LMFDB - Embedded newform 1840.2.e.g.369.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8 $ x^14 - 2x^11 + 39x^10 - 10x^9 + 2x^8 - 26x^7 + 297x^6 - 116x^5 + 24x^4 - 20x^3 + 64x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.4
Root			$-1.71470 - 1.71470i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.g.369.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58319i q^{3} +(-2.09277 + 0.787606i) q^{5} +2.84620i q^{7} +0.493499 q^{9} +O(q^{10})$	$q-1.58319i q^{3} +(-2.09277 + 0.787606i) q^{5} +2.84620i q^{7} +0.493499 q^{9} -1.98637 q^{11} -4.69204i q^{13} +(1.24693 + 3.31326i) q^{15} +3.16675i q^{17} -6.16110 q^{19} +4.50609 q^{21} +1.00000i q^{23} +(3.75935 - 3.29655i) q^{25} -5.53088i q^{27} +6.61816 q^{29} +8.29732 q^{31} +3.14481i q^{33} +(-2.24169 - 5.95644i) q^{35} -1.71181i q^{37} -7.42840 q^{39} +6.72440 q^{41} -0.177968i q^{43} +(-1.03278 + 0.388683i) q^{45} -11.4749i q^{47} -1.10086 q^{49} +5.01358 q^{51} -6.18762i q^{53} +(4.15701 - 1.56448i) q^{55} +9.75422i q^{57} -7.61559 q^{59} +11.8577 q^{61} +1.40460i q^{63} +(3.69548 + 9.81934i) q^{65} -3.07554i q^{67} +1.58319 q^{69} +1.96195 q^{71} -4.94732i q^{73} +(-5.21908 - 5.95178i) q^{75} -5.65361i q^{77} +8.53182 q^{79} -7.27596 q^{81} +3.49429i q^{83} +(-2.49415 - 6.62728i) q^{85} -10.4778i q^{87} -7.07506 q^{89} +13.3545 q^{91} -13.1363i q^{93} +(12.8938 - 4.85252i) q^{95} -7.00705i q^{97} -0.980272 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) $ 14 * q + 2 * q^5 - 4 * q^9	$ 14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100}) $ 14 * q + 2 * q^5 - 4 * q^9 + 14 * q^11 + 6 * q^15 - 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 + 20 * q^31 + 2 * q^35 - 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 + 14 * q^51 + 38 * q^55 - 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 + 28 * q^71 + 24 * q^75 - 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 + 14 * q^91 + 30 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	1.58319i		−	0.914057i	−0.889452	−	0.457029i	$-0.848914\pi$
$3$		−	1.58319i		−	0.914057i	0.889452	−	0.457029i	$-0.151086\pi$
$4$		0			0
$5$	−2.09277	+	0.787606i	−0.935914	+	0.352228i
$5$	−2.09277	+	0.787606i	−0.935914	+	0.352228i
$6$		0			0
$7$			2.84620i			1.07576i	0.843020	+	0.537881i	$0.180775\pi$
$7$			2.84620i			1.07576i	−0.843020	+	0.537881i	$0.819225\pi$
$8$		0			0
$9$	0.493499			0.164500
$10$		0			0
$11$	−1.98637			−0.598913			−0.299457	−	0.954110i	$-0.596805\pi$
$11$	−1.98637			−0.598913			−0.299457	+	0.954110i	$0.596805\pi$
$12$		0			0
$13$		−	4.69204i		−	1.30134i	−0.759362	−	0.650668i	$-0.774489\pi$
$13$		−	4.69204i		−	1.30134i	0.759362	−	0.650668i	$-0.225511\pi$
$14$		0			0
$15$	1.24693	+	3.31326i	0.321957	+	0.855479i
$16$		0			0
$17$			3.16675i			0.768050i	0.923323	+	0.384025i	$0.125462\pi$
$17$			3.16675i			0.768050i	−0.923323	+	0.384025i	$0.874538\pi$
$18$		0			0
$19$	−6.16110			−1.41345			−0.706727	−	0.707486i	$-0.749829\pi$
$19$	−6.16110			−1.41345			−0.706727	+	0.707486i	$0.749829\pi$
$20$		0			0
$21$	4.50609			0.983309
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	3.75935	−	3.29655i	0.751871	−	0.659311i
$26$		0			0
$27$		−	5.53088i		−	1.06442i
$28$		0			0
$29$	6.61816			1.22896			0.614481	−	0.788932i	$-0.289365\pi$
$29$	6.61816			1.22896			0.614481	+	0.788932i	$0.289365\pi$
$30$		0			0
$31$	8.29732			1.49024			0.745121	−	0.666929i	$-0.232392\pi$
$31$	8.29732			1.49024			0.745121	+	0.666929i	$0.232392\pi$
$32$		0			0
$33$			3.14481i			0.547441i
$34$		0			0
$35$	−2.24169	−	5.95644i	−0.378914	−	1.00682i
$36$		0			0
$37$		−	1.71181i		−	0.281420i	−0.990051	−	0.140710i	$-0.955061\pi$
$37$		−	1.71181i		−	0.281420i	0.990051	−	0.140710i	$-0.0449386\pi$
$38$		0			0
$39$	−7.42840			−1.18950
$40$		0			0
$41$	6.72440			1.05018			0.525088	−	0.851048i	$-0.324033\pi$
$41$	6.72440			1.05018			0.525088	+	0.851048i	$0.324033\pi$
$42$		0			0
$43$		−	0.177968i		−	0.0271399i	−0.999908	−	0.0135700i	$-0.995680\pi$
$43$		−	0.177968i		−	0.0271399i	0.999908	−	0.0135700i	$-0.00431958\pi$
$44$		0			0
$45$	−1.03278	+	0.388683i	−0.153958	+	0.0579414i
$46$		0			0
$47$		−	11.4749i		−	1.67378i	−0.547369	−	0.836891i	$-0.684371\pi$
$47$		−	11.4749i		−	1.67378i	0.547369	−	0.836891i	$-0.315629\pi$
$48$		0			0
$49$	−1.10086			−0.157266
$50$		0			0
$51$	5.01358			0.702042
$52$		0			0
$53$		−	6.18762i		−	0.849934i	−0.905209	−	0.424967i	$-0.860286\pi$
$53$		−	6.18762i		−	0.849934i	0.905209	−	0.424967i	$-0.139714\pi$
$54$		0			0
$55$	4.15701	−	1.56448i	0.560531	−	0.210954i
$56$		0			0
$57$			9.75422i			1.29198i
$58$		0			0
$59$	−7.61559			−0.991465			−0.495733	−	0.868475i	$-0.665100\pi$
$59$	−7.61559			−0.991465			−0.495733	+	0.868475i	$0.665100\pi$
$60$		0			0
$61$	11.8577			1.51822			0.759109	−	0.650964i	$-0.225635\pi$
$61$	11.8577			1.51822			0.759109	+	0.650964i	$0.225635\pi$
$62$		0			0
$63$			1.40460i			0.176963i
$64$		0			0
$65$	3.69548	+	9.81934i	0.458367	+	1.21794i
$66$		0			0
$67$		−	3.07554i		−	0.375737i	−0.982194	−	0.187868i	$-0.939842\pi$
$67$		−	3.07554i		−	0.375737i	0.982194	−	0.187868i	$-0.0601579\pi$
$68$		0			0
$69$	1.58319			0.190594
$70$		0			0
$71$	1.96195			0.232840			0.116420	−	0.993200i	$-0.462858\pi$
$71$	1.96195			0.232840			0.116420	+	0.993200i	$0.462858\pi$
$72$		0			0
$73$		−	4.94732i		−	0.579041i	−0.957172	−	0.289520i	$-0.906504\pi$
$73$		−	4.94732i		−	0.579041i	0.957172	−	0.289520i	$-0.0934958\pi$
$74$		0			0
$75$	−5.21908	−	5.95178i	−0.602647	−	0.687253i
$76$		0			0
$77$		−	5.65361i		−	0.644289i
$78$		0			0
