Properties

 Label 75.2.e.b.68.1 Level $75$ Weight $2$ Character 75.68 Analytic conductor $0.599$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

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Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 75.e (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.598878015160$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

 Embedding label 68.1 Root $$-1.22474 - 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.68 Dual form 75.2.e.b.32.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.22474 - 1.22474i) q^{3} -2.00000i q^{4} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +O(q^{10})$$ $$q+(-1.22474 - 1.22474i) q^{3} -2.00000i q^{4} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +(-2.44949 + 2.44949i) q^{12} +(3.67423 + 3.67423i) q^{13} -4.00000 q^{16} +1.00000i q^{19} -3.00000 q^{21} +(3.67423 - 3.67423i) q^{27} +(-2.44949 - 2.44949i) q^{28} +7.00000 q^{31} +6.00000 q^{36} +(-4.89898 + 4.89898i) q^{37} -9.00000i q^{39} +(-8.57321 - 8.57321i) q^{43} +(4.89898 + 4.89898i) q^{48} +4.00000i q^{49} +(7.34847 - 7.34847i) q^{52} +(1.22474 - 1.22474i) q^{57} -13.0000 q^{61} +(3.67423 + 3.67423i) q^{63} +8.00000i q^{64} +(-11.0227 + 11.0227i) q^{67} +(9.79796 + 9.79796i) q^{73} +2.00000 q^{76} -4.00000i q^{79} -9.00000 q^{81} +6.00000i q^{84} +9.00000 q^{91} +(-8.57321 - 8.57321i) q^{93} +(13.4722 - 13.4722i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 16 q^{16} - 12 q^{21} + 28 q^{31} + 24 q^{36} - 52 q^{61} + 8 q^{76} - 36 q^{81} + 36 q^{91}+O(q^{100})$$ 4 * q - 16 * q^16 - 12 * q^21 + 28 * q^31 + 24 * q^36 - 52 * q^61 + 8 * q^76 - 36 * q^81 + 36 * q^91

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$e\left(\frac{3}{4}\right)$$

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 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ −1.22474 1.22474i −0.707107 0.707107i
$$4$$ 2.00000i 1.00000i
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.22474 1.22474i 0.462910 0.462910i −0.436698 0.899608i $$-0.643852\pi$$
0.899608 + 0.436698i $$0.143852\pi$$
$$8$$ 0 0
$$9$$ 3.00000i 1.00000i
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −2.44949 + 2.44949i −0.707107 + 0.707107i
$$13$$ 3.67423 + 3.67423i 1.01905 + 1.01905i 0.999815 + 0.0192343i $$0.00612285\pi$$
0.0192343 + 0.999815i $$0.493877\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$18$$ 0 0
$$19$$ 1.00000i 0.229416i 0.993399 + 0.114708i $$0.0365932\pi$$
−0.993399 + 0.114708i $$0.963407\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.67423 3.67423i 0.707107 0.707107i
$$28$$ −2.44949 2.44949i −0.462910 0.462910i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 6.00000 1.00000
$$37$$ −4.89898 + 4.89898i −0.805387 + 0.805387i −0.983932 0.178545i $$-0.942861\pi$$
0.178545 + 0.983932i $$0.442861\pi$$
$$38$$ 0 0
$$39$$ 9.00000i 1.44115i
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ −8.57321 8.57321i −1.30740 1.30740i −0.923283 0.384120i $$-0.874505\pi$$
−0.384120 0.923283i $$-0.625495\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$48$$ 4.89898 + 4.89898i 0.707107 + 0.707107i
$$49$$ 4.00000i 0.571429i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 7.34847 7.34847i 1.01905 1.01905i
$$53$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.22474 1.22474i 0.162221 0.162221i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 0 0
$$63$$ 3.67423 + 3.67423i 0.462910 + 0.462910i
$$64$$ 8.00000i 1.00000i
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −11.0227 + 11.0227i −1.34664 + 1.34664i −0.457352 + 0.889286i $$0.651202\pi$$
−0.889286 + 0.457352i $$0.848798\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 9.79796 + 9.79796i 1.14676 + 1.14676i 0.987185 + 0.159579i $$0.0510137\pi$$
0.159579 + 0.987185i $$0.448986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000i 0.450035i −0.974355 0.225018i $$-0.927756\pi$$
0.974355 0.225018i $$-0.0722440\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$84$$ 6.00000i 0.654654i
