LMFDB - Embedded newform 3060.2.k.h.1189.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(1189,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.1189"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4441101041664.4
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 38x^{4} + 200x^{2} + 625 $ x^8 + 8x^6 + 38x^4 + 200*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1189.7
Root			$-1.68014 - 1.47551i$ of defining polynomial
Character	$\chi$	$=$	3060.1189
Dual form			3060.2.k.h.1189.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.68014 - 1.47551i) q^{5} -2.76510 q^{7} -5.37934i q^{11} +4.95813i q^{13} +(-0.595188 + 4.07992i) q^{17} +5.29150 q^{19} +4.93500 q^{23} +(0.645751 - 4.95813i) q^{25} +2.42832i q^{31} +(-4.64575 + 4.07992i) q^{35} +5.53019 q^{37} +2.95102i q^{41} +3.20172i q^{43} -6.71453i q^{47} +0.645751 q^{49} -8.15984i q^{53} +(-7.93725 - 9.03805i) q^{55} +9.29150 q^{59} -13.7097i q^{61} +(7.31575 + 8.33035i) q^{65} -6.71453i q^{67} -11.2814i q^{71} +15.6110 q^{73} +14.8744i q^{77} -2.42832i q^{79} -14.8744i q^{83} +(5.01996 + 7.73305i) q^{85} -14.2288 q^{89} -13.7097i q^{91} +(8.89047 - 7.80766i) q^{95} -9.10132 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{25} - 16 q^{35} - 16 q^{49} + 32 q^{59} - 8 q^{85} - 8 q^{89}+O(q^{100}) $ 8 * q - 16 * q^25 - 16 * q^35 - 16 * q^49 + 32 * q^59 - 8 * q^85 - 8 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.68014	−	1.47551i	0.751382	−	0.659867i
$5$	1.68014	−	1.47551i	0.751382	−	0.659867i
$6$		0			0
$7$	−2.76510			−1.04511			−0.522554	−	0.852606i	$-0.675021\pi$
$7$	−2.76510			−1.04511			−0.522554	+	0.852606i	$0.675021\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.37934i		−	1.62193i	−0.585094	−	0.810965i	$-0.698942\pi$
$11$		−	5.37934i		−	1.62193i	0.585094	−	0.810965i	$-0.301058\pi$
$12$		0			0
$13$			4.95813i			1.37514i	0.726120	+	0.687568i	$0.241322\pi$
$13$			4.95813i			1.37514i	−0.726120	+	0.687568i	$0.758678\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.595188	+	4.07992i	−0.144354	+	0.989526i
$17$	−0.595188	+	4.07992i	−0.144354	+	0.989526i
$18$		0			0
$19$	5.29150			1.21395			0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150			1.21395			0.606977	+	0.794719i	$0.292382\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.93500			1.02902			0.514510	−	0.857485i	$-0.327974\pi$
$23$	4.93500			1.02902			0.514510	+	0.857485i	$0.327974\pi$
$24$		0			0
$25$	0.645751	−	4.95813i	0.129150	−	0.991625i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			2.42832i			0.436139i	0.975933	+	0.218070i	$0.0699760\pi$
$31$			2.42832i			0.436139i	−0.975933	+	0.218070i	$0.930024\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.64575	+	4.07992i	−0.785275	+	0.689632i
$36$		0			0
$37$	5.53019			0.909158			0.454579	−	0.890707i	$-0.349790\pi$
$37$	5.53019			0.909158			0.454579	+	0.890707i	$0.349790\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.95102i			0.460871i	0.973088	+	0.230436i	$0.0740151\pi$
$41$			2.95102i			0.460871i	−0.973088	+	0.230436i	$0.925985\pi$
$42$		0			0
$43$			3.20172i			0.488257i	0.969743	+	0.244129i	$0.0785019\pi$
$43$			3.20172i			0.488257i	−0.969743	+	0.244129i	$0.921498\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.71453i		−	0.979416i	−0.871887	−	0.489708i	$-0.837103\pi$
$47$		−	6.71453i		−	0.979416i	0.871887	−	0.489708i	$-0.162897\pi$
$48$		0			0
$49$	0.645751			0.0922502
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.15984i		−	1.12084i	−0.828208	−	0.560420i	$-0.810640\pi$
$53$		−	8.15984i		−	1.12084i	0.828208	−	0.560420i	$-0.189360\pi$
$54$		0			0
$55$	−7.93725	−	9.03805i	−1.07026	−	1.21869i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.29150			1.20965			0.604825	−	0.796358i	$-0.293243\pi$
$59$	9.29150			1.20965			0.604825	+	0.796358i	$0.293243\pi$
$60$		0			0
$61$		−	13.7097i		−	1.75535i	−0.479260	−	0.877673i	$-0.659095\pi$
$61$		−	13.7097i		−	1.75535i	0.479260	−	0.877673i	$-0.340905\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.31575	+	8.33035i	0.907408	+	1.03325i
$66$		0			0
$67$		−	6.71453i		−	0.820311i	−0.912016	−	0.410155i	$-0.865474\pi$
$67$		−	6.71453i		−	0.820311i	0.912016	−	0.410155i	$-0.134526\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.2814i		−	1.33885i	−0.742879	−	0.669426i	$-0.766540\pi$
$71$		−	11.2814i		−	1.33885i	0.742879	−	0.669426i	$-0.233460\pi$
$72$		0			0
$73$	15.6110			1.82713			0.913567	−	0.406688i	$-0.133316\pi$
$73$	15.6110			1.82713			0.913567	+	0.406688i	$0.133316\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			14.8744i			1.69509i
$78$		0			0
$79$		−	2.42832i		−	0.273207i	−0.990626	−	0.136604i	$-0.956381\pi$
$79$		−	2.42832i		−	0.273207i	0.990626	−	0.136604i	$-0.0436187\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	14.8744i		−	1.63267i	−0.577575	−	0.816337i	$-0.696001\pi$
$83$		−	14.8744i		−	1.63267i	0.577575	−	0.816337i	$-0.303999\pi$
$84$		0			0
