# Properties

 Label 2550.2.d.m.2449.2 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ 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] = [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: no (minimal twist has level 102) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} -1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +10.0000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} -12.0000i q^{43} +4.00000 q^{44} -1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +10.0000i q^{58} -12.0000 q^{59} -10.0000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} -1.00000i q^{68} +1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +2.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} -4.00000i q^{83} +12.0000 q^{86} -10.0000i q^{87} +4.00000i q^{88} +6.00000 q^{89} -8.00000i q^{93} +1.00000 q^{96} -14.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} - 8 q^{19} - 2 q^{24} - 4 q^{26} + 20 q^{29} + 16 q^{31} - 2 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} + 8 q^{44} + 14 q^{49} + 2 q^{51} - 2 q^{54} - 24 q^{59} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 4 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} + 24 q^{86} + 12 q^{89} + 2 q^{96} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 - 8 * q^19 - 2 * q^24 - 4 * q^26 + 20 * q^29 + 16 * q^31 - 2 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 + 8 * q^44 + 14 * q^49 + 2 * q^51 - 2 * q^54 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 8 * q^66 + 4 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 + 24 * q^86 + 12 * q^89 + 2 * q^96 + 8 * 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
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 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
$$22$$ − 4.00000i − 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ − 12.0000i − 1.82998i −0.403473 0.914991i $$-0.632197\pi$$
0.403473 0.914991i $$-0.367803\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 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
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ 10.0000i 1.31306i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 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
$$82$$ 10.0000i 1.10432i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ − 10.0000i − 1.07211i
$$88$$ 4.00000i 0.426401i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ − 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
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ − 2.00000i − 0.184900i
$$118$$ − 12.0000i − 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 10.0000i − 0.905357i
$$123$$ − 10.0000i − 0.901670i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 8.00000i − 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ − 7.00000i − 0.577350i
$$148$$ 2.00000i 0.164399i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 12.0000i 0.914991i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 12.0000i 0.901975i
$$178$$ 6.00000i 0.449719i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 10.0000i 0.739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ − 4.00000i − 0.292509i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 18.0000i − 1.29567i −0.761781 0.647834i $$-0.775675\pi$$
0.761781 0.647834i $$-0.224325\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 14.0000i 0.997459i 0.866758 + 0.498729i $$0.166200\pi$$
−0.866758 + 0.498729i $$0.833800\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ − 10.0000i − 0.703598i
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 10.0000i 0.677285i
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ − 2.00000i − 0.134231i
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 10.0000i − 0.656532i
$$233$$ − 26.0000i − 1.70332i −0.524097 0.851658i $$-0.675597\pi$$
0.524097 0.851658i $$-0.324403\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ − 8.00000i − 0.509028i
$$248$$ − 8.00000i − 0.508001i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 2.00000i 0.124757i 0.998053 + 0.0623783i $$0.0198685\pi$$
−0.998053 + 0.0623783i $$0.980131\pi$$
$$258$$ − 12.0000i − 0.747087i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ − 12.0000i − 0.741362i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 12.0000i 0.733017i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 30.0000i 1.80253i 0.433273 + 0.901263i $$0.357359\pi$$
−0.433273 + 0.901263i $$0.642641\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 12.0000i − 0.713326i −0.934233 0.356663i $$-0.883914\pi$$
0.934233 0.356663i $$-0.116086\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 10.0000i 0.585206i
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ − 4.00000i − 0.232104i
$$298$$ 10.0000i 0.579284i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 24.0000i 1.38104i
$$303$$ 10.0000i 0.574485i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ −40.0000 −2.23957
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ − 10.0000i − 0.553001i
$$328$$ − 10.0000i − 0.552158i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ 2.00000i 0.109599i
$$334$$ −16.0000 −0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 4.00000i 0.216295i
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ 10.0000i 0.536056i
$$349$$ −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
$$352$$ − 4.00000i − 0.213201i
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 14.0000i 0.735824i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000i 0.414781i
$$373$$ − 6.00000i − 0.310668i −0.987862 0.155334i $$-0.950355\pi$$
0.987862 0.155334i $$-0.0496454\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 20.0000i 1.03005i
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 16.0000i − 0.818631i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 12.0000i 0.609994i
