# Properties

 Label 676.6.d.a.337.2 Level $676$ Weight $6$ Character 676.337 Analytic conductor $108.419$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 676.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$108.419462194$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 676.337 Dual form 676.6.d.a.337.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} +O(q^{10})$$ $$q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} -540.000i q^{11} -648.000i q^{15} -594.000 q^{17} +836.000i q^{19} -1056.00i q^{21} +4104.00 q^{23} +209.000 q^{25} +4104.00 q^{27} -594.000 q^{29} +4256.00i q^{31} +6480.00i q^{33} -4752.00 q^{35} +298.000i q^{37} +17226.0i q^{41} +12100.0 q^{43} -5346.00i q^{45} +1296.00i q^{47} +9063.00 q^{49} +7128.00 q^{51} +19494.0 q^{53} +29160.0 q^{55} -10032.0i q^{57} +7668.00i q^{59} -34738.0 q^{61} -8712.00i q^{63} +21812.0i q^{67} -49248.0 q^{69} -46872.0i q^{71} -67562.0i q^{73} -2508.00 q^{75} +47520.0 q^{77} -76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} -32076.0i q^{85} +7128.00 q^{87} -29754.0i q^{89} -51072.0i q^{93} -45144.0 q^{95} -122398. i q^{97} +53460.0i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 24 q^{3} - 198 q^{9} + O(q^{10})$$ $$2 q - 24 q^{3} - 198 q^{9} - 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8208 q^{27} - 1188 q^{29} - 9504 q^{35} + 24200 q^{43} + 18126 q^{49} + 14256 q^{51} + 38988 q^{53} + 58320 q^{55} - 69476 q^{61} - 98496 q^{69} - 5016 q^{75} + 95040 q^{77} - 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 90288 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$509$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 0
$$3$$ −12.0000 −0.769800 −0.384900 0.922958i $$-0.625764\pi$$
−0.384900 + 0.922958i $$0.625764\pi$$
$$4$$ 0 0
$$5$$ 54.0000i 0.965981i 0.875625 + 0.482991i $$0.160450\pi$$
−0.875625 + 0.482991i $$0.839550\pi$$
$$6$$ 0 0
$$7$$ 88.0000i 0.678793i 0.940643 + 0.339397i $$0.110223\pi$$
−0.940643 + 0.339397i $$0.889777\pi$$
$$8$$ 0 0
$$9$$ −99.0000 −0.407407
$$10$$ 0 0
$$11$$ − 540.000i − 1.34559i −0.739830 0.672794i $$-0.765094\pi$$
0.739830 0.672794i $$-0.234906\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 648.000i − 0.743613i
$$16$$ 0 0
$$17$$ −594.000 −0.498499 −0.249249 0.968439i $$-0.580184\pi$$
−0.249249 + 0.968439i $$0.580184\pi$$
$$18$$ 0 0
$$19$$ 836.000i 0.531279i 0.964072 + 0.265639i $$0.0855830\pi$$
−0.964072 + 0.265639i $$0.914417\pi$$
$$20$$ 0 0
$$21$$ − 1056.00i − 0.522535i
$$22$$ 0 0
$$23$$ 4104.00 1.61766 0.808831 0.588041i $$-0.200101\pi$$
0.808831 + 0.588041i $$0.200101\pi$$
$$24$$ 0 0
$$25$$ 209.000 0.0668800
$$26$$ 0 0
$$27$$ 4104.00 1.08342
$$28$$ 0 0
$$29$$ −594.000 −0.131157 −0.0655785 0.997847i $$-0.520889\pi$$
−0.0655785 + 0.997847i $$0.520889\pi$$
$$30$$ 0 0
$$31$$ 4256.00i 0.795422i 0.917511 + 0.397711i $$0.130195\pi$$
−0.917511 + 0.397711i $$0.869805\pi$$
$$32$$ 0 0
$$33$$ 6480.00i 1.03583i
$$34$$ 0 0
$$35$$ −4752.00 −0.655702
$$36$$ 0 0
$$37$$ 298.000i 0.0357859i 0.999840 + 0.0178930i $$0.00569581\pi$$
−0.999840 + 0.0178930i $$0.994304\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 17226.0i 1.60039i 0.599742 + 0.800193i $$0.295270\pi$$
−0.599742 + 0.800193i $$0.704730\pi$$
$$42$$ 0 0
$$43$$ 12100.0 0.997963 0.498981 0.866613i $$-0.333708\pi$$
0.498981 + 0.866613i $$0.333708\pi$$
$$44$$ 0 0
$$45$$ − 5346.00i − 0.393548i
$$46$$ 0 0
$$47$$ 1296.00i 0.0855777i 0.999084 + 0.0427888i $$0.0136243\pi$$
−0.999084 + 0.0427888i $$0.986376\pi$$
$$48$$ 0 0
$$49$$ 9063.00 0.539240
$$50$$ 0 0
$$51$$ 7128.00 0.383745
$$52$$ 0 0
$$53$$ 19494.0 0.953260 0.476630 0.879104i $$-0.341858\pi$$
0.476630 + 0.879104i $$0.341858\pi$$
$$54$$ 0 0
$$55$$ 29160.0 1.29981
$$56$$ 0 0
$$57$$ − 10032.0i − 0.408978i
$$58$$ 0 0
$$59$$ 7668.00i 0.286782i 0.989666 + 0.143391i $$0.0458007\pi$$
−0.989666 + 0.143391i $$0.954199\pi$$
$$60$$ 0 0
$$61$$ −34738.0 −1.19531 −0.597655 0.801754i $$-0.703901\pi$$
−0.597655 + 0.801754i $$0.703901\pi$$
$$62$$ 0 0
$$63$$ − 8712.00i − 0.276545i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 21812.0i 0.593620i 0.954937 + 0.296810i $$0.0959228\pi$$
−0.954937 + 0.296810i $$0.904077\pi$$
$$68$$ 0 0
$$69$$ −49248.0 −1.24528
$$70$$ 0 0
$$71$$ − 46872.0i − 1.10349i −0.834014 0.551744i $$-0.813963\pi$$
0.834014 0.551744i $$-0.186037\pi$$
$$72$$ 0 0
$$73$$ − 67562.0i − 1.48387i −0.670473 0.741934i $$-0.733909\pi$$
0.670473 0.741934i $$-0.266091\pi$$
$$74$$ 0 0
$$75$$ −2508.00 −0.0514842
$$76$$ 0 0
$$77$$ 47520.0 0.913376
$$78$$ 0 0
$$79$$ −76912.0 −1.38652 −0.693260 0.720687i $$-0.743826\pi$$