$85$	5.01996	+	7.73305i	0.544491	+	0.838767i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.2288			−1.50825			−0.754123	−	0.656734i	$-0.771938\pi$
$89$	−14.2288			−1.50825			−0.754123	+	0.656734i	$0.771938\pi$
$90$		0			0
$91$		−	13.7097i		−	1.43717i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.89047	−	7.80766i	0.912143	−	0.801049i
$96$		0			0
$97$	−9.10132			−0.924099			−0.462049	−	0.886854i	$-0.652886\pi$
$97$	−9.10132			−0.924099			−0.462049	+	0.886854i	$0.652886\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.35425			0.433264			0.216632	−	0.976253i	$-0.430493\pi$
$101$	4.35425			0.433264			0.216632	+	0.976253i	$0.430493\pi$
$102$		0			0
$103$			4.95813i			0.488539i	0.969707	+	0.244269i	$0.0785481\pi$
$103$			4.95813i			0.488539i	−0.969707	+	0.244269i	$0.921452\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.69651			−0.937397			−0.468698	−	0.883358i	$-0.655277\pi$
$107$	−9.69651			−0.937397			−0.468698	+	0.883358i	$0.655277\pi$
$108$		0			0
$109$		−	8.85305i		−	0.847968i	−0.905670	−	0.423984i	$-0.860631\pi$
$109$		−	8.85305i		−	0.847968i	0.905670	−	0.423984i	$-0.139369\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.74103			−0.540071			−0.270036	−	0.962850i	$-0.587035\pi$
$113$	−5.74103			−0.540071			−0.270036	+	0.962850i	$0.587035\pi$
$114$		0			0
$115$	8.29150	−	7.28164i	0.773187	−	0.679016i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.64575	−	11.2814i	0.150866	−	1.03416i
$120$		0			0
$121$	−17.9373			−1.63066
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.23080	−	9.28316i	−0.557300	−	0.830311i
$126$		0			0
$127$			13.1180i			1.16403i	0.813178	+	0.582016i	$0.197736\pi$
$127$			13.1180i			1.16403i	−0.813178	+	0.582016i	$0.802264\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	8.33035i		−	0.727826i	−0.931433	−	0.363913i	$-0.881441\pi$
$131$		−	8.33035i		−	0.727826i	0.931433	−	0.363913i	$-0.118559\pi$
$132$		0			0
$133$	−14.6315			−1.26871
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1.44531i			0.123481i	0.998092	+	0.0617404i	$0.0196651\pi$
$137$			1.44531i			0.123481i	−0.998092	+	0.0617404i	$0.980335\pi$
$138$		0			0
$139$			6.42473i			0.544938i	0.962164	+	0.272469i	$0.0878403\pi$
$139$			6.42473i			0.544938i	−0.962164	+	0.272469i	$0.912160\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	26.6714			2.23038
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.58301			0.539301			0.269650	−	0.962958i	$-0.413092\pi$
$149$	6.58301			0.539301			0.269650	+	0.962958i	$0.413092\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.58301	+	4.07992i	0.287794	+	0.327707i
$156$		0			0
$157$			11.6727i			0.931580i	0.884895	+	0.465790i	$0.154230\pi$
$157$			11.6727i			0.931580i	−0.884895	+	0.465790i	$0.845770\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.6458			−1.07544
$162$		0			0
$163$	13.8255			1.08289			0.541447	−	0.840735i	$-0.317877\pi$
$163$	13.8255			1.08289			0.541447	+	0.840735i	$0.317877\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.9749			1.31356			0.656779	−	0.754083i	$-0.271918\pi$
$167$	16.9749			1.31356			0.656779	+	0.754083i	$0.271918\pi$
$168$		0			0
$169$	−11.5830			−0.891000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.38075			−0.181005			−0.0905026	−	0.995896i	$-0.528847\pi$
$173$	−2.38075			−0.181005			−0.0905026	+	0.995896i	$0.528847\pi$
$174$		0			0
$175$	−1.78556	+	13.7097i	−0.134976	+	1.03635i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.29150			−0.246018			−0.123009	−	0.992406i	$-0.539254\pi$
$179$	−3.29150			−0.246018			−0.123009	+	0.992406i	$0.539254\pi$
$180$		0			0
$181$			22.5627i			1.67708i	0.544844	+	0.838538i	$0.316589\pi$
$181$			22.5627i			1.67708i	−0.544844	+	0.838538i	$0.683411\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.29150	−	8.15984i	0.683125	−	0.599923i
$186$		0			0
$187$	21.9473	+	3.20172i	1.60494	+	0.234133i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.00000			−0.434145			−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000			−0.434145			−0.217072	+	0.976156i	$0.569651\pi$
$192$		0			0
$193$	−12.0399			−0.866652			−0.433326	−	0.901237i	$-0.642660\pi$
$193$	−12.0399			−0.866652			−0.433326	+	0.901237i	$0.642660\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	21.3521			1.52127			0.760636	−	0.649178i	$-0.224887\pi$
$197$	21.3521			1.52127			0.760636	+	0.649178i	$0.224887\pi$
$198$		0			0
$199$		−	2.42832i		−	0.172139i	−0.996289	−	0.0860695i	$-0.972569\pi$
$199$		−	2.42832i		−	0.172139i	0.996289	−	0.0860695i	$-0.0274307\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.35425	+	4.95813i	0.304114	+	0.346290i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	28.4648i		−	1.96895i
$210$		0			0
$211$		−	20.1344i		−	1.38611i	−0.720884	−	0.693055i	$-0.756264\pi$
$211$		−	20.1344i		−	1.38611i	0.720884	−	0.693055i	$-0.243736\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.72416	+	5.37934i	0.322185	+	0.366868i
$216$		0			0
