# Properties

 Label 315.10.a.b.1.1 Level $315$ Weight $10$ Character 315.1 Self dual yes Analytic conductor $162.236$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,10,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("315.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 315.a (trivial)

## 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: yes Analytic conductor: $$162.236288392$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 315.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.17157 q^{2} -427.882 q^{4} -625.000 q^{5} +2401.00 q^{7} -8620.20 q^{8} +O(q^{10})$$ $$q+9.17157 q^{2} -427.882 q^{4} -625.000 q^{5} +2401.00 q^{7} -8620.20 q^{8} -5732.23 q^{10} -35089.6 q^{11} -77401.4 q^{13} +22020.9 q^{14} +140015. q^{16} +229907. q^{17} +16433.6 q^{19} +267426. q^{20} -321827. q^{22} +2.57284e6 q^{23} +390625. q^{25} -709892. q^{26} -1.02735e6 q^{28} +6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6 q^{32} +2.10861e6 q^{34} -1.50062e6 q^{35} +9.70272e6 q^{37} +150722. q^{38} +5.38762e6 q^{40} -2.98108e7 q^{41} -1.95343e7 q^{43} +1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6 q^{47} +5.76480e6 q^{49} +3.58265e6 q^{50} +3.31187e7 q^{52} +2.74263e7 q^{53} +2.19310e7 q^{55} -2.06971e7 q^{56} +6.07908e7 q^{58} -5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7 q^{62} -1.94308e7 q^{64} +4.83759e7 q^{65} +2.74351e8 q^{67} -9.83733e7 q^{68} -1.37631e7 q^{70} +3.63673e8 q^{71} +2.09245e7 q^{73} +8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7 q^{77} -2.65896e8 q^{79} -8.75093e7 q^{80} -2.73412e8 q^{82} +9.43764e6 q^{83} -1.43692e8 q^{85} -1.79160e8 q^{86} +3.02479e8 q^{88} +6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9 q^{92} -5.44603e7 q^{94} -1.02710e7 q^{95} -1.20731e9 q^{97} +5.28723e7 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8}+O(q^{10})$$ 2 * q + 24 * q^2 - 720 * q^4 - 1250 * q^5 + 4802 * q^7 - 20544 * q^8 $$2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22} + 498908 q^{23} + 781250 q^{25} - 319736 q^{26} - 1728720 q^{28} + 11577554 q^{29} - 3953760 q^{31} + 11398656 q^{32} + 4243944 q^{34} - 3001250 q^{35} - 3205412 q^{37} - 2217520 q^{38} + 12840000 q^{40} - 1058992 q^{41} + 15948180 q^{43} + 10187376 q^{44} - 7156176 q^{46} - 65501290 q^{47} + 11529602 q^{49} + 9375000 q^{50} + 25432656 q^{52} + 25114688 q^{53} + 11603750 q^{55} - 49326144 q^{56} + 134182296 q^{58} + 116159208 q^{59} - 44688544 q^{61} - 12388000 q^{62} + 79055872 q^{64} + 31931250 q^{65} + 118092496 q^{67} - 140439024 q^{68} - 36015000 q^{70} + 294165824 q^{71} - 57419332 q^{73} - 102418064 q^{74} + 39622368 q^{76} - 44576966 q^{77} - 692852854 q^{79} - 70480000 q^{80} + 152932128 q^{82} + 540679928 q^{83} - 233693750 q^{85} + 346989008 q^{86} + 105455296 q^{88} + 779043704 q^{89} - 122667090 q^{91} - 495040608 q^{92} - 937691032 q^{94} + 89547500 q^{95} - 2673039406 q^{97} + 138355224 q^{98}+O(q^{100})$$ 2 * q + 24 * q^2 - 720 * q^4 - 1250 * q^5 + 4802 * q^7 - 20544 * q^8 - 15000 * q^10 - 18566 * q^11 - 51090 * q^13 + 57624 * q^14 + 112768 * q^16 + 373910 * q^17 - 143276 * q^19 + 450000 * q^20 - 76808 * q^22 + 498908 * q^23 + 781250 * q^25 - 319736 * q^26 - 1728720 * q^28 + 11577554 * q^29 - 3953760 * q^31 + 11398656 * q^32 + 4243944 * q^34 - 3001250 * q^35 - 3205412 * q^37 - 2217520 * q^38 + 12840000 * q^40 - 1058992 * q^41 + 15948180 * q^43 + 10187376 * q^44 - 7156176 * q^46 - 65501290 * q^47 + 11529602 * q^49 + 9375000 * q^50 + 25432656 * q^52 + 25114688 * q^53 + 11603750 * q^55 - 49326144 * q^56 + 134182296 * q^58 + 116159208 * q^59 - 44688544 * q^61 - 12388000 * q^62 + 79055872 * q^64 + 31931250 * q^65 + 118092496 * q^67 - 140439024 * q^68 - 36015000 * q^70 + 294165824 * q^71 - 57419332 * q^73 - 102418064 * q^74 + 39622368 * q^76 - 44576966 * q^77 - 692852854 * q^79 - 70480000 * q^80 + 152932128 * q^82 + 540679928 * q^83 - 233693750 * q^85 + 346989008 * q^86 + 105455296 * q^88 + 779043704 * q^89 - 122667090 * q^91 - 495040608 * q^92 - 937691032 * q^94 + 89547500 * q^95 - 2673039406 * q^97 + 138355224 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 9.17157 0.405330 0.202665 0.979248i $$-0.435040\pi$$
0.202665 + 0.979248i $$0.435040\pi$$
$$3$$ 0 0
$$4$$ −427.882 −0.835708
$$5$$ −625.000 −0.447214
$$6$$ 0 0
$$7$$ 2401.00 0.377964
$$8$$ −8620.20 −0.744067
$$9$$ 0 0
$$10$$ −5732.23 −0.181269
$$11$$ −35089.6 −0.722622 −0.361311 0.932445i $$-0.617671\pi$$
−0.361311 + 0.932445i $$0.617671\pi$$
$$12$$ 0 0
$$13$$ −77401.4 −0.751629 −0.375815 0.926695i $$-0.622637\pi$$
−0.375815 + 0.926695i $$0.622637\pi$$
$$14$$ 22020.9 0.153200
$$15$$ 0 0
$$16$$ 140015. 0.534115
$$17$$ 229907. 0.667626 0.333813 0.942639i $$-0.391665\pi$$
0.333813 + 0.942639i $$0.391665\pi$$
