# Properties

 Label 1575.4.a.p.1.1 Level $1575$ Weight $4$ Character 1575.1 Self dual yes Analytic conductor $92.928$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.9280082590$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 Root $$4.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.27492 q^{2} +19.8248 q^{4} -7.00000 q^{7} -62.3746 q^{8} +O(q^{10})$$ $$q-5.27492 q^{2} +19.8248 q^{4} -7.00000 q^{7} -62.3746 q^{8} -34.7492 q^{11} +37.2990 q^{13} +36.9244 q^{14} +170.423 q^{16} -10.5498 q^{17} -58.5980 q^{19} +183.299 q^{22} -125.347 q^{23} -196.749 q^{26} -138.773 q^{28} +35.4020 q^{29} +291.794 q^{31} -399.969 q^{32} +55.6495 q^{34} +259.897 q^{37} +309.100 q^{38} +338.248 q^{41} -6.80397 q^{43} -688.894 q^{44} +661.196 q^{46} +250.694 q^{47} +49.0000 q^{49} +739.444 q^{52} -536.900 q^{53} +436.622 q^{56} -186.743 q^{58} +35.8904 q^{59} +57.7940 q^{61} -1539.19 q^{62} +746.423 q^{64} -481.691 q^{67} -209.148 q^{68} -363.752 q^{71} -581.299 q^{73} -1370.94 q^{74} -1161.69 q^{76} +243.244 q^{77} -693.691 q^{79} -1784.23 q^{82} +1334.39 q^{83} +35.8904 q^{86} +2167.47 q^{88} +353.038 q^{89} -261.093 q^{91} -2484.98 q^{92} -1322.39 q^{94} -1445.88 q^{97} -258.471 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + O(q^{10})$$ $$2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.27492 −1.86496 −0.932482 0.361215i $$-0.882362\pi$$
−0.932482 + 0.361215i $$0.882362\pi$$
$$3$$ 0 0
$$4$$ 19.8248 2.47809
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ −62.3746 −2.75659
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −34.7492 −0.952479 −0.476240 0.879316i $$-0.658000\pi$$
−0.476240 + 0.879316i $$0.658000\pi$$
$$12$$ 0 0
$$13$$ 37.2990 0.795760 0.397880 0.917437i $$-0.369746\pi$$
0.397880 + 0.917437i $$0.369746\pi$$
$$14$$ 36.9244 0.704890
$$15$$ 0 0
$$16$$ 170.423 2.66286
$$17$$ −10.5498 −0.150512 −0.0752562 0.997164i $$-0.523977\pi$$
−0.0752562 + 0.997164i $$0.523977\pi$$
$$18$$ 0 0
$$19$$ −58.5980 −0.707542 −0.353771 0.935332i $$-0.615101\pi$$
−0.353771 + 0.935332i $$0.615101\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 183.299 1.77634
$$23$$ −125.347 −1.13638 −0.568189 0.822898i $$-0.692356\pi$$
−0.568189 + 0.822898i $$0.692356\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −196.749 −1.48406
$$27$$ 0 0
$$28$$ −138.773 −0.936631
$$29$$ 35.4020 0.226689 0.113345 0.993556i $$-0.463844\pi$$
0.113345 + 0.993556i $$0.463844\pi$$
$$30$$ 0 0
$$31$$ 291.794 1.69057 0.845286 0.534313i $$-0.179430\pi$$
0.845286 + 0.534313i $$0.179430\pi$$
$$32$$ −399.969 −2.20954
$$33$$ 0 0
$$34$$ 55.6495 0.280700
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 259.897 1.15478 0.577389 0.816469i $$-0.304072\pi$$
0.577389 + 0.816469i $$0.304072\pi$$
$$38$$ 309.100 1.31954
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 338.248 1.28842 0.644212 0.764847i $$-0.277185\pi$$
0.644212 + 0.764847i $$0.277185\pi$$
$$42$$ 0 0
$$43$$ −6.80397 −0.0241301 −0.0120651 0.999927i $$-0.503841\pi$$
−0.0120651 + 0.999927i $$0.503841\pi$$
$$44$$ −688.894 −2.36033
$$45$$ 0 0
$$46$$ 661.196 2.11931
$$47$$ 250.694 0.778033 0.389016 0.921231i $$-0.372815\pi$$
0.389016 + 0.921231i $$0.372815\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 739.444 1.97197
$$53$$ −536.900 −1.39149 −0.695745 0.718289i $$-0.744925\pi$$
−0.695745 + 0.718289i $$0.744925\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 436.622 1.04189
$$57$$ 0 0
$$58$$ −186.743 −0.422767
$$59$$ 35.8904 0.0791955 0.0395977 0.999216i $$-0.487392\pi$$
0.0395977 + 0.999216i $$0.487392\pi$$
$$60$$ 0 0
$$61$$ 57.7940 0.121308 0.0606538 0.998159i $$-0.480681\pi$$
0.0606538 + 0.998159i $$0.480681\pi$$
$$62$$ −1539.19 −3.15286
$$63$$ 0 0
$$64$$ 746.423 1.45786
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −481.691 −0.878327 −0.439164 0.898407i $$-0.644725\pi$$
−0.439164 + 0.898407i $$0.644725\pi$$
$$68$$ −209.148 −0.372984
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −363.752 −0.608021 −0.304010 0.952669i $$-0.598326\pi$$
−0.304010 + 0.952669i $$0.598326\pi$$
$$72$$ 0 0
$$73$$ −581.299 −0.931999 −0.465999 0.884785i $$-0.654305\pi$$
−0.465999 + 0.884785i $$0.654305\pi$$
$$74$$ −1370.94 −2.15362
$$75$$ 0 0
$$76$$ −1161.69 −1.75336
$$77$$ 243.244 0.360003
$$78$$ 0 0
$$79$$ −693.691 −0.987928 −0.493964 0.869482i $$-0.664453\pi$$
−0.493964 + 0.869482i $$0.664453\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −1784.23 −2.40287
$$83$$ 1334.39 1.76468 0.882341 0.470611i $$-0.155967\pi$$
0.882341 + 0.470611i $$0.155967\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 35.8904 0.0450019
$$87$$ 0 0
$$88$$ 2167.47 2.62560
