LMFDB - Embedded newform 1840.2.e.g.369.13

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			369.13
Root			$0.285770 + 0.285770i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.g.369.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49931i q^{3} +(2.11714 - 0.719533i) q^{5} +2.92777i q^{7} -3.24657 q^{9} +O(q^{10})$	$q+2.49931i q^{3} +(2.11714 - 0.719533i) q^{5} +2.92777i q^{7} -3.24657 q^{9} +4.10909 q^{11} -0.0122971i q^{13} +(1.79834 + 5.29139i) q^{15} -0.155378i q^{17} +4.32471 q^{19} -7.31743 q^{21} +1.00000i q^{23} +(3.96454 - 3.04670i) q^{25} -0.616262i q^{27} +6.79978 q^{29} -2.20713 q^{31} +10.2699i q^{33} +(2.10663 + 6.19850i) q^{35} +4.60741i q^{37} +0.0307342 q^{39} -7.67826 q^{41} -8.38997i q^{43} +(-6.87344 + 2.33602i) q^{45} +6.38116i q^{47} -1.57186 q^{49} +0.388339 q^{51} -7.80358i q^{53} +(8.69951 - 2.95662i) q^{55} +10.8088i q^{57} -14.1275 q^{59} +7.05297 q^{61} -9.50523i q^{63} +(-0.00884814 - 0.0260346i) q^{65} +7.31363i q^{67} -2.49931 q^{69} +5.84061 q^{71} +0.727674i q^{73} +(7.61466 + 9.90864i) q^{75} +12.0305i q^{77} +4.81003 q^{79} -8.19948 q^{81} -7.75797i q^{83} +(-0.111800 - 0.328957i) q^{85} +16.9948i q^{87} -6.77560 q^{89} +0.0360030 q^{91} -5.51631i q^{93} +(9.15602 - 3.11177i) q^{95} -14.8705i q^{97} -13.3405 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			2.49931i			1.44298i	0.692425	+	0.721490i	$0.256542\pi$
$3$			2.49931i			1.44298i	−0.692425	+	0.721490i	$0.743458\pi$
$4$		0			0
$5$	2.11714	−	0.719533i	0.946813	−	0.321785i
$5$	2.11714	−	0.719533i	0.946813	−	0.321785i
$6$		0			0
$7$			2.92777i			1.10659i	0.832984	+	0.553297i	$0.186631\pi$
$7$			2.92777i			1.10659i	−0.832984	+	0.553297i	$0.813369\pi$
$8$		0			0
$9$	−3.24657			−1.08219
$10$		0			0
$11$	4.10909			1.23894			0.619468	−	0.785022i	$-0.287348\pi$
$11$	4.10909			1.23894			0.619468	+	0.785022i	$0.287348\pi$
$12$		0			0
$13$		−	0.0122971i		−	0.00341059i	−0.999999	−	0.00170530i	$-0.999457\pi$
$13$		−	0.0122971i		−	0.00341059i	0.999999	−	0.00170530i	$-0.000542813\pi$
$14$		0			0
$15$	1.79834	+	5.29139i	0.464329	+	1.36623i
$16$		0			0
$17$		−	0.155378i		−	0.0376847i	−0.999822	−	0.0188424i	$-0.994002\pi$
$17$		−	0.155378i		−	0.0376847i	0.999822	−	0.0188424i	$-0.00599807\pi$
$18$		0			0
$19$	4.32471			0.992158			0.496079	−	0.868277i	$-0.334773\pi$
$19$	4.32471			0.992158			0.496079	+	0.868277i	$0.334773\pi$
$20$		0			0
$21$	−7.31743			−1.59679
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	3.96454	−	3.04670i	0.792909	−	0.609340i
$26$		0			0
$27$		−	0.616262i		−	0.118600i
$28$		0			0
$29$	6.79978			1.26269			0.631344	−	0.775503i	$-0.282504\pi$
$29$	6.79978			1.26269			0.631344	+	0.775503i	$0.282504\pi$
$30$		0			0
$31$	−2.20713			−0.396412			−0.198206	−	0.980160i	$-0.563511\pi$
$31$	−2.20713			−0.396412			−0.198206	+	0.980160i	$0.563511\pi$
$32$		0			0
$33$			10.2699i			1.78776i
$34$		0			0
$35$	2.10663	+	6.19850i	0.356085	+	1.04774i
$36$		0			0
$37$			4.60741i			0.757453i	0.925509	+	0.378726i	$0.123638\pi$
$37$			4.60741i			0.757453i	−0.925509	+	0.378726i	$0.876362\pi$
$38$		0			0
$39$	0.0307342			0.00492142
$40$		0			0
$41$	−7.67826			−1.19914			−0.599571	−	0.800321i	$-0.704662\pi$
$41$	−7.67826			−1.19914			−0.599571	+	0.800321i	$0.704662\pi$
$42$		0			0
$43$		−	8.38997i		−	1.27946i	−0.768600	−	0.639729i	$-0.779046\pi$
$43$		−	8.38997i		−	1.27946i	0.768600	−	0.639729i	$-0.220954\pi$
$44$		0			0
$45$	−6.87344	+	2.33602i	−1.02463	+	0.348233i
$46$		0			0
$47$			6.38116i			0.930787i	0.885104	+	0.465394i	$0.154087\pi$
$47$			6.38116i			0.930787i	−0.885104	+	0.465394i	$0.845913\pi$
$48$		0			0
$49$	−1.57186			−0.224551
$50$		0			0
$51$	0.388339			0.0543783
$52$		0			0
$53$		−	7.80358i		−	1.07190i	−0.844248	−	0.535952i	$-0.819953\pi$
$53$		−	7.80358i		−	1.07190i	0.844248	−	0.535952i	$-0.180047\pi$
$54$		0			0
$55$	8.69951	−	2.95662i	1.17304	−	0.398671i
$56$		0			0
$57$			10.8088i			1.43166i
$58$		0			0
$59$	−14.1275			−1.83924			−0.919622	−	0.392805i	$-0.871505\pi$
$59$	−14.1275			−1.83924			−0.919622	+	0.392805i	$0.871505\pi$
$60$		0			0
$61$	7.05297			0.903040			0.451520	−	0.892261i	$-0.350882\pi$
$61$	7.05297			0.903040			0.451520	+	0.892261i	$0.350882\pi$
$62$		0			0
$63$		−	9.50523i		−	1.19755i
$64$		0			0
$65$	−0.00884814	−	0.0260346i	−0.00109748	−	0.00322919i
$66$		0			0
$67$			7.31363i			0.893502i	0.894658	+	0.446751i	$0.147419\pi$
$67$			7.31363i			0.893502i	−0.894658	+	0.446751i	$0.852581\pi$
$68$		0			0
$69$	−2.49931			−0.300882
$70$		0			0
$71$	5.84061			0.693153			0.346576	−	0.938022i	$-0.387344\pi$
$71$	5.84061			0.693153			0.346576	+	0.938022i	$0.387344\pi$
$72$		0			0
$73$			0.727674i			0.0851678i	0.999093	+	0.0425839i	$0.0135590\pi$
$73$			0.727674i			0.0851678i	−0.999093	+	0.0425839i	$0.986441\pi$
$74$		0			0
$75$	7.61466	+	9.90864i	0.879266	+	1.14415i
$76$		0			0
$77$			12.0305i			1.37100i
$78$		0			0
$79$	4.81003			0.541171			0.270586	−	0.962696i	$-0.412783\pi$
$79$	4.81003			0.541171			0.270586	+	0.962696i	$0.412783\pi$
$80$		0			0
$81$	−8.19948			−0.911054
$82$		0			0
$83$		−	7.75797i		−	0.851548i	−0.904830	−	0.425774i	$-0.860002\pi$
$83$		−	7.75797i		−	0.851548i	0.904830	−	0.425774i	$-0.139998\pi$
$84$		0			0
$85$	−0.111800	−	0.328957i	−0.0121264	−	0.0356804i
$86$		0			0
$87$			16.9948i			1.82203i
$88$		0			0
$89$	−6.77560			−0.718212			−0.359106	−	0.933297i	$-0.616918\pi$