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 9.00000 0.943456
$$92$$ 0 0
$$93$$ −8.57321 8.57321i −0.889001 0.889001i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 13.4722 13.4722i 1.36789 1.36789i 0.504457 0.863437i $$-0.331693\pi$$
0.863437 0.504457i $$-0.168307\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ −2.44949 2.44949i −0.241355 0.241355i 0.576055 0.817411i $$-0.304591\pi$$
−0.817411 + 0.576055i $$0.804591\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$108$$ −7.34847 7.34847i −0.707107 0.707107i
$$109$$ 19.0000i 1.81987i −0.414751 0.909935i $$-0.636131\pi$$
0.414751 0.909935i $$-0.363869\pi$$
$$110$$ 0 0
$$111$$ 12.0000 1.13899
$$112$$ −4.89898 + 4.89898i −0.462910 + 0.462910i
$$113$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −11.0227 + 11.0227i −1.01905 + 1.01905i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 14.0000i 1.25724i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 7.34847 7.34847i 0.652071 0.652071i −0.301420 0.953491i $$-0.597461\pi$$
0.953491 + 0.301420i $$0.0974607\pi$$
$$128$$ 0 0
$$129$$ 21.0000i 1.84895i
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 1.22474 + 1.22474i 0.106199 + 0.106199i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$138$$ 0 0
$$139$$ 16.0000i 1.35710i 0.734553 + 0.678551i $$0.237392\pi$$
−0.734553 + 0.678551i $$0.762608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 12.0000i 1.00000i
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.89898 4.89898i 0.404061 0.404061i
$$148$$ 9.79796 + 9.79796i 0.805387 + 0.805387i
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −23.0000 −1.87171 −0.935857 0.352381i $$-0.885372\pi$$
−0.935857 + 0.352381i $$0.885372\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −18.0000 −1.44115
$$157$$ 1.22474 1.22474i 0.0977453 0.0977453i −0.656543 0.754288i $$-0.727982\pi$$
0.754288 + 0.656543i $$0.227982\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3.67423 + 3.67423i 0.287788 + 0.287788i 0.836205 0.548417i $$-0.184769\pi$$
−0.548417 + 0.836205i $$0.684769\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$168$$ 0 0
$$169$$ 14.0000i 1.07692i
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ −17.1464 + 17.1464i −1.30740 + 1.30740i
$$173$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ 0 0
$$183$$ 15.9217 + 15.9217i 1.17696 + 1.17696i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 9.00000i 0.654654i
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 9.79796 9.79796i 0.707107 0.707107i
$$193$$ −8.57321 8.57321i −0.617113 0.617113i 0.327677 0.944790i $$-0.393734\pi$$
−0.944790 + 0.327677i $$0.893734\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 8.00000 0.571429
$$197$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$198$$ 0 0
$$199$$ 11.0000i 0.779769i 0.920864 + 0.389885i $$0.127485\pi$$
−0.920864 + 0.389885i $$0.872515\pi$$
$$200$$ 0 0
$$201$$ 27.0000 1.90443
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −14.6969 14.6969i −1.01905 1.01905i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 8.57321 8.57321i 0.581988 0.581988i
$$218$$ 0 0
$$219$$ 24.0000i 1.62177i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −20.8207 20.8207i −1.39425 1.39425i −0.815506 0.578749i $$-0.803541\pi$$
−0.578749 0.815506i $$-0.696459\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$228$$ −2.44949 2.44949i −0.162221 0.162221i
$$229$$ 29.0000i 1.91637i −0.286143 0.958187i $$-0.592373\pi$$
0.286143 0.958187i $$-0.407627\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4.89898 + 4.89898i −0.318223 + 0.318223i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 17.0000 1.09507 0.547533 0.836784i $$-0.315567\pi$$
0.547533 + 0.836784i $$0.315567\pi$$
$$242$$ 0 0
$$243$$ 11.0227 + 11.0227i 0.707107 + 0.707107i
$$244$$ 26.0000i 1.66448i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.67423 + 3.67423i −0.233786 + 0.233786i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 7.34847 7.34847i 0.462910 0.462910i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$258$$ 0 0
