# Properties

 Label 6525.2.a.bs.1.2 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.71250$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} +O(q^{10})$$ $$q-0.712495 q^{2} -1.49235 q^{4} -2.77986 q^{7} +2.48828 q^{8} -4.26814 q^{11} +0.779856 q^{13} +1.98063 q^{14} +1.21181 q^{16} +1.90354 q^{17} +6.72036 q^{19} +3.04103 q^{22} -2.17168 q^{23} -0.555643 q^{26} +4.14852 q^{28} -1.00000 q^{29} -8.82061 q^{31} -5.83998 q^{32} -1.35626 q^{34} +1.48402 q^{37} -4.78822 q^{38} +7.71389 q^{41} -8.19624 q^{43} +6.36956 q^{44} +1.54731 q^{46} +5.19381 q^{47} +0.727598 q^{49} -1.16382 q^{52} +11.7853 q^{53} -6.91707 q^{56} +0.712495 q^{58} +4.46028 q^{59} -5.24905 q^{61} +6.28464 q^{62} +1.73733 q^{64} -8.49375 q^{67} -2.84075 q^{68} -0.663102 q^{71} +16.5345 q^{73} -1.05736 q^{74} -10.0291 q^{76} +11.8648 q^{77} +9.54554 q^{79} -5.49611 q^{82} +0.0123998 q^{83} +5.83978 q^{86} -10.6203 q^{88} +5.46783 q^{89} -2.16789 q^{91} +3.24091 q^{92} -3.70056 q^{94} -0.952006 q^{97} -0.518410 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10})$$ 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 $$5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} - 12 q^{11} - 2 q^{13} - 6 q^{14} + q^{16} - 2 q^{19} - 14 q^{22} + 8 q^{23} + 6 q^{28} - 5 q^{29} + 2 q^{31} + q^{32} + 4 q^{34} - 16 q^{37} - 14 q^{38} + 14 q^{41} - 20 q^{44} - 6 q^{46} + 2 q^{47} - 7 q^{49} - 16 q^{52} + 26 q^{53} + 2 q^{56} - 3 q^{58} - 4 q^{59} - 12 q^{61} - 9 q^{64} - 12 q^{67} - 20 q^{68} - 30 q^{71} + 12 q^{73} - 2 q^{74} - 44 q^{76} + 18 q^{77} - 18 q^{79} - 10 q^{82} + 2 q^{83} - 30 q^{86} - 42 q^{88} + 22 q^{89} - 12 q^{91} + 20 q^{92} - 50 q^{94} - 20 q^{97} - 9 q^{98}+O(q^{100})$$ 5 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 - 12 * q^11 - 2 * q^13 - 6 * q^14 + q^16 - 2 * q^19 - 14 * q^22 + 8 * q^23 + 6 * q^28 - 5 * q^29 + 2 * q^31 + q^32 + 4 * q^34 - 16 * q^37 - 14 * q^38 + 14 * q^41 - 20 * q^44 - 6 * q^46 + 2 * q^47 - 7 * q^49 - 16 * q^52 + 26 * q^53 + 2 * q^56 - 3 * q^58 - 4 * q^59 - 12 * q^61 - 9 * q^64 - 12 * q^67 - 20 * q^68 - 30 * q^71 + 12 * q^73 - 2 * q^74 - 44 * q^76 + 18 * q^77 - 18 * q^79 - 10 * q^82 + 2 * q^83 - 30 * q^86 - 42 * q^88 + 22 * q^89 - 12 * q^91 + 20 * q^92 - 50 * q^94 - 20 * q^97 - 9 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.712495 −0.503810 −0.251905 0.967752i $$-0.581057\pi$$
−0.251905 + 0.967752i $$0.581057\pi$$
$$3$$ 0 0
$$4$$ −1.49235 −0.746175
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.77986 −1.05069 −0.525343 0.850890i $$-0.676063\pi$$
−0.525343 + 0.850890i $$0.676063\pi$$
$$8$$ 2.48828 0.879741
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.26814 −1.28689 −0.643446 0.765491i $$-0.722496\pi$$
−0.643446 + 0.765491i $$0.722496\pi$$
$$12$$ 0 0
$$13$$ 0.779856 0.216293 0.108147 0.994135i $$-0.465508\pi$$
0.108147 + 0.994135i $$0.465508\pi$$
$$14$$ 1.98063 0.529347
$$15$$ 0 0
$$16$$ 1.21181 0.302953
$$17$$ 1.90354 0.461677 0.230838 0.972992i $$-0.425853\pi$$
0.230838 + 0.972992i $$0.425853\pi$$
$$18$$ 0 0
$$19$$ 6.72036 1.54176 0.770878 0.636983i $$-0.219818\pi$$
0.770878 + 0.636983i $$0.219818\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.04103 0.648349
$$23$$ −2.17168 −0.452827 −0.226413 0.974031i $$-0.572700\pi$$
−0.226413 + 0.974031i $$0.572700\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −0.555643 −0.108971
$$27$$ 0 0
$$28$$ 4.14852 0.783997
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.82061 −1.58423 −0.792114 0.610373i $$-0.791020\pi$$
−0.792114 + 0.610373i $$0.791020\pi$$
$$32$$ −5.83998 −1.03237
$$33$$ 0 0
$$34$$ −1.35626 −0.232597
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.48402 0.243971 0.121986 0.992532i $$-0.461074\pi$$
0.121986 + 0.992532i $$0.461074\pi$$
$$38$$ −4.78822 −0.776752
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.71389 1.20471 0.602354 0.798229i $$-0.294230\pi$$
0.602354 + 0.798229i $$0.294230\pi$$
$$42$$ 0 0
$$43$$ −8.19624 −1.24991 −0.624957 0.780659i $$-0.714884\pi$$
−0.624957 + 0.780659i $$0.714884\pi$$
$$44$$ 6.36956 0.960247
$$45$$ 0 0
$$46$$ 1.54731 0.228139
$$47$$ 5.19381 0.757595 0.378798 0.925480i $$-0.376338\pi$$
0.378798 + 0.925480i $$0.376338\pi$$
$$48$$ 0 0
$$49$$ 0.727598 0.103943
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.16382 −0.161393
$$53$$ 11.7853 1.61884 0.809419 0.587231i $$-0.199782\pi$$
0.809419 + 0.587231i $$0.199782\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −6.91707 −0.924332
$$57$$ 0 0
$$58$$ 0.712495 0.0935552
$$59$$ 4.46028 0.580679 0.290339 0.956924i $$-0.406232\pi$$
0.290339 + 0.956924i $$0.406232\pi$$
$$60$$ 0 0
$$61$$ −5.24905 −0.672071 −0.336036 0.941849i $$-0.609086\pi$$
