# Properties

 Label 75.9.d.a.74.1 Level $75$ Weight $9$ Character 75.74 Analytic conductor $30.553$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 74.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.74 Dual form 75.9.d.a.74.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-81.0000i q^{3} -256.000 q^{4} -4273.00i q^{7} -6561.00 q^{9} +O(q^{10})$$ $$q-81.0000i q^{3} -256.000 q^{4} -4273.00i q^{7} -6561.00 q^{9} +20736.0i q^{12} -56447.0i q^{13} +65536.0 q^{16} -157967. q^{19} -346113. q^{21} +531441. i q^{27} +1.09389e6i q^{28} +1.22597e6 q^{31} +1.67962e6 q^{36} +503522. i q^{37} -4.57221e6 q^{39} +6.83707e6i q^{43} -5.30842e6i q^{48} -1.24937e7 q^{49} +1.44504e7i q^{52} +1.27953e7i q^{57} -307393. q^{61} +2.80352e7i q^{63} -1.67772e7 q^{64} -3.18748e7i q^{67} -1.61693e7i q^{73} +4.04396e7 q^{76} +1.88870e7 q^{79} +4.30467e7 q^{81} +8.86049e7 q^{84} -2.41198e8 q^{91} -9.93033e7i q^{93} -8.21325e7i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 512 q^{4} - 13122 q^{9}+O(q^{10})$$ 2 * q - 512 * q^4 - 13122 * q^9 $$2 q - 512 q^{4} - 13122 q^{9} + 131072 q^{16} - 315934 q^{19} - 692226 q^{21} + 2451934 q^{31} + 3359232 q^{36} - 9144414 q^{39} - 24987456 q^{49} - 614786 q^{61} - 33554432 q^{64} + 80879104 q^{76} + 37774076 q^{79} + 86093442 q^{81} + 177209856 q^{84} - 482396062 q^{91}+O(q^{100})$$ 2 * q - 512 * q^4 - 13122 * q^9 + 131072 * q^16 - 315934 * q^19 - 692226 * q^21 + 2451934 * q^31 + 3359232 * q^36 - 9144414 * q^39 - 24987456 * q^49 - 614786 * q^61 - 33554432 * q^64 + 80879104 * q^76 + 37774076 * q^79 + 86093442 * q^81 + 177209856 * q^84 - 482396062 * q^91

## Character values

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

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ − 81.0000i − 1.00000i
$$4$$ −256.000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4273.00i − 1.77968i −0.456277 0.889838i $$-0.650818\pi$$
0.456277 0.889838i $$-0.349182\pi$$
$$8$$ 0 0
$$9$$ −6561.00 −1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 20736.0i 1.00000i
$$13$$ − 56447.0i − 1.97637i −0.153277 0.988183i $$-0.548983\pi$$
0.153277 0.988183i $$-0.451017\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 65536.0 1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −157967. −1.21214 −0.606069 0.795412i $$-0.707254\pi$$
−0.606069 + 0.795412i $$0.707254\pi$$
$$20$$ 0 0
$$21$$ −346113. −1.77968
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 531441.i 1.00000i
$$28$$ 1.09389e6i 1.77968i
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 1.22597e6 1.32749 0.663746 0.747958i $$-0.268966\pi$$
0.663746 + 0.747958i $$0.268966\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.67962e6 1.00000
$$37$$ 503522.i 0.268665i 0.990936 + 0.134333i $$0.0428891\pi$$
−0.990936 + 0.134333i $$0.957111\pi$$
$$38$$ 0 0
$$39$$ −4.57221e6 −1.97637
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 6.83707e6i 1.99985i 0.0124389 + 0.999923i $$0.496040\pi$$
−0.0124389 + 0.999923i $$0.503960\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ − 5.30842e6i − 1.00000i
$$49$$ −1.24937e7 −2.16724
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.44504e7i 1.97637i
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.27953e7i 1.21214i
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ −307393. −0.0222011 −0.0111006 0.999938i $$-0.503533\pi$$
−0.0111006 + 0.999938i $$0.503533\pi$$
$$62$$ 0 0
$$63$$ 2.80352e7i 1.77968i
$$64$$ −1.67772e7 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 3.18748e7i − 1.58179i −0.611952 0.790895i $$-0.709616\pi$$
0.611952 0.790895i $$-0.290384\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ − 1.61693e7i − 0.569376i −0.958620 0.284688i $$-0.908110\pi$$
0.958620 0.284688i $$-0.0918900\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 4.04396e7 1.21214
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.88870e7 0.484904 0.242452 0.970163i $$-0.422048\pi$$
0.242452 + 0.970163i $$0.422048\pi$$
$$80$$ 0 0
$$81$$ 4.30467e7 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 8.86049e7 1.77968
