# Properties

 Label 6525.2.a.bw.1.1 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $7$ 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: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18$$ x^7 - x^6 - 13*x^5 + 12*x^4 + 47*x^3 - 37*x^2 - 35*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1305) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.66072$$ 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)

 $$f(q)$$ $$=$$ $$q-2.66072 q^{2} +5.07943 q^{4} -4.68271 q^{7} -8.19351 q^{8} +O(q^{10})$$ $$q-2.66072 q^{2} +5.07943 q^{4} -4.68271 q^{7} -8.19351 q^{8} +3.50383 q^{11} +4.56233 q^{13} +12.4594 q^{14} +11.6418 q^{16} +6.56233 q^{17} -4.10839 q^{19} -9.32272 q^{22} +6.92951 q^{23} -12.1391 q^{26} -23.7855 q^{28} +1.00000 q^{29} +8.10839 q^{31} -14.5885 q^{32} -17.4605 q^{34} -4.27156 q^{37} +10.9313 q^{38} +5.10898 q^{41} +8.47439 q^{43} +17.7975 q^{44} -18.4375 q^{46} +5.41114 q^{47} +14.9278 q^{49} +23.1741 q^{52} +4.20108 q^{53} +38.3678 q^{56} -2.66072 q^{58} -5.36542 q^{59} +1.27150 q^{61} -21.5741 q^{62} +15.5323 q^{64} -9.96017 q^{67} +33.3329 q^{68} +12.2168 q^{71} -8.37344 q^{73} +11.3654 q^{74} -20.8683 q^{76} -16.4074 q^{77} -11.1161 q^{79} -13.5936 q^{82} +14.6657 q^{83} -22.5480 q^{86} -28.7087 q^{88} -2.20459 q^{89} -21.3641 q^{91} +35.1980 q^{92} -14.3975 q^{94} -5.46673 q^{97} -39.7186 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} + 13 q^{4} - 10 q^{7}+O(q^{10})$$ 7 * q + q^2 + 13 * q^4 - 10 * q^7 $$7 q + q^{2} + 13 q^{4} - 10 q^{7} + 3 q^{11} - 6 q^{13} + 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} - 9 q^{22} + 11 q^{23} - 3 q^{26} - 25 q^{28} + 7 q^{29} + 18 q^{31} + q^{32} - q^{34} - 13 q^{37} + 12 q^{38} + 13 q^{41} - 9 q^{43} + 37 q^{44} - 8 q^{46} + 2 q^{47} + 21 q^{49} + q^{52} + 5 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 14 q^{61} - 8 q^{62} + 8 q^{64} - 14 q^{67} + 27 q^{68} + 8 q^{71} - 3 q^{73} + 34 q^{74} + 4 q^{76} - 28 q^{77} + 4 q^{79} + 20 q^{82} + 17 q^{83} - 4 q^{86} - 26 q^{88} + 20 q^{89} + 12 q^{91} + 60 q^{92} - 21 q^{94} - 13 q^{97} - 20 q^{98}+O(q^{100})$$ 7 * q + q^2 + 13 * q^4 - 10 * q^7 + 3 * q^11 - 6 * q^13 + 9 * q^14 + 21 * q^16 + 8 * q^17 + 10 * q^19 - 9 * q^22 + 11 * q^23 - 3 * q^26 - 25 * q^28 + 7 * q^29 + 18 * q^31 + q^32 - q^34 - 13 * q^37 + 12 * q^38 + 13 * q^41 - 9 * q^43 + 37 * q^44 - 8 * q^46 + 2 * q^47 + 21 * q^49 + q^52 + 5 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 14 * q^61 - 8 * q^62 + 8 * q^64 - 14 * q^67 + 27 * q^68 + 8 * q^71 - 3 * q^73 + 34 * q^74 + 4 * q^76 - 28 * q^77 + 4 * q^79 + 20 * q^82 + 17 * q^83 - 4 * q^86 - 26 * q^88 + 20 * q^89 + 12 * q^91 + 60 * q^92 - 21 * q^94 - 13 * q^97 - 20 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.66072 −1.88141 −0.940707 0.339221i $$-0.889837\pi$$
−0.940707 + 0.339221i $$0.889837\pi$$
$$3$$ 0 0
$$4$$ 5.07943 2.53972
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.68271 −1.76990 −0.884949 0.465689i $$-0.845807\pi$$
−0.884949 + 0.465689i $$0.845807\pi$$
$$8$$ −8.19351 −2.89684
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.50383 1.05645 0.528223 0.849106i $$-0.322859\pi$$
0.528223 + 0.849106i $$0.322859\pi$$
$$12$$ 0 0
$$13$$ 4.56233 1.26536 0.632682 0.774412i $$-0.281954\pi$$
0.632682 + 0.774412i $$0.281954\pi$$
$$14$$ 12.4594 3.32991
$$15$$ 0 0
$$16$$ 11.6418 2.91044
$$17$$ 6.56233 1.59160 0.795800 0.605560i $$-0.207051\pi$$
0.795800 + 0.605560i $$0.207051\pi$$
$$18$$ 0 0
$$19$$ −4.10839 −0.942529 −0.471264 0.881992i $$-0.656202\pi$$
−0.471264 + 0.881992i $$0.656202\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −9.32272 −1.98761
$$23$$ 6.92951 1.44490 0.722452 0.691422i $$-0.243015\pi$$
0.722452 + 0.691422i $$0.243015\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −12.1391 −2.38067
$$27$$ 0 0
$$28$$ −23.7855 −4.49504
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 8.10839 1.45631 0.728155 0.685413i $$-0.240378\pi$$
0.728155 + 0.685413i $$0.240378\pi$$
$$32$$ −14.5885 −2.57890
$$33$$ 0 0
$$34$$ −17.4605 −2.99446
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.27156 −0.702239 −0.351120 0.936331i $$-0.614199\pi$$
−0.351120 + 0.936331i $$0.614199\pi$$
$$38$$ 10.9313 1.77329
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.10898 0.797889 0.398944 0.916975i $$-0.369377\pi$$
0.398944 + 0.916975i $$0.369377\pi$$
$$42$$ 0 0
$$43$$ 8.47439 1.29233 0.646167 0.763196i $$-0.276371\pi$$
0.646167 + 0.763196i $$0.276371\pi$$
$$44$$ 17.7975 2.68307
$$45$$ 0 0
$$46$$ −18.4375 −2.71846
$$47$$ 5.41114 0.789295 0.394648 0.918833i $$-0.370867\pi$$
0.394648 + 0.918833i $$0.370867\pi$$
$$48$$ 0 0
$$49$$ 14.9278 2.13254
