# Properties

 Label 2175.2.c.b.349.1 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ 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] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.c (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: $$17.3674624396$$ 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 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.b.349.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -6.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} +6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} -12.0000 q^{56} +4.00000i q^{57} +1.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +8.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} -8.00000 q^{71} +3.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} +16.0000i q^{77} +6.00000i q^{78} +1.00000 q^{81} +6.00000i q^{82} +8.00000i q^{83} +4.00000 q^{84} +4.00000 q^{86} -1.00000i q^{87} +12.0000i q^{88} +6.00000 q^{89} +24.0000 q^{91} -4.00000i q^{92} -8.00000i q^{93} +5.00000 q^{96} +2.00000i q^{97} +9.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 8 q^{14} - 2 q^{16} + 8 q^{19} + 8 q^{21} + 6 q^{24} + 12 q^{26} - 2 q^{29} - 16 q^{31} - 12 q^{34} - 2 q^{36} - 12 q^{39} - 12 q^{41} - 8 q^{44} - 8 q^{46} - 18 q^{49} + 12 q^{51} - 2 q^{54} - 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{66} + 8 q^{69} - 16 q^{71} - 4 q^{74} + 8 q^{76} + 2 q^{81} + 8 q^{84} + 8 q^{86} + 12 q^{89} + 48 q^{91} + 10 q^{96} + 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 - 8 * q^14 - 2 * q^16 + 8 * q^19 + 8 * q^21 + 6 * q^24 + 12 * q^26 - 2 * q^29 - 16 * q^31 - 12 * q^34 - 2 * q^36 - 12 * q^39 - 12 * q^41 - 8 * q^44 - 8 * q^46 - 18 * q^49 + 12 * q^51 - 2 * q^54 - 24 * q^56 + 24 * q^59 - 20 * q^61 - 14 * q^64 - 8 * q^66 + 8 * q^69 - 16 * q^71 - 4 * q^74 + 8 * q^76 + 2 * q^81 + 8 * q^84 + 8 * q^86 + 12 * q^89 + 48 * q^91 + 10 * q^96 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 −0.935414 0.353553i $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ − 3.00000i − 1.06066i
$$9$$ −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
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 4.00000i 0.852803i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 5.00000i − 0.883883i
$$33$$ − 4.00000i − 0.696311i
$$34$$ −6.00000 −1.02899
$$35$$ 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
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 6.00000i 0.832050i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −12.0000 −1.60357
$$57$$ 4.00000i 0.529813i
$$58$$ 1.00000i 0.131306i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 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
$$63$$ 4.00000i 0.503953i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 3.00000i 0.353553i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 16.0000i 1.82337i
$$78$$ 6.00000i 0.679366i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 1.00000i − 0.107211i
$$88$$ 12.0000i 1.27920i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 24.0000 2.51588
$$92$$ − 4.00000i − 0.417029i
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 12.0000i − 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 18.0000 1.76505
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 4.00000i 0.377964i
$$113$$ − 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
$$116$$ −1.00000 −0.0928477
$$117$$ − 6.00000i − 0.554700i
$$118$$ − 12.0000i − 1.10469i
$$119$$ −24.0000 −2.20008
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000i 0.905357i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ − 3.00000i − 0.265165i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ − 16.0000i − 1.38738i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000i 0.671345i
$$143$$ − 24.0000i − 2.00698i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ − 9.00000i − 0.742307i
$$148$$ − 2.00000i − 0.164399i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ − 12.0000i − 0.973329i
$$153$$ 6.00000i 0.485071i
$$154$$ 16.0000 1.28932
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 12.0000i − 0.939913i −0.882690 0.469956i $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ − 12.0000i − 0.925820i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ −1.00000 −0.0758098
