# Properties

 Label 4225.2.a.bb.1.3 Level $4225$ Weight $2$ Character 4225.1 Self dual yes Analytic conductor $33.737$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4225 = 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4225.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: $$33.7367948540$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 169) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.445042$$ of defining polynomial Character $$\chi$$ $$=$$ 4225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.801938 q^{2} +2.24698 q^{3} -1.35690 q^{4} +1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +O(q^{10})$$ $$q+0.801938 q^{2} +2.24698 q^{3} -1.35690 q^{4} +1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +4.24698 q^{11} -3.04892 q^{12} -1.89008 q^{14} +0.554958 q^{16} -2.15883 q^{17} +1.64310 q^{18} +0.0881460 q^{19} -5.29590 q^{21} +3.40581 q^{22} -1.49396 q^{23} -6.04892 q^{24} -2.13706 q^{27} +3.19806 q^{28} +4.63102 q^{29} +6.63102 q^{31} +5.82908 q^{32} +9.54288 q^{33} -1.73125 q^{34} -2.78017 q^{36} +5.69202 q^{37} +0.0706876 q^{38} +11.5918 q^{41} -4.24698 q^{42} +0.295897 q^{43} -5.76271 q^{44} -1.19806 q^{46} -7.35690 q^{47} +1.24698 q^{48} -1.44504 q^{49} -4.85086 q^{51} +10.3937 q^{53} -1.71379 q^{54} +6.34481 q^{56} +0.198062 q^{57} +3.71379 q^{58} +6.78017 q^{59} +3.47219 q^{61} +5.31767 q^{62} -4.82908 q^{63} +3.56465 q^{64} +7.65279 q^{66} +7.67994 q^{67} +2.92931 q^{68} -3.35690 q^{69} +8.66487 q^{71} -5.51573 q^{72} +6.73556 q^{73} +4.56465 q^{74} -0.119605 q^{76} -10.0097 q^{77} +9.97046 q^{79} -10.9487 q^{81} +9.29590 q^{82} +1.60925 q^{83} +7.18598 q^{84} +0.237291 q^{86} +10.4058 q^{87} -11.4330 q^{88} +2.88471 q^{89} +2.02715 q^{92} +14.8998 q^{93} -5.89977 q^{94} +13.0978 q^{96} -8.05861 q^{97} -1.15883 q^{98} +8.70171 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 2 q^{3} + q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 2 * q^3 + q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 $$3 q - 2 q^{2} + 2 q^{3} + q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} + 9 q^{18} + 4 q^{19} - 2 q^{21} - 3 q^{22} + 5 q^{23} - 9 q^{24} - q^{27} + 14 q^{28} - q^{29} + 5 q^{31} + 7 q^{32} + 10 q^{33} - 13 q^{34} - 7 q^{36} + 12 q^{37} - 12 q^{38} + 7 q^{41} - 8 q^{42} - 13 q^{43} - 8 q^{46} - 18 q^{47} - q^{48} - 4 q^{49} - q^{51} - q^{53} + 3 q^{54} - 4 q^{56} + 5 q^{57} + 3 q^{58} + 19 q^{59} + 4 q^{61} - q^{62} - 4 q^{63} - 11 q^{64} + 5 q^{66} - q^{67} + 21 q^{68} - 6 q^{69} + 27 q^{71} - 4 q^{72} + 9 q^{73} - 8 q^{74} + 21 q^{76} - 8 q^{77} - 5 q^{79} - q^{81} + 14 q^{82} - 7 q^{83} + 7 q^{84} + 18 q^{86} + 18 q^{87} - 15 q^{88} + 11 q^{89} + 22 q^{93} + 5 q^{94} + 21 q^{96} + 7 q^{97} + 5 q^{98} - q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 2 * q^3 + q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 8 * q^11 - 5 * q^14 + 2 * q^16 + 2 * q^17 + 9 * q^18 + 4 * q^19 - 2 * q^21 - 3 * q^22 + 5 * q^23 - 9 * q^24 - q^27 + 14 * q^28 - q^29 + 5 * q^31 + 7 * q^32 + 10 * q^33 - 13 * q^34 - 7 * q^36 + 12 * q^37 - 12 * q^38 + 7 * q^41 - 8 * q^42 - 13 * q^43 - 8 * q^46 - 18 * q^47 - q^48 - 4 * q^49 - q^51 - q^53 + 3 * q^54 - 4 * q^56 + 5 * q^57 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 + 21 * q^68 - 6 * q^69 + 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 21 * q^76 - 8 * q^77 - 5 * q^79 - q^81 + 14 * q^82 - 7 * q^83 + 7 * q^84 + 18 * q^86 + 18 * q^87 - 15 * q^88 + 11 * q^89 + 22 * q^93 + 5 * q^94 + 21 * q^96 + 7 * q^97 + 5 * q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.801938 0.567056 0.283528 0.958964i $$-0.408495\pi$$
0.283528 + 0.958964i $$0.408495\pi$$
$$3$$ 2.24698 1.29729 0.648647 0.761089i $$-0.275335\pi$$
0.648647 + 0.761089i $$0.275335\pi$$
$$4$$ −1.35690 −0.678448
$$5$$ 0 0
$$6$$ 1.80194 0.735638
$$7$$ −2.35690 −0.890823 −0.445411 0.895326i $$-0.646943\pi$$
−0.445411 + 0.895326i $$0.646943\pi$$
$$8$$ −2.69202 −0.951773
$$9$$ 2.04892 0.682972
$$10$$ 0 0
$$11$$ 4.24698 1.28051 0.640256 0.768161i $$-0.278828\pi$$
0.640256 + 0.768161i $$0.278828\pi$$
$$12$$ −3.04892 −0.880147
$$13$$ 0 0
$$14$$ −1.89008 −0.505146
$$15$$ 0 0
$$16$$ 0.554958 0.138740
$$17$$ −2.15883 −0.523594 −0.261797 0.965123i $$-0.584315\pi$$
−0.261797 + 0.965123i $$0.584315\pi$$
$$18$$ 1.64310 0.387283
$$19$$ 0.0881460 0.0202221 0.0101110 0.999949i $$-0.496782\pi$$
0.0101110 + 0.999949i $$0.496782\pi$$
$$20$$ 0 0
$$21$$ −5.29590 −1.15566
$$22$$ 3.40581 0.726122
$$23$$ −1.49396 −0.311512 −0.155756 0.987796i $$-0.549781\pi$$
−0.155756 + 0.987796i $$0.549781\pi$$
$$24$$ −6.04892 −1.23473
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −2.13706 −0.411278
$$28$$ 3.19806 0.604377
$$29$$ 4.63102 0.859959 0.429980 0.902839i $$-0.358521\pi$$
0.429980 + 0.902839i $$0.358521\pi$$
$$30$$ 0 0
$$31$$ 6.63102 1.19097 0.595483 0.803368i $$-0.296961\pi$$
0.595483 + 0.803368i $$0.296961\pi$$
$$32$$ 5.82908 1.03045
$$33$$ 9.54288 1.66120
$$34$$ −1.73125 −0.296907
$$35$$ 0 0
$$36$$ −2.78017 −0.463361
$$37$$ 5.69202 0.935763 0.467881 0.883791i $$-0.345017\pi$$