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 23.1741 3.21367
$$53$$ 4.20108 0.577063 0.288532 0.957470i $$-0.406833\pi$$
0.288532 + 0.957470i $$0.406833\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 38.3678 5.12711
$$57$$ 0 0
$$58$$ −2.66072 −0.349370
$$59$$ −5.36542 −0.698518 −0.349259 0.937026i $$-0.613567\pi$$
−0.349259 + 0.937026i $$0.613567\pi$$
$$60$$ 0 0
$$61$$ 1.27150 0.162798 0.0813991 0.996682i $$-0.474061\pi$$
0.0813991 + 0.996682i $$0.474061\pi$$
$$62$$ −21.5741 −2.73992
$$63$$ 0 0
$$64$$ 15.5323 1.94154
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.96017 −1.21683 −0.608414 0.793619i $$-0.708194\pi$$
−0.608414 + 0.793619i $$0.708194\pi$$
$$68$$ 33.3329 4.04221
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.2168 1.44986 0.724932 0.688820i $$-0.241871\pi$$
0.724932 + 0.688820i $$0.241871\pi$$
$$72$$ 0 0
$$73$$ −8.37344 −0.980037 −0.490018 0.871712i $$-0.663010\pi$$
−0.490018 + 0.871712i $$0.663010\pi$$
$$74$$ 11.3654 1.32120
$$75$$ 0 0
$$76$$ −20.8683 −2.39375
$$77$$ −16.4074 −1.86980
$$78$$ 0 0
$$79$$ −11.1161 −1.25065 −0.625327 0.780363i $$-0.715034\pi$$
−0.625327 + 0.780363i $$0.715034\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −13.5936 −1.50116
$$83$$ 14.6657 1.60977 0.804884 0.593432i $$-0.202228\pi$$
0.804884 + 0.593432i $$0.202228\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −22.5480 −2.43141
$$87$$ 0 0
$$88$$ −28.7087 −3.06036
$$89$$ −2.20459 −0.233686 −0.116843 0.993150i $$-0.537277\pi$$
−0.116843 + 0.993150i $$0.537277\pi$$
$$90$$ 0 0
$$91$$ −21.3641 −2.23956
$$92$$ 35.1980 3.66964
$$93$$ 0 0
$$94$$ −14.3975 −1.48499
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −5.46673 −0.555062 −0.277531 0.960717i $$-0.589516\pi$$
−0.277531 + 0.960717i $$0.589516\pi$$
$$98$$ −39.7186 −4.01218
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.54728 −0.750982 −0.375491 0.926826i $$-0.622526\pi$$
−0.375491 + 0.926826i $$0.622526\pi$$
$$102$$ 0 0
$$103$$ 5.18595 0.510987 0.255493 0.966811i $$-0.417762\pi$$
0.255493 + 0.966811i $$0.417762\pi$$
$$104$$ −37.3815 −3.66556
$$105$$ 0 0
$$106$$ −11.1779 −1.08569
$$107$$ −4.39624 −0.425001 −0.212500 0.977161i $$-0.568161\pi$$
−0.212500 + 0.977161i $$0.568161\pi$$
$$108$$ 0 0
$$109$$ −3.48595 −0.333894 −0.166947 0.985966i $$-0.553391\pi$$
−0.166947 + 0.985966i $$0.553391\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −54.5150 −5.15118
$$113$$ −10.6642 −1.00321 −0.501603 0.865098i $$-0.667256\pi$$
−0.501603 + 0.865098i $$0.667256\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 5.07943 0.471613
$$117$$ 0 0
$$118$$ 14.2759 1.31420
$$119$$ −30.7295 −2.81697
$$120$$ 0 0
$$121$$ 1.27684 0.116077
$$122$$ −3.38309 −0.306291
$$123$$ 0 0
$$124$$ 41.1860 3.69861
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1.46673 0.130151 0.0650756 0.997880i $$-0.479271\pi$$
0.0650756 + 0.997880i $$0.479271\pi$$
$$128$$ −12.1502 −1.07393
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.43157 −0.474559 −0.237279 0.971441i $$-0.576256\pi$$
−0.237279 + 0.971441i $$0.576256\pi$$
$$132$$ 0 0
$$133$$ 19.2384 1.66818
$$134$$ 26.5012 2.28936
$$135$$ 0 0
$$136$$ −53.7685 −4.61061
$$137$$ 5.27747 0.450884 0.225442 0.974257i $$-0.427617\pi$$
0.225442 + 0.974257i $$0.427617\pi$$
$$138$$ 0 0
$$139$$ 10.7043 0.907929 0.453964 0.891020i $$-0.350009\pi$$
0.453964 + 0.891020i $$0.350009\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −32.5054 −2.72779
$$143$$ 15.9857 1.33679
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 22.2794 1.84385
$$147$$ 0 0
$$148$$ −21.6971 −1.78349
$$149$$ −10.7308 −0.879104 −0.439552 0.898217i $$-0.644863\pi$$
−0.439552 + 0.898217i $$0.644863\pi$$
$$150$$ 0 0
$$151$$ 20.2395 1.64707 0.823534 0.567268i $$-0.191999\pi$$
0.823534 + 0.567268i $$0.191999\pi$$
$$152$$ 33.6621 2.73036
$$153$$ 0 0
$$154$$ 43.6556 3.51787
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1.62924 0.130028 0.0650139 0.997884i $$-0.479291\pi$$
0.0650139 + 0.997884i $$0.479291\pi$$
$$158$$ 29.5767 2.35300
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −32.4489 −2.55733
$$162$$ 0 0
$$163$$ 16.6586 1.30480 0.652400 0.757875i $$-0.273762\pi$$
0.652400 + 0.757875i $$0.273762\pi$$
$$164$$ 25.9507 2.02641
$$165$$ 0 0
$$166$$ −39.0213 −3.02864
$$167$$ 10.7214 0.829646 0.414823 0.909902i $$-0.363843\pi$$
0.414823 + 0.909902i $$0.363843\pi$$
$$168$$ 0 0
$$169$$ 7.81490 0.601146
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 43.0451 3.28216
$$173$$ −12.7826 −0.971846 −0.485923 0.874002i $$-0.661517\pi$$
−0.485923 + 0.874002i $$0.661517\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 40.7908 3.07472
$$177$$ 0 0
$$178$$ 5.86579 0.439659