$$388$$ 14.0000i 0.710742i
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 7.00000i − 0.353553i
$$393$$ 12.0000i 0.605320i
$$394$$ −14.0000 −0.705310
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ − 26.0000i − 1.30490i −0.757831 0.652451i $$-0.773741\pi$$
0.757831 0.652451i $$-0.226259\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ 16.0000i 0.797017i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 8.00000i 0.396545i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ − 8.00000i − 0.394132i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 4.00000i − 0.195881i
$$418$$ 16.0000i 0.782586i
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ − 28.0000i − 1.36302i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4.00000i 0.193347i
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ − 10.0000i − 0.477818i
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 2.00000i − 0.0951303i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ − 10.0000i − 0.472984i
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 2.00000i 0.0940721i
$$453$$ − 24.0000i − 1.12762i
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ 26.0000i 1.21490i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ 26.0000 1.20443
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 12.0000i 0.552345i
$$473$$ 48.0000i 2.20704i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ 10.0000i 0.450377i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ − 4.00000i − 0.179244i
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ 28.0000i 1.24970i
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 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
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ − 10.0000i − 0.437688i
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 8.00000i 0.348485i
$$528$$ 4.00000i 0.174078i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 20.0000i 0.866296i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ − 12.0000i − 0.517838i
$$538$$ − 6.00000i − 0.258678i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ − 14.0000i − 0.600798i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 12.0000i − 0.513083i −0.966533 0.256541i $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ − 10.0000i − 0.427179i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −40.0000 −1.70406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −30.0000 −1.27458
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 34.0000i − 1.44063i −0.693649 0.720313i $$-0.743998\pi$$
0.693649 0.720313i $$-0.256002\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ − 6.00000i − 0.253095i
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ −18.0000 −0.748054
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 14.0000i − 0.580319i
$$583$$ 24.0000i 0.993978i
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ − 20.0000i − 0.825488i −0.910847 0.412744i $$-0.864570\pi$$
0.910847 0.412744i $$-0.135430\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 14.0000i 0.574911i 0.957794 + 0.287456i $$0.0928094\pi$$
−0.957794 + 0.287456i $$0.907191\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ −24.0000 −0.976546
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ − 24.0000i − 0.974130i −0.873366 0.487065i $$-0.838067\pi$$
0.873366 0.487065i $$-0.161933\pi$$
$$608$$ − 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 $$-0.721549\pi$$
0.641165 0.767403i $$-0.278451\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ − 16.0000i − 0.638978i
$$628$$ 2.00000i 0.0798087i
$$629$$ 2.00000 0.0797452
$$630$$ 0 0
$$631$$ −24.0000 −0.955425 −0.477712 0.878516i $$-0.658534\pi$$
−0.477712 + 0.878516i $$0.658534\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ 28.0000i 1.11290i
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 14.0000i 0.554700i
$$638$$ − 40.0000i − 1.58362i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ − 40.0000i − 1.57256i −0.617869 0.786281i $$-0.712004\pi$$
0.617869 0.786281i $$-0.287996\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 2.00000i 0.0776736i
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ − 16.0000i − 0.619059i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ 46.0000i 1.77317i 0.462566 + 0.886585i $$0.346929\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 46.0000i 1.76792i 0.467559 + 0.883962i $$0.345134\pi$$
−0.467559 + 0.883962i $$0.654866\pi$$
$$678$$ − 2.00000i − 0.0768095i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ − 32.0000i − 1.22534i
$$683$$ 20.0000i 0.765279i 0.923898 + 0.382639i $$0.124985\pi$$
−0.923898 + 0.382639i $$0.875015\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 26.0000i − 0.991962i
$$688$$ − 12.0000i − 0.457496i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ 10.0000i 0.378777i
$$698$$ − 14.0000i − 0.529908i
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 46.0000 1.73740 0.868698 0.495342i $$-0.164957\pi$$
0.868698 + 0.495342i $$0.164957\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 8.00000i 0.301726i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ − 12.0000i − 0.450988i
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ − 6.00000i − 0.224860i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 24.0000i 0.895672i
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ − 10.0000i − 0.369611i
$$733$$ − 46.0000i − 1.69905i −0.527549 0.849524i $$-0.676889\pi$$
0.527549 0.849524i $$-0.323111\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000i 1.76810i
$$738$$ − 10.0000i − 0.368105i
$$739$$ −52.0000 −1.91285 −0.956425 0.291977i $$-0.905687\pi$$
−0.956425 + 0.291977i $$0.905687\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 6.00000 0.219676
$$747$$ 4.00000i 0.146352i
