# Properties

 Label 63.4.c.a.62.2 Level $63$ Weight $4$ Character 63.62 Analytic conductor $3.717$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 62.2 Root $$2.57794i$$ of defining polynomial Character $$\chi$$ $$=$$ 63.62 Dual form 63.4.c.a.62.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.66471i q^{2} +5.22876 q^{4} +18.5203 q^{7} -22.0220i q^{8} +O(q^{10})$$ $$q-1.66471i q^{2} +5.22876 q^{4} +18.5203 q^{7} -22.0220i q^{8} -29.3750i q^{11} -30.8308i q^{14} +5.16995 q^{16} -48.9007 q^{22} +181.692i q^{23} -125.000 q^{25} +96.8379 q^{28} +304.463i q^{29} -184.782i q^{32} -10.5830 q^{37} -534.442 q^{43} -153.595i q^{44} +302.464 q^{46} +343.000 q^{49} +208.088i q^{50} -768.909i q^{53} -407.853i q^{56} +506.840 q^{58} -266.248 q^{64} +740.000 q^{67} +1178.70i q^{71} +17.6176i q^{74} -544.032i q^{77} -1384.00 q^{79} +889.688i q^{86} -646.895 q^{88} +950.024i q^{92} -570.994i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 32q^{4} + O(q^{10})$$ $$4q - 32q^{4} + 444q^{16} - 820q^{22} - 500q^{25} + 980q^{28} - 748q^{46} + 1372q^{49} + 260q^{58} - 3552q^{64} + 2960q^{67} - 5536q^{79} + 8260q^{88} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.66471i − 0.588562i −0.955719 0.294281i $$-0.904920\pi$$
0.955719 0.294281i $$-0.0950802\pi$$
$$3$$ 0 0
$$4$$ 5.22876 0.653595
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 18.5203 1.00000
$$8$$ − 22.0220i − 0.973243i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 29.3750i − 0.805172i −0.915382 0.402586i $$-0.868111\pi$$
0.915382 0.402586i $$-0.131889\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ − 30.8308i − 0.588562i
$$15$$ 0 0
$$16$$ 5.16995 0.0807804
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −48.9007 −0.473894
$$23$$ 181.692i 1.64719i 0.567176 + 0.823597i $$0.308036\pi$$
−0.567176 + 0.823597i $$0.691964\pi$$
$$24$$ 0 0
$$25$$ −125.000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 96.8379 0.653595
$$29$$ 304.463i 1.94956i 0.223165 + 0.974781i $$0.428361\pi$$
−0.223165 + 0.974781i $$0.571639\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ − 184.782i − 1.02079i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.5830 −0.0470226 −0.0235113 0.999724i $$-0.507485\pi$$
−0.0235113 + 0.999724i $$0.507485\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −534.442 −1.89539 −0.947693 0.319183i $$-0.896592\pi$$
−0.947693 + 0.319183i $$0.896592\pi$$
$$44$$ − 153.595i − 0.526256i
$$45$$ 0 0
$$46$$ 302.464 0.969476
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 343.000 1.00000
$$50$$ 208.088i 0.588562i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 768.909i − 1.99279i −0.0848489 0.996394i $$-0.527041\pi$$
0.0848489 0.996394i $$-0.472959\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ − 407.853i − 0.973243i
$$57$$ 0 0
$$58$$ 506.840 1.14744
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −266.248 −0.520016
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 740.000 1.34933 0.674667 0.738122i $$-0.264287\pi$$
0.674667 + 0.738122i $$0.264287\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1178.70i 1.97022i 0.171931 + 0.985109i $$0.445000\pi$$
−0.171931 + 0.985109i $$0.555000\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 17.6176i 0.0276757i
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 544.032i − 0.805172i
$$78$$ 0 0
$$79$$ −1384.00 −1.97104 −0.985520 0.169559i $$-0.945766\pi$$
−0.985520 + 0.169559i $$0.945766\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 889.688i 1.11555i
$$87$$ 0 0
$$88$$ −646.895 −0.783628
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 950.024i 1.07660i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ − 570.994i − 0.588562i
$$99$$ 0 0
$$100$$ −653.595 −0.653595
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −1280.01 −1.17288
$$107$$ − 20.9231i − 0.0189039i −0.999955 0.00945193i $$-0.996991\pi$$
0.999955 0.00945193i $$-0.00300869\pi$$
$$108$$ 0 0
$$109$$ 2275.35 1.99944 0.999718 0.0237260i $$-0.00755292\pi$$
