# Properties

 Label 1904.2.a.m.1.3 Level $1904$ Weight $2$ Character 1904.1 Self dual yes Analytic conductor $15.204$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1904,2,Mod(1,1904)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1904, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1904 = 2^{4} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1904.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: $$15.2035165449$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 952) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.11491$$ of defining polynomial Character $$\chi$$ $$=$$ 1904.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +O(q^{10})$$ $$q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +2.22982 q^{11} -1.64207 q^{15} -1.00000 q^{17} -4.94567 q^{19} -1.47283 q^{21} -9.17548 q^{23} -3.75698 q^{25} -5.64207 q^{27} +0.945668 q^{29} +1.11491 q^{31} +3.28415 q^{33} +1.11491 q^{35} -4.94567 q^{37} +10.2709 q^{41} +11.2166 q^{43} +0.926221 q^{45} -9.89134 q^{47} +1.00000 q^{49} -1.47283 q^{51} +7.83076 q^{53} -2.48604 q^{55} -7.28415 q^{57} -13.7438 q^{59} +0.904539 q^{61} +0.830760 q^{63} -9.91302 q^{67} -13.5140 q^{69} -14.6894 q^{71} +0.756981 q^{73} -5.53341 q^{75} -2.22982 q^{77} -8.00000 q^{79} -5.81756 q^{81} +11.9736 q^{83} +1.11491 q^{85} +1.39281 q^{87} +11.1755 q^{89} +1.64207 q^{93} +5.51396 q^{95} +10.0606 q^{97} -1.85244 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10})$$ 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 $$3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43} - 8 q^{47} + 3 q^{49} + q^{51} + 19 q^{53} - 22 q^{55} - 20 q^{57} - 14 q^{59} + q^{61} - 2 q^{63} - 7 q^{67} - 26 q^{69} - 6 q^{71} - 5 q^{73} + 6 q^{75} + 6 q^{77} - 24 q^{79} + 7 q^{81} - 4 q^{83} - 3 q^{85} + 24 q^{87} + 10 q^{89} + 4 q^{93} + 2 q^{95} + 13 q^{97}+O(q^{100})$$ 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^15 - 3 * q^17 - 4 * q^19 + q^21 - 4 * q^23 - 4 * q^25 - 16 * q^27 - 8 * q^29 - 3 * q^31 + 8 * q^33 - 3 * q^35 - 4 * q^37 + 9 * q^41 + q^43 - 8 * q^47 + 3 * q^49 + q^51 + 19 * q^53 - 22 * q^55 - 20 * q^57 - 14 * q^59 + q^61 - 2 * q^63 - 7 * q^67 - 26 * q^69 - 6 * q^71 - 5 * q^73 + 6 * q^75 + 6 * q^77 - 24 * q^79 + 7 * q^81 - 4 * q^83 - 3 * q^85 + 24 * q^87 + 10 * q^89 + 4 * q^93 + 2 * q^95 + 13 * q^97

## 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 0
$$3$$ 1.47283 0.850341 0.425171 0.905113i $$-0.360214\pi$$
0.425171 + 0.905113i $$0.360214\pi$$
$$4$$ 0 0
$$5$$ −1.11491 −0.498602 −0.249301 0.968426i $$-0.580201\pi$$
−0.249301 + 0.968426i $$0.580201\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −0.830760 −0.276920
$$10$$ 0 0
$$11$$ 2.22982 0.672315 0.336157 0.941806i $$-0.390873\pi$$
0.336157 + 0.941806i $$0.390873\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −1.64207 −0.423982
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ −4.94567 −1.13461 −0.567307 0.823506i $$-0.692015\pi$$
−0.567307 + 0.823506i $$0.692015\pi$$
$$20$$ 0 0
$$21$$ −1.47283 −0.321399
$$22$$ 0 0
$$23$$ −9.17548 −1.91322 −0.956610 0.291371i $$-0.905889\pi$$
−0.956610 + 0.291371i $$0.905889\pi$$
$$24$$ 0 0
$$25$$ −3.75698 −0.751396
$$26$$ 0 0
$$27$$ −5.64207 −1.08582
$$28$$ 0 0
$$29$$ 0.945668 0.175606 0.0878031 0.996138i $$-0.472015\pi$$
0.0878031 + 0.996138i $$0.472015\pi$$
$$30$$ 0 0
$$31$$ 1.11491 0.200243 0.100122 0.994975i $$-0.468077\pi$$
0.100122 + 0.994975i $$0.468077\pi$$
$$32$$ 0 0
$$33$$ 3.28415 0.571697
$$34$$ 0 0
$$35$$ 1.11491 0.188454
$$36$$ 0 0
$$37$$ −4.94567 −0.813063 −0.406531 0.913637i $$-0.633262\pi$$
−0.406531 + 0.913637i $$0.633262\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.2709 1.60405 0.802026 0.597289i $$-0.203756\pi$$
0.802026 + 0.597289i $$0.203756\pi$$
$$42$$ 0 0
$$43$$ 11.2166 1.71052 0.855259 0.518201i $$-0.173398\pi$$
0.855259 + 0.518201i $$0.173398\pi$$
$$44$$ 0 0
$$45$$ 0.926221 0.138073
$$46$$ 0 0
$$47$$ −9.89134 −1.44280 −0.721400 0.692519i $$-0.756501\pi$$
−0.721400 + 0.692519i $$0.756501\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −1.47283 −0.206238
$$52$$ 0 0
$$53$$ 7.83076 1.07564 0.537819 0.843060i $$-0.319248\pi$$
0.537819 + 0.843060i $$0.319248\pi$$
$$54$$ 0 0
$$55$$ −2.48604 −0.335217
