# Properties

 Label 175.10.b.c.99.3 Level $175$ Weight $10$ Character 175.99 Analytic conductor $90.131$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,10,Mod(99,175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$90.1312713287$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.3 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.99 Dual form 175.10.b.c.99.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.17157i q^{2} +65.7351i q^{3} +427.882 q^{4} -602.894 q^{6} -2401.00i q^{7} +8620.20i q^{8} +15361.9 q^{9} +O(q^{10})$$ $$q+9.17157i q^{2} +65.7351i q^{3} +427.882 q^{4} -602.894 q^{6} -2401.00i q^{7} +8620.20i q^{8} +15361.9 q^{9} +35089.6 q^{11} +28126.9i q^{12} -77401.4i q^{13} +22020.9 q^{14} +140015. q^{16} +229907. i q^{17} +140893. i q^{18} -16433.6 q^{19} +157830. q^{21} +321827. i q^{22} -2.57284e6i q^{23} -566649. q^{24} +709892. q^{26} +2.30368e6i q^{27} -1.02735e6i q^{28} +6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6i q^{32} +2.30662e6i q^{33} -2.10861e6 q^{34} +6.57308e6 q^{36} -9.70272e6i q^{37} -150722. i q^{38} +5.08798e6 q^{39} +2.98108e7 q^{41} +1.44755e6i q^{42} -1.95343e7i q^{43} +1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6i q^{47} +9.20389e6i q^{48} -5.76480e6 q^{49} -1.51130e7 q^{51} -3.31187e7i q^{52} -2.74263e7i q^{53} -2.11284e7 q^{54} +2.06971e7 q^{56} -1.08026e6i q^{57} +6.07908e7i q^{58} -5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7i q^{62} -3.68839e7i q^{63} +1.94308e7 q^{64} -2.11553e7 q^{66} -2.74351e8i q^{67} +9.83733e7i q^{68} +1.69126e8 q^{69} -3.63673e8 q^{71} +1.32423e8i q^{72} +2.09245e7i q^{73} +8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7i q^{77} +4.66648e7i q^{78} +2.65896e8 q^{79} +1.50936e8 q^{81} +2.73412e8i q^{82} -9.43764e6i q^{83} +6.75326e7 q^{84} +1.79160e8 q^{86} +4.35704e8i q^{87} +3.02479e8i q^{88} +6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9i q^{92} -5.37329e8i q^{93} +5.44603e7 q^{94} -3.74539e8 q^{96} +1.20731e9i q^{97} -5.28723e7i q^{98} +5.39042e8 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9}+O(q^{10})$$ 4 * q + 1440 * q^4 + 5904 * q^6 - 44856 * q^9 $$4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9} + 37132 q^{11} + 115248 q^{14} + 225536 q^{16} + 286552 q^{19} - 835548 q^{21} + 4583808 q^{24} + 639472 q^{26} + 23155108 q^{29} - 7907520 q^{31} - 8487888 q^{34} - 8932032 q^{36} + 22791492 q^{39} + 2117984 q^{41} + 20374752 q^{44} - 14312352 q^{46} - 23059204 q^{49} + 38818932 q^{51} - 170992944 q^{54} + 98652288 q^{56} + 232318416 q^{59} - 89377088 q^{61} - 158111744 q^{64} - 159789648 q^{66} + 1332641784 q^{69} - 588331648 q^{71} - 204836128 q^{74} + 79244736 q^{76} + 1385705708 q^{79} + 895546692 q^{81} - 201223008 q^{84} - 693978016 q^{86} + 1558087408 q^{89} - 245334180 q^{91} + 1875382064 q^{94} + 1984361472 q^{96} + 2326933080 q^{99}+O(q^{100})$$ 4 * q + 1440 * q^4 + 5904 * q^6 - 44856 * q^9 + 37132 * q^11 + 115248 * q^14 + 225536 * q^16 + 286552 * q^19 - 835548 * q^21 + 4583808 * q^24 + 639472 * q^26 + 23155108 * q^29 - 7907520 * q^31 - 8487888 * q^34 - 8932032 * q^36 + 22791492 * q^39 + 2117984 * q^41 + 20374752 * q^44 - 14312352 * q^46 - 23059204 * q^49 + 38818932 * q^51 - 170992944 * q^54 + 98652288 * q^56 + 232318416 * q^59 - 89377088 * q^61 - 158111744 * q^64 - 159789648 * q^66 + 1332641784 * q^69 - 588331648 * q^71 - 204836128 * q^74 + 79244736 * q^76 + 1385705708 * q^79 + 895546692 * q^81 - 201223008 * q^84 - 693978016 * q^86 + 1558087408 * q^89 - 245334180 * q^91 + 1875382064 * q^94 + 1984361472 * q^96 + 2326933080 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 9.17157i 0.405330i 0.979248 + 0.202665i $$0.0649603\pi$$
−0.979248 + 0.202665i $$0.935040\pi$$
$$3$$ 65.7351i 0.468545i 0.972171 + 0.234273i $$0.0752708\pi$$
−0.972171 + 0.234273i $$0.924729\pi$$
$$4$$ 427.882 0.835708
$$5$$ 0 0
$$6$$ −602.894 −0.189915
$$7$$ − 2401.00i − 0.377964i
$$8$$ 8620.20i 0.744067i
$$9$$ 15361.9 0.780465
$$10$$ 0 0
$$11$$ 35089.6 0.722622 0.361311 0.932445i $$-0.382329\pi$$
0.361311 + 0.932445i $$0.382329\pi$$
$$12$$ 28126.9i 0.391567i
$$13$$ − 77401.4i − 0.751629i −0.926695 0.375815i $$-0.877363\pi$$
0.926695 0.375815i $$-0.122637\pi$$
$$14$$ 22020.9 0.153200
$$15$$ 0 0
$$16$$ 140015. 0.534115
$$17$$ 229907.i 0.667626i 0.942639 + 0.333813i $$0.108335\pi$$
−0.942639 + 0.333813i $$0.891665\pi$$
$$18$$ 140893.i 0.316346i
$$19$$ −16433.6 −0.0289295 −0.0144647 0.999895i $$-0.504604\pi$$
−0.0144647 + 0.999895i $$0.504604\pi$$
$$20$$ 0 0
$$21$$ 157830. 0.177093
$$22$$ 321827.i 0.292900i
$$23$$ − 2.57284e6i − 1.91707i −0.284975 0.958535i $$-0.591985\pi$$
0.284975 0.958535i $$-0.408015\pi$$
$$24$$ −566649. −0.348629
$$25$$ 0 0
$$26$$ 709892. 0.304658
$$27$$ 2.30368e6i 0.834228i
$$28$$ − 1.02735e6i − 0.315868i
$$29$$ 6.62817e6 1.74022 0.870108 0.492862i $$-0.164049\pi$$
0.870108 + 0.492862i $$0.164049\pi$$
$$30$$ 0 0
$$31$$ −8.17416e6 −1.58970 −0.794851 0.606805i $$-0.792451\pi$$
−0.794851 + 0.606805i $$0.792451\pi$$
$$32$$ 5.69770e6i 0.960560i
$$33$$ 2.30662e6i 0.338581i
$$34$$ −2.10861e6 −0.270609
$$35$$ 0 0
$$36$$ 6.57308e6 0.652241
$$37$$ − 9.70272e6i − 0.851110i −0.904933 0.425555i $$-0.860079\pi$$
0.904933 0.425555i $$-0.139921\pi$$
$$38$$ − 150722.i − 0.0117260i
$$39$$ 5.08798e6 0.352172
$$40$$ 0 0
$$41$$ 2.98108e7 1.64758 0.823789 0.566896i $$-0.191856\pi$$
0.823789 + 0.566896i $$0.191856\pi$$
