# Properties

 Label 5625.2.a.z.1.4 Level $5625$ Weight $2$ Character 5625.1 Self dual yes Analytic conductor $44.916$ Analytic rank $1$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45$$ x^8 - 15*x^6 + 70*x^4 - 105*x^2 + 45 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-0.856773$$ of defining polynomial Character $$\chi$$ $$=$$ 5625.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.856773 q^{2} -1.26594 q^{4} +0.647907 q^{7} +2.79817 q^{8} +O(q^{10})$$ $$q-0.856773 q^{2} -1.26594 q^{4} +0.647907 q^{7} +2.79817 q^{8} +5.91382 q^{11} +2.88397 q^{13} -0.555109 q^{14} +0.134488 q^{16} -4.20027 q^{17} -1.64791 q^{19} -5.06680 q^{22} -6.42752 q^{23} -2.47091 q^{26} -0.820212 q^{28} -2.25888 q^{29} -5.28440 q^{31} -5.71156 q^{32} +3.59868 q^{34} -1.71560 q^{37} +1.41188 q^{38} +0.176662 q^{41} +7.95077 q^{43} -7.48654 q^{44} +5.50692 q^{46} -1.05903 q^{47} -6.58022 q^{49} -3.65094 q^{52} +4.48612 q^{53} +1.81295 q^{56} +1.93534 q^{58} -9.74542 q^{59} -11.2844 q^{61} +4.52753 q^{62} +4.62453 q^{64} -12.6171 q^{67} +5.31730 q^{68} +6.09048 q^{71} -7.08312 q^{73} +1.46988 q^{74} +2.08615 q^{76} +3.83160 q^{77} +1.58111 q^{79} -0.151359 q^{82} -11.5102 q^{83} -6.81200 q^{86} +16.5479 q^{88} +15.5918 q^{89} +1.86855 q^{91} +8.13685 q^{92} +0.907347 q^{94} -7.55034 q^{97} +5.63775 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 14 q^{4} - 10 q^{7}+O(q^{10})$$ 8 * q + 14 * q^4 - 10 * q^7 $$8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73} + 56 q^{76} + 28 q^{79} - 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100})$$ 8 * q + 14 * q^4 - 10 * q^7 - 10 * q^13 + 22 * q^16 + 2 * q^19 - 10 * q^22 - 70 * q^28 - 6 * q^31 - 50 * q^34 - 50 * q^37 + 14 * q^49 - 80 * q^52 + 30 * q^58 - 54 * q^61 + 36 * q^64 - 10 * q^67 - 30 * q^73 + 56 * q^76 + 28 * q^79 - 20 * q^88 + 60 * q^91 - 40 * q^94

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.856773 −0.605830 −0.302915 0.953018i $$-0.597960\pi$$
−0.302915 + 0.953018i $$0.597960\pi$$
$$3$$ 0 0
$$4$$ −1.26594 −0.632970
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.647907 0.244886 0.122443 0.992476i $$-0.460927\pi$$
0.122443 + 0.992476i $$0.460927\pi$$
$$8$$ 2.79817 0.989302
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.91382 1.78308 0.891542 0.452939i $$-0.149624\pi$$
0.891542 + 0.452939i $$0.149624\pi$$
$$12$$ 0 0
$$13$$ 2.88397 0.799871 0.399935 0.916543i $$-0.369033\pi$$
0.399935 + 0.916543i $$0.369033\pi$$
$$14$$ −0.555109 −0.148359
$$15$$ 0 0
$$16$$ 0.134488 0.0336219
$$17$$ −4.20027 −1.01872 −0.509358 0.860555i $$-0.670117\pi$$
−0.509358 + 0.860555i $$0.670117\pi$$
$$18$$ 0 0
$$19$$ −1.64791 −0.378056 −0.189028 0.981972i $$-0.560534\pi$$
−0.189028 + 0.981972i $$0.560534\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −5.06680 −1.08024
$$23$$ −6.42752 −1.34023 −0.670115 0.742257i $$-0.733755\pi$$
−0.670115 + 0.742257i $$0.733755\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.47091 −0.484585
$$27$$ 0 0
$$28$$ −0.820212 −0.155005
$$29$$ −2.25888 −0.419463 −0.209732 0.977759i $$-0.567259\pi$$
−0.209732 + 0.977759i $$0.567259\pi$$
$$30$$ 0 0
$$31$$ −5.28440 −0.949107 −0.474553 0.880227i $$-0.657390\pi$$
−0.474553 + 0.880227i $$0.657390\pi$$
$$32$$ −5.71156 −1.00967
$$33$$ 0 0
$$34$$ 3.59868 0.617168
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.71560 −0.282042 −0.141021 0.990007i $$-0.545039\pi$$
−0.141021 + 0.990007i $$0.545039\pi$$
$$38$$ 1.41188 0.229037
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.176662 0.0275900 0.0137950 0.999905i $$-0.495609\pi$$
0.0137950 + 0.999905i $$0.495609\pi$$
$$42$$ 0 0
$$43$$ 7.95077 1.21248 0.606241 0.795281i $$-0.292677\pi$$
0.606241 + 0.795281i $$0.292677\pi$$
$$44$$ −7.48654 −1.12864
$$45$$ 0 0
$$46$$ 5.50692 0.811951
$$47$$ −1.05903 −0.154475 −0.0772376 0.997013i $$-0.524610\pi$$
−0.0772376 + 0.997013i $$0.524610\pi$$
$$48$$ 0 0
$$49$$ −6.58022 −0.940031
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −3.65094 −0.506294
$$53$$ 4.48612 0.616216 0.308108 0.951351i $$-0.400304\pi$$
0.308108 + 0.951351i $$0.400304\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.81295 0.242266
$$57$$ 0 0
$$58$$ 1.93534 0.254123
$$59$$ −9.74542 −1.26875 −0.634373 0.773027i $$-0.718742\pi$$
−0.634373 + 0.773027i $$0.718742\pi$$
$$60$$ 0 0
$$61$$ −11.2844 −1.44482 −0.722410 0.691465i $$-0.756966\pi$$
−0.722410 + 0.691465i $$0.756966\pi$$
$$62$$ 4.52753 0.574997
$$63$$ 0 0
$$64$$ 4.62453 0.578067
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.6171 −1.54143 −0.770715 0.637181i $$-0.780101\pi$$
−0.770715 + 0.637181i $$0.780101\pi$$