$$56$$ 0 0
$$57$$ −7.28415 −0.964809
$$58$$ 0 0
$$59$$ −13.7438 −1.78929 −0.894644 0.446780i $$-0.852571\pi$$
−0.894644 + 0.446780i $$0.852571\pi$$
$$60$$ 0 0
$$61$$ 0.904539 0.115814 0.0579072 0.998322i $$-0.481557\pi$$
0.0579072 + 0.998322i $$0.481557\pi$$
$$62$$ 0 0
$$63$$ 0.830760 0.104666
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.91302 −1.21107 −0.605534 0.795820i $$-0.707040\pi$$
−0.605534 + 0.795820i $$0.707040\pi$$
$$68$$ 0 0
$$69$$ −13.5140 −1.62689
$$70$$ 0 0
$$71$$ −14.6894 −1.74332 −0.871658 0.490114i $$-0.836955\pi$$
−0.871658 + 0.490114i $$0.836955\pi$$
$$72$$ 0 0
$$73$$ 0.756981 0.0885979 0.0442990 0.999018i $$-0.485895\pi$$
0.0442990 + 0.999018i $$0.485895\pi$$
$$74$$ 0 0
$$75$$ −5.53341 −0.638943
$$76$$ 0 0
$$77$$ −2.22982 −0.254111
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −5.81756 −0.646395
$$82$$ 0 0
$$83$$ 11.9736 1.31427 0.657136 0.753772i $$-0.271768\pi$$
0.657136 + 0.753772i $$0.271768\pi$$
$$84$$ 0 0
$$85$$ 1.11491 0.120929
$$86$$ 0 0
$$87$$ 1.39281 0.149325
$$88$$ 0 0
$$89$$ 11.1755 1.18460 0.592299 0.805718i $$-0.298220\pi$$
0.592299 + 0.805718i $$0.298220\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 1.64207 0.170275
$$94$$ 0 0
$$95$$ 5.51396 0.565721
$$96$$ 0 0
$$97$$ 10.0606 1.02150 0.510748 0.859730i $$-0.329368\pi$$
0.510748 + 0.859730i $$0.329368\pi$$
$$98$$ 0 0
$$99$$ −1.85244 −0.186177
$$100$$ 0 0
$$101$$ −18.6894 −1.85967 −0.929835 0.367978i $$-0.880050\pi$$
−0.929835 + 0.367978i $$0.880050\pi$$
$$102$$ 0 0
$$103$$ −12.1212 −1.19433 −0.597166 0.802118i $$-0.703707\pi$$
−0.597166 + 0.802118i $$0.703707\pi$$
$$104$$ 0 0
$$105$$ 1.64207 0.160250
$$106$$ 0 0
$$107$$ 1.89134 0.182842 0.0914212 0.995812i $$-0.470859\pi$$
0.0914212 + 0.995812i $$0.470859\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −7.28415 −0.691381
$$112$$ 0 0
$$113$$ −3.85244 −0.362407 −0.181204 0.983446i $$-0.557999\pi$$
−0.181204 + 0.983446i $$0.557999\pi$$
$$114$$ 0 0
$$115$$ 10.2298 0.953935
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ −6.02792 −0.547993
$$122$$ 0 0
$$123$$ 15.1274 1.36399
$$124$$ 0 0
$$125$$ 9.76322 0.873249
$$126$$ 0 0
$$127$$ 21.2361 1.88440 0.942198 0.335057i $$-0.108756\pi$$
0.942198 + 0.335057i $$0.108756\pi$$
$$128$$ 0 0
$$129$$ 16.5202 1.45452
$$130$$ 0 0
$$131$$ 4.60719 0.402532 0.201266 0.979537i $$-0.435494\pi$$
0.201266 + 0.979537i $$0.435494\pi$$
$$132$$ 0 0
$$133$$ 4.94567 0.428844
$$134$$ 0 0
$$135$$ 6.29039 0.541391
$$136$$ 0 0
$$137$$ 0.379608 0.0324321 0.0162160 0.999869i $$-0.494838\pi$$
0.0162160 + 0.999869i $$0.494838\pi$$
$$138$$ 0 0
$$139$$ −6.06058 −0.514051 −0.257026 0.966405i $$-0.582742\pi$$
−0.257026 + 0.966405i $$0.582742\pi$$
$$140$$ 0 0
$$141$$ −14.5683 −1.22687
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −1.05433 −0.0875575
$$146$$ 0 0
$$147$$ 1.47283 0.121477
$$148$$ 0 0
$$149$$ −10.1887 −0.834690 −0.417345 0.908748i $$-0.637039\pi$$
−0.417345 + 0.908748i $$0.637039\pi$$
$$150$$ 0 0
$$151$$ 8.18869 0.666386 0.333193 0.942859i $$-0.391874\pi$$
0.333193 + 0.942859i $$0.391874\pi$$
$$152$$ 0 0
$$153$$ 0.830760 0.0671630
$$154$$ 0 0
$$155$$ −1.24302 −0.0998417
$$156$$ 0 0
$$157$$ −14.2687 −1.13877 −0.569383 0.822072i $$-0.692818\pi$$
−0.569383 + 0.822072i $$0.692818\pi$$
$$158$$ 0 0
$$159$$ 11.5334 0.914659
$$160$$ 0 0
$$161$$ 9.17548 0.723129
$$162$$ 0 0
$$163$$ −15.7438 −1.23315 −0.616574 0.787297i $$-0.711480\pi$$
−0.616574 + 0.787297i $$0.711480\pi$$
$$164$$ 0 0
$$165$$ −3.66152 −0.285049
$$166$$ 0 0
$$167$$ 0.648317 0.0501683 0.0250841 0.999685i $$-0.492015\pi$$
0.0250841 + 0.999685i $$0.492015\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 4.10866 0.314197
$$172$$ 0 0
$$173$$ 15.9325 1.21132 0.605661 0.795722i $$-0.292909\pi$$
0.605661 + 0.795722i $$0.292909\pi$$
$$174$$ 0 0
$$175$$ 3.75698 0.284001
$$176$$ 0 0
$$177$$ −20.2423 −1.52150
$$178$$ 0 0
$$179$$ 17.0257 1.27256 0.636280 0.771458i $$-0.280472\pi$$
0.636280 + 0.771458i $$0.280472\pi$$
$$180$$ 0 0
$$181$$ 5.63511 0.418855 0.209427 0.977824i $$-0.432840\pi$$
0.209427 + 0.977824i $$0.432840\pi$$
$$182$$ 0 0
$$183$$ 1.33224 0.0984817
$$184$$ 0 0
$$185$$ 5.51396 0.405395
$$186$$ 0 0
$$187$$ −2.22982 −0.163060
$$188$$ 0 0