$$18$$ 0 0
$$19$$ 16433.6 0.0289295 0.0144647 0.999895i $$-0.495396\pi$$
0.0144647 + 0.999895i $$0.495396\pi$$
$$20$$ 267426. 0.373740
$$21$$ 0 0
$$22$$ −321827. −0.292900
$$23$$ 2.57284e6 1.91707 0.958535 0.284975i $$-0.0919852\pi$$
0.958535 + 0.284975i $$0.0919852\pi$$
$$24$$ 0 0
$$25$$ 390625. 0.200000
$$26$$ −709892. −0.304658
$$27$$ 0 0
$$28$$ −1.02735e6 −0.315868
$$29$$ 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 $$0.792451\pi$$
$$32$$ 5.69770e6 0.960560
$$33$$ 0 0
$$34$$ 2.10861e6 0.270609
$$35$$ −1.50062e6 −0.169031
$$36$$ 0 0
$$37$$ 9.70272e6 0.851110 0.425555 0.904933i $$-0.360079\pi$$
0.425555 + 0.904933i $$0.360079\pi$$
$$38$$ 150722. 0.0117260
$$39$$ 0 0
$$40$$ 5.38762e6 0.332757
$$41$$ −2.98108e7 −1.64758 −0.823789 0.566896i $$-0.808144\pi$$
−0.823789 + 0.566896i $$0.808144\pi$$
$$42$$ 0 0
$$43$$ −1.95343e7 −0.871343 −0.435672 0.900106i $$-0.643489\pi$$
−0.435672 + 0.900106i $$0.643489\pi$$
$$44$$ 1.50142e7 0.603900
$$45$$ 0 0
$$46$$ 2.35970e7 0.777046
$$47$$ −5.93794e6 −0.177499 −0.0887494 0.996054i $$-0.528287\pi$$
−0.0887494 + 0.996054i $$0.528287\pi$$
$$48$$ 0 0
$$49$$ 5.76480e6 0.142857
$$50$$ 3.58265e6 0.0810660
$$51$$ 0 0
$$52$$ 3.31187e7 0.628142
$$53$$ 2.74263e7 0.477448 0.238724 0.971088i $$-0.423271\pi$$
0.238724 + 0.971088i $$0.423271\pi$$
$$54$$ 0 0
$$55$$ 2.19310e7 0.323166
$$56$$ −2.06971e7 −0.281231
$$57$$ 0 0
$$58$$ 6.07908e7 0.705362
$$59$$ −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 $$0.467080\pi$$
$$62$$ −7.49699e7 −0.644354
$$63$$ 0 0
$$64$$ −1.94308e7 −0.144771
$$65$$ 4.83759e7 0.336139
$$66$$ 0 0
$$67$$ 2.74351e8 1.66330 0.831649 0.555302i $$-0.187397\pi$$
0.831649 + 0.555302i $$0.187397\pi$$
$$68$$ −9.83733e7 −0.557940
$$69$$ 0 0
$$70$$ −1.37631e7 −0.0685133
$$71$$ 3.63673e8 1.69843 0.849216 0.528046i $$-0.177075\pi$$
0.849216 + 0.528046i $$0.177075\pi$$
$$72$$ 0 0
$$73$$ 2.09245e7 0.0862387 0.0431193 0.999070i $$-0.486270\pi$$
0.0431193 + 0.999070i $$0.486270\pi$$
$$74$$ 8.89892e7 0.344980
$$75$$ 0 0
$$76$$ −7.03163e6 −0.0241766
$$77$$ −8.42501e7 −0.273125
$$78$$ 0 0
$$79$$ −2.65896e8 −0.768051 −0.384025 0.923323i $$-0.625462\pi$$
−0.384025 + 0.923323i $$0.625462\pi$$
$$80$$ −8.75093e7 −0.238863
$$81$$ 0 0
$$82$$ −2.73412e8 −0.667813
$$83$$ 9.43764e6 0.0218279 0.0109140 0.999940i $$-0.496526\pi$$
0.0109140 + 0.999940i $$0.496526\pi$$
$$84$$ 0 0
$$85$$ −1.43692e8 −0.298571
$$86$$ −1.79160e8 −0.353182
$$87$$ 0 0
$$88$$ 3.02479e8 0.537679
$$89$$ 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
$$92$$ −1.10087e9 −1.60211
$$93$$ 0 0
$$94$$ −5.44603e7 −0.0719456
$$95$$ −1.02710e7 −0.0129377
$$96$$ 0 0
$$97$$ −1.20731e9 −1.38467 −0.692336 0.721575i $$-0.743418\pi$$
−0.692336 + 0.721575i $$0.743418\pi$$
$$98$$ 5.28723e7 0.0579043
$$99$$ 0 0
$$100$$ −1.67142e8 −0.167142
$$101$$ −1.18204e9 −1.13028 −0.565139 0.824996i $$-0.691177\pi$$
−0.565139 + 0.824996i $$0.691177\pi$$
$$102$$ 0 0
$$103$$ 1.97811e9 1.73174 0.865870 0.500268i $$-0.166765\pi$$
0.865870 + 0.500268i $$0.166765\pi$$
$$104$$ 6.67215e8 0.559263
$$105$$ 0 0
$$106$$ 2.51542e8 0.193524
$$107$$ −1.67828e8 −0.123776 −0.0618881 0.998083i $$-0.519712\pi$$
−0.0618881 + 0.998083i $$0.519712\pi$$
$$108$$ 0 0
$$109$$ −1.02540e9 −0.695784 −0.347892 0.937535i $$-0.613102\pi$$
−0.347892 + 0.937535i $$0.613102\pi$$
$$110$$ 2.01142e8 0.130989
$$111$$ 0 0
$$112$$ 3.36176e8 0.201876
$$113$$ −1.27533e9 −0.735814 −0.367907 0.929863i $$-0.619926\pi$$
−0.367907 + 0.929863i $$0.619926\pi$$
$$114$$ 0 0
$$115$$ −1.60803e9 −0.857340
$$116$$ −2.83608e9 −1.45431
$$117$$ 0 0
$$118$$ −4.81430e8 −0.228594
$$119$$ 5.52008e8 0.252339
$$120$$ 0 0
$$121$$ −1.12667e9 −0.477818
$$122$$ 2.04785e8 0.0836909
$$123$$ 0 0
$$124$$ 3.49758e9 1.32853
$$125$$ −2.44141e8 −0.0894427
$$126$$ 0 0
$$127$$ −2.90339e9 −0.990349 −0.495174 0.868794i $$-0.664896\pi$$
−0.495174 + 0.868794i $$0.664896\pi$$
$$128$$ −3.09543e9 −1.01924
$$129$$ 0 0
$$130$$ 4.43683e8 0.136247
$$131$$ −2.05173e9 −0.608694 −0.304347 0.952561i $$-0.598438\pi$$
−0.304347 + 0.952561i $$0.598438\pi$$
$$132$$ 0 0
$$133$$ 3.94570e7 0.0109343
$$134$$ 2.51623e9 0.674185
$$135$$ 0 0
$$136$$ −1.98185e9 −0.496759
$$137$$ −3.25539e9 −0.789514 −0.394757 0.918786i $$-0.629171\pi$$
−0.394757 + 0.918786i $$0.629171\pi$$
$$138$$ 0 0
$$139$$ −8.26776e9 −1.87854 −0.939272 0.343173i $$-0.888498\pi$$
−0.939272 + 0.343173i $$0.888498\pi$$
$$140$$ 6.42091e8 0.141260
$$141$$ 0 0
$$142$$ 3.33545e9 0.688425
$$143$$ 2.71598e9 0.543143
$$144$$ 0 0
$$145$$ −4.14261e9 −0.778248
$$146$$ 1.91910e8 0.0349551
$$147$$ 0 0
$$148$$ −4.15162e9 −0.711279
$$149$$ −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 $$0.450649\pi$$
$$152$$ −1.41661e8 −0.0215255
$$153$$ 0 0
$$154$$ −7.72706e8 −0.110706