$$175$$ 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
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ − 24.0000i − 1.77900i
$$183$$ − 10.0000i − 0.739221i
$$184$$ −12.0000 −0.884652
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 24.0000i 1.75505i
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ − 7.00000i − 0.505181i
$$193$$ 22.0000i 1.58359i 0.610784 + 0.791797i $$0.290854\pi$$
−0.610784 + 0.791797i $$0.709146\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ − 14.0000i − 0.997459i −0.866758 0.498729i $$-0.833800\pi$$
0.866758 0.498729i $$-0.166200\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 2.00000i 0.140720i
$$203$$ 4.00000i 0.280745i
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ −12.0000 −0.836080
$$207$$ 4.00000i 0.278019i
$$208$$ − 6.00000i − 0.416025i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ − 8.00000i − 0.548151i
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ 32.0000i 2.17230i
$$218$$ 14.0000i 0.948200i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 36.0000 2.42162
$$222$$ − 2.00000i − 0.134231i
$$223$$ 28.0000i 1.87502i 0.347960 + 0.937509i $$0.386874\pi$$
−0.347960 + 0.937509i $$0.613126\pi$$
$$224$$ −20.0000 −1.33631
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ − 24.0000i − 1.59294i −0.604681 0.796468i $$-0.706699\pi$$
0.604681 0.796468i $$-0.293301\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ 3.00000i 0.196960i
$$233$$ − 22.0000i − 1.44127i −0.693316 0.720634i $$-0.743851\pi$$
0.693316 0.720634i $$-0.256149\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 24.0000i 1.55569i
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 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
$$246$$ −6.00000 −0.382546
$$247$$ 24.0000i 1.52708i
$$248$$ 24.0000i 1.52400i
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 16.0000i 1.00591i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 22.0000i 1.37232i 0.727450 + 0.686161i $$0.240706\pi$$
−0.727450 + 0.686161i $$0.759294\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ − 12.0000i − 0.741362i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ −16.0000 −0.981023
$$267$$ 6.00000i 0.367194i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −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 $$0.338468\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 24.0000i 1.45255i
$$274$$ 2.00000 0.120824
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 8.00000i 0.475551i 0.971320 + 0.237775i $$0.0764182\pi$$
−0.971320 + 0.237775i $$0.923582\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ 24.0000i 1.41668i
$$288$$ 5.00000i 0.294628i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ − 2.00000i − 0.117041i
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 4.00000i 0.232104i
$$298$$ 6.00000i 0.347571i
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ − 16.0000i − 0.920697i
$$303$$ − 2.00000i − 0.114897i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 16.0000i 0.911685i
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 18.0000i 1.01905i
$$313$$ − 30.0000i − 1.69570i −0.530236 0.847850i $$-0.677897\pi$$
0.530236 0.847850i $$-0.322103\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ − 10.0000i − 0.560772i
$$319$$ 4.00000 0.223957
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 16.0000i 0.891645i
$$323$$ − 24.0000i − 1.33540i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ − 14.0000i − 0.774202i
$$328$$ 18.0000i 0.993884i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 8.00000i 0.439057i
$$333$$ 2.00000i 0.109599i
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ − 30.0000i − 1.63420i −0.576493 0.817102i $$-0.695579\pi$$
0.576493 0.817102i $$-0.304421\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 32.0000 1.73290
$$342$$ 4.00000i 0.216295i
$$343$$ 8.00000i 0.431959i
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ − 16.0000i − 0.858925i −0.903085 0.429463i $$-0.858703\pi$$
0.903085 0.429463i $$-0.141297\pi$$
$$348$$ − 1.00000i − 0.0536056i