−0.693260 + 0.720687i $$0.743826\pi$$
$$80$$ 0 0
$$81$$ −25191.0 −0.426612
$$82$$ 0 0
$$83$$ 67716.0i 1.07894i 0.842006 + 0.539468i $$0.181375\pi$$
−0.842006 + 0.539468i $$0.818625\pi$$
$$84$$ 0 0
$$85$$ − 32076.0i − 0.481541i
$$86$$ 0 0
$$87$$ 7128.00 0.100965
$$88$$ 0 0
$$89$$ − 29754.0i − 0.398172i −0.979982 0.199086i $$-0.936203\pi$$
0.979982 0.199086i $$-0.0637973\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 51072.0i − 0.612316i
$$94$$ 0 0
$$95$$ −45144.0 −0.513205
$$96$$ 0 0
$$97$$ − 122398.i − 1.32082i −0.750903 0.660412i $$-0.770382\pi$$
0.750903 0.660412i $$-0.229618\pi$$
$$98$$ 0 0
$$99$$ 53460.0i 0.548202i
$$100$$ 0 0
$$101$$ −11286.0 −0.110087 −0.0550436 0.998484i $$-0.517530\pi$$
−0.0550436 + 0.998484i $$0.517530\pi$$
$$102$$ 0 0
$$103$$ 27256.0 0.253145 0.126572 0.991957i $$-0.459602\pi$$
0.126572 + 0.991957i $$0.459602\pi$$
$$104$$ 0 0
$$105$$ 57024.0 0.504759
$$106$$ 0 0
$$107$$ 122364. 1.03322 0.516612 0.856220i $$-0.327193\pi$$
0.516612 + 0.856220i $$0.327193\pi$$
$$108$$ 0 0
$$109$$ 99902.0i 0.805393i 0.915334 + 0.402697i $$0.131927\pi$$
−0.915334 + 0.402697i $$0.868073\pi$$
$$110$$ 0 0
$$111$$ − 3576.00i − 0.0275480i
$$112$$ 0 0
$$113$$ −29646.0 −0.218409 −0.109204 0.994019i $$-0.534830\pi$$
−0.109204 + 0.994019i $$0.534830\pi$$
$$114$$ 0 0
$$115$$ 221616.i 1.56263i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 52272.0i − 0.338378i
$$120$$ 0 0
$$121$$ −130549. −0.810607
$$122$$ 0 0
$$123$$ − 206712.i − 1.23198i
$$124$$ 0 0
$$125$$ 180036.i 1.03059i
$$126$$ 0 0
$$127$$ −336512. −1.85136 −0.925681 0.378305i $$-0.876507\pi$$
−0.925681 + 0.378305i $$0.876507\pi$$
$$128$$ 0 0
$$129$$ −145200. −0.768232
$$130$$ 0 0
$$131$$ 100980. 0.514111 0.257056 0.966397i $$-0.417248\pi$$
0.257056 + 0.966397i $$0.417248\pi$$
$$132$$ 0 0
$$133$$ −73568.0 −0.360628
$$134$$ 0 0
$$135$$ 221616.i 1.04657i
$$136$$ 0 0
$$137$$ 317142.i 1.44362i 0.692092 + 0.721809i $$0.256689\pi$$
−0.692092 + 0.721809i $$0.743311\pi$$
$$138$$ 0 0
$$139$$ −148324. −0.651140 −0.325570 0.945518i $$-0.605556\pi$$
−0.325570 + 0.945518i $$0.605556\pi$$
$$140$$ 0 0
$$141$$ − 15552.0i − 0.0658777i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 32076.0i − 0.126695i
$$146$$ 0 0
$$147$$ −108756. −0.415107
$$148$$ 0 0
$$149$$ 196614.i 0.725519i 0.931883 + 0.362759i $$0.118165\pi$$
−0.931883 + 0.362759i $$0.881835\pi$$
$$150$$ 0 0
$$151$$ − 74360.0i − 0.265398i −0.991156 0.132699i $$-0.957636\pi$$
0.991156 0.132699i $$-0.0423643\pi$$
$$152$$ 0 0
$$153$$ 58806.0 0.203092
$$154$$ 0 0
$$155$$ −229824. −0.768362
$$156$$ 0 0
$$157$$ 120878. 0.391380 0.195690 0.980666i $$-0.437305\pi$$
0.195690 + 0.980666i $$0.437305\pi$$
$$158$$ 0 0
$$159$$ −233928. −0.733820
$$160$$ 0 0
$$161$$ 361152.i 1.09806i
$$162$$ 0 0
$$163$$ 111340.i 0.328233i 0.986441 + 0.164116i $$0.0524773\pi$$
−0.986441 + 0.164116i $$0.947523\pi$$
$$164$$ 0 0
$$165$$ −349920. −1.00060
$$166$$ 0 0
$$167$$ 491832.i 1.36466i 0.731043 + 0.682332i $$0.239034\pi$$
−0.731043 + 0.682332i $$0.760966\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 82764.0i − 0.216447i
$$172$$ 0 0
$$173$$ −707454. −1.79714 −0.898572 0.438826i $$-0.855395\pi$$
−0.898572 + 0.438826i $$0.855395\pi$$
$$174$$ 0 0
$$175$$ 18392.0i 0.0453977i
$$176$$ 0 0
$$177$$ − 92016.0i − 0.220765i
$$178$$ 0 0
$$179$$ −493668. −1.15160 −0.575801 0.817590i $$-0.695310\pi$$
−0.575801 + 0.817590i $$0.695310\pi$$
$$180$$ 0 0
$$181$$ 559450. 1.26930 0.634651 0.772799i $$-0.281144\pi$$
0.634651 + 0.772799i $$0.281144\pi$$
$$182$$ 0 0
$$183$$ 416856. 0.920149
$$184$$ 0 0
$$185$$ −16092.0 −0.0345685
$$186$$ 0 0
$$187$$ 320760.i 0.670774i
$$188$$ 0 0
$$189$$ 361152.i 0.735420i
$$190$$ 0 0
$$191$$ −724032. −1.43607 −0.718033 0.696009i $$-0.754957\pi$$
−0.718033 + 0.696009i $$0.754957\pi$$
$$192$$ 0 0
$$193$$ − 7106.00i − 0.0137319i −0.999976 0.00686597i $$-0.997814\pi$$
0.999976 0.00686597i $$-0.00218552\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 530442.i − 0.973806i −0.873456 0.486903i $$-0.838127\pi$$
0.873456 0.486903i $$-0.161873\pi$$
$$198$$ 0 0
$$199$$ −56168.0 −0.100544 −0.0502720 0.998736i $$-0.516009\pi$$
−0.0502720 + 0.998736i $$0.516009\pi$$
$$200$$ 0 0
$$201$$ − 261744.i − 0.456969i
$$202$$ 0 0
$$203$$ − 52272.0i − 0.0890285i
$$204$$ 0 0
$$205$$ −930204. −1.54594
$$206$$ 0 0
$$207$$ −406296. −0.659047
$$208$$ 0 0
$$209$$ 451440. 0.714882
$$210$$ 0 0
$$211$$ −339196. −0.524499 −0.262249 0.965000i $$-0.584464\pi$$
−0.262249 + 0.965000i $$0.584464\pi$$
$$212$$ 0 0
$$213$$ 562464.i 0.849465i
