# Properties

 Label 975.2.c.i.274.3 Level $975$ Weight $2$ Character 975.274 Analytic conductor $7.785$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.3 Root $$1.40680 + 0.144584i$$ of defining polynomial Character $$\chi$$ $$=$$ 975.274 Dual form 975.2.c.i.274.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.289169i q^{2} +1.00000i q^{3} +1.91638 q^{4} +0.289169 q^{6} +4.91638i q^{7} -1.13249i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-0.289169i q^{2} +1.00000i q^{3} +1.91638 q^{4} +0.289169 q^{6} +4.91638i q^{7} -1.13249i q^{8} -1.00000 q^{9} +4.91638 q^{11} +1.91638i q^{12} -1.00000i q^{13} +1.42166 q^{14} +3.50528 q^{16} -4.33804i q^{17} +0.289169i q^{18} -2.57834 q^{19} -4.91638 q^{21} -1.42166i q^{22} +6.33804i q^{23} +1.13249 q^{24} -0.289169 q^{26} -1.00000i q^{27} +9.42166i q^{28} -6.00000 q^{29} +1.42166 q^{31} -3.27861i q^{32} +4.91638i q^{33} -1.25443 q^{34} -1.91638 q^{36} +9.49472i q^{37} +0.745574i q^{38} +1.00000 q^{39} +4.33804 q^{41} +1.42166i q^{42} +1.15667i q^{43} +9.42166 q^{44} +1.83276 q^{46} -5.42166i q^{47} +3.50528i q^{48} -17.1708 q^{49} +4.33804 q^{51} -1.91638i q^{52} +0.338044i q^{53} -0.289169 q^{54} +5.56777 q^{56} -2.57834i q^{57} +1.73501i q^{58} +11.2544 q^{59} -10.1708 q^{61} -0.411100i q^{62} -4.91638i q^{63} +6.06249 q^{64} +1.42166 q^{66} -7.25443i q^{67} -8.31335i q^{68} -6.33804 q^{69} +0.916382 q^{71} +1.13249i q^{72} +3.15667i q^{73} +2.74557 q^{74} -4.94108 q^{76} +24.1708i q^{77} -0.289169i q^{78} +3.49472 q^{79} +1.00000 q^{81} -1.25443i q^{82} -11.2544i q^{83} -9.42166 q^{84} +0.334474 q^{86} -6.00000i q^{87} -5.56777i q^{88} +0.338044 q^{89} +4.91638 q^{91} +12.1461i q^{92} +1.42166i q^{93} -1.56777 q^{94} +3.27861 q^{96} -12.3380i q^{97} +4.96526i q^{98} -4.91638 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 16 q^{4} - 6 q^{9}+O(q^{10})$$ 6 * q - 16 * q^4 - 6 * q^9 $$6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100})$$ 6 * q - 16 * q^4 - 6 * q^9 + 2 * q^11 + 12 * q^14 + 52 * q^16 - 12 * q^19 - 2 * q^21 + 12 * q^24 - 36 * q^29 + 12 * q^31 + 44 * q^34 + 16 * q^36 + 6 * q^39 + 2 * q^41 + 60 * q^44 - 44 * q^46 - 24 * q^49 + 2 * q^51 - 32 * q^56 + 16 * q^59 + 18 * q^61 - 60 * q^64 + 12 * q^66 - 14 * q^69 - 22 * q^71 + 68 * q^74 + 8 * q^76 - 10 * q^79 + 6 * q^81 - 60 * q^84 + 112 * q^86 - 22 * q^89 + 2 * q^91 + 56 * q^94 - 44 * q^96 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 0.289169i − 0.204473i −0.994760 0.102237i $$-0.967400\pi$$
0.994760 0.102237i $$-0.0325999\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.91638 0.958191
$$5$$ 0 0
$$6$$ 0.289169 0.118053
$$7$$ 4.91638i 1.85822i 0.369807 + 0.929109i $$0.379424\pi$$
−0.369807 + 0.929109i $$0.620576\pi$$
$$8$$ − 1.13249i − 0.400397i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.91638 1.48234 0.741172 0.671315i $$-0.234270\pi$$
0.741172 + 0.671315i $$0.234270\pi$$
$$12$$ 1.91638i 0.553212i
$$13$$ − 1.00000i − 0.277350i
$$14$$ 1.42166 0.379955
$$15$$ 0 0
$$16$$ 3.50528 0.876320
$$17$$ − 4.33804i − 1.05213i −0.850444 0.526065i $$-0.823667\pi$$
0.850444 0.526065i $$-0.176333\pi$$
$$18$$ 0.289169i 0.0681577i
$$19$$ −2.57834 −0.591511 −0.295756 0.955264i $$-0.595571\pi$$
−0.295756 + 0.955264i $$0.595571\pi$$
$$20$$ 0 0
$$21$$ −4.91638 −1.07284
$$22$$ − 1.42166i − 0.303100i
$$23$$ 6.33804i 1.32157i 0.750574 + 0.660787i $$0.229777\pi$$
−0.750574 + 0.660787i $$0.770223\pi$$
$$24$$ 1.13249 0.231169
$$25$$ 0 0
$$26$$ −0.289169 −0.0567106
$$27$$ − 1.00000i − 0.192450i
$$28$$ 9.42166i 1.78053i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 1.42166 0.255338 0.127669 0.991817i $$-0.459250\pi$$
0.127669 + 0.991817i $$0.459250\pi$$
$$32$$ − 3.27861i − 0.579581i
$$33$$ 4.91638i 0.855832i
$$34$$ −1.25443 −0.215132
$$35$$ 0 0
$$36$$ −1.91638 −0.319397
$$37$$ 9.49472i 1.56092i 0.625204 + 0.780461i $$0.285016\pi$$
−0.625204 + 0.780461i $$0.714984\pi$$
$$38$$ 0.745574i 0.120948i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 4.33804 0.677489 0.338744 0.940878i $$-0.389998\pi$$
0.338744 + 0.940878i $$0.389998\pi$$
$$42$$ 1.42166i 0.219367i
$$43$$ 1.15667i 0.176391i 0.996103 + 0.0881956i $$0.0281100\pi$$
−0.996103 + 0.0881956i $$0.971890\pi$$
$$44$$ 9.42166 1.42037
$$45$$ 0 0
$$46$$ 1.83276 0.270226
$$47$$ − 5.42166i − 0.790831i −0.918502 0.395415i $$-0.870601\pi$$
0.918502 0.395415i $$-0.129399\pi$$
$$48$$ 3.50528i 0.505944i
$$49$$ −17.1708 −2.45297
$$50$$ 0 0
$$51$$ 4.33804 0.607448
$$52$$ − 1.91638i − 0.265754i
$$53$$ 0.338044i 0.0464340i 0.999730 + 0.0232170i $$0.00739086\pi$$
−0.999730 + 0.0232170i $$0.992609\pi$$
$$54$$ −0.289169 −0.0393509
$$55$$ 0 0
$$56$$ 5.56777 0.744025
$$57$$ − 2.57834i − 0.341509i
$$58$$ 1.73501i 0.227818i
$$59$$ 11.2544 1.46520 0.732601 0.680659i $$-0.238306\pi$$
0.732601 + 0.680659i $$0.238306\pi$$
$$60$$ 0 0
$$61$$ −10.1708 −1.30224 −0.651119 0.758975i $$-0.725700\pi$$
−0.651119 + 0.758975i $$0.725700\pi$$
$$62$$ − 0.411100i − 0.0522098i
$$63$$ − 4.91638i − 0.619406i
$$64$$ 6.06249 0.757812
