# Properties

 Label 2352.4.a.r.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +18.0000 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +18.0000 q^{5} +9.00000 q^{9} +36.0000 q^{11} +34.0000 q^{13} -54.0000 q^{15} -42.0000 q^{17} -124.000 q^{19} +199.000 q^{25} -27.0000 q^{27} +102.000 q^{29} -160.000 q^{31} -108.000 q^{33} +398.000 q^{37} -102.000 q^{39} +318.000 q^{41} +268.000 q^{43} +162.000 q^{45} +240.000 q^{47} +126.000 q^{51} -498.000 q^{53} +648.000 q^{55} +372.000 q^{57} -132.000 q^{59} -398.000 q^{61} +612.000 q^{65} -92.0000 q^{67} +720.000 q^{71} +502.000 q^{73} -597.000 q^{75} +1024.00 q^{79} +81.0000 q^{81} -204.000 q^{83} -756.000 q^{85} -306.000 q^{87} -354.000 q^{89} +480.000 q^{93} -2232.00 q^{95} +286.000 q^{97} +324.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ 18.0000 1.60997 0.804984 0.593296i $$-0.202174\pi$$
0.804984 + 0.593296i $$0.202174\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 36.0000 0.986764 0.493382 0.869813i $$-0.335760\pi$$
0.493382 + 0.869813i $$0.335760\pi$$
$$12$$ 0 0
$$13$$ 34.0000 0.725377 0.362689 0.931910i $$-0.381859\pi$$
0.362689 + 0.931910i $$0.381859\pi$$
$$14$$ 0 0
$$15$$ −54.0000 −0.929516
$$16$$ 0 0
$$17$$ −42.0000 −0.599206 −0.299603 0.954064i $$-0.596854\pi$$
−0.299603 + 0.954064i $$0.596854\pi$$
$$18$$ 0 0
$$19$$ −124.000 −1.49724 −0.748620 0.663000i $$-0.769283\pi$$
−0.748620 + 0.663000i $$0.769283\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 199.000 1.59200
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 102.000 0.653135 0.326568 0.945174i $$-0.394108\pi$$
0.326568 + 0.945174i $$0.394108\pi$$
$$30$$ 0 0
$$31$$ −160.000 −0.926995 −0.463498 0.886098i $$-0.653406\pi$$
−0.463498 + 0.886098i $$0.653406\pi$$
$$32$$ 0 0
$$33$$ −108.000 −0.569709
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 398.000 1.76840 0.884200 0.467109i $$-0.154704\pi$$
0.884200 + 0.467109i $$0.154704\pi$$
$$38$$ 0 0
$$39$$ −102.000 −0.418797
$$40$$ 0 0
$$41$$ 318.000 1.21130 0.605649 0.795732i $$-0.292913\pi$$
0.605649 + 0.795732i $$0.292913\pi$$
$$42$$ 0 0
$$43$$ 268.000 0.950456 0.475228 0.879863i $$-0.342366\pi$$
0.475228 + 0.879863i $$0.342366\pi$$
$$44$$ 0 0
$$45$$ 162.000 0.536656
$$46$$ 0 0
$$47$$ 240.000 0.744843 0.372421 0.928064i $$-0.378528\pi$$
0.372421 + 0.928064i $$0.378528\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 126.000 0.345952
$$52$$ 0 0
$$53$$ −498.000 −1.29067 −0.645335 0.763899i $$-0.723282\pi$$
−0.645335 + 0.763899i $$0.723282\pi$$
$$54$$ 0 0
$$55$$ 648.000 1.58866
$$56$$ 0 0
$$57$$ 372.000 0.864432
$$58$$ 0 0
$$59$$ −132.000 −0.291270 −0.145635 0.989338i $$-0.546523\pi$$
−0.145635 + 0.989338i $$0.546523\pi$$
$$60$$ 0 0
$$61$$ −398.000 −0.835388 −0.417694 0.908588i $$-0.637162\pi$$
−0.417694 + 0.908588i $$0.637162\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 612.000 1.16783
$$66$$ 0 0
$$67$$ −92.0000 −0.167755 −0.0838775 0.996476i $$-0.526730\pi$$
−0.0838775 + 0.996476i $$0.526730\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 720.000 1.20350 0.601748 0.798686i $$-0.294471\pi$$
0.601748 + 0.798686i $$0.294471\pi$$
$$72$$ 0 0
$$73$$ 502.000 0.804858 0.402429 0.915451i $$-0.368166\pi$$
0.402429 + 0.915451i $$0.368166\pi$$
$$74$$ 0 0
$$75$$ −597.000 −0.919142
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1024.00 1.45834 0.729171 0.684332i $$-0.239906\pi$$
0.729171 + 0.684332i $$0.239906\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −204.000 −0.269782 −0.134891 0.990860i $$-0.543068\pi$$
−0.134891 + 0.990860i $$0.543068\pi$$
$$84$$ 0 0
$$85$$ −756.000 −0.964703
$$86$$ 0 0
$$87$$ −306.000 −0.377088
$$88$$ 0 0
$$89$$ −354.000 −0.421617 −0.210809 0.977527i $$-0.567610\pi$$
−0.210809 + 0.977527i $$0.567610\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 480.000 0.535201
$$94$$ 0 0
$$95$$ −2232.00 −2.41051
$$96$$ 0 0
$$97$$ 286.000 0.299370 0.149685 0.988734i $$-0.452174\pi$$
0.149685 + 0.988734i $$0.452174\pi$$
$$98$$ 0 0
$$99$$ 324.000 0.328921
$$100$$ 0 0
$$101$$ −414.000 −0.407867 −0.203933 0.978985i $$-0.565373\pi$$
