LMFDB - Embedded newform 1150.2.b.f.599.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,2,Mod(599,1150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1150.599"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,-6,0,0,-10,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.18279623245$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.2.b.f.599.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} -0.302776i q^{7} +1.00000i q^{8} -7.90833 q^{9} -5.30278 q^{11} +3.30278i q^{12} +0.302776i q^{13} -0.302776 q^{14} +1.00000 q^{16} -3.90833i q^{17} +7.90833i q^{18} +4.90833 q^{19} -1.00000 q^{21} +5.30278i q^{22} +1.00000i q^{23} +3.30278 q^{24} +0.302776 q^{26} +16.2111i q^{27} +0.302776i q^{28} -4.60555 q^{29} +2.90833 q^{31} -1.00000i q^{32} +17.5139i q^{33} -3.90833 q^{34} +7.90833 q^{36} +8.00000i q^{37} -4.90833i q^{38} +1.00000 q^{39} -9.90833 q^{41} +1.00000i q^{42} -5.21110i q^{43} +5.30278 q^{44} +1.00000 q^{46} +4.60555i q^{47} -3.30278i q^{48} +6.90833 q^{49} -12.9083 q^{51} -0.302776i q^{52} -3.21110i q^{53} +16.2111 q^{54} +0.302776 q^{56} -16.2111i q^{57} +4.60555i q^{58} +10.6056 q^{59} -6.51388 q^{61} -2.90833i q^{62} +2.39445i q^{63} -1.00000 q^{64} +17.5139 q^{66} -4.00000i q^{67} +3.90833i q^{68} +3.30278 q^{69} -12.6972 q^{71} -7.90833i q^{72} -15.8167i q^{73} +8.00000 q^{74} -4.90833 q^{76} +1.60555i q^{77} -1.00000i q^{78} -14.4222 q^{79} +29.8167 q^{81} +9.90833i q^{82} +3.21110i q^{83} +1.00000 q^{84} -5.21110 q^{86} +15.2111i q^{87} -5.30278i q^{88} +0.0916731 q^{91} -1.00000i q^{92} -9.60555i q^{93} +4.60555 q^{94} -3.30278 q^{96} +2.69722i q^{97} -6.90833i q^{98} +41.9361 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 14 q^{11} + 6 q^{14} + 4 q^{16} - 2 q^{19} - 4 q^{21} + 6 q^{24} - 6 q^{26} - 4 q^{29} - 10 q^{31} + 6 q^{34} + 10 q^{36} + 4 q^{39} - 18 q^{41} + 14 q^{44} + 4 q^{46}+ \cdots + 74 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 - 14 * q^11 + 6 * q^14 + 4 * q^16 - 2 * q^19 - 4 * q^21 + 6 * q^24 - 6 * q^26 - 4 * q^29 - 10 * q^31 + 6 * q^34 + 10 * q^36 + 4 * q^39 - 18 * q^41 + 14 * q^44 + 4 * q^46 + 6 * q^49 - 30 * q^51 + 36 * q^54 - 6 * q^56 + 28 * q^59 + 10 * q^61 - 4 * q^64 + 34 * q^66 + 6 * q^69 - 58 * q^71 + 32 * q^74 + 2 * q^76 + 76 * q^81 + 4 * q^84 + 8 * q^86 + 22 * q^91 + 4 * q^94 - 6 * q^96 + 74 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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
$3$	− 3.30278i	− 1.90686i	−0.301617	−	0.953429i	$-0.597526\pi$
$3$	− 3.30278i	− 1.90686i	0.301617	−	0.953429i	$-0.402474\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.30278	−1.34835
$7$	− 0.302776i	− 0.114438i	−0.998362	−	0.0572192i	$-0.981777\pi$
$7$	− 0.302776i	− 0.114438i	0.998362	−	0.0572192i	$-0.0182234\pi$
$8$	1.00000i	0.353553i
$9$	−7.90833	−2.63611
$10$	0	0
$11$	−5.30278	−1.59885	−0.799424	−	0.600768i	$-0.794862\pi$
$11$	−5.30278	−1.59885	−0.799424	+	0.600768i	$0.794862\pi$
$12$	3.30278i	0.953429i
$13$	0.302776i	0.0839749i	0.999118	+	0.0419874i	$0.0133689\pi$
$13$	0.302776i	0.0839749i	−0.999118	+	0.0419874i	$0.986631\pi$
$14$	−0.302776	−0.0809202
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.90833i	− 0.947909i	−0.880549	−	0.473954i	$-0.842826\pi$
$17$	− 3.90833i	− 0.947909i	0.880549	−	0.473954i	$-0.157174\pi$
$18$	7.90833i	1.86401i
$19$	4.90833	1.12605	0.563024	−	0.826441i	$-0.309638\pi$
$19$	4.90833	1.12605	0.563024	+	0.826441i	$0.309638\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	5.30278i	1.13056i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	3.30278	0.674176
$25$	0	0
$26$	0.302776	0.0593792
$27$	16.2111i	3.11983i
$28$	0.302776i	0.0572192i
$29$	−4.60555	−0.855229	−0.427615	−	0.903961i	$-0.640646\pi$
$29$	−4.60555	−0.855229	−0.427615	+	0.903961i	$0.640646\pi$
$30$	0	0
$31$	2.90833	0.522351	0.261175	−	0.965291i	$-0.415890\pi$
$31$	2.90833	0.522351	0.261175	+	0.965291i	$0.415890\pi$
$32$	− 1.00000i	− 0.176777i
$33$	17.5139i	3.04877i
$34$	−3.90833	−0.670273
$35$	0	0
$36$	7.90833	1.31805
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	− 4.90833i	− 0.796236i
$39$	1.00000	0.160128
$40$	0	0
$41$	−9.90833	−1.54742	−0.773710	−	0.633540i	$-0.781601\pi$
$41$	−9.90833	−1.54742	−0.773710	+	0.633540i	$0.781601\pi$
$42$	1.00000i	0.154303i
$43$	− 5.21110i	− 0.794686i	−0.917670	−	0.397343i	$-0.869932\pi$
$43$	− 5.21110i	− 0.794686i	0.917670	−	0.397343i	$-0.130068\pi$
$44$	5.30278	0.799424
$45$	0	0
$46$	1.00000	0.147442
$47$	4.60555i	0.671789i	0.941900	+	0.335894i	$0.109039\pi$
$47$	4.60555i	0.671789i	−0.941900	+	0.335894i	$0.890961\pi$
$48$	− 3.30278i	− 0.476715i
$49$	6.90833	0.986904
$50$	0	0
$51$	−12.9083	−1.80753
$52$	− 0.302776i	− 0.0419874i
$53$	− 3.21110i	− 0.441079i	−0.975378	−	0.220539i	$-0.929218\pi$
$53$	− 3.21110i	− 0.441079i	0.975378	−	0.220539i	$-0.0707818\pi$
$54$	16.2111	2.20605
$55$	0	0
$56$	0.302776	0.0404601
$57$	− 16.2111i	− 2.14721i
$58$	4.60555i	0.604739i
$59$	10.6056	1.38073	0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056	1.38073	0.690363	+	0.723464i	$0.257451\pi$
$60$	0	0
$61$	−6.51388	−0.834017	−0.417008	−	0.908903i	$-0.636921\pi$
$61$	−6.51388	−0.834017	−0.417008	+	0.908903i	$0.636921\pi$
$62$	− 2.90833i	− 0.369358i
$63$	2.39445i	0.301672i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	17.5139	2.15581
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	3.90833i	0.473954i
