# Properties

 Label 725.2.b.a.349.1 Level $725$ Weight $2$ Character 725.349 Analytic conductor $5.789$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$5.78915414654$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 725.349 Dual form 725.2.b.a.349.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} -6.00000 q^{11} -2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -3.00000i q^{18} +2.00000 q^{19} +6.00000i q^{22} -2.00000i q^{23} -2.00000 q^{26} -2.00000i q^{28} +1.00000 q^{29} +2.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} +3.00000 q^{36} +10.0000i q^{37} -2.00000i q^{38} +2.00000 q^{41} -8.00000i q^{43} -6.00000 q^{44} -2.00000 q^{46} -12.0000i q^{47} +3.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -6.00000 q^{56} -1.00000i q^{58} +8.00000 q^{59} -6.00000 q^{61} -2.00000i q^{62} -6.00000i q^{63} -7.00000 q^{64} +2.00000i q^{67} -2.00000i q^{68} -12.0000 q^{71} -9.00000i q^{72} +6.00000i q^{73} +10.0000 q^{74} +2.00000 q^{76} +12.0000i q^{77} +10.0000 q^{79} +9.00000 q^{81} -2.00000i q^{82} +14.0000i q^{83} -8.00000 q^{86} +18.0000i q^{88} -18.0000 q^{89} -4.00000 q^{91} -2.00000i q^{92} -12.0000 q^{94} +2.00000i q^{97} -3.00000i q^{98} -18.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^9 $$2 q + 2 q^{4} + 6 q^{9} - 12 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{19} - 4 q^{26} + 2 q^{29} + 4 q^{31} - 4 q^{34} + 6 q^{36} + 4 q^{41} - 12 q^{44} - 4 q^{46} + 6 q^{49} - 12 q^{56} + 16 q^{59} - 12 q^{61} - 14 q^{64} - 24 q^{71} + 20 q^{74} + 4 q^{76} + 20 q^{79} + 18 q^{81} - 16 q^{86} - 36 q^{89} - 8 q^{91} - 24 q^{94} - 36 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^9 - 12 * q^11 - 4 * q^14 - 2 * q^16 + 4 * q^19 - 4 * q^26 + 2 * q^29 + 4 * q^31 - 4 * q^34 + 6 * q^36 + 4 * q^41 - 12 * q^44 - 4 * q^46 + 6 * q^49 - 12 * q^56 + 16 * q^59 - 12 * q^61 - 14 * q^64 - 24 * q^71 + 20 * q^74 + 4 * q^76 + 20 * q^79 + 18 * q^81 - 16 * q^86 - 36 * q^89 - 8 * q^91 - 24 * q^94 - 36 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i −0.935414 0.353553i $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 3.00000i − 1.06066i
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 3.00000i − 0.707107i
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 6.00000i 1.27920i
$$23$$ − 2.00000i − 0.417029i −0.978019 0.208514i $$-0.933137\pi$$
0.978019 0.208514i $$-0.0668628\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ − 2.00000i − 0.377964i
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ − 5.00000i − 0.883883i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 2.00000i − 0.324443i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −6.00000 −0.801784
$$57$$ 0 0
$$58$$ − 1.00000i − 0.131306i
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ − 2.00000i − 0.254000i
$$63$$ − 6.00000i − 0.755929i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ − 9.00000i − 1.06066i
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 12.0000i 1.36753i
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ − 2.00000i − 0.220863i
$$83$$ 14.0000i 1.53670i 0.640030 + 0.768350i $$0.278922\pi$$
−0.640030 + 0.768350i $$0.721078\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 18.0000i 1.91881i
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ − 2.00000i − 0.208514i
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ −18.0000 −1.80907
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 6.00000i 0.591198i 0.955312 + 0.295599i $$0.0955191\pi$$
−0.955312 + 0.295599i $$0.904481\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000i 0.188982i
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.00000 0.0928477
$$117$$ − 6.00000i − 0.554700i
$$118$$ − 8.00000i − 0.736460i
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 6.00000i 0.543214i
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ − 3.00000i − 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ − 4.00000i − 0.346844i
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 12.0000i 1.00702i
