Properties

 Label 3192.2.a.q.1.2 Level $3192$ Weight $2$ Character 3192.1 Self dual yes Analytic conductor $25.488$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3192.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: $$25.4882483252$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.46410 q^{13} -2.73205 q^{15} -1.26795 q^{17} -1.00000 q^{19} -1.00000 q^{21} +2.46410 q^{25} -1.00000 q^{27} -9.66025 q^{29} +2.00000 q^{31} +2.73205 q^{35} -10.0000 q^{37} +5.46410 q^{39} -8.92820 q^{41} +4.92820 q^{43} +2.73205 q^{45} -11.6603 q^{47} +1.00000 q^{49} +1.26795 q^{51} -6.73205 q^{53} +1.00000 q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} -14.9282 q^{65} -13.4641 q^{67} +4.73205 q^{71} +10.3923 q^{73} -2.46410 q^{75} +4.00000 q^{79} +1.00000 q^{81} +10.1962 q^{83} -3.46410 q^{85} +9.66025 q^{87} +4.92820 q^{89} -5.46410 q^{91} -2.00000 q^{93} -2.73205 q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 2 q^{63} - 16 q^{65} - 20 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{79} + 2 q^{81} + 10 q^{83} + 2 q^{87} - 4 q^{89} - 4 q^{91} - 4 q^{93} - 2 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 2 * q^35 - 20 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 2 * q^63 - 16 * q^65 - 20 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^79 + 2 * q^81 + 10 * q^83 + 2 * q^87 - 4 * q^89 - 4 * q^91 - 4 * q^93 - 2 * q^95 + 4 * 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 2.73205 1.22181 0.610905 0.791704i $$-0.290806\pi$$
0.610905 + 0.791704i $$0.290806\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −5.46410 −1.51547 −0.757735 0.652563i $$-0.773694\pi$$
−0.757735 + 0.652563i $$0.773694\pi$$
$$14$$ 0 0
$$15$$ −2.73205 −0.705412
$$16$$ 0 0
$$17$$ −1.26795 −0.307523 −0.153761 0.988108i $$-0.549139\pi$$
−0.153761 + 0.988108i $$0.549139\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 2.46410 0.492820
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −9.66025 −1.79386 −0.896932 0.442168i $$-0.854209\pi$$
−0.896932 + 0.442168i $$0.854209\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.73205 0.461801
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 5.46410 0.874957
$$40$$ 0 0
$$41$$ −8.92820 −1.39435 −0.697176 0.716900i $$-0.745560\pi$$
−0.697176 + 0.716900i $$0.745560\pi$$
$$42$$ 0 0
$$43$$ 4.92820 0.751544 0.375772 0.926712i $$-0.377378\pi$$
0.375772 + 0.926712i $$0.377378\pi$$
$$44$$ 0 0
$$45$$ 2.73205 0.407270
$$46$$ 0 0
$$47$$ −11.6603 −1.70082 −0.850411 0.526118i $$-0.823647\pi$$
−0.850411 + 0.526118i $$0.823647\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 1.26795 0.177548
$$52$$ 0 0
$$53$$ −6.73205 −0.924718 −0.462359 0.886693i $$-0.652997\pi$$
−0.462359 + 0.886693i $$0.652997\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −14.9282 −1.85162
$$66$$ 0 0
$$67$$ −13.4641 −1.64490 −0.822451 0.568836i $$-0.807394\pi$$
−0.822451 + 0.568836i $$0.807394\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.73205 0.561591 0.280796 0.959768i $$-0.409402\pi$$
0.280796 + 0.959768i $$0.409402\pi$$
$$72$$ 0 0
$$73$$ 10.3923 1.21633 0.608164 0.793812i $$-0.291906\pi$$
0.608164 + 0.793812i $$0.291906\pi$$
$$74$$ 0 0
$$75$$ −2.46410 −0.284530
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 10.1962 1.11917 0.559587 0.828772i $$-0.310960\pi$$
0.559587 + 0.828772i $$0.310960\pi$$
$$84$$ 0 0
$$85$$ −3.46410 −0.375735
$$86$$ 0 0
$$87$$ 9.66025 1.03569
$$88$$ 0 0
$$89$$ 4.92820 0.522388 0.261194 0.965286i $$-0.415884\pi$$
0.261194 + 0.965286i $$0.415884\pi$$
$$90$$ 0 0
$$91$$ −5.46410 −0.572793
$$92$$ 0 0
$$93$$ −2.00000 −0.207390
$$94$$ 0 0
$$95$$ −2.73205 −0.280302
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1.26795 0.126166 0.0630828 0.998008i $$-0.479907\pi$$