$$42$$ 1.44755e6i 0.0717813i
$$43$$ − 1.95343e7i − 0.871343i −0.900106 0.435672i $$-0.856511\pi$$
0.900106 0.435672i $$-0.143489\pi$$
$$44$$ 1.50142e7 0.603900
$$45$$ 0 0
$$46$$ 2.35970e7 0.777046
$$47$$ − 5.93794e6i − 0.177499i −0.996054 0.0887494i $$-0.971713\pi$$
0.996054 0.0887494i $$-0.0282870\pi$$
$$48$$ 9.20389e6i 0.250257i
$$49$$ −5.76480e6 −0.142857
$$50$$ 0 0
$$51$$ −1.51130e7 −0.312813
$$52$$ − 3.31187e7i − 0.628142i
$$53$$ − 2.74263e7i − 0.477448i −0.971088 0.238724i $$-0.923271\pi$$
0.971088 0.238724i $$-0.0767291\pi$$
$$54$$ −2.11284e7 −0.338138
$$55$$ 0 0
$$56$$ 2.06971e7 0.281231
$$57$$ − 1.08026e6i − 0.0135548i
$$58$$ 6.07908e7i 0.705362i
$$59$$ −5.24915e7 −0.563969 −0.281984 0.959419i $$-0.590993\pi$$
−0.281984 + 0.959419i $$0.590993\pi$$
$$60$$ 0 0
$$61$$ 2.23282e7 0.206476 0.103238 0.994657i $$-0.467080\pi$$
0.103238 + 0.994657i $$0.467080\pi$$
$$62$$ − 7.49699e7i − 0.644354i
$$63$$ − 3.68839e7i − 0.294988i
$$64$$ 1.94308e7 0.144771
$$65$$ 0 0
$$66$$ −2.11553e7 −0.137237
$$67$$ − 2.74351e8i − 1.66330i −0.555302 0.831649i $$-0.687397\pi$$
0.555302 0.831649i $$-0.312603\pi$$
$$68$$ 9.83733e7i 0.557940i
$$69$$ 1.69126e8 0.898234
$$70$$ 0 0
$$71$$ −3.63673e8 −1.69843 −0.849216 0.528046i $$-0.822925\pi$$
−0.849216 + 0.528046i $$0.822925\pi$$
$$72$$ 1.32423e8i 0.580719i
$$73$$ 2.09245e7i 0.0862387i 0.999070 + 0.0431193i $$0.0137296\pi$$
−0.999070 + 0.0431193i $$0.986270\pi$$
$$74$$ 8.89892e7 0.344980
$$75$$ 0 0
$$76$$ −7.03163e6 −0.0241766
$$77$$ − 8.42501e7i − 0.273125i
$$78$$ 4.66648e7i 0.142746i
$$79$$ 2.65896e8 0.768051 0.384025 0.923323i $$-0.374538\pi$$
0.384025 + 0.923323i $$0.374538\pi$$
$$80$$ 0 0
$$81$$ 1.50936e8 0.389592
$$82$$ 2.73412e8i 0.667813i
$$83$$ − 9.43764e6i − 0.0218279i −0.999940 0.0109140i $$-0.996526\pi$$
0.999940 0.0109140i $$-0.00347409\pi$$
$$84$$ 6.75326e7 0.147998
$$85$$ 0 0
$$86$$ 1.79160e8 0.353182
$$87$$ 4.35704e8i 0.815369i
$$88$$ 3.02479e8i 0.537679i
$$89$$ 6.64876e8 1.12327 0.561637 0.827384i $$-0.310172\pi$$
0.561637 + 0.827384i $$0.310172\pi$$
$$90$$ 0 0
$$91$$ −1.85841e8 −0.284089
$$92$$ − 1.10087e9i − 1.60211i
$$93$$ − 5.37329e8i − 0.744847i
$$94$$ 5.44603e7 0.0719456
$$95$$ 0 0
$$96$$ −3.74539e8 −0.450066
$$97$$ 1.20731e9i 1.38467i 0.721575 + 0.692336i $$0.243418\pi$$
−0.721575 + 0.692336i $$0.756582\pi$$
$$98$$ − 5.28723e7i − 0.0579043i
$$99$$ 5.39042e8 0.563981
$$100$$ 0 0
$$101$$ 1.18204e9 1.13028 0.565139 0.824996i $$-0.308823\pi$$
0.565139 + 0.824996i $$0.308823\pi$$
$$102$$ − 1.38610e8i − 0.126792i
$$103$$ 1.97811e9i 1.73174i 0.500268 + 0.865870i $$0.333235\pi$$
−0.500268 + 0.865870i $$0.666765\pi$$
$$104$$ 6.67215e8 0.559263
$$105$$ 0 0
$$106$$ 2.51542e8 0.193524
$$107$$ − 1.67828e8i − 0.123776i −0.998083 0.0618881i $$-0.980288\pi$$
0.998083 0.0618881i $$-0.0197122\pi$$
$$108$$ 9.85703e8i 0.697171i
$$109$$ 1.02540e9 0.695784 0.347892 0.937535i $$-0.386898\pi$$
0.347892 + 0.937535i $$0.386898\pi$$
$$110$$ 0 0
$$111$$ 6.37809e8 0.398783
$$112$$ − 3.36176e8i − 0.201876i
$$113$$ 1.27533e9i 0.735814i 0.929863 + 0.367907i $$0.119926\pi$$
−0.929863 + 0.367907i $$0.880074\pi$$
$$114$$ 9.90770e6 0.00549415
$$115$$ 0 0
$$116$$ 2.83608e9 1.45431
$$117$$ − 1.18903e9i − 0.586621i
$$118$$ − 4.81430e8i − 0.228594i
$$119$$ 5.52008e8 0.252339
$$120$$ 0 0
$$121$$ −1.12667e9 −0.477818
$$122$$ 2.04785e8i 0.0836909i
$$123$$ 1.95961e9i 0.771965i
$$124$$ −3.49758e9 −1.32853
$$125$$ 0 0
$$126$$ 3.38284e8 0.119568
$$127$$ 2.90339e9i 0.990349i 0.868794 + 0.495174i $$0.164896\pi$$
−0.868794 + 0.495174i $$0.835104\pi$$
$$128$$ 3.09543e9i 1.01924i
$$129$$ 1.28409e9 0.408264
$$130$$ 0 0
$$131$$ 2.05173e9 0.608694 0.304347 0.952561i $$-0.401562\pi$$
0.304347 + 0.952561i $$0.401562\pi$$
$$132$$ 9.86960e8i 0.282955i
$$133$$ 3.94570e7i 0.0109343i
$$134$$ 2.51623e9 0.674185
$$135$$ 0 0
$$136$$ −1.98185e9 −0.496759
$$137$$ − 3.25539e9i − 0.789514i −0.918786 0.394757i $$-0.870829\pi$$
0.918786 0.394757i $$-0.129171\pi$$
$$138$$ 1.55115e9i 0.364081i
$$139$$ 8.26776e9 1.87854 0.939272 0.343173i $$-0.111502\pi$$
0.939272 + 0.343173i $$0.111502\pi$$
$$140$$ 0 0
$$141$$ 3.90331e8 0.0831662
$$142$$ − 3.33545e9i − 0.688425i
$$143$$ − 2.71598e9i − 0.543143i
$$144$$ 2.15090e9 0.416858
$$145$$ 0 0
$$146$$ −1.91910e8 −0.0349551
$$147$$ − 3.78950e8i − 0.0669350i
$$148$$ − 4.15162e9i − 0.711279i
$$149$$ −1.07127e9 −0.178058 −0.0890289 0.996029i $$-0.528376\pi$$
−0.0890289 + 0.996029i $$0.528376\pi$$
$$150$$ 0 0
$$151$$ 1.97304e9 0.308844 0.154422 0.988005i $$-0.450649\pi$$
0.154422 + 0.988005i $$0.450649\pi$$
$$152$$ − 1.41661e8i − 0.0215255i
$$153$$ 3.53182e9i 0.521059i
$$154$$ 7.72706e8 0.110706
$$155$$ 0 0
$$156$$ 2.17706e9 0.294313
$$157$$ 4.61623e9i 0.606372i 0.952931 + 0.303186i $$0.0980503\pi$$
−0.952931 + 0.303186i $$0.901950\pi$$
$$158$$ 2.43868e9i 0.311314i
$$159$$ 1.80287e9 0.223706
$$160$$ 0 0
$$161$$ −6.17740e9 −0.724584
$$162$$ 1.38432e9i 0.157913i
$$163$$ 6.26525e9i 0.695175i 0.937648 + 0.347588i $$0.112999\pi$$
−0.937648 + 0.347588i $$0.887001\pi$$
$$164$$ 1.27555e10 1.37689
$$165$$ 0 0
$$166$$ 8.65580e7 0.00884751
$$167$$ 6.21672e9i 0.618496i 0.950981 + 0.309248i $$0.100077\pi$$
−0.950981 + 0.309248i $$0.899923\pi$$
$$168$$ 1.36052e9i 0.131769i
$$169$$ 4.61353e9 0.435054
$$170$$ 0 0
$$171$$ −2.52451e8 −0.0225785
$$172$$ − 8.35837e9i − 0.728188i
$$173$$ 8.97209e9i 0.761528i 0.924672 + 0.380764i $$0.124339\pi$$
−0.924672 + 0.380764i $$0.875661\pi$$