0.999718 + 0.0237260i $$0.00755292\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 95.7488 0.0807804
$$113$$ − 1157.60i − 0.963699i −0.876254 0.481849i $$-0.839965\pi$$
0.876254 0.481849i $$-0.160035\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1591.96i 1.27422i
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 468.111 0.351699
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2000.00 1.39741 0.698706 0.715409i $$-0.253760\pi$$
0.698706 + 0.715409i $$0.253760\pi$$
$$128$$ − 1035.03i − 0.714725i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ − 1231.88i − 0.794167i
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 2752.87i − 1.71674i −0.513031 0.858370i $$-0.671478\pi$$
0.513031 0.858370i $$-0.328522\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1962.18 1.15960
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −55.3360 −0.0307337
$$149$$ − 1931.33i − 1.06188i −0.847409 0.530941i $$-0.821839\pi$$
0.847409 0.530941i $$-0.178161\pi$$
$$150$$ 0 0
$$151$$ −2248.89 −1.21200 −0.606000 0.795465i $$-0.707227\pi$$
−0.606000 + 0.795465i $$0.707227\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −905.653 −0.473894
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 2303.95i 1.16008i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3364.99i 1.64719i
$$162$$ 0 0
$$163$$ −1780.00 −0.855340 −0.427670 0.903935i $$-0.640665\pi$$
−0.427670 + 0.903935i $$0.640665\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 2197.00 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −2794.47 −1.23881
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −2315.03 −1.00000
$$176$$ − 151.867i − 0.0650421i
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 1575.84i 0.658010i 0.944328 + 0.329005i $$0.106713\pi$$
−0.944328 + 0.329005i $$0.893287\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 4001.22 1.60312
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 5255.29i 1.99089i 0.0953502 + 0.995444i $$0.469603\pi$$
−0.0953502 + 0.995444i $$0.530397\pi$$
$$192$$ 0 0
$$193$$ −2772.75 −1.03413 −0.517064 0.855947i $$-0.672975\pi$$
−0.517064 + 0.855947i $$0.672975\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 1793.46 0.653595
$$197$$ − 5147.21i − 1.86154i −0.365603 0.930771i $$-0.619137\pi$$
0.365603 0.930771i $$-0.380863\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 2752.75i 0.973243i
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 5638.73i 1.94956i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1772.65 −0.578363 −0.289181 0.957274i $$-0.593383\pi$$
−0.289181 + 0.957274i $$0.593383\pi$$
$$212$$ − 4020.44i − 1.30248i
$$213$$ 0 0
$$214$$ −34.8308 −0.0111261
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ − 3787.78i − 1.17679i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ − 3422.22i − 1.02079i
$$225$$ 0 0
$$226$$ −1927.06 −0.567197
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6704.87 1.89740
$$233$$ 412.009i 0.115844i 0.998321 + 0.0579219i $$0.0184474\pi$$
−0.998321 + 0.0579219i $$0.981553\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 5533.66i − 1.49767i −0.662757 0.748834i $$-0.730614\pi$$
0.662757 0.748834i $$-0.269386\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ − 779.267i − 0.206997i
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 5337.20 1.32627
$$254$$ − 3329.41i − 0.822464i
$$255$$ 0 0
$$256$$ −3853.01 −0.940677
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ −196.000 −0.0470226
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 2470.04i − 0.579123i −0.957159 0.289561i $$-0.906491\pi$$
0.957159 0.289561i $$-0.0935094\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 3869.28 0.881917
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −4582.71 −1.01041
$$275$$ 3671.87i 0.805172i
$$276$$ 0 0
$$277$$ 7310.00 1.58561 0.792807 0.609472i $$-0.208619\pi$$
0.792807 + 0.609472i $$0.208619\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8982.08i 1.90685i 0.301625 + 0.953427i $$0.402471\pi$$