0.0630828 + 0.998008i $$0.479907\pi$$
$$102$$ 0 0
$$103$$ −2.92820 −0.288524 −0.144262 0.989539i $$-0.546081\pi$$
−0.144262 + 0.989539i $$0.546081\pi$$
$$104$$ 0 0
$$105$$ −2.73205 −0.266621
$$106$$ 0 0
$$107$$ −4.73205 −0.457465 −0.228732 0.973489i $$-0.573458\pi$$
−0.228732 + 0.973489i $$0.573458\pi$$
$$108$$ 0 0
$$109$$ 2.39230 0.229141 0.114571 0.993415i $$-0.463451\pi$$
0.114571 + 0.993415i $$0.463451\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ 0.196152 0.0184525 0.00922623 0.999957i $$-0.497063\pi$$
0.00922623 + 0.999957i $$0.497063\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −5.46410 −0.505156
$$118$$ 0 0
$$119$$ −1.26795 −0.116233
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 8.92820 0.805029
$$124$$ 0 0
$$125$$ −6.92820 −0.619677
$$126$$ 0 0
$$127$$ 20.3923 1.80952 0.904762 0.425917i $$-0.140048\pi$$
0.904762 + 0.425917i $$0.140048\pi$$
$$128$$ 0 0
$$129$$ −4.92820 −0.433904
$$130$$ 0 0
$$131$$ 14.5885 1.27460 0.637300 0.770616i $$-0.280051\pi$$
0.637300 + 0.770616i $$0.280051\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ −2.73205 −0.235137
$$136$$ 0 0
$$137$$ 12.9282 1.10453 0.552265 0.833668i $$-0.313763\pi$$
0.552265 + 0.833668i $$0.313763\pi$$
$$138$$ 0 0
$$139$$ −15.3205 −1.29947 −0.649734 0.760161i $$-0.725120\pi$$
−0.649734 + 0.760161i $$0.725120\pi$$
$$140$$ 0 0
$$141$$ 11.6603 0.981971
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −26.3923 −2.19176
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 14.3923 1.17906 0.589532 0.807745i $$-0.299312\pi$$
0.589532 + 0.807745i $$0.299312\pi$$
$$150$$ 0 0
$$151$$ −2.53590 −0.206368 −0.103184 0.994662i $$-0.532903\pi$$
−0.103184 + 0.994662i $$0.532903\pi$$
$$152$$ 0 0
$$153$$ −1.26795 −0.102508
$$154$$ 0 0
$$155$$ 5.46410 0.438887
$$156$$ 0 0
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 0 0
$$159$$ 6.73205 0.533886
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −7.07180 −0.553906 −0.276953 0.960883i $$-0.589325\pi$$
−0.276953 + 0.960883i $$0.589325\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 11.3205 0.876007 0.438004 0.898973i $$-0.355686\pi$$
0.438004 + 0.898973i $$0.355686\pi$$
$$168$$ 0 0
$$169$$ 16.8564 1.29665
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ −16.9282 −1.28703 −0.643514 0.765435i $$-0.722524\pi$$
−0.643514 + 0.765435i $$0.722524\pi$$
$$174$$ 0 0
$$175$$ 2.46410 0.186269
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ −14.1962 −1.06107 −0.530535 0.847663i $$-0.678009\pi$$
−0.530535 + 0.847663i $$0.678009\pi$$
$$180$$ 0 0
$$181$$ 4.92820 0.366310 0.183155 0.983084i $$-0.441369\pi$$
0.183155 + 0.983084i $$0.441369\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ −27.3205 −2.00864
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −14.9282 −1.08017 −0.540083 0.841611i $$-0.681607\pi$$
−0.540083 + 0.841611i $$0.681607\pi$$
$$192$$ 0 0
$$193$$ −23.8564 −1.71722 −0.858611 0.512628i $$-0.828672\pi$$
−0.858611 + 0.512628i $$0.828672\pi$$
$$194$$ 0 0
$$195$$ 14.9282 1.06903
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −12.3923 −0.878467 −0.439234 0.898373i $$-0.644750\pi$$
−0.439234 + 0.898373i $$0.644750\pi$$
$$200$$ 0 0
$$201$$ 13.4641 0.949685
$$202$$ 0 0
$$203$$ −9.66025 −0.678017
$$204$$ 0 0
$$205$$ −24.3923 −1.70363
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ −4.73205 −0.324235
$$214$$ 0 0
$$215$$ 13.4641 0.918244
$$216$$ 0 0
$$217$$ 2.00000 0.135769
$$218$$ 0 0
$$219$$ −10.3923 −0.702247
$$220$$ 0 0
$$221$$ 6.92820 0.466041
$$222$$ 0 0
$$223$$ −20.9282 −1.40146 −0.700728 0.713428i $$-0.747141\pi$$
−0.700728 + 0.713428i $$0.747141\pi$$
$$224$$ 0 0
$$225$$ 2.46410 0.164273
$$226$$ 0 0
$$227$$ 16.3923 1.08800 0.543998 0.839087i $$-0.316910\pi$$
0.543998 + 0.839087i $$0.316910\pi$$