$$179$$ −3.19306 −0.238660 −0.119330 0.992855i $$-0.538075\pi$$
−0.119330 + 0.992855i $$0.538075\pi$$
$$180$$ 0 0
$$181$$ −17.8264 −1.32503 −0.662514 0.749049i $$-0.730511\pi$$
−0.662514 + 0.749049i $$0.730511\pi$$
$$182$$ 56.8438 4.21355
$$183$$ 0 0
$$184$$ −56.7770 −4.18566
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 22.9933 1.68144
$$188$$ 27.4855 2.00459
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.37016 −0.243856 −0.121928 0.992539i $$-0.538908\pi$$
−0.121928 + 0.992539i $$0.538908\pi$$
$$192$$ 0 0
$$193$$ −11.0542 −0.795699 −0.397849 0.917451i $$-0.630243\pi$$
−0.397849 + 0.917451i $$0.630243\pi$$
$$194$$ 14.5454 1.04430
$$195$$ 0 0
$$196$$ 75.8245 5.41604
$$197$$ −7.99846 −0.569866 −0.284933 0.958547i $$-0.591971\pi$$
−0.284933 + 0.958547i $$0.591971\pi$$
$$198$$ 0 0
$$199$$ 8.30813 0.588948 0.294474 0.955660i $$-0.404856\pi$$
0.294474 + 0.955660i $$0.404856\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 20.0812 1.41291
$$203$$ −4.68271 −0.328662
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −13.7984 −0.961377
$$207$$ 0 0
$$208$$ 53.1136 3.68277
$$209$$ −14.3951 −0.995730
$$210$$ 0 0
$$211$$ 6.10957 0.420600 0.210300 0.977637i $$-0.432556\pi$$
0.210300 + 0.977637i $$0.432556\pi$$
$$212$$ 21.3391 1.46558
$$213$$ 0 0
$$214$$ 11.6972 0.799602
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −37.9692 −2.57752
$$218$$ 9.27514 0.628192
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 29.9396 2.01395
$$222$$ 0 0
$$223$$ −21.0235 −1.40784 −0.703920 0.710280i $$-0.748568\pi$$
−0.703920 + 0.710280i $$0.748568\pi$$
$$224$$ 68.3135 4.56439
$$225$$ 0 0
$$226$$ 28.3745 1.88744
$$227$$ −5.59548 −0.371385 −0.185692 0.982608i $$-0.559453\pi$$
−0.185692 + 0.982608i $$0.559453\pi$$
$$228$$ 0 0
$$229$$ 29.3568 1.93995 0.969975 0.243204i $$-0.0781985\pi$$
0.969975 + 0.243204i $$0.0781985\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −8.19351 −0.537930
$$233$$ −9.47855 −0.620960 −0.310480 0.950580i $$-0.600490\pi$$
−0.310480 + 0.950580i $$0.600490\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −27.2533 −1.77404
$$237$$ 0 0
$$238$$ 81.7626 5.29988
$$239$$ 5.45686 0.352975 0.176487 0.984303i $$-0.443527\pi$$
0.176487 + 0.984303i $$0.443527\pi$$
$$240$$ 0 0
$$241$$ −0.670684 −0.0432026 −0.0216013 0.999767i $$-0.506876\pi$$
−0.0216013 + 0.999767i $$0.506876\pi$$
$$242$$ −3.39733 −0.218388
$$243$$ 0 0
$$244$$ 6.45847 0.413461
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −18.7438 −1.19264
$$248$$ −66.4361 −4.21870
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 17.3401 1.09450 0.547250 0.836969i $$-0.315675\pi$$
0.547250 + 0.836969i $$0.315675\pi$$
$$252$$ 0 0
$$253$$ 24.2799 1.52646
$$254$$ −3.90256 −0.244868
$$255$$ 0 0
$$256$$ 1.26361 0.0789755
$$257$$ 15.8457 0.988425 0.494213 0.869341i $$-0.335456\pi$$
0.494213 + 0.869341i $$0.335456\pi$$
$$258$$ 0 0
$$259$$ 20.0024 1.24289
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 14.4519 0.892841
$$263$$ 9.70623 0.598512 0.299256 0.954173i $$-0.403262\pi$$
0.299256 + 0.954173i $$0.403262\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −51.1879 −3.13853
$$267$$ 0 0
$$268$$ −50.5920 −3.09040
$$269$$ 9.34619 0.569847 0.284924 0.958550i $$-0.408032\pi$$
0.284924 + 0.958550i $$0.408032\pi$$
$$270$$ 0 0
$$271$$ −32.4018 −1.96827 −0.984135 0.177421i $$-0.943225\pi$$
−0.984135 + 0.177421i $$0.943225\pi$$
$$272$$ 76.3972 4.63226
$$273$$ 0 0
$$274$$ −14.0419 −0.848300
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 21.4844 1.29087 0.645436 0.763815i $$-0.276676\pi$$
0.645436 + 0.763815i $$0.276676\pi$$
$$278$$ −28.4812 −1.70819
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.09329 −0.184530 −0.0922651 0.995734i $$-0.529411\pi$$
−0.0922651 + 0.995734i $$0.529411\pi$$
$$282$$ 0 0
$$283$$ −5.90145 −0.350805 −0.175402 0.984497i $$-0.556123\pi$$
−0.175402 + 0.984497i $$0.556123\pi$$
$$284$$ 62.0543 3.68224
$$285$$ 0 0
$$286$$ −42.5334 −2.51505
$$287$$ −23.9239 −1.41218
$$288$$ 0 0
$$289$$ 26.0642 1.53319
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −42.5323 −2.48902
$$293$$ −2.17061 −0.126808 −0.0634042 0.997988i $$-0.520196\pi$$
−0.0634042 + 0.997988i $$0.520196\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 34.9990 2.03428
$$297$$ 0 0
$$298$$ 28.5517 1.65396
$$299$$ 31.6148 1.82833
$$300$$ 0 0
$$301$$ −39.6831 −2.28730
$$302$$ −53.8516 −3.09881
$$303$$ 0 0
$$304$$ −47.8289 −2.74317
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −9.26410 −0.528730 −0.264365 0.964423i $$-0.585162\pi$$
−0.264365 + 0.964423i $$0.585162\pi$$
$$308$$ −83.3404 −4.74876
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 22.1337 1.25509 0.627544 0.778581i $$-0.284060\pi$$