$69$	3.30278	0.397607
$70$	0	0
$71$	−12.6972	−1.50688	−0.753442	−	0.657515i	$-0.771608\pi$
$71$	−12.6972	−1.50688	−0.753442	+	0.657515i	$0.771608\pi$
$72$	− 7.90833i	− 0.932005i
$73$	− 15.8167i	− 1.85120i	−0.378504	−	0.925600i	$-0.623561\pi$
$73$	− 15.8167i	− 1.85120i	0.378504	−	0.925600i	$-0.376439\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−4.90833	−0.563024
$77$	1.60555i	0.182970i
$78$	− 1.00000i	− 0.113228i
$79$	−14.4222	−1.62262	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	9.90833i	1.09419i
$83$	3.21110i	0.352464i	0.984349	+	0.176232i	$0.0563909\pi$
$83$	3.21110i	0.352464i	−0.984349	+	0.176232i	$0.943609\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−5.21110	−0.561928
$87$	15.2111i	1.63080i
$88$	− 5.30278i	− 0.565278i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0.0916731	0.00960995
$92$	− 1.00000i	− 0.104257i
$93$	− 9.60555i	− 0.996049i
$94$	4.60555	0.475026
$95$	0	0
$96$	−3.30278	−0.337088
$97$	2.69722i	0.273862i	0.990581	+	0.136931i	$0.0437238\pi$
$97$	2.69722i	0.273862i	−0.990581	+	0.136931i	$0.956276\pi$
$98$	− 6.90833i	− 0.697846i
$99$	41.9361	4.21473
$100$	0	0
$101$	−4.60555	−0.458269	−0.229135	−	0.973395i	$-0.573590\pi$
$101$	−4.60555	−0.458269	−0.229135	+	0.973395i	$0.573590\pi$
$102$	12.9083i	1.27811i
$103$	17.1194i	1.68683i	0.537265	+	0.843414i	$0.319458\pi$
$103$	17.1194i	1.68683i	−0.537265	+	0.843414i	$0.680542\pi$
$104$	−0.302776	−0.0296896
$105$	0	0
$106$	−3.21110	−0.311890
$107$	4.60555i	0.445235i	0.974906	+	0.222618i	$0.0714602\pi$
$107$	4.60555i	0.445235i	−0.974906	+	0.222618i	$0.928540\pi$
$108$	− 16.2111i	− 1.55991i
$109$	−19.5139	−1.86909	−0.934545	−	0.355844i	$-0.884193\pi$
$109$	−19.5139	−1.86909	−0.934545	+	0.355844i	$0.884193\pi$
$110$	0	0
$111$	26.4222	2.50788
$112$	− 0.302776i	− 0.0286096i
$113$	− 12.4222i	− 1.16858i	−0.811544	−	0.584291i	$-0.801373\pi$
$113$	− 12.4222i	− 1.16858i	0.811544	−	0.584291i	$-0.198627\pi$
$114$	−16.2111	−1.51831
$115$	0	0
$116$	4.60555	0.427615
$117$	− 2.39445i	− 0.221367i
$118$	− 10.6056i	− 0.976320i
$119$	−1.18335	−0.108477
$120$	0	0
$121$	17.1194	1.55631
$122$	6.51388i	0.589739i
$123$	32.7250i	2.95071i
$124$	−2.90833	−0.261175
$125$	0	0
$126$	2.39445	0.213314
$127$	− 11.8167i	− 1.04856i	−0.851546	−	0.524279i	$-0.824335\pi$
$127$	− 11.8167i	− 1.04856i	0.851546	−	0.524279i	$-0.175665\pi$
$128$	1.00000i	0.0883883i
$129$	−17.2111	−1.51535
$130$	0	0
$131$	3.21110	0.280555	0.140278	−	0.990112i	$-0.455200\pi$
$131$	3.21110	0.280555	0.140278	+	0.990112i	$0.455200\pi$
$132$	− 17.5139i	− 1.52439i
$133$	− 1.48612i	− 0.128863i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	3.90833	0.335136
$137$	− 6.90833i	− 0.590218i	−0.955464	−	0.295109i	$-0.904644\pi$
$137$	− 6.90833i	− 0.590218i	0.955464	−	0.295109i	$-0.0953560\pi$
$138$	− 3.30278i	− 0.281151i
$139$	5.39445	0.457551	0.228776	−	0.973479i	$-0.426528\pi$
$139$	5.39445	0.457551	0.228776	+	0.973479i	$0.426528\pi$
$140$	0	0
$141$	15.2111	1.28101
$142$	12.6972i	1.06553i
$143$	− 1.60555i	− 0.134263i
$144$	−7.90833	−0.659027
$145$	0	0
$146$	−15.8167	−1.30900
$147$	− 22.8167i	− 1.88189i
$148$	− 8.00000i	− 0.657596i
$149$	−9.69722	−0.794428	−0.397214	−	0.917726i	$-0.630023\pi$
$149$	−9.69722	−0.794428	−0.397214	+	0.917726i	$0.630023\pi$
$150$	0	0
$151$	−1.90833	−0.155297	−0.0776487	−	0.996981i	$-0.524741\pi$
$151$	−1.90833	−0.155297	−0.0776487	+	0.996981i	$0.524741\pi$
$152$	4.90833i	0.398118i
$153$	30.9083i	2.49879i
$154$	1.60555	0.129379
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	− 11.3944i	− 0.909376i	−0.890651	−	0.454688i	$-0.849751\pi$
$157$	− 11.3944i	− 0.909376i	0.890651	−	0.454688i	$-0.150249\pi$
$158$	14.4222i	1.14737i
$159$	−10.6056	−0.841075
$160$	0	0
$161$	0.302776	0.0238621
$162$	− 29.8167i	− 2.34262i
$163$	− 5.69722i	− 0.446241i	−0.974791	−	0.223121i	$-0.928376\pi$
$163$	− 5.69722i	− 0.446241i	0.974791	−	0.223121i	$-0.0716243\pi$
$164$	9.90833	0.773710
$165$	0	0
$166$	3.21110	0.249230
$167$	21.2111i	1.64136i	0.571385	+	0.820682i	$0.306406\pi$
$167$	21.2111i	1.64136i	−0.571385	+	0.820682i	$0.693594\pi$
$168$	− 1.00000i	− 0.0771517i
$169$	12.9083	0.992948
$170$	0	0
$171$	−38.8167	−2.96838
$172$	5.21110i	0.397343i
$173$	− 23.3028i	− 1.77168i	−0.463993	−	0.885839i	$-0.653584\pi$
$173$	− 23.3028i	− 1.77168i	0.463993	−	0.885839i	$-0.346416\pi$
$174$	15.2111	1.15315
$175$	0	0
$176$	−5.30278	−0.399712
$177$	− 35.0278i	− 2.63285i
$178$	0	0
$179$	−16.6056	−1.24116	−0.620579	−	0.784144i	$-0.713102\pi$
$179$	−16.6056	−1.24116	−0.620579	+	0.784144i	$0.713102\pi$
$180$	0	0
$181$	−8.11943	−0.603512	−0.301756	−	0.953385i	$-0.597573\pi$
$181$	−8.11943	−0.603512	−0.301756	+	0.953385i	$0.597573\pi$
$182$	− 0.0916731i	− 0.00679526i
$183$	21.5139i	1.59035i
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	−9.60555	−0.704313
$187$	20.7250i	1.51556i
$188$	− 4.60555i	− 0.335894i
$189$	4.90833	0.357028
$190$	0	0
$191$	−1.39445	−0.100899	−0.0504494	−	0.998727i	$-0.516065\pi$
$191$	−1.39445	−0.100899	−0.0504494	+	0.998727i	$0.516065\pi$
$192$	3.30278i	0.238357i
$193$	− 3.81665i	− 0.274729i	−0.990521	−	0.137364i	$-0.956137\pi$
$193$	− 3.81665i	− 0.274729i	0.990521	−	0.137364i	$-0.0438631\pi$
$194$	2.69722	0.193649
$195$	0	0
$196$	−6.90833	−0.493452
$197$	0.697224i	0.0496752i	0.999691	+	0.0248376i	$0.00790686\pi$
$197$	0.697224i	0.0496752i	−0.999691	+	0.0248376i	$0.992093\pi$
$198$	− 41.9361i	− 2.98027i
