LMFDB - Embedded newform 2850.2.d.p.799.2

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			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.p.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} +6.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -6.00000 q^{29} +2.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -6.00000 q^{46} +6.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} +12.0000 q^{59} +14.0000 q^{61} +2.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} -6.00000i q^{68} +6.00000 q^{69} +1.00000i q^{72} -14.0000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} -4.00000 q^{86} +6.00000i q^{87} +6.00000 q^{89} +16.0000 q^{91} -6.00000i q^{92} -2.00000i q^{93} -6.00000 q^{94} +1.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} - 2 q^{24} - 8 q^{26} - 12 q^{29} + 4 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} - 12 q^{46} - 18 q^{49} + 12 q^{51} - 2 q^{54} - 8 q^{56} + 24 q^{59} + 28 q^{61} - 2 q^{64} + 12 q^{69} + 8 q^{74} + 2 q^{76} + 20 q^{79} + 2 q^{81} + 8 q^{84} - 8 q^{86} + 12 q^{89} + 32 q^{91} - 12 q^{94} + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 - 2 * q^24 - 8 * q^26 - 12 * q^29 + 4 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 + 12 * q^41 - 12 * q^46 - 18 * q^49 + 12 * q^51 - 2 * q^54 - 8 * q^56 + 24 * q^59 + 28 * q^61 - 2 * q^64 + 12 * q^69 + 8 * q^74 + 2 * q^76 + 20 * q^79 + 2 * q^81 + 8 * q^84 - 8 * q^86 + 12 * q^89 + 32 * q^91 - 12 * q^94 + 2 * q^96

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	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	1.00000i	0.192450i
$28$	4.00000i	0.755929i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	− 1.00000i	− 0.162221i
$39$	4.00000	0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 4.00000i	− 0.617213i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	−6.00000	−0.884652
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	− 4.00000i	− 0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	1.00000i	0.132453i
$58$	− 6.00000i	− 0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	2.00000i	0.254000i
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	− 6.00000i	− 0.727607i
$69$	6.00000	0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000i	0.117851i
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	4.00000i	0.452911i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	−4.00000	−0.431331
$87$	6.00000i	0.643268i
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	− 6.00000i	− 0.625543i
$93$	− 2.00000i	− 0.207390i
$94$	−6.00000	−0.618853
$95$	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
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	6.00000i	0.594089i
$103$	10.0000i	0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$	10.0000i	0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 4.00000i	− 0.377964i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	6.00000	0.557086
$117$	− 4.00000i	− 0.369800i
$118$	12.0000i	1.10469i
$119$	24.0000	2.20008
$120$	0	0
$121$	−11.0000	−1.00000
$122$	14.0000i	1.26750i
$123$	− 6.00000i	− 0.541002i
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−4.00000	−0.356348
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	6.00000i	0.510754i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	9.00000i	0.742307i
$148$	4.00000i	0.328798i
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	1.00000i	0.0811107i
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	10.0000i	0.795557i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	24.0000	1.89146
$162$	1.00000i	0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	4.00000i	0.308607i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	− 4.00000i	− 0.304997i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	6.00000i	0.449719i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	16.0000i	1.18600i
$183$	− 14.0000i	− 1.03491i
$184$	6.00000	0.442326
$185$	0	0
$186$	2.00000	0.146647
$187$	0	0
$188$	− 6.00000i	− 0.437595i
$189$	4.00000	0.290957
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	1.00000i	0.0721688i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	24.0000i	1.68447i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−10.0000	−0.696733
$207$	− 6.00000i	− 0.417029i
$208$	4.00000i	0.277350i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 8.00000i	− 0.543075i
$218$	16.0000i	1.08366i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−24.0000	−1.61441
$222$	− 4.00000i	− 0.268462i