$79$	8.53182			0.959905			0.479952	−	0.877295i	$-0.340654\pi$
$79$	8.53182			0.959905			0.479952	+	0.877295i	$0.340654\pi$
$80$		0			0
$81$	−7.27596			−0.808440
$82$		0			0
$83$			3.49429i			0.383548i	0.981439	+	0.191774i	$0.0614241\pi$
$83$			3.49429i			0.383548i	−0.981439	+	0.191774i	$0.938576\pi$
$84$		0			0
$85$	−2.49415	−	6.62728i	−0.270529	−	0.718829i
$86$		0			0
$87$		−	10.4778i		−	1.12334i
$88$		0			0
$89$	−7.07506			−0.749954			−0.374977	−	0.927034i	$-0.622349\pi$
$89$	−7.07506			−0.749954			−0.374977	+	0.927034i	$0.622349\pi$
$90$		0			0
$91$	13.3545			1.39993
$92$		0			0
$93$		−	13.1363i		−	1.36217i
$94$		0			0
$95$	12.8938	−	4.85252i	1.32287	−	0.497858i
$96$		0			0
$97$		−	7.00705i		−	0.711458i	−0.934589	−	0.355729i	$-0.884233\pi$
$97$		−	7.00705i		−	0.711458i	0.934589	−	0.355729i	$-0.115767\pi$
$98$		0			0
$99$	−0.980272			−0.0985210
$100$		0			0
$101$	7.22362			0.718777			0.359389	−	0.933188i	$-0.382985\pi$
$101$	7.22362			0.718777			0.359389	+	0.933188i	$0.382985\pi$
$102$		0			0
$103$		−	12.8178i		−	1.26297i	−0.775386	−	0.631487i	$-0.782445\pi$
$103$		−	12.8178i		−	1.26297i	0.775386	−	0.631487i	$-0.217555\pi$
$104$		0			0
$105$	−9.43019	+	3.54902i	−0.920293	+	0.346349i
$106$		0			0
$107$			11.6841i			1.12954i	0.825247	+	0.564772i	$0.191036\pi$
$107$			11.6841i			1.12954i	−0.825247	+	0.564772i	$0.808964\pi$
$108$		0			0
$109$	13.6019			1.30283			0.651413	−	0.758723i	$-0.274176\pi$
$109$	13.6019			1.30283			0.651413	+	0.758723i	$0.274176\pi$
$110$		0			0
$111$	−2.71013			−0.257234
$112$		0			0
$113$		−	1.44526i		−	0.135959i	−0.997687	−	0.0679795i	$-0.978345\pi$
$113$		−	1.44526i		−	0.135959i	0.997687	−	0.0679795i	$-0.0216553\pi$
$114$		0			0
$115$	−0.787606	−	2.09277i	−0.0734446	−	0.195152i
$116$		0			0
$117$		−	2.31552i		−	0.214069i
$118$		0			0
$119$	−9.01322			−0.826240
$120$		0			0
$121$	−7.05433			−0.641303
$122$		0			0
$123$		−	10.6460i		−	0.959920i
$124$		0			0
$125$	−5.27107	+	9.85981i	−0.471459	+	0.881888i
$126$		0			0
$127$			9.12328i			0.809561i	0.914414	+	0.404780i	$0.132652\pi$
$127$			9.12328i			0.809561i	−0.914414	+	0.404780i	$0.867348\pi$
$128$		0			0
$129$	−0.281758			−0.0248074
$130$		0			0
$131$	−9.33180			−0.815323			−0.407661	−	0.913133i	$-0.633656\pi$
$131$	−9.33180			−0.815323			−0.407661	+	0.913133i	$0.633656\pi$
$132$		0			0
$133$		−	17.5357i		−	1.52054i
$134$		0			0
$135$	4.35616	+	11.5749i	0.374918	+	0.996205i
$136$		0			0
$137$		−	21.0013i		−	1.79426i	−0.441768	−	0.897129i	$-0.645649\pi$
$137$		−	21.0013i		−	1.79426i	0.441768	−	0.897129i	$-0.354351\pi$
$138$		0			0
$139$	−16.1299			−1.36812			−0.684061	−	0.729425i	$-0.739788\pi$
$139$	−16.1299			−1.36812			−0.684061	+	0.729425i	$0.739788\pi$
$140$		0			0
$141$	−18.1669			−1.52993
$142$		0			0
$143$			9.32012i			0.779388i
$144$		0			0
$145$	−13.8503	+	5.21251i	−1.15020	+	0.432875i
$146$		0			0
$147$			1.74288i			0.143750i
$148$		0			0
$149$	9.57227			0.784191			0.392096	−	0.919925i	$-0.371750\pi$
$149$	9.57227			0.784191			0.392096	+	0.919925i	$0.371750\pi$
$150$		0			0
$151$	−3.04120			−0.247489			−0.123745	−	0.992314i	$-0.539490\pi$
$151$	−3.04120			−0.247489			−0.123745	+	0.992314i	$0.539490\pi$
$152$		0			0
$153$			1.56279i			0.126344i
$154$		0			0
$155$	−17.3644	+	6.53502i	−1.39474	+	0.524905i
$156$		0			0
$157$		−	12.4786i		−	0.995900i	−0.867206	−	0.497950i	$-0.834086\pi$
$157$		−	12.4786i		−	0.995900i	0.867206	−	0.497950i	$-0.165914\pi$
$158$		0			0
$159$	−9.79619			−0.776889
$160$		0			0
$161$	−2.84620			−0.224312
$162$		0			0
$163$		−	16.3939i		−	1.28407i	−0.766676	−	0.642034i	$-0.778091\pi$
$163$		−	16.3939i		−	1.28407i	0.766676	−	0.642034i	$-0.221909\pi$
$164$		0			0
$165$	−2.47687	−	6.58135i	−0.192824	−	0.512358i
$166$		0			0
$167$			8.13265i			0.629323i	0.949204	+	0.314662i	$0.101891\pi$
$167$			8.13265i			0.629323i	−0.949204	+	0.314662i	$0.898109\pi$
$168$		0			0
$169$	−9.01521			−0.693477
$170$		0			0
$171$	−3.04050			−0.232513
$172$		0			0
$173$			10.5256i			0.800250i	0.916461	+	0.400125i	$0.131033\pi$
$173$			10.5256i			0.800250i	−0.916461	+	0.400125i	$0.868967\pi$
$174$		0			0
$175$	9.38265	+	10.6999i	0.709262	+	0.808835i
$176$		0			0
$177$			12.0570i			0.906256i
$178$		0			0
$179$	20.5421			1.53539			0.767695	−	0.640816i	$-0.221404\pi$
$179$	20.5421			1.53539			0.767695	+	0.640816i	$0.221404\pi$
$180$		0			0
$181$	−22.0754			−1.64085			−0.820424	−	0.571756i	$-0.806263\pi$
$181$	−22.0754			−1.64085			−0.820424	+	0.571756i	$0.806263\pi$
$182$		0			0
$183$		−	18.7730i		−	1.38774i
$184$		0			0
$185$	1.34823	+	3.58243i	0.0991242	+	0.263385i
$186$		0			0
$187$		−	6.29034i		−	0.459995i
$188$		0			0
$189$	15.7420			1.14506
$190$		0			0
$191$	4.50910			0.326267			0.163133	−	0.986604i	$-0.447840\pi$
$191$	4.50910			0.326267			0.163133	+	0.986604i	$0.447840\pi$
$192$		0			0
$193$		−	21.1534i		−	1.52266i	−0.648366	−	0.761329i	$-0.724547\pi$
$193$		−	21.1534i		−	1.52266i	0.648366	−	0.761329i	$-0.275453\pi$
$194$		0			0
$195$	15.5459	−	5.85065i	1.11327	−	0.418974i
$196$		0			0
$197$			10.1475i			0.722980i	0.932376	+	0.361490i	$0.117732\pi$
$197$			10.1475i			0.722980i	−0.932376	+	0.361490i	$0.882268\pi$
$198$		0			0
$199$	6.57598			0.466159			0.233079	−	0.972458i	$-0.425120\pi$
$199$	6.57598			0.466159			0.233079	+	0.972458i	$0.425120\pi$
$200$		0			0
$201$	−4.86917			−0.343445
$202$		0			0
$203$			18.8366i			1.32207i
$204$		0			0
$205$	−14.0726	+	5.29618i	−0.982874	+	0.369901i
$206$		0			0
$207$			0.493499i			0.0343005i
$208$		0			0
$209$	12.2382			0.846536
$210$		0			0
