# Properties

 Label 6525.2.a.bl.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 $1$

# Related objects

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 $$-1.15351$$ 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)

 $$f(q)$$ $$=$$ $$q-2.15351 q^{2} +2.63760 q^{4} +1.51591 q^{7} -1.37308 q^{8} +O(q^{10})$$ $$q-2.15351 q^{2} +2.63760 q^{4} +1.51591 q^{7} -1.37308 q^{8} -1.88899 q^{11} +0.484093 q^{13} -3.26452 q^{14} -2.31826 q^{16} -3.39257 q^{17} +2.85121 q^{19} +4.06795 q^{22} +1.28156 q^{23} -1.04250 q^{26} +3.99836 q^{28} -1.00000 q^{29} -5.47404 q^{31} +7.73856 q^{32} +7.30594 q^{34} +7.43995 q^{37} -6.14010 q^{38} -8.28237 q^{41} +1.43262 q^{43} -4.98240 q^{44} -2.75985 q^{46} +2.25328 q^{47} -4.70203 q^{49} +1.27684 q^{52} +5.62364 q^{53} -2.08146 q^{56} +2.15351 q^{58} +12.7907 q^{59} +7.40162 q^{61} +11.7884 q^{62} -12.0285 q^{64} -8.76646 q^{67} -8.94826 q^{68} -14.5705 q^{71} +2.80833 q^{73} -16.0220 q^{74} +7.52035 q^{76} -2.86353 q^{77} -9.34617 q^{79} +17.8362 q^{82} -1.79167 q^{83} -3.08516 q^{86} +2.59374 q^{88} -7.24703 q^{89} +0.733840 q^{91} +3.38025 q^{92} -4.85246 q^{94} +4.59510 q^{97} +10.1259 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.15351 −1.52276 −0.761380 0.648305i $$-0.775478\pi$$
−0.761380 + 0.648305i $$0.775478\pi$$
$$3$$ 0 0
$$4$$ 2.63760 1.31880
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.51591 0.572959 0.286480 0.958086i $$-0.407515\pi$$
0.286480 + 0.958086i $$0.407515\pi$$
$$8$$ −1.37308 −0.485458
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.88899 −0.569552 −0.284776 0.958594i $$-0.591919\pi$$
−0.284776 + 0.958594i $$0.591919\pi$$
$$12$$ 0 0
$$13$$ 0.484093 0.134263 0.0671316 0.997744i $$-0.478615\pi$$
0.0671316 + 0.997744i $$0.478615\pi$$
$$14$$ −3.26452 −0.872480
$$15$$ 0 0
$$16$$ −2.31826 −0.579565
$$17$$ −3.39257 −0.822820 −0.411410 0.911450i $$-0.634964\pi$$
−0.411410 + 0.911450i $$0.634964\pi$$
$$18$$ 0 0
$$19$$ 2.85121 0.654112 0.327056 0.945005i $$-0.393943\pi$$
0.327056 + 0.945005i $$0.393943\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.06795 0.867291
$$23$$ 1.28156 0.267224 0.133612 0.991034i $$-0.457342\pi$$
0.133612 + 0.991034i $$0.457342\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.04250 −0.204451
$$27$$ 0 0
$$28$$ 3.99836 0.755619
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −5.47404 −0.983166 −0.491583 0.870831i $$-0.663582\pi$$
−0.491583 + 0.870831i $$0.663582\pi$$
$$32$$ 7.73856 1.36800
$$33$$ 0 0
$$34$$ 7.30594 1.25296
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.43995 1.22312 0.611561 0.791198i $$-0.290542\pi$$
0.611561 + 0.791198i $$0.290542\pi$$
$$38$$ −6.14010 −0.996056
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.28237 −1.29349 −0.646745 0.762707i $$-0.723870\pi$$
−0.646745 + 0.762707i $$0.723870\pi$$
$$42$$ 0 0
$$43$$ 1.43262 0.218473 0.109236 0.994016i $$-0.465159\pi$$
0.109236 + 0.994016i $$0.465159\pi$$
$$44$$ −4.98240 −0.751125
$$45$$ 0 0
$$46$$ −2.75985 −0.406918
$$47$$ 2.25328 0.328674 0.164337 0.986404i $$-0.447451\pi$$
0.164337 + 0.986404i $$0.447451\pi$$
$$48$$ 0 0
$$49$$ −4.70203 −0.671718
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.27684 0.177066
$$53$$ 5.62364 0.772466 0.386233 0.922401i $$-0.373776\pi$$
0.386233 + 0.922401i $$0.373776\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.08146 −0.278147
$$57$$ 0 0
$$58$$ 2.15351 0.282770
$$59$$ 12.7907 1.66520 0.832601 0.553873i $$-0.186851\pi$$
0.832601 + 0.553873i $$0.186851\pi$$
$$60$$ 0 0
$$61$$ 7.40162 0.947680 0.473840 0.880611i $$-0.342868\pi$$
0.473840 + 0.880611i $$0.342868\pi$$
$$62$$ 11.7884 1.49713
$$63$$ 0 0
$$64$$ −12.0285 −1.50357
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.76646 −1.07099 −0.535497 0.844537i $$-0.679876\pi$$
−0.535497 + 0.844537i $$0.679876\pi$$
$$68$$ −8.94826 −1.08514
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −14.5705 −1.72920 −0.864598 0.502465i $$-0.832427\pi$$
−0.864598 + 0.502465i $$0.832427\pi$$
$$72$$ 0 0
$$73$$ 2.80833 0.328691 0.164345 0.986403i $$-0.447449\pi$$
0.164345 + 0.986403i $$0.447449\pi$$
$$74$$ −16.0220 −1.86252
$$75$$ 0 0
$$76$$ 7.52035 0.862644
$$77$$ −2.86353 −0.326330
$$78$$ 0 0
$$79$$ −9.34617 −1.05153 −0.525763 0.850631i $$-0.676220\pi$$
−0.525763 + 0.850631i $$0.676220\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 17.8362 1.96967
$$83$$ −1.79167 −0.196661 −0.0983306 0.995154i $$-0.531350\pi$$
−0.0983306 + 0.995154i $$0.531350\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −3.08516 −0.332682
