# Properties

 Label 2352.4.a.cf.1.2 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## 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: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +19.8995 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +19.8995 q^{5} +9.00000 q^{9} -23.9411 q^{11} +87.3553 q^{13} +59.6985 q^{15} -5.63961 q^{17} -64.8873 q^{19} +25.5980 q^{23} +270.990 q^{25} +27.0000 q^{27} +60.3188 q^{29} +122.711 q^{31} -71.8234 q^{33} -56.1177 q^{37} +262.066 q^{39} -299.713 q^{41} +501.421 q^{43} +179.095 q^{45} -305.553 q^{47} -16.9188 q^{51} -375.117 q^{53} -476.416 q^{55} -194.662 q^{57} +627.612 q^{59} +3.75736 q^{61} +1738.33 q^{65} +813.048 q^{67} +76.7939 q^{69} -165.902 q^{71} +619.100 q^{73} +812.970 q^{75} +138.246 q^{79} +81.0000 q^{81} -621.137 q^{83} -112.225 q^{85} +180.956 q^{87} +285.418 q^{89} +368.132 q^{93} -1291.22 q^{95} -603.114 q^{97} -215.470 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 20q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 20q^{5} + 18q^{9} + 20q^{11} + 104q^{13} + 60q^{15} + 116q^{17} - 192q^{19} - 28q^{23} + 146q^{25} + 54q^{27} + 296q^{29} + 104q^{31} + 60q^{33} - 248q^{37} + 312q^{39} + 20q^{41} + 720q^{43} + 180q^{45} + 96q^{47} + 348q^{51} + 268q^{53} - 472q^{55} - 576q^{57} + 616q^{59} + 16q^{61} + 1740q^{65} + 144q^{67} - 84q^{69} - 988q^{71} + 104q^{73} + 438q^{75} + 944q^{79} + 162q^{81} - 1016q^{83} - 100q^{85} + 888q^{87} - 388q^{89} + 312q^{93} - 1304q^{95} + 488q^{97} + 180q^{99} + O(q^{100})$$

## 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
$$5$$ 19.8995 1.77986 0.889932 0.456092i $$-0.150751\pi$$
0.889932 + 0.456092i $$0.150751\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −23.9411 −0.656229 −0.328115 0.944638i $$-0.606413\pi$$
−0.328115 + 0.944638i $$0.606413\pi$$
$$12$$ 0 0
$$13$$ 87.3553 1.86369 0.931847 0.362852i $$-0.118197\pi$$
0.931847 + 0.362852i $$0.118197\pi$$
$$14$$ 0 0
$$15$$ 59.6985 1.02761
$$16$$ 0 0
$$17$$ −5.63961 −0.0804592 −0.0402296 0.999190i $$-0.512809\pi$$
−0.0402296 + 0.999190i $$0.512809\pi$$
$$18$$ 0 0
$$19$$ −64.8873 −0.783483 −0.391741 0.920075i $$-0.628127\pi$$
−0.391741 + 0.920075i $$0.628127\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 25.5980 0.232067 0.116034 0.993245i $$-0.462982\pi$$
0.116034 + 0.993245i $$0.462982\pi$$
$$24$$ 0 0
$$25$$ 270.990 2.16792
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 60.3188 0.386238 0.193119 0.981175i $$-0.438140\pi$$
0.193119 + 0.981175i $$0.438140\pi$$
$$30$$ 0 0
$$31$$ 122.711 0.710951 0.355476 0.934686i $$-0.384319\pi$$
0.355476 + 0.934686i $$0.384319\pi$$
$$32$$ 0 0
$$33$$ −71.8234 −0.378874
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −56.1177 −0.249343 −0.124672 0.992198i $$-0.539788\pi$$
−0.124672 + 0.992198i $$0.539788\pi$$
$$38$$ 0 0
$$39$$ 262.066 1.07600
$$40$$ 0 0
$$41$$ −299.713 −1.14164 −0.570820 0.821075i $$-0.693375\pi$$
−0.570820 + 0.821075i $$0.693375\pi$$
$$42$$ 0 0
$$43$$ 501.421 1.77828 0.889140 0.457635i $$-0.151303\pi$$
0.889140 + 0.457635i $$0.151303\pi$$
$$44$$ 0 0
$$45$$ 179.095 0.593288
$$46$$ 0 0
$$47$$ −305.553 −0.948288 −0.474144 0.880447i $$-0.657242\pi$$
−0.474144 + 0.880447i $$0.657242\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −16.9188 −0.0464531
$$52$$ 0 0
$$53$$ −375.117 −0.972194 −0.486097 0.873905i $$-0.661580\pi$$
−0.486097 + 0.873905i $$0.661580\pi$$
$$54$$ 0 0
$$55$$ −476.416 −1.16800
$$56$$ 0 0
$$57$$ −194.662 −0.452344
$$58$$ 0 0
$$59$$ 627.612 1.38488 0.692442 0.721474i $$-0.256535\pi$$
0.692442 + 0.721474i $$0.256535\pi$$
$$60$$ 0 0
$$61$$ 3.75736 0.00788657 0.00394328 0.999992i $$-0.498745\pi$$
0.00394328 + 0.999992i $$0.498745\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1738.33 3.31712
$$66$$ 0 0
$$67$$ 813.048 1.48253 0.741266 0.671212i $$-0.234226\pi$$
0.741266 + 0.671212i $$0.234226\pi$$
$$68$$ 0 0
$$69$$ 76.7939 0.133984
$$70$$ 0 0
$$71$$ −165.902 −0.277310 −0.138655 0.990341i $$-0.544278\pi$$
−0.138655 + 0.990341i $$0.544278\pi$$
$$72$$ 0 0
$$73$$ 619.100 0.992605 0.496302 0.868150i $$-0.334691\pi$$
0.496302 + 0.868150i $$0.334691\pi$$
$$74$$ 0 0
$$75$$ 812.970 1.25165