$$228$$ 0 0
$$229$$ 28.2487 1.86673 0.933364 0.358932i $$-0.116859\pi$$
0.933364 + 0.358932i $$0.116859\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.39230 −0.156725 −0.0783626 0.996925i $$-0.524969\pi$$
−0.0783626 + 0.996925i $$0.524969\pi$$
$$234$$ 0 0
$$235$$ −31.8564 −2.07808
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ 0 0
$$239$$ −4.39230 −0.284115 −0.142057 0.989858i $$-0.545372\pi$$
−0.142057 + 0.989858i $$0.545372\pi$$
$$240$$ 0 0
$$241$$ −12.9282 −0.832779 −0.416389 0.909186i $$-0.636705\pi$$
−0.416389 + 0.909186i $$0.636705\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 2.73205 0.174544
$$246$$ 0 0
$$247$$ 5.46410 0.347672
$$248$$ 0 0
$$249$$ −10.1962 −0.646155
$$250$$ 0 0
$$251$$ 0.339746 0.0214446 0.0107223 0.999943i $$-0.496587\pi$$
0.0107223 + 0.999943i $$0.496587\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 3.46410 0.216930
$$256$$ 0 0
$$257$$ 11.4641 0.715111 0.357556 0.933892i $$-0.383610\pi$$
0.357556 + 0.933892i $$0.383610\pi$$
$$258$$ 0 0
$$259$$ −10.0000 −0.621370
$$260$$ 0 0
$$261$$ −9.66025 −0.597955
$$262$$ 0 0
$$263$$ −19.3205 −1.19135 −0.595677 0.803224i $$-0.703116\pi$$
−0.595677 + 0.803224i $$0.703116\pi$$
$$264$$ 0 0
$$265$$ −18.3923 −1.12983
$$266$$ 0 0
$$267$$ −4.92820 −0.301601
$$268$$ 0 0
$$269$$ 31.1769 1.90089 0.950445 0.310893i $$-0.100628\pi$$
0.950445 + 0.310893i $$0.100628\pi$$
$$270$$ 0 0
$$271$$ −28.7846 −1.74854 −0.874270 0.485440i $$-0.838660\pi$$
−0.874270 + 0.485440i $$0.838660\pi$$
$$272$$ 0 0
$$273$$ 5.46410 0.330702
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 12.3923 0.744581 0.372291 0.928116i $$-0.378572\pi$$
0.372291 + 0.928116i $$0.378572\pi$$
$$278$$ 0 0
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −4.19615 −0.250321 −0.125161 0.992136i $$-0.539945\pi$$
−0.125161 + 0.992136i $$0.539945\pi$$
$$282$$ 0 0
$$283$$ −20.3923 −1.21220 −0.606098 0.795390i $$-0.707266\pi$$
−0.606098 + 0.795390i $$0.707266\pi$$
$$284$$ 0 0
$$285$$ 2.73205 0.161833
$$286$$ 0 0
$$287$$ −8.92820 −0.527015
$$288$$ 0 0
$$289$$ −15.3923 −0.905430
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 21.8564 1.27253
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4.92820 0.284057
$$302$$ 0 0
$$303$$ −1.26795 −0.0728418
$$304$$ 0 0
$$305$$ −5.46410 −0.312874
$$306$$ 0 0
$$307$$ 18.7846 1.07209 0.536047 0.844188i $$-0.319917\pi$$
0.536047 + 0.844188i $$0.319917\pi$$
$$308$$ 0 0
$$309$$ 2.92820 0.166580
$$310$$ 0 0
$$311$$ 18.5885 1.05405 0.527027 0.849848i $$-0.323307\pi$$
0.527027 + 0.849848i $$0.323307\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ 0 0
$$315$$ 2.73205 0.153934
$$316$$ 0 0
$$317$$ 12.1962 0.685004 0.342502 0.939517i $$-0.388726\pi$$
0.342502 + 0.939517i $$0.388726\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 4.73205 0.264117
$$322$$ 0 0
$$323$$ 1.26795 0.0705506
$$324$$ 0 0
$$325$$ −13.4641 −0.746854
$$326$$ 0 0
$$327$$ −2.39230 −0.132295
$$328$$ 0 0
$$329$$ −11.6603 −0.642851
$$330$$ 0 0
$$331$$ 22.2487 1.22290 0.611450 0.791283i $$-0.290587\pi$$
0.611450 + 0.791283i $$0.290587\pi$$
$$332$$ 0 0
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ −36.7846 −2.00976
$$336$$ 0 0
$$337$$ 30.3923 1.65557 0.827787 0.561042i $$-0.189599\pi$$
0.827787 + 0.561042i $$0.189599\pi$$
$$338$$ 0 0
$$339$$ −0.196152 −0.0106535
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.3923 −0.879985 −0.439993 0.898001i $$-0.645019\pi$$
−0.439993 + 0.898001i $$0.645019\pi$$
$$348$$ 0 0
$$349$$ −19.0718 −1.02089 −0.510445 0.859910i $$-0.670519\pi$$
−0.510445 + 0.859910i $$0.670519\pi$$
$$350$$ 0 0
$$351$$ 5.46410 0.291652
$$352$$ 0 0
$$353$$ 16.1962 0.862034 0.431017 0.902344i $$-0.358155\pi$$
0.431017 + 0.902344i $$0.358155\pi$$
$$354$$ 0 0