$199$	−8.42221	−0.597034	−0.298517	−	0.954404i	$-0.596492\pi$
$199$	−8.42221	−0.597034	−0.298517	+	0.954404i	$0.596492\pi$
$200$	0	0
$201$	−13.2111	−0.931839
$202$	4.60555i	0.324045i
$203$	1.39445i	0.0978711i
$204$	12.9083	0.903764
$205$	0	0
$206$	17.1194	1.19277
$207$	− 7.90833i	− 0.549667i
$208$	0.302776i	0.0209937i
$209$	−26.0278	−1.80038
$210$	0	0
$211$	−7.21110	−0.496433	−0.248216	−	0.968705i	$-0.579844\pi$
$211$	−7.21110	−0.496433	−0.248216	+	0.968705i	$0.579844\pi$
$212$	3.21110i	0.220539i
$213$	41.9361i	2.87341i
$214$	4.60555	0.314829
$215$	0	0
$216$	−16.2111	−1.10303
$217$	− 0.880571i	− 0.0597770i
$218$	19.5139i	1.32165i
$219$	−52.2389	−3.52997
$220$	0	0
$221$	1.18335	0.0796005
$222$	− 26.4222i	− 1.77334i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−0.302776	−0.0202300
$225$	0	0
$226$	−12.4222	−0.826313
$227$	− 7.39445i	− 0.490787i	−0.969424	−	0.245393i	$-0.921083\pi$
$227$	− 7.39445i	− 0.490787i	0.969424	−	0.245393i	$-0.0789171\pi$
$228$	16.2111i	1.07361i
$229$	−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
$231$	5.30278	0.348897
$232$	− 4.60555i	− 0.302369i
$233$	− 4.18335i	− 0.274060i	−0.990567	−	0.137030i	$-0.956244\pi$
$233$	− 4.18335i	− 0.274060i	0.990567	−	0.137030i	$-0.0437557\pi$
$234$	−2.39445	−0.156530
$235$	0	0
$236$	−10.6056	−0.690363
$237$	47.6333i	3.09412i
$238$	1.18335i	0.0767049i
$239$	9.21110	0.595817	0.297908	−	0.954594i	$-0.403711\pi$
$239$	9.21110	0.595817	0.297908	+	0.954594i	$0.403711\pi$
$240$	0	0
$241$	14.4222	0.929016	0.464508	−	0.885569i	$-0.346231\pi$
$241$	14.4222	0.929016	0.464508	+	0.885569i	$0.346231\pi$
$242$	− 17.1194i	− 1.10048i
$243$	− 49.8444i	− 3.19752i
$244$	6.51388	0.417008
$245$	0	0
$246$	32.7250	2.08647
$247$	1.48612i	0.0945597i
$248$	2.90833i	0.184679i
$249$	10.6056	0.672100
$250$	0	0
$251$	−5.51388	−0.348033	−0.174016	−	0.984743i	$-0.555675\pi$
$251$	−5.51388	−0.348033	−0.174016	+	0.984743i	$0.555675\pi$
$252$	− 2.39445i	− 0.150836i
$253$	− 5.30278i	− 0.333383i
$254$	−11.8167	−0.741443
$255$	0	0
$256$	1.00000	0.0625000
$257$	19.8167i	1.23613i	0.786127	+	0.618064i	$0.212083\pi$
$257$	19.8167i	1.23613i	−0.786127	+	0.618064i	$0.787917\pi$
$258$	17.2111i	1.07152i
$259$	2.42221	0.150509
$260$	0	0
$261$	36.4222	2.25448
$262$	− 3.21110i	− 0.198383i
$263$	14.5139i	0.894964i	0.894293	+	0.447482i	$0.147679\pi$
$263$	14.5139i	0.894964i	−0.894293	+	0.447482i	$0.852321\pi$
$264$	−17.5139	−1.07790
$265$	0	0
$266$	−1.48612	−0.0911200
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−25.8167	−1.57407	−0.787035	−	0.616909i	$-0.788385\pi$
$269$	−25.8167	−1.57407	−0.787035	+	0.616909i	$0.788385\pi$
$270$	0	0
$271$	−6.30278	−0.382866	−0.191433	−	0.981506i	$-0.561314\pi$
$271$	−6.30278	−0.382866	−0.191433	+	0.981506i	$0.561314\pi$
$272$	− 3.90833i	− 0.236977i
$273$	− 0.302776i	− 0.0183248i
$274$	−6.90833	−0.417347
$275$	0	0
$276$	−3.30278	−0.198804
$277$	− 12.7889i	− 0.768410i	−0.923248	−	0.384205i	$-0.874476\pi$
$277$	− 12.7889i	− 0.768410i	0.923248	−	0.384205i	$-0.125524\pi$
$278$	− 5.39445i	− 0.323538i
$279$	−23.0000	−1.37697
$280$	0	0
$281$	−19.3944	−1.15698	−0.578488	−	0.815691i	$-0.696357\pi$
$281$	−19.3944	−1.15698	−0.578488	+	0.815691i	$0.696357\pi$
$282$	− 15.2111i	− 0.905808i
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	12.6972	0.753442
$285$	0	0
$286$	−1.60555	−0.0949382
$287$	3.00000i	0.177084i
$288$	7.90833i	0.466003i
$289$	1.72498	0.101469
$290$	0	0
$291$	8.90833	0.522215
$292$	15.8167i	0.925600i
$293$	− 8.78890i	− 0.513453i	−0.966484	−	0.256726i	$-0.917356\pi$
$293$	− 8.78890i	− 0.513453i	0.966484	−	0.256726i	$-0.0826439\pi$
$294$	−22.8167	−1.33069
$295$	0	0
$296$	−8.00000	−0.464991
$297$	− 85.9638i	− 4.98813i
$298$	9.69722i	0.561745i
$299$	−0.302776	−0.0175100
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	1.90833i	0.109812i
$303$	15.2111i	0.873855i
$304$	4.90833	0.281512
$305$	0	0
$306$	30.9083	1.76691
$307$	− 15.3028i	− 0.873376i	−0.899613	−	0.436688i	$-0.856151\pi$
$307$	− 15.3028i	− 0.873376i	0.899613	−	0.436688i	$-0.143849\pi$
$308$	− 1.60555i	− 0.0914848i
$309$	56.5416	3.21654
$310$	0	0
$311$	−6.42221	−0.364170	−0.182085	−	0.983283i	$-0.558285\pi$
$311$	−6.42221	−0.364170	−0.182085	+	0.983283i	$0.558285\pi$
$312$	1.00000i	0.0566139i
$313$	12.7250i	0.719258i	0.933095	+	0.359629i	$0.117097\pi$
$313$	12.7250i	0.719258i	−0.933095	+	0.359629i	$0.882903\pi$
$314$	−11.3944	−0.643026
$315$	0	0
$316$	14.4222	0.811312
$317$	− 14.7250i	− 0.827037i	−0.910496	−	0.413519i	$-0.864300\pi$
$317$	− 14.7250i	− 0.827037i	0.910496	−	0.413519i	$-0.135700\pi$
$318$	10.6056i	0.594730i
$319$	24.4222	1.36738
$320$	0	0
$321$	15.2111	0.849001
$322$	− 0.302776i	− 0.0168730i
$323$	− 19.1833i	− 1.06739i
$324$	−29.8167	−1.65648
$325$	0	0
$326$	−5.69722	−0.315540
$327$	64.4500i	3.56409i
$328$	− 9.90833i	− 0.547096i
$329$	1.39445	0.0768784
$330$	0	0
$331$	9.39445	0.516366	0.258183	−	0.966096i	$-0.416876\pi$
$331$	9.39445	0.516366	0.258183	+	0.966096i	$0.416876\pi$
$332$	− 3.21110i	− 0.176232i
$333$	− 63.2666i	− 3.46699i
$334$	21.2111	1.16062
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	− 4.48612i	− 0.244375i	−0.992507	−	0.122187i	$-0.961009\pi$
$337$	− 4.48612i	− 0.244375i	0.992507	−	0.122187i	$-0.0389909\pi$
$338$	− 12.9083i	− 0.702120i
$339$	−41.0278	−2.22832
$340$	0	0
$341$	−15.4222	−0.835159
$342$	38.8167i	2.09896i
$343$	− 4.21110i	− 0.227378i
$344$	5.21110	0.280964