−0.336036 + 0.941849i $$0.609086\pi$$
$$62$$ 6.28464 0.798150
$$63$$ 0 0
$$64$$ 1.73733 0.217166
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.49375 −1.03768 −0.518838 0.854872i $$-0.673635\pi$$
−0.518838 + 0.854872i $$0.673635\pi$$
$$68$$ −2.84075 −0.344492
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −0.663102 −0.0786957 −0.0393479 0.999226i $$-0.512528\pi$$
−0.0393479 + 0.999226i $$0.512528\pi$$
$$72$$ 0 0
$$73$$ 16.5345 1.93522 0.967609 0.252455i $$-0.0812380\pi$$
0.967609 + 0.252455i $$0.0812380\pi$$
$$74$$ −1.05736 −0.122915
$$75$$ 0 0
$$76$$ −10.0291 −1.15042
$$77$$ 11.8648 1.35212
$$78$$ 0 0
$$79$$ 9.54554 1.07396 0.536978 0.843596i $$-0.319566\pi$$
0.536978 + 0.843596i $$0.319566\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −5.49611 −0.606944
$$83$$ 0.0123998 0.00136106 0.000680528 1.00000i $$-0.499783\pi$$
0.000680528 1.00000i $$0.499783\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 5.83978 0.629720
$$87$$ 0 0
$$88$$ −10.6203 −1.13213
$$89$$ 5.46783 0.579588 0.289794 0.957089i $$-0.406413\pi$$
0.289794 + 0.957089i $$0.406413\pi$$
$$90$$ 0 0
$$91$$ −2.16789 −0.227256
$$92$$ 3.24091 0.337888
$$93$$ 0 0
$$94$$ −3.70056 −0.381684
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.952006 −0.0966615 −0.0483308 0.998831i $$-0.515390\pi$$
−0.0483308 + 0.998831i $$0.515390\pi$$
$$98$$ −0.518410 −0.0523673
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.31781 −0.827653 −0.413826 0.910356i $$-0.635808\pi$$
−0.413826 + 0.910356i $$0.635808\pi$$
$$102$$ 0 0
$$103$$ 12.4091 1.22271 0.611355 0.791357i $$-0.290625\pi$$
0.611355 + 0.791357i $$0.290625\pi$$
$$104$$ 1.94050 0.190282
$$105$$ 0 0
$$106$$ −8.39698 −0.815587
$$107$$ 17.5579 1.69739 0.848695 0.528883i $$-0.177389\pi$$
0.848695 + 0.528883i $$0.177389\pi$$
$$108$$ 0 0
$$109$$ 1.00973 0.0967146 0.0483573 0.998830i $$-0.484601\pi$$
0.0483573 + 0.998830i $$0.484601\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −3.36866 −0.318309
$$113$$ 15.5787 1.46552 0.732761 0.680487i $$-0.238232\pi$$
0.732761 + 0.680487i $$0.238232\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.49235 0.138561
$$117$$ 0 0
$$118$$ −3.17792 −0.292552
$$119$$ −5.29157 −0.485078
$$120$$ 0 0
$$121$$ 7.21701 0.656091
$$122$$ 3.73992 0.338596
$$123$$ 0 0
$$124$$ 13.1634 1.18211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −16.9536 −1.50438 −0.752192 0.658943i $$-0.771004\pi$$
−0.752192 + 0.658943i $$0.771004\pi$$
$$128$$ 10.4421 0.922961
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −11.9324 −1.04254 −0.521268 0.853393i $$-0.674541\pi$$
−0.521268 + 0.853393i $$0.674541\pi$$
$$132$$ 0 0
$$133$$ −18.6816 −1.61990
$$134$$ 6.05175 0.522792
$$135$$ 0 0
$$136$$ 4.73655 0.406156
$$137$$ 0.239785 0.0204862 0.0102431 0.999948i $$-0.496739\pi$$
0.0102431 + 0.999948i $$0.496739\pi$$
$$138$$ 0 0
$$139$$ −17.0255 −1.44408 −0.722040 0.691851i $$-0.756795\pi$$
−0.722040 + 0.691851i $$0.756795\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0.472457 0.0396477
$$143$$ −3.32853 −0.278346
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −11.7808 −0.974982
$$147$$ 0 0
$$148$$ −2.21468 −0.182045
$$149$$ 4.13046 0.338380 0.169190 0.985583i $$-0.445885\pi$$
0.169190 + 0.985583i $$0.445885\pi$$
$$150$$ 0 0
$$151$$ −3.21580 −0.261698 −0.130849 0.991402i $$-0.541770\pi$$
−0.130849 + 0.991402i $$0.541770\pi$$
$$152$$ 16.7221 1.35635
$$153$$ 0 0
$$154$$ −8.45362 −0.681212
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.84237 0.546081 0.273040 0.962003i $$-0.411971\pi$$
0.273040 + 0.962003i $$0.411971\pi$$
$$158$$ −6.80115 −0.541070
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.03696 0.475779
$$162$$ 0 0
$$163$$ −14.0502 −1.10050 −0.550248 0.835001i $$-0.685467\pi$$
−0.550248 + 0.835001i $$0.685467\pi$$
$$164$$ −11.5118 −0.898923
$$165$$ 0 0
$$166$$ −0.00883480 −0.000685713 0
$$167$$ −17.8725 −1.38301 −0.691507 0.722370i $$-0.743053\pi$$
−0.691507 + 0.722370i $$0.743053\pi$$
$$168$$ 0 0
$$169$$ −12.3918 −0.953217
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.2317 0.932656
$$173$$ 2.82518 0.214795 0.107397 0.994216i $$-0.465748\pi$$
0.107397 + 0.994216i $$0.465748\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −5.17218 −0.389868
$$177$$ 0 0
$$178$$ −3.89580 −0.292002
$$179$$ −20.2829 −1.51601 −0.758006 0.652247i $$-0.773826\pi$$
−0.758006 + 0.652247i $$0.773826\pi$$
$$180$$ 0 0
$$181$$ −8.63783 −0.642045 −0.321023 0.947072i $$-0.604027\pi$$
−0.321023 + 0.947072i $$0.604027\pi$$
$$182$$ 1.54461 0.114494
$$183$$ 0 0
$$184$$ −5.40376 −0.398370
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −8.12458 −0.594128
$$188$$ −7.75099 −0.565299
