# Properties

 Label 6525.2.a.bs.1.3 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 1.3 Root $$0.245526$$ 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)

 $$f(q)$$ $$=$$ $$q+0.754474 q^{2} -1.43077 q^{4} -4.18524 q^{7} -2.58843 q^{8} +O(q^{10})$$ $$q+0.754474 q^{2} -1.43077 q^{4} -4.18524 q^{7} -2.58843 q^{8} -0.596817 q^{11} +2.18524 q^{13} -3.15766 q^{14} +0.908639 q^{16} +2.81314 q^{17} +0.528904 q^{19} -0.450283 q^{22} +0.590044 q^{23} +1.64871 q^{26} +5.98812 q^{28} -1.00000 q^{29} +8.02005 q^{31} +5.86240 q^{32} +2.12244 q^{34} +2.52465 q^{37} +0.399044 q^{38} -1.57110 q^{41} +6.98484 q^{43} +0.853908 q^{44} +0.445173 q^{46} -2.57524 q^{47} +10.5163 q^{49} -3.12658 q^{52} +10.3878 q^{53} +10.8332 q^{56} -0.754474 q^{58} -13.9168 q^{59} -11.1942 q^{61} +6.05092 q^{62} +2.60575 q^{64} -0.614139 q^{67} -4.02495 q^{68} -9.28010 q^{71} -9.59116 q^{73} +1.90478 q^{74} -0.756739 q^{76} +2.49783 q^{77} -10.3396 q^{79} -1.18536 q^{82} +3.92531 q^{83} +5.26988 q^{86} +1.54482 q^{88} +12.3352 q^{89} -9.14577 q^{91} -0.844217 q^{92} -1.94295 q^{94} -3.21794 q^{97} +7.93424 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.754474 0.533494 0.266747 0.963767i $$-0.414051\pi$$
0.266747 + 0.963767i $$0.414051\pi$$
$$3$$ 0 0
$$4$$ −1.43077 −0.715385
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.18524 −1.58187 −0.790937 0.611898i $$-0.790406\pi$$
−0.790937 + 0.611898i $$0.790406\pi$$
$$8$$ −2.58843 −0.915147
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −0.596817 −0.179947 −0.0899736 0.995944i $$-0.528678\pi$$
−0.0899736 + 0.995944i $$0.528678\pi$$
$$12$$ 0 0
$$13$$ 2.18524 0.606077 0.303039 0.952978i $$-0.401999\pi$$
0.303039 + 0.952978i $$0.401999\pi$$
$$14$$ −3.15766 −0.843919
$$15$$ 0 0
$$16$$ 0.908639 0.227160
$$17$$ 2.81314 0.682286 0.341143 0.940011i $$-0.389186\pi$$
0.341143 + 0.940011i $$0.389186\pi$$
$$18$$ 0 0
$$19$$ 0.528904 0.121339 0.0606694 0.998158i $$-0.480676\pi$$
0.0606694 + 0.998158i $$0.480676\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −0.450283 −0.0960007
$$23$$ 0.590044 0.123033 0.0615163 0.998106i $$-0.480406\pi$$
0.0615163 + 0.998106i $$0.480406\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.64871 0.323338
$$27$$ 0 0
$$28$$ 5.98812 1.13165
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 8.02005 1.44044 0.720222 0.693744i $$-0.244040\pi$$
0.720222 + 0.693744i $$0.244040\pi$$
$$32$$ 5.86240 1.03633
$$33$$ 0 0
$$34$$ 2.12244 0.363995
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.52465 0.415050 0.207525 0.978230i $$-0.433459\pi$$
0.207525 + 0.978230i $$0.433459\pi$$
$$38$$ 0.399044 0.0647335
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.57110 −0.245365 −0.122683 0.992446i $$-0.539150\pi$$
−0.122683 + 0.992446i $$0.539150\pi$$
$$42$$ 0 0
$$43$$ 6.98484 1.06518 0.532589 0.846374i $$-0.321219\pi$$
0.532589 + 0.846374i $$0.321219\pi$$
$$44$$ 0.853908 0.128731
$$45$$ 0 0
$$46$$ 0.445173 0.0656372
$$47$$ −2.57524 −0.375638 −0.187819 0.982204i $$-0.560142\pi$$
−0.187819 + 0.982204i $$0.560142\pi$$
$$48$$ 0 0
$$49$$ 10.5163 1.50232
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −3.12658 −0.433578
$$53$$ 10.3878 1.42688 0.713438 0.700719i $$-0.247137\pi$$
0.713438 + 0.700719i $$0.247137\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 10.8332 1.44765
$$57$$ 0 0
$$58$$ −0.754474 −0.0990673
$$59$$ −13.9168 −1.81181 −0.905907 0.423477i $$-0.860810\pi$$
−0.905907 + 0.423477i $$0.860810\pi$$
$$60$$ 0 0
$$61$$ −11.1942 −1.43327 −0.716633 0.697450i $$-0.754318\pi$$
−0.716633 + 0.697450i $$0.754318\pi$$
$$62$$ 6.05092 0.768468
$$63$$ 0 0
$$64$$ 2.60575 0.325718
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −0.614139 −0.0750290 −0.0375145 0.999296i $$-0.511944\pi$$
−0.0375145 + 0.999296i $$0.511944\pi$$
$$68$$ −4.02495 −0.488097
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −9.28010 −1.10134 −0.550672 0.834721i $$-0.685629\pi$$
−0.550672 + 0.834721i $$0.685629\pi$$
$$72$$ 0 0
$$73$$ −9.59116 −1.12256 −0.561280 0.827626i $$-0.689691\pi$$
−0.561280 + 0.827626i $$0.689691\pi$$
$$74$$ 1.90478 0.221427
$$75$$ 0 0
$$76$$ −0.756739 −0.0868039
$$77$$ 2.49783 0.284654
$$78$$ 0 0
$$79$$ −10.3396 −1.16330 −0.581649 0.813440i $$-0.697592\pi$$
−0.581649 + 0.813440i $$0.697592\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −1.18536 −0.130901
$$83$$ 3.92531 0.430859 0.215430 0.976519i $$-0.430885\pi$$
0.215430 + 0.976519i $$0.430885\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 5.26988 0.568265
$$87$$ 0 0
$$88$$ 1.54482 0.164678