0.467881 + 0.883791i $$0.345017\pi$$
$$38$$ 0.0706876 0.0114670
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.5918 1.81033 0.905167 0.425056i $$-0.139746\pi$$
0.905167 + 0.425056i $$0.139746\pi$$
$$42$$ −4.24698 −0.655323
$$43$$ 0.295897 0.0451239 0.0225619 0.999745i $$-0.492818\pi$$
0.0225619 + 0.999745i $$0.492818\pi$$
$$44$$ −5.76271 −0.868761
$$45$$ 0 0
$$46$$ −1.19806 −0.176645
$$47$$ −7.35690 −1.07311 −0.536557 0.843864i $$-0.680275\pi$$
−0.536557 + 0.843864i $$0.680275\pi$$
$$48$$ 1.24698 0.179986
$$49$$ −1.44504 −0.206435
$$50$$ 0 0
$$51$$ −4.85086 −0.679256
$$52$$ 0 0
$$53$$ 10.3937 1.42769 0.713844 0.700304i $$-0.246952\pi$$
0.713844 + 0.700304i $$0.246952\pi$$
$$54$$ −1.71379 −0.233218
$$55$$ 0 0
$$56$$ 6.34481 0.847861
$$57$$ 0.198062 0.0262340
$$58$$ 3.71379 0.487645
$$59$$ 6.78017 0.882703 0.441351 0.897334i $$-0.354499\pi$$
0.441351 + 0.897334i $$0.354499\pi$$
$$60$$ 0 0
$$61$$ 3.47219 0.444568 0.222284 0.974982i $$-0.428649\pi$$
0.222284 + 0.974982i $$0.428649\pi$$
$$62$$ 5.31767 0.675344
$$63$$ −4.82908 −0.608407
$$64$$ 3.56465 0.445581
$$65$$ 0 0
$$66$$ 7.65279 0.941994
$$67$$ 7.67994 0.938254 0.469127 0.883131i $$-0.344569\pi$$
0.469127 + 0.883131i $$0.344569\pi$$
$$68$$ 2.92931 0.355231
$$69$$ −3.35690 −0.404123
$$70$$ 0 0
$$71$$ 8.66487 1.02833 0.514166 0.857691i $$-0.328102\pi$$
0.514166 + 0.857691i $$0.328102\pi$$
$$72$$ −5.51573 −0.650035
$$73$$ 6.73556 0.788338 0.394169 0.919038i $$-0.371032\pi$$
0.394169 + 0.919038i $$0.371032\pi$$
$$74$$ 4.56465 0.530629
$$75$$ 0 0
$$76$$ −0.119605 −0.0137196
$$77$$ −10.0097 −1.14071
$$78$$ 0 0
$$79$$ 9.97046 1.12176 0.560882 0.827896i $$-0.310462\pi$$
0.560882 + 0.827896i $$0.310462\pi$$
$$80$$ 0 0
$$81$$ −10.9487 −1.21652
$$82$$ 9.29590 1.02656
$$83$$ 1.60925 0.176638 0.0883192 0.996092i $$-0.471850\pi$$
0.0883192 + 0.996092i $$0.471850\pi$$
$$84$$ 7.18598 0.784055
$$85$$ 0 0
$$86$$ 0.237291 0.0255877
$$87$$ 10.4058 1.11562
$$88$$ −11.4330 −1.21876
$$89$$ 2.88471 0.305778 0.152889 0.988243i $$-0.451142\pi$$
0.152889 + 0.988243i $$0.451142\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2.02715 0.211345
$$93$$ 14.8998 1.54503
$$94$$ −5.89977 −0.608515
$$95$$ 0 0
$$96$$ 13.0978 1.33679
$$97$$ −8.05861 −0.818227 −0.409114 0.912483i $$-0.634162\pi$$
−0.409114 + 0.912483i $$0.634162\pi$$
$$98$$ −1.15883 −0.117060
$$99$$ 8.70171 0.874555
$$100$$ 0 0
$$101$$ −13.3545 −1.32882 −0.664411 0.747367i $$-0.731318\pi$$
−0.664411 + 0.747367i $$0.731318\pi$$
$$102$$ −3.89008 −0.385176
$$103$$ −1.36227 −0.134229 −0.0671144 0.997745i $$-0.521379\pi$$
−0.0671144 + 0.997745i $$0.521379\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 8.33513 0.809579
$$107$$ −3.26875 −0.316002 −0.158001 0.987439i $$-0.550505\pi$$
−0.158001 + 0.987439i $$0.550505\pi$$
$$108$$ 2.89977 0.279031
$$109$$ −15.7017 −1.50395 −0.751976 0.659191i $$-0.770899\pi$$
−0.751976 + 0.659191i $$0.770899\pi$$
$$110$$ 0 0
$$111$$ 12.7899 1.21396
$$112$$ −1.30798 −0.123592
$$113$$ −12.0489 −1.13347 −0.566733 0.823901i $$-0.691793\pi$$
−0.566733 + 0.823901i $$0.691793\pi$$
$$114$$ 0.158834 0.0148761
$$115$$ 0 0
$$116$$ −6.28382 −0.583438
$$117$$ 0 0
$$118$$ 5.43727 0.500541
$$119$$ 5.08815 0.466430
$$120$$ 0 0
$$121$$ 7.03684 0.639712
$$122$$ 2.78448 0.252095
$$123$$ 26.0465 2.34854
$$124$$ −8.99761 −0.808009
$$125$$ 0 0
$$126$$ −3.87263 −0.345001
$$127$$ 9.80731 0.870258 0.435129 0.900368i $$-0.356703\pi$$
0.435129 + 0.900368i $$0.356703\pi$$
$$128$$ −8.79954 −0.777777
$$129$$ 0.664874 0.0585389
$$130$$ 0 0
$$131$$ −6.57673 −0.574611 −0.287306 0.957839i $$-0.592760\pi$$
−0.287306 + 0.957839i $$0.592760\pi$$
$$132$$ −12.9487 −1.12704
$$133$$ −0.207751 −0.0180143
$$134$$ 6.15883 0.532042
$$135$$ 0 0
$$136$$ 5.81163 0.498343
$$137$$ 6.21983 0.531396 0.265698 0.964056i $$-0.414398\pi$$
0.265698 + 0.964056i $$0.414398\pi$$
$$138$$ −2.69202 −0.229160
$$139$$ −14.7071 −1.24744 −0.623719 0.781648i $$-0.714379\pi$$
−0.623719 + 0.781648i $$0.714379\pi$$
$$140$$ 0 0
$$141$$ −16.5308 −1.39214
$$142$$ 6.94869 0.583121
$$143$$ 0 0
$$144$$ 1.13706 0.0947553
$$145$$ 0 0
$$146$$ 5.40150 0.447031
$$147$$ −3.24698 −0.267806
$$148$$ −7.72348 −0.634866
$$149$$ −4.33513 −0.355147 −0.177574 0.984108i $$-0.556825\pi$$
−0.177574 + 0.984108i $$0.556825\pi$$
$$150$$ 0 0
$$151$$ −3.94438 −0.320989 −0.160494 0.987037i $$-0.551309\pi$$
−0.160494 + 0.987037i $$0.551309\pi$$
$$152$$ −0.237291 −0.0192468
$$153$$ −4.42327 −0.357600
$$154$$ −8.02715 −0.646846
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.45473 −0.355526 −0.177763 0.984073i $$-0.556886\pi$$
−0.177763 + 0.984073i $$0.556886\pi$$
$$158$$ 7.99569 0.636103
$$159$$ 23.3545 1.85213
$$160$$ 0 0
$$161$$ 3.52111 0.277502
$$162$$ −8.78017 −0.689835
$$163$$ 16.1588 1.26566 0.632829 0.774292i $$-0.281894\pi$$
0.632829 + 0.774292i $$0.281894\pi$$
$$164$$ −15.7289 −1.22822
$$165$$ 0 0
$$166$$ 1.29052 0.100164
$$167$$ 16.1172 1.24719 0.623594 0.781749i $$-0.285672\pi$$
0.623594 + 0.781749i $$0.285672\pi$$
$$168$$ 14.2567 1.09993
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 0.180604 0.0138111
