# Properties

 Label 1600.2.f.i.1249.1 Level $1600$ Weight $2$ Character 1600.1249 Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 320) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.1 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1600.1249 Dual form 1600.2.f.i.1249.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.732051 q^{3} -1.26795i q^{7} -2.46410 q^{9} +O(q^{10})$$ $$q-0.732051 q^{3} -1.26795i q^{7} -2.46410 q^{9} -3.46410i q^{11} -3.46410 q^{13} +3.46410i q^{17} +2.00000i q^{19} +0.928203i q^{21} +8.19615i q^{23} +4.00000 q^{27} +9.46410 q^{31} +2.53590i q^{33} -6.00000 q^{37} +2.53590 q^{39} +2.53590 q^{41} +10.1962 q^{43} +8.19615i q^{47} +5.39230 q^{49} -2.53590i q^{51} -10.3923 q^{53} -1.46410i q^{57} -6.00000i q^{59} +12.9282i q^{61} +3.12436i q^{63} -10.1962 q^{67} -6.00000i q^{69} +4.39230 q^{71} +14.3923i q^{73} -4.39230 q^{77} -12.0000 q^{79} +4.46410 q^{81} +4.73205 q^{83} -0.928203 q^{89} +4.39230i q^{91} -6.92820 q^{93} -6.39230i q^{97} +8.53590i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} + 24 q^{31} - 24 q^{37} + 24 q^{39} + 24 q^{41} + 20 q^{43} - 20 q^{49} - 20 q^{67} - 24 q^{71} + 24 q^{77} - 48 q^{79} + 4 q^{81} + 12 q^{83} + 24 q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ −0.732051 −0.422650 −0.211325 0.977416i $$-0.567778\pi$$
−0.211325 + 0.977416i $$0.567778\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.26795i − 0.479240i −0.970867 0.239620i $$-0.922977\pi$$
0.970867 0.239620i $$-0.0770228\pi$$
$$8$$ 0 0
$$9$$ −2.46410 −0.821367
$$10$$ 0 0
$$11$$ − 3.46410i − 1.04447i −0.852803 0.522233i $$-0.825099\pi$$
0.852803 0.522233i $$-0.174901\pi$$
$$12$$ 0 0
$$13$$ −3.46410 −0.960769 −0.480384 0.877058i $$-0.659503\pi$$
−0.480384 + 0.877058i $$0.659503\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.46410i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$18$$ 0 0
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 0 0
$$21$$ 0.928203i 0.202551i
$$22$$ 0 0
$$23$$ 8.19615i 1.70902i 0.519438 + 0.854508i $$0.326141\pi$$
−0.519438 + 0.854508i $$0.673859\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 9.46410 1.69980 0.849901 0.526942i $$-0.176661\pi$$
0.849901 + 0.526942i $$0.176661\pi$$
$$32$$ 0 0
$$33$$ 2.53590i 0.441443i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 2.53590 0.406069
$$40$$ 0 0
$$41$$ 2.53590 0.396041 0.198020 0.980198i $$-0.436549\pi$$
0.198020 + 0.980198i $$0.436549\pi$$
$$42$$ 0 0
$$43$$ 10.1962 1.55490 0.777449 0.628946i $$-0.216513\pi$$
0.777449 + 0.628946i $$0.216513\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.19615i 1.19553i 0.801671 + 0.597766i $$0.203945\pi$$
−0.801671 + 0.597766i $$0.796055\pi$$
$$48$$ 0 0
$$49$$ 5.39230 0.770329
$$50$$ 0 0
$$51$$ − 2.53590i − 0.355097i
$$52$$ 0 0
$$53$$ −10.3923 −1.42749 −0.713746 0.700404i $$-0.753003\pi$$
−0.713746 + 0.700404i $$0.753003\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 1.46410i − 0.193925i
$$58$$ 0 0
$$59$$ − 6.00000i − 0.781133i −0.920575 0.390567i $$-0.872279\pi$$
0.920575 0.390567i $$-0.127721\pi$$
$$60$$ 0 0
$$61$$ 12.9282i 1.65529i 0.561254 + 0.827643i $$0.310319\pi$$
−0.561254 + 0.827643i $$0.689681\pi$$
$$62$$ 0 0
$$63$$ 3.12436i 0.393632i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −10.1962 −1.24566 −0.622829 0.782358i $$-0.714017\pi$$
−0.622829 + 0.782358i $$0.714017\pi$$
$$68$$ 0 0
$$69$$ − 6.00000i − 0.722315i
$$70$$ 0 0
$$71$$ 4.39230 0.521271 0.260635 0.965437i $$-0.416068\pi$$
0.260635 + 0.965437i $$0.416068\pi$$
$$72$$ 0 0
$$73$$ 14.3923i 1.68449i 0.539093 + 0.842246i $$0.318767\pi$$
−0.539093 + 0.842246i $$0.681233\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −4.39230 −0.500550
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 4.46410 0.496011
$$82$$ 0 0
$$83$$ 4.73205 0.519410 0.259705 0.965688i $$-0.416375\pi$$
0.259705 + 0.965688i $$0.416375\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −0.928203 −0.0983893 −0.0491947 0.998789i $$-0.515665\pi$$
−0.0491947 + 0.998789i $$0.515665\pi$$
$$90$$ 0 0
$$91$$ 4.39230i 0.460439i
$$92$$ 0 0
$$93$$ −6.92820 −0.718421
