\pi$$
$$734$$ 13.1244 0.484429
$$735$$ 0 0
$$736$$ −6.19615 −0.228393
$$737$$ −13.7128 −0.505118
$$738$$ 0 0
$$739$$ −16.7321 −0.615498 −0.307749 0.951468i $$-0.599576\pi$$
−0.307749 + 0.951468i $$0.599576\pi$$
$$740$$ −24.8564 −0.913740
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 19.6077 0.719337 0.359668 0.933080i $$-0.382890\pi$$
0.359668 + 0.933080i $$0.382890\pi$$
$$744$$ 0 0
$$745$$ −33.5885 −1.23059
$$746$$ −33.8564 −1.23957
$$747$$ 0 0
$$748$$ 29.3731 1.07399
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −49.8564 −1.81929 −0.909643 0.415391i $$-0.863645\pi$$
−0.909643 + 0.415391i $$0.863645\pi$$
$$752$$ 1.26795 0.0462373
$$753$$ 0 0
$$754$$ 3.92820 0.143057
$$755$$ −60.4449 −2.19981
$$756$$ 0 0
$$757$$ −20.7846 −0.755429 −0.377715 0.925922i $$-0.623290\pi$$
−0.377715 + 0.925922i $$0.623290\pi$$
$$758$$ 17.5167 0.636234
$$759$$ 0 0
$$760$$ 10.1962 0.369853
$$761$$ 37.0000 1.34125 0.670624 0.741797i $$-0.266026\pi$$
0.670624 + 0.741797i $$0.266026\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −11.6603 −0.421853
$$765$$ 0 0
$$766$$ −35.7128 −1.29036
$$767$$ 2.87564 0.103833
$$768$$ 0 0
$$769$$ 35.5885 1.28335 0.641676 0.766976i $$-0.278239\pi$$
0.641676 + 0.766976i $$0.278239\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −8.85641 −0.318749
$$773$$ 20.1244 0.723823 0.361911 0.932213i $$-0.382124\pi$$
0.361911 + 0.932213i $$0.382124\pi$$
$$774$$ 0 0
$$775$$ 19.6077 0.704329
$$776$$ 2.92820 0.105116
$$777$$ 0 0
$$778$$ 6.53590 0.234323
$$779$$ 25.8564 0.926402
$$780$$ 0 0
$$781$$ −56.4974 −2.02164
$$782$$ −43.3731 −1.55102
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 3.73205 0.133203
$$786$$ 0 0
$$787$$ 7.60770 0.271185 0.135593 0.990765i $$-0.456706\pi$$
0.135593 + 0.990765i $$0.456706\pi$$
$$788$$ 25.7846 0.918539
$$789$$ 0 0
$$790$$ 56.4449 2.00822
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4.60770 −0.163624
$$794$$ 21.0000 0.745262
$$795$$ 0 0
$$796$$ −5.12436 −0.181628
$$797$$ 29.4449 1.04299 0.521495 0.853254i $$-0.325374\pi$$
0.521495 + 0.853254i $$0.325374\pi$$
$$798$$ 0 0
$$799$$ 8.87564 0.313998
$$800$$ 8.92820 0.315660
$$801$$ 0 0
$$802$$ 34.5167 1.21883
$$803$$ −49.2295 −1.73727
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.01924 −0.0359011
$$807$$ 0 0
$$808$$ −4.92820 −0.173374
$$809$$ −7.87564 −0.276893 −0.138446 0.990370i $$-0.544211\pi$$
−0.138446 + 0.990370i $$0.544211\pi$$
$$810$$ 0 0
$$811$$ −7.80385 −0.274030 −0.137015 0.990569i $$-0.543751\pi$$
−0.137015 + 0.990569i $$0.543751\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −27.9474 −0.979557
$$815$$ −24.3923 −0.854425
$$816$$ 0 0
$$817$$ 14.9282 0.522272
$$818$$ −34.6603 −1.21187
$$819$$ 0 0
$$820$$ 35.3205 1.23345
$$821$$ −12.0718 −0.421309 −0.210654 0.977561i $$-0.567559\pi$$
−0.210654 + 0.977561i $$0.567559\pi$$
$$822$$ 0 0
$$823$$ −40.7846 −1.42166 −0.710831 0.703363i $$-0.751681\pi$$
−0.710831 + 0.703363i $$0.751681\pi$$
$$824$$ −12.3923 −0.431706
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −11.3205 −0.393653 −0.196826 0.980438i $$-0.563064\pi$$
