LMFDB - Embedded newform 2300.2.i.f.1057.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1057,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2300.1057");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.i (of order $4$, degree $2$, 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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 74x^{12} + 1357x^{8} + 3177x^{4} + 1296 $ x^16 + 74x^12 + 1357x^8 + 3177*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.7
Root			$1.58699 - 1.58699i$ of defining polynomial
Character	$\chi$	$=$	2300.1057
Dual form			2300.2.i.f.1793.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58699 - 1.58699i) q^{3} +(0.216666 - 0.216666i) q^{7} -2.03710i q^{9} +O(q^{10})$	$q+(1.58699 - 1.58699i) q^{3} +(0.216666 - 0.216666i) q^{7} -2.03710i q^{9} -3.54348i q^{11} +(-2.55380 + 2.55380i) q^{13} +(4.67428 - 4.67428i) q^{17} +2.45652 q^{19} -0.687695i q^{21} +(-3.33178 - 3.44953i) q^{23} +(1.52812 + 1.52812i) q^{27} -0.693252i q^{29} -1.83747 q^{31} +(-5.62348 - 5.62348i) q^{33} +(7.00601 - 7.00601i) q^{37} +8.10574i q^{39} -2.34385 q^{41} +(-4.16644 - 4.16644i) q^{43} +(-3.83743 - 3.83743i) q^{47} +6.90611i q^{49} -14.8361i q^{51} +(-0.724498 - 0.724498i) q^{53} +(3.89849 - 3.89849i) q^{57} -13.3796i q^{59} +7.37539i q^{61} +(-0.441370 - 0.441370i) q^{63} +(5.28102 - 5.28102i) q^{67} +(-10.7619 - 0.186874i) q^{69} -2.63784 q^{71} +(4.02304 - 4.02304i) q^{73} +(-0.767751 - 0.767751i) q^{77} -11.3181 q^{79} +10.9615 q^{81} +(-1.40137 - 1.40137i) q^{83} +(-1.10019 - 1.10019i) q^{87} -2.83609 q^{89} +1.10664i q^{91} +(-2.91605 + 2.91605i) q^{93} +(3.72507 - 3.72507i) q^{97} -7.21842 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) $ 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	1.58699	−	1.58699i	0.916251	−	0.916251i	−0.0805030	−	0.996754i	$-0.525653\pi$
$3$	1.58699	−	1.58699i	0.916251	−	0.916251i	0.996754	+	0.0805030i	$0.0256527\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.216666	−	0.216666i	0.0818921	−	0.0818921i	−0.664974	−	0.746866i	$-0.731557\pi$
$7$	0.216666	−	0.216666i	0.0818921	−	0.0818921i	0.746866	+	0.664974i	$0.231557\pi$
$8$		0			0
$9$		−	2.03710i		−	0.679033i
$10$		0			0
$11$		−	3.54348i		−	1.06840i	−0.845359	−	0.534199i	$-0.820613\pi$
$11$		−	3.54348i		−	1.06840i	0.845359	−	0.534199i	$-0.179387\pi$
$12$		0			0
$13$	−2.55380	+	2.55380i	−0.708298	+	0.708298i	−0.966177	−	0.257879i	$-0.916976\pi$
$13$	−2.55380	+	2.55380i	−0.708298	+	0.708298i	0.257879	+	0.966177i	$0.416976\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.67428	−	4.67428i	1.13368	−	1.13368i	0.144118	−	0.989561i	$-0.453966\pi$
$17$	4.67428	−	4.67428i	1.13368	−	1.13368i	0.989561	−	0.144118i	$-0.0460344\pi$
$18$		0			0
$19$	2.45652			0.563565			0.281782	−	0.959478i	$-0.409074\pi$
$19$	2.45652			0.563565			0.281782	+	0.959478i	$0.409074\pi$
$20$		0			0
$21$		−	0.687695i		−	0.150067i
$22$		0			0
$23$	−3.33178	−	3.44953i	−0.694724	−	0.719277i
$23$	−3.33178	−	3.44953i	−0.694724	−	0.719277i
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.52812	+	1.52812i	0.294086	+	0.294086i
$28$		0			0
$29$		−	0.693252i		−	0.128734i	−0.997926	−	0.0643668i	$-0.979497\pi$
$29$		−	0.693252i		−	0.128734i	0.997926	−	0.0643668i	$-0.0205028\pi$
$30$		0			0
$31$	−1.83747			−0.330019			−0.165010	−	0.986292i	$-0.552766\pi$
$31$	−1.83747			−0.330019			−0.165010	+	0.986292i	$0.552766\pi$
$32$		0			0
$33$	−5.62348	−	5.62348i	−0.978922	−	0.978922i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.00601	−	7.00601i	1.15178	−	1.15178i	0.165586	−	0.986195i	$-0.447048\pi$
$37$	7.00601	−	7.00601i	1.15178	−	1.15178i	0.986195	−	0.165586i	$-0.0529515\pi$
$38$		0			0
$39$			8.10574i			1.29796i
$40$		0			0
$41$	−2.34385			−0.366048			−0.183024	−	0.983108i	$-0.558589\pi$
$41$	−2.34385			−0.366048			−0.183024	+	0.983108i	$0.558589\pi$
$42$		0			0
$43$	−4.16644	−	4.16644i	−0.635377	−	0.635377i	0.314035	−	0.949411i	$-0.398319\pi$
$43$	−4.16644	−	4.16644i	−0.635377	−	0.635377i	−0.949411	+	0.314035i	$0.898319\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.83743	−	3.83743i	−0.559746	−	0.559746i	0.369489	−	0.929235i	$-0.379533\pi$
$47$	−3.83743	−	3.83743i	−0.559746	−	0.559746i	−0.929235	+	0.369489i	$0.879533\pi$
$48$		0			0
$49$			6.90611i			0.986587i
$50$		0			0
$51$		−	14.8361i		−	2.07747i
$52$		0			0
$53$	−0.724498	−	0.724498i	−0.0995175	−	0.0995175i	0.655595	−	0.755113i	$-0.272418\pi$
$53$	−0.724498	−	0.724498i	−0.0995175	−	0.0995175i	−0.755113	+	0.655595i	$0.772418\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.89849	−	3.89849i	0.516367	−	0.516367i
$58$		0			0
$59$		−	13.3796i		−	1.74187i	−0.491397	−	0.870936i	$-0.663513\pi$
$59$		−	13.3796i		−	1.74187i	0.491397	−	0.870936i	$-0.336487\pi$
$60$		0			0
$61$			7.37539i			0.944322i	0.881512	+	0.472161i	$0.156526\pi$
$61$			7.37539i			0.944322i	−0.881512	+	0.472161i	$0.843474\pi$
$62$		0			0
$63$	−0.441370	−	0.441370i	−0.0556074	−	0.0556074i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.28102	−	5.28102i	0.645179	−	0.645179i	−0.306645	−	0.951824i	$-0.599206\pi$
$67$	5.28102	−	5.28102i	0.645179	−	0.645179i	0.951824	+	0.306645i	$0.0992063\pi$
$68$		0			0
$69$	−10.7619	−	0.186874i	−1.29558	−	0.0224969i
$70$		0			0
$71$	−2.63784			−0.313054			−0.156527	−	0.987674i	$-0.550030\pi$
$71$	−2.63784			−0.313054			−0.156527	+	0.987674i	$0.550030\pi$
$72$		0			0
$73$	4.02304	−	4.02304i	0.470862	−	0.470862i	−0.431332	−	0.902193i	$-0.641956\pi$
$73$	4.02304	−	4.02304i	0.470862	−	0.470862i	0.902193	+	0.431332i	$0.141956\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.767751	−	0.767751i	−0.0874934	−	0.0874934i
$78$		0			0
$79$	−11.3181			−1.27339			−0.636694	−	0.771116i	$-0.719699\pi$
$79$	−11.3181			−1.27339			−0.636694	+	0.771116i	$0.719699\pi$
