# Properties

 Label 112.10.a.c.1.2 Level $112$ Weight $10$ Character 112.1 Self dual yes Analytic conductor $57.684$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.6840136504$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2305})$$ Defining polynomial: $$x^{2} - x - 576$$ x^2 - x - 576 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$24.5052$$ of defining polynomial Character $$\chi$$ $$=$$ 112.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+247.052 q^{3} -2373.22 q^{5} -2401.00 q^{7} +41351.7 q^{9} +O(q^{10})$$ $$q+247.052 q^{3} -2373.22 q^{5} -2401.00 q^{7} +41351.7 q^{9} +27940.9 q^{11} +60943.3 q^{13} -586309. q^{15} -358867. q^{17} +391593. q^{19} -593172. q^{21} +302894. q^{23} +3.67904e6 q^{25} +5.35330e6 q^{27} +6.73993e6 q^{29} -2.98262e6 q^{31} +6.90287e6 q^{33} +5.69810e6 q^{35} -3.49582e6 q^{37} +1.50562e7 q^{39} +3.43724e7 q^{41} +1.45085e7 q^{43} -9.81367e7 q^{45} +2.76485e7 q^{47} +5.76480e6 q^{49} -8.86589e7 q^{51} -2.39217e7 q^{53} -6.63100e7 q^{55} +9.67439e7 q^{57} +1.20580e8 q^{59} +7.23140e7 q^{61} -9.92855e7 q^{63} -1.44632e8 q^{65} -8.70377e7 q^{67} +7.48307e7 q^{69} -2.19622e8 q^{71} +2.67792e8 q^{73} +9.08915e8 q^{75} -6.70862e7 q^{77} -2.85350e7 q^{79} +5.08619e8 q^{81} +3.83237e8 q^{83} +8.51671e8 q^{85} +1.66511e9 q^{87} +7.21581e8 q^{89} -1.46325e8 q^{91} -7.36861e8 q^{93} -9.29336e8 q^{95} -6.73736e8 q^{97} +1.15541e9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{3} - 2730 q^{5} - 4802 q^{7} + 75982 q^{9}+O(q^{10})$$ 2 * q + 14 * q^3 - 2730 * q^5 - 4802 * q^7 + 75982 * q^9 $$2 q + 14 q^{3} - 2730 q^{5} - 4802 q^{7} + 75982 q^{9} - 44940 q^{11} + 100282 q^{13} - 503160 q^{15} - 870408 q^{17} - 508774 q^{19} - 33614 q^{21} - 79800 q^{23} + 1853210 q^{25} + 1869812 q^{27} + 2006328 q^{29} - 2188732 q^{31} + 23887920 q^{33} + 6554730 q^{35} - 20723576 q^{37} + 5888224 q^{39} + 19016592 q^{41} - 4193716 q^{43} - 110492130 q^{45} + 74542524 q^{47} + 11529602 q^{49} + 30556644 q^{51} - 3239748 q^{53} - 40307400 q^{55} + 306576332 q^{57} + 133642362 q^{59} + 227801686 q^{61} - 182432782 q^{63} - 158667180 q^{65} - 332930272 q^{67} + 164018400 q^{69} + 167985720 q^{71} - 44684276 q^{73} + 1334428970 q^{75} + 107900940 q^{77} - 269642776 q^{79} + 638826478 q^{81} + 183105762 q^{83} + 1034179020 q^{85} + 2768288796 q^{87} + 791657748 q^{89} - 240777082 q^{91} - 921877624 q^{93} - 608102040 q^{95} - 4169480 q^{97} - 1368480540 q^{99}+O(q^{100})$$ 2 * q + 14 * q^3 - 2730 * q^5 - 4802 * q^7 + 75982 * q^9 - 44940 * q^11 + 100282 * q^13 - 503160 * q^15 - 870408 * q^17 - 508774 * q^19 - 33614 * q^21 - 79800 * q^23 + 1853210 * q^25 + 1869812 * q^27 + 2006328 * q^29 - 2188732 * q^31 + 23887920 * q^33 + 6554730 * q^35 - 20723576 * q^37 + 5888224 * q^39 + 19016592 * q^41 - 4193716 * q^43 - 110492130 * q^45 + 74542524 * q^47 + 11529602 * q^49 + 30556644 * q^51 - 3239748 * q^53 - 40307400 * q^55 + 306576332 * q^57 + 133642362 * q^59 + 227801686 * q^61 - 182432782 * q^63 - 158667180 * q^65 - 332930272 * q^67 + 164018400 * q^69 + 167985720 * q^71 - 44684276 * q^73 + 1334428970 * q^75 + 107900940 * q^77 - 269642776 * q^79 + 638826478 * q^81 + 183105762 * q^83 + 1034179020 * q^85 + 2768288796 * q^87 + 791657748 * q^89 - 240777082 * q^91 - 921877624 * q^93 - 608102040 * q^95 - 4169480 * q^97 - 1368480540 * q^99

## 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
$$3$$ 247.052 1.76093 0.880467 0.474108i $$-0.157229\pi$$
0.880467 + 0.474108i $$0.157229\pi$$
$$4$$ 0 0
$$5$$ −2373.22 −1.69814 −0.849069 0.528283i $$-0.822836\pi$$
−0.849069 + 0.528283i $$0.822836\pi$$
$$6$$ 0 0
$$7$$ −2401.00 −0.377964
$$8$$ 0 0
$$9$$ 41351.7 2.10089
$$10$$ 0 0
$$11$$ 27940.9 0.575405 0.287703 0.957720i $$-0.407109\pi$$
0.287703 + 0.957720i $$0.407109\pi$$
$$12$$ 0 0
$$13$$ 60943.3 0.591808 0.295904 0.955218i $$-0.404379\pi$$
0.295904 + 0.955218i $$0.404379\pi$$
$$14$$ 0 0
$$15$$ −586309. −2.99031
$$16$$ 0 0
$$17$$ −358867. −1.04211 −0.521055 0.853523i $$-0.674462\pi$$
−0.521055 + 0.853523i $$0.674462\pi$$
$$18$$ 0 0
$$19$$ 391593. 0.689356 0.344678 0.938721i $$-0.387988\pi$$
0.344678 + 0.938721i $$0.387988\pi$$
$$20$$ 0 0
$$21$$ −593172. −0.665570
$$22$$ 0 0
$$23$$ 302894. 0.225692 0.112846 0.993612i $$-0.464003\pi$$
0.112846 + 0.993612i $$0.464003\pi$$
$$24$$ 0 0
$$25$$ 3.67904e6 1.88367
$$26$$ 0 0
$$27$$ 5.35330e6 1.93859
$$28$$ 0 0
$$29$$ 6.73993e6 1.76956 0.884778 0.466013i $$-0.154310\pi$$
0.884778 + 0.466013i $$0.154310\pi$$
$$30$$ 0 0
$$31$$ −2.98262e6 −0.580056 −0.290028 0.957018i $$-0.593665\pi$$
−0.290028 + 0.957018i $$0.593665\pi$$
$$32$$ 0 0
