# Properties

 Label 144.7.e.d.17.2 Level $144$ Weight $7$ Character 144.17 Analytic conductor $33.128$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 144.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1277880413$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 17.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 144.17 Dual form 144.7.e.d.17.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+173.948i q^{5} +484.000 q^{7} +O(q^{10})$$ $$q+173.948i q^{5} +484.000 q^{7} -1340.67i q^{11} +3368.00 q^{13} +12.7279i q^{17} -5744.00 q^{19} +3377.14i q^{23} -14633.0 q^{25} +29354.8i q^{29} +39796.0 q^{31} +84191.0i q^{35} +52526.0 q^{37} -37042.5i q^{41} -3800.00 q^{43} -76791.8i q^{47} +116607. q^{49} +238738. i q^{53} +233208. q^{55} +249841. i q^{59} +13250.0 q^{61} +585858. i q^{65} -168968. q^{67} +531467. i q^{71} +236144. q^{73} -648886. i q^{77} +35116.0 q^{79} +10980.0i q^{83} -2214.00 q^{85} -129328. i q^{89} +1.63011e6 q^{91} -999159. i q^{95} -321424. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 968 q^{7}+O(q^{10})$$ 2 * q + 968 * q^7 $$2 q + 968 q^{7} + 6736 q^{13} - 11488 q^{19} - 29266 q^{25} + 79592 q^{31} + 105052 q^{37} - 7600 q^{43} + 233214 q^{49} + 466416 q^{55} + 26500 q^{61} - 337936 q^{67} + 472288 q^{73} + 70232 q^{79} - 4428 q^{85} + 3260224 q^{91} - 642848 q^{97}+O(q^{100})$$ 2 * q + 968 * q^7 + 6736 * q^13 - 11488 * q^19 - 29266 * q^25 + 79592 * q^31 + 105052 * q^37 - 7600 * q^43 + 233214 * q^49 + 466416 * q^55 + 26500 * q^61 - 337936 * q^67 + 472288 * q^73 + 70232 * q^79 - 4428 * q^85 + 3260224 * q^91 - 642848 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 173.948i 1.39159i 0.718242 + 0.695793i $$0.244947\pi$$
−0.718242 + 0.695793i $$0.755053\pi$$
$$6$$ 0 0
$$7$$ 484.000 1.41108 0.705539 0.708671i $$-0.250705\pi$$
0.705539 + 0.708671i $$0.250705\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 1340.67i − 1.00727i −0.863917 0.503634i $$-0.831996\pi$$
0.863917 0.503634i $$-0.168004\pi$$
$$12$$ 0 0
$$13$$ 3368.00 1.53300 0.766500 0.642245i $$-0.221997\pi$$
0.766500 + 0.642245i $$0.221997\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 12.7279i 0.00259066i 0.999999 + 0.00129533i $$0.000412317\pi$$
−0.999999 + 0.00129533i $$0.999588\pi$$
$$18$$ 0 0
$$19$$ −5744.00 −0.837440 −0.418720 0.908115i $$-0.637521\pi$$
−0.418720 + 0.908115i $$0.637521\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3377.14i 0.277566i 0.990323 + 0.138783i $$0.0443190\pi$$
−0.990323 + 0.138783i $$0.955681\pi$$
$$24$$ 0 0
$$25$$ −14633.0 −0.936512
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 29354.8i 1.20361i 0.798643 + 0.601805i $$0.205551\pi$$
−0.798643 + 0.601805i $$0.794449\pi$$
$$30$$ 0 0
$$31$$ 39796.0 1.33584 0.667920 0.744233i $$-0.267185\pi$$
0.667920 + 0.744233i $$0.267185\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 84191.0i 1.96364i
$$36$$ 0 0
$$37$$ 52526.0 1.03698 0.518489 0.855085i $$-0.326495\pi$$
0.518489 + 0.855085i $$0.326495\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 37042.5i − 0.537463i −0.963215 0.268732i $$-0.913396\pi$$
0.963215 0.268732i $$-0.0866045\pi$$
$$42$$ 0 0
$$43$$ −3800.00 −0.0477945 −0.0238973 0.999714i $$-0.507607\pi$$
−0.0238973 + 0.999714i $$0.507607\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 76791.8i − 0.739641i −0.929103 0.369821i $$-0.879419\pi$$
0.929103 0.369821i $$-0.120581\pi$$
$$48$$ 0 0
$$49$$ 116607. 0.991143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 238738.i 1.60359i 0.597599 + 0.801795i $$0.296121\pi$$
−0.597599 + 0.801795i $$0.703879\pi$$
$$54$$ 0 0
$$55$$ 233208. 1.40170
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 249841.i 1.21649i 0.793751 + 0.608243i $$0.208125\pi$$
−0.793751 + 0.608243i $$0.791875\pi$$
$$60$$ 0 0
$$61$$ 13250.0 0.0583749 0.0291875 0.999574i $$-0.490708\pi$$
0.0291875 + 0.999574i $$0.490708\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 585858.i 2.13330i
$$66$$ 0 0
$$67$$ −168968. −0.561798 −0.280899 0.959737i $$-0.590633\pi$$
−0.280899 + 0.959737i $$0.590633\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 531467.i 1.48491i 0.669894 + 0.742457i $$0.266340\pi$$
−0.669894 + 0.742457i $$0.733660\pi$$
$$72$$ 0 0
$$73$$ 236144. 0.607027 0.303514 0.952827i $$-0.401840\pi$$
