# Properties

 Label 225.12.b.d.199.1 Level $225$ Weight $12$ Character 225.199 Analytic conductor $172.877$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$172.877215626$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.12.b.d.199.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-24.0000i q^{2} +1472.00 q^{4} +16744.0i q^{7} -84480.0i q^{8} +O(q^{10})$$ $$q-24.0000i q^{2} +1472.00 q^{4} +16744.0i q^{7} -84480.0i q^{8} -534612. q^{11} -577738. i q^{13} +401856. q^{14} +987136. q^{16} -6.90593e6i q^{17} -1.06614e7 q^{19} +1.28307e7i q^{22} -1.86433e7i q^{23} -1.38657e7 q^{26} +2.46472e7i q^{28} +1.28407e8 q^{29} -5.28432e7 q^{31} -1.96706e8i q^{32} -1.65742e8 q^{34} +1.82213e8i q^{37} +2.55874e8i q^{38} -3.08120e8 q^{41} -1.71257e7i q^{43} -7.86949e8 q^{44} -4.47439e8 q^{46} +2.68735e9i q^{47} +1.69697e9 q^{49} -8.50430e8i q^{52} +1.59606e9i q^{53} +1.41453e9 q^{56} -3.08176e9i q^{58} -5.18920e9 q^{59} +6.95648e9 q^{61} +1.26824e9i q^{62} -2.69930e9 q^{64} +1.54818e10i q^{67} -1.01655e10i q^{68} -9.79149e9 q^{71} +1.46379e9i q^{73} +4.37312e9 q^{74} -1.56936e10 q^{76} -8.95154e9i q^{77} -3.81168e10 q^{79} +7.39489e9i q^{82} +2.93351e10i q^{83} -4.11017e8 q^{86} +4.51640e10i q^{88} -2.49929e10 q^{89} +9.67365e9 q^{91} -2.74429e10i q^{92} +6.44964e10 q^{94} -7.50136e10i q^{97} -4.07272e10i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2944q^{4} + O(q^{10})$$ $$2q + 2944q^{4} - 1069224q^{11} + 803712q^{14} + 1974272q^{16} - 21322840q^{19} - 27731424q^{26} + 256813260q^{29} - 105686336q^{31} - 331484832q^{34} - 616240884q^{41} - 1573897728q^{44} - 894877056q^{46} + 3393930414q^{49} + 2829066240q^{56} - 10378407480q^{59} + 13912957324q^{61} - 5398593536q^{64} - 19582970544q^{71} + 8746239072q^{74} - 31387220480q^{76} - 76233691360q^{79} - 822033984q^{86} - 49985834220q^{89} + 19347290144q^{91} + 128992727808q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 24.0000i − 0.530330i −0.964203 0.265165i $$-0.914574\pi$$
0.964203 0.265165i $$-0.0854264\pi$$
$$3$$ 0 0
$$4$$ 1472.00 0.718750
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 16744.0i 0.376548i 0.982117 + 0.188274i $$0.0602893\pi$$
−0.982117 + 0.188274i $$0.939711\pi$$
$$8$$ − 84480.0i − 0.911505i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −534612. −1.00087 −0.500436 0.865773i $$-0.666827\pi$$
−0.500436 + 0.865773i $$0.666827\pi$$
$$12$$ 0 0
$$13$$ − 577738.i − 0.431561i −0.976442 0.215781i $$-0.930770\pi$$
0.976442 0.215781i $$-0.0692296\pi$$
$$14$$ 401856. 0.199695
$$15$$ 0 0
$$16$$ 987136. 0.235352
$$17$$ − 6.90593e6i − 1.17965i −0.807531 0.589825i $$-0.799197\pi$$
0.807531 0.589825i $$-0.200803\pi$$
$$18$$ 0 0
$$19$$ −1.06614e7 −0.987803 −0.493901 0.869518i $$-0.664430\pi$$
−0.493901 + 0.869518i $$0.664430\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.28307e7i 0.530793i
$$23$$ − 1.86433e7i − 0.603975i −0.953312 0.301988i $$-0.902350\pi$$
0.953312 0.301988i $$-0.0976501\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.38657e7 −0.228870
$$27$$ 0 0
$$28$$ 2.46472e7i 0.270644i
$$29$$ 1.28407e8 1.16251 0.581257 0.813720i $$-0.302561\pi$$
0.581257 + 0.813720i $$0.302561\pi$$
$$30$$ 0 0
$$31$$ −5.28432e7 −0.331512 −0.165756 0.986167i $$-0.553006\pi$$
−0.165756 + 0.986167i $$0.553006\pi$$
$$32$$ − 1.96706e8i − 1.03632i
$$33$$ 0 0
$$34$$ −1.65742e8 −0.625604
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.82213e8i 0.431987i 0.976395 + 0.215993i $$0.0692990\pi$$
−0.976395 + 0.215993i $$0.930701\pi$$
$$38$$ 2.55874e8i 0.523862i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3.08120e8 −0.415345 −0.207673 0.978198i $$-0.566589\pi$$
−0.207673 + 0.978198i $$0.566589\pi$$
$$42$$ 0 0
$$43$$ − 1.71257e7i − 0.0177653i −0.999961 0.00888264i $$-0.997173\pi$$
0.999961 0.00888264i $$-0.00282747\pi$$
$$44$$ −7.86949e8 −0.719377
$$45$$ 0 0
$$46$$ −4.47439e8 −0.320306
$$47$$ 2.68735e9i 1.70917i 0.519310 + 0.854586i $$0.326189\pi$$
−0.519310 + 0.854586i $$0.673811\pi$$
$$48$$ 0 0
$$49$$ 1.69697e9 0.858212
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 8.50430e8i − 0.310185i
$$53$$ 1.59606e9i 0.524241i 0.965035 + 0.262120i $$0.0844217\pi$$
−0.965035 + 0.262120i $$0.915578\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.41453e9 0.343225
$$57$$ 0 0
$$58$$ − 3.08176e9i − 0.616517i
$$59$$ −5.18920e9 −0.944963 −0.472481 0.881341i $$-0.656642\pi$$
−0.472481 + 0.881341i $$0.656642\pi$$
$$60$$ 0 0
$$61$$ 6.95648e9 1.05457 0.527285 0.849689i $$-0.323210\pi$$
0.527285 + 0.849689i $$0.323210\pi$$
$$62$$ 1.26824e9i 0.175811i
$$63$$ 0 0
$$64$$ −2.69930e9 −0.314240
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.54818e10i 1.40091i 0.713696 + 0.700456i $$0.247020\pi$$
−0.713696 + 0.700456i $$0.752980\pi$$
$$68$$ − 1.01655e10i − 0.847874i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −9.79149e9 −0.644062 −0.322031 0.946729i $$-0.604366\pi$$
−0.322031 + 0.946729i $$0.604366\pi$$
$$72$$ 0 0
$$73$$ 1.46379e9i 0.0826425i 0.999146 + 0.0413212i $$0.0131567\pi$$
−0.999146 + 0.0413212i $$0.986843\pi$$
$$74$$ 4.37312e9 0.229096
$$75$$ 0 0