$223$	22.0000i	1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$	22.0000i	1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−18.0000	−1.19734
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	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	$0.937031\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−12.0000	−0.781133
$237$	− 10.0000i	− 0.649570i
$238$	24.0000i	1.55569i
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−14.0000	−0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	− 4.00000i	− 0.254514i
$248$	− 2.00000i	− 0.127000i
$249$	12.0000	0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	− 4.00000i	− 0.251976i
$253$	0	0
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	4.00000i	0.249029i
$259$	−16.0000	−0.994192
$260$	0	0
$261$	6.00000	0.371391
$262$	12.0000i	0.741362i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	− 6.00000i	− 0.367194i
$268$	− 8.00000i	− 0.488678i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000i	0.363803i
$273$	− 16.0000i	− 0.968364i
$274$	18.0000	1.08742
$275$	0	0
$276$	−6.00000	−0.361158
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 20.0000i	− 1.19952i
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	6.00000i	0.357295i
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	14.0000i	0.819288i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	− 12.0000i	− 0.695141i
$299$	−24.0000	−1.38796
$300$	0	0
$301$	16.0000	0.922225
$302$	− 10.0000i	− 0.575435i
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	6.00000	0.342997
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	− 4.00000i	− 0.226455i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	24.0000i	1.33747i
$323$	− 6.00000i	− 0.333849i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	− 16.0000i	− 0.884802i
$328$	− 6.00000i	− 0.331295i
$329$	24.0000	1.32316
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	− 12.0000i	− 0.658586i
$333$	4.00000i	0.219199i
$334$	12.0000	0.656611
$335$	0	0
$336$	−4.00000	−0.218218
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	− 3.00000i	− 0.163178i
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	1.00000i	0.0540738i
$343$	8.00000i	0.431959i
$344$	4.00000	0.215666
$345$	0	0
$346$	18.0000	0.967686
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 24.0000i	− 1.27021i
$358$	− 12.0000i	− 0.634220i
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 16.0000i	− 0.840941i
$363$	11.0000i	0.577350i
$364$	−16.0000	−0.838628
$365$	0	0
$366$	14.0000	0.731792
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	6.00000i	0.312772i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	2.00000i	0.103695i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	− 24.0000i	− 1.23606i
$378$	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
$381$	2.00000	0.102463
$382$	18.0000i	0.920960i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	− 4.00000i	− 0.203331i
$388$	10.0000i	0.507673i
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	9.00000i	0.454569i
$393$	− 12.0000i	− 0.605320i
$394$	−12.0000	−0.604551
$395$	0	0
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	− 20.0000i	− 1.00251i
$399$	4.00000	0.200250
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	8.00000i	0.399004i
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0	0
$406$	−24.0000	−1.19110
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	− 10.0000i	− 0.492665i
$413$	− 48.0000i	− 2.36193i
$414$	6.00000	0.294884
$415$	0	0
$416$	−4.00000	−0.196116
$417$	20.0000i	0.979404i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	− 4.00000i	− 0.194717i
$423$	− 6.00000i	− 0.291730i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 56.0000i	− 2.71003i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000i	0.0481125i
$433$	10.0000i	0.480569i	0.970702	+	0.240285i	$0.0772408\pi$
$433$	10.0000i	0.480569i	−0.970702	+	0.240285i	$0.922759\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−16.0000	−0.766261
$437$	− 6.00000i	− 0.287019i
$438$	− 14.0000i	− 0.668946i
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	− 24.0000i	− 1.14156i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−22.0000	−1.04173
$447$	12.0000i	0.567581i
$448$	4.00000i	0.188982i
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	− 18.0000i	− 0.846649i
$453$	10.0000i	0.469841i
$454$	−12.0000	−0.563188
$455$	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	$0.985105\pi$
$458$	10.0000i	0.467269i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	4.00000i	0.184900i
$469$	32.0000	1.47762
$470$	0	0
$471$	2.00000	0.0921551
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	10.0000	0.459315