$$87$$ 0 0
$$88$$ 2.59374 0.276493
$$89$$ −7.24703 −0.768183 −0.384092 0.923295i $$-0.625485\pi$$
−0.384092 + 0.923295i $$0.625485\pi$$
$$90$$ 0 0
$$91$$ 0.733840 0.0769273
$$92$$ 3.38025 0.352415
$$93$$ 0 0
$$94$$ −4.85246 −0.500493
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.59510 0.466562 0.233281 0.972409i $$-0.425054\pi$$
0.233281 + 0.972409i $$0.425054\pi$$
$$98$$ 10.1259 1.02287
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.8611 1.57823 0.789117 0.614243i $$-0.210538\pi$$
0.789117 + 0.614243i $$0.210538\pi$$
$$102$$ 0 0
$$103$$ −8.46995 −0.834569 −0.417284 0.908776i $$-0.637018\pi$$
−0.417284 + 0.908776i $$0.637018\pi$$
$$104$$ −0.664699 −0.0651791
$$105$$ 0 0
$$106$$ −12.1106 −1.17628
$$107$$ −0.445502 −0.0430684 −0.0215342 0.999768i $$-0.506855\pi$$
−0.0215342 + 0.999768i $$0.506855\pi$$
$$108$$ 0 0
$$109$$ −7.32651 −0.701752 −0.350876 0.936422i $$-0.614116\pi$$
−0.350876 + 0.936422i $$0.614116\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −3.51427 −0.332067
$$113$$ 9.41884 0.886050 0.443025 0.896509i $$-0.353905\pi$$
0.443025 + 0.896509i $$0.353905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.63760 −0.244895
$$117$$ 0 0
$$118$$ −27.5448 −2.53570
$$119$$ −5.14283 −0.471442
$$120$$ 0 0
$$121$$ −7.43172 −0.675611
$$122$$ −15.9394 −1.44309
$$123$$ 0 0
$$124$$ −14.4383 −1.29660
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.22656 0.197575 0.0987875 0.995109i $$-0.468504\pi$$
0.0987875 + 0.995109i $$0.468504\pi$$
$$128$$ 10.4264 0.921576
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −15.3567 −1.34172 −0.670859 0.741585i $$-0.734075\pi$$
−0.670859 + 0.741585i $$0.734075\pi$$
$$132$$ 0 0
$$133$$ 4.32217 0.374779
$$134$$ 18.8787 1.63087
$$135$$ 0 0
$$136$$ 4.65828 0.399444
$$137$$ −1.72253 −0.147166 −0.0735828 0.997289i $$-0.523443\pi$$
−0.0735828 + 0.997289i $$0.523443\pi$$
$$138$$ 0 0
$$139$$ −6.68293 −0.566838 −0.283419 0.958996i $$-0.591469\pi$$
−0.283419 + 0.958996i $$0.591469\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 31.3776 2.63315
$$143$$ −0.914446 −0.0764698
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −6.04777 −0.500517
$$147$$ 0 0
$$148$$ 19.6236 1.61305
$$149$$ −0.474203 −0.0388482 −0.0194241 0.999811i $$-0.506183\pi$$
−0.0194241 + 0.999811i $$0.506183\pi$$
$$150$$ 0 0
$$151$$ −12.5209 −1.01893 −0.509467 0.860490i $$-0.670157\pi$$
−0.509467 + 0.860490i $$0.670157\pi$$
$$152$$ −3.91494 −0.317544
$$153$$ 0 0
$$154$$ 6.16664 0.496922
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −23.8828 −1.90606 −0.953028 0.302881i $$-0.902051\pi$$
−0.953028 + 0.302881i $$0.902051\pi$$
$$158$$ 20.1271 1.60122
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.94273 0.153108
$$162$$ 0 0
$$163$$ 14.7254 1.15339 0.576693 0.816961i $$-0.304343\pi$$
0.576693 + 0.816961i $$0.304343\pi$$
$$164$$ −21.8456 −1.70585
$$165$$ 0 0
$$166$$ 3.85838 0.299468
$$167$$ −2.95547 −0.228701 −0.114351 0.993440i $$-0.536479\pi$$
−0.114351 + 0.993440i $$0.536479\pi$$
$$168$$ 0 0
$$169$$ −12.7657 −0.981973
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 3.77868 0.288122
$$173$$ 12.1974 0.927349 0.463675 0.886006i $$-0.346531\pi$$
0.463675 + 0.886006i $$0.346531\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.37917 0.330092
$$177$$ 0 0
$$178$$ 15.6065 1.16976
$$179$$ 12.3747 0.924929 0.462464 0.886638i $$-0.346965\pi$$
0.462464 + 0.886638i $$0.346965\pi$$
$$180$$ 0 0
$$181$$ −13.1895 −0.980367 −0.490183 0.871619i $$-0.663070\pi$$
−0.490183 + 0.871619i $$0.663070\pi$$
$$182$$ −1.58033 −0.117142
$$183$$ 0 0
$$184$$ −1.75969 −0.129726
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.40853 0.468638
$$188$$ 5.94325 0.433456
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.10636 0.441841 0.220920 0.975292i $$-0.429094\pi$$
0.220920 + 0.975292i $$0.429094\pi$$
$$192$$ 0 0
$$193$$ 19.6580 1.41501 0.707507 0.706707i $$-0.249820\pi$$
0.707507 + 0.706707i $$0.249820\pi$$
$$194$$ −9.89560 −0.710463
$$195$$ 0 0
$$196$$ −12.4021 −0.885862
$$197$$ −24.2075 −1.72471 −0.862357 0.506301i $$-0.831013\pi$$
−0.862357 + 0.506301i $$0.831013\pi$$
$$198$$ 0 0
$$199$$ 0.542480 0.0384554 0.0192277 0.999815i $$-0.493879\pi$$
0.0192277 + 0.999815i $$0.493879\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −34.1569 −2.40327
$$203$$ −1.51591 −0.106396
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 18.2401 1.27085
$$207$$ 0 0
$$208$$ −1.12225 −0.0778142
$$209$$ −5.38590 −0.372551
$$210$$ 0 0
$$211$$ −1.20999 −0.0832989 −0.0416495 0.999132i $$-0.513261\pi$$