$$143$$ 12.0000i 1.00349i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ 10.0000i 0.821995i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ − 6.00000i − 0.486664i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 12.0000 0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000i 1.75579i 0.478852 + 0.877896i $$0.341053\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ − 9.00000i − 0.707107i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 14.0000 1.08661
$$167$$ 18.0000i 1.39288i 0.717614 + 0.696441i $$0.245234\pi$$
−0.717614 + 0.696441i $$0.754766\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ − 8.00000i − 0.609994i
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ 0 0
$$178$$ 18.0000i 1.34916i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000i 0.877527i
$$188$$ − 12.0000i − 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −22.0000 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$192$$ 0 0
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 18.0000i 1.27920i
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 10.0000i − 0.703598i
$$203$$ − 2.00000i − 0.140372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 6.00000 0.418040
$$207$$ − 6.00000i − 0.417029i
$$208$$ 2.00000i 0.138675i
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ 6.00000 0.410152
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 4.00000i − 0.271538i
$$218$$ − 14.0000i − 0.948200i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 14.0000i 0.937509i 0.883328 + 0.468755i $$0.155297\pi$$
−0.883328 + 0.468755i $$0.844703\pi$$
$$224$$ −10.0000 −0.668153
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 22.0000i 1.46019i 0.683345 + 0.730096i $$0.260525\pi$$
−0.683345 + 0.730096i $$0.739475\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 3.00000i − 0.196960i
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 4.00000i 0.259281i
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ − 25.0000i − 1.60706i
$$243$$ 0 0
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4.00000i − 0.254514i
$$248$$ − 6.00000i − 0.381000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ − 6.00000i − 0.377964i
$$253$$ 12.0000i 0.754434i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 30.0000i − 1.87135i −0.352865 0.935674i $$-0.614792\pi$$
0.352865 0.935674i $$-0.385208\pi$$
$$258$$ 0 0
$$259$$ 20.0000 1.24274
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ − 14.0000i − 0.864923i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ 0 0
$$268$$ 2.00000i 0.122169i
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.0000i 1.08152i 0.841178 + 0.540758i $$0.181862\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ − 22.0000i − 1.30776i −0.756596 0.653882i $$-0.773139\pi$$
0.756596 0.653882i $$-0.226861\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ − 4.00000i − 0.236113i
$$288$$ − 15.0000i − 0.883883i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 6.00000i 0.351123i
$$293$$ 2.00000i 0.116841i 0.998292 + 0.0584206i $$0.0186065\pi$$
−0.998292 + 0.0584206i $$0.981394\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 30.0000 1.74371
$$297$$ 0 0
$$298$$ − 10.0000i − 0.579284i
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 4.00000i 0.230174i
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 12.0000i 0.683763i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −22.0000 −1.24751 −0.623753 0.781622i $$-0.714393\pi$$
−0.623753 + 0.781622i $$0.714393\pi$$
$$312$$ 0 0
$$313$$ − 2.00000i − 0.113047i −0.998401 0.0565233i $$-0.981998\pi$$
0.998401 0.0565233i $$-0.0180015\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 4.00000i 0.222911i
$$323$$ − 4.00000i − 0.222566i
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ − 6.00000i − 0.331295i
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ −18.0000 −0.989369 −0.494685 0.869072i $$-0.664716\pi$$
−0.494685 + 0.869072i $$0.664716\pi$$
$$332$$ 14.0000i 0.768350i
$$333$$ 30.0000i 1.64399i
$$334$$ 18.0000 0.984916
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ − 6.00000i − 0.324443i
$$343$$ − 20.0000i − 1.07990i
$$344$$ −24.0000 −1.29399
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 6.00000i − 0.322097i −0.986947 0.161048i $$-0.948512\pi$$
0.986947 0.161048i $$-0.0514875\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 30.0000i 1.59901i