$$189$$ 5.64207 0.410400
$$190$$ 0 0
$$191$$ −2.88509 −0.208758 −0.104379 0.994538i $$-0.533285\pi$$
−0.104379 + 0.994538i $$0.533285\pi$$
$$192$$ 0 0
$$193$$ 17.5264 1.26158 0.630791 0.775953i $$-0.282731\pi$$
0.630791 + 0.775953i $$0.282731\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.9193 1.06295 0.531477 0.847073i $$-0.321637\pi$$
0.531477 + 0.847073i $$0.321637\pi$$
$$198$$ 0 0
$$199$$ 16.6483 1.18017 0.590084 0.807342i $$-0.299095\pi$$
0.590084 + 0.807342i $$0.299095\pi$$
$$200$$ 0 0
$$201$$ −14.6002 −1.02982
$$202$$ 0 0
$$203$$ −0.945668 −0.0663729
$$204$$ 0 0
$$205$$ −11.4512 −0.799783
$$206$$ 0 0
$$207$$ 7.62263 0.529809
$$208$$ 0 0
$$209$$ −11.0279 −0.762818
$$210$$ 0 0
$$211$$ −7.40530 −0.509802 −0.254901 0.966967i $$-0.582043\pi$$
−0.254901 + 0.966967i $$0.582043\pi$$
$$212$$ 0 0
$$213$$ −21.6351 −1.48241
$$214$$ 0 0
$$215$$ −12.5055 −0.852867
$$216$$ 0 0
$$217$$ −1.11491 −0.0756849
$$218$$ 0 0
$$219$$ 1.11491 0.0753385
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −27.1491 −1.81804 −0.909018 0.416756i $$-0.863167\pi$$
−0.909018 + 0.416756i $$0.863167\pi$$
$$224$$ 0 0
$$225$$ 3.12115 0.208077
$$226$$ 0 0
$$227$$ 4.50772 0.299188 0.149594 0.988748i $$-0.452203\pi$$
0.149594 + 0.988748i $$0.452203\pi$$
$$228$$ 0 0
$$229$$ 10.2298 0.676005 0.338003 0.941145i $$-0.390249\pi$$
0.338003 + 0.941145i $$0.390249\pi$$
$$230$$ 0 0
$$231$$ −3.28415 −0.216081
$$232$$ 0 0
$$233$$ 11.5529 0.756853 0.378426 0.925631i $$-0.376465\pi$$
0.378426 + 0.925631i $$0.376465\pi$$
$$234$$ 0 0
$$235$$ 11.0279 0.719382
$$236$$ 0 0
$$237$$ −11.7827 −0.765367
$$238$$ 0 0
$$239$$ 30.7570 1.98950 0.994752 0.102317i $$-0.0326256\pi$$
0.994752 + 0.102317i $$0.0326256\pi$$
$$240$$ 0 0
$$241$$ −0.967349 −0.0623124 −0.0311562 0.999515i $$-0.509919\pi$$
−0.0311562 + 0.999515i $$0.509919\pi$$
$$242$$ 0 0
$$243$$ 8.35793 0.536161
$$244$$ 0 0
$$245$$ −1.11491 −0.0712288
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 17.6351 1.11758
$$250$$ 0 0
$$251$$ −0.0264075 −0.00166683 −0.000833413 1.00000i $$-0.500265\pi$$
−0.000833413 1.00000i $$0.500265\pi$$
$$252$$ 0 0
$$253$$ −20.4596 −1.28629
$$254$$ 0 0
$$255$$ 1.64207 0.102831
$$256$$ 0 0
$$257$$ 14.7159 0.917950 0.458975 0.888449i $$-0.348217\pi$$
0.458975 + 0.888449i $$0.348217\pi$$
$$258$$ 0 0
$$259$$ 4.94567 0.307309
$$260$$ 0 0
$$261$$ −0.785623 −0.0486289
$$262$$ 0 0
$$263$$ −24.9193 −1.53659 −0.768294 0.640097i $$-0.778894\pi$$
−0.768294 + 0.640097i $$0.778894\pi$$
$$264$$ 0 0
$$265$$ −8.73057 −0.536315
$$266$$ 0 0
$$267$$ 16.4596 1.00731
$$268$$ 0 0
$$269$$ −23.9861 −1.46246 −0.731229 0.682133i $$-0.761053\pi$$
−0.731229 + 0.682133i $$0.761053\pi$$
$$270$$ 0 0
$$271$$ 15.0668 0.915244 0.457622 0.889147i $$-0.348701\pi$$
0.457622 + 0.889147i $$0.348701\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −8.37737 −0.505175
$$276$$ 0 0
$$277$$ 0.350966 0.0210875 0.0105437 0.999944i $$-0.496644\pi$$
0.0105437 + 0.999944i $$0.496644\pi$$
$$278$$ 0 0
$$279$$ −0.926221 −0.0554514
$$280$$ 0 0
$$281$$ 5.30583 0.316519 0.158260 0.987398i $$-0.449412\pi$$
0.158260 + 0.987398i $$0.449412\pi$$
$$282$$ 0 0
$$283$$ −11.6134 −0.690347 −0.345173 0.938539i $$-0.612180\pi$$
−0.345173 + 0.938539i $$0.612180\pi$$
$$284$$ 0 0
$$285$$ 8.12115 0.481055
$$286$$ 0 0
$$287$$ −10.2709 −0.606275
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 14.8176 0.868621
$$292$$ 0 0
$$293$$ 20.2423 1.18257 0.591284 0.806463i $$-0.298621\pi$$
0.591284 + 0.806463i $$0.298621\pi$$
$$294$$ 0 0
$$295$$ 15.3230 0.892142
$$296$$ 0 0
$$297$$ −12.5808 −0.730011
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −11.2166 −0.646515
$$302$$ 0 0
$$303$$ −27.5264 −1.58135
$$304$$ 0 0
$$305$$ −1.00848 −0.0577453
$$306$$ 0 0
$$307$$ 1.20189 0.0685955 0.0342978 0.999412i $$-0.489081\pi$$
0.0342978 + 0.999412i $$0.489081\pi$$
$$308$$ 0 0
$$309$$ −17.8524 −1.01559
$$310$$ 0 0
$$311$$ 10.2081 0.578850 0.289425 0.957201i $$-0.406536\pi$$
0.289425 + 0.957201i $$0.406536\pi$$
$$312$$ 0 0
$$313$$ 18.7375 1.05911 0.529554 0.848276i $$-0.322359\pi$$
0.529554 + 0.848276i $$0.322359\pi$$
$$314$$ 0 0
$$315$$ −0.926221 −0.0521866
$$316$$ 0 0
$$317$$ −10.4985 −0.589656 −0.294828 0.955550i $$-0.595262\pi$$
