# Properties

 Label 2550.2.d.t.2449.1 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.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: $$20.3618525154$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.t.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -6.00000i q^{22} -5.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} -2.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -3.00000i q^{37} +4.00000i q^{38} +2.00000 q^{39} +5.00000 q^{41} +2.00000i q^{43} -6.00000 q^{44} -5.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} +2.00000i q^{52} +1.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} +3.00000 q^{59} +5.00000 q^{61} +2.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} +2.00000i q^{67} +1.00000i q^{68} +5.00000 q^{69} +5.00000 q^{71} -1.00000i q^{72} -3.00000 q^{74} +4.00000 q^{76} -2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -5.00000i q^{82} -1.00000i q^{83} +2.00000 q^{86} +6.00000i q^{88} -14.0000 q^{89} +5.00000i q^{92} -2.00000i q^{93} +1.00000 q^{96} -16.0000i q^{97} -7.00000i q^{98} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 12 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} - 4 q^{31} - 2 q^{34} + 2 q^{36} + 4 q^{39} + 10 q^{41} - 12 q^{44} - 10 q^{46} + 14 q^{49} + 2 q^{51} - 2 q^{54} + 6 q^{59} + 10 q^{61} - 2 q^{64} + 12 q^{66} + 10 q^{69} + 10 q^{71} - 6 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{86} - 28 q^{89} + 2 q^{96} - 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 12 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 - 4 * q^31 - 2 * q^34 + 2 * q^36 + 4 * q^39 + 10 * q^41 - 12 * q^44 - 10 * q^46 + 14 * q^49 + 2 * q^51 - 2 * q^54 + 6 * q^59 + 10 * q^61 - 2 * q^64 + 12 * q^66 + 10 * q^69 + 10 * q^71 - 6 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 + 4 * q^86 - 28 * q^89 + 2 * q^96 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 6.00000i − 1.27920i
$$23$$ − 5.00000i − 1.04257i −0.853382 0.521286i $$-0.825452\pi$$
0.853382 0.521286i $$-0.174548\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 6.00000i 1.04447i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 3.00000i − 0.493197i −0.969118 0.246598i $$-0.920687\pi$$
0.969118 0.246598i $$-0.0793129\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ −5.00000 −0.737210
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ 2.00000i 0.277350i
$$53$$ 1.00000i 0.137361i 0.997639 + 0.0686803i $$0.0218788\pi$$
−0.997639 + 0.0686803i $$0.978121\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 2.00000i 0.254000i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ 5.00000 0.601929
$$70$$ 0 0
$$71$$ 5.00000 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 5.00000i − 0.552158i
$$83$$ − 1.00000i − 0.109764i −0.998493 0.0548821i $$-0.982522\pi$$
0.998493 0.0548821i $$-0.0174783\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ 6.00000i 0.639602i
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 5.00000i 0.521286i
$$93$$ − 2.00000i − 0.207390i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 16.0000i − 1.62455i −0.583272 0.812277i $$-0.698228\pi$$
0.583272 0.812277i $$-0.301772\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ − 1.00000i − 0.0990148i
$$103$$ − 13.0000i − 1.28093i −0.767988 0.640464i $$-0.778742\pi$$
0.767988 0.640464i $$-0.221258\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 1.00000 0.0971286
$$107$$ − 16.0000i − 1.54678i −0.633932 0.773389i $$-0.718560\pi$$
0.633932 0.773389i $$-0.281440\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ − 13.0000i − 1.22294i −0.791269 0.611469i $$-0.790579\pi$$