$211$	26.5357			1.82680			0.913398	−	0.407069i	$-0.133449\pi$
$211$	26.5357			1.82680			0.913398	+	0.407069i	$0.133449\pi$
$212$		0			0
$213$		−	3.10614i		−	0.212829i
$214$		0			0
$215$	0.140169	+	0.372446i	0.00955944	+	0.0254006i
$216$		0			0
$217$			23.6158i			1.60315i
$218$		0			0
$219$	−7.83257			−0.529276
$220$		0			0
$221$	14.8585			0.999492
$222$		0			0
$223$		−	20.9231i		−	1.40111i	−0.713597	−	0.700557i	$-0.752935\pi$
$223$		−	20.9231i		−	1.40111i	0.713597	−	0.700557i	$-0.247065\pi$
$224$		0			0
$225$	1.85524	−	1.62685i	0.123682	−	0.108456i
$226$		0			0
$227$			9.32581i			0.618975i	0.950903	+	0.309488i	$0.100158\pi$
$227$			9.32581i			0.618975i	−0.950903	+	0.309488i	$0.899842\pi$
$228$		0			0
$229$	−29.5189			−1.95066			−0.975332	−	0.220742i	$-0.929152\pi$
$229$	−29.5189			−1.95066			−0.975332	+	0.220742i	$0.929152\pi$
$230$		0			0
$231$	−8.95076			−0.588917
$232$		0			0
$233$			28.9849i			1.89886i	0.313974	+	0.949431i	$0.398339\pi$
$233$			28.9849i			1.89886i	−0.313974	+	0.949431i	$0.601661\pi$
$234$		0			0
$235$	9.03768	+	24.0142i	0.589553	+	1.56652i
$236$		0			0
$237$		−	13.5075i		−	0.877408i
$238$		0			0
$239$	−9.62209			−0.622401			−0.311201	−	0.950344i	$-0.600731\pi$
$239$	−9.62209			−0.622401			−0.311201	+	0.950344i	$0.600731\pi$
$240$		0			0
$241$	15.5745			1.00324			0.501621	−	0.865088i	$-0.332737\pi$
$241$	15.5745			1.00324			0.501621	+	0.865088i	$0.332737\pi$
$242$		0			0
$243$		−	5.07340i		−	0.325459i
$244$		0			0
$245$	2.30385	−	0.867045i	0.147187	−	0.0553935i
$246$		0			0
$247$			28.9081i			1.83938i
$248$		0			0
$249$	5.53214			0.350585
$250$		0			0
$251$	22.8066			1.43954			0.719770	−	0.694213i	$-0.244247\pi$
$251$	22.8066			1.43954			0.719770	+	0.694213i	$0.244247\pi$
$252$		0			0
$253$		−	1.98637i		−	0.124882i
$254$		0			0
$255$	−10.4923	+	3.94873i	−0.657051	+	0.247279i
$256$		0			0
$257$			11.5591i			0.721038i	0.932752	+	0.360519i	$0.117400\pi$
$257$			11.5591i			0.721038i	−0.932752	+	0.360519i	$0.882600\pi$
$258$		0			0
$259$	4.87216			0.302742
$260$		0			0
$261$	3.26606			0.202164
$262$		0			0
$263$			6.57534i			0.405453i	0.979235	+	0.202726i	$0.0649802\pi$
$263$			6.57534i			0.405453i	−0.979235	+	0.202726i	$0.935020\pi$
$264$		0			0
$265$	4.87340	+	12.9492i	0.299371	+	0.795466i
$266$		0			0
$267$			11.2012i			0.685501i
$268$		0			0
$269$	16.6536			1.01539			0.507694	−	0.861537i	$-0.330498\pi$
$269$	16.6536			1.01539			0.507694	+	0.861537i	$0.330498\pi$
$270$		0			0
$271$	23.8675			1.44985			0.724925	−	0.688828i	$-0.241874\pi$
$271$	23.8675			1.44985			0.724925	+	0.688828i	$0.241874\pi$
$272$		0			0
$273$		−	21.1427i		−	1.27962i
$274$		0			0
$275$	−7.46747	+	6.54817i	−0.450305	+	0.394870i
$276$		0			0
$277$			5.65369i			0.339697i	0.985470	+	0.169849i	$0.0543279\pi$
$277$			5.65369i			0.339697i	−0.985470	+	0.169849i	$0.945672\pi$
$278$		0			0
$279$	4.09472			0.245144
$280$		0			0
$281$	−31.7408			−1.89350			−0.946750	−	0.321971i	$-0.895655\pi$
$281$	−31.7408			−1.89350			−0.946750	+	0.321971i	$0.895655\pi$
$282$		0			0
$283$		−	0.998092i		−	0.0593304i	−0.999560	−	0.0296652i	$-0.990556\pi$
$283$		−	0.998092i		−	0.0593304i	0.999560	−	0.0296652i	$-0.00944412\pi$
$284$		0			0
$285$	−7.68248	−	20.4133i	−0.455071	−	1.20918i
$286$		0			0
$287$			19.1390i			1.12974i
$288$		0			0
$289$	6.97168			0.410099
$290$		0			0
$291$	−11.0935			−0.650314
$292$		0			0
$293$		−	13.8947i		−	0.811739i	−0.913931	−	0.405869i	$-0.866969\pi$
$293$		−	13.8947i		−	0.811739i	0.913931	−	0.405869i	$-0.133031\pi$
$294$		0			0
$295$	15.9377	−	5.99808i	0.927927	−	0.349222i
$296$		0			0
$297$			10.9864i			0.637495i
$298$		0			0
$299$	4.69204			0.271347
$300$		0			0
$301$	0.506534			0.0291961
$302$		0			0
$303$		−	11.4364i		−	0.657003i
$304$		0			0
$305$	−24.8153	+	9.33916i	−1.42092	+	0.534759i
$306$		0			0
$307$		−	3.66633i		−	0.209249i	−0.994512	−	0.104624i	$-0.966636\pi$
$307$		−	3.66633i		−	0.209249i	0.994512	−	0.104624i	$-0.0333640\pi$
$308$		0			0
$309$	−20.2930			−1.15443
$310$		0			0
$311$	14.4210			0.817741			0.408870	−	0.912593i	$-0.365923\pi$
$311$	14.4210			0.817741			0.408870	+	0.912593i	$0.365923\pi$
$312$		0			0
$313$			10.5046i			0.593754i	0.954916	+	0.296877i	$0.0959451\pi$
$313$			10.5046i			0.593754i	−0.954916	+	0.296877i	$0.904055\pi$
$314$		0			0
$315$	−1.10627	−	2.93950i	−0.0623312	−	0.165622i
$316$		0			0
$317$			32.0818i			1.80189i	0.433931	+	0.900946i	$0.357126\pi$
$317$			32.0818i			1.80189i	−0.433931	+	0.900946i	$0.642874\pi$
$318$		0			0
$319$	−13.1461			−0.736042
$320$		0			0
$321$	18.4982			1.03247
$322$		0			0
$323$		−	19.5107i		−	1.08560i
$324$		0			0
$325$	−15.4675	−	17.6390i	−0.857985	−	0.978437i
$326$		0			0
$327$		−	21.5344i		−	1.19086i
$328$		0			0
$329$	32.6598			1.80059
$330$		0			0
$331$	−9.71307			−0.533879			−0.266939	−	0.963713i	$-0.586012\pi$
$331$	−9.71307			−0.533879			−0.266939	+	0.963713i	$0.586012\pi$
$332$		0			0
$333$		−	0.844778i		−	0.0462935i
$334$		0			0
$335$	2.42231	+	6.43639i	0.132345	+	0.351657i
$336$		0			0
$337$			0.131832i			0.00718136i	0.999994	+	0.00359068i	$0.00114295\pi$
$337$			0.131832i			0.00718136i	−0.999994	+	0.00359068i	$0.998857\pi$
$338$		0			0
$339$	−2.28813			−0.124274
$340$		0			0
$341$	−16.4815			−0.892526
$342$		0			0
$343$			16.7901i			0.906582i
$344$		0			0
$345$	−3.31326	+	1.24693i	−0.178380	+	0.0671326i
$346$		0			0
$347$		−	13.1981i		−	0.708512i	−0.935148	−	0.354256i	$-0.884734\pi$
$347$		−	13.1981i		−	0.708512i	0.935148	−	0.354256i	$-0.115266\pi$
$348$		0			0