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 138.246 0.196884 0.0984421 0.995143i $$-0.468614\pi$$
0.0984421 + 0.995143i $$0.468614\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −621.137 −0.821430 −0.410715 0.911764i $$-0.634721\pi$$
−0.410715 + 0.911764i $$0.634721\pi$$
$$84$$ 0 0
$$85$$ −112.225 −0.143207
$$86$$ 0 0
$$87$$ 180.956 0.222995
$$88$$ 0 0
$$89$$ 285.418 0.339936 0.169968 0.985450i $$-0.445634\pi$$
0.169968 + 0.985450i $$0.445634\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 368.132 0.410468
$$94$$ 0 0
$$95$$ −1291.22 −1.39449
$$96$$ 0 0
$$97$$ −603.114 −0.631309 −0.315654 0.948874i $$-0.602224\pi$$
−0.315654 + 0.948874i $$0.602224\pi$$
$$98$$ 0 0
$$99$$ −215.470 −0.218743
$$100$$ 0 0
$$101$$ 457.209 0.450436 0.225218 0.974308i $$-0.427691\pi$$
0.225218 + 0.974308i $$0.427691\pi$$
$$102$$ 0 0
$$103$$ −786.045 −0.751954 −0.375977 0.926629i $$-0.622693\pi$$
−0.375977 + 0.926629i $$0.622693\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 196.461 0.177501 0.0887504 0.996054i $$-0.471713\pi$$
0.0887504 + 0.996054i $$0.471713\pi$$
$$108$$ 0 0
$$109$$ −306.343 −0.269196 −0.134598 0.990900i $$-0.542974\pi$$
−0.134598 + 0.990900i $$0.542974\pi$$
$$110$$ 0 0
$$111$$ −168.353 −0.143958
$$112$$ 0 0
$$113$$ 1997.63 1.66302 0.831508 0.555512i $$-0.187478\pi$$
0.831508 + 0.555512i $$0.187478\pi$$
$$114$$ 0 0
$$115$$ 509.387 0.413048
$$116$$ 0 0
$$117$$ 786.198 0.621231
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −757.823 −0.569363
$$122$$ 0 0
$$123$$ −899.138 −0.659127
$$124$$ 0 0
$$125$$ 2905.13 2.07874
$$126$$ 0 0
$$127$$ 2311.40 1.61499 0.807494 0.589875i $$-0.200823\pi$$
0.807494 + 0.589875i $$0.200823\pi$$
$$128$$ 0 0
$$129$$ 1504.26 1.02669
$$130$$ 0 0
$$131$$ 155.018 0.103389 0.0516945 0.998663i $$-0.483538\pi$$
0.0516945 + 0.998663i $$0.483538\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 537.286 0.342535
$$136$$ 0 0
$$137$$ 516.936 0.322371 0.161186 0.986924i $$-0.448468\pi$$
0.161186 + 0.986924i $$0.448468\pi$$
$$138$$ 0 0
$$139$$ −958.067 −0.584620 −0.292310 0.956324i $$-0.594424\pi$$
−0.292310 + 0.956324i $$0.594424\pi$$
$$140$$ 0 0
$$141$$ −916.660 −0.547494
$$142$$ 0 0
$$143$$ −2091.39 −1.22301
$$144$$ 0 0
$$145$$ 1200.31 0.687452
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1770.63 −0.973526 −0.486763 0.873534i $$-0.661822\pi$$
−0.486763 + 0.873534i $$0.661822\pi$$
$$150$$ 0 0
$$151$$ 2540.24 1.36902 0.684508 0.729005i $$-0.260017\pi$$
0.684508 + 0.729005i $$0.260017\pi$$
$$152$$ 0 0
$$153$$ −50.7565 −0.0268197
$$154$$ 0 0
$$155$$ 2441.88 1.26540
$$156$$ 0 0
$$157$$ −1083.34 −0.550702 −0.275351 0.961344i $$-0.588794\pi$$
−0.275351 + 0.961344i $$0.588794\pi$$
$$158$$ 0 0
$$159$$ −1125.35 −0.561296
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −2968.72 −1.42655 −0.713277 0.700882i $$-0.752790\pi$$
−0.713277 + 0.700882i $$0.752790\pi$$
$$164$$ 0 0
$$165$$ −1429.25 −0.674345
$$166$$ 0 0
$$167$$ 2091.53 0.969149 0.484574 0.874750i $$-0.338975\pi$$
0.484574 + 0.874750i $$0.338975\pi$$
$$168$$ 0 0
$$169$$ 5433.96 2.47335
$$170$$ 0 0
$$171$$ −583.986 −0.261161
$$172$$ 0 0
$$173$$ −470.148 −0.206617 −0.103308 0.994649i $$-0.532943\pi$$
−0.103308 + 0.994649i $$0.532943\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1882.84 0.799563
$$178$$ 0 0
$$179$$ −1056.46 −0.441137 −0.220569 0.975371i $$-0.570791\pi$$
−0.220569 + 0.975371i $$0.570791\pi$$
$$180$$ 0 0
$$181$$ −406.470 −0.166921 −0.0834605 0.996511i $$-0.526597\pi$$
−0.0834605 + 0.996511i $$0.526597\pi$$
$$182$$ 0 0
$$183$$ 11.2721 0.00455331
$$184$$ 0 0
$$185$$ −1116.71 −0.443797
$$186$$ 0 0
$$187$$ 135.019 0.0527997
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −179.410 −0.0679666 −0.0339833 0.999422i $$-0.510819\pi$$
−0.0339833 + 0.999422i $$0.510819\pi$$
$$192$$ 0 0
$$193$$ −2388.37 −0.890769 −0.445385 0.895339i $$-0.646933\pi$$
−0.445385 + 0.895339i $$0.646933\pi$$
$$194$$ 0 0
$$195$$ 5214.98 1.91514
$$196$$ 0 0
$$197$$ −2665.35 −0.963952 −0.481976 0.876184i $$-0.660081\pi$$
−0.481976 + 0.876184i $$0.660081\pi$$
$$198$$ 0 0
$$199$$ 1342.31 0.478159 0.239079 0.971000i $$-0.423154\pi$$
