LMFDB - Embedded newform 2300.2.c.h.1749.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.2
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.2.c.h.1749.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +O(q^{10})$	$q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +2.00000 q^{11} -3.56155i q^{13} -2.56155i q^{17} -6.00000 q^{19} -4.00000 q^{21} +1.00000i q^{23} -5.56155i q^{27} -6.12311 q^{29} +7.24621 q^{31} -3.12311i q^{33} +4.56155i q^{37} -5.56155 q^{39} +4.12311 q^{41} -4.68466i q^{47} +0.438447 q^{49} -4.00000 q^{51} -4.56155i q^{53} +9.36932i q^{57} +3.68466 q^{59} -7.12311 q^{61} -1.43845i q^{63} +8.56155i q^{67} +1.56155 q^{69} +10.1231 q^{71} -4.43845i q^{73} -5.12311i q^{77} -4.87689 q^{79} -7.00000 q^{81} -13.9309i q^{83} +9.56155i q^{87} -14.2462 q^{89} -9.12311 q^{91} -11.3153i q^{93} +13.1231i q^{97} +1.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} + 8 q^{11} - 24 q^{19} - 16 q^{21} - 8 q^{29} - 4 q^{31} - 14 q^{39} + 10 q^{49} - 16 q^{51} - 10 q^{59} - 12 q^{61} - 2 q^{69} + 24 q^{71} - 36 q^{79} - 28 q^{81} - 24 q^{89} - 20 q^{91} - 12 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 + 8 * q^11 - 24 * q^19 - 16 * q^21 - 8 * q^29 - 4 * q^31 - 14 * q^39 + 10 * q^49 - 16 * q^51 - 10 * q^59 - 12 * q^61 - 2 * q^69 + 24 * q^71 - 36 * q^79 - 28 * q^81 - 24 * q^89 - 20 * q^91 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

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

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

$2$	0	0
$2$	0	0
$3$	− 1.56155i	− 0.901563i	−0.892634	−	0.450781i	$-0.851145\pi$
$3$	− 1.56155i	− 0.901563i	0.892634	−	0.450781i	$-0.148855\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.56155i	− 0.968176i	−0.875019	−	0.484088i	$-0.839151\pi$
$7$	− 2.56155i	− 0.968176i	0.875019	−	0.484088i	$-0.160849\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	− 3.56155i	− 0.987797i	−0.869520	−	0.493899i	$-0.835571\pi$
$13$	− 3.56155i	− 0.987797i	0.869520	−	0.493899i	$-0.164429\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.56155i	− 0.621268i	−0.950530	−	0.310634i	$-0.899459\pi$
$17$	− 2.56155i	− 0.621268i	0.950530	−	0.310634i	$-0.100541\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.56155i	− 1.07032i
$28$	0	0
$29$	−6.12311	−1.13703	−0.568516	−	0.822672i	$-0.692482\pi$
$29$	−6.12311	−1.13703	−0.568516	+	0.822672i	$0.692482\pi$
$30$	0	0
$31$	7.24621	1.30146	0.650729	−	0.759310i	$-0.274463\pi$
$31$	7.24621	1.30146	0.650729	+	0.759310i	$0.274463\pi$
$32$	0	0
$33$	− 3.12311i	− 0.543663i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.56155i	0.749915i	0.927042	+	0.374957i	$0.122343\pi$
$37$	4.56155i	0.749915i	−0.927042	+	0.374957i	$0.877657\pi$
$38$	0	0
$39$	−5.56155	−0.890561
$40$	0	0
$41$	4.12311	0.643921	0.321960	−	0.946753i	$-0.395658\pi$
$41$	4.12311	0.643921	0.321960	+	0.946753i	$0.395658\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.68466i	− 0.683328i	−0.939822	−	0.341664i	$-0.889010\pi$
$47$	− 4.68466i	− 0.683328i	0.939822	−	0.341664i	$-0.110990\pi$
$48$	0	0
$49$	0.438447	0.0626353
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	− 4.56155i	− 0.626577i	−0.949658	−	0.313289i	$-0.898569\pi$
$53$	− 4.56155i	− 0.626577i	0.949658	−	0.313289i	$-0.101431\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.36932i	1.24100i
$58$	0	0
$59$	3.68466	0.479702	0.239851	−	0.970810i	$-0.422901\pi$
$59$	3.68466	0.479702	0.239851	+	0.970810i	$0.422901\pi$
$60$	0	0
$61$	−7.12311	−0.912020	−0.456010	−	0.889975i	$-0.650722\pi$
$61$	−7.12311	−0.912020	−0.456010	+	0.889975i	$0.650722\pi$
$62$	0	0
$63$	− 1.43845i	− 0.181227i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.56155i	1.04596i	0.852345	+	0.522980i	$0.175180\pi$
$67$	8.56155i	1.04596i	−0.852345	+	0.522980i	$0.824820\pi$
$68$	0	0
$69$	1.56155	0.187989
$70$	0	0
$71$	10.1231	1.20139	0.600696	−	0.799478i	$-0.294890\pi$
$71$	10.1231	1.20139	0.600696	+	0.799478i	$0.294890\pi$
$72$	0	0
$73$	− 4.43845i	− 0.519481i	−0.965678	−	0.259740i	$-0.916363\pi$
$73$	− 4.43845i	− 0.519481i	0.965678	−	0.259740i	$-0.0836370\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.12311i	− 0.583832i
$78$	0	0
$79$	−4.87689	−0.548693	−0.274347	−	0.961631i	$-0.588462\pi$
$79$	−4.87689	−0.548693	−0.274347	+	0.961631i	$0.588462\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 13.9309i	− 1.52911i	−0.644558	−	0.764556i	$-0.722958\pi$
$83$	− 13.9309i	− 1.52911i	0.644558	−	0.764556i	$-0.277042\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.56155i	1.02511i
