# Properties

 Label 6525.2.a.bj.1.2 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $4$ Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.517638$$ 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)

 $$f(q)$$ $$=$$ $$q-0.517638 q^{2} -1.73205 q^{4} -2.44949 q^{7} +1.93185 q^{8} +O(q^{10})$$ $$q-0.517638 q^{2} -1.73205 q^{4} -2.44949 q^{7} +1.93185 q^{8} +1.26795 q^{11} -1.79315 q^{13} +1.26795 q^{14} +2.46410 q^{16} -1.41421 q^{17} -3.26795 q^{19} -0.656339 q^{22} +6.31319 q^{23} +0.928203 q^{26} +4.24264 q^{28} -1.00000 q^{29} -8.73205 q^{31} -5.13922 q^{32} +0.732051 q^{34} +9.14162 q^{37} +1.69161 q^{38} +6.92820 q^{41} +9.14162 q^{43} -2.19615 q^{44} -3.26795 q^{46} -1.41421 q^{47} -1.00000 q^{49} +3.10583 q^{52} -5.93426 q^{53} -4.73205 q^{56} +0.517638 q^{58} +10.3923 q^{59} +2.92820 q^{61} +4.52004 q^{62} -2.26795 q^{64} +4.24264 q^{67} +2.44949 q^{68} -3.46410 q^{71} -7.34847 q^{73} -4.73205 q^{74} +5.66025 q^{76} -3.10583 q^{77} -4.19615 q^{79} -3.58630 q^{82} -10.1769 q^{83} -4.73205 q^{86} +2.44949 q^{88} -10.3923 q^{89} +4.39230 q^{91} -10.9348 q^{92} +0.732051 q^{94} -10.9348 q^{97} +0.517638 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 12 q^{11} + 12 q^{14} - 4 q^{16} - 20 q^{19} - 24 q^{26} - 4 q^{29} - 28 q^{31} - 4 q^{34} + 12 q^{44} - 20 q^{46} - 4 q^{49} - 12 q^{56} - 16 q^{61} - 16 q^{64} - 12 q^{74} - 12 q^{76} + 4 q^{79} - 12 q^{86} - 24 q^{91} - 4 q^{94}+O(q^{100})$$ 4 * q + 12 * q^11 + 12 * q^14 - 4 * q^16 - 20 * q^19 - 24 * q^26 - 4 * q^29 - 28 * q^31 - 4 * q^34 + 12 * q^44 - 20 * q^46 - 4 * q^49 - 12 * q^56 - 16 * q^61 - 16 * q^64 - 12 * q^74 - 12 * q^76 + 4 * q^79 - 12 * q^86 - 24 * q^91 - 4 * q^94

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.517638 −0.366025 −0.183013 0.983111i $$-0.558585\pi$$
−0.183013 + 0.983111i $$0.558585\pi$$
$$3$$ 0 0
$$4$$ −1.73205 −0.866025
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.44949 −0.925820 −0.462910 0.886405i $$-0.653195\pi$$
−0.462910 + 0.886405i $$0.653195\pi$$
$$8$$ 1.93185 0.683013
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.26795 0.382301 0.191151 0.981561i $$-0.438778\pi$$
0.191151 + 0.981561i $$0.438778\pi$$
$$12$$ 0 0
$$13$$ −1.79315 −0.497331 −0.248665 0.968589i $$-0.579992\pi$$
−0.248665 + 0.968589i $$0.579992\pi$$
$$14$$ 1.26795 0.338874
$$15$$ 0 0
$$16$$ 2.46410 0.616025
$$17$$ −1.41421 −0.342997 −0.171499 0.985184i $$-0.554861\pi$$
−0.171499 + 0.985184i $$0.554861\pi$$
$$18$$ 0 0
$$19$$ −3.26795 −0.749719 −0.374859 0.927082i $$-0.622309\pi$$
−0.374859 + 0.927082i $$0.622309\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −0.656339 −0.139932
$$23$$ 6.31319 1.31639 0.658196 0.752847i $$-0.271320\pi$$
0.658196 + 0.752847i $$0.271320\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0.928203 0.182036
$$27$$ 0 0
$$28$$ 4.24264 0.801784
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.73205 −1.56832 −0.784161 0.620557i $$-0.786907\pi$$
−0.784161 + 0.620557i $$0.786907\pi$$
$$32$$ −5.13922 −0.908494
$$33$$ 0 0
$$34$$ 0.732051 0.125546
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.14162 1.50287 0.751437 0.659805i $$-0.229361\pi$$
0.751437 + 0.659805i $$0.229361\pi$$
$$38$$ 1.69161 0.274416
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.92820 1.08200 0.541002 0.841021i $$-0.318045\pi$$
0.541002 + 0.841021i $$0.318045\pi$$
$$42$$ 0 0
$$43$$ 9.14162 1.39408 0.697042 0.717030i $$-0.254499\pi$$
0.697042 + 0.717030i $$0.254499\pi$$
$$44$$ −2.19615 −0.331082
$$45$$ 0 0
$$46$$ −3.26795 −0.481833
$$47$$ −1.41421 −0.206284 −0.103142 0.994667i $$-0.532890\pi$$
−0.103142 + 0.994667i $$0.532890\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 3.10583 0.430701
$$53$$ −5.93426 −0.815133 −0.407566 0.913176i $$-0.633622\pi$$
−0.407566 + 0.913176i $$0.633622\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.73205 −0.632347
$$57$$ 0 0
$$58$$ 0.517638 0.0679692
$$59$$ 10.3923 1.35296 0.676481 0.736460i $$-0.263504\pi$$
0.676481 + 0.736460i $$0.263504\pi$$
$$60$$ 0 0
$$61$$ 2.92820 0.374918 0.187459 0.982272i $$-0.439975\pi$$
0.187459 + 0.982272i $$0.439975\pi$$
$$62$$ 4.52004 0.574046
$$63$$ 0 0
$$64$$ −2.26795 −0.283494
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.24264 0.518321 0.259161 0.965834i $$-0.416554\pi$$
0.259161 + 0.965834i $$0.416554\pi$$
$$68$$ 2.44949 0.297044
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.46410 −0.411113 −0.205557 0.978645i $$-0.565900\pi$$
−0.205557 + 0.978645i $$0.565900\pi$$
$$72$$ 0 0
$$73$$ −7.34847 −0.860073 −0.430037 0.902811i $$-0.641499\pi$$
