LMFDB - Embedded newform 2850.2.d.s.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, 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("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.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:	$22.7573645761$
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 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.s.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -4.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +6.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +1.00000i q^{38} +10.0000 q^{41} +4.00000i q^{42} +12.0000i q^{43} -4.00000 q^{44} +2.00000 q^{46} +10.0000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} -1.00000i q^{57} -6.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} -6.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +2.00000i q^{68} -2.00000 q^{69} -16.0000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -8.00000 q^{74} +1.00000 q^{76} +16.0000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +16.0000i q^{83} +4.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +4.00000i q^{88} +2.00000 q^{89} -2.00000i q^{92} +6.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{19} - 8 q^{21} - 2 q^{24} + 12 q^{29} + 12 q^{31} - 4 q^{34} + 2 q^{36} + 20 q^{41} - 8 q^{44} + 4 q^{46} - 18 q^{49} + 4 q^{51} - 2 q^{54} - 8 q^{56} - 8 q^{59} - 20 q^{61} - 2 q^{64} + 8 q^{66} - 4 q^{69} - 32 q^{71} - 16 q^{74} + 2 q^{76} - 20 q^{79} + 2 q^{81} + 8 q^{84} + 24 q^{86} + 4 q^{89} + 20 q^{94} + 2 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 - 2 * q^19 - 8 * q^21 - 2 * q^24 + 12 * q^29 + 12 * q^31 - 4 * q^34 + 2 * q^36 + 20 * q^41 - 8 * q^44 + 4 * q^46 - 18 * q^49 + 4 * q^51 - 2 * q^54 - 8 * q^56 - 8 * q^59 - 20 * q^61 - 2 * q^64 + 8 * q^66 - 4 * q^69 - 32 * q^71 - 16 * q^74 + 2 * q^76 - 20 * q^79 + 2 * q^81 + 8 * q^84 + 24 * q^86 + 4 * q^89 + 20 * q^94 + 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 1.00000i	− 0.288675i
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.00000	−0.872872
$22$	− 4.00000i	− 0.852803i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	− 4.00000i	− 0.755929i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	− 1.00000i	− 0.176777i
$33$	4.00000i	0.696311i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	1.00000i	0.162221i
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	4.00000i	0.617213i
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	2.00000	0.294884
$47$	10.0000i	1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$	10.0000i	1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 1.00000i	− 0.132453i
$58$	− 6.00000i	− 0.787839i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 6.00000i	− 0.762001i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	2.00000i	0.242536i
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	1.00000	0.114708
$77$	16.0000i	1.82337i
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.0000i	− 1.10432i
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	12.0000	1.29399
$87$	6.00000i	0.643268i
$88$	4.00000i	0.426401i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	− 2.00000i	− 0.208514i
$93$	6.00000i	0.622171i
$94$	10.0000	1.03142
$95$	0	0
$96$	1.00000	0.102062
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	9.00000i	0.909137i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	− 2.00000i	− 0.198030i
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	1.00000i	0.0962250i
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	4.00000i	0.377964i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	4.00000i	0.368230i
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000i	0.905357i
$123$	10.0000i	0.901670i
$124$	−6.00000	−0.538816
$125$	0	0
$126$	−4.00000	−0.356348
$127$	22.0000i	1.95218i	0.217357	+	0.976092i	$0.430256\pi$
$127$	22.0000i	1.95218i	−0.217357	+	0.976092i	$0.569744\pi$
$128$	1.00000i	0.0883883i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 4.00000i	− 0.348155i
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	2.00000	0.171499
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	2.00000i	0.170251i
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	16.0000i	1.34269i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	− 9.00000i	− 0.742307i
$148$	8.00000i	0.657596i
