LMFDB - Embedded newform 340.2.g.a.169.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [340,2,Mod(169,340)] 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("340.169"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 340 = 2^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	340.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.71491366872$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.5
Root			$1.68014 + 1.47551i$ of defining polynomial
Character	$\chi$	$=$	340.169
Dual form			340.2.g.a.169.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.595188 q^{3} +(-1.68014 - 1.47551i) q^{5} -2.76510 q^{7} -2.64575 q^{9} -5.37934i q^{11} -4.95813i q^{13} +(-1.00000 - 0.878205i) q^{15} +(0.595188 + 4.07992i) q^{17} +5.29150 q^{19} -1.64575 q^{21} -4.93500 q^{23} +(0.645751 + 4.95813i) q^{25} -3.36028 q^{27} -2.42832i q^{31} -3.20172i q^{33} +(4.64575 + 4.07992i) q^{35} +5.53019 q^{37} -2.95102i q^{39} +2.95102i q^{41} -3.20172i q^{43} +(4.44524 + 3.90383i) q^{45} -6.71453i q^{47} +0.645751 q^{49} +(0.354249 + 2.42832i) q^{51} -8.15984i q^{53} +(-7.93725 + 9.03805i) q^{55} +3.14944 q^{57} -9.29150 q^{59} +13.7097i q^{61} +7.31575 q^{63} +(-7.31575 + 8.33035i) q^{65} +6.71453i q^{67} -2.93725 q^{69} -11.2814i q^{71} +15.6110 q^{73} +(0.384343 + 2.95102i) q^{75} +14.8744i q^{77} +2.42832i q^{79} +5.93725 q^{81} -14.8744i q^{83} +(5.01996 - 7.73305i) q^{85} +14.2288 q^{89} +13.7097i q^{91} -1.44531i q^{93} +(-8.89047 - 7.80766i) q^{95} -9.10132 q^{97} +14.2324i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{15} + 8 q^{21} - 16 q^{25} + 16 q^{35} - 16 q^{49} + 24 q^{51} - 32 q^{59} + 40 q^{69} - 16 q^{81} - 8 q^{85} + 8 q^{89}+O(q^{100}) $ 8 * q - 8 * q^15 + 8 * q^21 - 16 * q^25 + 16 * q^35 - 16 * q^49 + 24 * q^51 - 32 * q^59 + 40 * q^69 - 16 * q^81 - 8 * q^85 + 8 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/340\mathbb{Z}\right)^\times$.

$n$	$137$	$171$	$241$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.595188			0.343632			0.171816	−	0.985129i	$-0.445037\pi$
$3$	0.595188			0.343632			0.171816	+	0.985129i	$0.445037\pi$
$4$		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
$9$	−2.64575			−0.881917
$10$		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	$-0.758678\pi$
$13$		−	4.95813i		−	1.37514i	0.726120	−	0.687568i	$-0.241322\pi$
$14$		0			0
$15$	−1.00000	−	0.878205i	−0.258199	−	0.226751i
$16$		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
$21$	−1.64575			−0.359132
$22$		0			0
$23$	−4.93500			−1.02902			−0.514510	−	0.857485i	$-0.672026\pi$
$23$	−4.93500			−1.02902			−0.514510	+	0.857485i	$0.672026\pi$
$24$		0			0
$25$	0.645751	+	4.95813i	0.129150	+	0.991625i
$26$		0			0
$27$	−3.36028			−0.646687
$28$		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	$-0.930024\pi$
$31$		−	2.42832i		−	0.436139i	0.975933	−	0.218070i	$-0.0699760\pi$
$32$		0			0
$33$		−	3.20172i		−	0.557347i
$34$		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
$39$		−	2.95102i		−	0.472541i
$40$		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	$-0.921498\pi$
$43$		−	3.20172i		−	0.488257i	0.969743	−	0.244129i	$-0.0785019\pi$
$44$		0			0
$45$	4.44524	+	3.90383i	0.662657	+	0.581948i
$46$		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
$51$	0.354249	+	2.42832i	0.0496047	+	0.340033i
$52$		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
$57$	3.14944			0.417153
$58$		0			0
$59$	−9.29150			−1.20965			−0.604825	−	0.796358i	$-0.706757\pi$
$59$	−9.29150			−1.20965			−0.604825	+	0.796358i	$0.706757\pi$
$60$		0			0
$61$			13.7097i			1.75535i	0.479260	+	0.877673i	$0.340905\pi$
$61$			13.7097i			1.75535i	−0.479260	+	0.877673i	$0.659095\pi$
$62$		0			0
$63$	7.31575			0.921698
$64$		0			0
$65$	−7.31575	+	8.33035i	−0.907408	+	1.03325i
$66$		0			0
$67$			6.71453i			0.820311i	0.912016	+	0.410155i	$0.134526\pi$
$67$			6.71453i			0.820311i	−0.912016	+	0.410155i	$0.865474\pi$
$68$		0			0
$69$	−2.93725			−0.353604