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ −2.41198e8 −3.51729
$$92$$ 0 0
$$93$$ − 9.93033e7i − 1.32749i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 8.21325e7i − 0.927744i −0.885902 0.463872i $$-0.846460\pi$$
0.885902 0.463872i $$-0.153540\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ − 4.44490e7i − 0.394923i −0.980311 0.197462i $$-0.936730\pi$$
0.980311 0.197462i $$-0.0632698\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ − 1.36049e8i − 1.00000i
$$109$$ 2.71340e8 1.92224 0.961122 0.276125i $$-0.0890504\pi$$
0.961122 + 0.276125i $$0.0890504\pi$$
$$110$$ 0 0
$$111$$ 4.07853e7 0.268665
$$112$$ − 2.80035e8i − 1.77968i
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 3.70349e8i 1.97637i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 2.14359e8 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −3.13848e8 −1.32749
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 4.00562e8i − 1.53977i −0.638185 0.769883i $$-0.720314\pi$$
0.638185 0.769883i $$-0.279686\pi$$
$$128$$ 0 0
$$129$$ 5.53803e8 1.99985
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 6.74993e8i 2.15721i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ −7.09431e8 −1.90043 −0.950213 0.311602i $$-0.899135\pi$$
−0.950213 + 0.311602i $$0.899135\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −4.29982e8 −1.00000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.01199e9i 2.16724i
$$148$$ − 1.28902e8i − 0.268665i
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ −1.00464e9 −1.93243 −0.966214 0.257740i $$-0.917022\pi$$
−0.966214 + 0.257740i $$0.917022\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 1.17048e9 1.97637
$$157$$ 1.03379e9i 1.70151i 0.525559 + 0.850757i $$0.323856\pi$$
−0.525559 + 0.850757i $$0.676144\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 9.68184e7i 0.137154i 0.997646 + 0.0685768i $$0.0218458\pi$$
−0.997646 + 0.0685768i $$0.978154\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −2.37053e9 −2.90602
$$170$$ 0 0
$$171$$ 1.03642e9 1.21214
$$172$$ − 1.75029e9i − 1.99985i
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ −1.09376e9 −1.01908 −0.509539 0.860448i $$-0.670184\pi$$
−0.509539 + 0.860448i $$0.670184\pi$$
$$182$$ 0 0
$$183$$ 2.48988e7i 0.0222011i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 2.27085e9 1.77968
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 1.35895e9i 1.00000i
$$193$$ 2.47302e9i 1.78237i 0.453641 + 0.891185i $$0.350125\pi$$
−0.453641 + 0.891185i $$0.649875\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 3.19839e9 2.16724
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 3.13036e9 1.99610 0.998050 0.0624175i $$-0.0198810\pi$$
0.998050 + 0.0624175i $$0.0198810\pi$$
$$200$$ 0 0
$$201$$ −2.58186e9 −1.58179
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ − 3.69931e9i − 1.97637i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −3.33759e9 −1.68385 −0.841924 0.539596i $$-0.818577\pi$$
−0.841924 + 0.539596i $$0.818577\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 5.23856e9i − 2.36251i
$$218$$ 0 0
$$219$$ −1.30971e9 −0.569376
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 1.10425e9i − 0.446527i −0.974758 0.223263i $$-0.928329\pi$$
0.974758 0.223263i $$-0.0716710\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ − 3.27560e9i − 1.21214i
$$229$$ 3.75229e9 1.36444 0.682220 0.731147i $$-0.261014\pi$$
0.682220 + 0.731147i $$0.261014\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 1.52985e9i − 0.484904i
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −5.19544e8 −0.154012 −0.0770059 0.997031i $$-0.524536\pi$$
−0.0770059 + 0.997031i $$0.524536\pi$$
$$242$$ 0 0
$$243$$ − 3.48678e9i − 1.00000i
$$244$$ 7.86926e7 0.0222011
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.91676e9i 2.39563i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ − 7.17700e9i − 1.77968i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 4.29497e9 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 2.15155e9 0.478137