0.791269 0.611469i $$-0.209421\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ − 3.00000i − 0.276172i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ − 5.00000i − 0.452679i
$$123$$ 5.00000i 0.450835i
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ − 6.00000i − 0.522233i
$$133$$ 0 0
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 20.0000i 1.70872i 0.519685 + 0.854358i $$0.326049\pi$$
−0.519685 + 0.854358i $$0.673951\pi$$
$$138$$ − 5.00000i − 0.425628i
$$139$$ −1.00000 −0.0848189 −0.0424094 0.999100i $$-0.513503\pi$$
−0.0424094 + 0.999100i $$0.513503\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 5.00000i − 0.419591i
$$143$$ − 12.0000i − 1.00349i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 7.00000i 0.577350i
$$148$$ 3.00000i 0.246598i
$$149$$ 5.00000 0.409616 0.204808 0.978802i $$-0.434343\pi$$
0.204808 + 0.978802i $$0.434343\pi$$
$$150$$ 0 0
$$151$$ 9.00000 0.732410 0.366205 0.930534i $$-0.380657\pi$$
0.366205 + 0.930534i $$0.380657\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 1.00000i 0.0808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −1.00000 −0.0793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 9.00000i 0.704934i 0.935824 + 0.352467i $$0.114657\pi$$
−0.935824 + 0.352467i $$0.885343\pi$$
$$164$$ −5.00000 −0.390434
$$165$$ 0 0
$$166$$ −1.00000 −0.0776151
$$167$$ 24.0000i 1.85718i 0.371113 + 0.928588i $$0.378976\pi$$
−0.371113 + 0.928588i $$0.621024\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ − 2.00000i − 0.152499i
$$173$$ − 24.0000i − 1.82469i −0.409426 0.912343i $$-0.634271\pi$$
0.409426 0.912343i $$-0.365729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ 3.00000i 0.225494i
$$178$$ 14.0000i 1.04934i
$$179$$ −3.00000 −0.224231 −0.112115 0.993695i $$-0.535763\pi$$
−0.112115 + 0.993695i $$0.535763\pi$$
$$180$$ 0 0
$$181$$ 19.0000 1.41226 0.706129 0.708083i $$-0.250440\pi$$
0.706129 + 0.708083i $$0.250440\pi$$
$$182$$ 0 0
$$183$$ 5.00000i 0.369611i
$$184$$ 5.00000 0.368605
$$185$$ 0 0
$$186$$ −2.00000 −0.146647
$$187$$ − 6.00000i − 0.438763i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 12.0000i − 0.863779i −0.901927 0.431889i $$-0.857847\pi$$
0.901927 0.431889i $$-0.142153\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 6.00000i 0.426401i
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ − 10.0000i − 0.703598i
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ −13.0000 −0.905753
$$207$$ 5.00000i 0.347524i
$$208$$ − 2.00000i − 0.138675i
$$209$$ −24.0000 −1.66011
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ − 1.00000i − 0.0686803i
$$213$$ 5.00000i 0.342594i
$$214$$ −16.0000 −1.09374
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ − 10.0000i − 0.677285i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ − 3.00000i − 0.201347i
$$223$$ 1.00000i 0.0669650i 0.999439 + 0.0334825i $$0.0106598\pi$$
−0.999439 + 0.0334825i $$0.989340\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −13.0000 −0.864747
$$227$$ 6.00000i 0.398234i 0.979976 + 0.199117i $$0.0638074\pi$$
−0.979976 + 0.199117i $$0.936193\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1.00000i 0.0655122i 0.999463 + 0.0327561i $$0.0104285\pi$$
−0.999463 + 0.0327561i $$0.989572\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −3.00000 −0.195283
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ 30.0000 1.94054 0.970269 0.242028i $$-0.0778125\pi$$
0.970269 + 0.242028i $$0.0778125\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ − 25.0000i − 1.60706i
$$243$$ 1.00000i 0.0641500i
$$244$$ −5.00000 −0.320092
$$245$$ 0 0
$$246$$ 5.00000 0.318788
$$247$$ 8.00000i 0.509028i
$$248$$ − 2.00000i − 0.127000i