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	2.00000i	0.161690i
$154$	16.0000	1.28932
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	10.0000i	0.795557i
$159$	2.00000	0.158610
$160$	0	0
$161$	−8.00000	−0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	16.0000	1.24184
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	− 4.00000i	− 0.308607i
$169$	13.0000	1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 12.0000i	− 0.914991i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	− 4.00000i	− 0.300658i
$178$	− 2.00000i	− 0.149906i
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	− 10.0000i	− 0.739221i
$184$	−2.00000	−0.147442
$185$	0	0
$186$	6.00000	0.439941
$187$	− 8.00000i	− 0.585018i
$188$	− 10.0000i	− 0.729325i
$189$	4.00000	0.290957
$190$	0	0
$191$	−2.00000	−0.144715	−0.0723575	−	0.997379i	$-0.523052\pi$
$191$	−2.00000	−0.144715	−0.0723575	+	0.997379i	$0.523052\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	− 20.0000i	− 1.42494i	−0.701702	−	0.712470i	$-0.747576\pi$
$197$	− 20.0000i	− 1.42494i	0.701702	−	0.712470i	$-0.252424\pi$
$198$	4.00000i	0.284268i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	− 8.00000i	− 0.562878i
$203$	24.0000i	1.68447i
$204$	−2.00000	−0.140028
$205$	0	0
$206$	6.00000	0.418040
$207$	− 2.00000i	− 0.139010i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	2.00000i	0.137361i
$213$	− 16.0000i	− 1.09630i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	24.0000i	1.62923i
$218$	− 4.00000i	− 0.270914i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	− 8.00000i	− 0.536925i
$223$	− 6.00000i	− 0.401790i	−0.979613	−	0.200895i	$-0.935615\pi$
$223$	− 6.00000i	− 0.401790i	0.979613	−	0.200895i	$-0.0643850\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	14.0000	0.931266
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	1.00000i	0.0662266i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	6.00000i	0.393919i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	− 10.0000i	− 0.649570i
$238$	− 8.00000i	− 0.518563i
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	− 5.00000i	− 0.321412i
$243$	1.00000i	0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	10.0000	0.637577
$247$	0	0
$248$	6.00000i	0.381000i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	4.00000i	0.251976i
$253$	8.00000i	0.502956i
$254$	22.0000	1.38040
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	12.0000i	0.747087i
$259$	32.0000	1.98838
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	−4.00000	−0.245256
$267$	2.00000i	0.122398i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	2.00000	0.120386
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 4.00000i	− 0.239904i
$279$	−6.00000	−0.359211
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	10.0000i	0.595491i
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	40.0000i	2.36113i
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	10.0000	0.586210
$292$	− 2.00000i	− 0.117041i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	8.00000	0.464991
$297$	− 4.00000i	− 0.232104i
$298$	20.0000i	1.15857i
$299$	0	0
$300$	0	0
$301$	−48.0000	−2.76667
$302$	− 10.0000i	− 0.575435i
$303$	8.00000i	0.459588i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	2.00000	0.114332
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	− 16.0000i	− 0.911685i
$309$	−6.00000	−0.341328
$310$	0	0
$311$	34.0000	1.92796	0.963982	−	0.265969i	$-0.0856919\pi$
$311$	34.0000	1.92796	0.963982	+	0.265969i	$0.0856919\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	10.0000	0.562544
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	− 2.00000i	− 0.112154i
$319$	24.0000	1.34374
$320$	0	0
$321$	4.00000	0.223258
$322$	8.00000i	0.445823i
$323$	2.00000i	0.111283i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	4.00000i	0.221201i
$328$	10.0000i	0.552158i
$329$	−40.0000	−2.20527
$330$	0	0
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	− 16.0000i	− 0.878114i
$333$	8.00000i	0.438397i
$334$	12.0000	0.656611
$335$	0	0
$336$	−4.00000	−0.218218
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	− 13.0000i	− 0.707107i
$339$	−14.0000	−0.760376
$340$	0	0
$341$	24.0000	1.29967
$342$	− 1.00000i	− 0.0540738i
$343$	− 8.00000i	− 0.431959i
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	− 4.00000i	− 0.213201i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−2.00000	−0.106000