−0.0416495 + 0.999132i $$0.513261\pi$$
$$212$$ 14.8329 1.01873
$$213$$ 0 0
$$214$$ 0.959394 0.0655828
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −8.29813 −0.563314
$$218$$ 15.7777 1.06860
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.64232 −0.110474
$$222$$ 0 0
$$223$$ 15.8267 1.05983 0.529917 0.848049i $$-0.322223\pi$$
0.529917 + 0.848049i $$0.322223\pi$$
$$224$$ 11.7309 0.783806
$$225$$ 0 0
$$226$$ −20.2836 −1.34924
$$227$$ 8.03996 0.533631 0.266815 0.963748i $$-0.414029\pi$$
0.266815 + 0.963748i $$0.414029\pi$$
$$228$$ 0 0
$$229$$ −0.0946339 −0.00625359 −0.00312679 0.999995i $$-0.500995\pi$$
−0.00312679 + 0.999995i $$0.500995\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.37308 0.0901472
$$233$$ −18.6579 −1.22232 −0.611159 0.791508i $$-0.709297\pi$$
−0.611159 + 0.791508i $$0.709297\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 33.7367 2.19607
$$237$$ 0 0
$$238$$ 11.0751 0.717894
$$239$$ 25.3243 1.63809 0.819047 0.573726i $$-0.194503\pi$$
0.819047 + 0.573726i $$0.194503\pi$$
$$240$$ 0 0
$$241$$ −28.5508 −1.83912 −0.919559 0.392951i $$-0.871454\pi$$
−0.919559 + 0.392951i $$0.871454\pi$$
$$242$$ 16.0043 1.02879
$$243$$ 0 0
$$244$$ 19.5225 1.24980
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.38025 0.0878232
$$248$$ 7.51630 0.477285
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.4952 1.04117 0.520584 0.853810i $$-0.325714\pi$$
0.520584 + 0.853810i $$0.325714\pi$$
$$252$$ 0 0
$$253$$ −2.42086 −0.152198
$$254$$ −4.79491 −0.300860
$$255$$ 0 0
$$256$$ 1.60362 0.100226
$$257$$ −13.7659 −0.858694 −0.429347 0.903140i $$-0.641256\pi$$
−0.429347 + 0.903140i $$0.641256\pi$$
$$258$$ 0 0
$$259$$ 11.2783 0.700798
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 33.0707 2.04312
$$263$$ 16.7039 1.03001 0.515004 0.857188i $$-0.327790\pi$$
0.515004 + 0.857188i $$0.327790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −9.30783 −0.570700
$$267$$ 0 0
$$268$$ −23.1224 −1.41243
$$269$$ 17.3518 1.05796 0.528979 0.848635i $$-0.322575\pi$$
0.528979 + 0.848635i $$0.322575\pi$$
$$270$$ 0 0
$$271$$ 8.27747 0.502821 0.251410 0.967881i $$-0.419106\pi$$
0.251410 + 0.967881i $$0.419106\pi$$
$$272$$ 7.86487 0.476878
$$273$$ 0 0
$$274$$ 3.70948 0.224098
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.0560 1.08488 0.542439 0.840095i $$-0.317501\pi$$
0.542439 + 0.840095i $$0.317501\pi$$
$$278$$ 14.3917 0.863159
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.6241 1.11102 0.555510 0.831510i $$-0.312523\pi$$
0.555510 + 0.831510i $$0.312523\pi$$
$$282$$ 0 0
$$283$$ 16.6920 0.992235 0.496117 0.868256i $$-0.334759\pi$$
0.496117 + 0.868256i $$0.334759\pi$$
$$284$$ −38.4311 −2.28046
$$285$$ 0 0
$$286$$ 1.96927 0.116445
$$287$$ −12.5553 −0.741116
$$288$$ 0 0
$$289$$ −5.49045 −0.322968
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 7.40727 0.433478
$$293$$ −32.9866 −1.92710 −0.963549 0.267532i $$-0.913792\pi$$
−0.963549 + 0.267532i $$0.913792\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.2157 −0.593773
$$297$$ 0 0
$$298$$ 1.02120 0.0591566
$$299$$ 0.620395 0.0358783
$$300$$ 0 0
$$301$$ 2.17172 0.125176
$$302$$ 26.9638 1.55159
$$303$$ 0 0
$$304$$ −6.60984 −0.379100
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5.20522 −0.297077 −0.148539 0.988907i $$-0.547457\pi$$
−0.148539 + 0.988907i $$0.547457\pi$$
$$308$$ −7.55286 −0.430364
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −15.1777 −0.860651 −0.430325 0.902674i $$-0.641601\pi$$
−0.430325 + 0.902674i $$0.641601\pi$$
$$312$$ 0 0
$$313$$ −6.91509 −0.390864 −0.195432 0.980717i $$-0.562611\pi$$
−0.195432 + 0.980717i $$0.562611\pi$$
$$314$$ 51.4319 2.90247
$$315$$ 0 0
$$316$$ −24.6515 −1.38675
$$317$$ −24.6694 −1.38557 −0.692786 0.721144i $$-0.743617\pi$$
−0.692786 + 0.721144i $$0.743617\pi$$
$$318$$ 0 0
$$319$$ 1.88899 0.105763
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −4.18368 −0.233148
$$323$$ −9.67293 −0.538216
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −31.7114 −1.75633
$$327$$ 0 0
$$328$$ 11.3724 0.627934
$$329$$ 3.41576 0.188317
$$330$$ 0 0
$$331$$ −4.24031 −0.233069 −0.116534 0.993187i $$-0.537178\pi$$
−0.116534 + 0.993187i $$0.537178\pi$$
$$332$$ −4.72571 −0.259357
$$333$$ 0 0
$$334$$ 6.36463 0.348257
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −26.2776 −1.43143 −0.715717 0.698390i $$-0.753900\pi$$
−0.715717 + 0.698390i $$0.753900\pi$$
$$338$$ 27.4910 1.49531
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 10.3404 0.559964
$$342$$ 0 0
$$343$$ −17.7392 −0.957826