−0.203933 + 0.978985i $$0.565373\pi$$
$$102$$ 0 0
$$103$$ 56.0000 0.0535713 0.0267857 0.999641i $$-0.491473\pi$$
0.0267857 + 0.999641i $$0.491473\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −0.0108419 −0.00542095 0.999985i $$-0.501726\pi$$
−0.00542095 + 0.999985i $$0.501726\pi$$
$$108$$ 0 0
$$109$$ 1478.00 1.29878 0.649389 0.760457i $$-0.275025\pi$$
0.649389 + 0.760457i $$0.275025\pi$$
$$110$$ 0 0
$$111$$ −1194.00 −1.02099
$$112$$ 0 0
$$113$$ 402.000 0.334664 0.167332 0.985901i $$-0.446485\pi$$
0.167332 + 0.985901i $$0.446485\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 306.000 0.241792
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −35.0000 −0.0262960
$$122$$ 0 0
$$123$$ −954.000 −0.699344
$$124$$ 0 0
$$125$$ 1332.00 0.953102
$$126$$ 0 0
$$127$$ −1280.00 −0.894344 −0.447172 0.894448i $$-0.647569\pi$$
−0.447172 + 0.894448i $$0.647569\pi$$
$$128$$ 0 0
$$129$$ −804.000 −0.548746
$$130$$ 0 0
$$131$$ 1764.00 1.17650 0.588250 0.808679i $$-0.299817\pi$$
0.588250 + 0.808679i $$0.299817\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −486.000 −0.309839
$$136$$ 0 0
$$137$$ −2358.00 −1.47049 −0.735246 0.677800i $$-0.762934\pi$$
−0.735246 + 0.677800i $$0.762934\pi$$
$$138$$ 0 0
$$139$$ −52.0000 −0.0317308 −0.0158654 0.999874i $$-0.505050\pi$$
−0.0158654 + 0.999874i $$0.505050\pi$$
$$140$$ 0 0
$$141$$ −720.000 −0.430035
$$142$$ 0 0
$$143$$ 1224.00 0.715776
$$144$$ 0 0
$$145$$ 1836.00 1.05153
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1746.00 −0.959986 −0.479993 0.877272i $$-0.659361\pi$$
−0.479993 + 0.877272i $$0.659361\pi$$
$$150$$ 0 0
$$151$$ 232.000 0.125032 0.0625162 0.998044i $$-0.480087\pi$$
0.0625162 + 0.998044i $$0.480087\pi$$
$$152$$ 0 0
$$153$$ −378.000 −0.199735
$$154$$ 0 0
$$155$$ −2880.00 −1.49243
$$156$$ 0 0
$$157$$ −1694.00 −0.861120 −0.430560 0.902562i $$-0.641684\pi$$
−0.430560 + 0.902562i $$0.641684\pi$$
$$158$$ 0 0
$$159$$ 1494.00 0.745169
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2932.00 1.40891 0.704454 0.709750i $$-0.251192\pi$$
0.704454 + 0.709750i $$0.251192\pi$$
$$164$$ 0 0
$$165$$ −1944.00 −0.917213
$$166$$ 0 0
$$167$$ 1176.00 0.544920 0.272460 0.962167i $$-0.412163\pi$$
0.272460 + 0.962167i $$0.412163\pi$$
$$168$$ 0 0
$$169$$ −1041.00 −0.473828
$$170$$ 0 0
$$171$$ −1116.00 −0.499080
$$172$$ 0 0
$$173$$ −870.000 −0.382340 −0.191170 0.981557i $$-0.561228\pi$$
−0.191170 + 0.981557i $$0.561228\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 396.000 0.168165
$$178$$ 0 0
$$179$$ 2316.00 0.967072 0.483536 0.875324i $$-0.339352\pi$$
0.483536 + 0.875324i $$0.339352\pi$$
$$180$$ 0 0
$$181$$ 106.000 0.0435299 0.0217650 0.999763i $$-0.493071\pi$$
0.0217650 + 0.999763i $$0.493071\pi$$
$$182$$ 0 0
$$183$$ 1194.00 0.482312
$$184$$ 0 0
$$185$$ 7164.00 2.84707
$$186$$ 0 0
$$187$$ −1512.00 −0.591275
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1128.00 0.427326 0.213663 0.976907i $$-0.431461\pi$$
0.213663 + 0.976907i $$0.431461\pi$$
$$192$$ 0 0
$$193$$ 4034.00 1.50453 0.752263 0.658862i $$-0.228962\pi$$
0.752263 + 0.658862i $$0.228962\pi$$
$$194$$ 0 0
$$195$$ −1836.00 −0.674250
$$196$$ 0 0
$$197$$ −1314.00 −0.475221 −0.237611 0.971360i $$-0.576364\pi$$
−0.237611 + 0.971360i $$0.576364\pi$$
$$198$$ 0 0
$$199$$ 5096.00 1.81531 0.907653 0.419722i $$-0.137872\pi$$
0.907653 + 0.419722i $$0.137872\pi$$
$$200$$ 0 0
$$201$$ 276.000 0.0968534
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 5724.00 1.95015
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4464.00 −1.47742
$$210$$ 0 0
$$211$$ 3076.00 1.00360 0.501802 0.864982i $$-0.332670\pi$$
0.501802 + 0.864982i $$0.332670\pi$$
$$212$$ 0 0
$$213$$ −2160.00 −0.694839
$$214$$ 0 0
$$215$$ 4824.00 1.53020
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1506.00 −0.464685
$$220$$ 0 0
$$221$$ −1428.00 −0.434650
$$222$$ 0 0
$$223$$ −1888.00 −0.566950 −0.283475 0.958980i $$-0.591487\pi$$
−0.283475 + 0.958980i $$0.591487\pi$$
$$224$$ 0 0
$$225$$ 1791.00 0.530667
$$226$$ 0 0
$$227$$ −4716.00 −1.37891 −0.689454 0.724330i $$-0.742149\pi$$