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −0.896850 −0.0648938 −0.0324469 0.999473i $$-0.510330\pi$$
−0.0324469 + 0.999473i $$0.510330\pi$$
$$192$$ 0 0
$$193$$ −10.4506 −0.752251 −0.376126 0.926569i $$-0.622744\pi$$
−0.376126 + 0.926569i $$0.622744\pi$$
$$194$$ 0.678299 0.0486991
$$195$$ 0 0
$$196$$ −1.08583 −0.0775594
$$197$$ −9.57740 −0.682361 −0.341181 0.939998i $$-0.610827\pi$$
−0.341181 + 0.939998i $$0.610827\pi$$
$$198$$ 0 0
$$199$$ 5.61768 0.398227 0.199113 0.979976i $$-0.436194\pi$$
0.199113 + 0.979976i $$0.436194\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 5.92640 0.416980
$$203$$ 2.77986 0.195108
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.84145 −0.616013
$$207$$ 0 0
$$208$$ 0.945039 0.0655267
$$209$$ −28.6834 −1.98407
$$210$$ 0 0
$$211$$ −0.605216 −0.0416648 −0.0208324 0.999783i $$-0.506632\pi$$
−0.0208324 + 0.999783i $$0.506632\pi$$
$$212$$ −17.5878 −1.20794
$$213$$ 0 0
$$214$$ −12.5099 −0.855162
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 24.5200 1.66453
$$218$$ −0.719428 −0.0487258
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.48449 0.0998575
$$222$$ 0 0
$$223$$ 16.5537 1.10852 0.554260 0.832344i $$-0.313001\pi$$
0.554260 + 0.832344i $$0.313001\pi$$
$$224$$ 16.2343 1.08470
$$225$$ 0 0
$$226$$ −11.0997 −0.738344
$$227$$ 22.0061 1.46060 0.730299 0.683128i $$-0.239381\pi$$
0.730299 + 0.683128i $$0.239381\pi$$
$$228$$ 0 0
$$229$$ −24.5647 −1.62328 −0.811641 0.584157i $$-0.801425\pi$$
−0.811641 + 0.584157i $$0.801425\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2.48828 −0.163364
$$233$$ 5.56824 0.364787 0.182394 0.983226i $$-0.441615\pi$$
0.182394 + 0.983226i $$0.441615\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.65630 −0.433288
$$237$$ 0 0
$$238$$ 3.77022 0.244387
$$239$$ −6.90640 −0.446738 −0.223369 0.974734i $$-0.571705\pi$$
−0.223369 + 0.974734i $$0.571705\pi$$
$$240$$ 0 0
$$241$$ −25.1989 −1.62320 −0.811602 0.584210i $$-0.801404\pi$$
−0.811602 + 0.584210i $$0.801404\pi$$
$$242$$ −5.14208 −0.330545
$$243$$ 0 0
$$244$$ 7.83342 0.501483
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.24091 0.333471
$$248$$ −21.9482 −1.39371
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −28.3687 −1.79062 −0.895309 0.445446i $$-0.853045\pi$$
−0.895309 + 0.445446i $$0.853045\pi$$
$$252$$ 0 0
$$253$$ 9.26903 0.582739
$$254$$ 12.0793 0.757924
$$255$$ 0 0
$$256$$ −10.9146 −0.682163
$$257$$ −8.71100 −0.543377 −0.271688 0.962385i $$-0.587582\pi$$
−0.271688 + 0.962385i $$0.587582\pi$$
$$258$$ 0 0
$$259$$ −4.12536 −0.256337
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 8.50175 0.525240
$$263$$ 13.2128 0.814736 0.407368 0.913264i $$-0.366447\pi$$
0.407368 + 0.913264i $$0.366447\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 13.3106 0.816123
$$267$$ 0 0
$$268$$ 12.6757 0.774289
$$269$$ 10.3447 0.630725 0.315363 0.948971i $$-0.397874\pi$$
0.315363 + 0.948971i $$0.397874\pi$$
$$270$$ 0 0
$$271$$ 9.76022 0.592891 0.296445 0.955050i $$-0.404199\pi$$
0.296445 + 0.955050i $$0.404199\pi$$
$$272$$ 2.30674 0.139866
$$273$$ 0 0
$$274$$ −0.170845 −0.0103212
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.87503 0.112659 0.0563297 0.998412i $$-0.482060\pi$$
0.0563297 + 0.998412i $$0.482060\pi$$
$$278$$ 12.1306 0.727542
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 21.2930 1.27024 0.635118 0.772415i $$-0.280951\pi$$
0.635118 + 0.772415i $$0.280951\pi$$
$$282$$ 0 0
$$283$$ −15.8729 −0.943544 −0.471772 0.881721i $$-0.656385\pi$$
−0.471772 + 0.881721i $$0.656385\pi$$
$$284$$ 0.989581 0.0587208
$$285$$ 0 0
$$286$$ 2.37156 0.140233
$$287$$ −21.4435 −1.26577
$$288$$ 0 0
$$289$$ −13.3765 −0.786855
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −24.6753 −1.44401
$$293$$ 33.1369 1.93588 0.967938 0.251189i $$-0.0808216\pi$$
0.967938 + 0.251189i $$0.0808216\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 3.69266 0.214631
$$297$$ 0 0
$$298$$ −2.94293 −0.170479
$$299$$ −1.69360 −0.0979433
$$300$$ 0 0
$$301$$ 22.7844 1.31327
$$302$$ 2.29124 0.131846
$$303$$ 0 0
$$304$$ 8.14381 0.467080
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −17.2605 −0.985109 −0.492555 0.870282i $$-0.663937\pi$$
−0.492555 + 0.870282i $$0.663937\pi$$
$$308$$ −17.7065 −1.00892
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 32.5116 1.84356 0.921781 0.387711i $$-0.126734\pi$$
0.921781 + 0.387711i $$0.126734\pi$$
$$312$$ 0 0
$$313$$ −6.26304 −0.354008 −0.177004 0.984210i $$-0.556641\pi$$
−0.177004 + 0.984210i $$0.556641\pi$$
$$314$$ −4.87516 −0.275121
$$315$$ 0 0
$$316$$ −14.2453 −0.801360
$$317$$ −18.8523 −1.05885 −0.529426 0.848356i $$-0.677593\pi$$
−0.529426 + 0.848356i $$0.677593\pi$$
$$318$$ 0 0
$$319$$ 4.26814 0.238970