$$76$$ −1.56936e10 −0.709983
$$77$$ − 8.95154e9i − 0.376876i
$$78$$ 0 0
$$79$$ −3.81168e10 −1.39370 −0.696848 0.717219i $$-0.745415\pi$$
−0.696848 + 0.717219i $$0.745415\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 7.39489e9i 0.220270i
$$83$$ 2.93351e10i 0.817444i 0.912659 + 0.408722i $$0.134025\pi$$
−0.912659 + 0.408722i $$0.865975\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.11017e8 −0.00942146
$$87$$ 0 0
$$88$$ 4.51640e10i 0.912300i
$$89$$ −2.49929e10 −0.474430 −0.237215 0.971457i $$-0.576235\pi$$
−0.237215 + 0.971457i $$0.576235\pi$$
$$90$$ 0 0
$$91$$ 9.67365e9 0.162503
$$92$$ − 2.74429e10i − 0.434107i
$$93$$ 0 0
$$94$$ 6.44964e10 0.906425
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 7.50136e10i − 0.886942i −0.896289 0.443471i $$-0.853747\pi$$
0.896289 0.443471i $$-0.146253\pi$$
$$98$$ − 4.07272e10i − 0.455136i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.17430e10 −0.773896 −0.386948 0.922101i $$-0.626471\pi$$
−0.386948 + 0.922101i $$0.626471\pi$$
$$102$$ 0 0
$$103$$ − 2.25755e11i − 1.91881i −0.282025 0.959407i $$-0.591006\pi$$
0.282025 0.959407i $$-0.408994\pi$$
$$104$$ −4.88073e10 −0.393370
$$105$$ 0 0
$$106$$ 3.83053e10 0.278021
$$107$$ 9.02413e10i 0.622006i 0.950409 + 0.311003i $$0.100665\pi$$
−0.950409 + 0.311003i $$0.899335\pi$$
$$108$$ 0 0
$$109$$ −7.34827e10 −0.457445 −0.228723 0.973492i $$-0.573455\pi$$
−0.228723 + 0.973492i $$0.573455\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.65286e10i 0.0886211i
$$113$$ 8.51469e10i 0.434748i 0.976088 + 0.217374i $$0.0697491\pi$$
−0.976088 + 0.217374i $$0.930251\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.89015e11 0.835557
$$117$$ 0 0
$$118$$ 1.24541e11i 0.501142i
$$119$$ 1.15633e11 0.444195
$$120$$ 0 0
$$121$$ 4.98320e8 0.00174658
$$122$$ − 1.66955e11i − 0.559270i
$$123$$ 0 0
$$124$$ −7.77851e10 −0.238274
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.62717e11i 0.705615i 0.935696 + 0.352808i $$0.114773\pi$$
−0.935696 + 0.352808i $$0.885227\pi$$
$$128$$ − 3.38071e11i − 0.869668i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −6.31529e11 −1.43021 −0.715107 0.699015i $$-0.753622\pi$$
−0.715107 + 0.699015i $$0.753622\pi$$
$$132$$ 0 0
$$133$$ − 1.78515e11i − 0.371955i
$$134$$ 3.71564e11 0.742946
$$135$$ 0 0
$$136$$ −5.83413e11 −1.07526
$$137$$ − 2.97199e11i − 0.526119i −0.964780 0.263059i $$-0.915268\pi$$
0.964780 0.263059i $$-0.0847315\pi$$
$$138$$ 0 0
$$139$$ −5.96794e11 −0.975535 −0.487767 0.872974i $$-0.662189\pi$$
−0.487767 + 0.872974i $$0.662189\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2.34996e11i 0.341565i
$$143$$ 3.08866e11i 0.431938i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.51310e10 0.0438278
$$147$$ 0 0
$$148$$ 2.68218e11i 0.310491i
$$149$$ −1.11543e12 −1.24428 −0.622142 0.782905i $$-0.713737\pi$$
−0.622142 + 0.782905i $$0.713737\pi$$
$$150$$ 0 0
$$151$$ −8.24447e11 −0.854653 −0.427326 0.904097i $$-0.640544\pi$$
−0.427326 + 0.904097i $$0.640544\pi$$
$$152$$ 9.00677e11i 0.900387i
$$153$$ 0 0
$$154$$ −2.14837e11 −0.199869
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 1.31512e12i − 1.10031i −0.835062 0.550156i $$-0.814568\pi$$
0.835062 0.550156i $$-0.185432\pi$$
$$158$$ 9.14804e11i 0.739119i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.12163e11 0.227425
$$162$$ 0 0
$$163$$ − 3.57833e11i − 0.243584i −0.992556 0.121792i $$-0.961136\pi$$
0.992556 0.121792i $$-0.0388640\pi$$
$$164$$ −4.53553e11 −0.298529
$$165$$ 0 0
$$166$$ 7.04042e11 0.433515
$$167$$ 2.75483e12i 1.64117i 0.571521 + 0.820587i $$0.306354\pi$$
−0.571521 + 0.820587i $$0.693646\pi$$
$$168$$ 0 0
$$169$$ 1.45838e12 0.813755
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 2.52090e10i − 0.0127688i
$$173$$ 9.50387e11i 0.466280i 0.972443 + 0.233140i $$0.0749001\pi$$
−0.972443 + 0.233140i $$0.925100\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −5.27735e11 −0.235557
$$177$$ 0 0
$$178$$ 5.99830e11i 0.251604i
$$179$$ 1.68138e12 0.683873 0.341936 0.939723i $$-0.388917\pi$$
0.341936 + 0.939723i $$0.388917\pi$$
$$180$$ 0 0
$$181$$ −9.96774e11 −0.381386 −0.190693 0.981650i $$-0.561073\pi$$
−0.190693 + 0.981650i $$0.561073\pi$$
$$182$$ − 2.32167e11i − 0.0861804i
$$183$$ 0 0
$$184$$ −1.57498e12 −0.550526
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.69200e12i 1.18068i
$$188$$ 3.95578e12i 1.22847i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2.76240e12 −0.786328 −0.393164 0.919468i $$-0.628619\pi$$
−0.393164 + 0.919468i $$0.628619\pi$$
$$192$$ 0 0
$$193$$ 5.44239e12i 1.46293i 0.681878 + 0.731466i $$0.261164\pi$$
−0.681878 + 0.731466i $$0.738836\pi$$
$$194$$ −1.80033e12 −0.470372
$$195$$ 0 0
$$196$$ 2.49793e12 0.616840
$$197$$ − 2.87609e12i − 0.690619i −0.938489 0.345309i $$-0.887774\pi$$
0.938489 0.345309i $$-0.112226\pi$$
$$198$$ 0 0
$$199$$ −7.28391e11 −0.165452 −0.0827262 0.996572i $$-0.526363\pi$$
−0.0827262 + 0.996572i $$0.526363\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 1.96183e12i 0.410421i
$$203$$ 2.15004e12i 0.437742i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −5.41812e12 −1.01760