−0.689454 + 0.724330i $$0.742149\pi$$
$$228$$ 0 0
$$229$$ 1690.00 0.487678 0.243839 0.969816i $$-0.421593\pi$$
0.243839 + 0.969816i $$0.421593\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 138.000 0.0388012 0.0194006 0.999812i $$-0.493824\pi$$
0.0194006 + 0.999812i $$0.493824\pi$$
$$234$$ 0 0
$$235$$ 4320.00 1.19917
$$236$$ 0 0
$$237$$ −3072.00 −0.841974
$$238$$ 0 0
$$239$$ −1896.00 −0.513147 −0.256573 0.966525i $$-0.582594\pi$$
−0.256573 + 0.966525i $$0.582594\pi$$
$$240$$ 0 0
$$241$$ 3598.00 0.961691 0.480846 0.876805i $$-0.340330\pi$$
0.480846 + 0.876805i $$0.340330\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4216.00 −1.08606
$$248$$ 0 0
$$249$$ 612.000 0.155759
$$250$$ 0 0
$$251$$ −3060.00 −0.769504 −0.384752 0.923020i $$-0.625713\pi$$
−0.384752 + 0.923020i $$0.625713\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 2268.00 0.556971
$$256$$ 0 0
$$257$$ 6822.00 1.65582 0.827908 0.560864i $$-0.189531\pi$$
0.827908 + 0.560864i $$0.189531\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 918.000 0.217712
$$262$$ 0 0
$$263$$ −2592.00 −0.607717 −0.303858 0.952717i $$-0.598275\pi$$
−0.303858 + 0.952717i $$0.598275\pi$$
$$264$$ 0 0
$$265$$ −8964.00 −2.07794
$$266$$ 0 0
$$267$$ 1062.00 0.243421
$$268$$ 0 0
$$269$$ −8214.00 −1.86177 −0.930886 0.365311i $$-0.880963\pi$$
−0.930886 + 0.365311i $$0.880963\pi$$
$$270$$ 0 0
$$271$$ −5344.00 −1.19788 −0.598939 0.800795i $$-0.704411\pi$$
−0.598939 + 0.800795i $$0.704411\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 7164.00 1.57093
$$276$$ 0 0
$$277$$ −6514.00 −1.41295 −0.706477 0.707736i $$-0.749717\pi$$
−0.706477 + 0.707736i $$0.749717\pi$$
$$278$$ 0 0
$$279$$ −1440.00 −0.308998
$$280$$ 0 0
$$281$$ 6618.00 1.40497 0.702485 0.711698i $$-0.252074\pi$$
0.702485 + 0.711698i $$0.252074\pi$$
$$282$$ 0 0
$$283$$ 3260.00 0.684759 0.342380 0.939562i $$-0.388767\pi$$
0.342380 + 0.939562i $$0.388767\pi$$
$$284$$ 0 0
$$285$$ 6696.00 1.39171
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −3149.00 −0.640953
$$290$$ 0 0
$$291$$ −858.000 −0.172841
$$292$$ 0 0
$$293$$ −5118.00 −1.02047 −0.510233 0.860036i $$-0.670441\pi$$
−0.510233 + 0.860036i $$0.670441\pi$$
$$294$$ 0 0
$$295$$ −2376.00 −0.468936
$$296$$ 0 0
$$297$$ −972.000 −0.189903
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1242.00 0.235482
$$304$$ 0 0
$$305$$ −7164.00 −1.34495
$$306$$ 0 0
$$307$$ 452.000 0.0840293 0.0420147 0.999117i $$-0.486622\pi$$
0.0420147 + 0.999117i $$0.486622\pi$$
$$308$$ 0 0
$$309$$ −168.000 −0.0309294
$$310$$ 0 0
$$311$$ 5016.00 0.914570 0.457285 0.889320i $$-0.348822\pi$$
0.457285 + 0.889320i $$0.348822\pi$$
$$312$$ 0 0
$$313$$ −5402.00 −0.975524 −0.487762 0.872977i $$-0.662187\pi$$
−0.487762 + 0.872977i $$0.662187\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10086.0 1.78702 0.893511 0.449041i $$-0.148234\pi$$
0.893511 + 0.449041i $$0.148234\pi$$
$$318$$ 0 0
$$319$$ 3672.00 0.644491
$$320$$ 0 0
$$321$$ 36.0000 0.00625958
$$322$$ 0 0
$$323$$ 5208.00 0.897154
$$324$$ 0 0
$$325$$ 6766.00 1.15480
$$326$$ 0 0
$$327$$ −4434.00 −0.749849
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8044.00 1.33577 0.667883 0.744267i $$-0.267201\pi$$
0.667883 + 0.744267i $$0.267201\pi$$
$$332$$ 0 0
$$333$$ 3582.00 0.589467
$$334$$ 0 0
$$335$$ −1656.00 −0.270080
$$336$$ 0 0
$$337$$ 4178.00 0.675342 0.337671 0.941264i $$-0.390361\pi$$
0.337671 + 0.941264i $$0.390361\pi$$
$$338$$ 0 0
$$339$$ −1206.00 −0.193218
$$340$$ 0 0
$$341$$ −5760.00 −0.914726
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −156.000 −0.0241341 −0.0120670 0.999927i $$-0.503841\pi$$
−0.0120670 + 0.999927i $$0.503841\pi$$
$$348$$ 0 0
$$349$$ 12418.0 1.90464 0.952321 0.305097i $$-0.0986888\pi$$
0.952321 + 0.305097i $$0.0986888\pi$$
$$350$$ 0 0
$$351$$ −918.000 −0.139599
$$352$$ 0 0
$$353$$ 7830.00 1.18059 0.590296 0.807187i $$-0.299011\pi$$
0.590296 + 0.807187i $$0.299011\pi$$
$$354$$ 0 0
$$355$$ 12960.0 1.93759
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 9312.00 1.36899 0.684497 0.729016i $$-0.260022\pi$$
