# Properties

 Label 6525.2.a.bx.1.6 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $7$ 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: $$0$$ 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: $$x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14$$ 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 1.6 Root $$2.26695$$ 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)

 $$f(q)$$ $$=$$ $$q+2.26695 q^{2} +3.13907 q^{4} -0.159887 q^{7} +2.58223 q^{8} +O(q^{10})$$ $$q+2.26695 q^{2} +3.13907 q^{4} -0.159887 q^{7} +2.58223 q^{8} +3.24930 q^{11} -7.05679 q^{13} -0.362456 q^{14} -0.424361 q^{16} +7.63808 q^{17} -3.99028 q^{19} +7.36601 q^{22} +2.06402 q^{23} -15.9974 q^{26} -0.501897 q^{28} +1.00000 q^{29} +10.2044 q^{31} -6.12646 q^{32} +17.3152 q^{34} +4.27815 q^{37} -9.04576 q^{38} +5.12330 q^{41} +3.94180 q^{43} +10.1998 q^{44} +4.67903 q^{46} +5.61626 q^{47} -6.97444 q^{49} -22.1518 q^{52} +12.0822 q^{53} -0.412864 q^{56} +2.26695 q^{58} +7.06781 q^{59} +11.4310 q^{61} +23.1329 q^{62} -13.0397 q^{64} +14.9512 q^{67} +23.9765 q^{68} -3.43347 q^{71} +12.0772 q^{73} +9.69836 q^{74} -12.5258 q^{76} -0.519521 q^{77} -12.7441 q^{79} +11.6143 q^{82} -2.59792 q^{83} +8.93586 q^{86} +8.39044 q^{88} -4.33165 q^{89} +1.12829 q^{91} +6.47910 q^{92} +12.7318 q^{94} +3.88355 q^{97} -15.8107 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 2 q^{2} + 10 q^{4} + q^{7} + 3 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.26695 1.60298 0.801489 0.598010i $$-0.204042\pi$$
0.801489 + 0.598010i $$0.204042\pi$$
$$3$$ 0 0
$$4$$ 3.13907 1.56954
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.159887 −0.0604316 −0.0302158 0.999543i $$-0.509619\pi$$
−0.0302158 + 0.999543i $$0.509619\pi$$
$$8$$ 2.58223 0.912955
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.24930 0.979701 0.489851 0.871806i $$-0.337051\pi$$
0.489851 + 0.871806i $$0.337051\pi$$
$$12$$ 0 0
$$13$$ −7.05679 −1.95720 −0.978600 0.205770i $$-0.934030\pi$$
−0.978600 + 0.205770i $$0.934030\pi$$
$$14$$ −0.362456 −0.0968704
$$15$$ 0 0
$$16$$ −0.424361 −0.106090
$$17$$ 7.63808 1.85251 0.926254 0.376901i $$-0.123010\pi$$
0.926254 + 0.376901i $$0.123010\pi$$
$$18$$ 0 0
$$19$$ −3.99028 −0.915432 −0.457716 0.889098i $$-0.651332\pi$$
−0.457716 + 0.889098i $$0.651332\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 7.36601 1.57044
$$23$$ 2.06402 0.430377 0.215189 0.976572i $$-0.430963\pi$$
0.215189 + 0.976572i $$0.430963\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −15.9974 −3.13735
$$27$$ 0 0
$$28$$ −0.501897 −0.0948496
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 10.2044 1.83277 0.916383 0.400303i $$-0.131095\pi$$
0.916383 + 0.400303i $$0.131095\pi$$
$$32$$ −6.12646 −1.08302
$$33$$ 0 0
$$34$$ 17.3152 2.96953
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.27815 0.703323 0.351662 0.936127i $$-0.385617\pi$$
0.351662 + 0.936127i $$0.385617\pi$$
$$38$$ −9.04576 −1.46742
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.12330 0.800125 0.400063 0.916488i $$-0.368988\pi$$
0.400063 + 0.916488i $$0.368988\pi$$
$$42$$ 0 0
$$43$$ 3.94180 0.601118 0.300559 0.953763i $$-0.402827\pi$$
0.300559 + 0.953763i $$0.402827\pi$$
$$44$$ 10.1998 1.53768
$$45$$ 0 0
$$46$$ 4.67903 0.689885
$$47$$ 5.61626 0.819215 0.409608 0.912262i $$-0.365666\pi$$
0.409608 + 0.912262i $$0.365666\pi$$
$$48$$ 0 0
$$49$$ −6.97444 −0.996348
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −22.1518 −3.07190
$$53$$ 12.0822 1.65962 0.829808 0.558049i $$-0.188450\pi$$
0.829808 + 0.558049i $$0.188450\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −0.412864 −0.0551713
$$57$$ 0 0
$$58$$ 2.26695 0.297665
$$59$$ 7.06781 0.920151 0.460075 0.887880i $$-0.347822\pi$$
0.460075 + 0.887880i $$0.347822\pi$$
$$60$$ 0 0
$$61$$ 11.4310 1.46359 0.731796 0.681524i $$-0.238682\pi$$
0.731796 + 0.681524i $$0.238682\pi$$
$$62$$ 23.1329 2.93788
$$63$$ 0 0
$$64$$ −13.0397 −1.62996
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 14.9512 1.82658 0.913289 0.407311i $$-0.133534\pi$$
0.913289 + 0.407311i $$0.133534\pi$$
$$68$$ 23.9765 2.90758
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.43347 −0.407478 −0.203739 0.979025i $$-0.565309\pi$$
−0.203739 + 0.979025i $$0.565309\pi$$
$$72$$ 0 0
$$73$$ 12.0772 1.41353 0.706764 0.707449i $$-0.250154\pi$$
0.706764 + 0.707449i $$0.250154\pi$$
$$74$$ 9.69836 1.12741
$$75$$ 0 0
$$76$$ −12.5258 −1.43680
$$77$$ −0.519521 −0.0592049
$$78$$ 0 0
$$79$$ −12.7441 −1.43383 −0.716913 0.697163i $$-0.754446\pi$$