$70$		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
$75$	0.384343	+	2.95102i	0.0443802	+	0.340754i
$76$		0			0
$77$			14.8744i			1.69509i
$78$		0			0
$79$			2.42832i			0.273207i	0.990626	+	0.136604i	$0.0436187\pi$
$79$			2.42832i			0.273207i	−0.990626	+	0.136604i	$0.956381\pi$
$80$		0			0
$81$	5.93725			0.659695
$82$		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	$-0.228062\pi$
$89$	14.2288			1.50825			0.754123	+	0.656734i	$0.228062\pi$
$90$		0			0
$91$			13.7097i			1.43717i
$92$		0			0
$93$		−	1.44531i		−	0.149871i
$94$		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
$99$			14.2324i			1.43041i
$100$		0			0
$101$	−4.35425			−0.433264			−0.216632	−	0.976253i	$-0.569507\pi$
$101$	−4.35425			−0.433264			−0.216632	+	0.976253i	$0.569507\pi$
$102$		0			0
$103$		−	4.95813i		−	0.488539i	−0.969707	−	0.244269i	$-0.921452\pi$
$103$		−	4.95813i		−	0.488539i	0.969707	−	0.244269i	$-0.0785481\pi$
$104$		0			0
$105$	2.76510	+	2.42832i	0.269846	+	0.236980i
$106$		0			0
$107$	9.69651			0.937397			0.468698	−	0.883358i	$-0.344723\pi$
$107$	9.69651			0.937397			0.468698	+	0.883358i	$0.344723\pi$
$108$		0			0
$109$			8.85305i			0.847968i	0.905670	+	0.423984i	$0.139369\pi$
$109$			8.85305i			0.847968i	−0.905670	+	0.423984i	$0.860631\pi$
$110$		0			0
$111$	3.29150			0.312416
$112$		0			0
$113$	5.74103			0.540071			0.270036	−	0.962850i	$-0.412965\pi$
$113$	5.74103			0.540071			0.270036	+	0.962850i	$0.412965\pi$
$114$		0			0
$115$	8.29150	+	7.28164i	0.773187	+	0.679016i
$116$		0			0
$117$			13.1180i			1.21276i
$118$		0			0
$119$	−1.64575	−	11.2814i	−0.150866	−	1.03416i
$120$		0			0
$121$	−17.9373			−1.63066
$122$		0			0
$123$			1.75641i			0.158370i
$124$		0			0
$125$	6.23080	−	9.28316i	0.557300	−	0.830311i
$126$		0			0
$127$		−	13.1180i		−	1.16403i	−0.813178	−	0.582016i	$-0.802264\pi$
$127$		−	13.1180i		−	1.16403i	0.813178	−	0.582016i	$-0.197736\pi$
$128$		0			0
$129$		−	1.90562i		−	0.167781i
$130$		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
$135$	5.64575	+	4.95813i	0.485909	+	0.426727i
$136$		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	$-0.912160\pi$
$139$		−	6.42473i		−	0.544938i	0.962164	−	0.272469i	$-0.0878403\pi$
$140$		0			0
$141$		−	3.99641i		−	0.336558i
$142$		0			0
$143$	−26.6714			−2.23038
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.384343			0.0317001
$148$		0			0
$149$	−6.58301			−0.539301			−0.269650	−	0.962958i	$-0.586908\pi$
$149$	−6.58301			−0.539301			−0.269650	+	0.962958i	$0.586908\pi$
$150$		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
$153$	−1.57472	−	10.7945i	−0.127309	−	0.872680i
$154$		0			0
$155$	−3.58301	+	4.07992i	−0.287794	+	0.327707i
$156$		0			0
$157$		−	11.6727i		−	0.931580i	−0.884895	−	0.465790i	$-0.845770\pi$
$157$		−	11.6727i		−	0.931580i	0.884895	−	0.465790i	$-0.154230\pi$
$158$		0			0
$159$		−	4.85664i		−	0.385157i
$160$		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
$165$	−4.72416	+	5.37934i	−0.367775	+	0.418781i
$166$		0			0
$167$	−16.9749			−1.31356			−0.656779	−	0.754083i	$-0.728082\pi$
$167$	−16.9749			−1.31356			−0.656779	+	0.754083i	$0.728082\pi$
$168$		0			0
$169$	−11.5830			−0.891000
$170$		0			0
$171$	−14.0000			−1.07061
$172$		0			0
$173$	2.38075			0.181005			0.0905026	−	0.995896i	$-0.471153\pi$
$173$	2.38075			0.181005			0.0905026	+	0.995896i	$0.471153\pi$
$174$		0			0
$175$	−1.78556	−	13.7097i	−0.134976	−	1.03635i
$176$		0			0
$177$	−5.53019			−0.415675
$178$		0			0
$179$	3.29150			0.246018			0.123009	−	0.992406i	$-0.460746\pi$
$179$	3.29150			0.246018			0.123009	+	0.992406i	$0.460746\pi$
$180$		0			0
$181$		−	22.5627i		−	1.67708i	−0.544844	−	0.838538i	$-0.683411\pi$
$181$		−	22.5627i		−	1.67708i	0.544844	−	0.838538i	$-0.316589\pi$
$182$		0			0
$183$			8.15984i			0.603193i
$184$		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
$189$	9.29150			0.675857
$190$		0			0
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$		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
