# Properties

 Label 336.4.a.f.1.1 Level $336$ Weight $4$ Character 336.1 Self dual yes Analytic conductor $19.825$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.8246417619$$ 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 $$\chi$$ $$=$$ 336.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -18.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -18.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +36.0000 q^{11} -34.0000 q^{13} -54.0000 q^{15} +42.0000 q^{17} +124.000 q^{19} -21.0000 q^{21} +199.000 q^{25} +27.0000 q^{27} +102.000 q^{29} +160.000 q^{31} +108.000 q^{33} +126.000 q^{35} +398.000 q^{37} -102.000 q^{39} -318.000 q^{41} +268.000 q^{43} -162.000 q^{45} -240.000 q^{47} +49.0000 q^{49} +126.000 q^{51} -498.000 q^{53} -648.000 q^{55} +372.000 q^{57} +132.000 q^{59} +398.000 q^{61} -63.0000 q^{63} +612.000 q^{65} -92.0000 q^{67} +720.000 q^{71} -502.000 q^{73} +597.000 q^{75} -252.000 q^{77} +1024.00 q^{79} +81.0000 q^{81} +204.000 q^{83} -756.000 q^{85} +306.000 q^{87} +354.000 q^{89} +238.000 q^{91} +480.000 q^{93} -2232.00 q^{95} -286.000 q^{97} +324.000 q^{99} +O(q^{100})$$

## 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 $$-0.797826\pi$$
−0.804984 + 0.593296i $$0.797826\pi$$
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 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 $$-0.618141\pi$$
−0.362689 + 0.931910i $$0.618141\pi$$
$$14$$ 0 0
$$15$$ −54.0000 −0.929516
$$16$$ 0 0
$$17$$ 42.0000 0.599206 0.299603 0.954064i $$-0.403146\pi$$
0.299603 + 0.954064i $$0.403146\pi$$
$$18$$ 0 0
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ 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 $$-0.346594\pi$$
0.463498 + 0.886098i $$0.346594\pi$$
$$32$$ 0 0
$$33$$ 108.000 0.569709
$$34$$ 0 0
$$35$$ 126.000 0.608511
$$36$$ 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 $$-0.707087\pi$$
−0.605649 + 0.795732i $$0.707087\pi$$
$$42$$ 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 $$-0.621472\pi$$
−0.372421 + 0.928064i $$0.621472\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 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 $$-0.453477\pi$$
0.145635 + 0.989338i $$0.453477\pi$$
$$60$$ 0 0
$$61$$ 398.000 0.835388 0.417694 0.908588i $$-0.362838\pi$$
0.417694 + 0.908588i $$0.362838\pi$$
$$62$$ 0 0
$$63$$ −63.0000 −0.125988
$$64$$ 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 $$-0.631834\pi$$
−0.402429 + 0.915451i $$0.631834\pi$$
$$74$$ 0 0
$$75$$ 597.000 0.919142
$$76$$ 0 0
$$77$$ −252.000 −0.372962
$$78$$ 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 $$-0.456932\pi$$
0.134891 + 0.990860i $$0.456932\pi$$
$$84$$ 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 $$-0.432390\pi$$
0.210809 + 0.977527i $$0.432390\pi$$
$$90$$ 0 0
$$91$$ 238.000 0.274167
$$92$$ 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 $$-0.547826\pi$$
−0.149685 + 0.988734i $$0.547826\pi$$
$$98$$ 0 0
$$99$$ 324.000 0.328921
$$100$$ 0 0
$$101$$ 414.000 0.407867 0.203933 0.978985i $$-0.434627\pi$$
0.203933 + 0.978985i $$0.434627\pi$$
$$102$$ 0 0
$$103$$ −56.0000 −0.0535713 −0.0267857 0.999641i $$-0.508527\pi$$
−0.0267857 + 0.999641i $$0.508527\pi$$
$$104$$ 0 0
$$105$$ 378.000 0.351324
$$106$$ 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
$$119$$ −294.000 −0.226478
$$120$$ 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 $$-0.700183\pi$$