$$344$$ −1.96711 −0.106059
$$345$$ 0 0
$$346$$ −26.2672 −1.41213
$$347$$ −21.7745 −1.16892 −0.584458 0.811424i $$-0.698693\pi$$
−0.584458 + 0.811424i $$0.698693\pi$$
$$348$$ 0 0
$$349$$ 8.94807 0.478979 0.239490 0.970899i $$-0.423020\pi$$
0.239490 + 0.970899i $$0.423020\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −14.6180 −0.779145
$$353$$ −7.14944 −0.380526 −0.190263 0.981733i $$-0.560934\pi$$
−0.190263 + 0.981733i $$0.560934\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −19.1148 −1.01308
$$357$$ 0 0
$$358$$ −26.6490 −1.40845
$$359$$ −31.4641 −1.66061 −0.830305 0.557309i $$-0.811834\pi$$
−0.830305 + 0.557309i $$0.811834\pi$$
$$360$$ 0 0
$$361$$ −10.8706 −0.572137
$$362$$ 28.4037 1.49286
$$363$$ 0 0
$$364$$ 1.93558 0.101452
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −27.7722 −1.44970 −0.724849 0.688908i $$-0.758090\pi$$
−0.724849 + 0.688908i $$0.758090\pi$$
$$368$$ −2.97099 −0.154874
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 8.52491 0.442592
$$372$$ 0 0
$$373$$ −15.9293 −0.824790 −0.412395 0.911005i $$-0.635308\pi$$
−0.412395 + 0.911005i $$0.635308\pi$$
$$374$$ −13.8008 −0.713624
$$375$$ 0 0
$$376$$ −3.09394 −0.159558
$$377$$ −0.484093 −0.0249320
$$378$$ 0 0
$$379$$ 18.4181 0.946073 0.473037 0.881043i $$-0.343158\pi$$
0.473037 + 0.881043i $$0.343158\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −13.1501 −0.672818
$$383$$ 14.1973 0.725447 0.362724 0.931897i $$-0.381847\pi$$
0.362724 + 0.931897i $$0.381847\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −42.3337 −2.15473
$$387$$ 0 0
$$388$$ 12.1201 0.615303
$$389$$ 25.0087 1.26799 0.633995 0.773337i $$-0.281414\pi$$
0.633995 + 0.773337i $$0.281414\pi$$
$$390$$ 0 0
$$391$$ −4.34779 −0.219877
$$392$$ 6.45626 0.326091
$$393$$ 0 0
$$394$$ 52.1311 2.62633
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −3.67651 −0.184519 −0.0922595 0.995735i $$-0.529409\pi$$
−0.0922595 + 0.995735i $$0.529409\pi$$
$$398$$ −1.16824 −0.0585584
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.68500 0.233958 0.116979 0.993134i $$-0.462679\pi$$
0.116979 + 0.993134i $$0.462679\pi$$
$$402$$ 0 0
$$403$$ −2.64994 −0.132003
$$404$$ 41.8352 2.08138
$$405$$ 0 0
$$406$$ 3.26452 0.162015
$$407$$ −14.0540 −0.696630
$$408$$ 0 0
$$409$$ 36.8798 1.82359 0.911794 0.410647i $$-0.134697\pi$$
0.911794 + 0.410647i $$0.134697\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −22.3403 −1.10063
$$413$$ 19.3895 0.954092
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 3.74618 0.183672
$$417$$ 0 0
$$418$$ 11.5986 0.567305
$$419$$ −19.1400 −0.935050 −0.467525 0.883980i $$-0.654854\pi$$
−0.467525 + 0.883980i $$0.654854\pi$$
$$420$$ 0 0
$$421$$ 29.5114 1.43830 0.719149 0.694856i $$-0.244532\pi$$
0.719149 + 0.694856i $$0.244532\pi$$
$$422$$ 2.60572 0.126844
$$423$$ 0 0
$$424$$ −7.72171 −0.375000
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 11.2202 0.542982
$$428$$ −1.17506 −0.0567986
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −13.6750 −0.658703 −0.329352 0.944207i $$-0.606830\pi$$
−0.329352 + 0.944207i $$0.606830\pi$$
$$432$$ 0 0
$$433$$ 34.7905 1.67192 0.835961 0.548788i $$-0.184911\pi$$
0.835961 + 0.548788i $$0.184911\pi$$
$$434$$ 17.8701 0.857793
$$435$$ 0 0
$$436$$ −19.3244 −0.925472
$$437$$ 3.65400 0.174794
$$438$$ 0 0
$$439$$ −17.0869 −0.815513 −0.407757 0.913091i $$-0.633689\pi$$
−0.407757 + 0.913091i $$0.633689\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 3.53675 0.168226
$$443$$ −10.2997 −0.489351 −0.244676 0.969605i $$-0.578682\pi$$
−0.244676 + 0.969605i $$0.578682\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −34.0830 −1.61388
$$447$$ 0 0
$$448$$ −18.2341 −0.861482
$$449$$ −15.3331 −0.723615 −0.361808 0.932253i $$-0.617840\pi$$
−0.361808 + 0.932253i $$0.617840\pi$$
$$450$$ 0 0
$$451$$ 15.6453 0.736709
$$452$$ 24.8431 1.16852
$$453$$ 0 0
$$454$$ −17.3141 −0.812592
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.1841 0.476394 0.238197 0.971217i $$-0.423444\pi$$
0.238197 + 0.971217i $$0.423444\pi$$
$$458$$ 0.203795 0.00952272
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 37.2567 1.73522 0.867609 0.497247i $$-0.165656\pi$$
0.867609 + 0.497247i $$0.165656\pi$$
$$462$$ 0 0
$$463$$ −24.2457 −1.12680 −0.563398 0.826186i $$-0.690506\pi$$
−0.563398 + 0.826186i $$0.690506\pi$$
$$464$$ 2.31826 0.107623
$$465$$ 0 0
$$466$$ 40.1799 1.86130
$$467$$ 28.5544 1.32134 0.660670 0.750676i $$-0.270272\pi$$
0.660670 + 0.750676i $$0.270272\pi$$
$$468$$ 0 0
$$469$$ −13.2891 −0.613636