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 8.15996e9i 1.58179i
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ −2.98709e9 −0.553824 −0.276912 0.960895i $$-0.589311\pi$$
−0.276912 + 0.960895i $$0.589311\pi$$
$$272$$ 0 0
$$273$$ 1.95370e10i 3.51729i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4.19798e9i 0.713053i 0.934285 + 0.356526i $$0.116039\pi$$
−0.934285 + 0.356526i $$0.883961\pi$$
$$278$$ 0 0
$$279$$ −8.04357e9 −1.32749
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ − 1.17255e10i − 1.82803i −0.405678 0.914016i $$-0.632964\pi$$
0.405678 0.914016i $$-0.367036\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −6.97576e9 −1.00000
$$290$$ 0 0
$$291$$ −6.65273e9 −0.927744
$$292$$ 4.13934e9i 0.569376i
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 2.92148e10 3.55907
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −1.03525e10 −1.21214
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.63938e10i 1.84555i 0.385338 + 0.922775i $$0.374085\pi$$
−0.385338 + 0.922775i $$0.625915\pi$$
$$308$$ 0 0
$$309$$ −3.60037e9 −0.394923
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ − 7.32851e9i − 0.763551i −0.924255 0.381776i $$-0.875313\pi$$
0.924255 0.381776i $$-0.124687\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −4.83508e9 −0.484904
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.10200e10 −1.00000
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2.19786e10i − 1.92224i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.62495e10 −1.35372 −0.676858 0.736113i $$-0.736659\pi$$
−0.676858 + 0.736113i $$0.736659\pi$$
$$332$$ 0 0
$$333$$ − 3.30361e9i − 0.268665i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −2.26829e10 −1.77968
$$337$$ 1.18646e10i 0.919886i 0.887948 + 0.459943i $$0.152130\pi$$
−0.887948 + 0.459943i $$0.847870\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 2.87527e10i 2.07731i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 2.96004e10 1.99524 0.997621 0.0689403i $$-0.0219618\pi$$
0.997621 + 0.0689403i $$0.0219618\pi$$
$$350$$ 0 0
$$351$$ 2.99983e10 1.97637
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 7.97001e9 0.469278
$$362$$ 0 0
$$363$$ − 1.73631e10i − 1.00000i
$$364$$ 6.17467e10 3.51729
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 3.54815e10i − 1.95586i −0.208929 0.977931i $$-0.566998\pi$$
0.208929 0.977931i $$-0.433002\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 2.54217e10i 1.32749i
$$373$$ − 3.07802e10i − 1.59014i −0.606517 0.795070i $$-0.707434\pi$$
0.606517 0.795070i $$-0.292566\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −3.73548e10 −1.81046 −0.905230 0.424921i $$-0.860302\pi$$
−0.905230 + 0.424921i $$0.860302\pi$$
$$380$$ 0 0
$$381$$ −3.24455e10 −1.53977
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 4.48580e10i − 1.99985i
$$388$$ 2.10259e10i 0.927744i
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3.29221e10i − 1.32533i −0.748914 0.662667i $$-0.769425\pi$$
0.748914 0.662667i $$-0.230575\pi$$
$$398$$ 0 0
$$399$$ 5.46744e10 2.15721
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ − 6.92022e10i − 2.62361i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −4.27011e10 −1.52597 −0.762985 0.646416i $$-0.776267\pi$$
−0.762985 + 0.646416i $$0.776267\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 1.13789e10i 0.394923i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 5.74639e10i 1.90043i
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ −1.16089e10 −0.369541 −0.184771 0.982782i $$-0.559154\pi$$
−0.184771 + 0.982782i $$0.559154\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.31349e9i 0.0395108i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 3.48285e10i 1.00000i
$$433$$ 9.74052e9i 0.277096i 0.990356 + 0.138548i $$0.0442435\pi$$
−0.990356 + 0.138548i $$0.955756\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.94631e10 −1.92224
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 7.01153e9 0.188779 0.0943897 0.995535i $$-0.469910\pi$$
0.0943897 + 0.995535i $$0.469910\pi$$
$$440$$ 0 0
$$441$$ 8.19713e10 2.16724