$$33$$ 6.90287e6 1.01325
$$34$$ 0 0
$$35$$ 5.69810e6 0.641835
$$36$$ 0 0
$$37$$ −3.49582e6 −0.306649 −0.153324 0.988176i $$-0.548998\pi$$
−0.153324 + 0.988176i $$0.548998\pi$$
$$38$$ 0 0
$$39$$ 1.50562e7 1.04214
$$40$$ 0 0
$$41$$ 3.43724e7 1.89969 0.949845 0.312722i $$-0.101241\pi$$
0.949845 + 0.312722i $$0.101241\pi$$
$$42$$ 0 0
$$43$$ 1.45085e7 0.647164 0.323582 0.946200i $$-0.395113\pi$$
0.323582 + 0.946200i $$0.395113\pi$$
$$44$$ 0 0
$$45$$ −9.81367e7 −3.56759
$$46$$ 0 0
$$47$$ 2.76485e7 0.826479 0.413239 0.910622i $$-0.364397\pi$$
0.413239 + 0.910622i $$0.364397\pi$$
$$48$$ 0 0
$$49$$ 5.76480e6 0.142857
$$50$$ 0 0
$$51$$ −8.86589e7 −1.83509
$$52$$ 0 0
$$53$$ −2.39217e7 −0.416438 −0.208219 0.978082i $$-0.566767\pi$$
−0.208219 + 0.978082i $$0.566767\pi$$
$$54$$ 0 0
$$55$$ −6.63100e7 −0.977117
$$56$$ 0 0
$$57$$ 9.67439e7 1.21391
$$58$$ 0 0
$$59$$ 1.20580e8 1.29551 0.647756 0.761848i $$-0.275707\pi$$
0.647756 + 0.761848i $$0.275707\pi$$
$$60$$ 0 0
$$61$$ 7.23140e7 0.668710 0.334355 0.942447i $$-0.391482\pi$$
0.334355 + 0.942447i $$0.391482\pi$$
$$62$$ 0 0
$$63$$ −9.92855e7 −0.794060
$$64$$ 0 0
$$65$$ −1.44632e8 −1.00497
$$66$$ 0 0
$$67$$ −8.70377e7 −0.527680 −0.263840 0.964566i $$-0.584989\pi$$
−0.263840 + 0.964566i $$0.584989\pi$$
$$68$$ 0 0
$$69$$ 7.48307e7 0.397428
$$70$$ 0 0
$$71$$ −2.19622e8 −1.02568 −0.512842 0.858483i $$-0.671407\pi$$
−0.512842 + 0.858483i $$0.671407\pi$$
$$72$$ 0 0
$$73$$ 2.67792e8 1.10368 0.551841 0.833949i $$-0.313926\pi$$
0.551841 + 0.833949i $$0.313926\pi$$
$$74$$ 0 0
$$75$$ 9.08915e8 3.31702
$$76$$ 0 0
$$77$$ −6.70862e7 −0.217483
$$78$$ 0 0
$$79$$ −2.85350e7 −0.0824243 −0.0412122 0.999150i $$-0.513122\pi$$
−0.0412122 + 0.999150i $$0.513122\pi$$
$$80$$ 0 0
$$81$$ 5.08619e8 1.31283
$$82$$ 0 0
$$83$$ 3.83237e8 0.886372 0.443186 0.896430i $$-0.353848\pi$$
0.443186 + 0.896430i $$0.353848\pi$$
$$84$$ 0 0
$$85$$ 8.51671e8 1.76965
$$86$$ 0 0
$$87$$ 1.66511e9 3.11607
$$88$$ 0 0
$$89$$ 7.21581e8 1.21907 0.609537 0.792757i $$-0.291355\pi$$
0.609537 + 0.792757i $$0.291355\pi$$
$$90$$ 0 0
$$91$$ −1.46325e8 −0.223683
$$92$$ 0 0
$$93$$ −7.36861e8 −1.02144
$$94$$ 0 0
$$95$$ −9.29336e8 −1.17062
$$96$$ 0 0
$$97$$ −6.73736e8 −0.772710 −0.386355 0.922350i $$-0.626266\pi$$
−0.386355 + 0.922350i $$0.626266\pi$$
$$98$$ 0 0
$$99$$ 1.15541e9 1.20886
$$100$$ 0 0
$$101$$ −1.04805e7 −0.0100216 −0.00501081 0.999987i $$-0.501595\pi$$
−0.00501081 + 0.999987i $$0.501595\pi$$
$$102$$ 0 0
$$103$$ −1.39782e9 −1.22373 −0.611864 0.790963i $$-0.709580\pi$$
−0.611864 + 0.790963i $$0.709580\pi$$
$$104$$ 0 0
$$105$$ 1.40773e9 1.13023
$$106$$ 0 0
$$107$$ 4.84414e8 0.357264 0.178632 0.983916i $$-0.442833\pi$$
0.178632 + 0.983916i $$0.442833\pi$$
$$108$$ 0 0
$$109$$ −1.96298e8 −0.133198 −0.0665989 0.997780i $$-0.521215\pi$$
−0.0665989 + 0.997780i $$0.521215\pi$$
$$110$$ 0 0
$$111$$ −8.63649e8 −0.539988
$$112$$ 0 0
$$113$$ −5.80849e8 −0.335128 −0.167564 0.985861i $$-0.553590\pi$$
−0.167564 + 0.985861i $$0.553590\pi$$
$$114$$ 0 0
$$115$$ −7.18835e8 −0.383256
$$116$$ 0 0
$$117$$ 2.52011e9 1.24332
$$118$$ 0 0
$$119$$ 8.61641e8 0.393881
$$120$$ 0 0
$$121$$ −1.57725e9 −0.668909
$$122$$ 0 0
$$123$$ 8.49178e9 3.34523
$$124$$ 0 0
$$125$$ −4.09598e9 −1.50059
$$126$$ 0 0
$$127$$ −2.59088e9 −0.883751 −0.441876 0.897076i $$-0.645687\pi$$
−0.441876 + 0.897076i $$0.645687\pi$$
$$128$$ 0 0
$$129$$ 3.58436e9 1.13961
$$130$$ 0 0
$$131$$ −2.45598e9 −0.728626 −0.364313 0.931277i $$-0.618696\pi$$
−0.364313 + 0.931277i $$0.618696\pi$$
$$132$$ 0 0
$$133$$ −9.40215e8 −0.260552
$$134$$ 0 0
$$135$$ −1.27046e10 −3.29198
$$136$$ 0 0
$$137$$ 6.08364e9 1.47544 0.737719 0.675108i $$-0.235903\pi$$
0.737719 + 0.675108i $$0.235903\pi$$
$$138$$ 0 0
$$139$$ −2.25249e9 −0.511796 −0.255898 0.966704i $$-0.582371\pi$$
−0.255898 + 0.966704i $$0.582371\pi$$
$$140$$ 0 0
$$141$$ 6.83063e9 1.45537
$$142$$ 0 0
$$143$$ 1.70281e9 0.340530
$$144$$ 0 0
$$145$$ −1.59953e10 −3.00495
$$146$$ 0 0
$$147$$ 1.42421e9 0.251562
$$148$$ 0 0
$$149$$ −3.13517e9 −0.521103 −0.260551 0.965460i $$-0.583904\pi$$
−0.260551 + 0.965460i $$0.583904\pi$$
$$150$$ 0 0
$$151$$ 6.20938e9 0.971969 0.485984 0.873968i $$-0.338461\pi$$
0.485984 + 0.873968i $$0.338461\pi$$
$$152$$ 0 0
$$153$$ −1.48398e10 −2.18936
$$154$$ 0 0
$$155$$ 7.07840e9 0.985014
$$156$$ 0 0
$$157$$ 1.33378e10 1.75201 0.876006 0.482300i $$-0.160198\pi$$
0.876006 + 0.482300i $$0.160198\pi$$
$$158$$ 0 0
$$159$$ −5.90990e9 −0.733319
$$160$$ 0 0
$$161$$ −7.27249e8 −0.0853035
$$162$$ 0 0
$$163$$ −7.33621e9 −0.814006 −0.407003 0.913427i $$-0.633426\pi$$
−0.407003 + 0.913427i $$0.633426\pi$$
$$164$$ 0 0
