# Properties

 Label 3700.2.d.a.149.2 Level $3700$ Weight $2$ Character 3700.149 Analytic conductor $29.545$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(149,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3700.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3700 = 2^{2} \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3700.d (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: $$29.5446487479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 740) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3700.149 Dual form 3700.2.d.a.149.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} +3.00000i q^{7} -6.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} +3.00000i q^{7} -6.00000 q^{9} +5.00000 q^{11} +2.00000i q^{13} -4.00000i q^{17} +4.00000 q^{19} -9.00000 q^{21} +6.00000i q^{23} -9.00000i q^{27} -6.00000 q^{29} -4.00000 q^{31} +15.0000i q^{33} +1.00000i q^{37} -6.00000 q^{39} -9.00000 q^{41} +10.0000i q^{43} +11.0000i q^{47} -2.00000 q^{49} +12.0000 q^{51} -11.0000i q^{53} +12.0000i q^{57} +8.00000 q^{59} -8.00000 q^{61} -18.0000i q^{63} +8.00000i q^{67} -18.0000 q^{69} +3.00000 q^{71} +7.00000i q^{73} +15.0000i q^{77} -8.00000 q^{79} +9.00000 q^{81} -9.00000i q^{83} -18.0000i q^{87} +16.0000 q^{89} -6.00000 q^{91} -12.0000i q^{93} -12.0000i q^{97} -30.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} + 10 q^{11} + 8 q^{19} - 18 q^{21} - 12 q^{29} - 8 q^{31} - 12 q^{39} - 18 q^{41} - 4 q^{49} + 24 q^{51} + 16 q^{59} - 16 q^{61} - 36 q^{69} + 6 q^{71} - 16 q^{79} + 18 q^{81} + 32 q^{89} - 12 q^{91} - 60 q^{99}+O(q^{100})$$ 2 * q - 12 * q^9 + 10 * q^11 + 8 * q^19 - 18 * q^21 - 12 * q^29 - 8 * q^31 - 12 * q^39 - 18 * q^41 - 4 * q^49 + 24 * q^51 + 16 * q^59 - 16 * q^61 - 36 * q^69 + 6 * q^71 - 16 * q^79 + 18 * q^81 + 32 * q^89 - 12 * q^91 - 60 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$1851$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000i 1.73205i 0.500000 + 0.866025i $$0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 0 0
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −9.00000 −1.96396
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 9.00000i − 1.73205i
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 15.0000i 2.61116i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.00000i 0.164399i
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ 10.0000i 1.52499i 0.646997 + 0.762493i $$0.276025\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.0000i 1.60451i 0.596978 + 0.802257i $$0.296368\pi$$
−0.596978 + 0.802257i $$0.703632\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 12.0000 1.68034
$$52$$ 0 0
$$53$$ − 11.0000i − 1.51097i −0.655168 0.755483i $$-0.727402\pi$$
0.655168 0.755483i $$-0.272598\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 12.0000i 1.58944i
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ − 18.0000i − 2.26779i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ −18.0000 −2.16695
$$70$$ 0 0
$$71$$ 3.00000 0.356034 0.178017 0.984027i $$-0.443032\pi$$
0.178017 + 0.984027i $$0.443032\pi$$
$$72$$ 0 0
$$73$$ 7.00000i 0.819288i 0.912245 + 0.409644i $$0.134347\pi$$
−0.912245 + 0.409644i $$0.865653\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 15.0000i 1.70941i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ − 9.00000i − 0.987878i −0.869496 0.493939i $$-0.835557\pi$$