$$65$$ 0 0
$$66$$ 1.42166 0.174995
$$67$$ − 7.25443i − 0.886269i −0.896455 0.443135i $$-0.853866\pi$$
0.896455 0.443135i $$-0.146134\pi$$
$$68$$ − 8.31335i − 1.00814i
$$69$$ −6.33804 −0.763011
$$70$$ 0 0
$$71$$ 0.916382 0.108754 0.0543772 0.998520i $$-0.482683\pi$$
0.0543772 + 0.998520i $$0.482683\pi$$
$$72$$ 1.13249i 0.133466i
$$73$$ 3.15667i 0.369461i 0.982789 + 0.184730i $$0.0591412\pi$$
−0.982789 + 0.184730i $$0.940859\pi$$
$$74$$ 2.74557 0.319166
$$75$$ 0 0
$$76$$ −4.94108 −0.566780
$$77$$ 24.1708i 2.75452i
$$78$$ − 0.289169i − 0.0327419i
$$79$$ 3.49472 0.393187 0.196593 0.980485i $$-0.437012\pi$$
0.196593 + 0.980485i $$0.437012\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 1.25443i − 0.138528i
$$83$$ − 11.2544i − 1.23533i −0.786440 0.617667i $$-0.788078\pi$$
0.786440 0.617667i $$-0.211922\pi$$
$$84$$ −9.42166 −1.02799
$$85$$ 0 0
$$86$$ 0.334474 0.0360672
$$87$$ − 6.00000i − 0.643268i
$$88$$ − 5.56777i − 0.593527i
$$89$$ 0.338044 0.0358326 0.0179163 0.999839i $$-0.494297\pi$$
0.0179163 + 0.999839i $$0.494297\pi$$
$$90$$ 0 0
$$91$$ 4.91638 0.515377
$$92$$ 12.1461i 1.26632i
$$93$$ 1.42166i 0.147420i
$$94$$ −1.56777 −0.161704
$$95$$ 0 0
$$96$$ 3.27861 0.334621
$$97$$ − 12.3380i − 1.25274i −0.779526 0.626369i $$-0.784540\pi$$
0.779526 0.626369i $$-0.215460\pi$$
$$98$$ 4.96526i 0.501567i
$$99$$ −4.91638 −0.494115
$$100$$ 0 0
$$101$$ 10.6761 1.06231 0.531155 0.847274i $$-0.321758\pi$$
0.531155 + 0.847274i $$0.321758\pi$$
$$102$$ − 1.25443i − 0.124207i
$$103$$ 14.5089i 1.42960i 0.699329 + 0.714800i $$0.253482\pi$$
−0.699329 + 0.714800i $$0.746518\pi$$
$$104$$ −1.13249 −0.111050
$$105$$ 0 0
$$106$$ 0.0977518 0.00949450
$$107$$ 4.17081i 0.403207i 0.979467 + 0.201604i $$0.0646153\pi$$
−0.979467 + 0.201604i $$0.935385\pi$$
$$108$$ − 1.91638i − 0.184404i
$$109$$ 3.83276 0.367112 0.183556 0.983009i $$-0.441239\pi$$
0.183556 + 0.983009i $$0.441239\pi$$
$$110$$ 0 0
$$111$$ −9.49472 −0.901199
$$112$$ 17.2333i 1.62839i
$$113$$ 0.843326i 0.0793334i 0.999213 + 0.0396667i $$0.0126296\pi$$
−0.999213 + 0.0396667i $$0.987370\pi$$
$$114$$ −0.745574 −0.0698294
$$115$$ 0 0
$$116$$ −11.4983 −1.06759
$$117$$ 1.00000i 0.0924500i
$$118$$ − 3.25443i − 0.299594i
$$119$$ 21.3275 1.95509
$$120$$ 0 0
$$121$$ 13.1708 1.19735
$$122$$ 2.94108i 0.266273i
$$123$$ 4.33804i 0.391148i
$$124$$ 2.72445 0.244663
$$125$$ 0 0
$$126$$ −1.42166 −0.126652
$$127$$ 1.83276i 0.162631i 0.996688 + 0.0813157i $$0.0259122\pi$$
−0.996688 + 0.0813157i $$0.974088\pi$$
$$128$$ − 8.31029i − 0.734533i
$$129$$ −1.15667 −0.101839
$$130$$ 0 0
$$131$$ 5.83276 0.509611 0.254805 0.966992i $$-0.417989\pi$$
0.254805 + 0.966992i $$0.417989\pi$$
$$132$$ 9.42166i 0.820050i
$$133$$ − 12.6761i − 1.09916i
$$134$$ −2.09775 −0.181218
$$135$$ 0 0
$$136$$ −4.91281 −0.421270
$$137$$ − 16.5089i − 1.41045i −0.708985 0.705223i $$-0.750847\pi$$
0.708985 0.705223i $$-0.249153\pi$$
$$138$$ 1.83276i 0.156015i
$$139$$ −7.49472 −0.635694 −0.317847 0.948142i $$-0.602960\pi$$
−0.317847 + 0.948142i $$0.602960\pi$$
$$140$$ 0 0
$$141$$ 5.42166 0.456586
$$142$$ − 0.264989i − 0.0222374i
$$143$$ − 4.91638i − 0.411128i
$$144$$ −3.50528 −0.292107
$$145$$ 0 0
$$146$$ 0.912811 0.0755448
$$147$$ − 17.1708i − 1.41622i
$$148$$ 18.1955i 1.49566i
$$149$$ −20.4842 −1.67813 −0.839064 0.544033i $$-0.816897\pi$$
−0.839064 + 0.544033i $$0.816897\pi$$
$$150$$ 0 0
$$151$$ −16.4111 −1.33552 −0.667758 0.744378i $$-0.732746\pi$$
−0.667758 + 0.744378i $$0.732746\pi$$
$$152$$ 2.91995i 0.236839i
$$153$$ 4.33804i 0.350710i
$$154$$ 6.98944 0.563225
$$155$$ 0 0
$$156$$ 1.91638 0.153433
$$157$$ − 21.6655i − 1.72910i −0.502549 0.864549i $$-0.667604\pi$$
0.502549 0.864549i $$-0.332396\pi$$
$$158$$ − 1.01056i − 0.0803961i
$$159$$ −0.338044 −0.0268087
$$160$$ 0 0
$$161$$ −31.1602 −2.45577
$$162$$ − 0.289169i − 0.0227192i
$$163$$ − 6.07306i − 0.475678i −0.971305 0.237839i $$-0.923561\pi$$
0.971305 0.237839i $$-0.0764391\pi$$
$$164$$ 8.31335 0.649163
$$165$$ 0 0
$$166$$ −3.25443 −0.252592
$$167$$ − 0.745574i − 0.0576942i −0.999584 0.0288471i $$-0.990816\pi$$
0.999584 0.0288471i $$-0.00918360\pi$$
$$168$$ 5.56777i 0.429563i
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 2.57834 0.197170
$$172$$ 2.21663i 0.169016i
$$173$$ − 0.843326i − 0.0641169i −0.999486 0.0320584i $$-0.989794\pi$$
0.999486 0.0320584i $$-0.0102063\pi$$
$$174$$ −1.73501 −0.131531
$$175$$ 0 0
$$176$$ 17.2333 1.29901
$$177$$ 11.2544i 0.845934i
$$178$$ − 0.0977518i − 0.00732681i
$$179$$ 18.9894 1.41934 0.709669 0.704536i $$-0.248845\pi$$
0.709669 + 0.704536i $$0.248845\pi$$
$$180$$ 0 0
$$181$$ 17.4947 1.30037 0.650186 0.759775i $$-0.274691\pi$$
0.650186 + 0.759775i $$0.274691\pi$$
$$182$$ − 1.42166i − 0.105381i
$$183$$ − 10.1708i − 0.751848i
$$184$$ 7.17780 0.529154
$$185$$ 0 0
$$186$$ 0.411100 0.0301433
$$187$$ − 21.3275i − 1.55962i
$$188$$ − 10.3900i − 0.757767i
$$189$$ 4.91638 0.357614
$$190$$ 0 0
$$191$$ −22.5089 −1.62868 −0.814342 0.580386i $$-0.802902\pi$$
−0.814342 + 0.580386i $$0.802902\pi$$
$$192$$ 6.06249i 0.437523i