$$355$$ 12.9282 0.686158
$$356$$ 0 0
$$357$$ 1.26795 0.0671070
$$358$$ 0 0
$$359$$ 30.9282 1.63233 0.816164 0.577820i $$-0.196097\pi$$
0.816164 + 0.577820i $$0.196097\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 28.3923 1.48612
$$366$$ 0 0
$$367$$ 18.5359 0.967566 0.483783 0.875188i $$-0.339262\pi$$
0.483783 + 0.875188i $$0.339262\pi$$
$$368$$ 0 0
$$369$$ −8.92820 −0.464784
$$370$$ 0 0
$$371$$ −6.73205 −0.349511
$$372$$ 0 0
$$373$$ 0.928203 0.0480605 0.0240303 0.999711i $$-0.492350\pi$$
0.0240303 + 0.999711i $$0.492350\pi$$
$$374$$ 0 0
$$375$$ 6.92820 0.357771
$$376$$ 0 0
$$377$$ 52.7846 2.71855
$$378$$ 0 0
$$379$$ −15.3205 −0.786962 −0.393481 0.919333i $$-0.628729\pi$$
−0.393481 + 0.919333i $$0.628729\pi$$
$$380$$ 0 0
$$381$$ −20.3923 −1.04473
$$382$$ 0 0
$$383$$ −11.3205 −0.578451 −0.289225 0.957261i $$-0.593398\pi$$
−0.289225 + 0.957261i $$0.593398\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.92820 0.250515
$$388$$ 0 0
$$389$$ −20.5359 −1.04121 −0.520606 0.853797i $$-0.674294\pi$$
−0.520606 + 0.853797i $$0.674294\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −14.5885 −0.735890
$$394$$ 0 0
$$395$$ 10.9282 0.549858
$$396$$ 0 0
$$397$$ −21.7128 −1.08973 −0.544867 0.838522i $$-0.683420\pi$$
−0.544867 + 0.838522i $$0.683420\pi$$
$$398$$ 0 0
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 0.196152 0.00979538 0.00489769 0.999988i $$-0.498441\pi$$
0.00489769 + 0.999988i $$0.498441\pi$$
$$402$$ 0 0
$$403$$ −10.9282 −0.544373
$$404$$ 0 0
$$405$$ 2.73205 0.135757
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −28.3923 −1.40391 −0.701955 0.712222i $$-0.747689\pi$$
−0.701955 + 0.712222i $$0.747689\pi$$
$$410$$ 0 0
$$411$$ −12.9282 −0.637701
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 27.8564 1.36742
$$416$$ 0 0
$$417$$ 15.3205 0.750249
$$418$$ 0 0
$$419$$ 20.7321 1.01283 0.506413 0.862291i $$-0.330971\pi$$
0.506413 + 0.862291i $$0.330971\pi$$
$$420$$ 0 0
$$421$$ −32.2487 −1.57171 −0.785853 0.618413i $$-0.787776\pi$$
−0.785853 + 0.618413i $$0.787776\pi$$
$$422$$ 0 0
$$423$$ −11.6603 −0.566941
$$424$$ 0 0
$$425$$ −3.12436 −0.151554
$$426$$ 0 0
$$427$$ −2.00000 −0.0967868
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0.0525589 0.00253167 0.00126584 0.999999i $$-0.499597\pi$$
0.00126584 + 0.999999i $$0.499597\pi$$
$$432$$ 0 0
$$433$$ −15.8564 −0.762010 −0.381005 0.924573i $$-0.624422\pi$$
−0.381005 + 0.924573i $$0.624422\pi$$
$$434$$ 0 0
$$435$$ 26.3923 1.26541
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 35.8564 1.71133 0.855666 0.517528i $$-0.173148\pi$$
0.855666 + 0.517528i $$0.173148\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −34.5359 −1.64085 −0.820425 0.571754i $$-0.806263\pi$$
−0.820425 + 0.571754i $$0.806263\pi$$
$$444$$ 0 0
$$445$$ 13.4641 0.638260
$$446$$ 0 0
$$447$$ −14.3923 −0.680733
$$448$$ 0 0
$$449$$ 5.26795 0.248610 0.124305 0.992244i $$-0.460330\pi$$
0.124305 + 0.992244i $$0.460330\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 2.53590 0.119147
$$454$$ 0 0
$$455$$ −14.9282 −0.699845
$$456$$ 0 0
$$457$$ −20.3923 −0.953912 −0.476956 0.878927i $$-0.658260\pi$$
−0.476956 + 0.878927i $$0.658260\pi$$
$$458$$ 0 0
$$459$$ 1.26795 0.0591828
$$460$$ 0 0
$$461$$ 14.3397 0.667869 0.333934 0.942596i $$-0.391624\pi$$
0.333934 + 0.942596i $$0.391624\pi$$
$$462$$ 0 0
$$463$$ 21.8564 1.01575 0.507877 0.861430i $$-0.330431\pi$$
0.507877 + 0.861430i $$0.330431\pi$$
$$464$$ 0 0
$$465$$ −5.46410 −0.253392
$$466$$ 0 0
$$467$$ −0.0525589 −0.00243214 −0.00121607 0.999999i $$-0.500387\pi$$
−0.00121607 + 0.999999i $$0.500387\pi$$
$$468$$ 0 0
$$469$$ −13.4641 −0.621714
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −2.46410 −0.113061
$$476$$ 0 0
$$477$$ −6.73205 −0.308239
$$478$$ 0 0