$$174$$ −3.99609e9 −0.330494
$$175$$ 0 0
$$176$$ 4.91306e9 0.385963
$$177$$ − 3.45053e9i − 0.264245i
$$178$$ 6.09796e9i 0.455297i
$$179$$ −1.76242e10 −1.28313 −0.641565 0.767069i $$-0.721714\pi$$
−0.641565 + 0.767069i $$0.721714\pi$$
$$180$$ 0 0
$$181$$ 1.62250e9 0.112365 0.0561824 0.998421i $$-0.482107\pi$$
0.0561824 + 0.998421i $$0.482107\pi$$
$$182$$ − 1.70445e9i − 0.115150i
$$183$$ 1.46775e9i 0.0967433i
$$184$$ 2.21784e10 1.42643
$$185$$ 0 0
$$186$$ 4.92815e9 0.301909
$$187$$ 8.06735e9i 0.482441i
$$188$$ − 2.54074e9i − 0.148337i
$$189$$ 5.53113e9 0.315309
$$190$$ 0 0
$$191$$ 1.66601e10 0.905788 0.452894 0.891564i $$-0.350392\pi$$
0.452894 + 0.891564i $$0.350392\pi$$
$$192$$ 1.27728e9i 0.0678316i
$$193$$ 2.41341e10i 1.25206i 0.779801 + 0.626028i $$0.215320\pi$$
−0.779801 + 0.626028i $$0.784680\pi$$
$$194$$ −1.10730e10 −0.561249
$$195$$ 0 0
$$196$$ −2.46666e9 −0.119387
$$197$$ 3.89843e9i 0.184413i 0.995740 + 0.0922066i $$0.0293920\pi$$
−0.995740 + 0.0922066i $$0.970608\pi$$
$$198$$ 4.94387e9i 0.228599i
$$199$$ 1.87489e10 0.847493 0.423747 0.905781i $$-0.360715\pi$$
0.423747 + 0.905781i $$0.360715\pi$$
$$200$$ 0 0
$$201$$ 1.80345e10 0.779330
$$202$$ 1.08411e10i 0.458135i
$$203$$ − 1.59142e10i − 0.657740i
$$204$$ −6.46658e9 −0.261420
$$205$$ 0 0
$$206$$ −1.81424e10 −0.701927
$$207$$ − 3.95238e10i − 1.49621i
$$208$$ − 1.08373e10i − 0.401456i
$$209$$ −5.76647e8 −0.0209051
$$210$$ 0 0
$$211$$ −2.20489e9 −0.0765801 −0.0382900 0.999267i $$-0.512191\pi$$
−0.0382900 + 0.999267i $$0.512191\pi$$
$$212$$ − 1.17352e10i − 0.399007i
$$213$$ − 2.39060e10i − 0.795792i
$$214$$ 1.53925e9 0.0501702
$$215$$ 0 0
$$216$$ −1.98582e10 −0.620722
$$217$$ 1.96262e10i 0.600851i
$$218$$ 9.40454e9i 0.282022i
$$219$$ −1.37547e9 −0.0404067
$$220$$ 0 0
$$221$$ 1.77952e10 0.501807
$$222$$ 5.84971e9i 0.161639i
$$223$$ − 2.65324e10i − 0.718463i −0.933249 0.359231i $$-0.883039\pi$$
0.933249 0.359231i $$-0.116961\pi$$
$$224$$ 1.36802e10 0.363058
$$225$$ 0 0
$$226$$ −1.16967e10 −0.298248
$$227$$ − 7.78091e10i − 1.94498i −0.232955 0.972488i $$-0.574839\pi$$
0.232955 0.972488i $$-0.425161\pi$$
$$228$$ − 4.62225e8i − 0.0113278i
$$229$$ −4.84637e10 −1.16455 −0.582274 0.812993i $$-0.697837\pi$$
−0.582274 + 0.812993i $$0.697837\pi$$
$$230$$ 0 0
$$231$$ 5.53818e9 0.127972
$$232$$ 5.71362e10i 1.29484i
$$233$$ − 2.38429e10i − 0.529978i −0.964251 0.264989i $$-0.914632\pi$$
0.964251 0.264989i $$-0.0853683\pi$$
$$234$$ 1.09053e10 0.237775
$$235$$ 0 0
$$236$$ −2.24602e10 −0.471313
$$237$$ 1.74787e10i 0.359866i
$$238$$ 5.06278e9i 0.102280i
$$239$$ −6.25895e10 −1.24083 −0.620413 0.784275i $$-0.713035\pi$$
−0.620413 + 0.784275i $$0.713035\pi$$
$$240$$ 0 0
$$241$$ −7.96605e10 −1.52113 −0.760565 0.649262i $$-0.775078\pi$$
−0.760565 + 0.649262i $$0.775078\pi$$
$$242$$ − 1.03333e10i − 0.193674i
$$243$$ 5.52651e10i 1.01677i
$$244$$ 9.55384e9 0.172554
$$245$$ 0 0
$$246$$ −1.79727e10 −0.312901
$$247$$ 1.27198e9i 0.0217442i
$$248$$ − 7.04629e10i − 1.18285i
$$249$$ 6.20384e8 0.0102274
$$250$$ 0 0
$$251$$ −5.44549e10 −0.865975 −0.432988 0.901400i $$-0.642541\pi$$
−0.432988 + 0.901400i $$0.642541\pi$$
$$252$$ − 1.57820e10i − 0.246524i
$$253$$ − 9.02799e10i − 1.38532i
$$254$$ −2.66286e10 −0.401418
$$255$$ 0 0
$$256$$ −1.84414e10 −0.268358
$$257$$ 5.35278e10i 0.765385i 0.923876 + 0.382693i $$0.125003\pi$$
−0.923876 + 0.382693i $$0.874997\pi$$
$$258$$ 1.17771e10i 0.165482i
$$259$$ −2.32962e10 −0.321689
$$260$$ 0 0
$$261$$ 1.01821e11 1.35818
$$262$$ 1.88176e10i 0.246722i
$$263$$ − 5.81425e10i − 0.749364i −0.927153 0.374682i $$-0.877752\pi$$
0.927153 0.374682i $$-0.122248\pi$$
$$264$$ −1.98835e10 −0.251927
$$265$$ 0 0
$$266$$ −3.61883e8 −0.00443201
$$267$$ 4.37057e10i 0.526304i
$$268$$ − 1.17390e11i − 1.39003i
$$269$$ −4.67380e10 −0.544233 −0.272116 0.962264i $$-0.587724\pi$$
−0.272116 + 0.962264i $$0.587724\pi$$
$$270$$ 0 0
$$271$$ 2.68147e10 0.302003 0.151001 0.988534i $$-0.451750\pi$$
0.151001 + 0.988534i $$0.451750\pi$$
$$272$$ 3.21905e10i 0.356589i
$$273$$ − 1.22163e10i − 0.133109i
$$274$$ 2.98570e10 0.320014
$$275$$ 0 0
$$276$$ 7.23660e10 0.750661
$$277$$ − 1.12549e11i − 1.14863i −0.818633 0.574316i $$-0.805268\pi$$
0.818633 0.574316i $$-0.194732\pi$$
$$278$$ 7.58284e10i 0.761431i
$$279$$ −1.25571e11 −1.24071
$$280$$ 0 0
$$281$$ −4.60761e10 −0.440857 −0.220428 0.975403i $$-0.570746\pi$$
−0.220428 + 0.975403i $$0.570746\pi$$
$$282$$ 3.57995e9i 0.0337098i
$$283$$ 7.94071e10i 0.735902i 0.929845 + 0.367951i $$0.119941\pi$$
−0.929845 + 0.367951i $$0.880059\pi$$
$$284$$ −1.55609e11 −1.41939
$$285$$ 0 0
$$286$$ 2.49098e10 0.220152
$$287$$ − 7.15757e10i − 0.622726i
$$288$$ 8.75275e10i 0.749684i
$$289$$ 6.57304e10 0.554276
$$290$$ 0 0
$$291$$ −7.93628e10 −0.648782
$$292$$ 8.95322e9i 0.0720703i
$$293$$ − 1.41265e11i − 1.11977i −0.828569 0.559887i $$-0.810844\pi$$
0.828569 0.559887i $$-0.189156\pi$$
$$294$$ 3.47556e9 0.0271308
$$295$$ 0 0
$$296$$ 8.36393e10 0.633283
$$297$$ 8.08351e10i 0.602832i
$$298$$ − 9.82523e9i − 0.0721721i
$$299$$ −1.99142e11 −1.44093
$$300$$ 0 0
$$301$$ −4.69018e10 −0.329337
$$302$$ 1.80958e10i 0.125184i
$$303$$ 7.77013e10i 0.529586i
$$304$$ −2.30094e9 −0.0154517
$$305$$ 0 0
$$306$$ −3.23923e10 −0.211201
$$307$$ 5.58349e10i 0.358742i 0.983781 + 0.179371i $$0.0574063\pi$$
−0.983781 + 0.179371i $$0.942594\pi$$
$$308$$ − 3.60491e10i − 0.228253i
$$309$$ −1.30031e11 −0.811399
$$310$$ 0 0