$475$	0	0
$476$	−24.0000	−1.10004
$477$	6.00000i	0.274721i
$478$	18.0000i	0.823301i
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	14.0000i	0.637683i
$483$	− 24.0000i	− 1.09204i
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	− 14.0000i	− 0.633750i
$489$	4.00000	0.180886
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	6.00000i	0.270501i
$493$	− 36.0000i	− 1.62136i
$494$	4.00000	0.179969
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	12.0000i	0.537733i
$499$	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
$502$	12.0000i	0.535586i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	− 2.00000i	− 0.0887357i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	1.00000i	0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 16.0000i	− 0.703000i
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	6.00000i	0.262613i
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−18.0000	−0.784837
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−12.0000	−0.520756
$532$	− 4.00000i	− 0.173422i
$533$	24.0000i	1.03956i
$534$	6.00000	0.259645
$535$	0	0
$536$	8.00000	0.345547
$537$	12.0000i	0.517838i
$538$	6.00000i	0.258678i
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	20.0000i	0.859074i
$543$	16.0000i	0.686626i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	16.0000	0.684737
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	18.0000i	0.768922i
$549$	−14.0000	−0.597505
$550$	0	0
$551$	6.00000	0.255609
$552$	− 6.00000i	− 0.255377i
$553$	− 40.0000i	− 1.70097i
$554$	10.0000	0.424859
$555$	0	0
$556$	20.0000	0.848189
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	− 2.00000i	− 0.0846668i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	− 18.0000i	− 0.759284i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−4.00000	−0.168133
$567$	− 4.00000i	− 0.167984i
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 19.0000i	− 0.790296i
$579$	10.0000	0.415586
$580$	0	0
$581$	48.0000	1.99138
$582$	− 10.0000i	− 0.414513i
$583$	0	0
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−6.00000	−0.247858
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	− 9.00000i	− 0.371154i
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	12.0000	0.493614
$592$	− 4.00000i	− 0.164399i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	20.0000i	0.818546i
$598$	− 24.0000i	− 0.981433i
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	16.0000i	0.652111i
$603$	− 8.00000i	− 0.325785i
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	− 34.0000i	− 1.38002i	−0.723801	−	0.690009i	$-0.757607\pi$
$607$	− 34.0000i	− 1.38002i	0.723801	−	0.690009i	$-0.242393\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	24.0000	0.972529
$610$	0	0
$611$	−24.0000	−0.970936
$612$	6.00000i	0.242536i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	10.0000i	0.402259i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	30.0000i	1.20289i
$623$	− 24.0000i	− 0.961540i
$624$	4.00000	0.160128
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	24.0000	0.956943
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 10.0000i	− 0.397779i
$633$	4.00000i	0.158986i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	− 36.0000i	− 1.42637i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	12.0000i	0.473602i
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	6.00000	0.236067
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	− 4.00000i	− 0.156652i
$653$	− 36.0000i	− 1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$	− 36.0000i	− 1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$	16.0000	0.625650
$655$	0	0
$656$	6.00000	0.234261
$657$	14.0000i	0.546192i
$658$	24.0000i	0.935617i
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	8.00000i	0.310929i
$663$	24.0000i	0.932083i
$664$	12.0000	0.465690
$665$	0	0
$666$	−4.00000	−0.154997
$667$	− 36.0000i	− 1.39393i
$668$	12.0000i	0.464294i
$669$	22.0000	0.850569
$670$	0	0
$671$	0	0
$672$	− 4.00000i	− 0.154303i
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	3.00000	0.115385
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	18.0000i	0.691286i
$679$	−40.0000	−1.53506
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 10.0000i	− 0.381524i
$688$	4.00000i	0.152499i
$689$	24.0000	0.914327
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000i	0.684257i
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	6.00000	0.227429
$697$	36.0000i	1.36360i
$698$	− 2.00000i	− 0.0757011i
$699$	6.00000	0.226941
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	− 4.00000i	− 0.150970i