$$207$$ 0 0
$$208$$ − 5.70306e11i − 0.101569i
$$209$$ 5.69972e12 0.988665
$$210$$ 0 0
$$211$$ −6.79317e12 −1.11820 −0.559099 0.829101i $$-0.688853\pi$$
−0.559099 + 0.829101i $$0.688853\pi$$
$$212$$ 2.34939e12i 0.376798i
$$213$$ 0 0
$$214$$ 2.16579e12 0.329868
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 8.84806e11i − 0.124830i
$$218$$ 1.76358e12i 0.242597i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3.98982e12 −0.509092
$$222$$ 0 0
$$223$$ 7.33486e12i 0.890667i 0.895365 + 0.445333i $$0.146915\pi$$
−0.895365 + 0.445333i $$0.853085\pi$$
$$224$$ 3.29365e12 0.390223
$$225$$ 0 0
$$226$$ 2.04352e12 0.230560
$$227$$ − 1.35984e12i − 0.149743i −0.997193 0.0748713i $$-0.976145\pi$$
0.997193 0.0748713i $$-0.0238546\pi$$
$$228$$ 0 0
$$229$$ 1.18244e13 1.24075 0.620375 0.784305i $$-0.286980\pi$$
0.620375 + 0.784305i $$0.286980\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 1.08478e13i − 1.05964i
$$233$$ 1.75634e13i 1.67552i 0.546038 + 0.837761i $$0.316135\pi$$
−0.546038 + 0.837761i $$0.683865\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −7.63851e12 −0.679192
$$237$$ 0 0
$$238$$ − 2.77519e12i − 0.235570i
$$239$$ −7.13958e12 −0.592221 −0.296111 0.955154i $$-0.595690\pi$$
−0.296111 + 0.955154i $$0.595690\pi$$
$$240$$ 0 0
$$241$$ −2.31307e11 −0.0183271 −0.00916357 0.999958i $$-0.502917\pi$$
−0.00916357 + 0.999958i $$0.502917\pi$$
$$242$$ − 1.19597e10i 0 0.000926264i
$$243$$ 0 0
$$244$$ 1.02399e13 0.757972
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.15951e12i 0.426297i
$$248$$ 4.46419e12i 0.302175i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.29831e13 −0.822567 −0.411284 0.911507i $$-0.634919\pi$$
−0.411284 + 0.911507i $$0.634919\pi$$
$$252$$ 0 0
$$253$$ 9.96692e12i 0.604502i
$$254$$ 6.30521e12 0.374209
$$255$$ 0 0
$$256$$ −1.36419e13 −0.775451
$$257$$ 2.39612e13i 1.33314i 0.745442 + 0.666571i $$0.232239\pi$$
−0.745442 + 0.666571i $$0.767761\pi$$
$$258$$ 0 0
$$259$$ −3.05098e12 −0.162664
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 1.51567e13i 0.758485i
$$263$$ 2.42737e13i 1.18954i 0.803895 + 0.594771i $$0.202757\pi$$
−0.803895 + 0.594771i $$0.797243\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.28436e12 −0.197259
$$267$$ 0 0
$$268$$ 2.27892e13i 1.00691i
$$269$$ 2.58377e13 1.11845 0.559225 0.829016i $$-0.311099\pi$$
0.559225 + 0.829016i $$0.311099\pi$$
$$270$$ 0 0
$$271$$ −3.76793e12 −0.156593 −0.0782964 0.996930i $$-0.524948\pi$$
−0.0782964 + 0.996930i $$0.524948\pi$$
$$272$$ − 6.81710e12i − 0.277633i
$$273$$ 0 0
$$274$$ −7.13277e12 −0.279017
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.64189e13i 0.604931i 0.953160 + 0.302466i $$0.0978098\pi$$
−0.953160 + 0.302466i $$0.902190\pi$$
$$278$$ 1.43230e13i 0.517355i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.10357e13 −0.716263 −0.358132 0.933671i $$-0.616586\pi$$
−0.358132 + 0.933671i $$0.616586\pi$$
$$282$$ 0 0
$$283$$ 1.67132e13i 0.547310i 0.961828 + 0.273655i $$0.0882327\pi$$
−0.961828 + 0.273655i $$0.911767\pi$$
$$284$$ −1.44131e13 −0.462920
$$285$$ 0 0
$$286$$ 7.41278e12 0.229070
$$287$$ − 5.15917e12i − 0.156397i
$$288$$ 0 0
$$289$$ −1.34200e13 −0.391575
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.15470e12i 0.0593993i
$$293$$ 2.39269e13i 0.647312i 0.946175 + 0.323656i $$0.104912\pi$$
−0.946175 + 0.323656i $$0.895088\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1.53934e13 0.393758
$$297$$ 0 0
$$298$$ 2.67704e13i 0.659881i
$$299$$ −1.07709e13 −0.260652
$$300$$ 0 0
$$301$$ 2.86753e11 0.00668947
$$302$$ 1.97867e13i 0.453248i
$$303$$ 0 0
$$304$$ −1.05243e13 −0.232481
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1.53111e13i − 0.320439i −0.987081 0.160219i $$-0.948780\pi$$
0.987081 0.160219i $$-0.0512202\pi$$
$$308$$ − 1.31767e13i − 0.270880i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.98752e13 −0.972080 −0.486040 0.873936i $$-0.661559\pi$$
−0.486040 + 0.873936i $$0.661559\pi$$
$$312$$ 0 0
$$313$$ − 9.94808e13i − 1.87174i −0.352345 0.935870i $$-0.614616\pi$$
0.352345 0.935870i $$-0.385384\pi$$
$$314$$ −3.15628e13 −0.583529
$$315$$ 0 0
$$316$$ −5.61080e13 −1.00172
$$317$$ 8.33692e13i 1.46278i 0.681958 + 0.731392i $$0.261129\pi$$
−0.681958 + 0.731392i $$0.738871\pi$$
$$318$$ 0 0
$$319$$ −6.86477e13 −1.16353
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 7.49191e12i − 0.120611i
$$323$$ 7.36271e13i 1.16526i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −8.58799e12 −0.129180
$$327$$ 0 0
$$328$$ 2.60300e13i 0.378589i
$$329$$ −4.49970e13 −0.643585
$$330$$ 0 0
$$331$$ −6.35840e13 −0.879618 −0.439809 0.898091i $$-0.644954\pi$$
−0.439809 + 0.898091i $$0.644954\pi$$
$$332$$ 4.31813e13i 0.587538i
$$333$$ 0 0
$$334$$ 6.61160e13 0.870364
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 1.21001e14i − 1.51644i −0.651997 0.758221i $$-0.726069\pi$$
0.651997 0.758221i $$-0.273931\pi$$
$$338$$ − 3.50011e13i − 0.431559i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2.82506e13 0.331802
$$342$$ 0 0
$$343$$ 6.15223e13i 0.699705i
$$344$$ −1.44678e12 −0.0161931
$$345$$ 0 0