0.627544 + 0.778581i $$0.284060\pi$$
$$312$$ 0 0
$$313$$ −0.982472 −0.0555326 −0.0277663 0.999614i $$-0.508839\pi$$
−0.0277663 + 0.999614i $$0.508839\pi$$
$$314$$ −4.33496 −0.244636
$$315$$ 0 0
$$316$$ −56.4632 −3.17631
$$317$$ −5.37998 −0.302170 −0.151085 0.988521i $$-0.548277\pi$$
−0.151085 + 0.988521i $$0.548277\pi$$
$$318$$ 0 0
$$319$$ 3.50383 0.196177
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 86.3374 4.81139
$$323$$ −26.9606 −1.50013
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −44.3238 −2.45487
$$327$$ 0 0
$$328$$ −41.8605 −2.31136
$$329$$ −25.3388 −1.39697
$$330$$ 0 0
$$331$$ 28.8813 1.58746 0.793729 0.608271i $$-0.208137\pi$$
0.793729 + 0.608271i $$0.208137\pi$$
$$332$$ 74.4933 4.08835
$$333$$ 0 0
$$334$$ −28.5266 −1.56091
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 17.6796 0.963071 0.481535 0.876427i $$-0.340079\pi$$
0.481535 + 0.876427i $$0.340079\pi$$
$$338$$ −20.7933 −1.13100
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 28.4104 1.53851
$$342$$ 0 0
$$343$$ −37.1233 −2.00447
$$344$$ −69.4350 −3.74369
$$345$$ 0 0
$$346$$ 34.0110 1.82844
$$347$$ 10.2375 0.549578 0.274789 0.961505i $$-0.411392\pi$$
0.274789 + 0.961505i $$0.411392\pi$$
$$348$$ 0 0
$$349$$ −7.50163 −0.401553 −0.200777 0.979637i $$-0.564347\pi$$
−0.200777 + 0.979637i $$0.564347\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −51.1156 −2.72447
$$353$$ −25.5268 −1.35865 −0.679327 0.733835i $$-0.737728\pi$$
−0.679327 + 0.733835i $$0.737728\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −11.1980 −0.593495
$$357$$ 0 0
$$358$$ 8.49584 0.449019
$$359$$ −26.6765 −1.40793 −0.703967 0.710232i $$-0.748590\pi$$
−0.703967 + 0.710232i $$0.748590\pi$$
$$360$$ 0 0
$$361$$ −2.12116 −0.111640
$$362$$ 47.4312 2.49293
$$363$$ 0 0
$$364$$ −108.517 −5.68786
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9.02936 0.471329 0.235664 0.971835i $$-0.424273\pi$$
0.235664 + 0.971835i $$0.424273\pi$$
$$368$$ 80.6718 4.20531
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −19.6724 −1.02134
$$372$$ 0 0
$$373$$ −22.4175 −1.16073 −0.580366 0.814356i $$-0.697091\pi$$
−0.580366 + 0.814356i $$0.697091\pi$$
$$374$$ −61.1788 −3.16348
$$375$$ 0 0
$$376$$ −44.3362 −2.28646
$$377$$ 4.56233 0.234972
$$378$$ 0 0
$$379$$ 9.65684 0.496038 0.248019 0.968755i $$-0.420220\pi$$
0.248019 + 0.968755i $$0.420220\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 8.96706 0.458795
$$383$$ −20.0311 −1.02354 −0.511771 0.859122i $$-0.671010\pi$$
−0.511771 + 0.859122i $$0.671010\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 29.4121 1.49704
$$387$$ 0 0
$$388$$ −27.7679 −1.40970
$$389$$ 19.7468 1.00120 0.500602 0.865678i $$-0.333112\pi$$
0.500602 + 0.865678i $$0.333112\pi$$
$$390$$ 0 0
$$391$$ 45.4738 2.29971
$$392$$ −122.311 −6.17762
$$393$$ 0 0
$$394$$ 21.2817 1.07215
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.39536 0.0700310 0.0350155 0.999387i $$-0.488852\pi$$
0.0350155 + 0.999387i $$0.488852\pi$$
$$398$$ −22.1056 −1.10805
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −25.3242 −1.26463 −0.632315 0.774711i $$-0.717895\pi$$
−0.632315 + 0.774711i $$0.717895\pi$$
$$402$$ 0 0
$$403$$ 36.9932 1.84276
$$404$$ −38.3359 −1.90728
$$405$$ 0 0
$$406$$ 12.4594 0.618348
$$407$$ −14.9668 −0.741878
$$408$$ 0 0
$$409$$ −5.44982 −0.269476 −0.134738 0.990881i $$-0.543019\pi$$
−0.134738 + 0.990881i $$0.543019\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 26.3417 1.29776
$$413$$ 25.1247 1.23630
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −66.5575 −3.26325
$$417$$ 0 0
$$418$$ 38.3013 1.87338
$$419$$ 26.6684 1.30284 0.651419 0.758719i $$-0.274174\pi$$
0.651419 + 0.758719i $$0.274174\pi$$
$$420$$ 0 0
$$421$$ 18.7599 0.914301 0.457150 0.889389i $$-0.348870\pi$$
0.457150 + 0.889389i $$0.348870\pi$$
$$422$$ −16.2559 −0.791323
$$423$$ 0 0
$$424$$ −34.4216 −1.67166
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −5.95404 −0.288136
$$428$$ −22.3304 −1.07938
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 21.5170 1.03644 0.518219 0.855248i $$-0.326595\pi$$
0.518219 + 0.855248i $$0.326595\pi$$
$$432$$ 0 0
$$433$$ −17.5946 −0.845544 −0.422772 0.906236i $$-0.638943\pi$$
−0.422772 + 0.906236i $$0.638943\pi$$
$$434$$ 101.025 4.84938
$$435$$ 0 0
$$436$$ −17.7067 −0.847995
$$437$$ −28.4691 −1.36186
$$438$$ 0 0
$$439$$ −28.3164 −1.35147 −0.675734 0.737146i $$-0.736173\pi$$
−0.675734 + 0.737146i $$0.736173\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −79.6608 −3.78908
$$443$$ 21.9916 1.04485 0.522427 0.852684i $$-0.325027\pi$$
0.522427 + 0.852684i $$0.325027\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 55.9377 2.64873