0.869496 0.493939i $$-0.164443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 18.0000i − 1.92980i
$$88$$ 0 0
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ 0 0
$$93$$ − 12.0000i − 1.24434i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 12.0000i − 1.21842i −0.793011 0.609208i $$-0.791488\pi$$
0.793011 0.609208i $$-0.208512\pi$$
$$98$$ 0 0
$$99$$ −30.0000 −3.01511
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ − 6.00000i − 0.591198i −0.955312 0.295599i $$-0.904481\pi$$
0.955312 0.295599i $$-0.0955191\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 12.0000i − 1.10940i
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ − 27.0000i − 2.43451i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 13.0000i 1.15356i 0.816898 + 0.576782i $$0.195692\pi$$
−0.816898 + 0.576782i $$0.804308\pi$$
$$128$$ 0 0
$$129$$ −30.0000 −2.64135
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 12.0000i 1.04053i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −33.0000 −2.77910
$$142$$ 0 0
$$143$$ 10.0000i 0.836242i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 6.00000i − 0.494872i
$$148$$ 0 0
$$149$$ 1.00000 0.0819232 0.0409616 0.999161i $$-0.486958\pi$$
0.0409616 + 0.999161i $$0.486958\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 24.0000i 1.94029i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 3.00000i − 0.239426i −0.992809 0.119713i $$-0.961803\pi$$
0.992809 0.119713i $$-0.0381975\pi$$
$$158$$ 0 0
$$159$$ 33.0000 2.61707
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ 0 0
$$163$$ − 10.0000i − 0.783260i −0.920123 0.391630i $$-0.871911\pi$$
0.920123 0.391630i $$-0.128089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −24.0000 −1.83533
$$172$$ 0 0
$$173$$ − 11.0000i − 0.836315i −0.908375 0.418157i $$-0.862676\pi$$
0.908375 0.418157i $$-0.137324\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 24.0000i 1.80395i
$$178$$ 0 0
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ 25.0000 1.85824 0.929118 0.369784i $$-0.120568\pi$$
0.929118 + 0.369784i $$0.120568\pi$$
$$182$$ 0 0
$$183$$ − 24.0000i − 1.77413i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 20.0000i − 1.46254i
$$188$$ 0 0
$$189$$ 27.0000 1.96396
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 26.0000i 1.87152i 0.352636 + 0.935760i $$0.385285\pi$$
−0.352636 + 0.935760i $$0.614715\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 23.0000i − 1.63868i −0.573306 0.819341i $$-0.694340\pi$$
0.573306 0.819341i $$-0.305660\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ −24.0000 −1.69283
$$202$$ 0 0
$$203$$ − 18.0000i − 1.26335i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 36.0000i − 2.50217i
$$208$$ 0 0
$$209$$ 20.0000 1.38343
$$210$$ 0 0
$$211$$ −11.0000 −0.757271 −0.378636 0.925546i $$-0.623607\pi$$
−0.378636 + 0.925546i $$0.623607\pi$$
$$212$$ 0 0
$$213$$ 9.00000i 0.616670i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 12.0000i − 0.814613i
$$218$$ 0 0
$$219$$ −21.0000 −1.41905
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ 5.00000i 0.334825i 0.985887 + 0.167412i $$0.0535411\pi$$
−0.985887 + 0.167412i $$0.946459\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 16.0000i 1.06196i 0.847385 + 0.530979i $$0.178176\pi$$
−0.847385 + 0.530979i $$0.821824\pi$$
$$228$$ 0 0
$$229$$ 21.0000 1.38772 0.693860 0.720110i $$-0.255909\pi$$
0.693860 + 0.720110i $$0.255909\pi$$
$$230$$ 0 0
$$231$$ −45.0000 −2.96078
$$232$$ 0 0