$195$	−4.35425	+	4.95813i	−0.311814	+	0.355059i
$196$		0			0
$197$	−21.3521			−1.52127			−0.760636	−	0.649178i	$-0.775113\pi$
$197$	−21.3521			−1.52127			−0.760636	+	0.649178i	$0.775113\pi$
$198$		0			0
$199$			2.42832i			0.172139i	0.996289	+	0.0860695i	$0.0274307\pi$
$199$			2.42832i			0.172139i	−0.996289	+	0.0860695i	$0.972569\pi$
$200$		0			0
$201$			3.99641i			0.281885i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.35425	−	4.95813i	0.304114	−	0.346290i
$206$		0			0
$207$	13.0568			0.907510
$208$		0			0
$209$		−	28.4648i		−	1.96895i
$210$		0			0
$211$			20.1344i			1.38611i	0.720884	+	0.693055i	$0.243736\pi$
$211$			20.1344i			1.38611i	−0.720884	+	0.693055i	$0.756264\pi$
$212$		0			0
$213$		−	6.71453i		−	0.460072i
$214$		0			0
$215$	−4.72416	+	5.37934i	−0.322185	+	0.366868i
$216$		0			0
$217$			6.71453i			0.455812i
$218$		0			0
$219$	9.29150			0.627862
$220$		0			0
$221$	20.2288	−	2.95102i	1.36073	−	0.198507i
$222$		0			0
$223$			21.2778i			1.42487i	0.701739	+	0.712434i	$0.252407\pi$
$223$			21.2778i			1.42487i	−0.701739	+	0.712434i	$0.747593\pi$
$224$		0			0
$225$	−1.70850	−	13.1180i	−0.113900	−	0.874531i
$226$		0			0
$227$	−19.7774			−1.31267			−0.656335	−	0.754470i	$-0.727894\pi$
$227$	−19.7774			−1.31267			−0.656335	+	0.754470i	$0.727894\pi$
$228$		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
$231$			8.85305i			0.582488i
$232$		0			0
$233$	12.2508			0.802574			0.401287	−	0.915952i	$-0.368563\pi$
$233$	12.2508			0.802574			0.401287	+	0.915952i	$0.368563\pi$
$234$		0			0
$235$	−9.90735	+	11.2814i	−0.646284	+	0.735915i
$236$		0			0
$237$			1.44531i			0.0938827i
$238$		0			0
$239$	9.87451			0.638729			0.319364	−	0.947632i	$-0.396531\pi$
$239$	9.87451			0.638729			0.319364	+	0.947632i	$0.396531\pi$
$240$		0			0
$241$		−	13.7097i		−	0.883119i	−0.897232	−	0.441559i	$-0.854425\pi$
$241$		−	13.7097i		−	0.883119i	0.897232	−	0.441559i	$-0.145575\pi$
$242$		0			0
$243$	13.6146			0.873379
$244$		0			0
$245$	−1.08495	−	0.952811i	−0.0693151	−	0.0608729i
$246$		0			0
$247$		−	26.2359i		−	1.66935i
$248$		0			0
$249$		−	8.85305i		−	0.561039i
$250$		0			0
$251$	−19.1660			−1.20975			−0.604874	−	0.796321i	$-0.706777\pi$
$251$	−19.1660			−1.20975			−0.604874	+	0.796321i	$0.706777\pi$
$252$		0			0
$253$			26.5470i			1.66900i
$254$		0			0
$255$	2.98782	−	4.60262i	0.187104	−	0.288227i
$256$		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
$267$	8.46878			0.518281
$268$		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
$273$			8.15984i			0.493856i
$274$		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
$279$			6.42473i			0.384638i
$280$		0			0
$281$	−13.1660			−0.785418			−0.392709	−	0.919663i	$-0.628462\pi$
$281$	−13.1660			−0.785418			−0.392709	+	0.919663i	$0.628462\pi$
$282$		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
$285$	−5.29150	−	4.64702i	−0.313442	−	0.275266i
$286$		0			0
$287$		−	8.15984i		−	0.481660i
$288$		0			0
$289$	−16.2915	+	4.85664i	−0.958324	+	0.285685i
$290$		0			0
$291$	−5.41699			−0.317550
$292$		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
$297$			18.0761i			1.04888i
$298$		0			0
$299$			24.4684i			1.41504i
$300$		0			0
$301$			8.85305i			0.510281i
$302$		0			0
$303$	−2.59160			−0.148883
$304$		0			0
$305$	20.2288	−	23.0342i	1.15830	−	1.31894i
$306$		0			0
$307$		−	17.7650i		−	1.01390i	−0.861975	−	0.506951i	$-0.830773\pi$
$307$		−	17.7650i		−	1.01390i	0.861975	−	0.506951i	$-0.169227\pi$
$308$		0			0
$309$		−	2.95102i		−	0.167877i
$310$		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
$315$	−12.2915	−	10.7945i	−0.692548	−	0.608199i
$316$		0			0
$317$	15.1894			0.853119			0.426559	−	0.904460i	$-0.359725\pi$
$317$	15.1894			0.853119			0.426559	+	0.904460i	$0.359725\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	5.77124			0.322119
$322$		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
$327$			5.26923i			0.291389i
$328$		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
$333$	−14.6315			−0.801802