$349$	34.0451			1.82239			0.911197	−	0.411971i	$-0.135159\pi$
$349$	34.0451			1.82239			0.911197	+	0.411971i	$0.135159\pi$
$350$		0			0
$351$	−25.9511			−1.38517
$352$		0			0
$353$			10.8593i			0.577984i	0.957331	+	0.288992i	$0.0933201\pi$
$353$			10.8593i			0.577984i	−0.957331	+	0.288992i	$0.906680\pi$
$354$		0			0
$355$	−4.10590	+	1.54524i	−0.217919	+	0.0820129i
$356$		0			0
$357$			14.2697i			0.755231i
$358$		0			0
$359$	−20.3091			−1.07188			−0.535938	−	0.844257i	$-0.680042\pi$
$359$	−20.3091			−1.07188			−0.535938	+	0.844257i	$0.680042\pi$
$360$		0			0
$361$	18.9592			0.997853
$362$		0			0
$363$			11.1684i			0.586188i
$364$		0			0
$365$	3.89654	+	10.3536i	0.203954	+	0.541932i
$366$		0			0
$367$		−	18.6731i		−	0.974727i	−0.873199	−	0.487364i	$-0.837959\pi$
$367$		−	18.6731i		−	0.974727i	0.873199	−	0.487364i	$-0.162041\pi$
$368$		0			0
$369$	3.31848			0.172753
$370$		0			0
$371$	17.6112			0.914328
$372$		0			0
$373$			9.51979i			0.492916i	0.969153	+	0.246458i	$0.0792667\pi$
$373$			9.51979i			0.492916i	−0.969153	+	0.246458i	$0.920733\pi$
$374$		0			0
$375$	15.6100	+	8.34512i	0.806096	+	0.430940i
$376$		0			0
$377$		−	31.0527i		−	1.59929i
$378$		0			0
$379$	18.8304			0.967251			0.483625	−	0.875275i	$-0.339320\pi$
$379$	18.8304			0.967251			0.483625	+	0.875275i	$0.339320\pi$
$380$		0			0
$381$	14.4439			0.739985
$382$		0			0
$383$			10.9401i			0.559015i	0.960144	+	0.279507i	$0.0901712\pi$
$383$			10.9401i			0.559015i	−0.960144	+	0.279507i	$0.909829\pi$
$384$		0			0
$385$	4.45282	+	11.8317i	0.226937	+	0.602999i
$386$		0			0
$387$		−	0.0878272i		−	0.00446451i
$388$		0			0
$389$	−22.2830			−1.12979			−0.564896	−	0.825162i	$-0.691084\pi$
$389$	−22.2830			−1.12979			−0.564896	+	0.825162i	$0.691084\pi$
$390$		0			0
$391$	−3.16675			−0.160150
$392$		0			0
$393$			14.7740i			0.745252i
$394$		0			0
$395$	−17.8551	+	6.71971i	−0.898389	+	0.338105i
$396$		0			0
$397$			10.8388i			0.543983i	0.962300	+	0.271991i	$0.0876822\pi$
$397$			10.8388i			0.543983i	−0.962300	+	0.271991i	$0.912318\pi$
$398$		0			0
$399$	−27.7625			−1.38986
$400$		0			0
$401$	−2.71329			−0.135495			−0.0677476	−	0.997702i	$-0.521581\pi$
$401$	−2.71329			−0.135495			−0.0677476	+	0.997702i	$0.521581\pi$
$402$		0			0
$403$		−	38.9313i		−	1.93931i
$404$		0			0
$405$	15.2269	−	5.73059i	0.756631	−	0.284755i
$406$		0			0
$407$			3.40029i			0.168546i
$408$		0			0
$409$	−21.6271			−1.06939			−0.534696	−	0.845044i	$-0.679574\pi$
$409$	−21.6271			−1.06939			−0.534696	+	0.845044i	$0.679574\pi$
$410$		0			0
$411$	−33.2490			−1.64005
$412$		0			0
$413$		−	21.6755i		−	1.06658i
$414$		0			0
$415$	−2.75212	−	7.31274i	−0.135097	−	0.358968i
$416$		0			0
$417$			25.5368i			1.25054i
$418$		0			0
$419$	4.00587			0.195700			0.0978498	−	0.995201i	$-0.468804\pi$
$419$	4.00587			0.195700			0.0978498	+	0.995201i	$0.468804\pi$
$420$		0			0
$421$	25.4251			1.23914			0.619572	−	0.784940i	$-0.287306\pi$
$421$	25.4251			1.23914			0.619572	+	0.784940i	$0.287306\pi$
$422$		0			0
$423$		−	5.66284i		−	0.275337i
$424$		0			0
$425$	10.4394	+	11.9049i	0.506384	+	0.577475i
$426$		0			0
$427$			33.7493i			1.63324i
$428$		0			0
$429$	14.7556			0.712405
$430$		0			0
$431$	−1.07679			−0.0518673			−0.0259336	−	0.999664i	$-0.508256\pi$
$431$	−1.07679			−0.0518673			−0.0259336	+	0.999664i	$0.508256\pi$
$432$		0			0
$433$		−	3.93871i		−	0.189283i	−0.995511	−	0.0946413i	$-0.969830\pi$
$433$		−	3.93871i		−	0.189283i	0.995511	−	0.0946413i	$-0.0301704\pi$
$434$		0			0
$435$	8.25240	+	21.9277i	0.395672	+	1.05135i
$436$		0			0
$437$		−	6.16110i		−	0.294726i
$438$		0			0
$439$	13.1539			0.627802			0.313901	−	0.949456i	$-0.398364\pi$
$439$	13.1539			0.627802			0.313901	+	0.949456i	$0.398364\pi$
$440$		0			0
$441$	−0.543274			−0.0258702
$442$		0			0
$443$			8.95394i			0.425415i	0.977116	+	0.212707i	$0.0682281\pi$
$443$			8.95394i			0.425415i	−0.977116	+	0.212707i	$0.931772\pi$
$444$		0			0
$445$	14.8064	−	5.57236i	0.701893	−	0.264155i
$446$		0			0
$447$		−	15.1548i		−	0.716795i
$448$		0			0
$449$	−8.23267			−0.388524			−0.194262	−	0.980950i	$-0.562231\pi$
$449$	−8.23267			−0.388524			−0.194262	+	0.980950i	$0.562231\pi$
$450$		0			0
$451$	−13.3571			−0.628964
$452$		0			0
$453$			4.81480i			0.226219i
$454$		0			0
$455$	−27.9478	+	10.5181i	−1.31021	+	0.493095i
$456$		0			0
$457$		−	36.3473i		−	1.70026i	−0.526576	−	0.850128i	$-0.676524\pi$
$457$		−	36.3473i		−	1.70026i	0.526576	−	0.850128i	$-0.323476\pi$
$458$		0			0
$459$	17.5149			0.817527
$460$		0			0
$461$	−15.2421			−0.709894			−0.354947	−	0.934886i	$-0.615501\pi$
$461$	−15.2421			−0.709894			−0.354947	+	0.934886i	$0.615501\pi$
$462$		0			0
$463$			5.57424i			0.259057i	0.991576	+	0.129528i	$0.0413463\pi$
$463$			5.57424i			0.259057i	−0.991576	+	0.129528i	$0.958654\pi$
$464$		0			0
$465$	10.3462	+	27.4911i	0.479793	+	1.27487i
$466$		0			0
$467$		−	6.21488i		−	0.287590i	−0.989607	−	0.143795i	$-0.954069\pi$
$467$		−	6.21488i		−	0.287590i	0.989607	−	0.143795i	$-0.0459306\pi$
$468$		0			0
$469$	8.75360			0.404204
$470$		0			0
$471$	−19.7560			−0.910309
$472$		0			0
$473$			0.353511i			0.0162545i
$474$		0			0
$475$	−23.1618	+	20.3104i	−1.06273	+	0.931905i
$476$		0			0
$477$		−	3.05358i		−	0.139814i
$478$		0			0
$479$	−34.4168			−1.57254			−0.786272	−	0.617880i	$-0.787992\pi$
$479$	−34.4168			−1.57254			−0.786272	+	0.617880i	$0.787992\pi$
$480$		0			0
$481$	−8.03189			−0.366223
$482$		0			0
$483$			4.50609i			0.205034i
$484$		0			0
$485$	5.51880	+	14.6641i	0.250596	+	0.665864i
$486$		0			0
$487$			22.7752i			1.03204i	0.856576	+	0.516021i	$0.172587\pi$