$$353$$ 14.0000i 0.745145i 0.928003 + 0.372572i $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ −22.0000 −1.16112 −0.580558 0.814219i $$-0.697165\pi$$
−0.580558 + 0.814219i $$0.697165\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 6.00000i − 0.315353i
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ 2.00000i 0.104257i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ − 18.0000i − 0.932005i −0.884783 0.466002i $$-0.845694\pi$$
0.884783 0.466002i $$-0.154306\pi$$
$$374$$ 12.0000 0.620505
$$375$$ 0 0
$$376$$ −36.0000 −1.85656
$$377$$ − 2.00000i − 0.103005i
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 22.0000i 1.12562i
$$383$$ − 14.0000i − 0.715367i −0.933843 0.357683i $$-0.883567\pi$$
0.933843 0.357683i $$-0.116433\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ − 24.0000i − 1.21999i
$$388$$ 2.00000i 0.101535i
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ − 9.00000i − 0.454569i
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ −18.0000 −0.904534
$$397$$ − 30.0000i − 1.50566i −0.658217 0.752828i $$-0.728689\pi$$
0.658217 0.752828i $$-0.271311\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ − 4.00000i − 0.199254i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ − 60.0000i − 2.97409i
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 6.00000i 0.295599i
$$413$$ − 16.0000i − 0.787309i
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ 12.0000i 0.586939i
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ − 14.0000i − 0.681509i
$$423$$ − 36.0000i − 1.75038i
$$424$$ 18.0000 0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 12.0000i 0.580721i
$$428$$ 6.00000i 0.290021i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ − 4.00000i − 0.191346i
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 4.00000i 0.190261i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ 0 0
$$448$$ 14.0000i 0.661438i
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ − 2.00000i − 0.0940721i
$$453$$ 0 0
$$454$$ 22.0000 1.03251
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 38.0000i − 1.77757i −0.458329 0.888783i $$-0.651552\pi$$
0.458329 0.888783i $$-0.348448\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ − 22.0000i − 1.02243i −0.859454 0.511213i $$-0.829196\pi$$
0.859454 0.511213i $$-0.170804\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ − 6.00000i − 0.277350i
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 24.0000i − 1.10469i
$$473$$ 48.0000i 2.20704i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ 18.0000i 0.824163i
$$478$$ − 12.0000i − 0.548867i
$$479$$ 14.0000 0.639676 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 26.0000i 1.18427i
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 34.0000i − 1.54069i −0.637629 0.770344i $$-0.720085\pi$$
0.637629 0.770344i $$-0.279915\pi$$
$$488$$ 18.0000i 0.814822i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −2.00000 −0.0902587 −0.0451294 0.998981i $$-0.514370\pi$$
−0.0451294 + 0.998981i $$0.514370\pi$$
$$492$$ 0 0
$$493$$ − 2.00000i − 0.0900755i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 24.0000i 1.07655i
$$498$$ 0 0
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 6.00000i 0.267793i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ −18.0000 −0.801784
$$505$$ 0 0
$$506$$ 12.0000 0.533465
$$507$$ 0 0
$$508$$ − 16.0000i − 0.709885i
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ 11.0000i 0.486136i
$$513$$ 0 0
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 72.0000i 3.16656i
$$518$$ − 20.0000i − 0.878750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ − 3.00000i − 0.131306i
$$523$$ − 42.0000i − 1.83653i −0.395964 0.918266i $$-0.629590\pi$$
0.395964 0.918266i $$-0.370410\pi$$
$$524$$ 14.0000 0.611593
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ − 4.00000i − 0.174243i
$$528$$ 0 0
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ − 4.00000i − 0.173422i
$$533$$ − 4.00000i − 0.173259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.00000 0.259161
$$537$$ 0 0
$$538$$ 26.0000i 1.12094i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ −14.0000 −0.601907 −0.300954 0.953639i $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ 2.00000i 0.0859074i
$$543$$ 0 0