−0.294828 + 0.955550i $$0.595262\pi$$
$$318$$ 0 0
$$319$$ 2.10866 0.118063
$$320$$ 0 0
$$321$$ 2.78562 0.155478
$$322$$ 0 0
$$323$$ 4.94567 0.275184
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −20.6197 −1.14027
$$328$$ 0 0
$$329$$ 9.89134 0.545327
$$330$$ 0 0
$$331$$ −27.0062 −1.48440 −0.742199 0.670180i $$-0.766217\pi$$
−0.742199 + 0.670180i $$0.766217\pi$$
$$332$$ 0 0
$$333$$ 4.10866 0.225153
$$334$$ 0 0
$$335$$ 11.0521 0.603841
$$336$$ 0 0
$$337$$ −10.4985 −0.571891 −0.285946 0.958246i $$-0.592308\pi$$
−0.285946 + 0.958246i $$0.592308\pi$$
$$338$$ 0 0
$$339$$ −5.67401 −0.308170
$$340$$ 0 0
$$341$$ 2.48604 0.134626
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 15.0668 0.811170
$$346$$ 0 0
$$347$$ −15.0668 −0.808829 −0.404415 0.914576i $$-0.632525\pi$$
−0.404415 + 0.914576i $$0.632525\pi$$
$$348$$ 0 0
$$349$$ 7.86493 0.421000 0.210500 0.977594i $$-0.432491\pi$$
0.210500 + 0.977594i $$0.432491\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 29.4053 1.56509 0.782543 0.622596i $$-0.213922\pi$$
0.782543 + 0.622596i $$0.213922\pi$$
$$354$$ 0 0
$$355$$ 16.3774 0.869221
$$356$$ 0 0
$$357$$ 1.47283 0.0779506
$$358$$ 0 0
$$359$$ 5.91302 0.312077 0.156039 0.987751i $$-0.450128\pi$$
0.156039 + 0.987751i $$0.450128\pi$$
$$360$$ 0 0
$$361$$ 5.45963 0.287349
$$362$$ 0 0
$$363$$ −8.87813 −0.465981
$$364$$ 0 0
$$365$$ −0.843964 −0.0441751
$$366$$ 0 0
$$367$$ 15.6762 0.818293 0.409147 0.912469i $$-0.365826\pi$$
0.409147 + 0.912469i $$0.365826\pi$$
$$368$$ 0 0
$$369$$ −8.53269 −0.444194
$$370$$ 0 0
$$371$$ −7.83076 −0.406553
$$372$$ 0 0
$$373$$ −9.81131 −0.508011 −0.254005 0.967203i $$-0.581748\pi$$
−0.254005 + 0.967203i $$0.581748\pi$$
$$374$$ 0 0
$$375$$ 14.3796 0.742560
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −11.6615 −0.599012 −0.299506 0.954094i $$-0.596822\pi$$
−0.299506 + 0.954094i $$0.596822\pi$$
$$380$$ 0 0
$$381$$ 31.2772 1.60238
$$382$$ 0 0
$$383$$ 27.6087 1.41074 0.705369 0.708840i $$-0.250781\pi$$
0.705369 + 0.708840i $$0.250781\pi$$
$$384$$ 0 0
$$385$$ 2.48604 0.126700
$$386$$ 0 0
$$387$$ −9.31832 −0.473677
$$388$$ 0 0
$$389$$ 11.8308 0.599843 0.299922 0.953964i $$-0.403039\pi$$
0.299922 + 0.953964i $$0.403039\pi$$
$$390$$ 0 0
$$391$$ 9.17548 0.464024
$$392$$ 0 0
$$393$$ 6.78562 0.342289
$$394$$ 0 0
$$395$$ 8.91926 0.448777
$$396$$ 0 0
$$397$$ −22.7111 −1.13984 −0.569919 0.821701i $$-0.693026\pi$$
−0.569919 + 0.821701i $$0.693026\pi$$
$$398$$ 0 0
$$399$$ 7.28415 0.364663
$$400$$ 0 0
$$401$$ 37.6087 1.87809 0.939045 0.343795i $$-0.111713\pi$$
0.939045 + 0.343795i $$0.111713\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 6.48604 0.322294
$$406$$ 0 0
$$407$$ −11.0279 −0.546634
$$408$$ 0 0
$$409$$ −22.5808 −1.11655 −0.558274 0.829657i $$-0.688536\pi$$
−0.558274 + 0.829657i $$0.688536\pi$$
$$410$$ 0 0
$$411$$ 0.559099 0.0275783
$$412$$ 0 0
$$413$$ 13.7438 0.676287
$$414$$ 0 0
$$415$$ −13.3494 −0.655299
$$416$$ 0 0
$$417$$ −8.92622 −0.437119
$$418$$ 0 0
$$419$$ −24.2904 −1.18666 −0.593332 0.804958i $$-0.702188\pi$$
−0.593332 + 0.804958i $$0.702188\pi$$
$$420$$ 0 0
$$421$$ 11.5676 0.563769 0.281885 0.959448i $$-0.409040\pi$$
0.281885 + 0.959448i $$0.409040\pi$$
$$422$$ 0 0
$$423$$ 8.21733 0.399540
$$424$$ 0 0
$$425$$ 3.75698 0.182240
$$426$$ 0 0
$$427$$ −0.904539 −0.0437737
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 22.9457 1.10525 0.552627 0.833429i $$-0.313625\pi$$
0.552627 + 0.833429i $$0.313625\pi$$
$$432$$ 0 0
$$433$$ 19.8913 0.955917 0.477958 0.878383i $$-0.341377\pi$$
0.477958 + 0.878383i $$0.341377\pi$$
$$434$$ 0 0
$$435$$ −1.55286 −0.0744538
$$436$$ 0 0
$$437$$ 45.3789 2.17077
$$438$$ 0 0
$$439$$ −19.8936 −0.949468 −0.474734 0.880129i $$-0.657456\pi$$
−0.474734 + 0.880129i $$0.657456\pi$$
$$440$$ 0 0
$$441$$ −0.830760 −0.0395600
$$442$$ 0 0
$$443$$ 15.7827 0.749857 0.374929 0.927054i $$-0.377667\pi$$
0.374929 + 0.927054i $$0.377667\pi$$
$$444$$ 0 0
$$445$$ −12.4596 −0.590643
$$446$$ 0 0
$$447$$ −15.0062 −0.709771
$$448$$ 0 0
$$449$$ −41.2702 −1.94766 −0.973831 0.227273i $$-0.927019\pi$$
−0.973831 + 0.227273i $$0.927019\pi$$
$$450$$ 0 0
$$451$$ 22.9023 1.07843
$$452$$ 0 0
$$453$$ 12.0606 0.566655