$$346$$ 2.28093e13 0.247283
$$347$$ − 1.55662e14i − 1.66100i −0.557020 0.830499i $$-0.688055\pi$$
0.557020 0.830499i $$-0.311945\pi$$
$$348$$ 0 0
$$349$$ 2.56430e13 0.265112 0.132556 0.991176i $$-0.457682\pi$$
0.132556 + 0.991176i $$0.457682\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.05162e14i 1.03722i
$$353$$ − 2.49098e13i − 0.241885i −0.992659 0.120943i $$-0.961408\pi$$
0.992659 0.120943i $$-0.0385917\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −3.67896e13 −0.340996
$$357$$ 0 0
$$358$$ − 4.03532e13i − 0.362678i
$$359$$ 1.57584e14 1.39474 0.697370 0.716712i $$-0.254354\pi$$
0.697370 + 0.716712i $$0.254354\pi$$
$$360$$ 0 0
$$361$$ −2.82438e12 −0.0242457
$$362$$ 2.39226e13i 0.202260i
$$363$$ 0 0
$$364$$ 1.42396e13 0.116799
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.77901e14i 1.39481i 0.716676 + 0.697406i $$0.245662\pi$$
−0.716676 + 0.697406i $$0.754338\pi$$
$$368$$ − 1.84034e13i − 0.142146i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.67244e13 −0.197402
$$372$$ 0 0
$$373$$ − 5.51617e13i − 0.395585i −0.980244 0.197792i $$-0.936623\pi$$
0.980244 0.197792i $$-0.0633772\pi$$
$$374$$ 8.86079e13 0.626150
$$375$$ 0 0
$$376$$ 2.27027e14 1.55792
$$377$$ − 7.41854e13i − 0.501696i
$$378$$ 0 0
$$379$$ −1.46463e14 −0.962083 −0.481042 0.876698i $$-0.659741\pi$$
−0.481042 + 0.876698i $$0.659741\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 6.62977e13i 0.417013i
$$383$$ − 2.31450e14i − 1.43504i −0.696539 0.717519i $$-0.745278\pi$$
0.696539 0.717519i $$-0.254722\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 1.30617e14 0.775837
$$387$$ 0 0
$$388$$ − 1.10420e14i − 0.637490i
$$389$$ −1.49872e14 −0.853093 −0.426547 0.904466i $$-0.640270\pi$$
−0.426547 + 0.904466i $$0.640270\pi$$
$$390$$ 0 0
$$391$$ −1.28749e14 −0.712480
$$392$$ − 1.43360e14i − 0.782264i
$$393$$ 0 0
$$394$$ −6.90262e13 −0.366256
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 2.08111e14i − 1.05912i −0.848271 0.529562i $$-0.822356\pi$$
0.848271 0.529562i $$-0.177644\pi$$
$$398$$ 1.74814e13i 0.0877443i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.33408e14 0.642521 0.321261 0.946991i $$-0.395893\pi$$
0.321261 + 0.946991i $$0.395893\pi$$
$$402$$ 0 0
$$403$$ 3.05295e13i 0.143068i
$$404$$ −1.20326e14 −0.556238
$$405$$ 0 0
$$406$$ 5.16010e13 0.232148
$$407$$ − 9.74134e13i − 0.432364i
$$408$$ 0 0
$$409$$ 2.06168e14 0.890722 0.445361 0.895351i $$-0.353075\pi$$
0.445361 + 0.895351i $$0.353075\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 3.32312e14i − 1.37915i
$$413$$ − 8.68880e13i − 0.355824i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.13645e14 −0.447235
$$417$$ 0 0
$$418$$ − 1.36793e14i − 0.524319i
$$419$$ 7.34035e13 0.277677 0.138838 0.990315i $$-0.455663\pi$$
0.138838 + 0.990315i $$0.455663\pi$$
$$420$$ 0 0
$$421$$ 1.71112e14 0.630563 0.315282 0.948998i $$-0.397901\pi$$
0.315282 + 0.948998i $$0.397901\pi$$
$$422$$ 1.63036e14i 0.593014i
$$423$$ 0 0
$$424$$ 1.34835e14 0.477848
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.16479e14i 0.397096i
$$428$$ 1.32835e14i 0.447067i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 7.17758e13 0.232463 0.116231 0.993222i $$-0.462919\pi$$
0.116231 + 0.993222i $$0.462919\pi$$
$$432$$ 0 0
$$433$$ 9.98812e13i 0.315356i 0.987491 + 0.157678i $$0.0504007\pi$$
−0.987491 + 0.157678i $$0.949599\pi$$
$$434$$ −2.12353e13 −0.0662012
$$435$$ 0 0
$$436$$ −1.08166e14 −0.328789
$$437$$ 1.98764e14i 0.596608i
$$438$$ 0 0
$$439$$ 2.90312e13 0.0849788 0.0424894 0.999097i $$-0.486471\pi$$
0.0424894 + 0.999097i $$0.486471\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 9.57557e13i 0.269987i
$$443$$ − 3.28370e14i − 0.914414i −0.889360 0.457207i $$-0.848850\pi$$
0.889360 0.457207i $$-0.151150\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 1.76037e14 0.472347
$$447$$ 0 0
$$448$$ − 4.51970e13i − 0.118326i
$$449$$ −6.12368e14 −1.58364 −0.791822 0.610752i $$-0.790867\pi$$
−0.791822 + 0.610752i $$0.790867\pi$$
$$450$$ 0 0
$$451$$ 1.64725e14 0.415708
$$452$$ 1.25336e14i 0.312475i
$$453$$ 0 0
$$454$$ −3.26361e13 −0.0794130
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 3.03483e14i − 0.712189i −0.934450 0.356095i $$-0.884108\pi$$
0.934450 0.356095i $$-0.115892\pi$$
$$458$$ − 2.83786e14i − 0.658007i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 7.29308e14 1.63138 0.815691 0.578487i $$-0.196357\pi$$
0.815691 + 0.578487i $$0.196357\pi$$
$$462$$ 0 0
$$463$$ 1.22188e14i 0.266891i 0.991056 + 0.133445i $$0.0426041\pi$$
−0.991056 + 0.133445i $$0.957396\pi$$
$$464$$ 1.26755e14 0.273600
$$465$$ 0 0
$$466$$ 4.21520e14 0.888579
$$467$$ − 6.17381e14i − 1.28621i −0.765780 0.643103i $$-0.777647\pi$$
0.765780 0.643103i $$-0.222353\pi$$
$$468$$ 0 0
$$469$$ −2.59228e14 −0.527510
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.38384e14i 0.861338i
$$473$$ 9.15561e12i 0.0177808i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1.70212e14 0.319265
$$477$$ 0 0
$$478$$ 1.71350e14i 0.314073i
$$479$$ 1.05084e15 1.90410 0.952052 0.305938i $$-0.0989700\pi$$
0.952052 + 0.305938i $$0.0989700\pi$$