$487$			22.7752i			1.03204i	−0.856576	+	0.516021i	$0.827413\pi$
$488$		0			0
$489$	−25.9547			−1.17371
$490$		0			0
$491$	−3.16911			−0.143020			−0.0715099	−	0.997440i	$-0.522782\pi$
$491$	−3.16911			−0.143020			−0.0715099	+	0.997440i	$0.522782\pi$
$492$		0			0
$493$			20.9581i			0.943905i
$494$		0			0
$495$	2.05148	−	0.772068i	0.0922072	−	0.0347019i
$496$		0			0
$497$			5.58410i			0.250481i
$498$		0			0
$499$	−14.0809			−0.630346			−0.315173	−	0.949034i	$-0.602063\pi$
$499$	−14.0809			−0.630346			−0.315173	+	0.949034i	$0.602063\pi$
$500$		0			0
$501$	12.8756			0.575237
$502$		0			0
$503$		−	21.4946i		−	0.958399i	−0.877706	−	0.479199i	$-0.840927\pi$
$503$		−	21.4946i		−	0.958399i	0.877706	−	0.479199i	$-0.159073\pi$
$504$		0			0
$505$	−15.1174	+	5.68937i	−0.672714	+	0.253173i
$506$		0			0
$507$			14.2728i			0.633878i
$508$		0			0
$509$	34.0952			1.51124			0.755622	−	0.655008i	$-0.227335\pi$
$509$	34.0952			1.51124			0.755622	+	0.655008i	$0.227335\pi$
$510$		0			0
$511$	14.0811			0.622910
$512$		0			0
$513$			34.0764i			1.50451i
$514$		0			0
$515$	10.0954	+	26.8247i	0.444855	+	1.18204i
$516$		0			0
$517$			22.7933i			1.00245i
$518$		0			0
$519$	16.6641			0.731474
$520$		0			0
$521$	−14.1367			−0.619339			−0.309669	−	0.950844i	$-0.600218\pi$
$521$	−14.1367			−0.619339			−0.309669	+	0.950844i	$0.600218\pi$
$522$		0			0
$523$			11.2020i			0.489828i	0.969545	+	0.244914i	$0.0787598\pi$
$523$			11.2020i			0.489828i	−0.969545	+	0.244914i	$0.921240\pi$
$524$		0			0
$525$	16.9400	−	14.8546i	0.739321	−	0.648306i
$526$		0			0
$527$			26.2756i			1.14458i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−3.75829			−0.163096
$532$		0			0
$533$		−	31.5511i		−	1.36663i
$534$		0			0
$535$	−9.20247	−	24.4521i	−0.397857	−	1.05716i
$536$		0			0
$537$		−	32.5221i		−	1.40343i
$538$		0			0
$539$	2.18672			0.0941887
$540$		0			0
$541$	−33.6704			−1.44760			−0.723802	−	0.690008i	$-0.757607\pi$
$541$	−33.6704			−1.44760			−0.723802	+	0.690008i	$0.757607\pi$
$542$		0			0
$543$			34.9496i			1.49983i
$544$		0			0
$545$	−28.4656	+	10.7129i	−1.21933	+	0.458892i
$546$		0			0
$547$			23.1062i			0.987949i	0.869476	+	0.493975i	$0.164456\pi$
$547$			23.1062i			0.987949i	−0.869476	+	0.493975i	$0.835544\pi$
$548$		0			0
$549$	5.85174			0.249746
$550$		0			0
$551$	−40.7752			−1.73708
$552$		0			0
$553$			24.2833i			1.03263i
$554$		0			0
$555$	5.67167	−	2.13452i	0.240749	−	0.0906051i
$556$		0			0
$557$			3.48937i			0.147849i	0.997264	+	0.0739246i	$0.0235524\pi$
$557$			3.48937i			0.147849i	−0.997264	+	0.0739246i	$0.976448\pi$
$558$		0			0
$559$	−0.835034			−0.0353182
$560$		0			0
$561$	−9.95883			−0.420462
$562$		0			0
$563$		−	38.8593i		−	1.63772i	−0.573991	−	0.818862i	$-0.694606\pi$
$563$		−	38.8593i		−	1.63772i	0.573991	−	0.818862i	$-0.305394\pi$
$564$		0			0
$565$	1.13830	+	3.02460i	0.0478886	+	0.127246i
$566$		0			0
$567$		−	20.7089i		−	0.869690i
$568$		0			0
$569$	23.2055			0.972823			0.486412	−	0.873730i	$-0.338306\pi$
$569$	23.2055			0.972823			0.486412	+	0.873730i	$0.338306\pi$
$570$		0			0
$571$	5.39836			0.225914			0.112957	−	0.993600i	$-0.463968\pi$
$571$	5.39836			0.225914			0.112957	+	0.993600i	$0.463968\pi$
$572$		0			0
$573$		−	7.13877i		−	0.298226i
$574$		0			0
$575$	3.29655	+	3.75935i	0.137476	+	0.156776i
$576$		0			0
$577$		−	13.1082i		−	0.545701i	−0.962056	−	0.272850i	$-0.912034\pi$
$577$		−	13.1082i		−	0.545701i	0.962056	−	0.272850i	$-0.0879664\pi$
$578$		0			0
$579$	−33.4900			−1.39180
$580$		0			0
$581$	−9.94546			−0.412607
$582$		0			0
$583$			12.2909i			0.509037i
$584$		0			0
$585$	1.82371	+	4.84584i	0.0754013	+	0.200351i
$586$		0			0
$587$		−	12.9646i		−	0.535108i	−0.963543	−	0.267554i	$-0.913785\pi$
$587$		−	12.9646i		−	0.535108i	0.963543	−	0.267554i	$-0.0862154\pi$
$588$		0			0
$589$	−51.1206			−2.10639
$590$		0			0
$591$	16.0655			0.660845
$592$		0			0
$593$			17.0225i			0.699030i	0.936931	+	0.349515i	$0.113654\pi$
$593$			17.0225i			0.699030i	−0.936931	+	0.349515i	$0.886346\pi$
$594$		0			0
$595$	18.8626	−	7.09886i	0.773290	−	0.291025i
$596$		0			0
$597$		−	10.4110i		−	0.426096i
$598$		0			0
$599$	−5.98111			−0.244381			−0.122191	−	0.992507i	$-0.538992\pi$
$599$	−5.98111			−0.244381			−0.122191	+	0.992507i	$0.538992\pi$
$600$		0			0
$601$	−21.3680			−0.871617			−0.435809	−	0.900039i	$-0.643538\pi$
$601$	−21.3680			−0.871617			−0.435809	+	0.900039i	$0.643538\pi$
$602$		0			0
$603$		−	1.51778i		−	0.0618086i
$604$		0			0
$605$	14.7631	−	5.55604i	0.600205	−	0.225885i
$606$		0			0
$607$		−	17.5464i		−	0.712186i	−0.934450	−	0.356093i	$-0.884109\pi$
$607$		−	17.5464i		−	0.712186i	0.934450	−	0.356093i	$-0.115891\pi$
$608$		0			0
$609$	29.8220			1.20845
$610$		0			0
$611$	−53.8405			−2.17815
$612$		0			0
$613$		−	45.4601i		−	1.83612i	−0.396444	−	0.918059i	$-0.629756\pi$
$613$		−	45.4601i		−	1.83612i	0.396444	−	0.918059i	$-0.370244\pi$
$614$		0			0
$615$	8.38487	+	22.2797i	0.338111	+	0.898403i
$616$		0			0
$617$			25.8720i			1.04157i	0.853688	+	0.520785i	$0.174360\pi$
$617$			25.8720i			1.04157i	−0.853688	+	0.520785i	$0.825640\pi$
$618$		0			0
$619$	−3.20381			−0.128772			−0.0643859	−	0.997925i	$-0.520509\pi$
$619$	−3.20381			−0.128772			−0.0643859	+	0.997925i	$0.520509\pi$
$620$		0			0
$621$	5.53088			0.221947
$622$		0			0
$623$		−	20.1370i		−	0.806773i
$624$		0			0
$625$	3.26548	−	24.7858i	0.130619	−	0.991433i
$626$		0			0
$627$		−	19.3755i		−	0.773783i
$628$		0			0
$629$	5.42089			0.216145
$630$		0			0
$631$	−0.0248650			−0.000989858			−0.000494929	−	1.00000i	$-0.500158\pi$