$89$	−6.77560			−0.718212			−0.359106	+	0.933297i	$0.616918\pi$
$90$		0			0
$91$	0.0360030			0.00377414
$92$		0			0
$93$		−	5.51631i		−	0.572014i
$94$		0			0
$95$	9.15602	−	3.11177i	0.939388	−	0.319261i
$96$		0			0
$97$		−	14.8705i		−	1.50987i	−0.655799	−	0.754935i	$-0.727668\pi$
$97$		−	14.8705i		−	1.50987i	0.655799	−	0.754935i	$-0.272332\pi$
$98$		0			0
$99$	−13.3405			−1.34077
$100$		0			0
$101$	−13.0023			−1.29378			−0.646889	−	0.762584i	$-0.723930\pi$
$101$	−13.0023			−1.29378			−0.646889	+	0.762584i	$0.723930\pi$
$102$		0			0
$103$			1.85814i			0.183088i	0.995801	+	0.0915442i	$0.0291803\pi$
$103$			1.85814i			0.183088i	−0.995801	+	0.0915442i	$0.970820\pi$
$104$		0			0
$105$	−15.4920	+	5.26513i	−1.51186	+	0.513824i
$106$		0			0
$107$			0.471267i			0.0455591i	0.999741	+	0.0227795i	$0.00725158\pi$
$107$			0.471267i			0.0455591i	−0.999741	+	0.0227795i	$0.992748\pi$
$108$		0			0
$109$	1.83022			0.175303			0.0876515	−	0.996151i	$-0.472064\pi$
$109$	1.83022			0.175303			0.0876515	+	0.996151i	$0.472064\pi$
$110$		0			0
$111$	−11.5154			−1.09299
$112$		0			0
$113$			7.81151i			0.734845i	0.930054	+	0.367422i	$0.119760\pi$
$113$			7.81151i			0.734845i	−0.930054	+	0.367422i	$0.880240\pi$
$114$		0			0
$115$	0.719533	+	2.11714i	0.0670968	+	0.197424i
$116$		0			0
$117$			0.0399233i			0.00369091i
$118$		0			0
$119$	0.454912			0.0417017
$120$		0			0
$121$	5.88461			0.534964
$122$		0			0
$123$		−	19.1904i		−	1.73034i
$124$		0			0
$125$	6.20129	−	9.30291i	0.554660	−	0.832077i
$126$		0			0
$127$		−	7.62237i		−	0.676376i	−0.941079	−	0.338188i	$-0.890186\pi$
$127$		−	7.62237i		−	0.676376i	0.941079	−	0.338188i	$-0.109814\pi$
$128$		0			0
$129$	20.9692			1.84623
$130$		0			0
$131$	−19.4575			−1.70001			−0.850007	−	0.526772i	$-0.823402\pi$
$131$	−19.4575			−1.70001			−0.850007	+	0.526772i	$0.823402\pi$
$132$		0			0
$133$			12.6618i			1.09792i
$134$		0			0
$135$	−0.443421	−	1.30471i	−0.0381636	−	0.112292i
$136$		0			0
$137$		−	16.3163i		−	1.39399i	−0.717074	−	0.696997i	$-0.754519\pi$
$137$		−	16.3163i		−	1.39399i	0.717074	−	0.696997i	$-0.245481\pi$
$138$		0			0
$139$	−19.9289			−1.69035			−0.845175	−	0.534490i	$-0.820504\pi$
$139$	−19.9289			−1.69035			−0.845175	+	0.534490i	$0.820504\pi$
$140$		0			0
$141$	−15.9485			−1.34311
$142$		0			0
$143$		−	0.0505297i		−	0.00422551i
$144$		0			0
$145$	14.3961	−	4.89267i	1.19553	−	0.406314i
$146$		0			0
$147$		−	3.92857i		−	0.324023i
$148$		0			0
$149$	8.04540			0.659105			0.329552	−	0.944137i	$-0.393102\pi$
$149$	8.04540			0.659105			0.329552	+	0.944137i	$0.393102\pi$
$150$		0			0
$151$	−3.40963			−0.277472			−0.138736	−	0.990329i	$-0.544304\pi$
$151$	−3.40963			−0.277472			−0.138736	+	0.990329i	$0.544304\pi$
$152$		0			0
$153$			0.504446i			0.0407821i
$154$		0			0
$155$	−4.67279	+	1.58810i	−0.375328	+	0.127559i
$156$		0			0
$157$			21.1868i			1.69089i	0.534061	+	0.845446i	$0.320665\pi$
$157$			21.1868i			1.69089i	−0.534061	+	0.845446i	$0.679335\pi$
$158$		0			0
$159$	19.5036			1.54674
$160$		0			0
$161$	−2.92777			−0.230741
$162$		0			0
$163$			20.9487i			1.64083i	0.571769	+	0.820415i	$0.306257\pi$
$163$			20.9487i			1.64083i	−0.571769	+	0.820415i	$0.693743\pi$
$164$		0			0
$165$	7.38953	+	21.7428i	0.575274	+	1.69267i
$166$		0			0
$167$			10.9861i			0.850131i	0.905163	+	0.425065i	$0.139749\pi$
$167$			10.9861i			0.850131i	−0.905163	+	0.425065i	$0.860251\pi$
$168$		0			0
$169$	12.9998			0.999988
$170$		0			0
$171$	−14.0405			−1.07370
$172$		0			0
$173$		−	4.31110i		−	0.327767i	−0.986480	−	0.163884i	$-0.947598\pi$
$173$		−	4.31110i		−	0.327767i	0.986480	−	0.163884i	$-0.0524021\pi$
$174$		0			0
$175$	8.92005	+	11.6073i	0.674292	+	0.877429i
$176$		0			0
$177$		−	35.3091i		−	2.65399i
$178$		0			0
$179$	−19.0791			−1.42604			−0.713018	−	0.701146i	$-0.752672\pi$
$179$	−19.0791			−1.42604			−0.713018	+	0.701146i	$0.752672\pi$
$180$		0			0
$181$	4.09312			0.304239			0.152120	−	0.988362i	$-0.451390\pi$
$181$	4.09312			0.304239			0.152120	+	0.988362i	$0.451390\pi$
$182$		0			0
$183$			17.6276i			1.30307i
$184$		0			0
$185$	3.31518	+	9.75451i	0.243737	+	0.717166i
$186$		0			0
$187$		−	0.638462i		−	0.0466890i
$188$		0			0
$189$	1.80428			0.131242
$190$		0			0
$191$	−6.21032			−0.449363			−0.224682	−	0.974432i	$-0.572134\pi$
$191$	−6.21032			−0.449363			−0.224682	+	0.974432i	$0.572134\pi$
$192$		0			0
$193$		−	13.2674i		−	0.955005i	−0.878630	−	0.477503i	$-0.841542\pi$
$193$		−	13.2674i		−	0.955005i	0.878630	−	0.477503i	$-0.158458\pi$
$194$		0			0
$195$	0.0650686	−	0.0221143i	0.00465966	−	0.00158364i
$196$		0			0
$197$			13.8836i			0.989163i	0.869131	+	0.494582i	$0.164679\pi$
$197$			13.8836i			0.989163i	−0.869131	+	0.494582i	$0.835321\pi$
$198$		0			0
$199$	−22.8004			−1.61628			−0.808139	−	0.588992i	$-0.799525\pi$
$199$	−22.8004			−1.61628			−0.808139	+	0.588992i	$0.799525\pi$
$200$		0			0
$201$	−18.2791			−1.28931
$202$		0			0
$203$			19.9082i			1.39728i
$204$		0			0
$205$	−16.2559	+	5.52476i	−1.13536	+	0.385866i
$206$		0			0
$207$		−	3.24657i		−	0.225652i
$208$		0			0
$209$	17.7706			1.22922
$210$		0			0
$211$	0.698944			0.0481173			0.0240587	−	0.999711i	$-0.492341\pi$
$211$	0.698944			0.0481173			0.0240587	+	0.999711i	$0.492341\pi$
$212$		0			0
$213$			14.5975i			1.00021i
$214$		0			0
$215$	−6.03686	−	17.7627i	−0.411711	−	1.21141i
$216$		0			0
$217$		−	6.46197i		−	0.438667i