$$447$$ 0 0
$$448$$ −72.7332 −3.43632
$$449$$ −6.08052 −0.286957 −0.143479 0.989653i $$-0.545829\pi$$
−0.143479 + 0.989653i $$0.545829\pi$$
$$450$$ 0 0
$$451$$ 17.9010 0.842926
$$452$$ −54.1682 −2.54786
$$453$$ 0 0
$$454$$ 14.8880 0.698728
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 27.5955 1.29086 0.645432 0.763818i $$-0.276677\pi$$
0.645432 + 0.763818i $$0.276677\pi$$
$$458$$ −78.1102 −3.64985
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −20.3657 −0.948525 −0.474262 0.880384i $$-0.657285\pi$$
−0.474262 + 0.880384i $$0.657285\pi$$
$$462$$ 0 0
$$463$$ −40.1133 −1.86422 −0.932111 0.362172i $$-0.882035\pi$$
−0.932111 + 0.362172i $$0.882035\pi$$
$$464$$ 11.6418 0.540455
$$465$$ 0 0
$$466$$ 25.2198 1.16828
$$467$$ 32.0467 1.48295 0.741473 0.670982i $$-0.234127\pi$$
0.741473 + 0.670982i $$0.234127\pi$$
$$468$$ 0 0
$$469$$ 46.6406 2.15366
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 43.9616 2.02350
$$473$$ 29.6929 1.36528
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −156.088 −7.15430
$$477$$ 0 0
$$478$$ −14.5192 −0.664091
$$479$$ 22.6657 1.03562 0.517811 0.855495i $$-0.326747\pi$$
0.517811 + 0.855495i $$0.326747\pi$$
$$480$$ 0 0
$$481$$ −19.4883 −0.888589
$$482$$ 1.78450 0.0812819
$$483$$ 0 0
$$484$$ 6.48565 0.294802
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 25.6034 1.16020 0.580101 0.814545i $$-0.303013\pi$$
0.580101 + 0.814545i $$0.303013\pi$$
$$488$$ −10.4180 −0.471601
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3.40078 −0.153475 −0.0767375 0.997051i $$-0.524450\pi$$
−0.0767375 + 0.997051i $$0.524450\pi$$
$$492$$ 0 0
$$493$$ 6.56233 0.295553
$$494$$ 49.8721 2.24385
$$495$$ 0 0
$$496$$ 94.3960 4.23850
$$497$$ −57.2076 −2.56611
$$498$$ 0 0
$$499$$ 5.58352 0.249953 0.124976 0.992160i $$-0.460114\pi$$
0.124976 + 0.992160i $$0.460114\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −46.1372 −2.05921
$$503$$ 7.58471 0.338185 0.169093 0.985600i $$-0.445916\pi$$
0.169093 + 0.985600i $$0.445916\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −64.6019 −2.87190
$$507$$ 0 0
$$508$$ 7.45015 0.330547
$$509$$ −9.47234 −0.419854 −0.209927 0.977717i $$-0.567323\pi$$
−0.209927 + 0.977717i $$0.567323\pi$$
$$510$$ 0 0
$$511$$ 39.2104 1.73456
$$512$$ 20.9382 0.925348
$$513$$ 0 0
$$514$$ −42.1609 −1.85964
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 18.9597 0.833847
$$518$$ −53.2209 −2.33839
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 15.9046 0.696794 0.348397 0.937347i $$-0.386726\pi$$
0.348397 + 0.937347i $$0.386726\pi$$
$$522$$ 0 0
$$523$$ −0.965392 −0.0422137 −0.0211068 0.999777i $$-0.506719\pi$$
−0.0211068 + 0.999777i $$0.506719\pi$$
$$524$$ −27.5893 −1.20524
$$525$$ 0 0
$$526$$ −25.8256 −1.12605
$$527$$ 53.2099 2.31786
$$528$$ 0 0
$$529$$ 25.0181 1.08775
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 97.7200 4.23670
$$533$$ 23.3089 1.00962
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 81.6088 3.52496
$$537$$ 0 0
$$538$$ −24.8676 −1.07212
$$539$$ 52.3043 2.25291
$$540$$ 0 0
$$541$$ −17.0885 −0.734694 −0.367347 0.930084i $$-0.619734\pi$$
−0.367347 + 0.930084i $$0.619734\pi$$
$$542$$ 86.2122 3.70313
$$543$$ 0 0
$$544$$ −95.7344 −4.10458
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −33.0004 −1.41100 −0.705498 0.708712i $$-0.749277\pi$$
−0.705498 + 0.708712i $$0.749277\pi$$
$$548$$ 26.8065 1.14512
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.10839 −0.175023
$$552$$ 0 0
$$553$$ 52.0532 2.21353
$$554$$ −57.1639 −2.42866
$$555$$ 0 0
$$556$$ 54.3719 2.30588
$$557$$ −34.0021 −1.44071 −0.720357 0.693603i $$-0.756022\pi$$
−0.720357 + 0.693603i $$0.756022\pi$$
$$558$$ 0 0
$$559$$ 38.6630 1.63527
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 8.23038 0.347178
$$563$$ 23.4343 0.987640 0.493820 0.869564i $$-0.335600\pi$$
0.493820 + 0.869564i $$0.335600\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 15.7021 0.660008
$$567$$ 0 0
$$568$$ −100.098 −4.20003
$$569$$ −1.69285 −0.0709681 −0.0354841 0.999370i $$-0.511297\pi$$
−0.0354841 + 0.999370i $$0.511297\pi$$
$$570$$ 0 0
$$571$$ −17.8528 −0.747117 −0.373559 0.927607i $$-0.621863\pi$$
−0.373559 + 0.927607i $$0.621863\pi$$
$$572$$ 81.1981 3.39506
$$573$$ 0 0
$$574$$ 63.6547 2.65690
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.78174 0.115805 0.0579027 0.998322i $$-0.481559\pi$$
0.0579027 + 0.998322i $$0.481559\pi$$
$$578$$ −69.3497 −2.88457
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −68.6751 −2.84912
$$582$$ 0 0
$$583$$ 14.7199 0.609636
$$584$$ 68.6078 2.83901
$$585$$ 0 0
$$586$$ 5.77539 0.238579
$$587$$ 11.8335 0.488422 0.244211 0.969722i $$-0.421471\pi$$
