LMFDB - Embedded newform 2760.2.k.c.2209.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(2209,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2760.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.k (of order $2$, degree $1$, 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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 31x^{10} + 359x^{8} + 1957x^{6} + 5132x^{4} + 5744x^{2} + 1600 $ x^12 + 31x^10 + 359x^8 + 1957x^6 + 5132x^4 + 5744*x^2 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.7
Root			$-1.70731i$ of defining polynomial
Character	$\chi$	$=$	2760.2209
Dual form			2760.2.k.c.2209.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.21432 + 0.311108i) q^{5} -1.70731i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-2.21432 + 0.311108i) q^{5} -1.70731i q^{7} -1.00000 q^{9} -3.78053 q^{11} +1.17615i q^{13} +(-0.311108 - 2.21432i) q^{15} -4.26125i q^{17} +6.98258 q^{19} +1.70731 q^{21} -1.00000i q^{23} +(4.80642 - 1.37778i) q^{25} -1.00000i q^{27} +4.26125 q^{29} -0.0889122 q^{31} -3.78053i q^{33} +(0.531157 + 3.78053i) q^{35} +1.72515i q^{37} -1.17615 q^{39} -7.69370 q^{41} +8.76311i q^{43} +(2.21432 - 0.311108i) q^{45} +2.25631i q^{47} +4.08509 q^{49} +4.26125 q^{51} +12.9673i q^{53} +(8.37130 - 1.17615i) q^{55} +6.98258i q^{57} -11.1431 q^{59} -11.0071 q^{61} +1.70731i q^{63} +(-0.365910 - 2.60438i) q^{65} +10.3880i q^{67} +1.00000 q^{69} +1.64458 q^{71} +9.70567i q^{73} +(1.37778 + 4.80642i) q^{75} +6.45453i q^{77} -6.07634 q^{79} +1.00000 q^{81} -6.53116i q^{83} +(1.32571 + 9.43576i) q^{85} +4.26125i q^{87} +15.6742 q^{89} +2.00806 q^{91} -0.0889122i q^{93} +(-15.4617 + 2.17233i) q^{95} +7.78476i q^{97} +3.78053 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} - 4 q^{15} + 12 q^{19} - 6 q^{21} + 4 q^{25} + 6 q^{29} - 2 q^{31} - 2 q^{35} + 4 q^{39} - 22 q^{41} + 22 q^{49} + 6 q^{51} - 16 q^{55} + 18 q^{59} - 60 q^{61} - 12 q^{65} + 12 q^{69} + 54 q^{71} + 16 q^{75} - 56 q^{79} + 12 q^{81} - 38 q^{85} + 28 q^{89} - 12 q^{91} - 52 q^{95}+O(q^{100}) $ 12 * q - 12 * q^9 - 4 * q^15 + 12 * q^19 - 6 * q^21 + 4 * q^25 + 6 * q^29 - 2 * q^31 - 2 * q^35 + 4 * q^39 - 22 * q^41 + 22 * q^49 + 6 * q^51 - 16 * q^55 + 18 * q^59 - 60 * q^61 - 12 * q^65 + 12 * q^69 + 54 * q^71 + 16 * q^75 - 56 * q^79 + 12 * q^81 - 38 * q^85 + 28 * q^89 - 12 * q^91 - 52 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1657$	$1841$	$2071$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$5$	−2.21432	+	0.311108i	−0.990274	+	0.139132i
$6$		0			0
$7$		−	1.70731i		−	0.645302i	−0.946518	−	0.322651i	$-0.895426\pi$
$7$		−	1.70731i		−	0.645302i	0.946518	−	0.322651i	$-0.104574\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.78053			−1.13987			−0.569936	−	0.821689i	$-0.693032\pi$
$11$	−3.78053			−1.13987			−0.569936	+	0.821689i	$0.693032\pi$
$12$		0			0
$13$			1.17615i			0.326206i	0.986609	+	0.163103i	$0.0521503\pi$
$13$			1.17615i			0.326206i	−0.986609	+	0.163103i	$0.947850\pi$
$14$		0			0
$15$	−0.311108	−	2.21432i	−0.0803277	−	0.571735i
$16$		0			0
$17$		−	4.26125i		−	1.03350i	−0.856135	−	0.516752i	$-0.827141\pi$
$17$		−	4.26125i		−	1.03350i	0.856135	−	0.516752i	$-0.172859\pi$
$18$		0			0
$19$	6.98258			1.60191			0.800956	−	0.598723i	$-0.204325\pi$
$19$	6.98258			1.60191			0.800956	+	0.598723i	$0.204325\pi$
$20$		0			0
$21$	1.70731			0.372565
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	4.80642	−	1.37778i	0.961285	−	0.275557i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	4.26125			0.791294			0.395647	−	0.918403i	$-0.370520\pi$
$29$	4.26125			0.791294			0.395647	+	0.918403i	$0.370520\pi$
$30$		0			0
$31$	−0.0889122			−0.0159691			−0.00798455	−	0.999968i	$-0.502542\pi$
$31$	−0.0889122			−0.0159691			−0.00798455	+	0.999968i	$0.502542\pi$
$32$		0			0
$33$		−	3.78053i		−	0.658106i
$34$		0			0
$35$	0.531157	+	3.78053i	0.0897820	+	0.639026i
$36$		0			0
$37$			1.72515i			0.283613i	0.989894	+	0.141806i	$0.0452910\pi$
$37$			1.72515i			0.283613i	−0.989894	+	0.141806i	$0.954709\pi$
$38$		0			0
$39$	−1.17615			−0.188335
$40$		0			0
$41$	−7.69370			−1.20155			−0.600777	−	0.799416i	$-0.705142\pi$
$41$	−7.69370			−1.20155			−0.600777	+	0.799416i	$0.705142\pi$
$42$		0			0
$43$			8.76311i			1.33636i	0.743999	+	0.668181i	$0.232927\pi$
$43$			8.76311i			1.33636i	−0.743999	+	0.668181i	$0.767073\pi$
$44$		0			0
$45$	2.21432	−	0.311108i	0.330091	−	0.0463772i
$46$		0			0
$47$			2.25631i			0.329116i	0.986367	+	0.164558i	$0.0526198\pi$
$47$			2.25631i			0.329116i	−0.986367	+	0.164558i	$0.947380\pi$
$48$		0			0
$49$	4.08509			0.583585
$50$		0			0
$51$	4.26125			0.596694
$52$		0			0
$53$			12.9673i			1.78120i	0.454787	+	0.890600i	$0.349715\pi$
$53$			12.9673i			1.78120i	−0.454787	+	0.890600i	$0.650285\pi$
$54$		0			0
$55$	8.37130	−	1.17615i	1.12879	−	0.158592i
$56$		0			0
$57$			6.98258i			0.924865i
$58$		0			0
$59$	−11.1431			−1.45070			−0.725352	−	0.688378i	$-0.758323\pi$
$59$	−11.1431			−1.45070			−0.725352	+	0.688378i	$0.758323\pi$
$60$		0			0
$61$	−11.0071			−1.40932			−0.704659	−	0.709546i	$-0.748900\pi$
$61$	−11.0071			−1.40932			−0.704659	+	0.709546i	$0.748900\pi$
$62$		0			0
$63$			1.70731i			0.215101i
$64$		0			0
$65$	−0.365910	−	2.60438i	−0.0453856	−	0.323033i
$66$		0			0
$67$			10.3880i			1.26910i	0.772882	+	0.634549i	$0.218814\pi$
$67$			10.3880i			1.26910i	−0.772882	+	0.634549i	$0.781186\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$	1.64458			0.195176			0.0975879	−	0.995227i	$-0.468887\pi$