0.239079 + 0.971000i $$0.423154\pi$$
$$200$$ 0 0
$$201$$ 2439.14 0.855940
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −5964.13 −2.03197
$$206$$ 0 0
$$207$$ 230.382 0.0773558
$$208$$ 0 0
$$209$$ 1553.48 0.514144
$$210$$ 0 0
$$211$$ −628.442 −0.205042 −0.102521 0.994731i $$-0.532691\pi$$
−0.102521 + 0.994731i $$0.532691\pi$$
$$212$$ 0 0
$$213$$ −497.707 −0.160105
$$214$$ 0 0
$$215$$ 9978.03 3.16510
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 1857.30 0.573081
$$220$$ 0 0
$$221$$ −492.650 −0.149951
$$222$$ 0 0
$$223$$ 969.970 0.291273 0.145637 0.989338i $$-0.453477\pi$$
0.145637 + 0.989338i $$0.453477\pi$$
$$224$$ 0 0
$$225$$ 2438.91 0.722640
$$226$$ 0 0
$$227$$ 4748.64 1.38845 0.694225 0.719758i $$-0.255747\pi$$
0.694225 + 0.719758i $$0.255747\pi$$
$$228$$ 0 0
$$229$$ −4801.99 −1.38570 −0.692848 0.721083i $$-0.743644\pi$$
−0.692848 + 0.721083i $$0.743644\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3155.29 −0.887166 −0.443583 0.896233i $$-0.646293\pi$$
−0.443583 + 0.896233i $$0.646293\pi$$
$$234$$ 0 0
$$235$$ −6080.36 −1.68782
$$236$$ 0 0
$$237$$ 414.737 0.113671
$$238$$ 0 0
$$239$$ −4241.93 −1.14806 −0.574032 0.818833i $$-0.694622\pi$$
−0.574032 + 0.818833i $$0.694622\pi$$
$$240$$ 0 0
$$241$$ −4342.99 −1.16081 −0.580407 0.814326i $$-0.697107\pi$$
−0.580407 + 0.814326i $$0.697107\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5668.25 −1.46017
$$248$$ 0 0
$$249$$ −1863.41 −0.474253
$$250$$ 0 0
$$251$$ −3003.01 −0.755172 −0.377586 0.925974i $$-0.623246\pi$$
−0.377586 + 0.925974i $$0.623246\pi$$
$$252$$ 0 0
$$253$$ −612.844 −0.152289
$$254$$ 0 0
$$255$$ −336.676 −0.0826803
$$256$$ 0 0
$$257$$ 4468.84 1.08466 0.542332 0.840164i $$-0.317541\pi$$
0.542332 + 0.840164i $$0.317541\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 542.869 0.128746
$$262$$ 0 0
$$263$$ −6160.13 −1.44430 −0.722148 0.691738i $$-0.756845\pi$$
−0.722148 + 0.691738i $$0.756845\pi$$
$$264$$ 0 0
$$265$$ −7464.64 −1.73037
$$266$$ 0 0
$$267$$ 856.255 0.196262
$$268$$ 0 0
$$269$$ 4988.90 1.13078 0.565388 0.824825i $$-0.308726\pi$$
0.565388 + 0.824825i $$0.308726\pi$$
$$270$$ 0 0
$$271$$ −4433.73 −0.993837 −0.496918 0.867797i $$-0.665535\pi$$
−0.496918 + 0.867797i $$0.665535\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −6487.80 −1.42265
$$276$$ 0 0
$$277$$ 1112.37 0.241284 0.120642 0.992696i $$-0.461505\pi$$
0.120642 + 0.992696i $$0.461505\pi$$
$$278$$ 0 0
$$279$$ 1104.40 0.236984
$$280$$ 0 0
$$281$$ 2813.22 0.597233 0.298616 0.954373i $$-0.403475\pi$$
0.298616 + 0.954373i $$0.403475\pi$$
$$282$$ 0 0
$$283$$ −3147.54 −0.661137 −0.330569 0.943782i $$-0.607241\pi$$
−0.330569 + 0.943782i $$0.607241\pi$$
$$284$$ 0 0
$$285$$ −3873.67 −0.805111
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4881.19 −0.993526
$$290$$ 0 0
$$291$$ −1809.34 −0.364486
$$292$$ 0 0
$$293$$ 9143.04 1.82301 0.911505 0.411289i $$-0.134921\pi$$
0.911505 + 0.411289i $$0.134921\pi$$
$$294$$ 0 0
$$295$$ 12489.2 2.46491
$$296$$ 0 0
$$297$$ −646.410 −0.126291
$$298$$ 0 0
$$299$$ 2236.12 0.432502
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1371.63 0.260059
$$304$$ 0 0
$$305$$ 74.7696 0.0140370
$$306$$ 0 0
$$307$$ 4648.90 0.864257 0.432129 0.901812i $$-0.357763\pi$$
0.432129 + 0.901812i $$0.357763\pi$$
$$308$$ 0 0
$$309$$ −2358.13 −0.434141
$$310$$ 0 0
$$311$$ 6417.18 1.17005 0.585024 0.811016i $$-0.301085\pi$$
0.585024 + 0.811016i $$0.301085\pi$$
$$312$$ 0 0
$$313$$ 5868.22 1.05972 0.529858 0.848086i $$-0.322245\pi$$
0.529858 + 0.848086i $$0.322245\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3974.52 −0.704200 −0.352100 0.935962i $$-0.614532\pi$$
−0.352100 + 0.935962i $$0.614532\pi$$
$$318$$ 0 0
$$319$$ −1444.10 −0.253461
$$320$$ 0 0
$$321$$ 589.383 0.102480
$$322$$ 0 0
$$323$$ 365.939 0.0630384
$$324$$ 0 0
$$325$$ 23672.4 4.04034
$$326$$ 0 0
$$327$$ −919.029 −0.155420
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8912.82 1.48004 0.740020 0.672585i $$-0.234816\pi$$
0.740020 + 0.672585i $$0.234816\pi$$
$$332$$ 0 0
$$333$$ −505.060 −0.0831144
$$334$$ 0 0
$$335$$ 16179.2 2.63871
$$336$$ 0 0
$$337$$ 3977.06 0.642862 0.321431 0.946933i $$-0.395836\pi$$