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ −1.04410e10 −0.268665
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 7.16890e10i 1.77968i
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 8.13760e10i 1.93243i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 2.72608e10i − 0.624991i −0.949919 0.312495i $$-0.898835\pi$$
0.949919 0.312495i $$-0.101165\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 6.87853e10i 1.49683i 0.663232 + 0.748413i $$0.269184\pi$$
−0.663232 + 0.748413i $$0.730816\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ − 9.48093e10i − 1.97637i
$$469$$ −1.36201e11 −2.81507
$$470$$ 0 0
$$471$$ 8.37374e10 1.70151
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 2.84223e10 0.530981
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −5.48759e10 −1.00000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 1.17841e10i 0.209498i 0.994499 + 0.104749i $$0.0334039\pi$$
−0.994499 + 0.104749i $$0.966596\pi$$
$$488$$ 0 0
$$489$$ 7.84229e9 0.137154
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.03450e10 1.32749
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 5.04931e10 0.814386 0.407193 0.913342i $$-0.366508\pi$$
0.407193 + 0.913342i $$0.366508\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.92013e11i 2.90602i
$$508$$ 1.02544e11i 1.53977i
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ −6.90913e10 −1.01330
$$512$$ 0 0
$$513$$ − 8.39501e10i − 1.21214i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ −1.41774e11 −1.99985
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ − 1.15606e11i − 1.54516i −0.634915 0.772582i $$-0.718965\pi$$
0.634915 0.772582i $$-0.281035\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.83110e10 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 1.72798e11i − 2.15721i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.69433e11 −1.97792 −0.988962 0.148171i $$-0.952661\pi$$
−0.988962 + 0.148171i $$0.952661\pi$$
$$542$$ 0 0
$$543$$ 8.85944e10i 1.01908i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.18266e10i − 0.690599i −0.938492 0.345300i $$-0.887777\pi$$
0.938492 0.345300i $$-0.112223\pi$$
$$548$$ 0 0
$$549$$ 2.01681e9 0.0222011
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ − 8.07043e10i − 0.862971i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 1.81614e11 1.90043
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 3.85932e11 3.95243
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 1.83939e11i − 1.77968i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 2.39863e10 0.225641 0.112821 0.993615i $$-0.464011\pi$$
0.112821 + 0.993615i $$0.464011\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.10075e11 1.00000
$$577$$ 2.14996e11i 1.93966i 0.243775 + 0.969832i $$0.421614\pi$$
−0.243775 + 0.969832i $$0.578386\pi$$
$$578$$ 0 0
$$579$$ 2.00314e11 1.78237
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ − 2.59070e11i − 2.16724i
$$589$$ −1.93662e11 −1.60910
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 3.29988e10i 0.268665i
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 2.53559e11i − 1.99610i
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −1.88317e11 −1.44341 −0.721707 0.692199i $$-0.756642\pi$$
−0.721707 + 0.692199i $$0.756642\pi$$
$$602$$ 0 0
$$603$$ 2.09131e11i 1.58179i
$$604$$ 2.57188e11 1.93243
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 2.65989e11i − 1.95933i −0.200634 0.979666i $$-0.564300\pi$$
0.200634 0.979666i $$-0.435700\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 1.57998e10i 0.111894i 0.998434 + 0.0559472i $$0.0178179\pi$$
−0.998434 + 0.0559472i $$0.982182\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ −5.00608e10 −0.340985 −0.170493 0.985359i $$-0.554536\pi$$
−0.170493 + 0.985359i $$0.554536\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ −2.99644e11 −1.97637
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ − 2.64651e11i − 1.70151i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −1.12984e11 −0.712686 −0.356343 0.934355i $$-0.615976\pi$$