$71$	1.64458			0.195176			0.0975879	+	0.995227i	$0.468887\pi$
$72$		0			0
$73$			9.70567i			1.13596i	0.823041	+	0.567982i	$0.192276\pi$
$73$			9.70567i			1.13596i	−0.823041	+	0.567982i	$0.807724\pi$
$74$		0			0
$75$	1.37778	+	4.80642i	0.159093	+	0.554998i
$76$		0			0
$77$			6.45453i			0.735562i
$78$		0			0
$79$	−6.07634			−0.683641			−0.341821	−	0.939765i	$-0.611044\pi$
$79$	−6.07634			−0.683641			−0.341821	+	0.939765i	$0.611044\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.53116i		−	0.716888i	−0.933551	−	0.358444i	$-0.883307\pi$
$83$		−	6.53116i		−	0.716888i	0.933551	−	0.358444i	$-0.116693\pi$
$84$		0			0
$85$	1.32571	+	9.43576i	0.143793	+	1.02345i
$86$		0			0
$87$			4.26125i			0.456854i
$88$		0			0
$89$	15.6742			1.66146			0.830732	−	0.556672i	$-0.187922\pi$
$89$	15.6742			1.66146			0.830732	+	0.556672i	$0.187922\pi$
$90$		0			0
$91$	2.00806			0.210501
$92$		0			0
$93$		−	0.0889122i		−	0.00921976i
$94$		0			0
$95$	−15.4617	+	2.17233i	−1.58633	+	0.222877i
$96$		0			0
$97$			7.78476i			0.790423i	0.918590	+	0.395211i	$0.129329\pi$
$97$			7.78476i			0.790423i	−0.918590	+	0.395211i	$0.870671\pi$
$98$		0			0
$99$	3.78053			0.379957
$100$		0			0
$101$	−6.35542			−0.632388			−0.316194	−	0.948695i	$-0.602405\pi$
$101$	−6.35542			−0.632388			−0.316194	+	0.948695i	$0.602405\pi$
$102$		0			0
$103$		−	2.62892i		−	0.259035i	−0.991577	−	0.129518i	$-0.958657\pi$
$103$		−	2.62892i		−	0.259035i	0.991577	−	0.129518i	$-0.0413429\pi$
$104$		0			0
$105$	−3.78053	+	0.531157i	−0.368942	+	0.0518357i
$106$		0			0
$107$		−	6.68513i		−	0.646276i	−0.946352	−	0.323138i	$-0.895262\pi$
$107$		−	6.68513i		−	0.646276i	0.946352	−	0.323138i	$-0.104738\pi$
$108$		0			0
$109$	17.5577			1.68172			0.840859	−	0.541254i	$-0.182050\pi$
$109$	17.5577			1.68172			0.840859	+	0.541254i	$0.182050\pi$
$110$		0			0
$111$	−1.72515			−0.163744
$112$		0			0
$113$		−	3.99688i		−	0.375995i	−0.982169	−	0.187998i	$-0.939800\pi$
$113$		−	3.99688i		−	0.375995i	0.982169	−	0.187998i	$-0.0601997\pi$
$114$		0			0
$115$	0.311108	+	2.21432i	0.0290110	+	0.206486i
$116$		0			0
$117$		−	1.17615i		−	0.108735i
$118$		0			0
$119$	−7.27527			−0.666923
$120$		0			0
$121$	3.29240			0.299309
$122$		0			0
$123$		−	7.69370i		−	0.693718i
$124$		0			0
$125$	−10.2143	+	4.54617i	−0.913597	+	0.406622i
$126$		0			0
$127$			2.61881i			0.232382i	0.993227	+	0.116191i	$0.0370685\pi$
$127$			2.61881i			0.232382i	−0.993227	+	0.116191i	$0.962932\pi$
$128$		0			0
$129$	−8.76311			−0.771548
$130$		0			0
$131$	15.3280			1.33921			0.669606	−	0.742716i	$-0.266463\pi$
$131$	15.3280			1.33921			0.669606	+	0.742716i	$0.266463\pi$
$132$		0			0
$133$		−	11.9214i		−	1.03372i
$134$		0			0
$135$	0.311108	+	2.21432i	0.0267759	+	0.190578i
$136$		0			0
$137$			8.01020i			0.684358i	0.939635	+	0.342179i	$0.111165\pi$
$137$			8.01020i			0.684358i	−0.939635	+	0.342179i	$0.888835\pi$
$138$		0			0
$139$	1.03954			0.0881722			0.0440861	−	0.999028i	$-0.485962\pi$
$139$	1.03954			0.0881722			0.0440861	+	0.999028i	$0.485962\pi$
$140$		0			0
$141$	−2.25631			−0.190015
$142$		0			0
$143$		−	4.44648i		−	0.371833i
$144$		0			0
$145$	−9.43576	+	1.32571i	−0.783597	+	0.110094i
$146$		0			0
$147$			4.08509i			0.336933i
$148$		0			0
$149$	17.0471			1.39656			0.698278	−	0.715827i	$-0.253950\pi$
$149$	17.0471			1.39656			0.698278	+	0.715827i	$0.253950\pi$
$150$		0			0
$151$	−15.0622			−1.22575			−0.612873	−	0.790182i	$-0.709986\pi$
$151$	−15.0622			−1.22575			−0.612873	+	0.790182i	$0.709986\pi$
$152$		0			0
$153$			4.26125i			0.344501i
$154$		0			0
$155$	0.196880	−	0.0276613i	0.0158138	−	0.00222181i
$156$		0			0
$157$			10.8678i			0.867342i	0.901071	+	0.433671i	$0.142782\pi$
$157$			10.8678i			0.867342i	−0.901071	+	0.433671i	$0.857218\pi$
$158$		0			0
$159$	−12.9673			−1.02838
$160$		0			0
$161$	−1.70731			−0.134555
$162$		0			0
$163$			12.8216i			1.00427i	0.864791	+	0.502133i	$0.167451\pi$
$163$			12.8216i			1.00427i	−0.864791	+	0.502133i	$0.832549\pi$
$164$		0			0
$165$	1.17615	+	8.37130i	0.0915633	+	0.651705i
$166$		0			0
$167$			16.3322i			1.26383i	0.775040	+	0.631913i	$0.217730\pi$
$167$			16.3322i			1.26383i	−0.775040	+	0.631913i	$0.782270\pi$
$168$		0			0
$169$	11.6167			0.893590
$170$		0			0
$171$	−6.98258			−0.533971
$172$		0			0
$173$			18.1255i			1.37805i	0.724736	+	0.689027i	$0.241962\pi$
$173$			18.1255i			1.37805i	−0.724736	+	0.689027i	$0.758038\pi$
$174$		0			0
$175$	−2.35230	−	8.20605i	−0.177818	−	0.620319i
$176$		0			0
$177$		−	11.1431i		−	0.837565i
$178$		0			0
$179$	−1.00382			−0.0750289			−0.0375144	−	0.999296i	$-0.511944\pi$
$179$	−1.00382			−0.0750289			−0.0375144	+	0.999296i	$0.511944\pi$
$180$		0			0
$181$	5.28047			0.392494			0.196247	−	0.980554i	$-0.437125\pi$
$181$	5.28047			0.392494			0.196247	+	0.980554i	$0.437125\pi$
$182$		0			0
$183$		−	11.0071i		−	0.813670i
$184$		0			0
$185$	−0.536707	−	3.82003i	−0.0394595	−	0.280854i
$186$		0			0
$187$			16.1098i			1.17806i
$188$		0			0
$189$	−1.70731			−0.124188
$190$		0			0
$191$	−8.55624			−0.619108			−0.309554	−	0.950882i	$-0.600180\pi$
$191$	−8.55624			−0.619108			−0.309554	+	0.950882i	$0.600180\pi$
$192$		0			0
$193$		−	13.2605i		−	0.954515i	−0.878764	−	0.477257i	$-0.841631\pi$
$193$		−	13.2605i		−	0.954515i	0.878764	−	0.477257i	$-0.158369\pi$
$194$		0			0
$195$	2.60438	−	0.365910i	0.186503	−	0.0262034i
$196$		0			0
$197$			25.0180i			1.78246i	0.453556	+	0.891228i	$0.350155\pi$
$197$			25.0180i			1.78246i	−0.453556	+	0.891228i	$0.649845\pi$
$198$		0			0