$$249$$ 1.00000 0.0633724
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ − 30.0000i − 1.88608i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 28.0000i 1.74659i 0.487190 + 0.873296i $$0.338022\pi$$
−0.487190 + 0.873296i $$0.661978\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.0000i 0.741362i
$$263$$ − 8.00000i − 0.493301i −0.969104 0.246651i $$-0.920670\pi$$
0.969104 0.246651i $$-0.0793300\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 14.0000i − 0.856786i
$$268$$ − 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
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 0 0
$$274$$ 20.0000 1.20824
$$275$$ 0 0
$$276$$ −5.00000 −0.300965
$$277$$ 25.0000i 1.50210i 0.660243 + 0.751052i $$0.270453\pi$$
−0.660243 + 0.751052i $$0.729547\pi$$
$$278$$ 1.00000i 0.0599760i
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 17.0000i 1.01055i 0.862960 + 0.505273i $$0.168608\pi$$
−0.862960 + 0.505273i $$0.831392\pi$$
$$284$$ −5.00000 −0.296695
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 16.0000 0.937937
$$292$$ 0 0
$$293$$ − 21.0000i − 1.22683i −0.789760 0.613417i $$-0.789795\pi$$
0.789760 0.613417i $$-0.210205\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ 3.00000 0.174371
$$297$$ − 6.00000i − 0.348155i
$$298$$ − 5.00000i − 0.289642i
$$299$$ −10.0000 −0.578315
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 9.00000i − 0.517892i
$$303$$ 10.0000i 0.574485i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 0 0
$$309$$ 13.0000 0.739544
$$310$$ 0 0
$$311$$ 15.0000 0.850572 0.425286 0.905059i $$-0.360174\pi$$
0.425286 + 0.905059i $$0.360174\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 20.0000i − 1.13047i −0.824931 0.565233i $$-0.808786\pi$$
0.824931 0.565233i $$-0.191214\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 14.0000i 0.786318i 0.919470 + 0.393159i $$0.128618\pi$$
−0.919470 + 0.393159i $$0.871382\pi$$
$$318$$ 1.00000i 0.0560772i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 16.0000 0.893033
$$322$$ 0 0
$$323$$ 4.00000i 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 9.00000 0.498464
$$327$$ 10.0000i 0.553001i
$$328$$ 5.00000i 0.276079i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ 1.00000i 0.0548821i
$$333$$ 3.00000i 0.164399i
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 6.00000i − 0.326841i −0.986557 0.163420i $$-0.947747\pi$$
0.986557 0.163420i $$-0.0522527\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 13.0000 0.706063
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ − 4.00000i − 0.216295i
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ −24.0000 −1.29025
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ − 6.00000i − 0.319801i
$$353$$ 20.0000i 1.06449i 0.846590 + 0.532246i $$0.178652\pi$$
−0.846590 + 0.532246i $$0.821348\pi$$
$$354$$ 3.00000 0.159448
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ 0 0
$$358$$ 3.00000i 0.158555i
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 19.0000i − 0.998618i
$$363$$ 25.0000i 1.31216i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 5.00000 0.261354
$$367$$ − 14.0000i − 0.730794i −0.930852 0.365397i $$-0.880933\pi$$
0.930852 0.365397i $$-0.119067\pi$$
$$368$$ − 5.00000i − 0.260643i
$$369$$ −5.00000 −0.260290
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 2.00000i 0.103695i
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −31.0000 −1.59236 −0.796182 0.605058i $$-0.793150\pi$$
−0.796182 + 0.605058i $$0.793150\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 6.00000i 0.306987i
$$383$$ 4.00000i 0.204390i 0.994764 + 0.102195i $$0.0325866\pi$$