$$89$$ 353.038 0.420472 0.210236 0.977651i $$-0.432577\pi$$
0.210236 + 0.977651i $$0.432577\pi$$
$$90$$ 0 0
$$91$$ −261.093 −0.300769
$$92$$ −2484.98 −2.81605
$$93$$ 0 0
$$94$$ −1322.39 −1.45100
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1445.88 −1.51347 −0.756735 0.653722i $$-0.773207\pi$$
−0.756735 + 0.653722i $$0.773207\pi$$
$$98$$ −258.471 −0.266424
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −474.852 −0.467817 −0.233909 0.972259i $$-0.575152\pi$$
−0.233909 + 0.972259i $$0.575152\pi$$
$$102$$ 0 0
$$103$$ 1999.59 1.91287 0.956433 0.291951i $$-0.0943044\pi$$
0.956433 + 0.291951i $$0.0943044\pi$$
$$104$$ −2326.51 −2.19359
$$105$$ 0 0
$$106$$ 2832.10 2.59508
$$107$$ 1166.74 1.05414 0.527068 0.849823i $$-0.323291\pi$$
0.527068 + 0.849823i $$0.323291\pi$$
$$108$$ 0 0
$$109$$ −1337.18 −1.17503 −0.587515 0.809213i $$-0.699894\pi$$
−0.587515 + 0.809213i $$0.699894\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1192.96 −1.00646
$$113$$ 906.578 0.754723 0.377361 0.926066i $$-0.376831\pi$$
0.377361 + 0.926066i $$0.376831\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 701.836 0.561757
$$117$$ 0 0
$$118$$ −189.319 −0.147697
$$119$$ 73.8488 0.0568883
$$120$$ 0 0
$$121$$ −123.495 −0.0927836
$$122$$ −304.859 −0.226235
$$123$$ 0 0
$$124$$ 5784.74 4.18940
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1714.89 1.19820 0.599101 0.800674i $$-0.295525\pi$$
0.599101 + 0.800674i $$0.295525\pi$$
$$128$$ −737.564 −0.509313
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −470.611 −0.313874 −0.156937 0.987609i $$-0.550162\pi$$
−0.156937 + 0.987609i $$0.550162\pi$$
$$132$$ 0 0
$$133$$ 410.186 0.267426
$$134$$ 2540.88 1.63805
$$135$$ 0 0
$$136$$ 658.042 0.414901
$$137$$ −443.910 −0.276831 −0.138415 0.990374i $$-0.544201\pi$$
−0.138415 + 0.990374i $$0.544201\pi$$
$$138$$ 0 0
$$139$$ 1669.98 1.01904 0.509518 0.860460i $$-0.329824\pi$$
0.509518 + 0.860460i $$0.329824\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1918.76 1.13394
$$143$$ −1296.11 −0.757945
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3066.30 1.73814
$$147$$ 0 0
$$148$$ 5152.39 2.86165
$$149$$ −743.871 −0.408995 −0.204497 0.978867i $$-0.565556\pi$$
−0.204497 + 0.978867i $$0.565556\pi$$
$$150$$ 0 0
$$151$$ 606.764 0.327005 0.163503 0.986543i $$-0.447721\pi$$
0.163503 + 0.986543i $$0.447721\pi$$
$$152$$ 3655.03 1.95041
$$153$$ 0 0
$$154$$ −1283.09 −0.671393
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3114.78 −1.58336 −0.791678 0.610939i $$-0.790792\pi$$
−0.791678 + 0.610939i $$0.790792\pi$$
$$158$$ 3659.16 1.84245
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 877.430 0.429511
$$162$$ 0 0
$$163$$ −2413.07 −1.15955 −0.579774 0.814777i $$-0.696859\pi$$
−0.579774 + 0.814777i $$0.696859\pi$$
$$164$$ 6705.67 3.19284
$$165$$ 0 0
$$166$$ −7038.81 −3.29107
$$167$$ −610.475 −0.282874 −0.141437 0.989947i $$-0.545172\pi$$
−0.141437 + 0.989947i $$0.545172\pi$$
$$168$$ 0 0
$$169$$ −805.784 −0.366766
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −134.887 −0.0597968
$$173$$ 3793.81 1.66727 0.833636 0.552315i $$-0.186255\pi$$
0.833636 + 0.552315i $$0.186255\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −5922.05 −2.53631
$$177$$ 0 0
$$178$$ −1862.25 −0.784165
$$179$$ 2804.68 1.17112 0.585562 0.810627i $$-0.300874\pi$$
0.585562 + 0.810627i $$0.300874\pi$$
$$180$$ 0 0
$$181$$ 3106.04 1.27553 0.637763 0.770232i $$-0.279860\pi$$
0.637763 + 0.770232i $$0.279860\pi$$
$$182$$ 1377.24 0.560924
$$183$$ 0 0
$$184$$ 7818.48 3.13253
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 366.598 0.143360
$$188$$ 4969.95 1.92804
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −261.952 −0.0992365 −0.0496182 0.998768i $$-0.515800\pi$$
−0.0496182 + 0.998768i $$0.515800\pi$$
$$192$$ 0 0
$$193$$ −4051.07 −1.51089 −0.755447 0.655210i $$-0.772580\pi$$
−0.755447 + 0.655210i $$0.772580\pi$$
$$194$$ 7626.88 2.82257
$$195$$ 0 0
$$196$$ 971.413 0.354013
$$197$$ −2874.83 −1.03971 −0.519855 0.854254i $$-0.674014\pi$$
−0.519855 + 0.854254i $$0.674014\pi$$
$$198$$ 0 0
$$199$$ −3066.97 −1.09252 −0.546261 0.837615i $$-0.683949\pi$$
−0.546261 + 0.837615i $$0.683949\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2504.81 0.872463
$$203$$ −247.814 −0.0856804
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −10547.7 −3.56743
$$207$$ 0 0
$$208$$ 6356.60 2.11899
$$209$$ 2036.23 0.673919
$$210$$ 0 0
$$211$$ 595.422 0.194268 0.0971340 0.995271i $$-0.469032\pi$$
0.0971340 + 0.995271i $$0.469032\pi$$
$$212$$ −10643.9 −3.44824
$$213$$ 0 0
$$214$$ −6154.44 −1.96593
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2042.56 −0.638976