$199$	−3.69059			−0.261619			−0.130809	−	0.991408i	$-0.541758\pi$
$199$	−3.69059			−0.261619			−0.130809	+	0.991408i	$0.541758\pi$
$200$		0			0
$201$	−10.3880			−0.732715
$202$		0			0
$203$		−	7.27527i		−	0.510624i
$204$		0			0
$205$	17.0363	−	2.39357i	1.18987	−	0.167174i
$206$		0			0
$207$			1.00000i			0.0695048i
$208$		0			0
$209$	−26.3978			−1.82598
$210$		0			0
$211$	−0.328696			−0.0226284			−0.0113142	−	0.999936i	$-0.503601\pi$
$211$	−0.328696			−0.0226284			−0.0113142	+	0.999936i	$0.503601\pi$
$212$		0			0
$213$			1.64458i			0.112685i
$214$		0			0
$215$	−2.72627	−	19.4043i	−0.185930	−	1.32336i
$216$		0			0
$217$			0.151801i			0.0103049i
$218$		0			0
$219$	−9.70567			−0.655849
$220$		0			0
$221$	5.01187			0.337135
$222$		0			0
$223$			18.0914i			1.21149i	0.795659	+	0.605745i	$0.207125\pi$
$223$			18.0914i			1.21149i	−0.795659	+	0.605745i	$0.792875\pi$
$224$		0			0
$225$	−4.80642	+	1.37778i	−0.320428	+	0.0918523i
$226$		0			0
$227$		−	25.1906i		−	1.67196i	−0.548763	−	0.835978i	$-0.684901\pi$
$227$		−	25.1906i		−	1.67196i	0.548763	−	0.835978i	$-0.315099\pi$
$228$		0			0
$229$	10.8968			0.720079			0.360040	−	0.932937i	$-0.382763\pi$
$229$	10.8968			0.720079			0.360040	+	0.932937i	$0.382763\pi$
$230$		0			0
$231$	−6.45453			−0.424677
$232$		0			0
$233$			8.87480i			0.581407i	0.956813	+	0.290704i	$0.0938894\pi$
$233$			8.87480i			0.581407i	−0.956813	+	0.290704i	$0.906111\pi$
$234$		0			0
$235$	−0.701954	−	4.99618i	−0.0457904	−	0.325915i
$236$		0			0
$237$		−	6.07634i		−	0.394700i
$238$		0			0
$239$	−29.6234			−1.91618			−0.958089	−	0.286470i	$-0.907518\pi$
$239$	−29.6234			−1.91618			−0.958089	+	0.286470i	$0.907518\pi$
$240$		0			0
$241$	29.8553			1.92315			0.961573	−	0.274549i	$-0.0885286\pi$
$241$	29.8553			1.92315			0.961573	+	0.274549i	$0.0885286\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−9.04570	+	1.27090i	−0.577909	+	0.0811951i
$246$		0			0
$247$			8.21257i			0.522553i
$248$		0			0
$249$	6.53116			0.413895
$250$		0			0
$251$	−4.15953			−0.262547			−0.131273	−	0.991346i	$-0.541907\pi$
$251$	−4.15953			−0.262547			−0.131273	+	0.991346i	$0.541907\pi$
$252$		0			0
$253$			3.78053i			0.237680i
$254$		0			0
$255$	−9.43576	+	1.32571i	−0.590890	+	0.0830190i
$256$		0			0
$257$			27.3784i			1.70782i	0.520421	+	0.853910i	$0.325775\pi$
$257$			27.3784i			1.70782i	−0.520421	+	0.853910i	$0.674225\pi$
$258$		0			0
$259$	2.94536			0.183016
$260$		0			0
$261$	−4.26125			−0.263765
$262$		0			0
$263$			8.66938i			0.534577i	0.963617	+	0.267288i	$0.0861277\pi$
$263$			8.66938i			0.534577i	−0.963617	+	0.267288i	$0.913872\pi$
$264$		0			0
$265$	−4.03424	−	28.7138i	−0.247821	−	1.76388i
$266$		0			0
$267$			15.6742i			0.959247i
$268$		0			0
$269$	12.8627			0.784255			0.392128	−	0.919911i	$-0.371739\pi$
$269$	12.8627			0.784255			0.392128	+	0.919911i	$0.371739\pi$
$270$		0			0
$271$	−24.5148			−1.48917			−0.744584	−	0.667529i	$-0.767352\pi$
$271$	−24.5148			−1.48917			−0.744584	+	0.667529i	$0.767352\pi$
$272$		0			0
$273$			2.00806i			0.121533i
$274$		0			0
$275$	−18.1708	+	5.20875i	−1.09574	+	0.314100i
$276$		0			0
$277$			1.80963i			0.108730i	0.998521	+	0.0543650i	$0.0173135\pi$
$277$			1.80963i			0.108730i	−0.998521	+	0.0543650i	$0.982687\pi$
$278$		0			0
$279$	0.0889122			0.00532303
$280$		0			0
$281$	17.8395			1.06422			0.532109	−	0.846676i	$-0.321400\pi$
$281$	17.8395			1.06422			0.532109	+	0.846676i	$0.321400\pi$
$282$		0			0
$283$		−	21.4440i		−	1.27471i	−0.770569	−	0.637357i	$-0.780028\pi$
$283$		−	21.4440i		−	1.27471i	0.770569	−	0.637357i	$-0.219972\pi$
$284$		0			0
$285$	−2.17233	−	15.4617i	−0.128678	−	0.915869i
$286$		0			0
$287$			13.1355i			0.775366i
$288$		0			0
$289$	−1.15822			−0.0681305
$290$		0			0
$291$	−7.78476			−0.456351
$292$		0			0
$293$			0.148076i			0.00865071i	0.999991	+	0.00432536i	$0.00137681\pi$
$293$			0.148076i			0.00865071i	−0.999991	+	0.00432536i	$0.998623\pi$
$294$		0			0
$295$	24.6743	−	3.46670i	1.43659	−	0.201839i
$296$		0			0
$297$			3.78053i			0.219369i
$298$		0			0
$299$	1.17615			0.0680186
$300$		0			0
$301$	14.9613			0.862357
$302$		0			0
$303$		−	6.35542i		−	0.365109i
$304$		0			0
$305$	24.3733	−	3.42440i	1.39561	−	0.196081i
$306$		0			0
$307$			18.7801i			1.07184i	0.844270	+	0.535919i	$0.180035\pi$
$307$			18.7801i			1.07184i	−0.844270	+	0.535919i	$0.819965\pi$
$308$		0			0
$309$	2.62892			0.149554
$310$		0			0
$311$	−1.41811			−0.0804139			−0.0402069	−	0.999191i	$-0.512802\pi$
$311$	−1.41811			−0.0804139			−0.0402069	+	0.999191i	$0.512802\pi$
$312$		0			0
$313$		−	31.6209i		−	1.78732i	−0.448744	−	0.893660i	$-0.648128\pi$
$313$		−	31.6209i		−	1.78732i	0.448744	−	0.893660i	$-0.351872\pi$
$314$		0			0
$315$	−0.531157	−	3.78053i	−0.0299273	−	0.213009i
$316$		0			0
$317$		−	27.2743i		−	1.53188i	−0.642915	−	0.765938i	$-0.722275\pi$
$317$		−	27.2743i		−	1.53188i	0.642915	−	0.765938i	$-0.277725\pi$
$318$		0			0
$319$	−16.1098			−0.901974
$320$		0			0
$321$	6.68513			0.373128
$322$		0			0
$323$		−	29.7545i		−	1.65558i
$324$		0			0
$325$	1.62048	+	5.65309i	0.0898883	+	0.313577i
$326$		0			0
$327$			17.5577i			0.970941i
$328$		0			0
$329$	3.85221			0.212379
$330$		0			0
$331$	−4.03982			−0.222049			−0.111024	−	0.993818i	$-0.535413\pi$
$331$	−4.03982			−0.222049			−0.111024	+	0.993818i	$0.535413\pi$
$332$		0			0
$333$		−	1.72515i		−	0.0945375i
$334$		0			0
$335$	−3.23179	−	23.0024i	−0.176572	−	1.25676i
$336$		0			0
$337$		−	29.1364i		−	1.58716i	−0.608466	−	0.793580i	$-0.708215\pi$
$337$		−	29.1364i		−	1.58716i	0.608466	−	0.793580i	$-0.291785\pi$