0.321431 + 0.946933i $$0.395836\pi$$
$$338$$ 0 0
$$339$$ 5992.88 0.960143
$$340$$ 0 0
$$341$$ −2937.83 −0.466547
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 1528.16 0.238474
$$346$$ 0 0
$$347$$ −6826.43 −1.05609 −0.528043 0.849218i $$-0.677074\pi$$
−0.528043 + 0.849218i $$0.677074\pi$$
$$348$$ 0 0
$$349$$ −807.342 −0.123828 −0.0619141 0.998081i $$-0.519720\pi$$
−0.0619141 + 0.998081i $$0.519720\pi$$
$$350$$ 0 0
$$351$$ 2358.59 0.358668
$$352$$ 0 0
$$353$$ −7919.20 −1.19404 −0.597020 0.802226i $$-0.703649\pi$$
−0.597020 + 0.802226i $$0.703649\pi$$
$$354$$ 0 0
$$355$$ −3301.38 −0.493574
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8819.21 1.29655 0.648273 0.761408i $$-0.275491\pi$$
0.648273 + 0.761408i $$0.275491\pi$$
$$360$$ 0 0
$$361$$ −2648.64 −0.386155
$$362$$ 0 0
$$363$$ −2273.47 −0.328722
$$364$$ 0 0
$$365$$ 12319.8 1.76670
$$366$$ 0 0
$$367$$ 11161.8 1.58758 0.793788 0.608194i $$-0.208106\pi$$
0.793788 + 0.608194i $$0.208106\pi$$
$$368$$ 0 0
$$369$$ −2697.41 −0.380547
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 2727.86 0.378668 0.189334 0.981913i $$-0.439367\pi$$
0.189334 + 0.981913i $$0.439367\pi$$
$$374$$ 0 0
$$375$$ 8715.38 1.20016
$$376$$ 0 0
$$377$$ 5269.17 0.719830
$$378$$ 0 0
$$379$$ −4086.49 −0.553849 −0.276924 0.960892i $$-0.589315\pi$$
−0.276924 + 0.960892i $$0.589315\pi$$
$$380$$ 0 0
$$381$$ 6934.20 0.932414
$$382$$ 0 0
$$383$$ −13032.9 −1.73878 −0.869389 0.494129i $$-0.835487\pi$$
−0.869389 + 0.494129i $$0.835487\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4512.79 0.592760
$$388$$ 0 0
$$389$$ −194.991 −0.0254151 −0.0127075 0.999919i $$-0.504045\pi$$
−0.0127075 + 0.999919i $$0.504045\pi$$
$$390$$ 0 0
$$391$$ −144.363 −0.0186719
$$392$$ 0 0
$$393$$ 465.053 0.0596916
$$394$$ 0 0
$$395$$ 2751.02 0.350427
$$396$$ 0 0
$$397$$ −14183.0 −1.79300 −0.896501 0.443042i $$-0.853899\pi$$
−0.896501 + 0.443042i $$0.853899\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10005.0 1.24596 0.622978 0.782239i $$-0.285923\pi$$
0.622978 + 0.782239i $$0.285923\pi$$
$$402$$ 0 0
$$403$$ 10719.4 1.32500
$$404$$ 0 0
$$405$$ 1611.86 0.197763
$$406$$ 0 0
$$407$$ 1343.52 0.163626
$$408$$ 0 0
$$409$$ 4634.93 0.560349 0.280174 0.959949i $$-0.409608\pi$$
0.280174 + 0.959949i $$0.409608\pi$$
$$410$$ 0 0
$$411$$ 1550.81 0.186121
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12360.3 −1.46203
$$416$$ 0 0
$$417$$ −2874.20 −0.337531
$$418$$ 0 0
$$419$$ −4998.31 −0.582777 −0.291388 0.956605i $$-0.594117\pi$$
−0.291388 + 0.956605i $$0.594117\pi$$
$$420$$ 0 0
$$421$$ −704.160 −0.0815170 −0.0407585 0.999169i $$-0.512977\pi$$
−0.0407585 + 0.999169i $$0.512977\pi$$
$$422$$ 0 0
$$423$$ −2749.98 −0.316096
$$424$$ 0 0
$$425$$ −1528.28 −0.174429
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −6274.16 −0.706105
$$430$$ 0 0
$$431$$ 10332.8 1.15479 0.577393 0.816466i $$-0.304070\pi$$
0.577393 + 0.816466i $$0.304070\pi$$
$$432$$ 0 0
$$433$$ 11106.8 1.23270 0.616348 0.787474i $$-0.288611\pi$$
0.616348 + 0.787474i $$0.288611\pi$$
$$434$$ 0 0
$$435$$ 3600.94 0.396901
$$436$$ 0 0
$$437$$ −1660.98 −0.181821
$$438$$ 0 0
$$439$$ 7299.28 0.793566 0.396783 0.917912i $$-0.370126\pi$$
0.396783 + 0.917912i $$0.370126\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16089.7 −1.72560 −0.862802 0.505542i $$-0.831293\pi$$
−0.862802 + 0.505542i $$0.831293\pi$$
$$444$$ 0 0
$$445$$ 5679.68 0.605040
$$446$$ 0 0
$$447$$ −5311.88 −0.562065
$$448$$ 0 0
$$449$$ 13561.7 1.42543 0.712715 0.701454i $$-0.247466\pi$$
0.712715 + 0.701454i $$0.247466\pi$$
$$450$$ 0 0
$$451$$ 7175.46 0.749178
$$452$$ 0 0
$$453$$ 7620.71 0.790402
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10848.6 1.11045 0.555224 0.831701i $$-0.312633\pi$$
0.555224 + 0.831701i $$0.312633\pi$$
$$458$$ 0 0
$$459$$ −152.269 −0.0154844
$$460$$ 0 0
$$461$$ −1758.69 −0.177679 −0.0888397 0.996046i $$-0.528316\pi$$
−0.0888397 + 0.996046i $$0.528316\pi$$
$$462$$ 0 0
$$463$$ 5411.95 0.543228 0.271614 0.962406i $$-0.412443\pi$$
0.271614 + 0.962406i $$0.412443\pi$$
$$464$$ 0 0
$$465$$ 7325.64 0.730577
$$466$$ 0 0
$$467$$ −8111.34 −0.803744 −0.401872 0.915696i $$-0.631640\pi$$