$88$	0	0
$89$	−14.2462	−1.51010	−0.755048	−	0.655670i	$-0.772386\pi$
$89$	−14.2462	−1.51010	−0.755048	+	0.655670i	$0.772386\pi$
$90$	0	0
$91$	−9.12311	−0.956361
$92$	0	0
$93$	− 11.3153i	− 1.17335i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.1231i	1.33245i	0.745751	+	0.666225i	$0.232091\pi$
$97$	13.1231i	1.33245i	−0.745751	+	0.666225i	$0.767909\pi$
$98$	0	0
$99$	1.12311	0.112876
$100$	0	0
$101$	−17.6847	−1.75969	−0.879845	−	0.475261i	$-0.842353\pi$
$101$	−17.6847	−1.75969	−0.879845	+	0.475261i	$0.842353\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.43845i	− 0.332407i	−0.986091	−	0.166204i	$-0.946849\pi$
$107$	− 3.43845i	− 0.332407i	0.986091	−	0.166204i	$-0.0531509\pi$
$108$	0	0
$109$	−15.3693	−1.47211	−0.736057	−	0.676920i	$-0.763314\pi$
$109$	−15.3693	−1.47211	−0.736057	+	0.676920i	$0.763314\pi$
$110$	0	0
$111$	7.12311	0.676095
$112$	0	0
$113$	12.8078i	1.20485i	0.798174	+	0.602427i	$0.205799\pi$
$113$	12.8078i	1.20485i	−0.798174	+	0.602427i	$0.794201\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−6.56155	−0.601497
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 6.43845i	− 0.580535i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.80776i	− 0.692827i	−0.938082	−	0.346414i	$-0.887399\pi$
$127$	− 7.80776i	− 0.692827i	0.938082	−	0.346414i	$-0.112601\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.192236	0.0167957	0.00839787	−	0.999965i	$-0.497327\pi$
$131$	0.192236	0.0167957	0.00839787	+	0.999965i	$0.497327\pi$
$132$	0	0
$133$	15.3693i	1.33269i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.87689i	− 0.587533i	−0.955877	−	0.293766i	$-0.905091\pi$
$137$	− 6.87689i	− 0.587533i	0.955877	−	0.293766i	$-0.0949088\pi$
$138$	0	0
$139$	11.2462	0.953891	0.476946	−	0.878933i	$-0.341744\pi$
$139$	11.2462	0.953891	0.476946	+	0.878933i	$0.341744\pi$
$140$	0	0
$141$	−7.31534	−0.616063
$142$	0	0
$143$	− 7.12311i	− 0.595664i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 0.684658i	− 0.0564697i
$148$	0	0
$149$	9.36932	0.767564	0.383782	−	0.923424i	$-0.374621\pi$
$149$	9.36932	0.767564	0.383782	+	0.923424i	$0.374621\pi$
$150$	0	0
$151$	−14.0540	−1.14370	−0.571848	−	0.820359i	$-0.693773\pi$
$151$	−14.0540	−1.14370	−0.571848	+	0.820359i	$0.693773\pi$
$152$	0	0
$153$	− 1.43845i	− 0.116292i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.31534i	0.344402i	0.985062	+	0.172201i	$0.0550878\pi$
$157$	4.31534i	0.344402i	−0.985062	+	0.172201i	$0.944912\pi$
$158$	0	0
$159$	−7.12311	−0.564899
$160$	0	0
$161$	2.56155	0.201879
$162$	0	0
$163$	16.9309i	1.32613i	0.748563	+	0.663064i	$0.230744\pi$
$163$	16.9309i	1.32613i	−0.748563	+	0.663064i	$0.769256\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	0.315342	0.0242570
$170$	0	0
$171$	−3.36932	−0.257658
$172$	0	0
$173$	5.36932i	0.408222i	0.978948	+	0.204111i	$0.0654303\pi$
$173$	5.36932i	0.408222i	−0.978948	+	0.204111i	$0.934570\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 5.75379i	− 0.432481i
$178$	0	0
$179$	6.93087	0.518038	0.259019	−	0.965872i	$-0.416601\pi$
$179$	6.93087	0.518038	0.259019	+	0.965872i	$0.416601\pi$
$180$	0	0
$181$	−5.36932	−0.399098	−0.199549	−	0.979888i	$-0.563948\pi$
$181$	−5.36932	−0.399098	−0.199549	+	0.979888i	$0.563948\pi$
$182$	0	0
$183$	11.1231i	0.822244i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.12311i	− 0.374639i
$188$	0	0
$189$	−14.2462	−1.03626
$190$	0	0
$191$	17.3693	1.25680	0.628400	−	0.777891i	$-0.283710\pi$
$191$	17.3693	1.25680	0.628400	+	0.777891i	$0.283710\pi$
$192$	0	0
$193$	2.43845i	0.175523i	0.996142	+	0.0877616i	$0.0279714\pi$
$193$	2.43845i	0.175523i	−0.996142	+	0.0877616i	$0.972029\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.6847i	1.33123i	0.746297	+	0.665613i	$0.231830\pi$
$197$	18.6847i	1.33123i	−0.746297	+	0.665613i	$0.768170\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	13.3693	0.942999
$202$	0	0
$203$	15.6847i	1.10085i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.561553i	0.0390306i
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	0.315342	0.0217090	0.0108545	−	0.999941i	$-0.496545\pi$
$211$	0.315342	0.0217090	0.0108545	+	0.999941i	$0.496545\pi$
$212$	0	0
$213$	− 15.8078i	− 1.08313i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 18.5616i	− 1.26004i
$218$	0	0
$219$	−6.93087	−0.468345
$220$	0	0
$221$	−9.12311	−0.613686
$222$	0	0
$223$	− 17.6155i	− 1.17962i	−0.807541	−	0.589812i	$-0.799202\pi$