$631$	−0.0248650			−0.000989858			−0.000494929		1.00000i	$0.500158\pi$
$632$		0			0
$633$		−	42.0112i		−	1.66980i
$634$		0			0
$635$	−7.18555	−	19.0929i	−0.285150	−	0.757679i
$636$		0			0
$637$			5.16528i			0.204656i
$638$		0			0
$639$	0.968219			0.0383022
$640$		0			0
$641$	−9.77145			−0.385949			−0.192975	−	0.981204i	$-0.561813\pi$
$641$	−9.77145			−0.385949			−0.192975	+	0.981204i	$0.561813\pi$
$642$		0			0
$643$		−	36.8211i		−	1.45208i	−0.687651	−	0.726041i	$-0.741358\pi$
$643$		−	36.8211i		−	1.45208i	0.687651	−	0.726041i	$-0.258642\pi$
$644$		0			0
$645$	0.589655	−	0.221914i	0.0232176	−	0.00873787i
$646$		0			0
$647$		−	46.5570i		−	1.83034i	−0.403064	−	0.915172i	$-0.632055\pi$
$647$		−	46.5570i		−	1.83034i	0.403064	−	0.915172i	$-0.367945\pi$
$648$		0			0
$649$	15.1274			0.593802
$650$		0			0
$651$	37.3884			1.46537
$652$		0			0
$653$		−	2.76029i		−	0.108018i	−0.998540	−	0.0540092i	$-0.982800\pi$
$653$		−	2.76029i		−	0.108018i	0.998540	−	0.0540092i	$-0.0172000\pi$
$654$		0			0
$655$	19.5293	−	7.34978i	0.763072	−	0.287180i
$656$		0			0
$657$		−	2.44150i		−	0.0952520i
$658$		0			0
$659$	39.3922			1.53450			0.767252	−	0.641346i	$-0.221624\pi$
$659$	39.3922			1.53450			0.767252	+	0.641346i	$0.221624\pi$
$660$		0			0
$661$	−17.7641			−0.690943			−0.345472	−	0.938429i	$-0.612281\pi$
$661$	−17.7641			−0.690943			−0.345472	+	0.938429i	$0.612281\pi$
$662$		0			0
$663$		−	23.5239i		−	0.913593i
$664$		0			0
$665$	13.8113	+	36.6982i	0.535578	+	1.42310i
$666$		0			0
$667$			6.61816i			0.256256i
$668$		0			0
$669$	−33.1253			−1.28070
$670$		0			0
$671$	−23.5537			−0.909280
$672$		0			0
$673$			23.7072i			0.913847i	0.889506	+	0.456923i	$0.151049\pi$
$673$			23.7072i			0.913847i	−0.889506	+	0.456923i	$0.848951\pi$
$674$		0			0
$675$	−18.2329	−	20.7925i	−0.701783	−	0.800306i
$676$		0			0
$677$			49.3507i			1.89670i	0.317221	+	0.948352i	$0.397250\pi$
$677$			49.3507i			1.89670i	−0.317221	+	0.948352i	$0.602750\pi$
$678$		0			0
$679$	19.9435			0.765361
$680$		0			0
$681$	14.7646			0.565779
$682$		0			0
$683$			15.6461i			0.598682i	0.954146	+	0.299341i	$0.0967669\pi$
$683$			15.6461i			0.598682i	−0.954146	+	0.299341i	$0.903233\pi$
$684$		0			0
$685$	16.5407	+	43.9507i	0.631988	+	1.67927i
$686$		0			0
$687$			46.7341i			1.78302i
$688$		0			0
$689$	−29.0325			−1.10605
$690$		0			0
$691$	−3.79498			−0.144368			−0.0721839	−	0.997391i	$-0.522997\pi$
$691$	−3.79498			−0.144368			−0.0721839	+	0.997391i	$0.522997\pi$
$692$		0			0
$693$		−	2.79005i		−	0.105985i
$694$		0			0
$695$	33.7562	−	12.7040i	1.28044	−	0.481891i
$696$		0			0
$697$			21.2945i			0.806587i
$698$		0			0
$699$	45.8887			1.73567
$700$		0			0
$701$	0.751725			0.0283923			0.0141961	−	0.999899i	$-0.495481\pi$
$701$	0.751725			0.0283923			0.0141961	+	0.999899i	$0.495481\pi$
$702$		0			0
$703$			10.5467i			0.397775i
$704$		0			0
$705$	38.0192	−	14.3084i	1.43189	−	0.538885i
$706$		0			0
$707$			20.5599i			0.773234i
$708$		0			0
$709$	−4.52035			−0.169765			−0.0848827	−	0.996391i	$-0.527052\pi$
$709$	−4.52035			−0.169765			−0.0848827	+	0.996391i	$0.527052\pi$
$710$		0			0
$711$	4.21044			0.157904
$712$		0			0
$713$			8.29732i			0.310737i
$714$		0			0
$715$	−7.34058	−	19.5048i	−0.274522	−	0.729440i
$716$		0			0
$717$			15.2336i			0.568910i
$718$		0			0
$719$	−36.6742			−1.36772			−0.683859	−	0.729614i	$-0.739700\pi$
$719$	−36.6742			−1.36772			−0.683859	+	0.729614i	$0.739700\pi$
$720$		0			0
$721$	36.4820			1.35866
$722$		0			0
$723$		−	24.6574i		−	0.917020i
$724$		0			0
$725$	24.8800	−	21.8171i	0.924021	−	0.810268i
$726$		0			0
$727$			33.6449i			1.24782i	0.781496	+	0.623910i	$0.214457\pi$
$727$			33.6449i			1.24782i	−0.781496	+	0.623910i	$0.785543\pi$
$728$		0			0
$729$	−29.8601			−1.10593
$730$		0			0
$731$	0.563582			0.0208448
$732$		0			0
$733$		−	47.5745i		−	1.75720i	−0.477555	−	0.878602i	$-0.658477\pi$
$733$		−	47.5745i		−	1.75720i	0.477555	−	0.878602i	$-0.341523\pi$
$734$		0			0
$735$	−1.37270	−	3.64744i	−0.0506328	−	0.134538i
$736$		0			0
$737$			6.10916i			0.225034i
$738$		0			0
$739$	28.8914			1.06279			0.531395	−	0.847124i	$-0.321668\pi$
$739$	28.8914			1.06279			0.531395	+	0.847124i	$0.321668\pi$
$740$		0			0
$741$	45.7672			1.68130
$742$		0			0
$743$			3.61970i			0.132794i	0.997793	+	0.0663969i	$0.0211504\pi$
$743$			3.61970i			0.132794i	−0.997793	+	0.0663969i	$0.978850\pi$
$744$		0			0
$745$	−20.0325	+	7.53918i	−0.733936	+	0.276214i
$746$		0			0
$747$			1.72443i			0.0630936i
$748$		0			0
$749$	−33.2553			−1.21512
$750$		0			0
$751$	−19.2050			−0.700801			−0.350401	−	0.936600i	$-0.613955\pi$
$751$	−19.2050			−0.700801			−0.350401	+	0.936600i	$0.613955\pi$
$752$		0			0
$753$		−	36.1073i		−	1.31582i
$754$		0			0
$755$	6.36452	−	2.39527i	0.231629	−	0.0871726i
$756$		0			0
$757$		−	12.6638i		−	0.460275i	−0.973158	−	0.230137i	$-0.926082\pi$
$757$		−	12.6638i		−	0.460275i	0.973158	−	0.230137i	$-0.0739176\pi$
$758$		0			0
$759$	−3.14481			−0.114149
$760$		0			0
$761$	−38.9053			−1.41032			−0.705159	−	0.709050i	$-0.749124\pi$
$761$	−38.9053			−1.41032			−0.705159	+	0.709050i	$0.749124\pi$
$762$		0			0
$763$			38.7138i			1.40153i
$764$		0			0
$765$	−1.23086	−	3.27055i	−0.0445019	−	0.118247i
$766$		0			0
$767$			35.7326i			1.29023i
$768$		0			0
$769$	−4.16816			−0.150308			−0.0751539	−	0.997172i	$-0.523945\pi$
$769$	−4.16816			−0.150308			−0.0751539	+	0.997172i	$0.523945\pi$
$770$		0			0
$771$	18.3003			0.659070
$772$		0			0
$773$		−	26.6030i		−	0.956842i	−0.878131	−	0.478421i	$-0.841209\pi$