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 20.0000i 1.06600i
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ − 24.0000i − 1.27021i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 22.0000i − 1.15629i
$$363$$ 5.00000i 0.262432i
$$364$$ 24.0000 1.25794
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −40.0000 −2.07670
$$372$$ − 8.00000i − 0.414781i
$$373$$ − 2.00000i − 0.103556i −0.998659 0.0517780i $$-0.983511\pi$$
0.998659 0.0517780i $$-0.0164888\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 6.00000i − 0.309016i
$$378$$ 4.00000i 0.205738i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 16.0000i 0.818631i
$$383$$ 20.0000i 1.02195i 0.859595 + 0.510976i $$0.170716\pi$$
−0.859595 + 0.510976i $$0.829284\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ − 4.00000i − 0.203331i
$$388$$ 2.00000i 0.101535i
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ 27.0000i 1.36371i
$$393$$ 12.0000i 0.605320i
$$394$$ −14.0000 −0.705310
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 10.0000i 0.501886i 0.968002 + 0.250943i $$0.0807406\pi$$
−0.968002 + 0.250943i $$0.919259\pi$$
$$398$$ − 24.0000i − 1.20301i
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ − 8.00000i − 0.399004i
$$403$$ − 48.0000i − 2.39105i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 8.00000i 0.396545i
$$408$$ − 18.0000i − 0.891133i
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ − 12.0000i − 0.591198i
$$413$$ − 48.0000i − 2.36193i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 30.0000 1.47087
$$417$$ − 4.00000i − 0.195881i
$$418$$ 16.0000i 0.782586i
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ 0 0
$$424$$ −30.0000 −1.45693
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 40.0000i 1.93574i
$$428$$ − 8.00000i − 0.386695i
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 34.0000i − 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ 32.0000 1.53605
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ − 16.0000i − 0.765384i
$$438$$ − 2.00000i − 0.0955637i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ − 36.0000i − 1.71235i
$$443$$ 20.0000i 0.950229i 0.879924 + 0.475114i $$0.157593\pi$$
−0.879924 + 0.475114i $$0.842407\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 28.0000 1.32584
$$447$$ − 6.00000i − 0.283790i
$$448$$ 28.0000i 1.32288i
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ − 2.00000i − 0.0940721i
$$453$$ 16.0000i 0.751746i
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ 12.0000 0.561951
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ − 2.00000i − 0.0934539i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 16.0000i 0.744387i
$$463$$ 36.0000i 1.67306i 0.547920 + 0.836531i $$0.315420\pi$$
−0.547920 + 0.836531i $$0.684580\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ 20.0000i 0.925490i 0.886492 + 0.462745i $$0.153135\pi$$
−0.886492 + 0.462745i $$0.846865\pi$$
$$468$$ − 6.00000i − 0.277350i
$$469$$ −32.0000 −1.47762
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ − 36.0000i − 1.65703i
$$473$$ − 16.0000i − 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −24.0000 −1.10004
$$477$$ 10.0000i 0.457869i
$$478$$ − 8.00000i − 0.365911i
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ − 2.00000i − 0.0910975i
$$483$$ − 16.0000i − 0.728025i
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 28.0000i 1.26880i 0.773004 + 0.634401i $$0.218753\pi$$
−0.773004 + 0.634401i $$0.781247\pi$$
$$488$$ 30.0000i 1.35804i
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −44.0000 −1.98569 −0.992846 0.119401i $$-0.961903\pi$$
−0.992846 + 0.119401i $$0.961903\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 6.00000i 0.270226i
$$494$$ 24.0000 1.07981
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 32.0000i 1.43540i
$$498$$ 8.00000i 0.358489i
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 20.0000i 0.892644i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 12.0000 0.534522
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ − 23.0000i − 1.02147i
$$508$$ 16.0000i 0.709885i
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 11.0000i 0.486136i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 22.0000 0.970378