$$68$$ 5.31730 0.644817
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.09048 0.722807 0.361404 0.932409i $$-0.382298\pi$$
0.361404 + 0.932409i $$0.382298\pi$$
$$72$$ 0 0
$$73$$ −7.08312 −0.829016 −0.414508 0.910046i $$-0.636046\pi$$
−0.414508 + 0.910046i $$0.636046\pi$$
$$74$$ 1.46988 0.170870
$$75$$ 0 0
$$76$$ 2.08615 0.239298
$$77$$ 3.83160 0.436652
$$78$$ 0 0
$$79$$ 1.58111 0.177889 0.0889443 0.996037i $$-0.471651\pi$$
0.0889443 + 0.996037i $$0.471651\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −0.151359 −0.0167149
$$83$$ −11.5102 −1.26340 −0.631702 0.775211i $$-0.717643\pi$$
−0.631702 + 0.775211i $$0.717643\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −6.81200 −0.734557
$$87$$ 0 0
$$88$$ 16.5479 1.76401
$$89$$ 15.5918 1.65272 0.826362 0.563140i $$-0.190407\pi$$
0.826362 + 0.563140i $$0.190407\pi$$
$$90$$ 0 0
$$91$$ 1.86855 0.195877
$$92$$ 8.13685 0.848326
$$93$$ 0 0
$$94$$ 0.907347 0.0935857
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.55034 −0.766621 −0.383311 0.923619i $$-0.625216\pi$$
−0.383311 + 0.923619i $$0.625216\pi$$
$$98$$ 5.63775 0.569499
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 17.1645 1.70793 0.853965 0.520330i $$-0.174191\pi$$
0.853965 + 0.520330i $$0.174191\pi$$
$$102$$ 0 0
$$103$$ −17.3561 −1.71015 −0.855074 0.518506i $$-0.826489\pi$$
−0.855074 + 0.518506i $$0.826489\pi$$
$$104$$ 8.06985 0.791314
$$105$$ 0 0
$$106$$ −3.84358 −0.373322
$$107$$ 16.2046 1.56656 0.783278 0.621672i $$-0.213546\pi$$
0.783278 + 0.621672i $$0.213546\pi$$
$$108$$ 0 0
$$109$$ 7.12709 0.682652 0.341326 0.939945i $$-0.389124\pi$$
0.341326 + 0.939945i $$0.389124\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0.0871355 0.00823353
$$113$$ −14.9372 −1.40518 −0.702589 0.711596i $$-0.747973\pi$$
−0.702589 + 0.711596i $$0.747973\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.85961 0.265508
$$117$$ 0 0
$$118$$ 8.34961 0.768644
$$119$$ −2.72139 −0.249469
$$120$$ 0 0
$$121$$ 23.9733 2.17939
$$122$$ 9.66817 0.875315
$$123$$ 0 0
$$124$$ 6.68974 0.600757
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −9.30722 −0.825883 −0.412941 0.910758i $$-0.635499\pi$$
−0.412941 + 0.910758i $$0.635499\pi$$
$$128$$ 7.46095 0.659461
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.47828 0.303899 0.151949 0.988388i $$-0.451445\pi$$
0.151949 + 0.988388i $$0.451445\pi$$
$$132$$ 0 0
$$133$$ −1.06769 −0.0925805
$$134$$ 10.8100 0.933844
$$135$$ 0 0
$$136$$ −11.7531 −1.00782
$$137$$ −8.17270 −0.698241 −0.349120 0.937078i $$-0.613520\pi$$
−0.349120 + 0.937078i $$0.613520\pi$$
$$138$$ 0 0
$$139$$ 2.93970 0.249342 0.124671 0.992198i $$-0.460212\pi$$
0.124671 + 0.992198i $$0.460212\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −5.21816 −0.437898
$$143$$ 17.0553 1.42624
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 6.06862 0.502243
$$147$$ 0 0
$$148$$ 2.17184 0.178524
$$149$$ 15.4826 1.26838 0.634191 0.773176i $$-0.281333\pi$$
0.634191 + 0.773176i $$0.281333\pi$$
$$150$$ 0 0
$$151$$ 10.8233 0.880791 0.440395 0.897804i $$-0.354838\pi$$
0.440395 + 0.897804i $$0.354838\pi$$
$$152$$ −4.61112 −0.374011
$$153$$ 0 0
$$154$$ −3.28281 −0.264537
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −16.9086 −1.34945 −0.674726 0.738068i $$-0.735738\pi$$
−0.674726 + 0.738068i $$0.735738\pi$$
$$158$$ −1.35465 −0.107770
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.16443 −0.328203
$$162$$ 0 0
$$163$$ 16.7750 1.31392 0.656960 0.753926i $$-0.271842\pi$$
0.656960 + 0.753926i $$0.271842\pi$$
$$164$$ −0.223644 −0.0174637
$$165$$ 0 0
$$166$$ 9.86159 0.765407
$$167$$ 10.1916 0.788653 0.394326 0.918970i $$-0.370978\pi$$
0.394326 + 0.918970i $$0.370978\pi$$
$$168$$ 0 0
$$169$$ −4.68269 −0.360207
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −10.0652 −0.767465
$$173$$ −14.3724 −1.09271 −0.546355 0.837554i $$-0.683985\pi$$
−0.546355 + 0.837554i $$0.683985\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.795336 0.0599507
$$177$$ 0 0
$$178$$ −13.3586 −1.00127
$$179$$ −7.59573 −0.567731 −0.283866 0.958864i $$-0.591617\pi$$
−0.283866 + 0.958864i $$0.591617\pi$$
$$180$$ 0 0
$$181$$ −21.8203 −1.62189 −0.810945 0.585122i $$-0.801047\pi$$
−0.810945 + 0.585122i $$0.801047\pi$$
$$182$$ −1.60092 −0.118668
$$183$$ 0 0
$$184$$ −17.9853 −1.32589
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −24.8397 −1.81646
$$188$$ 1.34067 0.0977783
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.0283 1.37684 0.688421 0.725311i $$-0.258304\pi$$
0.688421 + 0.725311i $$0.258304\pi$$
$$192$$ 0 0
$$193$$ 8.57675 0.617368 0.308684 0.951165i $$-0.400111\pi$$
0.308684 + 0.951165i $$0.400111\pi$$