$$479$$ 27.6603 1.26383 0.631915 0.775038i $$-0.282269\pi$$
0.631915 + 0.775038i $$0.282269\pi$$
$$480$$ 0 0
$$481$$ 54.6410 2.49142
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 5.46410 0.248112
$$486$$ 0 0
$$487$$ 25.8564 1.17167 0.585833 0.810432i $$-0.300768\pi$$
0.585833 + 0.810432i $$0.300768\pi$$
$$488$$ 0 0
$$489$$ 7.07180 0.319798
$$490$$ 0 0
$$491$$ 19.3205 0.871922 0.435961 0.899965i $$-0.356409\pi$$
0.435961 + 0.899965i $$0.356409\pi$$
$$492$$ 0 0
$$493$$ 12.2487 0.551654
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.73205 0.212261
$$498$$ 0 0
$$499$$ −9.85641 −0.441233 −0.220617 0.975361i $$-0.570807\pi$$
−0.220617 + 0.975361i $$0.570807\pi$$
$$500$$ 0 0
$$501$$ −11.3205 −0.505763
$$502$$ 0 0
$$503$$ 4.73205 0.210992 0.105496 0.994420i $$-0.466357\pi$$
0.105496 + 0.994420i $$0.466357\pi$$
$$504$$ 0 0
$$505$$ 3.46410 0.154150
$$506$$ 0 0
$$507$$ −16.8564 −0.748619
$$508$$ 0 0
$$509$$ −8.24871 −0.365618 −0.182809 0.983148i $$-0.558519\pi$$
−0.182809 + 0.983148i $$0.558519\pi$$
$$510$$ 0 0
$$511$$ 10.3923 0.459728
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 16.9282 0.743066
$$520$$ 0 0
$$521$$ 36.2487 1.58808 0.794042 0.607862i $$-0.207973\pi$$
0.794042 + 0.607862i $$0.207973\pi$$
$$522$$ 0 0
$$523$$ −22.0000 −0.961993 −0.480996 0.876723i $$-0.659725\pi$$
−0.480996 + 0.876723i $$0.659725\pi$$
$$524$$ 0 0
$$525$$ −2.46410 −0.107542
$$526$$ 0 0
$$527$$ −2.53590 −0.110465
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ 48.7846 2.11310
$$534$$ 0 0
$$535$$ −12.9282 −0.558935
$$536$$ 0 0
$$537$$ 14.1962 0.612609
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 0 0
$$543$$ −4.92820 −0.211489
$$544$$ 0 0
$$545$$ 6.53590 0.279967
$$546$$ 0 0
$$547$$ −11.7128 −0.500804 −0.250402 0.968142i $$-0.580563\pi$$
−0.250402 + 0.968142i $$0.580563\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 9.66025 0.411541
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ 0 0
$$555$$ 27.3205 1.15969
$$556$$ 0 0
$$557$$ 20.5359 0.870134 0.435067 0.900398i $$-0.356725\pi$$
0.435067 + 0.900398i $$0.356725\pi$$
$$558$$ 0 0
$$559$$ −26.9282 −1.13894
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −25.4641 −1.07318 −0.536592 0.843842i $$-0.680289\pi$$
−0.536592 + 0.843842i $$0.680289\pi$$
$$564$$ 0 0
$$565$$ 0.535898 0.0225454
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 28.5885 1.19849 0.599245 0.800566i $$-0.295467\pi$$
0.599245 + 0.800566i $$0.295467\pi$$
$$570$$ 0 0
$$571$$ −25.8564 −1.08206 −0.541028 0.841004i $$-0.681965\pi$$
−0.541028 + 0.841004i $$0.681965\pi$$
$$572$$ 0 0
$$573$$ 14.9282 0.623635
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ 23.8564 0.991438
$$580$$ 0 0
$$581$$ 10.1962 0.423008
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −14.9282 −0.617205
$$586$$ 0 0
$$587$$ 25.9090 1.06938 0.534689 0.845049i $$-0.320429\pi$$
0.534689 + 0.845049i $$0.320429\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ 27.8038 1.14177 0.570884 0.821031i $$-0.306601\pi$$
0.570884 + 0.821031i $$0.306601\pi$$
$$594$$ 0 0
$$595$$ −3.46410 −0.142014
$$596$$ 0 0
$$597$$ 12.3923 0.507183
$$598$$ 0 0
$$599$$ −7.26795 −0.296960 −0.148480 0.988915i $$-0.547438\pi$$
−0.148480 + 0.988915i $$0.547438\pi$$
$$600$$ 0 0
$$601$$ 34.0000 1.38689 0.693444 0.720510i $$-0.256092\pi$$
0.693444 + 0.720510i $$0.256092\pi$$
$$602$$ 0 0
$$603$$ −13.4641 −0.548301
$$604$$ 0 0
$$605$$ −30.0526 −1.22181
$$606$$ 0 0
$$607$$ −13.0718 −0.530568 −0.265284 0.964170i $$-0.585466\pi$$
−0.265284 + 0.964170i $$0.585466\pi$$
$$608$$ 0 0
$$609$$ 9.66025 0.391453
$$610$$ 0 0
$$611$$ 63.7128 2.57754
$$612$$ 0 0
$$613$$ −18.2487 −0.737059 −0.368529 0.929616i $$-0.620139\pi$$
−0.368529 + 0.929616i $$0.620139\pi$$
$$614$$ 0 0
$$615$$ 24.3923 0.983593