$334$		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
$339$	3.41699			0.185586
$340$		0			0
$341$	−13.0627			−0.707387
$342$		0			0
$343$	17.5701			0.948696
$344$		0			0
$345$	4.93500	+	4.33394i	0.265692	+	0.233332i
$346$		0			0
$347$	15.0159			0.806093			0.403047	−	0.915179i	$-0.367951\pi$
$347$	15.0159			0.806093			0.403047	+	0.915179i	$0.367951\pi$
$348$		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
$351$			16.6607i			0.889283i
$352$		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
$357$	−0.979531	−	6.71453i	−0.0518423	−	0.355371i
$358$		0			0
$359$	2.12549			0.112179			0.0560896	−	0.998426i	$-0.482137\pi$
$359$	2.12549			0.112179			0.0560896	+	0.998426i	$0.482137\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$		0			0
$363$	−10.6760			−0.560347
$364$		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
$369$		−	7.80766i		−	0.406450i
$370$		0			0
$371$			22.5627i			1.17140i
$372$		0			0
$373$			17.7650i			0.919836i	0.887961	+	0.459918i	$0.152121\pi$
$373$			17.7650i			0.919836i	−0.887961	+	0.459918i	$0.847879\pi$
$374$		0			0
$375$	3.70850	−	5.52523i	0.191506	−	0.285321i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			29.8477i			1.53317i	0.642141	+	0.766587i	$0.278046\pi$
$379$			29.8477i			1.53317i	−0.642141	+	0.766587i	$0.721954\pi$
$380$		0			0
$381$		−	7.80766i		−	0.399998i
$382$		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
$387$			8.47094i			0.430602i
$388$		0			0
$389$	−25.6458			−1.30029			−0.650146	−	0.759810i	$-0.725292\pi$
$389$	−25.6458			−1.30029			−0.650146	+	0.759810i	$0.725292\pi$
$390$		0			0
$391$	−2.93725	−	20.1344i	−0.148543	−	1.01824i
$392$		0			0
$393$		−	4.95813i		−	0.250104i
$394$		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
$399$	−8.70850			−0.435970
$400$		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
$405$	−9.97543	−	8.76047i	−0.495683	−	0.435311i
$406$		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
$411$			0.860229i			0.0424320i
$412$		0			0
$413$	25.6919			1.26422
$414$		0			0
$415$	−21.9473	+	24.9911i	−1.07735	+	1.22676i
$416$		0			0
$417$		−	3.82392i		−	0.187258i
$418$		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
$423$			17.7650i			0.863763i
$424$		0			0
$425$	−19.8444	+	5.58563i	−0.962595	+	0.270943i
$426$		0			0
$427$		−	37.9086i		−	1.83453i
$428$		0			0
$429$	−15.8745			−0.766428
$430$		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	$-0.984180\pi$
$433$		−	2.06751i		−	0.0993583i	0.998765	−	0.0496791i	$-0.0158199\pi$
$434$		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	$-0.874163\pi$
$439$		−	16.1380i		−	0.770225i	0.922870	−	0.385113i	$-0.125837\pi$
$440$		0			0
$441$	−1.70850			−0.0813570
$442$		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
$447$	−3.91813			−0.185321
$448$		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
$453$	5.95188			0.279644
$454$		0			0
$455$	20.2288	−	23.0342i	0.948339	−	1.07986i
$456$		0			0
$457$		−	27.6812i		−	1.29487i	−0.762119	−	0.647437i	$-0.775841\pi$
$457$		−	27.6812i		−	1.29487i	0.762119	−	0.647437i	$-0.224159\pi$
$458$		0			0
$459$	−2.00000	−	13.7097i	−0.0933520	−	0.639913i
$460$		0			0
$461$	−20.7085			−0.964491			−0.482245	−	0.876036i	$-0.660179\pi$
$461$	−20.7085			−0.964491			−0.482245	+	0.876036i	$0.660179\pi$
$462$		0			0
$463$			31.1941i			1.44971i	0.688901	+	0.724855i	$0.258093\pi$
$463$			31.1941i			1.44971i	−0.688901	+	0.724855i	$0.741907\pi$
$464$		0			0
$465$	−2.13256	+	2.42832i	−0.0988952	+	0.112611i
$466$		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
$471$		−	6.94743i		−	0.320121i
$472$		0			0
$473$	−17.2231			−0.791919
$474$		0			0
$475$	3.41699	+	26.2359i	0.156782	+	1.20379i
$476$		0			0
$477$			21.5889i			0.988488i
$478$		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
$483$	8.12179			0.369554
$484$		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
$489$	8.22876			0.372117
$490$		0			0
$491$	21.8745			0.987183			0.493591	−	0.869694i	$-0.335684\pi$