$345$	0	0
$346$	−23.3028	−1.25276
$347$	− 25.5416i	− 1.37115i	−0.728004	−	0.685573i	$-0.759552\pi$
$347$	− 25.5416i	− 1.37115i	0.728004	−	0.685573i	$-0.240448\pi$
$348$	− 15.2111i	− 0.815401i
$349$	12.7889	0.684574	0.342287	−	0.939595i	$-0.388798\pi$
$349$	12.7889	0.684574	0.342287	+	0.939595i	$0.388798\pi$
$350$	0	0
$351$	−4.90833	−0.261987
$352$	5.30278i	0.282639i
$353$	− 18.4222i	− 0.980515i	−0.871578	−	0.490258i	$-0.836903\pi$
$353$	− 18.4222i	− 0.980515i	0.871578	−	0.490258i	$-0.163097\pi$
$354$	−35.0278	−1.86170
$355$	0	0
$356$	0	0
$357$	3.90833i	0.206851i
$358$	16.6056i	0.877631i
$359$	3.21110	0.169476	0.0847378	−	0.996403i	$-0.472995\pi$
$359$	3.21110	0.169476	0.0847378	+	0.996403i	$0.472995\pi$
$360$	0	0
$361$	5.09167	0.267983
$362$	8.11943i	0.426748i
$363$	− 56.5416i	− 2.96767i
$364$	−0.0916731	−0.00480498
$365$	0	0
$366$	21.5139	1.12455
$367$	29.2111i	1.52481i	0.647102	+	0.762404i	$0.275981\pi$
$367$	29.2111i	1.52481i	−0.647102	+	0.762404i	$0.724019\pi$
$368$	1.00000i	0.0521286i
$369$	78.3583	4.07917
$370$	0	0
$371$	−0.972244	−0.0504764
$372$	9.60555i	0.498025i
$373$	2.60555i	0.134910i	0.997722	+	0.0674552i	$0.0214880\pi$
$373$	2.60555i	0.134910i	−0.997722	+	0.0674552i	$0.978512\pi$
$374$	20.7250	1.07166
$375$	0	0
$376$	−4.60555	−0.237513
$377$	− 1.39445i	− 0.0718178i
$378$	− 4.90833i	− 0.252457i
$379$	−4.09167	−0.210175	−0.105088	−	0.994463i	$-0.533512\pi$
$379$	−4.09167	−0.210175	−0.105088	+	0.994463i	$0.533512\pi$
$380$	0	0
$381$	−39.0278	−1.99945
$382$	1.39445i	0.0713462i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	3.30278	0.168544
$385$	0	0
$386$	−3.81665	−0.194263
$387$	41.2111i	2.09488i
$388$	− 2.69722i	− 0.136931i
$389$	−20.9361	−1.06150	−0.530751	−	0.847528i	$-0.678090\pi$
$389$	−20.9361	−1.06150	−0.530751	+	0.847528i	$0.678090\pi$
$390$	0	0
$391$	3.90833	0.197653
$392$	6.90833i	0.348923i
$393$	− 10.6056i	− 0.534979i
$394$	0.697224	0.0351257
$395$	0	0
$396$	−41.9361	−2.10737
$397$	− 21.7250i	− 1.09035i	−0.838324	−	0.545173i	$-0.816464\pi$
$397$	− 21.7250i	− 1.09035i	0.838324	−	0.545173i	$-0.183536\pi$
$398$	8.42221i	0.422167i
$399$	−4.90833	−0.245724
$400$	0	0
$401$	−1.39445	−0.0696354	−0.0348177	−	0.999394i	$-0.511085\pi$
$401$	−1.39445	−0.0696354	−0.0348177	+	0.999394i	$0.511085\pi$
$402$	13.2111i	0.658910i
$403$	0.880571i	0.0438643i
$404$	4.60555	0.229135
$405$	0	0
$406$	1.39445	0.0692053
$407$	− 42.4222i	− 2.10279i
$408$	− 12.9083i	− 0.639057i
$409$	15.0917	0.746235	0.373118	−	0.927784i	$-0.378289\pi$
$409$	15.0917	0.746235	0.373118	+	0.927784i	$0.378289\pi$
$410$	0	0
$411$	−22.8167	−1.12546
$412$	− 17.1194i	− 0.843414i
$413$	− 3.21110i	− 0.158008i
$414$	−7.90833	−0.388673
$415$	0	0
$416$	0.302776	0.0148448
$417$	− 17.8167i	− 0.872485i
$418$	26.0278i	1.27306i
$419$	39.6333	1.93621	0.968107	−	0.250538i	$-0.0806073\pi$
$419$	39.6333	1.93621	0.968107	+	0.250538i	$0.0806073\pi$
$420$	0	0
$421$	34.3028	1.67181	0.835907	−	0.548870i	$-0.184942\pi$
$421$	34.3028	1.67181	0.835907	+	0.548870i	$0.184942\pi$
$422$	7.21110i	0.351031i
$423$	− 36.4222i	− 1.77091i
$424$	3.21110	0.155945
$425$	0	0
$426$	41.9361	2.03181
$427$	1.97224i	0.0954436i
$428$	− 4.60555i	− 0.222618i
$429$	−5.30278	−0.256020
$430$	0	0
$431$	20.2389	0.974872	0.487436	−	0.873159i	$-0.337932\pi$
$431$	20.2389	0.974872	0.487436	+	0.873159i	$0.337932\pi$
$432$	16.2111i	0.779957i
$433$	34.9083i	1.67759i	0.544450	+	0.838794i	$0.316739\pi$
$433$	34.9083i	1.67759i	−0.544450	+	0.838794i	$0.683261\pi$
$434$	−0.880571	−0.0422687
$435$	0	0
$436$	19.5139	0.934545
$437$	4.90833i	0.234797i
$438$	52.2389i	2.49607i
$439$	18.3028	0.873544	0.436772	−	0.899572i	$-0.356122\pi$
$439$	18.3028	0.873544	0.436772	+	0.899572i	$0.356122\pi$
$440$	0	0
$441$	−54.6333	−2.60159
$442$	− 1.18335i	− 0.0562860i
$443$	− 35.5139i	− 1.68732i	−0.536882	−	0.843658i	$-0.680398\pi$
$443$	− 35.5139i	− 1.68732i	0.536882	−	0.843658i	$-0.319602\pi$
$444$	−26.4222	−1.25394
$445$	0	0
$446$	4.00000	0.189405
$447$	32.0278i	1.51486i
$448$	0.302776i	0.0143048i
$449$	12.9083	0.609182	0.304591	−	0.952483i	$-0.401480\pi$
$449$	12.9083	0.609182	0.304591	+	0.952483i	$0.401480\pi$
$450$	0	0
$451$	52.5416	2.47409
$452$	12.4222i	0.584291i
$453$	6.30278i	0.296130i
$454$	−7.39445	−0.347039
$455$	0	0
$456$	16.2111	0.759154
$457$	− 3.57779i	− 0.167362i	−0.996493	−	0.0836811i	$-0.973332\pi$
$457$	− 3.57779i	− 0.167362i	0.996493	−	0.0836811i	$-0.0266677\pi$
$458$	2.00000i	0.0934539i
$459$	63.3583	2.95731
$460$	0	0
$461$	31.8167	1.48185	0.740925	−	0.671588i	$-0.234388\pi$
$461$	31.8167	1.48185	0.740925	+	0.671588i	$0.234388\pi$
$462$	− 5.30278i	− 0.246707i
$463$	25.6333i	1.19128i	0.803251	+	0.595640i	$0.203102\pi$
$463$	25.6333i	1.19128i	−0.803251	+	0.595640i	$0.796898\pi$
$464$	−4.60555	−0.213807
$465$	0	0
$466$	−4.18335	−0.193790
$467$	19.8167i	0.917005i	0.888693	+	0.458503i	$0.151614\pi$
$467$	19.8167i	0.917005i	−0.888693	+	0.458503i	$0.848386\pi$
$468$	2.39445i	0.110683i
$469$	−1.21110	−0.0559235
$470$	0	0
$471$	−37.6333	−1.73405
$472$	10.6056i	0.488160i
$473$	27.6333i	1.27058i
$474$	47.6333	2.18787
$475$	0	0
$476$	1.18335	0.0542386
$477$	25.3944i	1.16273i
$478$	− 9.21110i	− 0.421306i
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	−2.42221	−0.110443
$482$	− 14.4222i	− 0.656913i
$483$	− 1.00000i	− 0.0455016i
$484$	−17.1194	−0.778156
$485$	0	0
$486$	−49.8444	−2.26099