−0.588250 + 0.808679i $$0.700183\pi$$
$$132$$ 0 0
$$133$$ −868.000 −0.565903
$$134$$ 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 $$-0.494950\pi$$
0.0158654 + 0.999874i $$0.494950\pi$$
$$140$$ 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
$$147$$ 147.000 0.0824786
$$148$$ 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 $$-0.358316\pi$$
0.430560 + 0.902562i $$0.358316\pi$$
$$158$$ 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 $$-0.587837\pi$$
−0.272460 + 0.962167i $$0.587837\pi$$
$$168$$ 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 $$-0.438772\pi$$
0.191170 + 0.981557i $$0.438772\pi$$
$$174$$ 0 0
$$175$$ −1393.00 −0.601719
$$176$$ 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 $$-0.506929\pi$$
−0.0217650 + 0.999763i $$0.506929\pi$$
$$182$$ 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
$$189$$ −189.000 −0.0727393
$$190$$ 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 $$-0.862128\pi$$
−0.907653 + 0.419722i $$0.862128\pi$$
$$200$$ 0 0
$$201$$ −276.000 −0.0968534
$$202$$ 0 0
$$203$$ −714.000 −0.246862
$$204$$ 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
$$217$$ −1120.00 −0.350371
$$218$$ 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 $$-0.408513\pi$$
0.283475 + 0.958980i $$0.408513\pi$$
$$224$$ 0 0
$$225$$ 1791.00 0.530667
$$226$$ 0 0
$$227$$ 4716.00 1.37891 0.689454 0.724330i $$-0.257851\pi$$
0.689454 + 0.724330i $$0.257851\pi$$
$$228$$ 0 0
$$229$$ −1690.00 −0.487678 −0.243839 0.969816i $$-0.578407\pi$$
−0.243839 + 0.969816i $$0.578407\pi$$
$$230$$ 0 0
$$231$$ −756.000 −0.215330
$$232$$ 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 $$-0.659670\pi$$
−0.480846 + 0.876805i $$0.659670\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −882.000 −0.229996
$$246$$ 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 $$-0.374287\pi$$
0.384752 + 0.923020i $$0.374287\pi$$
$$252$$ 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 $$-0.810469\pi$$
−0.827908 + 0.560864i $$0.810469\pi$$
$$258$$ 0 0
$$259$$ −2786.00 −0.668392
$$260$$ 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 $$-0.119037\pi$$
0.930886 + 0.365311i $$0.119037\pi$$
$$270$$ 0 0
$$271$$ 5344.00 1.19788 0.598939 0.800795i $$-0.295589\pi$$
0.598939 + 0.800795i $$0.295589\pi$$
$$272$$ 0 0
$$273$$ 714.000 0.158290
$$274$$ 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 $$-0.611233\pi$$
−0.342380 + 0.939562i $$0.611233\pi$$
$$284$$ 0 0
$$285$$ −6696.00 −1.39171
$$286$$ 0 0
$$287$$ 2226.00 0.457828
$$288$$ 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 $$-0.329559\pi$$
0.510233 + 0.860036i $$0.329559\pi$$
$$294$$ 0 0
$$295$$ −2376.00 −0.468936
$$296$$ 0 0
$$297$$ 972.000 0.189903
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1876.00 −0.359239
$$302$$ 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 $$-0.513378\pi$$
−0.0420147 + 0.999117i $$0.513378\pi$$
$$308$$ 0 0
$$309$$ −168.000 −0.0309294
$$310$$ 0 0
$$311$$ −5016.00 −0.914570 −0.457285 0.889320i $$-0.651178\pi$$
−0.457285 + 0.889320i $$0.651178\pi$$
$$312$$ 0 0
$$313$$ 5402.00 0.975524 0.487762 0.872977i $$-0.337813\pi$$
0.487762 + 0.872977i $$0.337813\pi$$
$$314$$ 0 0
$$315$$ 1134.00 0.202837
$$316$$ 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
$$329$$ 1680.00 0.281524
$$330$$ 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
$$343$$ −343.000 −0.0539949