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 6.39230i − 0.649040i −0.945879 0.324520i $$-0.894797\pi$$
0.945879 0.324520i $$-0.105203\pi$$
$$98$$ 0 0
$$99$$ 8.53590i 0.857890i
$$100$$ 0 0
$$101$$ 12.0000i 1.19404i 0.802225 + 0.597022i $$0.203650\pi$$
−0.802225 + 0.597022i $$0.796350\pi$$
$$102$$ 0 0
$$103$$ 8.19615i 0.807591i 0.914849 + 0.403795i $$0.132309\pi$$
−0.914849 + 0.403795i $$0.867691\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 16.7321 1.61755 0.808774 0.588119i $$-0.200131\pi$$
0.808774 + 0.588119i $$0.200131\pi$$
$$108$$ 0 0
$$109$$ − 0.928203i − 0.0889057i −0.999011 0.0444529i $$-0.985846\pi$$
0.999011 0.0444529i $$-0.0141545\pi$$
$$110$$ 0 0
$$111$$ 4.39230 0.416899
$$112$$ 0 0
$$113$$ − 0.928203i − 0.0873180i −0.999046 0.0436590i $$-0.986098\pi$$
0.999046 0.0436590i $$-0.0139015\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 8.53590 0.789144
$$118$$ 0 0
$$119$$ 4.39230 0.402642
$$120$$ 0 0
$$121$$ −1.00000 −0.0909091
$$122$$ 0 0
$$123$$ −1.85641 −0.167387
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 3.80385i 0.337537i 0.985656 + 0.168768i $$0.0539790\pi$$
−0.985656 + 0.168768i $$0.946021\pi$$
$$128$$ 0 0
$$129$$ −7.46410 −0.657178
$$130$$ 0 0
$$131$$ − 10.3923i − 0.907980i −0.891007 0.453990i $$-0.850000\pi$$
0.891007 0.453990i $$-0.150000\pi$$
$$132$$ 0 0
$$133$$ 2.53590 0.219890
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.9282i 1.10453i 0.833668 + 0.552265i $$0.186237\pi$$
−0.833668 + 0.552265i $$0.813763\pi$$
$$138$$ 0 0
$$139$$ − 10.0000i − 0.848189i −0.905618 0.424094i $$-0.860592\pi$$
0.905618 0.424094i $$-0.139408\pi$$
$$140$$ 0 0
$$141$$ − 6.00000i − 0.505291i
$$142$$ 0 0
$$143$$ 12.0000i 1.00349i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −3.94744 −0.325579
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i 0.675556 + 0.737309i $$0.263904\pi$$
−0.675556 + 0.737309i $$0.736096\pi$$
$$150$$ 0 0
$$151$$ −2.53590 −0.206368 −0.103184 0.994662i $$-0.532903\pi$$
−0.103184 + 0.994662i $$0.532903\pi$$
$$152$$ 0 0
$$153$$ − 8.53590i − 0.690086i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.928203 0.0740787 0.0370393 0.999314i $$-0.488207\pi$$
0.0370393 + 0.999314i $$0.488207\pi$$
$$158$$ 0 0
$$159$$ 7.60770 0.603329
$$160$$ 0 0
$$161$$ 10.3923 0.819028
$$162$$ 0 0
$$163$$ −5.80385 −0.454592 −0.227296 0.973826i $$-0.572989\pi$$
−0.227296 + 0.973826i $$0.572989\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.19615i 0.634237i 0.948386 + 0.317119i $$0.102715\pi$$
−0.948386 + 0.317119i $$0.897285\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ − 4.92820i − 0.376869i
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.39230i 0.330146i
$$178$$ 0 0
$$179$$ 19.8564i 1.48414i 0.670324 + 0.742069i $$0.266155\pi$$
−0.670324 + 0.742069i $$0.733845\pi$$
$$180$$ 0 0
$$181$$ − 6.92820i − 0.514969i −0.966282 0.257485i $$-0.917106\pi$$
0.966282 0.257485i $$-0.0828937\pi$$
$$182$$ 0 0
$$183$$ − 9.46410i − 0.699607i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ − 5.07180i − 0.368919i
$$190$$ 0 0
$$191$$ −16.3923 −1.18611 −0.593053 0.805164i $$-0.702077\pi$$
−0.593053 + 0.805164i $$0.702077\pi$$
$$192$$ 0 0
$$193$$ − 6.39230i − 0.460128i −0.973175 0.230064i $$-0.926106\pi$$
0.973175 0.230064i $$-0.0738936\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.3923 −0.740421 −0.370211 0.928948i $$-0.620714\pi$$
−0.370211 + 0.928948i $$0.620714\pi$$
$$198$$ 0 0
$$199$$ −6.92820 −0.491127 −0.245564 0.969380i $$-0.578973\pi$$
−0.245564 + 0.969380i $$0.578973\pi$$
$$200$$ 0 0
$$201$$ 7.46410 0.526477
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 20.1962i − 1.40373i
$$208$$ 0 0
$$209$$ 6.92820 0.479234
$$210$$ 0 0
$$211$$ 6.39230i 0.440064i 0.975493 + 0.220032i $$0.0706162\pi$$
−0.975493 + 0.220032i $$0.929384\pi$$
$$212$$ 0 0
$$213$$ −3.21539 −0.220315
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 12.0000i − 0.814613i
$$218$$ 0 0
$$219$$ − 10.5359i − 0.711950i
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 0 0
$$223$$ − 15.1244i − 1.01280i −0.862298 0.506401i $$-0.830976\pi$$