0.244211 + 0.969722i $$0.421471\pi$$
$$588$$ 0 0
$$589$$ −33.3124 −1.37261
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −49.7285 −2.04383
$$593$$ −28.2961 −1.16198 −0.580990 0.813910i $$-0.697335\pi$$
−0.580990 + 0.813910i $$0.697335\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −54.5065 −2.23267
$$597$$ 0 0
$$598$$ −84.1180 −3.43984
$$599$$ 7.78454 0.318068 0.159034 0.987273i $$-0.449162\pi$$
0.159034 + 0.987273i $$0.449162\pi$$
$$600$$ 0 0
$$601$$ −20.5514 −0.838311 −0.419155 0.907915i $$-0.637674\pi$$
−0.419155 + 0.907915i $$0.637674\pi$$
$$602$$ 105.586 4.30335
$$603$$ 0 0
$$604$$ 102.805 4.18308
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −17.4256 −0.707281 −0.353641 0.935381i $$-0.615056\pi$$
−0.353641 + 0.935381i $$0.615056\pi$$
$$608$$ 59.9351 2.43069
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.6874 0.998746
$$612$$ 0 0
$$613$$ 35.2749 1.42474 0.712369 0.701805i $$-0.247622\pi$$
0.712369 + 0.701805i $$0.247622\pi$$
$$614$$ 24.6492 0.994760
$$615$$ 0 0
$$616$$ 134.434 5.41651
$$617$$ −40.3548 −1.62462 −0.812311 0.583225i $$-0.801791\pi$$
−0.812311 + 0.583225i $$0.801791\pi$$
$$618$$ 0 0
$$619$$ 15.3790 0.618134 0.309067 0.951040i $$-0.399983\pi$$
0.309067 + 0.951040i $$0.399983\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −58.8916 −2.36134
$$623$$ 10.3234 0.413599
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 2.61408 0.104480
$$627$$ 0 0
$$628$$ 8.27564 0.330234
$$629$$ −28.0314 −1.11768
$$630$$ 0 0
$$631$$ −23.7917 −0.947131 −0.473565 0.880759i $$-0.657033\pi$$
−0.473565 + 0.880759i $$0.657033\pi$$
$$632$$ 91.0795 3.62295
$$633$$ 0 0
$$634$$ 14.3146 0.568506
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 68.1054 2.69843
$$638$$ −9.32272 −0.369090
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.4524 1.20280 0.601398 0.798949i $$-0.294610\pi$$
0.601398 + 0.798949i $$0.294610\pi$$
$$642$$ 0 0
$$643$$ 26.8637 1.05940 0.529700 0.848185i $$-0.322304\pi$$
0.529700 + 0.848185i $$0.322304\pi$$
$$644$$ −164.822 −6.49489
$$645$$ 0 0
$$646$$ 71.7346 2.82236
$$647$$ 7.95411 0.312708 0.156354 0.987701i $$-0.450026\pi$$
0.156354 + 0.987701i $$0.450026\pi$$
$$648$$ 0 0
$$649$$ −18.7995 −0.737946
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 84.6161 3.31382
$$653$$ −22.9890 −0.899627 −0.449814 0.893122i $$-0.648510\pi$$
−0.449814 + 0.893122i $$0.648510\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 59.4776 2.32221
$$657$$ 0 0
$$658$$ 67.4194 2.62828
$$659$$ 3.03695 0.118303 0.0591515 0.998249i $$-0.481161\pi$$
0.0591515 + 0.998249i $$0.481161\pi$$
$$660$$ 0 0
$$661$$ −21.2414 −0.826194 −0.413097 0.910687i $$-0.635553\pi$$
−0.413097 + 0.910687i $$0.635553\pi$$
$$662$$ −76.8450 −2.98667
$$663$$ 0 0
$$664$$ −120.163 −4.66324
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.92951 0.268312
$$668$$ 54.4586 2.10707
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.45511 0.171987
$$672$$ 0 0
$$673$$ 41.7638 1.60988 0.804938 0.593359i $$-0.202198\pi$$
0.804938 + 0.593359i $$0.202198\pi$$
$$674$$ −47.0406 −1.81193
$$675$$ 0 0
$$676$$ 39.6953 1.52674
$$677$$ −16.1894 −0.622210 −0.311105 0.950375i $$-0.600699\pi$$
−0.311105 + 0.950375i $$0.600699\pi$$
$$678$$ 0 0
$$679$$ 25.5991 0.982403
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −75.5922 −2.89458
$$683$$ 36.0012 1.37755 0.688774 0.724977i $$-0.258149\pi$$
0.688774 + 0.724977i $$0.258149\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 98.7748 3.77124
$$687$$ 0 0
$$688$$ 98.6569 3.76126
$$689$$ 19.1667 0.730195
$$690$$ 0 0
$$691$$ −18.5707 −0.706464 −0.353232 0.935536i $$-0.614917\pi$$
−0.353232 + 0.935536i $$0.614917\pi$$
$$692$$ −64.9286 −2.46821
$$693$$ 0 0
$$694$$ −27.2391 −1.03398
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 33.5268 1.26992
$$698$$ 19.9597 0.755488
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30.3600 1.14668 0.573342 0.819316i $$-0.305647\pi$$
0.573342 + 0.819316i $$0.305647\pi$$
$$702$$ 0 0
$$703$$ 17.5492 0.661881
$$704$$ 54.4226 2.05113
$$705$$ 0 0
$$706$$ 67.9197 2.55619
$$707$$ 35.3417 1.32916
$$708$$ 0 0
$$709$$ −31.8654 −1.19673 −0.598365 0.801224i $$-0.704183\pi$$
−0.598365 + 0.801224i $$0.704183\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 18.0633 0.676950
$$713$$ 56.1872 2.10423
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −16.2189 −0.606130
$$717$$ 0 0
$$718$$ 70.9788 2.64891
$$719$$ 29.9357 1.11641 0.558206 0.829702i $$-0.311490\pi$$
0.558206 + 0.829702i $$0.311490\pi$$
$$720$$ 0 0
$$721$$ −24.2843 −0.904394
$$722$$ 5.64381 0.210041
$$723$$ 0 0
$$724$$ −90.5482 −3.36520
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −36.9479 −1.37032 −0.685161 0.728392i $$-0.740268\pi$$