$$544$$ −10.0000 −0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 2.00000i − 0.0855138i −0.999086 0.0427569i $$-0.986386\pi$$
0.999086 0.0427569i $$-0.0136141\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ −18.0000 −0.768221
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ − 20.0000i − 0.850487i
$$554$$ 18.0000 0.764747
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 22.0000i − 0.932170i −0.884740 0.466085i $$-0.845664\pi$$
0.884740 0.466085i $$-0.154336\pi$$
$$558$$ − 6.00000i − 0.254000i
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 22.0000i − 0.928014i
$$563$$ − 20.0000i − 0.842900i −0.906852 0.421450i $$-0.861521\pi$$
0.906852 0.421450i $$-0.138479\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ − 18.0000i − 0.755929i
$$568$$ 36.0000i 1.51053i
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 12.0000i 0.501745i
$$573$$ 0 0
$$574$$ −4.00000 −0.166957
$$575$$ 0 0
$$576$$ −21.0000 −0.875000
$$577$$ 22.0000i 0.915872i 0.888985 + 0.457936i $$0.151411\pi$$
−0.888985 + 0.457936i $$0.848589\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 28.0000 1.16164
$$582$$ 0 0
$$583$$ − 36.0000i − 1.49097i
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ 2.00000 0.0826192
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 10.0000i − 0.410997i
$$593$$ 14.0000i 0.574911i 0.957794 + 0.287456i $$0.0928094\pi$$
−0.957794 + 0.287456i $$0.907191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 4.00000i 0.163572i
$$599$$ 18.0000 0.735460 0.367730 0.929933i $$-0.380135\pi$$
0.367730 + 0.929933i $$0.380135\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ 6.00000i 0.244339i
$$604$$ −4.00000 −0.162758
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 28.0000i 1.13648i 0.822861 + 0.568242i $$0.192376\pi$$
−0.822861 + 0.568242i $$0.807624\pi$$
$$608$$ − 10.0000i − 0.405554i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ − 6.00000i − 0.242536i
$$613$$ 6.00000i 0.242338i 0.992632 + 0.121169i $$0.0386643\pi$$
−0.992632 + 0.121169i $$0.961336\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 36.0000 1.45048
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 0 0
$$619$$ −26.0000 −1.04503 −0.522514 0.852631i $$-0.675006\pi$$
−0.522514 + 0.852631i $$0.675006\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 22.0000i 0.882120i
$$623$$ 36.0000i 1.44231i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ 22.0000i 0.877896i
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ − 30.0000i − 1.19334i
$$633$$ 0 0
$$634$$ −14.0000 −0.556011
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6.00000i − 0.237729i
$$638$$ 6.00000i 0.237542i
$$639$$ −36.0000 −1.42414
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ 26.0000i 1.02534i 0.858586 + 0.512670i $$0.171344\pi$$
−0.858586 + 0.512670i $$0.828656\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ − 6.00000i − 0.235884i −0.993020 0.117942i $$-0.962370\pi$$
0.993020 0.117942i $$-0.0376297\pi$$
$$648$$ − 27.0000i − 1.06066i
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.00000i − 0.156652i
$$653$$ 34.0000i 1.33052i 0.746611 + 0.665261i $$0.231680\pi$$
−0.746611 + 0.665261i $$0.768320\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 18.0000i 0.702247i
$$658$$ 24.0000i 0.935617i
$$659$$ 26.0000 1.01282 0.506408 0.862294i $$-0.330973\pi$$
0.506408 + 0.862294i $$0.330973\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 18.0000i 0.699590i
$$663$$ 0 0
$$664$$ 42.0000 1.62992
$$665$$ 0 0
$$666$$ 30.0000 1.16248
$$667$$ − 2.00000i − 0.0774403i
$$668$$ 18.0000i 0.696441i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 36.0000 1.38976
$$672$$ 0 0
$$673$$ 14.0000i 0.539660i 0.962908 + 0.269830i $$0.0869676\pi$$
−0.962908 + 0.269830i $$0.913032\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12.0000i 0.459504i
$$683$$ − 18.0000i − 0.688751i −0.938832 0.344375i $$-0.888091\pi$$
0.938832 0.344375i $$-0.111909\pi$$
$$684$$ 6.00000 0.229416
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 36.0000i 1.36753i
$$694$$ −6.00000 −0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 4.00000i − 0.151511i
$$698$$ 34.0000i 1.28692i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 20.0000i 0.754314i
$$704$$ 42.0000 1.58293