$$155$$ 5.10885e9 0.710936
$$156$$ 0 0
$$157$$ −4.61623e9 −0.606372 −0.303186 0.952931i $$-0.598050\pi$$
−0.303186 + 0.952931i $$0.598050\pi$$
$$158$$ −2.43868e9 −0.311314
$$159$$ 0 0
$$160$$ −3.56106e9 −0.429576
$$161$$ 6.17740e9 0.724584
$$162$$ 0 0
$$163$$ 6.26525e9 0.695175 0.347588 0.937648i $$-0.387001\pi$$
0.347588 + 0.937648i $$0.387001\pi$$
$$164$$ 1.27555e10 1.37689
$$165$$ 0 0
$$166$$ 8.65580e7 0.00884751
$$167$$ 6.21672e9 0.618496 0.309248 0.950981i $$-0.399923\pi$$
0.309248 + 0.950981i $$0.399923\pi$$
$$168$$ 0 0
$$169$$ −4.61353e9 −0.435054
$$170$$ −1.31788e9 −0.121020
$$171$$ 0 0
$$172$$ 8.35837e9 0.728188
$$173$$ −8.97209e9 −0.761528 −0.380764 0.924672i $$-0.624339\pi$$
−0.380764 + 0.924672i $$0.624339\pi$$
$$174$$ 0 0
$$175$$ 9.37891e8 0.0755929
$$176$$ −4.91306e9 −0.385963
$$177$$ 0 0
$$178$$ 6.09796e9 0.455297
$$179$$ −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 $$0.482107\pi$$
$$182$$ −1.70445e9 −0.115150
$$183$$ 0 0
$$184$$ −2.21784e10 −1.42643
$$185$$ −6.06420e9 −0.380628
$$186$$ 0 0
$$187$$ −8.06735e9 −0.482441
$$188$$ 2.54074e9 0.148337
$$189$$ 0 0
$$190$$ −9.42010e7 −0.00524402
$$191$$ −1.66601e10 −0.905788 −0.452894 0.891564i $$-0.649608\pi$$
−0.452894 + 0.891564i $$0.649608\pi$$
$$192$$ 0 0
$$193$$ 2.41341e10 1.25206 0.626028 0.779801i $$-0.284680\pi$$
0.626028 + 0.779801i $$0.284680\pi$$
$$194$$ −1.10730e10 −0.561249
$$195$$ 0 0
$$196$$ −2.46666e9 −0.119387
$$197$$ 3.89843e9 0.184413 0.0922066 0.995740i $$-0.470608\pi$$
0.0922066 + 0.995740i $$0.470608\pi$$
$$198$$ 0 0
$$199$$ −1.87489e10 −0.847493 −0.423747 0.905781i $$-0.639285\pi$$
−0.423747 + 0.905781i $$0.639285\pi$$
$$200$$ −3.36727e9 −0.148813
$$201$$ 0 0
$$202$$ −1.08411e10 −0.458135
$$203$$ 1.59142e10 0.657740
$$204$$ 0 0
$$205$$ 1.86317e10 0.736820
$$206$$ 1.81424e10 0.701927
$$207$$ 0 0
$$208$$ −1.08373e10 −0.401456
$$209$$ −5.76647e8 −0.0209051
$$210$$ 0 0
$$211$$ −2.20489e9 −0.0765801 −0.0382900 0.999267i $$-0.512191\pi$$
−0.0382900 + 0.999267i $$0.512191\pi$$
$$212$$ −1.17352e10 −0.399007
$$213$$ 0 0
$$214$$ −1.53925e9 −0.0501702
$$215$$ 1.22089e10 0.389677
$$216$$ 0 0
$$217$$ −1.96262e10 −0.600851
$$218$$ −9.40454e9 −0.282022
$$219$$ 0 0
$$220$$ −9.38388e9 −0.270072
$$221$$ −1.77952e10 −0.501807
$$222$$ 0 0
$$223$$ −2.65324e10 −0.718463 −0.359231 0.933249i $$-0.616961\pi$$
−0.359231 + 0.933249i $$0.616961\pi$$
$$224$$ 1.36802e10 0.363058
$$225$$ 0 0
$$226$$ −1.16967e10 −0.298248
$$227$$ −7.78091e10 −1.94498 −0.972488 0.232955i $$-0.925161\pi$$
−0.972488 + 0.232955i $$0.925161\pi$$
$$228$$ 0 0
$$229$$ 4.84637e10 1.16455 0.582274 0.812993i $$-0.302163\pi$$
0.582274 + 0.812993i $$0.302163\pi$$
$$230$$ −1.47481e10 −0.347506
$$231$$ 0 0
$$232$$ −5.71362e10 −1.29484
$$233$$ 2.38429e10 0.529978 0.264989 0.964251i $$-0.414632\pi$$
0.264989 + 0.964251i $$0.414632\pi$$
$$234$$ 0 0
$$235$$ 3.71121e9 0.0793799
$$236$$ 2.24602e10 0.471313
$$237$$ 0 0
$$238$$ 5.06278e9 0.102280
$$239$$ −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 $$0.775078\pi$$
$$242$$ −1.03333e10 −0.193674
$$243$$ 0 0
$$244$$ −9.55384e9 −0.172554
$$245$$ −3.60300e9 −0.0638877
$$246$$ 0 0
$$247$$ −1.27198e9 −0.0217442
$$248$$ 7.04629e10 1.18285
$$249$$ 0 0
$$250$$ −2.23915e9 −0.0362538
$$251$$ 5.44549e10 0.865975 0.432988 0.901400i $$-0.357459\pi$$
0.432988 + 0.901400i $$0.357459\pi$$
$$252$$ 0 0
$$253$$ −9.02799e10 −1.38532
$$254$$ −2.66286e10 −0.401418
$$255$$ 0 0
$$256$$ −1.84414e10 −0.268358
$$257$$ 5.35278e10 0.765385 0.382693 0.923876i $$-0.374997\pi$$
0.382693 + 0.923876i $$0.374997\pi$$
$$258$$ 0 0
$$259$$ 2.32962e10 0.321689
$$260$$ −2.06992e10 −0.280914
$$261$$ 0 0
$$262$$ −1.88176e10 −0.246722
$$263$$ 5.81425e10 0.749364 0.374682 0.927153i $$-0.377752\pi$$
0.374682 + 0.927153i $$0.377752\pi$$
$$264$$ 0 0
$$265$$ −1.71414e10 −0.213521
$$266$$ 3.61883e8 0.00443201
$$267$$ 0 0
$$268$$ −1.17390e11 −1.39003
$$269$$ −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 $$0.451750\pi$$
$$272$$ 3.21905e10 0.356589
$$273$$ 0 0
$$274$$ −2.98570e10 −0.320014
$$275$$ −1.37069e10 −0.144524
$$276$$ 0 0
$$277$$ 1.12549e11 1.14863 0.574316 0.818633i $$-0.305268\pi$$
0.574316 + 0.818633i $$0.305268\pi$$
$$278$$ −7.58284e10 −0.761431
$$279$$ 0 0
$$280$$ 1.29357e10 0.125770
$$281$$ 4.60761e10 0.440857 0.220428 0.975403i $$-0.429254\pi$$
0.220428 + 0.975403i $$0.429254\pi$$
$$282$$ 0 0
$$283$$ 7.94071e10 0.735902 0.367951 0.929845i $$-0.380059\pi$$
0.367951 + 0.929845i $$0.380059\pi$$
$$284$$ −1.55609e11 −1.41939
$$285$$ 0 0
$$286$$ 2.49098e10 0.220152
$$287$$ −7.15757e10 −0.622726
$$288$$ 0 0
$$289$$ −6.57304e10 −0.554276
$$290$$ −3.79942e10 −0.315447
$$291$$ 0 0
$$292$$ −8.95322e9 −0.0720703
$$293$$ 1.41265e11 1.11977 0.559887 0.828569i $$-0.310844\pi$$