$$311$$ 5.26501e10 0.319137 0.159569 0.987187i $$-0.448990\pi$$
0.159569 + 0.987187i $$0.448990\pi$$
$$312$$ 4.38594e10i 0.262040i
$$313$$ − 2.51256e11i − 1.47968i −0.672785 0.739838i $$-0.734902\pi$$
0.672785 0.739838i $$-0.265098\pi$$
$$314$$ −4.23381e10 −0.245781
$$315$$ 0 0
$$316$$ 1.13772e11 0.641866
$$317$$ 1.16999e11i 0.650749i 0.945585 + 0.325375i $$0.105490\pi$$
−0.945585 + 0.325375i $$0.894510\pi$$
$$318$$ 1.65351e10i 0.0906747i
$$319$$ 2.32580e11 1.25752
$$320$$ 0 0
$$321$$ 1.10322e10 0.0579948
$$322$$ − 5.66564e10i − 0.293696i
$$323$$ − 3.77820e9i − 0.0193141i
$$324$$ 6.45828e10 0.325585
$$325$$ 0 0
$$326$$ −5.74622e10 −0.281775
$$327$$ 6.74048e10i 0.326006i
$$328$$ 2.56975e11i 1.22591i
$$329$$ −1.42570e10 −0.0670883
$$330$$ 0 0
$$331$$ −2.51419e11 −1.15126 −0.575629 0.817711i $$-0.695243\pi$$
−0.575629 + 0.817711i $$0.695243\pi$$
$$332$$ − 4.03820e9i − 0.0182417i
$$333$$ − 1.49052e11i − 0.664262i
$$334$$ −5.70171e10 −0.250695
$$335$$ 0 0
$$336$$ 2.20985e10 0.0945882
$$337$$ − 6.11427e10i − 0.258232i −0.991630 0.129116i $$-0.958786\pi$$
0.991630 0.129116i $$-0.0412139\pi$$
$$338$$ 4.23133e10i 0.176340i
$$339$$ −8.38336e10 −0.344762
$$340$$ 0 0
$$341$$ −2.86828e11 −1.14875
$$342$$ − 2.31537e9i − 0.00915173i
$$343$$ 1.38413e10i 0.0539949i
$$344$$ 1.68389e11 0.648338
$$345$$ 0 0
$$346$$ −8.22882e10 −0.308670
$$347$$ 1.68668e11i 0.624524i 0.949996 + 0.312262i $$0.101087\pi$$
−0.949996 + 0.312262i $$0.898913\pi$$
$$348$$ 1.86430e11i 0.681410i
$$349$$ 3.31182e11 1.19496 0.597479 0.801885i $$-0.296169\pi$$
0.597479 + 0.801885i $$0.296169\pi$$
$$350$$ 0 0
$$351$$ 1.78308e11 0.627030
$$352$$ 1.99930e11i 0.694122i
$$353$$ 3.78560e11i 1.29762i 0.760949 + 0.648811i $$0.224734\pi$$
−0.760949 + 0.648811i $$0.775266\pi$$
$$354$$ 3.16468e10 0.107106
$$355$$ 0 0
$$356$$ 2.84489e11 0.938728
$$357$$ 3.62863e10i 0.118232i
$$358$$ − 1.61641e11i − 0.520091i
$$359$$ 1.60137e11 0.508822 0.254411 0.967096i $$-0.418118\pi$$
0.254411 + 0.967096i $$0.418118\pi$$
$$360$$ 0 0
$$361$$ −3.22418e11 −0.999163
$$362$$ 1.48809e10i 0.0455449i
$$363$$ − 7.40617e10i − 0.223879i
$$364$$ −7.95179e10 −0.237415
$$365$$ 0 0
$$366$$ −1.34615e10 −0.0392130
$$367$$ − 5.13837e11i − 1.47852i −0.673419 0.739261i $$-0.735175\pi$$
0.673419 0.739261i $$-0.264825\pi$$
$$368$$ − 3.60236e11i − 1.02394i
$$369$$ 4.57950e11 1.28588
$$370$$ 0 0
$$371$$ −6.58505e10 −0.180458
$$372$$ − 2.29914e11i − 0.622474i
$$373$$ − 6.70900e10i − 0.179460i −0.995966 0.0897301i $$-0.971400\pi$$
0.995966 0.0897301i $$-0.0286005\pi$$
$$374$$ −7.39903e10 −0.195548
$$375$$ 0 0
$$376$$ 5.11862e10 0.132071
$$377$$ − 5.13030e11i − 1.30800i
$$378$$ 5.07292e10i 0.127804i
$$379$$ −4.15471e11 −1.03434 −0.517171 0.855882i $$-0.673015\pi$$
−0.517171 + 0.855882i $$0.673015\pi$$
$$380$$ 0 0
$$381$$ −1.90854e11 −0.464023
$$382$$ 1.52799e11i 0.367143i
$$383$$ − 3.51976e11i − 0.835831i −0.908486 0.417915i $$-0.862761\pi$$
0.908486 0.417915i $$-0.137239\pi$$
$$384$$ −2.03478e11 −0.477560
$$385$$ 0 0
$$386$$ −2.21348e11 −0.507496
$$387$$ − 3.00084e11i − 0.680053i
$$388$$ 5.16588e11i 1.15718i
$$389$$ 2.60061e11 0.575840 0.287920 0.957654i $$-0.407036\pi$$
0.287920 + 0.957654i $$0.407036\pi$$
$$390$$ 0 0
$$391$$ 5.91516e11 1.27989
$$392$$ − 4.96937e10i − 0.106295i
$$393$$ 1.34871e11i 0.285201i
$$394$$ −3.57548e10 −0.0747482
$$395$$ 0 0
$$396$$ 2.30647e11 0.471323
$$397$$ − 7.34338e11i − 1.48367i −0.670580 0.741837i $$-0.733955\pi$$
0.670580 0.741837i $$-0.266045\pi$$
$$398$$ 1.71957e11i 0.343514i
$$399$$ −2.59371e9 −0.00512322
$$400$$ 0 0
$$401$$ 8.08296e11 1.56106 0.780532 0.625116i $$-0.214948\pi$$
0.780532 + 0.625116i $$0.214948\pi$$
$$402$$ 1.65405e11i 0.315886i
$$403$$ 6.32691e11i 1.19487i
$$404$$ 5.05773e11 0.944581
$$405$$ 0 0
$$406$$ 1.45959e11 0.266602
$$407$$ − 3.40464e11i − 0.615030i
$$408$$ − 1.30277e11i − 0.232754i
$$409$$ 9.11153e11 1.61004 0.805020 0.593248i $$-0.202155\pi$$
0.805020 + 0.593248i $$0.202155\pi$$
$$410$$ 0 0
$$411$$ 2.13993e11 0.369923
$$412$$ 8.46398e11i 1.44723i
$$413$$ 1.26032e11i 0.213160i
$$414$$ 3.62495e11 0.606458
$$415$$ 0 0
$$416$$ 4.41010e11 0.721985
$$417$$ 5.43482e11i 0.880183i
$$418$$ − 5.28876e9i − 0.00847345i
$$419$$ −4.94109e11 −0.783177 −0.391589 0.920140i $$-0.628074\pi$$
−0.391589 + 0.920140i $$0.628074\pi$$
$$420$$ 0 0
$$421$$ −1.15145e10 −0.0178639 −0.00893197 0.999960i $$-0.502843\pi$$
−0.00893197 + 0.999960i $$0.502843\pi$$
$$422$$ − 2.02223e10i − 0.0310402i
$$423$$ − 9.12181e10i − 0.138532i
$$424$$ 2.36420e11 0.355253
$$425$$ 0 0
$$426$$ 2.19256e11 0.322558
$$427$$ − 5.36100e10i − 0.0780406i
$$428$$ − 7.18106e10i − 0.103441i
$$429$$ 1.78535e11 0.254487
$$430$$ 0 0
$$431$$ −9.42534e11 −1.31568 −0.657839 0.753159i $$-0.728529\pi$$
−0.657839 + 0.753159i $$0.728529\pi$$
$$432$$ 3.22549e11i 0.445574i
$$433$$ 1.01849e12i 1.39239i 0.717852 + 0.696196i $$0.245125\pi$$
−0.717852 + 0.696196i $$0.754875\pi$$
$$434$$ −1.80003e11 −0.243543
$$435$$ 0 0
$$436$$ 4.38751e11 0.581472
$$437$$ 4.22810e10i 0.0554598i
$$438$$ − 1.26152e10i − 0.0163781i
$$439$$ 7.89357e11 1.01434 0.507169 0.861847i $$-0.330692\pi$$
0.507169 + 0.861847i $$0.330692\pi$$
$$440$$ 0 0
$$441$$ −8.85583e10 −0.111495
$$442$$ 1.63210e11i 0.203397i
$$443$$ 1.06770e12i 1.31714i 0.752518 + 0.658572i $$0.228839\pi$$
−0.752518 + 0.658572i $$0.771161\pi$$
$$444$$ 2.72907e11 0.333266
$$445$$ 0 0
$$446$$ 2.43344e11 0.291215
$$447$$ − 7.04200e10i − 0.0834281i