−0.401872 + 0.915696i $$0.631640\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −3250.03 −0.317948
$$472$$ 0 0
$$473$$ −12004.6 −1.16696
$$474$$ 0 0
$$475$$ −17583.8 −1.69853
$$476$$ 0 0
$$477$$ −3376.05 −0.324065
$$478$$ 0 0
$$479$$ −2095.76 −0.199912 −0.0999559 0.994992i $$-0.531870\pi$$
−0.0999559 + 0.994992i $$0.531870\pi$$
$$480$$ 0 0
$$481$$ −4902.18 −0.464699
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −12001.7 −1.12364
$$486$$ 0 0
$$487$$ −9610.08 −0.894197 −0.447099 0.894485i $$-0.647543\pi$$
−0.447099 + 0.894485i $$0.647543\pi$$
$$488$$ 0 0
$$489$$ −8906.17 −0.823622
$$490$$ 0 0
$$491$$ 11717.3 1.07698 0.538488 0.842633i $$-0.318996\pi$$
0.538488 + 0.842633i $$0.318996\pi$$
$$492$$ 0 0
$$493$$ −340.174 −0.0310764
$$494$$ 0 0
$$495$$ −4287.75 −0.389333
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 9195.19 0.824916 0.412458 0.910977i $$-0.364670\pi$$
0.412458 + 0.910977i $$0.364670\pi$$
$$500$$ 0 0
$$501$$ 6274.60 0.559538
$$502$$ 0 0
$$503$$ −16118.8 −1.42883 −0.714414 0.699724i $$-0.753307\pi$$
−0.714414 + 0.699724i $$0.753307\pi$$
$$504$$ 0 0
$$505$$ 9098.23 0.801715
$$506$$ 0 0
$$507$$ 16301.9 1.42799
$$508$$ 0 0
$$509$$ 4918.78 0.428333 0.214166 0.976797i $$-0.431297\pi$$
0.214166 + 0.976797i $$0.431297\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1751.96 −0.150781
$$514$$ 0 0
$$515$$ −15641.9 −1.33838
$$516$$ 0 0
$$517$$ 7315.29 0.622294
$$518$$ 0 0
$$519$$ −1410.44 −0.119290
$$520$$ 0 0
$$521$$ 13963.4 1.17418 0.587089 0.809522i $$-0.300274\pi$$
0.587089 + 0.809522i $$0.300274\pi$$
$$522$$ 0 0
$$523$$ 13755.3 1.15005 0.575024 0.818136i $$-0.304993\pi$$
0.575024 + 0.818136i $$0.304993\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −692.040 −0.0572026
$$528$$ 0 0
$$529$$ −11511.7 −0.946145
$$530$$ 0 0
$$531$$ 5648.51 0.461628
$$532$$ 0 0
$$533$$ −26181.5 −2.12767
$$534$$ 0 0
$$535$$ 3909.47 0.315928
$$536$$ 0 0
$$537$$ −3169.38 −0.254691
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14462.7 1.14935 0.574676 0.818381i $$-0.305128\pi$$
0.574676 + 0.818381i $$0.305128\pi$$
$$542$$ 0 0
$$543$$ −1219.41 −0.0963719
$$544$$ 0 0
$$545$$ −6096.07 −0.479132
$$546$$ 0 0
$$547$$ −13682.5 −1.06951 −0.534755 0.845007i $$-0.679596\pi$$
−0.534755 + 0.845007i $$0.679596\pi$$
$$548$$ 0 0
$$549$$ 33.8162 0.00262886
$$550$$ 0 0
$$551$$ −3913.92 −0.302611
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −3350.14 −0.256227
$$556$$ 0 0
$$557$$ −7663.13 −0.582939 −0.291470 0.956580i $$-0.594144\pi$$
−0.291470 + 0.956580i $$0.594144\pi$$
$$558$$ 0 0
$$559$$ 43801.8 3.31417
$$560$$ 0 0
$$561$$ 405.056 0.0304839
$$562$$ 0 0
$$563$$ −17470.5 −1.30780 −0.653902 0.756580i $$-0.726869\pi$$
−0.653902 + 0.756580i $$0.726869\pi$$
$$564$$ 0 0
$$565$$ 39751.8 2.95995
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −13873.9 −1.02219 −0.511094 0.859525i $$-0.670760\pi$$
−0.511094 + 0.859525i $$0.670760\pi$$
$$570$$ 0 0
$$571$$ 3777.52 0.276855 0.138428 0.990373i $$-0.455795\pi$$
0.138428 + 0.990373i $$0.455795\pi$$
$$572$$ 0 0
$$573$$ −538.229 −0.0392405
$$574$$ 0 0
$$575$$ 6936.79 0.503103
$$576$$ 0 0
$$577$$ 13880.5 1.00148 0.500738 0.865599i $$-0.333062\pi$$
0.500738 + 0.865599i $$0.333062\pi$$
$$578$$ 0 0
$$579$$ −7165.10 −0.514286
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 8980.72 0.637982
$$584$$ 0 0
$$585$$ 15644.9 1.10571
$$586$$ 0 0
$$587$$ 2395.61 0.168445 0.0842227 0.996447i $$-0.473159\pi$$
0.0842227 + 0.996447i $$0.473159\pi$$
$$588$$ 0 0
$$589$$ −7962.36 −0.557018
$$590$$ 0 0
$$591$$ −7996.06 −0.556538
$$592$$ 0 0
$$593$$ 6603.50 0.457290 0.228645 0.973510i $$-0.426570\pi$$
0.228645 + 0.973510i $$0.426570\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4026.92 0.276065
$$598$$ 0 0
$$599$$ −17252.1 −1.17680 −0.588399 0.808571i $$-0.700241\pi$$
−0.588399 + 0.808571i $$0.700241\pi$$
$$600$$ 0 0
$$601$$ 12833.1 0.871005 0.435503 0.900187i $$-0.356571\pi$$
0.435503 + 0.900187i $$0.356571\pi$$
$$602$$ 0 0
$$603$$ 7317.43 0.494177
$$604$$ 0 0
$$605$$ −15080.3 −1.01339
$$606$$ 0 0
$$607$$ −8620.16 −0.576411 −0.288206 0.957569i $$-0.593059\pi$$
−0.288206 + 0.957569i $$0.593059\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −26691.7 −1.76732