$$194$$ 6.46893 0.464442
$$195$$ 0 0
$$196$$ 8.33016 0.595012
$$197$$ −18.5726 −1.32325 −0.661623 0.749837i $$-0.730132\pi$$
−0.661623 + 0.749837i $$0.730132\pi$$
$$198$$ 0 0
$$199$$ −7.32927 −0.519558 −0.259779 0.965668i $$-0.583650\pi$$
−0.259779 + 0.965668i $$0.583650\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −14.7061 −1.03471
$$203$$ −1.46354 −0.102721
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 14.8702 1.03606
$$207$$ 0 0
$$208$$ 0.387859 0.0268932
$$209$$ −9.74542 −0.674105
$$210$$ 0 0
$$211$$ −14.5553 −1.00203 −0.501013 0.865440i $$-0.667039\pi$$
−0.501013 + 0.865440i $$0.667039\pi$$
$$212$$ −5.67916 −0.390046
$$213$$ 0 0
$$214$$ −13.8836 −0.949066
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.42380 −0.232423
$$218$$ −6.10630 −0.413571
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.1135 −0.814841
$$222$$ 0 0
$$223$$ −26.8711 −1.79942 −0.899712 0.436485i $$-0.856223\pi$$
−0.899712 + 0.436485i $$0.856223\pi$$
$$224$$ −3.70056 −0.247254
$$225$$ 0 0
$$226$$ 12.7978 0.851298
$$227$$ 4.20027 0.278782 0.139391 0.990237i $$-0.455486\pi$$
0.139391 + 0.990237i $$0.455486\pi$$
$$228$$ 0 0
$$229$$ 8.30722 0.548957 0.274478 0.961593i $$-0.411495\pi$$
0.274478 + 0.961593i $$0.411495\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.32072 −0.414976
$$233$$ 22.6158 1.48161 0.740805 0.671720i $$-0.234444\pi$$
0.740805 + 0.671720i $$0.234444\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.3371 0.803079
$$237$$ 0 0
$$238$$ 2.33161 0.151136
$$239$$ −1.28688 −0.0832413 −0.0416207 0.999133i $$-0.513252\pi$$
−0.0416207 + 0.999133i $$0.513252\pi$$
$$240$$ 0 0
$$241$$ −4.91599 −0.316667 −0.158333 0.987386i $$-0.550612\pi$$
−0.158333 + 0.987386i $$0.550612\pi$$
$$242$$ −20.5396 −1.32034
$$243$$ 0 0
$$244$$ 14.2854 0.914528
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.75252 −0.302396
$$248$$ −14.7867 −0.938953
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −18.9609 −1.19680 −0.598399 0.801198i $$-0.704196\pi$$
−0.598399 + 0.801198i $$0.704196\pi$$
$$252$$ 0 0
$$253$$ −38.0112 −2.38974
$$254$$ 7.97417 0.500344
$$255$$ 0 0
$$256$$ −15.6414 −0.977588
$$257$$ −24.1690 −1.50762 −0.753810 0.657093i $$-0.771786\pi$$
−0.753810 + 0.657093i $$0.771786\pi$$
$$258$$ 0 0
$$259$$ −1.11155 −0.0690682
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −2.98009 −0.184111
$$263$$ 17.6518 1.08846 0.544229 0.838937i $$-0.316822\pi$$
0.544229 + 0.838937i $$0.316822\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0.914768 0.0560880
$$267$$ 0 0
$$268$$ 15.9726 0.975679
$$269$$ −26.5149 −1.61664 −0.808320 0.588743i $$-0.799623\pi$$
−0.808320 + 0.588743i $$0.799623\pi$$
$$270$$ 0 0
$$271$$ 20.4180 1.24031 0.620153 0.784481i $$-0.287071\pi$$
0.620153 + 0.784481i $$0.287071\pi$$
$$272$$ −0.564885 −0.0342512
$$273$$ 0 0
$$274$$ 7.00214 0.423015
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 14.0643 0.845043 0.422521 0.906353i $$-0.361145\pi$$
0.422521 + 0.906353i $$0.361145\pi$$
$$278$$ −2.51866 −0.151059
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.19966 −0.369841 −0.184920 0.982753i $$-0.559203\pi$$
−0.184920 + 0.982753i $$0.559203\pi$$
$$282$$ 0 0
$$283$$ −12.9801 −0.771586 −0.385793 0.922585i $$-0.626072\pi$$
−0.385793 + 0.922585i $$0.626072\pi$$
$$284$$ −7.71019 −0.457516
$$285$$ 0 0
$$286$$ −14.6125 −0.864056
$$287$$ 0.114461 0.00675640
$$288$$ 0 0
$$289$$ 0.642299 0.0377823
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 8.96681 0.524743
$$293$$ −18.5563 −1.08407 −0.542037 0.840355i $$-0.682347\pi$$
−0.542037 + 0.840355i $$0.682347\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −4.80053 −0.279025
$$297$$ 0 0
$$298$$ −13.2650 −0.768424
$$299$$ −18.5368 −1.07201
$$300$$ 0 0
$$301$$ 5.15136 0.296919
$$302$$ −9.27314 −0.533609
$$303$$ 0 0
$$304$$ −0.221623 −0.0127110
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −24.0862 −1.37467 −0.687337 0.726338i $$-0.741221\pi$$
−0.687337 + 0.726338i $$0.741221\pi$$
$$308$$ −4.85058 −0.276388
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 25.3372 1.43674 0.718370 0.695661i $$-0.244889\pi$$
0.718370 + 0.695661i $$0.244889\pi$$
$$312$$ 0 0
$$313$$ 21.8224 1.23348 0.616739 0.787168i $$-0.288454\pi$$
0.616739 + 0.787168i $$0.288454\pi$$
$$314$$ 14.4868 0.817539
$$315$$ 0 0
$$316$$ −2.00159 −0.112598
$$317$$ −1.21940 −0.0684884 −0.0342442 0.999413i $$-0.510902\pi$$
−0.0342442 + 0.999413i $$0.510902\pi$$
$$318$$ 0 0
$$319$$ −13.3586 −0.747938
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3.56797 0.198835
$$323$$ 6.92166 0.385131