$$172$$ −0.401501 −0.0306142
$$173$$ 21.5362 1.63736 0.818682 0.574247i $$-0.194705\pi$$
0.818682 + 0.574247i $$0.194705\pi$$
$$174$$ 8.34481 0.632619
$$175$$ 0 0
$$176$$ 2.35690 0.177658
$$177$$ 15.2349 1.14513
$$178$$ 2.31336 0.173393
$$179$$ 11.4330 0.854540 0.427270 0.904124i $$-0.359475\pi$$
0.427270 + 0.904124i $$0.359475\pi$$
$$180$$ 0 0
$$181$$ 20.9705 1.55872 0.779361 0.626575i $$-0.215544\pi$$
0.779361 + 0.626575i $$0.215544\pi$$
$$182$$ 0 0
$$183$$ 7.80194 0.576736
$$184$$ 4.02177 0.296489
$$185$$ 0 0
$$186$$ 11.9487 0.876120
$$187$$ −9.16852 −0.670469
$$188$$ 9.98254 0.728052
$$189$$ 5.03684 0.366376
$$190$$ 0 0
$$191$$ −14.4373 −1.04464 −0.522322 0.852748i $$-0.674934\pi$$
−0.522322 + 0.852748i $$0.674934\pi$$
$$192$$ 8.00969 0.578049
$$193$$ −13.5797 −0.977489 −0.488745 0.872427i $$-0.662545\pi$$
−0.488745 + 0.872427i $$0.662545\pi$$
$$194$$ −6.46250 −0.463980
$$195$$ 0 0
$$196$$ 1.96077 0.140055
$$197$$ −0.560335 −0.0399222 −0.0199611 0.999801i $$-0.506354\pi$$
−0.0199611 + 0.999801i $$0.506354\pi$$
$$198$$ 6.97823 0.495921
$$199$$ 11.4916 0.814616 0.407308 0.913291i $$-0.366468\pi$$
0.407308 + 0.913291i $$0.366468\pi$$
$$200$$ 0 0
$$201$$ 17.2567 1.21719
$$202$$ −10.7095 −0.753516
$$203$$ −10.9148 −0.766071
$$204$$ 6.58211 0.460840
$$205$$ 0 0
$$206$$ −1.09246 −0.0761151
$$207$$ −3.06100 −0.212754
$$208$$ 0 0
$$209$$ 0.374354 0.0258946
$$210$$ 0 0
$$211$$ 8.78448 0.604748 0.302374 0.953189i $$-0.402221\pi$$
0.302374 + 0.953189i $$0.402221\pi$$
$$212$$ −14.1032 −0.968613
$$213$$ 19.4698 1.33405
$$214$$ −2.62133 −0.179191
$$215$$ 0 0
$$216$$ 5.75302 0.391443
$$217$$ −15.6286 −1.06094
$$218$$ −12.5918 −0.852824
$$219$$ 15.1347 1.02271
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 10.2567 0.688383
$$223$$ −2.25906 −0.151278 −0.0756390 0.997135i $$-0.524100\pi$$
−0.0756390 + 0.997135i $$0.524100\pi$$
$$224$$ −13.7385 −0.917945
$$225$$ 0 0
$$226$$ −9.66248 −0.642739
$$227$$ −6.96615 −0.462359 −0.231180 0.972911i $$-0.574259\pi$$
−0.231180 + 0.972911i $$0.574259\pi$$
$$228$$ −0.268750 −0.0177984
$$229$$ 24.1739 1.59746 0.798728 0.601692i $$-0.205507\pi$$
0.798728 + 0.601692i $$0.205507\pi$$
$$230$$ 0 0
$$231$$ −22.4916 −1.47984
$$232$$ −12.4668 −0.818486
$$233$$ 3.06100 0.200533 0.100266 0.994961i $$-0.468031\pi$$
0.100266 + 0.994961i $$0.468031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −9.19998 −0.598868
$$237$$ 22.4034 1.45526
$$238$$ 4.08038 0.264492
$$239$$ 25.1468 1.62661 0.813304 0.581839i $$-0.197667\pi$$
0.813304 + 0.581839i $$0.197667\pi$$
$$240$$ 0 0
$$241$$ 20.2664 1.30547 0.652735 0.757586i $$-0.273621\pi$$
0.652735 + 0.757586i $$0.273621\pi$$
$$242$$ 5.64310 0.362752
$$243$$ −18.1903 −1.16691
$$244$$ −4.71140 −0.301616
$$245$$ 0 0
$$246$$ 20.8877 1.33175
$$247$$ 0 0
$$248$$ −17.8509 −1.13353
$$249$$ 3.61596 0.229152
$$250$$ 0 0
$$251$$ −23.7211 −1.49726 −0.748631 0.662987i $$-0.769288\pi$$
−0.748631 + 0.662987i $$0.769288\pi$$
$$252$$ 6.55257 0.412773
$$253$$ −6.34481 −0.398895
$$254$$ 7.86486 0.493485
$$255$$ 0 0
$$256$$ −14.1860 −0.886624
$$257$$ −14.2241 −0.887278 −0.443639 0.896206i $$-0.646313\pi$$
−0.443639 + 0.896206i $$0.646313\pi$$
$$258$$ 0.533188 0.0331948
$$259$$ −13.4155 −0.833599
$$260$$ 0 0
$$261$$ 9.48858 0.587329
$$262$$ −5.27413 −0.325837
$$263$$ 17.0954 1.05415 0.527075 0.849819i $$-0.323289\pi$$
0.527075 + 0.849819i $$0.323289\pi$$
$$264$$ −25.6896 −1.58109
$$265$$ 0 0
$$266$$ −0.166603 −0.0102151
$$267$$ 6.48188 0.396684
$$268$$ −10.4209 −0.636556
$$269$$ −6.46681 −0.394288 −0.197144 0.980374i $$-0.563167\pi$$
−0.197144 + 0.980374i $$0.563167\pi$$
$$270$$ 0 0
$$271$$ −6.44803 −0.391690 −0.195845 0.980635i $$-0.562745\pi$$
−0.195845 + 0.980635i $$0.562745\pi$$
$$272$$ −1.19806 −0.0726432
$$273$$ 0 0
$$274$$ 4.98792 0.301331
$$275$$ 0 0
$$276$$ 4.55496 0.274176
$$277$$ −13.4601 −0.808739 −0.404370 0.914596i $$-0.632509\pi$$
−0.404370 + 0.914596i $$0.632509\pi$$
$$278$$ −11.7942 −0.707367
$$279$$ 13.5864 0.813398
$$280$$ 0 0
$$281$$ −5.03684 −0.300472 −0.150236 0.988650i $$-0.548003\pi$$
−0.150236 + 0.988650i $$0.548003\pi$$
$$282$$ −13.2567 −0.789423
$$283$$ −22.1280 −1.31537 −0.657686 0.753293i $$-0.728464\pi$$
−0.657686 + 0.753293i $$0.728464\pi$$
$$284$$ −11.7573 −0.697669
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −27.3207 −1.61269
$$288$$ 11.9433 0.703766
$$289$$ −12.3394 −0.725849
$$290$$ 0 0
$$291$$ −18.1075 −1.06148
$$292$$ −9.13946 −0.534846
$$293$$ 14.9463 0.873172 0.436586 0.899663i $$-0.356187\pi$$
0.436586 + 0.899663i $$0.356187\pi$$
$$294$$ −2.60388 −0.151861
$$295$$ 0 0
$$296$$ −15.3230 −0.890634
$$297$$ −9.07606 −0.526647
$$298$$ −3.47650 −0.201388
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −0.697398 −0.0401974
$$302$$ −3.16315 −0.182019
$$303$$ −30.0073 −1.72387
$$304$$ 0.0489173 0.00280560
$$305$$ 0 0
$$306$$ −3.54719 −0.202779
$$307$$ 19.1293 1.09177 0.545883 0.837861i $$-0.316194\pi$$
0.545883 + 0.837861i $$0.316194\pi$$
$$308$$ 13.5821 0.773912
$$309$$ −3.06100 −0.174134
$$310$$ 0 0
$$311$$ −0.269815 −0.0152998 −0.00764990 0.999971i $$-0.502435\pi$$