$$612$$ 0 0
$$613$$ 5136.73 0.338451 0.169226 0.985577i $$-0.445873\pi$$
0.169226 + 0.985577i $$0.445873\pi$$
$$614$$ 0 0
$$615$$ −17892.4 −1.17316
$$616$$ 0 0
$$617$$ −1759.82 −0.114826 −0.0574131 0.998351i $$-0.518285\pi$$
−0.0574131 + 0.998351i $$0.518285\pi$$
$$618$$ 0 0
$$619$$ 3560.24 0.231176 0.115588 0.993297i $$-0.463125\pi$$
0.115588 + 0.993297i $$0.463125\pi$$
$$620$$ 0 0
$$621$$ 691.145 0.0446614
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 23936.8 1.53195
$$626$$ 0 0
$$627$$ 4660.43 0.296841
$$628$$ 0 0
$$629$$ 316.482 0.0200620
$$630$$ 0 0
$$631$$ 27321.4 1.72369 0.861845 0.507172i $$-0.169309\pi$$
0.861845 + 0.507172i $$0.169309\pi$$
$$632$$ 0 0
$$633$$ −1885.33 −0.118381
$$634$$ 0 0
$$635$$ 45995.7 2.87446
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −1493.12 −0.0924366
$$640$$ 0 0
$$641$$ −21927.0 −1.35112 −0.675558 0.737307i $$-0.736097\pi$$
−0.675558 + 0.737307i $$0.736097\pi$$
$$642$$ 0 0
$$643$$ 5826.04 0.357320 0.178660 0.983911i $$-0.442824\pi$$
0.178660 + 0.983911i $$0.442824\pi$$
$$644$$ 0 0
$$645$$ 29934.1 1.82737
$$646$$ 0 0
$$647$$ −24210.7 −1.47113 −0.735565 0.677454i $$-0.763083\pi$$
−0.735565 + 0.677454i $$0.763083\pi$$
$$648$$ 0 0
$$649$$ −15025.7 −0.908801
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −25623.3 −1.53556 −0.767778 0.640716i $$-0.778637\pi$$
−0.767778 + 0.640716i $$0.778637\pi$$
$$654$$ 0 0
$$655$$ 3084.77 0.184018
$$656$$ 0 0
$$657$$ 5571.90 0.330868
$$658$$ 0 0
$$659$$ 23273.7 1.37574 0.687871 0.725833i $$-0.258545\pi$$
0.687871 + 0.725833i $$0.258545\pi$$
$$660$$ 0 0
$$661$$ −20036.5 −1.17902 −0.589508 0.807763i $$-0.700678\pi$$
−0.589508 + 0.807763i $$0.700678\pi$$
$$662$$ 0 0
$$663$$ −1477.95 −0.0865744
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1544.04 0.0896333
$$668$$ 0 0
$$669$$ 2909.91 0.168167
$$670$$ 0 0
$$671$$ −89.9554 −0.00517540
$$672$$ 0 0
$$673$$ −18127.8 −1.03830 −0.519149 0.854684i $$-0.673751\pi$$
−0.519149 + 0.854684i $$0.673751\pi$$
$$674$$ 0 0
$$675$$ 7316.73 0.417216
$$676$$ 0 0
$$677$$ 13815.5 0.784301 0.392150 0.919901i $$-0.371731\pi$$
0.392150 + 0.919901i $$0.371731\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 14245.9 0.801622
$$682$$ 0 0
$$683$$ 4604.23 0.257944 0.128972 0.991648i $$-0.458832\pi$$
0.128972 + 0.991648i $$0.458832\pi$$
$$684$$ 0 0
$$685$$ 10286.8 0.573777
$$686$$ 0 0
$$687$$ −14406.0 −0.800032
$$688$$ 0 0
$$689$$ −32768.5 −1.81187
$$690$$ 0 0
$$691$$ −17913.7 −0.986205 −0.493103 0.869971i $$-0.664137\pi$$
−0.493103 + 0.869971i $$0.664137\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −19065.1 −1.04054
$$696$$ 0 0
$$697$$ 1690.26 0.0918555
$$698$$ 0 0
$$699$$ −9465.86 −0.512206
$$700$$ 0 0
$$701$$ 11303.7 0.609035 0.304518 0.952507i $$-0.401505\pi$$
0.304518 + 0.952507i $$0.401505\pi$$
$$702$$ 0 0
$$703$$ 3641.33 0.195356
$$704$$ 0 0
$$705$$ −18241.1 −0.974466
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −16046.3 −0.849973 −0.424987 0.905200i $$-0.639721\pi$$
−0.424987 + 0.905200i $$0.639721\pi$$
$$710$$ 0 0
$$711$$ 1244.21 0.0656280
$$712$$ 0 0
$$713$$ 3141.15 0.164989
$$714$$ 0 0
$$715$$ −41617.5 −2.17679
$$716$$ 0 0
$$717$$ −12725.8 −0.662836
$$718$$ 0 0
$$719$$ 25190.5 1.30660 0.653300 0.757099i $$-0.273384\pi$$
0.653300 + 0.757099i $$0.273384\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −13029.0 −0.670197
$$724$$ 0 0
$$725$$ 16345.8 0.837334
$$726$$ 0 0
$$727$$ −11277.2 −0.575307 −0.287653 0.957735i $$-0.592875\pi$$
−0.287653 + 0.957735i $$0.592875\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −2827.82 −0.143079
$$732$$ 0 0
$$733$$ 23720.0 1.19525 0.597626 0.801775i $$-0.296111\pi$$
0.597626 + 0.801775i $$0.296111\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −19465.3 −0.972880
$$738$$ 0 0
$$739$$ −8124.72 −0.404428 −0.202214 0.979341i $$-0.564814\pi$$
−0.202214 + 0.979341i $$0.564814\pi$$
$$740$$ 0 0
$$741$$ −17004.8 −0.843030
$$742$$ 0 0
$$743$$ −20955.3 −1.03469 −0.517346 0.855777i $$-0.673080\pi$$
−0.517346 + 0.855777i $$0.673080\pi$$
$$744$$ 0 0
$$745$$ −35234.6 −1.73274
$$746$$ 0 0
$$747$$ −5590.23 −0.273810