$$218$$ 7053.49 2.19139
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −393.498 −0.119772
$$222$$ 0 0
$$223$$ 3779.79 1.13504 0.567520 0.823360i $$-0.307903\pi$$
0.567520 + 0.823360i $$0.307903\pi$$
$$224$$ 2799.79 0.835127
$$225$$ 0 0
$$226$$ −4782.12 −1.40753
$$227$$ 1827.62 0.534376 0.267188 0.963644i $$-0.413906\pi$$
0.267188 + 0.963644i $$0.413906\pi$$
$$228$$ 0 0
$$229$$ −850.249 −0.245354 −0.122677 0.992447i $$-0.539148\pi$$
−0.122677 + 0.992447i $$0.539148\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2208.18 −0.624890
$$233$$ −6591.10 −1.85321 −0.926604 0.376039i $$-0.877286\pi$$
−0.926604 + 0.376039i $$0.877286\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 711.518 0.196254
$$237$$ 0 0
$$238$$ −389.547 −0.106095
$$239$$ 182.556 0.0494083 0.0247042 0.999695i $$-0.492136\pi$$
0.0247042 + 0.999695i $$0.492136\pi$$
$$240$$ 0 0
$$241$$ 1523.90 0.407315 0.203657 0.979042i $$-0.434717\pi$$
0.203657 + 0.979042i $$0.434717\pi$$
$$242$$ 651.426 0.173038
$$243$$ 0 0
$$244$$ 1145.75 0.300612
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2185.65 −0.563034
$$248$$ −18200.5 −4.66022
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2357.73 −0.592903 −0.296451 0.955048i $$-0.595803\pi$$
−0.296451 + 0.955048i $$0.595803\pi$$
$$252$$ 0 0
$$253$$ 4355.71 1.08238
$$254$$ −9045.89 −2.23460
$$255$$ 0 0
$$256$$ −2080.79 −0.508006
$$257$$ 2782.55 0.675372 0.337686 0.941259i $$-0.390356\pi$$
0.337686 + 0.941259i $$0.390356\pi$$
$$258$$ 0 0
$$259$$ −1819.28 −0.436465
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2482.44 0.585364
$$263$$ 2043.78 0.479183 0.239591 0.970874i $$-0.422987\pi$$
0.239591 + 0.970874i $$0.422987\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2163.70 −0.498740
$$267$$ 0 0
$$268$$ −9549.41 −2.17658
$$269$$ −3452.84 −0.782614 −0.391307 0.920260i $$-0.627977\pi$$
−0.391307 + 0.920260i $$0.627977\pi$$
$$270$$ 0 0
$$271$$ 2644.29 0.592728 0.296364 0.955075i $$-0.404226\pi$$
0.296364 + 0.955075i $$0.404226\pi$$
$$272$$ −1797.93 −0.400793
$$273$$ 0 0
$$274$$ 2341.59 0.516280
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2679.49 −0.581208 −0.290604 0.956843i $$-0.593856\pi$$
−0.290604 + 0.956843i $$0.593856\pi$$
$$278$$ −8809.01 −1.90046
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1019.69 0.216476 0.108238 0.994125i $$-0.465479\pi$$
0.108238 + 0.994125i $$0.465479\pi$$
$$282$$ 0 0
$$283$$ −432.206 −0.0907844 −0.0453922 0.998969i $$-0.514454\pi$$
−0.0453922 + 0.998969i $$0.514454\pi$$
$$284$$ −7211.30 −1.50673
$$285$$ 0 0
$$286$$ 6836.87 1.41354
$$287$$ −2367.73 −0.486979
$$288$$ 0 0
$$289$$ −4801.70 −0.977346
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −11524.1 −2.30958
$$293$$ −2245.92 −0.447809 −0.223904 0.974611i $$-0.571880\pi$$
−0.223904 + 0.974611i $$0.571880\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −16211.0 −3.18325
$$297$$ 0 0
$$298$$ 3923.86 0.762761
$$299$$ −4675.33 −0.904284
$$300$$ 0 0
$$301$$ 47.6278 0.00912034
$$302$$ −3200.63 −0.609853
$$303$$ 0 0
$$304$$ −9986.44 −1.88408
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3197.08 0.594354 0.297177 0.954822i $$-0.403955\pi$$
0.297177 + 0.954822i $$0.403955\pi$$
$$308$$ 4822.26 0.892122
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 3355.60 0.611829 0.305915 0.952059i $$-0.401038\pi$$
0.305915 + 0.952059i $$0.401038\pi$$
$$312$$ 0 0
$$313$$ 2256.39 0.407472 0.203736 0.979026i $$-0.434692\pi$$
0.203736 + 0.979026i $$0.434692\pi$$
$$314$$ 16430.2 2.95290
$$315$$ 0 0
$$316$$ −13752.3 −2.44818
$$317$$ −6139.19 −1.08773 −0.543866 0.839172i $$-0.683040\pi$$
−0.543866 + 0.839172i $$0.683040\pi$$
$$318$$ 0 0
$$319$$ −1230.19 −0.215917
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −4628.37 −0.801022
$$323$$ 618.199 0.106494
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12728.8 2.16252
$$327$$ 0 0
$$328$$ −21098.0 −3.55166
$$329$$ −1754.86 −0.294069
$$330$$ 0 0
$$331$$ 7029.81 1.16735 0.583676 0.811987i $$-0.301614\pi$$
0.583676 + 0.811987i $$0.301614\pi$$
$$332$$ 26454.0 4.37305
$$333$$ 0 0
$$334$$ 3220.21 0.527550
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −10328.4 −1.66951 −0.834757 0.550619i $$-0.814392\pi$$
−0.834757 + 0.550619i $$0.814392\pi$$
$$338$$ 4250.44 0.684005
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10139.6 −1.61024
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 424.395 0.0665170
$$345$$ 0 0
$$346$$ −20012.0 −3.10940
$$347$$ 1967.54 0.304389 0.152194 0.988351i $$-0.451366\pi$$
0.152194 + 0.988351i $$0.451366\pi$$
$$348$$ 0 0
$$349$$ −4365.46 −0.669564 −0.334782 0.942296i $$-0.608663\pi$$