−0.00764990 + 0.999971i $$0.502435\pi$$
$$312$$ 0 0
$$313$$ 23.3937 1.32229 0.661146 0.750257i $$-0.270070\pi$$
0.661146 + 0.750257i $$0.270070\pi$$
$$314$$ −3.57242 −0.201603
$$315$$ 0 0
$$316$$ −13.5289 −0.761059
$$317$$ −13.9952 −0.786050 −0.393025 0.919528i $$-0.628571\pi$$
−0.393025 + 0.919528i $$0.628571\pi$$
$$318$$ 18.7289 1.05026
$$319$$ 19.6679 1.10119
$$320$$ 0 0
$$321$$ −7.34481 −0.409948
$$322$$ 2.82371 0.157359
$$323$$ −0.190293 −0.0105882
$$324$$ 14.8562 0.825346
$$325$$ 0 0
$$326$$ 12.9584 0.717698
$$327$$ −35.2814 −1.95107
$$328$$ −31.2054 −1.72303
$$329$$ 17.3394 0.955954
$$330$$ 0 0
$$331$$ −17.8213 −0.979548 −0.489774 0.871849i $$-0.662921\pi$$
−0.489774 + 0.871849i $$0.662921\pi$$
$$332$$ −2.18359 −0.119840
$$333$$ 11.6625 0.639100
$$334$$ 12.9250 0.707225
$$335$$ 0 0
$$336$$ −2.93900 −0.160336
$$337$$ 27.8485 1.51700 0.758501 0.651672i $$-0.225932\pi$$
0.758501 + 0.651672i $$0.225932\pi$$
$$338$$ 0 0
$$339$$ −27.0737 −1.47044
$$340$$ 0 0
$$341$$ 28.1618 1.52505
$$342$$ 0.144833 0.00783167
$$343$$ 19.9041 1.07472
$$344$$ −0.796561 −0.0429477
$$345$$ 0 0
$$346$$ 17.2707 0.928477
$$347$$ −1.50365 −0.0807200 −0.0403600 0.999185i $$-0.512850\pi$$
−0.0403600 + 0.999185i $$0.512850\pi$$
$$348$$ −14.1196 −0.756890
$$349$$ 14.1860 0.759358 0.379679 0.925118i $$-0.376034\pi$$
0.379679 + 0.925118i $$0.376034\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 24.7560 1.31950
$$353$$ −7.16852 −0.381542 −0.190771 0.981635i $$-0.561099\pi$$
−0.190771 + 0.981635i $$0.561099\pi$$
$$354$$ 12.2174 0.649350
$$355$$ 0 0
$$356$$ −3.91425 −0.207455
$$357$$ 11.4330 0.605096
$$358$$ 9.16852 0.484571
$$359$$ −19.8853 −1.04951 −0.524753 0.851255i $$-0.675842\pi$$
−0.524753 + 0.851255i $$0.675842\pi$$
$$360$$ 0 0
$$361$$ −18.9922 −0.999591
$$362$$ 16.8170 0.883882
$$363$$ 15.8116 0.829895
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 6.25667 0.327041
$$367$$ −1.08383 −0.0565757 −0.0282878 0.999600i $$-0.509006\pi$$
−0.0282878 + 0.999600i $$0.509006\pi$$
$$368$$ −0.829085 −0.0432190
$$369$$ 23.7506 1.23641
$$370$$ 0 0
$$371$$ −24.4969 −1.27182
$$372$$ −20.2174 −1.04823
$$373$$ 6.13036 0.317418 0.158709 0.987325i $$-0.449267\pi$$
0.158709 + 0.987325i $$0.449267\pi$$
$$374$$ −7.35258 −0.380193
$$375$$ 0 0
$$376$$ 19.8049 1.02136
$$377$$ 0 0
$$378$$ 4.03923 0.207756
$$379$$ 2.40880 0.123732 0.0618658 0.998084i $$-0.480295\pi$$
0.0618658 + 0.998084i $$0.480295\pi$$
$$380$$ 0 0
$$381$$ 22.0368 1.12898
$$382$$ −11.5778 −0.592371
$$383$$ −30.3913 −1.55292 −0.776462 0.630164i $$-0.782988\pi$$
−0.776462 + 0.630164i $$0.782988\pi$$
$$384$$ −19.7724 −1.00901
$$385$$ 0 0
$$386$$ −10.8901 −0.554291
$$387$$ 0.606268 0.0308184
$$388$$ 10.9347 0.555125
$$389$$ −15.9409 −0.808237 −0.404118 0.914707i $$-0.632422\pi$$
−0.404118 + 0.914707i $$0.632422\pi$$
$$390$$ 0 0
$$391$$ 3.22521 0.163106
$$392$$ 3.89008 0.196479
$$393$$ −14.7778 −0.745440
$$394$$ −0.449354 −0.0226381
$$395$$ 0 0
$$396$$ −11.8073 −0.593340
$$397$$ 16.9148 0.848931 0.424466 0.905444i $$-0.360462\pi$$
0.424466 + 0.905444i $$0.360462\pi$$
$$398$$ 9.21552 0.461932
$$399$$ −0.466812 −0.0233698
$$400$$ 0 0
$$401$$ −26.6625 −1.33146 −0.665730 0.746192i $$-0.731880\pi$$
−0.665730 + 0.746192i $$0.731880\pi$$
$$402$$ 13.8388 0.690215
$$403$$ 0 0
$$404$$ 18.1207 0.901537
$$405$$ 0 0
$$406$$ −8.75302 −0.434405
$$407$$ 24.1739 1.19826
$$408$$ 13.0586 0.646497
$$409$$ −28.5163 −1.41004 −0.705021 0.709187i $$-0.749062\pi$$
−0.705021 + 0.709187i $$0.749062\pi$$
$$410$$ 0 0
$$411$$ 13.9758 0.689377
$$412$$ 1.84846 0.0910672
$$413$$ −15.9801 −0.786332
$$414$$ −2.45473 −0.120643
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −33.0465 −1.61830
$$418$$ 0.300209 0.0146837
$$419$$ −29.6093 −1.44651 −0.723253 0.690583i $$-0.757354\pi$$
−0.723253 + 0.690583i $$0.757354\pi$$
$$420$$ 0 0
$$421$$ 11.6606 0.568301 0.284151 0.958780i $$-0.408288\pi$$
0.284151 + 0.958780i $$0.408288\pi$$
$$422$$ 7.04461 0.342926
$$423$$ −15.0737 −0.732907
$$424$$ −27.9801 −1.35884
$$425$$ 0 0
$$426$$ 15.6136 0.756480
$$427$$ −8.18359 −0.396032
$$428$$ 4.43535 0.214391
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −4.34913 −0.209490 −0.104745 0.994499i $$-0.533403\pi$$
−0.104745 + 0.994499i $$0.533403\pi$$
$$432$$ −1.18598 −0.0570605
$$433$$ 14.3884 0.691460 0.345730 0.938334i $$-0.387631\pi$$
0.345730 + 0.938334i $$0.387631\pi$$
$$434$$ −12.5332 −0.601612
$$435$$ 0 0
$$436$$ 21.3056 1.02035
$$437$$ −0.131687 −0.00629942
$$438$$ 12.1371 0.579931
$$439$$ −20.2325 −0.965645 −0.482822 0.875718i $$-0.660388\pi$$
−0.482822 + 0.875718i $$0.660388\pi$$
$$440$$ 0 0
$$441$$ −2.96077 −0.140989
$$442$$ 0 0
$$443$$ −8.12200 −0.385888 −0.192944 0.981210i $$-0.561804\pi$$
−0.192944 + 0.981210i $$0.561804\pi$$
$$444$$ −17.3545 −0.823608
$$445$$ 0 0
$$446$$ −1.81163 −0.0857830
$$447$$ −9.74094 −0.460731
$$448$$ −8.40150 −0.396934
$$449$$ −12.4916 −0.589513 −0.294757 0.955572i $$-0.595239\pi$$
−0.294757 + 0.955572i $$0.595239\pi$$
$$450$$ 0 0
$$451$$ 49.2301 2.31816
$$452$$ 16.3491 0.768998
$$453$$ −8.86294 −0.416417