$80$		0			0
$81$	10.9615			1.21795
$82$		0			0
$83$	−1.40137	−	1.40137i	−0.153821	−	0.153821i	0.626001	−	0.779822i	$-0.284690\pi$
$83$	−1.40137	−	1.40137i	−0.153821	−	0.153821i	−0.779822	+	0.626001i	$0.784690\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.10019	−	1.10019i	−0.117952	−	0.117952i
$88$		0			0
$89$	−2.83609			−0.300625			−0.150313	−	0.988639i	$-0.548028\pi$
$89$	−2.83609			−0.300625			−0.150313	+	0.988639i	$0.548028\pi$
$90$		0			0
$91$			1.10664i			0.116008i
$92$		0			0
$93$	−2.91605	+	2.91605i	−0.302381	+	0.302381i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.72507	−	3.72507i	0.378224	−	0.378224i	−0.492237	−	0.870461i	$-0.663821\pi$
$97$	3.72507	−	3.72507i	0.378224	−	0.378224i	0.870461	+	0.492237i	$0.163821\pi$
$98$		0			0
$99$	−7.21842			−0.725478
$100$		0			0
$101$	−0.469279			−0.0466950			−0.0233475	−	0.999727i	$-0.507432\pi$
$101$	−0.469279			−0.0466950			−0.0233475	+	0.999727i	$0.507432\pi$
$102$		0			0
$103$	5.74927	+	5.74927i	0.566492	+	0.566492i	0.931144	−	0.364652i	$-0.118812\pi$
$103$	5.74927	+	5.74927i	0.566492	+	0.566492i	−0.364652	+	0.931144i	$0.618812\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.67419	+	2.67419i	−0.258524	+	0.258524i	−0.824454	−	0.565930i	$-0.808517\pi$
$107$	−2.67419	+	2.67419i	−0.258524	+	0.258524i	0.565930	+	0.824454i	$0.308517\pi$
$108$		0			0
$109$	−5.16115			−0.494349			−0.247174	−	0.968971i	$-0.579502\pi$
$109$	−5.16115			−0.494349			−0.247174	+	0.968971i	$0.579502\pi$
$110$		0			0
$111$		−	22.2370i		−	2.11064i
$112$		0			0
$113$	10.6402	+	10.6402i	1.00095	+	1.00095i	1.00000		0.000947228i	$0.000301512\pi$
$113$	10.6402	+	10.6402i	1.00095	+	1.00095i	0.000947228		1.00000i	$0.499698\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	5.20235	+	5.20235i	0.480958	+	0.480958i
$118$		0			0
$119$		−	2.02551i		−	0.185678i
$120$		0			0
$121$	−1.55623			−0.141476
$122$		0			0
$123$	−3.71967	+	3.71967i	−0.335392	+	0.335392i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.57153	+	8.57153i	0.760600	+	0.760600i	0.976431	−	0.215831i	$-0.0692459\pi$
$127$	8.57153	+	8.57153i	0.760600	+	0.760600i	−0.215831	+	0.976431i	$0.569246\pi$
$128$		0			0
$129$	−13.2242			−1.16433
$130$		0			0
$131$	−4.24996			−0.371321			−0.185660	−	0.982614i	$-0.559442\pi$
$131$	−4.24996			−0.371321			−0.185660	+	0.982614i	$0.559442\pi$
$132$		0			0
$133$	0.532245	−	0.532245i	0.0461515	−	0.0461515i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.99529	−	8.99529i	0.768520	−	0.768520i	−0.209326	−	0.977846i	$-0.567127\pi$
$137$	8.99529	−	8.99529i	0.768520	−	0.768520i	0.977846	+	0.209326i	$0.0671271\pi$
$138$		0			0
$139$		−	1.36216i		−	0.115537i	−0.998330	−	0.0577685i	$-0.981601\pi$
$139$		−	1.36216i		−	0.115537i	0.998330	−	0.0577685i	$-0.0183985\pi$
$140$		0			0
$141$	−12.1799			−1.02574
$142$		0			0
$143$	9.04935	+	9.04935i	0.756744	+	0.756744i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	10.9600	+	10.9600i	0.903962	+	0.903962i
$148$		0			0
$149$	−13.7491			−1.12637			−0.563187	−	0.826330i	$-0.690425\pi$
$149$	−13.7491			−1.12637			−0.563187	+	0.826330i	$0.690425\pi$
$150$		0			0
$151$	12.8118			1.04261			0.521303	−	0.853372i	$-0.325446\pi$
$151$	12.8118			1.04261			0.521303	+	0.853372i	$0.325446\pi$
$152$		0			0
$153$	−9.52196	−	9.52196i	−0.769805	−	0.769805i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2.39820	+	2.39820i	−0.191397	+	0.191397i	−0.796300	−	0.604902i	$-0.793212\pi$
$157$	−2.39820	+	2.39820i	−0.191397	+	0.191397i	0.604902	+	0.796300i	$0.293212\pi$
$158$		0			0
$159$	−2.29955			−0.182366
$160$		0			0
$161$	−1.46928	−	0.0255131i	−0.115795	−	0.00201071i
$162$		0			0
$163$	−7.75616	+	7.75616i	−0.607509	+	0.607509i	−0.942294	−	0.334785i	$-0.891336\pi$
$163$	−7.75616	+	7.75616i	−0.607509	+	0.607509i	0.334785	+	0.942294i	$0.391336\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.85634	−	8.85634i	−0.685324	−	0.685324i	0.275870	−	0.961195i	$-0.411034\pi$
$167$	−8.85634	−	8.85634i	−0.685324	−	0.685324i	−0.961195	+	0.275870i	$0.911034\pi$
$168$		0			0
$169$		−	0.0438262i		−	0.00337125i
$170$		0			0
$171$		−	5.00418i		−	0.382679i
$172$		0			0
$173$	13.7717	−	13.7717i	1.04704	−	1.04704i	0.0482052	−	0.998837i	$-0.484650\pi$
$173$	13.7717	−	13.7717i	1.04704	−	1.04704i	0.998837	−	0.0482052i	$-0.0153501\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−21.2333	−	21.2333i	−1.59599	−	1.59599i
$178$		0			0
$179$			18.7733i			1.40318i	0.712581	+	0.701590i	$0.247526\pi$
$179$			18.7733i			1.40318i	−0.712581	+	0.701590i	$0.752474\pi$
$180$		0			0
$181$			24.3599i			1.81066i	0.424713	+	0.905328i	$0.360375\pi$
$181$			24.3599i			1.81066i	−0.424713	+	0.905328i	$0.639625\pi$
$182$		0			0
$183$	11.7047	+	11.7047i	0.865236	+	0.865236i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−16.5632	−	16.5632i	−1.21122	−	1.21122i
$188$		0			0
$189$	0.662182			0.0481667
$190$		0			0
$191$		−	16.4793i		−	1.19240i	−0.802836	−	0.596200i	$-0.796677\pi$
$191$		−	16.4793i		−	1.19240i	0.802836	−	0.596200i	$-0.203323\pi$
$192$		0			0
$193$	9.11721	−	9.11721i	0.656271	−	0.656271i	−0.298224	−	0.954496i	$-0.596394\pi$
$193$	9.11721	−	9.11721i	0.656271	−	0.656271i	0.954496	+	0.298224i	$0.0963944\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.6012	+	12.6012i	0.897799	+	0.897799i	0.995241	−	0.0974424i	$-0.0310662\pi$
$197$	12.6012	+	12.6012i	0.897799	+	0.897799i	−0.0974424	+	0.995241i	$0.531066\pi$
$198$		0			0
$199$	−21.6808			−1.53691			−0.768454	−	0.639905i	$-0.778974\pi$
$199$	−21.6808			−1.53691			−0.768454	+	0.639905i	$0.778974\pi$
$200$		0			0
$201$		−	16.7619i		−	1.18229i
$202$		0			0
$203$	−0.150204	−	0.150204i	−0.0105423	−	0.0105423i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−7.02704	+	6.78716i	−0.488413	+	0.471740i
$208$		0			0
$209$		−	8.70463i		−	0.602112i