0.303514 + 0.952827i $$0.401840\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 648886.i − 1.42134i
$$78$$ 0 0
$$79$$ 35116.0 0.0712236 0.0356118 0.999366i $$-0.488662\pi$$
0.0356118 + 0.999366i $$0.488662\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 10980.0i 0.0192029i 0.999954 + 0.00960144i $$0.00305628\pi$$
−0.999954 + 0.00960144i $$0.996944\pi$$
$$84$$ 0 0
$$85$$ −2214.00 −0.00360513
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 129328.i − 0.183453i −0.995784 0.0917263i $$-0.970762\pi$$
0.995784 0.0917263i $$-0.0292385\pi$$
$$90$$ 0 0
$$91$$ 1.63011e6 2.16318
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 999159.i − 1.16537i
$$96$$ 0 0
$$97$$ −321424. −0.352179 −0.176089 0.984374i $$-0.556345\pi$$
−0.176089 + 0.984374i $$0.556345\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 668780.i 0.649111i 0.945867 + 0.324556i $$0.105215\pi$$
−0.945867 + 0.324556i $$0.894785\pi$$
$$102$$ 0 0
$$103$$ −1.99341e6 −1.82425 −0.912127 0.409907i $$-0.865561\pi$$
−0.912127 + 0.409907i $$0.865561\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 260668.i 0.212783i 0.994324 + 0.106391i $$0.0339296\pi$$
−0.994324 + 0.106391i $$0.966070\pi$$
$$108$$ 0 0
$$109$$ 194456. 0.150156 0.0750779 0.997178i $$-0.476079\pi$$
0.0750779 + 0.997178i $$0.476079\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 821897.i − 0.569616i −0.958585 0.284808i $$-0.908070\pi$$
0.958585 0.284808i $$-0.0919298\pi$$
$$114$$ 0 0
$$115$$ −587448. −0.386257
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6160.31i 0.00365563i
$$120$$ 0 0
$$121$$ −25847.0 −0.0145900
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 172557.i 0.0883490i
$$126$$ 0 0
$$127$$ −3.05721e6 −1.49250 −0.746250 0.665666i $$-0.768148\pi$$
−0.746250 + 0.665666i $$0.768148\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 3.07388e6i − 1.36733i −0.729797 0.683664i $$-0.760385\pi$$
0.729797 0.683664i $$-0.239615\pi$$
$$132$$ 0 0
$$133$$ −2.78010e6 −1.18169
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 4.48412e6i − 1.74388i −0.489617 0.871938i $$-0.662863\pi$$
0.489617 0.871938i $$-0.337137\pi$$
$$138$$ 0 0
$$139$$ 1.09233e6 0.406732 0.203366 0.979103i $$-0.434812\pi$$
0.203366 + 0.979103i $$0.434812\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 4.51539e6i − 1.54414i
$$144$$ 0 0
$$145$$ −5.10622e6 −1.67493
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 2.22087e6i − 0.671375i −0.941973 0.335687i $$-0.891031\pi$$
0.941973 0.335687i $$-0.108969\pi$$
$$150$$ 0 0
$$151$$ 4.07871e6 1.18465 0.592327 0.805697i $$-0.298209\pi$$
0.592327 + 0.805697i $$0.298209\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6.92245e6i 1.85894i
$$156$$ 0 0
$$157$$ 6.15568e6 1.59066 0.795329 0.606178i $$-0.207298\pi$$
0.795329 + 0.606178i $$0.207298\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.63454e6i 0.391667i
$$162$$ 0 0
$$163$$ −800696. −0.184886 −0.0924432 0.995718i $$-0.529468\pi$$
−0.0924432 + 0.995718i $$0.529468\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 4.80467e6i − 1.03161i −0.856707 0.515804i $$-0.827493\pi$$
0.856707 0.515804i $$-0.172507\pi$$
$$168$$ 0 0
$$169$$ 6.51661e6 1.35009
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 3.56992e6i − 0.689478i −0.938699 0.344739i $$-0.887967\pi$$
0.938699 0.344739i $$-0.112033\pi$$
$$174$$ 0 0
$$175$$ −7.08237e6 −1.32149
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 7.43698e6i 1.29669i 0.761345 + 0.648347i $$0.224539\pi$$
−0.761345 + 0.648347i $$0.775461\pi$$
$$180$$ 0 0
$$181$$ −1.03812e7 −1.75070 −0.875350 0.483491i $$-0.839369\pi$$
−0.875350 + 0.483491i $$0.839369\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 9.13681e6i 1.44304i
$$186$$ 0 0
$$187$$ 17064.0 0.00260949
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 1.29941e7i − 1.86485i −0.361360 0.932426i $$-0.617687\pi$$
0.361360 0.932426i $$-0.382313\pi$$
$$192$$ 0 0
$$193$$ −3.93195e6 −0.546936 −0.273468 0.961881i $$-0.588171\pi$$
−0.273468 + 0.961881i $$0.588171\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.37967e6i 0.703651i 0.936066 + 0.351825i $$0.114439\pi$$
−0.936066 + 0.351825i $$0.885561\pi$$
$$198$$ 0 0
$$199$$ 565900. 0.0718093 0.0359046 0.999355i $$-0.488569\pi$$