$$89$$ 12.3352 1.30752 0.653762 0.756700i $$-0.273190\pi$$
0.653762 + 0.756700i $$0.273190\pi$$
$$90$$ 0 0
$$91$$ −9.14577 −0.958738
$$92$$ −0.844217 −0.0880157
$$93$$ 0 0
$$94$$ −1.94295 −0.200400
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −3.21794 −0.326732 −0.163366 0.986566i $$-0.552235\pi$$
−0.163366 + 0.986566i $$0.552235\pi$$
$$98$$ 7.93424 0.801480
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.31726 −0.927102 −0.463551 0.886070i $$-0.653425\pi$$
−0.463551 + 0.886070i $$0.653425\pi$$
$$102$$ 0 0
$$103$$ −20.1574 −1.98617 −0.993085 0.117393i $$-0.962546\pi$$
−0.993085 + 0.117393i $$0.962546\pi$$
$$104$$ −5.65634 −0.554650
$$105$$ 0 0
$$106$$ 7.83733 0.761229
$$107$$ 1.58569 0.153295 0.0766475 0.997058i $$-0.475578\pi$$
0.0766475 + 0.997058i $$0.475578\pi$$
$$108$$ 0 0
$$109$$ −7.91051 −0.757690 −0.378845 0.925460i $$-0.623679\pi$$
−0.378845 + 0.925460i $$0.623679\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −3.80287 −0.359338
$$113$$ −3.07328 −0.289110 −0.144555 0.989497i $$-0.546175\pi$$
−0.144555 + 0.989497i $$0.546175\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.43077 0.132844
$$117$$ 0 0
$$118$$ −10.4999 −0.966591
$$119$$ −11.7737 −1.07929
$$120$$ 0 0
$$121$$ −10.6438 −0.967619
$$122$$ −8.44571 −0.764639
$$123$$ 0 0
$$124$$ −11.4748 −1.03047
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.9254 1.14694 0.573472 0.819225i $$-0.305596\pi$$
0.573472 + 0.819225i $$0.305596\pi$$
$$128$$ −9.75882 −0.862566
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.69473 0.497551 0.248775 0.968561i $$-0.419972\pi$$
0.248775 + 0.968561i $$0.419972\pi$$
$$132$$ 0 0
$$133$$ −2.21359 −0.191943
$$134$$ −0.463352 −0.0400275
$$135$$ 0 0
$$136$$ −7.28160 −0.624392
$$137$$ 18.7274 1.59999 0.799996 0.600005i $$-0.204835\pi$$
0.799996 + 0.600005i $$0.204835\pi$$
$$138$$ 0 0
$$139$$ 1.34376 0.113976 0.0569880 0.998375i $$-0.481850\pi$$
0.0569880 + 0.998375i $$0.481850\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −7.00159 −0.587560
$$143$$ −1.30419 −0.109062
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −7.23628 −0.598879
$$147$$ 0 0
$$148$$ −3.61219 −0.296920
$$149$$ 15.9925 1.31016 0.655079 0.755561i $$-0.272635\pi$$
0.655079 + 0.755561i $$0.272635\pi$$
$$150$$ 0 0
$$151$$ 17.9595 1.46152 0.730760 0.682635i $$-0.239166\pi$$
0.730760 + 0.682635i $$0.239166\pi$$
$$152$$ −1.36903 −0.111043
$$153$$ 0 0
$$154$$ 1.88454 0.151861
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.78158 −0.541229 −0.270615 0.962688i $$-0.587227\pi$$
−0.270615 + 0.962688i $$0.587227\pi$$
$$158$$ −7.80097 −0.620612
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2.46948 −0.194622
$$162$$ 0 0
$$163$$ −20.9951 −1.64447 −0.822233 0.569151i $$-0.807272\pi$$
−0.822233 + 0.569151i $$0.807272\pi$$
$$164$$ 2.24789 0.175531
$$165$$ 0 0
$$166$$ 2.96155 0.229861
$$167$$ −14.9135 −1.15404 −0.577020 0.816730i $$-0.695784\pi$$
−0.577020 + 0.816730i $$0.695784\pi$$
$$168$$ 0 0
$$169$$ −8.22471 −0.632670
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −9.99369 −0.762011
$$173$$ −10.8685 −0.826318 −0.413159 0.910659i $$-0.635575\pi$$
−0.413159 + 0.910659i $$0.635575\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −0.542291 −0.0408767
$$177$$ 0 0
$$178$$ 9.30655 0.697556
$$179$$ −1.26613 −0.0946349 −0.0473175 0.998880i $$-0.515067\pi$$
−0.0473175 + 0.998880i $$0.515067\pi$$
$$180$$ 0 0
$$181$$ −5.80673 −0.431611 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$182$$ −6.90025 −0.511480
$$183$$ 0 0
$$184$$ −1.52729 −0.112593
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.67893 −0.122776
$$188$$ 3.68458 0.268725
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −15.3358 −1.10966 −0.554830 0.831964i $$-0.687217\pi$$
−0.554830 + 0.831964i $$0.687217\pi$$
$$192$$ 0 0
$$193$$ 16.4946 1.18731 0.593653 0.804721i $$-0.297685\pi$$
0.593653 + 0.804721i $$0.297685\pi$$
$$194$$ −2.42785 −0.174310
$$195$$ 0 0
$$196$$ −15.0463 −1.07474
$$197$$ 24.2353 1.72670 0.863348 0.504609i $$-0.168363\pi$$
0.863348 + 0.504609i $$0.168363\pi$$
$$198$$ 0 0
$$199$$ 17.4875 1.23965 0.619826 0.784739i $$-0.287203\pi$$
0.619826 + 0.784739i $$0.287203\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −7.02963 −0.494603
$$203$$ 4.18524 0.293746
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −15.2083 −1.05961
$$207$$ 0 0
$$208$$ 1.98560 0.137676
$$209$$ −0.315659 −0.0218346
$$210$$ 0 0
$$211$$ −26.0504 −1.79338 −0.896692 0.442655i $$-0.854037\pi$$
−0.896692 + 0.442655i $$0.854037\pi$$