$$344$$ 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 $$-0.901311\pi$$
−0.952321 + 0.305097i $$0.901311\pi$$
$$350$$ 0 0
$$351$$ −918.000 −0.139599
$$352$$ 0 0
$$353$$ −7830.00 −1.18059 −0.590296 0.807187i $$-0.700989\pi$$
−0.590296 + 0.807187i $$0.700989\pi$$
$$354$$ 0 0
$$355$$ −12960.0 −1.93759
$$356$$ 0 0
$$357$$ −882.000 −0.130757
$$358$$ 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 $$-0.413836\pi$$
0.267398 + 0.963586i $$0.413836\pi$$
$$368$$ 0 0
$$369$$ −2862.00 −0.403766
$$370$$ 0 0
$$371$$ 3486.00 0.487828
$$372$$ 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 $$-0.546025\pi$$
−0.144087 + 0.989565i $$0.546025\pi$$
$$384$$ 0 0
$$385$$ 4536.00 0.600457
$$386$$ 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 $$-0.635082\pi$$
−0.411748 + 0.911298i $$0.635082\pi$$
$$398$$ 0 0
$$399$$ −2604.00 −0.326724
$$400$$ 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 $$-0.605802\pi$$
−0.326301 + 0.945266i $$0.605802\pi$$
$$410$$ 0 0
$$411$$ −7074.00 −0.848990
$$412$$ 0 0
$$413$$ −924.000 −0.110090
$$414$$ 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 $$-0.776387\pi$$
−0.763229 + 0.646128i $$0.776387\pi$$
$$420$$ 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
$$427$$ −2786.00 −0.315747
$$428$$ 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 $$-0.422905\pi$$
0.239841 + 0.970812i $$0.422905\pi$$
$$434$$ 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 $$-0.336959\pi$$
0.490103 + 0.871665i $$0.336959\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 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
$$455$$ −4284.00 −0.441400
$$456$$ 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 $$-0.587442\pi$$
−0.271264 + 0.962505i $$0.587442\pi$$
$$462$$ 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 $$-0.572345\pi$$
−0.225328 + 0.974283i $$0.572345\pi$$
$$468$$ 0 0
$$469$$ 644.000 0.0634055
$$470$$ 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 $$-0.374337\pi$$
0.384607 + 0.923080i $$0.374337\pi$$
$$480$$ 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
$$497$$ −5040.00 −0.454879
$$498$$ 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 $$-0.406213\pi$$
0.290397 + 0.956906i $$0.406213\pi$$
$$504$$ 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 $$-0.461237\pi$$
0.121478 + 0.992594i $$0.461237\pi$$
$$510$$ 0 0
$$511$$ 3514.00 0.304208
$$512$$ 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 $$-0.714849\pi$$
−0.624871 + 0.780728i $$0.714849\pi$$
$$522$$ 0 0
$$523$$ −17660.0 −1.47652 −0.738258 0.674518i $$-0.764351\pi$$
−0.738258 + 0.674518i $$0.764351\pi$$
$$524$$ 0 0
$$525$$ −4179.00 −0.347403
$$526$$ 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
$$539$$ 1764.00 0.140966
$$540$$ 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
$$553$$ −7168.00 −0.551201
$$554$$ 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 $$-0.268352\pi$$
0.665187 + 0.746677i $$0.268352\pi$$
$$564$$ 0 0
$$565$$ −7236.00 −0.538798
$$566$$ 0 0
$$567$$ −567.000 −0.0419961
$$568$$ 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 $$-0.499977\pi$$
7.21500e−5 1.00000i $$0.499977\pi$$
$$578$$ 0 0
$$579$$ 12102.0 0.868639
$$580$$ 0 0
$$581$$ −1428.00 −0.101968
$$582$$ 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 $$-0.877521\pi$$
−0.926881 + 0.375354i $$0.877521\pi$$
$$588$$ 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 $$-0.474646\pi$$
0.0795679 + 0.996829i $$0.474646\pi$$