$210$		0			0
$211$	8.31813			0.572644			0.286322	−	0.958134i	$-0.407567\pi$
$211$	8.31813			0.572644			0.286322	+	0.958134i	$0.407567\pi$
$212$		0			0
$213$	−4.18623	+	4.18623i	−0.286836	+	0.286836i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.398117	+	0.398117i	−0.0270260	+	0.0270260i
$218$		0			0
$219$		−	12.7691i		−	0.862855i
$220$		0			0
$221$			23.8744i			1.60596i
$222$		0			0
$223$	−1.50787	+	1.50787i	−0.100975	+	0.100975i	−0.755789	−	0.654815i	$-0.772747\pi$
$223$	−1.50787	+	1.50787i	−0.100975	+	0.100975i	0.654815	+	0.755789i	$0.272747\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.8704	−	16.8704i	1.11973	−	1.11973i	0.127949	−	0.991781i	$-0.459161\pi$
$227$	16.8704	−	16.8704i	1.11973	−	1.11973i	0.991781	−	0.127949i	$-0.0408395\pi$
$228$		0			0
$229$	0.373747			0.0246979			0.0123489	−	0.999924i	$-0.496069\pi$
$229$	0.373747			0.0246979			0.0123489	+	0.999924i	$0.496069\pi$
$230$		0			0
$231$	−2.43683			−0.160332
$232$		0			0
$233$	−14.0094	+	14.0094i	−0.917785	+	0.917785i	−0.996868	−	0.0790826i	$-0.974801\pi$
$233$	−14.0094	+	14.0094i	−0.917785	+	0.917785i	0.0790826	+	0.996868i	$0.474801\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−17.9618	+	17.9618i	−1.16674	+	1.16674i
$238$		0			0
$239$			12.3608i			0.799553i	0.916613	+	0.399776i	$0.130912\pi$
$239$			12.3608i			0.799553i	−0.916613	+	0.399776i	$0.869088\pi$
$240$		0			0
$241$			16.9845i			1.09407i	0.837111	+	0.547034i	$0.184243\pi$
$241$			16.9845i			1.09407i	−0.837111	+	0.547034i	$0.815757\pi$
$242$		0			0
$243$	12.8115	−	12.8115i	0.821859	−	0.821859i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.27348	+	6.27348i	−0.399172	+	0.399172i
$248$		0			0
$249$	−4.44795			−0.281877
$250$		0			0
$251$		−	10.3626i		−	0.654084i	−0.945010	−	0.327042i	$-0.893948\pi$
$251$		−	10.3626i		−	0.654084i	0.945010	−	0.327042i	$-0.106052\pi$
$252$		0			0
$253$	−12.2233	+	11.8061i	−0.768474	+	0.742242i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.7324	+	16.7324i	1.04374	+	1.04374i	0.998999	+	0.0447420i	$0.0142466\pi$
$257$	16.7324	+	16.7324i	1.04374	+	1.04374i	0.0447420	+	0.998999i	$0.485753\pi$
$258$		0			0
$259$		−	3.03593i		−	0.188643i
$260$		0			0
$261$	−1.41222			−0.0874144
$262$		0			0
$263$	4.47398	+	4.47398i	0.275878	+	0.275878i	0.831461	−	0.555583i	$-0.187505\pi$
$263$	4.47398	+	4.47398i	0.275878	+	0.275878i	−0.555583	+	0.831461i	$0.687505\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.50086	+	4.50086i	−0.275448	+	0.275448i
$268$		0			0
$269$			20.2041i			1.23186i	0.787799	+	0.615932i	$0.211220\pi$
$269$			20.2041i			1.23186i	−0.787799	+	0.615932i	$0.788780\pi$
$270$		0			0
$271$	−15.2567			−0.926778			−0.463389	−	0.886155i	$-0.653367\pi$
$271$	−15.2567			−0.926778			−0.463389	+	0.886155i	$0.653367\pi$
$272$		0			0
$273$	1.75624	+	1.75624i	0.106292	+	0.106292i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.46771	−	9.46771i	−0.568859	−	0.568859i	0.362949	−	0.931809i	$-0.381770\pi$
$277$	−9.46771	−	9.46771i	−0.568859	−	0.568859i	−0.931809	+	0.362949i	$0.881770\pi$
$278$		0			0
$279$			3.74311i			0.224094i
$280$		0			0
$281$			10.9386i			0.652540i	0.945277	+	0.326270i	$0.105792\pi$
$281$			10.9386i			0.652540i	−0.945277	+	0.326270i	$0.894208\pi$
$282$		0			0
$283$	−10.0622	−	10.0622i	−0.598138	−	0.598138i	0.341678	−	0.939817i	$-0.389005\pi$
$283$	−10.0622	−	10.0622i	−0.598138	−	0.598138i	−0.939817	+	0.341678i	$0.889005\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.507832	+	0.507832i	−0.0299764	+	0.0299764i
$288$		0			0
$289$		−	26.6977i		−	1.57045i
$290$		0			0
$291$		−	11.8233i		−	0.693096i
$292$		0			0
$293$	6.57268	+	6.57268i	0.383980	+	0.383980i	0.872534	−	0.488554i	$-0.162475\pi$
$293$	6.57268	+	6.57268i	0.383980	+	0.383980i	−0.488554	+	0.872534i	$0.662475\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	5.41485	−	5.41485i	0.314201	−	0.314201i
$298$		0			0
$299$	17.3181	+	0.300718i	1.00153	+	0.0173910i
$300$		0			0
$301$	−1.80545			−0.104065
$302$		0			0
$303$	−0.744743	+	0.744743i	−0.0427843	+	0.0427843i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	22.6117	+	22.6117i	1.29052	+	1.29052i	0.934468	+	0.356048i	$0.115876\pi$
$307$	22.6117	+	22.6117i	1.29052	+	1.29052i	0.356048	+	0.934468i	$0.384124\pi$
$308$		0			0
$309$	18.2481			1.03810
$310$		0			0
$311$	16.9244			0.959696			0.479848	−	0.877352i	$-0.340692\pi$
$311$	16.9244			0.959696			0.479848	+	0.877352i	$0.340692\pi$
$312$		0			0
$313$	−10.1800	−	10.1800i	−0.575408	−	0.575408i	0.358227	−	0.933635i	$-0.383381\pi$
$313$	−10.1800	−	10.1800i	−0.575408	−	0.575408i	−0.933635	+	0.358227i	$0.883381\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.21773	+	4.21773i	0.236892	+	0.236892i	0.815562	−	0.578670i	$-0.196428\pi$
$317$	4.21773	+	4.21773i	0.236892	+	0.236892i	−0.578670	+	0.815562i	$0.696428\pi$
$318$		0			0
$319$	−2.45652			−0.137539
$320$		0			0
$321$			8.48786i			0.473746i
$322$		0			0
$323$	11.4825	−	11.4825i	0.638901	−	0.638901i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−8.19072	+	8.19072i	−0.452948	+	0.452948i
$328$		0			0
$329$	−1.66288			−0.0916775
$330$		0			0
$331$	12.3984			0.681475			0.340738	−	0.940158i	$-0.389323\pi$
$331$	12.3984			0.681475			0.340738	+	0.940158i	$0.389323\pi$
$332$		0			0
$333$	−14.2719	−	14.2719i	−0.782098	−	0.782098i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	4.39919	−	4.39919i	0.239639	−	0.239639i	−0.577062	−	0.816701i	$-0.695801\pi$
$337$	4.39919	−	4.39919i	0.239639	−	0.239639i	0.816701	+	0.577062i	$0.195801\pi$
$338$		0			0
$339$	33.7719			1.83424
$340$		0			0
$341$			6.51103i			0.352592i
$342$		0			0
$343$	3.01298	+	3.01298i	0.162686	+	0.162686i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.77561	+	3.77561i	0.202686	+	0.202686i	0.801150	−	0.598464i	$-0.204222\pi$