$$454$$ −5.58642 −0.262184
$$455$$ 0 0
$$456$$ −0.533188 −0.0249688
$$457$$ 5.98121 0.279789 0.139895 0.990166i $$-0.455324\pi$$
0.139895 + 0.990166i $$0.455324\pi$$
$$458$$ 19.3860 0.905847
$$459$$ 4.61356 0.215343
$$460$$ 0 0
$$461$$ 2.05669 0.0957895 0.0478947 0.998852i $$-0.484749\pi$$
0.0478947 + 0.998852i $$0.484749\pi$$
$$462$$ −18.0368 −0.839150
$$463$$ −8.44935 −0.392675 −0.196337 0.980536i $$-0.562905\pi$$
−0.196337 + 0.980536i $$0.562905\pi$$
$$464$$ 2.57002 0.119310
$$465$$ 0 0
$$466$$ 2.45473 0.113713
$$467$$ −33.5139 −1.55084 −0.775420 0.631446i $$-0.782462\pi$$
−0.775420 + 0.631446i $$0.782462\pi$$
$$468$$ 0 0
$$469$$ −18.1008 −0.835818
$$470$$ 0 0
$$471$$ −10.0097 −0.461222
$$472$$ −18.2524 −0.840133
$$473$$ 1.25667 0.0577817
$$474$$ 17.9661 0.825213
$$475$$ 0 0
$$476$$ −6.90408 −0.316448
$$477$$ 21.2959 0.975072
$$478$$ 20.1661 0.922377
$$479$$ 24.7313 1.13000 0.565000 0.825091i $$-0.308876\pi$$
0.565000 + 0.825091i $$0.308876\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 16.2524 0.740275
$$483$$ 7.91185 0.360002
$$484$$ −9.54825 −0.434012
$$485$$ 0 0
$$486$$ −14.5875 −0.661702
$$487$$ −37.7555 −1.71087 −0.855433 0.517913i $$-0.826709\pi$$
−0.855433 + 0.517913i $$0.826709\pi$$
$$488$$ −9.34721 −0.423128
$$489$$ 36.3086 1.64193
$$490$$ 0 0
$$491$$ 31.3110 1.41304 0.706522 0.707691i $$-0.250263\pi$$
0.706522 + 0.707691i $$0.250263\pi$$
$$492$$ −35.3424 −1.59336
$$493$$ −9.99761 −0.450270
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 3.67994 0.165234
$$497$$ −20.4222 −0.916061
$$498$$ 2.89977 0.129942
$$499$$ −21.4873 −0.961902 −0.480951 0.876748i $$-0.659708\pi$$
−0.480951 + 0.876748i $$0.659708\pi$$
$$500$$ 0 0
$$501$$ 36.2150 1.61797
$$502$$ −19.0228 −0.849031
$$503$$ −37.5924 −1.67616 −0.838081 0.545546i $$-0.816322\pi$$
−0.838081 + 0.545546i $$0.816322\pi$$
$$504$$ 13.0000 0.579066
$$505$$ 0 0
$$506$$ −5.08815 −0.226196
$$507$$ 0 0
$$508$$ −13.3075 −0.590425
$$509$$ 17.1075 0.758278 0.379139 0.925340i $$-0.376220\pi$$
0.379139 + 0.925340i $$0.376220\pi$$
$$510$$ 0 0
$$511$$ −15.8750 −0.702269
$$512$$ 6.22282 0.275012
$$513$$ −0.188374 −0.00831690
$$514$$ −11.4069 −0.503136
$$515$$ 0 0
$$516$$ −0.902165 −0.0397156
$$517$$ −31.2446 −1.37414
$$518$$ −10.7584 −0.472697
$$519$$ 48.3913 2.12414
$$520$$ 0 0
$$521$$ −19.8465 −0.869493 −0.434746 0.900553i $$-0.643162\pi$$
−0.434746 + 0.900553i $$0.643162\pi$$
$$522$$ 7.60925 0.333048
$$523$$ 11.4300 0.499798 0.249899 0.968272i $$-0.419603\pi$$
0.249899 + 0.968272i $$0.419603\pi$$
$$524$$ 8.92394 0.389844
$$525$$ 0 0
$$526$$ 13.7095 0.597762
$$527$$ −14.3153 −0.623583
$$528$$ 5.29590 0.230474
$$529$$ −20.7681 −0.902960
$$530$$ 0 0
$$531$$ 13.8920 0.602862
$$532$$ 0.281896 0.0122218
$$533$$ 0 0
$$534$$ 5.19806 0.224942
$$535$$ 0 0
$$536$$ −20.6746 −0.893005
$$537$$ 25.6896 1.10859
$$538$$ −5.18598 −0.223584
$$539$$ −6.13706 −0.264342
$$540$$ 0 0
$$541$$ −16.1884 −0.695993 −0.347996 0.937496i $$-0.613138\pi$$
−0.347996 + 0.937496i $$0.613138\pi$$
$$542$$ −5.17092 −0.222110
$$543$$ 47.1202 2.02212
$$544$$ −12.5840 −0.539536
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −5.33081 −0.227929 −0.113965 0.993485i $$-0.536355\pi$$
−0.113965 + 0.993485i $$0.536355\pi$$
$$548$$ −8.43967 −0.360525
$$549$$ 7.11423 0.303628
$$550$$ 0 0
$$551$$ 0.408206 0.0173902
$$552$$ 9.03684 0.384633
$$553$$ −23.4993 −0.999293
$$554$$ −10.7942 −0.458600
$$555$$ 0 0
$$556$$ 19.9560 0.846322
$$557$$ 7.39075 0.313156 0.156578 0.987666i $$-0.449954\pi$$
0.156578 + 0.987666i $$0.449954\pi$$
$$558$$ 10.8955 0.461242
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −20.6015 −0.869795
$$562$$ −4.03923 −0.170385
$$563$$ 9.47889 0.399488 0.199744 0.979848i $$-0.435989\pi$$
0.199744 + 0.979848i $$0.435989\pi$$
$$564$$ 22.4306 0.944497
$$565$$ 0 0
$$566$$ −17.7453 −0.745889
$$567$$ 25.8049 1.08370
$$568$$ −23.3260 −0.978738
$$569$$ −10.1438 −0.425249 −0.212624 0.977134i $$-0.568201\pi$$
−0.212624 + 0.977134i $$0.568201\pi$$
$$570$$ 0 0
$$571$$ −14.0925 −0.589751 −0.294876 0.955536i $$-0.595278\pi$$
−0.294876 + 0.955536i $$0.595278\pi$$
$$572$$ 0 0
$$573$$ −32.4403 −1.35521
$$574$$ −21.9095 −0.914483
$$575$$ 0 0
$$576$$ 7.30367 0.304319
$$577$$ 25.1545 1.04720 0.523598 0.851965i $$-0.324589\pi$$
0.523598 + 0.851965i $$0.324589\pi$$
$$578$$ −9.89546 −0.411597
$$579$$ −30.5133 −1.26809
$$580$$ 0 0
$$581$$ −3.79284 −0.157354
$$582$$ −14.5211 −0.601919
$$583$$ 44.1420 1.82817
$$584$$ −18.1323 −0.750319
$$585$$ 0 0
$$586$$ 11.9860 0.495137
$$587$$ 43.8353 1.80928 0.904639 0.426180i $$-0.140141\pi$$
0.904639 + 0.426180i $$0.140141\pi$$
$$588$$ 4.40581 0.181693
$$589$$ 0.584498 0.0240838
$$590$$ 0 0
$$591$$ −1.25906 −0.0517909
$$592$$ 3.15883 0.129827
$$593$$ −24.9965 −1.02648 −0.513242 0.858244i $$-0.671556\pi$$
−0.513242 + 0.858244i $$0.671556\pi$$
$$594$$ −7.27844 −0.298638
$$595$$ 0 0
$$596$$ 5.88231 0.240949
$$597$$ 25.8213 1.05680
$$598$$ 0 0
$$599$$ −6.24027 −0.254971 −0.127485 0.991840i $$-0.540691\pi$$
−0.127485 + 0.991840i $$0.540691\pi$$
$$600$$ 0 0
$$601$$ 6.32975 0.258196 0.129098 0.991632i $$-0.458792\pi$$