−0.716913 + 0.697163i $$0.754446\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 11.6143 1.28258
$$83$$ −2.59792 −0.285159 −0.142580 0.989783i $$-0.545540\pi$$
−0.142580 + 0.989783i $$0.545540\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.93586 0.963579
$$87$$ 0 0
$$88$$ 8.39044 0.894423
$$89$$ −4.33165 −0.459154 −0.229577 0.973290i $$-0.573734\pi$$
−0.229577 + 0.973290i $$0.573734\pi$$
$$90$$ 0 0
$$91$$ 1.12829 0.118277
$$92$$ 6.47910 0.675493
$$93$$ 0 0
$$94$$ 12.7318 1.31318
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.88355 0.394315 0.197158 0.980372i $$-0.436829\pi$$
0.197158 + 0.980372i $$0.436829\pi$$
$$98$$ −15.8107 −1.59712
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −15.4045 −1.53280 −0.766400 0.642363i $$-0.777954\pi$$
−0.766400 + 0.642363i $$0.777954\pi$$
$$102$$ 0 0
$$103$$ −7.68484 −0.757210 −0.378605 0.925558i $$-0.623596\pi$$
−0.378605 + 0.925558i $$0.623596\pi$$
$$104$$ −18.2222 −1.78684
$$105$$ 0 0
$$106$$ 27.3897 2.66033
$$107$$ −4.60924 −0.445592 −0.222796 0.974865i $$-0.571518\pi$$
−0.222796 + 0.974865i $$0.571518\pi$$
$$108$$ 0 0
$$109$$ −14.4509 −1.38415 −0.692073 0.721827i $$-0.743302\pi$$
−0.692073 + 0.721827i $$0.743302\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0.0678498 0.00641120
$$113$$ −4.08614 −0.384392 −0.192196 0.981357i $$-0.561561\pi$$
−0.192196 + 0.981357i $$0.561561\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3.13907 0.291456
$$117$$ 0 0
$$118$$ 16.0224 1.47498
$$119$$ −1.22123 −0.111950
$$120$$ 0 0
$$121$$ −0.442038 −0.0401853
$$122$$ 25.9136 2.34610
$$123$$ 0 0
$$124$$ 32.0324 2.87659
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 3.69650 0.328011 0.164006 0.986459i $$-0.447559\pi$$
0.164006 + 0.986459i $$0.447559\pi$$
$$128$$ −17.3074 −1.52977
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 9.92811 0.867423 0.433712 0.901052i $$-0.357204\pi$$
0.433712 + 0.901052i $$0.357204\pi$$
$$132$$ 0 0
$$133$$ 0.637993 0.0553210
$$134$$ 33.8936 2.92797
$$135$$ 0 0
$$136$$ 19.7233 1.69126
$$137$$ 3.82810 0.327057 0.163528 0.986539i $$-0.447712\pi$$
0.163528 + 0.986539i $$0.447712\pi$$
$$138$$ 0 0
$$139$$ −2.06987 −0.175564 −0.0877822 0.996140i $$-0.527978\pi$$
−0.0877822 + 0.996140i $$0.527978\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −7.78351 −0.653177
$$143$$ −22.9296 −1.91747
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 27.3784 2.26585
$$147$$ 0 0
$$148$$ 13.4294 1.10389
$$149$$ 1.46609 0.120107 0.0600536 0.998195i $$-0.480873\pi$$
0.0600536 + 0.998195i $$0.480873\pi$$
$$150$$ 0 0
$$151$$ −20.1279 −1.63799 −0.818995 0.573801i $$-0.805468\pi$$
−0.818995 + 0.573801i $$0.805468\pi$$
$$152$$ −10.3038 −0.835748
$$153$$ 0 0
$$154$$ −1.17773 −0.0949041
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −7.09106 −0.565928 −0.282964 0.959131i $$-0.591318\pi$$
−0.282964 + 0.959131i $$0.591318\pi$$
$$158$$ −28.8903 −2.29839
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −0.330009 −0.0260084
$$162$$ 0 0
$$163$$ 6.58715 0.515946 0.257973 0.966152i $$-0.416945\pi$$
0.257973 + 0.966152i $$0.416945\pi$$
$$164$$ 16.0824 1.25583
$$165$$ 0 0
$$166$$ −5.88937 −0.457104
$$167$$ −10.6638 −0.825192 −0.412596 0.910914i $$-0.635378\pi$$
−0.412596 + 0.910914i $$0.635378\pi$$
$$168$$ 0 0
$$169$$ 36.7983 2.83063
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.3736 0.943477
$$173$$ −13.8799 −1.05527 −0.527633 0.849472i $$-0.676920\pi$$
−0.527633 + 0.849472i $$0.676920\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.37888 −0.103937
$$177$$ 0 0
$$178$$ −9.81965 −0.736014
$$179$$ 11.1105 0.830439 0.415219 0.909721i $$-0.363705\pi$$
0.415219 + 0.909721i $$0.363705\pi$$
$$180$$ 0 0
$$181$$ 10.9453 0.813557 0.406778 0.913527i $$-0.366652\pi$$
0.406778 + 0.913527i $$0.366652\pi$$
$$182$$ 2.55777 0.189595
$$183$$ 0 0
$$184$$ 5.32976 0.392915
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 24.8184 1.81490
$$188$$ 17.6298 1.28579
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.1472 −1.31309 −0.656544 0.754287i $$-0.727983\pi$$
−0.656544 + 0.754287i $$0.727983\pi$$
$$192$$ 0 0
$$193$$ 17.3305 1.24747 0.623737 0.781634i $$-0.285614\pi$$
0.623737 + 0.781634i $$0.285614\pi$$
$$194$$ 8.80383 0.632078
$$195$$ 0 0
$$196$$ −21.8933 −1.56381
$$197$$ −10.4273 −0.742915 −0.371458 0.928450i $$-0.621142\pi$$
−0.371458 + 0.928450i $$0.621142\pi$$
$$198$$ 0 0
$$199$$ 11.5570 0.819256 0.409628 0.912253i $$-0.365659\pi$$
0.409628 + 0.912253i $$0.365659\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −34.9212 −2.45704
$$203$$ −0.159887 −0.0112219