$347$	3.77561	+	3.77561i	0.202686	+	0.202686i	−0.598464	+	0.801150i	$0.704222\pi$
$348$		0			0
$349$		−	23.5053i		−	1.25821i	−0.777321	−	0.629104i	$-0.783422\pi$
$349$		−	23.5053i		−	1.25821i	0.777321	−	0.629104i	$-0.216578\pi$
$350$		0			0
$351$	−7.80502			−0.416601
$352$		0			0
$353$	4.29273	−	4.29273i	0.228479	−	0.228479i	−0.583578	−	0.812057i	$-0.698348\pi$
$353$	4.29273	−	4.29273i	0.228479	−	0.228479i	0.812057	+	0.583578i	$0.198348\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.21448	−	3.21448i	−0.170128	−	0.170128i
$358$		0			0
$359$	32.9792			1.74058			0.870288	−	0.492543i	$-0.163933\pi$
$359$	32.9792			1.74058			0.870288	+	0.492543i	$0.163933\pi$
$360$		0			0
$361$	−12.9655			−0.682395
$362$		0			0
$363$	−2.46973	+	2.46973i	−0.129627	+	0.129627i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	11.6991	−	11.6991i	0.610689	−	0.610689i	−0.332436	−	0.943126i	$-0.607871\pi$
$367$	11.6991	−	11.6991i	0.610689	−	0.610689i	0.943126	+	0.332436i	$0.107871\pi$
$368$		0			0
$369$			4.77465i			0.248558i
$370$		0			0
$371$	−0.313948			−0.0162994
$372$		0			0
$373$	−14.8299	−	14.8299i	−0.767861	−	0.767861i	0.209868	−	0.977730i	$-0.432696\pi$
$373$	−14.8299	−	14.8299i	−0.767861	−	0.767861i	−0.977730	+	0.209868i	$0.932696\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.77043	+	1.77043i	0.0911817	+	0.0911817i
$378$		0			0
$379$	30.8505			1.58468			0.792342	−	0.610078i	$-0.208862\pi$
$379$	30.8505			1.58468			0.792342	+	0.610078i	$0.208862\pi$
$380$		0			0
$381$	27.2059			1.39380
$382$		0			0
$383$	8.76255	+	8.76255i	0.447745	+	0.447745i	0.894604	−	0.446859i	$-0.147457\pi$
$383$	8.76255	+	8.76255i	0.447745	+	0.447745i	−0.446859	+	0.894604i	$0.647457\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.48746	+	8.48746i	−0.431442	+	0.431442i
$388$		0			0
$389$	22.4735			1.13945			0.569725	−	0.821836i	$-0.307050\pi$
$389$	22.4735			1.13945			0.569725	+	0.821836i	$0.307050\pi$
$390$		0			0
$391$	−31.6977	−	0.550411i	−1.60302	−	0.0278355i
$392$		0			0
$393$	−6.74466	+	6.74466i	−0.340223	+	0.340223i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−17.6494	−	17.6494i	−0.885796	−	0.885796i	0.108320	−	0.994116i	$-0.465453\pi$
$397$	−17.6494	−	17.6494i	−0.885796	−	0.885796i	−0.994116	+	0.108320i	$0.965453\pi$
$398$		0			0
$399$		−	1.68934i		−	0.0845727i
$400$		0			0
$401$		−	16.7364i		−	0.835775i	−0.908499	−	0.417887i	$-0.862771\pi$
$401$		−	16.7364i		−	0.835775i	0.908499	−	0.417887i	$-0.137229\pi$
$402$		0			0
$403$	4.69254	−	4.69254i	0.233752	−	0.233752i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.8256	−	24.8256i	−1.23056	−	1.23056i
$408$		0			0
$409$			37.4425i			1.85141i	0.378246	+	0.925705i	$0.376527\pi$
$409$			37.4425i			1.85141i	−0.378246	+	0.925705i	$0.623473\pi$
$410$		0			0
$411$		−	28.5509i		−	1.40831i
$412$		0			0
$413$	−2.89890	−	2.89890i	−0.142645	−	0.142645i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.16174	−	2.16174i	−0.105861	−	0.105861i
$418$		0			0
$419$	20.5280			1.00286			0.501428	−	0.865199i	$-0.332808\pi$
$419$	20.5280			1.00286			0.501428	+	0.865199i	$0.332808\pi$
$420$		0			0
$421$			16.4596i			0.802191i	0.916036	+	0.401096i	$0.131371\pi$
$421$			16.4596i			0.802191i	−0.916036	+	0.401096i	$0.868629\pi$
$422$		0			0
$423$	−7.81722	+	7.81722i	−0.380086	+	0.380086i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.59800	+	1.59800i	0.0773325	+	0.0773325i
$428$		0			0
$429$	28.7225			1.38674
$430$		0			0
$431$			22.7364i			1.09517i	0.836749	+	0.547586i	$0.184453\pi$
$431$			22.7364i			1.09517i	−0.836749	+	0.547586i	$0.815547\pi$
$432$		0			0
$433$	−23.0577	−	23.0577i	−1.10808	−	1.10808i	−0.993403	−	0.114680i	$-0.963416\pi$
$433$	−23.0577	−	23.0577i	−1.10808	−	1.10808i	−0.114680	−	0.993403i	$-0.536584\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.18459	−	8.47385i	−0.391522	−	0.405359i
$438$		0			0
$439$		−	16.3984i		−	0.782651i	−0.920252	−	0.391325i	$-0.872017\pi$
$439$		−	16.3984i		−	0.782651i	0.920252	−	0.391325i	$-0.127983\pi$
$440$		0			0
$441$	14.0684			0.669926
$442$		0			0
$443$	−14.8220	+	14.8220i	−0.704214	+	0.704214i	−0.965312	−	0.261098i	$-0.915915\pi$
$443$	−14.8220	+	14.8220i	−0.704214	+	0.704214i	0.261098	+	0.965312i	$0.415915\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−21.8198	+	21.8198i	−1.03204	+	1.03204i
$448$		0			0
$449$		−	24.4554i		−	1.15412i	−0.816701	−	0.577061i	$-0.804199\pi$
$449$		−	24.4554i		−	1.15412i	0.816701	−	0.577061i	$-0.195801\pi$
$450$		0			0
$451$			8.30537i			0.391085i
$452$		0			0
$453$	20.3322	−	20.3322i	0.955289	−	0.955289i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−25.8209	+	25.8209i	−1.20785	+	1.20785i	−0.236127	+	0.971722i	$0.575878\pi$
$457$	−25.8209	+	25.8209i	−1.20785	+	1.20785i	−0.971722	+	0.236127i	$0.924122\pi$
$458$		0			0
$459$	14.2857			0.666799
$460$		0			0
$461$	28.3186			1.31893			0.659464	−	0.751736i	$-0.270783\pi$
$461$	28.3186			1.31893			0.659464	+	0.751736i	$0.270783\pi$
$462$		0			0
$463$	12.7236	−	12.7236i	0.591315	−	0.591315i	−0.346672	−	0.937987i	$-0.612688\pi$
$463$	12.7236	−	12.7236i	0.591315	−	0.591315i	0.937987	+	0.346672i	$0.112688\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	22.8132	−	22.8132i	1.05567	−	1.05567i	0.0573107	−	0.998356i	$-0.481747\pi$
$467$	22.8132	−	22.8132i	1.05567	−	1.05567i	0.998356	−	0.0573107i	$-0.0182526\pi$
$468$		0			0
$469$		−	2.28844i		−	0.105670i
$470$		0			0
$471$			7.61185i			0.350736i
$472$		0			0
$473$	−14.7637	+	14.7637i	−0.678836	+	0.678836i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.47587	+	1.47587i	−0.0675757	+	0.0675757i
$478$		0			0
$479$	−10.6963			−0.488725			−0.244363	−	0.969684i	$-0.578579\pi$
$479$	−10.6963			−0.488725			−0.244363	+	0.969684i	$0.578579\pi$
$480$		0			0
$481$			35.7840i			1.63161i
$482$		0			0