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −4.30130 −0.239702
$$323$$ 12.7925 0.711793
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 10.0107 0.554441
$$327$$ 0 0
$$328$$ 19.1943 1.05983
$$329$$ −14.4380 −0.795995
$$330$$ 0 0
$$331$$ −0.596245 −0.0327726 −0.0163863 0.999866i $$-0.505216\pi$$
−0.0163863 + 0.999866i $$0.505216\pi$$
$$332$$ −0.0185049 −0.00101559
$$333$$ 0 0
$$334$$ 12.7341 0.696776
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −32.6414 −1.77809 −0.889046 0.457818i $$-0.848631\pi$$
−0.889046 + 0.457818i $$0.848631\pi$$
$$338$$ 8.82911 0.480240
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 37.6476 2.03873
$$342$$ 0 0
$$343$$ 17.4364 0.941476
$$344$$ −20.3946 −1.09960
$$345$$ 0 0
$$346$$ −2.01293 −0.108216
$$347$$ −13.1985 −0.708531 −0.354265 0.935145i $$-0.615269\pi$$
−0.354265 + 0.935145i $$0.615269\pi$$
$$348$$ 0 0
$$349$$ 15.6412 0.837255 0.418628 0.908158i $$-0.362511\pi$$
0.418628 + 0.908158i $$0.362511\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 24.9258 1.32855
$$353$$ −5.65486 −0.300978 −0.150489 0.988612i $$-0.548085\pi$$
−0.150489 + 0.988612i $$0.548085\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −8.15991 −0.432475
$$357$$ 0 0
$$358$$ 14.4514 0.763782
$$359$$ −25.2693 −1.33366 −0.666832 0.745208i $$-0.732350\pi$$
−0.666832 + 0.745208i $$0.732350\pi$$
$$360$$ 0 0
$$361$$ 26.1632 1.37701
$$362$$ 6.15441 0.323469
$$363$$ 0 0
$$364$$ 3.23525 0.169573
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 26.2788 1.37174 0.685872 0.727722i $$-0.259421\pi$$
0.685872 + 0.727722i $$0.259421\pi$$
$$368$$ −2.63167 −0.137185
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −32.7615 −1.70089
$$372$$ 0 0
$$373$$ 1.43033 0.0740596 0.0370298 0.999314i $$-0.488210\pi$$
0.0370298 + 0.999314i $$0.488210\pi$$
$$374$$ 5.78872 0.299328
$$375$$ 0 0
$$376$$ 12.9237 0.666487
$$377$$ −0.779856 −0.0401646
$$378$$ 0 0
$$379$$ −35.2336 −1.80983 −0.904915 0.425591i $$-0.860066\pi$$
−0.904915 + 0.425591i $$0.860066\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0.639001 0.0326941
$$383$$ −32.0701 −1.63870 −0.819352 0.573291i $$-0.805666\pi$$
−0.819352 + 0.573291i $$0.805666\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 7.44601 0.378992
$$387$$ 0 0
$$388$$ 1.42073 0.0721265
$$389$$ −1.63922 −0.0831117 −0.0415558 0.999136i $$-0.513231\pi$$
−0.0415558 + 0.999136i $$0.513231\pi$$
$$390$$ 0 0
$$391$$ −4.13389 −0.209060
$$392$$ 1.81047 0.0914425
$$393$$ 0 0
$$394$$ 6.82385 0.343781
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.04098 −0.102434 −0.0512170 0.998688i $$-0.516310\pi$$
−0.0512170 + 0.998688i $$0.516310\pi$$
$$398$$ −4.00257 −0.200631
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −17.4108 −0.869455 −0.434727 0.900562i $$-0.643155\pi$$
−0.434727 + 0.900562i $$0.643155\pi$$
$$402$$ 0 0
$$403$$ −6.87880 −0.342658
$$404$$ 12.4131 0.617574
$$405$$ 0 0
$$406$$ −1.98063 −0.0982972
$$407$$ −6.33400 −0.313965
$$408$$ 0 0
$$409$$ −37.3010 −1.84441 −0.922207 0.386696i $$-0.873616\pi$$
−0.922207 + 0.386696i $$0.873616\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −18.5188 −0.912356
$$413$$ −12.3989 −0.610111
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −4.55434 −0.223295
$$417$$ 0 0
$$418$$ 20.4368 0.999596
$$419$$ −4.32941 −0.211506 −0.105753 0.994392i $$-0.533725\pi$$
−0.105753 + 0.994392i $$0.533725\pi$$
$$420$$ 0 0
$$421$$ −34.2505 −1.66927 −0.834634 0.550806i $$-0.814321\pi$$
−0.834634 + 0.550806i $$0.814321\pi$$
$$422$$ 0.431214 0.0209911
$$423$$ 0 0
$$424$$ 29.3252 1.42416
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 14.5916 0.706137
$$428$$ −26.2026 −1.26655
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.4885 −1.66125 −0.830626 0.556831i $$-0.812017\pi$$
−0.830626 + 0.556831i $$0.812017\pi$$
$$432$$ 0 0
$$433$$ −3.02258 −0.145256 −0.0726279 0.997359i $$-0.523139\pi$$
−0.0726279 + 0.997359i $$0.523139\pi$$
$$434$$ −17.4704 −0.838606
$$435$$ 0 0
$$436$$ −1.50687 −0.0721661
$$437$$ −14.5945 −0.698148
$$438$$ 0 0
$$439$$ 8.37392 0.399665 0.199833 0.979830i $$-0.435960\pi$$
0.199833 + 0.979830i $$0.435960\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −1.05769 −0.0503092
$$443$$ −29.8770 −1.41950 −0.709749 0.704454i $$-0.751192\pi$$
−0.709749 + 0.704454i $$0.751192\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −11.7945 −0.558484
$$447$$ 0 0
$$448$$ −4.82952 −0.228174
$$449$$ −12.0780 −0.569994 −0.284997 0.958528i $$-0.591993\pi$$
−0.284997 + 0.958528i $$0.591993\pi$$
$$450$$ 0 0
$$451$$ −32.9240 −1.55033
$$452$$ −23.2489 −1.09354
$$453$$ 0 0
$$454$$ −15.6793 −0.735864
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.03357 −0.141905 −0.0709523 0.997480i $$-0.522604\pi$$