$$480$$ 0 0
$$481$$ 1.05272e14 0.186429
$$482$$ 5.55137e12i 0.00971944i
$$483$$ 0 0
$$484$$ 7.33527e11 0.00125536
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.19910e14i 0.363777i 0.983319 + 0.181889i $$0.0582210\pi$$
−0.983319 + 0.181889i $$0.941779\pi$$
$$488$$ − 5.87683e14i − 0.961246i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.83863e14 0.765199 0.382599 0.923914i $$-0.375029\pi$$
0.382599 + 0.923914i $$0.375029\pi$$
$$492$$ 0 0
$$493$$ − 8.86768e14i − 1.37136i
$$494$$ 1.47828e14 0.226078
$$495$$ 0 0
$$496$$ −5.21634e13 −0.0780219
$$497$$ − 1.63949e14i − 0.242520i
$$498$$ 0 0
$$499$$ 1.08878e14 0.157538 0.0787691 0.996893i $$-0.474901\pi$$
0.0787691 + 0.996893i $$0.474901\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 3.11593e14i 0.436232i
$$503$$ − 5.06588e14i − 0.701506i −0.936468 0.350753i $$-0.885926\pi$$
0.936468 0.350753i $$-0.114074\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 2.39206e14 0.320586
$$507$$ 0 0
$$508$$ 3.86720e14i 0.507161i
$$509$$ 8.57534e13 0.111251 0.0556254 0.998452i $$-0.482285\pi$$
0.0556254 + 0.998452i $$0.482285\pi$$
$$510$$ 0 0
$$511$$ −2.45097e13 −0.0311188
$$512$$ − 3.64965e14i − 0.458423i
$$513$$ 0 0
$$514$$ 5.75069e14 0.707005
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 1.43669e15i − 1.71066i
$$518$$ 7.32235e13i 0.0862654i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −9.27575e14 −1.05862 −0.529312 0.848428i $$-0.677550\pi$$
−0.529312 + 0.848428i $$0.677550\pi$$
$$522$$ 0 0
$$523$$ − 2.18187e13i − 0.0243820i −0.999926 0.0121910i $$-0.996119\pi$$
0.999926 0.0121910i $$-0.00388061\pi$$
$$524$$ −9.29610e14 −1.02797
$$525$$ 0 0
$$526$$ 5.82569e14 0.630850
$$527$$ 3.64931e14i 0.391069i
$$528$$ 0 0
$$529$$ 6.05238e14 0.635214
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 2.62774e14i − 0.267343i
$$533$$ 1.78013e14i 0.179247i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 1.30790e15 1.27694
$$537$$ 0 0
$$538$$ − 6.20105e14i − 0.593147i
$$539$$ −9.07218e14 −0.858961
$$540$$ 0 0
$$541$$ −1.69527e15 −1.57273 −0.786363 0.617765i $$-0.788038\pi$$
−0.786363 + 0.617765i $$0.788038\pi$$
$$542$$ 9.04304e13i 0.0830459i
$$543$$ 0 0
$$544$$ −1.35844e15 −1.22249
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 7.52145e14i − 0.656706i −0.944555 0.328353i $$-0.893506\pi$$
0.944555 0.328353i $$-0.106494\pi$$
$$548$$ − 4.37477e14i − 0.378148i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.36900e15 −1.14834
$$552$$ 0 0
$$553$$ − 6.38228e14i − 0.524793i
$$554$$ 3.94054e14 0.320813
$$555$$ 0 0
$$556$$ −8.78480e14 −0.701166
$$557$$ 1.87489e14i 0.148174i 0.997252 + 0.0740870i $$0.0236043\pi$$
−0.997252 + 0.0740870i $$0.976396\pi$$
$$558$$ 0 0
$$559$$ −9.89417e12 −0.00766681
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5.04857e14i 0.379856i
$$563$$ − 2.44971e14i − 0.182524i −0.995827 0.0912618i $$-0.970910\pi$$
0.995827 0.0912618i $$-0.0290900\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.01116e14 0.290255
$$567$$ 0 0
$$568$$ 8.27185e14i 0.587066i
$$569$$ 1.35243e15 0.950596 0.475298 0.879825i $$-0.342340\pi$$
0.475298 + 0.879825i $$0.342340\pi$$
$$570$$ 0 0
$$571$$ 1.43223e15 0.987447 0.493723 0.869619i $$-0.335636\pi$$
0.493723 + 0.869619i $$0.335636\pi$$
$$572$$ 4.54650e14i 0.310455i
$$573$$ 0 0
$$574$$ −1.23820e14 −0.0829422
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8.77659e14i 0.571293i 0.958335 + 0.285647i $$0.0922083\pi$$
−0.958335 + 0.285647i $$0.907792\pi$$
$$578$$ 3.22081e14i 0.207664i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −4.91187e14 −0.307807
$$582$$ 0 0
$$583$$ − 8.53271e14i − 0.524698i
$$584$$ 1.23661e14 0.0753290
$$585$$ 0 0
$$586$$ 5.74245e14 0.343289
$$587$$ − 2.43425e15i − 1.44164i −0.693124 0.720818i $$-0.743766\pi$$
0.693124 0.720818i $$-0.256234\pi$$
$$588$$ 0 0
$$589$$ 5.63383e14 0.327469
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.79869e14i 0.101669i
$$593$$ 3.03318e14i 0.169863i 0.996387 + 0.0849313i $$0.0270671\pi$$
−0.996387 + 0.0849313i $$0.972933\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.64192e15 −0.894329
$$597$$ 0 0
$$598$$ 2.58502e14i 0.138232i
$$599$$ −1.70198e15 −0.901795 −0.450898 0.892576i $$-0.648896\pi$$
−0.450898 + 0.892576i $$0.648896\pi$$
$$600$$ 0 0
$$601$$ 2.33922e15 1.21692 0.608458 0.793586i $$-0.291788\pi$$
0.608458 + 0.793586i $$0.291788\pi$$
$$602$$ − 6.88207e12i − 0.00354763i
$$603$$ 0 0
$$604$$ −1.21359e15 −0.614282
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2.49607e15i 1.22947i 0.788732 + 0.614737i $$0.210738\pi$$
−0.788732 + 0.614737i $$0.789262\pi$$
$$608$$ 2.09717e15i 1.02368i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.55258e15 0.737612
$$612$$ 0 0
$$613$$ 2.47301e15i 1.15397i 0.816756 + 0.576983i $$0.195770\pi$$
−0.816756 + 0.576983i $$0.804230\pi$$
$$614$$ −3.67466e14 −0.169938
$$615$$ 0 0
$$616$$ −7.56226e14 −0.343525
$$617$$ 2.43368e13i 0.0109571i 0.999985 + 0.00547854i $$0.00174388\pi$$
−0.999985 + 0.00547854i $$0.998256\pi$$
$$618$$ 0 0
$$619$$ −4.22545e15 −1.86885 −0.934425 0.356160i $$-0.884086\pi$$