$$448$$ − 4.66533e10i − 0.0547182i
$$449$$ 3.31695e9 0.00385150 0.00192575 0.999998i $$-0.499387\pi$$
0.00192575 + 0.999998i $$0.499387\pi$$
$$450$$ 0 0
$$451$$ 1.04605e12 1.19058
$$452$$ 5.45689e11i 0.614926i
$$453$$ 1.29698e11i 0.144707i
$$454$$ 7.13632e11 0.788357
$$455$$ 0 0
$$456$$ 9.31207e9 0.0100857
$$457$$ 7.03146e11i 0.754089i 0.926195 + 0.377045i $$0.123060\pi$$
−0.926195 + 0.377045i $$0.876940\pi$$
$$458$$ − 4.44489e11i − 0.472026i
$$459$$ −5.29633e11 −0.556952
$$460$$ 0 0
$$461$$ −1.86192e12 −1.92003 −0.960015 0.279950i $$-0.909682\pi$$
−0.960015 + 0.279950i $$0.909682\pi$$
$$462$$ 5.07938e10i 0.0518707i
$$463$$ − 1.06950e11i − 0.108160i −0.998537 0.0540799i $$-0.982777\pi$$
0.998537 0.0540799i $$-0.0172226\pi$$
$$464$$ 9.28043e11 0.929474
$$465$$ 0 0
$$466$$ 2.18677e11 0.214816
$$467$$ − 4.13997e11i − 0.402783i −0.979511 0.201392i $$-0.935454\pi$$
0.979511 0.201392i $$-0.0645464\pi$$
$$468$$ − 5.08766e11i − 0.490243i
$$469$$ −6.58717e11 −0.628667
$$470$$ 0 0
$$471$$ −3.03448e11 −0.284113
$$472$$ − 4.52487e11i − 0.419631i
$$473$$ − 6.85450e11i − 0.629652i
$$474$$ −1.60307e11 −0.145865
$$475$$ 0 0
$$476$$ 2.36194e11 0.210881
$$477$$ − 4.21320e11i − 0.372631i
$$478$$ − 5.74044e11i − 0.502944i
$$479$$ 8.54131e10 0.0741336 0.0370668 0.999313i $$-0.488199\pi$$
0.0370668 + 0.999313i $$0.488199\pi$$
$$480$$ 0 0
$$481$$ −7.51004e11 −0.639719
$$482$$ − 7.30612e11i − 0.616560i
$$483$$ − 4.06072e11i − 0.339501i
$$484$$ −4.82082e11 −0.399316
$$485$$ 0 0
$$486$$ −5.06868e11 −0.412127
$$487$$ − 4.94789e11i − 0.398602i −0.979938 0.199301i $$-0.936133\pi$$
0.979938 0.199301i $$-0.0638672\pi$$
$$488$$ 1.92474e11i 0.153632i
$$489$$ −4.11847e11 −0.325721
$$490$$ 0 0
$$491$$ 1.11163e12 0.863168 0.431584 0.902073i $$-0.357955\pi$$
0.431584 + 0.902073i $$0.357955\pi$$
$$492$$ 8.38484e11i 0.645137i
$$493$$ 1.52387e12i 1.16181i
$$494$$ −1.16661e10 −0.00881359
$$495$$ 0 0
$$496$$ −1.14450e12 −0.849083
$$497$$ 8.73178e11i 0.641947i
$$498$$ 5.68990e9i 0.00414546i
$$499$$ −8.18377e11 −0.590882 −0.295441 0.955361i $$-0.595467\pi$$
−0.295441 + 0.955361i $$0.595467\pi$$
$$500$$ 0 0
$$501$$ −4.08656e11 −0.289793
$$502$$ − 4.99437e11i − 0.351006i
$$503$$ 3.13384e11i 0.218284i 0.994026 + 0.109142i $$0.0348102\pi$$
−0.994026 + 0.109142i $$0.965190\pi$$
$$504$$ 3.17947e11 0.219491
$$505$$ 0 0
$$506$$ 8.28009e11 0.561510
$$507$$ 3.03270e11i 0.203842i
$$508$$ 1.24231e12i 0.827642i
$$509$$ 1.02554e12 0.677211 0.338606 0.940928i $$-0.390045\pi$$
0.338606 + 0.940928i $$0.390045\pi$$
$$510$$ 0 0
$$511$$ 5.02397e10 0.0325951
$$512$$ 1.41572e12i 0.910467i
$$513$$ − 3.78577e10i − 0.0241338i
$$514$$ −4.90934e11 −0.310234
$$515$$ 0 0
$$516$$ 5.49438e11 0.341189
$$517$$ − 2.08360e11i − 0.128265i
$$518$$ − 2.13663e11i − 0.130390i
$$519$$ −5.89781e11 −0.356810
$$520$$ 0 0
$$521$$ −3.18952e12 −1.89651 −0.948255 0.317510i $$-0.897153\pi$$
−0.948255 + 0.317510i $$0.897153\pi$$
$$522$$ 9.33862e11i 0.550510i
$$523$$ 9.43708e11i 0.551544i 0.961223 + 0.275772i $$0.0889335\pi$$
−0.961223 + 0.275772i $$0.911067\pi$$
$$524$$ 8.77898e11 0.508690
$$525$$ 0 0
$$526$$ 5.33258e11 0.303740
$$527$$ − 1.87930e12i − 1.06133i
$$528$$ 3.22961e11i 0.180841i
$$529$$ −4.81837e12 −2.67516
$$530$$ 0 0
$$531$$ −8.06370e11 −0.440158
$$532$$ 1.68829e10i 0.00913789i
$$533$$ − 2.30740e12i − 1.23837i
$$534$$ −4.00850e11 −0.213327
$$535$$ 0 0
$$536$$ 2.36496e12 1.23761
$$537$$ − 1.15853e12i − 0.601204i
$$538$$ − 4.28661e11i − 0.220594i
$$539$$ −2.02284e11 −0.103232
$$540$$ 0 0
$$541$$ −8.78618e11 −0.440973 −0.220487 0.975390i $$-0.570765\pi$$
−0.220487 + 0.975390i $$0.570765\pi$$
$$542$$ 2.45933e11i 0.122411i
$$543$$ 1.06655e11i 0.0526480i
$$544$$ −1.30994e12 −0.641295
$$545$$ 0 0
$$546$$ 1.12042e11 0.0539529
$$547$$ − 4.56216e11i − 0.217885i −0.994048 0.108943i $$-0.965254\pi$$
0.994048 0.108943i $$-0.0347465\pi$$
$$548$$ − 1.39292e12i − 0.659803i
$$549$$ 3.43004e11 0.161147
$$550$$ 0 0
$$551$$ −1.08925e11 −0.0503435
$$552$$ 1.45790e12i 0.668347i
$$553$$ − 6.38416e11i − 0.290296i
$$554$$ 1.03225e12 0.465575
$$555$$ 0 0
$$556$$ 3.53763e12 1.56991
$$557$$ − 1.63980e12i − 0.721842i −0.932596 0.360921i $$-0.882462\pi$$
0.932596 0.360921i $$-0.117538\pi$$
$$558$$ − 1.15168e12i − 0.502896i
$$559$$ −1.51198e12 −0.654927
$$560$$ 0 0
$$561$$ −5.30308e11 −0.226045
$$562$$ − 4.22590e11i − 0.178692i
$$563$$ 4.36151e11i 0.182957i 0.995807 + 0.0914786i $$0.0291593\pi$$
−0.995807 + 0.0914786i $$0.970841\pi$$
$$564$$ 1.67016e11 0.0695026
$$565$$ 0 0
$$566$$ −7.28288e11 −0.298283
$$567$$ − 3.62397e11i − 0.147252i
$$568$$ − 3.13493e12i − 1.26375i
$$569$$ −1.76284e12 −0.705029 −0.352514 0.935806i $$-0.614673\pi$$
−0.352514 + 0.935806i $$0.614673\pi$$
$$570$$ 0 0
$$571$$ 2.37232e11 0.0933922 0.0466961 0.998909i $$-0.485131\pi$$
0.0466961 + 0.998909i $$0.485131\pi$$
$$572$$ − 1.16212e12i − 0.453909i
$$573$$ 1.09515e12i 0.424403i
$$574$$ 6.56462e11 0.252410
$$575$$ 0 0
$$576$$ 2.98494e11 0.112988
$$577$$ 3.72080e12i 1.39748i 0.715377 + 0.698739i $$0.246255\pi$$
−0.715377 + 0.698739i $$0.753745\pi$$
$$578$$ 6.02851e11i 0.224665i
$$579$$ −1.58646e12 −0.586644
$$580$$ 0 0
$$581$$ −2.26598e10 −0.00825017
$$582$$ − 7.27882e11i − 0.262971i
$$583$$ − 9.62377e11i − 0.345014i
$$584$$ −1.80373e11 −0.0641674
$$585$$ 0 0
$$586$$ 1.29562e12 0.453878
$$587$$ 6.46176e11i 0.224636i 0.993672 + 0.112318i $$0.0358275\pi$$
−0.993672 + 0.112318i $$0.964172\pi$$