$$214$$ 0 0
$$215$$ 653400.i 0.964013i
$$216$$ 0 0
$$217$$ −374528. −0.539927
$$218$$ 0 0
$$219$$ 810744.i 1.14228i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 779360.i 1.04948i 0.851261 + 0.524742i $$0.175838\pi$$
−0.851261 + 0.524742i $$0.824162\pi$$
$$224$$ 0 0
$$225$$ −20691.0 −0.0272474
$$226$$ 0 0
$$227$$ − 744876.i − 0.959443i −0.877421 0.479722i $$-0.840738\pi$$
0.877421 0.479722i $$-0.159262\pi$$
$$228$$ 0 0
$$229$$ 272746.i 0.343692i 0.985124 + 0.171846i $$0.0549732\pi$$
−0.985124 + 0.171846i $$0.945027\pi$$
$$230$$ 0 0
$$231$$ −570240. −0.703117
$$232$$ 0 0
$$233$$ 153846. 0.185651 0.0928253 0.995682i $$-0.470410\pi$$
0.0928253 + 0.995682i $$0.470410\pi$$
$$234$$ 0 0
$$235$$ −69984.0 −0.0826664
$$236$$ 0 0
$$237$$ 922944. 1.06734
$$238$$ 0 0
$$239$$ 1.15474e6i 1.30764i 0.756650 + 0.653820i $$0.226834\pi$$
−0.756650 + 0.653820i $$0.773166\pi$$
$$240$$ 0 0
$$241$$ − 657074.i − 0.728738i −0.931255 0.364369i $$-0.881285\pi$$
0.931255 0.364369i $$-0.118715\pi$$
$$242$$ 0 0
$$243$$ −694980. −0.755017
$$244$$ 0 0
$$245$$ 489402.i 0.520895i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ − 812592.i − 0.830566i
$$250$$ 0 0
$$251$$ −1.34190e6 −1.34442 −0.672211 0.740359i $$-0.734655\pi$$
−0.672211 + 0.740359i $$0.734655\pi$$
$$252$$ 0 0
$$253$$ − 2.21616e6i − 2.17671i
$$254$$ 0 0
$$255$$ 384912.i 0.370690i
$$256$$ 0 0
$$257$$ −132354. −0.124998 −0.0624992 0.998045i $$-0.519907\pi$$
−0.0624992 + 0.998045i $$0.519907\pi$$
$$258$$ 0 0
$$259$$ −26224.0 −0.0242912
$$260$$ 0 0
$$261$$ 58806.0 0.0534343
$$262$$ 0 0
$$263$$ 943272. 0.840906 0.420453 0.907314i $$-0.361871\pi$$
0.420453 + 0.907314i $$0.361871\pi$$
$$264$$ 0 0
$$265$$ 1.05268e6i 0.920831i
$$266$$ 0 0
$$267$$ 357048.i 0.306513i
$$268$$ 0 0
$$269$$ 967518. 0.815227 0.407613 0.913155i $$-0.366361\pi$$
0.407613 + 0.913155i $$0.366361\pi$$
$$270$$ 0 0
$$271$$ 518320.i 0.428721i 0.976755 + 0.214360i $$0.0687667\pi$$
−0.976755 + 0.214360i $$0.931233\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 112860.i − 0.0899929i
$$276$$ 0 0
$$277$$ −2.22273e6 −1.74055 −0.870275 0.492566i $$-0.836059\pi$$
−0.870275 + 0.492566i $$0.836059\pi$$
$$278$$ 0 0
$$279$$ − 421344.i − 0.324061i
$$280$$ 0 0
$$281$$ 196614.i 0.148542i 0.997238 + 0.0742709i $$0.0236629\pi$$
−0.997238 + 0.0742709i $$0.976337\pi$$
$$282$$ 0 0
$$283$$ 1.55228e6 1.15213 0.576067 0.817403i $$-0.304587\pi$$
0.576067 + 0.817403i $$0.304587\pi$$
$$284$$ 0 0
$$285$$ 541728. 0.395066
$$286$$ 0 0
$$287$$ −1.51589e6 −1.08633
$$288$$ 0 0
$$289$$ −1.06702e6 −0.751499
$$290$$ 0 0
$$291$$ 1.46878e6i 1.01677i
$$292$$ 0 0
$$293$$ 1.07217e6i 0.729616i 0.931083 + 0.364808i $$0.118865\pi$$
−0.931083 + 0.364808i $$0.881135\pi$$
$$294$$ 0 0
$$295$$ −414072. −0.277026
$$296$$ 0 0
$$297$$ − 2.21616e6i − 1.45784i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.06480e6i 0.677410i
$$302$$ 0 0
$$303$$ 135432. 0.0847451
$$304$$ 0 0
$$305$$ − 1.87585e6i − 1.15465i
$$306$$ 0 0
$$307$$ − 1.58589e6i − 0.960346i −0.877174 0.480173i $$-0.840574\pi$$
0.877174 0.480173i $$-0.159426\pi$$
$$308$$ 0 0
$$309$$ −327072. −0.194871
$$310$$ 0 0
$$311$$ 730728. 0.428405 0.214203 0.976789i $$-0.431285\pi$$
0.214203 + 0.976789i $$0.431285\pi$$
$$312$$ 0 0
$$313$$ 584858. 0.337435 0.168717 0.985664i $$-0.446038\pi$$
0.168717 + 0.985664i $$0.446038\pi$$
$$314$$ 0 0
$$315$$ 470448. 0.267138
$$316$$ 0 0
$$317$$ − 2.48287e6i − 1.38773i −0.720105 0.693865i $$-0.755906\pi$$
0.720105 0.693865i $$-0.244094\pi$$
$$318$$ 0 0
$$319$$ 320760.i 0.176483i
$$320$$ 0 0
$$321$$ −1.46837e6 −0.795376
$$322$$ 0 0
$$323$$ − 496584.i − 0.264842i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 1.19882e6i − 0.619992i
$$328$$ 0 0
$$329$$ −114048. −0.0580895
$$330$$ 0 0
$$331$$ 377948.i 0.189610i 0.995496 + 0.0948052i $$0.0302228\pi$$
−0.995496 + 0.0948052i $$0.969777\pi$$
$$332$$ 0 0
$$333$$ − 29502.0i − 0.0145794i
$$334$$ 0 0
$$335$$ −1.17785e6 −0.573426
$$336$$ 0 0
$$337$$ −639122. −0.306555 −0.153278 0.988183i $$-0.548983\pi$$
−0.153278 + 0.988183i $$0.548983\pi$$
$$338$$ 0 0
$$339$$ 355752. 0.168131
$$340$$ 0 0
$$341$$ 2.29824e6 1.07031
$$342$$ 0 0
$$343$$ 2.27656e6i 1.04483i
$$344$$ 0 0
$$345$$ − 2.65939e6i − 1.20291i
$$346$$ 0 0
$$347$$ −2.90466e6 −1.29501 −0.647503 0.762063i $$-0.724187\pi$$
−0.647503 + 0.762063i $$0.724187\pi$$
$$348$$ 0 0
$$349$$ 3.99157e6i 1.75420i 0.480304 + 0.877102i $$0.340526\pi$$
−0.480304 + 0.877102i $$0.659474\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.42922e6i 0.610466i 0.952278 + 0.305233i $$0.0987344\pi$$