$$193$$ − 2.65139i − 0.190851i −0.995437 0.0954257i $$-0.969579\pi$$
0.995437 0.0954257i $$-0.0304212\pi$$
$$194$$ −3.56777 −0.256151
$$195$$ 0 0
$$196$$ −32.9058 −2.35042
$$197$$ − 12.9894i − 0.925459i −0.886500 0.462730i $$-0.846870\pi$$
0.886500 0.462730i $$-0.153130\pi$$
$$198$$ 1.42166i 0.101033i
$$199$$ 2.84333 0.201558 0.100779 0.994909i $$-0.467866\pi$$
0.100779 + 0.994909i $$0.467866\pi$$
$$200$$ 0 0
$$201$$ 7.25443 0.511688
$$202$$ − 3.08719i − 0.217214i
$$203$$ − 29.4983i − 2.07037i
$$204$$ 8.31335 0.582051
$$205$$ 0 0
$$206$$ 4.19550 0.292315
$$207$$ − 6.33804i − 0.440525i
$$208$$ − 3.50528i − 0.243048i
$$209$$ −12.6761 −0.876823
$$210$$ 0 0
$$211$$ 6.31335 0.434629 0.217314 0.976102i $$-0.430270\pi$$
0.217314 + 0.976102i $$0.430270\pi$$
$$212$$ 0.647822i 0.0444926i
$$213$$ 0.916382i 0.0627894i
$$214$$ 1.20607 0.0824450
$$215$$ 0 0
$$216$$ −1.13249 −0.0770565
$$217$$ 6.98944i 0.474474i
$$218$$ − 1.10831i − 0.0750645i
$$219$$ −3.15667 −0.213308
$$220$$ 0 0
$$221$$ −4.33804 −0.291808
$$222$$ 2.74557i 0.184271i
$$223$$ − 19.2544i − 1.28937i −0.764448 0.644686i $$-0.776988\pi$$
0.764448 0.644686i $$-0.223012\pi$$
$$224$$ 16.1189 1.07699
$$225$$ 0 0
$$226$$ 0.243863 0.0162215
$$227$$ − 13.0872i − 0.868627i −0.900762 0.434314i $$-0.856991\pi$$
0.900762 0.434314i $$-0.143009\pi$$
$$228$$ − 4.94108i − 0.327231i
$$229$$ 24.5089 1.61959 0.809795 0.586713i $$-0.199578\pi$$
0.809795 + 0.586713i $$0.199578\pi$$
$$230$$ 0 0
$$231$$ −24.1708 −1.59032
$$232$$ 6.79497i 0.446111i
$$233$$ − 8.33804i − 0.546243i −0.961979 0.273122i $$-0.911944\pi$$
0.961979 0.273122i $$-0.0880562\pi$$
$$234$$ 0.289169 0.0189035
$$235$$ 0 0
$$236$$ 21.5678 1.40394
$$237$$ 3.49472i 0.227006i
$$238$$ − 6.16724i − 0.399763i
$$239$$ −8.91638 −0.576753 −0.288376 0.957517i $$-0.593115\pi$$
−0.288376 + 0.957517i $$0.593115\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ − 3.80858i − 0.244825i
$$243$$ 1.00000i 0.0641500i
$$244$$ −19.4911 −1.24779
$$245$$ 0 0
$$246$$ 1.25443 0.0799793
$$247$$ 2.57834i 0.164056i
$$248$$ − 1.61003i − 0.102237i
$$249$$ 11.2544 0.713220
$$250$$ 0 0
$$251$$ −6.31335 −0.398495 −0.199248 0.979949i $$-0.563850\pi$$
−0.199248 + 0.979949i $$0.563850\pi$$
$$252$$ − 9.42166i − 0.593509i
$$253$$ 31.1602i 1.95903i
$$254$$ 0.529977 0.0332537
$$255$$ 0 0
$$256$$ 9.72191 0.607619
$$257$$ − 11.1567i − 0.695934i −0.937507 0.347967i $$-0.886872\pi$$
0.937507 0.347967i $$-0.113128\pi$$
$$258$$ 0.334474i 0.0208234i
$$259$$ −46.6797 −2.90053
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ − 1.68665i − 0.104202i
$$263$$ − 8.00000i − 0.493301i −0.969104 0.246651i $$-0.920670\pi$$
0.969104 0.246651i $$-0.0793300\pi$$
$$264$$ 5.56777 0.342673
$$265$$ 0 0
$$266$$ −3.66553 −0.224748
$$267$$ 0.338044i 0.0206880i
$$268$$ − 13.9022i − 0.849215i
$$269$$ −18.6761 −1.13870 −0.569351 0.822095i $$-0.692805\pi$$
−0.569351 + 0.822095i $$0.692805\pi$$
$$270$$ 0 0
$$271$$ 6.57834 0.399606 0.199803 0.979836i $$-0.435970\pi$$
0.199803 + 0.979836i $$0.435970\pi$$
$$272$$ − 15.2061i − 0.922003i
$$273$$ 4.91638i 0.297553i
$$274$$ −4.77384 −0.288398
$$275$$ 0 0
$$276$$ −12.1461 −0.731110
$$277$$ 25.6655i 1.54209i 0.636779 + 0.771046i $$0.280266\pi$$
−0.636779 + 0.771046i $$0.719734\pi$$
$$278$$ 2.16724i 0.129982i
$$279$$ −1.42166 −0.0851127
$$280$$ 0 0
$$281$$ 3.15667 0.188311 0.0941557 0.995557i $$-0.469985\pi$$
0.0941557 + 0.995557i $$0.469985\pi$$
$$282$$ − 1.56777i − 0.0933596i
$$283$$ 3.47002i 0.206271i 0.994667 + 0.103136i $$0.0328876\pi$$
−0.994667 + 0.103136i $$0.967112\pi$$
$$284$$ 1.75614 0.104208
$$285$$ 0 0
$$286$$ −1.42166 −0.0840647
$$287$$ 21.3275i 1.25892i
$$288$$ 3.27861i 0.193194i
$$289$$ −1.81863 −0.106978
$$290$$ 0 0
$$291$$ 12.3380 0.723269
$$292$$ 6.04939i 0.354014i
$$293$$ − 28.6550i − 1.67404i −0.547172 0.837020i $$-0.684296\pi$$
0.547172 0.837020i $$-0.315704\pi$$
$$294$$ −4.96526 −0.289580
$$295$$ 0 0
$$296$$ 10.7527 0.624989
$$297$$ − 4.91638i − 0.285277i
$$298$$ 5.92337i 0.343132i
$$299$$ 6.33804 0.366539
$$300$$ 0 0
$$301$$ −5.68665 −0.327773
$$302$$ 4.74557i 0.273077i
$$303$$ 10.6761i 0.613325i
$$304$$ −9.03780 −0.518353
$$305$$ 0 0
$$306$$ 1.25443 0.0717108
$$307$$ − 1.92694i − 0.109977i −0.998487 0.0549883i $$-0.982488\pi$$
0.998487 0.0549883i $$-0.0175121\pi$$
$$308$$ 46.3205i 2.63935i
$$309$$ −14.5089 −0.825380
$$310$$ 0 0
$$311$$ −29.9789 −1.69995 −0.849973 0.526826i $$-0.823382\pi$$
−0.849973 + 0.526826i $$0.823382\pi$$
$$312$$ − 1.13249i − 0.0641149i
$$313$$ 16.3133i 0.922085i 0.887378 + 0.461042i $$0.152524\pi$$
−0.887378 + 0.461042i $$0.847476\pi$$
$$314$$ −6.26499 −0.353554
$$315$$ 0 0
$$316$$ 6.69721 0.376748
$$317$$ 30.6761i 1.72294i 0.507808 + 0.861470i $$0.330456\pi$$
−0.507808 + 0.861470i $$0.669544\pi$$
$$318$$ 0.0977518i 0.00548165i
$$319$$ −29.4983 −1.65159
$$320$$ 0 0
$$321$$ −4.17081 −0.232792
$$322$$ 9.01056i 0.502139i
$$323$$ 11.1849i 0.622347i
$$324$$ 1.91638 0.106466
$$325$$ 0 0
$$326$$ −1.75614 −0.0972634
$$327$$ 3.83276i 0.211952i
$$328$$ − 4.91281i − 0.271265i