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −17.5626 −0.808385
$$473$$ −2.70620 −0.124431
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −13.5647 −0.621738
$$477$$ 0 0
$$478$$ −54.5362 −2.49443
$$479$$ −20.8455 −0.952455 −0.476227 0.879322i $$-0.657996\pi$$
−0.476227 + 0.879322i $$0.657996\pi$$
$$480$$ 0 0
$$481$$ 3.60163 0.164220
$$482$$ 61.4844 2.80054
$$483$$ 0 0
$$484$$ −19.6019 −0.890996
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −5.31743 −0.240956 −0.120478 0.992716i $$-0.538443\pi$$
−0.120478 + 0.992716i $$0.538443\pi$$
$$488$$ −10.1630 −0.460058
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −34.6711 −1.56469 −0.782343 0.622848i $$-0.785975\pi$$
−0.782343 + 0.622848i $$0.785975\pi$$
$$492$$ 0 0
$$493$$ 3.39257 0.152794
$$494$$ −2.97238 −0.133734
$$495$$ 0 0
$$496$$ 12.6902 0.569809
$$497$$ −22.0875 −0.990758
$$498$$ 0 0
$$499$$ −37.2788 −1.66883 −0.834414 0.551138i $$-0.814194\pi$$
−0.834414 + 0.551138i $$0.814194\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −35.5226 −1.58545
$$503$$ −10.7589 −0.479717 −0.239858 0.970808i $$-0.577101\pi$$
−0.239858 + 0.970808i $$0.577101\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 5.21333 0.231761
$$507$$ 0 0
$$508$$ 5.87277 0.260562
$$509$$ 0.424406 0.0188115 0.00940573 0.999956i $$-0.497006\pi$$
0.00940573 + 0.999956i $$0.497006\pi$$
$$510$$ 0 0
$$511$$ 4.25717 0.188326
$$512$$ −24.3063 −1.07420
$$513$$ 0 0
$$514$$ 29.6450 1.30759
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4.25642 −0.187197
$$518$$ −24.2879 −1.06715
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.39104 −0.279996 −0.139998 0.990152i $$-0.544710\pi$$
−0.139998 + 0.990152i $$0.544710\pi$$
$$522$$ 0 0
$$523$$ 24.8564 1.08689 0.543446 0.839444i $$-0.317119\pi$$
0.543446 + 0.839444i $$0.317119\pi$$
$$524$$ −40.5048 −1.76946
$$525$$ 0 0
$$526$$ −35.9721 −1.56846
$$527$$ 18.5711 0.808968
$$528$$ 0 0
$$529$$ −21.3576 −0.928591
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 11.4002 0.494260
$$533$$ −4.00944 −0.173668
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0371 0.519922
$$537$$ 0 0
$$538$$ −37.3673 −1.61102
$$539$$ 8.88207 0.382578
$$540$$ 0 0
$$541$$ 13.2290 0.568759 0.284379 0.958712i $$-0.408212\pi$$
0.284379 + 0.958712i $$0.408212\pi$$
$$542$$ −17.8256 −0.765676
$$543$$ 0 0
$$544$$ −26.2536 −1.12561
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 43.7132 1.86904 0.934521 0.355908i $$-0.115828\pi$$
0.934521 + 0.355908i $$0.115828\pi$$
$$548$$ −4.54334 −0.194082
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.85121 −0.121466
$$552$$ 0 0
$$553$$ −14.1679 −0.602481
$$554$$ −38.8837 −1.65201
$$555$$ 0 0
$$556$$ −17.6269 −0.747547
$$557$$ −18.7407 −0.794067 −0.397033 0.917804i $$-0.629960\pi$$
−0.397033 + 0.917804i $$0.629960\pi$$
$$558$$ 0 0
$$559$$ 0.693521 0.0293328
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −40.1071 −1.69182
$$563$$ −6.52697 −0.275079 −0.137539 0.990496i $$-0.543919\pi$$
−0.137539 + 0.990496i $$0.543919\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −35.9463 −1.51094
$$567$$ 0 0
$$568$$ 20.0064 0.839451
$$569$$ −24.4702 −1.02585 −0.512923 0.858435i $$-0.671437\pi$$
−0.512923 + 0.858435i $$0.671437\pi$$
$$570$$ 0 0
$$571$$ 45.1164 1.88806 0.944031 0.329856i $$-0.107000\pi$$
0.944031 + 0.329856i $$0.107000\pi$$
$$572$$ −2.41194 −0.100848
$$573$$ 0 0
$$574$$ 27.0380 1.12854
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 25.4834 1.06089 0.530444 0.847720i $$-0.322025\pi$$
0.530444 + 0.847720i $$0.322025\pi$$
$$578$$ 11.8237 0.491803
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −2.71600 −0.112679
$$582$$ 0 0
$$583$$ −10.6230 −0.439959
$$584$$ −3.85607 −0.159565
$$585$$ 0 0
$$586$$ 71.0370 2.93451
$$587$$ 42.3930 1.74975 0.874873 0.484352i $$-0.160945\pi$$
0.874873 + 0.484352i $$0.160945\pi$$
$$588$$ 0 0
$$589$$ −15.6076 −0.643101
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −17.2477 −0.708878
$$593$$ −5.71303 −0.234606 −0.117303 0.993096i $$-0.537425\pi$$
−0.117303 + 0.993096i $$0.537425\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.25076 −0.0512331
$$597$$ 0 0
$$598$$ −1.33603 −0.0546341
$$599$$ 14.0026 0.572132 0.286066 0.958210i $$-0.407652\pi$$
0.286066 + 0.958210i $$0.407652\pi$$
$$600$$ 0 0
$$601$$ −22.6834 −0.925277 −0.462639 0.886547i $$-0.653097\pi$$
−0.462639 + 0.886547i $$0.653097\pi$$
$$602$$ −4.67682 −0.190613
$$603$$ 0 0
$$604$$ −33.0250 −1.34377
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −27.7581 −1.12667 −0.563333 0.826230i $$-0.690481\pi$$