$$165$$ −1.63820e10 −1.72064
$$166$$ 0 0
$$167$$ 6.42205e9 0.638925 0.319462 0.947599i $$-0.396498\pi$$
0.319462 + 0.947599i $$0.396498\pi$$
$$168$$ 0 0
$$169$$ −6.89041e9 −0.649763
$$170$$ 0 0
$$171$$ 1.61931e10 1.44826
$$172$$ 0 0
$$173$$ 1.91846e10 1.62834 0.814171 0.580626i $$-0.197192\pi$$
0.814171 + 0.580626i $$0.197192\pi$$
$$174$$ 0 0
$$175$$ −8.83338e9 −0.711960
$$176$$ 0 0
$$177$$ 2.97896e10 2.28131
$$178$$ 0 0
$$179$$ −1.53377e10 −1.11666 −0.558330 0.829619i $$-0.688558\pi$$
−0.558330 + 0.829619i $$0.688558\pi$$
$$180$$ 0 0
$$181$$ −1.73475e10 −1.20139 −0.600694 0.799479i $$-0.705109\pi$$
−0.600694 + 0.799479i $$0.705109\pi$$
$$182$$ 0 0
$$183$$ 1.78653e10 1.17755
$$184$$ 0 0
$$185$$ 8.29634e9 0.520732
$$186$$ 0 0
$$187$$ −1.00271e10 −0.599636
$$188$$ 0 0
$$189$$ −1.28533e10 −0.732717
$$190$$ 0 0
$$191$$ −2.70138e10 −1.46871 −0.734355 0.678765i $$-0.762515\pi$$
−0.734355 + 0.678765i $$0.762515\pi$$
$$192$$ 0 0
$$193$$ 2.70232e9 0.140194 0.0700970 0.997540i $$-0.477669\pi$$
0.0700970 + 0.997540i $$0.477669\pi$$
$$194$$ 0 0
$$195$$ −3.57316e10 −1.76969
$$196$$ 0 0
$$197$$ 2.39047e10 1.13080 0.565399 0.824818i $$-0.308722\pi$$
0.565399 + 0.824818i $$0.308722\pi$$
$$198$$ 0 0
$$199$$ 2.16111e9 0.0976872 0.0488436 0.998806i $$-0.484446\pi$$
0.0488436 + 0.998806i $$0.484446\pi$$
$$200$$ 0 0
$$201$$ −2.15028e10 −0.929209
$$202$$ 0 0
$$203$$ −1.61826e10 −0.668829
$$204$$ 0 0
$$205$$ −8.15732e10 −3.22593
$$206$$ 0 0
$$207$$ 1.25252e10 0.474153
$$208$$ 0 0
$$209$$ 1.09415e10 0.396659
$$210$$ 0 0
$$211$$ −3.61055e9 −0.125401 −0.0627007 0.998032i $$-0.519971\pi$$
−0.0627007 + 0.998032i $$0.519971\pi$$
$$212$$ 0 0
$$213$$ −5.42581e10 −1.80616
$$214$$ 0 0
$$215$$ −3.44319e10 −1.09897
$$216$$ 0 0
$$217$$ 7.16126e9 0.219240
$$218$$ 0 0
$$219$$ 6.61585e10 1.94351
$$220$$ 0 0
$$221$$ −2.18706e10 −0.616730
$$222$$ 0 0
$$223$$ 7.13001e10 1.93072 0.965358 0.260929i $$-0.0840287\pi$$
0.965358 + 0.260929i $$0.0840287\pi$$
$$224$$ 0 0
$$225$$ 1.52135e11 3.95737
$$226$$ 0 0
$$227$$ −7.15361e10 −1.78817 −0.894086 0.447896i $$-0.852173\pi$$
−0.894086 + 0.447896i $$0.852173\pi$$
$$228$$ 0 0
$$229$$ −3.56020e10 −0.855491 −0.427745 0.903899i $$-0.640692\pi$$
−0.427745 + 0.903899i $$0.640692\pi$$
$$230$$ 0 0
$$231$$ −1.65738e10 −0.382973
$$232$$ 0 0
$$233$$ −3.80069e10 −0.844814 −0.422407 0.906406i $$-0.638815\pi$$
−0.422407 + 0.906406i $$0.638815\pi$$
$$234$$ 0 0
$$235$$ −6.56160e10 −1.40347
$$236$$ 0 0
$$237$$ −7.04962e9 −0.145144
$$238$$ 0 0
$$239$$ 8.67126e10 1.71906 0.859531 0.511083i $$-0.170756\pi$$
0.859531 + 0.511083i $$0.170756\pi$$
$$240$$ 0 0
$$241$$ −8.18418e9 −0.156278 −0.0781391 0.996942i $$-0.524898\pi$$
−0.0781391 + 0.996942i $$0.524898\pi$$
$$242$$ 0 0
$$243$$ 2.02863e10 0.373228
$$244$$ 0 0
$$245$$ −1.36811e10 −0.242591
$$246$$ 0 0
$$247$$ 2.38650e10 0.407967
$$248$$ 0 0
$$249$$ 9.46795e10 1.56084
$$250$$ 0 0
$$251$$ −9.75467e10 −1.55125 −0.775624 0.631196i $$-0.782565\pi$$
−0.775624 + 0.631196i $$0.782565\pi$$
$$252$$ 0 0
$$253$$ 8.46315e9 0.129864
$$254$$ 0 0
$$255$$ 2.10407e11 3.11623
$$256$$ 0 0
$$257$$ −4.43042e9 −0.0633499 −0.0316750 0.999498i $$-0.510084\pi$$
−0.0316750 + 0.999498i $$0.510084\pi$$
$$258$$ 0 0
$$259$$ 8.39346e9 0.115902
$$260$$ 0 0
$$261$$ 2.78708e11 3.71763
$$262$$ 0 0
$$263$$ 1.20620e11 1.55460 0.777301 0.629129i $$-0.216588\pi$$
0.777301 + 0.629129i $$0.216588\pi$$
$$264$$ 0 0
$$265$$ 5.67714e10 0.707168
$$266$$ 0 0
$$267$$ 1.78268e11 2.14671
$$268$$ 0 0
$$269$$ 7.59025e10 0.883835 0.441917 0.897056i $$-0.354298\pi$$
0.441917 + 0.897056i $$0.354298\pi$$
$$270$$ 0 0
$$271$$ −7.11397e10 −0.801217 −0.400608 0.916249i $$-0.631201\pi$$
−0.400608 + 0.916249i $$0.631201\pi$$
$$272$$ 0 0
$$273$$ −3.61499e10 −0.393890
$$274$$ 0 0
$$275$$ 1.02796e11 1.08387
$$276$$ 0 0
$$277$$ 8.61542e10 0.879261 0.439630 0.898179i $$-0.355109\pi$$
0.439630 + 0.898179i $$0.355109\pi$$
$$278$$ 0 0
$$279$$ −1.23336e11 −1.21863
$$280$$ 0 0
$$281$$ −1.00179e11 −0.958511 −0.479256 0.877675i $$-0.659093\pi$$
−0.479256 + 0.877675i $$0.659093\pi$$
$$282$$ 0 0
$$283$$ 4.57444e10 0.423935 0.211967 0.977277i $$-0.432013\pi$$
0.211967 + 0.977277i $$0.432013\pi$$
$$284$$ 0 0
$$285$$ −2.29594e11 −2.06139
$$286$$ 0 0
$$287$$ −8.25282e10 −0.718015
$$288$$ 0 0
$$289$$ 1.01980e10 0.0859950
$$290$$ 0 0
$$291$$ −1.66448e11 −1.36069
$$292$$ 0 0
$$293$$ −1.01615e10 −0.0805476 −0.0402738 0.999189i $$-0.512823\pi$$
−0.0402738 + 0.999189i $$0.512823\pi$$
$$294$$ 0 0
$$295$$ −2.86163e11 −2.19996
$$296$$ 0 0
$$297$$ 1.49576e11 1.11547
$$298$$ 0 0
$$299$$ 1.84594e10 0.133566
$$300$$ 0 0
$$301$$ −3.48349e10 −0.244605