$218$		0			0
$219$	−1.81869			−0.122895
$220$		0			0
$221$	−0.00191069			−0.000128527
$222$		0			0
$223$		−	27.3719i		−	1.83296i	−0.400082	−	0.916480i	$-0.631018\pi$
$223$		−	27.3719i		−	1.83296i	0.400082	−	0.916480i	$-0.368982\pi$
$224$		0			0
$225$	−12.8712	+	9.89134i	−0.858079	+	0.659422i
$226$		0			0
$227$		−	10.6070i		−	0.704011i	−0.935998	−	0.352005i	$-0.885500\pi$
$227$		−	10.6070i		−	0.704011i	0.935998	−	0.352005i	$-0.114500\pi$
$228$		0			0
$229$	−3.11433			−0.205801			−0.102900	−	0.994692i	$-0.532812\pi$
$229$	−3.11433			−0.205801			−0.102900	+	0.994692i	$0.532812\pi$
$230$		0			0
$231$	−30.0680			−1.97833
$232$		0			0
$233$		−	23.5295i		−	1.54147i	−0.637156	−	0.770735i	$-0.719889\pi$
$233$		−	23.5295i		−	1.54147i	0.637156	−	0.770735i	$-0.280111\pi$
$234$		0			0
$235$	4.59145	+	13.5098i	0.299513	+	0.881281i
$236$		0			0
$237$			12.0218i			0.780899i
$238$		0			0
$239$	24.4214			1.57969			0.789844	−	0.613308i	$-0.210162\pi$
$239$	24.4214			1.57969			0.789844	+	0.613308i	$0.210162\pi$
$240$		0			0
$241$	27.5525			1.77481			0.887407	−	0.460986i	$-0.152504\pi$
$241$	27.5525			1.77481			0.887407	+	0.460986i	$0.152504\pi$
$242$		0			0
$243$		−	22.3419i		−	1.43323i
$244$		0			0
$245$	−3.32784	+	1.13101i	−0.212608	+	0.0722573i
$246$		0			0
$247$		−	0.0531813i		−	0.00338385i
$248$		0			0
$249$	19.3896			1.22877
$250$		0			0
$251$	25.3692			1.60129			0.800645	−	0.599139i	$-0.204490\pi$
$251$	25.3692			1.60129			0.800645	+	0.599139i	$0.204490\pi$
$252$		0			0
$253$			4.10909i			0.258336i
$254$		0			0
$255$	0.822167	−	0.279423i	0.0514861	−	0.0174981i
$256$		0			0
$257$			19.8501i			1.23822i	0.785306	+	0.619108i	$0.212506\pi$
$257$			19.8501i			1.23822i	−0.785306	+	0.619108i	$0.787494\pi$
$258$		0			0
$259$	−13.4894			−0.838193
$260$		0			0
$261$	−22.0760			−1.36647
$262$		0			0
$263$			22.9087i			1.41261i	0.707906	+	0.706306i	$0.249640\pi$
$263$			22.9087i			1.41261i	−0.707906	+	0.706306i	$0.750360\pi$
$264$		0			0
$265$	−5.61493	−	16.5213i	−0.344923	−	1.01489i
$266$		0			0
$267$		−	16.9344i		−	1.03637i
$268$		0			0
$269$	−1.03048			−0.0628295			−0.0314148	−	0.999506i	$-0.510001\pi$
$269$	−1.03048			−0.0628295			−0.0314148	+	0.999506i	$0.510001\pi$
$270$		0			0
$271$	3.79576			0.230576			0.115288	−	0.993332i	$-0.463221\pi$
$271$	3.79576			0.230576			0.115288	+	0.993332i	$0.463221\pi$
$272$		0			0
$273$			0.0899829i			0.00544601i
$274$		0			0
$275$	16.2907	−	12.5192i	0.982364	−	0.754934i
$276$		0			0
$277$		−	1.48782i		−	0.0893946i	−0.999001	−	0.0446973i	$-0.985768\pi$
$277$		−	1.48782i		−	0.0893946i	0.999001	−	0.0446973i	$-0.0142323\pi$
$278$		0			0
$279$	7.16560			0.428993
$280$		0			0
$281$	−10.2796			−0.613228			−0.306614	−	0.951834i	$-0.599196\pi$
$281$	−10.2796			−0.613228			−0.306614	+	0.951834i	$0.599196\pi$
$282$		0			0
$283$		−	4.62427i		−	0.274884i	−0.990510	−	0.137442i	$-0.956112\pi$
$283$		−	4.62427i		−	0.274884i	0.990510	−	0.137442i	$-0.0438881\pi$
$284$		0			0
$285$	7.77730	+	22.8838i	0.460688	+	1.35552i
$286$		0			0
$287$		−	22.4802i		−	1.32697i
$288$		0			0
$289$	16.9759			0.998580
$290$		0			0
$291$	37.1661			2.17871
$292$		0			0
$293$		−	16.3067i		−	0.952650i	−0.879269	−	0.476325i	$-0.841969\pi$
$293$		−	16.3067i		−	0.952650i	0.879269	−	0.476325i	$-0.158031\pi$
$294$		0			0
$295$	−29.9099	+	10.1652i	−1.74142	+	0.591841i
$296$		0			0
$297$		−	2.53228i		−	0.146938i
$298$		0			0
$299$	0.0122971			0.000711158
$300$		0			0
$301$	24.5639			1.41584
$302$		0			0
$303$		−	32.4969i		−	1.86690i
$304$		0			0
$305$	14.9321	−	5.07484i	0.855010	−	0.290585i
$306$		0			0
$307$		−	9.28659i		−	0.530014i	−0.964246	−	0.265007i	$-0.914626\pi$
$307$		−	9.28659i		−	0.530014i	0.964246	−	0.265007i	$-0.0853743\pi$
$308$		0			0
$309$	−4.64409			−0.264193
$310$		0			0
$311$	24.3713			1.38197			0.690984	−	0.722870i	$-0.257178\pi$
$311$	24.3713			1.38197			0.690984	+	0.722870i	$0.257178\pi$
$312$		0			0
$313$		−	15.0048i		−	0.848119i	−0.905634	−	0.424060i	$-0.860605\pi$
$313$		−	15.0048i		−	0.848119i	0.905634	−	0.424060i	$-0.139395\pi$
$314$		0			0
$315$	−6.83933	−	20.1239i	−0.385352	−	1.13385i
$316$		0			0
$317$		−	7.84747i		−	0.440758i	−0.975414	−	0.220379i	$-0.929271\pi$
$317$		−	7.84747i		−	0.440758i	0.975414	−	0.220379i	$-0.0707294\pi$
$318$		0			0
$319$	27.9409			1.56439
$320$		0			0
$321$	−1.17784			−0.0657408
$322$		0			0
$323$		−	0.671966i		−	0.0373892i
$324$		0			0
$325$	−0.0374655	−	0.0487523i	−0.00207821	−	0.00270429i
$326$		0			0
$327$			4.57429i			0.252959i
$328$		0			0
$329$	−18.6826			−1.03000
$330$		0			0
$331$	−9.28156			−0.510160			−0.255080	−	0.966920i	$-0.582102\pi$
$331$	−9.28156			−0.510160			−0.255080	+	0.966920i	$0.582102\pi$
$332$		0			0
$333$		−	14.9583i		−	0.819709i
$334$		0			0
$335$	5.26240	+	15.4840i	0.287515	+	0.845979i
$336$		0			0
$337$			26.3760i			1.43679i	0.695634	+	0.718397i	$0.255124\pi$
$337$			26.3760i			1.43679i	−0.695634	+	0.718397i	$0.744876\pi$
$338$		0			0
$339$	−19.5234			−1.06037
$340$		0			0
$341$	−9.06928			−0.491129
$342$		0			0
$343$			15.8924i			0.858107i
$344$		0			0
$345$	−5.29139	+	1.79834i	−0.284879	+	0.0968193i
$346$		0			0
$347$		−	18.3316i		−	0.984093i	−0.870569	−	0.492047i	$-0.836249\pi$
$347$		−	18.3316i		−	0.984093i	0.870569	−	0.492047i	$-0.163751\pi$
$348$		0			0
$349$	32.2759			1.72769			0.863845	−	0.503758i	$-0.168050\pi$