−0.430037 + 0.902811i $$0.641499\pi$$
$$74$$ −4.73205 −0.550090
$$75$$ 0 0
$$76$$ 5.66025 0.649276
$$77$$ −3.10583 −0.353942
$$78$$ 0 0
$$79$$ −4.19615 −0.472104 −0.236052 0.971740i $$-0.575854\pi$$
−0.236052 + 0.971740i $$0.575854\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −3.58630 −0.396041
$$83$$ −10.1769 −1.11706 −0.558530 0.829484i $$-0.688634\pi$$
−0.558530 + 0.829484i $$0.688634\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.73205 −0.510270
$$87$$ 0 0
$$88$$ 2.44949 0.261116
$$89$$ −10.3923 −1.10158 −0.550791 0.834643i $$-0.685674\pi$$
−0.550791 + 0.834643i $$0.685674\pi$$
$$90$$ 0 0
$$91$$ 4.39230 0.460439
$$92$$ −10.9348 −1.14003
$$93$$ 0 0
$$94$$ 0.732051 0.0755053
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.9348 −1.11026 −0.555129 0.831764i $$-0.687331\pi$$
−0.555129 + 0.831764i $$0.687331\pi$$
$$98$$ 0.517638 0.0522893
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 9.14162 0.900751 0.450375 0.892839i $$-0.351290\pi$$
0.450375 + 0.892839i $$0.351290\pi$$
$$104$$ −3.46410 −0.339683
$$105$$ 0 0
$$106$$ 3.07180 0.298359
$$107$$ 18.6622 1.80414 0.902070 0.431589i $$-0.142047\pi$$
0.902070 + 0.431589i $$0.142047\pi$$
$$108$$ 0 0
$$109$$ 17.8564 1.71033 0.855167 0.518353i $$-0.173455\pi$$
0.855167 + 0.518353i $$0.173455\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −6.03579 −0.570329
$$113$$ 13.0053 1.22344 0.611719 0.791075i $$-0.290478\pi$$
0.611719 + 0.791075i $$0.290478\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.73205 0.160817
$$117$$ 0 0
$$118$$ −5.37945 −0.495219
$$119$$ 3.46410 0.317554
$$120$$ 0 0
$$121$$ −9.39230 −0.853846
$$122$$ −1.51575 −0.137230
$$123$$ 0 0
$$124$$ 15.1244 1.35821
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.24264 0.376473 0.188237 0.982124i $$-0.439723\pi$$
0.188237 + 0.982124i $$0.439723\pi$$
$$128$$ 11.4524 1.01226
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 4.73205 0.413441 0.206721 0.978400i $$-0.433721\pi$$
0.206721 + 0.978400i $$0.433721\pi$$
$$132$$ 0 0
$$133$$ 8.00481 0.694105
$$134$$ −2.19615 −0.189719
$$135$$ 0 0
$$136$$ −2.73205 −0.234271
$$137$$ −14.7985 −1.26432 −0.632159 0.774838i $$-0.717831\pi$$
−0.632159 + 0.774838i $$0.717831\pi$$
$$138$$ 0 0
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1.79315 0.150478
$$143$$ −2.27362 −0.190130
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.80385 0.314809
$$147$$ 0 0
$$148$$ −15.8338 −1.30153
$$149$$ 0.928203 0.0760414 0.0380207 0.999277i $$-0.487895\pi$$
0.0380207 + 0.999277i $$0.487895\pi$$
$$150$$ 0 0
$$151$$ −18.7846 −1.52867 −0.764335 0.644819i $$-0.776933\pi$$
−0.764335 + 0.644819i $$0.776933\pi$$
$$152$$ −6.31319 −0.512068
$$153$$ 0 0
$$154$$ 1.60770 0.129552
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.9348 0.872690 0.436345 0.899780i $$-0.356273\pi$$
0.436345 + 0.899780i $$0.356273\pi$$
$$158$$ 2.17209 0.172802
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −15.4641 −1.21874
$$162$$ 0 0
$$163$$ 14.0406 1.09974 0.549872 0.835249i $$-0.314676\pi$$
0.549872 + 0.835249i $$0.314676\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 5.26795 0.408872
$$167$$ −11.2122 −0.867624 −0.433812 0.901003i $$-0.642832\pi$$
−0.433812 + 0.901003i $$0.642832\pi$$
$$168$$ 0 0
$$169$$ −9.78461 −0.752662
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −15.8338 −1.20731
$$173$$ −4.62158 −0.351372 −0.175686 0.984446i $$-0.556214\pi$$
−0.175686 + 0.984446i $$0.556214\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.12436 0.235507
$$177$$ 0 0
$$178$$ 5.37945 0.403207
$$179$$ −2.53590 −0.189542 −0.0947710 0.995499i $$-0.530212\pi$$
−0.0947710 + 0.995499i $$0.530212\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ −2.27362 −0.168532
$$183$$ 0 0
$$184$$ 12.1962 0.899112
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.79315 −0.131128
$$188$$ 2.44949 0.178647
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −15.1244 −1.09436 −0.547180 0.837015i $$-0.684299\pi$$
−0.547180 + 0.837015i $$0.684299\pi$$
$$192$$ 0 0
$$193$$ −0.656339 −0.0472443 −0.0236222 0.999721i $$-0.507520\pi$$
−0.0236222 + 0.999721i $$0.507520\pi$$
$$194$$ 5.66025 0.406383
$$195$$ 0 0
$$196$$ 1.73205 0.123718
$$197$$ 22.4243 1.59767 0.798834 0.601551i $$-0.205450\pi$$
0.798834 + 0.601551i $$0.205450\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −3.10583 −0.218525
$$203$$ 2.44949 0.171920
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.73205 −0.329698
$$207$$ 0 0
$$208$$ −4.41851 −0.306368
$$209$$ −4.14359 −0.286618