$$212$$ −14.8626 −1.02076
$$213$$ 0 0
$$214$$ 1.19637 0.0817819
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −33.5659 −2.27860
$$218$$ −5.96827 −0.404223
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.14739 0.413518
$$222$$ 0 0
$$223$$ −10.7895 −0.722519 −0.361259 0.932465i $$-0.617653\pi$$
−0.361259 + 0.932465i $$0.617653\pi$$
$$224$$ −24.5356 −1.63935
$$225$$ 0 0
$$226$$ −2.31871 −0.154238
$$227$$ −6.99182 −0.464063 −0.232032 0.972708i $$-0.574537\pi$$
−0.232032 + 0.972708i $$0.574537\pi$$
$$228$$ 0 0
$$229$$ −20.9503 −1.38444 −0.692218 0.721689i $$-0.743366\pi$$
−0.692218 + 0.721689i $$0.743366\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.58843 0.169938
$$233$$ −3.85567 −0.252593 −0.126297 0.991993i $$-0.540309\pi$$
−0.126297 + 0.991993i $$0.540309\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 19.9117 1.29614
$$237$$ 0 0
$$238$$ −8.88293 −0.575795
$$239$$ 25.3084 1.63706 0.818532 0.574461i $$-0.194788\pi$$
0.818532 + 0.574461i $$0.194788\pi$$
$$240$$ 0 0
$$241$$ −22.8510 −1.47196 −0.735981 0.677002i $$-0.763279\pi$$
−0.735981 + 0.677002i $$0.763279\pi$$
$$242$$ −8.03048 −0.516219
$$243$$ 0 0
$$244$$ 16.0163 1.02534
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.15578 0.0735407
$$248$$ −20.7593 −1.31822
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 7.37025 0.465206 0.232603 0.972572i $$-0.425276\pi$$
0.232603 + 0.972572i $$0.425276\pi$$
$$252$$ 0 0
$$253$$ −0.352149 −0.0221394
$$254$$ 9.75188 0.611888
$$255$$ 0 0
$$256$$ −12.5743 −0.785892
$$257$$ −3.21575 −0.200593 −0.100296 0.994958i $$-0.531979\pi$$
−0.100296 + 0.994958i $$0.531979\pi$$
$$258$$ 0 0
$$259$$ −10.5663 −0.656557
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4.29652 0.265440
$$263$$ −16.0190 −0.987774 −0.493887 0.869526i $$-0.664424\pi$$
−0.493887 + 0.869526i $$0.664424\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1.67010 −0.102400
$$267$$ 0 0
$$268$$ 0.878691 0.0536746
$$269$$ 19.9820 1.21832 0.609161 0.793047i $$-0.291506\pi$$
0.609161 + 0.793047i $$0.291506\pi$$
$$270$$ 0 0
$$271$$ −8.72743 −0.530153 −0.265077 0.964227i $$-0.585397\pi$$
−0.265077 + 0.964227i $$0.585397\pi$$
$$272$$ 2.55613 0.154988
$$273$$ 0 0
$$274$$ 14.1294 0.853585
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 30.5856 1.83771 0.918856 0.394594i $$-0.129115\pi$$
0.918856 + 0.394594i $$0.129115\pi$$
$$278$$ 1.01383 0.0608055
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.2570 −1.80498 −0.902490 0.430712i $$-0.858263\pi$$
−0.902490 + 0.430712i $$0.858263\pi$$
$$282$$ 0 0
$$283$$ 9.35107 0.555863 0.277932 0.960601i $$-0.410351\pi$$
0.277932 + 0.960601i $$0.410351\pi$$
$$284$$ 13.2777 0.787885
$$285$$ 0 0
$$286$$ −0.983978 −0.0581838
$$287$$ 6.57545 0.388137
$$288$$ 0 0
$$289$$ −9.08625 −0.534485
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 13.7227 0.803062
$$293$$ −8.47749 −0.495260 −0.247630 0.968855i $$-0.579652\pi$$
−0.247630 + 0.968855i $$0.579652\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.53487 −0.379832
$$297$$ 0 0
$$298$$ 12.0659 0.698961
$$299$$ 1.28939 0.0745673
$$300$$ 0 0
$$301$$ −29.2332 −1.68498
$$302$$ 13.5499 0.779711
$$303$$ 0 0
$$304$$ 0.480582 0.0275633
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −19.8740 −1.13427 −0.567135 0.823625i $$-0.691948\pi$$
−0.567135 + 0.823625i $$0.691948\pi$$
$$308$$ −3.57381 −0.203637
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.5476 0.711507 0.355754 0.934580i $$-0.384224\pi$$
0.355754 + 0.934580i $$0.384224\pi$$
$$312$$ 0 0
$$313$$ 2.82942 0.159928 0.0799640 0.996798i $$-0.474519\pi$$
0.0799640 + 0.996798i $$0.474519\pi$$
$$314$$ −5.11653 −0.288742
$$315$$ 0 0
$$316$$ 14.7936 0.832205
$$317$$ 6.27390 0.352377 0.176189 0.984356i $$-0.443623\pi$$
0.176189 + 0.984356i $$0.443623\pi$$
$$318$$ 0 0
$$319$$ 0.596817 0.0334154
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1.86316 −0.103830
$$323$$ 1.48788 0.0827878
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −15.8403 −0.877312
$$327$$ 0 0
$$328$$ 4.06669 0.224545
$$329$$ 10.7780 0.594211
$$330$$ 0 0
$$331$$ −4.47271 −0.245843 −0.122921 0.992416i $$-0.539226\pi$$
−0.122921 + 0.992416i $$0.539226\pi$$
$$332$$ −5.61622 −0.308230
$$333$$ 0 0
$$334$$ −11.2518 −0.615673
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3.62191 0.197298 0.0986489 0.995122i $$-0.468548\pi$$
0.0986489 + 0.995122i $$0.468548\pi$$
$$338$$ −6.20533 −0.337526
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −4.78651 −0.259204
$$342$$ 0 0
$$343$$ −14.7164 −0.794611
$$344$$ −18.0797 −0.974794