$$594$$ 0 0
$$595$$ 5292.00 0.364623
$$596$$ 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 $$-0.186472\pi$$
0.833260 + 0.552881i $$0.186472\pi$$
$$602$$ 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 $$-0.690260\pi$$
−0.562759 + 0.826621i $$0.690260\pi$$
$$608$$ 0 0
$$609$$ −2142.00 −0.142526
$$610$$ 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 $$-0.332623\pi$$
0.501930 + 0.864908i $$0.332623\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −2478.00 −0.159356
$$624$$ 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
$$637$$ −1666.00 −0.103625
$$638$$ 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 $$-0.377734\pi$$
0.374735 + 0.927132i $$0.377734\pi$$
$$644$$ 0 0
$$645$$ −14472.0 −0.883464
$$646$$ 0 0
$$647$$ −13560.0 −0.823955 −0.411977 0.911194i $$-0.635162\pi$$
−0.411977 + 0.911194i $$0.635162\pi$$
$$648$$ 0 0
$$649$$ 4752.00 0.287415
$$650$$ 0 0
$$651$$ −3360.00 −0.202287
$$652$$ 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 $$-0.328252\pi$$
0.513762 + 0.857933i $$0.328252\pi$$
$$662$$ 0 0
$$663$$ −4284.00 −0.250945
$$664$$ 0 0
$$665$$ 15624.0 0.911087
$$666$$ 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 $$-0.758321\pi$$
−0.725347 + 0.688383i $$0.758321\pi$$
$$678$$ 0 0
$$679$$ 2002.00 0.113151
$$680$$ 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 $$-0.771723\pi$$
−0.753679 + 0.657243i $$0.771723\pi$$
$$692$$ 0 0
$$693$$ −2268.00 −0.124321
$$694$$ 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
$$707$$ −2898.00 −0.154159
$$708$$ 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 $$-0.372169\pi$$
0.390884 + 0.920440i $$0.372169\pi$$
$$720$$ 0 0
$$721$$ 392.000 0.0202480
$$722$$ 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 $$-0.182726\pi$$
0.839708 + 0.543038i $$0.182726\pi$$
$$728$$ 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 $$-0.555994\pi$$
−0.175004 + 0.984568i $$0.555994\pi$$
$$734$$ 0 0
$$735$$ −2646.00 −0.132788
$$736$$ 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
$$749$$ 84.0000 0.00409785
$$750$$ 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 $$-0.411387\pi$$
0.274804 + 0.961500i $$0.411387\pi$$
$$762$$ 0 0
$$763$$ −10346.0 −0.490892
$$764$$ 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 $$-0.436150\pi$$
0.199249 + 0.979949i $$0.436150\pi$$
$$770$$ 0 0
$$771$$ −20466.0 −0.955986
$$772$$ 0 0
$$773$$ −32322.0 −1.50393 −0.751967 0.659200i $$-0.770895\pi$$
−0.751967 + 0.659200i $$0.770895\pi$$
$$774$$ 0 0
$$775$$ 31840.0 1.47578
$$776$$ 0 0
$$777$$ −8358.00 −0.385896
$$778$$ 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 $$-0.702445\pi$$
−0.593982 + 0.804479i $$0.702445\pi$$
$$788$$ 0 0
$$789$$ −7776.00 −0.350866
$$790$$ 0 0
$$791$$ −2814.00 −0.126491
$$792$$ 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 $$-0.913207\pi$$
−0.963056 + 0.269302i $$0.913207\pi$$
$$798$$ 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 $$-0.700114\pi$$
−0.588075 + 0.808807i $$0.700114\pi$$
$$812$$ 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
$$819$$ 2142.00 0.0913889
$$820$$ 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 $$-0.509830\pi$$
−0.0308770 + 0.999523i $$0.509830\pi$$
$$830$$ 0 0
$$831$$ −19542.0 −0.815770
$$832$$ 0 0
$$833$$ 2058.00 0.0856008
$$834$$ 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 $$-0.742310\pi$$
−0.689818 + 0.723983i $$0.742310\pi$$