−0.934425 + 0.356160i $$0.884086\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 1.19700e15i 0.515523i
$$623$$ − 4.18481e14i − 0.178645i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2.38754e15 −0.992640
$$627$$ 0 0
$$628$$ − 1.93585e15i − 0.790850i
$$629$$ 1.25835e15 0.509594
$$630$$ 0 0
$$631$$ −4.26326e15 −1.69660 −0.848302 0.529513i $$-0.822375\pi$$
−0.848302 + 0.529513i $$0.822375\pi$$
$$632$$ 3.22011e15i 1.27036i
$$633$$ 0 0
$$634$$ 2.00086e15 0.775758
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 9.80401e14i − 0.370371i
$$638$$ 1.64755e15i 0.617055i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.00830e15 −0.368018 −0.184009 0.982925i $$-0.558908\pi$$
−0.184009 + 0.982925i $$0.558908\pi$$
$$642$$ 0 0
$$643$$ 3.03982e14i 0.109066i 0.998512 + 0.0545328i $$0.0173670\pi$$
−0.998512 + 0.0545328i $$0.982633\pi$$
$$644$$ 4.59504e14 0.163462
$$645$$ 0 0
$$646$$ 1.76705e15 0.617974
$$647$$ 3.43583e15i 1.19140i 0.803207 + 0.595700i $$0.203125\pi$$
−0.803207 + 0.595700i $$0.796875\pi$$
$$648$$ 0 0
$$649$$ 2.77421e15 0.945788
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 5.26730e14i − 0.175076i
$$653$$ 1.18539e15i 0.390695i 0.980734 + 0.195347i $$0.0625834\pi$$
−0.980734 + 0.195347i $$0.937417\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3.04157e14 −0.0977522
$$657$$ 0 0
$$658$$ 1.07993e15i 0.341312i
$$659$$ −2.26510e15 −0.709934 −0.354967 0.934879i $$-0.615508\pi$$
−0.354967 + 0.934879i $$0.615508\pi$$
$$660$$ 0 0
$$661$$ −5.33012e15 −1.64297 −0.821484 0.570232i $$-0.806853\pi$$
−0.821484 + 0.570232i $$0.806853\pi$$
$$662$$ 1.52602e15i 0.466488i
$$663$$ 0 0
$$664$$ 2.47823e15 0.745104
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 2.39392e15i − 0.702130i
$$668$$ 4.05512e15i 1.17959i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −3.71902e15 −1.05549
$$672$$ 0 0
$$673$$ 4.74120e15i 1.32375i 0.749615 + 0.661874i $$0.230239\pi$$
−0.749615 + 0.661874i $$0.769761\pi$$
$$674$$ −2.90403e15 −0.804215
$$675$$ 0 0
$$676$$ 2.14673e15 0.584886
$$677$$ − 1.41307e15i − 0.381880i −0.981602 0.190940i $$-0.938846\pi$$
0.981602 0.190940i $$-0.0611535\pi$$
$$678$$ 0 0
$$679$$ 1.25603e15 0.333976
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 6.78014e14i − 0.175964i
$$683$$ 3.03116e15i 0.780359i 0.920739 + 0.390180i $$0.127587\pi$$
−0.920739 + 0.390180i $$0.872413\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.47654e15 0.371075
$$687$$ 0 0
$$688$$ − 1.69054e13i − 0.00418109i
$$689$$ 9.22102e14 0.226242
$$690$$ 0 0
$$691$$ −2.74731e15 −0.663405 −0.331703 0.943384i $$-0.607623\pi$$
−0.331703 + 0.943384i $$0.607623\pi$$
$$692$$ 1.39897e15i 0.335139i
$$693$$ 0 0
$$694$$ −3.73588e15 −0.880878
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.12786e15i 0.489962i
$$698$$ − 6.15433e14i − 0.140597i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −5.72747e15 −1.27795 −0.638974 0.769228i $$-0.720641\pi$$
−0.638974 + 0.769228i $$0.720641\pi$$
$$702$$ 0 0
$$703$$ − 1.94265e15i − 0.426718i
$$704$$ 1.44308e15 0.314514
$$705$$ 0 0
$$706$$ −5.97836e14 −0.128279
$$707$$ − 1.36870e15i − 0.291409i
$$708$$ 0 0
$$709$$ −6.98326e14 −0.146388 −0.0731938 0.997318i $$-0.523319\pi$$
−0.0731938 + 0.997318i $$0.523319\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 2.11140e15i 0.432445i
$$713$$ 9.85170e14i 0.200225i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 2.47500e15 0.491534
$$717$$ 0 0
$$718$$ − 3.78202e15i − 0.739672i
$$719$$ 9.70979e15 1.88452 0.942260 0.334882i $$-0.108696\pi$$
0.942260 + 0.334882i $$0.108696\pi$$
$$720$$ 0 0
$$721$$ 3.78004e15 0.722525
$$722$$ 6.77852e13i 0.0128582i
$$723$$ 0 0
$$724$$ −1.46725e15 −0.274121
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 2.46469e15i − 0.450114i −0.974346 0.225057i $$-0.927743\pi$$
0.974346 0.225057i $$-0.0722568\pi$$
$$728$$ − 8.17230e14i − 0.148123i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1.18269e14 −0.0209568
$$732$$ 0 0
$$733$$ 7.91285e15i 1.38121i 0.723230 + 0.690607i $$0.242657\pi$$
−0.723230 + 0.690607i $$0.757343\pi$$
$$734$$ 4.26963e15 0.739711
$$735$$ 0 0
$$736$$ −3.66725e15 −0.625911
$$737$$ − 8.27677e15i − 1.40213i
$$738$$ 0 0
$$739$$ 8.40694e15 1.40312 0.701558 0.712613i $$-0.252488\pi$$
0.701558 + 0.712613i $$0.252488\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 6.41385e14i 0.104688i
$$743$$ − 1.36287e15i − 0.220809i −0.993887 0.110404i $$-0.964785\pi$$
0.993887 0.110404i $$-0.0352146\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.32388e15 −0.209790
$$747$$ 0 0
$$748$$ 5.43462e15i 0.848614i
$$749$$ −1.51100e15 −0.234215
$$750$$ 0 0
$$751$$ 6.81722e15 1.04133 0.520664 0.853762i $$-0.325684\pi$$
0.520664 + 0.853762i $$0.325684\pi$$
$$752$$ 2.65278e15i 0.402256i
$$753$$ 0 0
$$754$$ −1.78045e15 −0.266065
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 6.67049e14i 0.0975282i 0.998810 + 0.0487641i $$0.0155283\pi$$
−0.998810 + 0.0487641i $$0.984472\pi$$
$$758$$ 3.51511e15i 0.510222i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 7.74408e15 1.09990 0.549951 0.835197i $$-0.314646\pi$$