−0.0709523 + 0.997480i $$0.522604\pi$$
$$458$$ 17.5022 0.817826
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 4.44508 0.207028 0.103514 0.994628i $$-0.466991\pi$$
0.103514 + 0.994628i $$0.466991\pi$$
$$462$$ 0 0
$$463$$ 9.04875 0.420531 0.210266 0.977644i $$-0.432567\pi$$
0.210266 + 0.977644i $$0.432567\pi$$
$$464$$ −1.21181 −0.0562570
$$465$$ 0 0
$$466$$ −3.96734 −0.183784
$$467$$ 8.11327 0.375437 0.187719 0.982223i $$-0.439891\pi$$
0.187719 + 0.982223i $$0.439891\pi$$
$$468$$ 0 0
$$469$$ 23.6114 1.09027
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 11.0984 0.510847
$$473$$ 34.9827 1.60851
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 7.89688 0.361953
$$477$$ 0 0
$$478$$ 4.92078 0.225071
$$479$$ −6.62947 −0.302908 −0.151454 0.988464i $$-0.548396\pi$$
−0.151454 + 0.988464i $$0.548396\pi$$
$$480$$ 0 0
$$481$$ 1.15732 0.0527693
$$482$$ 17.9541 0.817787
$$483$$ 0 0
$$484$$ −10.7703 −0.489559
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 29.2047 1.32339 0.661695 0.749773i $$-0.269837\pi$$
0.661695 + 0.749773i $$0.269837\pi$$
$$488$$ −13.0611 −0.591249
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.32464 0.195168 0.0975841 0.995227i $$-0.468889\pi$$
0.0975841 + 0.995227i $$0.468889\pi$$
$$492$$ 0 0
$$493$$ −1.90354 −0.0857312
$$494$$ −3.73412 −0.168006
$$495$$ 0 0
$$496$$ −10.6889 −0.479947
$$497$$ 1.84333 0.0826846
$$498$$ 0 0
$$499$$ 4.29688 0.192355 0.0961774 0.995364i $$-0.469338\pi$$
0.0961774 + 0.995364i $$0.469338\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 20.2126 0.902131
$$503$$ −14.2512 −0.635429 −0.317715 0.948186i $$-0.602915\pi$$
−0.317715 + 0.948186i $$0.602915\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −6.60414 −0.293590
$$507$$ 0 0
$$508$$ 25.3007 1.12254
$$509$$ 19.4974 0.864206 0.432103 0.901824i $$-0.357772\pi$$
0.432103 + 0.901824i $$0.357772\pi$$
$$510$$ 0 0
$$511$$ −45.9635 −2.03331
$$512$$ −13.1076 −0.579280
$$513$$ 0 0
$$514$$ 6.20654 0.273759
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −22.1679 −0.974943
$$518$$ 2.93930 0.129145
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −24.5229 −1.07437 −0.537185 0.843465i $$-0.680512\pi$$
−0.537185 + 0.843465i $$0.680512\pi$$
$$522$$ 0 0
$$523$$ 10.8566 0.474725 0.237362 0.971421i $$-0.423717\pi$$
0.237362 + 0.971421i $$0.423717\pi$$
$$524$$ 17.8073 0.777914
$$525$$ 0 0
$$526$$ −9.41406 −0.410472
$$527$$ −16.7904 −0.731401
$$528$$ 0 0
$$529$$ −18.2838 −0.794948
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 27.8795 1.20873
$$533$$ 6.01572 0.260570
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −21.1348 −0.912886
$$537$$ 0 0
$$538$$ −7.37052 −0.317766
$$539$$ −3.10549 −0.133763
$$540$$ 0 0
$$541$$ −22.7032 −0.976085 −0.488043 0.872820i $$-0.662289\pi$$
−0.488043 + 0.872820i $$0.662289\pi$$
$$542$$ −6.95410 −0.298704
$$543$$ 0 0
$$544$$ −11.1166 −0.476622
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 32.8229 1.40340 0.701702 0.712471i $$-0.252424\pi$$
0.701702 + 0.712471i $$0.252424\pi$$
$$548$$ −0.357843 −0.0152863
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.72036 −0.286297
$$552$$ 0 0
$$553$$ −26.5352 −1.12839
$$554$$ −1.33595 −0.0567590
$$555$$ 0 0
$$556$$ 25.4080 1.07754
$$557$$ 26.3234 1.11536 0.557679 0.830057i $$-0.311692\pi$$
0.557679 + 0.830057i $$0.311692\pi$$
$$558$$ 0 0
$$559$$ −6.39189 −0.270348
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −15.1712 −0.639958
$$563$$ 28.4750 1.20008 0.600039 0.799971i $$-0.295152\pi$$
0.600039 + 0.799971i $$0.295152\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 11.3093 0.475367
$$567$$ 0 0
$$568$$ −1.64999 −0.0692318
$$569$$ 14.7555 0.618582 0.309291 0.950967i $$-0.399908\pi$$
0.309291 + 0.950967i $$0.399908\pi$$
$$570$$ 0 0
$$571$$ −0.204397 −0.00855376 −0.00427688 0.999991i $$-0.501361\pi$$
−0.00427688 + 0.999991i $$0.501361\pi$$
$$572$$ 4.96734 0.207695
$$573$$ 0 0
$$574$$ 15.2784 0.637708
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 37.3300 1.55407 0.777035 0.629457i $$-0.216723\pi$$
0.777035 + 0.629457i $$0.216723\pi$$
$$578$$ 9.53071 0.396425
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −0.0344697 −0.00143004
$$582$$ 0 0
$$583$$ −50.3014 −2.08327
$$584$$ 41.1425 1.70249
$$585$$ 0 0
$$586$$ −23.6098 −0.975314
$$587$$ −10.8497 −0.447813 −0.223907 0.974611i $$-0.571881\pi$$
−0.223907 + 0.974611i $$0.571881\pi$$
$$588$$ 0 0
$$589$$ −59.2776 −2.44249
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.79835 0.0739119
$$593$$ 14.0575 0.577272 0.288636 0.957439i $$-0.406798\pi$$
0.288636 + 0.957439i $$0.406798\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.16409 −0.252491