$$302$$ 0 0
$$303$$ −2.58924e9 −0.0176474
$$304$$ 0 0
$$305$$ −1.71617e11 −1.13556
$$306$$ 0 0
$$307$$ 3.68957e10 0.237057 0.118529 0.992951i $$-0.462182\pi$$
0.118529 + 0.992951i $$0.462182\pi$$
$$308$$ 0 0
$$309$$ −3.45335e11 −2.15490
$$310$$ 0 0
$$311$$ −1.88558e11 −1.14294 −0.571469 0.820624i $$-0.693626\pi$$
−0.571469 + 0.820624i $$0.693626\pi$$
$$312$$ 0 0
$$313$$ 1.31778e11 0.776058 0.388029 0.921647i $$-0.373156\pi$$
0.388029 + 0.921647i $$0.373156\pi$$
$$314$$ 0 0
$$315$$ 2.35626e11 1.34842
$$316$$ 0 0
$$317$$ −1.50686e11 −0.838118 −0.419059 0.907959i $$-0.637640\pi$$
−0.419059 + 0.907959i $$0.637640\pi$$
$$318$$ 0 0
$$319$$ 1.88320e11 1.01821
$$320$$ 0 0
$$321$$ 1.19675e11 0.629118
$$322$$ 0 0
$$323$$ −1.40530e11 −0.718386
$$324$$ 0 0
$$325$$ 2.24213e11 1.11477
$$326$$ 0 0
$$327$$ −4.84959e10 −0.234552
$$328$$ 0 0
$$329$$ −6.63841e10 −0.312380
$$330$$ 0 0
$$331$$ −3.38877e10 −0.155173 −0.0775865 0.996986i $$-0.524721\pi$$
−0.0775865 + 0.996986i $$0.524721\pi$$
$$332$$ 0 0
$$333$$ −1.44558e11 −0.644234
$$334$$ 0 0
$$335$$ 2.06559e11 0.896073
$$336$$ 0 0
$$337$$ 1.98312e11 0.837555 0.418778 0.908089i $$-0.362459\pi$$
0.418778 + 0.908089i $$0.362459\pi$$
$$338$$ 0 0
$$339$$ −1.43500e11 −0.590138
$$340$$ 0 0
$$341$$ −8.33371e10 −0.333767
$$342$$ 0 0
$$343$$ −1.38413e10 −0.0539949
$$344$$ 0 0
$$345$$ −1.77590e11 −0.674888
$$346$$ 0 0
$$347$$ 1.71762e11 0.635981 0.317990 0.948094i $$-0.396992\pi$$
0.317990 + 0.948094i $$0.396992\pi$$
$$348$$ 0 0
$$349$$ −1.88189e11 −0.679014 −0.339507 0.940603i $$-0.610260\pi$$
−0.339507 + 0.940603i $$0.610260\pi$$
$$350$$ 0 0
$$351$$ 3.26248e11 1.14727
$$352$$ 0 0
$$353$$ 1.97995e11 0.678686 0.339343 0.940663i $$-0.389795\pi$$
0.339343 + 0.940663i $$0.389795\pi$$
$$354$$ 0 0
$$355$$ 5.21211e11 1.74175
$$356$$ 0 0
$$357$$ 2.12870e11 0.693598
$$358$$ 0 0
$$359$$ −4.34669e11 −1.38113 −0.690563 0.723272i $$-0.742637\pi$$
−0.690563 + 0.723272i $$0.742637\pi$$
$$360$$ 0 0
$$361$$ −1.69343e11 −0.524788
$$362$$ 0 0
$$363$$ −3.89663e11 −1.17790
$$364$$ 0 0
$$365$$ −6.35528e11 −1.87420
$$366$$ 0 0
$$367$$ −6.02066e10 −0.173240 −0.0866198 0.996241i $$-0.527607\pi$$
−0.0866198 + 0.996241i $$0.527607\pi$$
$$368$$ 0 0
$$369$$ 1.42136e12 3.99103
$$370$$ 0 0
$$371$$ 5.74359e10 0.157399
$$372$$ 0 0
$$373$$ −6.85521e11 −1.83371 −0.916855 0.399219i $$-0.869281\pi$$
−0.916855 + 0.399219i $$0.869281\pi$$
$$374$$ 0 0
$$375$$ −1.01192e12 −2.64244
$$376$$ 0 0
$$377$$ 4.10754e11 1.04724
$$378$$ 0 0
$$379$$ −5.05522e11 −1.25853 −0.629266 0.777190i $$-0.716644\pi$$
−0.629266 + 0.777190i $$0.716644\pi$$
$$380$$ 0 0
$$381$$ −6.40082e11 −1.55623
$$382$$ 0 0
$$383$$ −2.02054e11 −0.479814 −0.239907 0.970796i $$-0.577117\pi$$
−0.239907 + 0.970796i $$0.577117\pi$$
$$384$$ 0 0
$$385$$ 1.59210e11 0.369316
$$386$$ 0 0
$$387$$ 5.99952e11 1.35962
$$388$$ 0 0
$$389$$ 5.51954e11 1.22216 0.611082 0.791567i $$-0.290735\pi$$
0.611082 + 0.791567i $$0.290735\pi$$
$$390$$ 0 0
$$391$$ −1.08699e11 −0.235196
$$392$$ 0 0
$$393$$ −6.06756e11 −1.28306
$$394$$ 0 0
$$395$$ 6.77197e10 0.139968
$$396$$ 0 0
$$397$$ 1.26816e11 0.256222 0.128111 0.991760i $$-0.459109\pi$$
0.128111 + 0.991760i $$0.459109\pi$$
$$398$$ 0 0
$$399$$ −2.32282e11 −0.458815
$$400$$ 0 0
$$401$$ 5.86957e11 1.13359 0.566795 0.823859i $$-0.308183\pi$$
0.566795 + 0.823859i $$0.308183\pi$$
$$402$$ 0 0
$$403$$ −1.81771e11 −0.343282
$$404$$ 0 0
$$405$$ −1.20706e12 −2.22937
$$406$$ 0 0
$$407$$ −9.76764e10 −0.176447
$$408$$ 0 0
$$409$$ −9.08262e10 −0.160493 −0.0802465 0.996775i $$-0.525571\pi$$
−0.0802465 + 0.996775i $$0.525571\pi$$
$$410$$ 0 0
$$411$$ 1.50298e12 2.59815
$$412$$ 0 0
$$413$$ −2.89513e11 −0.489658
$$414$$ 0 0
$$415$$ −9.09505e11 −1.50518
$$416$$ 0 0
$$417$$ −5.56483e11 −0.901238
$$418$$ 0 0
$$419$$ 8.34669e11 1.32297 0.661487 0.749957i $$-0.269926\pi$$
0.661487 + 0.749957i $$0.269926\pi$$
$$420$$ 0 0
$$421$$ −1.61360e10 −0.0250337 −0.0125169 0.999922i $$-0.503984\pi$$
−0.0125169 + 0.999922i $$0.503984\pi$$
$$422$$ 0 0
$$423$$ 1.14331e12 1.73634
$$424$$ 0 0
$$425$$ −1.32029e12 −1.96299
$$426$$ 0 0
$$427$$ −1.73626e11 −0.252749
$$428$$ 0 0
$$429$$ 4.20684e11 0.599650
$$430$$ 0 0
$$431$$ 4.81935e11 0.672730 0.336365 0.941732i $$-0.390802\pi$$
0.336365 + 0.941732i $$0.390802\pi$$
$$432$$ 0 0
$$433$$ 6.68314e11 0.913661 0.456830 0.889554i $$-0.348985\pi$$
0.456830 + 0.889554i $$0.348985\pi$$
$$434$$ 0 0
$$435$$ −3.95168e12 −5.29151
$$436$$ 0 0
$$437$$ 1.18611e11 0.155582
$$438$$ 0 0
$$439$$ 4.79819e11 0.616577 0.308288 0.951293i $$-0.400244\pi$$
0.308288 + 0.951293i $$0.400244\pi$$
$$440$$ 0 0
$$441$$ 2.38384e11 0.300126
