# Properties

 Label 6525.2.a.bu.1.1 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $1$ 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: $$1$$ 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: $$x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14$$ x^7 - 2*x^6 - 10*x^5 + 19*x^4 + 24*x^3 - 44*x^2 - 3*x + 14 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2175) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.66356$$ 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)

 $$f(q)$$ $$=$$ $$q-2.66356 q^{2} +5.09453 q^{4} -0.0170416 q^{7} -8.24244 q^{8} +O(q^{10})$$ $$q-2.66356 q^{2} +5.09453 q^{4} -0.0170416 q^{7} -8.24244 q^{8} -1.70990 q^{11} -4.69566 q^{13} +0.0453913 q^{14} +11.7651 q^{16} +3.91494 q^{17} +6.02411 q^{19} +4.55440 q^{22} -4.82786 q^{23} +12.5071 q^{26} -0.0868190 q^{28} +1.00000 q^{29} +1.33709 q^{31} -14.8522 q^{32} -10.4277 q^{34} -8.18905 q^{37} -16.0455 q^{38} -8.36721 q^{41} +3.72560 q^{43} -8.71111 q^{44} +12.8593 q^{46} +5.36826 q^{47} -6.99971 q^{49} -23.9222 q^{52} +7.21637 q^{53} +0.140465 q^{56} -2.66356 q^{58} +8.65422 q^{59} +5.72637 q^{61} -3.56141 q^{62} +16.0294 q^{64} +3.87486 q^{67} +19.9448 q^{68} +4.32991 q^{71} -1.78245 q^{73} +21.8120 q^{74} +30.6900 q^{76} +0.0291394 q^{77} +0.233544 q^{79} +22.2865 q^{82} +6.15497 q^{83} -9.92334 q^{86} +14.0937 q^{88} +12.4053 q^{89} +0.0800217 q^{91} -24.5957 q^{92} -14.2986 q^{94} -19.2918 q^{97} +18.6441 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 2 q^{2} + 10 q^{4} - q^{7} - 3 q^{8}+O(q^{10})$$ 7 * q - 2 * q^2 + 10 * q^4 - q^7 - 3 * q^8 $$7 q - 2 q^{2} + 10 q^{4} - q^{7} - 3 q^{8} - 4 q^{11} - q^{13} - 15 q^{14} + 12 q^{16} - 8 q^{17} + 15 q^{19} + 3 q^{22} - 14 q^{23} - 6 q^{26} - 24 q^{28} + 7 q^{29} + 5 q^{31} - 18 q^{32} + 7 q^{34} - 6 q^{37} + 18 q^{38} - 22 q^{41} - 19 q^{43} - 15 q^{44} - 4 q^{46} - 22 q^{47} + 12 q^{49} + 11 q^{52} - 10 q^{53} - 14 q^{56} - 2 q^{58} - 6 q^{59} + 23 q^{61} - 40 q^{62} + 5 q^{64} - 13 q^{67} + 7 q^{68} - 26 q^{71} - 24 q^{73} + 10 q^{74} + 46 q^{76} - 4 q^{77} + 14 q^{79} + 16 q^{82} - 10 q^{83} - 44 q^{86} - 66 q^{88} - 14 q^{89} + 13 q^{91} - 58 q^{92} - 3 q^{94} - 31 q^{97} + 59 q^{98}+O(q^{100})$$ 7 * q - 2 * q^2 + 10 * q^4 - q^7 - 3 * q^8 - 4 * q^11 - q^13 - 15 * q^14 + 12 * q^16 - 8 * q^17 + 15 * q^19 + 3 * q^22 - 14 * q^23 - 6 * q^26 - 24 * q^28 + 7 * q^29 + 5 * q^31 - 18 * q^32 + 7 * q^34 - 6 * q^37 + 18 * q^38 - 22 * q^41 - 19 * q^43 - 15 * q^44 - 4 * q^46 - 22 * q^47 + 12 * q^49 + 11 * q^52 - 10 * q^53 - 14 * q^56 - 2 * q^58 - 6 * q^59 + 23 * q^61 - 40 * q^62 + 5 * q^64 - 13 * q^67 + 7 * q^68 - 26 * q^71 - 24 * q^73 + 10 * q^74 + 46 * q^76 - 4 * q^77 + 14 * q^79 + 16 * q^82 - 10 * q^83 - 44 * q^86 - 66 * q^88 - 14 * q^89 + 13 * q^91 - 58 * q^92 - 3 * q^94 - 31 * q^97 + 59 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.66356 −1.88342 −0.941709 0.336429i $$-0.890781\pi$$
−0.941709 + 0.336429i $$0.890781\pi$$
$$3$$ 0 0
$$4$$ 5.09453 2.54726
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.0170416 −0.00644113 −0.00322057 0.999995i $$-0.501025\pi$$
−0.00322057 + 0.999995i $$0.501025\pi$$
$$8$$ −8.24244 −2.91414
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.70990 −0.515553 −0.257777 0.966205i $$-0.582990\pi$$
−0.257777 + 0.966205i $$0.582990\pi$$
$$12$$ 0 0
$$13$$ −4.69566 −1.30234 −0.651171 0.758931i $$-0.725722\pi$$
−0.651171 + 0.758931i $$0.725722\pi$$
$$14$$ 0.0453913 0.0121313
$$15$$ 0 0
$$16$$ 11.7651 2.94129
$$17$$ 3.91494 0.949513 0.474757 0.880117i $$-0.342536\pi$$
0.474757 + 0.880117i $$0.342536\pi$$
$$18$$ 0 0
$$19$$ 6.02411 1.38202 0.691012 0.722843i $$-0.257165\pi$$
0.691012 + 0.722843i $$0.257165\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.55440 0.971002
$$23$$ −4.82786 −1.00668 −0.503339 0.864089i $$-0.667895\pi$$
−0.503339 + 0.864089i $$0.667895\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 12.5071 2.45285
$$27$$ 0 0
$$28$$ −0.0868190 −0.0164073
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 1.33709 0.240148 0.120074 0.992765i $$-0.461687\pi$$
0.120074 + 0.992765i $$0.461687\pi$$
$$32$$ −14.8522 −2.62553
$$33$$ 0 0
$$34$$ −10.4277 −1.78833
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.18905 −1.34627 −0.673136 0.739519i $$-0.735053\pi$$
−0.673136 + 0.739519i $$0.735053\pi$$
$$38$$ −16.0455 −2.60293
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.36721 −1.30674 −0.653369 0.757039i $$-0.726645\pi$$
−0.653369 + 0.757039i $$0.726645\pi$$
$$42$$ 0 0
$$43$$ 3.72560 0.568149 0.284074 0.958802i $$-0.408314\pi$$
0.284074 + 0.958802i $$0.408314\pi$$
$$44$$ −8.71111 −1.31325
$$45$$ 0 0
$$46$$ 12.8593 1.89600
$$47$$ 5.36826 0.783041 0.391520 0.920169i $$-0.371949\pi$$