$$259$$ 12.0000i 0.745644i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 22.0454 + 22.0454i 1.34664 + 1.34664i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 0 0
$$273$$ −11.0227 11.0227i −0.667124 0.667124i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −23.2702 + 23.2702i −1.39817 + 1.39817i −0.592869 + 0.805299i $$0.702005\pi$$
−0.805299 + 0.592869i $$0.797995\pi$$
$$278$$ 0 0
$$279$$ 21.0000i 1.25724i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 15.9217 + 15.9217i 0.946446 + 0.946446i 0.998637 0.0521913i $$-0.0166205\pi$$
−0.0521913 + 0.998637i $$0.516621\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000i 1.00000i
$$290$$ 0 0
$$291$$ −33.0000 −1.93449
$$292$$ 19.5959 19.5959i 1.14676 1.14676i
$$293$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −21.0000 −1.21042
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 4.00000i 0.229416i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.22474 1.22474i 0.0698999 0.0698999i −0.671293 0.741192i $$-0.734261\pi$$
0.741192 + 0.671293i $$0.234261\pi$$
$$308$$ 0 0
$$309$$ 6.00000i 0.341328i
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 3.67423 + 3.67423i 0.207680 + 0.207680i 0.803281 0.595601i $$-0.203086\pi$$
−0.595601 + 0.803281i $$0.703086\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 18.0000i 1.00000i
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −23.2702 + 23.2702i −1.28684 + 1.28684i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ 0 0
$$333$$ −14.6969 14.6969i −0.805387 0.805387i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 12.0000 0.654654
$$337$$ 25.7196 25.7196i 1.40104 1.40104i 0.604223 0.796815i $$-0.293484\pi$$
0.796815 0.604223i $$-0.206516\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 13.4722 + 13.4722i 0.727430 + 0.727430i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$348$$ 0 0
$$349$$ 14.0000i 0.749403i −0.927146 0.374701i $$-0.877745\pi$$
0.927146 0.374701i $$-0.122255\pi$$
$$350$$ 0 0
$$351$$ 27.0000 1.44115
$$352$$ 0 0
$$353$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 18.0000 0.947368
$$362$$ 0 0
$$363$$ −13.4722 13.4722i −0.707107 0.707107i
$$364$$ 18.0000i 0.943456i
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −11.0227 + 11.0227i −0.575380 + 0.575380i −0.933627 0.358247i $$-0.883375\pi$$
0.358247 + 0.933627i $$0.383375\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −17.1464 + 17.1464i −0.889001 + 0.889001i
$$373$$ −20.8207 20.8207i −1.07805 1.07805i −0.996684 0.0813690i $$-0.974071\pi$$
−0.0813690 0.996684i $$-0.525929\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 29.0000i 1.48963i −0.667271 0.744815i $$-0.732538\pi$$
0.667271 0.744815i $$-0.267462\pi$$
$$380$$ 0 0
$$381$$ −18.0000 −0.922168
$$382$$ 0 0
$$383$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 25.7196 25.7196i 1.30740 1.30740i
$$388$$ −26.9444 26.9444i −1.36789 1.36789i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 13.4722 13.4722i 0.676150 0.676150i −0.282977 0.959127i $$-0.591322\pi$$
0.959127 + 0.282977i $$0.0913219\pi$$
$$398$$ 0 0
$$399$$ 3.00000i 0.150188i
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 25.7196 + 25.7196i 1.28119 + 1.28119i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 31.0000i 1.53285i 0.642333 + 0.766426i $$0.277967\pi$$
−0.642333 + 0.766426i $$0.722033\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −4.89898 + 4.89898i −0.241355 + 0.241355i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 19.5959 19.5959i 0.959616 0.959616i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −15.9217 + 15.9217i −0.770504 + 0.770504i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ −14.6969 + 14.6969i −0.707107 + 0.707107i
$$433$$ 15.9217 + 15.9217i 0.765147 + 0.765147i 0.977248 0.212101i $$-0.0680304\pi$$
−0.212101 + 0.977248i $$0.568030\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −38.0000 −1.81987
