LMFDB - Embedded newform 1050.2.g.h.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,2,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1050.g (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:	$8.38429221223$
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 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.2.g.h.799.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	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
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−1.00000	−0.267261
$15$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	− 1.00000i	− 0.192450i
$28$	1.00000i	0.188982i
$29$	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
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 4.00000i	− 0.648886i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 1.00000i	− 0.154303i
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	1.00000i	0.144338i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−6.00000	−0.840168
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	4.00000i	0.529813i
$58$	− 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
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	4.00000i	0.508001i
$63$	1.00000i	0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	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	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	2.00000i	0.226455i
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	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	$0.771155\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	12.0000	1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000i	0.594089i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 1.00000i	− 0.0944911i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−6.00000	−0.557086
$117$	− 2.00000i	− 0.184900i
$118$	− 12.0000i	− 1.10469i
$119$	6.00000	0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 2.00000i	− 0.181071i
$123$	6.00000i	0.541002i
$124$	4.00000	0.359211
$125$	0	0
$126$	1.00000	0.0890871
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	1.00000i	0.0883883i
$129$	−8.00000	−0.704361
$130$	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
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	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
$141$	−12.0000	−1.01058
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	− 1.00000i	− 0.0824786i
$148$	2.00000i	0.164399i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000i	0.324443i
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	− 16.0000i	− 1.27289i
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	−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	$-0.153692\pi$
$168$	1.00000i	0.0771517i
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	− 8.00000i	− 0.609994i
$173$	− 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
$176$	0	0
$177$	12.0000i	0.901975i
$178$	6.00000i	0.449719i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 2.00000i	− 0.148250i
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	0	0
$188$	− 12.0000i	− 0.875190i
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	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
$201$	8.00000	0.564276
$202$	6.00000i	0.422159i
$203$	− 6.00000i	− 0.421117i
$204$	6.00000	0.420084
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	4.00000i	0.271538i
$218$	14.0000i	0.948200i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−12.0000	−0.807207
$222$	− 2.00000i	− 0.134231i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−1.00000	−0.0668153
$225$	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	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	16.0000i	1.03931i
$238$	− 6.00000i	− 0.388922i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	11.0000i	0.707107i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	6.00000	0.382546
$247$	8.00000i	0.509028i
$248$	− 4.00000i	− 0.254000i
$249$	−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	$0.376365\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	8.00000i	0.498058i
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	− 12.0000i	− 0.741362i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	− 6.00000i	− 0.367194i
$268$	8.00000i	0.488678i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	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
$273$	2.00000i	0.121046i
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 4.00000i	− 0.239904i
$279$	4.00000	0.239474
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	12.0000i	0.714590i
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.00000i	− 0.354169i
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	14.0000	0.820695
$292$	− 14.0000i	− 0.819288i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	18.0000i	1.04271i
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	16.0000i	0.920697i
$303$	− 6.00000i	− 0.344691i
$304$	4.00000	0.229416
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	− 2.00000i	− 0.113228i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	24.0000i	1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	− 14.0000i	− 0.774202i
$328$	6.00000i	0.331295i
$329$	12.0000	0.661581
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	− 12.0000i	− 0.658586i
$333$	2.00000i	0.109599i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	1.00000	0.0545545
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	1.00000i	0.0539949i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	6.00000	0.317999
$357$	6.00000i	0.317554i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 2.00000i	− 0.105118i
$363$	− 11.0000i	− 0.577350i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	2.00000	0.104542
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	6.00000	0.311504
$372$	4.00000i	0.207390i
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	12.0000i	0.618031i
$378$	1.00000i	0.0514344i
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	− 8.00000i	− 0.406663i
$388$	14.0000i	0.710742i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	− 1.00000i	− 0.0505076i