$$210$$ 0 0
$$211$$ −14.7321 −1.01420 −0.507098 0.861888i $$-0.669282\pi$$
−0.507098 + 0.861888i $$0.669282\pi$$
$$212$$ 10.2784 0.705926
$$213$$ 0 0
$$214$$ −9.66025 −0.660361
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 21.3891 1.45198
$$218$$ −9.24316 −0.626026
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2.53590 0.170583
$$222$$ 0 0
$$223$$ −15.3533 −1.02813 −0.514066 0.857751i $$-0.671861\pi$$
−0.514066 + 0.857751i $$0.671861\pi$$
$$224$$ 12.5885 0.841102
$$225$$ 0 0
$$226$$ −6.73205 −0.447809
$$227$$ 20.4553 1.35767 0.678834 0.734292i $$-0.262486\pi$$
0.678834 + 0.734292i $$0.262486\pi$$
$$228$$ 0 0
$$229$$ 18.7846 1.24132 0.620661 0.784079i $$-0.286864\pi$$
0.620661 + 0.784079i $$0.286864\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1.93185 −0.126832
$$233$$ 3.38323 0.221643 0.110821 0.993840i $$-0.464652\pi$$
0.110821 + 0.993840i $$0.464652\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −18.0000 −1.17170
$$237$$ 0 0
$$238$$ −1.79315 −0.116233
$$239$$ −10.3923 −0.672222 −0.336111 0.941822i $$-0.609112\pi$$
−0.336111 + 0.941822i $$0.609112\pi$$
$$240$$ 0 0
$$241$$ 11.4641 0.738468 0.369234 0.929337i $$-0.379620\pi$$
0.369234 + 0.929337i $$0.379620\pi$$
$$242$$ 4.86181 0.312529
$$243$$ 0 0
$$244$$ −5.07180 −0.324689
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.85993 0.372858
$$248$$ −16.8690 −1.07118
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −15.1244 −0.954641 −0.477320 0.878729i $$-0.658392\pi$$
−0.477320 + 0.878729i $$0.658392\pi$$
$$252$$ 0 0
$$253$$ 8.00481 0.503258
$$254$$ −2.19615 −0.137799
$$255$$ 0 0
$$256$$ −1.39230 −0.0870191
$$257$$ 9.52056 0.593876 0.296938 0.954897i $$-0.404035\pi$$
0.296938 + 0.954897i $$0.404035\pi$$
$$258$$ 0 0
$$259$$ −22.3923 −1.39139
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −2.44949 −0.151330
$$263$$ −4.79744 −0.295823 −0.147912 0.989001i $$-0.547255\pi$$
−0.147912 + 0.989001i $$0.547255\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.14359 −0.254060
$$267$$ 0 0
$$268$$ −7.34847 −0.448879
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ −26.7321 −1.62386 −0.811928 0.583757i $$-0.801582\pi$$
−0.811928 + 0.583757i $$0.801582\pi$$
$$272$$ −3.48477 −0.211295
$$273$$ 0 0
$$274$$ 7.66025 0.462773
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.7637 1.12740 0.563701 0.825979i $$-0.309377\pi$$
0.563701 + 0.825979i $$0.309377\pi$$
$$278$$ 4.14110 0.248367
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −32.7846 −1.95577 −0.977883 0.209153i $$-0.932929\pi$$
−0.977883 + 0.209153i $$0.932929\pi$$
$$282$$ 0 0
$$283$$ −27.4249 −1.63024 −0.815119 0.579293i $$-0.803329\pi$$
−0.815119 + 0.579293i $$0.803329\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 1.17691 0.0695924
$$287$$ −16.9706 −1.00174
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 12.7279 0.744845
$$293$$ −30.7338 −1.79549 −0.897743 0.440520i $$-0.854794\pi$$
−0.897743 + 0.440520i $$0.854794\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 17.6603 1.02648
$$297$$ 0 0
$$298$$ −0.480473 −0.0278331
$$299$$ −11.3205 −0.654682
$$300$$ 0 0
$$301$$ −22.3923 −1.29067
$$302$$ 9.72363 0.559532
$$303$$ 0 0
$$304$$ −8.05256 −0.461846
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 10.4543 0.596658 0.298329 0.954463i $$-0.403571\pi$$
0.298329 + 0.954463i $$0.403571\pi$$
$$308$$ 5.37945 0.306523
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6.58846 0.373597 0.186799 0.982398i $$-0.440189\pi$$
0.186799 + 0.982398i $$0.440189\pi$$
$$312$$ 0 0
$$313$$ −23.6627 −1.33749 −0.668747 0.743490i $$-0.733169\pi$$
−0.668747 + 0.743490i $$0.733169\pi$$
$$314$$ −5.66025 −0.319427
$$315$$ 0 0
$$316$$ 7.26795 0.408854
$$317$$ 5.75839 0.323423 0.161712 0.986838i $$-0.448299\pi$$
0.161712 + 0.986838i $$0.448299\pi$$
$$318$$ 0 0
$$319$$ −1.26795 −0.0709915
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8.00481 0.446091
$$323$$ 4.62158 0.257151
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −7.26795 −0.402534
$$327$$ 0 0
$$328$$ 13.3843 0.739022
$$329$$ 3.46410 0.190982
$$330$$ 0 0
$$331$$ −18.1962 −1.00015 −0.500075 0.865982i $$-0.666694\pi$$
−0.500075 + 0.865982i $$0.666694\pi$$
$$332$$ 17.6269 0.967402
$$333$$ 0 0
$$334$$ 5.80385 0.317572
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −27.9053 −1.52010 −0.760050 0.649864i $$-0.774826\pi$$
−0.760050 + 0.649864i $$0.774826\pi$$
$$338$$ 5.06489 0.275494
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −11.0718 −0.599571
$$342$$ 0 0
$$343$$ 19.5959 1.05808