0.0359046 + 0.999355i $$0.488569\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.42077e7i 1.69839i
$$204$$ 0 0
$$205$$ 6.44348e6 0.747926
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 7.70083e6i 0.843527i
$$210$$ 0 0
$$211$$ 1.35165e7 1.43885 0.719427 0.694568i $$-0.244404\pi$$
0.719427 + 0.694568i $$0.244404\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 661003.i − 0.0665102i
$$216$$ 0 0
$$217$$ 1.92613e7 1.88497
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 42867.6i 0.00397148i
$$222$$ 0 0
$$223$$ 5.35484e6 0.482872 0.241436 0.970417i $$-0.422382\pi$$
0.241436 + 0.970417i $$0.422382\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.36063e7i 1.16322i 0.813466 + 0.581612i $$0.197578\pi$$
−0.813466 + 0.581612i $$0.802422\pi$$
$$228$$ 0 0
$$229$$ 4.34641e6 0.361930 0.180965 0.983490i $$-0.442078\pi$$
0.180965 + 0.983490i $$0.442078\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 2.02333e7i − 1.59956i −0.600297 0.799778i $$-0.704951\pi$$
0.600297 0.799778i $$-0.295049\pi$$
$$234$$ 0 0
$$235$$ 1.33578e7 1.02927
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 2.03947e7i 1.49391i 0.664877 + 0.746953i $$0.268484\pi$$
−0.664877 + 0.746953i $$0.731516\pi$$
$$240$$ 0 0
$$241$$ −3.12093e6 −0.222963 −0.111481 0.993767i $$-0.535560\pi$$
−0.111481 + 0.993767i $$0.535560\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.02836e7i 1.37926i
$$246$$ 0 0
$$247$$ −1.93458e7 −1.28379
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 5.09519e6i − 0.322210i −0.986937 0.161105i $$-0.948494\pi$$
0.986937 0.161105i $$-0.0515058\pi$$
$$252$$ 0 0
$$253$$ 4.52765e6 0.279583
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1.44374e7i 0.850529i 0.905069 + 0.425264i $$0.139819\pi$$
−0.905069 + 0.425264i $$0.860181\pi$$
$$258$$ 0 0
$$259$$ 2.54226e7 1.46326
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 3.12567e7i − 1.71821i −0.511801 0.859104i $$-0.671022\pi$$
0.511801 0.859104i $$-0.328978\pi$$
$$264$$ 0 0
$$265$$ −4.15280e7 −2.23153
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 251338.i 0.0129122i 0.999979 + 0.00645612i $$0.00205506\pi$$
−0.999979 + 0.00645612i $$0.997945\pi$$
$$270$$ 0 0
$$271$$ −2.96399e7 −1.48925 −0.744627 0.667481i $$-0.767373\pi$$
−0.744627 + 0.667481i $$0.767373\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.96181e7i 0.943319i
$$276$$ 0 0
$$277$$ 1.32213e7 0.622062 0.311031 0.950400i $$-0.399326\pi$$
0.311031 + 0.950400i $$0.399326\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 6.12360e6i − 0.275987i −0.990433 0.137993i $$-0.955935\pi$$
0.990433 0.137993i $$-0.0440652\pi$$
$$282$$ 0 0
$$283$$ 6.74325e6 0.297516 0.148758 0.988874i $$-0.452473\pi$$
0.148758 + 0.988874i $$0.452473\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 1.79286e7i − 0.758403i
$$288$$ 0 0
$$289$$ 2.41374e7 0.999993
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 1.00239e7i − 0.398505i −0.979948 0.199253i $$-0.936149\pi$$
0.979948 0.199253i $$-0.0638514\pi$$
$$294$$ 0 0
$$295$$ −4.34593e7 −1.69284
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.13742e7i 0.425508i
$$300$$ 0 0
$$301$$ −1.83920e6 −0.0674418
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 2.30481e6i 0.0812337i
$$306$$ 0 0
$$307$$ 5.23060e6 0.180774 0.0903871 0.995907i $$-0.471190\pi$$
0.0903871 + 0.995907i $$0.471190\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 3.12221e7i − 1.03796i −0.854786 0.518981i $$-0.826312\pi$$
0.854786 0.518981i $$-0.173688\pi$$
$$312$$ 0 0
$$313$$ 2.24778e7 0.733029 0.366515 0.930412i $$-0.380551\pi$$
0.366515 + 0.930412i $$0.380551\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.76211e7i − 0.867088i −0.901132 0.433544i $$-0.857263\pi$$
0.901132 0.433544i $$-0.142737\pi$$
$$318$$ 0 0
$$319$$ 3.93553e7 1.21236
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 73109.2i − 0.00216952i
$$324$$ 0 0
$$325$$ −4.92839e7 −1.43567
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 3.71672e7i − 1.04369i
$$330$$ 0 0
$$331$$ 5.76138e6 0.158870 0.0794352 0.996840i $$-0.474688\pi$$
0.0794352 + 0.996840i $$0.474688\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 2.93917e7i − 0.781790i
$$336$$ 0 0
$$337$$ −4.01052e7 −1.04788 −0.523939 0.851756i $$-0.675538\pi$$