$$616$$ 0 0
$$617$$ −24.6410 −0.992010 −0.496005 0.868320i $$-0.665200\pi$$
−0.496005 + 0.868320i $$0.665200\pi$$
$$618$$ 0 0
$$619$$ −13.1769 −0.529625 −0.264812 0.964300i $$-0.585310\pi$$
−0.264812 + 0.964300i $$0.585310\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4.92820 0.197444
$$624$$ 0 0
$$625$$ −31.2487 −1.24995
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.6795 0.505564
$$630$$ 0 0
$$631$$ 40.9282 1.62933 0.814663 0.579935i $$-0.196922\pi$$
0.814663 + 0.579935i $$0.196922\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 55.7128 2.21090
$$636$$ 0 0
$$637$$ −5.46410 −0.216496
$$638$$ 0 0
$$639$$ 4.73205 0.187197
$$640$$ 0 0
$$641$$ −6.33975 −0.250405 −0.125202 0.992131i $$-0.539958\pi$$
−0.125202 + 0.992131i $$0.539958\pi$$
$$642$$ 0 0
$$643$$ 20.3923 0.804194 0.402097 0.915597i $$-0.368281\pi$$
0.402097 + 0.915597i $$0.368281\pi$$
$$644$$ 0 0
$$645$$ −13.4641 −0.530148
$$646$$ 0 0
$$647$$ −3.66025 −0.143899 −0.0719497 0.997408i $$-0.522922\pi$$
−0.0719497 + 0.997408i $$0.522922\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −2.00000 −0.0783862
$$652$$ 0 0
$$653$$ −42.7846 −1.67429 −0.837146 0.546980i $$-0.815777\pi$$
−0.837146 + 0.546980i $$0.815777\pi$$
$$654$$ 0 0
$$655$$ 39.8564 1.55732
$$656$$ 0 0
$$657$$ 10.3923 0.405442
$$658$$ 0 0
$$659$$ 32.0526 1.24859 0.624295 0.781189i $$-0.285386\pi$$
0.624295 + 0.781189i $$0.285386\pi$$
$$660$$ 0 0
$$661$$ −6.53590 −0.254217 −0.127108 0.991889i $$-0.540570\pi$$
−0.127108 + 0.991889i $$0.540570\pi$$
$$662$$ 0 0
$$663$$ −6.92820 −0.269069
$$664$$ 0 0
$$665$$ −2.73205 −0.105944
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 20.9282 0.809131
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −7.07180 −0.272598 −0.136299 0.990668i $$-0.543521\pi$$
−0.136299 + 0.990668i $$0.543521\pi$$
$$674$$ 0 0
$$675$$ −2.46410 −0.0948433
$$676$$ 0 0
$$677$$ 29.3205 1.12688 0.563439 0.826157i $$-0.309478\pi$$
0.563439 + 0.826157i $$0.309478\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ −16.3923 −0.628154
$$682$$ 0 0
$$683$$ 4.73205 0.181067 0.0905334 0.995893i $$-0.471143\pi$$
0.0905334 + 0.995893i $$0.471143\pi$$
$$684$$ 0 0
$$685$$ 35.3205 1.34953
$$686$$ 0 0
$$687$$ −28.2487 −1.07776
$$688$$ 0 0
$$689$$ 36.7846 1.40138
$$690$$ 0 0
$$691$$ −8.78461 −0.334182 −0.167091 0.985941i $$-0.553437\pi$$
−0.167091 + 0.985941i $$0.553437\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −41.8564 −1.58770
$$696$$ 0 0
$$697$$ 11.3205 0.428795
$$698$$ 0 0
$$699$$ 2.39230 0.0904853
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ 10.0000 0.377157
$$704$$ 0 0
$$705$$ 31.8564 1.19978
$$706$$ 0 0
$$707$$ 1.26795 0.0476861
$$708$$ 0 0
$$709$$ 0.392305 0.0147333 0.00736666 0.999973i $$-0.497655\pi$$
0.00736666 + 0.999973i $$0.497655\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 4.39230 0.164034
$$718$$ 0 0
$$719$$ 46.9808 1.75209 0.876043 0.482232i $$-0.160174\pi$$
0.876043 + 0.482232i $$0.160174\pi$$
$$720$$ 0 0
$$721$$ −2.92820 −0.109052
$$722$$ 0 0
$$723$$ 12.9282 0.480805
$$724$$ 0 0
$$725$$ −23.8038 −0.884053
$$726$$ 0 0
$$727$$ 18.9282 0.702008 0.351004 0.936374i $$-0.385840\pi$$
0.351004 + 0.936374i $$0.385840\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −6.24871 −0.231117
$$732$$ 0 0
$$733$$ 4.24871 0.156930 0.0784649 0.996917i $$-0.474998\pi$$
0.0784649 + 0.996917i $$0.474998\pi$$
$$734$$ 0 0
$$735$$ −2.73205 −0.100773
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −20.1436 −0.740994 −0.370497 0.928834i $$-0.620813\pi$$
−0.370497 + 0.928834i $$0.620813\pi$$
$$740$$ 0 0
$$741$$ −5.46410 −0.200729
$$742$$ 0 0
$$743$$ −8.73205 −0.320348 −0.160174 0.987089i $$-0.551206\pi$$
−0.160174 + 0.987089i $$0.551206\pi$$
$$744$$ 0 0
$$745$$ 39.3205 1.44059