$$344$$ 17.6603 0.952177
$$345$$ 0 0
$$346$$ 2.39230 0.128611
$$347$$ −21.4906 −1.15368 −0.576838 0.816859i $$-0.695714\pi$$
−0.576838 + 0.816859i $$0.695714\pi$$
$$348$$ 0 0
$$349$$ −22.7846 −1.21963 −0.609816 0.792543i $$-0.708757\pi$$
−0.609816 + 0.792543i $$0.708757\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −6.51626 −0.347318
$$353$$ 31.9449 1.70026 0.850128 0.526576i $$-0.176525\pi$$
0.850128 + 0.526576i $$0.176525\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 18.0000 0.953998
$$357$$ 0 0
$$358$$ 1.31268 0.0693772
$$359$$ 7.26795 0.383588 0.191794 0.981435i $$-0.438570\pi$$
0.191794 + 0.981435i $$0.438570\pi$$
$$360$$ 0 0
$$361$$ −8.32051 −0.437921
$$362$$ 8.28221 0.435303
$$363$$ 0 0
$$364$$ −7.60770 −0.398752
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −33.6365 −1.75581 −0.877906 0.478833i $$-0.841060\pi$$
−0.877906 + 0.478833i $$0.841060\pi$$
$$368$$ 15.5563 0.810931
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 14.5359 0.754666
$$372$$ 0 0
$$373$$ −19.5959 −1.01464 −0.507319 0.861758i $$-0.669363\pi$$
−0.507319 + 0.861758i $$0.669363\pi$$
$$374$$ 0.928203 0.0479962
$$375$$ 0 0
$$376$$ −2.73205 −0.140895
$$377$$ 1.79315 0.0923520
$$378$$ 0 0
$$379$$ −16.1962 −0.831940 −0.415970 0.909378i $$-0.636558\pi$$
−0.415970 + 0.909378i $$0.636558\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 7.82894 0.400564
$$383$$ 0.933740 0.0477119 0.0238559 0.999715i $$-0.492406\pi$$
0.0238559 + 0.999715i $$0.492406\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0.339746 0.0172926
$$387$$ 0 0
$$388$$ 18.9396 0.961511
$$389$$ 20.7846 1.05382 0.526911 0.849921i $$-0.323350\pi$$
0.526911 + 0.849921i $$0.323350\pi$$
$$390$$ 0 0
$$391$$ −8.92820 −0.451519
$$392$$ −1.93185 −0.0975732
$$393$$ 0 0
$$394$$ −11.6077 −0.584787
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4.89898 0.245873 0.122936 0.992415i $$-0.460769\pi$$
0.122936 + 0.992415i $$0.460769\pi$$
$$398$$ 7.24693 0.363256
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −37.8564 −1.89046 −0.945229 0.326407i $$-0.894162\pi$$
−0.945229 + 0.326407i $$0.894162\pi$$
$$402$$ 0 0
$$403$$ 15.6579 0.779975
$$404$$ −10.3923 −0.517036
$$405$$ 0 0
$$406$$ −1.26795 −0.0629273
$$407$$ 11.5911 0.574550
$$408$$ 0 0
$$409$$ 9.07180 0.448571 0.224286 0.974523i $$-0.427995\pi$$
0.224286 + 0.974523i $$0.427995\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −15.8338 −0.780073
$$413$$ −25.4558 −1.25260
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 9.21539 0.451822
$$417$$ 0 0
$$418$$ 2.14488 0.104910
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ 5.46410 0.266304 0.133152 0.991096i $$-0.457490\pi$$
0.133152 + 0.991096i $$0.457490\pi$$
$$422$$ 7.62587 0.371222
$$423$$ 0 0
$$424$$ −11.4641 −0.556746
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −7.17260 −0.347107
$$428$$ −32.3238 −1.56243
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 37.1769 1.79075 0.895374 0.445314i $$-0.146908\pi$$
0.895374 + 0.445314i $$0.146908\pi$$
$$432$$ 0 0
$$433$$ −2.92996 −0.140805 −0.0704025 0.997519i $$-0.522428\pi$$
−0.0704025 + 0.997519i $$0.522428\pi$$
$$434$$ −11.0718 −0.531463
$$435$$ 0 0
$$436$$ −30.9282 −1.48119
$$437$$ −20.6312 −0.986924
$$438$$ 0 0
$$439$$ −7.07180 −0.337518 −0.168759 0.985657i $$-0.553976\pi$$
−0.168759 + 0.985657i $$0.553976\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −1.31268 −0.0624377
$$443$$ −18.1817 −0.863839 −0.431919 0.901912i $$-0.642164\pi$$
−0.431919 + 0.901912i $$0.642164\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 7.94744 0.376322
$$447$$ 0 0
$$448$$ 5.55532 0.262464
$$449$$ 25.8564 1.22024 0.610120 0.792309i $$-0.291121\pi$$
0.610120 + 0.792309i $$0.291121\pi$$
$$450$$ 0 0
$$451$$ 8.78461 0.413651
$$452$$ −22.5259 −1.05953
$$453$$ 0 0
$$454$$ −10.5885 −0.496941
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −23.6627 −1.10689 −0.553447 0.832884i $$-0.686688\pi$$
−0.553447 + 0.832884i $$0.686688\pi$$
$$458$$ −9.72363 −0.454355
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −24.9282 −1.16102 −0.580511 0.814252i $$-0.697147\pi$$
−0.580511 + 0.814252i $$0.697147\pi$$
$$462$$ 0 0
$$463$$ 6.86800 0.319183 0.159591 0.987183i $$-0.448982\pi$$
0.159591 + 0.987183i $$0.448982\pi$$
$$464$$ −2.46410 −0.114393
$$465$$ 0 0
$$466$$ −1.75129 −0.0811269
$$467$$ −21.0101 −0.972233 −0.486116 0.873894i $$-0.661587\pi$$
−0.486116 + 0.873894i $$0.661587\pi$$
$$468$$ 0 0
$$469$$ −10.3923 −0.479872
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 20.0764 0.924091