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −14.3724 −0.796011
$$327$$ 0 0
$$328$$ 0.494331 0.0272949
$$329$$ −0.686152 −0.0378288
$$330$$ 0 0
$$331$$ −5.97470 −0.328399 −0.164200 0.986427i $$-0.552504\pi$$
−0.164200 + 0.986427i $$0.552504\pi$$
$$332$$ 14.5712 0.799697
$$333$$ 0 0
$$334$$ −8.73192 −0.477789
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −27.1718 −1.48014 −0.740072 0.672527i $$-0.765209\pi$$
−0.740072 + 0.672527i $$0.765209\pi$$
$$338$$ 4.01200 0.218224
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −31.2510 −1.69234
$$342$$ 0 0
$$343$$ −8.79871 −0.475086
$$344$$ 22.2476 1.19951
$$345$$ 0 0
$$346$$ 12.3138 0.661996
$$347$$ 12.4242 0.666964 0.333482 0.942757i $$-0.391776\pi$$
0.333482 + 0.942757i $$0.391776\pi$$
$$348$$ 0 0
$$349$$ 20.2826 1.08570 0.542852 0.839828i $$-0.317345\pi$$
0.542852 + 0.839828i $$0.317345\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −33.7771 −1.80033
$$353$$ 11.9880 0.638057 0.319029 0.947745i $$-0.396643\pi$$
0.319029 + 0.947745i $$0.396643\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −19.7382 −1.04613
$$357$$ 0 0
$$358$$ 6.50781 0.343949
$$359$$ 3.54576 0.187138 0.0935690 0.995613i $$-0.470172\pi$$
0.0935690 + 0.995613i $$0.470172\pi$$
$$360$$ 0 0
$$361$$ −16.2844 −0.857074
$$362$$ 18.6950 0.982589
$$363$$ 0 0
$$364$$ −2.36547 −0.123984
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −31.3333 −1.63558 −0.817792 0.575514i $$-0.804802\pi$$
−0.817792 + 0.575514i $$0.804802\pi$$
$$368$$ −0.864421 −0.0450611
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2.90659 0.150902
$$372$$ 0 0
$$373$$ 0.133595 0.00691730 0.00345865 0.999994i $$-0.498899\pi$$
0.00345865 + 0.999994i $$0.498899\pi$$
$$374$$ 21.2819 1.10046
$$375$$ 0 0
$$376$$ −2.96334 −0.152823
$$377$$ −6.51455 −0.335516
$$378$$ 0 0
$$379$$ −8.04343 −0.413163 −0.206582 0.978429i $$-0.566234\pi$$
−0.206582 + 0.978429i $$0.566234\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −16.3030 −0.834132
$$383$$ −12.2322 −0.625034 −0.312517 0.949912i $$-0.601172\pi$$
−0.312517 + 0.949912i $$0.601172\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −7.34832 −0.374020
$$387$$ 0 0
$$388$$ 9.55829 0.485249
$$389$$ −33.0004 −1.67319 −0.836593 0.547825i $$-0.815456\pi$$
−0.836593 + 0.547825i $$0.815456\pi$$
$$390$$ 0 0
$$391$$ 26.9973 1.36531
$$392$$ −18.4126 −0.929974
$$393$$ 0 0
$$394$$ 15.9125 0.801661
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 13.9249 0.698872 0.349436 0.936960i $$-0.386373\pi$$
0.349436 + 0.936960i $$0.386373\pi$$
$$398$$ 6.27952 0.314764
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 27.9494 1.39573 0.697863 0.716231i $$-0.254134\pi$$
0.697863 + 0.716231i $$0.254134\pi$$
$$402$$ 0 0
$$403$$ −15.2401 −0.759163
$$404$$ −21.7292 −1.08107
$$405$$ 0 0
$$406$$ 1.25392 0.0622311
$$407$$ −10.1457 −0.502905
$$408$$ 0 0
$$409$$ 25.9289 1.28210 0.641052 0.767498i $$-0.278498\pi$$
0.641052 + 0.767498i $$0.278498\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 21.9718 1.08247
$$413$$ −6.31413 −0.310698
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −16.4720 −0.807606
$$417$$ 0 0
$$418$$ 8.34961 0.408393
$$419$$ 1.21614 0.0594123 0.0297061 0.999559i $$-0.490543\pi$$
0.0297061 + 0.999559i $$0.490543\pi$$
$$420$$ 0 0
$$421$$ 20.4626 0.997286 0.498643 0.866807i $$-0.333832\pi$$
0.498643 + 0.866807i $$0.333832\pi$$
$$422$$ 12.4705 0.607056
$$423$$ 0 0
$$424$$ 12.5529 0.609624
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −7.31124 −0.353816
$$428$$ −20.5140 −0.991583
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −21.4381 −1.03264 −0.516319 0.856397i $$-0.672698\pi$$
−0.516319 + 0.856397i $$0.672698\pi$$
$$432$$ 0 0
$$433$$ −23.2844 −1.11898 −0.559489 0.828838i $$-0.689002\pi$$
−0.559489 + 0.828838i $$0.689002\pi$$
$$434$$ 2.93342 0.140809
$$435$$ 0 0
$$436$$ −9.02248 −0.432098
$$437$$ 10.5919 0.506681
$$438$$ 0 0
$$439$$ −12.5355 −0.598285 −0.299143 0.954208i $$-0.596701\pi$$
−0.299143 + 0.954208i $$0.596701\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 10.3785 0.493655
$$443$$ 13.3228 0.632986 0.316493 0.948595i $$-0.397495\pi$$
0.316493 + 0.948595i $$0.397495\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 23.0224 1.09014
$$447$$ 0 0
$$448$$ 2.99627 0.141560
$$449$$ −30.2500 −1.42758 −0.713792 0.700358i $$-0.753024\pi$$
−0.713792 + 0.700358i $$0.753024\pi$$
$$450$$ 0 0
$$451$$ 1.04475 0.0491953
$$452$$ 18.9097 0.889436
$$453$$ 0 0
$$454$$ −3.59868 −0.168894
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12.8391 0.600588 0.300294 0.953847i $$-0.402915\pi$$
0.300294 + 0.953847i $$0.402915\pi$$
$$458$$ −7.11740 −0.332574