0.559887 + 0.828569i $$0.310844\pi$$
$$294$$ 0 0
$$295$$ 3.28072e10 0.252215
$$296$$ −8.36393e10 −0.633283
$$297$$ 0 0
$$298$$ −9.82523e9 −0.0721721
$$299$$ −1.99142e11 −1.44093
$$300$$ 0 0
$$301$$ −4.69018e10 −0.329337
$$302$$ 1.80958e10 0.125184
$$303$$ 0 0
$$304$$ 2.30094e9 0.0154517
$$305$$ −1.39551e10 −0.0923389
$$306$$ 0 0
$$307$$ −5.58349e10 −0.358742 −0.179371 0.983781i $$-0.557406\pi$$
−0.179371 + 0.983781i $$0.557406\pi$$
$$308$$ 3.60491e10 0.228253
$$309$$ 0 0
$$310$$ 4.68562e10 0.288164
$$311$$ −5.26501e10 −0.319137 −0.159569 0.987187i $$-0.551010\pi$$
−0.159569 + 0.987187i $$0.551010\pi$$
$$312$$ 0 0
$$313$$ −2.51256e11 −1.47968 −0.739838 0.672785i $$-0.765098\pi$$
−0.739838 + 0.672785i $$0.765098\pi$$
$$314$$ −4.23381e10 −0.245781
$$315$$ 0 0
$$316$$ 1.13772e11 0.641866
$$317$$ 1.16999e11 0.650749 0.325375 0.945585i $$-0.394510\pi$$
0.325375 + 0.945585i $$0.394510\pi$$
$$318$$ 0 0
$$319$$ −2.32580e11 −1.25752
$$320$$ 1.21442e10 0.0647434
$$321$$ 0 0
$$322$$ 5.66564e10 0.293696
$$323$$ 3.77820e9 0.0193141
$$324$$ 0 0
$$325$$ −3.02349e10 −0.150326
$$326$$ 5.74622e10 0.281775
$$327$$ 0 0
$$328$$ 2.56975e11 1.22591
$$329$$ −1.42570e10 −0.0670883
$$330$$ 0 0
$$331$$ −2.51419e11 −1.15126 −0.575629 0.817711i $$-0.695243\pi$$
−0.575629 + 0.817711i $$0.695243\pi$$
$$332$$ −4.03820e9 −0.0182417
$$333$$ 0 0
$$334$$ 5.70171e10 0.250695
$$335$$ −1.71469e11 −0.743849
$$336$$ 0 0
$$337$$ 6.11427e10 0.258232 0.129116 0.991630i $$-0.458786\pi$$
0.129116 + 0.991630i $$0.458786\pi$$
$$338$$ −4.23133e10 −0.176340
$$339$$ 0 0
$$340$$ 6.14833e10 0.249518
$$341$$ 2.86828e11 1.14875
$$342$$ 0 0
$$343$$ 1.38413e10 0.0539949
$$344$$ 1.68389e11 0.648338
$$345$$ 0 0
$$346$$ −8.22882e10 −0.308670
$$347$$ 1.68668e11 0.624524 0.312262 0.949996i $$-0.398913\pi$$
0.312262 + 0.949996i $$0.398913\pi$$
$$348$$ 0 0
$$349$$ −3.31182e11 −1.19496 −0.597479 0.801885i $$-0.703831\pi$$
−0.597479 + 0.801885i $$0.703831\pi$$
$$350$$ 8.60193e9 0.0306401
$$351$$ 0 0
$$352$$ −1.99930e11 −0.694122
$$353$$ −3.78560e11 −1.29762 −0.648811 0.760949i $$-0.724734\pi$$
−0.648811 + 0.760949i $$0.724734\pi$$
$$354$$ 0 0
$$355$$ −2.27295e11 −0.759562
$$356$$ −2.84489e11 −0.938728
$$357$$ 0 0
$$358$$ −1.61641e11 −0.520091
$$359$$ 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
$$362$$ 1.48809e10 0.0455449
$$363$$ 0 0
$$364$$ 7.95179e10 0.237415
$$365$$ −1.30778e10 −0.0385671
$$366$$ 0 0
$$367$$ 5.13837e11 1.47852 0.739261 0.673419i $$-0.235175\pi$$
0.739261 + 0.673419i $$0.235175\pi$$
$$368$$ 3.60236e11 1.02394
$$369$$ 0 0
$$370$$ −5.56182e10 −0.154280
$$371$$ 6.58505e10 0.180458
$$372$$ 0 0
$$373$$ −6.70900e10 −0.179460 −0.0897301 0.995966i $$-0.528600\pi$$
−0.0897301 + 0.995966i $$0.528600\pi$$
$$374$$ −7.39903e10 −0.195548
$$375$$ 0 0
$$376$$ 5.11862e10 0.132071
$$377$$ −5.13030e11 −1.30800
$$378$$ 0 0
$$379$$ 4.15471e11 1.03434 0.517171 0.855882i $$-0.326985\pi$$
0.517171 + 0.855882i $$0.326985\pi$$
$$380$$ 4.39477e9 0.0108121
$$381$$ 0 0
$$382$$ −1.52799e11 −0.367143
$$383$$ 3.51976e11 0.835831 0.417915 0.908486i $$-0.362761\pi$$
0.417915 + 0.908486i $$0.362761\pi$$
$$384$$ 0 0
$$385$$ 5.26563e10 0.122145
$$386$$ 2.21348e11 0.507496
$$387$$ 0 0
$$388$$ 5.16588e11 1.15718
$$389$$ 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
$$392$$ −4.96937e10 −0.106295
$$393$$ 0 0
$$394$$ 3.57548e10 0.0747482
$$395$$ 1.66185e11 0.343483
$$396$$ 0 0
$$397$$ 7.34338e11 1.48367 0.741837 0.670580i $$-0.233955\pi$$
0.741837 + 0.670580i $$0.233955\pi$$
$$398$$ −1.71957e11 −0.343514
$$399$$ 0 0
$$400$$ 5.46933e10 0.106823
$$401$$ −8.08296e11 −1.56106 −0.780532 0.625116i $$-0.785052\pi$$
−0.780532 + 0.625116i $$0.785052\pi$$
$$402$$ 0 0
$$403$$ 6.32691e11 1.19487
$$404$$ 5.05773e11 0.944581
$$405$$ 0 0
$$406$$ 1.45959e11 0.266602
$$407$$ −3.40464e11 −0.615030
$$408$$ 0 0
$$409$$ −9.11153e11 −1.61004 −0.805020 0.593248i $$-0.797845\pi$$
−0.805020 + 0.593248i $$0.797845\pi$$
$$410$$ 1.70882e11 0.298655
$$411$$ 0 0
$$412$$ −8.46398e11 −1.44723
$$413$$ −1.26032e11 −0.213160
$$414$$ 0 0
$$415$$ −5.89853e9 −0.00976174
$$416$$ −4.41010e11 −0.721985
$$417$$ 0 0
$$418$$ −5.28876e9 −0.00847345
$$419$$ −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 $$0.502843\pi$$
$$422$$ −2.02223e10 −0.0310402
$$423$$ 0 0
$$424$$ −2.36420e11 −0.355253
$$425$$ 8.98076e10 0.133525
$$426$$ 0 0
$$427$$ 5.36100e10 0.0780406
$$428$$ 7.18106e10 0.103441
$$429$$ 0 0
$$430$$ 1.11975e11 0.157948
$$431$$ 9.42534e11 1.31568 0.657839 0.753159i $$-0.271471\pi$$
0.657839 + 0.753159i $$0.271471\pi$$
$$432$$ 0 0
$$433$$ 1.01849e12 1.39239 0.696196 0.717852i $$-0.254875\pi$$
0.696196 + 0.717852i $$0.254875\pi$$
$$434$$ −1.80003e11 −0.243543
$$435$$ 0 0
$$436$$ 4.38751e11 0.581472