−0.356343 + 0.934355i $$0.615976\pi$$
$$632$$ 0 0
$$633$$ 2.70345e11i 1.68385i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7.05233e11i 4.28327i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ − 1.88544e11i − 1.10298i −0.834181 0.551490i $$-0.814059\pi$$
0.834181 0.551490i $$-0.185941\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −4.24323e11 −2.36251
$$652$$ − 2.47855e10i − 0.137154i
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1.06087e11i 0.569376i
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 3.56009e11 1.86490 0.932449 0.361302i $$-0.117668\pi$$
0.932449 + 0.361302i $$0.117668\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −8.94441e10 −0.446527
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 2.28934e11i 1.11597i 0.829853 + 0.557983i $$0.188424\pi$$
−0.829853 + 0.557983i $$0.811576\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 6.06856e11 2.90602
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ −3.50952e11 −1.65108
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ −2.65324e11 −1.21214
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 3.03936e11i − 1.36444i
$$688$$ 4.48074e11i 1.99985i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1.55801e11 0.683374 0.341687 0.939814i $$-0.389002\pi$$
0.341687 + 0.939814i $$0.389002\pi$$
$$692$$ 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
$$703$$ − 7.95399e10i − 0.325659i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.29057e11 −0.510735 −0.255367 0.966844i $$-0.582196\pi$$
−0.255367 + 0.966844i $$0.582196\pi$$
$$710$$ 0 0
$$711$$ −1.23918e11 −0.484904
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ −1.89930e11 −0.702835
$$722$$ 0 0
$$723$$ 4.20830e10i 0.154012i
$$724$$ 2.80002e11 1.01908
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 4.11506e11i 1.47312i 0.676372 + 0.736560i $$0.263551\pi$$
−0.676372 + 0.736560i $$0.736449\pi$$
$$728$$ 0 0
$$729$$ −2.82430e11 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ − 6.37410e9i − 0.0222011i
$$733$$ 5.77330e11i 1.99990i 0.0100913 + 0.999949i $$0.496788\pi$$
−0.0100913 + 0.999949i $$0.503212\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 4.35662e11 1.46074 0.730368 0.683054i $$-0.239349\pi$$
0.730368 + 0.683054i $$0.239349\pi$$
$$740$$ 0 0
$$741$$ 7.22258e11 2.39563
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −5.35831e11 −1.68449 −0.842244 0.539097i $$-0.818766\pi$$
−0.842244 + 0.539097i $$0.818766\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −5.81337e11 −1.77968
$$757$$ 6.36051e11i 1.93691i 0.249197 + 0.968453i $$0.419833\pi$$
−0.249197 + 0.968453i $$0.580167\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ − 1.15944e12i − 3.42097i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ − 3.47892e11i − 1.00000i
$$769$$ 5.07613e11 1.45153 0.725767 0.687941i $$-0.241485\pi$$
0.725767 + 0.687941i $$0.241485\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 6.33092e11i − 1.78237i
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 1.74276e11i − 0.478137i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −8.18789e11 −2.16724
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 6.52899e11i − 1.70195i −0.525205 0.850976i $$-0.676011\pi$$
0.525205 0.850976i $$-0.323989\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.73514e10i 0.0438775i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −8.01373e11 −1.99610
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 6.60957e11 1.58179
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 7.00891e11 1.62019 0.810097 0.586296i $$-0.199415\pi$$
0.810097 + 0.586296i $$0.199415\pi$$
$$812$$ 0 0
$$813$$ 2.41954e11i 0.553824i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 1.08003e12i − 2.42409i
$$818$$ 0 0
$$819$$ 1.58250e12 3.51729
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 0 0
$$823$$ − 7.02937e11i − 1.53220i −0.642719 0.766102i $$-0.722194\pi$$
0.642719 0.766102i $$-0.277806\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ −4.11968e11 −0.872258 −0.436129 0.899884i $$-0.643651\pi$$