$487$	− 11.8167i	− 0.535464i	−0.963493	−	0.267732i	$-0.913726\pi$
$487$	− 11.8167i	− 0.535464i	0.963493	−	0.267732i	$-0.0862741\pi$
$488$	− 6.51388i	− 0.294869i
$489$	−18.8167	−0.850918
$490$	0	0
$491$	−25.8167	−1.16509	−0.582545	−	0.812799i	$-0.697943\pi$
$491$	−25.8167	−1.16509	−0.582545	+	0.812799i	$0.697943\pi$
$492$	− 32.7250i	− 1.47536i
$493$	18.0000i	0.810679i
$494$	1.48612	0.0668638
$495$	0	0
$496$	2.90833	0.130588
$497$	3.84441i	0.172445i
$498$	− 10.6056i	− 0.475246i
$499$	−11.6333	−0.520778	−0.260389	−	0.965504i	$-0.583851\pi$
$499$	−11.6333	−0.520778	−0.260389	+	0.965504i	$0.583851\pi$
$500$	0	0
$501$	70.0555	3.12985
$502$	5.51388i	0.246096i
$503$	2.72498i	0.121501i	0.998153	+	0.0607504i	$0.0193494\pi$
$503$	2.72498i	0.121501i	−0.998153	+	0.0607504i	$0.980651\pi$
$504$	−2.39445	−0.106657
$505$	0	0
$506$	−5.30278	−0.235737
$507$	− 42.6333i	− 1.89341i
$508$	11.8167i	0.524279i
$509$	−29.4500	−1.30535	−0.652673	−	0.757639i	$-0.726353\pi$
$509$	−29.4500	−1.30535	−0.652673	+	0.757639i	$0.726353\pi$
$510$	0	0
$511$	−4.78890	−0.211848
$512$	− 1.00000i	− 0.0441942i
$513$	79.5694i	3.51307i
$514$	19.8167	0.874075
$515$	0	0
$516$	17.2111	0.757677
$517$	− 24.4222i	− 1.07409i
$518$	− 2.42221i	− 0.106426i
$519$	−76.9638	−3.37834
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	− 36.4222i	− 1.59416i
$523$	− 8.42221i	− 0.368277i	−0.982900	−	0.184139i	$-0.941050\pi$
$523$	− 8.42221i	− 0.368277i	0.982900	−	0.184139i	$-0.0589495\pi$
$524$	−3.21110	−0.140278
$525$	0	0
$526$	14.5139	0.632835
$527$	− 11.3667i	− 0.495141i
$528$	17.5139i	0.762194i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−83.8722	−3.63974
$532$	1.48612i	0.0644316i
$533$	− 3.00000i	− 0.129944i
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	54.8444i	2.36671i
$538$	25.8167i	1.11303i
$539$	−36.6333	−1.57791
$540$	0	0
$541$	−28.8444	−1.24012	−0.620059	−	0.784555i	$-0.712891\pi$
$541$	−28.8444	−1.24012	−0.620059	+	0.784555i	$0.712891\pi$
$542$	6.30278i	0.270727i
$543$	26.8167i	1.15081i
$544$	−3.90833	−0.167568
$545$	0	0
$546$	−0.302776	−0.0129576
$547$	7.51388i	0.321270i	0.987014	+	0.160635i	$0.0513542\pi$
$547$	7.51388i	0.321270i	−0.987014	+	0.160635i	$0.948646\pi$
$548$	6.90833i	0.295109i
$549$	51.5139	2.19856
$550$	0	0
$551$	−22.6056	−0.963029
$552$	3.30278i	0.140575i
$553$	4.36669i	0.185691i
$554$	−12.7889	−0.543348
$555$	0	0
$556$	−5.39445	−0.228776
$557$	− 6.42221i	− 0.272118i	−0.990701	−	0.136059i	$-0.956556\pi$
$557$	− 6.42221i	− 0.272118i	0.990701	−	0.136059i	$-0.0434436\pi$
$558$	23.0000i	0.973668i
$559$	1.57779	0.0667336
$560$	0	0
$561$	68.4500	2.88996
$562$	19.3944i	0.818105i
$563$	− 39.6333i	− 1.67034i	−0.549988	−	0.835172i	$-0.685368\pi$
$563$	− 39.6333i	− 1.67034i	0.549988	−	0.835172i	$-0.314632\pi$
$564$	−15.2111	−0.640503
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	− 9.02776i	− 0.379130i
$568$	− 12.6972i	− 0.532764i
$569$	0.422205	0.0176998	0.00884988	−	0.999961i	$-0.497183\pi$
$569$	0.422205	0.0176998	0.00884988	+	0.999961i	$0.497183\pi$
$570$	0	0
$571$	9.11943	0.381636	0.190818	−	0.981625i	$-0.438886\pi$
$571$	9.11943	0.381636	0.190818	+	0.981625i	$0.438886\pi$
$572$	1.60555i	0.0671315i
$573$	4.60555i	0.192400i
$574$	3.00000	0.125218
$575$	0	0
$576$	7.90833	0.329514
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 1.72498i	− 0.0717497i
$579$	−12.6056	−0.523869
$580$	0	0
$581$	0.972244	0.0403355
$582$	− 8.90833i	− 0.369262i
$583$	17.0278i	0.705218i
$584$	15.8167	0.654498
$585$	0	0
$586$	−8.78890	−0.363066
$587$	− 37.5416i	− 1.54951i	−0.632262	−	0.774755i	$-0.717873\pi$
$587$	− 37.5416i	− 1.54951i	0.632262	−	0.774755i	$-0.282127\pi$
$588$	22.8167i	0.940943i
$589$	14.2750	0.588192
$590$	0	0
$591$	2.30278	0.0947235
$592$	8.00000i	0.328798i
$593$	− 19.8167i	− 0.813772i	−0.913479	−	0.406886i	$-0.866615\pi$
$593$	− 19.8167i	− 0.813772i	0.913479	−	0.406886i	$-0.133385\pi$
$594$	−85.9638	−3.52714
$595$	0	0
$596$	9.69722	0.397214
$597$	27.8167i	1.13846i
$598$	0.302776i	0.0123814i
$599$	−4.33053	−0.176941	−0.0884704	−	0.996079i	$-0.528198\pi$
$599$	−4.33053	−0.176941	−0.0884704	+	0.996079i	$0.528198\pi$
$600$	0	0
$601$	−3.93608	−0.160556	−0.0802781	−	0.996773i	$-0.525581\pi$
$601$	−3.93608	−0.160556	−0.0802781	+	0.996773i	$0.525581\pi$
$602$	1.57779i	0.0643061i
$603$	31.6333i	1.28821i
$604$	1.90833	0.0776487
$605$	0	0
$606$	15.2111	0.617909
$607$	− 26.0555i	− 1.05756i	−0.848759	−	0.528780i	$-0.822650\pi$
$607$	− 26.0555i	− 1.05756i	0.848759	−	0.528780i	$-0.177350\pi$
$608$	− 4.90833i	− 0.199059i
$609$	4.60555	0.186626
$610$	0	0
$611$	−1.39445	−0.0564134
$612$	− 30.9083i	− 1.24940i
$613$	− 32.4222i	− 1.30952i	−0.755837	−	0.654760i	$-0.772770\pi$
$613$	− 32.4222i	− 1.30952i	0.755837	−	0.654760i	$-0.227230\pi$
$614$	−15.3028	−0.617570
$615$	0	0
$616$	−1.60555	−0.0646895
$617$	8.09167i	0.325758i	0.986646	+	0.162879i	$0.0520781\pi$
$617$	8.09167i	0.325758i	−0.986646	+	0.162879i	$0.947922\pi$
$618$	− 56.5416i	− 2.27444i
$619$	−27.3305	−1.09851	−0.549253	−	0.835656i	$-0.685088\pi$
$619$	−27.3305	−1.09851	−0.549253	+	0.835656i	$0.685088\pi$
$620$	0	0
$621$	−16.2111	−0.650529
$622$	6.42221i	0.257507i
$623$	0	0
$624$	1.00000	0.0400320
$625$	0	0
$626$	12.7250	0.508593
$627$	85.9638i	3.43307i
$628$	11.3944i	0.454688i
$629$	31.2666	1.24668
$630$	0	0
$631$	30.6056	1.21839	0.609194	−	0.793021i	$-0.291493\pi$
$631$	30.6056	1.21839	0.609194	+	0.793021i	$0.291493\pi$