$$597$$ 0 0
$$598$$ 1.20668 0.0493448
$$599$$ 5.52446 0.225724 0.112862 0.993611i $$-0.463998\pi$$
0.112862 + 0.993611i $$0.463998\pi$$
$$600$$ 0 0
$$601$$ −6.85394 −0.279578 −0.139789 0.990181i $$-0.544642\pi$$
−0.139789 + 0.990181i $$0.544642\pi$$
$$602$$ −16.2337 −0.661638
$$603$$ 0 0
$$604$$ 4.79910 0.195273
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −30.4979 −1.23787 −0.618935 0.785442i $$-0.712436\pi$$
−0.618935 + 0.785442i $$0.712436\pi$$
$$608$$ −39.2467 −1.59166
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4.05042 0.163863
$$612$$ 0 0
$$613$$ −29.9211 −1.20850 −0.604250 0.796794i $$-0.706527\pi$$
−0.604250 + 0.796794i $$0.706527\pi$$
$$614$$ 12.2980 0.496308
$$615$$ 0 0
$$616$$ 29.5230 1.18952
$$617$$ −1.04305 −0.0419917 −0.0209959 0.999780i $$-0.506684\pi$$
−0.0209959 + 0.999780i $$0.506684\pi$$
$$618$$ 0 0
$$619$$ 22.1661 0.890930 0.445465 0.895299i $$-0.353038\pi$$
0.445465 + 0.895299i $$0.353038\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −23.1643 −0.928805
$$623$$ −15.1998 −0.608966
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 4.46238 0.178353
$$627$$ 0 0
$$628$$ −10.2112 −0.407472
$$629$$ 2.82489 0.112636
$$630$$ 0 0
$$631$$ 20.1089 0.800524 0.400262 0.916401i $$-0.368919\pi$$
0.400262 + 0.916401i $$0.368919\pi$$
$$632$$ 23.7520 0.944804
$$633$$ 0 0
$$634$$ 13.4322 0.533460
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.567422 0.0224821
$$638$$ −3.04103 −0.120395
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −5.93422 −0.234388 −0.117194 0.993109i $$-0.537390\pi$$
−0.117194 + 0.993109i $$0.537390\pi$$
$$642$$ 0 0
$$643$$ −0.951317 −0.0375163 −0.0187581 0.999824i $$-0.505971\pi$$
−0.0187581 + 0.999824i $$0.505971\pi$$
$$644$$ −9.00926 −0.355015
$$645$$ 0 0
$$646$$ −9.11458 −0.358608
$$647$$ −35.1384 −1.38143 −0.690716 0.723126i $$-0.742705\pi$$
−0.690716 + 0.723126i $$0.742705\pi$$
$$648$$ 0 0
$$649$$ −19.0371 −0.747271
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.9678 0.821163
$$653$$ −45.9930 −1.79984 −0.899922 0.436050i $$-0.856377\pi$$
−0.899922 + 0.436050i $$0.856377\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 9.34779 0.364970
$$657$$ 0 0
$$658$$ 10.2870 0.401030
$$659$$ 19.3162 0.752451 0.376226 0.926528i $$-0.377222\pi$$
0.376226 + 0.926528i $$0.377222\pi$$
$$660$$ 0 0
$$661$$ 44.3224 1.72394 0.861970 0.506959i $$-0.169230\pi$$
0.861970 + 0.506959i $$0.169230\pi$$
$$662$$ 0.424821 0.0165111
$$663$$ 0 0
$$664$$ 0.0308542 0.00119738
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.17168 0.0840878
$$668$$ 26.6720 1.03197
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 22.4037 0.864883
$$672$$ 0 0
$$673$$ −12.7421 −0.491170 −0.245585 0.969375i $$-0.578980\pi$$
−0.245585 + 0.969375i $$0.578980\pi$$
$$674$$ 23.2569 0.895821
$$675$$ 0 0
$$676$$ 18.4930 0.711267
$$677$$ −9.41654 −0.361907 −0.180954 0.983492i $$-0.557918\pi$$
−0.180954 + 0.983492i $$0.557918\pi$$
$$678$$ 0 0
$$679$$ 2.64644 0.101561
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −26.8237 −1.02713
$$683$$ 6.92316 0.264907 0.132454 0.991189i $$-0.457714\pi$$
0.132454 + 0.991189i $$0.457714\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −12.4233 −0.474325
$$687$$ 0 0
$$688$$ −9.93231 −0.378666
$$689$$ 9.19085 0.350144
$$690$$ 0 0
$$691$$ −48.7272 −1.85367 −0.926835 0.375469i $$-0.877482\pi$$
−0.926835 + 0.375469i $$0.877482\pi$$
$$692$$ −4.21616 −0.160274
$$693$$ 0 0
$$694$$ 9.40384 0.356965
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 14.6837 0.556186
$$698$$ −11.1443 −0.421818
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2.88580 −0.108995 −0.0544975 0.998514i $$-0.517356\pi$$
−0.0544975 + 0.998514i $$0.517356\pi$$
$$702$$ 0 0
$$703$$ 9.97314 0.376144
$$704$$ −7.41516 −0.279469
$$705$$ 0 0
$$706$$ 4.02906 0.151636
$$707$$ 23.1223 0.869604
$$708$$ 0 0
$$709$$ −1.73056 −0.0649924 −0.0324962 0.999472i $$-0.510346\pi$$
−0.0324962 + 0.999472i $$0.510346\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 13.6055 0.509887
$$713$$ 19.1555 0.717381
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 30.2692 1.13121
$$717$$ 0 0
$$718$$ 18.0043 0.671913
$$719$$ 12.9089 0.481420 0.240710 0.970597i $$-0.422620\pi$$
0.240710 + 0.970597i $$0.422620\pi$$
$$720$$ 0 0
$$721$$ −34.4956 −1.28468
$$722$$ −18.6412 −0.693752
$$723$$ 0 0
$$724$$ 12.8907 0.479078
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 38.3405 1.42197 0.710985 0.703207i $$-0.248249\pi$$
0.710985 + 0.703207i $$0.248249\pi$$
$$728$$ −5.39431 −0.199927
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −15.6019 −0.577057
$$732$$ 0 0
$$733$$ 34.8966 1.28894 0.644468 0.764631i $$-0.277079\pi$$
0.644468 + 0.764631i $$0.277079\pi$$