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −36.9923 −1.73043 −0.865214 0.501403i $$-0.832817\pi$$
−0.865214 + 0.501403i $$0.832817\pi$$
$$458$$ 0 0
$$459$$ 5.64207 0.263349
$$460$$ 0 0
$$461$$ 10.3649 0.482741 0.241370 0.970433i $$-0.422403\pi$$
0.241370 + 0.970433i $$0.422403\pi$$
$$462$$ 0 0
$$463$$ −12.2709 −0.570279 −0.285140 0.958486i $$-0.592040\pi$$
−0.285140 + 0.958486i $$0.592040\pi$$
$$464$$ 0 0
$$465$$ −1.83076 −0.0848995
$$466$$ 0 0
$$467$$ −8.86341 −0.410150 −0.205075 0.978746i $$-0.565744\pi$$
−0.205075 + 0.978746i $$0.565744\pi$$
$$468$$ 0 0
$$469$$ 9.91302 0.457741
$$470$$ 0 0
$$471$$ −21.0154 −0.968340
$$472$$ 0 0
$$473$$ 25.0110 1.15001
$$474$$ 0 0
$$475$$ 18.5808 0.852545
$$476$$ 0 0
$$477$$ −6.50548 −0.297866
$$478$$ 0 0
$$479$$ 4.56606 0.208629 0.104314 0.994544i $$-0.466735\pi$$
0.104314 + 0.994544i $$0.466735\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 13.5140 0.614907
$$484$$ 0 0
$$485$$ −11.2166 −0.509320
$$486$$ 0 0
$$487$$ −23.8649 −1.08142 −0.540712 0.841208i $$-0.681845\pi$$
−0.540712 + 0.841208i $$0.681845\pi$$
$$488$$ 0 0
$$489$$ −23.1880 −1.04860
$$490$$ 0 0
$$491$$ 7.00624 0.316187 0.158094 0.987424i $$-0.449465\pi$$
0.158094 + 0.987424i $$0.449465\pi$$
$$492$$ 0 0
$$493$$ −0.945668 −0.0425907
$$494$$ 0 0
$$495$$ 2.06530 0.0928284
$$496$$ 0 0
$$497$$ 14.6894 0.658912
$$498$$ 0 0
$$499$$ −10.2687 −0.459691 −0.229845 0.973227i $$-0.573822\pi$$
−0.229845 + 0.973227i $$0.573822\pi$$
$$500$$ 0 0
$$501$$ 0.954863 0.0426601
$$502$$ 0 0
$$503$$ −8.53413 −0.380518 −0.190259 0.981734i $$-0.560933\pi$$
−0.190259 + 0.981734i $$0.560933\pi$$
$$504$$ 0 0
$$505$$ 20.8370 0.927234
$$506$$ 0 0
$$507$$ −19.1468 −0.850341
$$508$$ 0 0
$$509$$ −23.8216 −1.05587 −0.527936 0.849284i $$-0.677034\pi$$
−0.527936 + 0.849284i $$0.677034\pi$$
$$510$$ 0 0
$$511$$ −0.756981 −0.0334869
$$512$$ 0 0
$$513$$ 27.9038 1.23198
$$514$$ 0 0
$$515$$ 13.5140 0.595496
$$516$$ 0 0
$$517$$ −22.0558 −0.970015
$$518$$ 0 0
$$519$$ 23.4659 1.03004
$$520$$ 0 0
$$521$$ −3.48532 −0.152695 −0.0763473 0.997081i $$-0.524326\pi$$
−0.0763473 + 0.997081i $$0.524326\pi$$
$$522$$ 0 0
$$523$$ 23.2144 1.01509 0.507547 0.861624i $$-0.330552\pi$$
0.507547 + 0.861624i $$0.330552\pi$$
$$524$$ 0 0
$$525$$ 5.53341 0.241498
$$526$$ 0 0
$$527$$ −1.11491 −0.0485661
$$528$$ 0 0
$$529$$ 61.1895 2.66041
$$530$$ 0 0
$$531$$ 11.4178 0.495490
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −2.10866 −0.0911655
$$536$$ 0 0
$$537$$ 25.0760 1.08211
$$538$$ 0 0
$$539$$ 2.22982 0.0960449
$$540$$ 0 0
$$541$$ −9.08074 −0.390411 −0.195206 0.980762i $$-0.562537\pi$$
−0.195206 + 0.980762i $$0.562537\pi$$
$$542$$ 0 0
$$543$$ 8.29959 0.356169
$$544$$ 0 0
$$545$$ 15.6087 0.668603
$$546$$ 0 0
$$547$$ −5.34945 −0.228726 −0.114363 0.993439i $$-0.536483\pi$$
−0.114363 + 0.993439i $$0.536483\pi$$
$$548$$ 0 0
$$549$$ −0.751455 −0.0320713
$$550$$ 0 0
$$551$$ −4.67696 −0.199245
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 0 0
$$555$$ 8.12115 0.344724
$$556$$ 0 0
$$557$$ −13.9472 −0.590961 −0.295481 0.955349i $$-0.595480\pi$$
−0.295481 + 0.955349i $$0.595480\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −3.28415 −0.138657
$$562$$ 0 0
$$563$$ −8.43322 −0.355418 −0.177709 0.984083i $$-0.556869\pi$$
−0.177709 + 0.984083i $$0.556869\pi$$
$$564$$ 0 0
$$565$$ 4.29512 0.180697
$$566$$ 0 0
$$567$$ 5.81756 0.244314
$$568$$ 0 0
$$569$$ −5.89357 −0.247071 −0.123536 0.992340i $$-0.539423\pi$$
−0.123536 + 0.992340i $$0.539423\pi$$
$$570$$ 0 0
$$571$$ −40.2857 −1.68590 −0.842951 0.537990i $$-0.819184\pi$$
−0.842951 + 0.537990i $$0.819184\pi$$
$$572$$ 0 0
$$573$$ −4.24926 −0.177515
$$574$$ 0 0
$$575$$ 34.4721 1.43759
$$576$$ 0 0
$$577$$ 27.1616 1.13075 0.565375 0.824834i $$-0.308731\pi$$
0.565375 + 0.824834i $$0.308731\pi$$
$$578$$ 0 0
$$579$$ 25.8135 1.07277
$$580$$ 0 0
$$581$$ −11.9736 −0.496748
$$582$$ 0 0
$$583$$ 17.4611 0.723167
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −10.2423 −0.422745 −0.211373 0.977406i $$-0.567793\pi$$
−0.211373 + 0.977406i $$0.567793\pi$$
$$588$$ 0 0
$$589$$ −5.51396 −0.227199
$$590$$ 0 0
$$591$$ 21.9736 0.903873
$$592$$ 0 0
$$593$$ 27.9736 1.14874 0.574369 0.818597i $$-0.305248\pi$$
0.574369 + 0.818597i $$0.305248\pi$$
$$594$$ 0 0