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 8.00000i 0.351500i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ − 1.00000i − 0.0437688i
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 48.0000i 2.09091i
$$528$$ 4.00000i 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ − 16.0000i − 0.693688i
$$533$$ − 36.0000i − 1.55933i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ −24.0000 −1.03664
$$537$$ 12.0000i 0.517838i
$$538$$ 6.00000i 0.258678i
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ 22.0000i 0.944110i
$$544$$ −30.0000 −1.28624
$$545$$ 0 0
$$546$$ 24.0000 1.02711
$$547$$ − 40.0000i − 1.71028i −0.518400 0.855138i $$-0.673472\pi$$
0.518400 0.855138i $$-0.326528\pi$$
$$548$$ 2.00000i 0.0854358i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ − 12.0000i − 0.510754i
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ − 10.0000i − 0.421825i
$$563$$ − 28.0000i − 1.18006i −0.807382 0.590030i $$-0.799116\pi$$
0.807382 0.590030i $$-0.200884\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 8.00000 0.336265
$$567$$ − 4.00000i − 0.167984i
$$568$$ 24.0000i 1.00702i
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ − 24.0000i − 1.00349i
$$573$$ − 16.0000i − 0.668410i
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 10.0000i 0.416305i 0.978096 + 0.208153i $$0.0667451\pi$$
−0.978096 + 0.208153i $$0.933255\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ −22.0000 −0.914289
$$580$$ 0 0
$$581$$ 32.0000 1.32758
$$582$$ 2.00000i 0.0829027i
$$583$$ 40.0000i 1.65663i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ 16.0000i 0.660391i 0.943913 + 0.330195i $$0.107115\pi$$
−0.943913 + 0.330195i $$0.892885\pi$$
$$588$$ − 9.00000i − 0.371154i
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ 2.00000i 0.0821995i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 24.0000i 0.982255i
$$598$$ − 24.0000i − 0.981433i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ − 16.0000i − 0.652111i
$$603$$ 8.00000i 0.325785i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ − 20.0000i − 0.811107i
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 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
$$616$$ 48.0000 1.93398
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ − 12.0000i − 0.482711i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ − 24.0000i − 0.961540i
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −30.0000 −1.19904
$$627$$ − 16.0000i − 0.638978i
$$628$$ − 18.0000i − 0.718278i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ 0 0
$$633$$ 20.0000i 0.794929i
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ − 54.0000i − 2.13956i
$$638$$ − 4.00000i − 0.158362i
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ − 8.00000i − 0.315735i
$$643$$ 16.0000i 0.630978i 0.948929 + 0.315489i $$0.102169\pi$$
−0.948929 + 0.315489i $$0.897831\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 28.0000i 1.10079i 0.834903 + 0.550397i $$0.185524\pi$$
−0.834903 + 0.550397i $$0.814476\pi$$
$$648$$ − 3.00000i − 0.117851i
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ − 12.0000i − 0.469956i
$$653$$ 50.0000i 1.95665i 0.207072 + 0.978326i $$0.433606\pi$$
−0.207072 + 0.978326i $$0.566394\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 36.0000i 1.39812i
$$664$$ 24.0000 0.931381
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 4.00000i 0.154881i
$$668$$ − 12.0000i − 0.464294i
$$669$$ −28.0000 −1.08254
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ − 20.0000i − 0.771517i
$$673$$ − 6.00000i − 0.231283i −0.993291 0.115642i $$-0.963108\pi$$
0.993291 0.115642i $$-0.0368924\pi$$
$$674$$ −30.0000 −1.15556
$$675$$ 0 0
$$676$$ −23.0000 −0.884615
$$677$$ − 50.0000i − 1.92166i −0.277145 0.960828i $$-0.589388\pi$$
0.277145 0.960828i $$-0.410612\pi$$
$$678$$ − 2.00000i − 0.0768095i
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ 24.0000 0.919682
$$682$$ − 32.0000i − 1.22534i
$$683$$ 40.0000i 1.53056i 0.643699 + 0.765279i $$0.277399\pi$$
−0.643699 + 0.765279i $$0.722601\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 2.00000i 0.0763048i