$$345$$ 0 0
$$346$$ −8.20002 −0.440836
$$347$$ −34.4636 −1.85010 −0.925051 0.379844i $$-0.875978\pi$$
−0.925051 + 0.379844i $$0.875978\pi$$
$$348$$ 0 0
$$349$$ −18.0401 −0.965665 −0.482832 0.875713i $$-0.660392\pi$$
−0.482832 + 0.875713i $$0.660392\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3.49878 −0.186486
$$353$$ 7.60470 0.404757 0.202379 0.979307i $$-0.435133\pi$$
0.202379 + 0.979307i $$0.435133\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −17.6488 −0.935382
$$357$$ 0 0
$$358$$ −0.955261 −0.0504871
$$359$$ 2.32286 0.122596 0.0612980 0.998120i $$-0.480476\pi$$
0.0612980 + 0.998120i $$0.480476\pi$$
$$360$$ 0 0
$$361$$ −18.7203 −0.985277
$$362$$ −4.38103 −0.230262
$$363$$ 0 0
$$364$$ 13.0855 0.685866
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 6.38930 0.333519 0.166759 0.985998i $$-0.446670\pi$$
0.166759 + 0.985998i $$0.446670\pi$$
$$368$$ 0.536137 0.0279481
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −43.4755 −2.25714
$$372$$ 0 0
$$373$$ 24.1627 1.25110 0.625549 0.780185i $$-0.284875\pi$$
0.625549 + 0.780185i $$0.284875\pi$$
$$374$$ −1.26671 −0.0654999
$$375$$ 0 0
$$376$$ 6.66583 0.343764
$$377$$ −2.18524 −0.112546
$$378$$ 0 0
$$379$$ −29.2093 −1.50038 −0.750191 0.661222i $$-0.770038\pi$$
−0.750191 + 0.661222i $$0.770038\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −11.5705 −0.591997
$$383$$ 31.7731 1.62353 0.811766 0.583983i $$-0.198507\pi$$
0.811766 + 0.583983i $$0.198507\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 12.4447 0.633420
$$387$$ 0 0
$$388$$ 4.60413 0.233739
$$389$$ −23.7158 −1.20244 −0.601220 0.799084i $$-0.705318\pi$$
−0.601220 + 0.799084i $$0.705318\pi$$
$$390$$ 0 0
$$391$$ 1.65988 0.0839435
$$392$$ −27.2206 −1.37485
$$393$$ 0 0
$$394$$ 18.2849 0.921181
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 32.8280 1.64759 0.823794 0.566889i $$-0.191853\pi$$
0.823794 + 0.566889i $$0.191853\pi$$
$$398$$ 13.1938 0.661347
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 8.21905 0.410440 0.205220 0.978716i $$-0.434209\pi$$
0.205220 + 0.978716i $$0.434209\pi$$
$$402$$ 0 0
$$403$$ 17.5258 0.873020
$$404$$ 13.3308 0.663234
$$405$$ 0 0
$$406$$ 3.15766 0.156712
$$407$$ −1.50676 −0.0746871
$$408$$ 0 0
$$409$$ 27.9114 1.38013 0.690065 0.723748i $$-0.257582\pi$$
0.690065 + 0.723748i $$0.257582\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 28.8406 1.42088
$$413$$ 58.2452 2.86606
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 12.8108 0.628099
$$417$$ 0 0
$$418$$ −0.238156 −0.0116486
$$419$$ −27.0379 −1.32089 −0.660445 0.750874i $$-0.729632\pi$$
−0.660445 + 0.750874i $$0.729632\pi$$
$$420$$ 0 0
$$421$$ 3.79559 0.184986 0.0924928 0.995713i $$-0.470516\pi$$
0.0924928 + 0.995713i $$0.470516\pi$$
$$422$$ −19.6544 −0.956759
$$423$$ 0 0
$$424$$ −26.8881 −1.30580
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 46.8503 2.26725
$$428$$ −2.26876 −0.109665
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0.405550 0.0195347 0.00976733 0.999952i $$-0.496891\pi$$
0.00976733 + 0.999952i $$0.496891\pi$$
$$432$$ 0 0
$$433$$ 25.5978 1.23015 0.615075 0.788468i $$-0.289126\pi$$
0.615075 + 0.788468i $$0.289126\pi$$
$$434$$ −25.3246 −1.21562
$$435$$ 0 0
$$436$$ 11.3181 0.542039
$$437$$ 0.312077 0.0149286
$$438$$ 0 0
$$439$$ −26.2378 −1.25226 −0.626132 0.779717i $$-0.715363\pi$$
−0.626132 + 0.779717i $$0.715363\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 4.63805 0.220609
$$443$$ 26.5050 1.25929 0.629645 0.776883i $$-0.283200\pi$$
0.629645 + 0.776883i $$0.283200\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −8.14040 −0.385459
$$447$$ 0 0
$$448$$ −10.9057 −0.515245
$$449$$ −5.57064 −0.262895 −0.131447 0.991323i $$-0.541962\pi$$
−0.131447 + 0.991323i $$0.541962\pi$$
$$450$$ 0 0
$$451$$ 0.937662 0.0441528
$$452$$ 4.39715 0.206825
$$453$$ 0 0
$$454$$ −5.27514 −0.247575
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −25.3774 −1.18710 −0.593551 0.804796i $$-0.702275\pi$$
−0.593551 + 0.804796i $$0.702275\pi$$
$$458$$ −15.8065 −0.738587
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2.20886 −0.102877 −0.0514384 0.998676i $$-0.516381\pi$$
−0.0514384 + 0.998676i $$0.516381\pi$$
$$462$$ 0 0
$$463$$ −7.56591 −0.351618 −0.175809 0.984424i $$-0.556254\pi$$
−0.175809 + 0.984424i $$0.556254\pi$$
$$464$$ −0.908639 −0.0421825
$$465$$ 0 0
$$466$$ −2.90900 −0.134757
$$467$$ 19.9056 0.921120 0.460560 0.887629i $$-0.347649\pi$$
0.460560 + 0.887629i $$0.347649\pi$$
$$468$$ 0 0
$$469$$ 2.57032 0.118686