−0.196826 + 0.980438i $$0.563064\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ −54.4449 −1.88981
$$831$$ 0 0
$$832$$ −0.464102 −0.0160898
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −45.5167 −1.57517
$$836$$ 11.4641 0.396494
$$837$$ 0 0
$$838$$ −2.53590 −0.0876012
$$839$$ −5.46410 −0.188642 −0.0943209 0.995542i $$-0.530068\pi$$
−0.0943209 + 0.995542i $$0.530068\pi$$
$$840$$ 0 0
$$841$$ 42.6410 1.47038
$$842$$ −24.1244 −0.831380
$$843$$ 0 0
$$844$$ −20.7321 −0.713627
$$845$$ −47.7128 −1.64137
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.53590 0.0870831
$$849$$ 0 0
$$850$$ 62.4974 2.14364
$$851$$ 41.2679 1.41465
$$852$$ 0 0
$$853$$ −5.71281 −0.195603 −0.0978015 0.995206i $$-0.531181\pi$$
−0.0978015 + 0.995206i $$0.531181\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 10.8564 0.370848 0.185424 0.982659i $$-0.440634\pi$$
0.185424 + 0.982659i $$0.440634\pi$$
$$858$$ 0 0
$$859$$ −3.60770 −0.123093 −0.0615465 0.998104i $$-0.519603\pi$$
−0.0615465 + 0.998104i $$0.519603\pi$$
$$860$$ 20.3923 0.695372
$$861$$ 0 0
$$862$$ 21.4641 0.731070
$$863$$ −17.1244 −0.582920 −0.291460 0.956583i $$-0.594141\pi$$
−0.291460 + 0.956583i $$0.594141\pi$$
$$864$$ 0 0
$$865$$ 36.3205 1.23493
$$866$$ −12.2679 −0.416882
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 63.4641 2.15287
$$870$$ 0 0
$$871$$ 1.51666 0.0513901
$$872$$ −7.19615 −0.243692
$$873$$ 0 0
$$874$$ −16.9282 −0.572605
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.51666 0.287587 0.143794 0.989608i $$-0.454070\pi$$
0.143794 + 0.989608i $$0.454070\pi$$
$$878$$ 11.3205 0.382049
$$879$$ 0 0
$$880$$ 15.6603 0.527907
$$881$$ −18.2487 −0.614815 −0.307407 0.951578i $$-0.599461\pi$$
−0.307407 + 0.951578i $$0.599461\pi$$
$$882$$ 0 0
$$883$$ 7.66025 0.257788 0.128894 0.991658i $$-0.458857\pi$$
0.128894 + 0.991658i $$0.458857\pi$$
$$884$$ −3.24871 −0.109266
$$885$$ 0 0
$$886$$ 18.7321 0.629316
$$887$$ 2.44486 0.0820905 0.0410452 0.999157i $$-0.486931\pi$$
0.0410452 + 0.999157i $$0.486931\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −14.6603 −0.491413
$$891$$ 0 0
$$892$$ 18.5359 0.620628
$$893$$ 3.46410 0.115922
$$894$$ 0 0
$$895$$ −30.5885 −1.02246
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 11.8564 0.395653
$$899$$ −18.5885 −0.619960
$$900$$ 0 0
$$901$$ 17.7513 0.591381
$$902$$ 39.7128 1.32229
$$903$$ 0 0
$$904$$ 2.26795 0.0754309
$$905$$ −16.3923 −0.544899
$$906$$ 0 0
$$907$$ −36.0000 −1.19536 −0.597680 0.801735i $$-0.703911\pi$$
−0.597680 + 0.801735i $$0.703911\pi$$
$$908$$ −5.07180 −0.168313
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6.24871 0.207029 0.103515 0.994628i $$-0.466991\pi$$
0.103515 + 0.994628i $$0.466991\pi$$
$$912$$ 0 0
$$913$$ −61.2154 −2.02593
$$914$$ 20.8564 0.689869
$$915$$ 0 0
$$916$$ −4.46410 −0.147498
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −24.9808 −0.824039 −0.412020 0.911175i $$-0.635176\pi$$
−0.412020 + 0.911175i $$0.635176\pi$$
$$920$$ −23.1244 −0.762387
$$921$$ 0 0
$$922$$ 34.7846 1.14557
$$923$$ 6.24871 0.205679
$$924$$ 0 0
$$925$$ −59.4641 −1.95517
$$926$$ −32.5885 −1.07092
$$927$$ 0 0