$223$	− 17.6155i	− 1.17962i	0.807541	−	0.589812i	$-0.200798\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 26.2462i	− 1.74202i	−0.491263	−	0.871011i	$-0.663465\pi$
$227$	− 26.2462i	− 1.74202i	0.491263	−	0.871011i	$-0.336535\pi$
$228$	0	0
$229$	26.7386	1.76694	0.883469	−	0.468489i	$-0.155201\pi$
$229$	26.7386	1.76694	0.883469	+	0.468489i	$0.155201\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	− 0.684658i	− 0.0448535i	−0.999748	−	0.0224267i	$-0.992861\pi$
$233$	− 0.684658i	− 0.0448535i	0.999748	−	0.0224267i	$-0.00713925\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.61553i	0.494682i
$238$	0	0
$239$	−2.75379	−0.178128	−0.0890639	−	0.996026i	$-0.528388\pi$
$239$	−2.75379	−0.178128	−0.0890639	+	0.996026i	$0.528388\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	− 5.75379i	− 0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21.3693i	1.35970i
$248$	0	0
$249$	−21.7538	−1.37859
$250$	0	0
$251$	−7.36932	−0.465147	−0.232574	−	0.972579i	$-0.574715\pi$
$251$	−7.36932	−0.465147	−0.232574	+	0.972579i	$0.574715\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 31.8078i	− 1.98411i	−0.125790	−	0.992057i	$-0.540147\pi$
$257$	− 31.8078i	− 1.98411i	0.125790	−	0.992057i	$-0.459853\pi$
$258$	0	0
$259$	11.6847	0.726049
$260$	0	0
$261$	−3.43845	−0.212835
$262$	0	0
$263$	0.0691303i	0.00426276i	0.999998	+	0.00213138i	$0.000678439\pi$
$263$	0.0691303i	0.00426276i	−0.999998	+	0.00213138i	$0.999322\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	22.2462i	1.36145i
$268$	0	0
$269$	−23.2462	−1.41735	−0.708673	−	0.705537i	$-0.750706\pi$
$269$	−23.2462	−1.41735	−0.708673	+	0.705537i	$0.750706\pi$
$270$	0	0
$271$	21.9309	1.33221	0.666103	−	0.745860i	$-0.267961\pi$
$271$	21.9309	1.33221	0.666103	+	0.745860i	$0.267961\pi$
$272$	0	0
$273$	14.2462i	0.862220i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.68466i	0.401642i	0.979628	+	0.200821i	$0.0643610\pi$
$277$	6.68466i	0.401642i	−0.979628	+	0.200821i	$0.935639\pi$
$278$	0	0
$279$	4.06913	0.243612
$280$	0	0
$281$	−6.87689	−0.410241	−0.205121	−	0.978737i	$-0.565759\pi$
$281$	−6.87689	−0.410241	−0.205121	+	0.978737i	$0.565759\pi$
$282$	0	0
$283$	14.8078i	0.880230i	0.897941	+	0.440115i	$0.145062\pi$
$283$	14.8078i	0.880230i	−0.897941	+	0.440115i	$0.854938\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 10.5616i	− 0.623429i
$288$	0	0
$289$	10.4384	0.614026
$290$	0	0
$291$	20.4924	1.20129
$292$	0	0
$293$	− 16.8078i	− 0.981920i	−0.871182	−	0.490960i	$-0.836646\pi$
$293$	− 16.8078i	− 0.981920i	0.871182	−	0.490960i	$-0.163354\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 11.1231i	− 0.645428i
$298$	0	0
$299$	3.56155	0.205970
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.6155i	1.58647i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	21.1231i	1.20556i	0.797908	+	0.602780i	$0.205940\pi$
$307$	21.1231i	1.20556i	−0.797908	+	0.602780i	$0.794060\pi$
$308$	0	0
$309$	−24.9848	−1.42134
$310$	0	0
$311$	8.68466	0.492462	0.246231	−	0.969211i	$-0.420808\pi$
$311$	8.68466	0.492462	0.246231	+	0.969211i	$0.420808\pi$
$312$	0	0
$313$	− 0.807764i	− 0.0456575i	−0.999739	−	0.0228288i	$-0.992733\pi$
$313$	− 0.807764i	− 0.0456575i	0.999739	−	0.0228288i	$-0.00726725\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.1231i	− 0.849398i	−0.905335	−	0.424699i	$-0.860380\pi$
$317$	− 15.1231i	− 0.849398i	0.905335	−	0.424699i	$-0.139620\pi$
$318$	0	0
$319$	−12.2462	−0.685656
$320$	0	0
$321$	−5.36932	−0.299686
$322$	0	0
$323$	15.3693i	0.855172i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	24.0000i	1.32720i
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	28.6155	1.57285	0.786426	−	0.617685i	$-0.211929\pi$
$331$	28.6155	1.57285	0.786426	+	0.617685i	$0.211929\pi$
$332$	0	0
$333$	2.56155i	0.140372i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.6155i	− 0.741685i	−0.928696	−	0.370843i	$-0.879069\pi$
$337$	− 13.6155i	− 0.741685i	0.928696	−	0.370843i	$-0.120931\pi$
$338$	0	0
$339$	20.0000	1.08625
$340$	0	0
$341$	14.4924	0.784809
$342$	0	0
$343$	− 19.0540i	− 1.02882i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.4924i	− 0.885360i	−0.896680	−	0.442680i	$-0.854028\pi$
$347$	− 16.4924i	− 0.885360i	0.896680	−	0.442680i	$-0.145972\pi$
$348$	0	0
$349$	26.3693	1.41152	0.705759	−	0.708452i	$-0.250606\pi$
$349$	26.3693	1.41152	0.705759	+	0.708452i	$0.250606\pi$
$350$	0	0
$351$	−19.8078	−1.05726
$352$	0	0
$353$	− 2.05398i	− 0.109322i	−0.998505	−	0.0546610i	$-0.982592\pi$
$353$	− 2.05398i	− 0.109322i	0.998505	−	0.0546610i	$-0.0174078\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.2462i	0.542287i