0.391520 + 0.920169i $$0.371949\pi$$
$$48$$ 0 0
$$49$$ −6.99971 −0.999959
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −23.9222 −3.31741
$$53$$ 7.21637 0.991245 0.495622 0.868538i $$-0.334940\pi$$
0.495622 + 0.868538i $$0.334940\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0.140465 0.0187704
$$57$$ 0 0
$$58$$ −2.66356 −0.349742
$$59$$ 8.65422 1.12668 0.563342 0.826224i $$-0.309515\pi$$
0.563342 + 0.826224i $$0.309515\pi$$
$$60$$ 0 0
$$61$$ 5.72637 0.733187 0.366594 0.930381i $$-0.380524\pi$$
0.366594 + 0.930381i $$0.380524\pi$$
$$62$$ −3.56141 −0.452299
$$63$$ 0 0
$$64$$ 16.0294 2.00368
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.87486 0.473390 0.236695 0.971584i $$-0.423936\pi$$
0.236695 + 0.971584i $$0.423936\pi$$
$$68$$ 19.9448 2.41866
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.32991 0.513866 0.256933 0.966429i $$-0.417288\pi$$
0.256933 + 0.966429i $$0.417288\pi$$
$$72$$ 0 0
$$73$$ −1.78245 −0.208620 −0.104310 0.994545i $$-0.533263\pi$$
−0.104310 + 0.994545i $$0.533263\pi$$
$$74$$ 21.8120 2.53559
$$75$$ 0 0
$$76$$ 30.6900 3.52038
$$77$$ 0.0291394 0.00332075
$$78$$ 0 0
$$79$$ 0.233544 0.0262757 0.0131379 0.999914i $$-0.495818\pi$$
0.0131379 + 0.999914i $$0.495818\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 22.2865 2.46114
$$83$$ 6.15497 0.675596 0.337798 0.941219i $$-0.390318\pi$$
0.337798 + 0.941219i $$0.390318\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −9.92334 −1.07006
$$87$$ 0 0
$$88$$ 14.0937 1.50240
$$89$$ 12.4053 1.31496 0.657478 0.753474i $$-0.271623\pi$$
0.657478 + 0.753474i $$0.271623\pi$$
$$90$$ 0 0
$$91$$ 0.0800217 0.00838855
$$92$$ −24.5957 −2.56427
$$93$$ 0 0
$$94$$ −14.2986 −1.47479
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −19.2918 −1.95879 −0.979395 0.201954i $$-0.935271\pi$$
−0.979395 + 0.201954i $$0.935271\pi$$
$$98$$ 18.6441 1.88334
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 8.74987 0.870644 0.435322 0.900275i $$-0.356634\pi$$
0.435322 + 0.900275i $$0.356634\pi$$
$$102$$ 0 0
$$103$$ 10.9615 1.08007 0.540036 0.841642i $$-0.318411\pi$$
0.540036 + 0.841642i $$0.318411\pi$$
$$104$$ 38.7037 3.79521
$$105$$ 0 0
$$106$$ −19.2212 −1.86693
$$107$$ −15.8139 −1.52879 −0.764393 0.644750i $$-0.776961\pi$$
−0.764393 + 0.644750i $$0.776961\pi$$
$$108$$ 0 0
$$109$$ −20.2768 −1.94216 −0.971082 0.238747i $$-0.923263\pi$$
−0.971082 + 0.238747i $$0.923263\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −0.200497 −0.0189452
$$113$$ −8.86901 −0.834326 −0.417163 0.908832i $$-0.636976\pi$$
−0.417163 + 0.908832i $$0.636976\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 5.09453 0.473015
$$117$$ 0 0
$$118$$ −23.0510 −2.12202
$$119$$ −0.0667170 −0.00611594
$$120$$ 0 0
$$121$$ −8.07625 −0.734205
$$122$$ −15.2525 −1.38090
$$123$$ 0 0
$$124$$ 6.81183 0.611721
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.88854 −0.433788 −0.216894 0.976195i $$-0.569593\pi$$
−0.216894 + 0.976195i $$0.569593\pi$$
$$128$$ −12.9909 −1.14824
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −14.0971 −1.23167 −0.615833 0.787876i $$-0.711181\pi$$
−0.615833 + 0.787876i $$0.711181\pi$$
$$132$$ 0 0
$$133$$ −0.102661 −0.00890180
$$134$$ −10.3209 −0.891591
$$135$$ 0 0
$$136$$ −32.2687 −2.76702
$$137$$ −3.57023 −0.305025 −0.152513 0.988302i $$-0.548736\pi$$
−0.152513 + 0.988302i $$0.548736\pi$$
$$138$$ 0 0
$$139$$ 11.7235 0.994371 0.497185 0.867644i $$-0.334367\pi$$
0.497185 + 0.867644i $$0.334367\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −11.5330 −0.967825
$$143$$ 8.02909 0.671426
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.74766 0.392919
$$147$$ 0 0
$$148$$ −41.7193 −3.42931
$$149$$ 0.672890 0.0551253 0.0275626 0.999620i $$-0.491225\pi$$
0.0275626 + 0.999620i $$0.491225\pi$$
$$150$$ 0 0
$$151$$ 24.2619 1.97441 0.987203 0.159468i $$-0.0509779\pi$$
0.987203 + 0.159468i $$0.0509779\pi$$
$$152$$ −49.6533 −4.02742
$$153$$ 0 0
$$154$$ −0.0776145 −0.00625435
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.0158 −1.11859 −0.559293 0.828970i $$-0.688927\pi$$
−0.559293 + 0.828970i $$0.688927\pi$$
$$158$$ −0.622057 −0.0494882
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0.0822746 0.00648415
$$162$$ 0 0
$$163$$ 9.77061 0.765293 0.382646 0.923895i $$-0.375013\pi$$
0.382646 + 0.923895i $$0.375013\pi$$
$$164$$ −42.6270 −3.32861
$$165$$ 0 0
$$166$$ −16.3941 −1.27243
$$167$$ −5.91586 −0.457783 −0.228891 0.973452i $$-0.573510\pi$$
−0.228891 + 0.973452i $$0.573510\pi$$
$$168$$ 0 0
$$169$$ 9.04921 0.696093
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 18.9802 1.44722
$$173$$ 20.1704 1.53353 0.766763 0.641930i $$-0.221866\pi$$
0.766763 + 0.641930i $$0.221866\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −20.1172 −1.51639