$349$	32.2759			1.72769			0.863845	+	0.503758i	$0.168050\pi$
$350$		0			0
$351$	−0.00757822			−0.000404495
$352$		0			0
$353$		−	29.5653i		−	1.57360i	−0.617208	−	0.786800i	$-0.711736\pi$
$353$		−	29.5653i		−	1.57360i	0.617208	−	0.786800i	$-0.288264\pi$
$354$		0			0
$355$	12.3654	−	4.20251i	0.656286	−	0.223046i
$356$		0			0
$357$			1.13697i			0.0601747i
$358$		0			0
$359$	33.4169			1.76368			0.881838	−	0.471552i	$-0.156306\pi$
$359$	33.4169			1.76368			0.881838	+	0.471552i	$0.156306\pi$
$360$		0			0
$361$	−0.296842			−0.0156232
$362$		0			0
$363$			14.7075i			0.771943i
$364$		0			0
$365$	0.523585	+	1.54059i	0.0274057	+	0.0806379i
$366$		0			0
$367$		−	25.4764i		−	1.32986i	−0.746908	−	0.664928i	$-0.768462\pi$
$367$		−	25.4764i		−	1.32986i	0.746908	−	0.664928i	$-0.231538\pi$
$368$		0			0
$369$	24.9280			1.29770
$370$		0			0
$371$	22.8471			1.18616
$372$		0			0
$373$		−	3.28172i		−	0.169921i	−0.996384	−	0.0849605i	$-0.972924\pi$
$373$		−	3.28172i		−	0.169921i	0.996384	−	0.0849605i	$-0.0270764\pi$
$374$		0			0
$375$	23.2509	+	15.4990i	1.20067	+	0.800363i
$376$		0			0
$377$		−	0.0836174i		−	0.00430651i
$378$		0			0
$379$	5.66762			0.291126			0.145563	−	0.989349i	$-0.453501\pi$
$379$	5.66762			0.291126			0.145563	+	0.989349i	$0.453501\pi$
$380$		0			0
$381$	19.0507			0.975997
$382$		0			0
$383$		−	34.4343i		−	1.75951i	−0.475428	−	0.879755i	$-0.657707\pi$
$383$		−	34.4343i		−	1.75951i	0.475428	−	0.879755i	$-0.342293\pi$
$384$		0			0
$385$	8.65633	+	25.4702i	0.441167	+	1.29808i
$386$		0			0
$387$			27.2387i			1.38462i
$388$		0			0
$389$	14.3737			0.728773			0.364387	−	0.931248i	$-0.381279\pi$
$389$	14.3737			0.728773			0.364387	+	0.931248i	$0.381279\pi$
$390$		0			0
$391$	0.155378			0.00785781
$392$		0			0
$393$		−	48.6305i		−	2.45309i
$394$		0			0
$395$	10.1835	−	3.46098i	0.512388	−	0.174141i
$396$		0			0
$397$			14.6149i			0.733500i	0.930319	+	0.366750i	$0.119530\pi$
$397$			14.6149i			0.733500i	−0.930319	+	0.366750i	$0.880470\pi$
$398$		0			0
$399$	−31.6458			−1.58427
$400$		0			0
$401$	−3.65553			−0.182548			−0.0912742	−	0.995826i	$-0.529094\pi$
$401$	−3.65553			−0.182548			−0.0912742	+	0.995826i	$0.529094\pi$
$402$		0			0
$403$			0.0271412i			0.00135200i
$404$		0			0
$405$	−17.3594	+	5.89980i	−0.862597	+	0.293163i
$406$		0			0
$407$			18.9322i			0.938436i
$408$		0			0
$409$	−4.40926			−0.218024			−0.109012	−	0.994040i	$-0.534769\pi$
$409$	−4.40926			−0.218024			−0.109012	+	0.994040i	$0.534769\pi$
$410$		0			0
$411$	40.7795			2.01150
$412$		0			0
$413$		−	41.3621i		−	2.03530i
$414$		0			0
$415$	−5.58212	−	16.4247i	−0.274015	−	0.806257i
$416$		0			0
$417$		−	49.8087i		−	2.43914i
$418$		0			0
$419$	32.5906			1.59216			0.796078	−	0.605195i	$-0.206905\pi$
$419$	32.5906			1.59216			0.796078	+	0.605195i	$0.206905\pi$
$420$		0			0
$421$	4.23954			0.206622			0.103311	−	0.994649i	$-0.467056\pi$
$421$	4.23954			0.206622			0.103311	+	0.994649i	$0.467056\pi$
$422$		0			0
$423$		−	20.7169i		−	1.00729i
$424$		0			0
$425$	−0.473391	−	0.616003i	−0.0229628	−	0.0298806i
$426$		0			0
$427$			20.6495i			0.999299i
$428$		0			0
$429$	0.126290			0.00609732
$430$		0			0
$431$	−1.81123			−0.0872436			−0.0436218	−	0.999048i	$-0.513890\pi$
$431$	−1.81123			−0.0872436			−0.0436218	+	0.999048i	$0.513890\pi$
$432$		0			0
$433$		−	1.76263i		−	0.0847068i	−0.999103	−	0.0423534i	$-0.986514\pi$
$433$		−	1.76263i		−	0.0847068i	0.999103	−	0.0423534i	$-0.0134855\pi$
$434$		0			0
$435$	12.2283	+	35.9803i	0.586303	+	1.72512i
$436$		0			0
$437$			4.32471i			0.206879i
$438$		0			0
$439$	−26.7848			−1.27837			−0.639184	−	0.769054i	$-0.720728\pi$
$439$	−26.7848			−1.27837			−0.639184	+	0.769054i	$0.720728\pi$
$440$		0			0
$441$	5.10316			0.243008
$442$		0			0
$443$		−	16.6631i		−	0.791686i	−0.918318	−	0.395843i	$-0.870452\pi$
$443$		−	16.6631i		−	0.791686i	0.918318	−	0.395843i	$-0.129548\pi$
$444$		0			0
$445$	−14.3449	+	4.87527i	−0.680012	+	0.231110i
$446$		0			0
$447$			20.1080i			0.951075i
$448$		0			0
$449$	34.1236			1.61039			0.805196	−	0.593008i	$-0.202060\pi$
$449$	34.1236			1.61039			0.805196	+	0.593008i	$0.202060\pi$
$450$		0			0
$451$	−31.5507			−1.48566
$452$		0			0
$453$		−	8.52174i		−	0.400386i
$454$		0			0
$455$	0.0762234	−	0.0259054i	0.00357341	−	0.00121446i
$456$		0			0
$457$			24.5980i			1.15065i	0.817926	+	0.575324i	$0.195124\pi$
$457$			24.5980i			1.15065i	−0.817926	+	0.575324i	$0.804876\pi$
$458$		0			0
$459$	−0.0957537			−0.00446940
$460$		0			0
$461$	−10.7297			−0.499733			−0.249866	−	0.968280i	$-0.580387\pi$
$461$	−10.7297			−0.499733			−0.249866	+	0.968280i	$0.580387\pi$
$462$		0			0
$463$			22.3124i			1.03695i	0.855094	+	0.518473i	$0.173499\pi$
$463$			22.3124i			1.03695i	−0.855094	+	0.518473i	$0.826501\pi$
$464$		0			0
$465$	−3.96917	−	11.6788i	−0.184066	−	0.541591i
$466$		0			0
$467$		−	20.0218i		−	0.926497i	−0.886229	−	0.463248i	$-0.846684\pi$
$467$		−	20.0218i		−	0.926497i	0.886229	−	0.463248i	$-0.153316\pi$
$468$		0			0
$469$	−21.4127			−0.988744
$470$		0			0
$471$	−52.9525			−2.43992
$472$		0			0
$473$		−	34.4751i		−	1.58517i
$474$		0			0
$475$	17.1455	−	13.1761i	0.786691	−	0.604561i
$476$		0			0
$477$			25.3349i			1.16001i
$478$		0			0
$479$	−7.61905			−0.348123			−0.174062	−	0.984735i	$-0.555689\pi$
$479$	−7.61905			−0.348123			−0.174062	+	0.984735i	$0.555689\pi$
$480$		0			0
$481$	0.0566576			0.00258336
$482$		0			0
$483$		−	7.31743i		−	0.332954i