−0.952278 + 0.305233i $$0.901266\pi$$
$$354$$ 0 0
$$355$$ 2.53109e6 1.06595
$$356$$ 0 0
$$357$$ 627264.i 0.260483i
$$358$$ 0 0
$$359$$ − 1.16186e6i − 0.475794i −0.971290 0.237897i $$-0.923542\pi$$
0.971290 0.237897i $$-0.0764581\pi$$
$$360$$ 0 0
$$361$$ 1.77720e6 0.717743
$$362$$ 0 0
$$363$$ 1.56659e6 0.624005
$$364$$ 0 0
$$365$$ 3.64835e6 1.43339
$$366$$ 0 0
$$367$$ −1.08923e6 −0.422139 −0.211069 0.977471i $$-0.567695\pi$$
−0.211069 + 0.977471i $$0.567695\pi$$
$$368$$ 0 0
$$369$$ − 1.70537e6i − 0.652009i
$$370$$ 0 0
$$371$$ 1.71547e6i 0.647066i
$$372$$ 0 0
$$373$$ 3.50577e6 1.30470 0.652350 0.757918i $$-0.273783\pi$$
0.652350 + 0.757918i $$0.273783\pi$$
$$374$$ 0 0
$$375$$ − 2.16043e6i − 0.793346i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 4.04385e6i 1.44610i 0.690798 + 0.723048i $$0.257260\pi$$
−0.690798 + 0.723048i $$0.742740\pi$$
$$380$$ 0 0
$$381$$ 4.03814e6 1.42518
$$382$$ 0 0
$$383$$ 5.18746e6i 1.80700i 0.428591 + 0.903499i $$0.359010\pi$$
−0.428591 + 0.903499i $$0.640990\pi$$
$$384$$ 0 0
$$385$$ 2.56608e6i 0.882304i
$$386$$ 0 0
$$387$$ −1.19790e6 −0.406577
$$388$$ 0 0
$$389$$ 950346. 0.318425 0.159213 0.987244i $$-0.449104\pi$$
0.159213 + 0.987244i $$0.449104\pi$$
$$390$$ 0 0
$$391$$ −2.43778e6 −0.806403
$$392$$ 0 0
$$393$$ −1.21176e6 −0.395763
$$394$$ 0 0
$$395$$ − 4.15325e6i − 1.33935i
$$396$$ 0 0
$$397$$ 520738.i 0.165822i 0.996557 + 0.0829112i $$0.0264218\pi$$
−0.996557 + 0.0829112i $$0.973578\pi$$
$$398$$ 0 0
$$399$$ 882816. 0.277612
$$400$$ 0 0
$$401$$ − 764370.i − 0.237379i −0.992931 0.118690i $$-0.962131\pi$$
0.992931 0.118690i $$-0.0378693\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 1.36031e6i − 0.412099i
$$406$$ 0 0
$$407$$ 160920. 0.0481531
$$408$$ 0 0
$$409$$ 2.64051e6i 0.780511i 0.920707 + 0.390255i $$0.127613\pi$$
−0.920707 + 0.390255i $$0.872387\pi$$
$$410$$ 0 0
$$411$$ − 3.80570e6i − 1.11130i
$$412$$ 0 0
$$413$$ −674784. −0.194666
$$414$$ 0 0
$$415$$ −3.65666e6 −1.04223
$$416$$ 0 0
$$417$$ 1.77989e6 0.501248
$$418$$ 0 0
$$419$$ −4.98020e6 −1.38584 −0.692918 0.721016i $$-0.743675\pi$$
−0.692918 + 0.721016i $$0.743675\pi$$
$$420$$ 0 0
$$421$$ − 237994.i − 0.0654426i −0.999465 0.0327213i $$-0.989583\pi$$
0.999465 0.0327213i $$-0.0104174\pi$$
$$422$$ 0 0
$$423$$ − 128304.i − 0.0348650i
$$424$$ 0 0
$$425$$ −124146. −0.0333396
$$426$$ 0 0
$$427$$ − 3.05694e6i − 0.811368i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 3.88238e6i − 1.00671i −0.864079 0.503356i $$-0.832098\pi$$
0.864079 0.503356i $$-0.167902\pi$$
$$432$$ 0 0
$$433$$ 66958.0 0.0171626 0.00858129 0.999963i $$-0.497268\pi$$
0.00858129 + 0.999963i $$0.497268\pi$$
$$434$$ 0 0
$$435$$ 384912.i 0.0975300i
$$436$$ 0 0
$$437$$ 3.43094e6i 0.859429i
$$438$$ 0 0
$$439$$ 6.50135e6 1.61006 0.805031 0.593233i $$-0.202149\pi$$
0.805031 + 0.593233i $$0.202149\pi$$
$$440$$ 0 0
$$441$$ −897237. −0.219690
$$442$$ 0 0
$$443$$ −4.60760e6 −1.11549 −0.557745 0.830012i $$-0.688333\pi$$
−0.557745 + 0.830012i $$0.688333\pi$$
$$444$$ 0 0
$$445$$ 1.60672e6 0.384626
$$446$$ 0 0
$$447$$ − 2.35937e6i − 0.558505i
$$448$$ 0 0
$$449$$ − 3.77671e6i − 0.884092i −0.896992 0.442046i $$-0.854253\pi$$
0.896992 0.442046i $$-0.145747\pi$$
$$450$$ 0 0
$$451$$ 9.30204e6 2.15346
$$452$$ 0 0
$$453$$ 892320.i 0.204303i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 3.18069e6i − 0.712412i −0.934407 0.356206i $$-0.884070\pi$$
0.934407 0.356206i $$-0.115930\pi$$
$$458$$ 0 0
$$459$$ −2.43778e6 −0.540085
$$460$$ 0 0
$$461$$ 6.68547e6i 1.46514i 0.680691 + 0.732571i $$0.261680\pi$$
−0.680691 + 0.732571i $$0.738320\pi$$
$$462$$ 0 0
$$463$$ 4.35122e6i 0.943318i 0.881781 + 0.471659i $$0.156345\pi$$
−0.881781 + 0.471659i $$0.843655\pi$$
$$464$$ 0 0
$$465$$ 2.75789e6 0.591486
$$466$$ 0 0
$$467$$ −7.07994e6 −1.50223 −0.751117 0.660170i $$-0.770484\pi$$
−0.751117 + 0.660170i $$0.770484\pi$$
$$468$$ 0 0
$$469$$ −1.91946e6 −0.402945
$$470$$ 0 0
$$471$$ −1.45054e6 −0.301284
$$472$$ 0 0
$$473$$ − 6.53400e6i − 1.34285i
$$474$$ 0 0
$$475$$ 174724.i 0.0355319i
$$476$$ 0 0
$$477$$ −1.92991e6 −0.388365
$$478$$ 0 0
$$479$$ − 3.22186e6i − 0.641604i −0.947146 0.320802i $$-0.896048\pi$$
0.947146 0.320802i $$-0.103952\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ − 4.33382e6i − 0.845286i
$$484$$ 0 0
$$485$$ 6.60949e6 1.27589
$$486$$ 0 0
$$487$$ 2.29710e6i 0.438891i 0.975625 + 0.219446i $$0.0704248\pi$$
−0.975625 + 0.219446i $$0.929575\pi$$
$$488$$ 0 0
$$489$$ − 1.33608e6i − 0.252674i
$$490$$ 0 0
$$491$$ −2.82150e6 −0.528173 −0.264087 0.964499i $$-0.585070\pi$$