−0.994764 + 0.102195i $$0.967413\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −12.0000 −0.610784
$$387$$ − 2.00000i − 0.101666i
$$388$$ 16.0000i 0.812277i
$$389$$ −19.0000 −0.963338 −0.481669 0.876353i $$-0.659969\pi$$
−0.481669 + 0.876353i $$0.659969\pi$$
$$390$$ 0 0
$$391$$ −5.00000 −0.252861
$$392$$ 7.00000i 0.353553i
$$393$$ − 12.0000i − 0.605320i
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 6.00000 0.301511
$$397$$ 21.0000i 1.05396i 0.849878 + 0.526980i $$0.176676\pi$$
−0.849878 + 0.526980i $$0.823324\pi$$
$$398$$ − 10.0000i − 0.501255i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −39.0000 −1.94757 −0.973784 0.227477i $$-0.926952\pi$$
−0.973784 + 0.227477i $$0.926952\pi$$
$$402$$ 2.00000i 0.0997509i
$$403$$ 4.00000i 0.199254i
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 18.0000i − 0.892227i
$$408$$ 1.00000i 0.0495074i
$$409$$ −11.0000 −0.543915 −0.271957 0.962309i $$-0.587671\pi$$
−0.271957 + 0.962309i $$0.587671\pi$$
$$410$$ 0 0
$$411$$ −20.0000 −0.986527
$$412$$ 13.0000i 0.640464i
$$413$$ 0 0
$$414$$ 5.00000 0.245737
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 1.00000i − 0.0489702i
$$418$$ 24.0000i 1.17388i
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 0 0
$$424$$ −1.00000 −0.0485643
$$425$$ 0 0
$$426$$ 5.00000 0.242251
$$427$$ 0 0
$$428$$ 16.0000i 0.773389i
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 6.00000i 0.288342i 0.989553 + 0.144171i $$0.0460515\pi$$
−0.989553 + 0.144171i $$0.953949\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 20.0000i 0.956730i
$$438$$ 0 0
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 2.00000i 0.0951303i
$$443$$ 11.0000i 0.522626i 0.965254 + 0.261313i $$0.0841554\pi$$
−0.965254 + 0.261313i $$0.915845\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 0 0
$$446$$ 1.00000 0.0473514
$$447$$ 5.00000i 0.236492i
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ 30.0000 1.41264
$$452$$ 13.0000i 0.611469i
$$453$$ 9.00000i 0.422857i
$$454$$ 6.00000 0.281594
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 25.0000i − 1.16945i −0.811231 0.584725i $$-0.801202\pi$$
0.811231 0.584725i $$-0.198798\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −35.0000 −1.63011 −0.815056 0.579382i $$-0.803294\pi$$
−0.815056 + 0.579382i $$0.803294\pi$$
$$462$$ 0 0
$$463$$ 27.0000i 1.25480i 0.778699 + 0.627398i $$0.215880\pi$$
−0.778699 + 0.627398i $$0.784120\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 1.00000 0.0463241
$$467$$ − 33.0000i − 1.52706i −0.645774 0.763529i $$-0.723465\pi$$
0.645774 0.763529i $$-0.276535\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 3.00000i 0.138086i
$$473$$ 12.0000i 0.551761i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 1.00000i − 0.0457869i
$$478$$ − 30.0000i − 1.37217i
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 18.0000i 0.819878i
$$483$$ 0 0
$$484$$ −25.0000 −1.13636
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 26.0000i 1.17817i 0.808070 + 0.589086i $$0.200512\pi$$
−0.808070 + 0.589086i $$0.799488\pi$$
$$488$$ 5.00000i 0.226339i
$$489$$ −9.00000 −0.406994
$$490$$ 0 0
$$491$$ −43.0000 −1.94056 −0.970281 0.241979i $$-0.922203\pi$$
−0.970281 + 0.241979i $$0.922203\pi$$
$$492$$ − 5.00000i − 0.225417i
$$493$$ 0 0
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ − 1.00000i − 0.0448111i
$$499$$ 41.0000 1.83541 0.917706 0.397260i $$-0.130039\pi$$
0.917706 + 0.397260i $$0.130039\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 12.0000i 0.535586i
$$503$$ 5.00000i 0.222939i 0.993768 + 0.111469i $$0.0355557\pi$$
−0.993768 + 0.111469i $$0.964444\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −30.0000 −1.33366