0.684497 + 0.729016i $$0.260022\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ 0 0
$$363$$ 105.000 0.0151820
$$364$$ 0 0
$$365$$ 9036.00 1.29580
$$366$$ 0 0
$$367$$ −3760.00 −0.534797 −0.267398 0.963586i $$-0.586164\pi$$
−0.267398 + 0.963586i $$0.586164\pi$$
$$368$$ 0 0
$$369$$ 2862.00 0.403766
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 5870.00 0.814845 0.407422 0.913240i $$-0.366428\pi$$
0.407422 + 0.913240i $$0.366428\pi$$
$$374$$ 0 0
$$375$$ −3996.00 −0.550273
$$376$$ 0 0
$$377$$ 3468.00 0.473769
$$378$$ 0 0
$$379$$ 1852.00 0.251005 0.125502 0.992093i $$-0.459946\pi$$
0.125502 + 0.992093i $$0.459946\pi$$
$$380$$ 0 0
$$381$$ 3840.00 0.516350
$$382$$ 0 0
$$383$$ 2160.00 0.288175 0.144087 0.989565i $$-0.453975\pi$$
0.144087 + 0.989565i $$0.453975\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2412.00 0.316819
$$388$$ 0 0
$$389$$ −6786.00 −0.884483 −0.442241 0.896896i $$-0.645817\pi$$
−0.442241 + 0.896896i $$0.645817\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −5292.00 −0.679252
$$394$$ 0 0
$$395$$ 18432.0 2.34788
$$396$$ 0 0
$$397$$ 6514.00 0.823497 0.411748 0.911298i $$-0.364918\pi$$
0.411748 + 0.911298i $$0.364918\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3330.00 0.414694 0.207347 0.978267i $$-0.433517\pi$$
0.207347 + 0.978267i $$0.433517\pi$$
$$402$$ 0 0
$$403$$ −5440.00 −0.672421
$$404$$ 0 0
$$405$$ 1458.00 0.178885
$$406$$ 0 0
$$407$$ 14328.0 1.74499
$$408$$ 0 0
$$409$$ 5398.00 0.652601 0.326301 0.945266i $$-0.394198\pi$$
0.326301 + 0.945266i $$0.394198\pi$$
$$410$$ 0 0
$$411$$ 7074.00 0.848990
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −3672.00 −0.434341
$$416$$ 0 0
$$417$$ 156.000 0.0183198
$$418$$ 0 0
$$419$$ 13092.0 1.52646 0.763229 0.646128i $$-0.223613\pi$$
0.763229 + 0.646128i $$0.223613\pi$$
$$420$$ 0 0
$$421$$ −322.000 −0.0372763 −0.0186381 0.999826i $$-0.505933\pi$$
−0.0186381 + 0.999826i $$0.505933\pi$$
$$422$$ 0 0
$$423$$ 2160.00 0.248281
$$424$$ 0 0
$$425$$ −8358.00 −0.953935
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −3672.00 −0.413254
$$430$$ 0 0
$$431$$ −2616.00 −0.292363 −0.146181 0.989258i $$-0.546698\pi$$
−0.146181 + 0.989258i $$0.546698\pi$$
$$432$$ 0 0
$$433$$ −4322.00 −0.479681 −0.239841 0.970812i $$-0.577095\pi$$
−0.239841 + 0.970812i $$0.577095\pi$$
$$434$$ 0 0
$$435$$ −5508.00 −0.607100
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −9016.00 −0.980205 −0.490103 0.871665i $$-0.663041\pi$$
−0.490103 + 0.871665i $$0.663041\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 5268.00 0.564989 0.282495 0.959269i $$-0.408838\pi$$
0.282495 + 0.959269i $$0.408838\pi$$
$$444$$ 0 0
$$445$$ −6372.00 −0.678790
$$446$$ 0 0
$$447$$ 5238.00 0.554248
$$448$$ 0 0
$$449$$ −5310.00 −0.558117 −0.279058 0.960274i $$-0.590022\pi$$
−0.279058 + 0.960274i $$0.590022\pi$$
$$450$$ 0 0
$$451$$ 11448.0 1.19527
$$452$$ 0 0
$$453$$ −696.000 −0.0721875
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 15770.0 1.61420 0.807100 0.590415i $$-0.201036\pi$$
0.807100 + 0.590415i $$0.201036\pi$$
$$458$$ 0 0
$$459$$ 1134.00 0.115317
$$460$$ 0 0
$$461$$ 5370.00 0.542529 0.271264 0.962505i $$-0.412558\pi$$
0.271264 + 0.962505i $$0.412558\pi$$
$$462$$ 0 0
$$463$$ 3328.00 0.334050 0.167025 0.985953i $$-0.446584\pi$$
0.167025 + 0.985953i $$0.446584\pi$$
$$464$$ 0 0
$$465$$ 8640.00 0.861657
$$466$$ 0 0
$$467$$ 4548.00 0.450656 0.225328 0.974283i $$-0.427655\pi$$
0.225328 + 0.974283i $$0.427655\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 5082.00 0.497168
$$472$$ 0 0
$$473$$ 9648.00 0.937876
$$474$$ 0 0
$$475$$ −24676.0 −2.38361
$$476$$ 0 0
$$477$$ −4482.00 −0.430224
$$478$$ 0 0
$$479$$ −8064.00 −0.769214 −0.384607 0.923080i $$-0.625663\pi$$
−0.384607 + 0.923080i $$0.625663\pi$$
$$480$$ 0 0
$$481$$ 13532.0 1.28276
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 5148.00 0.481977
$$486$$ 0 0
$$487$$ −16616.0 −1.54608 −0.773042 0.634355i $$-0.781266\pi$$
−0.773042 + 0.634355i $$0.781266\pi$$
$$488$$ 0 0
$$489$$ −8796.00 −0.813433
$$490$$ 0 0
$$491$$ 7140.00 0.656260 0.328130 0.944633i $$-0.393582\pi$$