0.129098 + 0.991632i $$0.458792\pi$$
$$602$$ −0.559270 −0.0227941
$$603$$ 15.7356 0.640802
$$604$$ 5.35211 0.217774
$$605$$ 0 0
$$606$$ −24.0640 −0.977532
$$607$$ 43.6480 1.77162 0.885809 0.464050i $$-0.153604\pi$$
0.885809 + 0.464050i $$0.153604\pi$$
$$608$$ 0.513811 0.0208378
$$609$$ −24.5254 −0.993820
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 6.00192 0.242613
$$613$$ −25.9541 −1.04827 −0.524137 0.851634i $$-0.675612\pi$$
−0.524137 + 0.851634i $$0.675612\pi$$
$$614$$ 15.3405 0.619092
$$615$$ 0 0
$$616$$ 26.9463 1.08570
$$617$$ 45.9396 1.84946 0.924729 0.380626i $$-0.124291\pi$$
0.924729 + 0.380626i $$0.124291\pi$$
$$618$$ −2.45473 −0.0987437
$$619$$ −6.73556 −0.270725 −0.135363 0.990796i $$-0.543220\pi$$
−0.135363 + 0.990796i $$0.543220\pi$$
$$620$$ 0 0
$$621$$ 3.19269 0.128118
$$622$$ −0.216375 −0.00867583
$$623$$ −6.79895 −0.272394
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 18.7603 0.749813
$$627$$ 0.841166 0.0335930
$$628$$ 6.04461 0.241206
$$629$$ −12.2881 −0.489960
$$630$$ 0 0
$$631$$ −45.0998 −1.79539 −0.897696 0.440614i $$-0.854761\pi$$
−0.897696 + 0.440614i $$0.854761\pi$$
$$632$$ −26.8407 −1.06767
$$633$$ 19.7385 0.784537
$$634$$ −11.2233 −0.445734
$$635$$ 0 0
$$636$$ −31.6896 −1.25658
$$637$$ 0 0
$$638$$ 15.7724 0.624435
$$639$$ 17.7536 0.702322
$$640$$ 0 0
$$641$$ 32.5821 1.28692 0.643458 0.765482i $$-0.277499\pi$$
0.643458 + 0.765482i $$0.277499\pi$$
$$642$$ −5.89008 −0.232463
$$643$$ 25.5754 1.00860 0.504298 0.863530i $$-0.331751\pi$$
0.504298 + 0.863530i $$0.331751\pi$$
$$644$$ −4.77777 −0.188271
$$645$$ 0 0
$$646$$ −0.152603 −0.00600408
$$647$$ 30.1715 1.18616 0.593082 0.805142i $$-0.297911\pi$$
0.593082 + 0.805142i $$0.297911\pi$$
$$648$$ 29.4741 1.15785
$$649$$ 28.7952 1.13031
$$650$$ 0 0
$$651$$ −35.1172 −1.37635
$$652$$ −21.9259 −0.858683
$$653$$ −36.9028 −1.44412 −0.722058 0.691832i $$-0.756804\pi$$
−0.722058 + 0.691832i $$0.756804\pi$$
$$654$$ −28.2935 −1.10636
$$655$$ 0 0
$$656$$ 6.43296 0.251165
$$657$$ 13.8006 0.538413
$$658$$ 13.9051 0.542079
$$659$$ 23.6866 0.922701 0.461350 0.887218i $$-0.347365\pi$$
0.461350 + 0.887218i $$0.347365\pi$$
$$660$$ 0 0
$$661$$ 31.7590 1.23528 0.617641 0.786460i $$-0.288089\pi$$
0.617641 + 0.786460i $$0.288089\pi$$
$$662$$ −14.2916 −0.555458
$$663$$ 0 0
$$664$$ −4.33214 −0.168120
$$665$$ 0 0
$$666$$ 9.35258 0.362405
$$667$$ −6.91856 −0.267888
$$668$$ −21.8694 −0.846152
$$669$$ −5.07606 −0.196252
$$670$$ 0 0
$$671$$ 14.7463 0.569275
$$672$$ −30.8702 −1.19085
$$673$$ 7.50232 0.289193 0.144597 0.989491i $$-0.453812\pi$$
0.144597 + 0.989491i $$0.453812\pi$$
$$674$$ 22.3327 0.860225
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 35.0315 1.34637 0.673184 0.739475i $$-0.264926\pi$$
0.673184 + 0.739475i $$0.264926\pi$$
$$678$$ −21.7114 −0.833821
$$679$$ 18.9933 0.728896
$$680$$ 0 0
$$681$$ −15.6528 −0.599816
$$682$$ 22.5840 0.864787
$$683$$ −24.0834 −0.921524 −0.460762 0.887524i $$-0.652424\pi$$
−0.460762 + 0.887524i $$0.652424\pi$$
$$684$$ −0.245061 −0.00937013
$$685$$ 0 0
$$686$$ 15.9618 0.609426
$$687$$ 54.3183 2.07237
$$688$$ 0.164210 0.00626046
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −2.01447 −0.0766342 −0.0383171 0.999266i $$-0.512200\pi$$
−0.0383171 + 0.999266i $$0.512200\pi$$
$$692$$ −29.2223 −1.11087
$$693$$ −20.5090 −0.779073
$$694$$ −1.20583 −0.0457728
$$695$$ 0 0
$$696$$ −28.0127 −1.06182
$$697$$ −25.0248 −0.947880
$$698$$ 11.3763 0.430598
$$699$$ 6.87800 0.260150
$$700$$ 0 0
$$701$$ −48.8189 −1.84387 −0.921933 0.387350i $$-0.873390\pi$$
−0.921933 + 0.387350i $$0.873390\pi$$
$$702$$ 0 0
$$703$$ 0.501729 0.0189231
$$704$$ 15.1390 0.570572
$$705$$ 0 0
$$706$$ −5.74871 −0.216355
$$707$$ 31.4752 1.18375
$$708$$ −20.6722 −0.776908
$$709$$ 20.8060 0.781385 0.390693 0.920521i $$-0.372236\pi$$
0.390693 + 0.920521i $$0.372236\pi$$
$$710$$ 0 0
$$711$$ 20.4286 0.766134
$$712$$ −7.76569 −0.291032
$$713$$ −9.90648 −0.371000
$$714$$ 9.16852 0.343123
$$715$$ 0 0
$$716$$ −15.5133 −0.579761
$$717$$ 56.5042 2.11019
$$718$$ −15.9468 −0.595128
$$719$$ 21.4306 0.799225 0.399613 0.916684i $$-0.369145\pi$$
0.399613 + 0.916684i $$0.369145\pi$$
$$720$$ 0 0
$$721$$ 3.21073 0.119574
$$722$$ −15.2306 −0.566824
$$723$$ 45.5381 1.69358
$$724$$ −28.4547 −1.05751
$$725$$ 0 0
$$726$$ 12.6799 0.470597
$$727$$ −13.4862 −0.500175 −0.250088 0.968223i $$-0.580459\pi$$
−0.250088 + 0.968223i $$0.580459\pi$$
$$728$$ 0 0
$$729$$ −8.02715 −0.297302
$$730$$ 0 0
$$731$$ −0.638792 −0.0236266
$$732$$ −10.5864 −0.391285
$$733$$ −43.5424 −1.60828 −0.804138 0.594443i $$-0.797373\pi$$
−0.804138 + 0.594443i $$0.797373\pi$$
$$734$$ −0.869167 −0.0320816
$$735$$ 0 0
$$736$$ −8.70841 −0.320996
$$737$$ 32.6165 1.20145
$$738$$ 19.0465 0.701112
$$739$$ −20.0543 −0.737709 −0.368855 0.929487i $$-0.620250\pi$$
−0.368855 + 0.929487i $$0.620250\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −19.6450 −0.721191
$$743$$ −33.1685 −1.21684 −0.608418 0.793617i $$-0.708195\pi$$
−0.608418 + 0.793617i $$0.708195\pi$$
$$744$$ −40.1105 −1.47052
$$745$$ 0 0
$$746$$ 4.91617 0.179994