−0.264087 + 0.964499i $$0.585070\pi$$
$$492$$ 0 0
$$493$$ 352836. 0.0653816
$$494$$ 0 0
$$495$$ −2.88684e6 −0.529553
$$496$$ 0 0
$$497$$ 4.12474e6 0.749040
$$498$$ 0 0
$$499$$ − 4.13628e6i − 0.743634i −0.928306 0.371817i $$-0.878735\pi$$
0.928306 0.371817i $$-0.121265\pi$$
$$500$$ 0 0
$$501$$ − 5.90198e6i − 1.05052i
$$502$$ 0 0
$$503$$ 8.33263e6 1.46846 0.734230 0.678901i $$-0.237543\pi$$
0.734230 + 0.678901i $$0.237543\pi$$
$$504$$ 0 0
$$505$$ − 609444.i − 0.106342i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 4.34101e6i 0.742670i 0.928499 + 0.371335i $$0.121100\pi$$
−0.928499 + 0.371335i $$0.878900\pi$$
$$510$$ 0 0
$$511$$ 5.94546e6 1.00724
$$512$$ 0 0
$$513$$ 3.43094e6i 0.575599i
$$514$$ 0 0
$$515$$ 1.47182e6i 0.244533i
$$516$$ 0 0
$$517$$ 699840. 0.115152
$$518$$ 0 0
$$519$$ 8.48945e6 1.38344
$$520$$ 0 0
$$521$$ −6.74185e6 −1.08814 −0.544070 0.839040i $$-0.683117\pi$$
−0.544070 + 0.839040i $$0.683117\pi$$
$$522$$ 0 0
$$523$$ −7.72196e6 −1.23445 −0.617224 0.786787i $$-0.711743\pi$$
−0.617224 + 0.786787i $$0.711743\pi$$
$$524$$ 0 0
$$525$$ − 220704.i − 0.0349472i
$$526$$ 0 0
$$527$$ − 2.52806e6i − 0.396517i
$$528$$ 0 0
$$529$$ 1.04065e7 1.61683
$$530$$ 0 0
$$531$$ − 759132.i − 0.116837i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 6.60766e6i 0.998075i
$$536$$ 0 0
$$537$$ 5.92402e6 0.886504
$$538$$ 0 0
$$539$$ − 4.89402e6i − 0.725594i
$$540$$ 0 0
$$541$$ 682066.i 0.100192i 0.998744 + 0.0500960i $$0.0159527\pi$$
−0.998744 + 0.0500960i $$0.984047\pi$$
$$542$$ 0 0
$$543$$ −6.71340e6 −0.977109
$$544$$ 0 0
$$545$$ −5.39471e6 −0.777995
$$546$$ 0 0
$$547$$ 2.15772e6 0.308337 0.154169 0.988045i $$-0.450730\pi$$
0.154169 + 0.988045i $$0.450730\pi$$
$$548$$ 0 0
$$549$$ 3.43906e6 0.486978
$$550$$ 0 0
$$551$$ − 496584.i − 0.0696809i
$$552$$ 0 0
$$553$$ − 6.76826e6i − 0.941161i
$$554$$ 0 0
$$555$$ 193104. 0.0266109
$$556$$ 0 0
$$557$$ 2.67597e6i 0.365463i 0.983163 + 0.182731i $$0.0584939\pi$$
−0.983163 + 0.182731i $$0.941506\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ − 3.84912e6i − 0.516362i
$$562$$ 0 0
$$563$$ 3.55331e6 0.472457 0.236228 0.971698i $$-0.424089\pi$$
0.236228 + 0.971698i $$0.424089\pi$$
$$564$$ 0 0
$$565$$ − 1.60088e6i − 0.210979i
$$566$$ 0 0
$$567$$ − 2.21681e6i − 0.289581i
$$568$$ 0 0
$$569$$ 1.29225e7 1.67327 0.836633 0.547764i $$-0.184521\pi$$
0.836633 + 0.547764i $$0.184521\pi$$
$$570$$ 0 0
$$571$$ 6.08357e6 0.780851 0.390426 0.920634i $$-0.372328\pi$$
0.390426 + 0.920634i $$0.372328\pi$$
$$572$$ 0 0
$$573$$ 8.68838e6 1.10548
$$574$$ 0 0
$$575$$ 857736. 0.108189
$$576$$ 0 0
$$577$$ − 1.58241e7i − 1.97869i −0.145579 0.989347i $$-0.546505\pi$$
0.145579 0.989347i $$-0.453495\pi$$
$$578$$ 0 0
$$579$$ 85272.0i 0.0105709i
$$580$$ 0 0
$$581$$ −5.95901e6 −0.732375
$$582$$ 0 0
$$583$$ − 1.05268e7i − 1.28269i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.60220e6i 0.551278i 0.961261 + 0.275639i $$0.0888894\pi$$
−0.961261 + 0.275639i $$0.911111\pi$$
$$588$$ 0 0
$$589$$ −3.55802e6 −0.422590
$$590$$ 0 0
$$591$$ 6.36530e6i 0.749636i
$$592$$ 0 0
$$593$$ − 8.61122e6i − 1.00561i −0.864401 0.502803i $$-0.832302\pi$$
0.864401 0.502803i $$-0.167698\pi$$
$$594$$ 0 0
$$595$$ 2.82269e6 0.326867
$$596$$ 0 0
$$597$$ 674016. 0.0773988
$$598$$ 0 0
$$599$$ −7.98228e6 −0.908992 −0.454496 0.890749i $$-0.650181\pi$$
−0.454496 + 0.890749i $$0.650181\pi$$
$$600$$ 0 0
$$601$$ 1.01740e7 1.14896 0.574481 0.818518i $$-0.305204\pi$$
0.574481 + 0.818518i $$0.305204\pi$$
$$602$$ 0 0
$$603$$ − 2.15939e6i − 0.241845i
$$604$$ 0 0
$$605$$ − 7.04965e6i − 0.783031i
$$606$$ 0 0
$$607$$ −9.95843e6 −1.09703 −0.548516 0.836140i $$-0.684807\pi$$
−0.548516 + 0.836140i $$0.684807\pi$$
$$608$$ 0 0
$$609$$ 627264.i 0.0685342i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 4.19586e6i 0.450993i 0.974244 + 0.225497i $$0.0724005\pi$$
−0.974244 + 0.225497i $$0.927600\pi$$
$$614$$ 0 0
$$615$$ 1.11624e7 1.19007
$$616$$ 0 0
$$617$$ 9.12551e6i 0.965038i 0.875885 + 0.482519i $$0.160278\pi$$
−0.875885 + 0.482519i $$0.839722\pi$$
$$618$$ 0 0
$$619$$ − 6.45734e6i − 0.677372i −0.940900 0.338686i $$-0.890018\pi$$
0.940900 0.338686i $$-0.109982\pi$$
$$620$$ 0 0
$$621$$ 1.68428e7 1.75261
$$622$$ 0 0
$$623$$ 2.61835e6 0.270276
$$624$$ 0 0
$$625$$ −9.06882e6 −0.928647
$$626$$ 0 0
$$627$$ −5.41728e6 −0.550316
$$628$$ 0 0
$$629$$ − 177012.i − 0.0178392i
$$630$$ 0 0
$$631$$ 1.40514e7i 1.40490i 0.711733 + 0.702450i $$0.247910\pi$$
−0.711733 + 0.702450i $$0.752090\pi$$
$$632$$ 0 0
$$633$$ 4.07035e6 0.403759
$$634$$ 0 0