0.328130 + 0.944633i $$0.393582\pi$$
$$492$$ 0 0
$$493$$ −4284.00 −0.391362
$$494$$ 0 0
$$495$$ 5832.00 0.529553
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 9124.00 0.818530 0.409265 0.912416i $$-0.365785\pi$$
0.409265 + 0.912416i $$0.365785\pi$$
$$500$$ 0 0
$$501$$ −3528.00 −0.314610
$$502$$ 0 0
$$503$$ −6552.00 −0.580794 −0.290397 0.956906i $$-0.593787\pi$$
−0.290397 + 0.956906i $$0.593787\pi$$
$$504$$ 0 0
$$505$$ −7452.00 −0.656653
$$506$$ 0 0
$$507$$ 3123.00 0.273565
$$508$$ 0 0
$$509$$ −2790.00 −0.242956 −0.121478 0.992594i $$-0.538763\pi$$
−0.121478 + 0.992594i $$0.538763\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 3348.00 0.288144
$$514$$ 0 0
$$515$$ 1008.00 0.0862481
$$516$$ 0 0
$$517$$ 8640.00 0.734984
$$518$$ 0 0
$$519$$ 2610.00 0.220744
$$520$$ 0 0
$$521$$ 14862.0 1.24974 0.624871 0.780728i $$-0.285151\pi$$
0.624871 + 0.780728i $$0.285151\pi$$
$$522$$ 0 0
$$523$$ 17660.0 1.47652 0.738258 0.674518i $$-0.235649\pi$$
0.738258 + 0.674518i $$0.235649\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6720.00 0.555461
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ −1188.00 −0.0970900
$$532$$ 0 0
$$533$$ 10812.0 0.878649
$$534$$ 0 0
$$535$$ −216.000 −0.0174551
$$536$$ 0 0
$$537$$ −6948.00 −0.558340
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −19834.0 −1.57621 −0.788106 0.615540i $$-0.788938\pi$$
−0.788106 + 0.615540i $$0.788938\pi$$
$$542$$ 0 0
$$543$$ −318.000 −0.0251320
$$544$$ 0 0
$$545$$ 26604.0 2.09099
$$546$$ 0 0
$$547$$ −20972.0 −1.63930 −0.819651 0.572863i $$-0.805833\pi$$
−0.819651 + 0.572863i $$0.805833\pi$$
$$548$$ 0 0
$$549$$ −3582.00 −0.278463
$$550$$ 0 0
$$551$$ −12648.0 −0.977900
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −21492.0 −1.64376
$$556$$ 0 0
$$557$$ 21174.0 1.61072 0.805360 0.592786i $$-0.201972\pi$$
0.805360 + 0.592786i $$0.201972\pi$$
$$558$$ 0 0
$$559$$ 9112.00 0.689439
$$560$$ 0 0
$$561$$ 4536.00 0.341373
$$562$$ 0 0
$$563$$ −17772.0 −1.33037 −0.665187 0.746677i $$-0.731648\pi$$
−0.665187 + 0.746677i $$0.731648\pi$$
$$564$$ 0 0
$$565$$ 7236.00 0.538798
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 8250.00 0.607835 0.303917 0.952698i $$-0.401705\pi$$
0.303917 + 0.952698i $$0.401705\pi$$
$$570$$ 0 0
$$571$$ −20756.0 −1.52121 −0.760606 0.649214i $$-0.775098\pi$$
−0.760606 + 0.649214i $$0.775098\pi$$
$$572$$ 0 0
$$573$$ −3384.00 −0.246717
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2.00000 −0.000144300 0 −7.21500e−5 1.00000i $$-0.500023\pi$$
−7.21500e−5 1.00000i $$0.500023\pi$$
$$578$$ 0 0
$$579$$ −12102.0 −0.868639
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −17928.0 −1.27359
$$584$$ 0 0
$$585$$ 5508.00 0.389278
$$586$$ 0 0
$$587$$ 26364.0 1.85376 0.926881 0.375354i $$-0.122479\pi$$
0.926881 + 0.375354i $$0.122479\pi$$
$$588$$ 0 0
$$589$$ 19840.0 1.38793
$$590$$ 0 0
$$591$$ 3942.00 0.274369
$$592$$ 0 0
$$593$$ −2298.00 −0.159136 −0.0795679 0.996829i $$-0.525354\pi$$
−0.0795679 + 0.996829i $$0.525354\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −15288.0 −1.04807
$$598$$ 0 0
$$599$$ −3072.00 −0.209547 −0.104773 0.994496i $$-0.533412\pi$$
−0.104773 + 0.994496i $$0.533412\pi$$
$$600$$ 0 0
$$601$$ −24554.0 −1.66652 −0.833260 0.552881i $$-0.813528\pi$$
−0.833260 + 0.552881i $$0.813528\pi$$
$$602$$ 0 0
$$603$$ −828.000 −0.0559184
$$604$$ 0 0
$$605$$ −630.000 −0.0423358
$$606$$ 0 0
$$607$$ 16832.0 1.12552 0.562759 0.826621i $$-0.309740\pi$$
0.562759 + 0.826621i $$0.309740\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8160.00 0.540292
$$612$$ 0 0
$$613$$ −2482.00 −0.163535 −0.0817676 0.996651i $$-0.526057\pi$$
−0.0817676 + 0.996651i $$0.526057\pi$$
$$614$$ 0 0
$$615$$ −17172.0 −1.12592
$$616$$ 0 0
$$617$$ −15798.0 −1.03080 −0.515400 0.856950i $$-0.672357\pi$$
−0.515400 + 0.856950i $$0.672357\pi$$
$$618$$ 0 0
$$619$$ −15460.0 −1.00386 −0.501930 0.864908i $$-0.667377\pi$$
−0.501930 + 0.864908i $$0.667377\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −899.000 −0.0575360
$$626$$ 0 0
$$627$$ 13392.0 0.852990
$$628$$ 0 0