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 36.0226 1.65808
$$473$$ −4.16867 −0.191676
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 16.8454 0.772108
$$477$$ 0 0
$$478$$ 19.0945 0.873363
$$479$$ −10.3910 −0.474778 −0.237389 0.971415i $$-0.576292\pi$$
−0.237389 + 0.971415i $$0.576292\pi$$
$$480$$ 0 0
$$481$$ 5.51698 0.251552
$$482$$ −17.2405 −0.785282
$$483$$ 0 0
$$484$$ 15.2288 0.692220
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −10.0574 −0.455744 −0.227872 0.973691i $$-0.573177\pi$$
−0.227872 + 0.973691i $$0.573177\pi$$
$$488$$ 28.9753 1.31165
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0.193599 0.00873699 0.00436849 0.999990i $$-0.498609\pi$$
0.00436849 + 0.999990i $$0.498609\pi$$
$$492$$ 0 0
$$493$$ −2.81314 −0.126697
$$494$$ 0.872008 0.0392335
$$495$$ 0 0
$$496$$ 7.28733 0.327211
$$497$$ 38.8395 1.74219
$$498$$ 0 0
$$499$$ 13.6181 0.609630 0.304815 0.952412i $$-0.401405\pi$$
0.304815 + 0.952412i $$0.401405\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 5.56066 0.248185
$$503$$ −30.7553 −1.37131 −0.685655 0.727926i $$-0.740484\pi$$
−0.685655 + 0.727926i $$0.740484\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −0.265687 −0.0118112
$$507$$ 0 0
$$508$$ −18.4933 −0.820506
$$509$$ 1.01272 0.0448881 0.0224441 0.999748i $$-0.492855\pi$$
0.0224441 + 0.999748i $$0.492855\pi$$
$$510$$ 0 0
$$511$$ 40.1413 1.77575
$$512$$ 10.0307 0.443298
$$513$$ 0 0
$$514$$ −2.42620 −0.107015
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.53695 0.0675950
$$518$$ −7.97198 −0.350269
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.02284 −0.263865 −0.131933 0.991259i $$-0.542118\pi$$
−0.131933 + 0.991259i $$0.542118\pi$$
$$522$$ 0 0
$$523$$ −20.3870 −0.891460 −0.445730 0.895167i $$-0.647056\pi$$
−0.445730 + 0.895167i $$0.647056\pi$$
$$524$$ −8.14784 −0.355940
$$525$$ 0 0
$$526$$ −12.0859 −0.526971
$$527$$ 22.5615 0.982795
$$528$$ 0 0
$$529$$ −22.6518 −0.984863
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 3.16714 0.137313
$$533$$ −3.43324 −0.148710
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 1.58965 0.0686625
$$537$$ 0 0
$$538$$ 15.0759 0.649967
$$539$$ −6.27629 −0.270339
$$540$$ 0 0
$$541$$ −16.4420 −0.706895 −0.353448 0.935454i $$-0.614991\pi$$
−0.353448 + 0.935454i $$0.614991\pi$$
$$542$$ −6.58461 −0.282833
$$543$$ 0 0
$$544$$ 16.4917 0.707077
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.2531 1.20802 0.604008 0.796979i $$-0.293570\pi$$
0.604008 + 0.796979i $$0.293570\pi$$
$$548$$ −26.7946 −1.14461
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −0.528904 −0.0225321
$$552$$ 0 0
$$553$$ 43.2738 1.84019
$$554$$ 23.0760 0.980407
$$555$$ 0 0
$$556$$ −1.92261 −0.0815367
$$557$$ 36.3662 1.54089 0.770443 0.637508i $$-0.220035\pi$$
0.770443 + 0.637508i $$0.220035\pi$$
$$558$$ 0 0
$$559$$ 15.2636 0.645580
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −22.8281 −0.962945
$$563$$ −5.24750 −0.221156 −0.110578 0.993867i $$-0.535270\pi$$
−0.110578 + 0.993867i $$0.535270\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 7.05514 0.296550
$$567$$ 0 0
$$568$$ 24.0208 1.00789
$$569$$ −17.3773 −0.728495 −0.364248 0.931302i $$-0.618674\pi$$
−0.364248 + 0.931302i $$0.618674\pi$$
$$570$$ 0 0
$$571$$ −22.2889 −0.932760 −0.466380 0.884584i $$-0.654442\pi$$
−0.466380 + 0.884584i $$0.654442\pi$$
$$572$$ 1.86600 0.0780212
$$573$$ 0 0
$$574$$ 4.96101 0.207068
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −26.1738 −1.08963 −0.544816 0.838556i $$-0.683400\pi$$
−0.544816 + 0.838556i $$0.683400\pi$$
$$578$$ −6.85534 −0.285145
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −16.4284 −0.681564
$$582$$ 0 0
$$583$$ −6.19962 −0.256762
$$584$$ 24.8260 1.02731
$$585$$ 0 0
$$586$$ −6.39604 −0.264218
$$587$$ −26.4002 −1.08965 −0.544827 0.838549i $$-0.683405\pi$$
−0.544827 + 0.838549i $$0.683405\pi$$
$$588$$ 0 0
$$589$$ 4.24184 0.174782
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.29400 0.0942826
$$593$$ 19.4942 0.800532 0.400266 0.916399i $$-0.368918\pi$$
0.400266 + 0.916399i $$0.368918\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −22.8816 −0.937266
$$597$$ 0 0
$$598$$ 0.972811 0.0397812
$$599$$ −20.9453 −0.855800 −0.427900 0.903826i $$-0.640746\pi$$
−0.427900 + 0.903826i $$0.640746\pi$$
$$600$$ 0 0
$$601$$ 14.3956 0.587208 0.293604 0.955927i $$-0.405145\pi$$
0.293604 + 0.955927i $$0.405145\pi$$
$$602$$ −22.0557 −0.898924
$$603$$ 0 0
$$604$$ −25.6958 −1.04555
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.53979 0.346620 0.173310 0.984867i $$-0.444554\pi$$