−0.523939 + 0.851756i $$0.675538\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 5.33535e7i − 1.34555i
$$342$$ 0 0
$$343$$ −504328. −0.0124977
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 6.78127e7i − 1.62302i −0.584341 0.811508i $$-0.698647\pi$$
0.584341 0.811508i $$-0.301353\pi$$
$$348$$ 0 0
$$349$$ −4.20638e7 −0.989538 −0.494769 0.869024i $$-0.664747\pi$$
−0.494769 + 0.869024i $$0.664747\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 1.75976e7i − 0.400063i −0.979789 0.200032i $$-0.935896\pi$$
0.979789 0.200032i $$-0.0641045\pi$$
$$354$$ 0 0
$$355$$ −9.24478e7 −2.06639
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 1.39920e7i 0.302410i 0.988502 + 0.151205i $$0.0483154\pi$$
−0.988502 + 0.151205i $$0.951685\pi$$
$$360$$ 0 0
$$361$$ −1.40523e7 −0.298694
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.10768e7i 0.844731i
$$366$$ 0 0
$$367$$ 2.65855e7 0.537832 0.268916 0.963164i $$-0.413335\pi$$
0.268916 + 0.963164i $$0.413335\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1.15549e8i 2.26279i
$$372$$ 0 0
$$373$$ 1.78829e7 0.344598 0.172299 0.985045i $$-0.444881\pi$$
0.172299 + 0.985045i $$0.444881\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.88671e7i 1.84513i
$$378$$ 0 0
$$379$$ −7.20978e7 −1.32435 −0.662177 0.749347i $$-0.730367\pi$$
−0.662177 + 0.749347i $$0.730367\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 8.68648e6i − 0.154614i −0.997007 0.0773068i $$-0.975368\pi$$
0.997007 0.0773068i $$-0.0246321\pi$$
$$384$$ 0 0
$$385$$ 1.12873e8 1.97791
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 4.94411e7i − 0.839923i −0.907542 0.419962i $$-0.862044\pi$$
0.907542 0.419962i $$-0.137956\pi$$
$$390$$ 0 0
$$391$$ −42984.0 −0.000719079 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 6.10837e6i 0.0991137i
$$396$$ 0 0
$$397$$ 1.56911e7 0.250774 0.125387 0.992108i $$-0.459983\pi$$
0.125387 + 0.992108i $$0.459983\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 4.74514e7i − 0.735895i −0.929847 0.367947i $$-0.880061\pi$$
0.929847 0.367947i $$-0.119939\pi$$
$$402$$ 0 0
$$403$$ 1.34033e8 2.04784
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 7.04203e7i − 1.04451i
$$408$$ 0 0
$$409$$ −1.15512e8 −1.68832 −0.844162 0.536088i $$-0.819901\pi$$
−0.844162 + 0.536088i $$0.819901\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1.20923e8i 1.71656i
$$414$$ 0 0
$$415$$ −1.90994e6 −0.0267225
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 1.46693e8i − 1.99420i −0.0761306 0.997098i $$-0.524257\pi$$
0.0761306 0.997098i $$-0.475743\pi$$
$$420$$ 0 0
$$421$$ 1.39239e8 1.86601 0.933005 0.359863i $$-0.117176\pi$$
0.933005 + 0.359863i $$0.117176\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 186248.i − 0.00242619i
$$426$$ 0 0
$$427$$ 6.41300e6 0.0823716
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1.00392e8i 1.25391i 0.779056 + 0.626954i $$0.215699\pi$$
−0.779056 + 0.626954i $$0.784301\pi$$
$$432$$ 0 0
$$433$$ −4.00631e7 −0.493493 −0.246747 0.969080i $$-0.579362\pi$$
−0.246747 + 0.969080i $$0.579362\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 1.93983e7i − 0.232445i
$$438$$ 0 0
$$439$$ 1.38592e8 1.63811 0.819057 0.573712i $$-0.194497\pi$$
0.819057 + 0.573712i $$0.194497\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 1.11443e8i 1.28186i 0.767600 + 0.640929i $$0.221451\pi$$
−0.767600 + 0.640929i $$0.778549\pi$$
$$444$$ 0 0
$$445$$ 2.24965e7 0.255290
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 6.11166e7i − 0.675181i −0.941293 0.337591i $$-0.890388\pi$$
0.941293 0.337591i $$-0.109612\pi$$
$$450$$ 0 0
$$451$$ −4.96619e7 −0.541370
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2.83555e8i 3.01026i
$$456$$ 0 0
$$457$$ 3.56665e7 0.373690 0.186845 0.982389i $$-0.440174\pi$$
0.186845 + 0.982389i $$0.440174\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 1.51983e8i 1.55128i 0.631173 + 0.775642i $$0.282574\pi$$
−0.631173 + 0.775642i $$0.717426\pi$$
$$462$$ 0 0
$$463$$ −1.14978e8 −1.15844 −0.579218 0.815173i $$-0.696642\pi$$
−0.579218 + 0.815173i $$0.696642\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 8.81705e7i − 0.865711i −0.901463 0.432855i $$-0.857506\pi$$
0.901463 0.432855i $$-0.142494\pi$$
$$468$$ 0 0
$$469$$ −8.17805e7 −0.792741