$$233$$ 14.0000i 0.917170i 0.888650 + 0.458585i $$0.151644\pi$$
−0.888650 + 0.458585i $$0.848356\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 24.0000i − 1.55897i
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000i 0.509028i
$$248$$ 0 0
$$249$$ 27.0000 1.71106
$$250$$ 0 0
$$251$$ −10.0000 −0.631194 −0.315597 0.948893i $$-0.602205\pi$$
−0.315597 + 0.948893i $$0.602205\pi$$
$$252$$ 0 0
$$253$$ 30.0000i 1.88608i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 16.0000i 0.998053i 0.866587 + 0.499026i $$0.166309\pi$$
−0.866587 + 0.499026i $$0.833691\pi$$
$$258$$ 0 0
$$259$$ −3.00000 −0.186411
$$260$$ 0 0
$$261$$ 36.0000 2.22834
$$262$$ 0 0
$$263$$ − 23.0000i − 1.41824i −0.705087 0.709120i $$-0.749092\pi$$
0.705087 0.709120i $$-0.250908\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 48.0000i 2.93755i
$$268$$ 0 0
$$269$$ 22.0000 1.34136 0.670682 0.741745i $$-0.266002\pi$$
0.670682 + 0.741745i $$0.266002\pi$$
$$270$$ 0 0
$$271$$ −5.00000 −0.303728 −0.151864 0.988401i $$-0.548528\pi$$
−0.151864 + 0.988401i $$0.548528\pi$$
$$272$$ 0 0
$$273$$ − 18.0000i − 1.08941i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4.00000i 0.240337i 0.992754 + 0.120168i $$0.0383434\pi$$
−0.992754 + 0.120168i $$0.961657\pi$$
$$278$$ 0 0
$$279$$ 24.0000 1.43684
$$280$$ 0 0
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 27.0000i − 1.59376i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 36.0000 2.11036
$$292$$ 0 0
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 45.0000i − 2.61116i
$$298$$ 0 0
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ −30.0000 −1.72917
$$302$$ 0 0
$$303$$ − 27.0000i − 1.55111i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 23.0000i 1.31268i 0.754466 + 0.656340i $$0.227896\pi$$
−0.754466 + 0.656340i $$0.772104\pi$$
$$308$$ 0 0
$$309$$ 18.0000 1.02398
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ − 22.0000i − 1.24351i −0.783210 0.621757i $$-0.786419\pi$$
0.783210 0.621757i $$-0.213581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ 0 0
$$319$$ −30.0000 −1.67968
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 16.0000i − 0.890264i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 36.0000i − 1.99080i
$$328$$ 0 0
$$329$$ −33.0000 −1.81935
$$330$$ 0 0
$$331$$ −22.0000 −1.20923 −0.604615 0.796518i $$-0.706673\pi$$
−0.604615 + 0.796518i $$0.706673\pi$$
$$332$$ 0 0
$$333$$ − 6.00000i − 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 7.00000i − 0.381314i −0.981657 0.190657i $$-0.938938\pi$$
0.981657 0.190657i $$-0.0610619\pi$$
$$338$$ 0 0
$$339$$ −42.0000 −2.28113
$$340$$ 0 0
$$341$$ −20.0000 −1.08306
$$342$$ 0 0
$$343$$ 15.0000i 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 10.0000i − 0.536828i −0.963304 0.268414i $$-0.913500\pi$$
0.963304 0.268414i $$-0.0864995\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 18.0000 0.960769
$$352$$ 0 0
$$353$$ − 12.0000i − 0.638696i −0.947638 0.319348i $$-0.896536\pi$$
0.947638 0.319348i $$-0.103464\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 36.0000i 1.90532i
$$358$$ 0 0
$$359$$ 21.0000 1.10834 0.554169 0.832404i $$-0.313036\pi$$
0.554169 + 0.832404i $$0.313036\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 42.0000i 2.20443i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ 54.0000 2.81113
$$370$$ 0 0
$$371$$ 33.0000 1.71327
$$372$$ 0 0
$$373$$ 1.00000i 0.0517780i 0.999665 + 0.0258890i $$0.00824165\pi$$