−0.301625 + 0.953427i $$0.597529\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 6163.11i 1.28772i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4913.00 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 233.059i 0.0457644i
$$297$$ 0 0
$$298$$ −3215.09 −0.624983
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −9898.00 −1.89539
$$302$$ 3743.74i 0.713337i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ − 2844.61i − 0.526256i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −7236.60 −1.28826
$$317$$ − 1349.97i − 0.239185i −0.992823 0.119593i $$-0.961841\pi$$
0.992823 0.119593i $$-0.0381588\pi$$
$$318$$ 0 0
$$319$$ 8943.58 1.56973
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 5601.71 0.969476
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 2963.18i 0.503421i
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −5106.30 −0.847938 −0.423969 0.905677i $$-0.639364\pi$$
−0.423969 + 0.905677i $$0.639364\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −11916.5 −1.92621 −0.963103 0.269135i $$-0.913262\pi$$
−0.963103 + 0.269135i $$0.913262\pi$$
$$338$$ − 3657.36i − 0.588562i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 6352.45 1.00000
$$344$$ 11769.5i 1.84467i
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 11568.6i − 1.78972i −0.446347 0.894860i $$-0.647275\pi$$
0.446347 0.894860i $$-0.352725\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 3853.85i 0.588562i
$$351$$ 0 0
$$352$$ −5427.97 −0.821909
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 2623.31 0.387280
$$359$$ − 1996.13i − 0.293459i −0.989177 0.146729i $$-0.953125\pi$$
0.989177 0.146729i $$-0.0468746\pi$$
$$360$$ 0 0
$$361$$ 6859.00 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 939.339i 0.133061i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 14240.4i − 1.99279i
$$372$$ 0 0
$$373$$ 13970.0 1.93925 0.969624 0.244602i $$-0.0786573\pi$$
0.969624 + 0.244602i $$0.0786573\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8704.52 1.17974 0.589870 0.807498i $$-0.299179\pi$$
0.589870 + 0.807498i $$0.299179\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 8748.51 1.17176
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4615.81i 0.608649i
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 15337.9i − 1.99913i −0.0294496 0.999566i $$-0.509375\pi$$
0.0294496 0.999566i $$-0.490625\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 7553.54i − 0.973243i
$$393$$ 0 0
$$394$$ −8568.59 −1.09563
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −646.243 −0.0807804
$$401$$ 10169.8i 1.26648i 0.773956 + 0.633239i $$0.218275\pi$$
−0.773956 + 0.633239i $$0.781725\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 9386.82 1.14744
$$407$$ 310.876i 0.0378612i
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −15262.0 −1.76680 −0.883402 0.468616i $$-0.844753\pi$$
−0.883402 + 0.468616i $$0.844753\pi$$
$$422$$ 2950.95i 0.340402i
$$423$$ 0 0
$$424$$ −16932.9 −1.93947
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 109.402i − 0.0123555i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 5007.24i − 0.559606i −0.960057 0.279803i $$-0.909731\pi$$
0.960057 0.279803i $$-0.0902692\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 11897.2 1.30682
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12011.8i 1.28826i 0.764917 + 0.644129i $$0.222780\pi$$
−0.764917 + 0.644129i $$0.777220\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −4930.99 −0.520016
$$449$$ 15219.6i 1.59968i 0.600213 + 0.799841i $$0.295083\pi$$
−0.600213 + 0.799841i $$0.704917\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6052.81i − 0.629868i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −17821.8 −1.82422 −0.912109 0.409947i $$-0.865547\pi$$
−0.912109 + 0.409947i $$0.865547\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ −8440.00 −0.847171 −0.423585 0.905856i $$-0.639229\pi$$
−0.423585 + 0.905856i $$0.639229\pi$$
$$464$$ 1574.06i 0.157486i
$$465$$ 0 0
$$466$$ 685.873 0.0681813