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 5.09456e6i 0.0481419i
$$474$$ 0 0
$$475$$ 8.40520e7 0.784272
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 8.94388e7i 0.813803i 0.913472 + 0.406902i $$0.133391\pi$$
−0.913472 + 0.406902i $$0.866609\pi$$
$$480$$ 0 0
$$481$$ 1.76908e8 1.58969
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 5.59111e7i − 0.490087i
$$486$$ 0 0
$$487$$ 7.51688e7 0.650805 0.325403 0.945576i $$-0.394500\pi$$
0.325403 + 0.945576i $$0.394500\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 4.50822e7i − 0.380856i −0.981701 0.190428i $$-0.939012\pi$$
0.981701 0.190428i $$-0.0609876\pi$$
$$492$$ 0 0
$$493$$ −373626. −0.00311815
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.57230e8i 2.09533i
$$498$$ 0 0
$$499$$ −9.15458e7 −0.736778 −0.368389 0.929672i $$-0.620091\pi$$
−0.368389 + 0.929672i $$0.620091\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 1.61043e8i 1.26543i 0.774386 + 0.632713i $$0.218059\pi$$
−0.774386 + 0.632713i $$0.781941\pi$$
$$504$$ 0 0
$$505$$ −1.16333e8 −0.903295
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 2.39995e7i 0.181990i 0.995851 + 0.0909951i $$0.0290048\pi$$
−0.995851 + 0.0909951i $$0.970995\pi$$
$$510$$ 0 0
$$511$$ 1.14294e8 0.856564
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 3.46751e8i − 2.53861i
$$516$$ 0 0
$$517$$ −1.02953e8 −0.745018
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 9.00897e7i − 0.637033i −0.947917 0.318517i $$-0.896815\pi$$
0.947917 0.318517i $$-0.103185\pi$$
$$522$$ 0 0
$$523$$ 3.77691e7 0.264016 0.132008 0.991249i $$-0.457857\pi$$
0.132008 + 0.991249i $$0.457857\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 506520.i 0.00346071i
$$528$$ 0 0
$$529$$ 1.36631e8 0.922957
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 1.24759e8i − 0.823931i
$$534$$ 0 0
$$535$$ −4.53427e7 −0.296105
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 1.56332e8i − 0.998347i
$$540$$ 0 0
$$541$$ 2.54800e7 0.160919 0.0804595 0.996758i $$-0.474361\pi$$
0.0804595 + 0.996758i $$0.474361\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 3.38253e7i 0.208955i
$$546$$ 0 0
$$547$$ −2.05216e8 −1.25386 −0.626930 0.779076i $$-0.715689\pi$$
−0.626930 + 0.779076i $$0.715689\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 1.68614e8i − 1.00795i
$$552$$ 0 0
$$553$$ 1.69961e7 0.100502
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 2.41143e8i − 1.39543i −0.716375 0.697715i $$-0.754200\pi$$
0.716375 0.697715i $$-0.245800\pi$$
$$558$$ 0 0
$$559$$ −1.27984e7 −0.0732690
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.68877e8i 0.946337i 0.880972 + 0.473168i $$0.156890\pi$$
−0.880972 + 0.473168i $$0.843110\pi$$
$$564$$ 0 0
$$565$$ 1.42968e8 0.792670
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 2.43995e8i − 1.32448i −0.749293 0.662238i $$-0.769607\pi$$
0.749293 0.662238i $$-0.230393\pi$$
$$570$$ 0 0
$$571$$ −2.41502e8 −1.29722 −0.648608 0.761123i $$-0.724648\pi$$
−0.648608 + 0.761123i $$0.724648\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 4.94177e7i − 0.259944i
$$576$$ 0 0
$$577$$ −4.93979e7 −0.257147 −0.128573 0.991700i $$-0.541040\pi$$
−0.128573 + 0.991700i $$0.541040\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5.31430e6i 0.0270968i
$$582$$ 0 0
$$583$$ 3.20069e8 1.61525
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.72052e8i 0.850639i 0.905043 + 0.425320i $$0.139838\pi$$
−0.905043 + 0.425320i $$0.860162\pi$$
$$588$$ 0 0
$$589$$ −2.28588e8 −1.11869
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 2.70643e8i 1.29788i 0.760841 + 0.648938i $$0.224787\pi$$
−0.760841 + 0.648938i $$0.775213\pi$$
$$594$$ 0 0
$$595$$ −1.07158e6 −0.00508712
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 1.73299e8i 0.806337i 0.915126 + 0.403169i $$0.132091\pi$$
−0.915126 + 0.403169i $$0.867909\pi$$
$$600$$ 0 0
$$601$$ −4.31090e8 −1.98584 −0.992921 0.118775i $$-0.962103\pi$$
−0.992921 + 0.118775i $$0.962103\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 4.49604e6i − 0.0203032i
$$606$$ 0 0
$$607$$ −1.66991e7 −0.0746665 −0.0373332 0.999303i $$-0.511886\pi$$
−0.0373332 + 0.999303i $$0.511886\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 2.58635e8i − 1.13387i
$$612$$ 0 0
$$613$$ −1.92321e8 −0.834920 −0.417460 0.908695i $$-0.637080\pi$$