$484$		0			0
$485$	−10.6998	−	31.4829i	−0.485854	−	1.42956i
$486$		0			0
$487$			21.5163i			0.974997i	0.873124	+	0.487499i	$0.162091\pi$
$487$			21.5163i			0.974997i	−0.873124	+	0.487499i	$0.837909\pi$
$488$		0			0
$489$	−52.3574			−2.36768
$490$		0			0
$491$	10.3705			0.468015			0.234007	−	0.972235i	$-0.424816\pi$
$491$	10.3705			0.468015			0.234007	+	0.972235i	$0.424816\pi$
$492$		0			0
$493$		−	1.05654i		−	0.0475841i
$494$		0			0
$495$	−28.2436	+	9.59890i	−1.26945	+	0.431438i
$496$		0			0
$497$			17.1000i			0.767039i
$498$		0			0
$499$	−16.2850			−0.729018			−0.364509	−	0.931200i	$-0.618763\pi$
$499$	−16.2850			−0.729018			−0.364509	+	0.931200i	$0.618763\pi$
$500$		0			0
$501$	−27.4577			−1.22672
$502$		0			0
$503$		−	16.9326i		−	0.754986i	−0.926012	−	0.377493i	$-0.876786\pi$
$503$		−	16.9326i		−	0.754986i	0.926012	−	0.377493i	$-0.123214\pi$
$504$		0			0
$505$	−27.5277	+	9.35559i	−1.22497	+	0.416318i
$506$		0			0
$507$			32.4907i			1.44296i
$508$		0			0
$509$	−39.0075			−1.72898			−0.864488	−	0.502654i	$-0.832357\pi$
$509$	−39.0075			−1.72898			−0.864488	+	0.502654i	$0.832357\pi$
$510$		0			0
$511$	−2.13046			−0.0942462
$512$		0			0
$513$		−	2.66516i		−	0.117670i
$514$		0			0
$515$	1.33700	+	3.93395i	0.0589151	+	0.173350i
$516$		0			0
$517$			26.2207i			1.15319i
$518$		0			0
$519$	10.7748			0.472961
$520$		0			0
$521$	33.1380			1.45180			0.725902	−	0.687798i	$-0.241423\pi$
$521$	33.1380			1.45180			0.725902	+	0.687798i	$0.241423\pi$
$522$		0			0
$523$		−	12.4056i		−	0.542460i	−0.962515	−	0.271230i	$-0.912570\pi$
$523$		−	12.4056i		−	0.542460i	0.962515	−	0.271230i	$-0.0874304\pi$
$524$		0			0
$525$	−29.0103	+	22.2940i	−1.26611	+	0.972990i
$526$		0			0
$527$			0.342939i			0.0149387i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	45.8660			1.99041
$532$		0			0
$533$			0.0944201i			0.00408979i
$534$		0			0
$535$	0.339092	+	0.997737i	0.0146602	+	0.0431359i
$536$		0			0
$537$		−	47.6846i		−	2.05774i
$538$		0			0
$539$	−6.45891			−0.278205
$540$		0			0
$541$	−8.53472			−0.366936			−0.183468	−	0.983026i	$-0.558732\pi$
$541$	−8.53472			−0.366936			−0.183468	+	0.983026i	$0.558732\pi$
$542$		0			0
$543$			10.2300i			0.439011i
$544$		0			0
$545$	3.87482	−	1.31690i	0.165979	−	0.0564099i
$546$		0			0
$547$			6.45828i			0.276136i	0.990423	+	0.138068i	$0.0440893\pi$
$547$			6.45828i			0.276136i	−0.990423	+	0.138068i	$0.955911\pi$
$548$		0			0
$549$	−22.8980			−0.977262
$550$		0			0
$551$	29.4071			1.25279
$552$		0			0
$553$			14.0827i			0.598857i
$554$		0			0
$555$	−24.3796	+	8.28568i	−1.03486	+	0.351708i
$556$		0			0
$557$			4.08413i			0.173050i	0.996250	+	0.0865251i	$0.0275763\pi$
$557$			4.08413i			0.173050i	−0.996250	+	0.0865251i	$0.972424\pi$
$558$		0			0
$559$	−0.103172			−0.00436371
$560$		0			0
$561$	1.59572			0.0673713
$562$		0			0
$563$			15.4494i			0.651114i	0.945522	+	0.325557i	$0.105552\pi$
$563$			15.4494i			0.651114i	−0.945522	+	0.325557i	$0.894448\pi$
$564$		0			0
$565$	5.62064	+	16.5380i	0.236462	+	0.695761i
$566$		0			0
$567$		−	24.0062i		−	1.00817i
$568$		0			0
$569$	−42.2633			−1.77177			−0.885886	−	0.463903i	$-0.846448\pi$
$569$	−42.2633			−1.77177			−0.885886	+	0.463903i	$0.846448\pi$
$570$		0			0
$571$	2.74497			0.114873			0.0574367	−	0.998349i	$-0.481707\pi$
$571$	2.74497			0.114873			0.0574367	+	0.998349i	$0.481707\pi$
$572$		0			0
$573$		−	15.5215i		−	0.648422i
$574$		0			0
$575$	3.04670	+	3.96454i	0.127056	+	0.165333i
$576$		0			0
$577$		−	8.93871i		−	0.372123i	−0.982538	−	0.186062i	$-0.940428\pi$
$577$		−	8.93871i		−	0.372123i	0.982538	−	0.186062i	$-0.0595724\pi$
$578$		0			0
$579$	33.1593			1.37805
$580$		0			0
$581$	22.7136			0.942318
$582$		0			0
$583$		−	32.0656i		−	1.32802i
$584$		0			0
$585$	0.0287261	+	0.0845232i	0.00118768	+	0.00349460i
$586$		0			0
$587$			5.90821i			0.243858i	0.992539	+	0.121929i	$0.0389080\pi$
$587$			5.90821i			0.243858i	−0.992539	+	0.121929i	$0.961092\pi$
$588$		0			0
$589$	−9.54520			−0.393303
$590$		0			0
$591$	−34.6994			−1.42734
$592$		0			0
$593$			27.8823i			1.14499i	0.819909	+	0.572494i	$0.194024\pi$
$593$			27.8823i			1.14499i	−0.819909	+	0.572494i	$0.805976\pi$
$594$		0			0
$595$	0.963111	−	0.327324i	0.0394837	−	0.0134190i
$596$		0			0
$597$		−	56.9854i		−	2.33226i
$598$		0			0
$599$	10.5644			0.431648			0.215824	−	0.976432i	$-0.430756\pi$
$599$	10.5644			0.431648			0.215824	+	0.976432i	$0.430756\pi$
$600$		0			0
$601$	35.8910			1.46403			0.732013	−	0.681291i	$-0.238581\pi$
$601$	35.8910			1.46403			0.732013	+	0.681291i	$0.238581\pi$
$602$		0			0
$603$		−	23.7442i		−	0.966940i
$604$		0			0
$605$	12.4585	−	4.23417i	0.506511	−	0.172143i
$606$		0			0
$607$		−	29.0640i		−	1.17967i	−0.807524	−	0.589835i	$-0.799193\pi$
$607$		−	29.0640i		−	1.17967i	0.807524	−	0.589835i	$-0.200807\pi$
$608$		0			0
$609$	−49.7569			−2.01625
$610$		0			0
$611$	0.0784695			0.00317454
$612$		0			0
$613$			24.4282i			0.986644i	0.869847	+	0.493322i	$0.164218\pi$
$613$			24.4282i			0.986644i	−0.869847	+	0.493322i	$0.835782\pi$
$614$		0			0
$615$	−13.8081	−	40.6287i	−0.556797	−	1.63831i
$616$		0			0
$617$		−	29.5614i		−	1.19010i	−0.803689	−	0.595049i	$-0.797133\pi$
$617$		−	29.5614i		−	1.19010i	0.803689	−	0.595049i	$-0.202867\pi$
$618$		0			0
$619$	−30.5880			−1.22944			−0.614718	−	0.788747i	$-0.710730\pi$
$619$	−30.5880			−1.22944			−0.614718	+	0.788747i	$0.710730\pi$