0.862298 0.506401i $$-0.169024\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −18.5885 −1.23376 −0.616880 0.787058i $$-0.711603\pi$$
−0.616880 + 0.787058i $$0.711603\pi$$
$$228$$ 0 0
$$229$$ − 18.9282i − 1.25081i −0.780300 0.625405i $$-0.784934\pi$$
0.780300 0.625405i $$-0.215066\pi$$
$$230$$ 0 0
$$231$$ 3.21539 0.211557
$$232$$ 0 0
$$233$$ 1.60770i 0.105324i 0.998612 + 0.0526618i $$0.0167705\pi$$
−0.998612 + 0.0526618i $$0.983229\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.78461 0.570622
$$238$$ 0 0
$$239$$ 20.7846 1.34444 0.672222 0.740349i $$-0.265340\pi$$
0.672222 + 0.740349i $$0.265340\pi$$
$$240$$ 0 0
$$241$$ −20.3923 −1.31358 −0.656792 0.754072i $$-0.728087\pi$$
−0.656792 + 0.754072i $$0.728087\pi$$
$$242$$ 0 0
$$243$$ −15.2679 −0.979439
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 6.92820i − 0.440831i
$$248$$ 0 0
$$249$$ −3.46410 −0.219529
$$250$$ 0 0
$$251$$ 15.4641i 0.976085i 0.872820 + 0.488043i $$0.162289\pi$$
−0.872820 + 0.488043i $$0.837711\pi$$
$$252$$ 0 0
$$253$$ 28.3923 1.78501
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 0 0
$$259$$ 7.60770i 0.472719i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 20.1962i − 1.24535i −0.782481 0.622674i $$-0.786046\pi$$
0.782481 0.622674i $$-0.213954\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0.679492 0.0415842
$$268$$ 0 0
$$269$$ 14.7846i 0.901434i 0.892667 + 0.450717i $$0.148832\pi$$
−0.892667 + 0.450717i $$0.851168\pi$$
$$270$$ 0 0
$$271$$ −4.39230 −0.266814 −0.133407 0.991061i $$-0.542592\pi$$
−0.133407 + 0.991061i $$0.542592\pi$$
$$272$$ 0 0
$$273$$ − 3.21539i − 0.194604i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 19.8564 1.19306 0.596528 0.802592i $$-0.296546\pi$$
0.596528 + 0.802592i $$0.296546\pi$$
$$278$$ 0 0
$$279$$ −23.3205 −1.39616
$$280$$ 0 0
$$281$$ −28.3923 −1.69374 −0.846871 0.531798i $$-0.821517\pi$$
−0.846871 + 0.531798i $$0.821517\pi$$
$$282$$ 0 0
$$283$$ 10.5885 0.629418 0.314709 0.949188i $$-0.398093\pi$$
0.314709 + 0.949188i $$0.398093\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 3.21539i − 0.189798i
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ 4.67949i 0.274317i
$$292$$ 0 0
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 13.8564i − 0.804030i
$$298$$ 0 0
$$299$$ − 28.3923i − 1.64197i
$$300$$ 0 0
$$301$$ − 12.9282i − 0.745169i
$$302$$ 0 0
$$303$$ − 8.78461i − 0.504663i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −34.1962 −1.95168 −0.975839 0.218492i $$-0.929886\pi$$
−0.975839 + 0.218492i $$0.929886\pi$$
$$308$$ 0 0
$$309$$ − 6.00000i − 0.341328i
$$310$$ 0 0
$$311$$ 7.60770 0.431393 0.215696 0.976460i $$-0.430798\pi$$
0.215696 + 0.976460i $$0.430798\pi$$
$$312$$ 0 0
$$313$$ − 22.0000i − 1.24351i −0.783210 0.621757i $$-0.786419\pi$$
0.783210 0.621757i $$-0.213581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −22.3923 −1.25768 −0.628839 0.777536i $$-0.716469\pi$$
−0.628839 + 0.777536i $$0.716469\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.2487 −0.683656
$$322$$ 0 0
$$323$$ −6.92820 −0.385496
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0.679492i 0.0375760i
$$328$$ 0 0
$$329$$ 10.3923 0.572946
$$330$$ 0 0
$$331$$ 5.60770i 0.308227i 0.988053 + 0.154113i $$0.0492521\pi$$
−0.988053 + 0.154113i $$0.950748\pi$$
$$332$$ 0 0
$$333$$ 14.7846 0.810192
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 0 0
$$339$$ 0.679492i 0.0369049i
$$340$$ 0 0
$$341$$ − 32.7846i − 1.77539i
$$342$$ 0 0
$$343$$ − 15.7128i − 0.848412i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 28.0526 1.50594 0.752970 0.658055i $$-0.228620\pi$$
0.752970 + 0.658055i $$0.228620\pi$$
$$348$$ 0 0
$$349$$ − 8.78461i − 0.470229i −0.971968 0.235115i $$-0.924453\pi$$
0.971968 0.235115i $$-0.0755466\pi$$
$$350$$ 0 0
$$351$$ −13.8564 −0.739600
$$352$$ 0 0
$$353$$ 14.7846i 0.786905i 0.919345 + 0.393453i $$0.128719\pi$$
−0.919345 + 0.393453i $$0.871281\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −3.21539 −0.170177
$$358$$ 0 0
$$359$$ 8.78461 0.463634 0.231817 0.972759i $$-0.425533\pi$$
0.231817 + 0.972759i $$0.425533\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ 0.732051 0.0384227