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −17.4212 −1.21379
$$207$$ 0 0
$$208$$ 2.99463 0.207640
$$209$$ −12.9656 −0.896850
$$210$$ 0 0
$$211$$ −21.1616 −1.45682 −0.728412 0.685140i $$-0.759741\pi$$
−0.728412 + 0.685140i $$0.759741\pi$$
$$212$$ 37.9269 2.60483
$$213$$ 0 0
$$214$$ −10.4489 −0.714274
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.63155 −0.110757
$$218$$ −32.7595 −2.21876
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −53.9003 −3.62573
$$222$$ 0 0
$$223$$ −9.46925 −0.634108 −0.317054 0.948408i $$-0.602694\pi$$
−0.317054 + 0.948408i $$0.602694\pi$$
$$224$$ 0.979541 0.0654483
$$225$$ 0 0
$$226$$ −9.26310 −0.616172
$$227$$ 22.4030 1.48694 0.743471 0.668768i $$-0.233178\pi$$
0.743471 + 0.668768i $$0.233178\pi$$
$$228$$ 0 0
$$229$$ 2.03117 0.134223 0.0671117 0.997745i $$-0.478622\pi$$
0.0671117 + 0.997745i $$0.478622\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.58223 0.169532
$$233$$ 11.0704 0.725249 0.362624 0.931935i $$-0.381881\pi$$
0.362624 + 0.931935i $$0.381881\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 22.1864 1.44421
$$237$$ 0 0
$$238$$ −2.76847 −0.179453
$$239$$ 2.24098 0.144957 0.0724786 0.997370i $$-0.476909\pi$$
0.0724786 + 0.997370i $$0.476909\pi$$
$$240$$ 0 0
$$241$$ 14.4838 0.932986 0.466493 0.884525i $$-0.345517\pi$$
0.466493 + 0.884525i $$0.345517\pi$$
$$242$$ −1.00208 −0.0644161
$$243$$ 0 0
$$244$$ 35.8828 2.29716
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 28.1585 1.79168
$$248$$ 26.3501 1.67323
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.2044 1.02281 0.511407 0.859339i $$-0.329125\pi$$
0.511407 + 0.859339i $$0.329125\pi$$
$$252$$ 0 0
$$253$$ 6.70661 0.421641
$$254$$ 8.37978 0.525794
$$255$$ 0 0
$$256$$ −13.1557 −0.822232
$$257$$ −3.36942 −0.210178 −0.105089 0.994463i $$-0.533513\pi$$
−0.105089 + 0.994463i $$0.533513\pi$$
$$258$$ 0 0
$$259$$ −0.684020 −0.0425029
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 22.5066 1.39046
$$263$$ 1.79671 0.110790 0.0553950 0.998465i $$-0.482358\pi$$
0.0553950 + 0.998465i $$0.482358\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 1.44630 0.0886783
$$267$$ 0 0
$$268$$ 46.9329 2.86688
$$269$$ 0.117055 0.00713699 0.00356849 0.999994i $$-0.498864\pi$$
0.00356849 + 0.999994i $$0.498864\pi$$
$$270$$ 0 0
$$271$$ −4.06294 −0.246806 −0.123403 0.992357i $$-0.539381\pi$$
−0.123403 + 0.992357i $$0.539381\pi$$
$$272$$ −3.24131 −0.196533
$$273$$ 0 0
$$274$$ 8.67813 0.524265
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 12.5436 0.753674 0.376837 0.926279i $$-0.377012\pi$$
0.376837 + 0.926279i $$0.377012\pi$$
$$278$$ −4.69231 −0.281426
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2.83163 0.168921 0.0844605 0.996427i $$-0.473083\pi$$
0.0844605 + 0.996427i $$0.473083\pi$$
$$282$$ 0 0
$$283$$ −28.2249 −1.67780 −0.838898 0.544288i $$-0.816800\pi$$
−0.838898 + 0.544288i $$0.816800\pi$$
$$284$$ −10.7779 −0.639551
$$285$$ 0 0
$$286$$ −51.9804 −3.07366
$$287$$ −0.819148 −0.0483528
$$288$$ 0 0
$$289$$ 41.3403 2.43178
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 37.9112 2.21859
$$293$$ 21.9337 1.28138 0.640691 0.767799i $$-0.278648\pi$$
0.640691 + 0.767799i $$0.278648\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 11.0472 0.642103
$$297$$ 0 0
$$298$$ 3.32357 0.192529
$$299$$ −14.5653 −0.842335
$$300$$ 0 0
$$301$$ −0.630241 −0.0363265
$$302$$ −45.6291 −2.62566
$$303$$ 0 0
$$304$$ 1.69332 0.0971185
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0.461574 0.0263434 0.0131717 0.999913i $$-0.495807\pi$$
0.0131717 + 0.999913i $$0.495807\pi$$
$$308$$ −1.63081 −0.0929243
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −27.5507 −1.56226 −0.781128 0.624371i $$-0.785355\pi$$
−0.781128 + 0.624371i $$0.785355\pi$$
$$312$$ 0 0
$$313$$ −14.9121 −0.842882 −0.421441 0.906856i $$-0.638476\pi$$
−0.421441 + 0.906856i $$0.638476\pi$$
$$314$$ −16.0751 −0.907170
$$315$$ 0 0
$$316$$ −40.0048 −2.25044
$$317$$ −22.9266 −1.28769 −0.643843 0.765157i $$-0.722661\pi$$
−0.643843 + 0.765157i $$0.722661\pi$$
$$318$$ 0 0
$$319$$ 3.24930 0.181926
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −0.748115 −0.0416908
$$323$$ −30.4781 −1.69584
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 14.9328 0.827049
$$327$$ 0 0
$$328$$ 13.2295 0.730478
$$329$$ −0.897966 −0.0495065
$$330$$ 0 0
$$331$$ −3.23147 −0.177618 −0.0888088 0.996049i $$-0.528306\pi$$
−0.0888088 + 0.996049i $$0.528306\pi$$
$$332$$ −8.15507 −0.447568
$$333$$ 0 0
$$334$$ −24.1744 −1.32277
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −28.8026 −1.56898 −0.784489 0.620143i $$-0.787075\pi$$