$$177$$ 0 0
$$178$$ −33.0421 −2.47661
$$179$$ −15.4896 −1.15775 −0.578873 0.815418i $$-0.696507\pi$$
−0.578873 + 0.815418i $$0.696507\pi$$
$$180$$ 0 0
$$181$$ 17.2509 1.28225 0.641126 0.767436i $$-0.278468\pi$$
0.641126 + 0.767436i $$0.278468\pi$$
$$182$$ −0.213142 −0.0157991
$$183$$ 0 0
$$184$$ 39.7933 2.93360
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.69415 −0.489525
$$188$$ 27.3487 1.99461
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −11.5130 −0.833048 −0.416524 0.909125i $$-0.636752\pi$$
−0.416524 + 0.909125i $$0.636752\pi$$
$$192$$ 0 0
$$193$$ 13.0261 0.937642 0.468821 0.883293i $$-0.344679\pi$$
0.468821 + 0.883293i $$0.344679\pi$$
$$194$$ 51.3849 3.68922
$$195$$ 0 0
$$196$$ −35.6602 −2.54716
$$197$$ −14.3280 −1.02083 −0.510415 0.859928i $$-0.670508\pi$$
−0.510415 + 0.859928i $$0.670508\pi$$
$$198$$ 0 0
$$199$$ −16.2728 −1.15355 −0.576775 0.816903i $$-0.695689\pi$$
−0.576775 + 0.816903i $$0.695689\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −23.3058 −1.63979
$$203$$ −0.0170416 −0.00119609
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −29.1966 −2.03423
$$207$$ 0 0
$$208$$ −55.2451 −3.83056
$$209$$ −10.3006 −0.712507
$$210$$ 0 0
$$211$$ −24.3854 −1.67876 −0.839380 0.543544i $$-0.817082\pi$$
−0.839380 + 0.543544i $$0.817082\pi$$
$$212$$ 36.7640 2.52496
$$213$$ 0 0
$$214$$ 42.1212 2.87934
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.0227862 −0.00154683
$$218$$ 54.0083 3.65791
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −18.3832 −1.23659
$$222$$ 0 0
$$223$$ 0.753555 0.0504617 0.0252309 0.999682i $$-0.491968\pi$$
0.0252309 + 0.999682i $$0.491968\pi$$
$$224$$ 0.253106 0.0169114
$$225$$ 0 0
$$226$$ 23.6231 1.57138
$$227$$ 21.1711 1.40518 0.702589 0.711596i $$-0.252027\pi$$
0.702589 + 0.711596i $$0.252027\pi$$
$$228$$ 0 0
$$229$$ −7.97533 −0.527025 −0.263512 0.964656i $$-0.584881\pi$$
−0.263512 + 0.964656i $$0.584881\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −8.24244 −0.541143
$$233$$ −14.0847 −0.922718 −0.461359 0.887214i $$-0.652638\pi$$
−0.461359 + 0.887214i $$0.652638\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 44.0892 2.86996
$$237$$ 0 0
$$238$$ 0.177705 0.0115189
$$239$$ −15.4813 −1.00140 −0.500701 0.865620i $$-0.666924\pi$$
−0.500701 + 0.865620i $$0.666924\pi$$
$$240$$ 0 0
$$241$$ 12.1294 0.781324 0.390662 0.920534i $$-0.372246\pi$$
0.390662 + 0.920534i $$0.372246\pi$$
$$242$$ 21.5115 1.38281
$$243$$ 0 0
$$244$$ 29.1732 1.86762
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −28.2871 −1.79987
$$248$$ −11.0209 −0.699826
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −14.1090 −0.890550 −0.445275 0.895394i $$-0.646894\pi$$
−0.445275 + 0.895394i $$0.646894\pi$$
$$252$$ 0 0
$$253$$ 8.25514 0.518996
$$254$$ 13.0209 0.817003
$$255$$ 0 0
$$256$$ 2.54296 0.158935
$$257$$ 26.6644 1.66328 0.831639 0.555317i $$-0.187403\pi$$
0.831639 + 0.555317i $$0.187403\pi$$
$$258$$ 0 0
$$259$$ 0.139555 0.00867151
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 37.5483 2.31974
$$263$$ −5.74517 −0.354263 −0.177131 0.984187i $$-0.556682\pi$$
−0.177131 + 0.984187i $$0.556682\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0.273442 0.0167658
$$267$$ 0 0
$$268$$ 19.7406 1.20585
$$269$$ −19.3477 −1.17965 −0.589826 0.807530i $$-0.700804\pi$$
−0.589826 + 0.807530i $$0.700804\pi$$
$$270$$ 0 0
$$271$$ −30.7485 −1.86784 −0.933920 0.357482i $$-0.883635\pi$$
−0.933920 + 0.357482i $$0.883635\pi$$
$$272$$ 46.0599 2.79279
$$273$$ 0 0
$$274$$ 9.50950 0.574490
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −24.0737 −1.44645 −0.723225 0.690613i $$-0.757341\pi$$
−0.723225 + 0.690613i $$0.757341\pi$$
$$278$$ −31.2261 −1.87282
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −23.4067 −1.39633 −0.698164 0.715938i $$-0.745999\pi$$
−0.698164 + 0.715938i $$0.745999\pi$$
$$282$$ 0 0
$$283$$ −20.8589 −1.23993 −0.619967 0.784628i $$-0.712854\pi$$
−0.619967 + 0.784628i $$0.712854\pi$$
$$284$$ 22.0589 1.30895
$$285$$ 0 0
$$286$$ −21.3859 −1.26458
$$287$$ 0.142591 0.00841688
$$288$$ 0 0
$$289$$ −1.67321 −0.0984242
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −9.08075 −0.531410
$$293$$ 12.0727 0.705294 0.352647 0.935756i $$-0.385282\pi$$
0.352647 + 0.935756i $$0.385282\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 67.4978 3.92323
$$297$$ 0 0
$$298$$ −1.79228 −0.103824
$$299$$ 22.6700 1.31104
$$300$$ 0 0
$$301$$ −0.0634903 −0.00365952
$$302$$ −64.6229 −3.71863
$$303$$ 0 0
$$304$$ 70.8745 4.06493
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.4035 1.16449 0.582245 0.813013i $$-0.302174\pi$$
0.582245 + 0.813013i $$0.302174\pi$$
$$308$$ 0.148452 0.00845881
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.4159 0.704040 0.352020 0.935993i $$-0.385495\pi$$