$$437$$ 4.22810e10 0.0554598
$$438$$ 0 0
$$439$$ −7.89357e11 −1.01434 −0.507169 0.861847i $$-0.669308\pi$$
−0.507169 + 0.861847i $$0.669308\pi$$
$$440$$ −1.89049e11 −0.240457
$$441$$ 0 0
$$442$$ −1.63210e11 −0.203397
$$443$$ −1.06770e12 −1.31714 −0.658572 0.752518i $$-0.728839\pi$$
−0.658572 + 0.752518i $$0.728839\pi$$
$$444$$ 0 0
$$445$$ −4.15547e11 −0.502343
$$446$$ −2.43344e11 −0.291215
$$447$$ 0 0
$$448$$ −4.66533e10 −0.0547182
$$449$$ 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
$$452$$ 5.45689e11 0.614926
$$453$$ 0 0
$$454$$ −7.13632e11 −0.788357
$$455$$ 1.16150e11 0.127049
$$456$$ 0 0
$$457$$ −7.03146e11 −0.754089 −0.377045 0.926195i $$-0.623060\pi$$
−0.377045 + 0.926195i $$0.623060\pi$$
$$458$$ 4.44489e11 0.472026
$$459$$ 0 0
$$460$$ 6.88046e11 0.716485
$$461$$ 1.86192e12 1.92003 0.960015 0.279950i $$-0.0903178\pi$$
0.960015 + 0.279950i $$0.0903178\pi$$
$$462$$ 0 0
$$463$$ −1.06950e11 −0.108160 −0.0540799 0.998537i $$-0.517223\pi$$
−0.0540799 + 0.998537i $$0.517223\pi$$
$$464$$ 9.28043e11 0.929474
$$465$$ 0 0
$$466$$ 2.18677e11 0.214816
$$467$$ −4.13997e11 −0.402783 −0.201392 0.979511i $$-0.564546\pi$$
−0.201392 + 0.979511i $$0.564546\pi$$
$$468$$ 0 0
$$469$$ 6.58717e11 0.628667
$$470$$ 3.40377e10 0.0321751
$$471$$ 0 0
$$472$$ 4.52487e11 0.419631
$$473$$ 6.85450e11 0.629652
$$474$$ 0 0
$$475$$ 6.41936e9 0.00578589
$$476$$ −2.36194e11 −0.210881
$$477$$ 0 0
$$478$$ −5.74044e11 −0.502944
$$479$$ 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
$$482$$ −7.30612e11 −0.616560
$$483$$ 0 0
$$484$$ 4.82082e11 0.399316
$$485$$ 7.54571e11 0.619244
$$486$$ 0 0
$$487$$ 4.94789e11 0.398602 0.199301 0.979938i $$-0.436133\pi$$
0.199301 + 0.979938i $$0.436133\pi$$
$$488$$ −1.92474e11 −0.153632
$$489$$ 0 0
$$490$$ −3.30452e10 −0.0258956
$$491$$ −1.11163e12 −0.863168 −0.431584 0.902073i $$-0.642045\pi$$
−0.431584 + 0.902073i $$0.642045\pi$$
$$492$$ 0 0
$$493$$ 1.52387e12 1.16181
$$494$$ −1.16661e10 −0.00881359
$$495$$ 0 0
$$496$$ −1.14450e12 −0.849083
$$497$$ 8.73178e11 0.641947
$$498$$ 0 0
$$499$$ 8.18377e11 0.590882 0.295441 0.955361i $$-0.404533\pi$$
0.295441 + 0.955361i $$0.404533\pi$$
$$500$$ 1.04463e11 0.0747480
$$501$$ 0 0
$$502$$ 4.99437e11 0.351006
$$503$$ −3.13384e11 −0.218284 −0.109142 0.994026i $$-0.534810\pi$$
−0.109142 + 0.994026i $$0.534810\pi$$
$$504$$ 0 0
$$505$$ 7.38773e11 0.505475
$$506$$ −8.28009e11 −0.561510
$$507$$ 0 0
$$508$$ 1.24231e12 0.827642
$$509$$ 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
$$512$$ 1.41572e12 0.910467
$$513$$ 0 0
$$514$$ 4.90934e11 0.310234
$$515$$ −1.23632e12 −0.774458
$$516$$ 0 0
$$517$$ 2.08360e11 0.128265
$$518$$ 2.13663e11 0.130390
$$519$$ 0 0
$$520$$ −4.17010e11 −0.250110
$$521$$ 3.18952e12 1.89651 0.948255 0.317510i $$-0.102847\pi$$
0.948255 + 0.317510i $$0.102847\pi$$
$$522$$ 0 0
$$523$$ 9.43708e11 0.551544 0.275772 0.961223i $$-0.411067\pi$$
0.275772 + 0.961223i $$0.411067\pi$$
$$524$$ 8.77898e11 0.508690
$$525$$ 0 0
$$526$$ 5.33258e11 0.303740
$$527$$ −1.87930e12 −1.06133
$$528$$ 0 0
$$529$$ 4.81837e12 2.67516
$$530$$ −1.57214e11 −0.0865465
$$531$$ 0 0
$$532$$ −1.68829e10 −0.00913789
$$533$$ 2.30740e12 1.23837
$$534$$ 0 0
$$535$$ 1.04892e11 0.0553544
$$536$$ −2.36496e12 −1.23761
$$537$$ 0 0
$$538$$ −4.28661e11 −0.220594
$$539$$ −2.02284e11 −0.103232
$$540$$ 0 0
$$541$$ −8.78618e11 −0.440973 −0.220487 0.975390i $$-0.570765\pi$$
−0.220487 + 0.975390i $$0.570765\pi$$
$$542$$ 2.45933e11 0.122411
$$543$$ 0 0
$$544$$ 1.30994e12 0.641295
$$545$$ 6.40875e11 0.311164
$$546$$ 0 0
$$547$$ 4.56216e11 0.217885 0.108943 0.994048i $$-0.465254\pi$$
0.108943 + 0.994048i $$0.465254\pi$$
$$548$$ 1.39292e12 0.659803
$$549$$ 0 0
$$550$$ −1.25713e11 −0.0585801
$$551$$ 1.08925e11 0.0503435
$$552$$ 0 0
$$553$$ −6.38416e11 −0.290296
$$554$$ 1.03225e12 0.465575
$$555$$ 0 0
$$556$$ 3.53763e12 1.56991
$$557$$ −1.63980e12 −0.721842 −0.360921 0.932596i $$-0.617538\pi$$
−0.360921 + 0.932596i $$0.617538\pi$$
$$558$$ 0 0
$$559$$ 1.51198e12 0.654927
$$560$$ −2.10110e11 −0.0902818
$$561$$ 0 0
$$562$$ 4.22590e11 0.178692
$$563$$ −4.36151e11 −0.182957 −0.0914786 0.995807i $$-0.529159\pi$$
−0.0914786 + 0.995807i $$0.529159\pi$$
$$564$$ 0 0
$$565$$ 7.97079e11 0.329066
$$566$$ 7.28288e11 0.298283
$$567$$ 0 0
$$568$$ −3.13493e12 −1.26375
$$569$$ −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 $$0.485131\pi$$
$$572$$ −1.16212e12 −0.453909
$$573$$ 0 0
$$574$$ −6.56462e11 −0.252410
$$575$$ 1.00502e12 0.383414
$$576$$ 0 0
$$577$$ −3.72080e12 −1.39748 −0.698739 0.715377i $$-0.746255\pi$$
−0.698739 + 0.715377i $$0.746255\pi$$
$$578$$ −6.02851e11 −0.224665
$$579$$ 0 0
$$580$$ 1.77255e12 0.650388
$$581$$ 2.26598e10 0.00825017
$$582$$ 0 0
$$583$$ −9.62377e11 −0.345014