$632$	− 14.4222i	− 0.573685i
$633$	23.8167i	0.946627i
$634$	−14.7250	−0.584804
$635$	0	0
$636$	10.6056	0.420537
$637$	2.09167i	0.0828751i
$638$	− 24.4222i	− 0.966884i
$639$	100.414	3.97231
$640$	0	0
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	− 15.2111i	− 0.600334i
$643$	− 16.2389i	− 0.640398i	−0.947350	−	0.320199i	$-0.896250\pi$
$643$	− 16.2389i	− 0.640398i	0.947350	−	0.320199i	$-0.103750\pi$
$644$	−0.302776	−0.0119310
$645$	0	0
$646$	−19.1833	−0.754759
$647$	− 30.8444i	− 1.21262i	−0.795229	−	0.606309i	$-0.792649\pi$
$647$	− 30.8444i	− 1.21262i	0.795229	−	0.606309i	$-0.207351\pi$
$648$	29.8167i	1.17131i
$649$	−56.2389	−2.20757
$650$	0	0
$651$	−2.90833	−0.113986
$652$	5.69722i	0.223121i
$653$	− 9.27502i	− 0.362960i	−0.983395	−	0.181480i	$-0.941911\pi$
$653$	− 9.27502i	− 0.362960i	0.983395	−	0.181480i	$-0.0580887\pi$
$654$	64.4500	2.52019
$655$	0	0
$656$	−9.90833	−0.386855
$657$	125.083i	4.87996i
$658$	− 1.39445i	− 0.0543613i
$659$	27.6333	1.07644	0.538220	−	0.842804i	$-0.319097\pi$
$659$	27.6333	1.07644	0.538220	+	0.842804i	$0.319097\pi$
$660$	0	0
$661$	−24.0917	−0.937057	−0.468529	−	0.883448i	$-0.655216\pi$
$661$	−24.0917	−0.937057	−0.468529	+	0.883448i	$0.655216\pi$
$662$	− 9.39445i	− 0.365126i
$663$	− 3.90833i	− 0.151787i
$664$	−3.21110	−0.124615
$665$	0	0
$666$	−63.2666	−2.45153
$667$	− 4.60555i	− 0.178328i
$668$	− 21.2111i	− 0.820682i
$669$	13.2111	0.510771
$670$	0	0
$671$	34.5416	1.33347
$672$	1.00000i	0.0385758i
$673$	− 5.63331i	− 0.217148i	−0.994088	−	0.108574i	$-0.965372\pi$
$673$	− 5.63331i	− 0.217148i	0.994088	−	0.108574i	$-0.0346284\pi$
$674$	−4.48612	−0.172799
$675$	0	0
$676$	−12.9083	−0.496474
$677$	− 12.4222i	− 0.477424i	−0.971090	−	0.238712i	$-0.923275\pi$
$677$	− 12.4222i	− 0.477424i	0.971090	−	0.238712i	$-0.0767252\pi$
$678$	41.0278i	1.57566i
$679$	0.816654	0.0313403
$680$	0	0
$681$	−24.4222	−0.935861
$682$	15.4222i	0.590547i
$683$	32.7250i	1.25219i	0.779748	+	0.626093i	$0.215347\pi$
$683$	32.7250i	1.25219i	−0.779748	+	0.626093i	$0.784653\pi$
$684$	38.8167	1.48419
$685$	0	0
$686$	−4.21110	−0.160781
$687$	6.60555i	0.252018i
$688$	− 5.21110i	− 0.198671i
$689$	0.972244	0.0370395
$690$	0	0
$691$	30.1833	1.14823	0.574114	−	0.818775i	$-0.305347\pi$
$691$	30.1833	1.14823	0.574114	+	0.818775i	$0.305347\pi$
$692$	23.3028i	0.885839i
$693$	− 12.6972i	− 0.482328i
$694$	−25.5416	−0.969547
$695$	0	0
$696$	−15.2111	−0.576575
$697$	38.7250i	1.46681i
$698$	− 12.7889i	− 0.484067i
$699$	−13.8167	−0.522594
$700$	0	0
$701$	42.9083	1.62063	0.810313	−	0.585998i	$-0.199297\pi$
$701$	42.9083	1.62063	0.810313	+	0.585998i	$0.199297\pi$
$702$	4.90833i	0.185253i
$703$	39.2666i	1.48097i
$704$	5.30278	0.199856
$705$	0	0
$706$	−18.4222	−0.693329
$707$	1.39445i	0.0524436i
$708$	35.0278i	1.31642i
$709$	41.1194	1.54427	0.772136	−	0.635457i	$-0.219188\pi$
$709$	41.1194	1.54427	0.772136	+	0.635457i	$0.219188\pi$
$710$	0	0
$711$	114.056	4.27742
$712$	0	0
$713$	2.90833i	0.108918i
$714$	3.90833	0.146265
$715$	0	0
$716$	16.6056	0.620579
$717$	− 30.4222i	− 1.13614i
$718$	− 3.21110i	− 0.119837i
$719$	−14.3028	−0.533404	−0.266702	−	0.963779i	$-0.585934\pi$
$719$	−14.3028	−0.533404	−0.266702	+	0.963779i	$0.585934\pi$
$720$	0	0
$721$	5.18335	0.193038
$722$	− 5.09167i	− 0.189492i
$723$	− 47.6333i	− 1.77150i
$724$	8.11943	0.301756
$725$	0	0
$726$	−56.5416	−2.09846
$727$	− 7.90833i	− 0.293304i	−0.989188	−	0.146652i	$-0.953150\pi$
$727$	− 7.90833i	− 0.293304i	0.989188	−	0.146652i	$-0.0468497\pi$
$728$	0.0916731i	0.00339763i
$729$	−75.1749	−2.78426
$730$	0	0
$731$	−20.3667	−0.753289
$732$	− 21.5139i	− 0.795176i
$733$	13.6333i	0.503558i	0.967785	+	0.251779i	$0.0810156\pi$
$733$	13.6333i	0.503558i	−0.967785	+	0.251779i	$0.918984\pi$
$734$	29.2111	1.07820
$735$	0	0
$736$	1.00000	0.0368605
$737$	21.2111i	0.781321i
$738$	− 78.3583i	− 2.88441i
$739$	7.63331	0.280796	0.140398	−	0.990095i	$-0.455162\pi$
$739$	7.63331	0.280796	0.140398	+	0.990095i	$0.455162\pi$
$740$	0	0
$741$	4.90833	0.180312
$742$	0.972244i	0.0356922i
$743$	− 7.33053i	− 0.268931i	−0.990918	−	0.134466i	$-0.957068\pi$
$743$	− 7.33053i	− 0.268931i	0.990918	−	0.134466i	$-0.0429318\pi$
$744$	9.60555	0.352157
$745$	0	0
$746$	2.60555	0.0953960
$747$	− 25.3944i	− 0.929134i
$748$	− 20.7250i	− 0.757780i
$749$	1.39445	0.0509520
$750$	0	0
$751$	0.183346	0.00669040	0.00334520	−	0.999994i	$-0.498935\pi$
$751$	0.183346	0.00669040	0.00334520	+	0.999994i	$0.498935\pi$
$752$	4.60555i	0.167947i
$753$	18.2111i	0.663649i
$754$	−1.39445	−0.0507828
$755$	0	0
$756$	−4.90833	−0.178514
$757$	− 1.21110i	− 0.0440183i	−0.999758	−	0.0220091i	$-0.992994\pi$
$757$	− 1.21110i	− 0.0440183i	0.999758	−	0.0220091i	$-0.00700629\pi$
$758$	4.09167i	0.148616i
$759$	−17.5139	−0.635714
$760$	0	0
$761$	4.54163	0.164634	0.0823171	−	0.996606i	$-0.473768\pi$
$761$	4.54163	0.164634	0.0823171	+	0.996606i	$0.473768\pi$
$762$	39.0278i	1.41383i
$763$	5.90833i	0.213896i
$764$	1.39445	0.0504494
$765$	0	0
$766$	0	0
$767$	3.21110i	0.115946i
$768$	− 3.30278i	− 0.119179i
$769$	41.2666	1.48811	0.744056	−	0.668117i	$-0.232900\pi$
$769$	41.2666	1.48811	0.744056	+	0.668117i	$0.232900\pi$
$770$	0	0
$771$	65.4500	2.35712
$772$	3.81665i	0.137364i
$773$	− 12.0000i	− 0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$	− 12.0000i	− 0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$	41.2111	1.48130
$775$	0	0
$776$	−2.69722	−0.0968247
$777$	− 8.00000i	− 0.286998i