$620$		0			0
$621$	0.616262			0.0247298
$622$		0			0
$623$		−	19.8374i		−	0.794769i
$624$		0			0
$625$	6.43523	−	24.1576i	0.257409	−	0.966303i
$626$		0			0
$627$			44.4144i			1.77374i
$628$		0			0
$629$	0.715890			0.0285444
$630$		0			0
$631$	−10.5713			−0.420835			−0.210418	−	0.977612i	$-0.567482\pi$
$631$	−10.5713			−0.420835			−0.210418	+	0.977612i	$0.567482\pi$
$632$		0			0
$633$			1.74688i			0.0694323i
$634$		0			0
$635$	−5.48455	−	16.1376i	−0.217648	−	0.640402i
$636$		0			0
$637$			0.0193293i			0.000765853i
$638$		0			0
$639$	−18.9620			−0.750123
$640$		0			0
$641$	−20.4738			−0.808666			−0.404333	−	0.914612i	$-0.632496\pi$
$641$	−20.4738			−0.808666			−0.404333	+	0.914612i	$0.632496\pi$
$642$		0			0
$643$		−	18.4989i		−	0.729525i	−0.931101	−	0.364763i	$-0.881150\pi$
$643$		−	18.4989i		−	0.729525i	0.931101	−	0.364763i	$-0.118850\pi$
$644$		0			0
$645$	44.3946	−	15.0880i	1.74804	−	0.594090i
$646$		0			0
$647$			18.6328i			0.732531i	0.930510	+	0.366266i	$0.119364\pi$
$647$			18.6328i			0.732531i	−0.930510	+	0.366266i	$0.880636\pi$
$648$		0			0
$649$	−58.0511			−2.27871
$650$		0			0
$651$	16.1505			0.632988
$652$		0			0
$653$			30.5290i			1.19469i	0.801983	+	0.597347i	$0.203778\pi$
$653$			30.5290i			1.19469i	−0.801983	+	0.597347i	$0.796222\pi$
$654$		0			0
$655$	−41.1943	+	14.0003i	−1.60959	+	0.547039i
$656$		0			0
$657$		−	2.36245i		−	0.0921678i
$658$		0			0
$659$	−26.6231			−1.03709			−0.518545	−	0.855051i	$-0.673526\pi$
$659$	−26.6231			−1.03709			−0.518545	+	0.855051i	$0.673526\pi$
$660$		0			0
$661$	−13.0529			−0.507700			−0.253850	−	0.967244i	$-0.581697\pi$
$661$	−13.0529			−0.507700			−0.253850	+	0.967244i	$0.581697\pi$
$662$		0			0
$663$		−	0.00477543i		−	0.000185462i
$664$		0			0
$665$	9.11057	+	26.8067i	0.353293	+	1.03952i
$666$		0			0
$667$			6.79978i			0.263289i
$668$		0			0
$669$	68.4110			2.64492
$670$		0			0
$671$	28.9813			1.11881
$672$		0			0
$673$			28.9120i			1.11448i	0.830353	+	0.557238i	$0.188139\pi$
$673$			28.9120i			1.11448i	−0.830353	+	0.557238i	$0.811861\pi$
$674$		0			0
$675$	−1.87757	−	2.44320i	−0.0722676	−	0.0940388i
$676$		0			0
$677$			12.4824i			0.479738i	0.970805	+	0.239869i	$0.0771045\pi$
$677$			12.4824i			0.479738i	−0.970805	+	0.239869i	$0.922896\pi$
$678$		0			0
$679$	43.5375			1.67081
$680$		0			0
$681$	26.5102			1.01587
$682$		0			0
$683$		−	40.0241i		−	1.53148i	−0.643150	−	0.765740i	$-0.722373\pi$
$683$		−	40.0241i		−	1.53148i	0.643150	−	0.765740i	$-0.277627\pi$
$684$		0			0
$685$	−11.7401	−	34.5438i	−0.448566	−	1.31985i
$686$		0			0
$687$		−	7.78369i		−	0.296966i
$688$		0			0
$689$	−0.0959612			−0.00365583
$690$		0			0
$691$	38.9938			1.48340			0.741698	−	0.670734i	$-0.234021\pi$
$691$	38.9938			1.48340			0.741698	+	0.670734i	$0.234021\pi$
$692$		0			0
$693$		−	39.0578i		−	1.48368i
$694$		0			0
$695$	−42.1923	+	14.3395i	−1.60044	+	0.543929i
$696$		0			0
$697$			1.19303i			0.0451894i
$698$		0			0
$699$	58.8077			2.22431
$700$		0			0
$701$	21.9042			0.827309			0.413655	−	0.910434i	$-0.364252\pi$
$701$	21.9042			0.827309			0.413655	+	0.910434i	$0.364252\pi$
$702$		0			0
$703$			19.9257i			0.751513i
$704$		0			0
$705$	−33.7652	+	11.4755i	−1.27167	+	0.432192i
$706$		0			0
$707$		−	38.0678i		−	1.43169i
$708$		0			0
$709$	−32.1360			−1.20689			−0.603447	−	0.797403i	$-0.706207\pi$
$709$	−32.1360			−1.20689			−0.603447	+	0.797403i	$0.706207\pi$
$710$		0			0
$711$	−15.6161			−0.585650
$712$		0			0
$713$		−	2.20713i		−	0.0826576i
$714$		0			0
$715$	−0.0363578	−	0.106978i	−0.00135970	−	0.00400077i
$716$		0			0
$717$			61.0367i			2.27946i
$718$		0			0
$719$	−24.4018			−0.910033			−0.455017	−	0.890483i	$-0.650367\pi$
$719$	−24.4018			−0.910033			−0.455017	+	0.890483i	$0.650367\pi$
$720$		0			0
$721$	−5.44023			−0.202605
$722$		0			0
$723$			68.8625i			2.56102i
$724$		0			0
$725$	26.9580	−	20.7169i	1.00120	−	0.769407i
$726$		0			0
$727$		−	34.8573i		−	1.29279i	−0.763005	−	0.646393i	$-0.776277\pi$
$727$		−	34.8573i		−	1.29279i	0.763005	−	0.646393i	$-0.223723\pi$
$728$		0			0
$729$	31.2409			1.15707
$730$		0			0
$731$	−1.30362			−0.0482161
$732$		0			0
$733$			30.7343i			1.13520i	0.823306	+	0.567598i	$0.192127\pi$
$733$			30.7343i			1.13520i	−0.823306	+	0.567598i	$0.807873\pi$
$734$		0			0
$735$	−2.82674	−	8.31733i	−0.104266	−	0.306789i
$736$		0			0
$737$			30.0523i			1.10699i
$738$		0			0
$739$	−38.7573			−1.42571			−0.712855	−	0.701312i	$-0.752598\pi$
$739$	−38.7573			−1.42571			−0.712855	+	0.701312i	$0.752598\pi$
$740$		0			0
$741$	0.132917			0.00488282
$742$		0			0
$743$		−	43.0371i		−	1.57888i	−0.613830	−	0.789438i	$-0.710372\pi$
$743$		−	43.0371i		−	1.57888i	0.613830	−	0.789438i	$-0.289628\pi$
$744$		0			0
$745$	17.0332	−	5.78893i	0.624049	−	0.212090i
$746$		0			0
$747$			25.1868i			0.921538i
$748$		0			0
$749$	−1.37976			−0.0504154
$750$		0			0
$751$	35.3761			1.29089			0.645447	−	0.763805i	$-0.276671\pi$
$751$	35.3761			1.29089			0.645447	+	0.763805i	$0.276671\pi$
$752$		0			0
$753$			63.4056i			2.31063i
$754$		0			0
$755$	−7.21866	+	2.45334i	−0.262714	+	0.0892862i
$756$		0			0
$757$			51.2605i			1.86309i	0.363621	+	0.931547i	$0.381540\pi$
$757$			51.2605i			1.86309i	−0.363621	+	0.931547i	$0.618460\pi$
$758$		0			0
$759$	−10.2699			−0.372774
$760$		0			0
$761$	−13.4331			−0.486950			−0.243475	−	0.969907i	$-0.578287\pi$