$$734$$ −18.7235 −0.691098
$$735$$ 0 0
$$736$$ 12.6826 0.467485
$$737$$ 36.2525 1.33538
$$738$$ 0 0
$$739$$ 14.2969 0.525921 0.262960 0.964807i $$-0.415301\pi$$
0.262960 + 0.964807i $$0.415301\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 23.3424 0.856927
$$743$$ −44.1357 −1.61918 −0.809591 0.586995i $$-0.800311\pi$$
−0.809591 + 0.586995i $$0.800311\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.01910 −0.0373120
$$747$$ 0 0
$$748$$ 12.1247 0.443324
$$749$$ −48.8085 −1.78343
$$750$$ 0 0
$$751$$ 9.56375 0.348986 0.174493 0.984658i $$-0.444171\pi$$
0.174493 + 0.984658i $$0.444171\pi$$
$$752$$ 6.29393 0.229516
$$753$$ 0 0
$$754$$ 0.555643 0.0202353
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −30.1910 −1.09731 −0.548656 0.836048i $$-0.684860\pi$$
−0.548656 + 0.836048i $$0.684860\pi$$
$$758$$ 25.1038 0.911811
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −8.27038 −0.299801 −0.149900 0.988701i $$-0.547895\pi$$
−0.149900 + 0.988701i $$0.547895\pi$$
$$762$$ 0 0
$$763$$ −2.80690 −0.101617
$$764$$ 1.33841 0.0484221
$$765$$ 0 0
$$766$$ 22.8498 0.825595
$$767$$ 3.47837 0.125597
$$768$$ 0 0
$$769$$ −22.4592 −0.809901 −0.404950 0.914339i $$-0.632711\pi$$
−0.404950 + 0.914339i $$0.632711\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 15.5960 0.561311
$$773$$ −26.9520 −0.969398 −0.484699 0.874681i $$-0.661071\pi$$
−0.484699 + 0.874681i $$0.661071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −2.36886 −0.0850371
$$777$$ 0 0
$$778$$ 1.16794 0.0418725
$$779$$ 51.8401 1.85737
$$780$$ 0 0
$$781$$ 2.83021 0.101273
$$782$$ 2.94537 0.105326
$$783$$ 0 0
$$784$$ 0.881713 0.0314897
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −1.16423 −0.0415004 −0.0207502 0.999785i $$-0.506605\pi$$
−0.0207502 + 0.999785i $$0.506605\pi$$
$$788$$ 14.2928 0.509161
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −43.3065 −1.53980
$$792$$ 0 0
$$793$$ −4.09350 −0.145364
$$794$$ 1.45419 0.0516073
$$795$$ 0 0
$$796$$ −8.38354 −0.297147
$$797$$ 15.9624 0.565418 0.282709 0.959206i $$-0.408767\pi$$
0.282709 + 0.959206i $$0.408767\pi$$
$$798$$ 0 0
$$799$$ 9.88664 0.349764
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 12.4051 0.438040
$$803$$ −70.5715 −2.49042
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.90111 0.172634
$$807$$ 0 0
$$808$$ −20.6971 −0.728120
$$809$$ 7.83605 0.275501 0.137750 0.990467i $$-0.456013\pi$$
0.137750 + 0.990467i $$0.456013\pi$$
$$810$$ 0 0
$$811$$ −23.9935 −0.842526 −0.421263 0.906939i $$-0.638413\pi$$
−0.421263 + 0.906939i $$0.638413\pi$$
$$812$$ −4.14852 −0.145585
$$813$$ 0 0
$$814$$ 4.51294 0.158179
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −55.0817 −1.92706
$$818$$ 26.5768 0.929235
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 12.8804 0.449528 0.224764 0.974413i $$-0.427839\pi$$
0.224764 + 0.974413i $$0.427839\pi$$
$$822$$ 0 0
$$823$$ −47.8165 −1.66678 −0.833389 0.552687i $$-0.813603\pi$$
−0.833389 + 0.552687i $$0.813603\pi$$
$$824$$ 30.8775 1.07567
$$825$$ 0 0
$$826$$ 8.83417 0.307380
$$827$$ −14.6873 −0.510726 −0.255363 0.966845i $$-0.582195\pi$$
−0.255363 + 0.966845i $$0.582195\pi$$
$$828$$ 0 0
$$829$$ 41.9876 1.45829 0.729145 0.684359i $$-0.239918\pi$$
0.729145 + 0.684359i $$0.239918\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.35487 0.0469715
$$833$$ 1.38501 0.0479879
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 42.8057 1.48047
$$837$$ 0 0
$$838$$ 3.08468 0.106559
$$839$$ 4.04408 0.139617 0.0698085 0.997560i $$-0.477761\pi$$
0.0698085 + 0.997560i $$0.477761\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 24.4033 0.840994
$$843$$ 0 0
$$844$$ 0.903195 0.0310892
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −20.0622 −0.689347
$$848$$ 14.2816 0.490432
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −3.22282 −0.110477
$$852$$ 0 0
$$853$$ 43.8299 1.50071 0.750353 0.661037i $$-0.229883\pi$$
0.750353 + 0.661037i $$0.229883\pi$$
$$854$$ −10.3964 −0.355759
$$855$$ 0 0
$$856$$ 43.6891 1.49326
$$857$$ 2.09275 0.0714868 0.0357434 0.999361i $$-0.488620\pi$$
0.0357434 + 0.999361i $$0.488620\pi$$
$$858$$ 0 0
$$859$$ 46.5885 1.58958 0.794790 0.606885i $$-0.207581\pi$$
0.794790 + 0.606885i $$0.207581\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 24.5729 0.836955
$$863$$ 32.1466 1.09428 0.547141 0.837040i $$-0.315716\pi$$
0.547141 + 0.837040i $$0.315716\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 2.15357 0.0731814
$$867$$ 0 0
$$868$$ −36.5925 −1.24203
$$869$$ −40.7417 −1.38207
$$870$$ 0 0
$$871$$ −6.62390 −0.224442
$$872$$ 2.51249 0.0850838
$$873$$ 0 0
$$874$$ 10.3985 0.351734