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −27.8435 −1.29680 −0.648400 0.761300i $$-0.724561\pi$$
−0.648400 + 0.761300i $$0.724561\pi$$
$$462$$ 0 0
$$463$$ −3.84616 −0.178746 −0.0893731 0.995998i $$-0.528486\pi$$
−0.0893731 + 0.995998i $$0.528486\pi$$
$$464$$ −0.303791 −0.0141031
$$465$$ 0 0
$$466$$ −19.3766 −0.897603
$$467$$ 40.6594 1.88149 0.940746 0.339112i $$-0.110126\pi$$
0.940746 + 0.339112i $$0.110126\pi$$
$$468$$ 0 0
$$469$$ −8.17473 −0.377474
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −27.2693 −1.25517
$$473$$ 47.0194 2.16196
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 3.44511 0.157907
$$477$$ 0 0
$$478$$ 1.10256 0.0504301
$$479$$ 15.2642 0.697440 0.348720 0.937227i $$-0.386616\pi$$
0.348720 + 0.937227i $$0.386616\pi$$
$$480$$ 0 0
$$481$$ −4.94774 −0.225597
$$482$$ 4.21189 0.191846
$$483$$ 0 0
$$484$$ −30.3487 −1.37949
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −8.06466 −0.365444 −0.182722 0.983165i $$-0.558491\pi$$
−0.182722 + 0.983165i $$0.558491\pi$$
$$488$$ −31.5757 −1.42936
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 9.50128 0.428787 0.214393 0.976747i $$-0.431223\pi$$
0.214393 + 0.976747i $$0.431223\pi$$
$$492$$ 0 0
$$493$$ 9.48790 0.427314
$$494$$ 4.07183 0.183200
$$495$$ 0 0
$$496$$ −0.710687 −0.0319108
$$497$$ 3.94606 0.177005
$$498$$ 0 0
$$499$$ −19.6405 −0.879230 −0.439615 0.898186i $$-0.644885\pi$$
−0.439615 + 0.898186i $$0.644885\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 16.2451 0.725056
$$503$$ −20.6428 −0.920415 −0.460208 0.887811i $$-0.652225\pi$$
−0.460208 + 0.887811i $$0.652225\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.5669 1.44778
$$507$$ 0 0
$$508$$ 11.7824 0.522759
$$509$$ 31.9372 1.41559 0.707795 0.706418i $$-0.249690\pi$$
0.707795 + 0.706418i $$0.249690\pi$$
$$510$$ 0 0
$$511$$ −4.58920 −0.203014
$$512$$ −1.52077 −0.0672093
$$513$$ 0 0
$$514$$ 20.7073 0.913360
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −6.26291 −0.275442
$$518$$ 0.952343 0.0418435
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 8.88261 0.389154 0.194577 0.980887i $$-0.437667\pi$$
0.194577 + 0.980887i $$0.437667\pi$$
$$522$$ 0 0
$$523$$ −28.3512 −1.23971 −0.619856 0.784716i $$-0.712809\pi$$
−0.619856 + 0.784716i $$0.712809\pi$$
$$524$$ −4.40329 −0.192359
$$525$$ 0 0
$$526$$ −15.1236 −0.659420
$$527$$ 22.1959 0.966870
$$528$$ 0 0
$$529$$ 18.3130 0.796215
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 1.35163 0.0586007
$$533$$ 0.509490 0.0220685
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −35.3049 −1.52494
$$537$$ 0 0
$$538$$ 22.7172 0.979409
$$539$$ −38.9142 −1.67615
$$540$$ 0 0
$$541$$ −13.8190 −0.594124 −0.297062 0.954858i $$-0.596007\pi$$
−0.297062 + 0.954858i $$0.596007\pi$$
$$542$$ −17.4936 −0.751414
$$543$$ 0 0
$$544$$ 23.9901 1.02857
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.0831 0.516637 0.258318 0.966060i $$-0.416832\pi$$
0.258318 + 0.966060i $$0.416832\pi$$
$$548$$ 10.3461 0.441966
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.72242 0.158580
$$552$$ 0 0
$$553$$ 1.02441 0.0435624
$$554$$ −12.0499 −0.511952
$$555$$ 0 0
$$556$$ −3.72149 −0.157826
$$557$$ 15.5918 0.660644 0.330322 0.943868i $$-0.392843\pi$$
0.330322 + 0.943868i $$0.392843\pi$$
$$558$$ 0 0
$$559$$ 22.9298 0.969828
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5.31170 0.224061
$$563$$ 19.1854 0.808570 0.404285 0.914633i $$-0.367520\pi$$
0.404285 + 0.914633i $$0.367520\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 11.1210 0.467450
$$567$$ 0 0
$$568$$ 17.0422 0.715074
$$569$$ 30.1698 1.26478 0.632392 0.774648i $$-0.282073\pi$$
0.632392 + 0.774648i $$0.282073\pi$$
$$570$$ 0 0
$$571$$ −25.2616 −1.05716 −0.528582 0.848882i $$-0.677276\pi$$
−0.528582 + 0.848882i $$0.677276\pi$$
$$572$$ −21.5910 −0.902765
$$573$$ 0 0
$$574$$ −0.0980668 −0.00409323
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 9.00948 0.375070 0.187535 0.982258i $$-0.439950\pi$$
0.187535 + 0.982258i $$0.439950\pi$$
$$578$$ −0.550304 −0.0228896
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −7.45751 −0.309390
$$582$$ 0 0
$$583$$ 26.5301 1.09876
$$584$$ −19.8198 −0.820147
$$585$$ 0 0
$$586$$ 15.8986 0.656764
$$587$$ −11.5059 −0.474901 −0.237451 0.971400i $$-0.576312\pi$$
−0.237451 + 0.971400i $$0.576312\pi$$
$$588$$ 0 0
$$589$$ 8.70820 0.358815
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −0.230727 −0.00948280
$$593$$ −23.0392 −0.946107 −0.473053 0.881034i $$-0.656848\pi$$
−0.473053 + 0.881034i $$0.656848\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −19.6000 −0.802848