$357$	8.00000i	0.423405i
$358$	− 20.0000i	− 1.05703i
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 12.0000i	− 0.630706i
$363$	5.00000i	0.262432i
$364$	0	0
$365$	0	0
$366$	−10.0000	−0.522708
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	2.00000i	0.104257i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	8.00000	0.415339
$372$	− 6.00000i	− 0.311086i
$373$	− 8.00000i	− 0.414224i	−0.978317	−	0.207112i	$-0.933593\pi$
$373$	− 8.00000i	− 0.414224i	0.978317	−	0.207112i	$-0.0664065\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	−10.0000	−0.515711
$377$	0	0
$378$	− 4.00000i	− 0.205738i
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−22.0000	−1.12709
$382$	2.00000i	0.102329i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	− 12.0000i	− 0.609994i
$388$	10.0000i	0.507673i
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	−20.0000	−1.00759
$395$	0	0
$396$	4.00000	0.201008
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	− 4.00000i	− 0.200502i
$399$	4.00000	0.200250
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	0	0
$404$	−8.00000	−0.398015
$405$	0	0
$406$	24.0000	1.19110
$407$	− 32.0000i	− 1.58618i
$408$	2.00000i	0.0990148i
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	− 6.00000i	− 0.295599i
$413$	− 16.0000i	− 0.787309i
$414$	−2.00000	−0.0982946
$415$	0	0
$416$	0	0
$417$	4.00000i	0.195881i
$418$	4.00000i	0.195646i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	20.0000i	0.973585i
$423$	− 10.0000i	− 0.486217i
$424$	2.00000	0.0971286
$425$	0	0
$426$	−16.0000	−0.775203
$427$	− 40.0000i	− 1.93574i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	18.0000i	0.865025i	0.901628	+	0.432512i	$0.142373\pi$
$433$	18.0000i	0.865025i	−0.901628	+	0.432512i	$0.857627\pi$
$434$	24.0000	1.15204
$435$	0	0
$436$	−4.00000	−0.191565
$437$	− 2.00000i	− 0.0956730i
$438$	2.00000i	0.0955637i
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−6.00000	−0.284108
$447$	− 20.0000i	− 0.945968i
$448$	− 4.00000i	− 0.188982i
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	− 14.0000i	− 0.658505i
$453$	10.0000i	0.469841i
$454$	20.0000	0.938647
$455$	0	0
$456$	1.00000	0.0468293
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	− 2.00000i	− 0.0934539i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	16.0000i	0.744387i
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−18.0000	−0.829396
$472$	− 4.00000i	− 0.184115i
$473$	48.0000i	2.20704i
$474$	−10.0000	−0.459315
$475$	0	0
$476$	−8.00000	−0.366679
$477$	2.00000i	0.0915737i
$478$	− 6.00000i	− 0.274434i
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	0	0
$482$	2.00000i	0.0910975i
$483$	− 8.00000i	− 0.364013i
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	− 10.0000i	− 0.452679i
$489$	20.0000	0.904431
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	− 10.0000i	− 0.450835i
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	6.00000	0.269408
$497$	− 64.0000i	− 2.87079i
$498$	16.0000i	0.716977i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	24.0000i	1.07117i
$503$	− 34.0000i	− 1.51599i	−0.652263	−	0.757993i	$-0.726180\pi$
$503$	− 34.0000i	− 1.51599i	0.652263	−	0.757993i	$-0.273820\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	8.00000	0.355643
$507$	13.0000i	0.577350i
$508$	− 22.0000i	− 0.976092i
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	1.00000i	0.0441511i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	12.0000	0.528271
$517$	40.0000i	1.75920i
$518$	− 32.0000i	− 1.40600i
$519$	6.00000	0.263371
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	6.00000i	0.262613i
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	0	0
$526$	6.00000	0.261612
$527$	− 12.0000i	− 0.522728i
$528$	4.00000i	0.174078i
$529$	19.0000	0.826087
$530$	0	0
$531$	4.00000	0.173585
$532$	4.00000i	0.173422i
$533$	0	0
$534$	2.00000	0.0865485
$535$	0	0
$536$	0	0
$537$	20.0000i	0.863064i
$538$	14.0000i	0.603583i
$539$	−36.0000	−1.55063
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	4.00000i	0.171815i
$543$	12.0000i	0.514969i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	− 16.0000i	− 0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$	− 16.0000i	− 0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$	− 6.00000i	− 0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	−6.00000	−0.255609