$$635$$ − 1.81716e7i − 1.78838i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 4.64033e6i 0.449569i
$$640$$ 0 0
$$641$$ −8.47168e6 −0.814375 −0.407188 0.913345i $$-0.633490\pi$$
−0.407188 + 0.913345i $$0.633490\pi$$
$$642$$ 0 0
$$643$$ 488564.i 0.0466009i 0.999729 + 0.0233004i $$0.00741743\pi$$
−0.999729 + 0.0233004i $$0.992583\pi$$
$$644$$ 0 0
$$645$$ − 7.84080e6i − 0.742098i
$$646$$ 0 0
$$647$$ −2.48119e6 −0.233023 −0.116512 0.993189i $$-0.537171\pi$$
−0.116512 + 0.993189i $$0.537171\pi$$
$$648$$ 0 0
$$649$$ 4.14072e6 0.385891
$$650$$ 0 0
$$651$$ 4.49434e6 0.415636
$$652$$ 0 0
$$653$$ −5.29130e6 −0.485601 −0.242800 0.970076i $$-0.578066\pi$$
−0.242800 + 0.970076i $$0.578066\pi$$
$$654$$ 0 0
$$655$$ 5.45292e6i 0.496622i
$$656$$ 0 0
$$657$$ 6.68864e6i 0.604539i
$$658$$ 0 0
$$659$$ 4.72468e6 0.423798 0.211899 0.977292i $$-0.432035\pi$$
0.211899 + 0.977292i $$0.432035\pi$$
$$660$$ 0 0
$$661$$ 6.17420e6i 0.549639i 0.961496 + 0.274819i $$0.0886180\pi$$
−0.961496 + 0.274819i $$0.911382\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 3.97267e6i − 0.348360i
$$666$$ 0 0
$$667$$ −2.43778e6 −0.212168
$$668$$ 0 0
$$669$$ − 9.35232e6i − 0.807893i
$$670$$ 0 0
$$671$$ 1.87585e7i 1.60839i
$$672$$ 0 0
$$673$$ 9.40925e6 0.800787 0.400394 0.916343i $$-0.368873\pi$$
0.400394 + 0.916343i $$0.368873\pi$$
$$674$$ 0 0
$$675$$ 857736. 0.0724593
$$676$$ 0 0
$$677$$ 1.50086e7 1.25854 0.629272 0.777185i $$-0.283353\pi$$
0.629272 + 0.777185i $$0.283353\pi$$
$$678$$ 0 0
$$679$$ 1.07710e7 0.896567
$$680$$ 0 0
$$681$$ 8.93851e6i 0.738580i
$$682$$ 0 0
$$683$$ 1.29707e7i 1.06393i 0.846768 + 0.531963i $$0.178545\pi$$
−0.846768 + 0.531963i $$0.821455\pi$$
$$684$$ 0 0
$$685$$ −1.71257e7 −1.39451
$$686$$ 0 0
$$687$$ − 3.27295e6i − 0.264574i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2.26556e7i 1.80501i 0.430677 + 0.902506i $$0.358275\pi$$
−0.430677 + 0.902506i $$0.641725\pi$$
$$692$$ 0 0
$$693$$ −4.70448e6 −0.372116
$$694$$ 0 0
$$695$$ − 8.00950e6i − 0.628989i
$$696$$ 0 0
$$697$$ − 1.02322e7i − 0.797791i
$$698$$ 0 0
$$699$$ −1.84615e6 −0.142914
$$700$$ 0 0
$$701$$ −1.90169e7 −1.46166 −0.730828 0.682562i $$-0.760866\pi$$
−0.730828 + 0.682562i $$0.760866\pi$$
$$702$$ 0 0
$$703$$ −249128. −0.0190123
$$704$$ 0 0
$$705$$ 839808. 0.0636366
$$706$$ 0 0
$$707$$ − 993168.i − 0.0747264i
$$708$$ 0 0
$$709$$ − 1.51311e7i − 1.13046i −0.824933 0.565231i $$-0.808787\pi$$
0.824933 0.565231i $$-0.191213\pi$$
$$710$$ 0 0
$$711$$ 7.61429e6 0.564879
$$712$$ 0 0
$$713$$ 1.74666e7i 1.28672i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 1.38568e7i − 1.00662i
$$718$$ 0 0
$$719$$ 1.50323e7 1.08443 0.542217 0.840238i $$-0.317585\pi$$
0.542217 + 0.840238i $$0.317585\pi$$
$$720$$ 0 0
$$721$$ 2.39853e6i 0.171833i
$$722$$ 0 0
$$723$$ 7.88489e6i 0.560983i
$$724$$ 0 0
$$725$$ −124146. −0.00877178
$$726$$ 0 0
$$727$$ 7.41230e6 0.520136 0.260068 0.965590i $$-0.416255\pi$$
0.260068 + 0.965590i $$0.416255\pi$$
$$728$$ 0 0
$$729$$ 1.44612e7 1.00782
$$730$$ 0 0
$$731$$ −7.18740e6 −0.497483
$$732$$ 0 0
$$733$$ − 2.77928e6i − 0.191061i −0.995426 0.0955306i $$-0.969545\pi$$
0.995426 0.0955306i $$-0.0304548\pi$$
$$734$$ 0 0
$$735$$ − 5.87282e6i − 0.400985i
$$736$$ 0 0
$$737$$ 1.17785e7 0.798768
$$738$$ 0 0
$$739$$ 1.21046e7i 0.815342i 0.913129 + 0.407671i $$0.133659\pi$$
−0.913129 + 0.407671i $$0.866341\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 4.46926e6i 0.297005i 0.988912 + 0.148502i $$0.0474452\pi$$
−0.988912 + 0.148502i $$0.952555\pi$$
$$744$$ 0 0
$$745$$ −1.06172e7 −0.700838
$$746$$ 0 0
$$747$$ − 6.70388e6i − 0.439567i
$$748$$ 0 0
$$749$$ 1.07680e7i 0.701345i
$$750$$ 0 0
$$751$$ −2.88463e7 −1.86634 −0.933168 0.359442i $$-0.882967\pi$$
−0.933168 + 0.359442i $$0.882967\pi$$
$$752$$ 0 0
$$753$$ 1.61028e7 1.03494
$$754$$ 0 0
$$755$$ 4.01544e6 0.256369
$$756$$ 0 0
$$757$$ 9.60868e6 0.609430 0.304715 0.952444i $$-0.401439\pi$$
0.304715 + 0.952444i $$0.401439\pi$$
$$758$$ 0 0
$$759$$ 2.65939e7i 1.67563i
$$760$$ 0 0
$$761$$ − 4.54588e6i − 0.284549i −0.989827 0.142274i $$-0.954558\pi$$
0.989827 0.142274i $$-0.0454415\pi$$
$$762$$ 0 0
$$763$$ −8.79138e6 −0.546696
$$764$$ 0 0
$$765$$ 3.17552e6i 0.196183i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ − 2.15923e7i − 1.31669i −0.752716 0.658345i $$-0.771257\pi$$
0.752716 0.658345i $$-0.228743\pi$$
$$770$$ 0 0
$$771$$ 1.58825e6 0.0962238
$$772$$ 0 0
$$773$$ − 1.48400e7i − 0.893276i −0.894715 0.446638i $$-0.852621\pi$$
0.894715 0.446638i $$-0.147379\pi$$
$$774$$ 0 0
$$775$$ 889504.i 0.0531978i
$$776$$ 0 0
$$777$$ 314688. 0.0186994
$$778$$ 0 0