$483$	−2.37223	+	2.29125i	−0.107940	+	0.104255i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.8814	+	22.8814i	1.03685	+	1.03685i	0.999294	+	0.0375586i	$0.0119581\pi$
$487$	22.8814	+	22.8814i	1.03685	+	1.03685i	0.0375586	+	0.999294i	$0.488042\pi$
$488$		0			0
$489$			24.6179i			1.11326i
$490$		0			0
$491$	−6.17549			−0.278696			−0.139348	−	0.990243i	$-0.544501\pi$
$491$	−6.17549			−0.278696			−0.139348	+	0.990243i	$0.544501\pi$
$492$		0			0
$493$	−3.24045	−	3.24045i	−0.145943	−	0.145943i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.571530	+	0.571530i	−0.0256366	+	0.0256366i
$498$		0			0
$499$			37.0601i			1.65904i	0.558478	+	0.829520i	$0.311386\pi$
$499$			37.0601i			1.65904i	−0.558478	+	0.829520i	$0.688614\pi$
$500$		0			0
$501$	−28.1099			−1.25586
$502$		0			0
$503$	8.02535	+	8.02535i	0.357832	+	0.357832i	0.863013	−	0.505181i	$-0.168574\pi$
$503$	8.02535	+	8.02535i	0.357832	+	0.357832i	−0.505181	+	0.863013i	$0.668574\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.0695519	−	0.0695519i	−0.00308891	−	0.00308891i
$508$		0			0
$509$			28.9970i			1.28527i	0.766172	+	0.642636i	$0.222159\pi$
$509$			28.9970i			1.28527i	−0.766172	+	0.642636i	$0.777841\pi$
$510$		0			0
$511$		−	1.74331i		−	0.0771197i
$512$		0			0
$513$	3.75385	+	3.75385i	0.165737	+	0.165737i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−13.5978	+	13.5978i	−0.598032	+	0.598032i
$518$		0			0
$519$		−	43.7112i		−	1.91871i
$520$		0			0
$521$			2.69881i			0.118237i	0.998251	+	0.0591185i	$0.0188290\pi$
$521$			2.69881i			0.118237i	−0.998251	+	0.0591185i	$0.981171\pi$
$522$		0			0
$523$	0.374908	+	0.374908i	0.0163936	+	0.0163936i	0.715256	−	0.698862i	$-0.246310\pi$
$523$	0.374908	+	0.374908i	0.0163936	+	0.0163936i	−0.698862	+	0.715256i	$0.746310\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.58884	+	8.58884i	−0.374136	+	0.374136i
$528$		0			0
$529$	−0.798520	+	22.9861i	−0.0347183	+	0.999397i
$530$		0			0
$531$	−27.2555			−1.18279
$532$		0			0
$533$	5.98573	−	5.98573i	0.259271	−	0.259271i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	29.7931	+	29.7931i	1.28567	+	1.28567i
$538$		0			0
$539$	24.4717			1.05407
$540$		0			0
$541$	24.9462			1.07252			0.536261	−	0.844052i	$-0.319836\pi$
$541$	24.9462			1.07252			0.536261	+	0.844052i	$0.319836\pi$
$542$		0			0
$543$	38.6590	+	38.6590i	1.65902	+	1.65902i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−20.2872	−	20.2872i	−0.867419	−	0.867419i	0.124767	−	0.992186i	$-0.460182\pi$
$547$	−20.2872	−	20.2872i	−0.867419	−	0.867419i	−0.992186	+	0.124767i	$0.960182\pi$
$548$		0			0
$549$	15.0244			0.641226
$550$		0			0
$551$		−	1.70299i		−	0.0725497i
$552$		0			0
$553$	−2.45225	+	2.45225i	−0.104280	+	0.104280i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.4666	+	23.4666i	−0.994313	+	0.994313i	−0.999984	−	0.00567133i	$-0.998195\pi$
$557$	−23.4666	+	23.4666i	−0.994313	+	0.994313i	0.00567133	+	0.999984i	$0.498195\pi$
$558$		0			0
$559$	21.2806			0.900072
$560$		0			0
$561$	−52.5714			−2.21956
$562$		0			0
$563$	15.3000	+	15.3000i	0.644819	+	0.644819i	0.951736	−	0.306917i	$-0.0992975\pi$
$563$	15.3000	+	15.3000i	0.644819	+	0.644819i	−0.306917	+	0.951736i	$0.599298\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.37499	−	2.37499i	0.0997402	−	0.0997402i
$568$		0			0
$569$	−34.1860			−1.43315			−0.716575	−	0.697510i	$-0.754291\pi$
$569$	−34.1860			−1.43315			−0.716575	+	0.697510i	$0.754291\pi$
$570$		0			0
$571$			43.8726i			1.83601i	0.396570	+	0.918005i	$0.370201\pi$
$571$			43.8726i			1.83601i	−0.396570	+	0.918005i	$0.629799\pi$
$572$		0			0
$573$	−26.1525	−	26.1525i	−1.09254	−	1.09254i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	31.3549	+	31.3549i	1.30532	+	1.30532i	0.924753	+	0.380567i	$0.124271\pi$
$577$	31.3549	+	31.3549i	1.30532	+	1.30532i	0.380567	+	0.924753i	$0.375729\pi$
$578$		0			0
$579$		−	28.9379i		−	1.20262i
$580$		0			0
$581$	−0.607260			−0.0251934
$582$		0			0
$583$	−2.56724	+	2.56724i	−0.106324	+	0.106324i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.83841	−	5.83841i	−0.240977	−	0.240977i	0.576277	−	0.817254i	$-0.304505\pi$
$587$	−5.83841	−	5.83841i	−0.240977	−	0.240977i	−0.817254	+	0.576277i	$0.804505\pi$
$588$		0			0
$589$	−4.51378			−0.185987
$590$		0			0
$591$	39.9961			1.64522
$592$		0			0
$593$	−9.36096	+	9.36096i	−0.384408	+	0.384408i	−0.872688	−	0.488279i	$-0.837625\pi$
$593$	−9.36096	+	9.36096i	−0.384408	+	0.384408i	0.488279	+	0.872688i	$0.337625\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−34.4072	+	34.4072i	−1.40819	+	1.40819i
$598$		0			0
$599$		−	26.1304i		−	1.06766i	−0.845593	−	0.533829i	$-0.820753\pi$
$599$		−	26.1304i		−	1.06766i	0.845593	−	0.533829i	$-0.179247\pi$
$600$		0			0
$601$	11.2017			0.456929			0.228464	−	0.973552i	$-0.426630\pi$
$601$	11.2017			0.456929			0.228464	+	0.973552i	$0.426630\pi$
$602$		0			0
$603$	−10.7580	−	10.7580i	−0.438098	−	0.438098i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.08024	−	4.08024i	−0.165612	−	0.165612i	0.619436	−	0.785047i	$-0.287361\pi$
$607$	−4.08024	−	4.08024i	−0.165612	−	0.165612i	−0.785047	+	0.619436i	$0.787361\pi$
$608$		0			0
$609$	−0.476746			−0.0193187
$610$		0			0
$611$	19.6001			0.792934
$612$		0			0
$613$	9.95093	+	9.95093i	0.401914	+	0.401914i	0.878907	−	0.476993i	$-0.158273\pi$
$613$	9.95093	+	9.95093i	0.401914	+	0.401914i	−0.476993	+	0.878907i	$0.658273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.50185	+	6.50185i	−0.261755	+	0.261755i	−0.825767	−	0.564012i	$-0.809257\pi$
$617$	−6.50185	+	6.50185i	−0.261755	+	0.261755i	0.564012	+	0.825767i	$0.309257\pi$
$618$		0			0
$619$	−15.6346			−0.628408			−0.314204	−	0.949355i	$-0.601738\pi$
$619$	−15.6346			−0.628408			−0.314204	+	0.949355i	$0.601738\pi$
$620$		0			0
$621$	0.179941	−	10.3626i	0.00722077	−	0.415838i