0.352020 + 0.935993i $$0.385495\pi$$
$$312$$ 0 0
$$313$$ 14.8518 0.839473 0.419737 0.907646i $$-0.362122\pi$$
0.419737 + 0.907646i $$0.362122\pi$$
$$314$$ 37.3320 2.10677
$$315$$ 0 0
$$316$$ 1.18980 0.0669312
$$317$$ 21.8783 1.22881 0.614404 0.788992i $$-0.289397\pi$$
0.614404 + 0.788992i $$0.289397\pi$$
$$318$$ 0 0
$$319$$ −1.70990 −0.0957358
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −0.219143 −0.0122124
$$323$$ 23.5840 1.31225
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −26.0245 −1.44137
$$327$$ 0 0
$$328$$ 68.9662 3.80802
$$329$$ −0.0914839 −0.00504367
$$330$$ 0 0
$$331$$ 28.6380 1.57409 0.787043 0.616898i $$-0.211611\pi$$
0.787043 + 0.616898i $$0.211611\pi$$
$$332$$ 31.3567 1.72092
$$333$$ 0 0
$$334$$ 15.7572 0.862196
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.75074 0.367736 0.183868 0.982951i $$-0.441138\pi$$
0.183868 + 0.982951i $$0.441138\pi$$
$$338$$ −24.1031 −1.31103
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.28628 −0.123809
$$342$$ 0 0
$$343$$ 0.238578 0.0128820
$$344$$ −30.7081 −1.65567
$$345$$ 0 0
$$346$$ −53.7249 −2.88827
$$347$$ 1.87591 0.100704 0.0503520 0.998732i $$-0.483966\pi$$
0.0503520 + 0.998732i $$0.483966\pi$$
$$348$$ 0 0
$$349$$ 24.6359 1.31873 0.659364 0.751824i $$-0.270826\pi$$
0.659364 + 0.751824i $$0.270826\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 25.3958 1.35360
$$353$$ 13.3468 0.710379 0.355189 0.934794i $$-0.384416\pi$$
0.355189 + 0.934794i $$0.384416\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 63.1989 3.34954
$$357$$ 0 0
$$358$$ 41.2574 2.18052
$$359$$ −21.5263 −1.13611 −0.568057 0.822990i $$-0.692305\pi$$
−0.568057 + 0.822990i $$0.692305\pi$$
$$360$$ 0 0
$$361$$ 17.2898 0.909992
$$362$$ −45.9488 −2.41502
$$363$$ 0 0
$$364$$ 0.407673 0.0213678
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9.36534 0.488867 0.244433 0.969666i $$-0.421398\pi$$
0.244433 + 0.969666i $$0.421398\pi$$
$$368$$ −56.8005 −2.96093
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −0.122979 −0.00638474
$$372$$ 0 0
$$373$$ 19.3366 1.00121 0.500605 0.865676i $$-0.333111\pi$$
0.500605 + 0.865676i $$0.333111\pi$$
$$374$$ 17.8302 0.921980
$$375$$ 0 0
$$376$$ −44.2475 −2.28189
$$377$$ −4.69566 −0.241839
$$378$$ 0 0
$$379$$ −12.4163 −0.637783 −0.318891 0.947791i $$-0.603311\pi$$
−0.318891 + 0.947791i $$0.603311\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 30.6654 1.56898
$$383$$ −24.5909 −1.25653 −0.628267 0.777998i $$-0.716236\pi$$
−0.628267 + 0.777998i $$0.716236\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −34.6958 −1.76597
$$387$$ 0 0
$$388$$ −98.2828 −4.98955
$$389$$ −17.1567 −0.869880 −0.434940 0.900460i $$-0.643230\pi$$
−0.434940 + 0.900460i $$0.643230\pi$$
$$390$$ 0 0
$$391$$ −18.9008 −0.955854
$$392$$ 57.6947 2.91402
$$393$$ 0 0
$$394$$ 38.1635 1.92265
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.90545 0.145820 0.0729102 0.997339i $$-0.476771\pi$$
0.0729102 + 0.997339i $$0.476771\pi$$
$$398$$ 43.3436 2.17262
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −20.4427 −1.02086 −0.510431 0.859919i $$-0.670514\pi$$
−0.510431 + 0.859919i $$0.670514\pi$$
$$402$$ 0 0
$$403$$ −6.27851 −0.312755
$$404$$ 44.5764 2.21776
$$405$$ 0 0
$$406$$ 0.0453913 0.00225273
$$407$$ 14.0024 0.694075
$$408$$ 0 0
$$409$$ −3.13657 −0.155093 −0.0775467 0.996989i $$-0.524709\pi$$
−0.0775467 + 0.996989i $$0.524709\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 55.8438 2.75123
$$413$$ −0.147482 −0.00725712
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 69.7410 3.41933
$$417$$ 0 0
$$418$$ 27.4362 1.34195
$$419$$ −36.6119 −1.78861 −0.894305 0.447458i $$-0.852330\pi$$
−0.894305 + 0.447458i $$0.852330\pi$$
$$420$$ 0 0
$$421$$ −13.8706 −0.676012 −0.338006 0.941144i $$-0.609752\pi$$
−0.338006 + 0.941144i $$0.609752\pi$$
$$422$$ 64.9519 3.16181
$$423$$ 0 0
$$424$$ −59.4805 −2.88863
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −0.0975868 −0.00472256
$$428$$ −80.5643 −3.89422
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 29.6786 1.42957 0.714785 0.699344i $$-0.246525\pi$$
0.714785 + 0.699344i $$0.246525\pi$$
$$432$$ 0 0
$$433$$ −32.1319 −1.54416 −0.772081 0.635525i $$-0.780784\pi$$
−0.772081 + 0.635525i $$0.780784\pi$$
$$434$$ 0.0606922 0.00291332
$$435$$ 0 0
$$436$$ −103.301 −4.94720
$$437$$ −29.0835 −1.39125
$$438$$ 0 0
$$439$$ 34.9322 1.66722 0.833611 0.552352i $$-0.186269\pi$$
0.833611 + 0.552352i $$0.186269\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 48.9648 2.32902
$$443$$ −30.7938 −1.46306 −0.731528 0.681812i $$-0.761192\pi$$
−0.731528 + 0.681812i $$0.761192\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −2.00713 −0.0950406
$$447$$ 0 0
$$448$$ −0.273168 −0.0129060