0.549951 + 0.835197i $$0.314646\pi$$
$$762$$ 0 0
$$763$$ − 1.23039e15i − 0.172250i
$$764$$ −4.06626e15 −0.565173
$$765$$ 0 0
$$766$$ −5.55479e15 −0.761043
$$767$$ 2.99800e15i 0.407809i
$$768$$ 0 0
$$769$$ −2.52411e15 −0.338465 −0.169232 0.985576i $$-0.554129\pi$$
−0.169232 + 0.985576i $$0.554129\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.01119e15i 1.05148i
$$773$$ 1.11453e16i 1.45246i 0.687453 + 0.726229i $$0.258729\pi$$
−0.687453 + 0.726229i $$0.741271\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.33715e15 −0.808452
$$777$$ 0 0
$$778$$ 3.59692e15i 0.452421i
$$779$$ 3.28500e15 0.410279
$$780$$ 0 0
$$781$$ 5.23465e15 0.644624
$$782$$ 3.08998e15i 0.377849i
$$783$$ 0 0
$$784$$ 1.67514e15 0.201981
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 1.32271e16i − 1.56172i −0.624705 0.780861i $$-0.714781\pi$$
0.624705 0.780861i $$-0.285219\pi$$
$$788$$ − 4.23361e15i − 0.496382i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −1.42570e15 −0.163703
$$792$$ 0 0
$$793$$ − 4.01902e15i − 0.455112i
$$794$$ −4.99466e15 −0.561685
$$795$$ 0 0
$$796$$ −1.07219e15 −0.118919
$$797$$ 2.30248e15i 0.253615i 0.991927 + 0.126807i $$0.0404730\pi$$
−0.991927 + 0.126807i $$0.959527\pi$$
$$798$$ 0 0
$$799$$ 1.85587e16 2.01623
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 3.20179e15i − 0.340748i
$$803$$ − 7.82560e14i − 0.0827146i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 7.32708e14 0.0758732
$$807$$ 0 0
$$808$$ 6.90565e15i 0.705410i
$$809$$ 5.60472e15 0.568639 0.284320 0.958730i $$-0.408232\pi$$
0.284320 + 0.958730i $$0.408232\pi$$
$$810$$ 0 0
$$811$$ −5.08516e15 −0.508968 −0.254484 0.967077i $$-0.581906\pi$$
−0.254484 + 0.967077i $$0.581906\pi$$
$$812$$ 3.16486e15i 0.314627i
$$813$$ 0 0
$$814$$ −2.33792e15 −0.229296
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.82584e14i 0.0175486i
$$818$$ − 4.94802e15i − 0.472377i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.79111e14 −0.0261150 −0.0130575 0.999915i $$-0.504156\pi$$
−0.0130575 + 0.999915i $$0.504156\pi$$
$$822$$ 0 0
$$823$$ − 1.35265e16i − 1.24878i −0.781112 0.624391i $$-0.785347\pi$$
0.781112 0.624391i $$-0.214653\pi$$
$$824$$ −1.90718e16 −1.74901
$$825$$ 0 0
$$826$$ −2.08531e15 −0.188704
$$827$$ 2.72544e14i 0.0244994i 0.999925 + 0.0122497i $$0.00389930\pi$$
−0.999925 + 0.0122497i $$0.996101\pi$$
$$828$$ 0 0
$$829$$ −1.80459e16 −1.60077 −0.800385 0.599486i $$-0.795372\pi$$
−0.800385 + 0.599486i $$0.795372\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.55949e15i 0.135614i
$$833$$ − 1.17191e16i − 1.01239i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 8.38999e15 0.710603
$$837$$ 0 0
$$838$$ − 1.76168e15i − 0.147260i
$$839$$ −7.96183e15 −0.661184 −0.330592 0.943774i $$-0.607248\pi$$
−0.330592 + 0.943774i $$0.607248\pi$$
$$840$$ 0 0
$$841$$ 4.28775e15 0.351440
$$842$$ − 4.10669e15i − 0.334407i
$$843$$ 0 0
$$844$$ −9.99954e15 −0.803705
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8.34387e12i 0 0.000657671i
$$848$$ 1.57552e15i 0.123381i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 3.39705e15 0.260909
$$852$$ 0 0
$$853$$ − 1.49826e16i − 1.13598i −0.823037 0.567988i $$-0.807722\pi$$
0.823037 0.567988i $$-0.192278\pi$$
$$854$$ 2.79550e15 0.210592
$$855$$ 0 0
$$856$$ 7.62358e15 0.566961
$$857$$ − 2.22561e16i − 1.64458i −0.569068 0.822290i $$-0.692696\pi$$
0.569068 0.822290i $$-0.307304\pi$$
$$858$$ 0 0
$$859$$ −5.44237e15 −0.397032 −0.198516 0.980098i $$-0.563612\pi$$
−0.198516 + 0.980098i $$0.563612\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 1.72262e15i − 0.123282i
$$863$$ − 1.08110e16i − 0.768787i −0.923169 0.384393i $$-0.874411\pi$$
0.923169 0.384393i $$-0.125589\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 2.39715e15 0.167243
$$867$$ 0 0
$$868$$ − 1.30243e15i − 0.0897217i
$$869$$ 2.03777e16 1.39491
$$870$$ 0 0
$$871$$ 8.94444e15 0.604579
$$872$$ 6.20782e15i 0.416964i
$$873$$ 0 0
$$874$$ 4.77033e15 0.316399
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.81024e16i 1.82914i 0.404431 + 0.914568i $$0.367470\pi$$
−0.404431 + 0.914568i $$0.632530\pi$$
$$878$$ − 6.96749e14i − 0.0450668i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −4.22209e15 −0.268016 −0.134008 0.990980i $$-0.542785\pi$$
−0.134008 + 0.990980i $$0.542785\pi$$
$$882$$ 0 0
$$883$$ 5.16092e14i 0.0323551i 0.999869 + 0.0161776i $$0.00514970\pi$$
−0.999869 + 0.0161776i $$0.994850\pi$$
$$884$$ −5.87302e15 −0.365910
$$885$$ 0 0
$$886$$ −7.88088e15 −0.484941
$$887$$ 5.71906e15i 0.349740i 0.984592 + 0.174870i $$0.0559504\pi$$
−0.984592 + 0.174870i $$0.944050\pi$$
$$888$$ 0 0
$$889$$ −4.39894e15 −0.265698
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 1.07969e16i 0.640167i
$$893$$ − 2.86510e16i − 1.68832i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 5.66067e15 0.327472
$$897$$ 0 0
$$898$$ 1.46968e16i 0.839854i
$$899$$ −6.78541e15 −0.385388
$$900$$ 0 0
$$901$$ 1.10223e16 0.618421
$$902$$ − 3.95340e15i − 0.220462i
$$903$$ 0 0
$$904$$ 7.19321e15 0.396275
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 8.43778e13i 0.00456445i 0.999997 + 0.00228222i $$0.000726455\pi$$