$$597$$ 0 0
$$598$$ 15.8818 0.649456
$$599$$ −2.07694 −0.0848614 −0.0424307 0.999099i $$-0.513510\pi$$
−0.0424307 + 0.999099i $$0.513510\pi$$
$$600$$ 0 0
$$601$$ 28.7922 1.17446 0.587230 0.809420i $$-0.300218\pi$$
0.587230 + 0.809420i $$0.300218\pi$$
$$602$$ −4.41354 −0.179883
$$603$$ 0 0
$$604$$ −13.7017 −0.557514
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 6.99573 0.283948 0.141974 0.989870i $$-0.454655\pi$$
0.141974 + 0.989870i $$0.454655\pi$$
$$608$$ 9.41212 0.381712
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.05421 −0.123560
$$612$$ 0 0
$$613$$ −34.6552 −1.39971 −0.699855 0.714285i $$-0.746752\pi$$
−0.699855 + 0.714285i $$0.746752\pi$$
$$614$$ 20.6364 0.832819
$$615$$ 0 0
$$616$$ 10.7215 0.431980
$$617$$ 14.9852 0.603280 0.301640 0.953422i $$-0.402466\pi$$
0.301640 + 0.953422i $$0.402466\pi$$
$$618$$ 0 0
$$619$$ −39.8216 −1.60056 −0.800282 0.599624i $$-0.795317\pi$$
−0.800282 + 0.599624i $$0.795317\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −21.7082 −0.870420
$$623$$ 10.1020 0.404728
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −18.6969 −0.747277
$$627$$ 0 0
$$628$$ 21.4053 0.854164
$$629$$ 7.20598 0.287321
$$630$$ 0 0
$$631$$ 1.66278 0.0661944 0.0330972 0.999452i $$-0.489463\pi$$
0.0330972 + 0.999452i $$0.489463\pi$$
$$632$$ 4.42421 0.175986
$$633$$ 0 0
$$634$$ 1.04475 0.0414923
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −18.9772 −0.751903
$$638$$ 11.4453 0.453123
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6.77663 −0.267661 −0.133830 0.991004i $$-0.542728\pi$$
−0.133830 + 0.991004i $$0.542728\pi$$
$$642$$ 0 0
$$643$$ −20.2108 −0.797035 −0.398517 0.917161i $$-0.630475\pi$$
−0.398517 + 0.917161i $$0.630475\pi$$
$$644$$ 5.27192 0.207743
$$645$$ 0 0
$$646$$ −5.93029 −0.233324
$$647$$ −5.05100 −0.198575 −0.0992877 0.995059i $$-0.531656\pi$$
−0.0992877 + 0.995059i $$0.531656\pi$$
$$648$$ 0 0
$$649$$ −57.6327 −2.26228
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −21.2362 −0.831672
$$653$$ 9.64210 0.377325 0.188662 0.982042i $$-0.439585\pi$$
0.188662 + 0.982042i $$0.439585\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0.0237589 0.000927629 0
$$657$$ 0 0
$$658$$ 0.587876 0.0229178
$$659$$ 26.7332 1.04138 0.520690 0.853746i $$-0.325675\pi$$
0.520690 + 0.853746i $$0.325675\pi$$
$$660$$ 0 0
$$661$$ −2.00159 −0.0778529 −0.0389264 0.999242i $$-0.512394\pi$$
−0.0389264 + 0.999242i $$0.512394\pi$$
$$662$$ 5.11896 0.198954
$$663$$ 0 0
$$664$$ −32.2074 −1.24989
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 14.5190 0.562177
$$668$$ −12.9020 −0.499194
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −66.7339 −2.57623
$$672$$ 0 0
$$673$$ 31.2362 1.20407 0.602033 0.798471i $$-0.294358\pi$$
0.602033 + 0.798471i $$0.294358\pi$$
$$674$$ 23.2801 0.896716
$$675$$ 0 0
$$676$$ 5.92801 0.228000
$$677$$ −42.4874 −1.63292 −0.816462 0.577400i $$-0.804067\pi$$
−0.816462 + 0.577400i $$0.804067\pi$$
$$678$$ 0 0
$$679$$ −4.89192 −0.187735
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 26.7750 1.02527
$$683$$ 5.20811 0.199283 0.0996415 0.995023i $$-0.468230\pi$$
0.0996415 + 0.995023i $$0.468230\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 7.53850 0.287821
$$687$$ 0 0
$$688$$ 1.06928 0.0407659
$$689$$ 12.9379 0.492893
$$690$$ 0 0
$$691$$ 8.07419 0.307157 0.153578 0.988136i $$-0.450920\pi$$
0.153578 + 0.988136i $$0.450920\pi$$
$$692$$ 18.1946 0.691653
$$693$$ 0 0
$$694$$ −10.6447 −0.404066
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −0.742030 −0.0281064
$$698$$ −17.3776 −0.657752
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −31.7552 −1.19938 −0.599689 0.800233i $$-0.704709\pi$$
−0.599689 + 0.800233i $$0.704709\pi$$
$$702$$ 0 0
$$703$$ 2.82714 0.106628
$$704$$ 27.3487 1.03074
$$705$$ 0 0
$$706$$ −10.2710 −0.386554
$$707$$ 11.1210 0.418248
$$708$$ 0 0
$$709$$ 9.71630 0.364903 0.182452 0.983215i $$-0.441597\pi$$
0.182452 + 0.983215i $$0.441597\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 43.6284 1.63504
$$713$$ 33.9656 1.27202
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 9.61574 0.359357
$$717$$ 0 0
$$718$$ −3.03791 −0.113374
$$719$$ 20.4661 0.763257 0.381628 0.924316i $$-0.375363\pi$$
0.381628 + 0.924316i $$0.375363\pi$$
$$720$$ 0 0
$$721$$ −11.2451 −0.418791
$$722$$ 13.9520 0.519241
$$723$$ 0 0
$$724$$ 27.6232 1.02661
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 50.2337 1.86306 0.931532 0.363659i $$-0.118473\pi$$
0.931532 + 0.363659i $$0.118473\pi$$
$$728$$ 5.22851 0.193781
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −33.3954 −1.23517
$$732$$ 0 0