−0.784489 + 0.620143i $$0.787075\pi$$
$$338$$ 83.4199 4.53744
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 33.1572 1.79556
$$342$$ 0 0
$$343$$ 2.23433 0.120642
$$344$$ 10.1786 0.548794
$$345$$ 0 0
$$346$$ −31.4650 −1.69157
$$347$$ −16.4175 −0.881337 −0.440669 0.897670i $$-0.645259\pi$$
−0.440669 + 0.897670i $$0.645259\pi$$
$$348$$ 0 0
$$349$$ −18.2107 −0.974796 −0.487398 0.873180i $$-0.662054\pi$$
−0.487398 + 0.873180i $$0.662054\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −19.9067 −1.06103
$$353$$ −18.8790 −1.00483 −0.502414 0.864627i $$-0.667555\pi$$
−0.502414 + 0.864627i $$0.667555\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −13.5974 −0.720660
$$357$$ 0 0
$$358$$ 25.1870 1.33117
$$359$$ −14.4500 −0.762640 −0.381320 0.924443i $$-0.624530\pi$$
−0.381320 + 0.924443i $$0.624530\pi$$
$$360$$ 0 0
$$361$$ −3.07770 −0.161984
$$362$$ 24.8125 1.30411
$$363$$ 0 0
$$364$$ 3.54178 0.185640
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 21.1171 1.10231 0.551153 0.834404i $$-0.314188\pi$$
0.551153 + 0.834404i $$0.314188\pi$$
$$368$$ −0.875889 −0.0456589
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.93178 −0.100293
$$372$$ 0 0
$$373$$ 17.1185 0.886361 0.443181 0.896432i $$-0.353850\pi$$
0.443181 + 0.896432i $$0.353850\pi$$
$$374$$ 56.2622 2.90925
$$375$$ 0 0
$$376$$ 14.5025 0.747907
$$377$$ −7.05679 −0.363443
$$378$$ 0 0
$$379$$ −0.206688 −0.0106169 −0.00530844 0.999986i $$-0.501690\pi$$
−0.00530844 + 0.999986i $$0.501690\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −41.1389 −2.10485
$$383$$ 16.3701 0.836474 0.418237 0.908338i $$-0.362648\pi$$
0.418237 + 0.908338i $$0.362648\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 39.2873 1.99967
$$387$$ 0 0
$$388$$ 12.1908 0.618892
$$389$$ −11.7627 −0.596393 −0.298197 0.954504i $$-0.596385\pi$$
−0.298197 + 0.954504i $$0.596385\pi$$
$$390$$ 0 0
$$391$$ 15.7651 0.797277
$$392$$ −18.0096 −0.909621
$$393$$ 0 0
$$394$$ −23.6382 −1.19088
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4.96134 0.249002 0.124501 0.992219i $$-0.460267\pi$$
0.124501 + 0.992219i $$0.460267\pi$$
$$398$$ 26.1992 1.31325
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −36.1967 −1.80758 −0.903789 0.427979i $$-0.859226\pi$$
−0.903789 + 0.427979i $$0.859226\pi$$
$$402$$ 0 0
$$403$$ −72.0103 −3.58709
$$404$$ −48.3557 −2.40579
$$405$$ 0 0
$$406$$ −0.362456 −0.0179884
$$407$$ 13.9010 0.689047
$$408$$ 0 0
$$409$$ −16.1607 −0.799094 −0.399547 0.916713i $$-0.630832\pi$$
−0.399547 + 0.916713i $$0.630832\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −24.1233 −1.18847
$$413$$ −1.13005 −0.0556061
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 43.2331 2.11968
$$417$$ 0 0
$$418$$ −29.3924 −1.43763
$$419$$ 33.5544 1.63924 0.819621 0.572906i $$-0.194184\pi$$
0.819621 + 0.572906i $$0.194184\pi$$
$$420$$ 0 0
$$421$$ 26.9847 1.31515 0.657577 0.753388i $$-0.271582\pi$$
0.657577 + 0.753388i $$0.271582\pi$$
$$422$$ −47.9723 −2.33526
$$423$$ 0 0
$$424$$ 31.1990 1.51515
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.82767 −0.0884471
$$428$$ −14.4687 −0.699373
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −0.832366 −0.0400937 −0.0200468 0.999799i $$-0.506382\pi$$
−0.0200468 + 0.999799i $$0.506382\pi$$
$$432$$ 0 0
$$433$$ 22.5647 1.08439 0.542195 0.840253i $$-0.317593\pi$$
0.542195 + 0.840253i $$0.317593\pi$$
$$434$$ −3.69865 −0.177541
$$435$$ 0 0
$$436$$ −45.3625 −2.17247
$$437$$ −8.23600 −0.393981
$$438$$ 0 0
$$439$$ 23.9583 1.14347 0.571734 0.820439i $$-0.306271\pi$$
0.571734 + 0.820439i $$0.306271\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −122.190 −5.81196
$$443$$ 17.4925 0.831092 0.415546 0.909572i $$-0.363591\pi$$
0.415546 + 0.909572i $$0.363591\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −21.4663 −1.01646
$$447$$ 0 0
$$448$$ 2.08487 0.0985010
$$449$$ −4.22200 −0.199249 −0.0996243 0.995025i $$-0.531764\pi$$
−0.0996243 + 0.995025i $$0.531764\pi$$
$$450$$ 0 0
$$451$$ 16.6471 0.783884
$$452$$ −12.8267 −0.603318
$$453$$ 0 0
$$454$$ 50.7866 2.38353
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.28562 0.0601389 0.0300694 0.999548i $$-0.490427\pi$$
0.0300694 + 0.999548i $$0.490427\pi$$
$$458$$ 4.60456 0.215157
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 23.8534 1.11097 0.555483 0.831528i $$-0.312534\pi$$
0.555483 + 0.831528i $$0.312534\pi$$
$$462$$ 0 0
$$463$$ −14.1402 −0.657152 −0.328576 0.944478i $$-0.606569\pi$$
−0.328576 + 0.944478i $$0.606569\pi$$
$$464$$ −0.424361 −0.0197005