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 13705.0 1.34933
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 15699.2i 1.52611i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −9211.91 −0.881471
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 2447.64 0.229868
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3296.61 −0.306742 −0.153371 0.988169i $$-0.549013\pi$$
−0.153371 + 0.988169i $$0.549013\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 8998.48i − 0.827079i −0.910486 0.413540i $$-0.864292\pi$$
0.910486 0.413540i $$-0.135708\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 21829.8i 1.97022i
$$498$$ 0 0
$$499$$ 21086.6 1.89172 0.945859 0.324577i $$-0.105222\pi$$
0.945859 + 0.324577i $$0.105222\pi$$
$$500$$ 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
$$506$$ − 8884.87i − 0.780594i
$$507$$ 0 0
$$508$$ 10457.5 0.913341
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1866.13i − 0.161079i
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 326.282i 0.0276757i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −4111.89 −0.340850
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −20845.1 −1.71325
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ − 16296.3i − 1.31323i
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 10075.6i − 0.805172i
$$540$$ 0 0
$$541$$ 15878.0 1.26183 0.630914 0.775853i $$-0.282680\pi$$
0.630914 + 0.775853i $$0.282680\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12980.0 1.01460 0.507299 0.861770i $$-0.330644\pi$$
0.507299 + 0.861770i $$0.330644\pi$$
$$548$$ − 14394.1i − 1.12205i
$$549$$ 0 0
$$550$$ 6112.58 0.473894
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −25632.0 −1.97104
$$554$$ − 12169.0i − 0.933233i
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 2807.99i 0.213606i 0.994280 + 0.106803i $$0.0340613\pi$$
−0.994280 + 0.106803i $$0.965939\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 14952.5 1.12230
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 25957.2 1.91750
$$569$$ − 16481.1i − 1.21428i −0.794596 0.607138i $$-0.792317\pi$$
0.794596 0.607138i $$-0.207683\pi$$
$$570$$ 0 0
$$571$$ 6788.00 0.497494 0.248747 0.968569i $$-0.419981\pi$$
0.248747 + 0.968569i $$0.419981\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 22711.5i − 1.64719i
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 8178.70i 0.588562i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −22586.7 −1.60454
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −54.7136 −0.00379850
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ − 10098.4i − 0.694040i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 28622.4i 1.95239i 0.216899 + 0.976194i $$0.430406\pi$$
−0.216899 + 0.976194i $$0.569594\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 16477.3i 1.11555i
$$603$$ 0 0
$$604$$ −11758.9 −0.792156
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −15010.0 −0.988986 −0.494493 0.869182i $$-0.664646\pi$$
−0.494493 + 0.869182i $$0.664646\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −11980.7 −0.783628
$$617$$ 19836.0i 1.29428i 0.762372 + 0.647139i $$0.224035\pi$$
−0.762372 + 0.647139i $$0.775965\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15625.0 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 26192.0 1.65244 0.826218 0.563351i $$-0.190488\pi$$
0.826218 + 0.563351i $$0.190488\pi$$
$$632$$ 30478.4i 1.91830i
$$633$$ 0 0
$$634$$ −2247.30 −0.140775
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ − 14888.4i − 0.923885i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 28353.5i 1.74711i 0.486729 + 0.873553i $$0.338190\pi$$
−0.486729 + 0.873553i $$0.661810\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 17594.7i 1.07660i
$$645$$ 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
$$651$$ 0 0
$$652$$ −9307.19 −0.559045
$$653$$ − 32948.9i − 1.97456i −0.158976 0.987282i $$-0.550819\pi$$
0.158976 0.987282i $$-0.449181\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 22614.9i − 1.33680i −0.743803 0.668399i $$-0.766980\pi$$