$552$	− 2.00000i	− 0.0851257i
$553$	− 40.0000i	− 1.70097i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	4.00000i	0.169485i	0.996403	+	0.0847427i	$0.0270068\pi$
$557$	4.00000i	0.169485i	−0.996403	+	0.0847427i	$0.972993\pi$
$558$	6.00000i	0.254000i
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	− 10.0000i	− 0.421825i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	10.0000	0.421076
$565$	0	0
$566$	−28.0000	−1.17693
$567$	4.00000i	0.167984i
$568$	− 16.0000i	− 0.671345i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	− 2.00000i	− 0.0835512i
$574$	40.0000	1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	− 13.0000i	− 0.540729i
$579$	14.0000	0.581820
$580$	0	0
$581$	−64.0000	−2.65517
$582$	− 10.0000i	− 0.414513i
$583$	− 8.00000i	− 0.331326i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−14.0000	−0.578335
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	9.00000i	0.371154i
$589$	−6.00000	−0.247226
$590$	0	0
$591$	20.0000	0.822690
$592$	− 8.00000i	− 0.328798i
$593$	− 10.0000i	− 0.410651i	−0.978694	−	0.205325i	$-0.934175\pi$
$593$	− 10.0000i	− 0.410651i	0.978694	−	0.205325i	$-0.0658253\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	20.0000	0.819232
$597$	4.00000i	0.163709i
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	48.0000i	1.95633i
$603$	0	0
$604$	−10.0000	−0.406894
$605$	0	0
$606$	8.00000	0.324978
$607$	− 22.0000i	− 0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	− 22.0000i	− 0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$	1.00000i	0.0405554i
$609$	−24.0000	−0.972529
$610$	0	0
$611$	0	0
$612$	− 2.00000i	− 0.0808452i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	−16.0000	−0.644658
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	6.00000i	0.241355i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	− 34.0000i	− 1.36328i
$623$	8.00000i	0.320513i
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	− 4.00000i	− 0.159745i
$628$	− 18.0000i	− 0.718278i
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 10.0000i	− 0.397779i
$633$	− 20.0000i	− 0.794929i
$634$	6.00000	0.238290
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	− 24.0000i	− 0.950169i
$639$	16.0000	0.632950
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	2.00000	0.0786889
$647$	− 18.0000i	− 0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$	− 18.0000i	− 0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$	1.00000i	0.0392837i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	−24.0000	−0.940634
$652$	20.0000i	0.783260i
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	10.0000	0.390434
$657$	− 2.00000i	− 0.0780274i
$658$	40.0000i	1.55936i
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	− 24.0000i	− 0.932786i
$663$	0	0
$664$	−16.0000	−0.620920
$665$	0	0
$666$	8.00000	0.309994
$667$	12.0000i	0.464642i
$668$	− 12.0000i	− 0.464294i
$669$	6.00000	0.231973
$670$	0	0
$671$	−40.0000	−1.54418
$672$	4.00000i	0.154303i
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	−13.0000	−0.500000
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	14.0000i	0.537667i
$679$	40.0000	1.53506
$680$	0	0
$681$	−20.0000	−0.766402
$682$	− 24.0000i	− 0.919007i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−8.00000	−0.305441
$687$	2.00000i	0.0763048i
$688$	12.0000i	0.457496i
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	6.00000i	0.228086i
$693$	− 16.0000i	− 0.607790i
$694$	28.0000	1.06287
$695$	0	0
$696$	−6.00000	−0.227429
$697$	− 20.0000i	− 0.757554i
$698$	10.0000i	0.378506i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	30.0000	1.12906
$707$	32.0000i	1.20348i
$708$	4.00000i	0.150329i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	2.00000i	0.0749532i
$713$	12.0000i	0.449404i
$714$	8.00000	0.299392
$715$	0	0
$716$	−20.0000	−0.747435
$717$	6.00000i	0.224074i
$718$	6.00000i	0.223918i
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	− 1.00000i	− 0.0372161i
$723$	− 2.00000i	− 0.0743808i
$724$	−12.0000	−0.445976
$725$	0	0
$726$	5.00000	0.185567
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	10.0000i	0.369611i
$733$	38.0000i	1.40356i	0.712393	+	0.701781i	$0.247612\pi$
$733$	38.0000i	1.40356i	−0.712393	+	0.701781i	$0.752388\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	2.00000	0.0737210