$$733$$ −37.8776 −1.39904 −0.699520 0.714613i $$-0.746603\pi$$
−0.699520 + 0.714613i $$0.746603\pi$$
$$734$$ 26.8455 0.990886
$$735$$ 0 0
$$736$$ 36.7112 1.35319
$$737$$ −74.6155 −2.74850
$$738$$ 0 0
$$739$$ 44.9709 1.65428 0.827141 0.561995i $$-0.189966\pi$$
0.827141 + 0.561995i $$0.189966\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −2.49028 −0.0914212
$$743$$ −29.4546 −1.08059 −0.540293 0.841477i $$-0.681687\pi$$
−0.540293 + 0.841477i $$0.681687\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −0.114461 −0.00419071
$$747$$ 0 0
$$748$$ 31.4455 1.14976
$$749$$ 10.4991 0.383627
$$750$$ 0 0
$$751$$ −34.1341 −1.24557 −0.622786 0.782392i $$-0.713999\pi$$
−0.622786 + 0.782392i $$0.713999\pi$$
$$752$$ −0.142426 −0.00519375
$$753$$ 0 0
$$754$$ 5.58148 0.203266
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 15.2875 0.555635 0.277817 0.960634i $$-0.410389\pi$$
0.277817 + 0.960634i $$0.410389\pi$$
$$758$$ 6.89139 0.250306
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 24.0119 0.870429 0.435215 0.900327i $$-0.356672\pi$$
0.435215 + 0.900327i $$0.356672\pi$$
$$762$$ 0 0
$$763$$ 4.61769 0.167172
$$764$$ −24.0887 −0.871500
$$765$$ 0 0
$$766$$ 10.4802 0.378664
$$767$$ −28.1056 −1.01483
$$768$$ 0 0
$$769$$ −0.656954 −0.0236904 −0.0118452 0.999930i $$-0.503771\pi$$
−0.0118452 + 0.999930i $$0.503771\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −10.8577 −0.390776
$$773$$ −28.1609 −1.01288 −0.506439 0.862276i $$-0.669039\pi$$
−0.506439 + 0.862276i $$0.669039\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −21.1271 −0.758420
$$777$$ 0 0
$$778$$ 28.2738 1.01367
$$779$$ −0.291123 −0.0104306
$$780$$ 0 0
$$781$$ 36.0180 1.28883
$$782$$ −23.1306 −0.827147
$$783$$ 0 0
$$784$$ −0.884958 −0.0316056
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −7.66560 −0.273249 −0.136625 0.990623i $$-0.543625\pi$$
−0.136625 + 0.990623i $$0.543625\pi$$
$$788$$ 23.5119 0.837575
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −9.67794 −0.344108
$$792$$ 0 0
$$793$$ −32.5439 −1.15567
$$794$$ −11.9305 −0.423397
$$795$$ 0 0
$$796$$ 9.27842 0.328865
$$797$$ −34.7609 −1.23129 −0.615647 0.788022i $$-0.711105\pi$$
−0.615647 + 0.788022i $$0.711105\pi$$
$$798$$ 0 0
$$799$$ 4.44821 0.157366
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −23.9463 −0.845572
$$803$$ −41.8883 −1.47821
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 13.0573 0.459923
$$807$$ 0 0
$$808$$ 48.0291 1.68966
$$809$$ −42.5017 −1.49428 −0.747140 0.664667i $$-0.768573\pi$$
−0.747140 + 0.664667i $$0.768573\pi$$
$$810$$ 0 0
$$811$$ 39.2810 1.37934 0.689672 0.724122i $$-0.257755\pi$$
0.689672 + 0.724122i $$0.257755\pi$$
$$812$$ 1.85276 0.0650191
$$813$$ 0 0
$$814$$ 8.69258 0.304675
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −13.1021 −0.458386
$$818$$ −22.2152 −0.776736
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −0.709911 −0.0247761 −0.0123880 0.999923i $$-0.503943\pi$$
−0.0123880 + 0.999923i $$0.503943\pi$$
$$822$$ 0 0
$$823$$ −10.6260 −0.370398 −0.185199 0.982701i $$-0.559293\pi$$
−0.185199 + 0.982701i $$0.559293\pi$$
$$824$$ −48.5653 −1.69185
$$825$$ 0 0
$$826$$ 5.40977 0.188230
$$827$$ −11.0597 −0.384585 −0.192292 0.981338i $$-0.561592\pi$$
−0.192292 + 0.981338i $$0.561592\pi$$
$$828$$ 0 0
$$829$$ −1.64791 −0.0572342 −0.0286171 0.999590i $$-0.509110\pi$$
−0.0286171 + 0.999590i $$0.509110\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 13.3370 0.462379
$$833$$ 27.6387 0.957625
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 12.3371 0.426688
$$837$$ 0 0
$$838$$ −1.04195 −0.0359937
$$839$$ 23.8789 0.824392 0.412196 0.911095i $$-0.364762\pi$$
0.412196 + 0.911095i $$0.364762\pi$$
$$840$$ 0 0
$$841$$ −23.8975 −0.824051
$$842$$ −17.5318 −0.604186
$$843$$ 0 0
$$844$$ 18.4261 0.634252
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 15.5324 0.533701
$$848$$ 0.603328 0.0207184
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 11.0270 0.378002
$$852$$ 0 0
$$853$$ 23.3570 0.799729 0.399864 0.916574i $$-0.369057\pi$$
0.399864 + 0.916574i $$0.369057\pi$$
$$854$$ 6.26407 0.214352
$$855$$ 0 0
$$856$$ 45.3431 1.54980
$$857$$ 2.70183 0.0922929 0.0461464 0.998935i $$-0.485306\pi$$
0.0461464 + 0.998935i $$0.485306\pi$$
$$858$$ 0 0
$$859$$ 23.5781 0.804474 0.402237 0.915536i $$-0.368233\pi$$
0.402237 + 0.915536i $$0.368233\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 18.3676 0.625602
$$863$$ −26.2965 −0.895144 −0.447572 0.894248i $$-0.647711\pi$$
−0.447572 + 0.894248i $$0.647711\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 19.9494 0.677909
$$867$$ 0 0
$$868$$ 4.33433 0.147117