$$746$$ 0 0
$$747$$ 10.1962 0.373058
$$748$$ 0 0
$$749$$ −4.73205 −0.172905
$$750$$ 0 0
$$751$$ −49.1769 −1.79449 −0.897246 0.441532i $$-0.854435\pi$$
−0.897246 + 0.441532i $$0.854435\pi$$
$$752$$ 0 0
$$753$$ −0.339746 −0.0123810
$$754$$ 0 0
$$755$$ −6.92820 −0.252143
$$756$$ 0 0
$$757$$ −22.7846 −0.828121 −0.414060 0.910249i $$-0.635890\pi$$
−0.414060 + 0.910249i $$0.635890\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.6603 1.22018 0.610092 0.792331i $$-0.291133\pi$$
0.610092 + 0.792331i $$0.291133\pi$$
$$762$$ 0 0
$$763$$ 2.39230 0.0866073
$$764$$ 0 0
$$765$$ −3.46410 −0.125245
$$766$$ 0 0
$$767$$ −43.7128 −1.57838
$$768$$ 0 0
$$769$$ 0.143594 0.00517812 0.00258906 0.999997i $$-0.499176\pi$$
0.00258906 + 0.999997i $$0.499176\pi$$
$$770$$ 0 0
$$771$$ −11.4641 −0.412870
$$772$$ 0 0
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 0 0
$$775$$ 4.92820 0.177026
$$776$$ 0 0
$$777$$ 10.0000 0.358748
$$778$$ 0 0
$$779$$ 8.92820 0.319886
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 9.66025 0.345229
$$784$$ 0 0
$$785$$ −38.2487 −1.36516
$$786$$ 0 0
$$787$$ 27.8564 0.992974 0.496487 0.868044i $$-0.334623\pi$$
0.496487 + 0.868044i $$0.334623\pi$$
$$788$$ 0 0
$$789$$ 19.3205 0.687828
$$790$$ 0 0
$$791$$ 0.196152 0.00697438
$$792$$ 0 0
$$793$$ 10.9282 0.388072
$$794$$ 0 0
$$795$$ 18.3923 0.652308
$$796$$ 0 0
$$797$$ −10.7846 −0.382010 −0.191005 0.981589i $$-0.561175\pi$$
−0.191005 + 0.981589i $$0.561175\pi$$
$$798$$ 0 0
$$799$$ 14.7846 0.523042
$$800$$ 0 0
$$801$$ 4.92820 0.174129
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −31.1769 −1.09748
$$808$$ 0 0
$$809$$ −10.7846 −0.379167 −0.189583 0.981865i $$-0.560714\pi$$
−0.189583 + 0.981865i $$0.560714\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 0 0
$$813$$ 28.7846 1.00952
$$814$$ 0 0
$$815$$ −19.3205 −0.676768
$$816$$ 0 0
$$817$$ −4.92820 −0.172416
$$818$$ 0 0
$$819$$ −5.46410 −0.190931
$$820$$ 0 0
$$821$$ 35.4641 1.23771 0.618853 0.785507i $$-0.287598\pi$$
0.618853 + 0.785507i $$0.287598\pi$$
$$822$$ 0 0
$$823$$ 20.6410 0.719501 0.359750 0.933049i $$-0.382862\pi$$
0.359750 + 0.933049i $$0.382862\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −53.9090 −1.87460 −0.937299 0.348526i $$-0.886682\pi$$
−0.937299 + 0.348526i $$0.886682\pi$$
$$828$$ 0 0
$$829$$ −37.7128 −1.30982 −0.654910 0.755707i $$-0.727294\pi$$
−0.654910 + 0.755707i $$0.727294\pi$$
$$830$$ 0 0
$$831$$ −12.3923 −0.429884
$$832$$ 0 0
$$833$$ −1.26795 −0.0439318
$$834$$ 0 0
$$835$$ 30.9282 1.07031
$$836$$ 0 0
$$837$$ −2.00000 −0.0691301
$$838$$ 0 0
$$839$$ −28.7846 −0.993755 −0.496878 0.867821i $$-0.665520\pi$$
−0.496878 + 0.867821i $$0.665520\pi$$
$$840$$ 0 0
$$841$$ 64.3205 2.21795
$$842$$ 0 0
$$843$$ 4.19615 0.144523
$$844$$ 0 0
$$845$$ 46.0526 1.58426
$$846$$ 0 0
$$847$$ −11.0000 −0.377964
$$848$$ 0 0
$$849$$ 20.3923 0.699862
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −16.1436 −0.552746 −0.276373 0.961050i $$-0.589133\pi$$
−0.276373 + 0.961050i $$0.589133\pi$$
$$854$$ 0 0
$$855$$ −2.73205 −0.0934342
$$856$$ 0 0
$$857$$ 16.9282 0.578256 0.289128 0.957290i $$-0.406635\pi$$
0.289128 + 0.957290i $$0.406635\pi$$
$$858$$ 0 0
$$859$$ 45.8564 1.56460 0.782300 0.622902i $$-0.214046\pi$$
0.782300 + 0.622902i $$0.214046\pi$$
$$860$$ 0 0
$$861$$ 8.92820 0.304272
$$862$$ 0 0
$$863$$ −42.9808 −1.46308 −0.731541 0.681797i $$-0.761199\pi$$
−0.731541 + 0.681797i $$0.761199\pi$$
$$864$$ 0 0
$$865$$ −46.2487 −1.57250
$$866$$ 0 0
$$867$$ 15.3923 0.522750
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 73.5692 2.49280
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ −6.92820 −0.234216
$$876$$ 0 0
$$877$$ −53.7128 −1.81375 −0.906876 0.421397i $$-0.861540\pi$$