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 41.0000i 1.95682i 0.206666 + 0.978412i $$0.433739\pi$$
−0.206666 + 0.978412i $$0.566261\pi$$
$$440$$ 0 0
$$441$$ −12.0000 −0.571429
$$442$$ 0 0
$$443$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$444$$ 24.0000i 1.13899i
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 9.79796 + 9.79796i 0.462910 + 0.462910i
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 28.1691 + 28.1691i 1.32350 + 1.32350i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −29.3939 + 29.3939i −1.37499 + 1.37499i −0.522108 + 0.852879i $$0.674854\pi$$
−0.852879 + 0.522108i $$0.825146\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ −26.9444 26.9444i −1.25221 1.25221i −0.954726 0.297486i $$-0.903852\pi$$
−0.297486 0.954726i $$-0.596148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$468$$ 22.0454 + 22.0454i 1.01905 + 1.01905i
$$469$$ 27.0000i 1.24674i
$$470$$ 0 0
$$471$$ −3.00000 −0.138233
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 22.0000i 1.00000i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 25.7196 25.7196i 1.16547 1.16547i 0.182208 0.983260i $$-0.441675\pi$$
0.983260 0.182208i $$-0.0583245\pi$$
$$488$$ 0 0
$$489$$ 9.00000i 0.406994i
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −28.0000 −1.25724
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 11.0000i 0.492428i 0.969216 + 0.246214i $$0.0791865\pi$$
−0.969216 + 0.246214i $$0.920813\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 17.1464 17.1464i 0.761500 0.761500i
$$508$$ −14.6969 14.6969i −0.652071 0.652071i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ 0 0
$$513$$ 3.67423 + 3.67423i 0.162221 + 0.162221i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 42.0000 1.84895
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ −20.8207 20.8207i −0.910424 0.910424i 0.0858814 0.996305i $$-0.472629\pi$$
−0.996305 + 0.0858814i $$0.972629\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000i 1.00000i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.44949 2.44949i 0.106199 0.106199i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ 0 0
$$543$$ −8.57321 8.57321i −0.367912 0.367912i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −17.1464 + 17.1464i −0.733128 + 0.733128i −0.971238 0.238110i $$-0.923472\pi$$
0.238110 + 0.971238i $$0.423472\pi$$
$$548$$ 0 0
$$549$$ 39.0000i 1.66448i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −4.89898 4.89898i −0.208326 0.208326i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 32.0000 1.35710
$$557$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$558$$ 0 0
$$559$$ 63.0000i 2.66462i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −11.0227 + 11.0227i −0.462910 + 0.462910i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 47.0000 1.96689 0.983444 0.181210i $$-0.0580014\pi$$
0.983444 + 0.181210i $$0.0580014\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −24.0000 −1.00000
$$577$$ −23.2702 + 23.2702i −0.968749 + 0.968749i −0.999526 0.0307771i $$-0.990202\pi$$
0.0307771 + 0.999526i $$0.490202\pi$$
$$578$$ 0 0
$$579$$ 21.0000i 0.872730i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$588$$ −9.79796 9.79796i −0.404061 0.404061i
$$589$$ 7.00000i 0.288430i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 19.5959 19.5959i 0.805387 0.805387i
$$593$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 13.4722 13.4722i 0.551380 0.551380i
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −23.0000 −0.938190 −0.469095 0.883148i $$-0.655420\pi$$
−0.469095 + 0.883148i $$0.655420\pi$$
$$602$$ 0 0
$$603$$ −33.0681 33.0681i −1.34664 1.34664i
$$604$$ 46.0000i 1.87171i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 31.8434 31.8434i 1.29248 1.29248i 0.359235 0.933247i $$-0.383038\pi$$
0.933247 0.359235i $$-0.116962\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 34.2929 + 34.2929i 1.38508 + 1.38508i 0.835337 + 0.549739i $$0.185273\pi$$