$338$		0			0
$339$	3.99688			0.217081
$340$		0			0
$341$	0.336135			0.0182027
$342$		0			0
$343$		−	18.9257i		−	1.02189i
$344$		0			0
$345$	−2.21432	+	0.311108i	−0.119215	+	0.0167495i
$346$		0			0
$347$			29.1459i			1.56463i	0.622882	+	0.782316i	$0.285962\pi$
$347$			29.1459i			1.56463i	−0.622882	+	0.782316i	$0.714038\pi$
$348$		0			0
$349$	15.5748			0.833699			0.416850	−	0.908975i	$-0.363134\pi$
$349$	15.5748			0.833699			0.416850	+	0.908975i	$0.363134\pi$
$350$		0			0
$351$	1.17615			0.0627784
$352$		0			0
$353$			18.0393i			0.960137i	0.877231	+	0.480069i	$0.159388\pi$
$353$			18.0393i			0.960137i	−0.877231	+	0.480069i	$0.840612\pi$
$354$		0			0
$355$	−3.64163	+	0.511642i	−0.193277	+	0.0271551i
$356$		0			0
$357$		−	7.27527i		−	0.385048i
$358$		0			0
$359$	3.40646			0.179786			0.0898931	−	0.995951i	$-0.471347\pi$
$359$	3.40646			0.179786			0.0898931	+	0.995951i	$0.471347\pi$
$360$		0			0
$361$	29.7564			1.56612
$362$		0			0
$363$			3.29240i			0.172806i
$364$		0			0
$365$	−3.01951	−	21.4915i	−0.158048	−	1.12491i
$366$		0			0
$367$		−	12.9298i		−	0.674928i	−0.941338	−	0.337464i	$-0.890431\pi$
$367$		−	12.9298i		−	0.674928i	0.941338	−	0.337464i	$-0.109569\pi$
$368$		0			0
$369$	7.69370			0.400518
$370$		0			0
$371$	22.1393			1.14941
$372$		0			0
$373$			3.94962i			0.204504i	0.994759	+	0.102252i	$0.0326047\pi$
$373$			3.94962i			0.204504i	−0.994759	+	0.102252i	$0.967395\pi$
$374$		0			0
$375$	−4.54617	−	10.2143i	−0.234763	−	0.527465i
$376$		0			0
$377$			5.01187i			0.258125i
$378$		0			0
$379$	31.8131			1.63413			0.817064	−	0.576547i	$-0.195600\pi$
$379$	31.8131			1.63413			0.817064	+	0.576547i	$0.195600\pi$
$380$		0			0
$381$	−2.61881			−0.134166
$382$		0			0
$383$			2.52446i			0.128994i	0.997918	+	0.0644969i	$0.0205443\pi$
$383$			2.52446i			0.128994i	−0.997918	+	0.0644969i	$0.979456\pi$
$384$		0			0
$385$	−2.00806	−	14.2924i	−0.102340	−	0.728408i
$386$		0			0
$387$		−	8.76311i		−	0.445454i
$388$		0			0
$389$	−9.08146			−0.460448			−0.230224	−	0.973138i	$-0.573946\pi$
$389$	−9.08146			−0.460448			−0.230224	+	0.973138i	$0.573946\pi$
$390$		0			0
$391$	−4.26125			−0.215500
$392$		0			0
$393$			15.3280i			0.773194i
$394$		0			0
$395$	13.4549	−	1.89040i	0.676992	−	0.0951161i
$396$		0			0
$397$			16.0939i			0.807727i	0.914819	+	0.403864i	$0.132333\pi$
$397$			16.0939i			0.807727i	−0.914819	+	0.403864i	$0.867667\pi$
$398$		0			0
$399$	11.9214			0.596817
$400$		0			0
$401$	−8.86110			−0.442502			−0.221251	−	0.975217i	$-0.571014\pi$
$401$	−8.86110			−0.442502			−0.221251	+	0.975217i	$0.571014\pi$
$402$		0			0
$403$		−	0.104574i		−	0.00520922i
$404$		0			0
$405$	−2.21432	+	0.311108i	−0.110030	+	0.0154591i
$406$		0			0
$407$		−	6.52197i		−	0.323282i
$408$		0			0
$409$	21.1315			1.04488			0.522442	−	0.852675i	$-0.325021\pi$
$409$	21.1315			1.04488			0.522442	+	0.852675i	$0.325021\pi$
$410$		0			0
$411$	−8.01020			−0.395114
$412$		0			0
$413$			19.0247i			0.936143i
$414$		0			0
$415$	2.03189	+	14.4621i	0.0997418	+	0.709915i
$416$		0			0
$417$			1.03954i			0.0509063i
$418$		0			0
$419$	36.9975			1.80745			0.903724	−	0.428115i	$-0.140822\pi$
$419$	36.9975			1.80745			0.903724	+	0.428115i	$0.140822\pi$
$420$		0			0
$421$	−38.0963			−1.85670			−0.928349	−	0.371710i	$-0.878772\pi$
$421$	−38.0963			−1.85670			−0.928349	+	0.371710i	$0.878772\pi$
$422$		0			0
$423$		−	2.25631i		−	0.109705i
$424$		0			0
$425$	−5.87108	−	20.4814i	−0.284789	−	0.993492i
$426$		0			0
$427$			18.7926i			0.909436i
$428$		0			0
$429$	4.44648			0.214678
$430$		0			0
$431$	2.24154			0.107971			0.0539856	−	0.998542i	$-0.482807\pi$
$431$	2.24154			0.107971			0.0539856	+	0.998542i	$0.482807\pi$
$432$		0			0
$433$			25.5322i			1.22700i	0.789696	+	0.613498i	$0.210238\pi$
$433$			25.5322i			1.22700i	−0.789696	+	0.613498i	$0.789762\pi$
$434$		0			0
$435$	−1.32571	−	9.43576i	−0.0635628	−	0.452410i
$436$		0			0
$437$		−	6.98258i		−	0.334022i
$438$		0			0
$439$	−21.4731			−1.02486			−0.512428	−	0.858730i	$-0.671254\pi$
$439$	−21.4731			−1.02486			−0.512428	+	0.858730i	$0.671254\pi$
$440$		0			0
$441$	−4.08509			−0.194528
$442$		0			0
$443$		−	10.5220i		−	0.499914i	−0.968257	−	0.249957i	$-0.919584\pi$
$443$		−	10.5220i		−	0.499914i	0.968257	−	0.249957i	$-0.0804165\pi$
$444$		0			0
$445$	−34.7078	+	4.87638i	−1.64531	+	0.231162i
$446$		0			0
$447$			17.0471i			0.806302i
$448$		0			0
$449$	−16.7516			−0.790555			−0.395277	−	0.918562i	$-0.629352\pi$
$449$	−16.7516			−0.790555			−0.395277	+	0.918562i	$0.629352\pi$
$450$		0			0
$451$	29.0863			1.36962
$452$		0			0
$453$		−	15.0622i		−	0.707685i
$454$		0			0
$455$	−4.44648	+	0.624722i	−0.208454	+	0.0292874i
$456$		0			0
$457$			10.4658i			0.489568i	0.969578	+	0.244784i	$0.0787170\pi$
$457$			10.4658i			0.489568i	−0.969578	+	0.244784i	$0.921283\pi$
$458$		0			0
$459$	−4.26125			−0.198898
$460$		0			0
$461$	3.37173			0.157037			0.0785185	−	0.996913i	$-0.474981\pi$
$461$	3.37173			0.157037			0.0785185	+	0.996913i	$0.474981\pi$
$462$		0			0
$463$			16.0419i			0.745531i	0.927926	+	0.372765i	$0.121590\pi$
$463$			16.0419i			0.745531i	−0.927926	+	0.372765i	$0.878410\pi$
$464$		0			0
$465$	0.0276613	+	0.196880i	0.00128276	+	0.00913009i
$466$		0			0
$467$		−	16.3217i		−	0.755280i	−0.925953	−	0.377640i	$-0.876736\pi$
$467$		−	16.3217i		−	0.755280i	0.925953	−	0.377640i	$-0.123264\pi$
$468$		0			0
$469$	17.7356			0.818952
$470$		0			0
$471$	−10.8678			−0.500760
$472$		0			0
$473$		−	33.1292i		−	1.52328i
$474$		0			0