$$779$$ −1.44009e7 −0.850251
$$780$$ 0 0
$$781$$ −2.53109e7 −1.48484
$$782$$ 0 0
$$783$$ −2.43778e6 −0.142098
$$784$$ 0 0
$$785$$ 6.52741e6i 0.378065i
$$786$$ 0 0
$$787$$ 2.48785e7i 1.43182i 0.698194 + 0.715909i $$0.253987\pi$$
−0.698194 + 0.715909i $$0.746013\pi$$
$$788$$ 0 0
$$789$$ −1.13193e7 −0.647330
$$790$$ 0 0
$$791$$ − 2.60885e6i − 0.148254i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ − 1.26321e7i − 0.708856i
$$796$$ 0 0
$$797$$ −3.16080e7 −1.76259 −0.881294 0.472568i $$-0.843327\pi$$
−0.881294 + 0.472568i $$0.843327\pi$$
$$798$$ 0 0
$$799$$ − 769824.i − 0.0426604i
$$800$$ 0 0
$$801$$ 2.94565e6i 0.162218i
$$802$$ 0 0
$$803$$ −3.64835e7 −1.99668
$$804$$ 0 0
$$805$$ −1.95022e7 −1.06070
$$806$$ 0 0
$$807$$ −1.16102e7 −0.627562
$$808$$ 0 0
$$809$$ −3.10009e6 −0.166534 −0.0832669 0.996527i $$-0.526535\pi$$
−0.0832669 + 0.996527i $$0.526535\pi$$
$$810$$ 0 0
$$811$$ 1.87180e6i 0.0999328i 0.998751 + 0.0499664i $$0.0159114\pi$$
−0.998751 + 0.0499664i $$0.984089\pi$$
$$812$$ 0 0
$$813$$ − 6.21984e6i − 0.330030i
$$814$$ 0 0
$$815$$ −6.01236e6 −0.317067
$$816$$ 0 0
$$817$$ 1.01156e7i 0.530196i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 2.00184e7i − 1.03650i −0.855228 0.518252i $$-0.826583\pi$$
0.855228 0.518252i $$-0.173417\pi$$
$$822$$ 0 0
$$823$$ −1.53118e7 −0.787999 −0.394000 0.919111i $$-0.628909\pi$$
−0.394000 + 0.919111i $$0.628909\pi$$
$$824$$ 0 0
$$825$$ 1.35432e6i 0.0692766i
$$826$$ 0 0
$$827$$ − 9.59310e6i − 0.487748i −0.969807 0.243874i $$-0.921582\pi$$
0.969807 0.243874i $$-0.0784183\pi$$
$$828$$ 0 0
$$829$$ −2.52209e7 −1.27460 −0.637302 0.770615i $$-0.719949\pi$$
−0.637302 + 0.770615i $$0.719949\pi$$
$$830$$ 0 0
$$831$$ 2.66727e7 1.33988
$$832$$ 0 0
$$833$$ −5.38342e6 −0.268810
$$834$$ 0 0
$$835$$ −2.65589e7 −1.31824
$$836$$ 0 0
$$837$$ 1.74666e7i 0.861778i
$$838$$ 0 0
$$839$$ 1.77623e7i 0.871154i 0.900151 + 0.435577i $$0.143456\pi$$
−0.900151 + 0.435577i $$0.856544\pi$$
$$840$$ 0 0
$$841$$ −2.01583e7 −0.982798
$$842$$ 0 0
$$843$$ − 2.35937e6i − 0.114348i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 1.14883e7i − 0.550234i
$$848$$ 0 0
$$849$$ −1.86273e7 −0.886913
$$850$$ 0 0
$$851$$ 1.22299e6i 0.0578895i
$$852$$ 0 0
$$853$$ 486970.i 0.0229155i 0.999934 + 0.0114578i $$0.00364720\pi$$
−0.999934 + 0.0114578i $$0.996353\pi$$
$$854$$ 0 0
$$855$$ 4.46926e6 0.209084
$$856$$ 0 0
$$857$$ 1.92634e6 0.0895945 0.0447972 0.998996i $$-0.485736\pi$$
0.0447972 + 0.998996i $$0.485736\pi$$
$$858$$ 0 0
$$859$$ 2.23538e7 1.03364 0.516820 0.856094i $$-0.327116\pi$$
0.516820 + 0.856094i $$0.327116\pi$$
$$860$$ 0 0
$$861$$ 1.81907e7 0.836258
$$862$$ 0 0
$$863$$ 1.85838e7i 0.849390i 0.905337 + 0.424695i $$0.139619\pi$$
−0.905337 + 0.424695i $$0.860381\pi$$
$$864$$ 0 0
$$865$$ − 3.82025e7i − 1.73601i
$$866$$ 0 0
$$867$$ 1.28043e7 0.578504
$$868$$ 0 0
$$869$$ 4.15325e7i 1.86569i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 1.21174e7i 0.538114i
$$874$$ 0 0
$$875$$ −1.58432e7 −0.699555
$$876$$ 0 0
$$877$$ − 2.91048e7i − 1.27781i −0.769286 0.638905i $$-0.779388\pi$$
0.769286 0.638905i $$-0.220612\pi$$
$$878$$ 0 0
$$879$$ − 1.28660e7i − 0.561659i
$$880$$ 0 0
$$881$$ 3.14696e6 0.136600 0.0683001 0.997665i $$-0.478242\pi$$
0.0683001 + 0.997665i $$0.478242\pi$$
$$882$$ 0 0
$$883$$ −1.59995e7 −0.690566 −0.345283 0.938499i $$-0.612217\pi$$
−0.345283 + 0.938499i $$0.612217\pi$$
$$884$$ 0 0
$$885$$ 4.96886e6 0.213255
$$886$$ 0 0
$$887$$ −3.45874e7 −1.47608 −0.738039 0.674758i $$-0.764248\pi$$
−0.738039 + 0.674758i $$0.764248\pi$$
$$888$$ 0 0
$$889$$ − 2.96131e7i − 1.25669i
$$890$$ 0 0
$$891$$ 1.36031e7i 0.574044i
$$892$$ 0 0
$$893$$ −1.08346e6 −0.0454656
$$894$$ 0 0
$$895$$ − 2.66581e7i − 1.11243i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 2.52806e6i − 0.104325i
$$900$$ 0 0
$$901$$ −1.15794e7 −0.475199
$$902$$ 0 0
$$903$$ − 1.27776e7i − 0.521471i
$$904$$ 0 0
$$905$$ 3.02103e7i 1.22612i
$$906$$ 0 0
$$907$$ −1.74396e7 −0.703914 −0.351957 0.936016i $$-0.614484\pi$$
−0.351957 + 0.936016i $$0.614484\pi$$
$$908$$ 0 0
$$909$$ 1.11731e6 0.0448503
$$910$$ 0 0
$$911$$ −2.59589e6 −0.103631 −0.0518155 0.998657i $$-0.516501\pi$$
−0.0518155 + 0.998657i $$0.516501\pi$$
$$912$$ 0 0
$$913$$ 3.65666e7 1.45180
$$914$$ 0 0
$$915$$ 2.25102e7i 0.888847i
$$916$$ 0 0
$$917$$ 8.88624e6i 0.348975i
$$918$$ 0 0
$$919$$ −1.76411e7 −0.689028 −0.344514 0.938781i $$-0.611956\pi$$
−0.344514 + 0.938781i $$0.611956\pi$$
$$920$$ 0 0
$$921$$ 1.90307e7i 0.739275i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 62282.0i 0.00239336i
$$926$$ 0 0