$$465$$ 0 0
$$466$$ 25.0962 1.16256
$$467$$ 7.06487 0.326923 0.163462 0.986550i $$-0.447734\pi$$
0.163462 + 0.986550i $$0.447734\pi$$
$$468$$ 0 0
$$469$$ −2.39050 −0.110383
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 18.2507 0.840056
$$473$$ 12.8081 0.588916
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.83353 −0.175710
$$477$$ 0 0
$$478$$ 5.08020 0.232363
$$479$$ −22.9561 −1.04889 −0.524444 0.851445i $$-0.675727\pi$$
−0.524444 + 0.851445i $$0.675727\pi$$
$$480$$ 0 0
$$481$$ −30.1900 −1.37654
$$482$$ 32.8342 1.49556
$$483$$ 0 0
$$484$$ −1.38759 −0.0630723
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 27.0319 1.22493 0.612467 0.790496i $$-0.290177\pi$$
0.612467 + 0.790496i $$0.290177\pi$$
$$488$$ 29.5175 1.33619
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 11.5498 0.521235 0.260617 0.965442i $$-0.416074\pi$$
0.260617 + 0.965442i $$0.416074\pi$$
$$492$$ 0 0
$$493$$ 7.63808 0.344002
$$494$$ 63.8340 2.87203
$$495$$ 0 0
$$496$$ −4.33036 −0.194439
$$497$$ 0.548966 0.0246245
$$498$$ 0 0
$$499$$ 3.77942 0.169190 0.0845950 0.996415i $$-0.473040\pi$$
0.0845950 + 0.996415i $$0.473040\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 36.7346 1.63955
$$503$$ −14.7684 −0.658489 −0.329244 0.944245i $$-0.606794\pi$$
−0.329244 + 0.944245i $$0.606794\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 15.2036 0.675881
$$507$$ 0 0
$$508$$ 11.6036 0.514826
$$509$$ 42.4182 1.88015 0.940077 0.340962i $$-0.110753\pi$$
0.940077 + 0.340962i $$0.110753\pi$$
$$510$$ 0 0
$$511$$ −1.93098 −0.0854217
$$512$$ 4.79143 0.211753
$$513$$ 0 0
$$514$$ −7.63831 −0.336911
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 18.2489 0.802586
$$518$$ −1.55064 −0.0681312
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −20.3409 −0.891152 −0.445576 0.895244i $$-0.647001\pi$$
−0.445576 + 0.895244i $$0.647001\pi$$
$$522$$ 0 0
$$523$$ −1.52190 −0.0665481 −0.0332740 0.999446i $$-0.510593\pi$$
−0.0332740 + 0.999446i $$0.510593\pi$$
$$524$$ 31.1651 1.36145
$$525$$ 0 0
$$526$$ 4.07306 0.177594
$$527$$ 77.9421 3.39521
$$528$$ 0 0
$$529$$ −18.7398 −0.814775
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00271 0.0868283
$$533$$ −36.1540 −1.56601
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 38.6074 1.66758
$$537$$ 0 0
$$538$$ 0.265359 0.0114404
$$539$$ −22.6620 −0.976123
$$540$$ 0 0
$$541$$ −9.40679 −0.404430 −0.202215 0.979341i $$-0.564814\pi$$
−0.202215 + 0.979341i $$0.564814\pi$$
$$542$$ −9.21049 −0.395624
$$543$$ 0 0
$$544$$ −46.7944 −2.00629
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −19.7232 −0.843304 −0.421652 0.906758i $$-0.638550\pi$$
−0.421652 + 0.906758i $$0.638550\pi$$
$$548$$ 12.0167 0.513328
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.99028 −0.169991
$$552$$ 0 0
$$553$$ 2.03762 0.0866483
$$554$$ 28.4358 1.20812
$$555$$ 0 0
$$556$$ −6.49749 −0.275555
$$557$$ 9.01986 0.382184 0.191092 0.981572i $$-0.438797\pi$$
0.191092 + 0.981572i $$0.438797\pi$$
$$558$$ 0 0
$$559$$ −27.8164 −1.17651
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.41918 0.270777
$$563$$ 17.5283 0.738728 0.369364 0.929285i $$-0.379576\pi$$
0.369364 + 0.929285i $$0.379576\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −63.9845 −2.68947
$$567$$ 0 0
$$568$$ −8.86599 −0.372009
$$569$$ −32.3981 −1.35820 −0.679099 0.734046i $$-0.737629\pi$$
−0.679099 + 0.734046i $$0.737629\pi$$
$$570$$ 0 0
$$571$$ −11.8393 −0.495458 −0.247729 0.968829i $$-0.579684\pi$$
−0.247729 + 0.968829i $$0.579684\pi$$
$$572$$ −71.9778 −3.00954
$$573$$ 0 0
$$574$$ −1.85697 −0.0775085
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0.192778 0.00802545 0.00401273 0.999992i $$-0.498723\pi$$
0.00401273 + 0.999992i $$0.498723\pi$$
$$578$$ 93.7165 3.89809
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0.415374 0.0172326
$$582$$ 0 0
$$583$$ 39.2587 1.62593
$$584$$ 31.1861 1.29049
$$585$$ 0 0
$$586$$ 49.7227 2.05403
$$587$$ 34.5089 1.42433 0.712167 0.702010i $$-0.247714\pi$$
0.712167 + 0.702010i $$0.247714\pi$$
$$588$$ 0 0
$$589$$ −40.7184 −1.67777
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −1.81548 −0.0746158
$$593$$ 2.42748 0.0996846 0.0498423 0.998757i $$-0.484128\pi$$
0.0498423 + 0.998757i $$0.484128\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.60218 0.188513
$$597$$ 0 0
$$598$$ −33.0189 −1.35024
$$599$$ −29.3312 −1.19844 −0.599220 0.800585i $$-0.704522\pi$$
−0.599220 + 0.800585i $$0.704522\pi$$
$$600$$ 0 0
$$601$$ 29.5631 1.20590 0.602952 0.797778i $$-0.293991\pi$$
0.602952 + 0.797778i $$0.293991\pi$$
$$602$$ −1.42873 −0.0582306