0.743803 0.668399i $$-0.233020\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 8500.48i 0.499065i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −55318.5 −3.21130
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 9609.37 0.550392 0.275196 0.961388i $$-0.411257\pi$$
0.275196 + 0.961388i $$0.411257\pi$$
$$674$$ 19837.4i 1.13369i
$$675$$ 0 0
$$676$$ 11487.6 0.653595
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 31645.9i 1.77291i 0.462813 + 0.886456i $$0.346840\pi$$
−0.462813 + 0.886456i $$0.653160\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ − 10575.0i − 0.588562i
$$687$$ 0 0
$$688$$ −2763.04 −0.153110
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −19258.2 −1.05336
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −12104.7 −0.653595
$$701$$ − 23110.9i − 1.24520i −0.782539 0.622602i $$-0.786076\pi$$
0.782539 0.622602i $$-0.213924\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 7821.04i 0.418703i
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 35611.8 1.88636 0.943180 0.332281i $$-0.107818\pi$$
0.943180 + 0.332281i $$0.107818\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 8239.68i 0.430072i
$$717$$ 0 0
$$718$$ −3322.97 −0.172719
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 11418.2i − 0.588562i
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 38057.8i − 1.94956i
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 33573.5 1.68143
$$737$$ − 21737.5i − 1.08645i
$$738$$ 0 0
$$739$$ −25324.0 −1.26057 −0.630283 0.776365i $$-0.717061\pi$$
−0.630283 + 0.776365i $$0.717061\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −23706.1 −1.17288
$$743$$ 4655.40i 0.229865i 0.993373 + 0.114933i $$0.0366652\pi$$
−0.993373 + 0.114933i $$0.963335\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ − 23255.9i − 1.14137i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 387.501i − 0.0189039i
$$750$$ 0 0
$$751$$ 41088.5 1.99646 0.998230 0.0594732i $$-0.0189421\pi$$
0.998230 + 0.0594732i $$0.0189421\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22848.7 1.09703 0.548514 0.836141i $$-0.315194\pi$$
0.548514 + 0.836141i $$0.315194\pi$$
$$758$$ − 14490.5i − 0.694350i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 42140.0 1.99944
$$764$$ 27478.6i 1.30123i
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14498.0 −0.675901
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −25533.1 −1.17661
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 34624.2 1.58636
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1773.29 0.0807804
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ − 26913.5i − 1.21669i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 21439.1i − 0.963699i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 23097.8i 1.02079i
$$801$$ 0 0
$$802$$ 16929.8 0.745402
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 7405.65i 0.321840i 0.986967 + 0.160920i $$0.0514461\pi$$
−0.986967 + 0.160920i $$0.948554\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 29483.5i 1.27422i
$$813$$ 0 0
$$814$$ 517.516 0.0222837
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 43395.4i − 1.84471i −0.386339 0.922357i $$-0.626261\pi$$
0.386339 0.922357i $$-0.373739\pi$$
$$822$$ 0 0
$$823$$ −46240.0 −1.95848 −0.979238 0.202716i $$-0.935023\pi$$
−0.979238 + 0.202716i $$0.935023\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 46002.9i 1.93431i 0.254179 + 0.967157i $$0.418195\pi$$
−0.254179 + 0.967157i $$0.581805\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −68308.5 −2.80079
$$842$$ 25406.7i 1.03987i
$$843$$ 0 0
$$844$$ −9268.77 −0.378015
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8669.54 0.351699
$$848$$ − 3975.22i − 0.160978i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 1922.85i − 0.0774553i
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −460.768 −0.0183980
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −8335.58 −0.329363