−0.999665 + 0.0258890i $$0.991758\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ −17.0000 −0.873231 −0.436616 0.899648i $$-0.643823\pi$$
−0.436616 + 0.899648i $$0.643823\pi$$
$$380$$ 0 0
$$381$$ −39.0000 −1.99803
$$382$$ 0 0
$$383$$ 36.0000i 1.83951i 0.392488 + 0.919757i $$0.371614\pi$$
−0.392488 + 0.919757i $$0.628386\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 60.0000i − 3.04997i
$$388$$ 0 0
$$389$$ −4.00000 −0.202808 −0.101404 0.994845i $$-0.532333\pi$$
−0.101404 + 0.994845i $$0.532333\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7.00000i − 0.351320i −0.984451 0.175660i $$-0.943794\pi$$
0.984451 0.175660i $$-0.0562059\pi$$
$$398$$ 0 0
$$399$$ −36.0000 −1.80225
$$400$$ 0 0
$$401$$ −10.0000 −0.499376 −0.249688 0.968326i $$-0.580328\pi$$
−0.249688 + 0.968326i $$0.580328\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 5.00000i 0.247841i
$$408$$ 0 0
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ 0 0
$$411$$ 54.0000 2.66362
$$412$$ 0 0
$$413$$ 24.0000i 1.18096i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ −1.00000 −0.0488532 −0.0244266 0.999702i $$-0.507776\pi$$
−0.0244266 + 0.999702i $$0.507776\pi$$
$$420$$ 0 0
$$421$$ 36.0000 1.75453 0.877266 0.480004i $$-0.159365\pi$$
0.877266 + 0.480004i $$0.159365\pi$$
$$422$$ 0 0
$$423$$ − 66.0000i − 3.20903i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 24.0000i − 1.16144i
$$428$$ 0 0
$$429$$ −30.0000 −1.44841
$$430$$ 0 0
$$431$$ −14.0000 −0.674356 −0.337178 0.941441i $$-0.609472\pi$$
−0.337178 + 0.941441i $$0.609472\pi$$
$$432$$ 0 0
$$433$$ − 23.0000i − 1.10531i −0.833410 0.552655i $$-0.813615\pi$$
0.833410 0.552655i $$-0.186385\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 24.0000i 1.14808i
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 0 0
$$443$$ − 9.00000i − 0.427603i −0.976877 0.213801i $$-0.931415\pi$$
0.976877 0.213801i $$-0.0685846\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 3.00000i 0.141895i
$$448$$ 0 0
$$449$$ 24.0000 1.13263 0.566315 0.824189i $$-0.308369\pi$$
0.566315 + 0.824189i $$0.308369\pi$$
$$450$$ 0 0
$$451$$ −45.0000 −2.11897
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 6.00000i − 0.280668i −0.990104 0.140334i $$-0.955182\pi$$
0.990104 0.140334i $$-0.0448177\pi$$
$$458$$ 0 0
$$459$$ −36.0000 −1.68034
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ 14.0000i 0.650635i 0.945605 + 0.325318i $$0.105471\pi$$
−0.945605 + 0.325318i $$0.894529\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 10.0000i − 0.462745i −0.972865 0.231372i $$-0.925678\pi$$
0.972865 0.231372i $$-0.0743216\pi$$
$$468$$ 0 0
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ 9.00000 0.414698
$$472$$ 0 0
$$473$$ 50.0000i 2.29900i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 66.0000i 3.02193i
$$478$$ 0 0
$$479$$ 14.0000 0.639676 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ − 54.0000i − 2.45709i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.00000i 0.181257i 0.995885 + 0.0906287i $$0.0288876\pi$$
−0.995885 + 0.0906287i $$0.971112\pi$$
$$488$$ 0 0
$$489$$ 30.0000 1.35665
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 24.0000i 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 9.00000i 0.403705i
$$498$$ 0 0
$$499$$ 44.0000 1.96971 0.984855 0.173379i $$-0.0554684\pi$$
0.984855 + 0.173379i $$0.0554684\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ 0 0