$$629$$ −16716.0 −1.05964
$$630$$ 0 0
$$631$$ 7720.00 0.487050 0.243525 0.969895i $$-0.421696\pi$$
0.243525 + 0.969895i $$0.421696\pi$$
$$632$$ 0 0
$$633$$ −9228.00 −0.579431
$$634$$ 0 0
$$635$$ −23040.0 −1.43987
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 6480.00 0.401166
$$640$$ 0 0
$$641$$ −17262.0 −1.06366 −0.531832 0.846850i $$-0.678496\pi$$
−0.531832 + 0.846850i $$0.678496\pi$$
$$642$$ 0 0
$$643$$ −12220.0 −0.749471 −0.374735 0.927132i $$-0.622266\pi$$
−0.374735 + 0.927132i $$0.622266\pi$$
$$644$$ 0 0
$$645$$ −14472.0 −0.883464
$$646$$ 0 0
$$647$$ 13560.0 0.823955 0.411977 0.911194i $$-0.364838\pi$$
0.411977 + 0.911194i $$0.364838\pi$$
$$648$$ 0 0
$$649$$ −4752.00 −0.287415
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 23094.0 1.38398 0.691989 0.721908i $$-0.256735\pi$$
0.691989 + 0.721908i $$0.256735\pi$$
$$654$$ 0 0
$$655$$ 31752.0 1.89413
$$656$$ 0 0
$$657$$ 4518.00 0.268286
$$658$$ 0 0
$$659$$ −22548.0 −1.33285 −0.666423 0.745574i $$-0.732175\pi$$
−0.666423 + 0.745574i $$0.732175\pi$$
$$660$$ 0 0
$$661$$ −17462.0 −1.02752 −0.513762 0.857933i $$-0.671748\pi$$
−0.513762 + 0.857933i $$0.671748\pi$$
$$662$$ 0 0
$$663$$ 4284.00 0.250945
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 5664.00 0.327329
$$670$$ 0 0
$$671$$ −14328.0 −0.824331
$$672$$ 0 0
$$673$$ −22462.0 −1.28655 −0.643274 0.765636i $$-0.722424\pi$$
−0.643274 + 0.765636i $$0.722424\pi$$
$$674$$ 0 0
$$675$$ −5373.00 −0.306381
$$676$$ 0 0
$$677$$ 25554.0 1.45069 0.725347 0.688383i $$-0.241679\pi$$
0.725347 + 0.688383i $$0.241679\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 14148.0 0.796112
$$682$$ 0 0
$$683$$ −9276.00 −0.519672 −0.259836 0.965653i $$-0.583669\pi$$
−0.259836 + 0.965653i $$0.583669\pi$$
$$684$$ 0 0
$$685$$ −42444.0 −2.36745
$$686$$ 0 0
$$687$$ −5070.00 −0.281561
$$688$$ 0 0
$$689$$ −16932.0 −0.936223
$$690$$ 0 0
$$691$$ 27380.0 1.50736 0.753679 0.657243i $$-0.228277\pi$$
0.753679 + 0.657243i $$0.228277\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −936.000 −0.0510856
$$696$$ 0 0
$$697$$ −13356.0 −0.725817
$$698$$ 0 0
$$699$$ −414.000 −0.0224019
$$700$$ 0 0
$$701$$ 25830.0 1.39171 0.695853 0.718184i $$-0.255027\pi$$
0.695853 + 0.718184i $$0.255027\pi$$
$$702$$ 0 0
$$703$$ −49352.0 −2.64772
$$704$$ 0 0
$$705$$ −12960.0 −0.692343
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −6226.00 −0.329792 −0.164896 0.986311i $$-0.552729\pi$$
−0.164896 + 0.986311i $$0.552729\pi$$
$$710$$ 0 0
$$711$$ 9216.00 0.486114
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 22032.0 1.15238
$$716$$ 0 0
$$717$$ 5688.00 0.296265
$$718$$ 0 0
$$719$$ −15072.0 −0.781767 −0.390884 0.920440i $$-0.627831\pi$$
−0.390884 + 0.920440i $$0.627831\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −10794.0 −0.555233
$$724$$ 0 0
$$725$$ 20298.0 1.03979
$$726$$ 0 0
$$727$$ −32920.0 −1.67942 −0.839708 0.543038i $$-0.817274\pi$$
−0.839708 + 0.543038i $$0.817274\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −11256.0 −0.569519
$$732$$ 0 0
$$733$$ 6946.00 0.350009 0.175004 0.984568i $$-0.444006\pi$$
0.175004 + 0.984568i $$0.444006\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −3312.00 −0.165535
$$738$$ 0 0
$$739$$ 2356.00 0.117276 0.0586379 0.998279i $$-0.481324\pi$$
0.0586379 + 0.998279i $$0.481324\pi$$
$$740$$ 0 0
$$741$$ 12648.0 0.627039
$$742$$ 0 0
$$743$$ 23520.0 1.16133 0.580663 0.814144i $$-0.302793\pi$$
0.580663 + 0.814144i $$0.302793\pi$$
$$744$$ 0 0
$$745$$ −31428.0 −1.54555
$$746$$ 0 0
$$747$$ −1836.00 −0.0899273
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −3008.00 −0.146156 −0.0730782 0.997326i $$-0.523282\pi$$
−0.0730782 + 0.997326i $$0.523282\pi$$
$$752$$ 0 0
$$753$$ 9180.00 0.444273
$$754$$ 0 0
$$755$$ 4176.00 0.201298
$$756$$ 0 0
$$757$$ −20770.0 −0.997224 −0.498612 0.866825i $$-0.666157\pi$$
−0.498612 + 0.866825i $$0.666157\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −11538.0 −0.549609 −0.274804 0.961500i $$-0.588613\pi$$
−0.274804 + 0.961500i $$0.588613\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −6804.00 −0.321568