$778$	20.9361i	0.750595i
$779$	−48.6333	−1.74247
$780$	0	0
$781$	67.3305	2.40928
$782$	− 3.90833i	− 0.139761i
$783$	− 74.6611i	− 2.66817i
$784$	6.90833	0.246726
$785$	0	0
$786$	−10.6056	−0.378287
$787$	− 27.4500i	− 0.978485i	−0.872148	−	0.489243i	$-0.837273\pi$
$787$	− 27.4500i	− 0.978485i	0.872148	−	0.489243i	$-0.162727\pi$
$788$	− 0.697224i	− 0.0248376i
$789$	47.9361	1.70657
$790$	0	0
$791$	−3.76114	−0.133731
$792$	41.9361i	1.49013i
$793$	− 1.97224i	− 0.0700364i
$794$	−21.7250	−0.770991
$795$	0	0
$796$	8.42221	0.298517
$797$	− 19.8167i	− 0.701942i	−0.936386	−	0.350971i	$-0.885852\pi$
$797$	− 19.8167i	− 0.701942i	0.936386	−	0.350971i	$-0.114148\pi$
$798$	4.90833i	0.173753i
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	1.39445i	0.0492397i
$803$	83.8722i	2.95978i
$804$	13.2111	0.465920
$805$	0	0
$806$	0.880571	0.0310168
$807$	85.2666i	3.00153i
$808$	− 4.60555i	− 0.162023i
$809$	−18.2750	−0.642515	−0.321258	−	0.946992i	$-0.604106\pi$
$809$	−18.2750	−0.642515	−0.321258	+	0.946992i	$0.604106\pi$
$810$	0	0
$811$	−4.97224	−0.174599	−0.0872995	−	0.996182i	$-0.527824\pi$
$811$	−4.97224	−0.174599	−0.0872995	+	0.996182i	$0.527824\pi$
$812$	− 1.39445i	− 0.0489356i
$813$	20.8167i	0.730072i
$814$	−42.4222	−1.48690
$815$	0	0
$816$	−12.9083	−0.451882
$817$	− 25.5778i	− 0.894854i
$818$	− 15.0917i	− 0.527668i
$819$	−0.724981	−0.0253329
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	22.8167i	0.795822i
$823$	0.788897i	0.0274992i	0.999905	+	0.0137496i	$0.00437678\pi$
$823$	0.788897i	0.0274992i	−0.999905	+	0.0137496i	$0.995623\pi$
$824$	−17.1194	−0.596384
$825$	0	0
$826$	−3.21110	−0.111729
$827$	35.4500i	1.23272i	0.787466	+	0.616358i	$0.211393\pi$
$827$	35.4500i	1.23272i	−0.787466	+	0.616358i	$0.788607\pi$
$828$	7.90833i	0.274833i
$829$	−16.7889	−0.583103	−0.291551	−	0.956555i	$-0.594171\pi$
$829$	−16.7889	−0.583103	−0.291551	+	0.956555i	$0.594171\pi$
$830$	0	0
$831$	−42.2389	−1.46525
$832$	− 0.302776i	− 0.0104969i
$833$	− 27.0000i	− 0.935495i
$834$	−17.8167	−0.616940
$835$	0	0
$836$	26.0278	0.900189
$837$	47.1472i	1.62965i
$838$	− 39.6333i	− 1.36911i
$839$	22.1833	0.765854	0.382927	−	0.923779i	$-0.374916\pi$
$839$	22.1833	0.765854	0.382927	+	0.923779i	$0.374916\pi$
$840$	0	0
$841$	−7.78890	−0.268583
$842$	− 34.3028i	− 1.18215i
$843$	64.0555i	2.20619i
$844$	7.21110	0.248216
$845$	0	0
$846$	−36.4222	−1.25222
$847$	− 5.18335i	− 0.178102i
$848$	− 3.21110i	− 0.110270i
$849$	−6.60555	−0.226702
$850$	0	0
$851$	−8.00000	−0.274236
$852$	− 41.9361i	− 1.43671i
$853$	− 10.7250i	− 0.367216i	−0.983000	−	0.183608i	$-0.941222\pi$
$853$	− 10.7250i	− 0.367216i	0.983000	−	0.183608i	$-0.0587778\pi$
$854$	1.97224	0.0674888
$855$	0	0
$856$	−4.60555	−0.157415
$857$	− 33.6333i	− 1.14889i	−0.818543	−	0.574446i	$-0.805218\pi$
$857$	− 33.6333i	− 1.14889i	0.818543	−	0.574446i	$-0.194782\pi$
$858$	5.30278i	0.181034i
$859$	14.1833	0.483930	0.241965	−	0.970285i	$-0.422208\pi$
$859$	14.1833	0.483930	0.241965	+	0.970285i	$0.422208\pi$
$860$	0	0
$861$	9.90833	0.337675
$862$	− 20.2389i	− 0.689338i
$863$	− 23.4500i	− 0.798246i	−0.916897	−	0.399123i	$-0.869315\pi$
$863$	− 23.4500i	− 0.798246i	0.916897	−	0.399123i	$-0.130685\pi$
$864$	16.2111	0.551513
$865$	0	0
$866$	34.9083	1.18623
$867$	− 5.69722i	− 0.193488i
$868$	0.880571i	0.0298885i
$869$	76.4777	2.59433
$870$	0	0
$871$	1.21110	0.0410366
$872$	− 19.5139i	− 0.660823i
$873$	− 21.3305i	− 0.721929i
$874$	4.90833	0.166027
$875$	0	0
$876$	52.2389	1.76499
$877$	49.1749i	1.66052i	0.557376	+	0.830260i	$0.311808\pi$
$877$	49.1749i	1.66052i	−0.557376	+	0.830260i	$0.688192\pi$
$878$	− 18.3028i	− 0.617689i
$879$	−29.0278	−0.979082
$880$	0	0
$881$	−31.2666	−1.05340	−0.526700	−	0.850052i	$-0.676571\pi$
$881$	−31.2666	−1.05340	−0.526700	+	0.850052i	$0.676571\pi$
$882$	54.6333i	1.83960i
$883$	− 40.7250i	− 1.37050i	−0.728306	−	0.685252i	$-0.759692\pi$
$883$	− 40.7250i	− 1.37050i	0.728306	−	0.685252i	$-0.240308\pi$
$884$	−1.18335	−0.0398002
$885$	0	0
$886$	−35.5139	−1.19311
$887$	− 15.6333i	− 0.524915i	−0.964944	−	0.262458i	$-0.915467\pi$
$887$	− 15.6333i	− 0.524915i	0.964944	−	0.262458i	$-0.0845330\pi$
$888$	26.4222i	0.886671i
$889$	−3.57779	−0.119995
$890$	0	0
$891$	−158.111	−5.29692
$892$	− 4.00000i	− 0.133930i
$893$	22.6056i	0.756466i
$894$	32.0278	1.07117
$895$	0	0
$896$	0.302776	0.0101150
$897$	1.00000i	0.0333890i
$898$	− 12.9083i	− 0.430756i
$899$	−13.3944	−0.446730
$900$	0	0
$901$	−12.5500	−0.418102
$902$	− 52.5416i	− 1.74945i
$903$	5.21110i	0.173415i
$904$	12.4222	0.413156
$905$	0	0
$906$	6.30278	0.209396
$907$	− 30.6611i	− 1.01808i	−0.860742	−	0.509042i	$-0.830000\pi$
$907$	− 30.6611i	− 1.01808i	0.860742	−	0.509042i	$-0.170000\pi$
$908$	7.39445i	0.245393i
$909$	36.4222	1.20805
$910$	0	0
$911$	−25.8167	−0.855344	−0.427672	−	0.903934i	$-0.640666\pi$
$911$	−25.8167	−0.855344	−0.427672	+	0.903934i	$0.640666\pi$
$912$	− 16.2111i	− 0.536803i
$913$	− 17.0278i	− 0.563536i
$914$	−3.57779	−0.118343
$915$	0	0
$916$	2.00000	0.0660819
$917$	− 0.972244i	− 0.0321063i
$918$	− 63.3583i	− 2.09114i
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	−50.5416	−1.66540
$922$	− 31.8167i	− 1.04783i
$923$	− 3.84441i	− 0.126540i
$924$	−5.30278	−0.174449
$925$	0	0
$926$	25.6333	0.842363
$927$	− 135.386i	− 4.44666i
$928$	4.60555i	0.151185i
$929$	57.6333	1.89089	0.945444	−	0.325785i	$-0.105629\pi$
$929$	57.6333	1.89089	0.945444	+	0.325785i	$0.105629\pi$