−0.436129 + 0.899884i $$0.643651\pi$$
$$830$$ 0 0
$$831$$ 3.40037e11 0.713053
$$832$$ 9.47024e11i 1.97637i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 6.51529e11i 1.32749i
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 5.00246e11 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 8.54422e11 1.68385
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 9.15955e11i − 1.77968i
$$848$$ 0 0
$$849$$ −9.49762e11 −1.82803
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 4.38996e11i − 0.829210i −0.910001 0.414605i $$-0.863920\pi$$
0.910001 0.414605i $$-0.136080\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 8.02752e11 1.47438 0.737189 0.675686i $$-0.236153\pi$$
0.737189 + 0.675686i $$0.236153\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 5.65036e11i 1.00000i
$$868$$ 1.34107e12i 2.36251i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −1.79924e12 −3.12620
$$872$$ 0 0
$$873$$ 5.38871e11i 0.927744i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 3.35286e11 0.569376
$$877$$ − 1.11362e12i − 1.88252i −0.337677 0.941262i $$-0.609641\pi$$
0.337677 0.941262i $$-0.390359\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 1.21517e12i 1.99891i 0.0329612 + 0.999457i $$0.489506\pi$$
−0.0329612 + 0.999457i $$0.510494\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −1.71160e12 −2.74028
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.82688e11i 0.446527i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ − 2.36640e12i − 3.55907i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.20490e12i 1.78042i 0.455547 + 0.890212i $$0.349444\pi$$
−0.455547 + 0.890212i $$0.650556\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 8.38555e11i 1.21214i
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −9.60587e11 −1.36444
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −1.41417e12 −1.98261 −0.991307 0.131571i $$-0.957998\pi$$
−0.991307 + 0.131571i $$0.957998\pi$$
$$920$$ 0 0
$$921$$ 1.32790e12 1.84555
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.91630e11i 0.394923i
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 1.97360e12 2.62700
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.39864e11i 0.311176i 0.987822 + 0.155588i $$0.0497272\pi$$
−0.987822 + 0.155588i $$0.950273\pi$$
$$938$$ 0 0
$$939$$ −5.93609e11 −0.763551
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 3.91642e11i 0.484904i
$$949$$ −9.12707e11 −1.12530
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 6.50104e11 0.762236
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 1.33003e11 0.154012
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.51552e12i − 1.73322i −0.498984 0.866611i $$-0.666293\pi$$
0.498984 0.866611i $$-0.333707\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 8.92617e11i 1.00000i
$$973$$ 3.03140e12i 3.38214i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −2.01453e10 −0.0222011
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −1.78026e12 −1.92224
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 2.28269e12i − 2.39563i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −1.11521e12 −1.15627 −0.578136 0.815940i $$-0.696220\pi$$
−0.578136 + 0.815940i $$0.696220\pi$$
$$992$$ 0 0
$$993$$ 1.31621e12i 1.35372i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 5.81390e11i 0.588420i 0.955741 + 0.294210i $$0.0950565\pi$$
−0.955741 + 0.294210i $$0.904944\pi$$
$$998$$ 0 0
$$999$$ −2.67592e11 −0.268665
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 75.9.d.a.74.1 2
3.2 odd 2 CM 75.9.d.a.74.1 2
5.2 odd 4 75.9.c.a.26.1 1
5.3 odd 4 75.9.c.b.26.1 yes 1
5.4 even 2 inner 75.9.d.a.74.2 2
15.2 even 4 75.9.c.a.26.1 1
15.8 even 4 75.9.c.b.26.1 yes 1
15.14 odd 2 inner 75.9.d.a.74.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
75.9.c.a.26.1 1 5.2 odd 4
75.9.c.a.26.1 1 15.2 even 4
75.9.c.b.26.1 yes 1 5.3 odd 4
75.9.c.b.26.1 yes 1 15.8 even 4
75.9.d.a.74.1 2 1.1 even 1 trivial
75.9.d.a.74.1 2 3.2 odd 2 CM
75.9.d.a.74.2 2 5.4 even 2 inner
75.9.d.a.74.2 2 15.14 odd 2 inner