$$329$$ 26.6550 1.46954
$$330$$ 0 0
$$331$$ −10.0978 −0.555023 −0.277511 0.960722i $$-0.589510\pi$$
−0.277511 + 0.960722i $$0.589510\pi$$
$$332$$ − 21.5678i − 1.18369i
$$333$$ − 9.49472i − 0.520307i
$$334$$ −0.215597 −0.0117969
$$335$$ 0 0
$$336$$ −17.2333 −0.940154
$$337$$ − 1.32391i − 0.0721180i −0.999350 0.0360590i $$-0.988520\pi$$
0.999350 0.0360590i $$-0.0114804\pi$$
$$338$$ 0.289169i 0.0157287i
$$339$$ −0.843326 −0.0458032
$$340$$ 0 0
$$341$$ 6.98944 0.378499
$$342$$ − 0.745574i − 0.0403160i
$$343$$ − 50.0036i − 2.69994i
$$344$$ 1.30993 0.0706265
$$345$$ 0 0
$$346$$ −0.243863 −0.0131102
$$347$$ − 7.49472i − 0.402338i −0.979557 0.201169i $$-0.935526\pi$$
0.979557 0.201169i $$-0.0644740\pi$$
$$348$$ − 11.4983i − 0.616373i
$$349$$ 22.1461 1.18545 0.592727 0.805403i $$-0.298051\pi$$
0.592727 + 0.805403i $$0.298051\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ − 16.1189i − 0.859139i
$$353$$ − 4.50885i − 0.239982i −0.992775 0.119991i $$-0.961713\pi$$
0.992775 0.119991i $$-0.0382866\pi$$
$$354$$ 3.25443 0.172971
$$355$$ 0 0
$$356$$ 0.647822 0.0343345
$$357$$ 21.3275i 1.12877i
$$358$$ − 5.49115i − 0.290216i
$$359$$ 20.4111 1.07726 0.538628 0.842543i $$-0.318943\pi$$
0.538628 + 0.842543i $$0.318943\pi$$
$$360$$ 0 0
$$361$$ −12.3522 −0.650115
$$362$$ − 5.05892i − 0.265891i
$$363$$ 13.1708i 0.691288i
$$364$$ 9.42166 0.493829
$$365$$ 0 0
$$366$$ −2.94108 −0.153733
$$367$$ − 10.3133i − 0.538352i −0.963091 0.269176i $$-0.913249\pi$$
0.963091 0.269176i $$-0.0867514\pi$$
$$368$$ 22.2166i 1.15812i
$$369$$ −4.33804 −0.225830
$$370$$ 0 0
$$371$$ −1.66196 −0.0862844
$$372$$ 2.72445i 0.141256i
$$373$$ − 18.6761i − 0.967011i −0.875341 0.483506i $$-0.839363\pi$$
0.875341 0.483506i $$-0.160637\pi$$
$$374$$ −6.16724 −0.318900
$$375$$ 0 0
$$376$$ −6.14000 −0.316646
$$377$$ 6.00000i 0.309016i
$$378$$ − 1.42166i − 0.0731224i
$$379$$ 28.7527 1.47693 0.738464 0.674293i $$-0.235552\pi$$
0.738464 + 0.674293i $$0.235552\pi$$
$$380$$ 0 0
$$381$$ −1.83276 −0.0938953
$$382$$ 6.50885i 0.333022i
$$383$$ 14.2439i 0.727827i 0.931433 + 0.363914i $$0.118560\pi$$
−0.931433 + 0.363914i $$0.881440\pi$$
$$384$$ 8.31029 0.424083
$$385$$ 0 0
$$386$$ −0.766699 −0.0390240
$$387$$ − 1.15667i − 0.0587971i
$$388$$ − 23.6444i − 1.20036i
$$389$$ −34.6761 −1.75815 −0.879074 0.476686i $$-0.841838\pi$$
−0.879074 + 0.476686i $$0.841838\pi$$
$$390$$ 0 0
$$391$$ 27.4947 1.39047
$$392$$ 19.4458i 0.982163i
$$393$$ 5.83276i 0.294224i
$$394$$ −3.75614 −0.189231
$$395$$ 0 0
$$396$$ −9.42166 −0.473456
$$397$$ 7.18137i 0.360423i 0.983628 + 0.180211i $$0.0576782\pi$$
−0.983628 + 0.180211i $$0.942322\pi$$
$$398$$ − 0.822200i − 0.0412132i
$$399$$ 12.6761 0.634598
$$400$$ 0 0
$$401$$ 37.8610 1.89069 0.945345 0.326072i $$-0.105725\pi$$
0.945345 + 0.326072i $$0.105725\pi$$
$$402$$ − 2.09775i − 0.104626i
$$403$$ − 1.42166i − 0.0708181i
$$404$$ 20.4595 1.01790
$$405$$ 0 0
$$406$$ −8.52998 −0.423336
$$407$$ 46.6797i 2.31382i
$$408$$ − 4.91281i − 0.243220i
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 16.5089 0.814322
$$412$$ 27.8045i 1.36983i
$$413$$ 55.3311i 2.72266i
$$414$$ −1.83276 −0.0900754
$$415$$ 0 0
$$416$$ −3.27861 −0.160747
$$417$$ − 7.49472i − 0.367018i
$$418$$ 3.66553i 0.179287i
$$419$$ 33.4983 1.63650 0.818249 0.574864i $$-0.194945\pi$$
0.818249 + 0.574864i $$0.194945\pi$$
$$420$$ 0 0
$$421$$ 13.5194 0.658896 0.329448 0.944174i $$-0.393137\pi$$
0.329448 + 0.944174i $$0.393137\pi$$
$$422$$ − 1.82562i − 0.0888699i
$$423$$ 5.42166i 0.263610i
$$424$$ 0.382833 0.0185920
$$425$$ 0 0
$$426$$ 0.264989 0.0128387
$$427$$ − 50.0036i − 2.41984i
$$428$$ 7.99286i 0.386349i
$$429$$ 4.91638 0.237365
$$430$$ 0 0
$$431$$ 12.4111 0.597822 0.298911 0.954281i $$-0.403377\pi$$
0.298911 + 0.954281i $$0.403377\pi$$
$$432$$ − 3.50528i − 0.168648i
$$433$$ 17.3239i 0.832534i 0.909242 + 0.416267i $$0.136662\pi$$
−0.909242 + 0.416267i $$0.863338\pi$$
$$434$$ 2.02113 0.0970171
$$435$$ 0 0
$$436$$ 7.34504 0.351763
$$437$$ − 16.3416i − 0.781725i
$$438$$ 0.912811i 0.0436158i
$$439$$ −0.651393 −0.0310893 −0.0155446 0.999879i $$-0.504948\pi$$
−0.0155446 + 0.999879i $$0.504948\pi$$
$$440$$ 0 0
$$441$$ 17.1708 0.817658
$$442$$ 1.25443i 0.0596670i
$$443$$ 8.84690i 0.420329i 0.977666 + 0.210164i $$0.0673999\pi$$
−0.977666 + 0.210164i $$0.932600\pi$$
$$444$$ −18.1955 −0.863520
$$445$$ 0 0
$$446$$ −5.56777 −0.263642
$$447$$ − 20.4842i − 0.968867i
$$448$$ 29.8055i 1.40818i
$$449$$ −4.33804 −0.204725 −0.102362 0.994747i $$-0.532640\pi$$
−0.102362 + 0.994747i $$0.532640\pi$$
$$450$$ 0 0
$$451$$ 21.3275 1.00427
$$452$$ 1.61613i 0.0760166i
$$453$$ − 16.4111i − 0.771061i
$$454$$ −3.78440 −0.177611
$$455$$ 0 0
$$456$$ −2.91995 −0.136739
$$457$$ 15.3275i 0.716989i 0.933532 + 0.358495i $$0.116710\pi$$
−0.933532 + 0.358495i $$0.883290\pi$$
$$458$$ − 7.08719i − 0.331163i
$$459$$ −4.33804 −0.202483
$$460$$ 0 0
$$461$$ −11.8575 −0.552257 −0.276128 0.961121i $$-0.589052\pi$$
−0.276128 + 0.961121i $$0.589052\pi$$
$$462$$ 6.98944i 0.325178i