$$766$$ 0 0
$$767$$ −4488.00 −0.211281
$$768$$ 0 0
$$769$$ −8498.00 −0.398499 −0.199249 0.979949i $$-0.563850\pi$$
−0.199249 + 0.979949i $$0.563850\pi$$
$$770$$ 0 0
$$771$$ −20466.0 −0.955986
$$772$$ 0 0
$$773$$ 32322.0 1.50393 0.751967 0.659200i $$-0.229105\pi$$
0.751967 + 0.659200i $$0.229105\pi$$
$$774$$ 0 0
$$775$$ −31840.0 −1.47578
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −39432.0 −1.81360
$$780$$ 0 0
$$781$$ 25920.0 1.18757
$$782$$ 0 0
$$783$$ −2754.00 −0.125696
$$784$$ 0 0
$$785$$ −30492.0 −1.38638
$$786$$ 0 0
$$787$$ 26228.0 1.18796 0.593982 0.804479i $$-0.297555\pi$$
0.593982 + 0.804479i $$0.297555\pi$$
$$788$$ 0 0
$$789$$ 7776.00 0.350866
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −13532.0 −0.605972
$$794$$ 0 0
$$795$$ 26892.0 1.19970
$$796$$ 0 0
$$797$$ 43338.0 1.92611 0.963056 0.269302i $$-0.0867931\pi$$
0.963056 + 0.269302i $$0.0867931\pi$$
$$798$$ 0 0
$$799$$ −10080.0 −0.446314
$$800$$ 0 0
$$801$$ −3186.00 −0.140539
$$802$$ 0 0
$$803$$ 18072.0 0.794206
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 24642.0 1.07489
$$808$$ 0 0
$$809$$ −28902.0 −1.25604 −0.628022 0.778195i $$-0.716135\pi$$
−0.628022 + 0.778195i $$0.716135\pi$$
$$810$$ 0 0
$$811$$ 27164.0 1.17615 0.588075 0.808807i $$-0.299886\pi$$
0.588075 + 0.808807i $$0.299886\pi$$
$$812$$ 0 0
$$813$$ 16032.0 0.691595
$$814$$ 0 0
$$815$$ 52776.0 2.26830
$$816$$ 0 0
$$817$$ −33232.0 −1.42306
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −17202.0 −0.731247 −0.365624 0.930763i $$-0.619144\pi$$
−0.365624 + 0.930763i $$0.619144\pi$$
$$822$$ 0 0
$$823$$ 5992.00 0.253789 0.126894 0.991916i $$-0.459499\pi$$
0.126894 + 0.991916i $$0.459499\pi$$
$$824$$ 0 0
$$825$$ −21492.0 −0.906976
$$826$$ 0 0
$$827$$ −25884.0 −1.08836 −0.544181 0.838968i $$-0.683159\pi$$
−0.544181 + 0.838968i $$0.683159\pi$$
$$828$$ 0 0
$$829$$ 1474.00 0.0617541 0.0308770 0.999523i $$-0.490170\pi$$
0.0308770 + 0.999523i $$0.490170\pi$$
$$830$$ 0 0
$$831$$ 19542.0 0.815770
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 21168.0 0.877304
$$836$$ 0 0
$$837$$ 4320.00 0.178400
$$838$$ 0 0
$$839$$ 33528.0 1.37964 0.689818 0.723983i $$-0.257690\pi$$
0.689818 + 0.723983i $$0.257690\pi$$
$$840$$ 0 0
$$841$$ −13985.0 −0.573414
$$842$$ 0 0
$$843$$ −19854.0 −0.811160
$$844$$ 0 0
$$845$$ −18738.0 −0.762848
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9780.00 −0.395346
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −1190.00 −0.0477665 −0.0238832 0.999715i $$-0.507603\pi$$
−0.0238832 + 0.999715i $$0.507603\pi$$
$$854$$ 0 0
$$855$$ −20088.0 −0.803503
$$856$$ 0 0
$$857$$ −34578.0 −1.37825 −0.689126 0.724642i $$-0.742005\pi$$
−0.689126 + 0.724642i $$0.742005\pi$$
$$858$$ 0 0
$$859$$ −44404.0 −1.76373 −0.881865 0.471501i $$-0.843712\pi$$
−0.881865 + 0.471501i $$0.843712\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 38328.0 1.51182 0.755910 0.654676i $$-0.227195\pi$$
0.755910 + 0.654676i $$0.227195\pi$$
$$864$$ 0 0
$$865$$ −15660.0 −0.615556
$$866$$ 0 0
$$867$$ 9447.00 0.370054
$$868$$ 0 0
$$869$$ 36864.0 1.43904
$$870$$ 0 0
$$871$$ −3128.00 −0.121686
$$872$$ 0 0
$$873$$ 2574.00 0.0997900
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −38842.0 −1.49555 −0.747777 0.663950i $$-0.768879\pi$$
−0.747777 + 0.663950i $$0.768879\pi$$
$$878$$ 0 0
$$879$$ 15354.0 0.589167
$$880$$ 0 0
$$881$$ 35046.0 1.34022 0.670108 0.742264i $$-0.266248\pi$$
0.670108 + 0.742264i $$0.266248\pi$$
$$882$$ 0 0
$$883$$ −14204.0 −0.541339 −0.270670 0.962672i $$-0.587245\pi$$
−0.270670 + 0.962672i $$0.587245\pi$$
$$884$$ 0 0
$$885$$ 7128.00 0.270740
$$886$$ 0 0
$$887$$ −26136.0 −0.989359 −0.494679 0.869076i $$-0.664714\pi$$
−0.494679 + 0.869076i $$0.664714\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 2916.00 0.109640
$$892$$ 0 0
$$893$$ −29760.0 −1.11521
$$894$$ 0 0
$$895$$ 41688.0 1.55696
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −16320.0 −0.605453
$$900$$ 0 0
$$901$$ 20916.0 0.773377
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 1908.00 0.0700818
$$906$$ 0 0