$$473$$ 11.5911 0.532960
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ 0 0
$$478$$ 5.37945 0.246050
$$479$$ −5.66025 −0.258624 −0.129312 0.991604i $$-0.541277\pi$$
−0.129312 + 0.991604i $$0.541277\pi$$
$$480$$ 0 0
$$481$$ −16.3923 −0.747425
$$482$$ −5.93426 −0.270298
$$483$$ 0 0
$$484$$ 16.2679 0.739452
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −10.9348 −0.495502 −0.247751 0.968824i $$-0.579691\pi$$
−0.247751 + 0.968824i $$0.579691\pi$$
$$488$$ 5.65685 0.256074
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7.26795 −0.327998 −0.163999 0.986461i $$-0.552439\pi$$
−0.163999 + 0.986461i $$0.552439\pi$$
$$492$$ 0 0
$$493$$ 1.41421 0.0636930
$$494$$ −3.03332 −0.136476
$$495$$ 0 0
$$496$$ −21.5167 −0.966127
$$497$$ 8.48528 0.380617
$$498$$ 0 0
$$499$$ −1.07180 −0.0479802 −0.0239901 0.999712i $$-0.507637\pi$$
−0.0239901 + 0.999712i $$0.507637\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 7.82894 0.349423
$$503$$ 12.1731 0.542773 0.271386 0.962471i $$-0.412518\pi$$
0.271386 + 0.962471i $$0.412518\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4.14359 −0.184205
$$507$$ 0 0
$$508$$ −7.34847 −0.326036
$$509$$ −33.4641 −1.48327 −0.741635 0.670804i $$-0.765949\pi$$
−0.741635 + 0.670804i $$0.765949\pi$$
$$510$$ 0 0
$$511$$ 18.0000 0.796273
$$512$$ −22.1841 −0.980408
$$513$$ 0 0
$$514$$ −4.92820 −0.217374
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.79315 −0.0788627
$$518$$ 11.5911 0.509284
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4.39230 −0.192430 −0.0962152 0.995361i $$-0.530674\pi$$
−0.0962152 + 0.995361i $$0.530674\pi$$
$$522$$ 0 0
$$523$$ 1.61729 0.0707190 0.0353595 0.999375i $$-0.488742\pi$$
0.0353595 + 0.999375i $$0.488742\pi$$
$$524$$ −8.19615 −0.358051
$$525$$ 0 0
$$526$$ 2.48334 0.108279
$$527$$ 12.3490 0.537930
$$528$$ 0 0
$$529$$ 16.8564 0.732887
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −13.8647 −0.601112
$$533$$ −12.4233 −0.538113
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.19615 0.354020
$$537$$ 0 0
$$538$$ −6.21166 −0.267804
$$539$$ −1.26795 −0.0546144
$$540$$ 0 0
$$541$$ 42.3923 1.82259 0.911294 0.411757i $$-0.135085\pi$$
0.911294 + 0.411757i $$0.135085\pi$$
$$542$$ 13.8375 0.594373
$$543$$ 0 0
$$544$$ 7.26795 0.311611
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −2.44949 −0.104733 −0.0523663 0.998628i $$-0.516676\pi$$
−0.0523663 + 0.998628i $$0.516676\pi$$
$$548$$ 25.6317 1.09493
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.26795 0.139219
$$552$$ 0 0
$$553$$ 10.2784 0.437083
$$554$$ −9.71281 −0.412658
$$555$$ 0 0
$$556$$ 13.8564 0.587643
$$557$$ 0.554803 0.0235078 0.0117539 0.999931i $$-0.496259\pi$$
0.0117539 + 0.999931i $$0.496259\pi$$
$$558$$ 0 0
$$559$$ −16.3923 −0.693321
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 16.9706 0.715860
$$563$$ −13.2827 −0.559800 −0.279900 0.960029i $$-0.590301\pi$$
−0.279900 + 0.960029i $$0.590301\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 14.1962 0.596709
$$567$$ 0 0
$$568$$ −6.69213 −0.280796
$$569$$ −21.4641 −0.899822 −0.449911 0.893073i $$-0.648544\pi$$
−0.449911 + 0.893073i $$0.648544\pi$$
$$570$$ 0 0
$$571$$ −39.3205 −1.64551 −0.822756 0.568395i $$-0.807565\pi$$
−0.822756 + 0.568395i $$0.807565\pi$$
$$572$$ 3.93803 0.164657
$$573$$ 0 0
$$574$$ 8.78461 0.366663
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 9.14162 0.380571 0.190285 0.981729i $$-0.439059\pi$$
0.190285 + 0.981729i $$0.439059\pi$$
$$578$$ 7.76457 0.322964
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.9282 1.03420
$$582$$ 0 0
$$583$$ −7.52433 −0.311626
$$584$$ −14.1962 −0.587441
$$585$$ 0 0
$$586$$ 15.9090 0.657193
$$587$$ 15.5563 0.642079 0.321040 0.947066i $$-0.395968\pi$$
0.321040 + 0.947066i $$0.395968\pi$$
$$588$$ 0 0
$$589$$ 28.5359 1.17580
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 22.5259 0.925808
$$593$$ −25.5302 −1.04840 −0.524199 0.851596i $$-0.675635\pi$$
−0.524199 + 0.851596i $$0.675635\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.60770 −0.0658538
$$597$$ 0 0
$$598$$ 5.85993 0.239630
$$599$$ −1.26795 −0.0518070 −0.0259035 0.999664i $$-0.508246\pi$$
−0.0259035 + 0.999664i $$0.508246\pi$$
$$600$$ 0 0
$$601$$ 16.7846 0.684659 0.342329 0.939580i $$-0.388784\pi$$
0.342329 + 0.939580i $$0.388784\pi$$
$$602$$ 11.5911 0.472418
$$603$$ 0 0
$$604$$ 32.5359 1.32387
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.3142 0.662174 0.331087 0.943600i $$-0.392585\pi$$
0.331087 + 0.943600i $$0.392585\pi$$
$$608$$ 16.7947 0.681115