$491$	21.8745			0.987183			0.493591	+	0.869694i	$0.335684\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	21.0000	−	23.9124i	0.943880	−	1.07478i
$496$		0			0
$497$			31.1941i			1.39924i
$498$		0			0
$499$		−	7.28496i		−	0.326120i	−0.986616	−	0.163060i	$-0.947864\pi$
$499$		−	7.28496i		−	0.326120i	0.986616	−	0.163060i	$-0.0521363\pi$
$500$		0			0
$501$	−10.1033			−0.451381
$502$		0			0
$503$	−12.6351			−0.563371			−0.281686	−	0.959507i	$-0.590894\pi$
$503$	−12.6351			−0.563371			−0.281686	+	0.959507i	$0.590894\pi$
$504$		0			0
$505$	7.31575	+	6.42473i	0.325547	+	0.285897i
$506$		0			0
$507$	−6.89407			−0.306176
$508$		0			0
$509$	34.4575			1.52730			0.763651	−	0.645629i	$-0.223405\pi$
$509$	34.4575			1.52730			0.763651	+	0.645629i	$0.223405\pi$
$510$		0			0
$511$	−43.1660			−1.90955
$512$		0			0
$513$	−17.7809			−0.785048
$514$		0			0
$515$	−7.31575	+	8.33035i	−0.322371	+	0.367079i
$516$		0			0
$517$	−36.1197			−1.58854
$518$		0			0
$519$	1.41699			0.0621992
$520$		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	$-0.953102\pi$
$523$		−	6.71453i		−	0.293606i	0.989166	−	0.146803i	$-0.0468984\pi$
$524$		0			0
$525$	−1.06275	−	8.15984i	−0.0463820	−	0.356125i
$526$		0			0
$527$	9.90735	−	1.44531i	0.431571	−	0.0629585i
$528$		0			0
$529$	1.35425			0.0588804
$530$		0			0
$531$	24.5830			1.06681
$532$		0			0
$533$	14.6315			0.633761
$534$		0			0
$535$	−16.2915	−	14.3073i	−0.704343	−	0.618557i
$536$		0			0
$537$	1.95906			0.0845398
$538$		0			0
$539$		−	3.47371i		−	0.149623i
$540$		0			0
$541$			37.1327i			1.59646i	0.602354	+	0.798229i	$0.294229\pi$
$541$			37.1327i			1.59646i	−0.602354	+	0.798229i	$0.705771\pi$
$542$		0			0
$543$		−	13.4291i		−	0.576297i
$544$		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
$549$		−	36.2724i		−	1.54807i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	6.71453i		−	0.285531i
$554$		0			0
$555$	−5.53019	−	4.85664i	−0.234744	−	0.206153i
$556$		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
$561$	13.0627	−	1.90562i	0.551510	−	0.0804555i
$562$		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
$567$	−16.4171			−0.689452
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$		−	47.5538i		−	1.99006i	−0.0995540	−	0.995032i	$-0.531742\pi$
$571$		−	47.5538i		−	1.99006i	0.0995540	−	0.995032i	$-0.468258\pi$
$572$		0			0
$573$	3.57113			0.149186
$574$		0			0
$575$	−3.18678	−	24.4684i	−0.132898	−	1.02040i
$576$		0			0
$577$		−	11.3616i		−	0.472988i	−0.971633	−	0.236494i	$-0.924002\pi$
$577$		−	11.3616i		−	0.472988i	0.971633	−	0.236494i	$-0.0759983\pi$
$578$		0			0
$579$	−7.16601			−0.297809
$580$		0			0
$581$			41.1291i			1.70632i
$582$		0			0
$583$	−43.8945			−1.81793
$584$		0			0
$585$	19.3557	−	22.0400i	0.800258	−	0.911244i
$586$		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
$591$	−12.7085			−0.522758
$592$		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
$597$			1.44531i			0.0591525i
$598$		0			0
$599$	−12.5830			−0.514128			−0.257064	−	0.966394i	$-0.582755\pi$
$599$	−12.5830			−0.514128			−0.257064	+	0.966394i	$0.582755\pi$
$600$		0			0
$601$		−	41.1291i		−	1.67769i	−0.544371	−	0.838845i	$-0.683232\pi$
$601$		−	41.1291i		−	1.67769i	0.544371	−	0.838845i	$-0.316768\pi$
$602$		0			0
$603$		−	17.7650i		−	0.723446i
$604$		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	$0.131168\pi$
$613$			19.8325i			0.801027i	−0.916291	+	0.400514i	$0.868832\pi$
$614$		0			0
$615$	2.59160	−	2.95102i	0.104503	−	0.118996i
$616$		0			0
$617$	47.6018			1.91638			0.958188	−	0.286138i	$-0.0923717\pi$
$617$	47.6018			1.91638			0.958188	+	0.286138i	$0.0923717\pi$
$618$		0			0
$619$		−	25.8513i		−	1.03905i	−0.854455	−	0.519525i	$-0.826109\pi$
$619$		−	25.8513i		−	1.03905i	0.854455	−	0.519525i	$-0.173891\pi$
$620$		0			0
$621$	16.5830			0.665453
$622$		0			0
$623$	−39.3439			−1.57628
$624$		0			0
$625$	−24.1660	+	6.40343i	−0.966640	+	0.256137i
$626$		0			0
$627$		−	16.9419i		−	0.676594i
$628$		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