−0.685161 + 0.728392i $$0.740268\pi$$
$$728$$ 175.047 6.48766
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 55.6118 2.05688
$$732$$ 0 0
$$733$$ 39.5741 1.46170 0.730851 0.682537i $$-0.239123\pi$$
0.730851 + 0.682537i $$0.239123\pi$$
$$734$$ −24.0246 −0.886764
$$735$$ 0 0
$$736$$ −101.091 −3.72626
$$737$$ −34.8988 −1.28551
$$738$$ 0 0
$$739$$ 6.40567 0.235636 0.117818 0.993035i $$-0.462410\pi$$
0.117818 + 0.993035i $$0.462410\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 52.3429 1.92157
$$743$$ −1.37536 −0.0504570 −0.0252285 0.999682i $$-0.508031\pi$$
−0.0252285 + 0.999682i $$0.508031\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 59.6466 2.18382
$$747$$ 0 0
$$748$$ 116.793 4.27038
$$749$$ 20.5863 0.752207
$$750$$ 0 0
$$751$$ 27.5712 1.00609 0.503044 0.864261i $$-0.332213\pi$$
0.503044 + 0.864261i $$0.332213\pi$$
$$752$$ 62.9952 2.29720
$$753$$ 0 0
$$754$$ −12.1391 −0.442080
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 31.8451 1.15743 0.578714 0.815530i $$-0.303555\pi$$
0.578714 + 0.815530i $$0.303555\pi$$
$$758$$ −25.6941 −0.933253
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −20.2493 −0.734038 −0.367019 0.930213i $$-0.619622\pi$$
−0.367019 + 0.930213i $$0.619622\pi$$
$$762$$ 0 0
$$763$$ 16.3237 0.590957
$$764$$ −17.1185 −0.619326
$$765$$ 0 0
$$766$$ 53.2971 1.92570
$$767$$ −24.4788 −0.883879
$$768$$ 0 0
$$769$$ 14.9837 0.540325 0.270162 0.962815i $$-0.412923\pi$$
0.270162 + 0.962815i $$0.412923\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −56.1490 −2.02085
$$773$$ −10.7077 −0.385131 −0.192565 0.981284i $$-0.561681\pi$$
−0.192565 + 0.981284i $$0.561681\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 44.7917 1.60793
$$777$$ 0 0
$$778$$ −52.5407 −1.88368
$$779$$ −20.9897 −0.752033
$$780$$ 0 0
$$781$$ 42.8055 1.53170
$$782$$ −120.993 −4.32670
$$783$$ 0 0
$$784$$ 173.785 6.20662
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 9.72906 0.346804 0.173402 0.984851i $$-0.444524\pi$$
0.173402 + 0.984851i $$0.444524\pi$$
$$788$$ −40.6276 −1.44730
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 49.9374 1.77557
$$792$$ 0 0
$$793$$ 5.80099 0.205999
$$794$$ −3.71266 −0.131757
$$795$$ 0 0
$$796$$ 42.2006 1.49576
$$797$$ −9.52385 −0.337352 −0.168676 0.985672i $$-0.553949\pi$$
−0.168676 + 0.985672i $$0.553949\pi$$
$$798$$ 0 0
$$799$$ 35.5097 1.25624
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 67.3806 2.37929
$$803$$ −29.3391 −1.03536
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −98.4285 −3.46700
$$807$$ 0 0
$$808$$ 61.8387 2.17548
$$809$$ 43.2976 1.52226 0.761131 0.648598i $$-0.224644\pi$$
0.761131 + 0.648598i $$0.224644\pi$$
$$810$$ 0 0
$$811$$ −12.4910 −0.438619 −0.219310 0.975655i $$-0.570381\pi$$
−0.219310 + 0.975655i $$0.570381\pi$$
$$812$$ −23.7855 −0.834707
$$813$$ 0 0
$$814$$ 39.8225 1.39578
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −34.8161 −1.21806
$$818$$ 14.5005 0.506996
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16.3447 −0.570434 −0.285217 0.958463i $$-0.592066\pi$$
−0.285217 + 0.958463i $$0.592066\pi$$
$$822$$ 0 0
$$823$$ −24.4971 −0.853916 −0.426958 0.904271i $$-0.640415\pi$$
−0.426958 + 0.904271i $$0.640415\pi$$
$$824$$ −42.4911 −1.48025
$$825$$ 0 0
$$826$$ −66.8497 −2.32600
$$827$$ −44.6342 −1.55208 −0.776042 0.630681i $$-0.782776\pi$$
−0.776042 + 0.630681i $$0.782776\pi$$
$$828$$ 0 0
$$829$$ 19.8536 0.689543 0.344772 0.938687i $$-0.387956\pi$$
0.344772 + 0.938687i $$0.387956\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 70.8636 2.45675
$$833$$ 97.9609 3.39414
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −73.1189 −2.52887
$$837$$ 0 0
$$838$$ −70.9572 −2.45118
$$839$$ 1.62403 0.0560679 0.0280339 0.999607i $$-0.491075\pi$$
0.0280339 + 0.999607i $$0.491075\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −49.9148 −1.72018
$$843$$ 0 0
$$844$$ 31.0332 1.06821
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −5.97909 −0.205444
$$848$$ 48.9080 1.67951
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −29.5998 −1.01467
$$852$$ 0 0
$$853$$ −51.1148 −1.75014 −0.875069 0.483998i $$-0.839184\pi$$
−0.875069 + 0.483998i $$0.839184\pi$$
$$854$$ 15.8420 0.542103
$$855$$ 0 0
$$856$$ 36.0206 1.23116
$$857$$ −33.4832 −1.14376 −0.571882 0.820336i $$-0.693786\pi$$
−0.571882 + 0.820336i $$0.693786\pi$$
$$858$$ 0 0
$$859$$ 20.6161 0.703414 0.351707 0.936110i $$-0.385601\pi$$
0.351707 + 0.936110i $$0.385601\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −57.2507 −1.94997
$$863$$ 5.87941 0.200137 0.100069 0.994981i $$-0.468094\pi$$