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 28.0000 1.23503
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 24.0000 1.05348
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 26.0000i 1.13690i 0.822718 + 0.568450i $$0.192457\pi$$
−0.822718 + 0.568450i $$0.807543\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 2.00000i 0.0871214i
$$528$$ 6.00000i 0.261116i
$$529$$ −2.00000 −0.0869565
$$530$$ 0 0
$$531$$ −3.00000 −0.130189
$$532$$ 0 0
$$533$$ − 10.0000i − 0.433148i
$$534$$ −14.0000 −0.605839
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ − 3.00000i − 0.129460i
$$538$$ 26.0000i 1.12094i
$$539$$ 42.0000 1.80907
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ − 11.0000i − 0.472490i
$$543$$ 19.0000i 0.815368i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 13.0000i − 0.555840i −0.960604 0.277920i $$-0.910355\pi$$
0.960604 0.277920i $$-0.0896450\pi$$
$$548$$ − 20.0000i − 0.854358i
$$549$$ −5.00000 −0.213395
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 5.00000i 0.212814i
$$553$$ 0 0
$$554$$ 25.0000 1.06215
$$555$$ 0 0
$$556$$ 1.00000 0.0424094
$$557$$ − 11.0000i − 0.466085i −0.972467 0.233042i $$-0.925132\pi$$
0.972467 0.233042i $$-0.0748681\pi$$
$$558$$ − 2.00000i − 0.0846668i
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ 6.00000i 0.253095i
$$563$$ 21.0000i 0.885044i 0.896758 + 0.442522i $$0.145916\pi$$
−0.896758 + 0.442522i $$0.854084\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 17.0000 0.714563
$$567$$ 0 0
$$568$$ 5.00000i 0.209795i
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 23.0000 0.962520 0.481260 0.876578i $$-0.340179\pi$$
0.481260 + 0.876578i $$0.340179\pi$$
$$572$$ 12.0000i 0.501745i
$$573$$ − 6.00000i − 0.250654i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 7.00000i − 0.291414i −0.989328 0.145707i $$-0.953454\pi$$
0.989328 0.145707i $$-0.0465456\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 12.0000 0.498703
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 16.0000i − 0.663221i
$$583$$ 6.00000i 0.248495i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −21.0000 −0.867502
$$587$$ 45.0000i 1.85735i 0.370896 + 0.928674i $$0.379051\pi$$
−0.370896 + 0.928674i $$0.620949\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ − 3.00000i − 0.123299i
$$593$$ − 24.0000i − 0.985562i −0.870153 0.492781i $$-0.835980\pi$$
0.870153 0.492781i $$-0.164020\pi$$
$$594$$ −6.00000 −0.246183
$$595$$ 0 0
$$596$$ −5.00000 −0.204808
$$597$$ 10.0000i 0.409273i
$$598$$ 10.0000i 0.408930i
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ − 2.00000i − 0.0814463i
$$604$$ −9.00000 −0.366205
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ 34.0000i 1.38002i 0.723801 + 0.690009i $$0.242393\pi$$
−0.723801 + 0.690009i $$0.757607\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 1.00000i − 0.0404226i
$$613$$ 38.0000i 1.53481i 0.641165 + 0.767403i $$0.278451\pi$$
−0.641165 + 0.767403i $$0.721549\pi$$
$$614$$ 2.00000 0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ − 13.0000i − 0.522937i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ − 15.0000i − 0.601445i
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ − 24.0000i − 0.958468i
$$628$$ 18.0000i 0.718278i
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 41.0000 1.63218 0.816092 0.577922i $$-0.196136\pi$$
0.816092 + 0.577922i $$0.196136\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 12.0000i 0.476957i
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ 1.00000 0.0396526
$$637$$ − 14.0000i − 0.554700i
$$638$$ 0 0
$$639$$ −5.00000 −0.197797
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ − 16.0000i − 0.631470i