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 10.0526i − 0.524739i −0.964967 0.262370i $$-0.915496\pi$$
0.964967 0.262370i $$-0.0845040\pi$$
$$368$$ 0 0
$$369$$ −6.24871 −0.325295
$$370$$ 0 0
$$371$$ 13.1769i 0.684111i
$$372$$ 0 0
$$373$$ 7.85641 0.406789 0.203395 0.979097i $$-0.434803\pi$$
0.203395 + 0.979097i $$0.434803\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 2.00000i − 0.102733i −0.998680 0.0513665i $$-0.983642\pi$$
0.998680 0.0513665i $$-0.0163577\pi$$
$$380$$ 0 0
$$381$$ − 2.78461i − 0.142660i
$$382$$ 0 0
$$383$$ 3.80385i 0.194368i 0.995266 + 0.0971838i $$0.0309835\pi$$
−0.995266 + 0.0971838i $$0.969017\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −25.1244 −1.27714
$$388$$ 0 0
$$389$$ 26.7846i 1.35803i 0.734123 + 0.679017i $$0.237594\pi$$
−0.734123 + 0.679017i $$0.762406\pi$$
$$390$$ 0 0
$$391$$ −28.3923 −1.43586
$$392$$ 0 0
$$393$$ 7.60770i 0.383757i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 27.4641 1.37838 0.689192 0.724579i $$-0.257966\pi$$
0.689192 + 0.724579i $$0.257966\pi$$
$$398$$ 0 0
$$399$$ −1.85641 −0.0929366
$$400$$ 0 0
$$401$$ −4.14359 −0.206921 −0.103461 0.994634i $$-0.532992\pi$$
−0.103461 + 0.994634i $$0.532992\pi$$
$$402$$ 0 0
$$403$$ −32.7846 −1.63312
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.7846i 1.03025i
$$408$$ 0 0
$$409$$ −3.60770 −0.178389 −0.0891945 0.996014i $$-0.528429\pi$$
−0.0891945 + 0.996014i $$0.528429\pi$$
$$410$$ 0 0
$$411$$ − 9.46410i − 0.466830i
$$412$$ 0 0
$$413$$ −7.60770 −0.374350
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 7.32051i 0.358487i
$$418$$ 0 0
$$419$$ 0.928203i 0.0453457i 0.999743 + 0.0226728i $$0.00721761\pi$$
−0.999743 + 0.0226728i $$0.992782\pi$$
$$420$$ 0 0
$$421$$ 6.00000i 0.292422i 0.989253 + 0.146211i $$0.0467079\pi$$
−0.989253 + 0.146211i $$0.953292\pi$$
$$422$$ 0 0
$$423$$ − 20.1962i − 0.981971i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 16.3923 0.793279
$$428$$ 0 0
$$429$$ − 8.78461i − 0.424125i
$$430$$ 0 0
$$431$$ −28.3923 −1.36761 −0.683805 0.729665i $$-0.739676\pi$$
−0.683805 + 0.729665i $$0.739676\pi$$
$$432$$ 0 0
$$433$$ − 26.3923i − 1.26833i −0.773196 0.634167i $$-0.781343\pi$$
0.773196 0.634167i $$-0.218657\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −16.3923 −0.784150
$$438$$ 0 0
$$439$$ 18.9282 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$440$$ 0 0
$$441$$ −13.2872 −0.632723
$$442$$ 0 0
$$443$$ −0.339746 −0.0161418 −0.00807091 0.999967i $$-0.502569\pi$$
−0.00807091 + 0.999967i $$0.502569\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 13.1769i − 0.623247i
$$448$$ 0 0
$$449$$ 2.53590 0.119676 0.0598382 0.998208i $$-0.480942\pi$$
0.0598382 + 0.998208i $$0.480942\pi$$
$$450$$ 0 0
$$451$$ − 8.78461i − 0.413651i
$$452$$ 0 0
$$453$$ 1.85641 0.0872216
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.7846i 1.06582i 0.846172 + 0.532910i $$0.178901\pi$$
−0.846172 + 0.532910i $$0.821099\pi$$
$$458$$ 0 0
$$459$$ 13.8564i 0.646762i
$$460$$ 0 0
$$461$$ 12.0000i 0.558896i 0.960161 + 0.279448i $$0.0901514\pi$$
−0.960161 + 0.279448i $$0.909849\pi$$
$$462$$ 0 0
$$463$$ − 15.8038i − 0.734467i −0.930129 0.367234i $$-0.880305\pi$$
0.930129 0.367234i $$-0.119695\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.9808 1.06342 0.531711 0.846926i $$-0.321549\pi$$
0.531711 + 0.846926i $$0.321549\pi$$
$$468$$ 0 0
$$469$$ 12.9282i 0.596969i
$$470$$ 0 0
$$471$$ −0.679492 −0.0313093
$$472$$ 0 0
$$473$$ − 35.3205i − 1.62404i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 25.6077 1.17250
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.7846 0.947697
$$482$$ 0 0
$$483$$ −7.60770 −0.346162
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 39.1244i − 1.77289i −0.462830 0.886447i $$-0.653166\pi$$
0.462830 0.886447i $$-0.346834\pi$$
$$488$$ 0 0
$$489$$ 4.24871 0.192133
$$490$$ 0 0
$$491$$ 22.3923i 1.01055i 0.862958 + 0.505275i $$0.168609\pi$$
−0.862958 + 0.505275i $$0.831391\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 5.56922i − 0.249814i
$$498$$ 0 0
$$499$$ 39.5692i 1.77136i 0.464295 + 0.885681i $$0.346308\pi$$
−0.464295 + 0.885681i $$0.653692\pi$$