$$869$$ 9.35039 0.317190
$$870$$ 0 0
$$871$$ −36.3875 −1.23294
$$872$$ 19.9428 0.675349
$$873$$ 0 0
$$874$$ −9.07489 −0.306963
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 33.4164 1.12839 0.564196 0.825641i $$-0.309186\pi$$
0.564196 + 0.825641i $$0.309186\pi$$
$$878$$ 10.7400 0.362459
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 21.0398 0.708849 0.354425 0.935085i $$-0.384677\pi$$
0.354425 + 0.935085i $$0.384677\pi$$
$$882$$ 0 0
$$883$$ 14.5435 0.489428 0.244714 0.969595i $$-0.421306\pi$$
0.244714 + 0.969595i $$0.421306\pi$$
$$884$$ 15.3350 0.515770
$$885$$ 0 0
$$886$$ −11.4146 −0.383482
$$887$$ 55.3525 1.85855 0.929277 0.369382i $$-0.120431\pi$$
0.929277 + 0.369382i $$0.120431\pi$$
$$888$$ 0 0
$$889$$ −6.03021 −0.202247
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 34.0172 1.13898
$$893$$ 1.74518 0.0584003
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.83400 0.161493
$$897$$ 0 0
$$898$$ 25.9173 0.864873
$$899$$ 11.9368 0.398115
$$900$$ 0 0
$$901$$ −18.8429 −0.627749
$$902$$ −0.895112 −0.0298040
$$903$$ 0 0
$$904$$ −41.7969 −1.39015
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 25.9003 0.860006 0.430003 0.902828i $$-0.358513\pi$$
0.430003 + 0.902828i $$0.358513\pi$$
$$908$$ −5.31730 −0.176461
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10.6499 −0.352848 −0.176424 0.984314i $$-0.556453\pi$$
−0.176424 + 0.984314i $$0.556453\pi$$
$$912$$ 0 0
$$913$$ −68.0690 −2.25275
$$914$$ −11.0002 −0.363854
$$915$$ 0 0
$$916$$ −10.5165 −0.347473
$$917$$ 2.25360 0.0744204
$$918$$ 0 0
$$919$$ 11.1958 0.369314 0.184657 0.982803i $$-0.440883\pi$$
0.184657 + 0.982803i $$0.440883\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 23.8555 0.785640
$$923$$ 17.5648 0.578152
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 3.29528 0.108290
$$927$$ 0 0
$$928$$ 12.9017 0.423520
$$929$$ −58.0987 −1.90616 −0.953078 0.302723i $$-0.902104\pi$$
−0.953078 + 0.302723i $$0.902104\pi$$
$$930$$ 0 0
$$931$$ 10.8436 0.355384
$$932$$ −28.6303 −0.937815
$$933$$ 0 0
$$934$$ −34.8359 −1.13986
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 40.7577 1.33150 0.665749 0.746176i $$-0.268112\pi$$
0.665749 + 0.746176i $$0.268112\pi$$
$$938$$ 7.00389 0.228685
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 2.47724 0.0807559 0.0403779 0.999184i $$-0.487144\pi$$
0.0403779 + 0.999184i $$0.487144\pi$$
$$942$$ 0 0
$$943$$ −1.13550 −0.0369770
$$944$$ −1.31064 −0.0426577
$$945$$ 0 0
$$946$$ −40.2850 −1.30978
$$947$$ −18.3448 −0.596125 −0.298063 0.954546i $$-0.596340\pi$$
−0.298063 + 0.954546i $$0.596340\pi$$
$$948$$ 0 0
$$949$$ −20.4275 −0.663106
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −7.61490 −0.246800
$$953$$ −13.8466 −0.448535 −0.224267 0.974528i $$-0.571999\pi$$
−0.224267 + 0.974528i $$0.571999\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 1.62911 0.0526893
$$957$$ 0 0
$$958$$ −13.0780 −0.422530
$$959$$ −5.29515 −0.170989
$$960$$ 0 0
$$961$$ −3.07508 −0.0991962
$$962$$ 4.23909 0.136674
$$963$$ 0 0
$$964$$ 6.22335 0.200441
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −7.61678 −0.244939 −0.122470 0.992472i $$-0.539081\pi$$
−0.122470 + 0.992472i $$0.539081\pi$$
$$968$$ 67.0812 2.15607
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 40.9964 1.31564 0.657819 0.753176i $$-0.271479\pi$$
0.657819 + 0.753176i $$0.271479\pi$$
$$972$$ 0 0
$$973$$ 1.90465 0.0610604
$$974$$ 6.90958 0.221397
$$975$$ 0 0
$$976$$ −1.51761 −0.0485776
$$977$$ 57.8650 1.85127 0.925633 0.378423i $$-0.123534\pi$$
0.925633 + 0.378423i $$0.123534\pi$$
$$978$$ 0 0
$$979$$ 92.2069 2.94694
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −8.14044 −0.259772
$$983$$ 8.89795 0.283801 0.141900 0.989881i $$-0.454679\pi$$
0.141900 + 0.989881i $$0.454679\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −8.12898 −0.258879
$$987$$ 0 0
$$988$$ 6.01641 0.191408
$$989$$ −51.1037 −1.62500
$$990$$ 0 0
$$991$$ −7.01945 −0.222980 −0.111490 0.993766i $$-0.535562\pi$$
−0.111490 + 0.993766i $$0.535562\pi$$
$$992$$ 30.1822 0.958286
$$993$$ 0 0
$$994$$ −3.38088 −0.107235
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 49.6542 1.57256 0.786281 0.617868i $$-0.212004\pi$$
0.786281 + 0.617868i $$0.212004\pi$$
$$998$$ 16.8275 0.532664
$$999$$ 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 5625.2.a.z.1.4 8
3.2 odd 2 inner 5625.2.a.z.1.5 yes 8
5.4 even 2 5625.2.a.bb.1.5 yes 8
15.14 odd 2 5625.2.a.bb.1.4 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.4 8 1.1 even 1 trivial
5625.2.a.z.1.5 yes 8 3.2 odd 2 inner
5625.2.a.bb.1.4 yes 8 15.14 odd 2
5625.2.a.bb.1.5 yes 8 5.4 even 2