$$588$$ − 1.62146e11i − 0.0559381i
$$589$$ 1.34331e11 0.0459892
$$590$$ 0 0
$$591$$ −2.56264e11 −0.0864059
$$592$$ − 1.35853e12i − 0.454590i
$$593$$ − 5.05774e12i − 1.67962i −0.542883 0.839808i $$-0.682667\pi$$
0.542883 0.839808i $$-0.317333\pi$$
$$594$$ −7.41385e11 −0.244346
$$595$$ 0 0
$$596$$ −4.58377e11 −0.148804
$$597$$ 1.23246e12i 0.397089i
$$598$$ − 1.82644e12i − 0.584051i
$$599$$ 4.61588e11 0.146499 0.0732494 0.997314i $$-0.476663\pi$$
0.0732494 + 0.997314i $$0.476663\pi$$
$$600$$ 0 0
$$601$$ −6.31800e12 −1.97535 −0.987677 0.156509i $$-0.949976\pi$$
−0.987677 + 0.156509i $$0.949976\pi$$
$$602$$ − 4.30163e11i − 0.133490i
$$603$$ − 4.21455e12i − 1.29815i
$$604$$ 8.44227e11 0.258103
$$605$$ 0 0
$$606$$ −7.12643e11 −0.214657
$$607$$ 1.45276e12i 0.434356i 0.976132 + 0.217178i $$0.0696852\pi$$
−0.976132 + 0.217178i $$0.930315\pi$$
$$608$$ − 9.36335e10i − 0.0277885i
$$609$$ 1.04612e12 0.308181
$$610$$ 0 0
$$611$$ −4.59605e11 −0.133413
$$612$$ 1.51120e12i 0.435453i
$$613$$ 1.20124e12i 0.343603i 0.985132 + 0.171802i $$0.0549588\pi$$
−0.985132 + 0.171802i $$0.945041\pi$$
$$614$$ −5.12094e11 −0.145409
$$615$$ 0 0
$$616$$ 7.26252e11 0.203224
$$617$$ − 3.13545e12i − 0.870997i −0.900189 0.435498i $$-0.856572\pi$$
0.900189 0.435498i $$-0.143428\pi$$
$$618$$ − 1.19259e12i − 0.328884i
$$619$$ 5.17127e12 1.41576 0.707880 0.706333i $$-0.249652\pi$$
0.707880 + 0.706333i $$0.249652\pi$$
$$620$$ 0 0
$$621$$ 5.92700e12 1.59927
$$622$$ 4.82884e11i 0.129356i
$$623$$ − 1.59637e12i − 0.424557i
$$624$$ 7.12394e11 0.188100
$$625$$ 0 0
$$626$$ 2.30441e12 0.599757
$$627$$ − 3.79059e10i − 0.00979497i
$$628$$ 1.97520e12i 0.506749i
$$629$$ 2.23073e12 0.568223
$$630$$ 0 0
$$631$$ −6.37331e12 −1.60042 −0.800208 0.599723i $$-0.795278\pi$$
−0.800208 + 0.599723i $$0.795278\pi$$
$$632$$ 2.29208e12i 0.571481i
$$633$$ − 1.44939e11i − 0.0358812i
$$634$$ −1.07306e12 −0.263768
$$635$$ 0 0
$$636$$ 7.71416e11 0.186953
$$637$$ 4.46204e11i 0.107376i
$$638$$ 2.13312e12i 0.509710i
$$639$$ −5.58670e12 −1.32557
$$640$$ 0 0
$$641$$ −5.74174e12 −1.34333 −0.671665 0.740855i $$-0.734421\pi$$
−0.671665 + 0.740855i $$0.734421\pi$$
$$642$$ 1.01182e11i 0.0235070i
$$643$$ − 5.85135e11i − 0.134992i −0.997720 0.0674958i $$-0.978499\pi$$
0.997720 0.0674958i $$-0.0215009\pi$$
$$644$$ −2.64320e12 −0.605541
$$645$$ 0 0
$$646$$ 3.46520e10 0.00782857
$$647$$ 1.80915e12i 0.405887i 0.979190 + 0.202943i $$0.0650507\pi$$
−0.979190 + 0.202943i $$0.934949\pi$$
$$648$$ 1.30110e12i 0.289883i
$$649$$ −1.84191e12 −0.407536
$$650$$ 0 0
$$651$$ −1.29013e12 −0.281526
$$652$$ 2.68079e12i 0.580963i
$$653$$ 2.43900e12i 0.524932i 0.964941 + 0.262466i $$0.0845357\pi$$
−0.964941 + 0.262466i $$0.915464\pi$$
$$654$$ −6.18208e11 −0.132140
$$655$$ 0 0
$$656$$ 4.17396e12 0.879996
$$657$$ 3.21440e11i 0.0673063i
$$658$$ − 1.30759e11i − 0.0271929i
$$659$$ −7.42836e12 −1.53429 −0.767147 0.641471i $$-0.778324\pi$$
−0.767147 + 0.641471i $$0.778324\pi$$
$$660$$ 0 0
$$661$$ −6.34861e12 −1.29352 −0.646759 0.762695i $$-0.723876\pi$$
−0.646759 + 0.762695i $$0.723876\pi$$
$$662$$ − 2.30591e12i − 0.466640i
$$663$$ 1.16977e12i 0.235119i
$$664$$ 8.13544e10 0.0162414
$$665$$ 0 0
$$666$$ 1.36704e12 0.269245
$$667$$ − 1.70533e13i − 3.33611i
$$668$$ 2.66002e12i 0.516882i
$$669$$ 1.74411e12 0.336632
$$670$$ 0 0
$$671$$ 7.83487e11 0.149204
$$672$$ 8.99267e11i 0.170109i
$$673$$ 3.17186e12i 0.596000i 0.954566 + 0.298000i $$0.0963196\pi$$
−0.954566 + 0.298000i $$0.903680\pi$$
$$674$$ 5.60774e11 0.104669
$$675$$ 0 0
$$676$$ 1.97405e12 0.363578
$$677$$ 2.32240e12i 0.424901i 0.977172 + 0.212450i $$0.0681444\pi$$
−0.977172 + 0.212450i $$0.931856\pi$$
$$678$$ − 7.68886e11i − 0.139742i
$$679$$ 2.89876e12 0.523357
$$680$$ 0 0
$$681$$ 5.11479e12 0.911309
$$682$$ − 2.63066e12i − 0.465624i
$$683$$ 3.78639e12i 0.665782i 0.942965 + 0.332891i $$0.108024\pi$$
−0.942965 + 0.332891i $$0.891976\pi$$
$$684$$ −1.08019e11 −0.0188690
$$685$$ 0 0
$$686$$ −1.26946e11 −0.0218858
$$687$$ − 3.18577e12i − 0.545643i
$$688$$ − 2.73509e12i − 0.465397i
$$689$$ −2.12283e12 −0.358864
$$690$$ 0 0
$$691$$ −5.70532e12 −0.951982 −0.475991 0.879450i $$-0.657911\pi$$
−0.475991 + 0.879450i $$0.657911\pi$$
$$692$$ 3.83900e12i 0.636415i
$$693$$ − 1.29424e12i − 0.213165i
$$694$$ −1.54695e12 −0.253138
$$695$$ 0 0
$$696$$ −3.75585e12 −0.606690
$$697$$ 6.85372e12i 1.09997i
$$698$$ 3.03746e12i 0.484352i
$$699$$ 1.56732e12 0.248318
$$700$$ 0 0
$$701$$ 6.25160e12 0.977823 0.488912 0.872333i $$-0.337394\pi$$
0.488912 + 0.872333i $$0.337394\pi$$
$$702$$ 1.63536e12i 0.254154i
$$703$$ 1.59450e11i 0.0246222i
$$704$$ 6.81818e11 0.104614
$$705$$ 0 0
$$706$$ −3.47199e12 −0.525965
$$707$$ − 2.83807e12i − 0.427205i
$$708$$ − 1.47642e12i − 0.220831i
$$709$$ 8.01996e12 1.19197 0.595983 0.802997i $$-0.296762\pi$$
0.595983 + 0.802997i $$0.296762\pi$$
$$710$$ 0 0
$$711$$ 4.08467e12 0.599437
$$712$$ 5.73136e12i 0.835791i
$$713$$ 2.10308e13i 3.04757i
$$714$$ −3.32802e11 −0.0479230
$$715$$ 0 0
$$716$$ −7.54107e12 −1.07232
$$717$$ − 4.11433e12i − 0.581383i
$$718$$ 1.46871e12i 0.206241i
$$719$$ −1.35002e12 −0.188391 −0.0941953 0.995554i $$-0.530028\pi$$
−0.0941953 + 0.995554i $$0.530028\pi$$
$$720$$ 0 0
$$721$$ 4.74944e12 0.654537
$$722$$ − 2.95708e12i − 0.404991i
$$723$$ − 5.23649e12i − 0.712718i
$$724$$ 6.94238e11 0.0939042
$$725$$ 0 0
$$726$$ 6.79262e11 0.0907450
$$727$$ 1.47778e13i 1.96203i 0.193941 + 0.981013i $$0.437873\pi$$