$475$	33.5612	−	9.62048i	1.53989	−	0.441418i
$476$		0			0
$477$		−	12.9673i		−	0.593733i
$478$		0			0
$479$	3.56594			0.162932			0.0814660	−	0.996676i	$-0.474040\pi$
$479$	3.56594			0.162932			0.0814660	+	0.996676i	$0.474040\pi$
$480$		0			0
$481$	−2.02904			−0.0925161
$482$		0			0
$483$		−	1.70731i		−	0.0776853i
$484$		0			0
$485$	−2.42190	−	17.2380i	−0.109973	−	0.782735i
$486$		0			0
$487$		−	8.96355i		−	0.406177i	−0.979160	−	0.203089i	$-0.934902\pi$
$487$		−	8.96355i		−	0.406177i	0.979160	−	0.203089i	$-0.0650980\pi$
$488$		0			0
$489$	−12.8216			−0.579813
$490$		0			0
$491$	−23.7934			−1.07378			−0.536890	−	0.843652i	$-0.680401\pi$
$491$	−23.7934			−1.07378			−0.536890	+	0.843652i	$0.680401\pi$
$492$		0			0
$493$		−	18.1582i		−	0.817805i
$494$		0			0
$495$	−8.37130	+	1.17615i	−0.376262	+	0.0528641i
$496$		0			0
$497$		−	2.80781i		−	0.125947i
$498$		0			0
$499$	−5.26199			−0.235559			−0.117779	−	0.993040i	$-0.537578\pi$
$499$	−5.26199			−0.235559			−0.117779	+	0.993040i	$0.537578\pi$
$500$		0			0
$501$	−16.3322			−0.729670
$502$		0			0
$503$			0.583883i			0.0260341i	0.999915	+	0.0130170i	$0.00414357\pi$
$503$			0.583883i			0.0260341i	−0.999915	+	0.0130170i	$0.995856\pi$
$504$		0			0
$505$	14.0729	−	1.97722i	0.626237	−	0.0879852i
$506$		0			0
$507$			11.6167i			0.515914i
$508$		0			0
$509$	−7.81717			−0.346490			−0.173245	−	0.984879i	$-0.555425\pi$
$509$	−7.81717			−0.346490			−0.173245	+	0.984879i	$0.555425\pi$
$510$		0			0
$511$	16.5706			0.733040
$512$		0			0
$513$		−	6.98258i		−	0.308288i
$514$		0			0
$515$	0.817879	+	5.82128i	0.0360400	+	0.256516i
$516$		0			0
$517$		−	8.53003i		−	0.375150i
$518$		0			0
$519$	−18.1255			−0.795619
$520$		0			0
$521$	36.9167			1.61735			0.808676	−	0.588255i	$-0.200185\pi$
$521$	36.9167			1.61735			0.808676	+	0.588255i	$0.200185\pi$
$522$		0			0
$523$		−	12.5244i		−	0.547652i	−0.961779	−	0.273826i	$-0.911711\pi$
$523$		−	12.5244i		−	0.547652i	0.961779	−	0.273826i	$-0.0882892\pi$
$524$		0			0
$525$	8.20605	−	2.35230i	0.358142	−	0.102663i
$526$		0			0
$527$			0.378877i			0.0165041i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	11.1431			0.483568
$532$		0			0
$533$		−	9.04897i		−	0.391954i
$534$		0			0
$535$	2.07980	+	14.8030i	0.0899175	+	0.639991i
$536$		0			0
$537$		−	1.00382i		−	0.0433180i
$538$		0			0
$539$	−15.4438			−0.665212
$540$		0			0
$541$	13.3636			0.574544			0.287272	−	0.957849i	$-0.407252\pi$
$541$	13.3636			0.574544			0.287272	+	0.957849i	$0.407252\pi$
$542$		0			0
$543$			5.28047i			0.226607i
$544$		0			0
$545$	−38.8783	+	5.46232i	−1.66536	+	0.233980i
$546$		0			0
$547$			40.2133i			1.71940i	0.510802	+	0.859698i	$0.329348\pi$
$547$			40.2133i			1.71940i	−0.510802	+	0.859698i	$0.670652\pi$
$548$		0			0
$549$	11.0071			0.469773
$550$		0			0
$551$	29.7545			1.26758
$552$		0			0
$553$			10.3742i			0.441155i
$554$		0			0
$555$	3.82003	−	0.536707i	0.162151	−	0.0227819i
$556$		0			0
$557$			39.1840i			1.66028i	0.557556	+	0.830139i	$0.311739\pi$
$557$			39.1840i			1.66028i	−0.557556	+	0.830139i	$0.688261\pi$
$558$		0			0
$559$	−10.3067			−0.435929
$560$		0			0
$561$	−16.1098			−0.680155
$562$		0			0
$563$			22.9184i			0.965895i	0.875649	+	0.482948i	$0.160434\pi$
$563$			22.9184i			0.965895i	−0.875649	+	0.482948i	$0.839566\pi$
$564$		0			0
$565$	1.24346	+	8.85038i	0.0523128	+	0.372338i
$566$		0			0
$567$		−	1.70731i		−	0.0717003i
$568$		0			0
$569$	−10.6447			−0.446250			−0.223125	−	0.974790i	$-0.571626\pi$
$569$	−10.6447			−0.446250			−0.223125	+	0.974790i	$0.571626\pi$
$570$		0			0
$571$	−1.70163			−0.0712109			−0.0356055	−	0.999366i	$-0.511336\pi$
$571$	−1.70163			−0.0712109			−0.0356055	+	0.999366i	$0.511336\pi$
$572$		0			0
$573$		−	8.55624i		−	0.357442i
$574$		0			0
$575$	−1.37778	−	4.80642i	−0.0574576	−	0.200442i
$576$		0			0
$577$		−	13.0240i		−	0.542196i	−0.962552	−	0.271098i	$-0.912613\pi$
$577$		−	13.0240i		−	0.542196i	0.962552	−	0.271098i	$-0.0873868\pi$
$578$		0			0
$579$	13.2605			0.551089
$580$		0			0
$581$	−11.1507			−0.462609
$582$		0			0
$583$		−	49.0234i		−	2.03034i
$584$		0			0
$585$	0.365910	+	2.60438i	0.0151285	+	0.107678i
$586$		0			0
$587$		−	9.46660i		−	0.390728i	−0.980731	−	0.195364i	$-0.937411\pi$
$587$		−	9.46660i		−	0.390728i	0.980731	−	0.195364i	$-0.0625889\pi$
$588$		0			0
$589$	−0.620836			−0.0255811
$590$		0			0
$591$	−25.0180			−1.02910
$592$		0			0
$593$			4.13745i			0.169905i	0.996385	+	0.0849524i	$0.0270738\pi$
$593$			4.13745i			0.169905i	−0.996385	+	0.0849524i	$0.972926\pi$
$594$		0			0
$595$	16.1098	−	2.26339i	0.660436	−	0.0927900i
$596$		0			0
$597$		−	3.69059i		−	0.151046i
$598$		0			0
$599$	−42.5801			−1.73978			−0.869888	−	0.493250i	$-0.835809\pi$
$599$	−42.5801			−1.73978			−0.869888	+	0.493250i	$0.835809\pi$
$600$		0			0
$601$	−22.9351			−0.935542			−0.467771	−	0.883850i	$-0.654943\pi$
$601$	−22.9351			−0.935542			−0.467771	+	0.883850i	$0.654943\pi$
$602$		0			0
$603$		−	10.3880i		−	0.423033i
$604$		0			0
$605$	−7.29043	+	1.02429i	−0.296398	+	0.0416434i
$606$		0			0
$607$		−	14.4452i		−	0.586311i	−0.956065	−	0.293156i	$-0.905295\pi$
$607$		−	14.4452i		−	0.586311i	0.956065	−	0.293156i	$-0.0947054\pi$
$608$		0			0
$609$	7.27527			0.294809
$610$		0			0
$611$	−2.65376			−0.107360
$612$		0			0
$613$		−	14.2815i		−	0.576824i	−0.957506	−	0.288412i	$-0.906873\pi$
$613$		−	14.2815i		−	0.576824i	0.957506	−	0.288412i	$-0.0931273\pi$
$614$		0			0
$615$	2.39357	+	17.0363i	0.0965181	+	0.686971i
$616$		0			0
$617$		−	6.28172i		−	0.252893i	−0.991973	−	0.126446i	$-0.959643\pi$