0.549739 + 0.835337i $$0.314727\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$618$$ 0 0
$$619$$ 49.0000i 1.96948i −0.174042 0.984738i $$-0.555683\pi$$
0.174042 0.984738i $$-0.444317\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 36.0000i 1.44115i
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −2.44949 2.44949i −0.0977453 0.0977453i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −43.0000 −1.71180 −0.855901 0.517139i $$-0.826997\pi$$
−0.855901 + 0.517139i $$0.826997\pi$$
$$632$$ 0 0
$$633$$ 15.9217 + 15.9217i 0.632830 + 0.632830i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −14.6969 + 14.6969i −0.582314 + 0.582314i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 22.0454 + 22.0454i 0.869386 + 0.869386i 0.992404 0.123018i $$-0.0392574\pi$$
−0.123018 + 0.992404i $$0.539257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −21.0000 −0.823055
$$652$$ 7.34847 7.34847i 0.287788 0.287788i
$$653$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −29.3939 + 29.3939i −1.14676 + 1.14676i
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 51.0000i 1.97177i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 9.79796 + 9.79796i 0.377684 + 0.377684i 0.870266 0.492582i $$-0.163947\pi$$
−0.492582 + 0.870266i $$0.663947\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 28.0000 1.07692
$$677$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$678$$ 0 0
$$679$$ 33.0000i 1.26642i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$684$$ 6.00000i 0.229416i
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −35.5176 + 35.5176i −1.35508 + 1.35508i
$$688$$ 34.2929 + 34.2929i 1.30740 + 1.30740i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ −4.89898 4.89898i −0.184769 0.184769i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 31.0000i 1.16423i 0.813107 + 0.582115i $$0.197775\pi$$
−0.813107 + 0.582115i $$0.802225\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −6.00000 −0.223452
$$722$$ 0 0
$$723$$ −20.8207 20.8207i −0.774329 0.774329i
$$724$$ 14.0000i 0.520306i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 37.9671 37.9671i 1.40812 1.40812i 0.638498 0.769624i $$-0.279556\pi$$
0.769624 0.638498i $$-0.220444\pi$$
$$728$$ 0 0
$$729$$ 27.0000i 1.00000i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 31.8434 31.8434i 1.17696 1.17696i
$$733$$ −14.6969 14.6969i −0.542844 0.542844i 0.381518 0.924362i $$-0.375402\pi$$
−0.924362 + 0.381518i $$0.875402\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 16.0000i 0.588570i 0.955718 + 0.294285i $$0.0950814\pi$$
−0.955718 + 0.294285i $$0.904919\pi$$
$$740$$ 0 0
$$741$$ 9.00000 0.330623
$$742$$ 0 0
$$743$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 52.0000 1.89751 0.948753 0.316017i $$-0.102346\pi$$
0.948753 + 0.316017i $$0.102346\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −18.0000 −0.654654
$$757$$ 1.22474 1.22474i 0.0445141 0.0445141i −0.684499 0.729013i $$-0.739979\pi$$
0.729013 + 0.684499i $$0.239979\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ −23.2702 23.2702i −0.842436 0.842436i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −19.5959 19.5959i −0.707107 0.707107i
$$769$$ 49.0000i 1.76699i −0.468445 0.883493i $$-0.655186\pi$$
0.468445 0.883493i $$-0.344814\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −17.1464 + 17.1464i −0.617113 + 0.617113i
$$773$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 14.6969 14.6969i 0.527250 0.527250i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 16.0000i 0.571429i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −35.5176 + 35.5176i −1.26607 + 1.26607i −0.317962 + 0.948103i $$0.602999\pi$$
−0.948103 + 0.317962i $$0.897001\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −47.7650 47.7650i −1.69619 1.69619i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 22.0000 0.779769
$$797$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 54.0000i 1.90443i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0