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 27.0000i 1.19911i
$$508$$ 0 0
$$509$$ −21.0000 −0.930809 −0.465404 0.885098i $$-0.654091\pi$$
−0.465404 + 0.885098i $$0.654091\pi$$
$$510$$ 0 0
$$511$$ −21.0000 −0.928985
$$512$$ 0 0
$$513$$ − 36.0000i − 1.58944i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 55.0000i 2.41890i
$$518$$ 0 0
$$519$$ 33.0000 1.44854
$$520$$ 0 0
$$521$$ −41.0000 −1.79624 −0.898121 0.439748i $$-0.855068\pi$$
−0.898121 + 0.439748i $$0.855068\pi$$
$$522$$ 0 0
$$523$$ − 6.00000i − 0.262362i −0.991358 0.131181i $$-0.958123\pi$$
0.991358 0.131181i $$-0.0418769\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.0000i 0.696971i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −48.0000 −2.08302
$$532$$ 0 0
$$533$$ − 18.0000i − 0.779667i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 6.00000i 0.258919i
$$538$$ 0 0
$$539$$ −10.0000 −0.430730
$$540$$ 0 0
$$541$$ −36.0000 −1.54776 −0.773880 0.633332i $$-0.781687\pi$$
−0.773880 + 0.633332i $$0.781687\pi$$
$$542$$ 0 0
$$543$$ 75.0000i 3.21856i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 16.0000i − 0.684111i −0.939680 0.342055i $$-0.888877\pi$$
0.939680 0.342055i $$-0.111123\pi$$
$$548$$ 0 0
$$549$$ 48.0000 2.04859
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ − 24.0000i − 1.02058i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10.0000i 0.423714i 0.977301 + 0.211857i $$0.0679510\pi$$
−0.977301 + 0.211857i $$0.932049\pi$$
$$558$$ 0 0
$$559$$ −20.0000 −0.845910
$$560$$ 0 0
$$561$$ 60.0000 2.53320
$$562$$ 0 0
$$563$$ 42.0000i 1.77009i 0.465506 + 0.885044i $$0.345872\pi$$
−0.465506 + 0.885044i $$0.654128\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 27.0000i 1.13389i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 33.0000 1.38101 0.690504 0.723329i $$-0.257389\pi$$
0.690504 + 0.723329i $$0.257389\pi$$
$$572$$ 0 0
$$573$$ − 36.0000i − 1.50392i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 28.0000i − 1.16566i −0.812596 0.582828i $$-0.801946\pi$$
0.812596 0.582828i $$-0.198054\pi$$
$$578$$ 0 0
$$579$$ −78.0000 −3.24157
$$580$$ 0 0
$$581$$ 27.0000 1.12015
$$582$$ 0 0
$$583$$ − 55.0000i − 2.27787i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 36.0000i 1.48588i 0.669359 + 0.742940i $$0.266569\pi$$
−0.669359 + 0.742940i $$0.733431\pi$$
$$588$$ 0 0
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 69.0000 2.83828
$$592$$ 0 0
$$593$$ 43.0000i 1.76580i 0.469563 + 0.882899i $$0.344412\pi$$
−0.469563 + 0.882899i $$0.655588\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 6.00000i 0.245564i
$$598$$ 0 0
$$599$$ 45.0000 1.83865 0.919325 0.393499i $$-0.128735\pi$$
0.919325 + 0.393499i $$0.128735\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ − 48.0000i − 1.95471i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 12.0000i 0.487065i 0.969893 + 0.243532i $$0.0783062\pi$$
−0.969893 + 0.243532i $$0.921694\pi$$
$$608$$ 0 0
$$609$$ 54.0000 2.18819
$$610$$ 0 0
$$611$$ −22.0000 −0.890025
$$612$$ 0 0
$$613$$ 27.0000i 1.09052i 0.838267 + 0.545260i $$0.183569\pi$$
−0.838267 + 0.545260i $$0.816431\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 9.00000i − 0.362326i −0.983453 0.181163i $$-0.942014\pi$$
0.983453 0.181163i $$-0.0579862\pi$$
$$618$$ 0 0
$$619$$ −25.0000 −1.00483 −0.502417 0.864625i $$-0.667556\pi$$
−0.502417 + 0.864625i $$0.667556\pi$$
$$620$$ 0 0
$$621$$ 54.0000 2.16695
$$622$$ 0 0