$358$	0	0
$359$	−15.6155	−0.824156	−0.412078	−	0.911149i	$-0.635197\pi$
$359$	−15.6155	−0.824156	−0.412078	+	0.911149i	$0.635197\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	10.9309i	0.573722i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.5616i	1.17770i	0.808241	+	0.588852i	$0.200420\pi$
$367$	22.5616i	1.17770i	−0.808241	+	0.588852i	$0.799580\pi$
$368$	0	0
$369$	2.31534	0.120532
$370$	0	0
$371$	−11.6847	−0.606637
$372$	0	0
$373$	− 20.2462i	− 1.04831i	−0.851623	−	0.524155i	$-0.824381\pi$
$373$	− 20.2462i	− 1.04831i	0.851623	−	0.524155i	$-0.175619\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21.8078i	1.12316i
$378$	0	0
$379$	36.4924	1.87449	0.937245	−	0.348672i	$-0.113367\pi$
$379$	36.4924	1.87449	0.937245	+	0.348672i	$0.113367\pi$
$380$	0	0
$381$	−12.1922	−0.624627
$382$	0	0
$383$	3.19224i	0.163116i	0.996669	+	0.0815578i	$0.0259895\pi$
$383$	3.19224i	0.163116i	−0.996669	+	0.0815578i	$0.974010\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.8769	0.551480	0.275740	−	0.961232i	$-0.411077\pi$
$389$	10.8769	0.551480	0.275740	+	0.961232i	$0.411077\pi$
$390$	0	0
$391$	2.56155	0.129543
$392$	0	0
$393$	− 0.300187i	− 0.0151424i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.5464i	− 1.73383i	−0.498453	−	0.866917i	$-0.666098\pi$
$397$	− 34.5464i	− 1.73383i	0.498453	−	0.866917i	$-0.333902\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	− 25.8078i	− 1.28558i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.12311i	0.452216i
$408$	0	0
$409$	35.9848	1.77934	0.889668	−	0.456608i	$-0.150936\pi$
$409$	35.9848	1.77934	0.889668	+	0.456608i	$0.150936\pi$
$410$	0	0
$411$	−10.7386	−0.529698
$412$	0	0
$413$	− 9.43845i	− 0.464436i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 17.5616i	− 0.859993i
$418$	0	0
$419$	18.8769	0.922197	0.461098	−	0.887349i	$-0.347456\pi$
$419$	18.8769	0.922197	0.461098	+	0.887349i	$0.347456\pi$
$420$	0	0
$421$	25.1231	1.22443	0.612213	−	0.790693i	$-0.290280\pi$
$421$	25.1231	1.22443	0.612213	+	0.790693i	$0.290280\pi$
$422$	0	0
$423$	− 2.63068i	− 0.127908i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.2462i	0.882996i
$428$	0	0
$429$	−11.1231	−0.537029
$430$	0	0
$431$	−10.2462	−0.493543	−0.246771	−	0.969074i	$-0.579370\pi$
$431$	−10.2462	−0.493543	−0.246771	+	0.969074i	$0.579370\pi$
$432$	0	0
$433$	39.9309i	1.91896i	0.281785	+	0.959478i	$0.409074\pi$
$433$	39.9309i	1.91896i	−0.281785	+	0.959478i	$0.590926\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	−19.8078	−0.945373	−0.472686	−	0.881231i	$-0.656716\pi$
$439$	−19.8078	−0.945373	−0.472686	+	0.881231i	$0.656716\pi$
$440$	0	0
$441$	0.246211	0.0117243
$442$	0	0
$443$	− 11.8078i	− 0.561004i	−0.959853	−	0.280502i	$-0.909499\pi$
$443$	− 11.8078i	− 0.561004i	0.959853	−	0.280502i	$-0.0905009\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 14.6307i	− 0.692008i
$448$	0	0
$449$	21.6847	1.02336	0.511681	−	0.859175i	$-0.329023\pi$
$449$	21.6847	1.02336	0.511681	+	0.859175i	$0.329023\pi$
$450$	0	0
$451$	8.24621	0.388299
$452$	0	0
$453$	21.9460i	1.03111i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9.43845i	− 0.441512i	−0.975329	−	0.220756i	$-0.929148\pi$
$457$	− 9.43845i	− 0.441512i	0.975329	−	0.220756i	$-0.0708524\pi$
$458$	0	0
$459$	−14.2462	−0.664956
$460$	0	0
$461$	−40.9309	−1.90634	−0.953170	−	0.302434i	$-0.902201\pi$
$461$	−40.9309	−1.90634	−0.953170	+	0.302434i	$0.902201\pi$
$462$	0	0
$463$	− 31.3693i	− 1.45786i	−0.684590	−	0.728928i	$-0.740019\pi$
$463$	− 31.3693i	− 1.45786i	0.684590	−	0.728928i	$-0.259981\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.3153i	1.03263i	0.856398	+	0.516315i	$0.172697\pi$
$467$	22.3153i	1.03263i	−0.856398	+	0.516315i	$0.827303\pi$
$468$	0	0
$469$	21.9309	1.01267
$470$	0	0
$471$	6.73863	0.310500
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.56155i	− 0.117285i
$478$	0	0
$479$	43.2311	1.97528	0.987639	−	0.156748i	$-0.0501009\pi$
$479$	43.2311	1.97528	0.987639	+	0.156748i	$0.0501009\pi$
$480$	0	0
$481$	16.2462	0.740763
$482$	0	0
$483$	− 4.00000i	− 0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7.80776i	− 0.353804i	−0.984229	−	0.176902i	$-0.943393\pi$
$487$	− 7.80776i	− 0.353804i	0.984229	−	0.176902i	$-0.0566075\pi$
$488$	0	0
$489$	26.4384	1.19559
$490$	0	0
$491$	11.4924	0.518646	0.259323	−	0.965791i	$-0.416501\pi$
$491$	11.4924	0.518646	0.259323	+	0.965791i	$0.416501\pi$
$492$	0	0
$493$	15.6847i	0.706401i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 25.9309i	− 1.16316i