−0.334782 + 0.942296i $$0.608663\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 13898.6 2.10454
$$353$$ −6071.59 −0.915462 −0.457731 0.889091i $$-0.651338\pi$$
−0.457731 + 0.889091i $$0.651338\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6998.90 1.04197
$$357$$ 0 0
$$358$$ −14794.4 −2.18411
$$359$$ −9638.04 −1.41693 −0.708463 0.705748i $$-0.750611\pi$$
−0.708463 + 0.705748i $$0.750611\pi$$
$$360$$ 0 0
$$361$$ −3425.27 −0.499384
$$362$$ −16384.1 −2.37881
$$363$$ 0 0
$$364$$ −5176.10 −0.745334
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −522.725 −0.0743488 −0.0371744 0.999309i $$-0.511836\pi$$
−0.0371744 + 0.999309i $$0.511836\pi$$
$$368$$ −21362.0 −3.02601
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3758.30 0.525934
$$372$$ 0 0
$$373$$ −3229.84 −0.448351 −0.224175 0.974549i $$-0.571969\pi$$
−0.224175 + 0.974549i $$0.571969\pi$$
$$374$$ −1933.77 −0.267361
$$375$$ 0 0
$$376$$ −15637.0 −2.14472
$$377$$ 1320.46 0.180390
$$378$$ 0 0
$$379$$ 6639.71 0.899892 0.449946 0.893056i $$-0.351443\pi$$
0.449946 + 0.893056i $$0.351443\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1381.77 0.185073
$$383$$ −14224.4 −1.89774 −0.948871 0.315664i $$-0.897773\pi$$
−0.948871 + 0.315664i $$0.897773\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 21369.1 2.81777
$$387$$ 0 0
$$388$$ −28664.2 −3.75052
$$389$$ −2921.82 −0.380828 −0.190414 0.981704i $$-0.560983\pi$$
−0.190414 + 0.981704i $$0.560983\pi$$
$$390$$ 0 0
$$391$$ 1322.39 0.171039
$$392$$ −3056.35 −0.393799
$$393$$ 0 0
$$394$$ 15164.5 1.93902
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −811.940 −0.102645 −0.0513226 0.998682i $$-0.516344\pi$$
−0.0513226 + 0.998682i $$0.516344\pi$$
$$398$$ 16178.0 2.03751
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2338.63 −0.291237 −0.145618 0.989341i $$-0.546517\pi$$
−0.145618 + 0.989341i $$0.546517\pi$$
$$402$$ 0 0
$$403$$ 10883.6 1.34529
$$404$$ −9413.83 −1.15930
$$405$$ 0 0
$$406$$ 1307.20 0.159791
$$407$$ −9031.21 −1.09990
$$408$$ 0 0
$$409$$ −2727.57 −0.329755 −0.164877 0.986314i $$-0.552723\pi$$
−0.164877 + 0.986314i $$0.552723\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 39641.3 4.74026
$$413$$ −251.233 −0.0299331
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −14918.5 −1.75826
$$417$$ 0 0
$$418$$ −10741.0 −1.25684
$$419$$ −13306.3 −1.55144 −0.775721 0.631076i $$-0.782614\pi$$
−0.775721 + 0.631076i $$0.782614\pi$$
$$420$$ 0 0
$$421$$ −11007.5 −1.27428 −0.637138 0.770750i $$-0.719882\pi$$
−0.637138 + 0.770750i $$0.719882\pi$$
$$422$$ −3140.80 −0.362303
$$423$$ 0 0
$$424$$ 33488.9 3.83577
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −404.558 −0.0458500
$$428$$ 23130.2 2.61225
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6525.62 0.729300 0.364650 0.931145i $$-0.381189\pi$$
0.364650 + 0.931145i $$0.381189\pi$$
$$432$$ 0 0
$$433$$ 11716.3 1.30034 0.650171 0.759788i $$-0.274697\pi$$
0.650171 + 0.759788i $$0.274697\pi$$
$$434$$ 10774.3 1.19167
$$435$$ 0 0
$$436$$ −26509.2 −2.91183
$$437$$ 7345.10 0.804036
$$438$$ 0 0
$$439$$ −14611.4 −1.58853 −0.794264 0.607573i $$-0.792143\pi$$
−0.794264 + 0.607573i $$0.792143\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 2075.67 0.223370
$$443$$ −15239.8 −1.63446 −0.817228 0.576314i $$-0.804490\pi$$
−0.817228 + 0.576314i $$0.804490\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −19938.1 −2.11681
$$447$$ 0 0
$$448$$ −5224.96 −0.551018
$$449$$ −10678.8 −1.12241 −0.561206 0.827676i $$-0.689662\pi$$
−0.561206 + 0.827676i $$0.689662\pi$$
$$450$$ 0 0
$$451$$ −11753.8 −1.22720
$$452$$ 17972.7 1.87027
$$453$$ 0 0
$$454$$ −9640.53 −0.996592
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4228.23 −0.432797 −0.216399 0.976305i $$-0.569431\pi$$
−0.216399 + 0.976305i $$0.569431\pi$$
$$458$$ 4484.99 0.457577
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −910.121 −0.0919492 −0.0459746 0.998943i $$-0.514639\pi$$
−0.0459746 + 0.998943i $$0.514639\pi$$
$$462$$ 0 0
$$463$$ −4456.16 −0.447290 −0.223645 0.974671i $$-0.571796\pi$$
−0.223645 + 0.974671i $$0.571796\pi$$
$$464$$ 6033.30 0.603640
$$465$$ 0 0
$$466$$ 34767.5 3.45617
$$467$$ −4429.42 −0.438907 −0.219453 0.975623i $$-0.570427\pi$$
−0.219453 + 0.975623i $$0.570427\pi$$
$$468$$ 0 0
$$469$$ 3371.84 0.331977
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −2238.65 −0.218310
$$473$$ 236.432 0.0229835
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1464.03 0.140975
$$477$$ 0 0
$$478$$ −962.970 −0.0921448
$$479$$ −2752.85 −0.262591 −0.131296 0.991343i $$-0.541914\pi$$
−0.131296 + 0.991343i $$0.541914\pi$$
$$480$$ 0 0
$$481$$ 9693.90 0.918927