$$840$$ 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
$$847$$ 245.000 0.00993896
$$848$$ 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 $$-0.492397\pi$$
0.0238832 + 0.999715i $$0.492397\pi$$
$$854$$ 0 0
$$855$$ −20088.0 −0.803503
$$856$$ 0 0
$$857$$ 34578.0 1.37825 0.689126 0.724642i $$-0.257995\pi$$
0.689126 + 0.724642i $$0.257995\pi$$
$$858$$ 0 0
$$859$$ 44404.0 1.76373 0.881865 0.471501i $$-0.156288\pi$$
0.881865 + 0.471501i $$0.156288\pi$$
$$860$$ 0 0
$$861$$ 6678.00 0.264327
$$862$$ 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
$$875$$ 9324.00 0.360239
$$876$$ 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 $$-0.733752\pi$$
−0.670108 + 0.742264i $$0.733752\pi$$
$$882$$ 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 $$-0.335286\pi$$
0.494679 + 0.869076i $$0.335286\pi$$
$$888$$ 0 0
$$889$$ 8960.00 0.338030
$$890$$ 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
$$903$$ −5628.00 −0.207407
$$904$$ 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
$$917$$ 12348.0 0.444675
$$918$$ 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 $$-0.359243\pi$$
0.427929 + 0.903812i $$0.359243\pi$$
$$930$$ 0 0
$$931$$ 6076.00 0.213891
$$932$$ 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 $$-0.577872\pi$$
−0.242208 + 0.970224i $$0.577872\pi$$
$$938$$ 0 0
$$939$$ 16206.0 0.563219
$$940$$ 0 0
$$941$$ 46758.0 1.61984 0.809919 0.586542i $$-0.199511\pi$$
0.809919 + 0.586542i $$0.199511\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 3402.00 0.117108
$$946$$ 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
$$959$$ 16506.0 0.555794
$$960$$ 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 $$-0.0862200\pi$$
0.963539 + 0.267568i $$0.0862200\pi$$
$$972$$ 0 0
$$973$$ −364.000 −0.0119931
$$974$$ 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 $$-0.582342\pi$$
−0.255809 + 0.966727i $$0.582342\pi$$
$$984$$ 0 0
$$985$$ 23652.0 0.765092
$$986$$ 0 0
$$987$$ 5040.00 0.162538
$$988$$ 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 $$-0.654294\pi$$
−0.465970 + 0.884801i $$0.654294\pi$$
$$998$$ 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 336.4.a.f.1.1 1
3.2 odd 2 1008.4.a.v.1.1 1
4.3 odd 2 21.4.a.a.1.1 1
7.6 odd 2 2352.4.a.r.1.1 1
8.3 odd 2 1344.4.a.ba.1.1 1
8.5 even 2 1344.4.a.n.1.1 1
12.11 even 2 63.4.a.c.1.1 1
20.3 even 4 525.4.d.c.274.2 2
20.7 even 4 525.4.d.c.274.1 2
20.19 odd 2 525.4.a.g.1.1 1
28.3 even 6 147.4.e.g.79.1 2
28.11 odd 6 147.4.e.i.79.1 2
28.19 even 6 147.4.e.g.67.1 2
28.23 odd 6 147.4.e.i.67.1 2
28.27 even 2 147.4.a.c.1.1 1
60.59 even 2 1575.4.a.b.1.1 1
84.11 even 6 441.4.e.b.226.1 2
84.23 even 6 441.4.e.b.361.1 2
84.47 odd 6 441.4.e.d.361.1 2
84.59 odd 6 441.4.e.d.226.1 2
84.83 odd 2 441.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 4.3 odd 2
63.4.a.c.1.1 1 12.11 even 2
147.4.a.c.1.1 1 28.27 even 2
147.4.e.g.67.1 2 28.19 even 6
147.4.e.g.79.1 2 28.3 even 6
147.4.e.i.67.1 2 28.23 odd 6
147.4.e.i.79.1 2 28.11 odd 6
336.4.a.f.1.1 1 1.1 even 1 trivial
441.4.a.j.1.1 1 84.83 odd 2
441.4.e.b.226.1 2 84.11 even 6
441.4.e.b.361.1 2 84.23 even 6
441.4.e.d.226.1 2 84.59 odd 6
441.4.e.d.361.1 2 84.47 odd 6
525.4.a.g.1.1 1 20.19 odd 2
525.4.d.c.274.1 2 20.7 even 4
525.4.d.c.274.2 2 20.3 even 4
1008.4.a.v.1.1 1 3.2 odd 2
1344.4.a.n.1.1 1 8.5 even 2
1344.4.a.ba.1.1 1 8.3 odd 2
1575.4.a.b.1.1 1 60.59 even 2
2352.4.a.r.1.1 1 7.6 odd 2