$217$		−	6.71453i		−	0.455812i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−20.2288	−	2.95102i	−1.36073	−	0.198507i
$222$		0			0
$223$		−	21.2778i		−	1.42487i	−0.701739	−	0.712434i	$-0.747593\pi$
$223$		−	21.2778i		−	1.42487i	0.701739	−	0.712434i	$-0.252407\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	19.7774			1.31267			0.656335	−	0.754470i	$-0.272106\pi$
$227$	19.7774			1.31267			0.656335	+	0.754470i	$0.272106\pi$
$228$		0			0
$229$	3.64575			0.240918			0.120459	−	0.992718i	$-0.461563\pi$
$229$	3.64575			0.240918			0.120459	+	0.992718i	$0.461563\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.2508			−0.802574			−0.401287	−	0.915952i	$-0.631437\pi$
$233$	−12.2508			−0.802574			−0.401287	+	0.915952i	$0.631437\pi$
$234$		0			0
$235$	−9.90735	−	11.2814i	−0.646284	−	0.735915i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.87451			−0.638729			−0.319364	−	0.947632i	$-0.603469\pi$
$239$	−9.87451			−0.638729			−0.319364	+	0.947632i	$0.603469\pi$
$240$		0			0
$241$			13.7097i			0.883119i	0.897232	+	0.441559i	$0.145575\pi$
$241$			13.7097i			0.883119i	−0.897232	+	0.441559i	$0.854425\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.08495	−	0.952811i	0.0693151	−	0.0608729i
$246$		0			0
$247$			26.2359i			1.66935i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.1660			1.20975			0.604874	−	0.796321i	$-0.293223\pi$
$251$	19.1660			1.20975			0.604874	+	0.796321i	$0.293223\pi$
$252$		0			0
$253$		−	26.5470i		−	1.66900i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.8744i			0.927838i	0.885878	+	0.463919i	$0.153557\pi$
$257$			14.8744i			0.927838i	−0.885878	+	0.463919i	$0.846443\pi$
$258$		0			0
$259$	−15.2915			−0.950168
$260$		0			0
$261$		0			0
$262$		0			0
$263$			9.60515i			0.592279i	0.955145	+	0.296139i	$0.0956993\pi$
$263$			9.60515i			0.592279i	−0.955145	+	0.296139i	$0.904301\pi$
$264$		0			0
$265$	−12.0399	−	13.7097i	−0.739606	−	0.842179i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			14.7551i			0.899633i	0.893121	+	0.449817i	$0.148511\pi$
$269$			14.7551i			0.899633i	−0.893121	+	0.449817i	$0.851489\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−26.6714	−	3.47371i	−1.60835	−	0.209473i
$276$		0			0
$277$	17.2231			1.03484			0.517418	−	0.855733i	$-0.326893\pi$
$277$	17.2231			1.03484			0.517418	+	0.855733i	$0.326893\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.1660			0.785418			0.392709	−	0.919663i	$-0.371538\pi$
$281$	13.1660			0.785418			0.392709	+	0.919663i	$0.371538\pi$
$282$		0			0
$283$	8.29529			0.493104			0.246552	−	0.969130i	$-0.420702\pi$
$283$	8.29529			0.493104			0.246552	+	0.969130i	$0.420702\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	8.15984i		−	0.481660i
$288$		0			0
$289$	−16.2915	−	4.85664i	−0.958324	−	0.285685i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	13.4291i		−	0.784535i	−0.919851	−	0.392267i	$-0.871691\pi$
$293$		−	13.4291i		−	0.784535i	0.919851	−	0.392267i	$-0.128309\pi$
$294$		0			0
$295$	15.6110	−	13.7097i	0.908910	−	0.798209i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			24.4684i			1.41504i
$300$		0			0
$301$		−	8.85305i		−	0.510281i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−20.2288	−	23.0342i	−1.15830	−	1.31894i
$306$		0			0
$307$			17.7650i			1.01390i	0.861975	+	0.506951i	$0.169227\pi$
$307$			17.7650i			1.01390i	−0.861975	+	0.506951i	$0.830773\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	11.2814i		−	0.639708i	−0.947467	−	0.319854i	$-0.896366\pi$
$311$		−	11.2814i		−	0.639708i	0.947467	−	0.319854i	$-0.103634\pi$
$312$		0			0
$313$	−22.7533			−1.28609			−0.643046	−	0.765827i	$-0.722330\pi$
$313$	−22.7533			−1.28609			−0.643046	+	0.765827i	$0.722330\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.1894			−0.853119			−0.426559	−	0.904460i	$-0.640275\pi$
$317$	−15.1894			−0.853119			−0.426559	+	0.904460i	$0.640275\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.14944	+	21.5889i	−0.175239	+	1.20124i
$324$		0			0
$325$	24.5830	+	3.20172i	1.36362	+	0.177599i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			18.5663i			1.02359i
$330$		0			0
$331$	29.2915			1.61001			0.805003	−	0.593270i	$-0.202163\pi$
$331$	29.2915			1.61001			0.805003	+	0.593270i	$0.202163\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.90735	−	11.2814i	−0.541296	−	0.616367i
$336$		0			0
$337$	1.61206			0.0878148			0.0439074	−	0.999036i	$-0.486019\pi$
$337$	1.61206			0.0878148			0.0439074	+	0.999036i	$0.486019\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	13.0627			0.707387
$342$		0			0
$343$	17.5701			0.948696
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.0159			−0.806093			−0.403047	−	0.915179i	$-0.632049\pi$
$347$	−15.0159			−0.806093			−0.403047	+	0.915179i	$0.632049\pi$
$348$		0			0
$349$	13.8745			0.742685			0.371343	−	0.928496i	$-0.378898\pi$
$349$	13.8745			0.742685			0.371343	+	0.928496i	$0.378898\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			13.4291i			0.714757i	0.933960	+	0.357379i	$0.116329\pi$