−0.193941 + 0.981013i $$0.562127\pi$$
$$728$$ − 1.60198e12i − 0.211381i
$$729$$ −6.61984e11 −0.0868108
$$730$$ 0 0
$$731$$ 4.49108e12 0.581731
$$732$$ 6.28022e11i 0.0808491i
$$733$$ − 6.70116e12i − 0.857398i −0.903447 0.428699i $$-0.858972\pi$$
0.903447 0.428699i $$-0.141028\pi$$
$$734$$ 4.71269e12 0.599290
$$735$$ 0 0
$$736$$ 1.46593e13 1.84146
$$737$$ − 9.62686e12i − 1.20193i
$$738$$ 4.20013e12i 0.521205i
$$739$$ 1.39054e13 1.71508 0.857540 0.514417i $$-0.171992\pi$$
0.857540 + 0.514417i $$0.171992\pi$$
$$740$$ 0 0
$$741$$ −8.36137e10 −0.0101882
$$742$$ − 6.03953e11i − 0.0731452i
$$743$$ − 3.43953e12i − 0.414047i −0.978336 0.207024i $$-0.933622\pi$$
0.978336 0.207024i $$-0.0663777\pi$$
$$744$$ 4.63188e12 0.554216
$$745$$ 0 0
$$746$$ 6.15321e11 0.0727407
$$747$$ − 1.44980e11i − 0.0170359i
$$748$$ 3.45188e12i 0.403179i
$$749$$ −4.02955e11 −0.0467830
$$750$$ 0 0
$$751$$ 1.00823e13 1.15660 0.578298 0.815826i $$-0.303717\pi$$
0.578298 + 0.815826i $$0.303717\pi$$
$$752$$ − 8.31401e11i − 0.0948047i
$$753$$ − 3.57960e12i − 0.405748i
$$754$$ 4.70529e12 0.530170
$$755$$ 0 0
$$756$$ 2.36667e12 0.263506
$$757$$ 1.02703e13i 1.13672i 0.822780 + 0.568360i $$0.192422\pi$$
−0.822780 + 0.568360i $$0.807578\pi$$
$$758$$ − 3.81052e12i − 0.419250i
$$759$$ 5.93456e12 0.649083
$$760$$ 0 0
$$761$$ −3.20840e12 −0.346783 −0.173391 0.984853i $$-0.555473\pi$$
−0.173391 + 0.984853i $$0.555473\pi$$
$$762$$ − 1.75043e12i − 0.188083i
$$763$$ − 2.46199e12i − 0.262982i
$$764$$ 7.12855e12 0.756974
$$765$$ 0 0
$$766$$ 3.22817e12 0.338787
$$767$$ 4.06292e12i 0.423896i
$$768$$ − 1.21225e12i − 0.125738i
$$769$$ −1.52349e13 −1.57098 −0.785491 0.618872i $$-0.787590\pi$$
−0.785491 + 0.618872i $$0.787590\pi$$
$$770$$ 0 0
$$771$$ −3.51865e12 −0.358618
$$772$$ 1.03266e13i 1.04635i
$$773$$ − 8.54195e11i − 0.0860497i −0.999074 0.0430249i $$-0.986301\pi$$
0.999074 0.0430249i $$-0.0136995\pi$$
$$774$$ 2.75224e12 0.275646
$$775$$ 0 0
$$776$$ −1.04073e13 −1.03029
$$777$$ − 1.53138e12i − 0.150726i
$$778$$ 2.38517e12i 0.233405i
$$779$$ −4.89898e11 −0.0476636
$$780$$ 0 0
$$781$$ −1.27611e13 −1.22732
$$782$$ 5.42513e12i 0.518776i
$$783$$ 1.52692e13i 1.45174i
$$784$$ −8.07158e11 −0.0763021
$$785$$ 0 0
$$786$$ −1.23697e12 −0.115600
$$787$$ − 4.25016e12i − 0.394929i −0.980310 0.197465i $$-0.936729\pi$$
0.980310 0.197465i $$-0.0632707\pi$$
$$788$$ 1.66807e12i 0.154116i
$$789$$ 3.82200e12 0.351111
$$790$$ 0 0
$$791$$ 3.06206e12 0.278112
$$792$$ 4.64665e12i 0.419640i
$$793$$ − 1.72823e12i − 0.155193i
$$794$$ 6.73503e12 0.601378
$$795$$ 0 0
$$796$$ 8.02231e12 0.708256
$$797$$ − 2.02157e13i − 1.77470i −0.461093 0.887352i $$-0.652542\pi$$
0.461093 0.887352i $$-0.347458\pi$$
$$798$$ − 2.37884e10i − 0.00207660i
$$799$$ 1.36518e12 0.118503
$$800$$ 0 0
$$801$$ 1.02138e13 0.876676
$$802$$ 7.41334e12i 0.632746i
$$803$$ 7.34231e11i 0.0623179i
$$804$$ 7.71664e12 0.651292
$$805$$ 0 0
$$806$$ −5.80278e12 −0.484315
$$807$$ − 3.07232e12i − 0.254998i
$$808$$ 1.01894e13i 0.841002i
$$809$$ 6.65781e12 0.546466 0.273233 0.961948i $$-0.411907\pi$$
0.273233 + 0.961948i $$0.411907\pi$$
$$810$$ 0 0
$$811$$ −9.35525e12 −0.759384 −0.379692 0.925113i $$-0.623970\pi$$
−0.379692 + 0.925113i $$0.623970\pi$$
$$812$$ − 6.80942e12i − 0.549678i
$$813$$ 1.76267e12i 0.141502i
$$814$$ 3.12259e12 0.249290
$$815$$ 0 0
$$816$$ −2.11604e12 −0.167078
$$817$$ 3.21018e11i 0.0252075i
$$818$$ 8.35671e12i 0.652597i
$$819$$ −2.85487e12 −0.221722
$$820$$ 0 0
$$821$$ 1.61631e13 1.24160 0.620798 0.783971i $$-0.286809\pi$$
0.620798 + 0.783971i $$0.286809\pi$$
$$822$$ 1.96265e12i 0.149941i
$$823$$ 5.97042e12i 0.453634i 0.973937 + 0.226817i $$0.0728319\pi$$
−0.973937 + 0.226817i $$0.927168\pi$$
$$824$$ −1.70517e13 −1.28853
$$825$$ 0 0
$$826$$ −1.15591e12 −0.0864003
$$827$$ − 7.76424e12i − 0.577197i −0.957450 0.288599i $$-0.906811\pi$$
0.957450 0.288599i $$-0.0931893\pi$$
$$828$$ − 1.69115e13i − 1.25039i
$$829$$ 4.09585e12 0.301195 0.150598 0.988595i $$-0.451880\pi$$
0.150598 + 0.988595i $$0.451880\pi$$
$$830$$ 0 0
$$831$$ 7.39839e12 0.538186
$$832$$ − 1.50397e12i − 0.108814i
$$833$$ − 1.32537e12i − 0.0953751i
$$834$$ −4.98458e12 −0.356765
$$835$$ 0 0
$$836$$ −2.46737e11 −0.0174705
$$837$$ − 1.88306e13i − 1.32617i
$$838$$ − 4.53176e12i − 0.317445i
$$839$$ −4.46741e11 −0.0311263 −0.0155631 0.999879i $$-0.504954\pi$$
−0.0155631 + 0.999879i $$0.504954\pi$$
$$840$$ 0 0
$$841$$ 2.94256e13 2.02835
$$842$$ − 1.05606e11i − 0.00724079i
$$843$$ − 3.02882e12i − 0.206561i
$$844$$ −9.43433e11 −0.0639985
$$845$$ 0 0
$$846$$ 8.36613e11 0.0561511
$$847$$ 2.70513e12i 0.180598i
$$848$$ − 3.84009e12i − 0.255012i
$$849$$ −5.21983e12 −0.344803
$$850$$ 0 0
$$851$$ −2.49636e13 −1.63164
$$852$$ − 1.02290e13i − 0.665049i
$$853$$ 2.72968e13i 1.76539i 0.469945 + 0.882696i $$0.344274\pi$$
−0.469945 + 0.882696i $$0.655726\pi$$
$$854$$ 4.91688e11 0.0316322
$$855$$ 0 0
$$856$$ 1.44671e12 0.0920979
$$857$$ − 7.67571e12i − 0.486077i −0.970017 0.243038i $$-0.921856\pi$$
0.970017 0.243038i $$-0.0781441\pi$$
$$858$$ 1.63745e12i 0.103151i
$$859$$ 1.67172e13 1.04759 0.523797 0.851843i $$-0.324515\pi$$
0.523797 + 0.851843i $$0.324515\pi$$
$$860$$ 0 0
$$861$$ 4.70503e12 0.291775
$$862$$ − 8.64452e12i − 0.533284i
$$863$$ − 2.85615e13i − 1.75280i −0.481582 0.876401i $$-0.659938\pi$$
0.481582 0.876401i $$-0.340062\pi$$
$$864$$ −1.31257e13 −0.801327
$$865$$ 0 0
$$866$$ −9.34116e12 −0.564378