$$463$$ 26.4147i 1.22759i 0.789464 + 0.613797i $$0.210359\pi$$
−0.789464 + 0.613797i $$0.789641\pi$$
$$464$$ −21.0317 −0.976372
$$465$$ 0 0
$$466$$ −2.41110 −0.111692
$$467$$ − 33.6691i − 1.55802i −0.627012 0.779010i $$-0.715722\pi$$
0.627012 0.779010i $$-0.284278\pi$$
$$468$$ 1.91638i 0.0885848i
$$469$$ 35.6655 1.64688
$$470$$ 0 0
$$471$$ 21.6655 0.998295
$$472$$ − 12.7456i − 0.586663i
$$473$$ 5.68665i 0.261473i
$$474$$ 1.01056 0.0464167
$$475$$ 0 0
$$476$$ 40.8716 1.87335
$$477$$ − 0.338044i − 0.0154780i
$$478$$ 2.57834i 0.117930i
$$479$$ −10.7491 −0.491141 −0.245570 0.969379i $$-0.578975\pi$$
−0.245570 + 0.969379i $$0.578975\pi$$
$$480$$ 0 0
$$481$$ 9.49472 0.432922
$$482$$ 1.73501i 0.0790276i
$$483$$ − 31.1602i − 1.41784i
$$484$$ 25.2403 1.14729
$$485$$ 0 0
$$486$$ 0.289169 0.0131170
$$487$$ 22.7491i 1.03086i 0.856931 + 0.515431i $$0.172368\pi$$
−0.856931 + 0.515431i $$0.827632\pi$$
$$488$$ 11.5184i 0.521413i
$$489$$ 6.07306 0.274633
$$490$$ 0 0
$$491$$ −17.6867 −0.798187 −0.399094 0.916910i $$-0.630675\pi$$
−0.399094 + 0.916910i $$0.630675\pi$$
$$492$$ 8.31335i 0.374795i
$$493$$ 26.0283i 1.17225i
$$494$$ 0.745574 0.0335450
$$495$$ 0 0
$$496$$ 4.98333 0.223758
$$497$$ 4.50528i 0.202089i
$$498$$ − 3.25443i − 0.145834i
$$499$$ −19.9305 −0.892212 −0.446106 0.894980i $$-0.647190\pi$$
−0.446106 + 0.894980i $$0.647190\pi$$
$$500$$ 0 0
$$501$$ 0.745574 0.0333098
$$502$$ 1.82562i 0.0814815i
$$503$$ − 33.3522i − 1.48710i −0.668680 0.743550i $$-0.733140\pi$$
0.668680 0.743550i $$-0.266860\pi$$
$$504$$ −5.56777 −0.248008
$$505$$ 0 0
$$506$$ 9.01056 0.400568
$$507$$ − 1.00000i − 0.0444116i
$$508$$ 3.51227i 0.155832i
$$509$$ −13.8363 −0.613285 −0.306642 0.951825i $$-0.599206\pi$$
−0.306642 + 0.951825i $$0.599206\pi$$
$$510$$ 0 0
$$511$$ −15.5194 −0.686538
$$512$$ − 19.4319i − 0.858775i
$$513$$ 2.57834i 0.113836i
$$514$$ −3.22616 −0.142300
$$515$$ 0 0
$$516$$ −2.21663 −0.0975817
$$517$$ − 26.6550i − 1.17228i
$$518$$ 13.4983i 0.593081i
$$519$$ 0.843326 0.0370179
$$520$$ 0 0
$$521$$ −23.3522 −1.02308 −0.511539 0.859260i $$-0.670924\pi$$
−0.511539 + 0.859260i $$0.670924\pi$$
$$522$$ − 1.73501i − 0.0759394i
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 11.1778 0.488304
$$525$$ 0 0
$$526$$ −2.31335 −0.100867
$$527$$ − 6.16724i − 0.268649i
$$528$$ 17.2333i 0.749983i
$$529$$ −17.1708 −0.746557
$$530$$ 0 0
$$531$$ −11.2544 −0.488400
$$532$$ − 24.2922i − 1.05320i
$$533$$ − 4.33804i − 0.187902i
$$534$$ 0.0977518 0.00423014
$$535$$ 0 0
$$536$$ −8.21560 −0.354860
$$537$$ 18.9894i 0.819455i
$$538$$ 5.40054i 0.232834i
$$539$$ −84.4182 −3.63615
$$540$$ 0 0
$$541$$ 16.1744 0.695391 0.347695 0.937608i $$-0.386964\pi$$
0.347695 + 0.937608i $$0.386964\pi$$
$$542$$ − 1.90225i − 0.0817086i
$$543$$ 17.4947i 0.750770i
$$544$$ −14.2227 −0.609795
$$545$$ 0 0
$$546$$ 1.42166 0.0608416
$$547$$ 9.68665i 0.414171i 0.978323 + 0.207086i $$0.0663979\pi$$
−0.978323 + 0.207086i $$0.933602\pi$$
$$548$$ − 31.6373i − 1.35148i
$$549$$ 10.1708 0.434079
$$550$$ 0 0
$$551$$ 15.4700 0.659045
$$552$$ 7.17780i 0.305507i
$$553$$ 17.1814i 0.730626i
$$554$$ 7.42166 0.315316
$$555$$ 0 0
$$556$$ −14.3627 −0.609116
$$557$$ 0.647822i 0.0274491i 0.999906 + 0.0137246i $$0.00436880\pi$$
−0.999906 + 0.0137246i $$0.995631\pi$$
$$558$$ 0.411100i 0.0174033i
$$559$$ 1.15667 0.0489221
$$560$$ 0 0
$$561$$ 21.3275 0.900447
$$562$$ − 0.912811i − 0.0385046i
$$563$$ 16.3169i 0.687676i 0.939029 + 0.343838i $$0.111727\pi$$
−0.939029 + 0.343838i $$0.888273\pi$$
$$564$$ 10.3900 0.437497
$$565$$ 0 0
$$566$$ 1.00342 0.0421769
$$567$$ 4.91638i 0.206469i
$$568$$ − 1.03780i − 0.0435450i
$$569$$ −44.6550 −1.87203 −0.936017 0.351956i $$-0.885517\pi$$
−0.936017 + 0.351956i $$0.885517\pi$$
$$570$$ 0 0
$$571$$ −6.67252 −0.279236 −0.139618 0.990205i $$-0.544587\pi$$
−0.139618 + 0.990205i $$0.544587\pi$$
$$572$$ − 9.42166i − 0.393940i
$$573$$ − 22.5089i − 0.940321i
$$574$$ 6.16724 0.257415
$$575$$ 0 0
$$576$$ −6.06249 −0.252604
$$577$$ 15.3275i 0.638091i 0.947739 + 0.319046i $$0.103362\pi$$
−0.947739 + 0.319046i $$0.896638\pi$$
$$578$$ 0.525891i 0.0218742i
$$579$$ 2.65139 0.110188
$$580$$ 0 0
$$581$$ 55.3311 2.29552
$$582$$ − 3.56777i − 0.147889i
$$583$$ 1.66196i 0.0688312i
$$584$$ 3.57492 0.147931
$$585$$ 0 0
$$586$$ −8.28611 −0.342296
$$587$$ 12.2650i 0.506230i 0.967436 + 0.253115i $$0.0814551\pi$$
−0.967436 + 0.253115i $$0.918545\pi$$
$$588$$ − 32.9058i − 1.35701i
$$589$$ −3.66553 −0.151035
$$590$$ 0 0
$$591$$ 12.9894 0.534314
$$592$$ 33.2817i 1.36787i
$$593$$ 5.85389i 0.240390i 0.992750 + 0.120195i $$0.0383520\pi$$
−0.992750 + 0.120195i $$0.961648\pi$$
$$594$$ −1.42166 −0.0583315
$$595$$ 0 0
$$596$$ −39.2555 −1.60797
$$597$$ 2.84333i 0.116370i
$$598$$ − 1.83276i − 0.0749473i
$$599$$ −27.1355 −1.10873 −0.554364 0.832274i $$-0.687039\pi$$
−0.554364 + 0.832274i $$0.687039\pi$$
$$600$$ 0 0
$$601$$ 34.1708 1.39386 0.696928 0.717141i $$-0.254550\pi$$
0.696928 + 0.717141i $$0.254550\pi$$
$$602$$ 1.64440i 0.0670208i
$$603$$ 7.25443i 0.295423i
$$604$$ −31.4499 −1.27968