$$449$$ −26.6304 −1.25677 −0.628383 0.777904i $$-0.716283\pi$$
−0.628383 + 0.777904i $$0.716283\pi$$
$$450$$ 0 0
$$451$$ 14.3071 0.673693
$$452$$ −45.1834 −2.12525
$$453$$ 0 0
$$454$$ −56.3905 −2.64654
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9.03169 0.422485 0.211242 0.977434i $$-0.432249\pi$$
0.211242 + 0.977434i $$0.432249\pi$$
$$458$$ 21.2427 0.992608
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 17.6240 0.820832 0.410416 0.911898i $$-0.365384\pi$$
0.410416 + 0.911898i $$0.365384\pi$$
$$462$$ 0 0
$$463$$ −11.1431 −0.517865 −0.258933 0.965895i $$-0.583371\pi$$
−0.258933 + 0.965895i $$0.583371\pi$$
$$464$$ 11.7651 0.546183
$$465$$ 0 0
$$466$$ 37.5153 1.73786
$$467$$ 19.8932 0.920548 0.460274 0.887777i $$-0.347751\pi$$
0.460274 + 0.887777i $$0.347751\pi$$
$$468$$ 0 0
$$469$$ −0.0660340 −0.00304917
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −71.3319 −3.28332
$$473$$ −6.37039 −0.292911
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −0.339892 −0.0155789
$$477$$ 0 0
$$478$$ 41.2353 1.88606
$$479$$ 21.6917 0.991118 0.495559 0.868574i $$-0.334963\pi$$
0.495559 + 0.868574i $$0.334963\pi$$
$$480$$ 0 0
$$481$$ 38.4530 1.75331
$$482$$ −32.3073 −1.47156
$$483$$ 0 0
$$484$$ −41.1447 −1.87021
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 11.1364 0.504639 0.252320 0.967644i $$-0.418807\pi$$
0.252320 + 0.967644i $$0.418807\pi$$
$$488$$ −47.1993 −2.13661
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −27.7172 −1.25086 −0.625431 0.780280i $$-0.715077\pi$$
−0.625431 + 0.780280i $$0.715077\pi$$
$$492$$ 0 0
$$493$$ 3.91494 0.176320
$$494$$ 75.3444 3.38990
$$495$$ 0 0
$$496$$ 15.7310 0.706344
$$497$$ −0.0737888 −0.00330988
$$498$$ 0 0
$$499$$ −29.8657 −1.33697 −0.668485 0.743725i $$-0.733057\pi$$
−0.668485 + 0.743725i $$0.733057\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 37.5800 1.67728
$$503$$ −28.6721 −1.27843 −0.639213 0.769030i $$-0.720740\pi$$
−0.639213 + 0.769030i $$0.720740\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −21.9880 −0.977487
$$507$$ 0 0
$$508$$ −24.9048 −1.10497
$$509$$ 22.7354 1.00773 0.503864 0.863783i $$-0.331911\pi$$
0.503864 + 0.863783i $$0.331911\pi$$
$$510$$ 0 0
$$511$$ 0.0303759 0.00134375
$$512$$ 19.2084 0.848899
$$513$$ 0 0
$$514$$ −71.0220 −3.13265
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −9.17916 −0.403699
$$518$$ −0.371712 −0.0163321
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −15.8392 −0.693926 −0.346963 0.937879i $$-0.612787\pi$$
−0.346963 + 0.937879i $$0.612787\pi$$
$$522$$ 0 0
$$523$$ 28.1248 1.22981 0.614905 0.788601i $$-0.289194\pi$$
0.614905 + 0.788601i $$0.289194\pi$$
$$524$$ −71.8179 −3.13738
$$525$$ 0 0
$$526$$ 15.3026 0.667224
$$527$$ 5.23463 0.228024
$$528$$ 0 0
$$529$$ 0.308221 0.0134009
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −0.523007 −0.0226752
$$533$$ 39.2896 1.70182
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −31.9383 −1.37953
$$537$$ 0 0
$$538$$ 51.5338 2.22178
$$539$$ 11.9688 0.515532
$$540$$ 0 0
$$541$$ −23.9993 −1.03181 −0.515906 0.856645i $$-0.672545\pi$$
−0.515906 + 0.856645i $$0.672545\pi$$
$$542$$ 81.9004 3.51792
$$543$$ 0 0
$$544$$ −58.1457 −2.49297
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −24.1285 −1.03166 −0.515831 0.856691i $$-0.672517\pi$$
−0.515831 + 0.856691i $$0.672517\pi$$
$$548$$ −18.1886 −0.776980
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.02411 0.256636
$$552$$ 0 0
$$553$$ −0.00397997 −0.000169245 0
$$554$$ 64.1217 2.72427
$$555$$ 0 0
$$556$$ 59.7255 2.53292
$$557$$ −29.8474 −1.26467 −0.632337 0.774693i $$-0.717904\pi$$
−0.632337 + 0.774693i $$0.717904\pi$$
$$558$$ 0 0
$$559$$ −17.4942 −0.739924
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 62.3451 2.62987
$$563$$ −9.29980 −0.391940 −0.195970 0.980610i $$-0.562786\pi$$
−0.195970 + 0.980610i $$0.562786\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 55.5589 2.33531
$$567$$ 0 0
$$568$$ −35.6891 −1.49748
$$569$$ 10.7468 0.450529 0.225264 0.974298i $$-0.427675\pi$$
0.225264 + 0.974298i $$0.427675\pi$$
$$570$$ 0 0
$$571$$ −22.3620 −0.935822 −0.467911 0.883776i $$-0.654993\pi$$
−0.467911 + 0.883776i $$0.654993\pi$$
$$572$$ 40.9044 1.71030
$$573$$ 0 0
$$574$$ −0.379799 −0.0158525
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −29.7336 −1.23783 −0.618913 0.785459i $$-0.712427\pi$$
−0.618913 + 0.785459i $$0.712427\pi$$
$$578$$ 4.45669 0.185374
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −0.104891 −0.00435160
$$582$$ 0 0
$$583$$ −12.3392 −0.511039
$$584$$ 14.6918 0.607949
$$585$$ 0 0
$$586$$ −32.1563 −1.32836
$$587$$ −25.9770 −1.07219 −0.536093 0.844159i $$-0.680101\pi$$
−0.536093 + 0.844159i $$0.680101\pi$$
$$588$$ 0 0
$$589$$ 8.05476 0.331891