0.100069 + 0.994981i $$0.468094\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 46.8144 1.59082
$$867$$ 0 0
$$868$$ −192.862 −6.54616
$$869$$ −38.9488 −1.32125
$$870$$ 0 0
$$871$$ −45.4416 −1.53973
$$872$$ 28.5622 0.967237
$$873$$ 0 0
$$874$$ 75.7484 2.56223
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 42.2191 1.42564 0.712818 0.701349i $$-0.247418\pi$$
0.712818 + 0.701349i $$0.247418\pi$$
$$878$$ 75.3420 2.54267
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 34.8813 1.17518 0.587590 0.809159i $$-0.300077\pi$$
0.587590 + 0.809159i $$0.300077\pi$$
$$882$$ 0 0
$$883$$ 44.5605 1.49958 0.749790 0.661675i $$-0.230154\pi$$
0.749790 + 0.661675i $$0.230154\pi$$
$$884$$ 152.076 5.11487
$$885$$ 0 0
$$886$$ −58.5136 −1.96580
$$887$$ 24.0830 0.808627 0.404313 0.914620i $$-0.367510\pi$$
0.404313 + 0.914620i $$0.367510\pi$$
$$888$$ 0 0
$$889$$ −6.86826 −0.230354
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −106.788 −3.57551
$$893$$ −22.2310 −0.743933
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 56.8957 1.90075
$$897$$ 0 0
$$898$$ 16.1786 0.539885
$$899$$ 8.10839 0.270430
$$900$$ 0 0
$$901$$ 27.5689 0.918454
$$902$$ −47.6296 −1.58589
$$903$$ 0 0
$$904$$ 87.3774 2.90613
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 22.3174 0.741037 0.370519 0.928825i $$-0.379180\pi$$
0.370519 + 0.928825i $$0.379180\pi$$
$$908$$ −28.4218 −0.943212
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 45.7050 1.51427 0.757137 0.653256i $$-0.226597\pi$$
0.757137 + 0.653256i $$0.226597\pi$$
$$912$$ 0 0
$$913$$ 51.3861 1.70063
$$914$$ −73.4240 −2.42865
$$915$$ 0 0
$$916$$ 149.116 4.92692
$$917$$ 25.4345 0.839920
$$918$$ 0 0
$$919$$ −29.5423 −0.974509 −0.487255 0.873260i $$-0.662002\pi$$
−0.487255 + 0.873260i $$0.662002\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 54.1874 1.78457
$$923$$ 55.7370 1.83461
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 106.730 3.50737
$$927$$ 0 0
$$928$$ −14.5885 −0.478890
$$929$$ 55.6718 1.82653 0.913267 0.407362i $$-0.133551\pi$$
0.913267 + 0.407362i $$0.133551\pi$$
$$930$$ 0 0
$$931$$ −61.3290 −2.00998
$$932$$ −48.1456 −1.57706
$$933$$ 0 0
$$934$$ −85.2674 −2.79004
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −36.7113 −1.19931 −0.599654 0.800260i $$-0.704695\pi$$
−0.599654 + 0.800260i $$0.704695\pi$$
$$938$$ −124.098 −4.05193
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 23.3426 0.760948 0.380474 0.924792i $$-0.375761\pi$$
0.380474 + 0.924792i $$0.375761\pi$$
$$942$$ 0 0
$$943$$ 35.4027 1.15287
$$944$$ −62.4629 −2.03299
$$945$$ 0 0
$$946$$ −79.0044 −2.56865
$$947$$ 51.6216 1.67748 0.838739 0.544534i $$-0.183293\pi$$
0.838739 + 0.544534i $$0.183293\pi$$
$$948$$ 0 0
$$949$$ −38.2024 −1.24010
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 251.782 8.16031
$$953$$ 37.5233 1.21550 0.607750 0.794129i $$-0.292072\pi$$
0.607750 + 0.794129i $$0.292072\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 27.7177 0.896456
$$957$$ 0 0
$$958$$ −60.3070 −1.94843
$$959$$ −24.7128 −0.798019
$$960$$ 0 0
$$961$$ 34.7459 1.12084
$$962$$ 51.8528 1.67180
$$963$$ 0 0
$$964$$ −3.40670 −0.109722
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −51.5532 −1.65784 −0.828919 0.559369i $$-0.811044\pi$$
−0.828919 + 0.559369i $$0.811044\pi$$
$$968$$ −10.4618 −0.336256
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −19.8516 −0.637068 −0.318534 0.947911i $$-0.603191\pi$$
−0.318534 + 0.947911i $$0.603191\pi$$
$$972$$ 0 0
$$973$$ −50.1252 −1.60694
$$974$$ −68.1235 −2.18282
$$975$$ 0 0
$$976$$ 14.8024 0.473815
$$977$$ 8.90236 0.284812 0.142406 0.989808i $$-0.454516\pi$$
0.142406 + 0.989808i $$0.454516\pi$$
$$978$$ 0 0
$$979$$ −7.72450 −0.246876
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 9.04853 0.288750
$$983$$ −3.44153 −0.109768 −0.0548838 0.998493i $$-0.517479\pi$$
−0.0548838 + 0.998493i $$0.517479\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −17.4605 −0.556057
$$987$$ 0 0
$$988$$ −95.2080 −3.02897
$$989$$ 58.7234 1.86730
$$990$$ 0 0
$$991$$ −26.7962 −0.851209 −0.425605 0.904909i $$-0.639939\pi$$
−0.425605 + 0.904909i $$0.639939\pi$$
$$992$$ −118.289 −3.75568
$$993$$ 0 0
$$994$$ 152.213 4.82791
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −0.913058 −0.0289168 −0.0144584 0.999895i $$-0.504602\pi$$
−0.0144584 + 0.999895i $$0.504602\pi$$
$$998$$ −14.8562 −0.470265
$$999$$ 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.bw.1.1 7
3.2 odd 2 6525.2.a.bv.1.7 7
5.4 even 2 1305.2.a.s.1.7 7
15.14 odd 2 1305.2.a.t.1.1 yes 7

By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.s.1.7 7 5.4 even 2
1305.2.a.t.1.1 yes 7 15.14 odd 2
6525.2.a.bv.1.7 7 3.2 odd 2
6525.2.a.bw.1.1 7 1.1 even 1 trivial