$$500$$ 0 0
$$501$$ − 6.00000i − 0.268060i
$$502$$ 0 0
$$503$$ 8.19615i 0.365448i 0.983164 + 0.182724i $$0.0584915\pi$$
−0.983164 + 0.182724i $$0.941508\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0.732051 0.0325115
$$508$$ 0 0
$$509$$ 8.78461i 0.389371i 0.980866 + 0.194685i $$0.0623686\pi$$
−0.980866 + 0.194685i $$0.937631\pi$$
$$510$$ 0 0
$$511$$ 18.2487 0.807275
$$512$$ 0 0
$$513$$ 8.00000i 0.353209i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 28.3923 1.24869
$$518$$ 0 0
$$519$$ 4.39230 0.192801
$$520$$ 0 0
$$521$$ 31.8564 1.39565 0.697827 0.716266i $$-0.254150\pi$$
0.697827 + 0.716266i $$0.254150\pi$$
$$522$$ 0 0
$$523$$ −14.9808 −0.655063 −0.327531 0.944840i $$-0.606217\pi$$
−0.327531 + 0.944840i $$0.606217\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 32.7846i 1.42812i
$$528$$ 0 0
$$529$$ −44.1769 −1.92074
$$530$$ 0 0
$$531$$ 14.7846i 0.641597i
$$532$$ 0 0
$$533$$ −8.78461 −0.380504
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 14.5359i − 0.627270i
$$538$$ 0 0
$$539$$ − 18.6795i − 0.804583i
$$540$$ 0 0
$$541$$ − 15.7128i − 0.675547i −0.941227 0.337773i $$-0.890326\pi$$
0.941227 0.337773i $$-0.109674\pi$$
$$542$$ 0 0
$$543$$ 5.07180i 0.217652i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.80385 −0.0771270 −0.0385635 0.999256i $$-0.512278\pi$$
−0.0385635 + 0.999256i $$0.512278\pi$$
$$548$$ 0 0
$$549$$ − 31.8564i − 1.35960i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 15.2154i 0.647024i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −14.7846 −0.626444 −0.313222 0.949680i $$-0.601408\pi$$
−0.313222 + 0.949680i $$0.601408\pi$$
$$558$$ 0 0
$$559$$ −35.3205 −1.49390
$$560$$ 0 0
$$561$$ −8.78461 −0.370887
$$562$$ 0 0
$$563$$ 26.1962 1.10404 0.552018 0.833832i $$-0.313858\pi$$
0.552018 + 0.833832i $$0.313858\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 5.66025i − 0.237708i
$$568$$ 0 0
$$569$$ 42.2487 1.77116 0.885579 0.464489i $$-0.153762\pi$$
0.885579 + 0.464489i $$0.153762\pi$$
$$570$$ 0 0
$$571$$ − 30.3923i − 1.27188i −0.771739 0.635939i $$-0.780613\pi$$
0.771739 0.635939i $$-0.219387\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 0 0
$$579$$ 4.67949i 0.194473i
$$580$$ 0 0
$$581$$ − 6.00000i − 0.248922i
$$582$$ 0 0
$$583$$ 36.0000i 1.49097i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −13.5167 −0.557892 −0.278946 0.960307i $$-0.589985\pi$$
−0.278946 + 0.960307i $$0.589985\pi$$
$$588$$ 0 0
$$589$$ 18.9282i 0.779923i
$$590$$ 0 0
$$591$$ 7.60770 0.312939
$$592$$ 0 0
$$593$$ 0.928203i 0.0381167i 0.999818 + 0.0190584i $$0.00606683\pi$$
−0.999818 + 0.0190584i $$0.993933\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.07180 0.207575
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −20.3923 −0.831819 −0.415910 0.909406i $$-0.636537\pi$$
−0.415910 + 0.909406i $$0.636537\pi$$
$$602$$ 0 0
$$603$$ 25.1244 1.02314
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.19615i 0.332672i 0.986069 + 0.166336i $$0.0531936\pi$$
−0.986069 + 0.166336i $$0.946806\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 28.3923i − 1.14863i
$$612$$ 0 0
$$613$$ −13.6077 −0.549610 −0.274805 0.961500i $$-0.588613\pi$$
−0.274805 + 0.961500i $$0.588613\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 13.6077i 0.547825i 0.961755 + 0.273913i $$0.0883179\pi$$
−0.961755 + 0.273913i $$0.911682\pi$$
$$618$$ 0 0
$$619$$ − 6.78461i − 0.272696i −0.990661 0.136348i $$-0.956463\pi$$
0.990661 0.136348i $$-0.0435366\pi$$
$$620$$ 0 0
$$621$$ 32.7846i 1.31560i
$$622$$ 0 0
$$623$$ 1.17691i 0.0471521i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −5.07180 −0.202548
$$628$$ 0 0
$$629$$ − 20.7846i − 0.828737i
$$630$$ 0 0
$$631$$ −21.4641 −0.854472 −0.427236 0.904140i $$-0.640513\pi$$
−0.427236 + 0.904140i $$0.640513\pi$$
$$632$$ 0 0
$$633$$ − 4.67949i − 0.185993i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −18.6795 −0.740108
$$638$$ 0 0
$$639$$ −10.8231 −0.428155
$$640$$ 0 0
$$641$$ −4.39230 −0.173486 −0.0867428 0.996231i $$-0.527646\pi$$
−0.0867428 + 0.996231i $$0.527646\pi$$
$$642$$ 0 0
$$643$$ 10.5885 0.417568 0.208784 0.977962i $$-0.433049\pi$$