$$603$$ 0 0
$$604$$ −63.1831 −2.57089
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −33.1756 −1.34656 −0.673278 0.739390i $$-0.735114\pi$$
−0.673278 + 0.739390i $$0.735114\pi$$
$$608$$ 24.4463 0.991427
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −39.6327 −1.60337
$$612$$ 0 0
$$613$$ 13.9979 0.565371 0.282686 0.959213i $$-0.408775\pi$$
0.282686 + 0.959213i $$0.408775\pi$$
$$614$$ 1.04637 0.0422279
$$615$$ 0 0
$$616$$ −1.34152 −0.0540514
$$617$$ −29.9344 −1.20511 −0.602556 0.798077i $$-0.705851\pi$$
−0.602556 + 0.798077i $$0.705851\pi$$
$$618$$ 0 0
$$619$$ −14.9032 −0.599009 −0.299505 0.954095i $$-0.596821\pi$$
−0.299505 + 0.954095i $$0.596821\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −62.4561 −2.50426
$$623$$ 0.692574 0.0277474
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −33.8050 −1.35112
$$627$$ 0 0
$$628$$ −22.2594 −0.888245
$$629$$ 32.6769 1.30291
$$630$$ 0 0
$$631$$ −28.8643 −1.14907 −0.574534 0.818480i $$-0.694817\pi$$
−0.574534 + 0.818480i $$0.694817\pi$$
$$632$$ −32.9082 −1.30902
$$633$$ 0 0
$$634$$ −51.9735 −2.06413
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 49.2171 1.95005
$$638$$ 7.36601 0.291623
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −44.0167 −1.73856 −0.869278 0.494324i $$-0.835415\pi$$
−0.869278 + 0.494324i $$0.835415\pi$$
$$642$$ 0 0
$$643$$ 29.6239 1.16825 0.584125 0.811663i $$-0.301438\pi$$
0.584125 + 0.811663i $$0.301438\pi$$
$$644$$ −1.03592 −0.0408211
$$645$$ 0 0
$$646$$ −69.0923 −2.71840
$$647$$ −39.3943 −1.54875 −0.774375 0.632727i $$-0.781936\pi$$
−0.774375 + 0.632727i $$0.781936\pi$$
$$648$$ 0 0
$$649$$ 22.9654 0.901473
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.6776 0.809796
$$653$$ 22.3518 0.874692 0.437346 0.899293i $$-0.355918\pi$$
0.437346 + 0.899293i $$0.355918\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.17413 −0.0848855
$$657$$ 0 0
$$658$$ −2.03565 −0.0793577
$$659$$ −42.1979 −1.64380 −0.821898 0.569634i $$-0.807085\pi$$
−0.821898 + 0.569634i $$0.807085\pi$$
$$660$$ 0 0
$$661$$ 8.38035 0.325958 0.162979 0.986630i $$-0.447890\pi$$
0.162979 + 0.986630i $$0.447890\pi$$
$$662$$ −7.32559 −0.284717
$$663$$ 0 0
$$664$$ −6.70843 −0.260337
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.06402 0.0799191
$$668$$ −33.4746 −1.29517
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 37.1428 1.43388
$$672$$ 0 0
$$673$$ 3.13725 0.120932 0.0604660 0.998170i $$-0.480741\pi$$
0.0604660 + 0.998170i $$0.480741\pi$$
$$674$$ −65.2941 −2.51504
$$675$$ 0 0
$$676$$ 115.512 4.44279
$$677$$ 16.2776 0.625600 0.312800 0.949819i $$-0.398733\pi$$
0.312800 + 0.949819i $$0.398733\pi$$
$$678$$ 0 0
$$679$$ −0.620929 −0.0238291
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 75.1658 2.87825
$$683$$ −1.78249 −0.0682050 −0.0341025 0.999418i $$-0.510857\pi$$
−0.0341025 + 0.999418i $$0.510857\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 5.06512 0.193387
$$687$$ 0 0
$$688$$ −1.67275 −0.0637728
$$689$$ −85.2614 −3.24820
$$690$$ 0 0
$$691$$ −20.8504 −0.793187 −0.396594 0.917994i $$-0.629808\pi$$
−0.396594 + 0.917994i $$0.629808\pi$$
$$692$$ −43.5699 −1.65628
$$693$$ 0 0
$$694$$ −37.2177 −1.41276
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 39.1322 1.48224
$$698$$ −41.2828 −1.56258
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −32.5504 −1.22941 −0.614707 0.788756i $$-0.710726\pi$$
−0.614707 + 0.788756i $$0.710726\pi$$
$$702$$ 0 0
$$703$$ −17.0710 −0.643845
$$704$$ −42.3698 −1.59687
$$705$$ 0 0
$$706$$ −42.7978 −1.61072
$$707$$ 2.46297 0.0926295
$$708$$ 0 0
$$709$$ 5.93195 0.222779 0.111390 0.993777i $$-0.464470\pi$$
0.111390 + 0.993777i $$0.464470\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −11.1853 −0.419187
$$713$$ 21.0621 0.788781
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 34.8767 1.30340
$$717$$ 0 0
$$718$$ −32.7574 −1.22250
$$719$$ 13.5279 0.504505 0.252252 0.967661i $$-0.418829\pi$$
0.252252 + 0.967661i $$0.418829\pi$$
$$720$$ 0 0
$$721$$ 1.22871 0.0457594
$$722$$ −6.97701 −0.259657
$$723$$ 0 0
$$724$$ 34.3581 1.27691
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 12.7407 0.472525 0.236263 0.971689i $$-0.424077\pi$$
0.236263 + 0.971689i $$0.424077\pi$$
$$728$$ 2.91350 0.107981
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 30.1078 1.11358
$$732$$ 0 0
$$733$$ −30.9422 −1.14288 −0.571438 0.820645i $$-0.693614\pi$$
−0.571438 + 0.820645i $$0.693614\pi$$
$$734$$ 47.8716 1.76697
$$735$$ 0 0
$$736$$ −12.6451 −0.466105
$$737$$ 48.5809 1.78950
$$738$$ 0 0