$$595$$ −1.11491 −0.0457068
$$596$$ 0 0
$$597$$ 24.5202 1.00355
$$598$$ 0 0
$$599$$ −6.00223 −0.245245 −0.122622 0.992453i $$-0.539130\pi$$
−0.122622 + 0.992453i $$0.539130\pi$$
$$600$$ 0 0
$$601$$ −19.8913 −0.811385 −0.405692 0.914010i $$-0.632970\pi$$
−0.405692 + 0.914010i $$0.632970\pi$$
$$602$$ 0 0
$$603$$ 8.23534 0.335369
$$604$$ 0 0
$$605$$ 6.72058 0.273230
$$606$$ 0 0
$$607$$ 29.4784 1.19649 0.598245 0.801313i $$-0.295865\pi$$
0.598245 + 0.801313i $$0.295865\pi$$
$$608$$ 0 0
$$609$$ −1.39281 −0.0564396
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −36.4115 −1.47065 −0.735324 0.677716i $$-0.762970\pi$$
−0.735324 + 0.677716i $$0.762970\pi$$
$$614$$ 0 0
$$615$$ −16.8656 −0.680088
$$616$$ 0 0
$$617$$ −7.51396 −0.302501 −0.151250 0.988495i $$-0.548330\pi$$
−0.151250 + 0.988495i $$0.548330\pi$$
$$618$$ 0 0
$$619$$ −21.7438 −0.873956 −0.436978 0.899472i $$-0.643951\pi$$
−0.436978 + 0.899472i $$0.643951\pi$$
$$620$$ 0 0
$$621$$ 51.7688 2.07741
$$622$$ 0 0
$$623$$ −11.1755 −0.447736
$$624$$ 0 0
$$625$$ 7.89981 0.315993
$$626$$ 0 0
$$627$$ −16.2423 −0.648655
$$628$$ 0 0
$$629$$ 4.94567 0.197197
$$630$$ 0 0
$$631$$ 9.57454 0.381156 0.190578 0.981672i $$-0.438964\pi$$
0.190578 + 0.981672i $$0.438964\pi$$
$$632$$ 0 0
$$633$$ −10.9068 −0.433505
$$634$$ 0 0
$$635$$ −23.6762 −0.939563
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 12.2034 0.482759
$$640$$ 0 0
$$641$$ 22.8759 0.903544 0.451772 0.892134i $$-0.350792\pi$$
0.451772 + 0.892134i $$0.350792\pi$$
$$642$$ 0 0
$$643$$ −28.2834 −1.11539 −0.557695 0.830046i $$-0.688314\pi$$
−0.557695 + 0.830046i $$0.688314\pi$$
$$644$$ 0 0
$$645$$ −18.4185 −0.725228
$$646$$ 0 0
$$647$$ 10.0125 0.393631 0.196816 0.980440i $$-0.436940\pi$$
0.196816 + 0.980440i $$0.436940\pi$$
$$648$$ 0 0
$$649$$ −30.6461 −1.20296
$$650$$ 0 0
$$651$$ −1.64207 −0.0643579
$$652$$ 0 0
$$653$$ −2.45963 −0.0962528 −0.0481264 0.998841i $$-0.515325\pi$$
−0.0481264 + 0.998841i $$0.515325\pi$$
$$654$$ 0 0
$$655$$ −5.13659 −0.200703
$$656$$ 0 0
$$657$$ −0.628870 −0.0245346
$$658$$ 0 0
$$659$$ −33.0187 −1.28623 −0.643114 0.765771i $$-0.722358\pi$$
−0.643114 + 0.765771i $$0.722358\pi$$
$$660$$ 0 0
$$661$$ 5.97359 0.232346 0.116173 0.993229i $$-0.462937\pi$$
0.116173 + 0.993229i $$0.462937\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −5.51396 −0.213822
$$666$$ 0 0
$$667$$ −8.67696 −0.335973
$$668$$ 0 0
$$669$$ −39.9861 −1.54595
$$670$$ 0 0
$$671$$ 2.01696 0.0778637
$$672$$ 0 0
$$673$$ 6.29512 0.242659 0.121329 0.992612i $$-0.461284\pi$$
0.121329 + 0.992612i $$0.461284\pi$$
$$674$$ 0 0
$$675$$ 21.1972 0.815879
$$676$$ 0 0
$$677$$ 46.5544 1.78923 0.894615 0.446838i $$-0.147450\pi$$
0.894615 + 0.446838i $$0.147450\pi$$
$$678$$ 0 0
$$679$$ −10.0606 −0.386089
$$680$$ 0 0
$$681$$ 6.63912 0.254412
$$682$$ 0 0
$$683$$ −43.9861 −1.68308 −0.841540 0.540194i $$-0.818351\pi$$
−0.841540 + 0.540194i $$0.818351\pi$$
$$684$$ 0 0
$$685$$ −0.423228 −0.0161707
$$686$$ 0 0
$$687$$ 15.0668 0.574835
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 9.17772 0.349137 0.174568 0.984645i $$-0.444147\pi$$
0.174568 + 0.984645i $$0.444147\pi$$
$$692$$ 0 0
$$693$$ 1.85244 0.0703684
$$694$$ 0 0
$$695$$ 6.75698 0.256307
$$696$$ 0 0
$$697$$ −10.2709 −0.389040
$$698$$ 0 0
$$699$$ 17.0154 0.643583
$$700$$ 0 0
$$701$$ −7.59622 −0.286905 −0.143453 0.989657i $$-0.545820\pi$$
−0.143453 + 0.989657i $$0.545820\pi$$
$$702$$ 0 0
$$703$$ 24.4596 0.922512
$$704$$ 0 0
$$705$$ 16.2423 0.611720
$$706$$ 0 0
$$707$$ 18.6894 0.702889
$$708$$ 0 0
$$709$$ 7.63511 0.286743 0.143371 0.989669i $$-0.454206\pi$$
0.143371 + 0.989669i $$0.454206\pi$$
$$710$$ 0 0
$$711$$ 6.64608 0.249248
$$712$$ 0 0
$$713$$ −10.2298 −0.383110
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 45.2999 1.69176
$$718$$ 0 0
$$719$$ 13.6204 0.507955 0.253977 0.967210i $$-0.418261\pi$$
0.253977 + 0.967210i $$0.418261\pi$$
$$720$$ 0 0
$$721$$ 12.1212 0.451415
$$722$$ 0 0
$$723$$ −1.42474 −0.0529868
$$724$$ 0 0
$$725$$ −3.55286 −0.131950
$$726$$ 0 0
$$727$$ −10.9457 −0.405952 −0.202976 0.979184i $$-0.565061\pi$$
−0.202976 + 0.979184i $$0.565061\pi$$
$$728$$ 0 0
$$729$$ 29.7625 1.10232
$$730$$ 0 0
$$731$$ −11.2166 −0.414861
$$732$$ 0 0
$$733$$ 7.28415 0.269046 0.134523 0.990910i $$-0.457050\pi$$