0.208784 + 0.977962i $$0.433049\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 36.5885i 1.43844i 0.694782 + 0.719220i $$0.255501\pi$$
−0.694782 + 0.719220i $$0.744499\pi$$
$$648$$ 0 0
$$649$$ −20.7846 −0.815867
$$650$$ 0 0
$$651$$ 8.78461i 0.344296i
$$652$$ 0 0
$$653$$ −19.1769 −0.750451 −0.375225 0.926934i $$-0.622435\pi$$
−0.375225 + 0.926934i $$0.622435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 35.4641i − 1.38359i
$$658$$ 0 0
$$659$$ − 40.6410i − 1.58315i −0.611073 0.791575i $$-0.709262\pi$$
0.611073 0.791575i $$-0.290738\pi$$
$$660$$ 0 0
$$661$$ 35.5692i 1.38348i 0.722146 + 0.691741i $$0.243156\pi$$
−0.722146 + 0.691741i $$0.756844\pi$$
$$662$$ 0 0
$$663$$ 8.78461i 0.341166i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 11.0718i 0.428060i
$$670$$ 0 0
$$671$$ 44.7846 1.72889
$$672$$ 0 0
$$673$$ 39.1769i 1.51016i 0.655633 + 0.755080i $$0.272402\pi$$
−0.655633 + 0.755080i $$0.727598\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −43.1769 −1.65942 −0.829712 0.558192i $$-0.811495\pi$$
−0.829712 + 0.558192i $$0.811495\pi$$
$$678$$ 0 0
$$679$$ −8.10512 −0.311046
$$680$$ 0 0
$$681$$ 13.6077 0.521448
$$682$$ 0 0
$$683$$ −11.6603 −0.446167 −0.223084 0.974799i $$-0.571612\pi$$
−0.223084 + 0.974799i $$0.571612\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 13.8564i 0.528655i
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 5.60770i 0.213327i 0.994295 + 0.106663i $$0.0340167\pi$$
−0.994295 + 0.106663i $$0.965983\pi$$
$$692$$ 0 0
$$693$$ 10.8231 0.411135
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 8.78461i 0.332741i
$$698$$ 0 0
$$699$$ − 1.17691i − 0.0445150i
$$700$$ 0 0
$$701$$ 14.7846i 0.558407i 0.960232 + 0.279204i $$0.0900704\pi$$
−0.960232 + 0.279204i $$0.909930\pi$$
$$702$$ 0 0
$$703$$ − 12.0000i − 0.452589i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 15.2154 0.572234
$$708$$ 0 0
$$709$$ 10.1436i 0.380951i 0.981692 + 0.190475i $$0.0610029\pi$$
−0.981692 + 0.190475i $$0.938997\pi$$
$$710$$ 0 0
$$711$$ 29.5692 1.10893
$$712$$ 0 0
$$713$$ 77.5692i 2.90499i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −15.2154 −0.568229
$$718$$ 0 0
$$719$$ 44.7846 1.67018 0.835092 0.550110i $$-0.185414\pi$$
0.835092 + 0.550110i $$0.185414\pi$$
$$720$$ 0 0
$$721$$ 10.3923 0.387030
$$722$$ 0 0
$$723$$ 14.9282 0.555186
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 34.0526i − 1.26294i −0.775401 0.631470i $$-0.782452\pi$$
0.775401 0.631470i $$-0.217548\pi$$
$$728$$ 0 0
$$729$$ −2.21539 −0.0820515
$$730$$ 0 0
$$731$$ 35.3205i 1.30638i
$$732$$ 0 0
$$733$$ −38.7846 −1.43254 −0.716271 0.697822i $$-0.754153\pi$$
−0.716271 + 0.697822i $$0.754153\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 35.3205i 1.30105i
$$738$$ 0 0
$$739$$ − 2.00000i − 0.0735712i −0.999323 0.0367856i $$-0.988288\pi$$
0.999323 0.0367856i $$-0.0117119\pi$$
$$740$$ 0 0
$$741$$ 5.07180i 0.186317i
$$742$$ 0 0
$$743$$ 36.5885i 1.34230i 0.741321 + 0.671150i $$0.234199\pi$$
−0.741321 + 0.671150i $$0.765801\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −11.6603 −0.426626
$$748$$ 0 0
$$749$$ − 21.2154i − 0.775193i
$$750$$ 0 0
$$751$$ 40.3923 1.47394 0.736968 0.675928i $$-0.236257\pi$$
0.736968 + 0.675928i $$0.236257\pi$$
$$752$$ 0 0
$$753$$ − 11.3205i − 0.412542i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −23.0718 −0.838559 −0.419279 0.907857i $$-0.637717\pi$$
−0.419279 + 0.907857i $$0.637717\pi$$
$$758$$ 0 0
$$759$$ −20.7846 −0.754434
$$760$$ 0 0
$$761$$ −21.7128 −0.787089 −0.393544 0.919306i $$-0.628751\pi$$
−0.393544 + 0.919306i $$0.628751\pi$$
$$762$$ 0 0
$$763$$ −1.17691 −0.0426072
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 20.7846i 0.750489i
$$768$$ 0 0
$$769$$ 6.78461 0.244659 0.122330 0.992490i $$-0.460963\pi$$
0.122330 + 0.992490i $$0.460963\pi$$
$$770$$ 0 0
$$771$$ − 4.39230i − 0.158185i
$$772$$ 0 0
$$773$$ −46.3923 −1.66862 −0.834308 0.551299i $$-0.814132\pi$$
−0.834308 + 0.551299i $$0.814132\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 5.56922i − 0.199795i
$$778$$ 0 0
$$779$$ 5.07180i 0.181716i
$$780$$ 0 0