$773$		−	26.6030i		−	0.956842i	0.878131	−	0.478421i	$-0.158791\pi$
$774$		0			0
$775$	31.1926	−	27.3525i	1.12047	−	0.982533i
$776$		0			0
$777$		−	7.71358i		−	0.276723i
$778$		0			0
$779$	−41.4297			−1.48437
$780$		0			0
$781$	−3.89716			−0.139451
$782$		0			0
$783$		−	36.6043i		−	1.30813i
$784$		0			0
$785$	9.82821	+	26.1148i	0.350784	+	0.932077i
$786$		0			0
$787$		−	21.7538i		−	0.775440i	−0.921777	−	0.387720i	$-0.873263\pi$
$787$		−	21.7538i		−	0.775440i	0.921777	−	0.387720i	$-0.126737\pi$
$788$		0			0
$789$	10.4100			0.370607
$790$		0			0
$791$	4.11351			0.146260
$792$		0			0
$793$		−	55.6366i		−	1.97571i
$794$		0			0
$795$	20.5012	−	7.71554i	0.727101	−	0.273642i
$796$		0			0
$797$			10.2490i			0.363039i	0.983387	+	0.181519i	$0.0581015\pi$
$797$			10.2490i			0.363039i	−0.983387	+	0.181519i	$0.941898\pi$
$798$		0			0
$799$	36.3381			1.28555
$800$		0			0
$801$	−3.49153			−0.123367
$802$		0			0
$803$			9.82722i			0.346795i
$804$		0			0
$805$	5.95644	−	2.24169i	0.209937	−	0.0790090i
$806$		0			0
$807$		−	26.3659i		−	0.928123i
$808$		0			0
$809$	−37.0993			−1.30434			−0.652170	−	0.758072i	$-0.726141\pi$
$809$	−37.0993			−1.30434			−0.652170	+	0.758072i	$0.726141\pi$
$810$		0			0
$811$	−35.8084			−1.25740			−0.628701	−	0.777647i	$-0.716413\pi$
$811$	−35.8084			−1.25740			−0.628701	+	0.777647i	$0.716413\pi$
$812$		0			0
$813$		−	37.7869i		−	1.32525i
$814$		0			0
$815$	12.9119	+	34.3086i	0.452285	+	1.20178i
$816$		0			0
$817$			1.09648i			0.0383610i
$818$		0			0
$819$	6.59042			0.230288
$820$		0			0
$821$	−21.5343			−0.751551			−0.375775	−	0.926711i	$-0.622624\pi$
$821$	−21.5343			−0.751551			−0.375775	+	0.926711i	$0.622624\pi$
$822$		0			0
$823$		−	41.1466i		−	1.43428i	−0.696929	−	0.717140i	$-0.745451\pi$
$823$		−	41.1466i		−	1.43428i	0.696929	−	0.717140i	$-0.254549\pi$
$824$		0			0
$825$	10.3670	+	11.8224i	0.360933	+	0.411605i
$826$		0			0
$827$			9.48319i			0.329763i	0.986313	+	0.164881i	$0.0527241\pi$
$827$			9.48319i			0.329763i	−0.986313	+	0.164881i	$0.947276\pi$
$828$		0			0
$829$	−24.0091			−0.833872			−0.416936	−	0.908936i	$-0.636896\pi$
$829$	−24.0091			−0.833872			−0.416936	+	0.908936i	$0.636896\pi$
$830$		0			0
$831$	8.95088			0.310503
$832$		0			0
$833$		−	3.48616i		−	0.120788i
$834$		0			0
$835$	−6.40532	−	17.0197i	−0.221665	−	0.588992i
$836$		0			0
$837$		−	45.8915i		−	1.58624i
$838$		0			0
$839$	9.89039			0.341454			0.170727	−	0.985318i	$-0.445388\pi$
$839$	9.89039			0.341454			0.170727	+	0.985318i	$0.445388\pi$
$840$		0			0
$841$	14.8001			0.510348
$842$		0			0
$843$			50.2519i			1.73077i
$844$		0			0
$845$	18.8667	−	7.10043i	0.649035	−	0.244262i
$846$		0			0
$847$		−	20.0781i		−	0.689890i
$848$		0			0
$849$	−1.58017			−0.0542314
$850$		0			0
$851$	1.71181			0.0586802
$852$		0			0
$853$		−	5.02358i		−	0.172004i	−0.996295	−	0.0860021i	$-0.972591\pi$
$853$		−	5.02358i		−	0.172004i	0.996295	−	0.0860021i	$-0.0274092\pi$
$854$		0			0
$855$	6.36306	−	2.39471i	0.217612	−	0.0818975i
$856$		0			0
$857$		−	26.8011i		−	0.915509i	−0.889079	−	0.457754i	$-0.848654\pi$
$857$		−	26.8011i		−	0.915509i	0.889079	−	0.457754i	$-0.151346\pi$
$858$		0			0
$859$	27.3018			0.931525			0.465763	−	0.884910i	$-0.345780\pi$
$859$	27.3018			0.931525			0.465763	+	0.884910i	$0.345780\pi$
$860$		0			0
$861$	30.3007			1.03265
$862$		0			0
$863$			2.22910i			0.0758796i	0.999280	+	0.0379398i	$0.0120795\pi$
$863$			2.22910i			0.0758796i	−0.999280	+	0.0379398i	$0.987920\pi$
$864$		0			0
$865$	−8.29006	−	22.0277i	−0.281871	−	0.748965i
$866$		0			0
$867$		−	11.0375i		−	0.374854i
$868$		0			0
$869$	−16.9474			−0.574900
$870$		0			0
$871$	−14.4305			−0.488960
$872$		0			0
$873$		−	3.45797i		−	0.117035i
$874$		0			0
$875$	−28.0630	−	15.0025i	−0.948703	−	0.507178i
$876$		0			0
$877$		−	46.4517i		−	1.56856i	−0.620407	−	0.784280i	$-0.713032\pi$
$877$		−	46.4517i		−	1.56856i	0.620407	−	0.784280i	$-0.286968\pi$
$878$		0			0
$879$	−21.9980			−0.741975
$880$		0			0
$881$	−25.6878			−0.865444			−0.432722	−	0.901527i	$-0.642447\pi$
$881$	−25.6878			−0.865444			−0.432722	+	0.901527i	$0.642447\pi$
$882$		0			0
$883$			48.8481i			1.64387i	0.569582	+	0.821934i	$0.307105\pi$
$883$			48.8481i			1.64387i	−0.569582	+	0.821934i	$0.692895\pi$
$884$		0			0
$885$	−9.49613	−	25.2324i	−0.319209	−	0.848178i
$886$		0			0
$887$		−	45.1121i		−	1.51472i	−0.653000	−	0.757358i	$-0.726490\pi$
$887$		−	45.1121i		−	1.51472i	0.653000	−	0.757358i	$-0.273510\pi$
$888$		0			0
$889$	−25.9667			−0.870895
$890$		0			0
$891$	14.4528			0.484185
$892$		0			0
$893$			70.6979i			2.36581i
$894$		0			0
$895$	−42.9899	+	16.1791i	−1.43699	+	0.540807i
$896$		0			0
$897$		−	7.42840i		−	0.248027i
$898$		0			0
$899$	54.9130			1.83145
$900$		0			0
$901$	19.5947			0.652792
$902$		0			0
$903$		−	0.801941i		−	0.0266869i
$904$		0			0
$905$	46.1986	−	17.3867i	1.53569	−	0.577953i
$906$		0			0
$907$		−	15.5516i		−	0.516382i	−0.966094	−	0.258191i	$-0.916874\pi$
$907$		−	15.5516i		−	0.516382i	0.966094	−	0.258191i	$-0.0831264\pi$
$908$		0			0
$909$	3.56485			0.118239
$910$		0			0
$911$	30.1283			0.998195			0.499097	−	0.866546i	$-0.333665\pi$
$911$	30.1283			0.998195			0.499097	+	0.866546i	$0.333665\pi$
$912$		0			0
$913$		−	6.94096i		−	0.229712i
$914$		0			0
$915$	14.7857	+	39.2874i	0.488800	+	1.29880i
$916$		0			0
$917$		−	26.5602i		−	0.877094i
$918$		0			0
$919$	34.2483			1.12975			0.564874	−	0.825177i	$-0.308925\pi$
$919$	34.2483			1.12975			0.564874	+	0.825177i	$0.308925\pi$
$920$		0			0
$921$	−5.80452			−0.191265
$922$		0			0