$$747$$ 3.29722 0.120639
$$748$$ 12.4407 0.454878
$$749$$ 7.70410 0.281502
$$750$$ 0 0
$$751$$ 39.2814 1.43340 0.716700 0.697382i $$-0.245652\pi$$
0.716700 + 0.697382i $$0.245652\pi$$
$$752$$ −4.08277 −0.148883
$$753$$ −53.3008 −1.94239
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −6.83446 −0.248567
$$757$$ 46.6426 1.69526 0.847628 0.530592i $$-0.178030\pi$$
0.847628 + 0.530592i $$0.178030\pi$$
$$758$$ 1.93171 0.0701627
$$759$$ −14.2567 −0.517484
$$760$$ 0 0
$$761$$ 21.8984 0.793818 0.396909 0.917858i $$-0.370083\pi$$
0.396909 + 0.917858i $$0.370083\pi$$
$$762$$ 17.6722 0.640195
$$763$$ 37.0073 1.33975
$$764$$ 19.5899 0.708737
$$765$$ 0 0
$$766$$ −24.3720 −0.880595
$$767$$ 0 0
$$768$$ −31.8756 −1.15021
$$769$$ −46.7096 −1.68439 −0.842196 0.539172i $$-0.818737\pi$$
−0.842196 + 0.539172i $$0.818737\pi$$
$$770$$ 0 0
$$771$$ −31.9614 −1.15106
$$772$$ 18.4263 0.663175
$$773$$ −30.2416 −1.08771 −0.543857 0.839178i $$-0.683037\pi$$
−0.543857 + 0.839178i $$0.683037\pi$$
$$774$$ 0.486189 0.0174757
$$775$$ 0 0
$$776$$ 21.6939 0.778767
$$777$$ −30.1444 −1.08142
$$778$$ −12.7836 −0.458315
$$779$$ 1.02177 0.0366087
$$780$$ 0 0
$$781$$ 36.7995 1.31679
$$782$$ 2.58642 0.0924901
$$783$$ −9.89679 −0.353682
$$784$$ −0.801938 −0.0286406
$$785$$ 0 0
$$786$$ −11.8509 −0.422706
$$787$$ 28.7023 1.02313 0.511563 0.859246i $$-0.329067\pi$$
0.511563 + 0.859246i $$0.329067\pi$$
$$788$$ 0.760316 0.0270851
$$789$$ 38.4131 1.36754
$$790$$ 0 0
$$791$$ 28.3980 1.00972
$$792$$ −23.4252 −0.832378
$$793$$ 0 0
$$794$$ 13.5646 0.481391
$$795$$ 0 0
$$796$$ −15.5929 −0.552674
$$797$$ 18.5418 0.656785 0.328392 0.944541i $$-0.393493\pi$$
0.328392 + 0.944541i $$0.393493\pi$$
$$798$$ −0.374354 −0.0132520
$$799$$ 15.8823 0.561876
$$800$$ 0 0
$$801$$ 5.91053 0.208838
$$802$$ −21.3817 −0.755012
$$803$$ 28.6058 1.00948
$$804$$ −23.4155 −0.825801
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.5308 −0.511508
$$808$$ 35.9506 1.26474
$$809$$ −10.0677 −0.353962 −0.176981 0.984214i $$-0.556633\pi$$
−0.176981 + 0.984214i $$0.556633\pi$$
$$810$$ 0 0
$$811$$ 10.0285 0.352147 0.176074 0.984377i $$-0.443660\pi$$
0.176074 + 0.984377i $$0.443660\pi$$
$$812$$ 14.8103 0.519740
$$813$$ −14.4886 −0.508137
$$814$$ 19.3860 0.679478
$$815$$ 0 0
$$816$$ −2.69202 −0.0942396
$$817$$ 0.0260821 0.000912498 0
$$818$$ −22.8683 −0.799572
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 26.1704 0.913355 0.456677 0.889632i $$-0.349039\pi$$
0.456677 + 0.889632i $$0.349039\pi$$
$$822$$ 11.2078 0.390915
$$823$$ −1.82238 −0.0635242 −0.0317621 0.999495i $$-0.510112\pi$$
−0.0317621 + 0.999495i $$0.510112\pi$$
$$824$$ 3.66727 0.127755
$$825$$ 0 0
$$826$$ −12.8151 −0.445894
$$827$$ −32.2941 −1.12298 −0.561488 0.827485i $$-0.689771\pi$$
−0.561488 + 0.827485i $$0.689771\pi$$
$$828$$ 4.15346 0.144343
$$829$$ 15.1002 0.524453 0.262226 0.965006i $$-0.415543\pi$$
0.262226 + 0.965006i $$0.415543\pi$$
$$830$$ 0 0
$$831$$ −30.2446 −1.04917
$$832$$ 0 0
$$833$$ 3.11960 0.108088
$$834$$ −26.5013 −0.917663
$$835$$ 0 0
$$836$$ −0.507960 −0.0175682
$$837$$ −14.1709 −0.489818
$$838$$ −23.7448 −0.820250
$$839$$ 32.9965 1.13917 0.569584 0.821933i $$-0.307104\pi$$
0.569584 + 0.821933i $$0.307104\pi$$
$$840$$ 0 0
$$841$$ −7.55363 −0.260470
$$842$$ 9.35105 0.322258
$$843$$ −11.3177 −0.389801
$$844$$ −11.9196 −0.410290
$$845$$ 0 0
$$846$$ −12.0881 −0.415599
$$847$$ −16.5851 −0.569870
$$848$$ 5.76809 0.198077
$$849$$ −49.7211 −1.70642
$$850$$ 0 0
$$851$$ −8.50365 −0.291501
$$852$$ −26.4185 −0.905082
$$853$$ −37.7802 −1.29357 −0.646784 0.762673i $$-0.723887\pi$$
−0.646784 + 0.762673i $$0.723887\pi$$
$$854$$ −6.56273 −0.224572
$$855$$ 0 0
$$856$$ 8.79954 0.300762
$$857$$ −27.3623 −0.934677 −0.467339 0.884078i $$-0.654787\pi$$
−0.467339 + 0.884078i $$0.654787\pi$$
$$858$$ 0 0
$$859$$ −20.0629 −0.684538 −0.342269 0.939602i $$-0.611195\pi$$
−0.342269 + 0.939602i $$0.611195\pi$$
$$860$$ 0 0
$$861$$ −61.3889 −2.09213
$$862$$ −3.48773 −0.118792
$$863$$ −6.14483 −0.209173 −0.104586 0.994516i $$-0.533352\pi$$
−0.104586 + 0.994516i $$0.533352\pi$$
$$864$$ −12.4571 −0.423800
$$865$$ 0 0
$$866$$ 11.5386 0.392096
$$867$$ −27.7265 −0.941640
$$868$$ 21.2064 0.719793
$$869$$ 42.3443 1.43643
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 42.2693 1.43142
$$873$$ −16.5114 −0.558827
$$874$$ −0.105604 −0.00357212
$$875$$ 0 0
$$876$$ −20.5362 −0.693853
$$877$$ −13.5077 −0.456123 −0.228061 0.973647i $$-0.573239\pi$$
−0.228061 + 0.973647i $$0.573239\pi$$
$$878$$ −16.2252 −0.547574
$$879$$ 33.5840 1.13276
$$880$$ 0 0
$$881$$ −5.23431 −0.176348 −0.0881741 0.996105i $$-0.528103\pi$$
−0.0881741 + 0.996105i $$0.528103\pi$$
$$882$$ −2.37435 −0.0799487
$$883$$ 4.57301 0.153894 0.0769470 0.997035i $$-0.475483\pi$$
0.0769470 + 0.997035i $$0.475483\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −6.51334 −0.218820
$$887$$ 1.64071 0.0550897 0.0275448 0.999621i $$-0.491231\pi$$
0.0275448 + 0.999621i $$0.491231\pi$$
$$888$$ −34.4306 −1.15541
$$889$$ −23.1148 −0.775246
$$890$$ 0 0
$$891$$ −46.4989 −1.55777
$$892$$ 3.06531 0.102634
$$893$$ −0.648481 −0.0217006