$$623$$ 48.0000i 1.92308i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 60.0000i 2.39617i
$$628$$ 0 0
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −48.0000 −1.91085 −0.955425 0.295234i $$-0.904602\pi$$
−0.955425 + 0.295234i $$0.904602\pi$$
$$632$$ 0 0
$$633$$ − 33.0000i − 1.31163i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 4.00000i − 0.158486i
$$638$$ 0 0
$$639$$ −18.0000 −0.712069
$$640$$ 0 0
$$641$$ 47.0000 1.85639 0.928194 0.372096i $$-0.121361\pi$$
0.928194 + 0.372096i $$0.121361\pi$$
$$642$$ 0 0
$$643$$ 34.0000i 1.34083i 0.741987 + 0.670415i $$0.233884\pi$$
−0.741987 + 0.670415i $$0.766116\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.0000i 1.10079i 0.834903 + 0.550397i $$0.185524\pi$$
−0.834903 + 0.550397i $$0.814476\pi$$
$$648$$ 0 0
$$649$$ 40.0000 1.57014
$$650$$ 0 0
$$651$$ 36.0000 1.41095
$$652$$ 0 0
$$653$$ − 8.00000i − 0.313064i −0.987673 0.156532i $$-0.949969\pi$$
0.987673 0.156532i $$-0.0500315\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 42.0000i − 1.63858i
$$658$$ 0 0
$$659$$ 49.0000 1.90877 0.954384 0.298580i $$-0.0965131\pi$$
0.954384 + 0.298580i $$0.0965131\pi$$
$$660$$ 0 0
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ 0 0
$$663$$ 24.0000i 0.932083i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 36.0000i − 1.39393i
$$668$$ 0 0
$$669$$ −15.0000 −0.579934
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 19.0000i 0.732396i 0.930537 + 0.366198i $$0.119341\pi$$
−0.930537 + 0.366198i $$0.880659\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 7.00000i 0.269032i 0.990911 + 0.134516i $$0.0429479\pi$$
−0.990911 + 0.134516i $$0.957052\pi$$
$$678$$ 0 0
$$679$$ 36.0000 1.38155
$$680$$ 0 0
$$681$$ −48.0000 −1.83936
$$682$$ 0 0
$$683$$ 18.0000i 0.688751i 0.938832 + 0.344375i $$0.111909\pi$$
−0.938832 + 0.344375i $$0.888091\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 63.0000i 2.40360i
$$688$$ 0 0
$$689$$ 22.0000 0.838133
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ − 90.0000i − 3.41882i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000i 1.36360i
$$698$$ 0 0
$$699$$ −42.0000 −1.58859
$$700$$ 0 0
$$701$$ 20.0000 0.755390 0.377695 0.925930i $$-0.376717\pi$$
0.377695 + 0.925930i $$0.376717\pi$$
$$702$$ 0 0
$$703$$ 4.00000i 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 27.0000i − 1.01544i
$$708$$ 0 0
$$709$$ −36.0000 −1.35201 −0.676004 0.736898i $$-0.736290\pi$$
−0.676004 + 0.736898i $$0.736290\pi$$
$$710$$ 0 0
$$711$$ 48.0000 1.80014
$$712$$ 0 0
$$713$$ − 24.0000i − 0.898807i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 18.0000i − 0.672222i
$$718$$ 0 0
$$719$$ 19.0000 0.708580 0.354290 0.935136i $$-0.384723\pi$$
0.354290 + 0.935136i $$0.384723\pi$$
$$720$$ 0 0
$$721$$ 18.0000 0.670355
$$722$$ 0 0
$$723$$ 42.0000i 1.56200i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 4.00000i − 0.148352i −0.997245 0.0741759i $$-0.976367\pi$$
0.997245 0.0741759i $$-0.0236326\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 40.0000 1.47945
$$732$$ 0 0
$$733$$ 43.0000i 1.58824i 0.607760 + 0.794121i $$0.292068\pi$$
−0.607760 + 0.794121i $$0.707932\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 40.0000i 1.47342i
$$738$$ 0 0
$$739$$ 15.0000 0.551784 0.275892 0.961189i $$-0.411027\pi$$
0.275892 + 0.961189i $$0.411027\pi$$
$$740$$ 0 0
$$741$$ −24.0000 −0.881662
$$742$$ 0 0
$$743$$ − 9.00000i − 0.330178i −0.986279 0.165089i $$-0.947209\pi$$