−0.417460 + 0.908695i $$0.637080\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 1.87023e8i − 0.796233i −0.917335 0.398117i $$-0.869664\pi$$
0.917335 0.398117i $$-0.130336\pi$$
$$618$$ 0 0
$$619$$ −2.54873e8 −1.07461 −0.537307 0.843387i $$-0.680558\pi$$
−0.537307 + 0.843387i $$0.680558\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 6.25950e7i − 0.258866i
$$624$$ 0 0
$$625$$ −2.58657e8 −1.05946
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 668547.i 0.00268646i
$$630$$ 0 0
$$631$$ −9.23602e7 −0.367618 −0.183809 0.982962i $$-0.558843\pi$$
−0.183809 + 0.982962i $$0.558843\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 5.31797e8i − 2.07694i
$$636$$ 0 0
$$637$$ 3.92732e8 1.51942
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 4.24666e8i − 1.61240i −0.591643 0.806200i $$-0.701520\pi$$
0.591643 0.806200i $$-0.298480\pi$$
$$642$$ 0 0
$$643$$ −3.75946e8 −1.41414 −0.707071 0.707143i $$-0.749984\pi$$
−0.707071 + 0.707143i $$0.749984\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 2.63747e7i − 0.0973813i −0.998814 0.0486906i $$-0.984495\pi$$
0.998814 0.0486906i $$-0.0155048\pi$$
$$648$$ 0 0
$$649$$ 3.34955e8 1.22533
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.58756e8i 0.929291i 0.885497 + 0.464645i $$0.153818\pi$$
−0.885497 + 0.464645i $$0.846182\pi$$
$$654$$ 0 0
$$655$$ 5.34696e8 1.90275
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.39345e8i 0.486895i 0.969914 + 0.243447i $$0.0782783\pi$$
−0.969914 + 0.243447i $$0.921722\pi$$
$$660$$ 0 0
$$661$$ −4.72545e8 −1.63621 −0.818104 0.575070i $$-0.804975\pi$$
−0.818104 + 0.575070i $$0.804975\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 4.83593e8i − 1.64443i
$$666$$ 0 0
$$667$$ −9.91354e7 −0.334081
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 1.77639e7i − 0.0587992i
$$672$$ 0 0
$$673$$ 5.48833e8 1.80051 0.900254 0.435364i $$-0.143380\pi$$
0.900254 + 0.435364i $$0.143380\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 1.00760e8i − 0.324731i −0.986731 0.162365i $$-0.948088\pi$$
0.986731 0.162365i $$-0.0519123\pi$$
$$678$$ 0 0
$$679$$ −1.55569e8 −0.496952
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 313056.i 0 0.000982562i 1.00000 0.000491281i $$0.000156380\pi$$
−1.00000 0.000491281i $$0.999844\pi$$
$$684$$ 0 0
$$685$$ 7.80005e8 2.42675
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 8.04068e8i 2.45830i
$$690$$ 0 0
$$691$$ 3.72812e8 1.12994 0.564971 0.825111i $$-0.308887\pi$$
0.564971 + 0.825111i $$0.308887\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1.90009e8i 0.566003i
$$696$$ 0 0
$$697$$ 471474. 0.00139239
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.21170e8i 1.80325i 0.432517 + 0.901626i $$0.357626\pi$$
−0.432517 + 0.901626i $$0.642374\pi$$
$$702$$ 0 0
$$703$$ −3.01709e8 −0.868406
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 3.23690e8i 0.915947i
$$708$$ 0 0
$$709$$ −2.46510e8 −0.691666 −0.345833 0.938296i $$-0.612404\pi$$
−0.345833 + 0.938296i $$0.612404\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.34397e8i 0.370783i
$$714$$ 0 0
$$715$$ 7.85445e8 2.14881
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 9.60389e7i 0.258381i 0.991620 + 0.129191i $$0.0412379\pi$$
−0.991620 + 0.129191i $$0.958762\pi$$
$$720$$ 0 0
$$721$$ −9.64811e8 −2.57417
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 4.29549e8i − 1.12719i
$$726$$ 0 0
$$727$$ −3.91371e8 −1.01856 −0.509278 0.860602i $$-0.670088\pi$$
−0.509278 + 0.860602i $$0.670088\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 48366.1i 0 0.000123819i
$$732$$ 0 0
$$733$$ 3.49078e7 0.0886361 0.0443181 0.999017i $$-0.485889\pi$$
0.0443181 + 0.999017i $$0.485889\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2.26531e8i 0.565881i
$$738$$ 0 0
$$739$$ 3.02999e8 0.750773 0.375386 0.926868i $$-0.377510\pi$$
0.375386 + 0.926868i $$0.377510\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 2.45628e8i − 0.598842i −0.954121 0.299421i $$-0.903207\pi$$
0.954121 0.299421i $$-0.0967935\pi$$
$$744$$ 0 0
$$745$$ 3.86317e8 0.934276
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 1.26163e8i 0.300253i
$$750$$ 0 0
$$751$$ 8.23270e7 0.194367 0.0971835 0.995266i $$-0.469017\pi$$