−0.563333 + 0.826230i $$0.690481\pi$$
$$608$$ 22.0642 0.894823
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.09080 0.0441289
$$612$$ 0 0
$$613$$ −12.0619 −0.487176 −0.243588 0.969879i $$-0.578325\pi$$
−0.243588 + 0.969879i $$0.578325\pi$$
$$614$$ 11.2095 0.452378
$$615$$ 0 0
$$616$$ 3.93186 0.158419
$$617$$ −45.0988 −1.81561 −0.907804 0.419394i $$-0.862243\pi$$
−0.907804 + 0.419394i $$0.862243\pi$$
$$618$$ 0 0
$$619$$ −35.5206 −1.42769 −0.713846 0.700302i $$-0.753049\pi$$
−0.713846 + 0.700302i $$0.753049\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 32.6854 1.31057
$$623$$ −10.9858 −0.440138
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 14.8917 0.595192
$$627$$ 0 0
$$628$$ −62.9934 −2.51371
$$629$$ −25.2406 −1.00641
$$630$$ 0 0
$$631$$ −32.0402 −1.27550 −0.637750 0.770243i $$-0.720135\pi$$
−0.637750 + 0.770243i $$0.720135\pi$$
$$632$$ 12.8330 0.510471
$$633$$ 0 0
$$634$$ 53.1258 2.10989
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.27622 −0.0901870
$$638$$ −4.06795 −0.161052
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 28.4963 1.12554 0.562769 0.826614i $$-0.309736\pi$$
0.562769 + 0.826614i $$0.309736\pi$$
$$642$$ 0 0
$$643$$ −43.1625 −1.70216 −0.851081 0.525035i $$-0.824052\pi$$
−0.851081 + 0.525035i $$0.824052\pi$$
$$644$$ 5.12414 0.201920
$$645$$ 0 0
$$646$$ 20.8307 0.819575
$$647$$ −32.5875 −1.28115 −0.640573 0.767898i $$-0.721303\pi$$
−0.640573 + 0.767898i $$0.721303\pi$$
$$648$$ 0 0
$$649$$ −24.1614 −0.948418
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 38.8398 1.52109
$$653$$ −28.1951 −1.10336 −0.551679 0.834057i $$-0.686013\pi$$
−0.551679 + 0.834057i $$0.686013\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 19.2007 0.749661
$$657$$ 0 0
$$658$$ −7.35587 −0.286762
$$659$$ −42.7564 −1.66555 −0.832776 0.553611i $$-0.813250\pi$$
−0.832776 + 0.553611i $$0.813250\pi$$
$$660$$ 0 0
$$661$$ 17.4726 0.679606 0.339803 0.940497i $$-0.389640\pi$$
0.339803 + 0.940497i $$0.389640\pi$$
$$662$$ 9.13155 0.354908
$$663$$ 0 0
$$664$$ 2.46011 0.0954707
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.28156 −0.0496223
$$668$$ −7.79535 −0.301611
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −13.9816 −0.539753
$$672$$ 0 0
$$673$$ 12.5821 0.485006 0.242503 0.970151i $$-0.422032\pi$$
0.242503 + 0.970151i $$0.422032\pi$$
$$674$$ 56.5891 2.17973
$$675$$ 0 0
$$676$$ −33.6707 −1.29503
$$677$$ −27.7829 −1.06778 −0.533892 0.845553i $$-0.679271\pi$$
−0.533892 + 0.845553i $$0.679271\pi$$
$$678$$ 0 0
$$679$$ 6.96575 0.267321
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −22.2681 −0.852691
$$683$$ −41.1587 −1.57489 −0.787446 0.616383i $$-0.788597\pi$$
−0.787446 + 0.616383i $$0.788597\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 38.2015 1.45854
$$687$$ 0 0
$$688$$ −3.32119 −0.126619
$$689$$ 2.72236 0.103714
$$690$$ 0 0
$$691$$ −6.51859 −0.247979 −0.123989 0.992284i $$-0.539569\pi$$
−0.123989 + 0.992284i $$0.539569\pi$$
$$692$$ 32.1718 1.22299
$$693$$ 0 0
$$694$$ 46.8916 1.77998
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 28.0985 1.06431
$$698$$ −19.2698 −0.729371
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3.82210 −0.144359 −0.0721793 0.997392i $$-0.522995\pi$$
−0.0721793 + 0.997392i $$0.522995\pi$$
$$702$$ 0 0
$$703$$ 21.2129 0.800058
$$704$$ 22.7218 0.856359
$$705$$ 0 0
$$706$$ 15.3964 0.579450
$$707$$ 24.0439 0.904264
$$708$$ 0 0
$$709$$ 4.67767 0.175673 0.0878367 0.996135i $$-0.472005\pi$$
0.0878367 + 0.996135i $$0.472005\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 9.95076 0.372920
$$713$$ −7.01532 −0.262726
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 32.6395 1.21980
$$717$$ 0 0
$$718$$ 67.7582 2.52871
$$719$$ −26.0976 −0.973276 −0.486638 0.873604i $$-0.661777\pi$$
−0.486638 + 0.873604i $$0.661777\pi$$
$$720$$ 0 0
$$721$$ −12.8397 −0.478174
$$722$$ 23.4100 0.871228
$$723$$ 0 0
$$724$$ −34.7886 −1.29291
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −42.0811 −1.56070 −0.780352 0.625341i $$-0.784960\pi$$
−0.780352 + 0.625341i $$0.784960\pi$$
$$728$$ −1.00762 −0.0373449
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −4.86027 −0.179764
$$732$$ 0 0
$$733$$ 5.06122 0.186940 0.0934702 0.995622i $$-0.470204\pi$$
0.0934702 + 0.995622i $$0.470204\pi$$
$$734$$ 59.8077 2.20754
$$735$$ 0 0
$$736$$ 9.91744 0.365562
$$737$$ 16.5598 0.609986
$$738$$ 0 0
$$739$$ −26.3474 −0.969203 −0.484601 0.874735i $$-0.661035\pi$$
−0.484601 + 0.874735i $$0.661035\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −18.3585 −0.673961