0.986279 0.165089i $$-0.0527911\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 54.0000i 1.97576i
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 11.0000 0.401396 0.200698 0.979653i $$-0.435679\pi$$
0.200698 + 0.979653i $$0.435679\pi$$
$$752$$ 0 0
$$753$$ − 30.0000i − 1.09326i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 0 0
$$759$$ −90.0000 −3.26679
$$760$$ 0 0
$$761$$ −3.00000 −0.108750 −0.0543750 0.998521i $$-0.517317\pi$$
−0.0543750 + 0.998521i $$0.517317\pi$$
$$762$$ 0 0
$$763$$ − 36.0000i − 1.30329i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 16.0000i 0.577727i
$$768$$ 0 0
$$769$$ −42.0000 −1.51456 −0.757279 0.653091i $$-0.773472\pi$$
−0.757279 + 0.653091i $$0.773472\pi$$
$$770$$ 0 0
$$771$$ −48.0000 −1.72868
$$772$$ 0 0
$$773$$ 35.0000i 1.25886i 0.777056 + 0.629431i $$0.216712\pi$$
−0.777056 + 0.629431i $$0.783288\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 9.00000i − 0.322873i
$$778$$ 0 0
$$779$$ −36.0000 −1.28983
$$780$$ 0 0
$$781$$ 15.0000 0.536742
$$782$$ 0 0
$$783$$ 54.0000i 1.92980i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 21.0000i − 0.748569i −0.927314 0.374285i $$-0.877888\pi$$
0.927314 0.374285i $$-0.122112\pi$$
$$788$$ 0 0
$$789$$ 69.0000 2.45647
$$790$$ 0 0
$$791$$ −42.0000 −1.49335
$$792$$ 0 0
$$793$$ − 16.0000i − 0.568177i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 12.0000i − 0.425062i −0.977154 0.212531i $$-0.931829\pi$$
0.977154 0.212531i $$-0.0681706\pi$$
$$798$$ 0 0
$$799$$ 44.0000 1.55661
$$800$$ 0 0
$$801$$ −96.0000 −3.39199
$$802$$ 0 0
$$803$$ 35.0000i 1.23512i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 66.0000i 2.32331i
$$808$$ 0 0
$$809$$ 46.0000 1.61727 0.808637 0.588308i $$-0.200206\pi$$
0.808637 + 0.588308i $$0.200206\pi$$
$$810$$ 0 0
$$811$$ 49.0000 1.72062 0.860311 0.509769i $$-0.170269\pi$$
0.860311 + 0.509769i $$0.170269\pi$$
$$812$$ 0 0
$$813$$ − 15.0000i − 0.526073i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 40.0000i 1.39942i
$$818$$ 0 0
$$819$$ 36.0000 1.25794
$$820$$ 0 0
$$821$$ 21.0000 0.732905 0.366453 0.930437i $$-0.380572\pi$$
0.366453 + 0.930437i $$0.380572\pi$$
$$822$$ 0 0
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30.0000i 1.04320i 0.853189 + 0.521601i $$0.174665\pi$$
−0.853189 + 0.521601i $$0.825335\pi$$
$$828$$ 0 0
$$829$$ −52.0000 −1.80603 −0.903017 0.429604i $$-0.858653\pi$$
−0.903017 + 0.429604i $$0.858653\pi$$
$$830$$ 0 0
$$831$$ −12.0000 −0.416275
$$832$$ 0 0
$$833$$ 8.00000i 0.277184i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 36.0000i 1.24434i
$$838$$ 0 0
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ − 12.0000i − 0.413302i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 42.0000i 1.44314i
$$848$$ 0 0
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 12.0000i 0.409912i 0.978771 + 0.204956i $$0.0657052\pi$$
−0.978771 + 0.204956i $$0.934295\pi$$
$$858$$ 0 0
$$859$$ −56.0000 −1.91070 −0.955348 0.295484i $$-0.904519\pi$$
−0.955348 + 0.295484i $$0.904519\pi$$
$$860$$ 0 0
$$861$$ 81.0000 2.76047
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.00000i 0.101885i
$$868$$ 0 0
$$869$$ −40.0000 −1.35691
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 0 0
$$873$$ 72.0000i 2.43683i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 54.0000i 1.82345i 0.410801 + 0.911725i $$0.365249\pi$$
−0.410801 + 0.911725i $$0.634751\pi$$
$$878$$ 0 0
$$879$$ −90.0000 −3.03562