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ − 20.0000i − 0.752177i
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ 30.0000 1.12509
$$712$$ 54.0000i 2.02374i
$$713$$ − 4.00000i − 0.149801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 22.0000i 0.821033i
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 15.0000i 0.558242i
$$723$$ 0 0
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 32.0000i − 1.18681i −0.804902 0.593407i $$-0.797782\pi$$
0.804902 0.593407i $$-0.202218\pi$$
$$728$$ 12.0000i 0.444750i
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ 0 0
$$733$$ 30.0000i 1.10808i 0.832492 + 0.554038i $$0.186914\pi$$
−0.832492 + 0.554038i $$0.813086\pi$$
$$734$$ 24.0000 0.885856
$$735$$ 0 0
$$736$$ −10.0000 −0.368605
$$737$$ − 12.0000i − 0.442026i
$$738$$ − 6.00000i − 0.220863i
$$739$$ −46.0000 −1.69214 −0.846069 0.533074i $$-0.821037\pi$$
−0.846069 + 0.533074i $$0.821037\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 12.0000i − 0.440534i
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −18.0000 −0.659027
$$747$$ 42.0000i 1.53670i
$$748$$ 12.0000i 0.438763i
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ 0 0
$$754$$ −2.00000 −0.0728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 26.0000i − 0.944986i −0.881334 0.472493i $$-0.843354\pi$$
0.881334 0.472493i $$-0.156646\pi$$
$$758$$ 6.00000i 0.217930i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ − 28.0000i − 1.01367i
$$764$$ −22.0000 −0.795932
$$765$$ 0 0
$$766$$ −14.0000 −0.505841
$$767$$ − 16.0000i − 0.577727i
$$768$$ 0 0
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 10.0000i 0.359908i
$$773$$ − 10.0000i − 0.359675i −0.983696 0.179838i $$-0.942443\pi$$
0.983696 0.179838i $$-0.0575572\pi$$
$$774$$ −24.0000 −0.862662
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ − 14.0000i − 0.501924i
$$779$$ 4.00000 0.143315
$$780$$ 0 0
$$781$$ 72.0000 2.57636
$$782$$ 4.00000i 0.143040i
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 18.0000i 0.641631i 0.947142 + 0.320815i $$0.103957\pi$$
−0.947142 + 0.320815i $$0.896043\pi$$
$$788$$ 2.00000i 0.0712470i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 54.0000i 1.91881i
$$793$$ 12.0000i 0.426132i
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ − 34.0000i − 1.20434i −0.798367 0.602171i $$-0.794303\pi$$
0.798367 0.602171i $$-0.205697\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ −54.0000 −1.90800
$$802$$ 14.0000i 0.494357i
$$803$$ − 36.0000i − 1.27041i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ − 30.0000i − 1.05540i
$$809$$ 14.0000 0.492214 0.246107 0.969243i $$-0.420849\pi$$
0.246107 + 0.969243i $$0.420849\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ − 2.00000i − 0.0701862i
$$813$$ 0 0
$$814$$ −60.0000 −2.10300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 22.0000i − 0.769212i
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 0 0
$$823$$ − 4.00000i − 0.139431i −0.997567 0.0697156i $$-0.977791\pi$$
0.997567 0.0697156i $$-0.0222092\pi$$
$$824$$ 18.0000 0.627060
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ − 48.0000i − 1.66912i −0.550914 0.834562i $$-0.685721\pi$$
0.550914 0.834562i $$-0.314279\pi$$
$$828$$ − 6.00000i − 0.208514i
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 14.0000i 0.485363i
$$833$$ − 6.00000i − 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ 0 0
$$838$$ 4.00000i 0.138178i
$$839$$ 14.0000 0.483334 0.241667 0.970359i $$-0.422306\pi$$
0.241667 + 0.970359i $$0.422306\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 2.00000i − 0.0689246i
$$843$$ 0 0
$$844$$ 14.0000 0.481900
$$845$$ 0 0
$$846$$ −36.0000 −1.23771
$$847$$ − 50.0000i − 1.71802i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 20.0000 0.685591
$$852$$ 0 0
$$853$$ 22.0000i 0.753266i 0.926363 + 0.376633i $$0.122918\pi$$
−0.926363 + 0.376633i $$0.877082\pi$$
$$854$$ 12.0000 0.410632
$$855$$ 0 0
$$856$$ 18.0000 0.615227
$$857$$ 58.0000i 1.98124i 0.136637 + 0.990621i $$0.456370\pi$$
−0.136637 + 0.990621i $$0.543630\pi$$
$$858$$ 0 0
$$859$$ −18.0000 −0.614152 −0.307076 0.951685i $$-0.599351\pi$$
−0.307076 + 0.951685i $$0.599351\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 24.0000i 0.817443i