$353$			13.4291i			0.714757i	−0.933960	+	0.357379i	$0.883671\pi$
$354$		0			0
$355$	−16.6458	−	18.9543i	−0.883465	−	1.00599i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.12549			−0.112179			−0.0560896	−	0.998426i	$-0.517863\pi$
$359$	−2.12549			−0.112179			−0.0560896	+	0.998426i	$0.517863\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$		0			0
$363$		0			0
$364$		0			0
$365$	26.2288	−	23.0342i	1.37288	−	1.20567i
$366$		0			0
$367$	−2.76510			−0.144337			−0.0721684	−	0.997392i	$-0.522992\pi$
$367$	−2.76510			−0.144337			−0.0721684	+	0.997392i	$0.522992\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			22.5627i			1.17140i
$372$		0			0
$373$		−	17.7650i		−	0.919836i	−0.887961	−	0.459918i	$-0.847879\pi$
$373$		−	17.7650i		−	0.919836i	0.887961	−	0.459918i	$-0.152121\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	29.8477i		−	1.53317i	−0.642141	−	0.766587i	$-0.721954\pi$
$379$		−	29.8477i		−	1.53317i	0.642141	−	0.766587i	$-0.278046\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			31.1941i			1.59394i	0.604018	+	0.796971i	$0.293566\pi$
$383$			31.1941i			1.59394i	−0.604018	+	0.796971i	$0.706434\pi$
$384$		0			0
$385$	21.9473	+	24.9911i	1.11854	+	1.27366i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.6458			1.30029			0.650146	−	0.759810i	$-0.274708\pi$
$389$	25.6458			1.30029			0.650146	+	0.759810i	$0.274708\pi$
$390$		0			0
$391$	−2.93725	+	20.1344i	−0.148543	+	1.01824i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.58301	−	4.07992i	−0.180281	−	0.205283i
$396$		0			0
$397$	2.93859			0.147484			0.0737419	−	0.997277i	$-0.476506\pi$
$397$	2.93859			0.147484			0.0737419	+	0.997277i	$0.476506\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.04539i			0.0522045i	0.999659	+	0.0261022i	$0.00830954\pi$
$401$			1.04539i			0.0522045i	−0.999659	+	0.0261022i	$0.991690\pi$
$402$		0			0
$403$	−12.0399			−0.599751
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	29.7488i		−	1.47459i
$408$		0			0
$409$	28.5830			1.41334			0.706669	−	0.707544i	$-0.250197\pi$
$409$	28.5830			1.41334			0.706669	+	0.707544i	$0.250197\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−25.6919			−1.26422
$414$		0			0
$415$	−21.9473	−	24.9911i	−1.07735	−	1.22676i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	7.28496i		−	0.355894i	−0.984040	−	0.177947i	$-0.943055\pi$
$419$		−	7.28496i		−	0.355894i	0.984040	−	0.177947i	$-0.0569455\pi$
$420$		0			0
$421$	3.52026			0.171567			0.0857835	−	0.996314i	$-0.472661\pi$
$421$	3.52026			0.171567			0.0857835	+	0.996314i	$0.472661\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	19.8444	+	5.58563i	0.962595	+	0.270943i
$426$		0			0
$427$			37.9086i			1.83453i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	7.28496i		−	0.350904i	−0.984488	−	0.175452i	$-0.943861\pi$
$431$		−	7.28496i		−	0.350904i	0.984488	−	0.175452i	$-0.0561387\pi$
$432$		0			0
$433$			2.06751i			0.0993583i	0.998765	+	0.0496791i	$0.0158199\pi$
$433$			2.06751i			0.0993583i	−0.998765	+	0.0496791i	$0.984180\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	26.1136			1.24918
$438$		0			0
$439$			16.1380i			0.770225i	0.922870	+	0.385113i	$0.125837\pi$
$439$			16.1380i			0.770225i	−0.922870	+	0.385113i	$0.874163\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	1.44531i		−	0.0686686i	−0.999410	−	0.0343343i	$-0.989069\pi$
$443$		−	1.44531i		−	0.0686686i	0.999410	−	0.0343343i	$-0.0109311\pi$
$444$		0			0
$445$	−23.9063	+	20.9946i	−1.13327	+	0.995242i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18.5663i			0.876199i	0.898926	+	0.438100i	$0.144348\pi$
$449$			18.5663i			0.876199i	−0.898926	+	0.438100i	$0.855652\pi$
$450$		0			0
$451$	15.8745			0.747501
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−20.2288	−	23.0342i	−0.948339	−	1.07986i
$456$		0			0
$457$			27.6812i			1.29487i	0.762119	+	0.647437i	$0.224159\pi$
$457$			27.6812i			1.29487i	−0.762119	+	0.647437i	$0.775841\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	20.7085			0.964491			0.482245	−	0.876036i	$-0.339821\pi$
$461$	20.7085			0.964491			0.482245	+	0.876036i	$0.339821\pi$
$462$		0			0
$463$		−	31.1941i		−	1.44971i	−0.688901	−	0.724855i	$-0.741907\pi$
$463$		−	31.1941i		−	1.44971i	0.688901	−	0.724855i	$-0.258093\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			9.60515i			0.444473i	0.974993	+	0.222237i	$0.0713357\pi$
$467$			9.60515i			0.444473i	−0.974993	+	0.222237i	$0.928664\pi$
$468$		0			0
$469$			18.5663i			0.857313i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	17.2231			0.791919
$474$		0			0
$475$	3.41699	−	26.2359i	0.156782	−	1.20379i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	35.7497i		−	1.63345i	−0.577029	−	0.816723i	$-0.695788\pi$
$479$		−	35.7497i		−	1.63345i	0.577029	−	0.816723i	$-0.304212\pi$
$480$		0			0
$481$			27.4194i			1.25022i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.2915	+	13.4291i	−0.694351	+	0.609783i
$486$		0			0