$737$	0	0
$738$	10.0000i	0.368105i
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	0	0
$742$	− 8.00000i	− 0.293689i
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	−8.00000	−0.292901
$747$	− 16.0000i	− 0.585409i
$748$	8.00000i	0.292509i
$749$	16.0000	0.584627
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	10.0000i	0.364662i
$753$	− 24.0000i	− 0.874609i
$754$	0	0
$755$	0	0
$756$	−4.00000	−0.145479
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	8.00000i	0.290573i
$759$	−8.00000	−0.290382
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	22.0000i	0.796976i
$763$	16.0000i	0.579239i
$764$	2.00000	0.0723575
$765$	0	0
$766$	−8.00000	−0.289052
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	14.0000i	0.503871i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	10.0000	0.358979
$777$	32.0000i	1.14799i
$778$	8.00000i	0.286814i
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−64.0000	−2.29010
$782$	− 4.00000i	− 0.143040i
$783$	− 6.00000i	− 0.214423i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	20.0000i	0.712470i
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−56.0000	−1.99113
$792$	− 4.00000i	− 0.142134i
$793$	0	0
$794$	−30.0000	−1.06466
$795$	0	0
$796$	−4.00000	−0.141776
$797$	− 34.0000i	− 1.20434i	−0.798367	−	0.602171i	$-0.794303\pi$
$797$	− 34.0000i	− 1.20434i	0.798367	−	0.602171i	$-0.205697\pi$
$798$	− 4.00000i	− 0.141598i
$799$	20.0000	0.707549
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	30.0000i	1.05934i
$803$	8.00000i	0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 14.0000i	− 0.492823i
$808$	8.00000i	0.281439i
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	− 24.0000i	− 0.842235i
$813$	− 4.00000i	− 0.140286i
$814$	−32.0000	−1.12160
$815$	0	0
$816$	2.00000	0.0700140
$817$	− 12.0000i	− 0.419827i
$818$	− 18.0000i	− 0.629355i
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	6.00000i	0.209274i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	−16.0000	−0.556711
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	2.00000i	0.0695048i
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	18.0000i	0.623663i
$834$	4.00000	0.138509
$835$	0	0
$836$	4.00000	0.138343
$837$	− 6.00000i	− 0.207390i
$838$	− 12.0000i	− 0.414533i
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	8.00000i	0.275698i
$843$	10.0000i	0.344418i
$844$	20.0000	0.688428
$845$	0	0
$846$	−10.0000	−0.343807
$847$	20.0000i	0.687208i
$848$	− 2.00000i	− 0.0686803i
$849$	28.0000	0.960958
$850$	0	0
$851$	16.0000	0.548473
$852$	16.0000i	0.548151i
$853$	− 54.0000i	− 1.84892i	−0.381273	−	0.924462i	$-0.624514\pi$
$853$	− 54.0000i	− 1.84892i	0.381273	−	0.924462i	$-0.375486\pi$
$854$	−40.0000	−1.36877
$855$	0	0
$856$	4.00000	0.136717
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−40.0000	−1.36320
$862$	4.00000i	0.136241i
$863$	28.0000i	0.953131i	0.879139	+	0.476566i	$0.158119\pi$
$863$	28.0000i	0.953131i	−0.879139	+	0.476566i	$0.841881\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	18.0000	0.611665
$867$	13.0000i	0.441503i
$868$	− 24.0000i	− 0.814613i
$869$	−40.0000	−1.35691
$870$	0	0
$871$	0	0
$872$	4.00000i	0.135457i
$873$	10.0000i	0.338449i
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	2.00000	0.0675737
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	22.0000i	0.742464i
$879$	14.0000	0.472208
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	− 9.00000i	− 0.303046i
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	8.00000i	0.268462i
$889$	−88.0000	−2.95143
$890$	0	0
$891$	4.00000	0.134005
$892$	6.00000i	0.200895i
$893$	− 10.0000i	− 0.334637i
$894$	−20.0000	−0.668900
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	36.0000	1.20067
$900$	0	0
$901$	−4.00000	−0.133259
$902$	− 40.0000i	− 1.33185i
$903$	− 48.0000i	− 1.59734i
$904$	−14.0000	−0.465633
$905$	0	0
$906$	10.0000	0.332228
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	− 20.0000i	− 0.663723i
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	64.0000i	2.11809i
$914$	2.00000	0.0661541
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	2.00000i	0.0660098i
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	12.0000i	0.395199i
$923$	0	0
$924$	16.0000	0.526361
$925$	0	0
$926$	−36.0000	−1.18303
$927$	− 6.00000i	− 0.197066i
$928$	− 6.00000i	− 0.196960i