$703$	4.00000i	0.150863i
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	12.0000i	0.450988i
$709$	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
$712$	− 6.00000i	− 0.224860i
$713$	12.0000i	0.449404i
$714$	24.0000	0.898177
$715$	0	0
$716$	12.0000	0.448461
$717$	− 18.0000i	− 0.672222i
$718$	6.00000i	0.223918i
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	40.0000	1.48968
$722$	1.00000i	0.0372161i
$723$	− 14.0000i	− 0.520666i
$724$	16.0000	0.594635
$725$	0	0
$726$	−11.0000	−0.408248
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	− 16.0000i	− 0.592999i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	14.0000i	0.517455i
$733$	− 50.0000i	− 1.84679i	−0.383849	−	0.923396i	$-0.625402\pi$
$733$	− 50.0000i	− 1.84679i	0.383849	−	0.923396i	$-0.374598\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	− 6.00000i	− 0.220863i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	− 24.0000i	− 0.881068i
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	−4.00000	−0.146450
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	6.00000i	0.218797i
$753$	− 12.0000i	− 0.437304i
$754$	24.0000	0.874028
$755$	0	0
$756$	−4.00000	−0.145479
$757$	− 46.0000i	− 1.67190i	−0.548807	−	0.835949i	$-0.684918\pi$
$757$	− 46.0000i	− 1.67190i	0.548807	−	0.835949i	$-0.315082\pi$
$758$	− 8.00000i	− 0.290573i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	2.00000i	0.0724524i
$763$	− 64.0000i	− 2.31696i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	24.0000	0.867155
$767$	48.0000i	1.73318i
$768$	− 1.00000i	− 0.0360844i
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 10.0000i	− 0.359908i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−10.0000	−0.358979
$777$	16.0000i	0.573997i
$778$	− 24.0000i	− 0.860442i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	− 36.0000i	− 1.28736i
$783$	− 6.00000i	− 0.214423i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	20.0000i	0.712923i	0.934310	+	0.356462i	$0.116017\pi$
$787$	20.0000i	0.712923i	−0.934310	+	0.356462i	$0.883983\pi$
$788$	− 12.0000i	− 0.427482i
$789$	18.0000	0.640817
$790$	0	0
$791$	72.0000	2.56003
$792$	0	0
$793$	56.0000i	1.98862i
$794$	−26.0000	−0.922705
$795$	0	0
$796$	20.0000	0.708881
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	4.00000i	0.141598i
$799$	−36.0000	−1.27359
$800$	0	0
$801$	−6.00000	−0.212000
$802$	30.0000i	1.05934i
$803$	0	0
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−8.00000	−0.281788
$807$	− 6.00000i	− 0.211210i
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	− 24.0000i	− 0.842235i
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	− 4.00000i	− 0.139942i
$818$	− 14.0000i	− 0.489499i
$819$	−16.0000	−0.559085
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	− 18.0000i	− 0.627822i
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	48.0000	1.67013
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	6.00000i	0.208514i
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	− 4.00000i	− 0.138675i
$833$	− 54.0000i	− 1.87099i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	0	0
$837$	2.00000i	0.0691301i
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 28.0000i	− 0.964944i
$843$	18.0000i	0.619953i
$844$	4.00000	0.137686
$845$	0	0
$846$	6.00000	0.206284
$847$	44.0000i	1.51186i
$848$	− 6.00000i	− 0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	− 38.0000i	− 1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$	− 38.0000i	− 1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$	56.0000	1.91628
$855$	0	0
$856$	12.0000	0.410152
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	− 12.0000i	− 0.408722i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−10.0000	−0.339814
$867$	19.0000i	0.645274i
$868$	8.00000i	0.271538i
$869$	0	0
$870$	0	0
$871$	−32.0000	−1.08428
$872$	− 16.0000i	− 0.541828i
$873$	10.0000i	0.338449i
$874$	6.00000	0.202953
$875$	0	0
$876$	14.0000	0.473016
$877$	44.0000i	1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$	44.0000i	1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$	− 26.0000i	− 0.877457i
$879$	6.00000	0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	9.00000i	0.303046i
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	24.0000	0.806296
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	4.00000i	0.134231i
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	− 22.0000i	− 0.736614i
$893$	− 6.00000i	− 0.200782i
$894$	−12.0000	−0.401340
$895$	0	0
$896$	−4.00000	−0.133631
$897$	24.0000i	0.801337i
$898$	30.0000i	1.00111i
$899$	−12.0000	−0.400222