$617$		−	6.28172i		−	0.252893i	0.991973	−	0.126446i	$-0.0403571\pi$
$618$		0			0
$619$	34.9744			1.40574			0.702870	−	0.711319i	$-0.251902\pi$
$619$	34.9744			1.40574			0.702870	+	0.711319i	$0.251902\pi$
$620$		0			0
$621$	−1.00000			−0.0401286
$622$		0			0
$623$		−	26.7608i		−	1.07215i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$		−	26.3978i		−	1.05423i
$628$		0			0
$629$	7.35128			0.293115
$630$		0			0
$631$	−5.25411			−0.209163			−0.104581	−	0.994516i	$-0.533350\pi$
$631$	−5.25411			−0.209163			−0.104581	+	0.994516i	$0.533350\pi$
$632$		0			0
$633$		−	0.328696i		−	0.0130645i
$634$		0			0
$635$	−0.814733	−	5.79889i	−0.0323317	−	0.230122i
$636$		0			0
$637$			4.80469i			0.190369i
$638$		0			0
$639$	−1.64458			−0.0650586
$640$		0			0
$641$	29.1863			1.15279			0.576394	−	0.817172i	$-0.304459\pi$
$641$	29.1863			1.15279			0.576394	+	0.817172i	$0.304459\pi$
$642$		0			0
$643$			16.1607i			0.637316i	0.947870	+	0.318658i	$0.103232\pi$
$643$			16.1607i			0.637316i	−0.947870	+	0.318658i	$0.896768\pi$
$644$		0			0
$645$	19.4043	−	2.72627i	0.764044	−	0.107347i
$646$		0			0
$647$		−	43.7900i		−	1.72156i	−0.508974	−	0.860782i	$-0.669975\pi$
$647$		−	43.7900i		−	1.72156i	0.508974	−	0.860782i	$-0.330025\pi$
$648$		0			0
$649$	42.1267			1.65362
$650$		0			0
$651$	−0.151801			−0.00594954
$652$		0			0
$653$		−	20.4492i		−	0.800239i	−0.916463	−	0.400120i	$-0.868969\pi$
$653$		−	20.4492i		−	0.800239i	0.916463	−	0.400120i	$-0.131031\pi$
$654$		0			0
$655$	−33.9411	+	4.76866i	−1.32619	+	0.186327i
$656$		0			0
$657$		−	9.70567i		−	0.378654i
$658$		0			0
$659$	−19.9431			−0.776872			−0.388436	−	0.921476i	$-0.626985\pi$
$659$	−19.9431			−0.776872			−0.388436	+	0.921476i	$0.626985\pi$
$660$		0			0
$661$	−5.21668			−0.202905			−0.101453	−	0.994840i	$-0.532349\pi$
$661$	−5.21668			−0.202905			−0.101453	+	0.994840i	$0.532349\pi$
$662$		0			0
$663$			5.01187i			0.194645i
$664$		0			0
$665$	3.70885	+	26.3978i	0.143823	+	1.02366i
$666$		0			0
$667$		−	4.26125i		−	0.164996i
$668$		0			0
$669$	−18.0914			−0.699454
$670$		0			0
$671$	41.6127			1.60644
$672$		0			0
$673$		−	14.8037i		−	0.570639i	−0.958432	−	0.285320i	$-0.907900\pi$
$673$		−	14.8037i		−	0.570639i	0.958432	−	0.285320i	$-0.0920998\pi$
$674$		0			0
$675$	−1.37778	−	4.80642i	−0.0530309	−	0.184999i
$676$		0			0
$677$			31.0622i			1.19382i	0.802309	+	0.596909i	$0.203605\pi$
$677$			31.0622i			1.19382i	−0.802309	+	0.596909i	$0.796395\pi$
$678$		0			0
$679$	13.2910			0.510062
$680$		0			0
$681$	25.1906			0.965304
$682$		0			0
$683$		−	9.90002i		−	0.378814i	−0.981899	−	0.189407i	$-0.939344\pi$
$683$		−	9.90002i		−	0.378814i	0.981899	−	0.189407i	$-0.0606565\pi$
$684$		0			0
$685$	−2.49204	−	17.7371i	−0.0952158	−	0.677702i
$686$		0			0
$687$			10.8968i			0.415738i
$688$		0			0
$689$	−15.2516			−0.581038
$690$		0			0
$691$	8.56571			0.325855			0.162927	−	0.986638i	$-0.447906\pi$
$691$	8.56571			0.325855			0.162927	+	0.986638i	$0.447906\pi$
$692$		0			0
$693$		−	6.45453i		−	0.245187i
$694$		0			0
$695$	−2.30186	+	0.323408i	−0.0873147	+	0.0122676i
$696$		0			0
$697$			32.7848i			1.24181i
$698$		0			0
$699$	−8.87480			−0.335676
$700$		0			0
$701$	−24.2891			−0.917387			−0.458694	−	0.888594i	$-0.651682\pi$
$701$	−24.2891			−0.917387			−0.458694	+	0.888594i	$0.651682\pi$
$702$		0			0
$703$			12.0460i			0.454323i
$704$		0			0
$705$	4.99618	−	0.701954i	0.188167	−	0.0264371i
$706$		0			0
$707$			10.8507i			0.408081i
$708$		0			0
$709$	−2.56299			−0.0962551			−0.0481275	−	0.998841i	$-0.515325\pi$
$709$	−2.56299			−0.0962551			−0.0481275	+	0.998841i	$0.515325\pi$
$710$		0			0
$711$	6.07634			0.227880
$712$		0			0
$713$			0.0889122i			0.00332979i
$714$		0			0
$715$	1.38333	+	9.84592i	0.0517338	+	0.368217i
$716$		0			0
$717$		−	29.6234i		−	1.10631i
$718$		0			0
$719$	−25.8195			−0.962904			−0.481452	−	0.876472i	$-0.659890\pi$
$719$	−25.8195			−0.962904			−0.481452	+	0.876472i	$0.659890\pi$
$720$		0			0
$721$	−4.48839			−0.167156
$722$		0			0
$723$			29.8553i			1.11033i
$724$		0			0
$725$	20.4814	−	5.87108i	0.760658	−	0.218046i
$726$		0			0
$727$			8.49856i			0.315194i	0.987504	+	0.157597i	$0.0503747\pi$
$727$			8.49856i			0.315194i	−0.987504	+	0.157597i	$0.949625\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	37.3417			1.38113
$732$		0			0
$733$		−	15.4601i		−	0.571032i	−0.958374	−	0.285516i	$-0.907835\pi$
$733$		−	15.4601i		−	0.571032i	0.958374	−	0.285516i	$-0.0921650\pi$
$734$		0			0
$735$	−1.27090	−	9.04570i	−0.0468780	−	0.333656i
$736$		0			0
$737$		−	39.2722i		−	1.44661i
$738$		0			0
$739$	1.29086			0.0474850			0.0237425	−	0.999718i	$-0.492442\pi$
$739$	1.29086			0.0474850			0.0237425	+	0.999718i	$0.492442\pi$
$740$		0			0
$741$	−8.21257			−0.301696
$742$		0			0
$743$		−	2.32278i		−	0.0852144i	−0.999092	−	0.0426072i	$-0.986434\pi$
$743$		−	2.32278i		−	0.0852144i	0.999092	−	0.0426072i	$-0.0135664\pi$
$744$		0			0
$745$	−37.7478	+	5.30350i	−1.38297	+	0.194305i
$746$		0			0
$747$			6.53116i			0.238963i
$748$		0			0
$749$	−11.4136			−0.417044
$750$		0			0
$751$	52.1277			1.90217			0.951083	−	0.308935i	$-0.0999724\pi$
$751$	52.1277			1.90217			0.951083	+	0.308935i	$0.0999724\pi$
$752$		0			0
$753$		−	4.15953i		−	0.151582i
$754$		0			0
$755$	33.3526	−	4.68597i	1.21382	−	0.170540i
$756$		0			0
$757$		−	1.72645i		−	0.0627490i	−0.999508	−	0.0313745i	$-0.990012\pi$
$757$		−	1.72645i		−	0.0627490i	0.999508	−	0.0313745i	$-0.00998846\pi$
$758$		0			0
$759$	−3.78053			−0.137225
$760$		0			0