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −96.3454 −3.95977
$$593$$ 30.1217 1.23695 0.618476 0.785804i $$-0.287750\pi$$
0.618476 + 0.785804i $$0.287750\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 3.42805 0.140419
$$597$$ 0 0
$$598$$ −60.3827 −2.46923
$$599$$ 27.4666 1.12226 0.561128 0.827729i $$-0.310368\pi$$
0.561128 + 0.827729i $$0.310368\pi$$
$$600$$ 0 0
$$601$$ −6.66716 −0.271959 −0.135980 0.990712i $$-0.543418\pi$$
−0.135980 + 0.990712i $$0.543418\pi$$
$$602$$ 0.169110 0.00689241
$$603$$ 0 0
$$604$$ 123.603 5.02933
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −15.4510 −0.627138 −0.313569 0.949565i $$-0.601525\pi$$
−0.313569 + 0.949565i $$0.601525\pi$$
$$608$$ −89.4714 −3.62854
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −25.2075 −1.01979
$$612$$ 0 0
$$613$$ 23.4474 0.947031 0.473516 0.880785i $$-0.342985\pi$$
0.473516 + 0.880785i $$0.342985\pi$$
$$614$$ −54.3459 −2.19322
$$615$$ 0 0
$$616$$ −0.240180 −0.00967713
$$617$$ −36.8162 −1.48216 −0.741082 0.671415i $$-0.765687\pi$$
−0.741082 + 0.671415i $$0.765687\pi$$
$$618$$ 0 0
$$619$$ 24.5202 0.985551 0.492776 0.870156i $$-0.335982\pi$$
0.492776 + 0.870156i $$0.335982\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −33.0704 −1.32600
$$623$$ −0.211406 −0.00846980
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −39.5586 −1.58108
$$627$$ 0 0
$$628$$ −71.4041 −2.84933
$$629$$ −32.0597 −1.27830
$$630$$ 0 0
$$631$$ 11.1275 0.442978 0.221489 0.975163i $$-0.428908\pi$$
0.221489 + 0.975163i $$0.428908\pi$$
$$632$$ −1.92497 −0.0765712
$$633$$ 0 0
$$634$$ −58.2740 −2.31436
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 32.8682 1.30229
$$638$$ 4.55440 0.180311
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −26.2776 −1.03790 −0.518951 0.854804i $$-0.673677\pi$$
−0.518951 + 0.854804i $$0.673677\pi$$
$$642$$ 0 0
$$643$$ −6.11885 −0.241304 −0.120652 0.992695i $$-0.538498\pi$$
−0.120652 + 0.992695i $$0.538498\pi$$
$$644$$ 0.419150 0.0165168
$$645$$ 0 0
$$646$$ −62.8174 −2.47152
$$647$$ −26.3758 −1.03694 −0.518470 0.855096i $$-0.673498\pi$$
−0.518470 + 0.855096i $$0.673498\pi$$
$$648$$ 0 0
$$649$$ −14.7978 −0.580865
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 49.7766 1.94940
$$653$$ 47.9227 1.87536 0.937679 0.347502i $$-0.112970\pi$$
0.937679 + 0.347502i $$0.112970\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −98.4415 −3.84349
$$657$$ 0 0
$$658$$ 0.243672 0.00949934
$$659$$ −33.0636 −1.28798 −0.643988 0.765036i $$-0.722721\pi$$
−0.643988 + 0.765036i $$0.722721\pi$$
$$660$$ 0 0
$$661$$ 20.6186 0.801970 0.400985 0.916085i $$-0.368668\pi$$
0.400985 + 0.916085i $$0.368668\pi$$
$$662$$ −76.2789 −2.96466
$$663$$ 0 0
$$664$$ −50.7320 −1.96878
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.82786 −0.186935
$$668$$ −30.1385 −1.16609
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −9.79151 −0.377997
$$672$$ 0 0
$$673$$ 28.1489 1.08506 0.542529 0.840037i $$-0.317467\pi$$
0.542529 + 0.840037i $$0.317467\pi$$
$$674$$ −17.9810 −0.692601
$$675$$ 0 0
$$676$$ 46.1014 1.77313
$$677$$ −4.01689 −0.154382 −0.0771908 0.997016i $$-0.524595\pi$$
−0.0771908 + 0.997016i $$0.524595\pi$$
$$678$$ 0 0
$$679$$ 0.328764 0.0126168
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 6.08964 0.233184
$$683$$ 2.61163 0.0999312 0.0499656 0.998751i $$-0.484089\pi$$
0.0499656 + 0.998751i $$0.484089\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −0.635465 −0.0242622
$$687$$ 0 0
$$688$$ 43.8322 1.67109
$$689$$ −33.8856 −1.29094
$$690$$ 0 0
$$691$$ −19.9707 −0.759723 −0.379861 0.925043i $$-0.624028\pi$$
−0.379861 + 0.925043i $$0.624028\pi$$
$$692$$ 102.759 3.90629
$$693$$ 0 0
$$694$$ −4.99658 −0.189668
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −32.7572 −1.24077
$$698$$ −65.6190 −2.48372
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −17.4402 −0.658709 −0.329355 0.944206i $$-0.606831\pi$$
−0.329355 + 0.944206i $$0.606831\pi$$
$$702$$ 0 0
$$703$$ −49.3317 −1.86058
$$704$$ −27.4087 −1.03300
$$705$$ 0 0
$$706$$ −35.5500 −1.33794
$$707$$ −0.149112 −0.00560793
$$708$$ 0 0
$$709$$ −27.8452 −1.04575 −0.522874 0.852410i $$-0.675140\pi$$
−0.522874 + 0.852410i $$0.675140\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −102.250 −3.83197
$$713$$ −6.45527 −0.241752
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −78.9121 −2.94908
$$717$$ 0 0
$$718$$ 57.3364 2.13978
$$719$$ 23.3574 0.871085 0.435542 0.900168i $$-0.356557\pi$$
0.435542 + 0.900168i $$0.356557\pi$$
$$720$$ 0 0
$$721$$ −0.186802 −0.00695688
$$722$$ −46.0525 −1.71389
$$723$$ 0 0
$$724$$ 87.8854 3.26623
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −22.8360 −0.846939 −0.423470 0.905910i $$-0.639188\pi$$
−0.423470 + 0.905910i $$0.639188\pi$$