$$605$$ 0 0
$$606$$ 3.08719 0.125408
$$607$$ 10.3133i 0.418606i 0.977851 + 0.209303i $$0.0671195\pi$$
−0.977851 + 0.209303i $$0.932881\pi$$
$$608$$ 8.45335i 0.342829i
$$609$$ 29.4983 1.19533
$$610$$ 0 0
$$611$$ −5.42166 −0.219337
$$612$$ 8.31335i 0.336047i
$$613$$ − 0.484156i − 0.0195549i −0.999952 0.00977744i $$-0.996888\pi$$
0.999952 0.00977744i $$-0.00311230\pi$$
$$614$$ −0.557212 −0.0224872
$$615$$ 0 0
$$616$$ 27.3733 1.10290
$$617$$ − 7.15667i − 0.288117i −0.989569 0.144058i $$-0.953985\pi$$
0.989569 0.144058i $$-0.0460153\pi$$
$$618$$ 4.19550i 0.168768i
$$619$$ −5.42166 −0.217915 −0.108958 0.994046i $$-0.534751\pi$$
−0.108958 + 0.994046i $$0.534751\pi$$
$$620$$ 0 0
$$621$$ 6.33804 0.254337
$$622$$ 8.66895i 0.347593i
$$623$$ 1.66196i 0.0665848i
$$624$$ 3.50528 0.140324
$$625$$ 0 0
$$626$$ 4.71731 0.188542
$$627$$ − 12.6761i − 0.506234i
$$628$$ − 41.5194i − 1.65681i
$$629$$ 41.1885 1.64229
$$630$$ 0 0
$$631$$ −10.7244 −0.426934 −0.213467 0.976950i $$-0.568476\pi$$
−0.213467 + 0.976950i $$0.568476\pi$$
$$632$$ − 3.95775i − 0.157431i
$$633$$ 6.31335i 0.250933i
$$634$$ 8.87056 0.352295
$$635$$ 0 0
$$636$$ −0.647822 −0.0256878
$$637$$ 17.1708i 0.680332i
$$638$$ 8.52998i 0.337705i
$$639$$ −0.916382 −0.0362515
$$640$$ 0 0
$$641$$ −0.362741 −0.0143274 −0.00716370 0.999974i $$-0.502280\pi$$
−0.00716370 + 0.999974i $$0.502280\pi$$
$$642$$ 1.20607i 0.0475996i
$$643$$ 9.39697i 0.370580i 0.982684 + 0.185290i $$0.0593225\pi$$
−0.982684 + 0.185290i $$0.940678\pi$$
$$644$$ −59.7149 −2.35310
$$645$$ 0 0
$$646$$ 3.23433 0.127253
$$647$$ 18.0036i 0.707793i 0.935285 + 0.353897i $$0.115144\pi$$
−0.935285 + 0.353897i $$0.884856\pi$$
$$648$$ − 1.13249i − 0.0444886i
$$649$$ 55.3311 2.17193
$$650$$ 0 0
$$651$$ −6.98944 −0.273938
$$652$$ − 11.6383i − 0.455791i
$$653$$ 34.8222i 1.36270i 0.731959 + 0.681349i $$0.238606\pi$$
−0.731959 + 0.681349i $$0.761394\pi$$
$$654$$ 1.10831 0.0433385
$$655$$ 0 0
$$656$$ 15.2061 0.593697
$$657$$ − 3.15667i − 0.123154i
$$658$$ − 7.70778i − 0.300480i
$$659$$ −11.4700 −0.446809 −0.223404 0.974726i $$-0.571717\pi$$
−0.223404 + 0.974726i $$0.571717\pi$$
$$660$$ 0 0
$$661$$ 12.1672 0.473251 0.236625 0.971601i $$-0.423959\pi$$
0.236625 + 0.971601i $$0.423959\pi$$
$$662$$ 2.91995i 0.113487i
$$663$$ − 4.33804i − 0.168476i
$$664$$ −12.7456 −0.494624
$$665$$ 0 0
$$666$$ −2.74557 −0.106389
$$667$$ − 38.0283i − 1.47246i
$$668$$ − 1.42880i − 0.0552821i
$$669$$ 19.2544 0.744419
$$670$$ 0 0
$$671$$ −50.0036 −1.93037
$$672$$ 16.1189i 0.621799i
$$673$$ 27.9789i 1.07851i 0.842144 + 0.539253i $$0.181293\pi$$
−0.842144 + 0.539253i $$0.818707\pi$$
$$674$$ −0.382833 −0.0147462
$$675$$ 0 0
$$676$$ −1.91638 −0.0737070
$$677$$ 22.9930i 0.883693i 0.897091 + 0.441847i $$0.145676\pi$$
−0.897091 + 0.441847i $$0.854324\pi$$
$$678$$ 0.243863i 0.00936551i
$$679$$ 60.6585 2.32786
$$680$$ 0 0
$$681$$ 13.0872 0.501502
$$682$$ − 2.02113i − 0.0773929i
$$683$$ 28.6066i 1.09460i 0.836936 + 0.547301i $$0.184345\pi$$
−0.836936 + 0.547301i $$0.815655\pi$$
$$684$$ 4.94108 0.188927
$$685$$ 0 0
$$686$$ −14.4595 −0.552065
$$687$$ 24.5089i 0.935071i
$$688$$ 4.05447i 0.154575i
$$689$$ 0.338044 0.0128785
$$690$$ 0 0
$$691$$ −19.4005 −0.738031 −0.369016 0.929423i $$-0.620305\pi$$
−0.369016 + 0.929423i $$0.620305\pi$$
$$692$$ − 1.61613i − 0.0614362i
$$693$$ − 24.1708i − 0.918173i
$$694$$ −2.16724 −0.0822672
$$695$$ 0 0
$$696$$ −6.79497 −0.257563
$$697$$ − 18.8186i − 0.712806i
$$698$$ − 6.40396i − 0.242393i
$$699$$ 8.33804 0.315374
$$700$$ 0 0
$$701$$ −38.9683 −1.47181 −0.735906 0.677083i $$-0.763244\pi$$
−0.735906 + 0.677083i $$0.763244\pi$$
$$702$$ 0.289169i 0.0109140i
$$703$$ − 24.4806i − 0.923303i
$$704$$ 29.8055 1.12334
$$705$$ 0 0
$$706$$ −1.30382 −0.0490698
$$707$$ 52.4877i 1.97400i
$$708$$ 21.5678i 0.810567i
$$709$$ 17.5194 0.657955 0.328978 0.944338i $$-0.393296\pi$$
0.328978 + 0.944338i $$0.393296\pi$$
$$710$$ 0 0
$$711$$ −3.49472 −0.131062
$$712$$ − 0.382833i − 0.0143473i
$$713$$ 9.01056i 0.337448i
$$714$$ 6.16724 0.230803
$$715$$ 0 0
$$716$$ 36.3910 1.36000
$$717$$ − 8.91638i − 0.332988i
$$718$$ − 5.90225i − 0.220270i
$$719$$ 4.33447 0.161649 0.0808243 0.996728i $$-0.474245\pi$$
0.0808243 + 0.996728i $$0.474245\pi$$
$$720$$ 0 0
$$721$$ −71.3311 −2.65651
$$722$$ 3.57186i 0.132931i
$$723$$ − 6.00000i − 0.223142i
$$724$$ 33.5266 1.24600
$$725$$ 0 0
$$726$$ 3.80858 0.141350
$$727$$ 22.1672i 0.822137i 0.911604 + 0.411069i $$0.134844\pi$$
−0.911604 + 0.411069i $$0.865156\pi$$
$$728$$ − 5.56777i − 0.206355i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 5.01770 0.185586
$$732$$ − 19.4911i − 0.720414i
$$733$$ 2.83976i 0.104889i 0.998624 + 0.0524444i $$0.0167012\pi$$
−0.998624 + 0.0524444i $$0.983299\pi$$
$$734$$ −2.98230 −0.110079
$$735$$ 0 0
$$736$$ 20.7799 0.765959
$$737$$ − 35.6655i − 1.31376i
$$738$$ 1.25443i 0.0461761i
$$739$$ 43.9305 1.61601 0.808005 0.589176i $$-0.200547\pi$$
0.808005 + 0.589176i $$0.200547\pi$$
$$740$$ 0 0
$$741$$ −2.57834 −0.0947176
$$742$$ 0.480585i 0.0176428i