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2.53590 0.102591
$$612$$ 0 0
$$613$$ 18.2832 0.738453 0.369227 0.929339i $$-0.379623\pi$$
0.369227 + 0.929339i $$0.379623\pi$$
$$614$$ −5.41154 −0.218392
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ 32.0464 1.29014 0.645071 0.764123i $$-0.276828\pi$$
0.645071 + 0.764123i $$0.276828\pi$$
$$618$$ 0 0
$$619$$ −23.8038 −0.956757 −0.478379 0.878154i $$-0.658775\pi$$
−0.478379 + 0.878154i $$0.658775\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −3.41044 −0.136746
$$623$$ 25.4558 1.01987
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 12.2487 0.489557
$$627$$ 0 0
$$628$$ −18.9396 −0.755771
$$629$$ −12.9282 −0.515481
$$630$$ 0 0
$$631$$ 15.6077 0.621333 0.310666 0.950519i $$-0.399448\pi$$
0.310666 + 0.950519i $$0.399448\pi$$
$$632$$ −8.10634 −0.322453
$$633$$ 0 0
$$634$$ −2.98076 −0.118381
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.79315 0.0710472
$$638$$ 0.656339 0.0259847
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −34.6410 −1.36824 −0.684119 0.729370i $$-0.739813\pi$$
−0.684119 + 0.729370i $$0.739813\pi$$
$$642$$ 0 0
$$643$$ −10.4543 −0.412277 −0.206139 0.978523i $$-0.566090\pi$$
−0.206139 + 0.978523i $$0.566090\pi$$
$$644$$ 26.7846 1.05546
$$645$$ 0 0
$$646$$ −2.39230 −0.0941240
$$647$$ −14.3180 −0.562899 −0.281449 0.959576i $$-0.590815\pi$$
−0.281449 + 0.959576i $$0.590815\pi$$
$$648$$ 0 0
$$649$$ 13.1769 0.517239
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −24.3190 −0.952407
$$653$$ 13.0053 0.508938 0.254469 0.967081i $$-0.418099\pi$$
0.254469 + 0.967081i $$0.418099\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 17.0718 0.666542
$$657$$ 0 0
$$658$$ −1.79315 −0.0699043
$$659$$ −32.4449 −1.26387 −0.631936 0.775020i $$-0.717740\pi$$
−0.631936 + 0.775020i $$0.717740\pi$$
$$660$$ 0 0
$$661$$ −0.784610 −0.0305178 −0.0152589 0.999884i $$-0.504857\pi$$
−0.0152589 + 0.999884i $$0.504857\pi$$
$$662$$ 9.41902 0.366081
$$663$$ 0 0
$$664$$ −19.6603 −0.762966
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.31319 −0.244448
$$668$$ 19.4201 0.751384
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 3.71281 0.143332
$$672$$ 0 0
$$673$$ 46.3644 1.78722 0.893609 0.448846i $$-0.148165\pi$$
0.893609 + 0.448846i $$0.148165\pi$$
$$674$$ 14.4449 0.556395
$$675$$ 0 0
$$676$$ 16.9474 0.651825
$$677$$ −32.2495 −1.23945 −0.619725 0.784819i $$-0.712756\pi$$
−0.619725 + 0.784819i $$0.712756\pi$$
$$678$$ 0 0
$$679$$ 26.7846 1.02790
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 5.73118 0.219458
$$683$$ 21.9711 0.840700 0.420350 0.907362i $$-0.361907\pi$$
0.420350 + 0.907362i $$0.361907\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −10.1436 −0.387284
$$687$$ 0 0
$$688$$ 22.5259 0.858791
$$689$$ 10.6410 0.405390
$$690$$ 0 0
$$691$$ 41.7128 1.58683 0.793415 0.608681i $$-0.208301\pi$$
0.793415 + 0.608681i $$0.208301\pi$$
$$692$$ 8.00481 0.304297
$$693$$ 0 0
$$694$$ 11.1244 0.422275
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −9.79796 −0.371124
$$698$$ 11.7942 0.446416
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −27.7128 −1.04670 −0.523349 0.852118i $$-0.675318\pi$$
−0.523349 + 0.852118i $$0.675318\pi$$
$$702$$ 0 0
$$703$$ −29.8744 −1.12673
$$704$$ −2.87564 −0.108380
$$705$$ 0 0
$$706$$ −16.5359 −0.622337
$$707$$ −14.6969 −0.552735
$$708$$ 0 0
$$709$$ 20.3923 0.765849 0.382925 0.923780i $$-0.374917\pi$$
0.382925 + 0.923780i $$0.374917\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −20.0764 −0.752395
$$713$$ −55.1271 −2.06453
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.39230 0.164148
$$717$$ 0 0
$$718$$ −3.76217 −0.140403
$$719$$ 19.8564 0.740519 0.370260 0.928928i $$-0.379269\pi$$
0.370260 + 0.928928i $$0.379269\pi$$
$$720$$ 0 0
$$721$$ −22.3923 −0.833933
$$722$$ 4.30701 0.160290
$$723$$ 0 0
$$724$$ 27.7128 1.02994
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −46.6690 −1.73086 −0.865430 0.501031i $$-0.832954\pi$$
−0.865430 + 0.501031i $$0.832954\pi$$
$$728$$ 8.48528 0.314485
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −12.9282 −0.478167
$$732$$ 0 0
$$733$$ 12.7279 0.470117 0.235058 0.971981i $$-0.424472\pi$$
0.235058 + 0.971981i $$0.424472\pi$$
$$734$$ 17.4115 0.642672
$$735$$ 0 0
$$736$$ −32.4449 −1.19593
$$737$$ 5.37945 0.198155
$$738$$ 0 0
$$739$$ −18.9808 −0.698219 −0.349109 0.937082i $$-0.613516\pi$$
−0.349109 + 0.937082i $$0.613516\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −7.52433 −0.276227
$$743$$ 9.89949 0.363177 0.181589 0.983375i $$-0.441876\pi$$