$487$	−2.76510			−0.125298			−0.0626492	−	0.998036i	$-0.519955\pi$
$487$	−2.76510			−0.125298			−0.0626492	+	0.998036i	$0.519955\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.8745			−0.987183			−0.493591	−	0.869694i	$-0.664316\pi$
$491$	−21.8745			−0.987183			−0.493591	+	0.869694i	$0.664316\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			31.1941i			1.39924i
$498$		0			0
$499$			7.28496i			0.326120i	0.986616	+	0.163060i	$0.0521363\pi$
$499$			7.28496i			0.326120i	−0.986616	+	0.163060i	$0.947864\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.6351			0.563371			0.281686	−	0.959507i	$-0.409106\pi$
$503$	12.6351			0.563371			0.281686	+	0.959507i	$0.409106\pi$
$504$		0			0
$505$	7.31575	−	6.42473i	0.325547	−	0.285897i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−34.4575			−1.52730			−0.763651	−	0.645629i	$-0.776595\pi$
$509$	−34.4575			−1.52730			−0.763651	+	0.645629i	$0.776595\pi$
$510$		0			0
$511$	−43.1660			−1.90955
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.31575	+	8.33035i	0.322371	+	0.367079i
$516$		0			0
$517$	−36.1197			−1.58854
$518$		0			0
$519$		0			0
$520$		0			0
$521$			26.3740i			1.15547i	0.816226	+	0.577733i	$0.196062\pi$
$521$			26.3740i			1.15547i	−0.816226	+	0.577733i	$0.803938\pi$
$522$		0			0
$523$			6.71453i			0.293606i	0.989166	+	0.146803i	$0.0468984\pi$
$523$			6.71453i			0.293606i	−0.989166	+	0.146803i	$0.953102\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.90735	−	1.44531i	−0.431571	−	0.0629585i
$528$		0			0
$529$	1.35425			0.0588804
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−14.6315			−0.633761
$534$		0			0
$535$	−16.2915	+	14.3073i	−0.704343	+	0.618557i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	3.47371i		−	0.149623i
$540$		0			0
$541$		−	37.1327i		−	1.59646i	−0.602354	−	0.798229i	$-0.705771\pi$
$541$		−	37.1327i		−	1.59646i	0.602354	−	0.798229i	$-0.294229\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−13.0627	−	14.8744i	−0.559547	−	0.637148i
$546$		0			0
$547$	10.8869			0.465489			0.232745	−	0.972538i	$-0.425229\pi$
$547$	10.8869			0.465489			0.232745	+	0.972538i	$0.425229\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			6.71453i			0.285531i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			14.8744i			0.630248i	0.949051	+	0.315124i	$0.102046\pi$
$557$			14.8744i			0.630248i	−0.949051	+	0.315124i	$0.897954\pi$
$558$		0			0
$559$	−15.8745			−0.671420
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	14.8744i		−	0.626880i	−0.949608	−	0.313440i	$-0.898519\pi$
$563$		−	14.8744i		−	0.626880i	0.949608	−	0.313440i	$-0.101481\pi$
$564$		0			0
$565$	−9.64575	+	8.47094i	−0.405800	+	0.356375i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$			47.5538i			1.99006i	0.0995540	+	0.995032i	$0.468258\pi$
$571$			47.5538i			1.99006i	−0.0995540	+	0.995032i	$0.531742\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.18678	−	24.4684i	0.132898	−	1.02040i
$576$		0			0
$577$			11.3616i			0.472988i	0.971633	+	0.236494i	$0.0759983\pi$
$577$			11.3616i			0.472988i	−0.971633	+	0.236494i	$0.924002\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			41.1291i			1.70632i
$582$		0			0
$583$	−43.8945			−1.81793
$584$		0			0
$585$		0			0
$586$		0			0
$587$			14.8744i			0.613931i	0.951721	+	0.306966i	$0.0993137\pi$
$587$			14.8744i			0.613931i	−0.951721	+	0.306966i	$0.900686\pi$
$588$		0			0
$589$			12.8495i			0.529453i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	24.4795i		−	1.00525i	−0.864504	−	0.502627i	$-0.832367\pi$
$593$		−	24.4795i		−	1.00525i	0.864504	−	0.502627i	$-0.167633\pi$
$594$		0			0
$595$	−13.8807	−	21.3826i	−0.569051	−	0.876602i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.5830			0.514128			0.257064	−	0.966394i	$-0.417245\pi$
$599$	12.5830			0.514128			0.257064	+	0.966394i	$0.417245\pi$
$600$		0			0
$601$			41.1291i			1.67769i	0.544371	+	0.838845i	$0.316768\pi$
$601$			41.1291i			1.67769i	−0.544371	+	0.838845i	$0.683232\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−30.1371	+	26.4666i	−1.22525	+	1.07602i
$606$		0			0
$607$	9.27482			0.376453			0.188227	−	0.982126i	$-0.439726\pi$
$607$	9.27482			0.376453			0.188227	+	0.982126i	$0.439726\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	33.2915			1.34683
$612$		0			0
$613$		−	19.8325i		−	0.801027i	−0.916291	−	0.400514i	$-0.868832\pi$
$613$		−	19.8325i		−	0.801027i	0.916291	−	0.400514i	$-0.131168\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−47.6018			−1.91638			−0.958188	−	0.286138i	$-0.907628\pi$
$617$	−47.6018			−1.91638			−0.958188	+	0.286138i	$0.907628\pi$
$618$		0			0
$619$			25.8513i			1.03905i	0.854455	+	0.519525i	$0.173891\pi$
$619$			25.8513i			1.03905i	−0.854455	+	0.519525i	$0.826109\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	39.3439			1.57628
$624$		0			0
$625$	−24.1660	−	6.40343i	−0.966640	−	0.256137i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.29150	+	22.5627i	−0.131241	+	0.899635i
$630$		0			0