$$482$$ −8038.43 −0.759628
$$483$$ 0 0
$$484$$ −2448.26 −0.229927
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 670.598 0.0623977 0.0311989 0.999513i $$-0.490067\pi$$
0.0311989 + 0.999513i $$0.490067\pi$$
$$488$$ −3604.88 −0.334396
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8244.70 0.757797 0.378898 0.925438i $$-0.376303\pi$$
0.378898 + 0.925438i $$0.376303\pi$$
$$492$$ 0 0
$$493$$ −373.485 −0.0341195
$$494$$ 11529.1 1.05004
$$495$$ 0 0
$$496$$ 49728.3 4.50175
$$497$$ 2546.27 0.229810
$$498$$ 0 0
$$499$$ 8164.91 0.732488 0.366244 0.930519i $$-0.380644\pi$$
0.366244 + 0.930519i $$0.380644\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12436.8 1.10574
$$503$$ 8175.59 0.724715 0.362357 0.932039i $$-0.381972\pi$$
0.362357 + 0.932039i $$0.381972\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −22976.0 −2.01859
$$507$$ 0 0
$$508$$ 33997.2 2.96926
$$509$$ 878.448 0.0764961 0.0382480 0.999268i $$-0.487822\pi$$
0.0382480 + 0.999268i $$0.487822\pi$$
$$510$$ 0 0
$$511$$ 4069.09 0.352262
$$512$$ 16876.5 1.45673
$$513$$ 0 0
$$514$$ −14677.7 −1.25955
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8711.42 −0.741060
$$518$$ 9596.55 0.813992
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11712.6 −0.984910 −0.492455 0.870338i $$-0.663900\pi$$
−0.492455 + 0.870338i $$0.663900\pi$$
$$522$$ 0 0
$$523$$ 7341.82 0.613834 0.306917 0.951736i $$-0.400703\pi$$
0.306917 + 0.951736i $$0.400703\pi$$
$$524$$ −9329.75 −0.777809
$$525$$ 0 0
$$526$$ −10780.8 −0.893659
$$527$$ −3078.38 −0.254452
$$528$$ 0 0
$$529$$ 3544.92 0.291355
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 8131.84 0.662707
$$533$$ 12616.3 1.02528
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 30045.3 2.42119
$$537$$ 0 0
$$538$$ 18213.4 1.45955
$$539$$ −1702.71 −0.136068
$$540$$ 0 0
$$541$$ −15868.7 −1.26109 −0.630545 0.776153i $$-0.717169\pi$$
−0.630545 + 0.776153i $$0.717169\pi$$
$$542$$ −13948.4 −1.10542
$$543$$ 0 0
$$544$$ 4219.61 0.332563
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −2315.26 −0.180975 −0.0904875 0.995898i $$-0.528843\pi$$
−0.0904875 + 0.995898i $$0.528843\pi$$
$$548$$ −8800.41 −0.686013
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2074.49 −0.160392
$$552$$ 0 0
$$553$$ 4855.84 0.373402
$$554$$ 14134.1 1.08393
$$555$$ 0 0
$$556$$ 33106.9 2.52526
$$557$$ −4819.05 −0.366588 −0.183294 0.983058i $$-0.558676\pi$$
−0.183294 + 0.983058i $$0.558676\pi$$
$$558$$ 0 0
$$559$$ −253.781 −0.0192018
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −5378.79 −0.403720
$$563$$ 2540.86 0.190203 0.0951017 0.995468i $$-0.469682\pi$$
0.0951017 + 0.995468i $$0.469682\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 2279.85 0.169310
$$567$$ 0 0
$$568$$ 22688.9 1.67607
$$569$$ 24220.0 1.78445 0.892227 0.451587i $$-0.149142\pi$$
0.892227 + 0.451587i $$0.149142\pi$$
$$570$$ 0 0
$$571$$ −11772.1 −0.862778 −0.431389 0.902166i $$-0.641976\pi$$
−0.431389 + 0.902166i $$0.641976\pi$$
$$572$$ −25695.1 −1.87826
$$573$$ 0 0
$$574$$ 12489.6 0.908198
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10584.3 −0.763655 −0.381827 0.924234i $$-0.624705\pi$$
−0.381827 + 0.924234i $$0.624705\pi$$
$$578$$ 25328.6 1.82272
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −9340.74 −0.666987
$$582$$ 0 0
$$583$$ 18656.8 1.32536
$$584$$ 36258.3 2.56914
$$585$$ 0 0
$$586$$ 11847.0 0.835148
$$587$$ −8712.63 −0.612621 −0.306311 0.951932i $$-0.599095\pi$$
−0.306311 + 0.951932i $$0.599095\pi$$
$$588$$ 0 0
$$589$$ −17098.6 −1.19615
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 44292.4 3.07501
$$593$$ −15362.9 −1.06387 −0.531937 0.846784i $$-0.678536\pi$$
−0.531937 + 0.846784i $$0.678536\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −14747.0 −1.01353
$$597$$ 0 0
$$598$$ 24662.0 1.68646
$$599$$ −26003.8 −1.77377 −0.886883 0.461994i $$-0.847134\pi$$
−0.886883 + 0.461994i $$0.847134\pi$$
$$600$$ 0 0
$$601$$ 20567.7 1.39596 0.697982 0.716115i $$-0.254082\pi$$
0.697982 + 0.716115i $$0.254082\pi$$
$$602$$ −251.233 −0.0170091
$$603$$ 0 0
$$604$$ 12029.0 0.810349
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −19642.1 −1.31342 −0.656711 0.754142i $$-0.728053\pi$$
−0.656711 + 0.754142i $$0.728053\pi$$
$$608$$ 23437.4 1.56334
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 9350.65 0.619127
$$612$$ 0 0
$$613$$ −8454.59 −0.557060 −0.278530 0.960428i $$-0.589847\pi$$
−0.278530 + 0.960428i $$0.589847\pi$$
$$614$$ −16864.3 −1.10845
$$615$$ 0 0
$$616$$ −15172.3 −0.992383
$$617$$ −24168.4 −1.57696 −0.788479 0.615061i $$-0.789131\pi$$
−0.788479 + 0.615061i $$0.789131\pi$$
$$618$$ 0 0
$$619$$ −2037.56 −0.132305 −0.0661523 0.997810i $$-0.521072\pi$$