$$748$$ 4.00000i 0.146254i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ − 28.0000i − 1.02038i
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ − 24.0000i − 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ 18.0000i 0.647834i
$$773$$ − 38.0000i − 1.36677i −0.730061 0.683383i $$-0.760508\pi$$
0.730061 0.683383i $$-0.239492\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 26.0000i 0.932145i
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 10.0000i 0.357371i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ − 14.0000i − 0.498729i
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 4.00000i − 0.142134i
$$793$$ − 20.0000i − 0.710221i
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 14.0000i 0.495905i 0.968772 + 0.247953i $$0.0797578\pi$$
−0.968772 + 0.247953i $$0.920242\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 14.0000i − 0.494357i
$$803$$ 40.0000i 1.41157i
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 6.00000i 0.211210i
$$808$$ 10.0000i 0.351799i
$$809$$ 22.0000 0.773479 0.386739 0.922189i $$-0.373601\pi$$
0.386739 + 0.922189i $$0.373601\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ 48.0000i 1.67931i
$$818$$ − 26.0000i − 0.909069i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 10.0000i 0.348790i
$$823$$ − 32.0000i − 1.11545i −0.830026 0.557725i $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 20.0000i − 0.695468i −0.937593 0.347734i $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 30.0000 1.04069
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 7.00000i 0.242536i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 8.00000i 0.276520i
$$838$$ − 4.00000i − 0.138178i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 22.0000i 0.758170i
$$843$$ 6.00000i 0.206651i
$$844$$ 28.0000 0.963800
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 2.00000i 0.0684787i 0.999414 + 0.0342393i $$0.0109009\pi$$
−0.999414 + 0.0342393i $$0.989099\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ − 6.00000i − 0.204956i −0.994735 0.102478i $$-0.967323\pi$$
0.994735 0.102478i $$-0.0326771\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 8.00000i − 0.272481i
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 1.00000i 0.0339618i
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ − 10.0000i − 0.338643i
$$873$$ 14.0000i 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ − 16.0000i − 0.539974i
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ − 36.0000i − 1.21150i −0.795656 0.605748i $$-0.792874\pi$$
0.795656 0.605748i $$-0.207126\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 80.0000 2.66815
$$900$$ 0 0
$$901$$ 6.00000 0.199889
$$902$$ − 40.0000i − 1.33185i
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 0 0
$$906$$ 24.0000 0.797347
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 16.0000i 0.529523i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 30.0000i 0.987997i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 32.0000 1.05159
$$927$$ − 8.00000i − 0.262754i
$$928$$ 10.0000i 0.328266i
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ 26.0000i 0.851658i
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 22.0000i − 0.718709i −0.933201 0.359354i $$-0.882997\pi$$
0.933201 0.359354i $$-0.117003\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ − 28.0000i − 0.909878i −0.890523 0.454939i $$-0.849661\pi$$
0.890523 0.454939i $$-0.150339\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 40.0000i 1.29302i
$$958$$ − 24.0000i − 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 4.00000i 0.128965i
$$963$$ 4.00000i 0.128898i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ −4.00000 −0.128366 −0.0641831 0.997938i $$-0.520444\pi$$
−0.0641831 + 0.997938i $$0.520444\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ − 46.0000i − 1.47167i −0.677161 0.735835i $$-0.736790\pi$$
0.677161 0.735835i $$-0.263210\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 12.0000i 0.382935i
$$983$$ − 48.0000i − 1.53096i −0.643458 0.765481i $$-0.722501\pi$$
0.643458 0.765481i $$-0.277499\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ −10.0000 −0.318465
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 20.0000i 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ − 2.00000i − 0.0633406i −0.999498 0.0316703i $$-0.989917\pi$$
0.999498 0.0316703i $$-0.0100827\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 2.00000 0.0632772
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.m.2449.2 2
5.2 odd 4 2550.2.a.c.1.1 1
5.3 odd 4 102.2.a.c.1.1 1
5.4 even 2 inner 2550.2.d.m.2449.1 2
15.2 even 4 7650.2.a.ca.1.1 1
15.8 even 4 306.2.a.b.1.1 1
20.3 even 4 816.2.a.b.1.1 1
35.13 even 4 4998.2.a.be.1.1 1
40.3 even 4 3264.2.a.bc.1.1 1
40.13 odd 4 3264.2.a.m.1.1 1
60.23 odd 4 2448.2.a.p.1.1 1
85.8 odd 8 1734.2.f.e.829.1 4
85.13 odd 4 1734.2.b.b.577.2 2
85.33 odd 4 1734.2.a.j.1.1 1
85.38 odd 4 1734.2.b.b.577.1 2
85.43 odd 8 1734.2.f.e.829.2 4
85.53 odd 8 1734.2.f.e.1483.2 4
85.83 odd 8 1734.2.f.e.1483.1 4
120.53 even 4 9792.2.a.k.1.1 1
120.83 odd 4 9792.2.a.l.1.1 1
255.203 even 4 5202.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.c.1.1 1 5.3 odd 4
306.2.a.b.1.1 1 15.8 even 4
816.2.a.b.1.1 1 20.3 even 4
1734.2.a.j.1.1 1 85.33 odd 4
1734.2.b.b.577.1 2 85.38 odd 4
1734.2.b.b.577.2 2 85.13 odd 4
1734.2.f.e.829.1 4 85.8 odd 8
1734.2.f.e.829.2 4 85.43 odd 8
1734.2.f.e.1483.1 4 85.83 odd 8
1734.2.f.e.1483.2 4 85.53 odd 8
2448.2.a.p.1.1 1 60.23 odd 4
2550.2.a.c.1.1 1 5.2 odd 4
2550.2.d.m.2449.1 2 5.4 even 2 inner
2550.2.d.m.2449.2 2 1.1 even 1 trivial
3264.2.a.m.1.1 1 40.13 odd 4
3264.2.a.bc.1.1 1 40.3 even 4
4998.2.a.be.1.1 1 35.13 even 4
5202.2.a.c.1.1 1 255.203 even 4
7650.2.a.ca.1.1 1 15.2 even 4
9792.2.a.k.1.1 1 120.53 even 4
9792.2.a.l.1.1 1 120.83 odd 4