$761$	−13.4331			−0.486950			−0.243475	+	0.969907i	$0.578287\pi$
$762$		0			0
$763$			5.35846i			0.193989i
$764$		0			0
$765$	0.362966	+	1.06798i	0.0131231	+	0.0386130i
$766$		0			0
$767$			0.173727i			0.00627291i
$768$		0			0
$769$	24.7779			0.893512			0.446756	−	0.894656i	$-0.352579\pi$
$769$	24.7779			0.893512			0.446756	+	0.894656i	$0.352579\pi$
$770$		0			0
$771$	−49.6117			−1.78672
$772$		0			0
$773$			11.9946i			0.431417i	0.976458	+	0.215708i	$0.0692061\pi$
$773$			11.9946i			0.431417i	−0.976458	+	0.215708i	$0.930794\pi$
$774$		0			0
$775$	−8.75026	+	6.72446i	−0.314319	+	0.241550i
$776$		0			0
$777$		−	33.7144i		−	1.20950i
$778$		0			0
$779$	−33.2063			−1.18974
$780$		0			0
$781$	23.9996			0.858772
$782$		0			0
$783$		−	4.19045i		−	0.149755i
$784$		0			0
$785$	15.2446	+	44.8554i	0.544103	+	1.60096i
$786$		0			0
$787$			10.4110i			0.371114i	0.982634	+	0.185557i	$0.0594089\pi$
$787$			10.4110i			0.371114i	−0.982634	+	0.185557i	$0.940591\pi$
$788$		0			0
$789$	−57.2561			−2.03837
$790$		0			0
$791$	−22.8703			−0.813175
$792$		0			0
$793$		−	0.0867308i		−	0.00307990i
$794$		0			0
$795$	41.2918	−	14.0335i	1.46447	−	0.497717i
$796$		0			0
$797$		−	28.0848i		−	0.994814i	−0.867517	−	0.497407i	$-0.834286\pi$
$797$		−	28.0848i		−	0.994814i	0.867517	−	0.497407i	$-0.165714\pi$
$798$		0			0
$799$	0.991492			0.0350765
$800$		0			0
$801$	21.9975			0.777243
$802$		0			0
$803$			2.99008i			0.105517i
$804$		0			0
$805$	−6.19850	+	2.10663i	−0.218468	+	0.0742489i
$806$		0			0
$807$		−	2.57550i		−	0.0906618i
$808$		0			0
$809$	11.1238			0.391092			0.195546	−	0.980695i	$-0.437352\pi$
$809$	11.1238			0.391092			0.195546	+	0.980695i	$0.437352\pi$
$810$		0			0
$811$	28.7572			1.00980			0.504900	−	0.863178i	$-0.331529\pi$
$811$	28.7572			1.00980			0.504900	+	0.863178i	$0.331529\pi$
$812$		0			0
$813$			9.48679i			0.332716i
$814$		0			0
$815$	15.0733	+	44.3513i	0.527994	+	1.55356i
$816$		0			0
$817$		−	36.2842i		−	1.26942i
$818$		0			0
$819$	−0.116886			−0.00408434
$820$		0			0
$821$	37.0808			1.29413			0.647064	−	0.762435i	$-0.275996\pi$
$821$	37.0808			1.29413			0.647064	+	0.762435i	$0.275996\pi$
$822$		0			0
$823$			10.9501i			0.381698i	0.981619	+	0.190849i	$0.0611241\pi$
$823$			10.9501i			0.381698i	−0.981619	+	0.190849i	$0.938876\pi$
$824$		0			0
$825$	31.2893	+	40.7155i	1.08935	+	1.41753i
$826$		0			0
$827$			13.9973i			0.486732i	0.969934	+	0.243366i	$0.0782517\pi$
$827$			13.9973i			0.486732i	−0.969934	+	0.243366i	$0.921748\pi$
$828$		0			0
$829$	−16.0622			−0.557864			−0.278932	−	0.960311i	$-0.589980\pi$
$829$	−16.0622			−0.557864			−0.278932	+	0.960311i	$0.589980\pi$
$830$		0			0
$831$	3.71853			0.128995
$832$		0			0
$833$			0.244233i			0.00846216i
$834$		0			0
$835$	7.90487	+	23.2591i	0.273559	+	0.804914i
$836$		0			0
$837$			1.36017i			0.0470144i
$838$		0			0
$839$	−9.49187			−0.327696			−0.163848	−	0.986486i	$-0.552391\pi$
$839$	−9.49187			−0.327696			−0.163848	+	0.986486i	$0.552391\pi$
$840$		0			0
$841$	17.2371			0.594381
$842$		0			0
$843$		−	25.6919i		−	0.884876i
$844$		0			0
$845$	27.5225	−	9.35382i	0.946802	−	0.321781i
$846$		0			0
$847$			17.2288i			0.591988i
$848$		0			0
$849$	11.5575			0.396653
$850$		0			0
$851$	−4.60741			−0.157940
$852$		0			0
$853$		−	3.45965i		−	0.118456i	−0.998244	−	0.0592280i	$-0.981136\pi$
$853$		−	3.45965i		−	0.118456i	0.998244	−	0.0592280i	$-0.0188639\pi$
$854$		0			0
$855$	−29.7257	+	10.1026i	−1.01660	+	0.345502i
$856$		0			0
$857$			0.729973i			0.0249354i	0.999922	+	0.0124677i	$0.00396869\pi$
$857$			0.729973i			0.0249354i	−0.999922	+	0.0124677i	$0.996031\pi$
$858$		0			0
$859$	0.357241			0.0121889			0.00609446	−	0.999981i	$-0.498060\pi$
$859$	0.357241			0.0121889			0.00609446	+	0.999981i	$0.498060\pi$
$860$		0			0
$861$	56.1851			1.91478
$862$		0			0
$863$			16.8358i			0.573098i	0.958066	+	0.286549i	$0.0925082\pi$
$863$			16.8358i			0.573098i	−0.958066	+	0.286549i	$0.907492\pi$
$864$		0			0
$865$	−3.10198	−	9.12720i	−0.105470	−	0.310334i
$866$		0			0
$867$			42.4280i			1.44093i
$868$		0			0
$869$	19.7649			0.670477
$870$		0			0
$871$	0.0899362			0.00304737
$872$		0			0
$873$			48.2782i			1.63397i
$874$		0			0
$875$	27.2368	+	18.1560i	0.920772	+	0.613784i
$876$		0			0
$877$		−	8.06147i		−	0.272217i	−0.990694	−	0.136108i	$-0.956540\pi$
$877$		−	8.06147i		−	0.272217i	0.990694	−	0.136108i	$-0.0434595\pi$
$878$		0			0
$879$	40.7557			1.37466
$880$		0			0
$881$	−37.7548			−1.27199			−0.635996	−	0.771693i	$-0.719410\pi$
$881$	−37.7548			−1.27199			−0.635996	+	0.771693i	$0.719410\pi$
$882$		0			0
$883$		−	35.2103i		−	1.18492i	−0.805600	−	0.592460i	$-0.798157\pi$
$883$		−	35.2103i		−	1.18492i	0.805600	−	0.592460i	$-0.201843\pi$
$884$		0			0
$885$	−25.4060	−	74.7541i	−0.854014	−	2.51283i
$886$		0			0
$887$		−	21.4225i		−	0.719297i	−0.933088	−	0.359649i	$-0.882897\pi$
$887$		−	21.4225i		−	0.719297i	0.933088	−	0.359649i	$-0.117103\pi$
$888$		0			0
$889$	22.3166			0.748474
$890$		0			0
$891$	−33.6924			−1.12874
$892$		0			0
$893$			27.5967i			0.923488i
$894$		0			0
$895$	−40.3930	+	13.7280i	−1.35019	+	0.458877i
$896$		0			0
$897$			0.0307342i			0.00102619i
$898$		0			0
$899$	−15.0080			−0.500545
$900$		0			0
$901$	−1.21251			−0.0403944
$902$		0			0
$903$			61.3930i			2.04303i
$904$		0			0
$905$	8.66571	−	2.94514i	0.288058	−	0.0978997i
$906$		0			0
$907$			38.7517i			1.28673i	0.765560	+	0.643364i	$0.222462\pi$