−0.0661523 + 0.997810i $$0.521072\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −17700.5 −1.14104
$$623$$ −2471.27 −0.158923
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −11902.3 −0.759921
$$627$$ 0 0
$$628$$ −61749.8 −3.92370
$$629$$ −2741.87 −0.173808
$$630$$ 0 0
$$631$$ 12339.5 0.778489 0.389244 0.921135i $$-0.372736\pi$$
0.389244 + 0.921135i $$0.372736\pi$$
$$632$$ 43268.7 2.72332
$$633$$ 0 0
$$634$$ 32383.7 2.02858
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1827.65 0.113680
$$638$$ 6489.15 0.402677
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 10222.6 0.629906 0.314953 0.949107i $$-0.398011\pi$$
0.314953 + 0.949107i $$0.398011\pi$$
$$642$$ 0 0
$$643$$ 1211.75 0.0743187 0.0371594 0.999309i $$-0.488169\pi$$
0.0371594 + 0.999309i $$0.488169\pi$$
$$644$$ 17394.8 1.06437
$$645$$ 0 0
$$646$$ −3260.95 −0.198607
$$647$$ −2817.22 −0.171184 −0.0855922 0.996330i $$-0.527278\pi$$
−0.0855922 + 0.996330i $$0.527278\pi$$
$$648$$ 0 0
$$649$$ −1247.16 −0.0754320
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −47838.6 −2.87347
$$653$$ 20986.2 1.25766 0.628831 0.777542i $$-0.283534\pi$$
0.628831 + 0.777542i $$0.283534\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 57645.1 3.43089
$$657$$ 0 0
$$658$$ 9256.74 0.548428
$$659$$ 2384.09 0.140927 0.0704635 0.997514i $$-0.477552\pi$$
0.0704635 + 0.997514i $$0.477552\pi$$
$$660$$ 0 0
$$661$$ −7577.10 −0.445862 −0.222931 0.974834i $$-0.571562\pi$$
−0.222931 + 0.974834i $$0.571562\pi$$
$$662$$ −37081.7 −2.17707
$$663$$ 0 0
$$664$$ −83232.2 −4.86451
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4437.54 −0.257605
$$668$$ −12102.5 −0.700989
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2008.30 −0.115543
$$672$$ 0 0
$$673$$ −11724.6 −0.671547 −0.335774 0.941943i $$-0.608998\pi$$
−0.335774 + 0.941943i $$0.608998\pi$$
$$674$$ 54481.7 3.11358
$$675$$ 0 0
$$676$$ −15974.5 −0.908880
$$677$$ 32304.3 1.83390 0.916952 0.398997i $$-0.130642\pi$$
0.916952 + 0.398997i $$0.130642\pi$$
$$678$$ 0 0
$$679$$ 10121.1 0.572038
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 53485.6 3.00303
$$683$$ 33367.1 1.86934 0.934669 0.355519i $$-0.115696\pi$$
0.934669 + 0.355519i $$0.115696\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1809.30 0.100699
$$687$$ 0 0
$$688$$ −1159.55 −0.0642551
$$689$$ −20025.8 −1.10729
$$690$$ 0 0
$$691$$ −1043.67 −0.0574577 −0.0287288 0.999587i $$-0.509146\pi$$
−0.0287288 + 0.999587i $$0.509146\pi$$
$$692$$ 75211.3 4.13166
$$693$$ 0 0
$$694$$ −10378.6 −0.567674
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3568.46 −0.193924
$$698$$ 23027.4 1.24871
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 11305.7 0.609143 0.304572 0.952489i $$-0.401487\pi$$
0.304572 + 0.952489i $$0.401487\pi$$
$$702$$ 0 0
$$703$$ −15229.4 −0.817055
$$704$$ −25937.6 −1.38858
$$705$$ 0 0
$$706$$ 32027.1 1.70730
$$707$$ 3323.97 0.176818
$$708$$ 0 0
$$709$$ −13306.8 −0.704860 −0.352430 0.935838i $$-0.614645\pi$$
−0.352430 + 0.935838i $$0.614645\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −22020.6 −1.15907
$$713$$ −36575.6 −1.92113
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 55602.0 2.90216
$$717$$ 0 0
$$718$$ 50839.9 2.64252
$$719$$ −10701.2 −0.555062 −0.277531 0.960717i $$-0.589516\pi$$
−0.277531 + 0.960717i $$0.589516\pi$$
$$720$$ 0 0
$$721$$ −13997.1 −0.722996
$$722$$ 18068.0 0.931333
$$723$$ 0 0
$$724$$ 61576.5 3.16088
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2121.14 0.108210 0.0541051 0.998535i $$-0.482769\pi$$
0.0541051 + 0.998535i $$0.482769\pi$$
$$728$$ 16285.6 0.829098
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 71.7808 0.00363189
$$732$$ 0 0
$$733$$ 21584.0 1.08762 0.543809 0.839209i $$-0.316981\pi$$
0.543809 + 0.839209i $$0.316981\pi$$
$$734$$ 2757.33 0.138658
$$735$$ 0 0
$$736$$ 50135.0 2.51087
$$737$$ 16738.4 0.836588
$$738$$ 0 0
$$739$$ −9945.21 −0.495048 −0.247524 0.968882i $$-0.579617\pi$$
−0.247524 + 0.968882i $$0.579617\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −19824.7 −0.980848
$$743$$ 2867.01 0.141562 0.0707808 0.997492i $$-0.477451\pi$$
0.0707808 + 0.997492i $$0.477451\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 17037.1 0.836158
$$747$$ 0 0
$$748$$ 7267.71 0.355259
$$749$$ −8167.15 −0.398426
$$750$$ 0 0
$$751$$ −10824.1 −0.525934 −0.262967 0.964805i $$-0.584701\pi$$
−0.262967 + 0.964805i $$0.584701\pi$$
$$752$$ 42724.0 2.07179
$$753$$ 0 0
$$754$$ −6965.31 −0.336421
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14512.0 0.696761 0.348381 0.937353i $$-0.386732\pi$$
0.348381 + 0.937353i $$0.386732\pi$$