$$688$$ − 4.00000i − 0.152499i
$$689$$ 60.0000 2.28582
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ − 16.0000i − 0.607790i
$$694$$ −16.0000 −0.607352
$$695$$ 0 0
$$696$$ −3.00000 −0.113715
$$697$$ 36.0000i 1.36360i
$$698$$ − 34.0000i − 1.28692i
$$699$$ 22.0000 0.832116
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ − 6.00000i − 0.226455i
$$703$$ − 8.00000i − 0.301726i
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 8.00000i 0.300871i
$$708$$ 12.0000i 0.450988i
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 18.0000i − 0.674579i
$$713$$ 32.0000i 1.19841i
$$714$$ −24.0000 −0.898177
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 8.00000i 0.298765i
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ 3.00000i 0.111648i
$$723$$ 2.00000i 0.0743808i
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ − 72.0000i − 2.66850i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ − 10.0000i − 0.369611i
$$733$$ 2.00000i 0.0738717i 0.999318 + 0.0369358i $$0.0117597\pi$$
−0.999318 + 0.0369358i $$0.988240\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ 32.0000i 1.17874i
$$738$$ − 6.00000i − 0.220863i
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ −24.0000 −0.881662
$$742$$ 40.0000i 1.46845i
$$743$$ − 8.00000i − 0.293492i −0.989174 0.146746i $$-0.953120\pi$$
0.989174 0.146746i $$-0.0468799\pi$$
$$744$$ −24.0000 −0.879883
$$745$$ 0 0
$$746$$ −2.00000 −0.0732252
$$747$$ − 8.00000i − 0.292705i
$$748$$ 24.0000i 0.877527i
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 0 0
$$753$$ − 20.0000i − 0.728841i
$$754$$ −6.00000 −0.218507
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 14.0000i 0.508839i 0.967094 + 0.254419i $$0.0818843\pi$$
−0.967094 + 0.254419i $$0.918116\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ 56.0000i 2.02734i
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ 72.0000i 2.59977i
$$768$$ − 17.0000i − 0.613435i
$$769$$ −42.0000 −1.51456 −0.757279 0.653091i $$-0.773472\pi$$
−0.757279 + 0.653091i $$0.773472\pi$$
$$770$$ 0 0
$$771$$ −22.0000 −0.792311
$$772$$ 22.0000i 0.791797i
$$773$$ − 38.0000i − 1.36677i −0.730061 0.683383i $$-0.760508\pi$$
0.730061 0.683383i $$-0.239492\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ − 8.00000i − 0.286998i
$$778$$ − 2.00000i − 0.0717035i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 24.0000i 0.858238i
$$783$$ 1.00000i 0.0357371i
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 48.0000i 1.71102i 0.517790 + 0.855508i $$0.326755\pi$$
−0.517790 + 0.855508i $$0.673245\pi$$
$$788$$ − 14.0000i − 0.498729i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ − 12.0000i − 0.426401i
$$793$$ − 60.0000i − 2.13066i
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ − 50.0000i − 1.77109i −0.464553 0.885545i $$-0.653785\pi$$
0.464553 0.885545i $$-0.346215\pi$$
$$798$$ − 16.0000i − 0.566394i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 2.00000i − 0.0706225i
$$803$$ 8.00000i 0.282314i
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ − 6.00000i − 0.211210i
$$808$$ 6.00000i 0.211079i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 4.00000i 0.140372i
$$813$$ 16.0000i 0.561144i
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ 16.0000i 0.559769i
$$818$$ − 22.0000i − 0.769212i
$$819$$ −24.0000 −0.838628
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 2.00000i 0.0697580i
$$823$$ 16.0000i 0.557725i 0.960331 + 0.278862i $$0.0899574\pi$$
−0.960331 + 0.278862i $$0.910043\pi$$
$$824$$ −36.0000 −1.25412
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ − 20.0000i − 0.695468i −0.937593 0.347734i $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ − 42.0000i − 1.45609i
$$833$$ 54.0000i 1.87099i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 8.00000i 0.276520i
$$838$$ − 4.00000i − 0.138178i
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 18.0000i 0.620321i
$$843$$ 10.0000i 0.344418i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 20.0000i − 0.687208i
$$848$$ 10.0000i 0.343401i
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 38.0000i − 1.30110i −0.759465 0.650548i $$-0.774539\pi$$
0.759465 0.650548i $$-0.225461\pi$$