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	− 6.00000i	− 0.196537i
$933$	34.0000i	1.11311i
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	6.00000	0.195803
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	18.0000i	0.586472i
$943$	20.0000i	0.651290i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	48.0000	1.56061
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	10.0000i	0.324785i
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	8.00000i	0.259281i
$953$	30.0000i	0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$	30.0000i	0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−6.00000	−0.194054
$957$	24.0000i	0.775810i
$958$	6.00000i	0.193851i
$959$	−24.0000	−0.775000
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	4.00000i	0.128898i
$964$	2.00000	0.0644157
$965$	0	0
$966$	−8.00000	−0.257396
$967$	12.0000i	0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	12.0000i	0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$	5.00000i	0.160706i
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	16.0000i	0.512936i
$974$	2.00000	0.0640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	− 20.0000i	− 0.639529i
$979$	8.00000	0.255681
$980$	0	0
$981$	−4.00000	−0.127710
$982$	− 24.0000i	− 0.765871i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	−12.0000	−0.382158
$987$	− 40.0000i	− 1.27321i
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−50.0000	−1.58830	−0.794151	−	0.607720i	$-0.792084\pi$
$991$	−50.0000	−1.58830	−0.794151	+	0.607720i	$0.792084\pi$
$992$	− 6.00000i	− 0.190500i
$993$	24.0000i	0.761617i
$994$	−64.0000	−2.02996
$995$	0	0
$996$	16.0000	0.506979
$997$	− 22.0000i	− 0.696747i	−0.937356	−	0.348373i	$-0.886734\pi$
$997$	− 22.0000i	− 0.696747i	0.937356	−	0.348373i	$-0.113266\pi$
$998$	− 4.00000i	− 0.126618i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.s.799.1		2
5.2	odd	4		2850.2.a.x.1.1		1
5.3	odd	4		114.2.a.a.1.1	✓	1
5.4	even	2	inner	2850.2.d.s.799.2		2
15.2	even	4		8550.2.a.a.1.1		1
15.8	even	4		342.2.a.f.1.1		1
20.3	even	4		912.2.a.h.1.1		1
35.13	even	4		5586.2.a.p.1.1		1
40.3	even	4		3648.2.a.j.1.1		1
40.13	odd	4		3648.2.a.bb.1.1		1
60.23	odd	4		2736.2.a.j.1.1		1
95.18	even	4		2166.2.a.i.1.1		1
285.113	odd	4		6498.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.a.a.1.1	✓	1	5.3	odd	4
342.2.a.f.1.1		1	15.8	even	4
912.2.a.h.1.1		1	20.3	even	4
2166.2.a.i.1.1		1	95.18	even	4
2736.2.a.j.1.1		1	60.23	odd	4
2850.2.a.x.1.1		1	5.2	odd	4
2850.2.d.s.799.1		2	1.1	even	1	trivial
2850.2.d.s.799.2		2	5.4	even	2	inner
3648.2.a.j.1.1		1	40.3	even	4
3648.2.a.bb.1.1		1	40.13	odd	4
5586.2.a.p.1.1		1	35.13	even	4
6498.2.a.h.1.1		1	285.113	odd	4
8550.2.a.a.1.1		1	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -4.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +6.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +1.00000i q^{38} +10.0000 q^{41} +4.00000i q^{42} +12.0000i q^{43} -4.00000 q^{44} +2.00000 q^{46} +10.0000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} -1.00000i q^{57} -6.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} -6.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +2.00000i q^{68} -2.00000 q^{69} -16.0000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -8.00000 q^{74} +1.00000 q^{76} +16.0000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +16.0000i q^{83} +4.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +4.00000i q^{88} +2.00000 q^{89} -2.00000i q^{92} +6.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 8 q^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} - 2 q^{24} + 12 q^{29} + 12 q^{31} - 4 q^{34} + 2 q^{36} + 20 q^{41} - 8 q^{44} + 4 q^{46} - 18 q^{49} + 4 q^{51} - 2 q^{54} - 8 q^{56} - 8 q^{59} - 20 q^{61} - 2 q^{64} + 8 q^{66} - 4 q^{69} - 32 q^{71} - 16 q^{74} + 2 q^{76} - 20 q^{79} + 2 q^{81} + 8 q^{84} + 24 q^{86} + 4 q^{89} + 20 q^{94} + 2 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 - 2 * q^24 + 12 * q^29 + 12 * q^31 - 4 * q^34 + 2 * q^36 + 20 * q^41 - 8 * q^44 + 4 * q^46 - 18 * q^49 + 4 * q^51 - 2 * q^54 - 8 * q^56 - 8 * q^59 - 20 * q^61 - 2 * q^64 + 8 * q^66 - 4 * q^69 - 32 * q^71 - 16 * q^74 + 2 * q^76 - 20 * q^79 + 2 * q^81 + 8 * q^84 + 24 * q^86 + 4 * q^89 + 20 * q^94 + 2 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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