$907$			38.7517i			1.28673i	−0.765560	+	0.643364i	$0.777538\pi$
$908$		0			0
$909$	42.2129			1.40012
$910$		0			0
$911$	47.0069			1.55741			0.778704	−	0.627391i	$-0.215877\pi$
$911$	47.0069			1.55741			0.778704	+	0.627391i	$0.215877\pi$
$912$		0			0
$913$		−	31.8782i		−	1.05501i
$914$		0			0
$915$	12.6836	+	37.3200i	0.419308	+	1.23376i
$916$		0			0
$917$		−	56.9673i		−	1.88123i
$918$		0			0
$919$	12.2360			0.403629			0.201814	−	0.979424i	$-0.435316\pi$
$919$	12.2360			0.403629			0.201814	+	0.979424i	$0.435316\pi$
$920$		0			0
$921$	23.2101			0.764799
$922$		0			0
$923$		−	0.0718223i		−	0.00236406i
$924$		0			0
$925$	14.0374	+	18.2663i	0.461547	+	0.600591i
$926$		0			0
$927$		−	6.03260i		−	0.198137i
$928$		0			0
$929$	13.0143			0.426984			0.213492	−	0.976945i	$-0.431516\pi$
$929$	13.0143			0.426984			0.213492	+	0.976945i	$0.431516\pi$
$930$		0			0
$931$	−6.79785			−0.222790
$932$		0			0
$933$			60.9114i			1.99415i
$934$		0			0
$935$	−0.459395	−	1.35171i	−0.0150238	−	0.0442057i
$936$		0			0
$937$			38.6856i			1.26380i	0.775048	+	0.631902i	$0.217726\pi$
$937$			38.6856i			1.26380i	−0.775048	+	0.631902i	$0.782274\pi$
$938$		0			0
$939$	37.5016			1.22382
$940$		0			0
$941$	−43.8241			−1.42862			−0.714312	−	0.699828i	$-0.753260\pi$
$941$	−43.8241			−1.42862			−0.714312	+	0.699828i	$0.753260\pi$
$942$		0			0
$943$		−	7.67826i		−	0.250039i
$944$		0			0
$945$	3.81990	−	1.29824i	0.124261	−	0.0422316i
$946$		0			0
$947$		−	43.3854i		−	1.40984i	−0.709289	−	0.704918i	$-0.750984\pi$
$947$		−	43.3854i		−	1.40984i	0.709289	−	0.704918i	$-0.249016\pi$
$948$		0			0
$949$	0.00894825			0.000290473
$950$		0			0
$951$	19.6133			0.636005
$952$		0			0
$953$			2.80245i			0.0907802i	0.998969	+	0.0453901i	$0.0144531\pi$
$953$			2.80245i			0.0907802i	−0.998969	+	0.0453901i	$0.985547\pi$
$954$		0			0
$955$	−13.1481	+	4.46853i	−0.425463	+	0.144598i
$956$		0			0
$957$			69.8331i			2.25738i
$958$		0			0
$959$	47.7704			1.54259
$960$		0			0
$961$	−26.1286			−0.842858
$962$		0			0
$963$		−	1.53000i		−	0.0493036i
$964$		0			0
$965$	−9.54630	−	28.0888i	−0.307306	−	0.904211i
$966$		0			0
$967$			30.0015i			0.964783i	0.875956	+	0.482391i	$0.160232\pi$
$967$			30.0015i			0.964783i	−0.875956	+	0.482391i	$0.839768\pi$
$968$		0			0
$969$	1.67945			0.0539518
$970$		0			0
$971$	−53.6632			−1.72213			−0.861067	−	0.508492i	$-0.830203\pi$
$971$	−53.6632			−1.72213			−0.861067	+	0.508492i	$0.830203\pi$
$972$		0			0
$973$		−	58.3474i		−	1.87053i
$974$		0			0
$975$	0.121847	−	0.0936380i	0.00390223	−	0.00299882i
$976$		0			0
$977$		−	46.4590i		−	1.48636i	−0.669094	−	0.743178i	$-0.733318\pi$
$977$		−	46.4590i		−	1.48636i	0.669094	−	0.743178i	$-0.266682\pi$
$978$		0			0
$979$	−27.8415			−0.889819
$980$		0			0
$981$	−5.94193			−0.189711
$982$		0			0
$983$			50.8346i			1.62137i	0.585481	+	0.810686i	$0.300906\pi$
$983$			50.8346i			1.62137i	−0.585481	+	0.810686i	$0.699094\pi$
$984$		0			0
$985$	9.98969	+	29.3934i	0.318298	+	0.936552i
$986$		0			0
$987$		−	46.6936i		−	1.48627i
$988$		0			0
$989$	8.38997			0.266786
$990$		0			0
$991$	−4.30720			−0.136823			−0.0684114	−	0.997657i	$-0.521793\pi$
$991$	−4.30720			−0.136823			−0.0684114	+	0.997657i	$0.521793\pi$
$992$		0			0
$993$		−	23.1975i		−	0.736151i
$994$		0			0
$995$	−48.2716	+	16.4056i	−1.53031	+	0.520094i
$996$		0			0
$997$			13.2358i			0.419181i	0.977789	+	0.209590i	$0.0672131\pi$
$997$			13.2358i			0.419181i	−0.977789	+	0.209590i	$0.932787\pi$
$998$		0			0
$999$	2.83937			0.0898337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.g.369.13		14
4.3	odd	2		920.2.e.b.369.2	✓	14
5.2	odd	4		9200.2.a.cz.1.7		7
5.3	odd	4		9200.2.a.dc.1.1		7
5.4	even	2	inner	1840.2.e.g.369.2		14
20.3	even	4		4600.2.a.bh.1.7		7
20.7	even	4		4600.2.a.bi.1.1		7
20.19	odd	2		920.2.e.b.369.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.2	✓	14	4.3	odd	2
920.2.e.b.369.13	yes	14	20.19	odd	2
1840.2.e.g.369.2		14	5.4	even	2	inner
1840.2.e.g.369.13		14	1.1	even	1	trivial
4600.2.a.bh.1.7		7	20.3	even	4
4600.2.a.bi.1.1		7	20.7	even	4
9200.2.a.cz.1.7		7	5.2	odd	4
9200.2.a.dc.1.1		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)

\(f(q)\)	\(=\)	\(q+2.49931i q^{3} +(2.11714 - 0.719533i) q^{5} +2.92777i q^{7} -3.24657 q^{9} +O(q^{10})\)	\(q+2.49931i q^{3} +(2.11714 - 0.719533i) q^{5} +2.92777i q^{7} -3.24657 q^{9} +4.10909 q^{11} -0.0122971i q^{13} +(1.79834 + 5.29139i) q^{15} -0.155378i q^{17} +4.32471 q^{19} -7.31743 q^{21} +1.00000i q^{23} +(3.96454 - 3.04670i) q^{25} -0.616262i q^{27} +6.79978 q^{29} -2.20713 q^{31} +10.2699i q^{33} +(2.10663 + 6.19850i) q^{35} +4.60741i q^{37} +0.0307342 q^{39} -7.67826 q^{41} -8.38997i q^{43} +(-6.87344 + 2.33602i) q^{45} +6.38116i q^{47} -1.57186 q^{49} +0.388339 q^{51} -7.80358i q^{53} +(8.69951 - 2.95662i) q^{55} +10.8088i q^{57} -14.1275 q^{59} +7.05297 q^{61} -9.50523i q^{63} +(-0.00884814 - 0.0260346i) q^{65} +7.31363i q^{67} -2.49931 q^{69} +5.84061 q^{71} +0.727674i q^{73} +(7.61466 + 9.90864i) q^{75} +12.0305i q^{77} +4.81003 q^{79} -8.19948 q^{81} -7.75797i q^{83} +(-0.111800 - 0.328957i) q^{85} +16.9948i q^{87} -6.77560 q^{89} +0.0360030 q^{91} -5.51631i q^{93} +(9.15602 - 3.11177i) q^{95} -14.8705i q^{97} -13.3405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) 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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