$498$	0	0
$499$	−22.6155	−1.01241	−0.506205	−	0.862413i	$-0.668952\pi$
$499$	−22.6155	−1.01241	−0.506205	+	0.862413i	$0.668952\pi$
$500$	0	0
$501$	12.4924	0.558120
$502$	0	0
$503$	− 3.93087i	− 0.175269i	−0.996153	−	0.0876344i	$-0.972069\pi$
$503$	− 3.93087i	− 0.175269i	0.996153	−	0.0876344i	$-0.0279307\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.492423i	− 0.0218693i
$508$	0	0
$509$	27.1771	1.20460	0.602301	−	0.798269i	$-0.294251\pi$
$509$	27.1771	1.20460	0.602301	+	0.798269i	$0.294251\pi$
$510$	0	0
$511$	−11.3693	−0.502949
$512$	0	0
$513$	33.3693i	1.47329i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.36932i	− 0.412062i
$518$	0	0
$519$	8.38447	0.368037
$520$	0	0
$521$	−13.6155	−0.596507	−0.298254	−	0.954487i	$-0.596404\pi$
$521$	−13.6155	−0.596507	−0.298254	+	0.954487i	$0.596404\pi$
$522$	0	0
$523$	− 14.7386i	− 0.644475i	−0.946659	−	0.322238i	$-0.895565\pi$
$523$	− 14.7386i	− 0.644475i	0.946659	−	0.322238i	$-0.104435\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 18.5616i	− 0.808554i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	2.06913	0.0897926
$532$	0	0
$533$	− 14.6847i	− 0.636063i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 10.8229i	− 0.467043i
$538$	0	0
$539$	0.876894	0.0377705
$540$	0	0
$541$	−2.68466	−0.115422	−0.0577112	−	0.998333i	$-0.518380\pi$
$541$	−2.68466	−0.115422	−0.0577112	+	0.998333i	$0.518380\pi$
$542$	0	0
$543$	8.38447i	0.359812i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.1771i	0.905467i	0.891646	+	0.452733i	$0.149551\pi$
$547$	21.1771i	0.905467i	−0.891646	+	0.452733i	$0.850449\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	36.7386	1.56512
$552$	0	0
$553$	12.4924i	0.531232i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.6847i	0.664580i	0.943177	+	0.332290i	$0.107821\pi$
$557$	15.6847i	0.664580i	−0.943177	+	0.332290i	$0.892179\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	32.6695i	1.37686i	0.725305	+	0.688428i	$0.241699\pi$
$563$	32.6695i	1.37686i	−0.725305	+	0.688428i	$0.758301\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	17.9309i	0.753026i
$568$	0	0
$569$	−16.7386	−0.701720	−0.350860	−	0.936428i	$-0.614111\pi$
$569$	−16.7386	−0.701720	−0.350860	+	0.936428i	$0.614111\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	− 27.1231i	− 1.13308i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.5616i	1.23066i	0.788268	+	0.615332i	$0.210978\pi$
$577$	29.5616i	1.23066i	−0.788268	+	0.615332i	$0.789022\pi$
$578$	0	0
$579$	3.80776	0.158245
$580$	0	0
$581$	−35.6847	−1.48045
$582$	0	0
$583$	− 9.12311i	− 0.377840i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.06913i	− 0.291774i	−0.989301	−	0.145887i	$-0.953396\pi$
$587$	− 7.06913i	− 0.291774i	0.989301	−	0.145887i	$-0.0466036\pi$
$588$	0	0
$589$	−43.4773	−1.79145
$590$	0	0
$591$	29.1771	1.20018
$592$	0	0
$593$	15.6155i	0.641253i	0.947206	+	0.320626i	$0.103893\pi$
$593$	15.6155i	0.641253i	−0.947206	+	0.320626i	$0.896107\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 15.6155i	− 0.639101i
$598$	0	0
$599$	32.9848	1.34772	0.673862	−	0.738857i	$-0.264634\pi$
$599$	32.9848	1.34772	0.673862	+	0.738857i	$0.264634\pi$
$600$	0	0
$601$	36.3693	1.48354	0.741768	−	0.670657i	$-0.233988\pi$
$601$	36.3693	1.48354	0.741768	+	0.670657i	$0.233988\pi$
$602$	0	0
$603$	4.80776i	0.195787i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.49242i	− 0.344697i	−0.985036	−	0.172348i	$-0.944865\pi$
$607$	− 8.49242i	− 0.344697i	0.985036	−	0.172348i	$-0.0551355\pi$
$608$	0	0
$609$	24.4924	0.992483
$610$	0	0
$611$	−16.6847	−0.674989
$612$	0	0
$613$	− 5.61553i	− 0.226809i	−0.993549	−	0.113405i	$-0.963824\pi$
$613$	− 5.61553i	− 0.226809i	0.993549	−	0.113405i	$-0.0361756\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 0.561553i	− 0.0226073i	−0.999936	−	0.0113036i	$-0.996402\pi$
$617$	− 0.561553i	− 0.0226073i	0.999936	−	0.0113036i	$-0.00359813\pi$
$618$	0	0
$619$	−12.4924	−0.502113	−0.251056	−	0.967972i	$-0.580778\pi$
$619$	−12.4924	−0.502113	−0.251056	+	0.967972i	$0.580778\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	36.4924i	1.46204i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	18.7386i	0.748349i
$628$	0	0
$629$	11.6847	0.465898
$630$	0	0
$631$	30.2462	1.20408	0.602041	−	0.798465i	$-0.294354\pi$
$631$	30.2462	1.20408	0.602041	+	0.798465i	$0.294354\pi$
$632$	0	0
$633$	− 0.492423i	− 0.0195720i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.56155i	− 0.0618710i