$$781$$ − 15.2154i − 0.544449i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 5.80385 0.206885 0.103442 0.994635i $$-0.467014\pi$$
0.103442 + 0.994635i $$0.467014\pi$$
$$788$$ 0 0
$$789$$ 14.7846i 0.526346i
$$790$$ 0 0
$$791$$ −1.17691 −0.0418463
$$792$$ 0 0
$$793$$ − 44.7846i − 1.59035i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.60770 0.0569475 0.0284737 0.999595i $$-0.490935\pi$$
0.0284737 + 0.999595i $$0.490935\pi$$
$$798$$ 0 0
$$799$$ −28.3923 −1.00445
$$800$$ 0 0
$$801$$ 2.28719 0.0808138
$$802$$ 0 0
$$803$$ 49.8564 1.75939
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 10.8231i − 0.380991i
$$808$$ 0 0
$$809$$ −35.5692 −1.25055 −0.625274 0.780406i $$-0.715013\pi$$
−0.625274 + 0.780406i $$0.715013\pi$$
$$810$$ 0 0
$$811$$ − 38.3923i − 1.34814i −0.738669 0.674068i $$-0.764545\pi$$
0.738669 0.674068i $$-0.235455\pi$$
$$812$$ 0 0
$$813$$ 3.21539 0.112769
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 20.3923i 0.713436i
$$818$$ 0 0
$$819$$ − 10.8231i − 0.378189i
$$820$$ 0 0
$$821$$ 50.7846i 1.77240i 0.463308 + 0.886198i $$0.346663\pi$$
−0.463308 + 0.886198i $$0.653337\pi$$
$$822$$ 0 0
$$823$$ 30.8372i 1.07492i 0.843290 + 0.537458i $$0.180615\pi$$
−0.843290 + 0.537458i $$0.819385\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −4.73205 −0.164550 −0.0822748 0.996610i $$-0.526219\pi$$
−0.0822748 + 0.996610i $$0.526219\pi$$
$$828$$ 0 0
$$829$$ − 50.7846i − 1.76382i −0.471416 0.881911i $$-0.656257\pi$$
0.471416 0.881911i $$-0.343743\pi$$
$$830$$ 0 0
$$831$$ −14.5359 −0.504245
$$832$$ 0 0
$$833$$ 18.6795i 0.647206i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 37.8564 1.30851
$$838$$ 0 0
$$839$$ 8.78461 0.303278 0.151639 0.988436i $$-0.451545\pi$$
0.151639 + 0.988436i $$0.451545\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ 20.7846 0.715860
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.26795i 0.0435672i
$$848$$ 0 0
$$849$$ −7.75129 −0.266024
$$850$$ 0 0
$$851$$ − 49.1769i − 1.68576i
$$852$$ 0 0
$$853$$ −3.46410 −0.118609 −0.0593043 0.998240i $$-0.518888\pi$$
−0.0593043 + 0.998240i $$0.518888\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 16.1436i − 0.551455i −0.961236 0.275727i $$-0.911081\pi$$
0.961236 0.275727i $$-0.0889187\pi$$
$$858$$ 0 0
$$859$$ 51.5692i 1.75952i 0.475419 + 0.879760i $$0.342296\pi$$
−0.475419 + 0.879760i $$0.657704\pi$$
$$860$$ 0 0
$$861$$ 2.35383i 0.0802183i
$$862$$ 0 0
$$863$$ 40.9808i 1.39500i 0.716584 + 0.697501i $$0.245705\pi$$
−0.716584 + 0.697501i $$0.754295\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −3.66025 −0.124309
$$868$$ 0 0
$$869$$ 41.5692i 1.41014i
$$870$$ 0 0
$$871$$ 35.3205 1.19679
$$872$$ 0 0
$$873$$ 15.7513i 0.533100i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7.85641 −0.265292 −0.132646 0.991163i $$-0.542347\pi$$
−0.132646 + 0.991163i $$0.542347\pi$$
$$878$$ 0 0
$$879$$ −21.9615 −0.740744
$$880$$ 0 0
$$881$$ −7.60770 −0.256310 −0.128155 0.991754i $$-0.540905\pi$$
−0.128155 + 0.991754i $$0.540905\pi$$
$$882$$ 0 0
$$883$$ −5.80385 −0.195315 −0.0976575 0.995220i $$-0.531135\pi$$
−0.0976575 + 0.995220i $$0.531135\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 21.3731i 0.717637i 0.933407 + 0.358819i $$0.116820\pi$$
−0.933407 + 0.358819i $$0.883180\pi$$
$$888$$ 0 0
$$889$$ 4.82309 0.161761
$$890$$ 0 0
$$891$$ − 15.4641i − 0.518067i
$$892$$ 0 0
$$893$$ −16.3923 −0.548548
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 20.7846i 0.693978i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ − 36.0000i − 1.19933i
$$902$$ 0 0
$$903$$ 9.46410i 0.314946i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −1.41154 −0.0468695 −0.0234348 0.999725i $$-0.507460\pi$$
−0.0234348 + 0.999725i $$0.507460\pi$$
$$908$$ 0 0
$$909$$ − 29.5692i − 0.980749i
$$910$$ 0 0
$$911$$ 37.1769 1.23173 0.615863 0.787853i $$-0.288807\pi$$
0.615863 + 0.787853i $$0.288807\pi$$
$$912$$ 0 0
$$913$$ − 16.3923i − 0.542506i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −13.1769 −0.435140
$$918$$ 0 0
$$919$$ 39.7128 1.31000 0.655002 0.755627i $$-0.272668\pi$$
0.655002 + 0.755627i $$0.272668\pi$$
$$920$$ 0 0
$$921$$ 25.0333 0.824876
$$922$$ 0 0