$$643$$ 37.0000i 1.45914i 0.683907 + 0.729569i $$0.260279\pi$$
−0.683907 + 0.729569i $$0.739721\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ 10.0000i 0.393141i 0.980490 + 0.196570i $$0.0629804\pi$$
−0.980490 + 0.196570i $$0.937020\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 18.0000 0.706562
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 9.00000i − 0.352467i
$$653$$ − 24.0000i − 0.939193i −0.882881 0.469596i $$-0.844399\pi$$
0.882881 0.469596i $$-0.155601\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 5.00000 0.195217
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ − 10.0000i − 0.388661i
$$663$$ − 2.00000i − 0.0776736i
$$664$$ 1.00000 0.0388075
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 0 0
$$668$$ − 24.0000i − 0.928588i
$$669$$ −1.00000 −0.0386622
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ − 16.0000i − 0.616755i −0.951264 0.308377i $$-0.900214\pi$$
0.951264 0.308377i $$-0.0997859\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 36.0000i − 1.38359i −0.722093 0.691796i $$-0.756820\pi$$
0.722093 0.691796i $$-0.243180\pi$$
$$678$$ − 13.0000i − 0.499262i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −6.00000 −0.229920
$$682$$ 12.0000i 0.459504i
$$683$$ − 30.0000i − 1.14792i −0.818884 0.573959i $$-0.805407\pi$$
0.818884 0.573959i $$-0.194593\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 6.00000i 0.228914i
$$688$$ 2.00000i 0.0762493i
$$689$$ 2.00000 0.0761939
$$690$$ 0 0
$$691$$ −33.0000 −1.25538 −0.627690 0.778464i $$-0.715999\pi$$
−0.627690 + 0.778464i $$0.715999\pi$$
$$692$$ 24.0000i 0.912343i
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 5.00000i − 0.189389i
$$698$$ 14.0000i 0.529908i
$$699$$ −1.00000 −0.0378235
$$700$$ 0 0
$$701$$ 31.0000 1.17085 0.585427 0.810725i $$-0.300927\pi$$
0.585427 + 0.810725i $$0.300927\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 12.0000i 0.452589i
$$704$$ −6.00000 −0.226134
$$705$$ 0 0
$$706$$ 20.0000 0.752710
$$707$$ 0 0
$$708$$ − 3.00000i − 0.112747i
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ − 14.0000i − 0.524672i
$$713$$ 10.0000i 0.374503i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3.00000 0.112115
$$717$$ 30.0000i 1.12037i
$$718$$ − 4.00000i − 0.149279i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000i 0.111648i
$$723$$ − 18.0000i − 0.669427i
$$724$$ −19.0000 −0.706129
$$725$$ 0 0
$$726$$ 25.0000 0.927837
$$727$$ − 52.0000i − 1.92857i −0.264861 0.964287i $$-0.585326\pi$$
0.264861 0.964287i $$-0.414674\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ − 5.00000i − 0.184805i
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ −5.00000 −0.184302
$$737$$ 12.0000i 0.442026i
$$738$$ 5.00000i 0.184053i
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ 19.0000i 0.697042i 0.937301 + 0.348521i $$0.113316\pi$$
−0.937301 + 0.348521i $$0.886684\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ 1.00000i 0.0365881i
$$748$$ 6.00000i 0.219382i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ 0 0
$$753$$ − 12.0000i − 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 40.0000i 1.45382i 0.686730 + 0.726912i $$0.259045\pi$$
−0.686730 + 0.726912i $$0.740955\pi$$
$$758$$ 31.0000i 1.12597i
$$759$$ 30.0000 1.08893
$$760$$ 0 0
$$761$$ −36.0000 −1.30500 −0.652499 0.757789i $$-0.726280\pi$$
−0.652499 + 0.757789i $$0.726280\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ 4.00000 0.144526
$$767$$ − 6.00000i − 0.216647i
$$768$$ 1.00000i 0.0360844i
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 0 0
$$771$$ −28.0000 −1.00840
$$772$$ 12.0000i 0.431889i