$$739$$ 35.0047 1.28767 0.643834 0.765166i $$-0.277343\pi$$
0.643834 + 0.765166i $$0.277343\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −4.37926 −0.160768
$$743$$ −12.5515 −0.460468 −0.230234 0.973135i $$-0.573949\pi$$
−0.230234 + 0.973135i $$0.573949\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 38.8068 1.42082
$$747$$ 0 0
$$748$$ 77.9069 2.84856
$$749$$ 0.736956 0.0269278
$$750$$ 0 0
$$751$$ −33.8278 −1.23439 −0.617197 0.786809i $$-0.711732\pi$$
−0.617197 + 0.786809i $$0.711732\pi$$
$$752$$ −2.38332 −0.0869108
$$753$$ 0 0
$$754$$ −15.9974 −0.582591
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 6.60568 0.240088 0.120044 0.992769i $$-0.461697\pi$$
0.120044 + 0.992769i $$0.461697\pi$$
$$758$$ −0.468553 −0.0170186
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 22.8638 0.828814 0.414407 0.910092i $$-0.363989\pi$$
0.414407 + 0.910092i $$0.363989\pi$$
$$762$$ 0 0
$$763$$ 2.31051 0.0836461
$$764$$ −56.9656 −2.06094
$$765$$ 0 0
$$766$$ 37.1103 1.34085
$$767$$ −49.8760 −1.80092
$$768$$ 0 0
$$769$$ −35.0078 −1.26241 −0.631206 0.775615i $$-0.717440\pi$$
−0.631206 + 0.775615i $$0.717440\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 54.4016 1.95796
$$773$$ 17.7019 0.636691 0.318346 0.947975i $$-0.396873\pi$$
0.318346 + 0.947975i $$0.396873\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 10.0282 0.359992
$$777$$ 0 0
$$778$$ −26.6655 −0.956005
$$779$$ −20.4434 −0.732460
$$780$$ 0 0
$$781$$ −11.1564 −0.399206
$$782$$ 35.7388 1.27802
$$783$$ 0 0
$$784$$ 2.95968 0.105703
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −5.13176 −0.182928 −0.0914638 0.995808i $$-0.529155\pi$$
−0.0914638 + 0.995808i $$0.529155\pi$$
$$788$$ −32.7321 −1.16603
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0.653321 0.0232294
$$792$$ 0 0
$$793$$ −80.6662 −2.86454
$$794$$ 11.2471 0.399145
$$795$$ 0 0
$$796$$ 36.2783 1.28585
$$797$$ 27.5492 0.975842 0.487921 0.872888i $$-0.337755\pi$$
0.487921 + 0.872888i $$0.337755\pi$$
$$798$$ 0 0
$$799$$ 42.8974 1.51760
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −82.0562 −2.89751
$$803$$ 39.2424 1.38484
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −163.244 −5.75002
$$807$$ 0 0
$$808$$ −39.7778 −1.39938
$$809$$ 5.51555 0.193917 0.0969583 0.995288i $$-0.469089\pi$$
0.0969583 + 0.995288i $$0.469089\pi$$
$$810$$ 0 0
$$811$$ −17.1711 −0.602959 −0.301479 0.953473i $$-0.597480\pi$$
−0.301479 + 0.953473i $$0.597480\pi$$
$$812$$ −0.501897 −0.0176131
$$813$$ 0 0
$$814$$ 31.5129 1.10453
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −15.7288 −0.550283
$$818$$ −36.6355 −1.28093
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.53740 −0.228157 −0.114078 0.993472i $$-0.536391\pi$$
−0.114078 + 0.993472i $$0.536391\pi$$
$$822$$ 0 0
$$823$$ 35.8588 1.24996 0.624979 0.780641i $$-0.285107\pi$$
0.624979 + 0.780641i $$0.285107\pi$$
$$824$$ −19.8440 −0.691299
$$825$$ 0 0
$$826$$ −2.56177 −0.0891354
$$827$$ −2.36177 −0.0821269 −0.0410635 0.999157i $$-0.513075\pi$$
−0.0410635 + 0.999157i $$0.513075\pi$$
$$828$$ 0 0
$$829$$ 54.0332 1.87665 0.938325 0.345754i $$-0.112377\pi$$
0.938325 + 0.345754i $$0.112377\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 92.0182 3.19016
$$833$$ −53.2713 −1.84574
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −40.7000 −1.40764
$$837$$ 0 0
$$838$$ 76.0663 2.62767
$$839$$ −27.6639 −0.955063 −0.477532 0.878615i $$-0.658468\pi$$
−0.477532 + 0.878615i $$0.658468\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 61.1730 2.10816
$$843$$ 0 0
$$844$$ −66.4278 −2.28654
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.0706761 0.00242846
$$848$$ −5.12721 −0.176069
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.83017 0.302694
$$852$$ 0 0
$$853$$ −16.7180 −0.572413 −0.286207 0.958168i $$-0.592394\pi$$
−0.286207 + 0.958168i $$0.592394\pi$$
$$854$$ −4.14324 −0.141779
$$855$$ 0 0
$$856$$ −11.9021 −0.406805
$$857$$ −50.6304 −1.72950 −0.864750 0.502202i $$-0.832523\pi$$
−0.864750 + 0.502202i $$0.832523\pi$$
$$858$$ 0 0
$$859$$ −23.0473 −0.786363 −0.393182 0.919461i $$-0.628626\pi$$
−0.393182 + 0.919461i $$0.628626\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −1.88693 −0.0642692
$$863$$ 45.4971 1.54874 0.774370 0.632733i $$-0.218067\pi$$
0.774370 + 0.632733i $$0.218067\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 51.1531 1.73825
$$867$$ 0 0
$$868$$ −5.12156 −0.173837
$$869$$ −41.4095 −1.40472
$$870$$ 0 0
$$871$$ −105.507 −3.57498
$$872$$ −37.3155 −1.26366
$$873$$ 0 0
$$874$$ −18.6706 −0.631543