$631$	−41.2915			−1.64379			−0.821894	−	0.569640i	$-0.807083\pi$
$631$	−41.2915			−1.64379			−0.821894	+	0.569640i	$0.807083\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19.3557	+	22.0400i	0.768106	+	0.874632i
$636$		0			0
$637$			3.20172i			0.126857i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			24.4684i			0.966442i	0.875498	+	0.483221i	$0.160533\pi$
$641$			24.4684i			0.966442i	−0.875498	+	0.483221i	$0.839467\pi$
$642$		0			0
$643$	−7.66275			−0.302189			−0.151095	−	0.988519i	$-0.548280\pi$
$643$	−7.66275			−0.302189			−0.151095	+	0.988519i	$0.548280\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			23.0342i			0.905568i	0.891620	+	0.452784i	$0.149569\pi$
$647$			23.0342i			0.905568i	−0.891620	+	0.452784i	$0.850431\pi$
$648$		0			0
$649$		−	49.9821i		−	1.96197i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−41.8608			−1.63814			−0.819069	−	0.573695i	$-0.805510\pi$
$653$	−41.8608			−1.63814			−0.819069	+	0.573695i	$0.805510\pi$
$654$		0			0
$655$	−12.2915	−	13.9962i	−0.480269	−	0.546876i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−28.4575			−1.10855			−0.554274	−	0.832334i	$-0.687004\pi$
$659$	−28.4575			−1.10855			−0.554274	+	0.832334i	$0.687004\pi$
$660$		0			0
$661$	45.1660			1.75675			0.878377	−	0.477968i	$-0.158627\pi$
$661$	45.1660			1.75675			0.878377	+	0.477968i	$0.158627\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−24.5830	+	21.5889i	−0.953288	+	0.837182i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−73.7490			−2.84705
$672$		0			0
$673$	0.979531			0.0377582			0.0188791	−	0.999822i	$-0.493990\pi$
$673$	0.979531			0.0377582			0.0188791	+	0.999822i	$0.493990\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−27.8618			−1.07082			−0.535408	−	0.844594i	$-0.679842\pi$
$677$	−27.8618			−1.07082			−0.535408	+	0.844594i	$0.679842\pi$
$678$		0			0
$679$	25.1660			0.965783
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−26.2871			−1.00585			−0.502924	−	0.864331i	$-0.667742\pi$
$683$	−26.2871			−1.00585			−0.502924	+	0.864331i	$0.667742\pi$
$684$		0			0
$685$	2.13256	+	2.42832i	0.0814810	+	0.0927813i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	40.4575			1.54131
$690$		0			0
$691$		−	16.1380i		−	0.613919i	−0.951723	−	0.306959i	$-0.900688\pi$
$691$		−	16.1380i		−	0.613919i	0.951723	−	0.306959i	$-0.0993116\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.47974	+	10.7945i	0.359587	+	0.409457i
$696$		0			0
$697$	−12.0399	−	1.75641i	−0.456044	−	0.0665287i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.06275			−0.266756			−0.133378	−	0.991065i	$-0.542582\pi$
$701$	−7.06275			−0.266756			−0.133378	+	0.991065i	$0.542582\pi$
$702$		0			0
$703$	29.2630			1.10368
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.0399			−0.452807
$708$		0			0
$709$		−	12.8495i		−	0.482572i	−0.970454	−	0.241286i	$-0.922431\pi$
$709$		−	12.8495i		−	0.482572i	0.970454	−	0.241286i	$-0.0775691\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			11.9838i			0.448795i
$714$		0			0
$715$	44.8118	−	39.3539i	1.67586	−	1.47175i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			23.0854i			0.860941i	0.902605	+	0.430471i	$0.141652\pi$
$719$			23.0854i			0.860941i	−0.902605	+	0.430471i	$0.858348\pi$
$720$		0			0
$721$		−	13.7097i		−	0.510575i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			44.6231i			1.65498i	0.561480	+	0.827490i	$0.310232\pi$
$727$			44.6231i			1.65498i	−0.561480	+	0.827490i	$0.689768\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−13.0627	−	1.90562i	−0.483143	−	0.0704820i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36.1197			−1.33049
$738$		0			0
$739$	−10.7085			−0.393918			−0.196959	−	0.980412i	$-0.563107\pi$
$739$	−10.7085			−0.393918			−0.196959	+	0.980412i	$0.563107\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−31.8173			−1.16726			−0.583631	−	0.812019i	$-0.698369\pi$
$743$	−31.8173			−1.16726			−0.583631	+	0.812019i	$0.698369\pi$
$744$		0			0
$745$	11.0604	−	9.71328i	0.405221	−	0.355867i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	26.8118			0.979680
$750$		0			0
$751$		−	6.42473i		−	0.234442i	−0.993106	−	0.117221i	$-0.962601\pi$
$751$		−	6.42473i		−	0.234442i	0.993106	−	0.117221i	$-0.0373985\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.8014	−	14.7551i	0.611466	−	0.536992i
$756$		0			0
$757$			24.1684i			0.878416i	0.898385	+	0.439208i	$0.144741\pi$
$757$			24.1684i			0.878416i	−0.898385	+	0.439208i	$0.855259\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	26.2288			0.950792			0.475396	−	0.879772i	$-0.342305\pi$
$761$	26.2288			0.950792			0.475396	+	0.879772i	$0.342305\pi$
$762$		0			0
$763$			24.4795i			0.886218i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			46.0684i			1.66343i
$768$		0			0
$769$	−2.93725			−0.105920			−0.0529600	−	0.998597i	$-0.516866\pi$
$769$	−2.93725			−0.105920			−0.0529600	+	0.998597i	$0.516866\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1.44531i		−	0.0519841i	−0.999662	−	0.0259920i	$-0.991726\pi$