$$758$$ −35023.9 −1.67827
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33075.8 1.57556 0.787778 0.615959i $$-0.211231\pi$$
0.787778 + 0.615959i $$0.211231\pi$$
$$762$$ 0 0
$$763$$ 9360.23 0.444120
$$764$$ −5193.13 −0.245917
$$765$$ 0 0
$$766$$ 75032.8 3.53922
$$767$$ 1338.68 0.0630206
$$768$$ 0 0
$$769$$ 6728.44 0.315518 0.157759 0.987478i $$-0.449573\pi$$
0.157759 + 0.987478i $$0.449573\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −80311.5 −3.74414
$$773$$ −24233.3 −1.12757 −0.563784 0.825922i $$-0.690655\pi$$
−0.563784 + 0.825922i $$0.690655\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 90186.0 4.17202
$$777$$ 0 0
$$778$$ 15412.4 0.710231
$$779$$ −19820.6 −0.911615
$$780$$ 0 0
$$781$$ 12640.1 0.579127
$$782$$ −6975.51 −0.318982
$$783$$ 0 0
$$784$$ 8350.72 0.380408
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 17200.4 0.779069 0.389535 0.921012i $$-0.372636\pi$$
0.389535 + 0.921012i $$0.372636\pi$$
$$788$$ −56992.7 −2.57650
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6346.05 −0.285258
$$792$$ 0 0
$$793$$ 2155.66 0.0965318
$$794$$ 4282.92 0.191430
$$795$$ 0 0
$$796$$ −60801.9 −2.70737
$$797$$ −4208.87 −0.187059 −0.0935295 0.995617i $$-0.529815\pi$$
−0.0935295 + 0.995617i $$0.529815\pi$$
$$798$$ 0 0
$$799$$ −2644.78 −0.117104
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 12336.1 0.543146
$$803$$ 20199.7 0.887709
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −57410.2 −2.50892
$$807$$ 0 0
$$808$$ 29618.7 1.28958
$$809$$ 23632.1 1.02702 0.513511 0.858083i $$-0.328344\pi$$
0.513511 + 0.858083i $$0.328344\pi$$
$$810$$ 0 0
$$811$$ 28425.1 1.23075 0.615377 0.788233i $$-0.289004\pi$$
0.615377 + 0.788233i $$0.289004\pi$$
$$812$$ −4912.85 −0.212324
$$813$$ 0 0
$$814$$ 47638.9 2.05128
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 398.699 0.0170731
$$818$$ 14387.7 0.614981
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −39409.6 −1.67528 −0.837640 0.546223i $$-0.816065\pi$$
−0.837640 + 0.546223i $$0.816065\pi$$
$$822$$ 0 0
$$823$$ −16346.6 −0.692352 −0.346176 0.938170i $$-0.612520\pi$$
−0.346176 + 0.938170i $$0.612520\pi$$
$$824$$ −124723. −5.27300
$$825$$ 0 0
$$826$$ 1325.23 0.0558241
$$827$$ −3738.87 −0.157211 −0.0786054 0.996906i $$-0.525047\pi$$
−0.0786054 + 0.996906i $$0.525047\pi$$
$$828$$ 0 0
$$829$$ −45196.2 −1.89352 −0.946761 0.321937i $$-0.895666\pi$$
−0.946761 + 0.321937i $$0.895666\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 27840.8 1.16010
$$833$$ −516.942 −0.0215018
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 40367.8 1.67004
$$837$$ 0 0
$$838$$ 70189.5 2.89338
$$839$$ −15899.7 −0.654254 −0.327127 0.944980i $$-0.606080\pi$$
−0.327127 + 0.944980i $$0.606080\pi$$
$$840$$ 0 0
$$841$$ −23135.7 −0.948612
$$842$$ 58063.4 2.37648
$$843$$ 0 0
$$844$$ 11804.1 0.481414
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 864.465 0.0350689
$$848$$ −91500.0 −3.70534
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −32577.4 −1.31227
$$852$$ 0 0
$$853$$ −33926.7 −1.36182 −0.680908 0.732369i $$-0.738415\pi$$
−0.680908 + 0.732369i $$0.738415\pi$$
$$854$$ 2134.01 0.0855086
$$855$$ 0 0
$$856$$ −72774.7 −2.90583
$$857$$ −35432.4 −1.41231 −0.706154 0.708058i $$-0.749572\pi$$
−0.706154 + 0.708058i $$0.749572\pi$$
$$858$$ 0 0
$$859$$ −6780.17 −0.269309 −0.134655 0.990893i $$-0.542992\pi$$
−0.134655 + 0.990893i $$0.542992\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −34422.1 −1.36012
$$863$$ −30675.1 −1.20995 −0.604977 0.796243i $$-0.706818\pi$$
−0.604977 + 0.796243i $$0.706818\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −61802.4 −2.42509
$$867$$ 0 0
$$868$$ −40493.2 −1.58344
$$869$$ 24105.2 0.940981
$$870$$ 0 0
$$871$$ −17966.6 −0.698938
$$872$$ 83405.8 3.23908
$$873$$ 0 0
$$874$$ −38744.8 −1.49950
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −40861.3 −1.57330 −0.786652 0.617397i $$-0.788187\pi$$
−0.786652 + 0.617397i $$0.788187\pi$$
$$878$$ 77073.9 2.96255
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 43839.0 1.67647 0.838236 0.545308i $$-0.183587\pi$$
0.838236 + 0.545308i $$0.183587\pi$$
$$882$$ 0 0
$$883$$ −44625.1 −1.70074 −0.850371 0.526183i $$-0.823623\pi$$
−0.850371 + 0.526183i $$0.823623\pi$$
$$884$$ −7801.01 −0.296806
$$885$$ 0 0
$$886$$ 80388.6 3.04820
$$887$$ −43967.5 −1.66436 −0.832178 0.554509i $$-0.812906\pi$$
−0.832178 + 0.554509i $$0.812906\pi$$
$$888$$ 0 0
$$889$$ −12004.2 −0.452878
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 74933.5 2.81273
$$893$$ −14690.2 −0.550491
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 5162.95 0.192502
$$897$$ 0 0
$$898$$ 56329.7 2.09326