$$442$$ 0 0
$$443$$ 6.53598e11 0.806295 0.403148 0.915135i $$-0.367916\pi$$
0.403148 + 0.915135i $$0.367916\pi$$
$$444$$ 0 0
$$445$$ −1.71247e12 −2.07016
$$446$$ 0 0
$$447$$ −7.74551e11 −0.917627
$$448$$ 0 0
$$449$$ −1.88660e11 −0.219064 −0.109532 0.993983i $$-0.534935\pi$$
−0.109532 + 0.993983i $$0.534935\pi$$
$$450$$ 0 0
$$451$$ 9.60397e11 1.09309
$$452$$ 0 0
$$453$$ 1.53404e12 1.71157
$$454$$ 0 0
$$455$$ 3.47261e11 0.379844
$$456$$ 0 0
$$457$$ 1.31205e12 1.40711 0.703556 0.710639i $$-0.251594\pi$$
0.703556 + 0.710639i $$0.251594\pi$$
$$458$$ 0 0
$$459$$ −1.92113e12 −2.02022
$$460$$ 0 0
$$461$$ 1.16594e12 1.20233 0.601165 0.799125i $$-0.294703\pi$$
0.601165 + 0.799125i $$0.294703\pi$$
$$462$$ 0 0
$$463$$ 3.87318e10 0.0391700 0.0195850 0.999808i $$-0.493766\pi$$
0.0195850 + 0.999808i $$0.493766\pi$$
$$464$$ 0 0
$$465$$ 1.74873e12 1.73454
$$466$$ 0 0
$$467$$ −1.17150e11 −0.113977 −0.0569883 0.998375i $$-0.518150\pi$$
−0.0569883 + 0.998375i $$0.518150\pi$$
$$468$$ 0 0
$$469$$ 2.08977e11 0.199444
$$470$$ 0 0
$$471$$ 3.29514e12 3.08518
$$472$$ 0 0
$$473$$ 4.05381e11 0.372382
$$474$$ 0 0
$$475$$ 1.44069e12 1.29852
$$476$$ 0 0
$$477$$ −9.89203e11 −0.874888
$$478$$ 0 0
$$479$$ 1.94734e12 1.69018 0.845089 0.534626i $$-0.179548\pi$$
0.845089 + 0.534626i $$0.179548\pi$$
$$480$$ 0 0
$$481$$ −2.13047e11 −0.181477
$$482$$ 0 0
$$483$$ −1.79668e11 −0.150214
$$484$$ 0 0
$$485$$ 1.59892e12 1.31217
$$486$$ 0 0
$$487$$ 1.95983e12 1.57884 0.789421 0.613852i $$-0.210381\pi$$
0.789421 + 0.613852i $$0.210381\pi$$
$$488$$ 0 0
$$489$$ −1.81243e12 −1.43341
$$490$$ 0 0
$$491$$ 2.93589e11 0.227967 0.113984 0.993483i $$-0.463639\pi$$
0.113984 + 0.993483i $$0.463639\pi$$
$$492$$ 0 0
$$493$$ −2.41874e12 −1.84407
$$494$$ 0 0
$$495$$ −2.74203e12 −2.05281
$$496$$ 0 0
$$497$$ 5.27313e11 0.387672
$$498$$ 0 0
$$499$$ −1.71390e12 −1.23746 −0.618732 0.785602i $$-0.712353\pi$$
−0.618732 + 0.785602i $$0.712353\pi$$
$$500$$ 0 0
$$501$$ 1.58658e12 1.12510
$$502$$ 0 0
$$503$$ −1.61385e12 −1.12411 −0.562053 0.827101i $$-0.689988\pi$$
−0.562053 + 0.827101i $$0.689988\pi$$
$$504$$ 0 0
$$505$$ 2.48726e10 0.0170181
$$506$$ 0 0
$$507$$ −1.70229e12 −1.14419
$$508$$ 0 0
$$509$$ 1.17088e12 0.773181 0.386590 0.922252i $$-0.373653\pi$$
0.386590 + 0.922252i $$0.373653\pi$$
$$510$$ 0 0
$$511$$ −6.42967e11 −0.417153
$$512$$ 0 0
$$513$$ 2.09632e12 1.33638
$$514$$ 0 0
$$515$$ 3.31734e12 2.07806
$$516$$ 0 0
$$517$$ 7.72526e11 0.475560
$$518$$ 0 0
$$519$$ 4.73960e12 2.86740
$$520$$ 0 0
$$521$$ 2.91339e12 1.73232 0.866161 0.499765i $$-0.166580\pi$$
0.866161 + 0.499765i $$0.166580\pi$$
$$522$$ 0 0
$$523$$ −2.53787e12 −1.48324 −0.741620 0.670820i $$-0.765942\pi$$
−0.741620 + 0.670820i $$0.765942\pi$$
$$524$$ 0 0
$$525$$ −2.18230e12 −1.25371
$$526$$ 0 0
$$527$$ 1.07036e12 0.604482
$$528$$ 0 0
$$529$$ −1.70941e12 −0.949063
$$530$$ 0 0
$$531$$ 4.98620e12 2.72172
$$532$$ 0 0
$$533$$ 2.09477e12 1.12425
$$534$$ 0 0
$$535$$ −1.14962e12 −0.606683
$$536$$ 0 0
$$537$$ −3.78920e12 −1.96636
$$538$$ 0 0
$$539$$ 1.61074e11 0.0822008
$$540$$ 0 0
$$541$$ 1.11804e12 0.561136 0.280568 0.959834i $$-0.409477\pi$$
0.280568 + 0.959834i $$0.409477\pi$$
$$542$$ 0 0
$$543$$ −4.28574e12 −2.11556
$$544$$ 0 0
$$545$$ 4.65859e11 0.226188
$$546$$ 0 0
$$547$$ −1.99438e12 −0.952500 −0.476250 0.879310i $$-0.658004\pi$$
−0.476250 + 0.879310i $$0.658004\pi$$
$$548$$ 0 0
$$549$$ 2.99031e12 1.40488
$$550$$ 0 0
$$551$$ 2.63931e12 1.21985
$$552$$ 0 0
$$553$$ 6.85125e10 0.0311535
$$554$$ 0 0
$$555$$ 2.04963e12 0.916973
$$556$$ 0 0
$$557$$ −4.69088e11 −0.206493 −0.103247 0.994656i $$-0.532923\pi$$
−0.103247 + 0.994656i $$0.532923\pi$$
$$558$$ 0 0
$$559$$ 8.84197e11 0.382997
$$560$$ 0 0
$$561$$ −2.47721e12 −1.05592
$$562$$ 0 0
$$563$$ −1.74518e12 −0.732069 −0.366034 0.930601i $$-0.619285\pi$$
−0.366034 + 0.930601i $$0.619285\pi$$
$$564$$ 0 0
$$565$$ 1.37848e12 0.569093
$$566$$ 0 0
$$567$$ −1.22119e12 −0.496205
$$568$$ 0 0
$$569$$ −4.13341e12 −1.65311 −0.826557 0.562853i $$-0.809704\pi$$
−0.826557 + 0.562853i $$0.809704\pi$$
$$570$$ 0 0
$$571$$ −4.14262e12 −1.63084 −0.815422 0.578867i $$-0.803495\pi$$
−0.815422 + 0.578867i $$0.803495\pi$$
$$572$$ 0 0
$$573$$ −6.67383e12 −2.58630
$$574$$ 0 0
$$575$$ 1.11436e12 0.425129
$$576$$ 0 0
$$577$$ 4.27130e12 1.60424 0.802120 0.597164i $$-0.203706\pi$$
0.802120 + 0.597164i $$0.203706\pi$$
$$578$$ 0 0
$$579$$ 6.67614e11 0.246872
$$580$$ 0 0
$$581$$ −9.20152e11 −0.335017
$$582$$ 0 0
$$583$$ −6.68394e11 −0.239621
$$584$$ 0 0
$$585$$ −5.98078e12 −2.11133
$$586$$ 0 0
$$587$$ −5.32282e12 −1.85042 −0.925210 0.379455i $$-0.876111\pi$$