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −48.0544 −1.62268 −0.811342 0.584572i $$-0.801262\pi$$
−0.811342 + 0.584572i $$0.801262\pi$$
$$878$$ −5.96637 −0.201355
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.9846 1.04390 0.521950 0.852976i $$-0.325205\pi$$
0.521950 + 0.852976i $$0.325205\pi$$
$$882$$ 0 0
$$883$$ −9.60378 −0.323193 −0.161597 0.986857i $$-0.551664\pi$$
−0.161597 + 0.986857i $$0.551664\pi$$
$$884$$ −2.21538 −0.0745112
$$885$$ 0 0
$$886$$ 21.2872 0.715158
$$887$$ −29.7279 −0.998165 −0.499083 0.866554i $$-0.666330\pi$$
−0.499083 + 0.866554i $$0.666330\pi$$
$$888$$ 0 0
$$889$$ 47.1284 1.58064
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −24.7040 −0.827150
$$893$$ 34.9043 1.16803
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −29.0276 −0.969743
$$897$$ 0 0
$$898$$ 8.60548 0.287169
$$899$$ 8.82061 0.294184
$$900$$ 0 0
$$901$$ 22.4339 0.747380
$$902$$ 23.4582 0.781071
$$903$$ 0 0
$$904$$ 38.7642 1.28928
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −4.35876 −0.144730 −0.0723651 0.997378i $$-0.523055\pi$$
−0.0723651 + 0.997378i $$0.523055\pi$$
$$908$$ −32.8409 −1.08986
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0.389317 0.0128986 0.00644932 0.999979i $$-0.497947\pi$$
0.00644932 + 0.999979i $$0.497947\pi$$
$$912$$ 0 0
$$913$$ −0.0529241 −0.00175153
$$914$$ 2.16141 0.0714930
$$915$$ 0 0
$$916$$ 36.6592 1.21125
$$917$$ 33.1703 1.09538
$$918$$ 0 0
$$919$$ −54.9915 −1.81400 −0.907001 0.421128i $$-0.861634\pi$$
−0.907001 + 0.421128i $$0.861634\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −3.16710 −0.104303
$$923$$ −0.517124 −0.0170213
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −6.44719 −0.211868
$$927$$ 0 0
$$928$$ 5.83998 0.191707
$$929$$ −17.4617 −0.572901 −0.286450 0.958095i $$-0.592475\pi$$
−0.286450 + 0.958095i $$0.592475\pi$$
$$930$$ 0 0
$$931$$ 4.88972 0.160254
$$932$$ −8.30977 −0.272195
$$933$$ 0 0
$$934$$ −5.78066 −0.189149
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −27.4261 −0.895970 −0.447985 0.894041i $$-0.647858\pi$$
−0.447985 + 0.894041i $$0.647858\pi$$
$$938$$ −16.8230 −0.549291
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 35.1587 1.14614 0.573070 0.819506i $$-0.305752\pi$$
0.573070 + 0.819506i $$0.305752\pi$$
$$942$$ 0 0
$$943$$ −16.7521 −0.545524
$$944$$ 5.40502 0.175918
$$945$$ 0 0
$$946$$ −24.9250 −0.810381
$$947$$ −44.1438 −1.43448 −0.717240 0.696827i $$-0.754595\pi$$
−0.717240 + 0.696827i $$0.754595\pi$$
$$948$$ 0 0
$$949$$ 12.8945 0.418574
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −13.1669 −0.426743
$$953$$ −27.3169 −0.884881 −0.442441 0.896798i $$-0.645887\pi$$
−0.442441 + 0.896798i $$0.645887\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 10.3068 0.333345
$$957$$ 0 0
$$958$$ 4.72346 0.152608
$$959$$ −0.666567 −0.0215246
$$960$$ 0 0
$$961$$ 46.8031 1.50978
$$962$$ −0.824585 −0.0265857
$$963$$ 0 0
$$964$$ 37.6056 1.21120
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −54.3910 −1.74910 −0.874548 0.484939i $$-0.838842\pi$$
−0.874548 + 0.484939i $$0.838842\pi$$
$$968$$ 17.9579 0.577190
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 14.2906 0.458606 0.229303 0.973355i $$-0.426355\pi$$
0.229303 + 0.973355i $$0.426355\pi$$
$$972$$ 0 0
$$973$$ 47.3283 1.51728
$$974$$ −20.8082 −0.666737
$$975$$ 0 0
$$976$$ −6.36086 −0.203606
$$977$$ −8.94905 −0.286306 −0.143153 0.989701i $$-0.545724\pi$$
−0.143153 + 0.989701i $$0.545724\pi$$
$$978$$ 0 0
$$979$$ −23.3374 −0.745868
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −3.08128 −0.0983277
$$983$$ −22.1981 −0.708010 −0.354005 0.935244i $$-0.615180\pi$$
−0.354005 + 0.935244i $$0.615180\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 1.35626 0.0431923
$$987$$ 0 0
$$988$$ −7.82128 −0.248828
$$989$$ 17.7996 0.565995
$$990$$ 0 0
$$991$$ −13.2764 −0.421740 −0.210870 0.977514i $$-0.567630\pi$$
−0.210870 + 0.977514i $$0.567630\pi$$
$$992$$ 51.5121 1.63551
$$993$$ 0 0
$$994$$ −1.31336 −0.0416573
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 19.6929 0.623682 0.311841 0.950134i $$-0.399054\pi$$
0.311841 + 0.950134i $$0.399054\pi$$
$$998$$ −3.06150 −0.0969102
$$999$$ 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 6525.2.a.bs.1.2 5
3.2 odd 2 2175.2.a.w.1.4 5
5.2 odd 4 1305.2.c.j.784.5 10
5.3 odd 4 1305.2.c.j.784.6 10
5.4 even 2 6525.2.a.bl.1.4 5
15.2 even 4 435.2.c.e.349.6 yes 10
15.8 even 4 435.2.c.e.349.5 10
15.14 odd 2 2175.2.a.z.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.e.349.5 10 15.8 even 4
435.2.c.e.349.6 yes 10 15.2 even 4
1305.2.c.j.784.5 10 5.2 odd 4
1305.2.c.j.784.6 10 5.3 odd 4
2175.2.a.w.1.4 5 3.2 odd 2
2175.2.a.z.1.2 5 15.14 odd 2
6525.2.a.bl.1.4 5 5.4 even 2
6525.2.a.bs.1.2 5 1.1 even 1 trivial