0.173310 + 0.984867i $$0.444554\pi$$
$$608$$ 3.10064 0.125748
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5.62753 −0.227666
$$612$$ 0 0
$$613$$ −36.7173 −1.48300 −0.741499 0.670954i $$-0.765885\pi$$
−0.741499 + 0.670954i $$0.765885\pi$$
$$614$$ −14.9944 −0.605125
$$615$$ 0 0
$$616$$ −6.46544 −0.260500
$$617$$ −45.1304 −1.81688 −0.908441 0.418014i $$-0.862726\pi$$
−0.908441 + 0.418014i $$0.862726\pi$$
$$618$$ 0 0
$$619$$ −18.9737 −0.762618 −0.381309 0.924448i $$-0.624527\pi$$
−0.381309 + 0.924448i $$0.624527\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 9.46681 0.379585
$$623$$ −51.6256 −2.06834
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 2.13472 0.0853206
$$627$$ 0 0
$$628$$ 9.70288 0.387187
$$629$$ 7.10219 0.283183
$$630$$ 0 0
$$631$$ −18.5514 −0.738518 −0.369259 0.929327i $$-0.620388\pi$$
−0.369259 + 0.929327i $$0.620388\pi$$
$$632$$ 26.7633 1.06459
$$633$$ 0 0
$$634$$ 4.73349 0.187991
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 22.9806 0.910524
$$638$$ 0.450283 0.0178269
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 34.5735 1.36557 0.682785 0.730619i $$-0.260769\pi$$
0.682785 + 0.730619i $$0.260769\pi$$
$$642$$ 0 0
$$643$$ −6.58510 −0.259691 −0.129846 0.991534i $$-0.541448\pi$$
−0.129846 + 0.991534i $$0.541448\pi$$
$$644$$ 3.53325 0.139230
$$645$$ 0 0
$$646$$ 1.12257 0.0441668
$$647$$ 10.0375 0.394614 0.197307 0.980342i $$-0.436780\pi$$
0.197307 + 0.980342i $$0.436780\pi$$
$$648$$ 0 0
$$649$$ 8.30579 0.326031
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 30.0392 1.17643
$$653$$ 3.25188 0.127256 0.0636280 0.997974i $$-0.479733\pi$$
0.0636280 + 0.997974i $$0.479733\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −1.42757 −0.0557371
$$657$$ 0 0
$$658$$ 8.13173 0.317008
$$659$$ 9.93148 0.386875 0.193438 0.981113i $$-0.438036\pi$$
0.193438 + 0.981113i $$0.438036\pi$$
$$660$$ 0 0
$$661$$ −9.44925 −0.367533 −0.183767 0.982970i $$-0.558829\pi$$
−0.183767 + 0.982970i $$0.558829\pi$$
$$662$$ −3.37455 −0.131155
$$663$$ 0 0
$$664$$ −10.1604 −0.394299
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.590044 −0.0228466
$$668$$ 21.3377 0.825582
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6.68088 0.257912
$$672$$ 0 0
$$673$$ −11.1154 −0.428468 −0.214234 0.976782i $$-0.568725\pi$$
−0.214234 + 0.976782i $$0.568725\pi$$
$$674$$ 2.73263 0.105257
$$675$$ 0 0
$$676$$ 11.7677 0.452603
$$677$$ −44.6045 −1.71429 −0.857145 0.515075i $$-0.827764\pi$$
−0.857145 + 0.515075i $$0.827764\pi$$
$$678$$ 0 0
$$679$$ 13.4679 0.516849
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −3.61129 −0.138284
$$683$$ 0.490486 0.0187679 0.00938396 0.999956i $$-0.497013\pi$$
0.00938396 + 0.999956i $$0.497013\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −11.1031 −0.423920
$$687$$ 0 0
$$688$$ 6.34669 0.241965
$$689$$ 22.6999 0.864797
$$690$$ 0 0
$$691$$ −6.72252 −0.255737 −0.127868 0.991791i $$-0.540814\pi$$
−0.127868 + 0.991791i $$0.540814\pi$$
$$692$$ 15.5503 0.591135
$$693$$ 0 0
$$694$$ −26.0019 −0.987017
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4.41973 −0.167409
$$698$$ −13.6108 −0.515176
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 32.2913 1.21963 0.609813 0.792546i $$-0.291245\pi$$
0.609813 + 0.792546i $$0.291245\pi$$
$$702$$ 0 0
$$703$$ 1.33530 0.0503617
$$704$$ −1.55515 −0.0586121
$$705$$ 0 0
$$706$$ 5.73755 0.215936
$$707$$ 38.9950 1.46656
$$708$$ 0 0
$$709$$ 9.75887 0.366502 0.183251 0.983066i $$-0.441338\pi$$
0.183251 + 0.983066i $$0.441338\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −31.9286 −1.19658
$$713$$ 4.73218 0.177222
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.81154 0.0677004
$$717$$ 0 0
$$718$$ 1.75254 0.0654042
$$719$$ −44.2063 −1.64862 −0.824308 0.566142i $$-0.808435\pi$$
−0.824308 + 0.566142i $$0.808435\pi$$
$$720$$ 0 0
$$721$$ 84.3638 3.14187
$$722$$ −14.1239 −0.525639
$$723$$ 0 0
$$724$$ 8.30809 0.308768
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −44.9849 −1.66840 −0.834199 0.551463i $$-0.814070\pi$$
−0.834199 + 0.551463i $$0.814070\pi$$
$$728$$ 23.6732 0.877385
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 19.6493 0.726756
$$732$$ 0 0
$$733$$ −1.59233 −0.0588142 −0.0294071 0.999568i $$-0.509362\pi$$
−0.0294071 + 0.999568i $$0.509362\pi$$
$$734$$ 4.82056 0.177930
$$735$$ 0 0
$$736$$ 3.45907 0.127503
$$737$$ 0.366529 0.0135013
$$738$$ 0 0
$$739$$ −37.2457 −1.37011 −0.685053 0.728493i $$-0.740221\pi$$
−0.685053 + 0.728493i $$0.740221\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −32.8011 −1.20417