$773$		−	1.44531i		−	0.0519841i	0.999662	−	0.0259920i	$-0.00827445\pi$
$774$		0			0
$775$	12.0399	+	1.56809i	0.432486	+	0.0563275i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			15.6153i			0.559477i
$780$		0			0
$781$	−60.6863			−2.17153
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.2231	+	19.6117i	0.614719	+	0.699972i
$786$		0			0
$787$	−50.5777			−1.80290			−0.901451	−	0.432881i	$-0.857497\pi$
$787$	−50.5777			−1.80290			−0.901451	+	0.432881i	$0.857497\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	15.8745			0.564433
$792$		0			0
$793$	67.9743			2.41384
$794$		0			0
$795$		0			0
$796$		0			0
$797$			16.3197i			0.578073i	0.957318	+	0.289036i	$0.0933349\pi$
$797$			16.3197i			0.578073i	−0.957318	+	0.289036i	$0.906665\pi$
$798$		0			0
$799$	27.3948	+	3.99641i	0.969157	+	0.141383i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	83.9770i		−	2.96348i
$804$		0			0
$805$	−22.9268	+	20.1344i	−0.808063	+	0.709645i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			10.7587i			0.378255i	0.981953	+	0.189127i	$0.0605659\pi$
$809$			10.7587i			0.378255i	−0.981953	+	0.189127i	$0.939434\pi$
$810$		0			0
$811$		−	1.56809i		−	0.0550631i	−0.999621	−	0.0275316i	$-0.991235\pi$
$811$		−	1.56809i		−	0.0550631i	0.999621	−	0.0275316i	$-0.00876467\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.2288	−	20.3996i	0.813668	−	0.714567i
$816$		0			0
$817$			16.9419i			0.592722i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			12.6643i			0.441987i	0.975275	+	0.220993i	$0.0709299\pi$
$821$			12.6643i			0.441987i	−0.975275	+	0.220993i	$0.929070\pi$
$822$		0			0
$823$	9.27482			0.323300			0.161650	−	0.986848i	$-0.448318\pi$
$823$	9.27482			0.323300			0.161650	+	0.986848i	$0.448318\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.01688			0.0353603			0.0176801	−	0.999844i	$-0.494372\pi$
$827$	1.01688			0.0353603			0.0176801	+	0.999844i	$0.494372\pi$
$828$		0			0
$829$	7.41699			0.257603			0.128801	−	0.991670i	$-0.458887\pi$
$829$	7.41699			0.257603			0.128801	+	0.991670i	$0.458887\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.384343	+	2.63461i	−0.0133167	+	0.0912840i
$834$		0			0
$835$	28.5203	−	25.0466i	0.986984	−	0.866774i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	5.37934i		−	0.185715i	−0.995679	−	0.0928576i	$-0.970400\pi$
$839$		−	5.37934i		−	0.185715i	0.995679	−	0.0928576i	$-0.0296002\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.4611	+	17.0908i	−0.669482	+	0.587942i
$846$		0			0
$847$	49.5982			1.70421
$848$		0			0
$849$		0			0
$850$		0			0
$851$	27.2915			0.935541
$852$		0			0
$853$	−18.5496			−0.635127			−0.317564	−	0.948237i	$-0.602865\pi$
$853$	−18.5496			−0.635127			−0.317564	+	0.948237i	$0.602865\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−27.0931			−0.925483			−0.462741	−	0.886493i	$-0.653134\pi$
$857$	−27.0931			−0.925483			−0.462741	+	0.886493i	$0.653134\pi$
$858$		0			0
$859$	−23.7490			−0.810306			−0.405153	−	0.914249i	$-0.632782\pi$
$859$	−23.7490			−0.810306			−0.405153	+	0.914249i	$0.632782\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			9.60515i			0.326963i	0.986546	+	0.163482i	$0.0522724\pi$
$863$			9.60515i			0.326963i	−0.986546	+	0.163482i	$0.947728\pi$
$864$		0			0
$865$	−4.00000	+	3.51282i	−0.136004	+	0.119439i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.0627			−0.443123
$870$		0			0
$871$	33.2915			1.12804
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17.2288	+	25.6688i	0.582438	+	0.867765i
$876$		0			0
$877$	22.7533			0.768324			0.384162	−	0.923266i	$-0.374490\pi$
$877$	22.7533			0.768324			0.384162	+	0.923266i	$0.374490\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	41.1291i		−	1.38567i	−0.721095	−	0.692837i	$-0.756361\pi$
$881$		−	41.1291i		−	1.38567i	0.721095	−	0.692837i	$-0.243639\pi$
$882$		0			0
$883$			24.1684i			0.813332i	0.913577	+	0.406666i	$0.133309\pi$
$883$			24.1684i			0.813332i	−0.913577	+	0.406666i	$0.866691\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	44.8367			1.50547			0.752735	−	0.658324i	$-0.228734\pi$
$887$	44.8367			1.50547			0.752735	+	0.658324i	$0.228734\pi$
$888$		0			0
$889$		−	36.2724i		−	1.21654i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	35.5300i		−	1.18897i
$894$		0			0
$895$	−5.53019	+	4.85664i	−0.184854	+	0.162340i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	33.2915	+	4.85664i	1.10910	+	0.161798i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	33.2915	+	37.9086i	1.10665	+	1.26012i
$906$		0			0
$907$	−50.5777			−1.67941			−0.839703	−	0.543046i	$-0.817271\pi$
$907$	−50.5777			−1.67941			−0.839703	+	0.543046i	$0.817271\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			5.37934i			0.178225i	0.996022	+	0.0891127i	$0.0284031\pi$
$911$			5.37934i			0.178225i	−0.996022	+	0.0891127i	$0.971597\pi$
$912$		0			0
$913$	−80.0143			−2.64809
$914$		0			0
$915$		0			0
$916$		0			0