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −6.44199 −0.217531 −0.108765 0.994067i $$-0.534690\pi$$
−0.108765 + 0.994067i $$0.534690\pi$$
$$878$$ 54.3124 1.83295
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 43.3346 1.45998 0.729990 0.683458i $$-0.239525\pi$$
0.729990 + 0.683458i $$0.239525\pi$$
$$882$$ 0 0
$$883$$ 24.6736 0.830333 0.415166 0.909745i $$-0.363723\pi$$
0.415166 + 0.909745i $$0.363723\pi$$
$$884$$ −169.197 −5.69072
$$885$$ 0 0
$$886$$ 39.6546 1.33222
$$887$$ 37.1527 1.24747 0.623733 0.781637i $$-0.285615\pi$$
0.623733 + 0.781637i $$0.285615\pi$$
$$888$$ 0 0
$$889$$ −0.591021 −0.0198222
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −29.7247 −0.995255
$$893$$ −22.4104 −0.749936
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 2.76723 0.0924466
$$897$$ 0 0
$$898$$ −9.57108 −0.319391
$$899$$ 10.2044 0.340336
$$900$$ 0 0
$$901$$ 92.2847 3.07445
$$902$$ 37.7383 1.25655
$$903$$ 0 0
$$904$$ −10.5514 −0.350933
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −7.49457 −0.248853 −0.124427 0.992229i $$-0.539709\pi$$
−0.124427 + 0.992229i $$0.539709\pi$$
$$908$$ 70.3248 2.33381
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −48.6818 −1.61290 −0.806449 0.591303i $$-0.798614\pi$$
−0.806449 + 0.591303i $$0.798614\pi$$
$$912$$ 0 0
$$913$$ −8.44143 −0.279371
$$914$$ 2.91444 0.0964012
$$915$$ 0 0
$$916$$ 6.37598 0.210668
$$917$$ −1.58737 −0.0524197
$$918$$ 0 0
$$919$$ −26.1563 −0.862816 −0.431408 0.902157i $$-0.641983\pi$$
−0.431408 + 0.902157i $$0.641983\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 54.0746 1.78085
$$923$$ 24.2293 0.797515
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −32.0552 −1.05340
$$927$$ 0 0
$$928$$ −6.12646 −0.201111
$$929$$ 8.15366 0.267513 0.133756 0.991014i $$-0.457296\pi$$
0.133756 + 0.991014i $$0.457296\pi$$
$$930$$ 0 0
$$931$$ 27.8299 0.912089
$$932$$ 34.7509 1.13830
$$933$$ 0 0
$$934$$ 16.0157 0.524051
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −41.9169 −1.36936 −0.684682 0.728842i $$-0.740059\pi$$
−0.684682 + 0.728842i $$0.740059\pi$$
$$938$$ −5.41915 −0.176941
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −14.4789 −0.471998 −0.235999 0.971753i $$-0.575836\pi$$
−0.235999 + 0.971753i $$0.575836\pi$$
$$942$$ 0 0
$$943$$ 10.5746 0.344356
$$944$$ −2.99931 −0.0976191
$$945$$ 0 0
$$946$$ 29.0353 0.944020
$$947$$ 1.83838 0.0597392 0.0298696 0.999554i $$-0.490491\pi$$
0.0298696 + 0.999554i $$0.490491\pi$$
$$948$$ 0 0
$$949$$ −85.2262 −2.76656
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −3.15349 −0.102205
$$953$$ −49.5140 −1.60391 −0.801957 0.597382i $$-0.796208\pi$$
−0.801957 + 0.597382i $$0.796208\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 7.03461 0.227516
$$957$$ 0 0
$$958$$ −52.0403 −1.68135
$$959$$ −0.612063 −0.0197646
$$960$$ 0 0
$$961$$ 73.1299 2.35903
$$962$$ −68.4393 −2.20657
$$963$$ 0 0
$$964$$ 45.4658 1.46436
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −5.90299 −0.189827 −0.0949137 0.995486i $$-0.530258\pi$$
−0.0949137 + 0.995486i $$0.530258\pi$$
$$968$$ −1.14144 −0.0366874
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 35.7964 1.14876 0.574380 0.818588i $$-0.305243\pi$$
0.574380 + 0.818588i $$0.305243\pi$$
$$972$$ 0 0
$$973$$ 0.330946 0.0106096
$$974$$ 61.2801 1.96354
$$975$$ 0 0
$$976$$ −4.85088 −0.155273
$$977$$ −1.79673 −0.0574825 −0.0287412 0.999587i $$-0.509150\pi$$
−0.0287412 + 0.999587i $$0.509150\pi$$
$$978$$ 0 0
$$979$$ −14.0748 −0.449834
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 26.1828 0.835528
$$983$$ 6.24151 0.199073 0.0995365 0.995034i $$-0.468264\pi$$
0.0995365 + 0.995034i $$0.468264\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 17.3152 0.551427
$$987$$ 0 0
$$988$$ 88.3917 2.81211
$$989$$ 8.13593 0.258708
$$990$$ 0 0
$$991$$ 41.0410 1.30371 0.651855 0.758344i $$-0.273991\pi$$
0.651855 + 0.758344i $$0.273991\pi$$
$$992$$ −62.5169 −1.98491
$$993$$ 0 0
$$994$$ 1.24448 0.0394725
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −29.8910 −0.946657 −0.473329 0.880886i $$-0.656948\pi$$
−0.473329 + 0.880886i $$0.656948\pi$$
$$998$$ 8.56776 0.271208
$$999$$ 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.bx.1.6 7
3.2 odd 2 2175.2.a.ba.1.2 7
5.4 even 2 6525.2.a.bu.1.2 7
15.2 even 4 2175.2.c.o.349.3 14
15.8 even 4 2175.2.c.o.349.12 14
15.14 odd 2 2175.2.a.bb.1.6 yes 7

By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.ba.1.2 7 3.2 odd 2
2175.2.a.bb.1.6 yes 7 15.14 odd 2
2175.2.c.o.349.3 14 15.2 even 4
2175.2.c.o.349.12 14 15.8 even 4
6525.2.a.bu.1.2 7 5.4 even 2
6525.2.a.bx.1.6 7 1.1 even 1 trivial