$$743$$ 41.1086 1.50813 0.754065 0.656800i $$-0.228091\pi$$
0.754065 + 0.656800i $$0.228091\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 18.2301 0.667452
$$747$$ 0 0
$$748$$ 2.40216 0.0878317
$$749$$ −6.63652 −0.242493
$$750$$ 0 0
$$751$$ −44.6707 −1.63006 −0.815029 0.579420i $$-0.803279\pi$$
−0.815029 + 0.579420i $$0.803279\pi$$
$$752$$ −2.33997 −0.0853298
$$753$$ 0 0
$$754$$ −1.64871 −0.0600424
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −13.8828 −0.504577 −0.252289 0.967652i $$-0.581183\pi$$
−0.252289 + 0.967652i $$0.581183\pi$$
$$758$$ −22.0377 −0.800444
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.9137 1.12062 0.560310 0.828283i $$-0.310682\pi$$
0.560310 + 0.828283i $$0.310682\pi$$
$$762$$ 0 0
$$763$$ 33.1074 1.19857
$$764$$ 21.9420 0.793834
$$765$$ 0 0
$$766$$ 23.9720 0.866144
$$767$$ −30.4116 −1.09810
$$768$$ 0 0
$$769$$ 31.1489 1.12326 0.561630 0.827389i $$-0.310175\pi$$
0.561630 + 0.827389i $$0.310175\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −23.5999 −0.849380
$$773$$ −25.7705 −0.926902 −0.463451 0.886123i $$-0.653389\pi$$
−0.463451 + 0.886123i $$0.653389\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 8.32940 0.299008
$$777$$ 0 0
$$778$$ −17.8930 −0.641494
$$779$$ −0.830963 −0.0297723
$$780$$ 0 0
$$781$$ 5.53852 0.198184
$$782$$ 1.25233 0.0447833
$$783$$ 0 0
$$784$$ 9.55548 0.341267
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −26.2806 −0.936802 −0.468401 0.883516i $$-0.655170\pi$$
−0.468401 + 0.883516i $$0.655170\pi$$
$$788$$ −34.6752 −1.23525
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.8624 0.457335
$$792$$ 0 0
$$793$$ −24.4620 −0.868671
$$794$$ 24.7678 0.878978
$$795$$ 0 0
$$796$$ −25.0205 −0.886828
$$797$$ 53.7239 1.90300 0.951500 0.307650i $$-0.0995427\pi$$
0.951500 + 0.307650i $$0.0995427\pi$$
$$798$$ 0 0
$$799$$ −7.24452 −0.256293
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 6.20106 0.218967
$$803$$ 5.72417 0.202002
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 13.2227 0.465751
$$807$$ 0 0
$$808$$ 24.1170 0.848434
$$809$$ 6.02488 0.211824 0.105912 0.994376i $$-0.466224\pi$$
0.105912 + 0.994376i $$0.466224\pi$$
$$810$$ 0 0
$$811$$ −7.70554 −0.270578 −0.135289 0.990806i $$-0.543196\pi$$
−0.135289 + 0.990806i $$0.543196\pi$$
$$812$$ −5.98812 −0.210142
$$813$$ 0 0
$$814$$ −1.13681 −0.0398451
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3.69431 0.129247
$$818$$ 21.0584 0.736290
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16.4338 −0.573543 −0.286772 0.957999i $$-0.592582\pi$$
−0.286772 + 0.957999i $$0.592582\pi$$
$$822$$ 0 0
$$823$$ −45.4920 −1.58575 −0.792876 0.609383i $$-0.791417\pi$$
−0.792876 + 0.609383i $$0.791417\pi$$
$$824$$ 52.1760 1.81764
$$825$$ 0 0
$$826$$ 43.9445 1.52902
$$827$$ −33.1498 −1.15273 −0.576367 0.817191i $$-0.695530\pi$$
−0.576367 + 0.817191i $$0.695530\pi$$
$$828$$ 0 0
$$829$$ 27.9403 0.970407 0.485203 0.874401i $$-0.338746\pi$$
0.485203 + 0.874401i $$0.338746\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 5.69419 0.197411
$$833$$ 29.5837 1.02501
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0.451635 0.0156201
$$837$$ 0 0
$$838$$ −20.3994 −0.704686
$$839$$ −27.5639 −0.951610 −0.475805 0.879551i $$-0.657843\pi$$
−0.475805 + 0.879551i $$0.657843\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 2.86367 0.0986886
$$843$$ 0 0
$$844$$ 37.2721 1.28296
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 44.5469 1.53065
$$848$$ 9.43876 0.324128
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.48966 0.0510647
$$852$$ 0 0
$$853$$ 14.4467 0.494645 0.247323 0.968933i $$-0.420449\pi$$
0.247323 + 0.968933i $$0.420449\pi$$
$$854$$ 35.3474 1.20956
$$855$$ 0 0
$$856$$ −4.10445 −0.140287
$$857$$ −20.6065 −0.703904 −0.351952 0.936018i $$-0.614482\pi$$
−0.351952 + 0.936018i $$0.614482\pi$$
$$858$$ 0 0
$$859$$ −22.6473 −0.772717 −0.386358 0.922349i $$-0.626267\pi$$
−0.386358 + 0.922349i $$0.626267\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0.305977 0.0104216
$$863$$ −18.3880 −0.625935 −0.312967 0.949764i $$-0.601323\pi$$
−0.312967 + 0.949764i $$0.601323\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 19.3129 0.656278
$$867$$ 0 0
$$868$$ 48.0250 1.63007
$$869$$ 6.17086 0.209332
$$870$$ 0 0
$$871$$ −1.34204 −0.0454734
$$872$$ 20.4758 0.693397
$$873$$ 0 0
$$874$$ 0.235454 0.00796434
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.7681 0.701289 0.350645 0.936509i $$-0.385962\pi$$
0.350645 + 0.936509i $$0.385962\pi$$
$$878$$ −19.7958 −0.668075