$$867$$ 4.32079e12i 0.259703i
$$868$$ 8.39769e12i 0.502135i
$$869$$ 9.33018e12 0.555010
$$870$$ 0 0
$$871$$ −2.12351e13 −1.25018
$$872$$ 8.83916e12i 0.517710i
$$873$$ 1.85466e13i 1.08069i
$$874$$ −3.87783e11 −0.0224795
$$875$$ 0 0
$$876$$ −5.88540e11 −0.0337682
$$877$$ 8.40714e12i 0.479899i 0.970785 + 0.239950i $$0.0771309\pi$$
−0.970785 + 0.239950i $$0.922869\pi$$
$$878$$ 7.23964e12i 0.411142i
$$879$$ 9.28607e12 0.524665
$$880$$ 0 0
$$881$$ −1.99694e13 −1.11680 −0.558398 0.829573i $$-0.688584\pi$$
−0.558398 + 0.829573i $$0.688584\pi$$
$$882$$ − 8.12219e11i − 0.0451923i
$$883$$ − 2.36498e13i − 1.30919i −0.755978 0.654597i $$-0.772838\pi$$
0.755978 0.654597i $$-0.227162\pi$$
$$884$$ 7.61423e12 0.419364
$$885$$ 0 0
$$886$$ −9.79250e12 −0.533878
$$887$$ 3.83859e12i 0.208217i 0.994566 + 0.104108i $$0.0331989\pi$$
−0.994566 + 0.104108i $$0.966801\pi$$
$$888$$ 5.49804e12i 0.296722i
$$889$$ 6.97103e12 0.374317
$$890$$ 0 0
$$891$$ 5.29627e12 0.281527
$$892$$ − 1.13527e13i − 0.600425i
$$893$$ 9.75816e10i 0.00513495i
$$894$$ 6.45862e11 0.0338159
$$895$$ 0 0
$$896$$ 7.43213e12 0.385237
$$897$$ − 1.30906e13i − 0.675139i
$$898$$ 3.04216e10i 0.00156113i
$$899$$ −5.41798e13 −2.76642
$$900$$ 0 0
$$901$$ 6.30551e12 0.318756
$$902$$ 9.59390e12i 0.482576i
$$903$$ − 3.08309e12i − 0.154309i
$$904$$ −1.09936e13 −0.547495
$$905$$ 0 0
$$906$$ −1.18953e12 −0.0586542
$$907$$ 2.31944e13i 1.13802i 0.822331 + 0.569009i $$0.192673\pi$$
−0.822331 + 0.569009i $$0.807327\pi$$
$$908$$ − 3.32931e13i − 1.62543i
$$909$$ 1.81583e13 0.882142
$$910$$ 0 0
$$911$$ 1.70743e13 0.821317 0.410659 0.911789i $$-0.365299\pi$$
0.410659 + 0.911789i $$0.365299\pi$$
$$912$$ − 1.51253e11i − 0.00723980i
$$913$$ − 3.31163e11i − 0.0157733i
$$914$$ −6.44896e12 −0.305655
$$915$$ 0 0
$$916$$ −2.07368e13 −0.973221
$$917$$ − 4.92620e12i − 0.230065i
$$918$$ − 4.85757e12i − 0.225750i
$$919$$ −1.49265e13 −0.690301 −0.345151 0.938547i $$-0.612172\pi$$
−0.345151 + 0.938547i $$0.612172\pi$$
$$920$$ 0 0
$$921$$ −3.67031e12 −0.168087
$$922$$ − 1.70768e13i − 0.778246i
$$923$$ 2.81488e13i 1.27659i
$$924$$ 2.36969e12 0.106947
$$925$$ 0 0
$$926$$ 9.80898e11 0.0438404
$$927$$ 3.03875e13i 1.35156i
$$928$$ 3.77653e13i 1.67158i
$$929$$ −2.92103e12 −0.128666 −0.0643332 0.997928i $$-0.520492\pi$$
−0.0643332 + 0.997928i $$0.520492\pi$$
$$930$$ 0 0
$$931$$ 9.47362e10 0.00413278
$$932$$ − 1.02020e13i − 0.442906i
$$933$$ 3.46096e12i 0.149530i
$$934$$ 3.79701e12 0.163260
$$935$$ 0 0
$$936$$ 1.02497e13 0.436485
$$937$$ 3.53996e13i 1.50027i 0.661284 + 0.750135i $$0.270012\pi$$
−0.661284 + 0.750135i $$0.729988\pi$$
$$938$$ − 6.04147e12i − 0.254818i
$$939$$ 1.65163e13 0.693295
$$940$$ 0 0
$$941$$ −4.67286e13 −1.94280 −0.971402 0.237439i $$-0.923692\pi$$
−0.971402 + 0.237439i $$0.923692\pi$$
$$942$$ − 2.78310e12i − 0.115159i
$$943$$ − 7.66985e13i − 3.15852i
$$944$$ −7.34960e12 −0.301224
$$945$$ 0 0
$$946$$ 6.28665e12 0.255217
$$947$$ 5.22799e12i 0.211232i 0.994407 + 0.105616i $$0.0336815\pi$$
−0.994407 + 0.105616i $$0.966319\pi$$
$$948$$ 7.47882e12i 0.300743i
$$949$$ 1.61958e12 0.0648195
$$950$$ 0 0
$$951$$ −7.69091e12 −0.304905
$$952$$ 4.75842e12i 0.187757i
$$953$$ − 2.46017e13i − 0.966156i −0.875577 0.483078i $$-0.839519\pi$$
0.875577 0.483078i $$-0.160481\pi$$
$$954$$ 3.86417e12 0.151039
$$955$$ 0 0
$$956$$ −2.67809e13 −1.03697
$$957$$ 1.52886e13i 0.589204i
$$958$$ 7.83373e11i 0.0300486i
$$959$$ −7.81618e12 −0.298408
$$960$$ 0 0
$$961$$ 4.03773e13 1.52715
$$962$$ − 6.88788e12i − 0.259297i
$$963$$ − 2.57816e12i − 0.0966031i
$$964$$ −3.40853e13 −1.27122
$$965$$ 0 0
$$966$$ 3.72431e12 0.137610
$$967$$ − 1.24062e13i − 0.456267i −0.973630 0.228134i $$-0.926738\pi$$
0.973630 0.228134i $$-0.0732623\pi$$
$$968$$ − 9.71212e12i − 0.355529i
$$969$$ 2.48360e11 0.00904951
$$970$$ 0 0
$$971$$ 5.30324e12 0.191450 0.0957248 0.995408i $$-0.469483\pi$$
0.0957248 + 0.995408i $$0.469483\pi$$
$$972$$ 2.36469e13i 0.849722i
$$973$$ − 1.98509e13i − 0.710023i
$$974$$ 4.53799e12 0.161565
$$975$$ 0 0
$$976$$ 3.12628e12 0.110282
$$977$$ 1.45131e13i 0.509606i 0.966993 + 0.254803i $$0.0820107\pi$$
−0.966993 + 0.254803i $$0.917989\pi$$
$$978$$ − 3.77728e12i − 0.132024i
$$979$$ 2.33302e13 0.811702
$$980$$ 0 0
$$981$$ 1.57521e13 0.543035
$$982$$ 1.01954e13i 0.349868i
$$983$$ − 3.61534e13i − 1.23498i −0.786580 0.617488i $$-0.788150\pi$$
0.786580 0.617488i $$-0.211850\pi$$
$$984$$ −1.68923e13 −0.574394
$$985$$ 0 0
$$986$$ −1.39763e13 −0.470917
$$987$$ − 9.37185e11i − 0.0314339i
$$988$$ 5.44258e11i 0.0181718i
$$989$$ −5.02586e13 −1.67043
$$990$$ 0 0
$$991$$ 4.94162e13 1.62756 0.813782 0.581170i $$-0.197405\pi$$
0.813782 + 0.581170i $$0.197405\pi$$
$$992$$ − 4.65739e13i − 1.52700i
$$993$$ − 1.65271e13i − 0.539417i
$$994$$ −8.00841e12 −0.260200
$$995$$ 0 0
$$996$$ 2.65451e11 0.00854708
$$997$$ 3.03522e13i 0.972886i 0.873712 + 0.486443i $$0.161706\pi$$
−0.873712 + 0.486443i $$0.838294\pi$$
$$998$$ − 7.50580e12i − 0.239502i
$$999$$ 2.23519e13 0.710020
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 175.10.b.c.99.3 4
5.2 odd 4 35.10.a.b.1.2 2
5.3 odd 4 175.10.a.c.1.1 2
5.4 even 2 inner 175.10.b.c.99.2 4
15.2 even 4 315.10.a.b.1.1 2
35.27 even 4 245.10.a.c.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 2 5.2 odd 4
175.10.a.c.1.1 2 5.3 odd 4
175.10.b.c.99.2 4 5.4 even 2 inner
175.10.b.c.99.3 4 1.1 even 1 trivial
245.10.a.c.1.2 2 35.27 even 4
315.10.a.b.1.1 2 15.2 even 4