$$923$$ −15.2154 −0.500821
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 20.1962i − 0.663329i
$$928$$ 0 0
$$929$$ −7.60770 −0.249600 −0.124800 0.992182i $$-0.539829\pi$$
−0.124800 + 0.992182i $$0.539829\pi$$
$$930$$ 0 0
$$931$$ 10.7846i 0.353451i
$$932$$ 0 0
$$933$$ −5.56922 −0.182328
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 9.60770i − 0.313870i −0.987609 0.156935i $$-0.949839\pi$$
0.987609 0.156935i $$-0.0501613\pi$$
$$938$$ 0 0
$$939$$ 16.1051i 0.525571i
$$940$$ 0 0
$$941$$ − 20.7846i − 0.677559i −0.940866 0.338779i $$-0.889986\pi$$
0.940866 0.338779i $$-0.110014\pi$$
$$942$$ 0 0
$$943$$ 20.7846i 0.676840i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 45.8038 1.48843 0.744213 0.667943i $$-0.232825\pi$$
0.744213 + 0.667943i $$0.232825\pi$$
$$948$$ 0 0
$$949$$ − 49.8564i − 1.61841i
$$950$$ 0 0
$$951$$ 16.3923 0.531557
$$952$$ 0 0
$$953$$ − 24.9282i − 0.807504i −0.914869 0.403752i $$-0.867706\pi$$
0.914869 0.403752i $$-0.132294\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 16.3923 0.529335
$$960$$ 0 0
$$961$$ 58.5692 1.88933
$$962$$ 0 0
$$963$$ −41.2295 −1.32860
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.19615i 0.263570i 0.991278 + 0.131785i $$0.0420709\pi$$
−0.991278 + 0.131785i $$0.957929\pi$$
$$968$$ 0 0
$$969$$ 5.07180 0.162930
$$970$$ 0 0
$$971$$ − 33.0333i − 1.06009i −0.847970 0.530045i $$-0.822175\pi$$
0.847970 0.530045i $$-0.177825\pi$$
$$972$$ 0 0
$$973$$ −12.6795 −0.406486
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 5.32051i 0.170218i 0.996372 + 0.0851091i $$0.0271239\pi$$
−0.996372 + 0.0851091i $$0.972876\pi$$
$$978$$ 0 0
$$979$$ 3.21539i 0.102764i
$$980$$ 0 0
$$981$$ 2.28719i 0.0730243i
$$982$$ 0 0
$$983$$ − 11.4115i − 0.363972i −0.983301 0.181986i $$-0.941747\pi$$
0.983301 0.181986i $$-0.0582525\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −7.60770 −0.242156
$$988$$ 0 0
$$989$$ 83.5692i 2.65735i
$$990$$ 0 0
$$991$$ −44.1051 −1.40105 −0.700523 0.713630i $$-0.747050\pi$$
−0.700523 + 0.713630i $$0.747050\pi$$
$$992$$ 0 0
$$993$$ − 4.10512i − 0.130272i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 25.6077 0.811004 0.405502 0.914094i $$-0.367097\pi$$
0.405502 + 0.914094i $$0.367097\pi$$
$$998$$ 0 0
$$999$$ −24.0000 −0.759326
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 1600.2.f.i.1249.1 4
4.3 odd 2 1600.2.f.d.1249.4 4
5.2 odd 4 1600.2.d.h.801.3 4
5.3 odd 4 320.2.d.a.161.2 4
5.4 even 2 1600.2.f.e.1249.4 4
8.3 odd 2 1600.2.f.h.1249.2 4
8.5 even 2 1600.2.f.e.1249.3 4
15.8 even 4 2880.2.k.e.1441.4 4
20.3 even 4 320.2.d.b.161.3 yes 4
20.7 even 4 1600.2.d.b.801.2 4
20.19 odd 2 1600.2.f.h.1249.1 4
40.3 even 4 320.2.d.b.161.2 yes 4
40.13 odd 4 320.2.d.a.161.3 yes 4
40.19 odd 2 1600.2.f.d.1249.3 4
40.27 even 4 1600.2.d.b.801.3 4
40.29 even 2 inner 1600.2.f.i.1249.2 4
40.37 odd 4 1600.2.d.h.801.2 4
60.23 odd 4 2880.2.k.l.1441.3 4
80.3 even 4 1280.2.a.m.1.1 2
80.13 odd 4 1280.2.a.b.1.2 2
80.27 even 4 6400.2.a.ck.1.1 2
80.37 odd 4 6400.2.a.y.1.2 2
80.43 even 4 1280.2.a.c.1.2 2
80.53 odd 4 1280.2.a.p.1.1 2
80.67 even 4 6400.2.a.bf.1.2 2
80.77 odd 4 6400.2.a.cd.1.1 2
120.53 even 4 2880.2.k.e.1441.2 4
120.83 odd 4 2880.2.k.l.1441.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.2 4 5.3 odd 4
320.2.d.a.161.3 yes 4 40.13 odd 4
320.2.d.b.161.2 yes 4 40.3 even 4
320.2.d.b.161.3 yes 4 20.3 even 4
1280.2.a.b.1.2 2 80.13 odd 4
1280.2.a.c.1.2 2 80.43 even 4
1280.2.a.m.1.1 2 80.3 even 4
1280.2.a.p.1.1 2 80.53 odd 4
1600.2.d.b.801.2 4 20.7 even 4
1600.2.d.b.801.3 4 40.27 even 4
1600.2.d.h.801.2 4 40.37 odd 4
1600.2.d.h.801.3 4 5.2 odd 4
1600.2.f.d.1249.3 4 40.19 odd 2
1600.2.f.d.1249.4 4 4.3 odd 2
1600.2.f.e.1249.3 4 8.5 even 2
1600.2.f.e.1249.4 4 5.4 even 2
1600.2.f.h.1249.1 4 20.19 odd 2
1600.2.f.h.1249.2 4 8.3 odd 2
1600.2.f.i.1249.1 4 1.1 even 1 trivial
1600.2.f.i.1249.2 4 40.29 even 2 inner
2880.2.k.e.1441.2 4 120.53 even 4
2880.2.k.e.1441.4 4 15.8 even 4
2880.2.k.l.1441.1 4 120.83 odd 4
2880.2.k.l.1441.3 4 60.23 odd 4
6400.2.a.y.1.2 2 80.37 odd 4
6400.2.a.bf.1.2 2 80.67 even 4
6400.2.a.cd.1.1 2 80.77 odd 4
6400.2.a.ck.1.1 2 80.27 even 4