$$743$$ −1.17798 −0.0432158 −0.0216079 0.999767i $$-0.506879\pi$$
−0.0216079 + 0.999767i $$0.506879\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 34.3040 1.25596
$$747$$ 0 0
$$748$$ 16.9032 0.618041
$$749$$ −0.675340 −0.0246764
$$750$$ 0 0
$$751$$ −18.1100 −0.660844 −0.330422 0.943833i $$-0.607191\pi$$
−0.330422 + 0.943833i $$0.607191\pi$$
$$752$$ −5.22369 −0.190488
$$753$$ 0 0
$$754$$ 1.04250 0.0379655
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 4.02431 0.146266 0.0731331 0.997322i $$-0.476700\pi$$
0.0731331 + 0.997322i $$0.476700\pi$$
$$758$$ −39.6635 −1.44064
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −35.2780 −1.27883 −0.639414 0.768863i $$-0.720823\pi$$
−0.639414 + 0.768863i $$0.720823\pi$$
$$762$$ 0 0
$$763$$ −11.1063 −0.402075
$$764$$ 16.1062 0.582700
$$765$$ 0 0
$$766$$ −30.5740 −1.10468
$$767$$ 6.19186 0.223575
$$768$$ 0 0
$$769$$ 37.6829 1.35888 0.679440 0.733731i $$-0.262223\pi$$
0.679440 + 0.733731i $$0.262223\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 51.8500 1.86612
$$773$$ −25.4551 −0.915558 −0.457779 0.889066i $$-0.651355\pi$$
−0.457779 + 0.889066i $$0.651355\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.30945 −0.226496
$$777$$ 0 0
$$778$$ −53.8564 −1.93085
$$779$$ −23.6148 −0.846087
$$780$$ 0 0
$$781$$ 27.5234 0.984866
$$782$$ 9.36301 0.334820
$$783$$ 0 0
$$784$$ 10.9005 0.389304
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −27.0582 −0.964521 −0.482261 0.876028i $$-0.660184\pi$$
−0.482261 + 0.876028i $$0.660184\pi$$
$$788$$ −63.8498 −2.27455
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 14.2781 0.507670
$$792$$ 0 0
$$793$$ 3.58307 0.127238
$$794$$ 7.91741 0.280978
$$795$$ 0 0
$$796$$ 1.43085 0.0507150
$$797$$ 43.5501 1.54262 0.771312 0.636457i $$-0.219601\pi$$
0.771312 + 0.636457i $$0.219601\pi$$
$$798$$ 0 0
$$799$$ −7.64441 −0.270440
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −10.0892 −0.356262
$$803$$ −5.30491 −0.187206
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5.70667 0.201009
$$807$$ 0 0
$$808$$ −21.7785 −0.766166
$$809$$ −32.0173 −1.12567 −0.562834 0.826570i $$-0.690289\pi$$
−0.562834 + 0.826570i $$0.690289\pi$$
$$810$$ 0 0
$$811$$ 4.19698 0.147376 0.0736880 0.997281i $$-0.476523\pi$$
0.0736880 + 0.997281i $$0.476523\pi$$
$$812$$ −3.99836 −0.140315
$$813$$ 0 0
$$814$$ 30.2654 1.06080
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.08470 0.142906
$$818$$ −79.4210 −2.77689
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −47.5461 −1.65937 −0.829686 0.558230i $$-0.811481\pi$$
−0.829686 + 0.558230i $$0.811481\pi$$
$$822$$ 0 0
$$823$$ 13.3520 0.465422 0.232711 0.972546i $$-0.425240\pi$$
0.232711 + 0.972546i $$0.425240\pi$$
$$824$$ 11.6299 0.405148
$$825$$ 0 0
$$826$$ −41.7554 −1.45285
$$827$$ 1.07408 0.0373493 0.0186747 0.999826i $$-0.494055\pi$$
0.0186747 + 0.999826i $$0.494055\pi$$
$$828$$ 0 0
$$829$$ −53.3629 −1.85337 −0.926684 0.375840i $$-0.877354\pi$$
−0.926684 + 0.375840i $$0.877354\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −5.82293 −0.201874
$$833$$ 15.9520 0.552703
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −14.2059 −0.491320
$$837$$ 0 0
$$838$$ 41.2182 1.42386
$$839$$ −1.82410 −0.0629751 −0.0314875 0.999504i $$-0.510024\pi$$
−0.0314875 + 0.999504i $$0.510024\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −63.5531 −2.19019
$$843$$ 0 0
$$844$$ −3.19146 −0.109855
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −11.2658 −0.387097
$$848$$ −13.0371 −0.447694
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 9.53476 0.326847
$$852$$ 0 0
$$853$$ −26.3499 −0.902203 −0.451102 0.892473i $$-0.648969\pi$$
−0.451102 + 0.892473i $$0.648969\pi$$
$$854$$ −24.1627 −0.826831
$$855$$ 0 0
$$856$$ 0.611711 0.0209079
$$857$$ −39.7239 −1.35694 −0.678472 0.734627i $$-0.737357\pi$$
−0.678472 + 0.734627i $$0.737357\pi$$
$$858$$ 0 0
$$859$$ −39.1847 −1.33696 −0.668482 0.743729i $$-0.733055\pi$$
−0.668482 + 0.743729i $$0.733055\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 29.4493 1.00305
$$863$$ −23.6601 −0.805399 −0.402699 0.915332i $$-0.631928\pi$$
−0.402699 + 0.915332i $$0.631928\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −74.9216 −2.54594
$$867$$ 0 0
$$868$$ −21.8872 −0.742899
$$869$$ 17.6548 0.598898
$$870$$ 0 0
$$871$$ −4.24378 −0.143795
$$872$$ 10.0599 0.340671
$$873$$ 0 0
$$874$$ −7.86892 −0.266170
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −52.7791 −1.78222 −0.891112 0.453783i $$-0.850074\pi$$
−0.891112 + 0.453783i $$0.850074\pi$$