0.0971835 + 0.995266i $$0.469017\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 7.09484e8i 1.64855i
$$756$$ 0 0
$$757$$ −6.03579e8 −1.39138 −0.695691 0.718341i $$-0.744902\pi$$
−0.695691 + 0.718341i $$0.744902\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 2.32982e8i − 0.528651i −0.964434 0.264325i $$-0.914851\pi$$
0.964434 0.264325i $$-0.0851493\pi$$
$$762$$ 0 0
$$763$$ 9.41167e7 0.211882
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.41463e8i 1.86487i
$$768$$ 0 0
$$769$$ 8.15796e8 1.79392 0.896958 0.442115i $$-0.145772\pi$$
0.896958 + 0.442115i $$0.145772\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 3.66587e8i − 0.793667i −0.917891 0.396833i $$-0.870109\pi$$
0.917891 0.396833i $$-0.129891\pi$$
$$774$$ 0 0
$$775$$ −5.82335e8 −1.25103
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.12772e8i 0.450093i
$$780$$ 0 0
$$781$$ 7.12524e8 1.49571
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.07077e9i 2.21354i
$$786$$ 0 0
$$787$$ 4.02462e8 0.825659 0.412830 0.910808i $$-0.364540\pi$$
0.412830 + 0.910808i $$0.364540\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 3.97798e8i − 0.803773i
$$792$$ 0 0
$$793$$ 4.46260e7 0.0894887
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 5.18940e8i − 1.02504i −0.858675 0.512521i $$-0.828712\pi$$
0.858675 0.512521i $$-0.171288\pi$$
$$798$$ 0 0
$$799$$ 977400. 0.00191616
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 3.16592e8i − 0.611440i
$$804$$ 0 0
$$805$$ −2.84325e8 −0.545038
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 3.04036e6i − 0.00574221i −0.999996 0.00287110i $$-0.999086\pi$$
0.999996 0.00287110i $$-0.000913902\pi$$
$$810$$ 0 0
$$811$$ −2.25521e8 −0.422790 −0.211395 0.977401i $$-0.567801\pi$$
−0.211395 + 0.977401i $$0.567801\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1.39280e8i − 0.257285i
$$816$$ 0 0
$$817$$ 2.18272e7 0.0400250
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.77035e8i 0.500617i 0.968166 + 0.250309i $$0.0805321\pi$$
−0.968166 + 0.250309i $$0.919468\pi$$
$$822$$ 0 0
$$823$$ 7.07336e8 1.26890 0.634448 0.772965i $$-0.281227\pi$$
0.634448 + 0.772965i $$0.281227\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.66346e8i 0.470900i 0.971886 + 0.235450i $$0.0756564\pi$$
−0.971886 + 0.235450i $$0.924344\pi$$
$$828$$ 0 0
$$829$$ 5.03826e8 0.884336 0.442168 0.896932i $$-0.354209\pi$$
0.442168 + 0.896932i $$0.354209\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.48416e6i 0.00256772i
$$834$$ 0 0
$$835$$ 8.35764e8 1.43557
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 7.63364e8i − 1.29255i −0.763106 0.646273i $$-0.776327\pi$$
0.763106 0.646273i $$-0.223673\pi$$
$$840$$ 0 0
$$841$$ −2.66883e8 −0.448676
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.13355e9i 1.87876i
$$846$$ 0 0
$$847$$ −1.25099e7 −0.0205876
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.77388e8i 0.287829i
$$852$$ 0 0
$$853$$ −1.87985e7 −0.0302884 −0.0151442 0.999885i $$-0.504821\pi$$
−0.0151442 + 0.999885i $$0.504821\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 6.86427e8i − 1.09057i −0.838252 0.545283i $$-0.816422\pi$$
0.838252 0.545283i $$-0.183578\pi$$
$$858$$ 0 0
$$859$$ −5.51932e8 −0.870775 −0.435387 0.900243i $$-0.643389\pi$$
−0.435387 + 0.900243i $$0.643389\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 3.65665e8i − 0.568920i −0.958688 0.284460i $$-0.908186\pi$$
0.958688 0.284460i $$-0.0918142\pi$$
$$864$$ 0 0
$$865$$ 6.20982e8 0.959468
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 4.70791e7i − 0.0717413i
$$870$$ 0 0
$$871$$ −5.69084e8 −0.861236
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 8.35174e7i 0.124667i
$$876$$ 0 0
$$877$$ −5.85387e8 −0.867849 −0.433925 0.900949i $$-0.642872\pi$$
−0.433925 + 0.900949i $$0.642872\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 4.29761e8i − 0.628491i −0.949342 0.314246i $$-0.898248\pi$$
0.949342 0.314246i $$-0.101752\pi$$
$$882$$ 0 0
$$883$$ −2.20085e8 −0.319675 −0.159837 0.987143i $$-0.551097\pi$$
−0.159837 + 0.987143i $$0.551097\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 1.17196e9i − 1.67936i −0.543084 0.839678i $$-0.682744\pi$$
0.543084 0.839678i $$-0.317256\pi$$
$$888$$ 0 0
$$889$$ −1.47969e9 −2.10604
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 4.41092e8i 0.619405i