$$928$$ −8.46410 −0.277848
$$929$$ 45.4974 1.49272 0.746361 0.665541i $$-0.231799\pi$$
0.746361 + 0.665541i $$0.231799\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 13.1962 0.432254
$$933$$ 0 0
$$934$$ 14.5885 0.477349
$$935$$ 109.622 3.58502
$$936$$ 0 0
$$937$$ −53.8372 −1.75878 −0.879392 0.476099i $$-0.842050\pi$$
−0.879392 + 0.476099i $$0.842050\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 4.73205 0.154342
$$941$$ −9.87564 −0.321937 −0.160968 0.986960i $$-0.551462\pi$$
−0.160968 + 0.986960i $$0.551462\pi$$
$$942$$ 0 0
$$943$$ −58.6410 −1.90961
$$944$$ −6.19615 −0.201668
$$945$$ 0 0
$$946$$ 22.9282 0.745460
$$947$$ −2.24871 −0.0730733 −0.0365366 0.999332i $$-0.511633\pi$$
−0.0365366 + 0.999332i $$0.511633\pi$$
$$948$$ 0 0
$$949$$ 5.44486 0.176748
$$950$$ 24.3923 0.791391
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 10.4115 0.337263 0.168631 0.985679i $$-0.446065\pi$$
0.168631 + 0.985679i $$0.446065\pi$$
$$954$$ 0 0
$$955$$ −43.5167 −1.40817
$$956$$ −28.0526 −0.907285
$$957$$ 0 0
$$958$$ 23.5167 0.759789
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −26.1769 −0.844417
$$962$$ 3.09103 0.0996590
$$963$$ 0 0
$$964$$ 17.7321 0.571111
$$965$$ −33.0526 −1.06400
$$966$$ 0 0
$$967$$ 13.6603 0.439284 0.219642 0.975581i $$-0.429511\pi$$
0.219642 + 0.975581i $$0.429511\pi$$
$$968$$ 6.60770 0.212379
$$969$$ 0 0
$$970$$ 10.9282 0.350883
$$971$$ 33.1244 1.06301 0.531506 0.847055i $$-0.321626\pi$$
0.531506 + 0.847055i $$0.321626\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −28.5885 −0.916033
$$975$$ 0 0
$$976$$ 9.92820 0.317794
$$977$$ −2.28719 −0.0731736 −0.0365868 0.999330i $$-0.511649\pi$$
−0.0365868 + 0.999330i $$0.511649\pi$$
$$978$$ 0 0
$$979$$ −16.4833 −0.526810
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −33.4641 −1.06788
$$983$$ −10.6410 −0.339396 −0.169698 0.985496i $$-0.554279\pi$$
−0.169698 + 0.985496i $$0.554279\pi$$
$$984$$ 0 0
$$985$$ 96.2295 3.06613
$$986$$ −59.2487 −1.88686
$$987$$ 0 0
$$988$$ −1.26795 −0.0403388
$$989$$ −33.8564 −1.07657
$$990$$ 0 0
$$991$$ −10.3397 −0.328453 −0.164226 0.986423i $$-0.552513\pi$$
−0.164226 + 0.986423i $$0.552513\pi$$
$$992$$ 2.19615 0.0697279
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −19.1244 −0.606283
$$996$$ 0 0
$$997$$ 7.24871 0.229569 0.114784 0.993390i $$-0.463382\pi$$
0.114784 + 0.993390i $$0.463382\pi$$
$$998$$ 30.1962 0.955843
$$999$$ 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 7938.2.a.bt.1.2 2
3.2 odd 2 7938.2.a.bg.1.1 2
7.6 odd 2 1134.2.a.m.1.1 yes 2
21.20 even 2 1134.2.a.l.1.2 2
28.27 even 2 9072.2.a.y.1.1 2
63.13 odd 6 1134.2.f.r.379.2 4
63.20 even 6 1134.2.f.s.757.1 4
63.34 odd 6 1134.2.f.r.757.2 4
63.41 even 6 1134.2.f.s.379.1 4
84.83 odd 2 9072.2.a.bp.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.2 2 21.20 even 2
1134.2.a.m.1.1 yes 2 7.6 odd 2
1134.2.f.r.379.2 4 63.13 odd 6
1134.2.f.r.757.2 4 63.34 odd 6
1134.2.f.s.379.1 4 63.41 even 6
1134.2.f.s.757.1 4 63.20 even 6
7938.2.a.bg.1.1 2 3.2 odd 2
7938.2.a.bt.1.2 2 1.1 even 1 trivial
9072.2.a.y.1.1 2 28.27 even 2
9072.2.a.bp.1.2 2 84.83 odd 2