$930$	0	0
$931$	33.9083	1.11130
$932$	4.18335i	0.137030i
$933$	21.2111i	0.694420i
$934$	19.8167	0.648421
$935$	0	0
$936$	2.39445	0.0782650
$937$	− 44.9638i	− 1.46890i	−0.678660	−	0.734452i	$-0.737439\pi$
$937$	− 44.9638i	− 1.46890i	0.678660	−	0.734452i	$-0.262561\pi$
$938$	1.21110i	0.0395439i
$939$	42.0278	1.37152
$940$	0	0
$941$	−20.9361	−0.682497	−0.341248	−	0.939973i	$-0.610850\pi$
$941$	−20.9361	−0.682497	−0.341248	+	0.939973i	$0.610850\pi$
$942$	37.6333i	1.22616i
$943$	− 9.90833i	− 0.322660i
$944$	10.6056	0.345181
$945$	0	0
$946$	27.6333	0.898436
$947$	41.9361i	1.36274i	0.731939	+	0.681370i	$0.238615\pi$
$947$	41.9361i	1.36274i	−0.731939	+	0.681370i	$0.761385\pi$
$948$	− 47.6333i	− 1.54706i
$949$	4.78890	0.155454
$950$	0	0
$951$	−48.6333	−1.57704
$952$	− 1.18335i	− 0.0383525i
$953$	1.66947i	0.0540794i	0.999634	+	0.0270397i	$0.00860805\pi$
$953$	1.66947i	0.0540794i	−0.999634	+	0.0270397i	$0.991392\pi$
$954$	25.3944	0.822176
$955$	0	0
$956$	−9.21110	−0.297908
$957$	− 80.6611i	− 2.60740i
$958$	− 30.0000i	− 0.969256i
$959$	−2.09167	−0.0675436
$960$	0	0
$961$	−22.5416	−0.727150
$962$	2.42221i	0.0780950i
$963$	− 36.4222i	− 1.17369i
$964$	−14.4222	−0.464508
$965$	0	0
$966$	−1.00000	−0.0321745
$967$	− 5.39445i	− 0.173474i	−0.996231	−	0.0867369i	$-0.972356\pi$
$967$	− 5.39445i	− 0.173474i	0.996231	−	0.0867369i	$-0.0276439\pi$
$968$	17.1194i	0.550239i
$969$	−63.3583	−2.03536
$970$	0	0
$971$	−27.9083	−0.895621	−0.447810	−	0.894129i	$-0.647796\pi$
$971$	−27.9083	−0.895621	−0.447810	+	0.894129i	$0.647796\pi$
$972$	49.8444i	1.59876i
$973$	− 1.63331i	− 0.0523614i
$974$	−11.8167	−0.378630
$975$	0	0
$976$	−6.51388	−0.208504
$977$	− 11.5139i	− 0.368362i	−0.982892	−	0.184181i	$-0.941037\pi$
$977$	− 11.5139i	− 0.368362i	0.982892	−	0.184181i	$-0.0589632\pi$
$978$	18.8167i	0.601690i
$979$	0	0
$980$	0	0
$981$	154.322	4.92713
$982$	25.8167i	0.823843i
$983$	19.5416i	0.623281i	0.950200	+	0.311641i	$0.100879\pi$
$983$	19.5416i	0.623281i	−0.950200	+	0.311641i	$0.899121\pi$
$984$	−32.7250	−1.04323
$985$	0	0
$986$	18.0000	0.573237
$987$	− 4.60555i	− 0.146596i
$988$	− 1.48612i	− 0.0472798i
$989$	5.21110	0.165703
$990$	0	0
$991$	24.3305	0.772885	0.386442	−	0.922314i	$-0.373704\pi$
$991$	24.3305	0.772885	0.386442	+	0.922314i	$0.373704\pi$
$992$	− 2.90833i	− 0.0923395i
$993$	− 31.0278i	− 0.984636i
$994$	3.84441	0.121937
$995$	0	0
$996$	−10.6056	−0.336050
$997$	− 31.2111i	− 0.988466i	−0.869330	−	0.494233i	$-0.835449\pi$
$997$	− 31.2111i	− 0.988466i	0.869330	−	0.494233i	$-0.164551\pi$
$998$	11.6333i	0.368246i
$999$	−129.689	−4.10317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.2.b.f.599.1		4
5.2	odd	4		1150.2.a.m.1.1		2
5.3	odd	4		230.2.a.b.1.2	✓	2
5.4	even	2	inner	1150.2.b.f.599.4		4
15.8	even	4		2070.2.a.w.1.1		2
20.3	even	4		1840.2.a.j.1.1		2
20.7	even	4		9200.2.a.ca.1.2		2
40.3	even	4		7360.2.a.bu.1.2		2
40.13	odd	4		7360.2.a.bc.1.1		2
115.68	even	4		5290.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.b.1.2	✓	2	5.3	odd	4
1150.2.a.m.1.1		2	5.2	odd	4
1150.2.b.f.599.1		4	1.1	even	1	trivial
1150.2.b.f.599.4		4	5.4	even	2	inner
1840.2.a.j.1.1		2	20.3	even	4
2070.2.a.w.1.1		2	15.8	even	4
5290.2.a.j.1.2		2	115.68	even	4
7360.2.a.bc.1.1		2	40.13	odd	4
7360.2.a.bu.1.2		2	40.3	even	4
9200.2.a.ca.1.2		2	20.7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} -0.302776i q^{7} +1.00000i q^{8} -7.90833 q^{9} -5.30278 q^{11} +3.30278i q^{12} +0.302776i q^{13} -0.302776 q^{14} +1.00000 q^{16} -3.90833i q^{17} +7.90833i q^{18} +4.90833 q^{19} -1.00000 q^{21} +5.30278i q^{22} +1.00000i q^{23} +3.30278 q^{24} +0.302776 q^{26} +16.2111i q^{27} +0.302776i q^{28} -4.60555 q^{29} +2.90833 q^{31} -1.00000i q^{32} +17.5139i q^{33} -3.90833 q^{34} +7.90833 q^{36} +8.00000i q^{37} -4.90833i q^{38} +1.00000 q^{39} -9.90833 q^{41} +1.00000i q^{42} -5.21110i q^{43} +5.30278 q^{44} +1.00000 q^{46} +4.60555i q^{47} -3.30278i q^{48} +6.90833 q^{49} -12.9083 q^{51} -0.302776i q^{52} -3.21110i q^{53} +16.2111 q^{54} +0.302776 q^{56} -16.2111i q^{57} +4.60555i q^{58} +10.6056 q^{59} -6.51388 q^{61} -2.90833i q^{62} +2.39445i q^{63} -1.00000 q^{64} +17.5139 q^{66} -4.00000i q^{67} +3.90833i q^{68} +3.30278 q^{69} -12.6972 q^{71} -7.90833i q^{72} -15.8167i q^{73} +8.00000 q^{74} -4.90833 q^{76} +1.60555i q^{77} -1.00000i q^{78} -14.4222 q^{79} +29.8167 q^{81} +9.90833i q^{82} +3.21110i q^{83} +1.00000 q^{84} -5.21110 q^{86} +15.2111i q^{87} -5.30278i q^{88} +0.0916731 q^{91} -1.00000i q^{92} -9.60555i q^{93} +4.60555 q^{94} -3.30278 q^{96} +2.69722i q^{97} -6.90833i q^{98} +41.9361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 14 q^{11} + 6 q^{14} + 4 q^{16} - 2 q^{19} - 4 q^{21} + 6 q^{24} - 6 q^{26} - 4 q^{29} - 10 q^{31} + 6 q^{34} + 10 q^{36} + 4 q^{39} - 18 q^{41} + 14 q^{44} + 4 q^{46}+ \cdots + 74 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 6 * q^6 - 10 * q^9 - 14 * q^11 + 6 * q^14 + 4 * q^16 - 2 * q^19 - 4 * q^21 + 6 * q^24 - 6 * q^26 - 4 * q^29 - 10 * q^31 + 6 * q^34 + 10 * q^36 + 4 * q^39 - 18 * q^41 + 14 * q^44 + 4 * q^46 + 6 * q^49 - 30 * q^51 + 36 * q^54 - 6 * q^56 + 28 * q^59 + 10 * q^61 - 4 * q^64 + 34 * q^66 + 6 * q^69 - 58 * q^71 + 32 * q^74 + 2 * q^76 + 76 * q^81 + 4 * q^84 + 8 * q^86 + 22 * q^91 + 4 * q^94 - 6 * q^96 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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