$638$	0	0
$639$	5.68466	0.224882
$640$	0	0
$641$	6.87689	0.271621	0.135810	−	0.990735i	$-0.456636\pi$
$641$	6.87689	0.271621	0.135810	+	0.990735i	$0.456636\pi$
$642$	0	0
$643$	20.4233i	0.805416i	0.915328	+	0.402708i	$0.131931\pi$
$643$	20.4233i	0.805416i	−0.915328	+	0.402708i	$0.868069\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.31534i	− 0.208968i	−0.994527	−	0.104484i	$-0.966681\pi$
$647$	− 5.31534i	− 0.208968i	0.994527	−	0.104484i	$-0.0333190\pi$
$648$	0	0
$649$	7.36932	0.289271
$650$	0	0
$651$	−28.9848	−1.13601
$652$	0	0
$653$	39.6695i	1.55239i	0.630494	+	0.776194i	$0.282852\pi$
$653$	39.6695i	1.55239i	−0.630494	+	0.776194i	$0.717148\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.49242i	− 0.0972387i
$658$	0	0
$659$	−15.3693	−0.598704	−0.299352	−	0.954143i	$-0.596770\pi$
$659$	−15.3693	−0.598704	−0.299352	+	0.954143i	$0.596770\pi$
$660$	0	0
$661$	−6.49242	−0.252526	−0.126263	−	0.991997i	$-0.540298\pi$
$661$	−6.49242	−0.252526	−0.126263	+	0.991997i	$0.540298\pi$
$662$	0	0
$663$	14.2462i	0.553277i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6.12311i	− 0.237088i
$668$	0	0
$669$	−27.5076	−1.06350
$670$	0	0
$671$	−14.2462	−0.549969
$672$	0	0
$673$	34.5464i	1.33167i	0.746101	+	0.665833i	$0.231924\pi$
$673$	34.5464i	1.33167i	−0.746101	+	0.665833i	$0.768076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 26.8078i	− 1.03031i	−0.857098	−	0.515153i	$-0.827735\pi$
$677$	− 26.8078i	− 1.03031i	0.857098	−	0.515153i	$-0.172265\pi$
$678$	0	0
$679$	33.6155	1.29005
$680$	0	0
$681$	−40.9848	−1.57054
$682$	0	0
$683$	42.0540i	1.60915i	0.593851	+	0.804575i	$0.297607\pi$
$683$	42.0540i	1.60915i	−0.593851	+	0.804575i	$0.702393\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 41.7538i	− 1.59301i
$688$	0	0
$689$	−16.2462	−0.618931
$690$	0	0
$691$	16.4924	0.627401	0.313701	−	0.949522i	$-0.398431\pi$
$691$	16.4924	0.627401	0.313701	+	0.949522i	$0.398431\pi$
$692$	0	0
$693$	− 2.87689i	− 0.109284i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10.5616i	− 0.400047i
$698$	0	0
$699$	−1.06913	−0.0404382
$700$	0	0
$701$	−1.75379	−0.0662397	−0.0331198	−	0.999451i	$-0.510544\pi$
$701$	−1.75379	−0.0662397	−0.0331198	+	0.999451i	$0.510544\pi$
$702$	0	0
$703$	− 27.3693i	− 1.03225i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45.3002i	1.70369i
$708$	0	0
$709$	−29.7538	−1.11743	−0.558713	−	0.829361i	$-0.688705\pi$
$709$	−29.7538	−1.11743	−0.558713	+	0.829361i	$0.688705\pi$
$710$	0	0
$711$	−2.73863	−0.102707
$712$	0	0
$713$	7.24621i	0.271373i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.30019i	0.160593i
$718$	0	0
$719$	−19.0540	−0.710593	−0.355297	−	0.934754i	$-0.615620\pi$
$719$	−19.0540	−0.710593	−0.355297	+	0.934754i	$0.615620\pi$
$720$	0	0
$721$	−40.9848	−1.52636
$722$	0	0
$723$	− 9.36932i	− 0.348449i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21.1922i	0.785977i	0.919543	+	0.392988i	$0.128559\pi$
$727$	21.1922i	0.785977i	−0.919543	+	0.392988i	$0.871441\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.315342i	0.0116474i	0.999983	+	0.00582370i	$0.00185375\pi$
$733$	0.315342i	0.0116474i	−0.999983	+	0.00582370i	$0.998146\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.1231i	0.630738i
$738$	0	0
$739$	−0.615528	−0.0226426	−0.0113213	−	0.999936i	$-0.503604\pi$
$739$	−0.615528	−0.0226426	−0.0113213	+	0.999936i	$0.503604\pi$
$740$	0	0
$741$	33.3693	1.22585
$742$	0	0
$743$	3.50758i	0.128681i	0.997928	+	0.0643403i	$0.0204943\pi$
$743$	3.50758i	0.128681i	−0.997928	+	0.0643403i	$0.979506\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 7.82292i	− 0.286226i
$748$	0	0
$749$	−8.80776	−0.321829
$750$	0	0
$751$	−13.1231	−0.478869	−0.239434	−	0.970913i	$-0.576962\pi$
$751$	−13.1231	−0.478869	−0.239434	+	0.970913i	$0.576962\pi$
$752$	0	0
$753$	11.5076i	0.419359i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 12.5616i	− 0.456557i	−0.973596	−	0.228279i	$-0.926690\pi$
$757$	− 12.5616i	− 0.456557i	0.973596	−	0.228279i	$-0.0733097\pi$
$758$	0	0
$759$	3.12311	0.113362
$760$	0	0
$761$	−43.9848	−1.59445	−0.797225	−	0.603683i	$-0.793699\pi$
$761$	−43.9848	−1.59445	−0.797225	+	0.603683i	$0.793699\pi$
$762$	0	0
$763$	39.3693i	1.42526i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 13.1231i	− 0.473848i
$768$	0	0
$769$	10.6307	0.383352	0.191676	−	0.981458i	$-0.438608\pi$
$769$	10.6307	0.383352	0.191676	+	0.981458i	$0.438608\pi$
$770$	0	0
$771$	−49.6695	−1.78880
$772$	0	0
$773$	− 41.1231i	− 1.47910i	−0.673104	−	0.739548i	$-0.735039\pi$