$$907$$ 9052.00 0.331386 0.165693 0.986177i $$-0.447014\pi$$
0.165693 + 0.986177i $$0.447014\pi$$
$$908$$ 0 0
$$909$$ −3726.00 −0.135956
$$910$$ 0 0
$$911$$ −5016.00 −0.182423 −0.0912116 0.995832i $$-0.529074\pi$$
−0.0912116 + 0.995832i $$0.529074\pi$$
$$912$$ 0 0
$$913$$ −7344.00 −0.266211
$$914$$ 0 0
$$915$$ 21492.0 0.776507
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −44552.0 −1.59917 −0.799584 0.600555i $$-0.794946\pi$$
−0.799584 + 0.600555i $$0.794946\pi$$
$$920$$ 0 0
$$921$$ −1356.00 −0.0485144
$$922$$ 0 0
$$923$$ 24480.0 0.872989
$$924$$ 0 0
$$925$$ 79202.0 2.81529
$$926$$ 0 0
$$927$$ 504.000 0.0178571
$$928$$ 0 0
$$929$$ −24234.0 −0.855858 −0.427929 0.903812i $$-0.640757\pi$$
−0.427929 + 0.903812i $$0.640757\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −15048.0 −0.528027
$$934$$ 0 0
$$935$$ −27216.0 −0.951934
$$936$$ 0 0
$$937$$ 13894.0 0.484415 0.242208 0.970224i $$-0.422128\pi$$
0.242208 + 0.970224i $$0.422128\pi$$
$$938$$ 0 0
$$939$$ 16206.0 0.563219
$$940$$ 0 0
$$941$$ −46758.0 −1.61984 −0.809919 0.586542i $$-0.800489\pi$$
−0.809919 + 0.586542i $$0.800489\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −13812.0 −0.473949 −0.236974 0.971516i $$-0.576156\pi$$
−0.236974 + 0.971516i $$0.576156\pi$$
$$948$$ 0 0
$$949$$ 17068.0 0.583826
$$950$$ 0 0
$$951$$ −30258.0 −1.03174
$$952$$ 0 0
$$953$$ −58518.0 −1.98907 −0.994535 0.104402i $$-0.966707\pi$$
−0.994535 + 0.104402i $$0.966707\pi$$
$$954$$ 0 0
$$955$$ 20304.0 0.687981
$$956$$ 0 0
$$957$$ −11016.0 −0.372097
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −4191.00 −0.140680
$$962$$ 0 0
$$963$$ −108.000 −0.00361397
$$964$$ 0 0
$$965$$ 72612.0 2.42224
$$966$$ 0 0
$$967$$ −19640.0 −0.653133 −0.326567 0.945174i $$-0.605892\pi$$
−0.326567 + 0.945174i $$0.605892\pi$$
$$968$$ 0 0
$$969$$ −15624.0 −0.517972
$$970$$ 0 0
$$971$$ −58308.0 −1.92708 −0.963539 0.267568i $$-0.913780\pi$$
−0.963539 + 0.267568i $$0.913780\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −20298.0 −0.666724
$$976$$ 0 0
$$977$$ −23550.0 −0.771168 −0.385584 0.922673i $$-0.626000\pi$$
−0.385584 + 0.922673i $$0.626000\pi$$
$$978$$ 0 0
$$979$$ −12744.0 −0.416037
$$980$$ 0 0
$$981$$ 13302.0 0.432926
$$982$$ 0 0
$$983$$ 15768.0 0.511619 0.255809 0.966727i $$-0.417658\pi$$
0.255809 + 0.966727i $$0.417658\pi$$
$$984$$ 0 0
$$985$$ −23652.0 −0.765092
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −35264.0 −1.13037 −0.565186 0.824964i $$-0.691195\pi$$
−0.565186 + 0.824964i $$0.691195\pi$$
$$992$$ 0 0
$$993$$ −24132.0 −0.771204
$$994$$ 0 0
$$995$$ 91728.0 2.92259
$$996$$ 0 0
$$997$$ 29338.0 0.931940 0.465970 0.884801i $$-0.345706\pi$$
0.465970 + 0.884801i $$0.345706\pi$$
$$998$$ 0 0
$$999$$ −10746.0 −0.340329
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 2352.4.a.r.1.1 1
4.3 odd 2 147.4.a.c.1.1 1
7.6 odd 2 336.4.a.f.1.1 1
12.11 even 2 441.4.a.j.1.1 1
21.20 even 2 1008.4.a.v.1.1 1
28.3 even 6 147.4.e.i.79.1 2
28.11 odd 6 147.4.e.g.79.1 2
28.19 even 6 147.4.e.i.67.1 2
28.23 odd 6 147.4.e.g.67.1 2
28.27 even 2 21.4.a.a.1.1 1
56.13 odd 2 1344.4.a.n.1.1 1
56.27 even 2 1344.4.a.ba.1.1 1
84.11 even 6 441.4.e.d.226.1 2
84.23 even 6 441.4.e.d.361.1 2
84.47 odd 6 441.4.e.b.361.1 2
84.59 odd 6 441.4.e.b.226.1 2
84.83 odd 2 63.4.a.c.1.1 1
140.27 odd 4 525.4.d.c.274.1 2
140.83 odd 4 525.4.d.c.274.2 2
140.139 even 2 525.4.a.g.1.1 1
420.419 odd 2 1575.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 28.27 even 2
63.4.a.c.1.1 1 84.83 odd 2
147.4.a.c.1.1 1 4.3 odd 2
147.4.e.g.67.1 2 28.23 odd 6
147.4.e.g.79.1 2 28.11 odd 6
147.4.e.i.67.1 2 28.19 even 6
147.4.e.i.79.1 2 28.3 even 6
336.4.a.f.1.1 1 7.6 odd 2
441.4.a.j.1.1 1 12.11 even 2
441.4.e.b.226.1 2 84.59 odd 6
441.4.e.b.361.1 2 84.47 odd 6
441.4.e.d.226.1 2 84.11 even 6
441.4.e.d.361.1 2 84.23 even 6
525.4.a.g.1.1 1 140.139 even 2
525.4.d.c.274.1 2 140.27 odd 4
525.4.d.c.274.2 2 140.83 odd 4
1008.4.a.v.1.1 1 21.20 even 2
1344.4.a.n.1.1 1 56.13 odd 2
1344.4.a.ba.1.1 1 56.27 even 2
1575.4.a.b.1.1 1 420.419 odd 2
2352.4.a.r.1.1 1 1.1 even 1 trivial