$622$		0			0
$623$	−0.614485	+	0.614485i	−0.0246188	+	0.0246188i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−13.8142	−	13.8142i	−0.551686	−	0.551686i
$628$		0			0
$629$		−	65.4961i		−	2.61150i
$630$		0			0
$631$			32.7046i			1.30195i	0.759099	+	0.650975i	$0.225640\pi$
$631$			32.7046i			1.30195i	−0.759099	+	0.650975i	$0.774360\pi$
$632$		0			0
$633$	13.2008	−	13.2008i	0.524685	−	0.524685i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−17.6369	−	17.6369i	−0.698798	−	0.698798i
$638$		0			0
$639$			5.37354i			0.212574i
$640$		0			0
$641$		−	6.51103i		−	0.257170i	−0.991698	−	0.128585i	$-0.958956\pi$
$641$		−	6.51103i		−	0.257170i	0.991698	−	0.128585i	$-0.0410436\pi$
$642$		0			0
$643$	−11.4636	−	11.4636i	−0.452081	−	0.452081i	0.443964	−	0.896045i	$-0.353572\pi$
$643$	−11.4636	−	11.4636i	−0.452081	−	0.452081i	−0.896045	+	0.443964i	$0.853572\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.24557	+	1.24557i	0.0489684	+	0.0489684i	0.731167	−	0.682199i	$-0.238976\pi$
$647$	1.24557	+	1.24557i	0.0489684	+	0.0489684i	−0.682199	+	0.731167i	$0.738976\pi$
$648$		0			0
$649$	−47.4102			−1.86101
$650$		0			0
$651$			1.26362i			0.0495251i
$652$		0			0
$653$	−10.0403	+	10.0403i	−0.392906	+	0.392906i	−0.875722	−	0.482816i	$-0.839614\pi$
$653$	−10.0403	+	10.0403i	−0.392906	+	0.392906i	0.482816	+	0.875722i	$0.339614\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−8.19534	−	8.19534i	−0.319731	−	0.319731i
$658$		0			0
$659$	31.0673			1.21021			0.605104	−	0.796146i	$-0.293131\pi$
$659$	31.0673			1.21021			0.605104	+	0.796146i	$0.293131\pi$
$660$		0			0
$661$			41.9458i			1.63150i	0.578403	+	0.815751i	$0.303676\pi$
$661$			41.9458i			1.63150i	−0.578403	+	0.815751i	$0.696324\pi$
$662$		0			0
$663$	37.8885	+	37.8885i	1.47147	+	1.47147i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.39139	+	2.30976i	−0.0925951	+	0.0894343i
$668$		0			0
$669$			4.78597i			0.185036i
$670$		0			0
$671$	26.1345			1.00891
$672$		0			0
$673$	21.3214	−	21.3214i	0.821880	−	0.821880i	−0.164498	−	0.986377i	$-0.552600\pi$
$673$	21.3214	−	21.3214i	0.821880	−	0.821880i	0.986377	+	0.164498i	$0.0526003\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.3686	−	10.3686i	0.398497	−	0.398497i	−0.479206	−	0.877703i	$-0.659075\pi$
$677$	10.3686	−	10.3686i	0.398497	−	0.398497i	0.877703	+	0.479206i	$0.159075\pi$
$678$		0			0
$679$		−	1.61419i		−	0.0619471i
$680$		0			0
$681$		−	53.5465i		−	2.05191i
$682$		0			0
$683$	−8.72053	+	8.72053i	−0.333682	+	0.333682i	−0.853983	−	0.520301i	$-0.825820\pi$
$683$	−8.72053	+	8.72053i	−0.333682	+	0.333682i	0.520301	+	0.853983i	$0.325820\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	0.593134	−	0.593134i	0.0226295	−	0.0226295i
$688$		0			0
$689$	3.70045			0.140976
$690$		0			0
$691$	2.59492			0.0987153			0.0493576	−	0.998781i	$-0.484283\pi$
$691$	2.59492			0.0987153			0.0493576	+	0.998781i	$0.484283\pi$
$692$		0			0
$693$	−1.56399	+	1.56399i	−0.0594109	+	0.0594109i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−10.9558	+	10.9558i	−0.414980	+	0.414980i
$698$		0			0
$699$			44.4656i			1.68184i
$700$		0			0
$701$		−	40.3881i		−	1.52544i	−0.646729	−	0.762720i	$-0.723864\pi$
$701$		−	40.3881i		−	1.52544i	0.646729	−	0.762720i	$-0.276136\pi$
$702$		0			0
$703$	17.2104	−	17.2104i	0.649103	−	0.649103i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.101677	+	0.101677i	−0.00382395	+	0.00382395i
$708$		0			0
$709$	29.3495			1.10224			0.551121	−	0.834425i	$-0.314200\pi$
$709$	29.3495			1.10224			0.551121	+	0.834425i	$0.314200\pi$
$710$		0			0
$711$			23.0562i			0.864673i
$712$		0			0
$713$	6.12204	+	6.33841i	0.229272	+	0.237375i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	19.6165	+	19.6165i	0.732591	+	0.732591i
$718$		0			0
$719$			40.4568i			1.50879i	0.656423	+	0.754393i	$0.272069\pi$
$719$			40.4568i			1.50879i	−0.656423	+	0.754393i	$0.727931\pi$
$720$		0			0
$721$	2.49134			0.0927824
$722$		0			0
$723$	26.9543	+	26.9543i	1.00244	+	1.00244i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.42458	−	1.42458i	0.0528349	−	0.0528349i	−0.680196	−	0.733031i	$-0.738105\pi$
$727$	1.42458	−	1.42458i	0.0528349	−	0.0528349i	0.733031	+	0.680196i	$0.238105\pi$
$728$		0			0
$729$		−	7.77904i		−	0.288112i
$730$		0			0
$731$	−38.9502			−1.44063
$732$		0			0
$733$	8.76868	+	8.76868i	0.323879	+	0.323879i	0.850253	−	0.526374i	$-0.176449\pi$
$733$	8.76868	+	8.76868i	0.323879	+	0.323879i	−0.526374	+	0.850253i	$0.676449\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−18.7132	−	18.7132i	−0.689309	−	0.689309i
$738$		0			0
$739$		−	0.285415i		−	0.0104992i	−0.999986	−	0.00524959i	$-0.998329\pi$
$739$		−	0.285415i		−	0.0104992i	0.999986	−	0.00524959i	$-0.00167100\pi$
$740$		0			0
$741$			19.9119i			0.731483i
$742$		0			0
$743$	23.8792	+	23.8792i	0.876043	+	0.876043i	0.993123	−	0.117079i	$-0.0373532\pi$
$743$	23.8792	+	23.8792i	0.876043	+	0.876043i	−0.117079	+	0.993123i	$0.537353\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.85474	+	2.85474i	−0.104449	+	0.104449i
$748$		0			0
$749$			1.15881i			0.0423421i
$750$		0			0
$751$		−	23.3191i		−	0.850925i	−0.904976	−	0.425463i	$-0.860111\pi$
$751$		−	23.3191i		−	0.850925i	0.904976	−	0.425463i	$-0.139889\pi$
$752$		0			0
$753$	−16.4454	−	16.4454i	−0.599305	−	0.599305i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−17.6429	+	17.6429i	−0.641240	+	0.641240i	−0.950860	−	0.309620i	$-0.899798\pi$
$757$	−17.6429	+	17.6429i	−0.641240	+	0.641240i	0.309620	+	0.950860i	$0.399798\pi$
$758$		0			0
$759$	−0.662182	+	38.1345i	−0.0240357	+	1.38420i
$760$		0			0
$761$	−23.5697			−0.854400			−0.427200	−	0.904157i	$-0.640500\pi$
$761$	−23.5697			−0.854400			−0.427200	+	0.904157i	$0.640500\pi$
$762$		0			0