$$863$$ 42.0000i 1.42970i 0.699280 + 0.714848i $$0.253504\pi$$
−0.699280 + 0.714848i $$0.746496\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ − 4.00000i − 0.135769i
$$869$$ −60.0000 −2.03536
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ − 42.0000i − 1.42230i
$$873$$ 6.00000i 0.203069i
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 38.0000i − 1.28317i −0.767052 0.641584i $$-0.778277\pi$$
0.767052 0.641584i $$-0.221723\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ 50.0000i 1.68263i 0.540542 + 0.841317i $$0.318219\pi$$
−0.540542 + 0.841317i $$0.681781\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ 0 0
$$889$$ −32.0000 −1.07325
$$890$$ 0 0
$$891$$ −54.0000 −1.80907
$$892$$ 14.0000i 0.468755i
$$893$$ − 24.0000i − 0.803129i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ 10.0000i 0.333704i
$$899$$ 2.00000 0.0667037
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 12.0000i 0.399556i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 24.0000i 0.796907i 0.917189 + 0.398453i $$0.130453\pi$$
−0.917189 + 0.398453i $$0.869547\pi$$
$$908$$ 22.0000i 0.730096i
$$909$$ 30.0000 0.995037
$$910$$ 0 0
$$911$$ 14.0000 0.463841 0.231920 0.972735i $$-0.425499\pi$$
0.231920 + 0.972735i $$0.425499\pi$$
$$912$$ 0 0
$$913$$ − 84.0000i − 2.77999i
$$914$$ −38.0000 −1.25693
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ − 28.0000i − 0.924641i
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 18.0000i − 0.592798i
$$923$$ 24.0000i 0.789970i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −22.0000 −0.722965
$$927$$ 18.0000i 0.591198i
$$928$$ − 5.00000i − 0.164133i
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ − 18.0000i − 0.589610i
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ −18.0000 −0.588348
$$937$$ 18.0000i 0.588034i 0.955800 + 0.294017i $$0.0949923\pi$$
−0.955800 + 0.294017i $$0.905008\pi$$
$$938$$ − 4.00000i − 0.130605i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −26.0000 −0.847576 −0.423788 0.905761i $$-0.639300\pi$$
−0.423788 + 0.905761i $$0.639300\pi$$
$$942$$ 0 0
$$943$$ − 4.00000i − 0.130258i
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ 60.0000i 1.94974i 0.222779 + 0.974869i $$0.428487\pi$$
−0.222779 + 0.974869i $$0.571513\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 12.0000i 0.388922i
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ 18.0000 0.582772
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ − 14.0000i − 0.452319i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 20.0000i − 0.644826i
$$963$$ 18.0000i 0.580042i
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 20.0000i − 0.643157i −0.946883 0.321578i $$-0.895787\pi$$
0.946883 0.321578i $$-0.104213\pi$$
$$968$$ − 75.0000i − 2.41059i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −34.0000 −1.08943
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 0 0
$$979$$ 108.000 3.45169
$$980$$ 0 0
$$981$$ 42.0000 1.34096
$$982$$ 2.00000i 0.0638226i
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.00000 −0.0636930
$$987$$ 0 0
$$988$$ − 4.00000i − 0.127257i
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ 28.0000 0.889449 0.444725 0.895667i $$-0.353302\pi$$
0.444725 + 0.895667i $$0.353302\pi$$
$$992$$ − 10.0000i − 0.317500i
$$993$$ 0 0
$$994$$ 24.0000 0.761234
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 58.0000i − 1.83688i −0.395562 0.918439i $$-0.629450\pi$$
0.395562 0.918439i $$-0.370550\pi$$
$$998$$ − 12.0000i − 0.379853i
$$999$$ 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 725.2.b.a.349.1 2
5.2 odd 4 725.2.a.a.1.1 1
5.3 odd 4 145.2.a.a.1.1 1
5.4 even 2 inner 725.2.b.a.349.2 2
15.2 even 4 6525.2.a.d.1.1 1
15.8 even 4 1305.2.a.f.1.1 1
20.3 even 4 2320.2.a.e.1.1 1
35.13 even 4 7105.2.a.b.1.1 1
40.3 even 4 9280.2.a.o.1.1 1
40.13 odd 4 9280.2.a.l.1.1 1
145.28 odd 4 4205.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.a.1.1 1 5.3 odd 4
725.2.a.a.1.1 1 5.2 odd 4
725.2.b.a.349.1 2 1.1 even 1 trivial
725.2.b.a.349.2 2 5.4 even 2 inner
1305.2.a.f.1.1 1 15.8 even 4
2320.2.a.e.1.1 1 20.3 even 4
4205.2.a.a.1.1 1 145.28 odd 4
6525.2.a.d.1.1 1 15.2 even 4
7105.2.a.b.1.1 1 35.13 even 4
9280.2.a.l.1.1 1 40.13 odd 4
9280.2.a.o.1.1 1 40.3 even 4