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −25.7085 −0.866140 −0.433070 0.901360i $$-0.642570\pi$$
−0.433070 + 0.901360i $$0.642570\pi$$
$$882$$ 0 0
$$883$$ −48.3574 −1.62736 −0.813679 0.581315i $$-0.802539\pi$$
−0.813679 + 0.581315i $$0.802539\pi$$
$$884$$ −8.79550 −0.295825
$$885$$ 0 0
$$886$$ 19.9973 0.671824
$$887$$ −48.7809 −1.63790 −0.818951 0.573864i $$-0.805444\pi$$
−0.818951 + 0.573864i $$0.805444\pi$$
$$888$$ 0 0
$$889$$ −54.0960 −1.81432
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 15.4373 0.516879
$$893$$ −1.36206 −0.0455795
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 40.8430 1.36447
$$897$$ 0 0
$$898$$ −4.20291 −0.140253
$$899$$ −8.02005 −0.267484
$$900$$ 0 0
$$901$$ 29.2223 0.973537
$$902$$ 0.707442 0.0235552
$$903$$ 0 0
$$904$$ 7.95495 0.264578
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −10.8048 −0.358767 −0.179383 0.983779i $$-0.557410\pi$$
−0.179383 + 0.983779i $$0.557410\pi$$
$$908$$ 10.0037 0.331984
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −22.2530 −0.737274 −0.368637 0.929573i $$-0.620175\pi$$
−0.368637 + 0.929573i $$0.620175\pi$$
$$912$$ 0 0
$$913$$ −2.34269 −0.0775319
$$914$$ −19.1466 −0.633312
$$915$$ 0 0
$$916$$ 29.9751 0.990404
$$917$$ −23.8338 −0.787062
$$918$$ 0 0
$$919$$ 47.1476 1.55526 0.777628 0.628724i $$-0.216423\pi$$
0.777628 + 0.628724i $$0.216423\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −1.66653 −0.0548841
$$923$$ −20.2793 −0.667500
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −5.70828 −0.187586
$$927$$ 0 0
$$928$$ −5.86240 −0.192443
$$929$$ −13.0115 −0.426893 −0.213446 0.976955i $$-0.568469\pi$$
−0.213446 + 0.976955i $$0.568469\pi$$
$$930$$ 0 0
$$931$$ 5.56209 0.182290
$$932$$ 5.51657 0.180701
$$933$$ 0 0
$$934$$ 15.0182 0.491412
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −19.4077 −0.634021 −0.317011 0.948422i $$-0.602679\pi$$
−0.317011 + 0.948422i $$0.602679\pi$$
$$938$$ 1.93924 0.0633184
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −16.3192 −0.531990 −0.265995 0.963974i $$-0.585701\pi$$
−0.265995 + 0.963974i $$0.585701\pi$$
$$942$$ 0 0
$$943$$ −0.927021 −0.0301880
$$944$$ −12.6453 −0.411571
$$945$$ 0 0
$$946$$ −3.14515 −0.102258
$$947$$ 20.0894 0.652819 0.326409 0.945229i $$-0.394161\pi$$
0.326409 + 0.945229i $$0.394161\pi$$
$$948$$ 0 0
$$949$$ −20.9590 −0.680358
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 30.4753 0.987709
$$953$$ 23.6518 0.766158 0.383079 0.923716i $$-0.374864\pi$$
0.383079 + 0.923716i $$0.374864\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −36.2105 −1.17113
$$957$$ 0 0
$$958$$ −7.83975 −0.253291
$$959$$ −78.3788 −2.53098
$$960$$ 0 0
$$961$$ 33.3212 1.07488
$$962$$ 4.16241 0.134202
$$963$$ 0 0
$$964$$ 32.6945 1.05302
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 5.19177 0.166956 0.0834780 0.996510i $$-0.473397\pi$$
0.0834780 + 0.996510i $$0.473397\pi$$
$$968$$ 27.5507 0.885513
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 31.0165 0.995366 0.497683 0.867359i $$-0.334184\pi$$
0.497683 + 0.867359i $$0.334184\pi$$
$$972$$ 0 0
$$973$$ −5.62395 −0.180296
$$974$$ −7.58804 −0.243137
$$975$$ 0 0
$$976$$ −10.1715 −0.325580
$$977$$ 35.0578 1.12160 0.560798 0.827952i $$-0.310494\pi$$
0.560798 + 0.827952i $$0.310494\pi$$
$$978$$ 0 0
$$979$$ −7.36183 −0.235285
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0.146065 0.00466113
$$983$$ −11.3717 −0.362702 −0.181351 0.983418i $$-0.558047\pi$$
−0.181351 + 0.983418i $$0.558047\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.12244 −0.0675922
$$987$$ 0 0
$$988$$ −1.65366 −0.0526099
$$989$$ 4.12136 0.131052
$$990$$ 0 0
$$991$$ −20.5008 −0.651230 −0.325615 0.945503i $$-0.605571\pi$$
−0.325615 + 0.945503i $$0.605571\pi$$
$$992$$ 47.0167 1.49278
$$993$$ 0 0
$$994$$ 29.3034 0.929446
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −8.50559 −0.269375 −0.134687 0.990888i $$-0.543003\pi$$
−0.134687 + 0.990888i $$0.543003\pi$$
$$998$$ 10.2745 0.325234
$$999$$ 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.1.3 5
3.2 odd 2 2175.2.a.w.1.3 5
5.2 odd 4 1305.2.c.j.784.7 10
5.3 odd 4 1305.2.c.j.784.4 10
5.4 even 2 6525.2.a.bl.1.3 5
15.2 even 4 435.2.c.e.349.4 10
15.8 even 4 435.2.c.e.349.7 yes 10
15.14 odd 2 2175.2.a.z.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.e.349.4 10 15.2 even 4
435.2.c.e.349.7 yes 10 15.8 even 4
1305.2.c.j.784.4 10 5.3 odd 4
1305.2.c.j.784.7 10 5.2 odd 4
2175.2.a.w.1.3 5 3.2 odd 2
2175.2.a.z.1.3 5 15.14 odd 2
6525.2.a.bl.1.3 5 5.4 even 2
6525.2.a.bs.1.3 5 1.1 even 1 trivial