$$728$$ −0.659574 −0.0244454
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 14.5855 0.539465
$$732$$ 0 0
$$733$$ −1.44837 −0.0534967 −0.0267484 0.999642i $$-0.508515\pi$$
−0.0267484 + 0.999642i $$0.508515\pi$$
$$734$$ −24.9451 −0.920741
$$735$$ 0 0
$$736$$ 71.7045 2.64306
$$737$$ −6.62561 −0.244058
$$738$$ 0 0
$$739$$ −21.4167 −0.787824 −0.393912 0.919148i $$-0.628879\pi$$
−0.393912 + 0.919148i $$0.628879\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.327561 0.0120251
$$743$$ 16.7848 0.615776 0.307888 0.951423i $$-0.400378\pi$$
0.307888 + 0.951423i $$0.400378\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −51.5040 −1.88570
$$747$$ 0 0
$$748$$ −34.1035 −1.24695
$$749$$ 0.269495 0.00984712
$$750$$ 0 0
$$751$$ −13.1194 −0.478733 −0.239366 0.970929i $$-0.576940\pi$$
−0.239366 + 0.970929i $$0.576940\pi$$
$$752$$ 63.1583 2.30315
$$753$$ 0 0
$$754$$ 12.5071 0.455483
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 15.3278 0.557099 0.278549 0.960422i $$-0.410146\pi$$
0.278549 + 0.960422i $$0.410146\pi$$
$$758$$ 33.0715 1.20121
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −16.7398 −0.606818 −0.303409 0.952860i $$-0.598125\pi$$
−0.303409 + 0.952860i $$0.598125\pi$$
$$762$$ 0 0
$$763$$ 0.345549 0.0125097
$$764$$ −58.6531 −2.12199
$$765$$ 0 0
$$766$$ 65.4991 2.36658
$$767$$ −40.6373 −1.46733
$$768$$ 0 0
$$769$$ −19.4502 −0.701391 −0.350696 0.936489i $$-0.614055\pi$$
−0.350696 + 0.936489i $$0.614055\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 66.3620 2.38842
$$773$$ −34.5781 −1.24369 −0.621844 0.783142i $$-0.713616\pi$$
−0.621844 + 0.783142i $$0.713616\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 159.012 5.70819
$$777$$ 0 0
$$778$$ 45.6978 1.63835
$$779$$ −50.4050 −1.80595
$$780$$ 0 0
$$781$$ −7.40370 −0.264925
$$782$$ 50.3433 1.80027
$$783$$ 0 0
$$784$$ −82.3526 −2.94116
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −39.5669 −1.41041 −0.705204 0.709004i $$-0.749145\pi$$
−0.705204 + 0.709004i $$0.749145\pi$$
$$788$$ −72.9946 −2.60033
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0.151142 0.00537400
$$792$$ 0 0
$$793$$ −26.8891 −0.954860
$$794$$ −7.73883 −0.274641
$$795$$ 0 0
$$796$$ −82.9023 −2.93839
$$797$$ −32.0055 −1.13369 −0.566847 0.823823i $$-0.691837\pi$$
−0.566847 + 0.823823i $$0.691837\pi$$
$$798$$ 0 0
$$799$$ 21.0164 0.743508
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 54.4504 1.92271
$$803$$ 3.04781 0.107555
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 16.7232 0.589048
$$807$$ 0 0
$$808$$ −72.1203 −2.53718
$$809$$ −27.5627 −0.969052 −0.484526 0.874777i $$-0.661008\pi$$
−0.484526 + 0.874777i $$0.661008\pi$$
$$810$$ 0 0
$$811$$ −25.3316 −0.889514 −0.444757 0.895651i $$-0.646710\pi$$
−0.444757 + 0.895651i $$0.646710\pi$$
$$812$$ −0.0868190 −0.00304675
$$813$$ 0 0
$$814$$ −37.2963 −1.30723
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 22.4434 0.785196
$$818$$ 8.35442 0.292106
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 46.4854 1.62235 0.811177 0.584801i $$-0.198827\pi$$
0.811177 + 0.584801i $$0.198827\pi$$
$$822$$ 0 0
$$823$$ −16.9991 −0.592552 −0.296276 0.955102i $$-0.595745\pi$$
−0.296276 + 0.955102i $$0.595745\pi$$
$$824$$ −90.3497 −3.14748
$$825$$ 0 0
$$826$$ 0.392827 0.0136682
$$827$$ −14.1835 −0.493208 −0.246604 0.969116i $$-0.579315\pi$$
−0.246604 + 0.969116i $$0.579315\pi$$
$$828$$ 0 0
$$829$$ 28.9652 1.00600 0.503001 0.864286i $$-0.332229\pi$$
0.503001 + 0.864286i $$0.332229\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −75.2688 −2.60948
$$833$$ −27.4035 −0.949474
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −52.4767 −1.81494
$$837$$ 0 0
$$838$$ 97.5179 3.36870
$$839$$ −22.6082 −0.780523 −0.390262 0.920704i $$-0.627615\pi$$
−0.390262 + 0.920704i $$0.627615\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 36.9451 1.27321
$$843$$ 0 0
$$844$$ −124.232 −4.27625
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.137633 0.00472911
$$848$$ 84.9017 2.91554
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 39.5356 1.35526
$$852$$ 0 0
$$853$$ 35.6577 1.22090 0.610448 0.792057i $$-0.290990\pi$$
0.610448 + 0.792057i $$0.290990\pi$$
$$854$$ 0.259928 0.00889455
$$855$$ 0 0
$$856$$ 130.345 4.45510
$$857$$ −3.41032 −0.116494 −0.0582472 0.998302i $$-0.518551\pi$$
−0.0582472 + 0.998302i $$0.518551\pi$$
$$858$$ 0 0
$$859$$ −28.5267 −0.973318 −0.486659 0.873592i $$-0.661785\pi$$
−0.486659 + 0.873592i $$0.661785\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −79.0507 −2.69248
$$863$$ −24.5873 −0.836962 −0.418481 0.908226i $$-0.637437\pi$$
−0.418481 + 0.908226i $$0.637437\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 85.5851 2.90830
$$867$$ 0 0
$$868$$ −0.116085 −0.00394017