$$894$$ −7.81163 −0.261260
$$895$$ 0 0
$$896$$ 20.7396 0.692862
$$897$$ 0 0
$$898$$ −10.0175 −0.334287
$$899$$ 30.7084 1.02418
$$900$$ 0 0
$$901$$ −22.4383 −0.747529
$$902$$ 39.4795 1.31452
$$903$$ −1.56704 −0.0521478
$$904$$ 32.4359 1.07880
$$905$$ 0 0
$$906$$ −7.10752 −0.236132
$$907$$ −8.10215 −0.269027 −0.134514 0.990912i $$-0.542947\pi$$
−0.134514 + 0.990912i $$0.542947\pi$$
$$908$$ 9.45234 0.313687
$$909$$ −27.3623 −0.907549
$$910$$ 0 0
$$911$$ −9.18119 −0.304187 −0.152093 0.988366i $$-0.548601\pi$$
−0.152093 + 0.988366i $$0.548601\pi$$
$$912$$ 0.109916 0.00363969
$$913$$ 6.83446 0.226188
$$914$$ 4.79656 0.158656
$$915$$ 0 0
$$916$$ −32.8015 −1.08379
$$917$$ 15.5007 0.511877
$$918$$ 3.69979 0.122111
$$919$$ 27.5036 0.907262 0.453631 0.891190i $$-0.350128\pi$$
0.453631 + 0.891190i $$0.350128\pi$$
$$920$$ 0 0
$$921$$ 42.9831 1.41634
$$922$$ 1.64933 0.0543180
$$923$$ 0 0
$$924$$ 30.5187 1.00399
$$925$$ 0 0
$$926$$ −6.77586 −0.222668
$$927$$ −2.79118 −0.0916745
$$928$$ 26.9946 0.886142
$$929$$ −24.2131 −0.794407 −0.397203 0.917731i $$-0.630019\pi$$
−0.397203 + 0.917731i $$0.630019\pi$$
$$930$$ 0 0
$$931$$ −0.127375 −0.00417454
$$932$$ −4.15346 −0.136051
$$933$$ −0.606268 −0.0198483
$$934$$ −26.8761 −0.879412
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −11.1830 −0.365333 −0.182666 0.983175i $$-0.558473\pi$$
−0.182666 + 0.983175i $$0.558473\pi$$
$$938$$ −14.5157 −0.473955
$$939$$ 52.5652 1.71540
$$940$$ 0 0
$$941$$ 15.9638 0.520404 0.260202 0.965554i $$-0.416211\pi$$
0.260202 + 0.965554i $$0.416211\pi$$
$$942$$ −8.02715 −0.261539
$$943$$ −17.3177 −0.563941
$$944$$ 3.76271 0.122466
$$945$$ 0 0
$$946$$ 1.00777 0.0327654
$$947$$ 6.51466 0.211698 0.105849 0.994382i $$-0.466244\pi$$
0.105849 + 0.994382i $$0.466244\pi$$
$$948$$ −30.3991 −0.987317
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −31.4470 −1.01974
$$952$$ −13.6974 −0.443935
$$953$$ −47.6469 −1.54344 −0.771718 0.635965i $$-0.780602\pi$$
−0.771718 + 0.635965i $$0.780602\pi$$
$$954$$ 17.0780 0.552920
$$955$$ 0 0
$$956$$ −34.1215 −1.10357
$$957$$ 44.1933 1.42857
$$958$$ 19.8329 0.640773
$$959$$ −14.6595 −0.473380
$$960$$ 0 0
$$961$$ 12.9705 0.418402
$$962$$ 0 0
$$963$$ −6.69740 −0.215821
$$964$$ −27.4993 −0.885694
$$965$$ 0 0
$$966$$ 6.34481 0.204141
$$967$$ −43.8122 −1.40891 −0.704453 0.709751i $$-0.748808\pi$$
−0.704453 + 0.709751i $$0.748808\pi$$
$$968$$ −18.9433 −0.608861
$$969$$ −0.427583 −0.0137360
$$970$$ 0 0
$$971$$ 4.29483 0.137828 0.0689139 0.997623i $$-0.478047\pi$$
0.0689139 + 0.997623i $$0.478047\pi$$
$$972$$ 24.6823 0.791686
$$973$$ 34.6631 1.11125
$$974$$ −30.2776 −0.970156
$$975$$ 0 0
$$976$$ 1.92692 0.0616792
$$977$$ 26.8019 0.857470 0.428735 0.903430i $$-0.358959\pi$$
0.428735 + 0.903430i $$0.358959\pi$$
$$978$$ 29.1172 0.931066
$$979$$ 12.2513 0.391553
$$980$$ 0 0
$$981$$ −32.1715 −1.02716
$$982$$ 25.1094 0.801275
$$983$$ 27.2495 0.869124 0.434562 0.900642i $$-0.356903\pi$$
0.434562 + 0.900642i $$0.356903\pi$$
$$984$$ −70.1178 −2.23527
$$985$$ 0 0
$$986$$ −8.01746 −0.255328
$$987$$ 38.9614 1.24015
$$988$$ 0 0
$$989$$ −0.442058 −0.0140566
$$990$$ 0 0
$$991$$ 24.3889 0.774740 0.387370 0.921924i $$-0.373384\pi$$
0.387370 + 0.921924i $$0.373384\pi$$
$$992$$ 38.6528 1.22723
$$993$$ −40.0441 −1.27076
$$994$$ −16.3773 −0.519458
$$995$$ 0 0
$$996$$ −4.90648 −0.155468
$$997$$ −31.3207 −0.991935 −0.495967 0.868341i $$-0.665186\pi$$
−0.495967 + 0.868341i $$0.665186\pi$$
$$998$$ −17.2314 −0.545452
$$999$$ −12.1642 −0.384859
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 4225.2.a.bb.1.3 3
5.4 even 2 169.2.a.c.1.1 yes 3
13.12 even 2 4225.2.a.bg.1.1 3
15.14 odd 2 1521.2.a.o.1.3 3
20.19 odd 2 2704.2.a.ba.1.3 3
35.34 odd 2 8281.2.a.bj.1.1 3
65.4 even 6 169.2.c.c.146.1 6
65.9 even 6 169.2.c.b.146.3 6
65.19 odd 12 169.2.e.b.23.5 12
65.24 odd 12 169.2.e.b.147.5 12
65.29 even 6 169.2.c.b.22.3 6
65.34 odd 4 169.2.b.b.168.2 6
65.44 odd 4 169.2.b.b.168.5 6
65.49 even 6 169.2.c.c.22.1 6
65.54 odd 12 169.2.e.b.147.2 12
65.59 odd 12 169.2.e.b.23.2 12
65.64 even 2 169.2.a.b.1.3 3
195.44 even 4 1521.2.b.l.1351.2 6
195.164 even 4 1521.2.b.l.1351.5 6
195.194 odd 2 1521.2.a.r.1.1 3
260.99 even 4 2704.2.f.o.337.5 6
260.239 even 4 2704.2.f.o.337.6 6
260.259 odd 2 2704.2.a.z.1.3 3
455.454 odd 2 8281.2.a.bf.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 65.64 even 2
169.2.a.c.1.1 yes 3 5.4 even 2
169.2.b.b.168.2 6 65.34 odd 4
169.2.b.b.168.5 6 65.44 odd 4
169.2.c.b.22.3 6 65.29 even 6
169.2.c.b.146.3 6 65.9 even 6
169.2.c.c.22.1 6 65.49 even 6
169.2.c.c.146.1 6 65.4 even 6
169.2.e.b.23.2 12 65.59 odd 12
169.2.e.b.23.5 12 65.19 odd 12
169.2.e.b.147.2 12 65.54 odd 12
169.2.e.b.147.5 12 65.24 odd 12
1521.2.a.o.1.3 3 15.14 odd 2
1521.2.a.r.1.1 3 195.194 odd 2
1521.2.b.l.1351.2 6 195.44 even 4
1521.2.b.l.1351.5 6 195.164 even 4
2704.2.a.z.1.3 3 260.259 odd 2
2704.2.a.ba.1.3 3 20.19 odd 2
2704.2.f.o.337.5 6 260.99 even 4
2704.2.f.o.337.6 6 260.239 even 4
4225.2.a.bb.1.3 3 1.1 even 1 trivial
4225.2.a.bg.1.1 3 13.12 even 2
8281.2.a.bf.1.3 3 455.454 odd 2
8281.2.a.bj.1.1 3 35.34 odd 2