0.181589 + 0.983375i $$0.441876\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 10.1436 0.371383
$$747$$ 0 0
$$748$$ 3.10583 0.113560
$$749$$ −45.7128 −1.67031
$$750$$ 0 0
$$751$$ 49.9090 1.82120 0.910602 0.413284i $$-0.135618\pi$$
0.910602 + 0.413284i $$0.135618\pi$$
$$752$$ −3.48477 −0.127076
$$753$$ 0 0
$$754$$ −0.928203 −0.0338032
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.9348 0.397431 0.198716 0.980057i $$-0.436323\pi$$
0.198716 + 0.980057i $$0.436323\pi$$
$$758$$ 8.38375 0.304511
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 35.5692 1.28938 0.644692 0.764443i $$-0.276986\pi$$
0.644692 + 0.764443i $$0.276986\pi$$
$$762$$ 0 0
$$763$$ −43.7391 −1.58346
$$764$$ 26.1962 0.947744
$$765$$ 0 0
$$766$$ −0.483340 −0.0174638
$$767$$ −18.6350 −0.672870
$$768$$ 0 0
$$769$$ −19.0718 −0.687747 −0.343873 0.939016i $$-0.611739\pi$$
−0.343873 + 0.939016i $$0.611739\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.13681 0.0409148
$$773$$ 11.2122 0.403274 0.201637 0.979460i $$-0.435374\pi$$
0.201637 + 0.979460i $$0.435374\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −21.1244 −0.758320
$$777$$ 0 0
$$778$$ −10.7589 −0.385725
$$779$$ −22.6410 −0.811199
$$780$$ 0 0
$$781$$ −4.39230 −0.157169
$$782$$ 4.62158 0.165267
$$783$$ 0 0
$$784$$ −2.46410 −0.0880036
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 35.0779 1.25039 0.625197 0.780467i $$-0.285019\pi$$
0.625197 + 0.780467i $$0.285019\pi$$
$$788$$ −38.8401 −1.38362
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −31.8564 −1.13268
$$792$$ 0 0
$$793$$ −5.25071 −0.186458
$$794$$ −2.53590 −0.0899957
$$795$$ 0 0
$$796$$ 24.2487 0.859473
$$797$$ −29.0149 −1.02776 −0.513881 0.857862i $$-0.671793\pi$$
−0.513881 + 0.857862i $$0.671793\pi$$
$$798$$ 0 0
$$799$$ 2.00000 0.0707549
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 19.5959 0.691956
$$803$$ −9.31749 −0.328807
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.10512 −0.285491
$$807$$ 0 0
$$808$$ 11.5911 0.407774
$$809$$ −13.6077 −0.478421 −0.239211 0.970968i $$-0.576889\pi$$
−0.239211 + 0.970968i $$0.576889\pi$$
$$810$$ 0 0
$$811$$ −36.7846 −1.29168 −0.645841 0.763472i $$-0.723493\pi$$
−0.645841 + 0.763472i $$0.723493\pi$$
$$812$$ −4.24264 −0.148888
$$813$$ 0 0
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −29.8744 −1.04517
$$818$$ −4.69591 −0.164189
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −9.21539 −0.321619 −0.160810 0.986985i $$-0.551411\pi$$
−0.160810 + 0.986985i $$0.551411\pi$$
$$822$$ 0 0
$$823$$ −40.8091 −1.42252 −0.711258 0.702931i $$-0.751874\pi$$
−0.711258 + 0.702931i $$0.751874\pi$$
$$824$$ 17.6603 0.615224
$$825$$ 0 0
$$826$$ 13.1769 0.458483
$$827$$ −3.68784 −0.128239 −0.0641193 0.997942i $$-0.520424\pi$$
−0.0641193 + 0.997942i $$0.520424\pi$$
$$828$$ 0 0
$$829$$ −16.7846 −0.582954 −0.291477 0.956578i $$-0.594147\pi$$
−0.291477 + 0.956578i $$0.594147\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 4.06678 0.140990
$$833$$ 1.41421 0.0489996
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 7.17691 0.248219
$$837$$ 0 0
$$838$$ −3.10583 −0.107289
$$839$$ −40.9808 −1.41481 −0.707407 0.706807i $$-0.750135\pi$$
−0.707407 + 0.706807i $$0.750135\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −2.82843 −0.0974740
$$843$$ 0 0
$$844$$ 25.5167 0.878320
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 23.0064 0.790508
$$848$$ −14.6226 −0.502142
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 57.7128 1.97837
$$852$$ 0 0
$$853$$ 32.3238 1.10675 0.553374 0.832933i $$-0.313340\pi$$
0.553374 + 0.832933i $$0.313340\pi$$
$$854$$ 3.71281 0.127050
$$855$$ 0 0
$$856$$ 36.0526 1.23225
$$857$$ −33.2576 −1.13606 −0.568029 0.823009i $$-0.692294\pi$$
−0.568029 + 0.823009i $$0.692294\pi$$
$$858$$ 0 0
$$859$$ −18.7321 −0.639129 −0.319565 0.947564i $$-0.603537\pi$$
−0.319565 + 0.947564i $$0.603537\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −19.2442 −0.655460
$$863$$ −1.69161 −0.0575832 −0.0287916 0.999585i $$-0.509166\pi$$
−0.0287916 + 0.999585i $$0.509166\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 1.51666 0.0515382
$$867$$ 0 0
$$868$$ −37.0470 −1.25746
$$869$$ −5.32051 −0.180486
$$870$$ 0 0
$$871$$ −7.60770 −0.257777
$$872$$ 34.4959 1.16818
$$873$$ 0 0
$$874$$ 10.6795 0.361239
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 24.9754 0.843358 0.421679 0.906745i $$-0.361441\pi$$
0.421679 + 0.906745i $$0.361441\pi$$
$$878$$ 3.66063 0.123540
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 50.5359 1.70260 0.851299 0.524681i $$-0.175815\pi$$