$$584$$ −1.80373e11 −0.0641674
$$585$$ 0 0
$$586$$ 1.29562e12 0.453878
$$587$$ 6.46176e11 0.224636 0.112318 0.993672i $$-0.464172\pi$$
0.112318 + 0.993672i $$0.464172\pi$$
$$588$$ 0 0
$$589$$ −1.34331e11 −0.0459892
$$590$$ 3.00894e11 0.102230
$$591$$ 0 0
$$592$$ 1.35853e12 0.454590
$$593$$ 5.05774e12 1.67962 0.839808 0.542883i $$-0.182667\pi$$
0.839808 + 0.542883i $$0.182667\pi$$
$$594$$ 0 0
$$595$$ −3.45005e11 −0.112849
$$596$$ 4.58377e11 0.148804
$$597$$ 0 0
$$598$$ −1.82644e12 −0.584051
$$599$$ 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 $$0.949976\pi$$
$$602$$ −4.30163e11 −0.133490
$$603$$ 0 0
$$604$$ −8.44227e11 −0.258103
$$605$$ 7.04169e11 0.213687
$$606$$ 0 0
$$607$$ −1.45276e12 −0.434356 −0.217178 0.976132i $$-0.569685\pi$$
−0.217178 + 0.976132i $$0.569685\pi$$
$$608$$ 9.36335e10 0.0277885
$$609$$ 0 0
$$610$$ −1.27990e11 −0.0374277
$$611$$ 4.59605e11 0.133413
$$612$$ 0 0
$$613$$ 1.20124e12 0.343603 0.171802 0.985132i $$-0.445041\pi$$
0.171802 + 0.985132i $$0.445041\pi$$
$$614$$ −5.12094e11 −0.145409
$$615$$ 0 0
$$616$$ 7.26252e11 0.203224
$$617$$ −3.13545e12 −0.870997 −0.435498 0.900189i $$-0.643428\pi$$
−0.435498 + 0.900189i $$0.643428\pi$$
$$618$$ 0 0
$$619$$ −5.17127e12 −1.41576 −0.707880 0.706333i $$-0.750348\pi$$
−0.707880 + 0.706333i $$0.750348\pi$$
$$620$$ −2.18599e12 −0.594135
$$621$$ 0 0
$$622$$ −4.82884e11 −0.129356
$$623$$ 1.59637e12 0.424557
$$624$$ 0 0
$$625$$ 1.52588e11 0.0400000
$$626$$ −2.30441e12 −0.599757
$$627$$ 0 0
$$628$$ 1.97520e12 0.506749
$$629$$ 2.23073e12 0.568223
$$630$$ 0 0
$$631$$ −6.37331e12 −1.60042 −0.800208 0.599723i $$-0.795278\pi$$
−0.800208 + 0.599723i $$0.795278\pi$$
$$632$$ 2.29208e12 0.571481
$$633$$ 0 0
$$634$$ 1.07306e12 0.263768
$$635$$ 1.81462e12 0.442897
$$636$$ 0 0
$$637$$ −4.46204e11 −0.107376
$$638$$ −2.13312e12 −0.509710
$$639$$ 0 0
$$640$$ 1.93465e12 0.455818
$$641$$ 5.74174e12 1.34333 0.671665 0.740855i $$-0.265579\pi$$
0.671665 + 0.740855i $$0.265579\pi$$
$$642$$ 0 0
$$643$$ −5.85135e11 −0.134992 −0.0674958 0.997720i $$-0.521501\pi$$
−0.0674958 + 0.997720i $$0.521501\pi$$
$$644$$ −2.64320e12 −0.605541
$$645$$ 0 0
$$646$$ 3.46520e10 0.00782857
$$647$$ 1.80915e12 0.405887 0.202943 0.979190i $$-0.434949\pi$$
0.202943 + 0.979190i $$0.434949\pi$$
$$648$$ 0 0
$$649$$ 1.84191e12 0.407536
$$650$$ −2.77302e11 −0.0609316
$$651$$ 0 0
$$652$$ −2.68079e12 −0.580963
$$653$$ −2.43900e12 −0.524932 −0.262466 0.964941i $$-0.584536\pi$$
−0.262466 + 0.964941i $$0.584536\pi$$
$$654$$ 0 0
$$655$$ 1.28233e12 0.272216
$$656$$ −4.17396e12 −0.879996
$$657$$ 0 0
$$658$$ −1.30759e11 −0.0271929
$$659$$ −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 $$0.723876\pi$$
$$662$$ −2.30591e12 −0.466640
$$663$$ 0 0
$$664$$ −8.13544e10 −0.0162414
$$665$$ −2.46606e10 −0.00488997
$$666$$ 0 0
$$667$$ 1.70533e13 3.33611
$$668$$ −2.66002e12 −0.516882
$$669$$ 0 0
$$670$$ −1.57264e12 −0.301505
$$671$$ −7.83487e11 −0.149204
$$672$$ 0 0
$$673$$ 3.17186e12 0.596000 0.298000 0.954566i $$-0.403680\pi$$
0.298000 + 0.954566i $$0.403680\pi$$
$$674$$ 5.60774e11 0.104669
$$675$$ 0 0
$$676$$ 1.97405e12 0.363578
$$677$$ 2.32240e12 0.424901 0.212450 0.977172i $$-0.431856\pi$$
0.212450 + 0.977172i $$0.431856\pi$$
$$678$$ 0 0
$$679$$ −2.89876e12 −0.523357
$$680$$ 1.23866e12 0.222157
$$681$$ 0 0
$$682$$ 2.63066e12 0.465624
$$683$$ −3.78639e12 −0.665782 −0.332891 0.942965i $$-0.608024\pi$$
−0.332891 + 0.942965i $$0.608024\pi$$
$$684$$ 0 0
$$685$$ 2.03462e12 0.353081
$$686$$ 1.26946e11 0.0218858
$$687$$ 0 0
$$688$$ −2.73509e12 −0.465397
$$689$$ −2.12283e12 −0.358864
$$690$$ 0 0
$$691$$ −5.70532e12 −0.951982 −0.475991 0.879450i $$-0.657911\pi$$
−0.475991 + 0.879450i $$0.657911\pi$$
$$692$$ 3.83900e12 0.636415
$$693$$ 0 0
$$694$$ 1.54695e12 0.253138
$$695$$ 5.16735e12 0.840111
$$696$$ 0 0
$$697$$ −6.85372e12 −1.09997
$$698$$ −3.03746e12 −0.484352
$$699$$ 0 0
$$700$$ −4.01307e11 −0.0631736
$$701$$ −6.25160e12 −0.977823 −0.488912 0.872333i $$-0.662606\pi$$
−0.488912 + 0.872333i $$0.662606\pi$$
$$702$$ 0 0
$$703$$ 1.59450e11 0.0246222
$$704$$ 6.81818e11 0.104614
$$705$$ 0 0
$$706$$ −3.47199e12 −0.525965
$$707$$ −2.83807e12 −0.427205
$$708$$ 0 0
$$709$$ −8.01996e12 −1.19197 −0.595983 0.802997i $$-0.703238\pi$$
−0.595983 + 0.802997i $$0.703238\pi$$
$$710$$ −2.08466e12 −0.307873
$$711$$ 0 0
$$712$$ −5.73136e12 −0.835791
$$713$$ −2.10308e13 −3.04757
$$714$$ 0 0
$$715$$ −1.69749e12 −0.242901
$$716$$ 7.54107e12 1.07232
$$717$$ 0 0
$$718$$ 1.46871e12 0.206241
$$719$$ −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
$$722$$ −2.95708e12 −0.404991
$$723$$ 0 0
$$724$$ −6.94238e11 −0.0939042
$$725$$ 2.58913e12 0.348043
$$726$$ 0 0
$$727$$ −1.47778e13 −1.96203 −0.981013 0.193941i $$-0.937873\pi$$