$$743$$ − 41.0872i − 1.50734i −0.657251 0.753671i $$-0.728281\pi$$
0.657251 0.753671i $$-0.271719\pi$$
$$744$$ 1.61003 0.0590264
$$745$$ 0 0
$$746$$ −5.40054 −0.197728
$$747$$ 11.2544i 0.411778i
$$748$$ − 40.8716i − 1.49441i
$$749$$ −20.5053 −0.749247
$$750$$ 0 0
$$751$$ 23.6902 0.864468 0.432234 0.901761i $$-0.357725\pi$$
0.432234 + 0.901761i $$0.357725\pi$$
$$752$$ − 19.0045i − 0.693021i
$$753$$ − 6.31335i − 0.230071i
$$754$$ 1.73501 0.0631854
$$755$$ 0 0
$$756$$ 9.42166 0.342663
$$757$$ 9.32391i 0.338883i 0.985540 + 0.169442i $$0.0541964\pi$$
−0.985540 + 0.169442i $$0.945804\pi$$
$$758$$ − 8.31438i − 0.301992i
$$759$$ −31.1602 −1.13105
$$760$$ 0 0
$$761$$ −42.8222 −1.55230 −0.776152 0.630546i $$-0.782831\pi$$
−0.776152 + 0.630546i $$0.782831\pi$$
$$762$$ 0.529977i 0.0191991i
$$763$$ 18.8433i 0.682174i
$$764$$ −43.1355 −1.56059
$$765$$ 0 0
$$766$$ 4.11888 0.148821
$$767$$ − 11.2544i − 0.406374i
$$768$$ 9.72191i 0.350809i
$$769$$ 17.3239 0.624716 0.312358 0.949964i $$-0.398881\pi$$
0.312358 + 0.949964i $$0.398881\pi$$
$$770$$ 0 0
$$771$$ 11.1567 0.401798
$$772$$ − 5.08108i − 0.182872i
$$773$$ 11.6373i 0.418563i 0.977855 + 0.209282i $$0.0671125\pi$$
−0.977855 + 0.209282i $$0.932887\pi$$
$$774$$ −0.334474 −0.0120224
$$775$$ 0 0
$$776$$ −13.9728 −0.501593
$$777$$ − 46.6797i − 1.67462i
$$778$$ 10.0272i 0.359494i
$$779$$ −11.1849 −0.400742
$$780$$ 0 0
$$781$$ 4.50528 0.161212
$$782$$ − 7.95061i − 0.284313i
$$783$$ 6.00000i 0.214423i
$$784$$ −60.1885 −2.14959
$$785$$ 0 0
$$786$$ 1.68665 0.0601609
$$787$$ 20.9411i 0.746469i 0.927737 + 0.373234i $$0.121751\pi$$
−0.927737 + 0.373234i $$0.878249\pi$$
$$788$$ − 24.8927i − 0.886766i
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ −4.14611 −0.147419
$$792$$ 5.56777i 0.197842i
$$793$$ 10.1708i 0.361176i
$$794$$ 2.07663 0.0736967
$$795$$ 0 0
$$796$$ 5.44890 0.193131
$$797$$ 20.3380i 0.720410i 0.932873 + 0.360205i $$0.117293\pi$$
−0.932873 + 0.360205i $$0.882707\pi$$
$$798$$ − 3.66553i − 0.129758i
$$799$$ −23.5194 −0.832057
$$800$$ 0 0
$$801$$ −0.338044 −0.0119442
$$802$$ − 10.9482i − 0.386595i
$$803$$ 15.5194i 0.547668i
$$804$$ 13.9022 0.490294
$$805$$ 0 0
$$806$$ −0.411100 −0.0144804
$$807$$ − 18.6761i − 0.657429i
$$808$$ − 12.0906i − 0.425346i
$$809$$ −7.68665 −0.270248 −0.135124 0.990829i $$-0.543143\pi$$
−0.135124 + 0.990829i $$0.543143\pi$$
$$810$$ 0 0
$$811$$ −44.4111 −1.55948 −0.779742 0.626101i $$-0.784650\pi$$
−0.779742 + 0.626101i $$0.784650\pi$$
$$812$$ − 56.5300i − 1.98381i
$$813$$ 6.57834i 0.230712i
$$814$$ 13.4983 0.473115
$$815$$ 0 0
$$816$$ 15.2061 0.532319
$$817$$ − 2.98230i − 0.104337i
$$818$$ − 4.04836i − 0.141548i
$$819$$ −4.91638 −0.171792
$$820$$ 0 0
$$821$$ −46.4630 −1.62157 −0.810785 0.585343i $$-0.800960\pi$$
−0.810785 + 0.585343i $$0.800960\pi$$
$$822$$ − 4.77384i − 0.166507i
$$823$$ − 46.5089i − 1.62120i −0.585603 0.810598i $$-0.699142\pi$$
0.585603 0.810598i $$-0.300858\pi$$
$$824$$ 16.4312 0.572408
$$825$$ 0 0
$$826$$ 16.0000 0.556711
$$827$$ 39.4005i 1.37009i 0.728500 + 0.685045i $$0.240218\pi$$
−0.728500 + 0.685045i $$0.759782\pi$$
$$828$$ − 12.1461i − 0.422107i
$$829$$ −47.6444 −1.65476 −0.827379 0.561644i $$-0.810169\pi$$
−0.827379 + 0.561644i $$0.810169\pi$$
$$830$$ 0 0
$$831$$ −25.6655 −0.890327
$$832$$ − 6.06249i − 0.210179i
$$833$$ 74.4877i 2.58085i
$$834$$ −2.16724 −0.0750453
$$835$$ 0 0
$$836$$ −24.2922 −0.840164
$$837$$ − 1.42166i − 0.0491399i
$$838$$ − 9.68665i − 0.334620i
$$839$$ 39.9058 1.37770 0.688851 0.724903i $$-0.258115\pi$$
0.688851 + 0.724903i $$0.258115\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 3.90939i − 0.134726i
$$843$$ 3.15667i 0.108722i
$$844$$ 12.0988 0.416457
$$845$$ 0 0
$$846$$ 1.56777 0.0539012
$$847$$ 64.7527i 2.22493i
$$848$$ 1.18494i 0.0406910i
$$849$$ −3.47002 −0.119091
$$850$$ 0 0
$$851$$ −60.1779 −2.06287
$$852$$ 1.75614i 0.0601643i
$$853$$ − 29.5019i − 1.01012i −0.863083 0.505062i $$-0.831470\pi$$
0.863083 0.505062i $$-0.168530\pi$$
$$854$$ −14.4595 −0.494793
$$855$$ 0 0
$$856$$ 4.72342 0.161443
$$857$$ 8.33804i 0.284822i 0.989808 + 0.142411i $$0.0454855\pi$$
−0.989808 + 0.142411i $$0.954515\pi$$
$$858$$ − 1.42166i − 0.0485348i
$$859$$ 4.17081 0.142306 0.0711531 0.997465i $$-0.477332\pi$$
0.0711531 + 0.997465i $$0.477332\pi$$
$$860$$ 0 0
$$861$$ −21.3275 −0.726839
$$862$$ − 3.58890i − 0.122238i
$$863$$ 3.93051i 0.133796i 0.997760 + 0.0668981i $$0.0213103\pi$$
−0.997760 + 0.0668981i $$0.978690\pi$$
$$864$$ −3.27861 −0.111540
$$865$$ 0 0
$$866$$ 5.00953 0.170231
$$867$$ − 1.81863i − 0.0617639i
$$868$$ 13.3944i 0.454637i
$$869$$ 17.1814 0.582838
$$870$$ 0 0
$$871$$ −7.25443 −0.245807
$$872$$ − 4.34058i − 0.146991i
$$873$$ 12.3380i 0.417580i
$$874$$ −4.72548 −0.159842
$$875$$ 0 0
$$876$$ −6.04939 −0.204390
$$877$$ − 23.0177i − 0.777253i −0.921395 0.388626i $$-0.872950\pi$$
0.921395 0.388626i $$-0.127050\pi$$
$$878$$ 0.188362i 0.00635692i
$$879$$ 28.6550 0.966508
$$880$$ 0 0
$$881$$ −15.3522 −0.517228 −0.258614 0.965981i $$-0.583266\pi$$
−0.258614 + 0.965981i $$0.583266\pi$$