−0.925210 + 0.379455i $$0.876111\pi$$
$$588$$ 0 0
$$589$$ −1.16797e12 −0.399865
$$590$$ 0 0
$$591$$ 5.90570e12 1.99126
$$592$$ 0 0
$$593$$ −2.69262e11 −0.0894190 −0.0447095 0.999000i $$-0.514236\pi$$
−0.0447095 + 0.999000i $$0.514236\pi$$
$$594$$ 0 0
$$595$$ −2.04486e12 −0.668864
$$596$$ 0 0
$$597$$ 5.33906e11 0.172021
$$598$$ 0 0
$$599$$ −3.51807e12 −1.11657 −0.558283 0.829651i $$-0.688540\pi$$
−0.558283 + 0.829651i $$0.688540\pi$$
$$600$$ 0 0
$$601$$ −1.97549e12 −0.617645 −0.308823 0.951120i $$-0.599935\pi$$
−0.308823 + 0.951120i $$0.599935\pi$$
$$602$$ 0 0
$$603$$ −3.59916e12 −1.10860
$$604$$ 0 0
$$605$$ 3.74316e12 1.13590
$$606$$ 0 0
$$607$$ 1.20562e12 0.360464 0.180232 0.983624i $$-0.442315\pi$$
0.180232 + 0.983624i $$0.442315\pi$$
$$608$$ 0 0
$$609$$ −3.99794e12 −1.17776
$$610$$ 0 0
$$611$$ 1.68499e12 0.489117
$$612$$ 0 0
$$613$$ −5.02026e11 −0.143600 −0.0717999 0.997419i $$-0.522874\pi$$
−0.0717999 + 0.997419i $$0.522874\pi$$
$$614$$ 0 0
$$615$$ −2.01528e13 −5.68065
$$616$$ 0 0
$$617$$ 5.17852e12 1.43854 0.719271 0.694730i $$-0.244476\pi$$
0.719271 + 0.694730i $$0.244476\pi$$
$$618$$ 0 0
$$619$$ 4.69963e12 1.28664 0.643318 0.765599i $$-0.277557\pi$$
0.643318 + 0.765599i $$0.277557\pi$$
$$620$$ 0 0
$$621$$ 1.62149e12 0.437523
$$622$$ 0 0
$$623$$ −1.73252e12 −0.460767
$$624$$ 0 0
$$625$$ 2.53502e12 0.664541
$$626$$ 0 0
$$627$$ 2.70312e12 0.698491
$$628$$ 0 0
$$629$$ 1.25454e12 0.319562
$$630$$ 0 0
$$631$$ 5.53678e12 1.39035 0.695177 0.718839i $$-0.255326\pi$$
0.695177 + 0.718839i $$0.255326\pi$$
$$632$$ 0 0
$$633$$ −8.91994e11 −0.220823
$$634$$ 0 0
$$635$$ 6.14872e12 1.50073
$$636$$ 0 0
$$637$$ 3.51326e11 0.0845441
$$638$$ 0 0
$$639$$ −9.08175e12 −2.15484
$$640$$ 0 0
$$641$$ −2.85506e12 −0.667966 −0.333983 0.942579i $$-0.608393\pi$$
−0.333983 + 0.942579i $$0.608393\pi$$
$$642$$ 0 0
$$643$$ 7.48209e12 1.72613 0.863065 0.505093i $$-0.168542\pi$$
0.863065 + 0.505093i $$0.168542\pi$$
$$644$$ 0 0
$$645$$ −8.50646e12 −1.93522
$$646$$ 0 0
$$647$$ 3.83872e12 0.861225 0.430613 0.902537i $$-0.358297\pi$$
0.430613 + 0.902537i $$0.358297\pi$$
$$648$$ 0 0
$$649$$ 3.36912e12 0.745445
$$650$$ 0 0
$$651$$ 1.76920e12 0.386068
$$652$$ 0 0
$$653$$ −1.09261e11 −0.0235157 −0.0117578 0.999931i $$-0.503743\pi$$
−0.0117578 + 0.999931i $$0.503743\pi$$
$$654$$ 0 0
$$655$$ 5.82859e12 1.23731
$$656$$ 0 0
$$657$$ 1.10736e13 2.31871
$$658$$ 0 0
$$659$$ 2.10319e12 0.434403 0.217202 0.976127i $$-0.430307\pi$$
0.217202 + 0.976127i $$0.430307\pi$$
$$660$$ 0 0
$$661$$ −1.76045e12 −0.358688 −0.179344 0.983786i $$-0.557397\pi$$
−0.179344 + 0.983786i $$0.557397\pi$$
$$662$$ 0 0
$$663$$ −5.40317e12 −1.08602
$$664$$ 0 0
$$665$$ 2.23134e12 0.442453
$$666$$ 0 0
$$667$$ 2.04149e12 0.399374
$$668$$ 0 0
$$669$$ 1.76148e13 3.39986
$$670$$ 0 0
$$671$$ 2.02052e12 0.384779
$$672$$ 0 0
$$673$$ 8.52389e12 1.60166 0.800830 0.598892i $$-0.204392\pi$$
0.800830 + 0.598892i $$0.204392\pi$$
$$674$$ 0 0
$$675$$ 1.96950e13 3.65165
$$676$$ 0 0
$$677$$ −2.88497e12 −0.527828 −0.263914 0.964546i $$-0.585014\pi$$
−0.263914 + 0.964546i $$0.585014\pi$$
$$678$$ 0 0
$$679$$ 1.61764e12 0.292057
$$680$$ 0 0
$$681$$ −1.76732e13 −3.14885
$$682$$ 0 0
$$683$$ −1.03086e13 −1.81262 −0.906309 0.422616i $$-0.861112\pi$$
−0.906309 + 0.422616i $$0.861112\pi$$
$$684$$ 0 0
$$685$$ −1.44378e13 −2.50550
$$686$$ 0 0
$$687$$ −8.79556e12 −1.50646
$$688$$ 0 0
$$689$$ −1.45787e12 −0.246451
$$690$$ 0 0
$$691$$ −7.85439e12 −1.31057 −0.655287 0.755380i $$-0.727452\pi$$
−0.655287 + 0.755380i $$0.727452\pi$$
$$692$$ 0 0
$$693$$ −2.77413e12 −0.456906
$$694$$ 0 0
$$695$$ 5.34566e12 0.869099
$$696$$ 0 0
$$697$$ −1.23351e13 −1.97969
$$698$$ 0 0
$$699$$ −9.38969e12 −1.48766
$$700$$ 0 0
$$701$$ 1.69579e12 0.265242 0.132621 0.991167i $$-0.457661\pi$$
0.132621 + 0.991167i $$0.457661\pi$$
$$702$$ 0 0
$$703$$ −1.36894e12 −0.211390
$$704$$ 0 0
$$705$$ −1.62106e13 −2.47142
$$706$$ 0 0
$$707$$ 2.51638e10 0.00378781
$$708$$ 0 0
$$709$$ −2.20703e12 −0.328020 −0.164010 0.986459i $$-0.552443\pi$$
−0.164010 + 0.986459i $$0.552443\pi$$
$$710$$ 0 0
$$711$$ −1.17997e12 −0.173164
$$712$$ 0 0
$$713$$ −9.03417e11 −0.130914
$$714$$ 0 0
$$715$$ −4.04115e12 −0.578266
$$716$$ 0 0
$$717$$ 2.14225e13 3.02715
$$718$$ 0 0
$$719$$ 1.46869e12 0.204952 0.102476 0.994735i $$-0.467324\pi$$
0.102476 + 0.994735i $$0.467324\pi$$
$$720$$ 0 0
$$721$$ 3.35617e12 0.462525
$$722$$ 0 0
$$723$$ −2.02192e12 −0.275195
$$724$$ 0 0
$$725$$ 2.47965e13 3.33326
$$726$$ 0 0
$$727$$ 4.97511e11 0.0660538 0.0330269 0.999454i $$-0.489485\pi$$
0.0330269 + 0.999454i $$0.489485\pi$$
$$728$$ 0 0
$$729$$ −4.99938e12 −0.655605
$$730$$ 0 0