$$854$$ 40.0000 1.36877
$$855$$ 0 0
$$856$$ −24.0000 −0.820303
$$857$$ − 2.00000i − 0.0683187i −0.999416 0.0341593i $$-0.989125\pi$$
0.999416 0.0341593i $$-0.0108754\pi$$
$$858$$ − 24.0000i − 0.819346i
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ − 24.0000i − 0.817443i
$$863$$ 4.00000i 0.136162i 0.997680 + 0.0680808i $$0.0216876\pi$$
−0.997680 + 0.0680808i $$0.978312\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ − 19.0000i − 0.645274i
$$868$$ 32.0000i 1.08615i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ 42.0000i 1.42230i
$$873$$ − 2.00000i − 0.0676897i
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ − 24.0000i − 0.809961i
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ − 56.0000i − 1.88455i −0.334840 0.942275i $$-0.608682\pi$$
0.334840 0.942275i $$-0.391318\pi$$
$$884$$ 36.0000 1.21081
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ − 6.00000i − 0.201347i
$$889$$ 64.0000 2.14649
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 28.0000i 0.937509i
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ −12.0000 −0.400892
$$897$$ 24.0000i 0.801337i
$$898$$ 26.0000i 0.867631i
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ −60.0000 −1.99889
$$902$$ − 24.0000i − 0.799113i
$$903$$ 16.0000i 0.532447i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ − 24.0000i − 0.796468i
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ −56.0000 −1.85536 −0.927681 0.373373i $$-0.878201\pi$$
−0.927681 + 0.373373i $$0.878201\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 32.0000i − 1.05905i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ − 48.0000i − 1.58510i
$$918$$ 6.00000i 0.198030i
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 6.00000i − 0.197599i
$$923$$ − 48.0000i − 1.57994i
$$924$$ −16.0000 −0.526361
$$925$$ 0 0
$$926$$ 36.0000 1.18303
$$927$$ 12.0000i 0.394132i
$$928$$ 5.00000i 0.164133i
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ − 22.0000i − 0.720634i
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ 0 0
$$936$$ −18.0000 −0.588348
$$937$$ − 10.0000i − 0.326686i −0.986569 0.163343i $$-0.947772\pi$$
0.986569 0.163343i $$-0.0522277\pi$$
$$938$$ 32.0000i 1.04484i
$$939$$ 30.0000 0.979013
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ − 18.0000i − 0.586472i
$$943$$ 24.0000i 0.781548i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ − 28.0000i − 0.909878i −0.890523 0.454939i $$-0.849661\pi$$
0.890523 0.454939i $$-0.150339\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 72.0000i 2.33353i
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ 4.00000i 0.129302i
$$958$$ − 16.0000i − 0.516937i
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 12.0000i − 0.386896i
$$963$$ 8.00000i 0.257796i
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ − 15.0000i − 0.482118i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 16.0000i 0.512936i
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 6.00000i 0.191957i 0.995383 + 0.0959785i $$0.0305980\pi$$
−0.995383 + 0.0959785i $$0.969402\pi$$
$$978$$ − 12.0000i − 0.383718i
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 44.0000i 1.40410i
$$983$$ 40.0000i 1.27580i 0.770118 + 0.637901i $$0.220197\pi$$
−0.770118 + 0.637901i $$0.779803\pi$$
$$984$$ −18.0000 −0.573819
$$985$$ 0 0
$$986$$ 6.00000 0.191079
$$987$$ 0 0
$$988$$ 24.0000i 0.763542i
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 40.0000i 1.27000i
$$993$$ 4.00000i 0.126936i
$$994$$ 32.0000 1.01498
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ − 50.0000i − 1.58352i −0.610835 0.791758i $$-0.709166\pi$$
0.610835 0.791758i $$-0.290834\pi$$
$$998$$ − 36.0000i − 1.13956i
$$999$$ −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 2175.2.c.b.349.1 2
5.2 odd 4 435.2.a.d.1.1 1
5.3 odd 4 2175.2.a.b.1.1 1
5.4 even 2 inner 2175.2.c.b.349.2 2
15.2 even 4 1305.2.a.b.1.1 1
15.8 even 4 6525.2.a.j.1.1 1
20.7 even 4 6960.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.d.1.1 1 5.2 odd 4
1305.2.a.b.1.1 1 15.2 even 4
2175.2.a.b.1.1 1 5.3 odd 4
2175.2.c.b.349.1 2 1.1 even 1 trivial
2175.2.c.b.349.2 2 5.4 even 2 inner
6525.2.a.j.1.1 1 15.8 even 4
6960.2.a.l.1.1 1 20.7 even 4