$393$	12.0000i	0.605320i
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 4.00000i	− 0.200502i
$399$	4.00000	0.200250
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	− 8.00000i	− 0.399004i
$403$	− 8.00000i	− 0.398508i
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	16.0000i	0.788263i
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	4.00000i	0.195881i
$418$	0	0
$419$	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
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	28.0000i	1.36302i
$423$	− 12.0000i	− 0.583460i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	14.0000i	0.668946i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	12.0000i	0.570782i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−16.0000	−0.757622
$447$	− 18.0000i	− 0.851371i
$448$	1.00000i	0.0472456i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	− 18.0000i	− 0.846649i
$453$	− 16.0000i	− 0.751746i
$454$	12.0000	0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	2.00000i	0.0934539i
$459$	6.00000	0.280056
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	2.00000i	0.0924500i
$469$	−8.00000	−0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	12.0000i	0.552345i
$473$	0	0
$474$	16.0000	0.734904
$475$	0	0
$476$	−6.00000	−0.275010
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	− 26.0000i	− 1.18427i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	2.00000i	0.0905357i
$489$	16.0000	0.723545
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	− 6.00000i	− 0.270501i
$493$	36.0000i	1.62136i
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	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
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	8.00000i	0.354943i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	6.00000	0.264649
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	2.00000i	0.0878750i
$519$	6.00000	0.263371
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	6.00000i	0.262613i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	4.00000i	0.173422i
$533$	12.0000i	0.519778i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	18.0000i	0.776035i
$539$	0	0
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	− 20.0000i	− 0.859074i
$543$	2.00000i	0.0858282i
$544$	6.00000	0.257248
$545$	0	0
$546$	2.00000	0.0855921
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	22.0000	0.934690
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	30.0000i	1.26547i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	20.0000	0.840663
$567$	− 1.00000i	− 0.0419961i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	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
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	19.0000i	0.790296i
$579$	22.0000	0.914289
$580$	0	0
$581$	12.0000	0.497844
$582$	− 14.0000i	− 0.580319i
$583$	0	0
$584$	−14.0000	−0.579324
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	1.00000i	0.0412393i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−18.0000	−0.740421
$592$	− 2.00000i	− 0.0821995i
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	4.00000i	0.163709i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	− 8.00000i	− 0.326056i
$603$	8.00000i	0.325785i
$604$	16.0000	0.651031
$605$	0	0
$606$	−6.00000	−0.243733
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	− 4.00000i	− 0.162221i
$609$	6.00000	0.243132
$610$	0	0
$611$	−24.0000	−0.970936
$612$	6.00000i	0.242536i
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	− 16.0000i	− 0.643614i
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000i	0.240385i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	12.0000	0.478471
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	16.0000i	0.636446i
$633$	− 28.0000i	− 1.11290i
$634$	−30.0000	−1.19145
$635$	0	0
$636$	6.00000	0.237915
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−4.00000	−0.156772
$652$	16.0000i	0.626608i
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	6.00000	0.234261
$657$	− 14.0000i	− 0.546192i
$658$	− 12.0000i	− 0.467809i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	4.00000i	0.155464i
$663$	− 12.0000i	− 0.466041i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	12.0000i	0.464294i
$669$	16.0000	0.618596
$670$	0	0
$671$	0	0
$672$	− 1.00000i	− 0.0385758i
$673$	50.0000i	1.92736i	0.267063	+	0.963679i	$0.413947\pi$
$673$	50.0000i	1.92736i	−0.267063	+	0.963679i	$0.586053\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	18.0000i	0.691286i
$679$	−14.0000	−0.537271
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	1.00000	0.0381802
$687$	− 2.00000i	− 0.0763048i
$688$	8.00000i	0.304997i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	36.0000i	1.36360i
$698$	26.0000i	0.984115i
$699$	6.00000	0.226941
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	6.00000i	0.225653i
$708$	− 12.0000i	− 0.450988i
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	6.00000	0.224544
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	3.00000i	0.111648i
$723$	26.0000i	0.966950i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	2.00000i	0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−48.0000	−1.77534
$732$	− 2.00000i	− 0.0739221i
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	0	0
$738$	6.00000i	0.220863i
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	− 6.00000i	− 0.220267i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	26.0000	0.951928
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	12.0000i	0.437595i
$753$	12.0000i	0.437304i
$754$	12.0000	0.437014
$755$	0	0
$756$	1.00000	0.0363696
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	− 8.00000i	− 0.289809i
$763$	14.0000i	0.506834i
$764$	0	0
$765$	0	0
$766$	−36.0000	−1.30073
$767$	24.0000i	0.866590i
$768$	1.00000i	0.0360844i
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	22.0000i	0.791797i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	14.0000	0.502571
$777$	− 2.00000i	− 0.0717496i
$778$	− 30.0000i	− 1.07555i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	12.0000	0.428026