$773$	− 41.1231i	− 1.47910i	0.673104	−	0.739548i	$-0.264961\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 18.2462i	− 0.654579i
$778$	0	0
$779$	−24.7386	−0.886354
$780$	0	0
$781$	20.2462	0.724466
$782$	0	0
$783$	34.0540i	1.21699i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.6847i	1.41461i	0.706911	+	0.707303i	$0.250088\pi$
$787$	39.6847i	1.41461i	−0.706911	+	0.707303i	$0.749912\pi$
$788$	0	0
$789$	0.107951	0.00384314
$790$	0	0
$791$	32.8078	1.16651
$792$	0	0
$793$	25.3693i	0.900891i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.4233i	1.85693i	0.371422	+	0.928464i	$0.378870\pi$
$797$	52.4233i	1.85693i	−0.371422	+	0.928464i	$0.621130\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	− 8.87689i	− 0.313259i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	36.3002i	1.27783i
$808$	0	0
$809$	54.1771	1.90476	0.952382	−	0.304906i	$-0.0986251\pi$
$809$	54.1771	1.90476	0.952382	+	0.304906i	$0.0986251\pi$
$810$	0	0
$811$	47.2462	1.65904	0.829519	−	0.558478i	$-0.188614\pi$
$811$	47.2462	1.65904	0.829519	+	0.558478i	$0.188614\pi$
$812$	0	0
$813$	− 34.2462i	− 1.20107i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−5.12311	−0.179016
$820$	0	0
$821$	−34.4924	−1.20379	−0.601897	−	0.798574i	$-0.705588\pi$
$821$	−34.4924	−1.20379	−0.601897	+	0.798574i	$0.705588\pi$
$822$	0	0
$823$	− 38.0540i	− 1.32648i	−0.748408	−	0.663239i	$-0.769181\pi$
$823$	− 38.0540i	− 1.32648i	0.748408	−	0.663239i	$-0.230819\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 19.6847i	− 0.684503i	−0.939608	−	0.342251i	$-0.888811\pi$
$827$	− 19.6847i	− 0.684503i	0.939608	−	0.342251i	$-0.111189\pi$
$828$	0	0
$829$	29.5464	1.02619	0.513094	−	0.858332i	$-0.328499\pi$
$829$	29.5464	1.02619	0.513094	+	0.858332i	$0.328499\pi$
$830$	0	0
$831$	10.4384	0.362106
$832$	0	0
$833$	− 1.12311i	− 0.0389133i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 40.3002i	− 1.39298i
$838$	0	0
$839$	42.8769	1.48027	0.740137	−	0.672456i	$-0.234760\pi$
$839$	42.8769	1.48027	0.740137	+	0.672456i	$0.234760\pi$
$840$	0	0
$841$	8.49242	0.292842
$842$	0	0
$843$	10.7386i	0.369858i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.9309i	0.616112i
$848$	0	0
$849$	23.1231	0.793583
$850$	0	0
$851$	−4.56155	−0.156368
$852$	0	0
$853$	− 20.2462i	− 0.693217i	−0.938010	−	0.346609i	$-0.887333\pi$
$853$	− 20.2462i	− 0.693217i	0.938010	−	0.346609i	$-0.112667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.3153i	1.13803i	0.822327	+	0.569015i	$0.192675\pi$
$857$	33.3153i	1.13803i	−0.822327	+	0.569015i	$0.807325\pi$
$858$	0	0
$859$	14.5076	0.494992	0.247496	−	0.968889i	$-0.420392\pi$
$859$	14.5076	0.494992	0.247496	+	0.968889i	$0.420392\pi$
$860$	0	0
$861$	−16.4924	−0.562060
$862$	0	0
$863$	− 7.56155i	− 0.257398i	−0.991684	−	0.128699i	$-0.958920\pi$
$863$	− 7.56155i	− 0.257398i	0.991684	−	0.128699i	$-0.0410801\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16.3002i	− 0.553583i
$868$	0	0
$869$	−9.75379	−0.330875
$870$	0	0
$871$	30.4924	1.03320
$872$	0	0
$873$	7.36932i	0.249414i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.9848i	0.641073i	0.947236	+	0.320536i	$0.103863\pi$
$877$	18.9848i	0.641073i	−0.947236	+	0.320536i	$0.896137\pi$
$878$	0	0
$879$	−26.2462	−0.885263
$880$	0	0
$881$	41.4773	1.39740	0.698702	−	0.715413i	$-0.253761\pi$
$881$	41.4773	1.39740	0.698702	+	0.715413i	$0.253761\pi$
$882$	0	0
$883$	10.7386i	0.361384i	0.983540	+	0.180692i	$0.0578337\pi$
$883$	10.7386i	0.361384i	−0.983540	+	0.180692i	$0.942166\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 42.7926i	− 1.43684i	−0.695612	−	0.718418i	$-0.744867\pi$
$887$	− 42.7926i	− 1.43684i	0.695612	−	0.718418i	$-0.255133\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	−14.0000	−0.469018
$892$	0	0
$893$	28.1080i	0.940597i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.56155i	− 0.185695i
$898$	0	0
$899$	−44.3693	−1.47980
$900$	0	0
$901$	−11.6847	−0.389272
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 21.6847i	− 0.720027i	−0.932947	−	0.360014i	$-0.882772\pi$
$907$	− 21.6847i	− 0.720027i	0.932947	−	0.360014i	$-0.117228\pi$
$908$	0	0
$909$	−9.93087	−0.329386
$910$	0	0
$911$	−3.12311	−0.103473	−0.0517366	−	0.998661i	$-0.516476\pi$
$911$	−3.12311	−0.103473	−0.0517366	+	0.998661i	$0.516476\pi$
$912$	0	0
$913$	− 27.8617i	− 0.922089i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.492423i	− 0.0162612i
$918$	0	0
$919$	−46.7386	−1.54177	−0.770883	−	0.636977i	$-0.780185\pi$
$919$	−46.7386	−1.54177	−0.770883	+	0.636977i	$0.780185\pi$