$$869$$ −0.399336 −0.0135465
$$870$$ 0 0
$$871$$ −18.1950 −0.616515
$$872$$ 167.130 5.65974
$$873$$ 0 0
$$874$$ 77.4656 2.62031
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 58.0640 1.96068 0.980341 0.197310i $$-0.0632204\pi$$
0.980341 + 0.197310i $$0.0632204\pi$$
$$878$$ −93.0438 −3.14008
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −52.7466 −1.77708 −0.888538 0.458802i $$-0.848279\pi$$
−0.888538 + 0.458802i $$0.848279\pi$$
$$882$$ 0 0
$$883$$ 38.1567 1.28408 0.642038 0.766672i $$-0.278089\pi$$
0.642038 + 0.766672i $$0.278089\pi$$
$$884$$ −93.6539 −3.14992
$$885$$ 0 0
$$886$$ 82.0209 2.75555
$$887$$ −22.8088 −0.765845 −0.382922 0.923781i $$-0.625082\pi$$
−0.382922 + 0.923781i $$0.625082\pi$$
$$888$$ 0 0
$$889$$ 0.0833087 0.00279408
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 3.83900 0.128539
$$893$$ 32.3389 1.08218
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0.221385 0.00739596
$$897$$ 0 0
$$898$$ 70.9316 2.36702
$$899$$ 1.33709 0.0445944
$$900$$ 0 0
$$901$$ 28.2517 0.941200
$$902$$ −38.1077 −1.26885
$$903$$ 0 0
$$904$$ 73.1023 2.43135
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −33.4371 −1.11026 −0.555131 0.831763i $$-0.687332\pi$$
−0.555131 + 0.831763i $$0.687332\pi$$
$$908$$ 107.857 3.57936
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.0752 0.598857 0.299428 0.954119i $$-0.403204\pi$$
0.299428 + 0.954119i $$0.403204\pi$$
$$912$$ 0 0
$$913$$ −10.5244 −0.348305
$$914$$ −24.0564 −0.795715
$$915$$ 0 0
$$916$$ −40.6305 −1.34247
$$917$$ 0.240237 0.00793333
$$918$$ 0 0
$$919$$ 38.7798 1.27923 0.639614 0.768696i $$-0.279094\pi$$
0.639614 + 0.768696i $$0.279094\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −46.9425 −1.54597
$$923$$ −20.3318 −0.669229
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 29.6804 0.975357
$$927$$ 0 0
$$928$$ −14.8522 −0.487548
$$929$$ −8.16319 −0.267826 −0.133913 0.990993i $$-0.542754\pi$$
−0.133913 + 0.990993i $$0.542754\pi$$
$$930$$ 0 0
$$931$$ −42.1670 −1.38197
$$932$$ −71.7547 −2.35040
$$933$$ 0 0
$$934$$ −52.9866 −1.73378
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −16.7498 −0.547192 −0.273596 0.961845i $$-0.588213\pi$$
−0.273596 + 0.961845i $$0.588213\pi$$
$$938$$ 0.175885 0.00574285
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1.27331 −0.0415088 −0.0207544 0.999785i $$-0.506607\pi$$
−0.0207544 + 0.999785i $$0.506607\pi$$
$$942$$ 0 0
$$943$$ 40.3957 1.31547
$$944$$ 101.818 3.31390
$$945$$ 0 0
$$946$$ 16.9679 0.551674
$$947$$ −38.5681 −1.25329 −0.626647 0.779303i $$-0.715573\pi$$
−0.626647 + 0.779303i $$0.715573\pi$$
$$948$$ 0 0
$$949$$ 8.36978 0.271695
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0.549911 0.0178227
$$953$$ 47.2051 1.52912 0.764561 0.644551i $$-0.222956\pi$$
0.764561 + 0.644551i $$0.222956\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −78.8699 −2.55084
$$957$$ 0 0
$$958$$ −57.7770 −1.86669
$$959$$ 0.0608425 0.00196471
$$960$$ 0 0
$$961$$ −29.2122 −0.942329
$$962$$ −102.422 −3.30221
$$963$$ 0 0
$$964$$ 61.7936 1.99024
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14.9495 0.480743 0.240371 0.970681i $$-0.422731\pi$$
0.240371 + 0.970681i $$0.422731\pi$$
$$968$$ 66.5680 2.13958
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −30.4517 −0.977240 −0.488620 0.872497i $$-0.662500\pi$$
−0.488620 + 0.872497i $$0.662500\pi$$
$$972$$ 0 0
$$973$$ −0.199787 −0.00640487
$$974$$ −29.6625 −0.950447
$$975$$ 0 0
$$976$$ 67.3716 2.15651
$$977$$ 0.336322 0.0107599 0.00537995 0.999986i $$-0.498287\pi$$
0.00537995 + 0.999986i $$0.498287\pi$$
$$978$$ 0 0
$$979$$ −21.2117 −0.677929
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 73.8264 2.35590
$$983$$ 1.44050 0.0459449 0.0229724 0.999736i $$-0.492687\pi$$
0.0229724 + 0.999736i $$0.492687\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −10.4277 −0.332085
$$987$$ 0 0
$$988$$ −144.110 −4.58474
$$989$$ −17.9867 −0.571943
$$990$$ 0 0
$$991$$ 40.0507 1.27225 0.636126 0.771585i $$-0.280536\pi$$
0.636126 + 0.771585i $$0.280536\pi$$
$$992$$ −19.8587 −0.630516
$$993$$ 0 0
$$994$$ 0.196541 0.00623389
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −24.3419 −0.770916 −0.385458 0.922725i $$-0.625957\pi$$
−0.385458 + 0.922725i $$0.625957\pi$$
$$998$$ 79.5488 2.51807
$$999$$ 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.bu.1.1 7
3.2 odd 2 2175.2.a.bb.1.7 yes 7
5.4 even 2 6525.2.a.bx.1.7 7
15.2 even 4 2175.2.c.o.349.14 14
15.8 even 4 2175.2.c.o.349.1 14
15.14 odd 2 2175.2.a.ba.1.1 7

By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.ba.1.1 7 15.14 odd 2
2175.2.a.bb.1.7 yes 7 3.2 odd 2
2175.2.c.o.349.1 14 15.8 even 4
2175.2.c.o.349.14 14 15.2 even 4
6525.2.a.bu.1.1 7 1.1 even 1 trivial
6525.2.a.bx.1.7 7 5.4 even 2