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 38208.1 1.85650 0.928251 0.371954i $$-0.121312\pi$$
0.928251 + 0.371954i $$0.121312\pi$$
$$752$$ 0 0
$$753$$ −9009.02 −0.435999
$$754$$ 0 0
$$755$$ 50549.4 2.43666
$$756$$ 0 0
$$757$$ 30958.1 1.48638 0.743191 0.669079i $$-0.233311\pi$$
0.743191 + 0.669079i $$0.233311\pi$$
$$758$$ 0 0
$$759$$ −1838.53 −0.0879243
$$760$$ 0 0
$$761$$ −40049.4 −1.90774 −0.953871 0.300218i $$-0.902941\pi$$
−0.953871 + 0.300218i $$0.902941\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −1010.03 −0.0477355
$$766$$ 0 0
$$767$$ 54825.3 2.58100
$$768$$ 0 0
$$769$$ −8002.01 −0.375240 −0.187620 0.982242i $$-0.560077\pi$$
−0.187620 + 0.982242i $$0.560077\pi$$
$$770$$ 0 0
$$771$$ 13406.5 0.626231
$$772$$ 0 0
$$773$$ −13933.4 −0.648316 −0.324158 0.946003i $$-0.605081\pi$$
−0.324158 + 0.946003i $$0.605081\pi$$
$$774$$ 0 0
$$775$$ 33253.4 1.54128
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 19447.6 0.894456
$$780$$ 0 0
$$781$$ 3971.89 0.181979
$$782$$ 0 0
$$783$$ 1628.61 0.0743316
$$784$$ 0 0
$$785$$ −21558.0 −0.980175
$$786$$ 0 0
$$787$$ −37581.7 −1.70222 −0.851108 0.524991i $$-0.824069\pi$$
−0.851108 + 0.524991i $$0.824069\pi$$
$$788$$ 0 0
$$789$$ −18480.4 −0.833865
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 328.225 0.0146981
$$794$$ 0 0
$$795$$ −22393.9 −0.999032
$$796$$ 0 0
$$797$$ −9458.78 −0.420385 −0.210193 0.977660i $$-0.567409\pi$$
−0.210193 + 0.977660i $$0.567409\pi$$
$$798$$ 0 0
$$799$$ 1723.20 0.0762985
$$800$$ 0 0
$$801$$ 2568.77 0.113312
$$802$$ 0 0
$$803$$ −14821.9 −0.651376
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14966.7 0.652854
$$808$$ 0 0
$$809$$ −1909.94 −0.0830034 −0.0415017 0.999138i $$-0.513214\pi$$
−0.0415017 + 0.999138i $$0.513214\pi$$
$$810$$ 0 0
$$811$$ −43110.6 −1.86661 −0.933303 0.359091i $$-0.883087\pi$$
−0.933303 + 0.359091i $$0.883087\pi$$
$$812$$ 0 0
$$813$$ −13301.2 −0.573792
$$814$$ 0 0
$$815$$ −59076.1 −2.53907
$$816$$ 0 0
$$817$$ −32535.9 −1.39325
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 4026.56 0.171167 0.0855834 0.996331i $$-0.472725\pi$$
0.0855834 + 0.996331i $$0.472725\pi$$
$$822$$ 0 0
$$823$$ −39668.1 −1.68012 −0.840062 0.542491i $$-0.817481\pi$$
−0.840062 + 0.542491i $$0.817481\pi$$
$$824$$ 0 0
$$825$$ −19463.4 −0.821368
$$826$$ 0 0
$$827$$ −30137.7 −1.26722 −0.633611 0.773652i $$-0.718428\pi$$
−0.633611 + 0.773652i $$0.718428\pi$$
$$828$$ 0 0
$$829$$ 23278.0 0.975245 0.487622 0.873055i $$-0.337864\pi$$
0.487622 + 0.873055i $$0.337864\pi$$
$$830$$ 0 0
$$831$$ 3337.10 0.139306
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 41620.5 1.72495
$$836$$ 0 0
$$837$$ 3313.19 0.136823
$$838$$ 0 0
$$839$$ 9494.43 0.390684 0.195342 0.980735i $$-0.437418\pi$$
0.195342 + 0.980735i $$0.437418\pi$$
$$840$$ 0 0
$$841$$ −20750.6 −0.850820
$$842$$ 0 0
$$843$$ 8439.65 0.344813
$$844$$ 0 0
$$845$$ 108133. 4.40223
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9442.62 −0.381708
$$850$$ 0 0
$$851$$ −1436.50 −0.0578644
$$852$$ 0 0
$$853$$ −12692.4 −0.509471 −0.254736 0.967011i $$-0.581988\pi$$
−0.254736 + 0.967011i $$0.581988\pi$$
$$854$$ 0 0
$$855$$ −11621.0 −0.464831
$$856$$ 0 0
$$857$$ 22206.0 0.885114 0.442557 0.896740i $$-0.354071\pi$$
0.442557 + 0.896740i $$0.354071\pi$$
$$858$$ 0 0
$$859$$ −19820.5 −0.787271 −0.393636 0.919267i $$-0.628783\pi$$
−0.393636 + 0.919267i $$0.628783\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −36413.8 −1.43631 −0.718157 0.695881i $$-0.755014\pi$$
−0.718157 + 0.695881i $$0.755014\pi$$
$$864$$ 0 0
$$865$$ −9355.70 −0.367749
$$866$$ 0 0
$$867$$ −14643.6 −0.573613
$$868$$ 0 0
$$869$$ −3309.76 −0.129201
$$870$$ 0 0
$$871$$ 71024.1 2.76298
$$872$$ 0 0
$$873$$ −5428.03 −0.210436
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 19442.6 0.748609 0.374305 0.927306i $$-0.377881\pi$$
0.374305 + 0.927306i $$0.377881\pi$$
$$878$$ 0 0
$$879$$ 27429.1 1.05252
$$880$$ 0 0
$$881$$ −25184.2 −0.963082 −0.481541 0.876423i $$-0.659923\pi$$
−0.481541 + 0.876423i $$0.659923\pi$$
$$882$$ 0 0
$$883$$ 4050.03 0.154354 0.0771769 0.997017i $$-0.475409\pi$$
0.0771769 + 0.997017i $$0.475409\pi$$
$$884$$ 0 0
$$885$$ 37467.5 1.42311
$$886$$ 0 0