−0.981013 + 0.193941i $$0.937873\pi$$
$$728$$ 1.60198e12 0.211381
$$729$$ 0 0
$$730$$ −1.19944e11 −0.0156324
$$731$$ −4.49108e12 −0.581731
$$732$$ 0 0
$$733$$ −6.70116e12 −0.857398 −0.428699 0.903447i $$-0.641028\pi$$
−0.428699 + 0.903447i $$0.641028\pi$$
$$734$$ 4.71269e12 0.599290
$$735$$ 0 0
$$736$$ 1.46593e13 1.84146
$$737$$ −9.62686e12 −1.20193
$$738$$ 0 0
$$739$$ −1.39054e13 −1.71508 −0.857540 0.514417i $$-0.828008\pi$$
−0.857540 + 0.514417i $$0.828008\pi$$
$$740$$ 2.59476e12 0.318094
$$741$$ 0 0
$$742$$ 6.03953e11 0.0731452
$$743$$ 3.43953e12 0.414047 0.207024 0.978336i $$-0.433622\pi$$
0.207024 + 0.978336i $$0.433622\pi$$
$$744$$ 0 0
$$745$$ 6.69544e11 0.0796298
$$746$$ −6.15321e11 −0.0727407
$$747$$ 0 0
$$748$$ 3.45188e12 0.403179
$$749$$ −4.02955e11 −0.0467830
$$750$$ 0 0
$$751$$ 1.00823e13 1.15660 0.578298 0.815826i $$-0.303717\pi$$
0.578298 + 0.815826i $$0.303717\pi$$
$$752$$ −8.31401e11 −0.0948047
$$753$$ 0 0
$$754$$ −4.70529e12 −0.530170
$$755$$ −1.23315e12 −0.138119
$$756$$ 0 0
$$757$$ −1.02703e13 −1.13672 −0.568360 0.822780i $$-0.692422\pi$$
−0.568360 + 0.822780i $$0.692422\pi$$
$$758$$ 3.81052e12 0.419250
$$759$$ 0 0
$$760$$ 8.85379e10 0.00962649
$$761$$ 3.20840e12 0.346783 0.173391 0.984853i $$-0.444527\pi$$
0.173391 + 0.984853i $$0.444527\pi$$
$$762$$ 0 0
$$763$$ −2.46199e12 −0.262982
$$764$$ 7.12855e12 0.756974
$$765$$ 0 0
$$766$$ 3.22817e12 0.338787
$$767$$ 4.06292e12 0.423896
$$768$$ 0 0
$$769$$ 1.52349e13 1.57098 0.785491 0.618872i $$-0.212410\pi$$
0.785491 + 0.618872i $$0.212410\pi$$
$$770$$ 4.82941e11 0.0495092
$$771$$ 0 0
$$772$$ −1.03266e13 −1.04635
$$773$$ 8.54195e11 0.0860497 0.0430249 0.999074i $$-0.486301\pi$$
0.0430249 + 0.999074i $$0.486301\pi$$
$$774$$ 0 0
$$775$$ −3.19303e12 −0.317940
$$776$$ 1.04073e13 1.03029
$$777$$ 0 0
$$778$$ 2.38517e12 0.233405
$$779$$ −4.89898e11 −0.0476636
$$780$$ 0 0
$$781$$ −1.27611e13 −1.22732
$$782$$ 5.42513e12 0.518776
$$783$$ 0 0
$$784$$ 8.07158e11 0.0763021
$$785$$ 2.88514e12 0.271178
$$786$$ 0 0
$$787$$ 4.25016e12 0.394929 0.197465 0.980310i $$-0.436729\pi$$
0.197465 + 0.980310i $$0.436729\pi$$
$$788$$ −1.66807e12 −0.154116
$$789$$ 0 0
$$790$$ 1.52418e12 0.139224
$$791$$ −3.06206e12 −0.278112
$$792$$ 0 0
$$793$$ −1.72823e12 −0.155193
$$794$$ 6.73503e12 0.601378
$$795$$ 0 0
$$796$$ 8.02231e12 0.708256
$$797$$ −2.02157e13 −1.77470 −0.887352 0.461093i $$-0.847458\pi$$
−0.887352 + 0.461093i $$0.847458\pi$$
$$798$$ 0 0
$$799$$ −1.36518e12 −0.118503
$$800$$ 2.22566e12 0.192112
$$801$$ 0 0
$$802$$ −7.41334e12 −0.632746
$$803$$ −7.34231e11 −0.0623179
$$804$$ 0 0
$$805$$ −3.86087e12 −0.324044
$$806$$ 5.80278e12 0.484315
$$807$$ 0 0
$$808$$ 1.01894e13 0.841002
$$809$$ 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 $$0.623970\pi$$
$$812$$ −6.80942e12 −0.549678
$$813$$ 0 0
$$814$$ −3.12259e12 −0.249290
$$815$$ −3.91578e12 −0.310892
$$816$$ 0 0
$$817$$ −3.21018e11 −0.0252075
$$818$$ −8.35671e12 −0.652597
$$819$$ 0 0
$$820$$ −7.97219e12 −0.615766
$$821$$ −1.61631e13 −1.24160 −0.620798 0.783971i $$-0.713191\pi$$
−0.620798 + 0.783971i $$0.713191\pi$$
$$822$$ 0 0
$$823$$ 5.97042e12 0.453634 0.226817 0.973937i $$-0.427168\pi$$
0.226817 + 0.973937i $$0.427168\pi$$
$$824$$ −1.70517e13 −1.28853
$$825$$ 0 0
$$826$$ −1.15591e12 −0.0864003
$$827$$ −7.76424e12 −0.577197 −0.288599 0.957450i $$-0.593189\pi$$
−0.288599 + 0.957450i $$0.593189\pi$$
$$828$$ 0 0
$$829$$ −4.09585e12 −0.301195 −0.150598 0.988595i $$-0.548120\pi$$
−0.150598 + 0.988595i $$0.548120\pi$$
$$830$$ −5.40988e10 −0.00395673
$$831$$ 0 0
$$832$$ 1.50397e12 0.108814
$$833$$ 1.32537e12 0.0953751
$$834$$ 0 0
$$835$$ −3.88545e12 −0.276600
$$836$$ 2.46737e11 0.0174705
$$837$$ 0 0
$$838$$ −4.53176e12 −0.317445
$$839$$ −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
$$842$$ −1.05606e11 −0.00724079
$$843$$ 0 0
$$844$$ 9.43433e11 0.0639985
$$845$$ 2.88345e12 0.194562
$$846$$ 0 0
$$847$$ −2.70513e12 −0.180598
$$848$$ 3.84009e12 0.255012
$$849$$ 0 0
$$850$$ 8.23677e11 0.0541217
$$851$$ 2.49636e13 1.63164
$$852$$ 0 0
$$853$$ 2.72968e13 1.76539 0.882696 0.469945i $$-0.155726\pi$$
0.882696 + 0.469945i $$0.155726\pi$$
$$854$$ 4.91688e11 0.0316322
$$855$$ 0 0
$$856$$ 1.44671e12 0.0920979
$$857$$ −7.67571e12 −0.486077 −0.243038 0.970017i $$-0.578144\pi$$
−0.243038 + 0.970017i $$0.578144\pi$$
$$858$$ 0 0
$$859$$ −1.67172e13 −1.04759 −0.523797 0.851843i $$-0.675485\pi$$
−0.523797 + 0.851843i $$0.675485\pi$$
$$860$$ −5.22398e12 −0.325656
$$861$$ 0 0
$$862$$ 8.64452e12 0.533284
$$863$$ 2.85615e13 1.75280 0.876401 0.481582i $$-0.159938\pi$$
0.876401 + 0.481582i $$0.159938\pi$$
$$864$$ 0 0
$$865$$ 5.60756e12 0.340566
$$866$$ 9.34116e12 0.564378
$$867$$ 0 0
$$868$$ 8.39769e12 0.502135
$$869$$ 9.33018e12 0.555010