0.134523 + 0.990910i $$0.457050\pi$$
$$734$$ 0 0
$$735$$ −1.64207 −0.0605688
$$736$$ 0 0
$$737$$ −22.1042 −0.814218
$$738$$ 0 0
$$739$$ 50.3268 1.85130 0.925651 0.378380i $$-0.123519\pi$$
0.925651 + 0.378380i $$0.123519\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −14.9317 −0.547793 −0.273896 0.961759i $$-0.588313\pi$$
−0.273896 + 0.961759i $$0.588313\pi$$
$$744$$ 0 0
$$745$$ 11.3594 0.416178
$$746$$ 0 0
$$747$$ −9.94719 −0.363948
$$748$$ 0 0
$$749$$ −1.89134 −0.0691079
$$750$$ 0 0
$$751$$ −29.9347 −1.09233 −0.546166 0.837677i $$-0.683913\pi$$
−0.546166 + 0.837677i $$0.683913\pi$$
$$752$$ 0 0
$$753$$ −0.0388938 −0.00141737
$$754$$ 0 0
$$755$$ −9.12963 −0.332261
$$756$$ 0 0
$$757$$ −6.48827 −0.235820 −0.117910 0.993024i $$-0.537619\pi$$
−0.117910 + 0.993024i $$0.537619\pi$$
$$758$$ 0 0
$$759$$ −30.1336 −1.09378
$$760$$ 0 0
$$761$$ −28.2159 −1.02283 −0.511413 0.859335i $$-0.670878\pi$$
−0.511413 + 0.859335i $$0.670878\pi$$
$$762$$ 0 0
$$763$$ 14.0000 0.506834
$$764$$ 0 0
$$765$$ −0.926221 −0.0334876
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −21.8260 −0.787067 −0.393533 0.919310i $$-0.628747\pi$$
−0.393533 + 0.919310i $$0.628747\pi$$
$$770$$ 0 0
$$771$$ 21.6740 0.780570
$$772$$ 0 0
$$773$$ 26.5683 0.955595 0.477798 0.878470i $$-0.341435\pi$$
0.477798 + 0.878470i $$0.341435\pi$$
$$774$$ 0 0
$$775$$ −4.18869 −0.150462
$$776$$ 0 0
$$777$$ 7.28415 0.261317
$$778$$ 0 0
$$779$$ −50.7967 −1.81998
$$780$$ 0 0
$$781$$ −32.7547 −1.17206
$$782$$ 0 0
$$783$$ −5.33553 −0.190676
$$784$$ 0 0
$$785$$ 15.9083 0.567791
$$786$$ 0 0
$$787$$ −4.98903 −0.177840 −0.0889199 0.996039i $$-0.528342\pi$$
−0.0889199 + 0.996039i $$0.528342\pi$$
$$788$$ 0 0
$$789$$ −36.7019 −1.30662
$$790$$ 0 0
$$791$$ 3.85244 0.136977
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −12.8587 −0.456051
$$796$$ 0 0
$$797$$ −47.0963 −1.66824 −0.834118 0.551587i $$-0.814023\pi$$
−0.834118 + 0.551587i $$0.814023\pi$$
$$798$$ 0 0
$$799$$ 9.89134 0.349930
$$800$$ 0 0
$$801$$ −9.28415 −0.328039
$$802$$ 0 0
$$803$$ 1.68793 0.0595657
$$804$$ 0 0
$$805$$ −10.2298 −0.360554
$$806$$ 0 0
$$807$$ −35.3275 −1.24359
$$808$$ 0 0
$$809$$ −10.2423 −0.360100 −0.180050 0.983657i $$-0.557626\pi$$
−0.180050 + 0.983657i $$0.557626\pi$$
$$810$$ 0 0
$$811$$ 2.98680 0.104881 0.0524403 0.998624i $$-0.483300\pi$$
0.0524403 + 0.998624i $$0.483300\pi$$
$$812$$ 0 0
$$813$$ 22.1909 0.778270
$$814$$ 0 0
$$815$$ 17.5529 0.614850
$$816$$ 0 0
$$817$$ −55.4736 −1.94078
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −52.7283 −1.84023 −0.920116 0.391646i $$-0.871906\pi$$
−0.920116 + 0.391646i $$0.871906\pi$$
$$822$$ 0 0
$$823$$ 6.68945 0.233179 0.116590 0.993180i $$-0.462804\pi$$
0.116590 + 0.993180i $$0.462804\pi$$
$$824$$ 0 0
$$825$$ −12.3385 −0.429571
$$826$$ 0 0
$$827$$ −41.5140 −1.44358 −0.721791 0.692111i $$-0.756681\pi$$
−0.721791 + 0.692111i $$0.756681\pi$$
$$828$$ 0 0
$$829$$ −48.4596 −1.68307 −0.841536 0.540201i $$-0.818348\pi$$
−0.841536 + 0.540201i $$0.818348\pi$$
$$830$$ 0 0
$$831$$ 0.516914 0.0179316
$$832$$ 0 0
$$833$$ −1.00000 −0.0346479
$$834$$ 0 0
$$835$$ −0.722813 −0.0250140
$$836$$ 0 0
$$837$$ −6.29039 −0.217428
$$838$$ 0 0
$$839$$ 9.64903 0.333122 0.166561 0.986031i $$-0.446734\pi$$
0.166561 + 0.986031i $$0.446734\pi$$
$$840$$ 0 0
$$841$$ −28.1057 −0.969162
$$842$$ 0 0
$$843$$ 7.81460 0.269149
$$844$$ 0 0
$$845$$ 14.4938 0.498602
$$846$$ 0 0
$$847$$ 6.02792 0.207122
$$848$$ 0 0
$$849$$ −17.1047 −0.587030
$$850$$ 0 0
$$851$$ 45.3789 1.55557
$$852$$ 0 0
$$853$$ −35.3619 −1.21077 −0.605385 0.795933i $$-0.706981\pi$$
−0.605385 + 0.795933i $$0.706981\pi$$
$$854$$ 0 0
$$855$$ −4.58078 −0.156659
$$856$$ 0 0
$$857$$ 29.0451 0.992163 0.496081 0.868276i $$-0.334772\pi$$
0.496081 + 0.868276i $$0.334772\pi$$
$$858$$ 0 0
$$859$$ 43.7996 1.49442 0.747212 0.664586i $$-0.231392\pi$$
0.747212 + 0.664586i $$0.231392\pi$$
$$860$$ 0 0
$$861$$ −15.1274 −0.515540
$$862$$ 0 0
$$863$$ 3.39754 0.115654 0.0578268 0.998327i $$-0.481583\pi$$
0.0578268 + 0.998327i $$0.481583\pi$$
$$864$$ 0 0
$$865$$ −17.7632 −0.603968
$$866$$ 0 0
$$867$$ 1.47283 0.0500201
$$868$$ 0 0
$$869$$ −17.8385 −0.605130
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −8.35793 −0.282873