$$894$$ 0 0
$$895$$ −1.29365e9 −1.80446
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.16820e9i 1.60783i
$$900$$ 0 0
$$901$$ −3.03863e6 −0.00415436
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 1.80579e9i − 2.43625i
$$906$$ 0 0
$$907$$ −7.31614e8 −0.980529 −0.490264 0.871574i $$-0.663100\pi$$
−0.490264 + 0.871574i $$0.663100\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 9.18595e8i − 1.21498i −0.794327 0.607490i $$-0.792177\pi$$
0.794327 0.607490i $$-0.207823\pi$$
$$912$$ 0 0
$$913$$ 1.47205e7 0.0193425
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 1.48776e9i − 1.92941i
$$918$$ 0 0
$$919$$ 2.15987e8 0.278279 0.139139 0.990273i $$-0.455566\pi$$
0.139139 + 0.990273i $$0.455566\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1.78998e9i 2.27637i
$$924$$ 0 0
$$925$$ −7.68613e8 −0.971141
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 3.10124e8i 0.386802i 0.981120 + 0.193401i $$0.0619518\pi$$
−0.981120 + 0.193401i $$0.938048\pi$$
$$930$$ 0 0
$$931$$ −6.69791e8 −0.830023
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2.96825e6i 0.00363133i
$$936$$ 0 0
$$937$$ −7.42448e8 −0.902501 −0.451250 0.892397i $$-0.649022\pi$$
−0.451250 + 0.892397i $$0.649022\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1.81766e8i 0.218144i 0.994034 + 0.109072i $$0.0347879\pi$$
−0.994034 + 0.109072i $$0.965212\pi$$
$$942$$ 0 0
$$943$$ 1.25098e8 0.149181
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 8.59189e8i − 1.01167i −0.862630 0.505835i $$-0.831184\pi$$
0.862630 0.505835i $$-0.168816\pi$$
$$948$$ 0 0
$$949$$ 7.95333e8 0.930573
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.86819e8i 0.793530i 0.917920 + 0.396765i $$0.129867\pi$$
−0.917920 + 0.396765i $$0.870133\pi$$
$$954$$ 0 0
$$955$$ 2.26029e9 2.59510
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 2.17031e9i − 2.46075i
$$960$$ 0 0
$$961$$ 6.96218e8 0.784468
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 6.83957e8i − 0.761109i
$$966$$ 0 0
$$967$$ 1.09411e9 1.20999 0.604995 0.796230i $$-0.293175\pi$$
0.604995 + 0.796230i $$0.293175\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 4.43115e8i − 0.484014i −0.970274 0.242007i $$-0.922194\pi$$
0.970274 0.242007i $$-0.0778058\pi$$
$$972$$ 0 0
$$973$$ 5.28687e8 0.573931
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1.19004e9i 1.27608i 0.770001 + 0.638042i $$0.220256\pi$$
−0.770001 + 0.638042i $$0.779744\pi$$
$$978$$ 0 0
$$979$$ −1.73387e8 −0.184786
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1.18187e9i 1.24425i 0.782918 + 0.622125i $$0.213730\pi$$
−0.782918 + 0.622125i $$0.786270\pi$$
$$984$$ 0 0
$$985$$ −9.35785e8 −0.979191
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 1.28331e7i − 0.0132661i
$$990$$ 0 0
$$991$$ −5.09602e8 −0.523613 −0.261806 0.965120i $$-0.584318\pi$$
−0.261806 + 0.965120i $$0.584318\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 9.84373e7i 0.0999288i
$$996$$ 0 0
$$997$$ −9.90780e8 −0.999751 −0.499875 0.866097i $$-0.666621\pi$$
−0.499875 + 0.866097i $$0.666621\pi$$
$$998$$ 0 0
$$999$$ 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 144.7.e.d.17.2 2
3.2 odd 2 inner 144.7.e.d.17.1 2
4.3 odd 2 18.7.b.a.17.2 yes 2
8.3 odd 2 576.7.e.b.449.1 2
8.5 even 2 576.7.e.k.449.1 2
12.11 even 2 18.7.b.a.17.1 2
20.3 even 4 450.7.b.a.449.4 4
20.7 even 4 450.7.b.a.449.1 4
20.19 odd 2 450.7.d.a.251.1 2
24.5 odd 2 576.7.e.k.449.2 2
24.11 even 2 576.7.e.b.449.2 2
36.7 odd 6 162.7.d.d.53.2 4
36.11 even 6 162.7.d.d.53.1 4
36.23 even 6 162.7.d.d.107.2 4
36.31 odd 6 162.7.d.d.107.1 4
60.23 odd 4 450.7.b.a.449.2 4
60.47 odd 4 450.7.b.a.449.3 4
60.59 even 2 450.7.d.a.251.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.b.a.17.1 2 12.11 even 2
18.7.b.a.17.2 yes 2 4.3 odd 2
144.7.e.d.17.1 2 3.2 odd 2 inner
144.7.e.d.17.2 2 1.1 even 1 trivial
162.7.d.d.53.1 4 36.11 even 6
162.7.d.d.53.2 4 36.7 odd 6
162.7.d.d.107.1 4 36.31 odd 6
162.7.d.d.107.2 4 36.23 even 6
450.7.b.a.449.1 4 20.7 even 4
450.7.b.a.449.2 4 60.23 odd 4
450.7.b.a.449.3 4 60.47 odd 4
450.7.b.a.449.4 4 20.3 even 4
450.7.d.a.251.1 2 20.19 odd 2
450.7.d.a.251.2 2 60.59 even 2
576.7.e.b.449.1 2 8.3 odd 2
576.7.e.b.449.2 2 24.11 even 2
576.7.e.k.449.1 2 8.5 even 2
576.7.e.k.449.2 2 24.5 odd 2