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	18.0000	0.598671
$905$	0	0
$906$	−10.0000	−0.332228
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	− 12.0000i	− 0.398234i
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	1.00000i	0.0331133i
$913$	0	0
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−10.0000	−0.330409
$917$	− 48.0000i	− 1.58510i
$918$	− 6.00000i	− 0.198030i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	12.0000i	0.395199i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	− 10.0000i	− 0.328443i
$928$	− 6.00000i	− 0.196960i
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	− 6.00000i	− 0.196537i
$933$	− 30.0000i	− 0.982156i
$934$	36.0000	1.17796
$935$	0	0
$936$	−4.00000	−0.130744
$937$	− 58.0000i	− 1.89478i	−0.320085	−	0.947389i	$-0.603712\pi$
$937$	− 58.0000i	− 1.89478i	0.320085	−	0.947389i	$-0.396288\pi$
$938$	32.0000i	1.04484i
$939$	10.0000	0.326338
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	2.00000i	0.0651635i
$943$	36.0000i	1.17232i
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	10.0000i	0.324785i
$949$	56.0000	1.81784
$950$	0	0
$951$	18.0000	0.583690
$952$	− 24.0000i	− 0.777844i
$953$	− 54.0000i	− 1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$	− 54.0000i	− 1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−18.0000	−0.582162
$957$	0	0
$958$	6.00000i	0.193851i
$959$	−72.0000	−2.32500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	16.0000i	0.515861i
$963$	− 12.0000i	− 0.386695i
$964$	−14.0000	−0.450910
$965$	0	0
$966$	24.0000	0.772187
$967$	− 28.0000i	− 0.900419i	−0.892923	−	0.450210i	$-0.851349\pi$
$967$	− 28.0000i	− 0.900419i	0.892923	−	0.450210i	$-0.148651\pi$
$968$	11.0000i	0.353553i
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000i	0.0320750i
$973$	80.0000i	2.56468i
$974$	−38.0000	−1.21760
$975$	0	0
$976$	14.0000	0.448129
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	4.00000i	0.127906i
$979$	0	0
$980$	0	0
$981$	−16.0000	−0.510841
$982$	36.0000i	1.14881i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	− 24.0000i	− 0.763928i
$988$	4.00000i	0.127257i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	2.00000i	0.0635001i
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	4.00000i	0.126618i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.p.799.2		2
5.2	odd	4		2850.2.a.g.1.1		1
5.3	odd	4		114.2.a.c.1.1	✓	1
5.4	even	2	inner	2850.2.d.p.799.1		2
15.2	even	4		8550.2.a.bj.1.1		1
15.8	even	4		342.2.a.c.1.1		1
20.3	even	4		912.2.a.c.1.1		1
35.13	even	4		5586.2.a.u.1.1		1
40.3	even	4		3648.2.a.bc.1.1		1
40.13	odd	4		3648.2.a.i.1.1		1
60.23	odd	4		2736.2.a.o.1.1		1
95.18	even	4		2166.2.a.a.1.1		1
285.113	odd	4		6498.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.a.c.1.1	✓	1	5.3	odd	4
342.2.a.c.1.1		1	15.8	even	4
912.2.a.c.1.1		1	20.3	even	4
2166.2.a.a.1.1		1	95.18	even	4
2736.2.a.o.1.1		1	60.23	odd	4
2850.2.a.g.1.1		1	5.2	odd	4
2850.2.d.p.799.1		2	5.4	even	2	inner
2850.2.d.p.799.2		2	1.1	even	1	trivial
3648.2.a.i.1.1		1	40.13	odd	4
3648.2.a.bc.1.1		1	40.3	even	4
5586.2.a.u.1.1		1	35.13	even	4
6498.2.a.t.1.1		1	285.113	odd	4
8550.2.a.bj.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} +6.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -6.00000 q^{29} +2.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -6.00000 q^{46} +6.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} +12.0000 q^{59} +14.0000 q^{61} +2.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} -6.00000i q^{68} +6.00000 q^{69} +1.00000i q^{72} -14.0000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} -4.00000 q^{86} +6.00000i q^{87} +6.00000 q^{89} +16.0000 q^{91} -6.00000i q^{92} -2.00000i q^{93} -6.00000 q^{94} +1.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} - 2 q^{24} - 8 q^{26} - 12 q^{29} + 4 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} - 12 q^{46} - 18 q^{49} + 12 q^{51} - 2 q^{54} - 8 q^{56} + 24 q^{59} + 28 q^{61} - 2 q^{64} + 12 q^{69} + 8 q^{74} + 2 q^{76} + 20 q^{79} + 2 q^{81} + 8 q^{84} - 8 q^{86} + 12 q^{89} + 32 q^{91} - 12 q^{94} + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 - 2 * q^24 - 8 * q^26 - 12 * q^29 + 4 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 + 12 * q^41 - 12 * q^46 - 18 * q^49 + 12 * q^51 - 2 * q^54 - 8 * q^56 + 24 * q^59 + 28 * q^61 - 2 * q^64 + 12 * q^69 + 8 * q^74 + 2 * q^76 + 20 * q^79 + 2 * q^81 + 8 * q^84 - 8 * q^86 + 12 * q^89 + 32 * q^91 - 12 * q^94 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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