$$731$$ −5.20663e12 −0.674417
$$732$$ 0 0
$$733$$ −5.22437e12 −0.668445 −0.334223 0.942494i $$-0.608474\pi$$
−0.334223 + 0.942494i $$0.608474\pi$$
$$734$$ 0 0
$$735$$ −3.37995e12 −0.427187
$$736$$ 0 0
$$737$$ −2.43191e12 −0.303630
$$738$$ 0 0
$$739$$ −7.35495e11 −0.0907152 −0.0453576 0.998971i $$-0.514443\pi$$
−0.0453576 + 0.998971i $$0.514443\pi$$
$$740$$ 0 0
$$741$$ 5.89590e12 0.718403
$$742$$ 0 0
$$743$$ 1.12076e12 0.134916 0.0674580 0.997722i $$-0.478511\pi$$
0.0674580 + 0.997722i $$0.478511\pi$$
$$744$$ 0 0
$$745$$ 7.44045e12 0.884904
$$746$$ 0 0
$$747$$ 1.58475e13 1.86217
$$748$$ 0 0
$$749$$ −1.16308e12 −0.135033
$$750$$ 0 0
$$751$$ 1.62183e12 0.186049 0.0930244 0.995664i $$-0.470347\pi$$
0.0930244 + 0.995664i $$0.470347\pi$$
$$752$$ 0 0
$$753$$ −2.40991e13 −2.73164
$$754$$ 0 0
$$755$$ −1.47362e13 −1.65054
$$756$$ 0 0
$$757$$ −1.55150e12 −0.171720 −0.0858600 0.996307i $$-0.527364\pi$$
−0.0858600 + 0.996307i $$0.527364\pi$$
$$758$$ 0 0
$$759$$ 2.09084e12 0.228682
$$760$$ 0 0
$$761$$ 1.62863e13 1.76032 0.880159 0.474679i $$-0.157436\pi$$
0.880159 + 0.474679i $$0.157436\pi$$
$$762$$ 0 0
$$763$$ 4.71312e11 0.0503440
$$764$$ 0 0
$$765$$ 3.52181e13 3.71783
$$766$$ 0 0
$$767$$ 7.34856e12 0.766695
$$768$$ 0 0
$$769$$ −1.32915e13 −1.37059 −0.685293 0.728267i $$-0.740326\pi$$
−0.685293 + 0.728267i $$0.740326\pi$$
$$770$$ 0 0
$$771$$ −1.09455e12 −0.111555
$$772$$ 0 0
$$773$$ −7.58593e12 −0.764190 −0.382095 0.924123i $$-0.624797\pi$$
−0.382095 + 0.924123i $$0.624797\pi$$
$$774$$ 0 0
$$775$$ −1.09732e13 −1.09263
$$776$$ 0 0
$$777$$ 2.07362e12 0.204096
$$778$$ 0 0
$$779$$ 1.34600e13 1.30956
$$780$$ 0 0
$$781$$ −6.13645e12 −0.590184
$$782$$ 0 0
$$783$$ 3.60809e13 3.43044
$$784$$ 0 0
$$785$$ −3.16536e13 −2.97516
$$786$$ 0 0
$$787$$ −3.30985e12 −0.307554 −0.153777 0.988106i $$-0.549144\pi$$
−0.153777 + 0.988106i $$0.549144\pi$$
$$788$$ 0 0
$$789$$ 2.97995e13 2.73755
$$790$$ 0 0
$$791$$ 1.39462e12 0.126666
$$792$$ 0 0
$$793$$ 4.40706e12 0.395748
$$794$$ 0 0
$$795$$ 1.40255e13 1.24528
$$796$$ 0 0
$$797$$ −3.26446e12 −0.286582 −0.143291 0.989681i $$-0.545768\pi$$
−0.143291 + 0.989681i $$0.545768\pi$$
$$798$$ 0 0
$$799$$ −9.92216e12 −0.861283
$$800$$ 0 0
$$801$$ 2.98386e13 2.56114
$$802$$ 0 0
$$803$$ 7.48235e12 0.635064
$$804$$ 0 0
$$805$$ 1.72592e12 0.144857
$$806$$ 0 0
$$807$$ 1.87519e13 1.55637
$$808$$ 0 0
$$809$$ 3.46768e12 0.284624 0.142312 0.989822i $$-0.454546\pi$$
0.142312 + 0.989822i $$0.454546\pi$$
$$810$$ 0 0
$$811$$ −1.02915e13 −0.835381 −0.417690 0.908589i $$-0.637160\pi$$
−0.417690 + 0.908589i $$0.637160\pi$$
$$812$$ 0 0
$$813$$ −1.75752e13 −1.41089
$$814$$ 0 0
$$815$$ 1.74104e13 1.38229
$$816$$ 0 0
$$817$$ 5.68143e12 0.446127
$$818$$ 0 0
$$819$$ −6.05079e12 −0.469931
$$820$$ 0 0
$$821$$ 6.23700e12 0.479106 0.239553 0.970883i $$-0.422999\pi$$
0.239553 + 0.970883i $$0.422999\pi$$
$$822$$ 0 0
$$823$$ −2.46689e13 −1.87435 −0.937175 0.348860i $$-0.886569\pi$$
−0.937175 + 0.348860i $$0.886569\pi$$
$$824$$ 0 0
$$825$$ 2.53959e13 1.90863
$$826$$ 0 0
$$827$$ 1.45027e13 1.07814 0.539068 0.842262i $$-0.318776\pi$$
0.539068 + 0.842262i $$0.318776\pi$$
$$828$$ 0 0
$$829$$ −1.07681e13 −0.791854 −0.395927 0.918282i $$-0.629577\pi$$
−0.395927 + 0.918282i $$0.629577\pi$$
$$830$$ 0 0
$$831$$ 2.12846e13 1.54832
$$832$$ 0 0
$$833$$ −2.06880e12 −0.148873
$$834$$ 0 0
$$835$$ −1.52409e13 −1.08498
$$836$$ 0 0
$$837$$ −1.59668e13 −1.12449
$$838$$ 0 0
$$839$$ 2.36129e13 1.64521 0.822603 0.568616i $$-0.192521\pi$$
0.822603 + 0.568616i $$0.192521\pi$$
$$840$$ 0 0
$$841$$ 3.09195e13 2.13133
$$842$$ 0 0
$$843$$ −2.47494e13 −1.68787
$$844$$ 0 0
$$845$$ 1.63524e13 1.10339
$$846$$ 0 0
$$847$$ 3.78698e12 0.252824
$$848$$ 0 0
$$849$$ 1.13013e13 0.746521
$$850$$ 0 0
$$851$$ −1.05886e12 −0.0692081
$$852$$ 0 0
$$853$$ −1.67503e13 −1.08331 −0.541655 0.840601i $$-0.682202\pi$$
−0.541655 + 0.840601i $$0.682202\pi$$
$$854$$ 0 0
$$855$$ −3.84297e13 −2.45934
$$856$$ 0 0
$$857$$ −1.16182e13 −0.735742 −0.367871 0.929877i $$-0.619913\pi$$
−0.367871 + 0.929877i $$0.619913\pi$$
$$858$$ 0 0
$$859$$ −1.58818e13 −0.995246 −0.497623 0.867393i $$-0.665794\pi$$
−0.497623 + 0.867393i $$0.665794\pi$$
$$860$$ 0 0
$$861$$ −2.03888e13 −1.26438
$$862$$ 0 0
$$863$$ −1.35971e13 −0.834446 −0.417223 0.908804i $$-0.636997\pi$$
−0.417223 + 0.908804i $$0.636997\pi$$
$$864$$ 0 0
$$865$$ −4.55293e13 −2.76515
$$866$$ 0 0
$$867$$ 2.51943e12 0.151431
$$868$$ 0 0
$$869$$ −7.97294e11 −0.0474274
$$870$$ 0 0
$$871$$ −5.30437e12 −0.312285
$$872$$ 0 0
$$873$$ −2.78601e13 −1.62338
$$874$$ 0 0
$$875$$ 9.83445e12 0.567170
$$876$$ 0 0