$$887$$ −41604.2 −1.57489 −0.787447 0.616383i $$-0.788598\pi$$
−0.787447 + 0.616383i $$0.788598\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1939.23 −0.0729144
$$892$$ 0 0
$$893$$ 19826.5 0.742967
$$894$$ 0 0
$$895$$ −21023.0 −0.785165
$$896$$ 0 0
$$897$$ 6708.36 0.249705
$$898$$ 0 0
$$899$$ 7401.76 0.274597
$$900$$ 0 0
$$901$$ 2115.51 0.0782219
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8088.56 −0.297097
$$906$$ 0 0
$$907$$ 3572.60 0.130790 0.0653949 0.997859i $$-0.479169\pi$$
0.0653949 + 0.997859i $$0.479169\pi$$
$$908$$ 0 0
$$909$$ 4114.88 0.150145
$$910$$ 0 0
$$911$$ −29457.7 −1.07133 −0.535663 0.844432i $$-0.679938\pi$$
−0.535663 + 0.844432i $$0.679938\pi$$
$$912$$ 0 0
$$913$$ 14870.7 0.539046
$$914$$ 0 0
$$915$$ 224.309 0.00810428
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 3310.65 0.118834 0.0594168 0.998233i $$-0.481076\pi$$
0.0594168 + 0.998233i $$0.481076\pi$$
$$920$$ 0 0
$$921$$ 13946.7 0.498979
$$922$$ 0 0
$$923$$ −14492.5 −0.516820
$$924$$ 0 0
$$925$$ −15207.3 −0.540556
$$926$$ 0 0
$$927$$ −7074.40 −0.250651
$$928$$ 0 0
$$929$$ −31467.5 −1.11132 −0.555660 0.831410i $$-0.687534\pi$$
−0.555660 + 0.831410i $$0.687534\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 19251.5 0.675527
$$934$$ 0 0
$$935$$ 2686.80 0.0939763
$$936$$ 0 0
$$937$$ −17363.4 −0.605375 −0.302688 0.953090i $$-0.597884\pi$$
−0.302688 + 0.953090i $$0.597884\pi$$
$$938$$ 0 0
$$939$$ 17604.6 0.611828
$$940$$ 0 0
$$941$$ −5547.77 −0.192192 −0.0960958 0.995372i $$-0.530636\pi$$
−0.0960958 + 0.995372i $$0.530636\pi$$
$$942$$ 0 0
$$943$$ −7672.04 −0.264937
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 37960.3 1.30258 0.651290 0.758829i $$-0.274228\pi$$
0.651290 + 0.758829i $$0.274228\pi$$
$$948$$ 0 0
$$949$$ 54081.7 1.84991
$$950$$ 0 0
$$951$$ −11923.6 −0.406570
$$952$$ 0 0
$$953$$ −10019.3 −0.340563 −0.170282 0.985395i $$-0.554468\pi$$
−0.170282 + 0.985395i $$0.554468\pi$$
$$954$$ 0 0
$$955$$ −3570.16 −0.120971
$$956$$ 0 0
$$957$$ −4332.30 −0.146336
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −14733.1 −0.494548
$$962$$ 0 0
$$963$$ 1768.15 0.0591670
$$964$$ 0 0
$$965$$ −47527.3 −1.58545
$$966$$ 0 0
$$967$$ −27834.4 −0.925641 −0.462820 0.886452i $$-0.653163\pi$$
−0.462820 + 0.886452i $$0.653163\pi$$
$$968$$ 0 0
$$969$$ 1097.82 0.0363952
$$970$$ 0 0
$$971$$ 18275.3 0.604000 0.302000 0.953308i $$-0.402346\pi$$
0.302000 + 0.953308i $$0.402346\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 71017.2 2.33269
$$976$$ 0 0
$$977$$ −43028.9 −1.40903 −0.704513 0.709691i $$-0.748834\pi$$
−0.704513 + 0.709691i $$0.748834\pi$$
$$978$$ 0 0
$$979$$ −6833.24 −0.223076
$$980$$ 0 0
$$981$$ −2757.09 −0.0897320
$$982$$ 0 0
$$983$$ −30559.2 −0.991544 −0.495772 0.868453i $$-0.665115\pi$$
−0.495772 + 0.868453i $$0.665115\pi$$
$$984$$ 0 0
$$985$$ −53039.2 −1.71571
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 12835.4 0.412681
$$990$$ 0 0
$$991$$ 44945.9 1.44072 0.720360 0.693600i $$-0.243976\pi$$
0.720360 + 0.693600i $$0.243976\pi$$
$$992$$ 0 0
$$993$$ 26738.5 0.854501
$$994$$ 0 0
$$995$$ 26711.2 0.851058
$$996$$ 0 0
$$997$$ 29006.1 0.921397 0.460698 0.887557i $$-0.347599\pi$$
0.460698 + 0.887557i $$0.347599\pi$$
$$998$$ 0 0
$$999$$ −1515.18 −0.0479861
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.cf.1.2 2
4.3 odd 2 147.4.a.j.1.2 2
7.6 odd 2 2352.4.a.bl.1.1 2
12.11 even 2 441.4.a.n.1.1 2
28.3 even 6 147.4.e.j.79.1 4
28.11 odd 6 147.4.e.k.79.1 4
28.19 even 6 147.4.e.j.67.1 4
28.23 odd 6 147.4.e.k.67.1 4
28.27 even 2 147.4.a.k.1.2 yes 2
84.11 even 6 441.4.e.v.226.2 4
84.23 even 6 441.4.e.v.361.2 4
84.47 odd 6 441.4.e.u.361.2 4
84.59 odd 6 441.4.e.u.226.2 4
84.83 odd 2 441.4.a.o.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.2 2 4.3 odd 2
147.4.a.k.1.2 yes 2 28.27 even 2
147.4.e.j.67.1 4 28.19 even 6
147.4.e.j.79.1 4 28.3 even 6
147.4.e.k.67.1 4 28.23 odd 6
147.4.e.k.79.1 4 28.11 odd 6
441.4.a.n.1.1 2 12.11 even 2
441.4.a.o.1.1 2 84.83 odd 2
441.4.e.u.226.2 4 84.59 odd 6
441.4.e.u.361.2 4 84.47 odd 6
441.4.e.v.226.2 4 84.11 even 6
441.4.e.v.361.2 4 84.23 even 6
2352.4.a.bl.1.1 2 7.6 odd 2
2352.4.a.cf.1.2 2 1.1 even 1 trivial