$633$			11.9838i			0.476312i
$634$		0			0
$635$	−19.3557	+	22.0400i	−0.768106	+	0.874632i
$636$		0			0
$637$		−	3.20172i		−	0.126857i
$638$		0			0
$639$			29.8477i			1.18076i
$640$		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
$645$	−2.81176	+	3.20172i	−0.110713	+	0.126067i
$646$		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
$651$			3.99641i			0.156632i
$652$		0			0
$653$	41.8608			1.63814			0.819069	−	0.573695i	$-0.194490\pi$
$653$	41.8608			1.63814			0.819069	+	0.573695i	$0.194490\pi$
$654$		0			0
$655$	−12.2915	+	13.9962i	−0.480269	+	0.546876i
$656$		0			0
$657$	−41.3029			−1.61138
$658$		0			0
$659$	28.4575			1.10855			0.554274	−	0.832334i	$-0.312996\pi$
$659$	28.4575			1.10855			0.554274	+	0.832334i	$0.312996\pi$
$660$		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
$663$	12.0399	−	1.75641i	0.467591	−	0.0682133i
$664$		0			0
$665$	24.5830	+	21.5889i	0.953288	+	0.837182i
$666$		0			0
$667$		0			0
$668$		0			0
$669$			12.6643i			0.489630i
$670$		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
$675$	−2.16991	−	16.6607i	−0.0835198	−	0.641271i
$676$		0			0
$677$	27.8618			1.07082			0.535408	−	0.844594i	$-0.320158\pi$
$677$	27.8618			1.07082			0.535408	+	0.844594i	$0.320158\pi$
$678$		0			0
$679$	25.1660			0.965783
$680$		0			0
$681$	−11.7712			−0.451075
$682$		0			0
$683$	26.2871			1.00585			0.502924	−	0.864331i	$-0.332258\pi$
$683$	26.2871			1.00585			0.502924	+	0.864331i	$0.332258\pi$
$684$		0			0
$685$	2.13256	−	2.42832i	0.0814810	−	0.0927813i
$686$		0			0
$687$	2.16991			0.0827871
$688$		0			0
$689$	−40.4575			−1.54131
$690$		0			0
$691$			16.1380i			0.613919i	0.951723	+	0.306959i	$0.0993116\pi$
$691$			16.1380i			0.613919i	−0.951723	+	0.306959i	$0.900688\pi$
$692$		0			0
$693$		−	39.3539i		−	1.49493i
$694$		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
$699$	7.29150			0.275790
$700$		0			0
$701$	7.06275			0.266756			0.133378	−	0.991065i	$-0.457418\pi$
$701$	7.06275			0.266756			0.133378	+	0.991065i	$0.457418\pi$
$702$		0			0
$703$	29.2630			1.10368
$704$		0			0
$705$	−5.89674	+	6.71453i	−0.222084	+	0.252884i
$706$		0			0
$707$	12.0399			0.452807
$708$		0			0
$709$			12.8495i			0.482572i	0.970454	+	0.241286i	$0.0775691\pi$
$709$			12.8495i			0.482572i	−0.970454	+	0.241286i	$0.922431\pi$
$710$		0			0
$711$		−	6.42473i		−	0.240946i
$712$		0			0
$713$			11.9838i			0.448795i
$714$		0			0
$715$	44.8118	+	39.3539i	1.67586	+	1.47175i
$716$		0			0
$717$	5.87719			0.219488
$718$		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
$723$		−	8.15984i		−	0.303468i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	44.6231i		−	1.65498i	−0.561480	−	0.827490i	$-0.689768\pi$
$727$		−	44.6231i		−	1.65498i	0.561480	−	0.827490i	$-0.310232\pi$
$728$		0			0
$729$	−9.70850			−0.359574
$730$		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
$735$	−0.645751	−	0.567102i	−0.0238189	−	0.0209179i
$736$		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
$741$		−	15.6153i		−	0.573643i
$742$		0			0
$743$	31.8173			1.16726			0.583631	−	0.812019i	$-0.301631\pi$
$743$	31.8173			1.16726			0.583631	+	0.812019i	$0.301631\pi$
$744$		0			0
$745$	11.0604	+	9.71328i	0.405221	+	0.355867i
$746$		0			0
$747$			39.3539i			1.43988i
$748$		0			0
$749$	−26.8118			−0.979680
$750$		0			0
$751$			6.42473i			0.234442i	0.993106	+	0.117221i	$0.0373985\pi$
$751$			6.42473i			0.234442i	−0.993106	+	0.117221i	$0.962601\pi$
$752$		0			0
$753$	−11.4074			−0.415708
$754$		0			0
$755$	−16.8014	−	14.7551i	−0.611466	−	0.536992i
$756$		0			0
$757$		−	24.1684i		−	0.878416i	−0.898385	−	0.439208i	$-0.855259\pi$
$757$		−	24.1684i		−	0.878416i	0.898385	−	0.439208i	$-0.144741\pi$
$758$		0			0
$759$			15.8005i			0.573521i
$760$		0			0
$761$	−26.2288			−0.950792			−0.475396	−	0.879772i	$-0.657695\pi$
$761$	−26.2288			−0.950792			−0.475396	+	0.879772i	$0.657695\pi$
$762$		0			0
$763$		−	24.4795i		−	0.886218i
$764$		0			0
$765$	−13.2816	+	20.4597i	−0.480196	+	0.739723i