$$874$$ 0 0
$$875$$ −9.76322 −0.330057
$$876$$ 0 0
$$877$$ −29.0529 −0.981047 −0.490523 0.871428i $$-0.663194\pi$$
−0.490523 + 0.871428i $$0.663194\pi$$
$$878$$ 0 0
$$879$$ 29.8135 1.00559
$$880$$ 0 0
$$881$$ −11.4923 −0.387185 −0.193592 0.981082i $$-0.562014\pi$$
−0.193592 + 0.981082i $$0.562014\pi$$
$$882$$ 0 0
$$883$$ −25.1927 −0.847802 −0.423901 0.905709i $$-0.639340\pi$$
−0.423901 + 0.905709i $$0.639340\pi$$
$$884$$ 0 0
$$885$$ 22.5683 0.758625
$$886$$ 0 0
$$887$$ 35.1957 1.18176 0.590878 0.806761i $$-0.298781\pi$$
0.590878 + 0.806761i $$0.298781\pi$$
$$888$$ 0 0
$$889$$ −21.2361 −0.712235
$$890$$ 0 0
$$891$$ −12.9721 −0.434581
$$892$$ 0 0
$$893$$ 48.9193 1.63702
$$894$$ 0 0
$$895$$ −18.9821 −0.634501
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.05433 0.0351639
$$900$$ 0 0
$$901$$ −7.83076 −0.260880
$$902$$ 0 0
$$903$$ −16.5202 −0.549758
$$904$$ 0 0
$$905$$ −6.28263 −0.208842
$$906$$ 0 0
$$907$$ 25.1227 0.834184 0.417092 0.908864i $$-0.363049\pi$$
0.417092 + 0.908864i $$0.363049\pi$$
$$908$$ 0 0
$$909$$ 15.5264 0.514980
$$910$$ 0 0
$$911$$ −22.5110 −0.745823 −0.372912 0.927867i $$-0.621641\pi$$
−0.372912 + 0.927867i $$0.621641\pi$$
$$912$$ 0 0
$$913$$ 26.6989 0.883605
$$914$$ 0 0
$$915$$ −1.48532 −0.0491032
$$916$$ 0 0
$$917$$ −4.60719 −0.152143
$$918$$ 0 0
$$919$$ −13.5676 −0.447553 −0.223777 0.974640i $$-0.571839\pi$$
−0.223777 + 0.974640i $$0.571839\pi$$
$$920$$ 0 0
$$921$$ 1.77018 0.0583296
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 18.5808 0.610932
$$926$$ 0 0
$$927$$ 10.0698 0.330735
$$928$$ 0 0
$$929$$ −11.9519 −0.392129 −0.196065 0.980591i $$-0.562816\pi$$
−0.196065 + 0.980591i $$0.562816\pi$$
$$930$$ 0 0
$$931$$ −4.94567 −0.162088
$$932$$ 0 0
$$933$$ 15.0349 0.492220
$$934$$ 0 0
$$935$$ 2.48604 0.0813021
$$936$$ 0 0
$$937$$ 53.0140 1.73189 0.865946 0.500138i $$-0.166717\pi$$
0.865946 + 0.500138i $$0.166717\pi$$
$$938$$ 0 0
$$939$$ 27.5973 0.900603
$$940$$ 0 0
$$941$$ −27.1204 −0.884101 −0.442050 0.896990i $$-0.645749\pi$$
−0.442050 + 0.896990i $$0.645749\pi$$
$$942$$ 0 0
$$943$$ −94.2409 −3.06890
$$944$$ 0 0
$$945$$ −6.29039 −0.204626
$$946$$ 0 0
$$947$$ 44.5808 1.44868 0.724340 0.689443i $$-0.242144\pi$$
0.724340 + 0.689443i $$0.242144\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −15.4626 −0.501409
$$952$$ 0 0
$$953$$ −51.8991 −1.68118 −0.840588 0.541675i $$-0.817791\pi$$
−0.840588 + 0.541675i $$0.817791\pi$$
$$954$$ 0 0
$$955$$ 3.21661 0.104087
$$956$$ 0 0
$$957$$ 3.10571 0.100393
$$958$$ 0 0
$$959$$ −0.379608 −0.0122582
$$960$$ 0 0
$$961$$ −29.7570 −0.959903
$$962$$ 0 0
$$963$$ −1.57125 −0.0506327
$$964$$ 0 0
$$965$$ −19.5404 −0.629027
$$966$$ 0 0
$$967$$ −37.2680 −1.19846 −0.599229 0.800578i $$-0.704526\pi$$
−0.599229 + 0.800578i $$0.704526\pi$$
$$968$$ 0 0
$$969$$ 7.28415 0.234001
$$970$$ 0 0
$$971$$ 3.05433 0.0980182 0.0490091 0.998798i $$-0.484394\pi$$
0.0490091 + 0.998798i $$0.484394\pi$$
$$972$$ 0 0
$$973$$ 6.06058 0.194293
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −47.3961 −1.51634 −0.758168 0.652059i $$-0.773905\pi$$
−0.758168 + 0.652059i $$0.773905\pi$$
$$978$$ 0 0
$$979$$ 24.9193 0.796423
$$980$$ 0 0
$$981$$ 11.6306 0.371338
$$982$$ 0 0
$$983$$ 51.1957 1.63289 0.816445 0.577423i $$-0.195942\pi$$
0.816445 + 0.577423i $$0.195942\pi$$
$$984$$ 0 0
$$985$$ −16.6336 −0.529990
$$986$$ 0 0
$$987$$ 14.5683 0.463714
$$988$$ 0 0
$$989$$ −102.918 −3.27260
$$990$$ 0 0
$$991$$ −16.9332 −0.537900 −0.268950 0.963154i $$-0.586677\pi$$
−0.268950 + 0.963154i $$0.586677\pi$$
$$992$$ 0 0
$$993$$ −39.7757 −1.26224
$$994$$ 0 0
$$995$$ −18.5613 −0.588434
$$996$$ 0 0
$$997$$ −53.3069 −1.68825 −0.844123 0.536150i $$-0.819878\pi$$
−0.844123 + 0.536150i $$0.819878\pi$$
$$998$$ 0 0
$$999$$ 27.9038 0.882838
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 1904.2.a.m.1.3 3
4.3 odd 2 952.2.a.f.1.1 3
8.3 odd 2 7616.2.a.bc.1.3 3
8.5 even 2 7616.2.a.be.1.1 3
12.11 even 2 8568.2.a.x.1.3 3
28.27 even 2 6664.2.a.j.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.f.1.1 3 4.3 odd 2
1904.2.a.m.1.3 3 1.1 even 1 trivial
6664.2.a.j.1.3 3 28.27 even 2
7616.2.a.bc.1.3 3 8.3 odd 2
7616.2.a.be.1.1 3 8.5 even 2
8568.2.a.x.1.3 3 12.11 even 2