−0.999997 + 0.00228222i $$0.999274\pi$$
$$908$$ − 2.00168e15i − 0.107628i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1.10091e16 0.581298 0.290649 0.956830i $$-0.406129\pi$$
0.290649 + 0.956830i $$0.406129\pi$$
$$912$$ 0 0
$$913$$ − 1.56829e16i − 0.818158i
$$914$$ −7.28359e15 −0.377695
$$915$$ 0 0
$$916$$ 1.74055e16 0.891789
$$917$$ − 1.05743e16i − 0.538544i
$$918$$ 0 0
$$919$$ 4.86351e15 0.244746 0.122373 0.992484i $$-0.460950\pi$$
0.122373 + 0.992484i $$0.460950\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 1.75034e16i − 0.865171i
$$923$$ 5.65691e15i 0.277952i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 2.93252e15 0.141540
$$927$$ 0 0
$$928$$ − 2.52584e16i − 1.20474i
$$929$$ 3.57534e15 0.169524 0.0847620 0.996401i $$-0.472987\pi$$
0.0847620 + 0.996401i $$0.472987\pi$$
$$930$$ 0 0
$$931$$ −1.80921e16 −0.847744
$$932$$ 2.58533e16i 1.20428i
$$933$$ 0 0
$$934$$ −1.48171e16 −0.682113
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 3.86373e16i − 1.74759i −0.486295 0.873795i $$-0.661652\pi$$
0.486295 0.873795i $$-0.338348\pi$$
$$938$$ 6.22147e15i 0.279754i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 3.48997e16 1.54198 0.770991 0.636846i $$-0.219761\pi$$
0.770991 + 0.636846i $$0.219761\pi$$
$$942$$ 0 0
$$943$$ 5.74437e15i 0.250858i
$$944$$ −5.12245e15 −0.222398
$$945$$ 0 0
$$946$$ 2.19735e14 0.00942969
$$947$$ − 2.85123e16i − 1.21649i −0.793751 0.608243i $$-0.791875\pi$$
0.793751 0.608243i $$-0.208125\pi$$
$$948$$ 0 0
$$949$$ 8.45688e14 0.0356653
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 9.76867e15i − 0.404886i
$$953$$ − 4.00334e16i − 1.64973i −0.565332 0.824863i $$-0.691252\pi$$
0.565332 0.824863i $$-0.308748\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1.05095e16 −0.425659
$$957$$ 0 0
$$958$$ − 2.52201e16i − 1.00980i
$$959$$ 4.97630e15 0.198109
$$960$$ 0 0
$$961$$ −2.26161e16 −0.890100
$$962$$ − 2.52652e15i − 0.0988688i
$$963$$ 0 0
$$964$$ −3.40484e14 −0.0131726
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.84953e16i − 0.703422i −0.936109 0.351711i $$-0.885600\pi$$
0.936109 0.351711i $$-0.114400\pi$$
$$968$$ − 4.20981e13i − 0.00159202i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 2.14877e16 0.798884 0.399442 0.916759i $$-0.369204\pi$$
0.399442 + 0.916759i $$0.369204\pi$$
$$972$$ 0 0
$$973$$ − 9.99271e15i − 0.367335i
$$974$$ 5.27784e15 0.192922
$$975$$ 0 0
$$976$$ 6.86699e15 0.248195
$$977$$ − 8.73880e15i − 0.314074i −0.987593 0.157037i $$-0.949806\pi$$
0.987593 0.157037i $$-0.0501942\pi$$
$$978$$ 0 0
$$979$$ 1.33615e16 0.474844
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 1.16127e16i − 0.405808i
$$983$$ − 1.18924e16i − 0.413263i −0.978419 0.206631i $$-0.933750\pi$$
0.978419 0.206631i $$-0.0662501\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.12824e16 −0.727274
$$987$$ 0 0
$$988$$ 9.06679e15i 0.306401i
$$989$$ −3.19279e14 −0.0107298
$$990$$ 0 0
$$991$$ 2.34409e16 0.779056 0.389528 0.921015i $$-0.372638\pi$$
0.389528 + 0.921015i $$0.372638\pi$$
$$992$$ 1.03946e16i 0.343552i
$$993$$ 0 0
$$994$$ −3.93477e15 −0.128616
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.14004e16i 0.688016i 0.938967 + 0.344008i $$0.111785\pi$$
−0.938967 + 0.344008i $$0.888215\pi$$
$$998$$ − 2.61307e15i − 0.0835473i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 225.12.b.d.199.1 2
3.2 odd 2 25.12.b.b.24.2 2
5.2 odd 4 9.12.a.b.1.1 1
5.3 odd 4 225.12.a.b.1.1 1
5.4 even 2 inner 225.12.b.d.199.2 2
15.2 even 4 1.12.a.a.1.1 1
15.8 even 4 25.12.a.b.1.1 1
15.14 odd 2 25.12.b.b.24.1 2
20.7 even 4 144.12.a.d.1.1 1
45.2 even 12 81.12.c.d.28.1 2
45.7 odd 12 81.12.c.b.28.1 2
45.22 odd 12 81.12.c.b.55.1 2
45.32 even 12 81.12.c.d.55.1 2
60.47 odd 4 16.12.a.a.1.1 1
105.2 even 12 49.12.c.b.18.1 2
105.17 odd 12 49.12.c.c.30.1 2
105.32 even 12 49.12.c.b.30.1 2
105.47 odd 12 49.12.c.c.18.1 2
105.62 odd 4 49.12.a.a.1.1 1
120.77 even 4 64.12.a.b.1.1 1
120.107 odd 4 64.12.a.f.1.1 1
165.32 odd 4 121.12.a.b.1.1 1
195.77 even 4 169.12.a.a.1.1 1
240.77 even 4 256.12.b.e.129.2 2
240.107 odd 4 256.12.b.c.129.2 2
240.197 even 4 256.12.b.e.129.1 2
240.227 odd 4 256.12.b.c.129.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1.12.a.a.1.1 1 15.2 even 4
9.12.a.b.1.1 1 5.2 odd 4
16.12.a.a.1.1 1 60.47 odd 4
25.12.a.b.1.1 1 15.8 even 4
25.12.b.b.24.1 2 15.14 odd 2
25.12.b.b.24.2 2 3.2 odd 2
49.12.a.a.1.1 1 105.62 odd 4
49.12.c.b.18.1 2 105.2 even 12
49.12.c.b.30.1 2 105.32 even 12
49.12.c.c.18.1 2 105.47 odd 12
49.12.c.c.30.1 2 105.17 odd 12
64.12.a.b.1.1 1 120.77 even 4
64.12.a.f.1.1 1 120.107 odd 4
81.12.c.b.28.1 2 45.7 odd 12
81.12.c.b.55.1 2 45.22 odd 12
81.12.c.d.28.1 2 45.2 even 12
81.12.c.d.55.1 2 45.32 even 12
121.12.a.b.1.1 1 165.32 odd 4
144.12.a.d.1.1 1 20.7 even 4
169.12.a.a.1.1 1 195.77 even 4
225.12.a.b.1.1 1 5.3 odd 4
225.12.b.d.199.1 2 1.1 even 1 trivial
225.12.b.d.199.2 2 5.4 even 2 inner
256.12.b.c.129.1 2 240.227 odd 4
256.12.b.c.129.2 2 240.107 odd 4
256.12.b.e.129.1 2 240.197 even 4
256.12.b.e.129.2 2 240.77 even 4