$$863$$ 47169.0i 1.86055i 0.366868 + 0.930273i $$0.380430\pi$$
−0.366868 + 0.930273i $$0.619570\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 40655.0i 1.58703i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ − 50107.6i − 1.94594i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −6550.00 −0.252198 −0.126099 0.992018i $$-0.540246\pi$$
−0.126099 + 0.992018i $$0.540246\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ −43014.6 −1.63936 −0.819681 0.572820i $$-0.805850\pi$$
−0.819681 + 0.572820i $$0.805850\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 19996.1 0.758220
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 37040.5 1.39741
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ − 19169.1i − 0.714725i
$$897$$ 0 0
$$898$$ 25336.1 0.941512
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −25492.7 −0.937913
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −14250.0 −0.521680 −0.260840 0.965382i $$-0.584000\pi$$
−0.260840 + 0.965382i $$0.584000\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 1065.41i − 0.0387473i −0.999812 0.0193736i $$-0.993833\pi$$
0.999812 0.0193736i $$-0.00616720\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 29668.0i 1.07367i
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −51301.1 −1.84142 −0.920711 0.390244i $$-0.872391\pi$$
−0.920711 + 0.390244i $$0.872391\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 1322.88 0.0470226
$$926$$ 14050.1i 0.498613i
$$927$$ 0 0
$$928$$ 56259.3 1.99009
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 2154.29i 0.0757149i
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ − 22814.8i − 0.794167i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 26134.6 0.898211
$$947$$ 57034.5i 1.95710i 0.206013 + 0.978549i $$0.433951\pi$$
−0.206013 + 0.978549i $$0.566049\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 56795.7i 1.93053i 0.261276 + 0.965264i $$0.415857\pi$$
−0.261276 + 0.965264i $$0.584143\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ − 28934.2i − 0.978868i
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 50983.8i − 1.71674i
$$960$$ 0 0
$$961$$ 29791.0 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 52040.0 1.73060 0.865302 0.501251i $$-0.167127\pi$$
0.865302 + 0.501251i $$0.167127\pi$$
$$968$$ − 10308.7i − 0.342288i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 5487.88i 0.180537i
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 60603.4i − 1.98452i −0.124182 0.992259i $$-0.539631\pi$$
0.124182 0.992259i $$-0.460369\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −14979.8 −0.486788
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 97103.9i − 3.12207i
$$990$$ 0 0
$$991$$ −24155.7 −0.774300 −0.387150 0.922017i $$-0.626540\pi$$
−0.387150 + 0.922017i $$0.626540\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 36340.1 1.15960
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ − 35103.0i − 1.11339i
$$999$$ 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 63.4.c.a.62.2 4
3.2 odd 2 inner 63.4.c.a.62.3 yes 4
4.3 odd 2 1008.4.k.a.881.2 4
7.2 even 3 441.4.p.b.80.2 8
7.3 odd 6 441.4.p.b.215.3 8
7.4 even 3 441.4.p.b.215.3 8
7.5 odd 6 441.4.p.b.80.2 8
7.6 odd 2 CM 63.4.c.a.62.2 4
12.11 even 2 1008.4.k.a.881.1 4
21.2 odd 6 441.4.p.b.80.3 8
21.5 even 6 441.4.p.b.80.3 8
21.11 odd 6 441.4.p.b.215.2 8
21.17 even 6 441.4.p.b.215.2 8
21.20 even 2 inner 63.4.c.a.62.3 yes 4
28.27 even 2 1008.4.k.a.881.2 4
84.83 odd 2 1008.4.k.a.881.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.c.a.62.2 4 1.1 even 1 trivial
63.4.c.a.62.2 4 7.6 odd 2 CM
63.4.c.a.62.3 yes 4 3.2 odd 2 inner
63.4.c.a.62.3 yes 4 21.20 even 2 inner
441.4.p.b.80.2 8 7.2 even 3
441.4.p.b.80.2 8 7.5 odd 6
441.4.p.b.80.3 8 21.2 odd 6
441.4.p.b.80.3 8 21.5 even 6
441.4.p.b.215.2 8 21.11 odd 6
441.4.p.b.215.2 8 21.17 even 6
441.4.p.b.215.3 8 7.3 odd 6
441.4.p.b.215.3 8 7.4 even 3
1008.4.k.a.881.1 4 12.11 even 2
1008.4.k.a.881.1 4 84.83 odd 2
1008.4.k.a.881.2 4 4.3 odd 2
1008.4.k.a.881.2 4 28.27 even 2