$917$			23.0342i			0.760657i
$918$		0			0
$919$	25.2915			0.834290			0.417145	−	0.908840i	$-0.363031\pi$
$919$	25.2915			0.834290			0.417145	+	0.908840i	$0.363031\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	55.9344			1.84110
$924$		0			0
$925$	3.57113	−	27.4194i	0.117418	−	0.901544i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	12.6643i		−	0.415502i	−0.978182	−	0.207751i	$-0.933386\pi$
$929$		−	12.6643i		−	0.415502i	0.978182	−	0.207751i	$-0.0666144\pi$
$930$		0			0
$931$	3.41699			0.111987
$932$		0			0
$933$		0			0
$934$		0			0
$935$	41.5987	−	27.0040i	1.36042	−	0.883126i
$936$		0			0
$937$			11.0505i			0.361003i	0.983575	+	0.180501i	$0.0577720\pi$
$937$			11.0505i			0.361003i	−0.983575	+	0.180501i	$0.942228\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			14.5699i			0.474966i	0.971392	+	0.237483i	$0.0763224\pi$
$941$			14.5699i			0.474966i	−0.971392	+	0.237483i	$0.923678\pi$
$942$		0			0
$943$			14.5633i			0.474245i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.95547			−0.128536			−0.0642678	−	0.997933i	$-0.520471\pi$
$947$	−3.95547			−0.128536			−0.0642678	+	0.997933i	$0.520471\pi$
$948$		0			0
$949$			77.4015i			2.51256i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	58.0522i		−	1.88050i	−0.340492	−	0.940248i	$-0.610594\pi$
$953$		−	58.0522i		−	1.88050i	0.340492	−	0.940248i	$-0.389406\pi$
$954$		0			0
$955$	−10.0808	+	8.85305i	−0.326209	+	0.286478i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	3.99641i		−	0.129051i
$960$		0			0
$961$	25.1033			0.809783
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−20.2288	+	17.7650i	−0.651187	+	0.571875i
$966$		0			0
$967$		−	46.3795i		−	1.49147i	−0.666245	−	0.745733i	$-0.732100\pi$
$967$		−	46.3795i		−	1.49147i	0.666245	−	0.745733i	$-0.267900\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$		−	17.7650i		−	0.569519i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	21.5889i		−	0.690690i	−0.938476	−	0.345345i	$-0.887762\pi$
$977$		−	21.5889i		−	0.690690i	0.938476	−	0.345345i	$-0.112238\pi$
$978$		0			0
$979$			76.5413i			2.44627i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−47.0813			−1.50166			−0.750830	−	0.660495i	$-0.770346\pi$
$983$	−47.0813			−1.50166			−0.750830	+	0.660495i	$0.770346\pi$
$984$		0			0
$985$	35.8745	−	31.5052i	1.14306	−	1.00384i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			15.8005i			0.502426i
$990$		0			0
$991$		−	46.6936i		−	1.48327i	−0.670804	−	0.741635i	$-0.734051\pi$
$991$		−	46.6936i		−	1.48327i	0.670804	−	0.741635i	$-0.265949\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.58301	−	4.07992i	−0.113589	−	0.129342i
$996$		0			0
$997$	27.3040			0.864725			0.432362	−	0.901700i	$-0.357680\pi$
$997$	27.3040			0.864725			0.432362	+	0.901700i	$0.357680\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.k.h.1189.7		8
3.2	odd	2		340.2.g.a.169.6	yes	8
5.4	even	2	inner	3060.2.k.h.1189.1		8
12.11	even	2		1360.2.o.e.849.4		8
15.2	even	4		1700.2.c.e.101.4		8
15.8	even	4		1700.2.c.e.101.6		8
15.14	odd	2		340.2.g.a.169.4	yes	8
17.16	even	2	inner	3060.2.k.h.1189.2		8
51.50	odd	2		340.2.g.a.169.3	✓	8
60.59	even	2		1360.2.o.e.849.6		8
85.84	even	2	inner	3060.2.k.h.1189.8		8
204.203	even	2		1360.2.o.e.849.5		8
255.152	even	4		1700.2.c.e.101.5		8
255.203	even	4		1700.2.c.e.101.3		8
255.254	odd	2		340.2.g.a.169.5	yes	8
1020.1019	even	2		1360.2.o.e.849.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.g.a.169.3	✓	8	51.50	odd	2
340.2.g.a.169.4	yes	8	15.14	odd	2
340.2.g.a.169.5	yes	8	255.254	odd	2
340.2.g.a.169.6	yes	8	3.2	odd	2
1360.2.o.e.849.3		8	1020.1019	even	2
1360.2.o.e.849.4		8	12.11	even	2
1360.2.o.e.849.5		8	204.203	even	2
1360.2.o.e.849.6		8	60.59	even	2
1700.2.c.e.101.3		8	255.203	even	4
1700.2.c.e.101.4		8	15.2	even	4
1700.2.c.e.101.5		8	255.152	even	4
1700.2.c.e.101.6		8	15.8	even	4
3060.2.k.h.1189.1		8	5.4	even	2	inner
3060.2.k.h.1189.2		8	17.16	even	2	inner
3060.2.k.h.1189.7		8	1.1	even	1	trivial
3060.2.k.h.1189.8		8	85.84	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.68014 - 1.47551i) q^{5} -2.76510 q^{7} -5.37934i q^{11} +4.95813i q^{13} +(-0.595188 + 4.07992i) q^{17} +5.29150 q^{19} +4.93500 q^{23} +(0.645751 - 4.95813i) q^{25} +2.42832i q^{31} +(-4.64575 + 4.07992i) q^{35} +5.53019 q^{37} +2.95102i q^{41} +3.20172i q^{43} -6.71453i q^{47} +0.645751 q^{49} -8.15984i q^{53} +(-7.93725 - 9.03805i) q^{55} +9.29150 q^{59} -13.7097i q^{61} +(7.31575 + 8.33035i) q^{65} -6.71453i q^{67} -11.2814i q^{71} +15.6110 q^{73} +14.8744i q^{77} -2.42832i q^{79} -14.8744i q^{83} +(5.01996 + 7.73305i) q^{85} -14.2288 q^{89} -13.7097i q^{91} +(8.89047 - 7.80766i) q^{95} -9.10132 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{25} - 16 q^{35} - 16 q^{49} + 32 q^{59} - 8 q^{85} - 8 q^{89}+O(q^{100}) \) 8 * q - 16 * q^25 - 16 * q^35 - 16 * q^49 + 32 * q^59 - 8 * q^85 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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