$$899$$ 10330.1 0.383234
$$900$$ 0 0
$$901$$ 5664.21 0.209436
$$902$$ 62000.4 2.28868
$$903$$ 0 0
$$904$$ −56547.4 −2.08046
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 13584.3 0.497309 0.248654 0.968592i $$-0.420012\pi$$
0.248654 + 0.968592i $$0.420012\pi$$
$$908$$ 36232.1 1.32423
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 16421.6 0.597226 0.298613 0.954374i $$-0.403476\pi$$
0.298613 + 0.954374i $$0.403476\pi$$
$$912$$ 0 0
$$913$$ −46369.0 −1.68082
$$914$$ 22303.6 0.807152
$$915$$ 0 0
$$916$$ −16856.0 −0.608010
$$917$$ 3294.28 0.118633
$$918$$ 0 0
$$919$$ −29487.3 −1.05843 −0.529214 0.848488i $$-0.677513\pi$$
−0.529214 + 0.848488i $$0.677513\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 4800.81 0.171482
$$923$$ −13567.6 −0.483839
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 23505.9 0.834181
$$927$$ 0 0
$$928$$ −14159.7 −0.500878
$$929$$ −3441.85 −0.121554 −0.0607769 0.998151i $$-0.519358\pi$$
−0.0607769 + 0.998151i $$0.519358\pi$$
$$930$$ 0 0
$$931$$ −2871.30 −0.101077
$$932$$ −130667. −4.59242
$$933$$ 0 0
$$934$$ 23364.8 0.818545
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −5646.60 −0.196869 −0.0984346 0.995144i $$-0.531384\pi$$
−0.0984346 + 0.995144i $$0.531384\pi$$
$$938$$ −17786.2 −0.619125
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 44680.1 1.54785 0.773927 0.633275i $$-0.218290\pi$$
0.773927 + 0.633275i $$0.218290\pi$$
$$942$$ 0 0
$$943$$ −42398.4 −1.46414
$$944$$ 6116.54 0.210886
$$945$$ 0 0
$$946$$ −1247.16 −0.0428633
$$947$$ −48924.6 −1.67881 −0.839406 0.543505i $$-0.817097\pi$$
−0.839406 + 0.543505i $$0.817097\pi$$
$$948$$ 0 0
$$949$$ −21681.9 −0.741647
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4606.29 −0.156818
$$953$$ 52014.3 1.76801 0.884003 0.467482i $$-0.154839\pi$$
0.884003 + 0.467482i $$0.154839\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 3619.14 0.122439
$$957$$ 0 0
$$958$$ 14521.1 0.489723
$$959$$ 3107.37 0.104632
$$960$$ 0 0
$$961$$ 55352.8 1.85804
$$962$$ −51134.5 −1.71377
$$963$$ 0 0
$$964$$ 30210.9 1.00936
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 47117.7 1.56691 0.783456 0.621448i $$-0.213455\pi$$
0.783456 + 0.621448i $$0.213455\pi$$
$$968$$ 7702.95 0.255767
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 8195.04 0.270846 0.135423 0.990788i $$-0.456761\pi$$
0.135423 + 0.990788i $$0.456761\pi$$
$$972$$ 0 0
$$973$$ −11689.9 −0.385159
$$974$$ −3537.35 −0.116370
$$975$$ 0 0
$$976$$ 9849.42 0.323025
$$977$$ 4643.51 0.152056 0.0760282 0.997106i $$-0.475776\pi$$
0.0760282 + 0.997106i $$0.475776\pi$$
$$978$$ 0 0
$$979$$ −12267.8 −0.400490
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −43490.1 −1.41326
$$983$$ 43986.5 1.42721 0.713607 0.700546i $$-0.247060\pi$$
0.713607 + 0.700546i $$0.247060\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 1970.10 0.0636317
$$987$$ 0 0
$$988$$ −43329.9 −1.39525
$$989$$ 852.859 0.0274210
$$990$$ 0 0
$$991$$ 1595.21 0.0511337 0.0255668 0.999673i $$-0.491861\pi$$
0.0255668 + 0.999673i $$0.491861\pi$$
$$992$$ −116709. −3.73539
$$993$$ 0 0
$$994$$ −13431.3 −0.428588
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −21501.2 −0.682998 −0.341499 0.939882i $$-0.610935\pi$$
−0.341499 + 0.939882i $$0.610935\pi$$
$$998$$ −43069.2 −1.36606
$$999$$ 0 0
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 1575.4.a.p.1.1 2
3.2 odd 2 525.4.a.n.1.2 2
5.4 even 2 63.4.a.e.1.2 2
15.2 even 4 525.4.d.g.274.4 4
15.8 even 4 525.4.d.g.274.1 4
15.14 odd 2 21.4.a.c.1.1 2
20.19 odd 2 1008.4.a.ba.1.1 2
35.4 even 6 441.4.e.q.226.1 4
35.9 even 6 441.4.e.q.361.1 4
35.19 odd 6 441.4.e.p.361.1 4
35.24 odd 6 441.4.e.p.226.1 4
35.34 odd 2 441.4.a.r.1.2 2
60.59 even 2 336.4.a.m.1.2 2
105.44 odd 6 147.4.e.l.67.2 4
105.59 even 6 147.4.e.m.79.2 4
105.74 odd 6 147.4.e.l.79.2 4
105.89 even 6 147.4.e.m.67.2 4
105.104 even 2 147.4.a.i.1.1 2
120.29 odd 2 1344.4.a.bg.1.1 2
120.59 even 2 1344.4.a.bo.1.1 2
420.419 odd 2 2352.4.a.bz.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.1 2 15.14 odd 2
63.4.a.e.1.2 2 5.4 even 2
147.4.a.i.1.1 2 105.104 even 2
147.4.e.l.67.2 4 105.44 odd 6
147.4.e.l.79.2 4 105.74 odd 6
147.4.e.m.67.2 4 105.89 even 6
147.4.e.m.79.2 4 105.59 even 6
336.4.a.m.1.2 2 60.59 even 2
441.4.a.r.1.2 2 35.34 odd 2
441.4.e.p.226.1 4 35.24 odd 6
441.4.e.p.361.1 4 35.19 odd 6
441.4.e.q.226.1 4 35.4 even 6
441.4.e.q.361.1 4 35.9 even 6
525.4.a.n.1.2 2 3.2 odd 2
525.4.d.g.274.1 4 15.8 even 4
525.4.d.g.274.4 4 15.2 even 4
1008.4.a.ba.1.1 2 20.19 odd 2
1344.4.a.bg.1.1 2 120.29 odd 2
1344.4.a.bo.1.1 2 120.59 even 2
1575.4.a.p.1.1 2 1.1 even 1 trivial
2352.4.a.bz.1.1 2 420.419 odd 2