$766$		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
$771$			8.85305i			0.318835i
$772$		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
$777$	−9.10132			−0.326508
$778$		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
$789$			5.71687i			0.203526i
$790$		0			0
$791$	−15.8745			−0.564433
$792$		0			0
$793$	67.9743			2.41384
$794$		0			0
$795$	−7.16601	+	8.15984i	−0.254152	+	0.289400i
$796$		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
$801$	−37.6458			−1.33015
$802$		0			0
$803$		−	83.9770i		−	2.96348i
$804$		0			0
$805$	−22.9268	−	20.1344i	−0.808063	−	0.709645i
$806$		0			0
$807$			8.78205i			0.309143i
$808$		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	$0.00876467\pi$
$811$			1.56809i			0.0550631i	−0.999621	+	0.0275316i	$0.991235\pi$
$812$		0			0
$813$	−4.76150			−0.166993
$814$		0			0
$815$	−23.2288	−	20.3996i	−0.813668	−	0.714567i
$816$		0			0
$817$		−	16.9419i		−	0.592722i
$818$		0			0
$819$		−	36.2724i		−	1.26746i
$820$		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
$825$	15.8745	−	2.06751i	0.552679	−	0.0719815i
$826$		0			0
$827$	−1.01688			−0.0353603			−0.0176801	−	0.999844i	$-0.505628\pi$
$827$	−1.01688			−0.0353603			−0.0176801	+	0.999844i	$0.505628\pi$
$828$		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
$831$	10.2510			0.355603
$832$		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
$837$			8.15984i			0.282045i
$838$		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
$843$	−7.83625			−0.269895
$844$		0			0
$845$	19.4611	+	17.0908i	0.669482	+	0.587942i
$846$		0			0
$847$	49.5982			1.70421
$848$		0			0
$849$	4.93725			0.169446
$850$		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
$855$	23.5220	+	20.6571i	0.804435	+	0.706458i
$856$		0			0
$857$	27.0931			0.925483			0.462741	−	0.886493i	$-0.346866\pi$
$857$	27.0931			0.925483			0.462741	+	0.886493i	$0.346866\pi$
$858$		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
$861$		−	4.85664i		−	0.165514i
$862$		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
$867$	−9.69651	+	2.89061i	−0.329311	+	0.0981704i
$868$		0			0
$869$	13.0627			0.443123
$870$		0			0
$871$	33.2915			1.12804
$872$		0			0
$873$	24.0798			0.814979
$874$		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
$879$		−	7.99282i		−	0.269591i
$880$		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	$-0.866691\pi$
$883$		−	24.1684i		−	0.813332i	0.913577	−	0.406666i	$-0.133309\pi$
$884$		0			0
$885$	9.29150	+	8.15984i	0.312330	+	0.274290i
$886$		0			0
$887$	−44.8367			−1.50547			−0.752735	−	0.658324i	$-0.771266\pi$
$887$	−44.8367			−1.50547			−0.752735	+	0.658324i	$0.771266\pi$
$888$		0			0
$889$			36.2724i			1.21654i
$890$		0			0
$891$		−	31.9385i		−	1.06998i
$892$		0			0
$893$		−	35.5300i		−	1.18897i
$894$		0			0
$895$	−5.53019	−	4.85664i	−0.184854	−	0.162340i
$896$		0			0
$897$			14.5633i			0.486254i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	33.2915	−	4.85664i	1.10910	−	0.161798i
$902$		0			0
$903$			5.26923i			0.175349i
$904$		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
$909$	11.5203			0.382103
$910$		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
$915$	12.0399	−	13.7097i	0.398027	−	0.453228i
$916$		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
$921$		−	10.5735i		−	0.348409i
$922$		0			0
$923$	−55.9344			−1.84110
$924$		0			0
$925$	3.57113	+	27.4194i	0.117418	+	0.901544i
$926$		0			0
$927$			13.1180i			0.430851i
$928$		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
$933$		−	6.71453i		−	0.219824i
$934$		0			0
$935$	−41.5987	−	27.0040i	−1.36042	−	0.883126i
$936$		0			0
$937$		−	11.0505i		−	0.361003i	−0.983575	−	0.180501i	$-0.942228\pi$
$937$		−	11.0505i		−	0.361003i	0.983575	−	0.180501i	$-0.0577720\pi$
$938$		0			0
$939$	−13.5425			−0.441942
$940$		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
$945$	−15.6110	−	13.7097i	−0.507827	−	0.445976i
$946$		0			0
$947$	3.95547			0.128536			0.0642678	−	0.997933i	$-0.479529\pi$