$761$	5.84922			0.212034			0.106017	−	0.994364i	$-0.466190\pi$
$761$	5.84922			0.212034			0.106017	+	0.994364i	$0.466190\pi$
$762$		0			0
$763$		−	29.9764i		−	1.08522i
$764$		0			0
$765$	−1.32571	−	9.43576i	−0.0479310	−	0.341151i
$766$		0			0
$767$		−	13.1059i		−	0.473228i
$768$		0			0
$769$	−15.7788			−0.568997			−0.284498	−	0.958677i	$-0.591827\pi$
$769$	−15.7788			−0.568997			−0.284498	+	0.958677i	$0.591827\pi$
$770$		0			0
$771$	−27.3784			−0.986010
$772$		0			0
$773$		−	28.1883i		−	1.01386i	−0.861987	−	0.506931i	$-0.830780\pi$
$773$		−	28.1883i		−	1.01386i	0.861987	−	0.506931i	$-0.169220\pi$
$774$		0			0
$775$	−0.427350	+	0.122502i	−0.0153509	+	0.00440040i
$776$		0			0
$777$			2.94536i			0.105664i
$778$		0			0
$779$	−53.7219			−1.92479
$780$		0			0
$781$	−6.21738			−0.222475
$782$		0			0
$783$		−	4.26125i		−	0.152285i
$784$		0			0
$785$	−3.38105	−	24.0647i	−0.120675	−	0.858907i
$786$		0			0
$787$		−	29.2245i		−	1.04174i	−0.853636	−	0.520871i	$-0.825607\pi$
$787$		−	29.2245i		−	1.04174i	0.853636	−	0.520871i	$-0.174393\pi$
$788$		0			0
$789$	−8.66938			−0.308638
$790$		0			0
$791$	−6.82392			−0.242631
$792$		0			0
$793$		−	12.9461i		−	0.459728i
$794$		0			0
$795$	28.7138	−	4.03424i	1.01837	−	0.143080i
$796$		0			0
$797$			42.3799i			1.50117i	0.660772	+	0.750587i	$0.270229\pi$
$797$			42.3799i			1.50117i	−0.660772	+	0.750587i	$0.729771\pi$
$798$		0			0
$799$	9.61467			0.340143
$800$		0			0
$801$	−15.6742			−0.553822
$802$		0			0
$803$		−	36.6926i		−	1.29485i
$804$		0			0
$805$	3.78053	−	0.531157i	0.133246	−	0.0187208i
$806$		0			0
$807$			12.8627i			0.452790i
$808$		0			0
$809$	31.6479			1.11268			0.556341	−	0.830954i	$-0.312205\pi$
$809$	31.6479			1.11268			0.556341	+	0.830954i	$0.312205\pi$
$810$		0			0
$811$	−34.8706			−1.22447			−0.612237	−	0.790675i	$-0.709730\pi$
$811$	−34.8706			−1.22447			−0.612237	+	0.790675i	$0.709730\pi$
$812$		0			0
$813$		−	24.5148i		−	0.859771i
$814$		0			0
$815$	−3.98890	−	28.3911i	−0.139725	−	0.994498i
$816$		0			0
$817$			61.1891i			2.14073i
$818$		0			0
$819$	−2.00806			−0.0701672
$820$		0			0
$821$	−8.08387			−0.282129			−0.141065	−	0.990000i	$-0.545053\pi$
$821$	−8.08387			−0.282129			−0.141065	+	0.990000i	$0.545053\pi$
$822$		0			0
$823$		−	27.3511i		−	0.953398i	−0.879067	−	0.476699i	$-0.841833\pi$
$823$		−	27.3511i		−	0.953398i	0.879067	−	0.476699i	$-0.158167\pi$
$824$		0			0
$825$	−5.20875	−	18.1708i	−0.181346	−	0.632627i
$826$		0			0
$827$			22.6936i			0.789134i	0.918867	+	0.394567i	$0.129105\pi$
$827$			22.6936i			0.789134i	−0.918867	+	0.394567i	$0.870895\pi$
$828$		0			0
$829$	−23.7763			−0.825784			−0.412892	−	0.910780i	$-0.635481\pi$
$829$	−23.7763			−0.825784			−0.412892	+	0.910780i	$0.635481\pi$
$830$		0			0
$831$	−1.80963			−0.0627753
$832$		0			0
$833$		−	17.4076i		−	0.603137i
$834$		0			0
$835$	−5.08108	−	36.1648i	−0.175838	−	1.25153i
$836$		0			0
$837$			0.0889122i			0.00307325i
$838$		0			0
$839$	36.9716			1.27640			0.638201	−	0.769869i	$-0.279679\pi$
$839$	36.9716			1.27640			0.638201	+	0.769869i	$0.279679\pi$
$840$		0			0
$841$	−10.8418			−0.373855
$842$		0			0
$843$			17.8395i			0.614426i
$844$		0			0
$845$	−25.7230	+	3.61404i	−0.884899	+	0.124327i
$846$		0			0
$847$		−	5.62115i		−	0.193145i
$848$		0			0
$849$	21.4440			0.735956
$850$		0			0
$851$	1.72515			0.0591373
$852$		0			0
$853$			25.3940i			0.869476i	0.900557	+	0.434738i	$0.143159\pi$
$853$			25.3940i			0.869476i	−0.900557	+	0.434738i	$0.856841\pi$
$854$		0			0
$855$	15.4617	−	2.17233i	0.528777	−	0.0742923i
$856$		0			0
$857$			13.3746i			0.456869i	0.973559	+	0.228435i	$0.0733607\pi$
$857$			13.3746i			0.456869i	−0.973559	+	0.228435i	$0.926639\pi$
$858$		0			0
$859$	12.8387			0.438050			0.219025	−	0.975719i	$-0.429712\pi$
$859$	12.8387			0.438050			0.219025	+	0.975719i	$0.429712\pi$
$860$		0			0
$861$	−13.1355			−0.447658
$862$		0			0
$863$		−	6.35630i		−	0.216371i	−0.994131	−	0.108185i	$-0.965496\pi$
$863$		−	6.35630i		−	0.216371i	0.994131	−	0.108185i	$-0.0345040\pi$
$864$		0			0
$865$	−5.63897	−	40.1356i	−0.191731	−	1.36465i
$866$		0			0
$867$		−	1.15822i		−	0.0393352i
$868$		0			0
$869$	22.9718			0.779264
$870$		0			0
$871$	−12.2179			−0.413988
$872$		0			0
$873$		−	7.78476i		−	0.263474i
$874$		0			0
$875$	7.76172	+	17.4390i	0.262394	+	0.589546i
$876$		0			0
$877$			37.0865i			1.25232i	0.779694	+	0.626161i	$0.215375\pi$
$877$			37.0865i			1.25232i	−0.779694	+	0.626161i	$0.784625\pi$
$878$		0			0
$879$	−0.148076			−0.00499449
$880$		0			0
$881$	−27.8509			−0.938320			−0.469160	−	0.883113i	$-0.655443\pi$
$881$	−27.8509			−0.938320			−0.469160	+	0.883113i	$0.655443\pi$
$882$		0			0
$883$		−	54.8709i		−	1.84655i	−0.384136	−	0.923276i	$-0.625501\pi$
$883$		−	54.8709i		−	1.84655i	0.384136	−	0.923276i	$-0.374499\pi$
$884$		0			0
$885$	3.46670	+	24.6743i	0.116532	+	0.829418i
$886$		0			0
$887$		−	56.4462i		−	1.89528i	−0.319341	−	0.947640i	$-0.603462\pi$
$887$		−	56.4462i		−	1.89528i	0.319341	−	0.947640i	$-0.396538\pi$
$888$		0			0
$889$	4.47112			0.149957
$890$		0			0
$891$	−3.78053			−0.126652
$892$		0			0
$893$			15.7548i			0.527215i
$894$		0			0
$895$	2.22277	−	0.312296i	0.0742992	−	0.0104389i
$896$		0			0
$897$			1.17615i			0.0392706i
$898$		0			0
$899$	−0.378877			−0.0126362
$900$		0			0
$901$	55.2570			1.84088
$902$		0			0
$903$			14.9613i			0.497882i
$904$		0			0
$905$	−11.6926	+	1.64280i	−0.388677	+	0.0546084i
$906$		0			0
$907$		−	56.5469i		−	1.87761i	−0.344450	−	0.938805i	$-0.611935\pi$
$907$		−	56.5469i		−	1.87761i	0.344450	−	0.938805i	$-0.388065\pi$