0.851299 + 0.524681i $$0.175815\pi$$
$$882$$ 0 0
$$883$$ 1.48854 0.0500935 0.0250467 0.999686i $$-0.492027\pi$$
0.0250467 + 0.999686i $$0.492027\pi$$
$$884$$ −4.39230 −0.147729
$$885$$ 0 0
$$886$$ 9.41154 0.316187
$$887$$ −21.0101 −0.705451 −0.352726 0.935727i $$-0.614745\pi$$
−0.352726 + 0.935727i $$0.614745\pi$$
$$888$$ 0 0
$$889$$ −10.3923 −0.348547
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 26.5927 0.890388
$$893$$ 4.62158 0.154655
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −28.0526 −0.937170
$$897$$ 0 0
$$898$$ −13.3843 −0.446639
$$899$$ 8.73205 0.291230
$$900$$ 0 0
$$901$$ 8.39230 0.279588
$$902$$ −4.54725 −0.151407
$$903$$ 0 0
$$904$$ 25.1244 0.835624
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −45.7081 −1.51771 −0.758856 0.651258i $$-0.774242\pi$$
−0.758856 + 0.651258i $$0.774242\pi$$
$$908$$ −35.4297 −1.17577
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9.80385 0.324816 0.162408 0.986724i $$-0.448074\pi$$
0.162408 + 0.986724i $$0.448074\pi$$
$$912$$ 0 0
$$913$$ −12.9038 −0.427053
$$914$$ 12.2487 0.405151
$$915$$ 0 0
$$916$$ −32.5359 −1.07502
$$917$$ −11.5911 −0.382772
$$918$$ 0 0
$$919$$ 43.9615 1.45016 0.725078 0.688666i $$-0.241803\pi$$
0.725078 + 0.688666i $$0.241803\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 12.9038 0.424964
$$923$$ 6.21166 0.204459
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −3.55514 −0.116829
$$927$$ 0 0
$$928$$ 5.13922 0.168703
$$929$$ −55.8564 −1.83259 −0.916295 0.400505i $$-0.868835\pi$$
−0.916295 + 0.400505i $$0.868835\pi$$
$$930$$ 0 0
$$931$$ 3.26795 0.107103
$$932$$ −5.85993 −0.191948
$$933$$ 0 0
$$934$$ 10.8756 0.355862
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.75410 0.0899724 0.0449862 0.998988i $$-0.485676\pi$$
0.0449862 + 0.998988i $$0.485676\pi$$
$$938$$ 5.37945 0.175645
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 7.85641 0.256112 0.128056 0.991767i $$-0.459126\pi$$
0.128056 + 0.991767i $$0.459126\pi$$
$$942$$ 0 0
$$943$$ 43.7391 1.42434
$$944$$ 25.6077 0.833459
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ 27.9797 0.909217 0.454608 0.890691i $$-0.349779\pi$$
0.454608 + 0.890691i $$0.349779\pi$$
$$948$$ 0 0
$$949$$ 13.1769 0.427741
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 6.69213 0.216893
$$953$$ 5.65685 0.183243 0.0916217 0.995794i $$-0.470795\pi$$
0.0916217 + 0.995794i $$0.470795\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 18.0000 0.582162
$$957$$ 0 0
$$958$$ 2.92996 0.0946628
$$959$$ 36.2487 1.17053
$$960$$ 0 0
$$961$$ 45.2487 1.45964
$$962$$ 8.48528 0.273576
$$963$$ 0 0
$$964$$ −19.8564 −0.639532
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1.96902 −0.0633193 −0.0316596 0.999499i $$-0.510079\pi$$
−0.0316596 + 0.999499i $$0.510079\pi$$
$$968$$ −18.1445 −0.583188
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 49.5167 1.58907 0.794533 0.607221i $$-0.207716\pi$$
0.794533 + 0.607221i $$0.207716\pi$$
$$972$$ 0 0
$$973$$ 19.5959 0.628216
$$974$$ 5.66025 0.181366
$$975$$ 0 0
$$976$$ 7.21539 0.230959
$$977$$ 22.9048 0.732790 0.366395 0.930459i $$-0.380592\pi$$
0.366395 + 0.930459i $$0.380592\pi$$
$$978$$ 0 0
$$979$$ −13.1769 −0.421136
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 3.76217 0.120056
$$983$$ −30.2533 −0.964930 −0.482465 0.875915i $$-0.660258\pi$$
−0.482465 + 0.875915i $$0.660258\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −0.732051 −0.0233132
$$987$$ 0 0
$$988$$ −10.1497 −0.322905
$$989$$ 57.7128 1.83516
$$990$$ 0 0
$$991$$ 34.7846 1.10497 0.552485 0.833523i $$-0.313680\pi$$
0.552485 + 0.833523i $$0.313680\pi$$
$$992$$ 44.8759 1.42481
$$993$$ 0 0
$$994$$ −4.39230 −0.139315
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 26.5927 0.842198 0.421099 0.907015i $$-0.361645\pi$$
0.421099 + 0.907015i $$0.361645\pi$$
$$998$$ 0.554803 0.0175620
$$999$$ 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.bj.1.2 4
3.2 odd 2 725.2.a.f.1.3 4
5.2 odd 4 1305.2.c.f.784.2 4
5.3 odd 4 1305.2.c.f.784.3 4
5.4 even 2 inner 6525.2.a.bj.1.3 4
15.2 even 4 145.2.b.b.59.3 yes 4
15.8 even 4 145.2.b.b.59.2 4
15.14 odd 2 725.2.a.f.1.2 4
60.23 odd 4 2320.2.d.f.929.1 4
60.47 odd 4 2320.2.d.f.929.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.b.59.2 4 15.8 even 4
145.2.b.b.59.3 yes 4 15.2 even 4
725.2.a.f.1.2 4 15.14 odd 2
725.2.a.f.1.3 4 3.2 odd 2
1305.2.c.f.784.2 4 5.2 odd 4
1305.2.c.f.784.3 4 5.3 odd 4
2320.2.d.f.929.1 4 60.23 odd 4
2320.2.d.f.929.3 4 60.47 odd 4
6525.2.a.bj.1.2 4 1.1 even 1 trivial
6525.2.a.bj.1.3 4 5.4 even 2 inner