−0.906876 + 0.421397i $$0.861540\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −2.73205 −0.0920451 −0.0460226 0.998940i $$-0.514655\pi$$
−0.0460226 + 0.998940i $$0.514655\pi$$
$$882$$ 0 0
$$883$$ 9.07180 0.305290 0.152645 0.988281i $$-0.451221\pi$$
0.152645 + 0.988281i $$0.451221\pi$$
$$884$$ 0 0
$$885$$ −21.8564 −0.734695
$$886$$ 0 0
$$887$$ 42.2487 1.41857 0.709286 0.704920i $$-0.249017\pi$$
0.709286 + 0.704920i $$0.249017\pi$$
$$888$$ 0 0
$$889$$ 20.3923 0.683936
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 11.6603 0.390196
$$894$$ 0 0
$$895$$ −38.7846 −1.29643
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −19.3205 −0.644375
$$900$$ 0 0
$$901$$ 8.53590 0.284372
$$902$$ 0 0
$$903$$ −4.92820 −0.164000
$$904$$ 0 0
$$905$$ 13.4641 0.447562
$$906$$ 0 0
$$907$$ 14.6410 0.486147 0.243073 0.970008i $$-0.421844\pi$$
0.243073 + 0.970008i $$0.421844\pi$$
$$908$$ 0 0
$$909$$ 1.26795 0.0420552
$$910$$ 0 0
$$911$$ −12.7321 −0.421832 −0.210916 0.977504i $$-0.567645\pi$$
−0.210916 + 0.977504i $$0.567645\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 5.46410 0.180638
$$916$$ 0 0
$$917$$ 14.5885 0.481753
$$918$$ 0 0
$$919$$ −29.8564 −0.984872 −0.492436 0.870349i $$-0.663893\pi$$
−0.492436 + 0.870349i $$0.663893\pi$$
$$920$$ 0 0
$$921$$ −18.7846 −0.618974
$$922$$ 0 0
$$923$$ −25.8564 −0.851074
$$924$$ 0 0
$$925$$ −24.6410 −0.810192
$$926$$ 0 0
$$927$$ −2.92820 −0.0961748
$$928$$ 0 0
$$929$$ −4.98076 −0.163414 −0.0817068 0.996656i $$-0.526037\pi$$
−0.0817068 + 0.996656i $$0.526037\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ −18.5885 −0.608559
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 48.6410 1.58903 0.794516 0.607243i $$-0.207724\pi$$
0.794516 + 0.607243i $$0.207724\pi$$
$$938$$ 0 0
$$939$$ 2.00000 0.0652675
$$940$$ 0 0
$$941$$ −51.4641 −1.67768 −0.838841 0.544377i $$-0.816766\pi$$
−0.838841 + 0.544377i $$0.816766\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −2.73205 −0.0888736
$$946$$ 0 0
$$947$$ 9.07180 0.294794 0.147397 0.989077i $$-0.452911\pi$$
0.147397 + 0.989077i $$0.452911\pi$$
$$948$$ 0 0
$$949$$ −56.7846 −1.84331
$$950$$ 0 0
$$951$$ −12.1962 −0.395487
$$952$$ 0 0
$$953$$ 40.1962 1.30208 0.651041 0.759043i $$-0.274333\pi$$
0.651041 + 0.759043i $$0.274333\pi$$
$$954$$ 0 0
$$955$$ −40.7846 −1.31976
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12.9282 0.417473
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −4.73205 −0.152488
$$964$$ 0 0
$$965$$ −65.1769 −2.09812
$$966$$ 0 0
$$967$$ −36.6410 −1.17830 −0.589148 0.808025i $$-0.700536\pi$$
−0.589148 + 0.808025i $$0.700536\pi$$
$$968$$ 0 0
$$969$$ −1.26795 −0.0407324
$$970$$ 0 0
$$971$$ 4.28719 0.137582 0.0687912 0.997631i $$-0.478086\pi$$
0.0687912 + 0.997631i $$0.478086\pi$$
$$972$$ 0 0
$$973$$ −15.3205 −0.491153
$$974$$ 0 0
$$975$$ 13.4641 0.431196
$$976$$ 0 0
$$977$$ −28.3013 −0.905438 −0.452719 0.891653i $$-0.649546\pi$$
−0.452719 + 0.891653i $$0.649546\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.39230 0.0763804
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ −49.1769 −1.56691
$$986$$ 0 0
$$987$$ 11.6603 0.371150
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −12.7846 −0.406117 −0.203058 0.979167i $$-0.565088\pi$$
−0.203058 + 0.979167i $$0.565088\pi$$
$$992$$ 0 0
$$993$$ −22.2487 −0.706042
$$994$$ 0 0
$$995$$ −33.8564 −1.07332
$$996$$ 0 0
$$997$$ −19.8564 −0.628859 −0.314429 0.949281i $$-0.601813\pi$$
−0.314429 + 0.949281i $$0.601813\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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 3192.2.a.q.1.2 2
3.2 odd 2 9576.2.a.bj.1.1 2
4.3 odd 2 6384.2.a.bs.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.q.1.2 2 1.1 even 1 trivial
6384.2.a.bs.1.2 2 4.3 odd 2
9576.2.a.bj.1.1 2 3.2 odd 2