$920$	0	0
$921$	32.9848	1.08689
$922$	0	0
$923$	− 36.0540i	− 1.18673i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.98485i	− 0.295101i
$928$	0	0
$929$	56.7235	1.86104	0.930518	−	0.366245i	$-0.119357\pi$
$929$	56.7235	1.86104	0.930518	+	0.366245i	$0.119357\pi$
$930$	0	0
$931$	−2.63068	−0.0862172
$932$	0	0
$933$	− 13.5616i	− 0.443985i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.2462i	0.530741i	0.964147	+	0.265370i	$0.0854942\pi$
$937$	16.2462i	0.530741i	−0.964147	+	0.265370i	$0.914506\pi$
$938$	0	0
$939$	−1.26137	−0.0411631
$940$	0	0
$941$	−37.7538	−1.23074	−0.615369	−	0.788239i	$-0.710993\pi$
$941$	−37.7538	−1.23074	−0.615369	+	0.788239i	$0.710993\pi$
$942$	0	0
$943$	4.12311i	0.134267i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 56.6847i	− 1.84200i	−0.389558	−	0.921002i	$-0.627372\pi$
$947$	− 56.6847i	− 1.84200i	0.389558	−	0.921002i	$-0.372628\pi$
$948$	0	0
$949$	−15.8078	−0.513142
$950$	0	0
$951$	−23.6155	−0.765786
$952$	0	0
$953$	− 38.4924i	− 1.24689i	−0.781866	−	0.623446i	$-0.785732\pi$
$953$	− 38.4924i	− 1.24689i	0.781866	−	0.623446i	$-0.214268\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.1231i	0.618162i
$958$	0	0
$959$	−17.6155	−0.568835
$960$	0	0
$961$	21.5076	0.693793
$962$	0	0
$963$	− 1.93087i	− 0.0622214i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56.6847i	1.82286i	0.411460	+	0.911428i	$0.365019\pi$
$967$	56.6847i	1.82286i	−0.411460	+	0.911428i	$0.634981\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	−22.6307	−0.726253	−0.363127	−	0.931740i	$-0.618291\pi$
$971$	−22.6307	−0.726253	−0.363127	+	0.931740i	$0.618291\pi$
$972$	0	0
$973$	− 28.8078i	− 0.923535i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 59.7926i	− 1.91294i	−0.291839	−	0.956468i	$-0.594267\pi$
$977$	− 59.7926i	− 1.91294i	0.291839	−	0.956468i	$-0.405733\pi$
$978$	0	0
$979$	−28.4924	−0.910622
$980$	0	0
$981$	−8.63068	−0.275557
$982$	0	0
$983$	− 37.0540i	− 1.18184i	−0.806731	−	0.590919i	$-0.798765\pi$
$983$	− 37.0540i	− 1.18184i	0.806731	−	0.590919i	$-0.201235\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	18.7386i	0.596457i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	53.3002	1.69314	0.846568	−	0.532280i	$-0.178665\pi$
$991$	53.3002	1.69314	0.846568	+	0.532280i	$0.178665\pi$
$992$	0	0
$993$	− 44.6847i	− 1.41802i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 55.6155i	− 1.76136i	−0.473710	−	0.880681i	$-0.657086\pi$
$997$	− 55.6155i	− 1.76136i	0.473710	−	0.880681i	$-0.342914\pi$
$998$	0	0
$999$	25.3693	0.802650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.c.h.1749.2		4
5.2	odd	4		460.2.a.e.1.1	✓	2
5.3	odd	4		2300.2.a.i.1.2		2
5.4	even	2	inner	2300.2.c.h.1749.3		4
15.2	even	4		4140.2.a.m.1.2		2
20.3	even	4		9200.2.a.bv.1.1		2
20.7	even	4		1840.2.a.m.1.2		2
40.27	even	4		7360.2.a.bo.1.1		2
40.37	odd	4		7360.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.a.e.1.1	✓	2	5.2	odd	4
1840.2.a.m.1.2		2	20.7	even	4
2300.2.a.i.1.2		2	5.3	odd	4
2300.2.c.h.1749.2		4	1.1	even	1	trivial
2300.2.c.h.1749.3		4	5.4	even	2	inner
4140.2.a.m.1.2		2	15.2	even	4
7360.2.a.bi.1.2		2	40.37	odd	4
7360.2.a.bo.1.1		2	40.27	even	4
9200.2.a.bv.1.1		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +O(q^{10})\)	\(q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +2.00000 q^{11} -3.56155i q^{13} -2.56155i q^{17} -6.00000 q^{19} -4.00000 q^{21} +1.00000i q^{23} -5.56155i q^{27} -6.12311 q^{29} +7.24621 q^{31} -3.12311i q^{33} +4.56155i q^{37} -5.56155 q^{39} +4.12311 q^{41} -4.68466i q^{47} +0.438447 q^{49} -4.00000 q^{51} -4.56155i q^{53} +9.36932i q^{57} +3.68466 q^{59} -7.12311 q^{61} -1.43845i q^{63} +8.56155i q^{67} +1.56155 q^{69} +10.1231 q^{71} -4.43845i q^{73} -5.12311i q^{77} -4.87689 q^{79} -7.00000 q^{81} -13.9309i q^{83} +9.56155i q^{87} -14.2462 q^{89} -9.12311 q^{91} -11.3153i q^{93} +13.1231i q^{97} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} + 8 q^{11} - 24 q^{19} - 16 q^{21} - 8 q^{29} - 4 q^{31} - 14 q^{39} + 10 q^{49} - 16 q^{51} - 10 q^{59} - 12 q^{61} - 2 q^{69} + 24 q^{71} - 36 q^{79} - 28 q^{81} - 24 q^{89} - 20 q^{91} - 12 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 + 8 * q^11 - 24 * q^19 - 16 * q^21 - 8 * q^29 - 4 * q^31 - 14 * q^39 + 10 * q^49 - 16 * q^51 - 10 * q^59 - 12 * q^61 - 2 * q^69 + 24 * q^71 - 36 * q^79 - 28 * q^81 - 24 * q^89 - 20 * q^91 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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