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	4.00000i	0.142044i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−4.00000	−0.141776
$797$	6.00000i	0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$	6.00000i	0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$	− 4.00000i	− 0.141598i
$799$	−72.0000	−2.54718
$800$	0	0
$801$	6.00000	0.212000
$802$	− 18.0000i	− 0.635602i
$803$	0	0
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−8.00000	−0.281788
$807$	− 18.0000i	− 0.633630i
$808$	− 6.00000i	− 0.211079i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	6.00000i	0.210559i
$813$	20.0000i	0.701431i
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	32.0000i	1.11954i
$818$	− 22.0000i	− 0.769212i
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	6.00000i	0.209274i
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−12.0000	−0.417533
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	− 2.00000i	− 0.0693375i
$833$	− 6.00000i	− 0.207888i
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	4.00000i	0.138260i
$838$	− 12.0000i	− 0.414533i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000i	0.344623i
$843$	− 30.0000i	− 1.03325i
$844$	28.0000	0.963800
$845$	0	0
$846$	−12.0000	−0.412568
$847$	11.0000i	0.377964i
$848$	6.00000i	0.206041i
$849$	−20.0000	−0.686398
$850$	0	0
$851$	0	0
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	12.0000	0.410152
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	− 24.0000i	− 0.817443i
$863$	− 48.0000i	− 1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$	− 48.0000i	− 1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−34.0000	−1.15537
$867$	− 19.0000i	− 0.645274i
$868$	− 4.00000i	− 0.135769i
$869$	0	0
$870$	0	0
$871$	16.0000	0.542139
$872$	− 14.0000i	− 0.474100i
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	14.0000	0.473016
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	− 28.0000i	− 0.944954i
$879$	6.00000	0.202375
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	− 40.0000i	− 1.34611i	−0.739594	−	0.673054i	$-0.764982\pi$
$883$	− 40.0000i	− 1.34611i	0.739594	−	0.673054i	$-0.235018\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	2.00000i	0.0671156i
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	16.0000i	0.535720i
$893$	48.0000i	1.60626i
$894$	−18.0000	−0.602010
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	− 6.00000i	− 0.200223i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	8.00000i	0.266223i
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−16.0000	−0.531564
$907$	16.0000i	0.531271i	0.964073	+	0.265636i	$0.0855818\pi$
$907$	16.0000i	0.531271i	−0.964073	+	0.265636i	$0.914418\pi$
$908$	− 12.0000i	− 0.398234i
$909$	6.00000	0.199007
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	2.00000	0.0660819
$917$	− 12.0000i	− 0.396275i
$918$	− 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
$921$	20.0000	0.659022
$922$	30.0000i	0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	16.0000i	0.525509i
$928$	− 6.00000i	− 0.196960i
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	6.00000i	0.196537i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	34.0000i	1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$	34.0000i	1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$	8.00000i	0.261209i
$939$	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
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	− 16.0000i	− 0.519656i
$949$	−28.0000	−0.908918
$950$	0	0
$951$	30.0000	0.972817
$952$	6.00000i	0.194461i
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	6.00000	0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 4.00000i	− 0.128965i
$963$	12.0000i	0.386695i
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	− 56.0000i	− 1.80084i	−0.435023	−	0.900419i	$-0.643260\pi$
$967$	− 56.0000i	− 1.80084i	0.435023	−	0.900419i	$-0.356740\pi$
$968$	− 11.0000i	− 0.353553i
$969$	−24.0000	−0.770991
$970$	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
$973$	− 4.00000i	− 0.128234i
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	− 16.0000i	− 0.511624i
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	− 24.0000i	− 0.765871i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	12.0000i	0.381964i
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	4.00000i	0.127000i
$993$	− 4.00000i	− 0.126936i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 4.00000i	− 0.126618i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.2.g.h.799.1		2
3.2	odd	2		3150.2.g.o.2899.2		2
5.2	odd	4		210.2.a.d.1.1	✓	1
5.3	odd	4		1050.2.a.a.1.1		1
5.4	even	2	inner	1050.2.g.h.799.2		2
15.2	even	4		630.2.a.f.1.1		1
15.8	even	4		3150.2.a.ba.1.1		1
15.14	odd	2		3150.2.g.o.2899.1		2
20.3	even	4		8400.2.a.cn.1.1		1
20.7	even	4		1680.2.a.b.1.1		1
35.2	odd	12		1470.2.i.d.361.1		2
35.12	even	12		1470.2.i.h.361.1		2
35.13	even	4		7350.2.a.bd.1.1		1
35.17	even	12		1470.2.i.h.961.1		2
35.27	even	4		1470.2.a.m.1.1		1
35.32	odd	12		1470.2.i.d.961.1		2
40.27	even	4		6720.2.a.cc.1.1		1
40.37	odd	4		6720.2.a.bb.1.1		1
60.47	odd	4		5040.2.a.ba.1.1		1
105.62	odd	4		4410.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.a.d.1.1	✓	1	5.2	odd	4
630.2.a.f.1.1		1	15.2	even	4
1050.2.a.a.1.1		1	5.3	odd	4
1050.2.g.h.799.1		2	1.1	even	1	trivial
1050.2.g.h.799.2		2	5.4	even	2	inner
1470.2.a.m.1.1		1	35.27	even	4
1470.2.i.d.361.1		2	35.2	odd	12
1470.2.i.d.961.1		2	35.32	odd	12
1470.2.i.h.361.1		2	35.12	even	12
1470.2.i.h.961.1		2	35.17	even	12
1680.2.a.b.1.1		1	20.7	even	4
3150.2.a.ba.1.1		1	15.8	even	4
3150.2.g.o.2899.1		2	15.14	odd	2
3150.2.g.o.2899.2		2	3.2	odd	2
4410.2.a.f.1.1		1	105.62	odd	4
5040.2.a.ba.1.1		1	60.47	odd	4
6720.2.a.bb.1.1		1	40.37	odd	4
6720.2.a.cc.1.1		1	40.27	even	4
7350.2.a.bd.1.1		1	35.13	even	4
8400.2.a.cn.1.1		1	20.3	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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

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Database

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