$$773$$ 3.00000i 0.107903i 0.998544 + 0.0539513i $$0.0171816\pi$$
−0.998544 + 0.0539513i $$0.982818\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ 16.0000 0.574367
$$777$$ 0 0
$$778$$ 19.0000i 0.681183i
$$779$$ −20.0000 −0.716574
$$780$$ 0 0
$$781$$ 30.0000 1.07348
$$782$$ 5.00000i 0.178800i
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 21.0000i 0.748569i 0.927314 + 0.374285i $$0.122112\pi$$
−0.927314 + 0.374285i $$0.877888\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 6.00000i − 0.213201i
$$793$$ − 10.0000i − 0.355110i
$$794$$ 21.0000 0.745262
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ − 19.0000i − 0.673015i −0.941681 0.336507i $$-0.890754\pi$$
0.941681 0.336507i $$-0.109246\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 14.0000 0.494666
$$802$$ 39.0000i 1.37714i
$$803$$ 0 0
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ − 26.0000i − 0.915243i
$$808$$ 10.0000i 0.351799i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 11.0000i 0.385787i
$$814$$ −18.0000 −0.630900
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ − 8.00000i − 0.279885i
$$818$$ 11.0000i 0.384606i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 36.0000 1.25641 0.628204 0.778048i $$-0.283790\pi$$
0.628204 + 0.778048i $$0.283790\pi$$
$$822$$ 20.0000i 0.697580i
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 13.0000 0.452876
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 10.0000i − 0.347734i −0.984769 0.173867i $$-0.944374\pi$$
0.984769 0.173867i $$-0.0556263\pi$$
$$828$$ − 5.00000i − 0.173762i
$$829$$ −16.0000 −0.555703 −0.277851 0.960624i $$-0.589622\pi$$
−0.277851 + 0.960624i $$0.589622\pi$$
$$830$$ 0 0
$$831$$ −25.0000 −0.867240
$$832$$ 2.00000i 0.0693375i
$$833$$ − 7.00000i − 0.242536i
$$834$$ −1.00000 −0.0346272
$$835$$ 0 0
$$836$$ 24.0000 0.830057
$$837$$ 2.00000i 0.0691301i
$$838$$ 4.00000i 0.138178i
$$839$$ 15.0000 0.517858 0.258929 0.965896i $$-0.416631\pi$$
0.258929 + 0.965896i $$0.416631\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 28.0000i 0.964944i
$$843$$ − 6.00000i − 0.206651i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 1.00000i 0.0343401i
$$849$$ −17.0000 −0.583438
$$850$$ 0 0
$$851$$ −15.0000 −0.514193
$$852$$ − 5.00000i − 0.171297i
$$853$$ − 42.0000i − 1.43805i −0.694983 0.719026i $$-0.744588\pi$$
0.694983 0.719026i $$-0.255412\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 16.0000 0.546869
$$857$$ 11.0000i 0.375753i 0.982193 + 0.187876i $$0.0601604\pi$$
−0.982193 + 0.187876i $$0.939840\pi$$
$$858$$ − 12.0000i − 0.409673i
$$859$$ 16.0000 0.545913 0.272956 0.962026i $$-0.411998\pi$$
0.272956 + 0.962026i $$0.411998\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 28.0000i 0.953684i
$$863$$ 28.0000i 0.953131i 0.879139 + 0.476566i $$0.158119\pi$$
−0.879139 + 0.476566i $$0.841881\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 6.00000 0.203888
$$867$$ − 1.00000i − 0.0339618i
$$868$$ 0 0
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 10.0000i 0.338643i
$$873$$ 16.0000i 0.541518i
$$874$$ 20.0000 0.676510
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ − 4.00000i − 0.134993i
$$879$$ 21.0000 0.708312
$$880$$ 0 0
$$881$$ −53.0000 −1.78562 −0.892808 0.450438i $$-0.851268\pi$$
−0.892808 + 0.450438i $$0.851268\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ 6.00000i 0.201916i 0.994891 + 0.100958i $$0.0321908\pi$$
−0.994891 + 0.100958i $$0.967809\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ 11.0000 0.369552
$$887$$ 3.00000i 0.100730i 0.998731 + 0.0503651i $$0.0160385\pi$$
−0.998731 + 0.0503651i $$0.983962\pi$$
$$888$$ 3.00000i 0.100673i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ − 1.00000i − 0.0334825i