$908$		0			0
$909$	6.35542			0.210796
$910$		0			0
$911$	−6.12565			−0.202952			−0.101476	−	0.994838i	$-0.532356\pi$
$911$	−6.12565			−0.202952			−0.101476	+	0.994838i	$0.532356\pi$
$912$		0			0
$913$			24.6912i			0.817161i
$914$		0			0
$915$	3.42440	+	24.3733i	0.113207	+	0.805756i
$916$		0			0
$917$		−	26.1696i		−	0.864197i
$918$		0			0
$919$	−26.6564			−0.879313			−0.439657	−	0.898166i	$-0.644900\pi$
$919$	−26.6564			−0.879313			−0.439657	+	0.898166i	$0.644900\pi$
$920$		0			0
$921$	−18.7801			−0.618825
$922$		0			0
$923$			1.93428i			0.0636675i
$924$		0			0
$925$	2.37688	+	8.29179i	0.0781514	+	0.272632i
$926$		0			0
$927$			2.62892i			0.0863452i
$928$		0			0
$929$	−41.7919			−1.37115			−0.685573	−	0.728004i	$-0.740448\pi$
$929$	−41.7919			−1.37115			−0.685573	+	0.728004i	$0.740448\pi$
$930$		0			0
$931$	28.5245			0.934852
$932$		0			0
$933$		−	1.41811i		−	0.0464270i
$934$		0			0
$935$	−5.01187	−	35.6722i	−0.163906	−	1.16660i
$936$		0			0
$937$		−	9.88647i		−	0.322977i	−0.986875	−	0.161488i	$-0.948371\pi$
$937$		−	9.88647i		−	0.322977i	0.986875	−	0.161488i	$-0.0516295\pi$
$938$		0			0
$939$	31.6209			1.03191
$940$		0			0
$941$	−29.7766			−0.970689			−0.485344	−	0.874323i	$-0.661306\pi$
$941$	−29.7766			−0.970689			−0.485344	+	0.874323i	$0.661306\pi$
$942$		0			0
$943$			7.69370i			0.250541i
$944$		0			0
$945$	3.78053	−	0.531157i	0.122981	−	0.0172786i
$946$		0			0
$947$			40.6289i			1.32026i	0.751150	+	0.660132i	$0.229500\pi$
$947$			40.6289i			1.32026i	−0.751150	+	0.660132i	$0.770500\pi$
$948$		0			0
$949$	−11.4153			−0.370558
$950$		0			0
$951$	27.2743			0.884429
$952$		0			0
$953$			44.3652i			1.43713i	0.695459	+	0.718566i	$0.255201\pi$
$953$			44.3652i			1.43713i	−0.695459	+	0.718566i	$0.744799\pi$
$954$		0			0
$955$	18.9463	−	2.66191i	0.613087	−	0.0861375i
$956$		0			0
$957$		−	16.1098i		−	0.520755i
$958$		0			0
$959$	13.6759			0.441618
$960$		0			0
$961$	−30.9921			−0.999745
$962$		0			0
$963$			6.68513i			0.215425i
$964$		0			0
$965$	4.12546	+	29.3631i	0.132803	+	0.945231i
$966$		0			0
$967$		−	3.40923i		−	0.109633i	−0.998496	−	0.0548167i	$-0.982543\pi$
$967$		−	3.40923i		−	0.109633i	0.998496	−	0.0548167i	$-0.0174574\pi$
$968$		0			0
$969$	29.7545			0.955851
$970$		0			0
$971$	37.7110			1.21020			0.605101	−	0.796149i	$-0.293133\pi$
$971$	37.7110			1.21020			0.605101	+	0.796149i	$0.293133\pi$
$972$		0			0
$973$		−	1.77481i		−	0.0568978i
$974$		0			0
$975$	−5.65309	+	1.62048i	−0.181044	+	0.0518970i
$976$		0			0
$977$			24.8097i			0.793734i	0.917876	+	0.396867i	$0.129903\pi$
$977$			24.8097i			0.793734i	−0.917876	+	0.396867i	$0.870097\pi$
$978$		0			0
$979$	−59.2569			−1.89386
$980$		0			0
$981$	−17.5577			−0.560573
$982$		0			0
$983$		−	18.0835i		−	0.576775i	−0.957514	−	0.288387i	$-0.906881\pi$
$983$		−	18.0835i		−	0.576775i	0.957514	−	0.288387i	$-0.0931191\pi$
$984$		0			0
$985$	−7.78328	−	55.3978i	−0.247996	−	1.76512i
$986$		0			0
$987$			3.85221i			0.122617i
$988$		0			0
$989$	8.76311			0.278651
$990$		0			0
$991$	21.9496			0.697251			0.348625	−	0.937262i	$-0.386649\pi$
$991$	21.9496			0.697251			0.348625	+	0.937262i	$0.386649\pi$
$992$		0			0
$993$		−	4.03982i		−	0.128200i
$994$		0			0
$995$	8.17214	−	1.14817i	0.259074	−	0.0363995i
$996$		0			0
$997$		−	30.9724i		−	0.980905i	−0.871468	−	0.490452i	$-0.836832\pi$
$997$		−	30.9724i		−	0.980905i	0.871468	−	0.490452i	$-0.163168\pi$
$998$		0			0
$999$	1.72515			0.0545813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.k.c.2209.7	yes	12
5.4	even	2	inner	2760.2.k.c.2209.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.k.c.2209.2	✓	12	5.4	even	2	inner
2760.2.k.c.2209.7	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.21432 + 0.311108i) q^{5} -1.70731i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-2.21432 + 0.311108i) q^{5} -1.70731i q^{7} -1.00000 q^{9} -3.78053 q^{11} +1.17615i q^{13} +(-0.311108 - 2.21432i) q^{15} -4.26125i q^{17} +6.98258 q^{19} +1.70731 q^{21} -1.00000i q^{23} +(4.80642 - 1.37778i) q^{25} -1.00000i q^{27} +4.26125 q^{29} -0.0889122 q^{31} -3.78053i q^{33} +(0.531157 + 3.78053i) q^{35} +1.72515i q^{37} -1.17615 q^{39} -7.69370 q^{41} +8.76311i q^{43} +(2.21432 - 0.311108i) q^{45} +2.25631i q^{47} +4.08509 q^{49} +4.26125 q^{51} +12.9673i q^{53} +(8.37130 - 1.17615i) q^{55} +6.98258i q^{57} -11.1431 q^{59} -11.0071 q^{61} +1.70731i q^{63} +(-0.365910 - 2.60438i) q^{65} +10.3880i q^{67} +1.00000 q^{69} +1.64458 q^{71} +9.70567i q^{73} +(1.37778 + 4.80642i) q^{75} +6.45453i q^{77} -6.07634 q^{79} +1.00000 q^{81} -6.53116i q^{83} +(1.32571 + 9.43576i) q^{85} +4.26125i q^{87} +15.6742 q^{89} +2.00806 q^{91} -0.0889122i q^{93} +(-15.4617 + 2.17233i) q^{95} +7.78476i q^{97} +3.78053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} - 4 q^{15} + 12 q^{19} - 6 q^{21} + 4 q^{25} + 6 q^{29} - 2 q^{31} - 2 q^{35} + 4 q^{39} - 22 q^{41} + 22 q^{49} + 6 q^{51} - 16 q^{55} + 18 q^{59} - 60 q^{61} - 12 q^{65} + 12 q^{69} + 54 q^{71} + 16 q^{75} - 56 q^{79} + 12 q^{81} - 38 q^{85} + 28 q^{89} - 12 q^{91} - 52 q^{95}+O(q^{100}) \) 12 * q - 12 * q^9 - 4 * q^15 + 12 * q^19 - 6 * q^21 + 4 * q^25 + 6 * q^29 - 2 * q^31 - 2 * q^35 + 4 * q^39 - 22 * q^41 + 22 * q^49 + 6 * q^51 - 16 * q^55 + 18 * q^59 - 60 * q^61 - 12 * q^65 + 12 * q^69 + 54 * q^71 + 16 * q^75 - 56 * q^79 + 12 * q^81 - 38 * q^85 + 28 * q^89 - 12 * q^91 - 52 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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