$763$	−1.11825	+	1.11825i	−0.0404832	+	0.0404832i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	34.1688	+	34.1688i	1.23376	+	1.23376i
$768$		0			0
$769$	−52.9706			−1.91017			−0.955085	−	0.296333i	$-0.904236\pi$
$769$	−52.9706			−1.91017			−0.955085	+	0.296333i	$0.904236\pi$
$770$		0			0
$771$	53.1085			1.91266
$772$		0			0
$773$	18.4220	+	18.4220i	0.662594	+	0.662594i	0.955991	−	0.293397i	$-0.0947858\pi$
$773$	18.4220	+	18.4220i	0.662594	+	0.662594i	−0.293397	+	0.955991i	$0.594786\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−4.81800	−	4.81800i	−0.172845	−	0.172845i
$778$		0			0
$779$	−5.75771			−0.206292
$780$		0			0
$781$			9.34712i			0.334466i
$782$		0			0
$783$	1.05937	−	1.05937i	0.0378588	−	0.0378588i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	32.0879	−	32.0879i	1.14381	−	1.14381i	0.156063	−	0.987747i	$-0.450120\pi$
$787$	32.0879	−	32.0879i	1.14381	−	1.14381i	0.987747	−	0.156063i	$-0.0498804\pi$
$788$		0			0
$789$	14.2004			0.505547
$790$		0			0
$791$	4.61074			0.163939
$792$		0			0
$793$	−18.8353	−	18.8353i	−0.668861	−	0.668861i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−29.9128	+	29.9128i	−1.05957	+	1.05957i	−0.0614563	+	0.998110i	$0.519574\pi$
$797$	−29.9128	+	29.9128i	−1.05957	+	1.05957i	−0.998110	+	0.0614563i	$0.980426\pi$
$798$		0			0
$799$	−35.8744			−1.26914
$800$		0			0
$801$			5.77740i			0.204135i
$802$		0			0
$803$	−14.2556	−	14.2556i	−0.503068	−	0.503068i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	32.0637	+	32.0637i	1.12870	+	1.12870i
$808$		0			0
$809$		−	26.8971i		−	0.945652i	−0.881156	−	0.472826i	$-0.843234\pi$
$809$		−	26.8971i		−	0.945652i	0.881156	−	0.472826i	$-0.156766\pi$
$810$		0			0
$811$	−19.1827			−0.673597			−0.336799	−	0.941577i	$-0.609344\pi$
$811$	−19.1827			−0.673597			−0.336799	+	0.941577i	$0.609344\pi$
$812$		0			0
$813$	−24.2123	+	24.2123i	−0.849161	+	0.849161i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.2350	−	10.2350i	−0.358076	−	0.358076i
$818$		0			0
$819$	2.25435			0.0787732
$820$		0			0
$821$	−37.1207			−1.29552			−0.647760	−	0.761844i	$-0.724294\pi$
$821$	−37.1207			−1.29552			−0.647760	+	0.761844i	$0.724294\pi$
$822$		0			0
$823$	−26.1279	+	26.1279i	−0.910762	+	0.910762i	−0.996332	−	0.0855705i	$-0.972729\pi$
$823$	−26.1279	+	26.1279i	−0.910762	+	0.910762i	0.0855705	+	0.996332i	$0.472729\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.28023	+	4.28023i	−0.148838	+	0.148838i	−0.777599	−	0.628761i	$-0.783563\pi$
$827$	−4.28023	+	4.28023i	−0.148838	+	0.148838i	0.628761	+	0.777599i	$0.283563\pi$
$828$		0			0
$829$		−	18.5238i		−	0.643358i	−0.946849	−	0.321679i	$-0.895753\pi$
$829$		−	18.5238i		−	0.643358i	0.946849	−	0.321679i	$-0.104247\pi$
$830$		0			0
$831$	−30.0504			−1.04244
$832$		0			0
$833$	32.2811	+	32.2811i	1.11847	+	1.11847i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2.80787	−	2.80787i	−0.0970541	−	0.0970541i
$838$		0			0
$839$	−25.5518			−0.882147			−0.441074	−	0.897471i	$-0.645402\pi$
$839$	−25.5518			−0.882147			−0.441074	+	0.897471i	$0.645402\pi$
$840$		0			0
$841$	28.5194			0.983428
$842$		0			0
$843$	17.3594	+	17.3594i	0.597890	+	0.597890i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.337183	+	0.337183i	−0.0115857	+	0.0115857i
$848$		0			0
$849$	−31.9374			−1.09609
$850$		0			0
$851$	−47.5099	−	0.824980i	−1.62862	−	0.0282799i
$852$		0			0
$853$	−8.46948	+	8.46948i	−0.289989	+	0.289989i	−0.837076	−	0.547087i	$-0.815737\pi$
$853$	−8.46948	+	8.46948i	−0.289989	+	0.289989i	0.547087	+	0.837076i	$0.315737\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−33.8682	−	33.8682i	−1.15692	−	1.15692i	−0.985135	−	0.171780i	$-0.945048\pi$
$857$	−33.8682	−	33.8682i	−1.15692	−	1.15692i	−0.171780	−	0.985135i	$-0.554952\pi$
$858$		0			0
$859$		−	28.9189i		−	0.986701i	−0.869831	−	0.493350i	$-0.835772\pi$
$859$		−	28.9189i		−	0.986701i	0.869831	−	0.493350i	$-0.164228\pi$
$860$		0			0
$861$			1.61185i			0.0549318i
$862$		0			0
$863$	26.2769	−	26.2769i	0.894476	−	0.894476i	−0.100464	−	0.994941i	$-0.532033\pi$
$863$	26.2769	−	26.2769i	0.894476	−	0.894476i	0.994941	+	0.100464i	$0.0320328\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−42.3691	−	42.3691i	−1.43893	−	1.43893i
$868$		0			0
$869$			40.1055i			1.36049i
$870$		0			0
$871$			26.9734i			0.913958i
$872$		0			0
$873$	−7.58834	−	7.58834i	−0.256827	−	0.256827i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.62273	−	5.62273i	−0.189866	−	0.189866i	0.605772	−	0.795638i	$-0.292864\pi$
$877$	−5.62273	−	5.62273i	−0.189866	−	0.189866i	−0.795638	+	0.605772i	$0.792864\pi$
$878$		0			0
$879$	20.8616			0.703645
$880$		0			0
$881$		−	1.77465i		−	0.0597895i	−0.999553	−	0.0298948i	$-0.990483\pi$
$881$		−	1.77465i		−	0.0597895i	0.999553	−	0.0298948i	$-0.00951721\pi$
$882$		0			0
$883$	−14.1726	+	14.1726i	−0.476945	+	0.476945i	−0.904153	−	0.427208i	$-0.859497\pi$
$883$	−14.1726	+	14.1726i	−0.476945	+	0.476945i	0.427208	+	0.904153i	$0.359497\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.1647	+	13.1647i	0.442027	+	0.442027i	0.892693	−	0.450666i	$-0.148813\pi$
$887$	13.1647	+	13.1647i	0.442027	+	0.442027i	−0.450666	+	0.892693i	$0.648813\pi$
$888$		0			0
$889$	3.71432			0.124574
$890$		0			0
$891$		−	38.8419i		−	1.30125i
$892$		0			0
$893$	−9.42672	−	9.42672i	−0.315453	−	0.315453i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	27.9610	−	27.0065i	0.933591	−	0.901722i
$898$		0			0
$899$			1.27383i			0.0424846i
$900$		0			0
$901$	−6.77301			−0.225642
$902$		0			0
$903$	−2.86524	+	2.86524i	−0.0953493	+	0.0953493i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	24.5641	−	24.5641i	0.815638	−	0.815638i	−0.169834	−	0.985473i	$-0.554323\pi$
$907$	24.5641	−	24.5641i	0.815638	−	0.815638i	0.985473	+	0.169834i	$0.0543233\pi$
$908$		0			0
$909$			0.955968i			0.0317074i
$910$		0			0