$$878$$ 36.7968 1.24183
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 15.0520 0.507115 0.253558 0.967320i $$-0.418399\pi$$
0.253558 + 0.967320i $$0.418399\pi$$
$$882$$ 0 0
$$883$$ 37.7972 1.27198 0.635989 0.771698i $$-0.280592\pi$$
0.635989 + 0.771698i $$0.280592\pi$$
$$884$$ −4.33179 −0.145694
$$885$$ 0 0
$$886$$ 22.1804 0.745165
$$887$$ 4.81353 0.161622 0.0808112 0.996729i $$-0.474249\pi$$
0.0808112 + 0.996729i $$0.474249\pi$$
$$888$$ 0 0
$$889$$ 3.37526 0.113202
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 41.7446 1.39771
$$893$$ 6.42457 0.214990
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 15.8055 0.528026
$$897$$ 0 0
$$898$$ 33.0201 1.10189
$$899$$ 5.47404 0.182569
$$900$$ 0 0
$$901$$ −19.0786 −0.635600
$$902$$ −33.6923 −1.12183
$$903$$ 0 0
$$904$$ −12.9328 −0.430140
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 43.5796 1.44704 0.723518 0.690305i $$-0.242524\pi$$
0.723518 + 0.690305i $$0.242524\pi$$
$$908$$ 21.2062 0.703753
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 15.9296 0.527770 0.263885 0.964554i $$-0.414996\pi$$
0.263885 + 0.964554i $$0.414996\pi$$
$$912$$ 0 0
$$913$$ 3.38444 0.112009
$$914$$ −21.9316 −0.725434
$$915$$ 0 0
$$916$$ −0.249607 −0.00824724
$$917$$ −23.2793 −0.768750
$$918$$ 0 0
$$919$$ 44.8752 1.48030 0.740148 0.672444i $$-0.234755\pi$$
0.740148 + 0.672444i $$0.234755\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −80.2327 −2.64232
$$923$$ −7.05345 −0.232167
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 52.2134 1.71584
$$927$$ 0 0
$$928$$ −7.73856 −0.254031
$$929$$ 34.3242 1.12614 0.563071 0.826409i $$-0.309620\pi$$
0.563071 + 0.826409i $$0.309620\pi$$
$$930$$ 0 0
$$931$$ −13.4065 −0.439379
$$932$$ −49.2121 −1.61200
$$933$$ 0 0
$$934$$ −61.4922 −2.01209
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 21.0494 0.687653 0.343827 0.939033i $$-0.388277\pi$$
0.343827 + 0.939033i $$0.388277\pi$$
$$938$$ 28.6183 0.934421
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −46.2494 −1.50769 −0.753844 0.657053i $$-0.771803\pi$$
−0.753844 + 0.657053i $$0.771803\pi$$
$$942$$ 0 0
$$943$$ −10.6144 −0.345651
$$944$$ −29.6521 −0.965093
$$945$$ 0 0
$$946$$ 5.82784 0.189479
$$947$$ 21.7174 0.705720 0.352860 0.935676i $$-0.385209\pi$$
0.352860 + 0.935676i $$0.385209\pi$$
$$948$$ 0 0
$$949$$ 1.35949 0.0441310
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 7.06152 0.228865
$$953$$ 16.6122 0.538120 0.269060 0.963123i $$-0.413287\pi$$
0.269060 + 0.963123i $$0.413287\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 66.7955 2.16032
$$957$$ 0 0
$$958$$ 44.8910 1.45036
$$959$$ −2.61119 −0.0843198
$$960$$ 0 0
$$961$$ −1.03492 −0.0333845
$$962$$ −7.75614 −0.250068
$$963$$ 0 0
$$964$$ −75.3056 −2.42543
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 12.1886 0.391960 0.195980 0.980608i $$-0.437211\pi$$
0.195980 + 0.980608i $$0.437211\pi$$
$$968$$ 10.2044 0.327981
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 10.0067 0.321131 0.160566 0.987025i $$-0.448668\pi$$
0.160566 + 0.987025i $$0.448668\pi$$
$$972$$ 0 0
$$973$$ −10.1307 −0.324775
$$974$$ 11.4511 0.366918
$$975$$ 0 0
$$976$$ −17.1589 −0.549242
$$977$$ −23.0188 −0.736435 −0.368218 0.929740i $$-0.620032\pi$$
−0.368218 + 0.929740i $$0.620032\pi$$
$$978$$ 0 0
$$979$$ 13.6895 0.437520
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 74.6645 2.38264
$$983$$ −54.9480 −1.75257 −0.876285 0.481793i $$-0.839986\pi$$
−0.876285 + 0.481793i $$0.839986\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −7.30594 −0.232668
$$987$$ 0 0
$$988$$ 3.64055 0.115821
$$989$$ 1.83599 0.0583811
$$990$$ 0 0
$$991$$ −8.69215 −0.276115 −0.138058 0.990424i $$-0.544086\pi$$
−0.138058 + 0.990424i $$0.544086\pi$$
$$992$$ −42.3612 −1.34497
$$993$$ 0 0
$$994$$ 47.5656 1.50869
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −18.8275 −0.596274 −0.298137 0.954523i $$-0.596365\pi$$
−0.298137 + 0.954523i $$0.596365\pi$$
$$998$$ 80.2802 2.54123
$$999$$ 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.bl.1.2 5
3.2 odd 2 2175.2.a.z.1.4 5
5.2 odd 4 1305.2.c.j.784.2 10
5.3 odd 4 1305.2.c.j.784.9 10
5.4 even 2 6525.2.a.bs.1.4 5
15.2 even 4 435.2.c.e.349.9 yes 10
15.8 even 4 435.2.c.e.349.2 10
15.14 odd 2 2175.2.a.w.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.e.349.2 10 15.8 even 4
435.2.c.e.349.9 yes 10 15.2 even 4
1305.2.c.j.784.2 10 5.2 odd 4
1305.2.c.j.784.9 10 5.3 odd 4
2175.2.a.w.1.2 5 15.14 odd 2
2175.2.a.z.1.4 5 3.2 odd 2
6525.2.a.bl.1.2 5 1.1 even 1 trivial
6525.2.a.bs.1.4 5 5.4 even 2