$$870$$ 0 0
$$871$$ −2.12351e13 −1.25018
$$872$$ 8.83916e12 0.517710
$$873$$ 0 0
$$874$$ 3.87783e11 0.0224795
$$875$$ −5.86182e11 −0.0338062
$$876$$ 0 0
$$877$$ −8.40714e12 −0.479899 −0.239950 0.970785i $$-0.577131\pi$$
−0.239950 + 0.970785i $$0.577131\pi$$
$$878$$ −7.23964e12 −0.411142
$$879$$ 0 0
$$880$$ 3.07066e12 0.172608
$$881$$ 1.99694e13 1.11680 0.558398 0.829573i $$-0.311416\pi$$
0.558398 + 0.829573i $$0.311416\pi$$
$$882$$ 0 0
$$883$$ −2.36498e13 −1.30919 −0.654597 0.755978i $$-0.727162\pi$$
−0.654597 + 0.755978i $$0.727162\pi$$
$$884$$ 7.61423e12 0.419364
$$885$$ 0 0
$$886$$ −9.79250e12 −0.533878
$$887$$ 3.83859e12 0.208217 0.104108 0.994566i $$-0.466801\pi$$
0.104108 + 0.994566i $$0.466801\pi$$
$$888$$ 0 0
$$889$$ −6.97103e12 −0.374317
$$890$$ −3.81122e12 −0.203615
$$891$$ 0 0
$$892$$ 1.13527e13 0.600425
$$893$$ −9.75816e10 −0.00513495
$$894$$ 0 0
$$895$$ 1.10151e13 0.573833
$$896$$ −7.43213e12 −0.385237
$$897$$ 0 0
$$898$$ 3.04216e10 0.00156113
$$899$$ −5.41798e13 −2.76642
$$900$$ 0 0
$$901$$ 6.30551e12 0.318756
$$902$$ 9.59390e12 0.482576
$$903$$ 0 0
$$904$$ 1.09936e13 0.547495
$$905$$ −1.01406e12 −0.0502511
$$906$$ 0 0
$$907$$ −2.31944e13 −1.13802 −0.569009 0.822331i $$-0.692673\pi$$
−0.569009 + 0.822331i $$0.692673\pi$$
$$908$$ 3.32931e13 1.62543
$$909$$ 0 0
$$910$$ 1.06528e12 0.0514966
$$911$$ −1.70743e13 −0.821317 −0.410659 0.911789i $$-0.634701\pi$$
−0.410659 + 0.911789i $$0.634701\pi$$
$$912$$ 0 0
$$913$$ −3.31163e11 −0.0157733
$$914$$ −6.44896e12 −0.305655
$$915$$ 0 0
$$916$$ −2.07368e13 −0.973221
$$917$$ −4.92620e12 −0.230065
$$918$$ 0 0
$$919$$ 1.49265e13 0.690301 0.345151 0.938547i $$-0.387828\pi$$
0.345151 + 0.938547i $$0.387828\pi$$
$$920$$ 1.38615e13 0.637919
$$921$$ 0 0
$$922$$ 1.70768e13 0.778246
$$923$$ −2.81488e13 −1.27659
$$924$$ 0 0
$$925$$ 3.79012e12 0.170222
$$926$$ −9.80898e11 −0.0438404
$$927$$ 0 0
$$928$$ 3.77653e13 1.67158
$$929$$ −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
$$932$$ −1.02020e13 −0.442906
$$933$$ 0 0
$$934$$ −3.79701e12 −0.163260
$$935$$ 5.04210e12 0.215754
$$936$$ 0 0
$$937$$ −3.53996e13 −1.50027 −0.750135 0.661284i $$-0.770012\pi$$
−0.750135 + 0.661284i $$0.770012\pi$$
$$938$$ 6.04147e12 0.254818
$$939$$ 0 0
$$940$$ −1.58796e12 −0.0663384
$$941$$ 4.67286e13 1.94280 0.971402 0.237439i $$-0.0763080\pi$$
0.971402 + 0.237439i $$0.0763080\pi$$
$$942$$ 0 0
$$943$$ −7.66985e13 −3.15852
$$944$$ −7.34960e12 −0.301224
$$945$$ 0 0
$$946$$ 6.28665e12 0.255217
$$947$$ 5.22799e12 0.211232 0.105616 0.994407i $$-0.466319\pi$$
0.105616 + 0.994407i $$0.466319\pi$$
$$948$$ 0 0
$$949$$ −1.61958e12 −0.0648195
$$950$$ 5.88756e10 0.00234520
$$951$$ 0 0
$$952$$ −4.75842e12 −0.187757
$$953$$ 2.46017e13 0.966156 0.483078 0.875577i $$-0.339519\pi$$
0.483078 + 0.875577i $$0.339519\pi$$
$$954$$ 0 0
$$955$$ 1.04125e13 0.405081
$$956$$ 2.67809e13 1.03697
$$957$$ 0 0
$$958$$ 7.83373e11 0.0300486
$$959$$ −7.81618e12 −0.298408
$$960$$ 0 0
$$961$$ 4.03773e13 1.52715
$$962$$ −6.88788e12 −0.259297
$$963$$ 0 0
$$964$$ 3.40853e13 1.27122
$$965$$ −1.50838e13 −0.559936
$$966$$ 0 0
$$967$$ 1.24062e13 0.456267 0.228134 0.973630i $$-0.426738\pi$$
0.228134 + 0.973630i $$0.426738\pi$$
$$968$$ 9.71212e12 0.355529
$$969$$ 0 0
$$970$$ 6.92060e12 0.250998
$$971$$ −5.30324e12 −0.191450 −0.0957248 0.995408i $$-0.530517\pi$$
−0.0957248 + 0.995408i $$0.530517\pi$$
$$972$$ 0 0
$$973$$ −1.98509e13 −0.710023
$$974$$ 4.53799e12 0.161565
$$975$$ 0 0
$$976$$ 3.12628e12 0.110282
$$977$$ 1.45131e13 0.509606 0.254803 0.966993i $$-0.417989\pi$$
0.254803 + 0.966993i $$0.417989\pi$$
$$978$$ 0 0
$$979$$ −2.33302e13 −0.811702
$$980$$ 1.54166e12 0.0533914
$$981$$ 0 0
$$982$$ −1.01954e13 −0.349868
$$983$$ 3.61534e13 1.23498 0.617488 0.786580i $$-0.288150\pi$$
0.617488 + 0.786580i $$0.288150\pi$$
$$984$$ 0 0
$$985$$ −2.43652e12 −0.0824721
$$986$$ 1.39763e13 0.470917
$$987$$ 0 0
$$988$$ 5.44258e11 0.0181718
$$989$$ −5.02586e13 −1.67043
$$990$$ 0 0
$$991$$ 4.94162e13 1.62756 0.813782 0.581170i $$-0.197405\pi$$
0.813782 + 0.581170i $$0.197405\pi$$
$$992$$ −4.65739e13 −1.52700
$$993$$ 0 0
$$994$$ 8.00841e12 0.260200
$$995$$ 1.17180e13 0.379010
$$996$$ 0 0
$$997$$ −3.03522e13 −0.972886 −0.486443 0.873712i $$-0.661706\pi$$
−0.486443 + 0.873712i $$0.661706\pi$$
$$998$$ 7.50580e12 0.239502
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.10.a.b.1.1 2
3.2 odd 2 35.10.a.b.1.2 2
15.2 even 4 175.10.b.c.99.2 4
15.8 even 4 175.10.b.c.99.3 4
15.14 odd 2 175.10.a.c.1.1 2
21.20 even 2 245.10.a.c.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 2 3.2 odd 2
175.10.a.c.1.1 2 15.14 odd 2
175.10.b.c.99.2 4 15.2 even 4
175.10.b.c.99.3 4 15.8 even 4
245.10.a.c.1.2 2 21.20 even 2
315.10.a.b.1.1 2 1.1 even 1 trivial