$$877$$ 3.40277e12 0.194238 0.0971189 0.995273i $$-0.469037\pi$$
0.0971189 + 0.995273i $$0.469037\pi$$
$$878$$ 0 0
$$879$$ −2.51041e12 −0.141839
$$880$$ 0 0
$$881$$ 9.30779e12 0.520541 0.260270 0.965536i $$-0.416188\pi$$
0.260270 + 0.965536i $$0.416188\pi$$
$$882$$ 0 0
$$883$$ −1.00154e13 −0.554430 −0.277215 0.960808i $$-0.589411\pi$$
−0.277215 + 0.960808i $$0.589411\pi$$
$$884$$ 0 0
$$885$$ −7.06972e13 −3.87398
$$886$$ 0 0
$$887$$ 2.58440e13 1.40186 0.700928 0.713232i $$-0.252769\pi$$
0.700928 + 0.713232i $$0.252769\pi$$
$$888$$ 0 0
$$889$$ 6.22070e12 0.334027
$$890$$ 0 0
$$891$$ 1.42113e13 0.755412
$$892$$ 0 0
$$893$$ 1.08270e13 0.569739
$$894$$ 0 0
$$895$$ 3.63997e13 1.89624
$$896$$ 0 0
$$897$$ 4.56043e12 0.235201
$$898$$ 0 0
$$899$$ −2.01026e13 −1.02644
$$900$$ 0 0
$$901$$ 8.58471e12 0.433974
$$902$$ 0 0
$$903$$ −8.60604e12 −0.430733
$$904$$ 0 0
$$905$$ 4.11694e13 2.04012
$$906$$ 0 0
$$907$$ −3.82049e13 −1.87451 −0.937253 0.348650i $$-0.886640\pi$$
−0.937253 + 0.348650i $$0.886640\pi$$
$$908$$ 0 0
$$909$$ −4.33389e11 −0.0210543
$$910$$ 0 0
$$911$$ −9.85478e12 −0.474039 −0.237020 0.971505i $$-0.576171\pi$$
−0.237020 + 0.971505i $$0.576171\pi$$
$$912$$ 0 0
$$913$$ 1.07080e13 0.510023
$$914$$ 0 0
$$915$$ −4.23983e13 −1.99965
$$916$$ 0 0
$$917$$ 5.89682e12 0.275395
$$918$$ 0 0
$$919$$ −2.60834e13 −1.20627 −0.603135 0.797639i $$-0.706082\pi$$
−0.603135 + 0.797639i $$0.706082\pi$$
$$920$$ 0 0
$$921$$ 9.11517e12 0.417442
$$922$$ 0 0
$$923$$ −1.33845e13 −0.607008
$$924$$ 0 0
$$925$$ −1.28613e13 −0.577625
$$926$$ 0 0
$$927$$ −5.78024e13 −2.57091
$$928$$ 0 0
$$929$$ −3.60592e13 −1.58835 −0.794174 0.607690i $$-0.792096\pi$$
−0.794174 + 0.607690i $$0.792096\pi$$
$$930$$ 0 0
$$931$$ 2.25746e12 0.0984795
$$932$$ 0 0
$$933$$ −4.65836e13 −2.01264
$$934$$ 0 0
$$935$$ 2.37965e13 1.01826
$$936$$ 0 0
$$937$$ 2.75358e13 1.16700 0.583499 0.812114i $$-0.301683\pi$$
0.583499 + 0.812114i $$0.301683\pi$$
$$938$$ 0 0
$$939$$ 3.25561e13 1.36659
$$940$$ 0 0
$$941$$ −7.60157e12 −0.316046 −0.158023 0.987435i $$-0.550512\pi$$
−0.158023 + 0.987435i $$0.550512\pi$$
$$942$$ 0 0
$$943$$ 1.04112e13 0.428745
$$944$$ 0 0
$$945$$ 3.05037e13 1.24425
$$946$$ 0 0
$$947$$ 2.35269e13 0.950582 0.475291 0.879829i $$-0.342343\pi$$
0.475291 + 0.879829i $$0.342343\pi$$
$$948$$ 0 0
$$949$$ 1.63201e13 0.653168
$$950$$ 0 0
$$951$$ −3.72272e13 −1.47587
$$952$$ 0 0
$$953$$ 3.76056e12 0.147684 0.0738422 0.997270i $$-0.476474\pi$$
0.0738422 + 0.997270i $$0.476474\pi$$
$$954$$ 0 0
$$955$$ 6.41098e13 2.49407
$$956$$ 0 0
$$957$$ 4.65248e13 1.79300
$$958$$ 0 0
$$959$$ −1.46068e13 −0.557663
$$960$$ 0 0
$$961$$ −1.75436e13 −0.663535
$$962$$ 0 0
$$963$$ 2.00313e13 0.750571
$$964$$ 0 0
$$965$$ −6.41320e12 −0.238068
$$966$$ 0 0
$$967$$ −3.59678e13 −1.32280 −0.661400 0.750033i $$-0.730037\pi$$
−0.661400 + 0.750033i $$0.730037\pi$$
$$968$$ 0 0
$$969$$ −3.47182e13 −1.26503
$$970$$ 0 0
$$971$$ 1.20049e13 0.433383 0.216692 0.976240i $$-0.430473\pi$$
0.216692 + 0.976240i $$0.430473\pi$$
$$972$$ 0 0
$$973$$ 5.40823e12 0.193441
$$974$$ 0 0
$$975$$ 5.53923e13 1.96304
$$976$$ 0 0
$$977$$ −4.59954e12 −0.161506 −0.0807530 0.996734i $$-0.525732\pi$$
−0.0807530 + 0.996734i $$0.525732\pi$$
$$978$$ 0 0
$$979$$ 2.01617e13 0.701462
$$980$$ 0 0
$$981$$ −8.11727e12 −0.279833
$$982$$ 0 0
$$983$$ 3.13830e12 0.107202 0.0536011 0.998562i $$-0.482930\pi$$
0.0536011 + 0.998562i $$0.482930\pi$$
$$984$$ 0 0
$$985$$ −5.67310e13 −1.92025
$$986$$ 0 0
$$987$$ −1.64003e13 −0.550080
$$988$$ 0 0
$$989$$ 4.39454e12 0.146060
$$990$$ 0 0
$$991$$ −1.38736e13 −0.456939 −0.228470 0.973551i $$-0.573372\pi$$
−0.228470 + 0.973551i $$0.573372\pi$$
$$992$$ 0 0
$$993$$ −8.37203e12 −0.273249
$$994$$ 0 0
$$995$$ −5.12878e12 −0.165886
$$996$$ 0 0
$$997$$ −1.65453e13 −0.530331 −0.265166 0.964203i $$-0.585427\pi$$
−0.265166 + 0.964203i $$0.585427\pi$$
$$998$$ 0 0
$$999$$ −1.87142e13 −0.594465
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 112.10.a.c.1.2 2
4.3 odd 2 14.10.a.c.1.1 2
12.11 even 2 126.10.a.o.1.2 2
20.3 even 4 350.10.c.j.99.3 4
20.7 even 4 350.10.c.j.99.2 4
20.19 odd 2 350.10.a.j.1.2 2
28.3 even 6 98.10.c.h.79.1 4
28.11 odd 6 98.10.c.j.79.2 4
28.19 even 6 98.10.c.h.67.1 4
28.23 odd 6 98.10.c.j.67.2 4
28.27 even 2 98.10.a.e.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
14.10.a.c.1.1 2 4.3 odd 2
98.10.a.e.1.2 2 28.27 even 2
98.10.c.h.67.1 4 28.19 even 6
98.10.c.h.79.1 4 28.3 even 6
98.10.c.j.67.2 4 28.23 odd 6
98.10.c.j.79.2 4 28.11 odd 6
112.10.a.c.1.2 2 1.1 even 1 trivial
126.10.a.o.1.2 2 12.11 even 2
350.10.a.j.1.2 2 20.19 odd 2
350.10.c.j.99.2 4 20.7 even 4
350.10.c.j.99.3 4 20.3 even 4