$923$		−	9.20553i		−	0.303004i
$924$		0			0
$925$	−5.64308	−	6.43531i	−0.185543	−	0.211592i
$926$		0			0
$927$		−	6.32556i		−	0.207759i
$928$		0			0
$929$	54.6989			1.79461			0.897307	−	0.441408i	$-0.145521\pi$
$929$	54.6989			1.79461			0.897307	+	0.441408i	$0.145521\pi$
$930$		0			0
$931$	6.78252			0.222288
$932$		0			0
$933$		−	22.8312i		−	0.747462i
$934$		0			0
$935$	4.95431	+	13.1642i	0.162023	+	0.430516i
$936$		0			0
$937$		−	12.1635i		−	0.397364i	−0.980064	−	0.198682i	$-0.936334\pi$
$937$		−	12.1635i		−	0.397364i	0.980064	−	0.198682i	$-0.0636661\pi$
$938$		0			0
$939$	16.6308			0.542725
$940$		0			0
$941$	−27.9048			−0.909672			−0.454836	−	0.890575i	$-0.650302\pi$
$941$	−27.9048			−0.909672			−0.454836	+	0.890575i	$0.650302\pi$
$942$		0			0
$943$			6.72440i			0.218977i
$944$		0			0
$945$	−32.9444	+	12.3985i	−1.07168	+	0.403323i
$946$		0			0
$947$			10.5104i			0.341541i	0.985311	+	0.170770i	$0.0546256\pi$
$947$			10.5104i			0.341541i	−0.985311	+	0.170770i	$0.945374\pi$
$948$		0			0
$949$	−23.2130			−0.753527
$950$		0			0
$951$	50.7916			1.64703
$952$		0			0
$953$		−	11.7005i		−	0.379017i	−0.981879	−	0.189508i	$-0.939311\pi$
$953$		−	11.7005i		−	0.379017i	0.981879	−	0.189508i	$-0.0606894\pi$
$954$		0			0
$955$	−9.43649	+	3.55139i	−0.305358	+	0.114920i
$956$		0			0
$957$			20.8129i			0.672784i
$958$		0			0
$959$	59.7738			1.93020
$960$		0			0
$961$	37.8455			1.22082
$962$		0			0
$963$			5.76609i			0.185810i
$964$		0			0
$965$	16.6606	+	44.2692i	0.536323	+	1.42508i
$966$		0			0
$967$			11.7133i			0.376673i	0.982105	+	0.188337i	$0.0603096\pi$
$967$			11.7133i			0.376673i	−0.982105	+	0.188337i	$0.939690\pi$
$968$		0			0
$969$	−30.8892			−0.992304
$970$		0			0
$971$	37.8963			1.21615			0.608075	−	0.793879i	$-0.291942\pi$
$971$	37.8963			1.21615			0.608075	+	0.793879i	$0.291942\pi$
$972$		0			0
$973$		−	45.9090i		−	1.47178i
$974$		0			0
$975$	−27.9260	+	24.4881i	−0.894347	+	0.784247i
$976$		0			0
$977$			2.83500i			0.0906997i	0.998971	+	0.0453499i	$0.0144403\pi$
$977$			2.83500i			0.0906997i	−0.998971	+	0.0453499i	$0.985560\pi$
$978$		0			0
$979$	14.0537			0.449158
$980$		0			0
$981$	6.71253			0.214314
$982$		0			0
$983$			4.47696i			0.142793i	0.997448	+	0.0713964i	$0.0227455\pi$
$983$			4.47696i			0.142793i	−0.997448	+	0.0713964i	$0.977254\pi$
$984$		0			0
$985$	−7.99224	−	21.2364i	−0.254654	−	0.676647i
$986$		0			0
$987$		−	51.7068i		−	1.64584i
$988$		0			0
$989$	0.177968			0.00565906
$990$		0			0
$991$	32.2466			1.02435			0.512173	−	0.858882i	$-0.328841\pi$
$991$	32.2466			1.02435			0.512173	+	0.858882i	$0.328841\pi$
$992$		0			0
$993$			15.3777i			0.487996i
$994$		0			0
$995$	−13.7620	+	5.17928i	−0.436285	+	0.164194i
$996$		0			0
$997$			27.5708i			0.873177i	0.899661	+	0.436589i	$0.143813\pi$
$997$			27.5708i			0.873177i	−0.899661	+	0.436589i	$0.856187\pi$
$998$		0			0
$999$	−9.46784			−0.299549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.g.369.4		14
4.3	odd	2		920.2.e.b.369.11	yes	14
5.2	odd	4		9200.2.a.cz.1.3		7
5.3	odd	4		9200.2.a.dc.1.5		7
5.4	even	2	inner	1840.2.e.g.369.11		14
20.3	even	4		4600.2.a.bh.1.3		7
20.7	even	4		4600.2.a.bi.1.5		7
20.19	odd	2		920.2.e.b.369.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.4	✓	14	20.19	odd	2
920.2.e.b.369.11	yes	14	4.3	odd	2
1840.2.e.g.369.4		14	1.1	even	1	trivial
1840.2.e.g.369.11		14	5.4	even	2	inner
4600.2.a.bh.1.3		7	20.3	even	4
4600.2.a.bi.1.5		7	20.7	even	4
9200.2.a.cz.1.3		7	5.2	odd	4
9200.2.a.dc.1.5		7	5.3	odd	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.58319i q^{3} +(-2.09277 + 0.787606i) q^{5} +2.84620i q^{7} +0.493499 q^{9} +O(q^{10})\)	\(q-1.58319i q^{3} +(-2.09277 + 0.787606i) q^{5} +2.84620i q^{7} +0.493499 q^{9} -1.98637 q^{11} -4.69204i q^{13} +(1.24693 + 3.31326i) q^{15} +3.16675i q^{17} -6.16110 q^{19} +4.50609 q^{21} +1.00000i q^{23} +(3.75935 - 3.29655i) q^{25} -5.53088i q^{27} +6.61816 q^{29} +8.29732 q^{31} +3.14481i q^{33} +(-2.24169 - 5.95644i) q^{35} -1.71181i q^{37} -7.42840 q^{39} +6.72440 q^{41} -0.177968i q^{43} +(-1.03278 + 0.388683i) q^{45} -11.4749i q^{47} -1.10086 q^{49} +5.01358 q^{51} -6.18762i q^{53} +(4.15701 - 1.56448i) q^{55} +9.75422i q^{57} -7.61559 q^{59} +11.8577 q^{61} +1.40460i q^{63} +(3.69548 + 9.81934i) q^{65} -3.07554i q^{67} +1.58319 q^{69} +1.96195 q^{71} -4.94732i q^{73} +(-5.21908 - 5.95178i) q^{75} -5.65361i q^{77} +8.53182 q^{79} -7.27596 q^{81} +3.49429i q^{83} +(-2.49415 - 6.62728i) q^{85} -10.4778i q^{87} -7.07506 q^{89} +13.3545 q^{91} -13.1363i q^{93} +(12.8938 - 4.85252i) q^{95} -7.00705i q^{97} -0.980272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) 14 * q + 2 * q^5 - 4 * q^9	\( 14 q + 2 q^{5} - 4 q^{9} + 14 q^{11} + 6 q^{15} - 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} + 20 q^{31} + 2 q^{35} - 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} + 14 q^{51} + 38 q^{55} - 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} + 28 q^{71} + 24 q^{75} - 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} + 14 q^{91} + 30 q^{95} - 122 q^{99}+O(q^{100}) \) 14 * q + 2 * q^5 - 4 * q^9 + 14 * q^11 + 6 * q^15 - 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 + 20 * q^31 + 2 * q^35 - 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 + 14 * q^51 + 38 * q^55 - 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 + 28 * q^71 + 24 * q^75 - 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 + 14 * q^91 + 30 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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