$$893$$ 0 0
$$894$$ 5.00000 0.167225
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 10.0000i − 0.333890i
$$898$$ 2.00000i 0.0667409i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 1.00000 0.0333148
$$902$$ − 30.0000i − 0.998891i
$$903$$ 0 0
$$904$$ 13.0000 0.432374
$$905$$ 0 0
$$906$$ 9.00000 0.299005
$$907$$ 17.0000i 0.564476i 0.959344 + 0.282238i $$0.0910767\pi$$
−0.959344 + 0.282238i $$0.908923\pi$$
$$908$$ − 6.00000i − 0.199117i
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 6.00000i − 0.198571i
$$914$$ −25.0000 −0.826927
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ 1.00000i 0.0330049i
$$919$$ −49.0000 −1.61636 −0.808180 0.588935i $$-0.799547\pi$$
−0.808180 + 0.588935i $$0.799547\pi$$
$$920$$ 0 0
$$921$$ −2.00000 −0.0659022
$$922$$ 35.0000i 1.15266i
$$923$$ − 10.0000i − 0.329154i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 27.0000 0.887275
$$927$$ 13.0000i 0.426976i
$$928$$ 0 0
$$929$$ 13.0000 0.426516 0.213258 0.976996i $$-0.431592\pi$$
0.213258 + 0.976996i $$0.431592\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ − 1.00000i − 0.0327561i
$$933$$ 15.0000i 0.491078i
$$934$$ −33.0000 −1.07979
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 23.0000i − 0.751377i −0.926746 0.375689i $$-0.877406\pi$$
0.926746 0.375689i $$-0.122594\pi$$
$$938$$ 0 0
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ − 18.0000i − 0.586472i
$$943$$ − 25.0000i − 0.814112i
$$944$$ 3.00000 0.0976417
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ 48.0000i 1.55979i 0.625910 + 0.779895i $$0.284728\pi$$
−0.625910 + 0.779895i $$0.715272\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −14.0000 −0.453981
$$952$$ 0 0
$$953$$ 14.0000i 0.453504i 0.973952 + 0.226752i $$0.0728108\pi$$
−0.973952 + 0.226752i $$0.927189\pi$$
$$954$$ −1.00000 −0.0323762
$$955$$ 0 0
$$956$$ −30.0000 −0.970269
$$957$$ 0 0
$$958$$ − 16.0000i − 0.516937i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 6.00000i 0.193448i
$$963$$ 16.0000i 0.515593i
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 57.0000i − 1.83300i −0.400039 0.916498i $$-0.631003\pi$$
0.400039 0.916498i $$-0.368997\pi$$
$$968$$ 25.0000i 0.803530i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ 51.0000 1.63667 0.818334 0.574743i $$-0.194898\pi$$
0.818334 + 0.574743i $$0.194898\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ − 34.0000i − 1.08776i −0.839164 0.543878i $$-0.816955\pi$$
0.839164 0.543878i $$-0.183045\pi$$
$$978$$ 9.00000i 0.287788i
$$979$$ −84.0000 −2.68465
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 43.0000i 1.37219i
$$983$$ 8.00000i 0.255160i 0.991828 + 0.127580i $$0.0407210\pi$$
−0.991828 + 0.127580i $$0.959279\pi$$
$$984$$ −5.00000 −0.159394
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ 10.0000 0.317982
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 2.00000i 0.0635001i
$$993$$ 10.0000i 0.317340i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −1.00000 −0.0316862
$$997$$ 22.0000i 0.696747i 0.937356 + 0.348373i $$0.113266\pi$$
−0.937356 + 0.348373i $$0.886734\pi$$
$$998$$ − 41.0000i − 1.29783i
$$999$$ −3.00000 −0.0949158
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 2550.2.d.t.2449.1 2
5.2 odd 4 2550.2.a.bd.1.1 yes 1
5.3 odd 4 2550.2.a.e.1.1 1
5.4 even 2 inner 2550.2.d.t.2449.2 2
15.2 even 4 7650.2.a.p.1.1 1
15.8 even 4 7650.2.a.bv.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.e.1.1 1 5.3 odd 4
2550.2.a.bd.1.1 yes 1 5.2 odd 4
2550.2.d.t.2449.1 2 1.1 even 1 trivial
2550.2.d.t.2449.2 2 5.4 even 2 inner
7650.2.a.p.1.1 1 15.2 even 4
7650.2.a.bv.1.1 1 15.8 even 4