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ − 16.0000i − 0.538443i −0.963078 0.269221i $$-0.913234\pi$$
0.963078 0.269221i $$-0.0867663\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 37.0000i 1.24234i 0.783676 + 0.621169i $$0.213342\pi$$
−0.783676 + 0.621169i $$0.786658\pi$$
$$888$$ 0 0
$$889$$ −39.0000 −1.30802
$$890$$ 0 0
$$891$$ 45.0000 1.50756
$$892$$ 0 0
$$893$$ 44.0000i 1.47240i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 36.0000i − 1.20201i
$$898$$ 0 0
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −44.0000 −1.46585
$$902$$ 0 0
$$903$$ − 90.0000i − 2.99501i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ 0 0
$$909$$ 54.0000 1.79107
$$910$$ 0 0
$$911$$ 42.0000 1.39152 0.695761 0.718273i $$-0.255067\pi$$
0.695761 + 0.718273i $$0.255067\pi$$
$$912$$ 0 0
$$913$$ − 45.0000i − 1.48928i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −14.0000 −0.461817 −0.230909 0.972975i $$-0.574170\pi$$
−0.230909 + 0.972975i $$0.574170\pi$$
$$920$$ 0 0
$$921$$ −69.0000 −2.27363
$$922$$ 0 0
$$923$$ 6.00000i 0.197492i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 36.0000i 1.18240i
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −8.00000 −0.262189
$$932$$ 0 0
$$933$$ 36.0000i 1.17859i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 13.0000i − 0.424691i −0.977195 0.212346i $$-0.931890\pi$$
0.977195 0.212346i $$-0.0681103\pi$$
$$938$$ 0 0
$$939$$ 66.0000 2.15383
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ − 54.0000i − 1.75848i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 8.00000i − 0.259965i −0.991516 0.129983i $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 0 0
$$951$$ 42.0000 1.36194
$$952$$ 0 0
$$953$$ − 51.0000i − 1.65205i −0.563632 0.826026i $$-0.690596\pi$$
0.563632 0.826026i $$-0.309404\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 90.0000i − 2.90929i
$$958$$ 0 0
$$959$$ 54.0000 1.74375
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 24.0000i 0.773389i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2.00000i 0.0643157i 0.999483 + 0.0321578i $$0.0102379\pi$$
−0.999483 + 0.0321578i $$0.989762\pi$$
$$968$$ 0 0
$$969$$ 48.0000 1.54198
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 0 0
$$973$$ 12.0000i 0.384702i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 28.0000i − 0.895799i −0.894084 0.447900i $$-0.852172\pi$$
0.894084 0.447900i $$-0.147828\pi$$
$$978$$ 0 0
$$979$$ 80.0000 2.55681
$$980$$ 0 0
$$981$$ 72.0000 2.29878
$$982$$ 0 0
$$983$$ 27.0000i 0.861166i 0.902551 + 0.430583i $$0.141692\pi$$
−0.902551 + 0.430583i $$0.858308\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 99.0000i − 3.15120i
$$988$$ 0 0
$$989$$ −60.0000 −1.90789
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 0 0
$$993$$ − 66.0000i − 2.09445i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 46.0000i − 1.45683i −0.685134 0.728417i $$-0.740256\pi$$
0.685134 0.728417i $$-0.259744\pi$$
$$998$$ 0 0
$$999$$ 9.00000 0.284747
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 3700.2.d.a.149.2 2
5.2 odd 4 740.2.a.c.1.1 1
5.3 odd 4 3700.2.a.a.1.1 1
5.4 even 2 inner 3700.2.d.a.149.1 2
15.2 even 4 6660.2.a.b.1.1 1
20.7 even 4 2960.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
740.2.a.c.1.1 1 5.2 odd 4
2960.2.a.a.1.1 1 20.7 even 4
3700.2.a.a.1.1 1 5.3 odd 4
3700.2.d.a.149.1 2 5.4 even 2 inner
3700.2.d.a.149.2 2 1.1 even 1 trivial
6660.2.a.b.1.1 1 15.2 even 4