$947$	3.95547			0.128536			0.0642678	+	0.997933i	$0.479529\pi$
$948$		0			0
$949$		−	77.4015i		−	2.51256i
$950$		0			0
$951$	9.04052			0.293159
$952$		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
$963$	−25.6545			−0.826706
$964$		0			0
$965$	20.2288	+	17.7650i	0.651187	+	0.571875i
$966$		0			0
$967$			46.3795i			1.49147i	0.666245	+	0.745733i	$0.267900\pi$
$967$			46.3795i			1.49147i	−0.666245	+	0.745733i	$0.732100\pi$
$968$		0			0
$969$	1.87451	+	12.8495i	0.0602179	+	0.412784i
$970$		0			0
$971$	−36.0000			−1.15529			−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		0			0
$973$			17.7650i			0.569519i
$974$		0			0
$975$	14.6315	−	1.90562i	0.468583	−	0.0610288i
$976$		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
$981$		−	23.4230i		−	0.747838i
$982$		0			0
$983$	47.0813			1.50166			0.750830	−	0.660495i	$-0.229654\pi$
$983$	47.0813			1.50166			0.750830	+	0.660495i	$0.229654\pi$
$984$		0			0
$985$	35.8745	+	31.5052i	1.14306	+	1.00384i
$986$		0			0
$987$			11.0505i			0.351740i
$988$		0			0
$989$			15.8005i			0.502426i
$990$		0			0
$991$			46.6936i			1.48327i	0.670804	+	0.741635i	$0.265949\pi$
$991$			46.6936i			1.48327i	−0.670804	+	0.741635i	$0.734051\pi$
$992$		0			0
$993$	17.4339			0.553250
$994$		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
$999$	−18.5830			−0.587940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	340.2.g.a.169.5	yes	8
3.2	odd	2		3060.2.k.h.1189.8		8
4.3	odd	2		1360.2.o.e.849.3		8
5.2	odd	4		1700.2.c.e.101.3		8
5.3	odd	4		1700.2.c.e.101.5		8
5.4	even	2	inner	340.2.g.a.169.3	✓	8
15.14	odd	2		3060.2.k.h.1189.2		8
17.16	even	2	inner	340.2.g.a.169.4	yes	8
20.19	odd	2		1360.2.o.e.849.5		8
51.50	odd	2		3060.2.k.h.1189.1		8
68.67	odd	2		1360.2.o.e.849.6		8
85.33	odd	4		1700.2.c.e.101.4		8
85.67	odd	4		1700.2.c.e.101.6		8
85.84	even	2	inner	340.2.g.a.169.6	yes	8
255.254	odd	2		3060.2.k.h.1189.7		8
340.339	odd	2		1360.2.o.e.849.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.g.a.169.3	✓	8	5.4	even	2	inner
340.2.g.a.169.4	yes	8	17.16	even	2	inner
340.2.g.a.169.5	yes	8	1.1	even	1	trivial
340.2.g.a.169.6	yes	8	85.84	even	2	inner
1360.2.o.e.849.3		8	4.3	odd	2
1360.2.o.e.849.4		8	340.339	odd	2
1360.2.o.e.849.5		8	20.19	odd	2
1360.2.o.e.849.6		8	68.67	odd	2
1700.2.c.e.101.3		8	5.2	odd	4
1700.2.c.e.101.4		8	85.33	odd	4
1700.2.c.e.101.5		8	5.3	odd	4
1700.2.c.e.101.6		8	85.67	odd	4
3060.2.k.h.1189.1		8	51.50	odd	2
3060.2.k.h.1189.2		8	15.14	odd	2
3060.2.k.h.1189.7		8	255.254	odd	2
3060.2.k.h.1189.8		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	340.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.595188 q^{3} +(-1.68014 - 1.47551i) q^{5} -2.76510 q^{7} -2.64575 q^{9} -5.37934i q^{11} -4.95813i q^{13} +(-1.00000 - 0.878205i) q^{15} +(0.595188 + 4.07992i) q^{17} +5.29150 q^{19} -1.64575 q^{21} -4.93500 q^{23} +(0.645751 + 4.95813i) q^{25} -3.36028 q^{27} -2.42832i q^{31} -3.20172i q^{33} +(4.64575 + 4.07992i) q^{35} +5.53019 q^{37} -2.95102i q^{39} +2.95102i q^{41} -3.20172i q^{43} +(4.44524 + 3.90383i) q^{45} -6.71453i q^{47} +0.645751 q^{49} +(0.354249 + 2.42832i) q^{51} -8.15984i q^{53} +(-7.93725 + 9.03805i) q^{55} +3.14944 q^{57} -9.29150 q^{59} +13.7097i q^{61} +7.31575 q^{63} +(-7.31575 + 8.33035i) q^{65} +6.71453i q^{67} -2.93725 q^{69} -11.2814i q^{71} +15.6110 q^{73} +(0.384343 + 2.95102i) q^{75} +14.8744i q^{77} +2.42832i q^{79} +5.93725 q^{81} -14.8744i q^{83} +(5.01996 - 7.73305i) q^{85} +14.2288 q^{89} +13.7097i q^{91} -1.44531i q^{93} +(-8.89047 - 7.80766i) q^{95} -9.10132 q^{97} +14.2324i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{15} + 8 q^{21} - 16 q^{25} + 16 q^{35} - 16 q^{49} + 24 q^{51} - 32 q^{59} + 40 q^{69} - 16 q^{81} - 8 q^{85} + 8 q^{89}+O(q^{100}) \) 8 * q - 8 * q^15 + 8 * q^21 - 16 * q^25 + 16 * q^35 - 16 * q^49 + 24 * q^51 - 32 * q^59 + 40 * q^69 - 16 * q^81 - 8 * q^85 + 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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