$$882$$ − 4.96526i − 0.167189i
$$883$$ − 42.8011i − 1.44037i −0.693782 0.720185i $$-0.744057\pi$$
0.693782 0.720185i $$-0.255943\pi$$
$$884$$ −8.31335 −0.279608
$$885$$ 0 0
$$886$$ 2.55824 0.0859459
$$887$$ − 53.1885i − 1.78590i −0.450160 0.892948i $$-0.648633\pi$$
0.450160 0.892948i $$-0.351367\pi$$
$$888$$ 10.7527i 0.360837i
$$889$$ −9.01056 −0.302205
$$890$$ 0 0
$$891$$ 4.91638 0.164705
$$892$$ − 36.8988i − 1.23546i
$$893$$ 13.9789i 0.467785i
$$894$$ −5.92337 −0.198107
$$895$$ 0 0
$$896$$ 40.8566 1.36492
$$897$$ 6.33804i 0.211621i
$$898$$ 1.25443i 0.0418607i
$$899$$ −8.52998 −0.284491
$$900$$ 0 0
$$901$$ 1.46645 0.0488546
$$902$$ − 6.16724i − 0.205347i
$$903$$ − 5.68665i − 0.189240i
$$904$$ 0.955062 0.0317649
$$905$$ 0 0
$$906$$ −4.74557 −0.157661
$$907$$ 11.8116i 0.392199i 0.980584 + 0.196099i $$0.0628276\pi$$
−0.980584 + 0.196099i $$0.937172\pi$$
$$908$$ − 25.0800i − 0.832311i
$$909$$ −10.6761 −0.354104
$$910$$ 0 0
$$911$$ 44.1955 1.46426 0.732131 0.681164i $$-0.238526\pi$$
0.732131 + 0.681164i $$0.238526\pi$$
$$912$$ − 9.03780i − 0.299271i
$$913$$ − 55.3311i − 1.83119i
$$914$$ 4.43223 0.146605
$$915$$ 0 0
$$916$$ 46.9683 1.55188
$$917$$ 28.6761i 0.946968i
$$918$$ 1.25443i 0.0414022i
$$919$$ −55.2096 −1.82120 −0.910599 0.413291i $$-0.864379\pi$$
−0.910599 + 0.413291i $$0.864379\pi$$
$$920$$ 0 0
$$921$$ 1.92694 0.0634950
$$922$$ 3.42880i 0.112922i
$$923$$ − 0.916382i − 0.0301631i
$$924$$ −46.3205 −1.52383
$$925$$ 0 0
$$926$$ 7.63829 0.251010
$$927$$ − 14.5089i − 0.476533i
$$928$$ 19.6716i 0.645753i
$$929$$ −22.9930 −0.754376 −0.377188 0.926137i $$-0.623109\pi$$
−0.377188 + 0.926137i $$0.623109\pi$$
$$930$$ 0 0
$$931$$ 44.2721 1.45096
$$932$$ − 15.9789i − 0.523405i
$$933$$ − 29.9789i − 0.981464i
$$934$$ −9.73604 −0.318573
$$935$$ 0 0
$$936$$ 1.13249 0.0370167
$$937$$ 7.97887i 0.260658i 0.991471 + 0.130329i $$0.0416034\pi$$
−0.991471 + 0.130329i $$0.958397\pi$$
$$938$$ − 10.3133i − 0.336743i
$$939$$ −16.3133 −0.532366
$$940$$ 0 0
$$941$$ 41.5019 1.35292 0.676461 0.736478i $$-0.263513\pi$$
0.676461 + 0.736478i $$0.263513\pi$$
$$942$$ − 6.26499i − 0.204124i
$$943$$ 27.4947i 0.895351i
$$944$$ 39.4499 1.28399
$$945$$ 0 0
$$946$$ 1.64440 0.0534641
$$947$$ 47.4499i 1.54192i 0.636886 + 0.770958i $$0.280222\pi$$
−0.636886 + 0.770958i $$0.719778\pi$$
$$948$$ 6.69721i 0.217515i
$$949$$ 3.15667 0.102470
$$950$$ 0 0
$$951$$ −30.6761 −0.994740
$$952$$ − 24.1533i − 0.782811i
$$953$$ − 30.3663i − 0.983661i −0.870691 0.491831i $$-0.836328\pi$$
0.870691 0.491831i $$-0.163672\pi$$
$$954$$ −0.0977518 −0.00316483
$$955$$ 0 0
$$956$$ −17.0872 −0.552639
$$957$$ − 29.4983i − 0.953544i
$$958$$ 3.10831i 0.100425i
$$959$$ 81.1638 2.62092
$$960$$ 0 0
$$961$$ −28.9789 −0.934802
$$962$$ − 2.74557i − 0.0885209i
$$963$$ − 4.17081i − 0.134402i
$$964$$ −11.4983 −0.370335
$$965$$ 0 0
$$966$$ −9.01056 −0.289910
$$967$$ − 41.0943i − 1.32150i −0.750604 0.660752i $$-0.770237\pi$$
0.750604 0.660752i $$-0.229763\pi$$
$$968$$ − 14.9159i − 0.479414i
$$969$$ −11.1849 −0.359312
$$970$$ 0 0
$$971$$ 38.1744 1.22507 0.612537 0.790442i $$-0.290149\pi$$
0.612537 + 0.790442i $$0.290149\pi$$
$$972$$ 1.91638i 0.0614680i
$$973$$ − 36.8469i − 1.18126i
$$974$$ 6.57834 0.210784
$$975$$ 0 0
$$976$$ −35.6515 −1.14118
$$977$$ − 10.4806i − 0.335304i −0.985846 0.167652i $$-0.946382\pi$$
0.985846 0.167652i $$-0.0536184\pi$$
$$978$$ − 1.75614i − 0.0561551i
$$979$$ 1.66196 0.0531163
$$980$$ 0 0
$$981$$ −3.83276 −0.122371
$$982$$ 5.11442i 0.163208i
$$983$$ 0.0766264i 0.00244400i 0.999999 + 0.00122200i $$0.000388975\pi$$
−0.999999 + 0.00122200i $$0.999611\pi$$
$$984$$ 4.91281 0.156615
$$985$$ 0 0
$$986$$ 7.52656 0.239694
$$987$$ 26.6550i 0.848437i
$$988$$ 4.94108i 0.157197i
$$989$$ −7.33105 −0.233114
$$990$$ 0 0
$$991$$ −13.8575 −0.440197 −0.220098 0.975478i $$-0.570638\pi$$
−0.220098 + 0.975478i $$0.570638\pi$$
$$992$$ − 4.66107i − 0.147989i
$$993$$ − 10.0978i − 0.320442i
$$994$$ 1.30279 0.0413219
$$995$$ 0 0
$$996$$ 21.5678 0.683401
$$997$$ − 10.3416i − 0.327522i −0.986500 0.163761i $$-0.947637\pi$$
0.986500 0.163761i $$-0.0523626\pi$$
$$998$$ 5.76328i 0.182433i
$$999$$ 9.49472 0.300400
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 975.2.c.i.274.3 6
3.2 odd 2 2925.2.c.w.2224.4 6
5.2 odd 4 975.2.a.o.1.2 3
5.3 odd 4 195.2.a.e.1.2 3
5.4 even 2 inner 975.2.c.i.274.4 6
15.2 even 4 2925.2.a.bh.1.2 3
15.8 even 4 585.2.a.n.1.2 3
15.14 odd 2 2925.2.c.w.2224.3 6
20.3 even 4 3120.2.a.bj.1.1 3
35.13 even 4 9555.2.a.bq.1.2 3
60.23 odd 4 9360.2.a.dd.1.1 3
65.38 odd 4 2535.2.a.bc.1.2 3
195.38 even 4 7605.2.a.bx.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.2 3 5.3 odd 4
585.2.a.n.1.2 3 15.8 even 4
975.2.a.o.1.2 3 5.2 odd 4
975.2.c.i.274.3 6 1.1 even 1 trivial
975.2.c.i.274.4 6 5.4 even 2 inner
2535.2.a.bc.1.2 3 65.38 odd 4
2925.2.a.bh.1.2 3 15.2 even 4
2925.2.c.w.2224.3 6 15.14 odd 2
2925.2.c.w.2224.4 6 3.2 odd 2
3120.2.a.bj.1.1 3 20.3 even 4
7605.2.a.bx.1.2 3 195.38 even 4
9360.2.a.dd.1.1 3 60.23 odd 4
9555.2.a.bq.1.2 3 35.13 even 4