$911$		−	55.8519i		−	1.85046i	−0.379411	−	0.925228i	$-0.623873\pi$
$911$		−	55.8519i		−	1.85046i	0.379411	−	0.925228i	$-0.376127\pi$
$912$		0			0
$913$	−4.96574	+	4.96574i	−0.164342	+	0.164342i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.920822	+	0.920822i	−0.0304082	+	0.0304082i
$918$		0			0
$919$	−49.0592			−1.61831			−0.809157	−	0.587592i	$-0.800076\pi$
$919$	−49.0592			−1.61831			−0.809157	+	0.587592i	$0.800076\pi$
$920$		0			0
$921$	71.7691			2.36487
$922$		0			0
$923$	6.73652	−	6.73652i	0.221735	−	0.221735i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	11.7118	−	11.7118i	0.384667	−	0.384667i
$928$		0			0
$929$		−	12.1164i		−	0.397526i	−0.980048	−	0.198763i	$-0.936308\pi$
$929$		−	12.1164i		−	0.397526i	0.980048	−	0.198763i	$-0.0636923\pi$
$930$		0			0
$931$			16.9650i			0.556006i
$932$		0			0
$933$	26.8590	−	26.8590i	0.879323	−	0.879323i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−25.2751	+	25.2751i	−0.825700	+	0.825700i	−0.986919	−	0.161219i	$-0.948458\pi$
$937$	−25.2751	+	25.2751i	−0.825700	+	0.825700i	0.161219	+	0.986919i	$0.448458\pi$
$938$		0			0
$939$	−32.3112			−1.05444
$940$		0			0
$941$		−	22.2968i		−	0.726855i	−0.931623	−	0.363427i	$-0.881606\pi$
$941$		−	22.2968i		−	0.726855i	0.931623	−	0.363427i	$-0.118394\pi$
$942$		0			0
$943$	7.80918	+	8.08517i	0.254302	+	0.263289i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	34.4481	+	34.4481i	1.11941	+	1.11941i	0.991828	+	0.127584i	$0.0407221\pi$
$947$	34.4481	+	34.4481i	1.11941	+	1.11941i	0.127584	+	0.991828i	$0.459278\pi$
$948$		0			0
$949$			20.5481i			0.667021i
$950$		0			0
$951$	13.3870			0.434104
$952$		0			0
$953$	−18.5954	−	18.5954i	−0.602365	−	0.602365i	0.338575	−	0.940940i	$-0.390055\pi$
$953$	−18.5954	−	18.5954i	−0.602365	−	0.602365i	−0.940940	+	0.338575i	$0.890055\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−3.89849	+	3.89849i	−0.126020	+	0.126020i
$958$		0			0
$959$		−	3.89795i		−	0.125871i
$960$		0			0
$961$	−27.6237			−0.891087
$962$		0			0
$963$	5.44760	+	5.44760i	0.175546	+	0.175546i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−16.0209	−	16.0209i	−0.515197	−	0.515197i	0.400917	−	0.916114i	$-0.368691\pi$
$967$	−16.0209	−	16.0209i	−0.515197	−	0.515197i	−0.916114	+	0.400917i	$0.868691\pi$
$968$		0			0
$969$		−	36.4452i		−	1.17079i
$970$		0			0
$971$			29.9293i			0.960476i	0.877138	+	0.480238i	$0.159450\pi$
$971$			29.9293i			0.960476i	−0.877138	+	0.480238i	$0.840550\pi$
$972$		0			0
$973$	−0.295134	−	0.295134i	−0.00946156	−	0.00946156i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	37.4894	−	37.4894i	1.19939	−	1.19939i	0.225045	−	0.974348i	$-0.427747\pi$
$977$	37.4894	−	37.4894i	1.19939	−	1.19939i	0.974348	−	0.225045i	$-0.0722530\pi$
$978$		0			0
$979$			10.0496i			0.321188i
$980$		0			0
$981$			10.5138i			0.335679i
$982$		0			0
$983$	−25.2157	−	25.2157i	−0.804257	−	0.804257i	0.179501	−	0.983758i	$-0.442552\pi$
$983$	−25.2157	−	25.2157i	−0.804257	−	0.804257i	−0.983758	+	0.179501i	$0.942552\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2.63898	+	2.63898i	−0.0839996	+	0.0839996i
$988$		0			0
$989$	−0.490612	+	28.2539i	−0.0156005	+	0.898423i
$990$		0			0
$991$	33.0634			1.05029			0.525147	−	0.851012i	$-0.324010\pi$
$991$	33.0634			1.05029			0.525147	+	0.851012i	$0.324010\pi$
$992$		0			0
$993$	19.6761	−	19.6761i	0.624403	−	0.624403i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	35.4366	+	35.4366i	1.12229	+	1.12229i	0.991397	+	0.130893i	$0.0417843\pi$
$997$	35.4366	+	35.4366i	1.12229	+	1.12229i	0.130893	+	0.991397i	$0.458216\pi$
$998$		0			0
$999$	21.4120			0.677446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.i.f.1057.7	yes	16
5.2	odd	4		2300.2.i.e.1793.2	yes	16
5.3	odd	4		2300.2.i.e.1793.7	yes	16
5.4	even	2	inner	2300.2.i.f.1057.2	yes	16
23.22	odd	2		2300.2.i.e.1057.7	yes	16
115.22	even	4	inner	2300.2.i.f.1793.2	yes	16
115.68	even	4	inner	2300.2.i.f.1793.7	yes	16
115.114	odd	2		2300.2.i.e.1057.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.i.e.1057.2	✓	16	115.114	odd	2
2300.2.i.e.1057.7	yes	16	23.22	odd	2
2300.2.i.e.1793.2	yes	16	5.2	odd	4
2300.2.i.e.1793.7	yes	16	5.3	odd	4
2300.2.i.f.1057.2	yes	16	5.4	even	2	inner
2300.2.i.f.1057.7	yes	16	1.1	even	1	trivial
2300.2.i.f.1793.2	yes	16	115.22	even	4	inner
2300.2.i.f.1793.7	yes	16	115.68	even	4	inner

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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.58699 - 1.58699i) q^{3} +(0.216666 - 0.216666i) q^{7} -2.03710i q^{9} +O(q^{10})\)	\(q+(1.58699 - 1.58699i) q^{3} +(0.216666 - 0.216666i) q^{7} -2.03710i q^{9} -3.54348i q^{11} +(-2.55380 + 2.55380i) q^{13} +(4.67428 - 4.67428i) q^{17} +2.45652 q^{19} -0.687695i q^{21} +(-3.33178 - 3.44953i) q^{23} +(1.52812 + 1.52812i) q^{27} -0.693252i q^{29} -1.83747 q^{31} +(-5.62348 - 5.62348i) q^{33} +(7.00601 - 7.00601i) q^{37} +8.10574i q^{39} -2.34385 q^{41} +(-4.16644 - 4.16644i) q^{43} +(-3.83743 - 3.83743i) q^{47} +6.90611i q^{49} -14.8361i q^{51} +(-0.724498 - 0.724498i) q^{53} +(3.89849 - 3.89849i) q^{57} -13.3796i q^{59} +7.37539i q^{61} +(-0.441370 - 0.441370i) q^{63} +(5.28102 - 5.28102i) q^{67} +(-10.7619 - 0.186874i) q^{69} -2.63784 q^{71} +(4.02304 - 4.02304i) q^{73} +(-0.767751 - 0.767751i) q^{77} -11.3181 q^{79} +10.9615 q^{81} +(-1.40137 - 1.40137i) q^{83} +(-1.10019 - 1.10019i) q^{87} -2.83609 q^{89} +1.10664i q^{91} +(-2.91605 + 2.91605i) q^{93} +(3.72507 - 3.72507i) q^{97} -7.21842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 68 q^{19} - 4 q^{41} - 56 q^{69} - 8 q^{71} - 28 q^{79} + 64 q^{81} + 120 q^{89} - 28 q^{99}+O(q^{100}) \) 16 * q + 68 * q^19 - 4 * q^41 - 56 * q^69 - 8 * q^71 - 28 * q^79 + 64 * q^81 + 120 * q^89 - 28 * q^99

Introduction

L-functions

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