$$927$$ −2.69834e6 −0.103133
$$928$$ 0 0
$$929$$ 3.96785e7i 1.50840i 0.656646 + 0.754199i $$0.271975\pi$$
−0.656646 + 0.754199i $$0.728025\pi$$
$$930$$ 0 0
$$931$$ 7.57667e6i 0.286486i
$$932$$ 0 0
$$933$$ −8.76874e6 −0.329787
$$934$$ 0 0
$$935$$ −1.73210e7 −0.647955
$$936$$ 0 0
$$937$$ 3.93413e7 1.46386 0.731930 0.681380i $$-0.238620\pi$$
0.731930 + 0.681380i $$0.238620\pi$$
$$938$$ 0 0
$$939$$ −7.01830e6 −0.259757
$$940$$ 0 0
$$941$$ 4.62506e7i 1.70272i 0.524581 + 0.851361i $$0.324222\pi$$
−0.524581 + 0.851361i $$0.675778\pi$$
$$942$$ 0 0
$$943$$ 7.06955e7i 2.58888i
$$944$$ 0 0
$$945$$ −1.95022e7 −0.710402
$$946$$ 0 0
$$947$$ 3.79025e7i 1.37339i 0.726947 + 0.686693i $$0.240938\pi$$
−0.726947 + 0.686693i $$0.759062\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 2.97944e7i 1.06828i
$$952$$ 0 0
$$953$$ 2.66462e7 0.950394 0.475197 0.879879i $$-0.342377\pi$$
0.475197 + 0.879879i $$0.342377\pi$$
$$954$$ 0 0
$$955$$ − 3.90977e7i − 1.38721i
$$956$$ 0 0
$$957$$ − 3.84912e6i − 0.135857i
$$958$$ 0 0
$$959$$ −2.79085e7 −0.979918
$$960$$ 0 0
$$961$$ 1.05156e7 0.367304
$$962$$ 0 0
$$963$$ −1.21140e7 −0.420943
$$964$$ 0 0
$$965$$ 383724. 0.0132648
$$966$$ 0 0
$$967$$ 4.09790e7i 1.40927i 0.709568 + 0.704637i $$0.248890\pi$$
−0.709568 + 0.704637i $$0.751110\pi$$
$$968$$ 0 0
$$969$$ 5.95901e6i 0.203875i
$$970$$ 0 0
$$971$$ −2.72034e7 −0.925922 −0.462961 0.886379i $$-0.653213\pi$$
−0.462961 + 0.886379i $$0.653213\pi$$
$$972$$ 0 0
$$973$$ − 1.30525e7i − 0.441990i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.53555e7i 0.849839i 0.905231 + 0.424919i $$0.139698\pi$$
−0.905231 + 0.424919i $$0.860302\pi$$
$$978$$ 0 0
$$979$$ −1.60672e7 −0.535775
$$980$$ 0 0
$$981$$ − 9.89030e6i − 0.328123i
$$982$$ 0 0
$$983$$ − 1.19139e7i − 0.393252i −0.980479 0.196626i $$-0.937002\pi$$
0.980479 0.196626i $$-0.0629984\pi$$
$$984$$ 0 0
$$985$$ 2.86439e7 0.940678
$$986$$ 0 0
$$987$$ 1.36858e6 0.0447173
$$988$$ 0 0
$$989$$ 4.96584e7 1.61437
$$990$$ 0 0
$$991$$ 2.91931e7 0.944268 0.472134 0.881527i $$-0.343484\pi$$
0.472134 + 0.881527i $$0.343484\pi$$
$$992$$ 0 0
$$993$$ − 4.53538e6i − 0.145962i
$$994$$ 0 0
$$995$$ − 3.03307e6i − 0.0971237i
$$996$$ 0 0
$$997$$ −1.73001e7 −0.551201 −0.275601 0.961272i $$-0.588877\pi$$
−0.275601 + 0.961272i $$0.588877\pi$$
$$998$$ 0 0
$$999$$ 1.22299e6i 0.0387713i
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 676.6.d.a.337.2 2
13.5 odd 4 4.6.a.a.1.1 1
13.8 odd 4 676.6.a.a.1.1 1
13.12 even 2 inner 676.6.d.a.337.1 2
39.5 even 4 36.6.a.a.1.1 1
52.31 even 4 16.6.a.b.1.1 1
65.18 even 4 100.6.c.b.49.1 2
65.44 odd 4 100.6.a.b.1.1 1
65.57 even 4 100.6.c.b.49.2 2
91.5 even 12 196.6.e.d.165.1 2
91.18 odd 12 196.6.e.g.177.1 2
91.31 even 12 196.6.e.d.177.1 2
91.44 odd 12 196.6.e.g.165.1 2
91.83 even 4 196.6.a.e.1.1 1
104.5 odd 4 64.6.a.f.1.1 1
104.83 even 4 64.6.a.b.1.1 1
117.5 even 12 324.6.e.d.217.1 2
117.31 odd 12 324.6.e.a.217.1 2
117.70 odd 12 324.6.e.a.109.1 2
117.83 even 12 324.6.e.d.109.1 2
143.109 even 4 484.6.a.a.1.1 1
156.83 odd 4 144.6.a.c.1.1 1
195.44 even 4 900.6.a.h.1.1 1
195.83 odd 4 900.6.d.a.649.2 2
195.122 odd 4 900.6.d.a.649.1 2
208.5 odd 4 256.6.b.g.129.2 2
208.83 even 4 256.6.b.c.129.2 2
208.109 odd 4 256.6.b.g.129.1 2
208.187 even 4 256.6.b.c.129.1 2
260.83 odd 4 400.6.c.f.49.2 2
260.187 odd 4 400.6.c.f.49.1 2
260.239 even 4 400.6.a.d.1.1 1
312.5 even 4 576.6.a.bc.1.1 1
312.83 odd 4 576.6.a.bd.1.1 1
364.83 odd 4 784.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 13.5 odd 4
16.6.a.b.1.1 1 52.31 even 4
36.6.a.a.1.1 1 39.5 even 4
64.6.a.b.1.1 1 104.83 even 4
64.6.a.f.1.1 1 104.5 odd 4
100.6.a.b.1.1 1 65.44 odd 4
100.6.c.b.49.1 2 65.18 even 4
100.6.c.b.49.2 2 65.57 even 4
144.6.a.c.1.1 1 156.83 odd 4
196.6.a.e.1.1 1 91.83 even 4
196.6.e.d.165.1 2 91.5 even 12
196.6.e.d.177.1 2 91.31 even 12
196.6.e.g.165.1 2 91.44 odd 12
196.6.e.g.177.1 2 91.18 odd 12
256.6.b.c.129.1 2 208.187 even 4
256.6.b.c.129.2 2 208.83 even 4
256.6.b.g.129.1 2 208.109 odd 4
256.6.b.g.129.2 2 208.5 odd 4
324.6.e.a.109.1 2 117.70 odd 12
324.6.e.a.217.1 2 117.31 odd 12
324.6.e.d.109.1 2 117.83 even 12
324.6.e.d.217.1 2 117.5 even 12
400.6.a.d.1.1 1 260.239 even 4
400.6.c.f.49.1 2 260.187 odd 4
400.6.c.f.49.2 2 260.83 odd 4
484.6.a.a.1.1 1 143.109 even 4
576.6.a.bc.1.1 1 312.5 even 4
576.6.a.bd.1.1 1 312.83 odd 4
676.6.a.a.1.1 1 13.8 odd 4
676.6.d.a.337.1 2 13.12 even 2 inner
676.6.d.a.337.2 2 1.1 even 1 trivial
784.6.a.d.1.1 1 364.83 odd 4
900.6.a.h.1.1 1 195.44 even 4
900.6.d.a.649.1 2 195.122 odd 4
900.6.d.a.649.2 2 195.83 odd 4