LMFDB - Embedded newform 1440.2.bj.a.17.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(17,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1440.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bj (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.12
Character	$\chi$	$=$	1440.17
Dual form			1440.2.bj.a.593.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.215413 + 2.22567i) q^{5} +(2.32063 + 2.32063i) q^{7} +O(q^{10})$	$q+(-0.215413 + 2.22567i) q^{5} +(2.32063 + 2.32063i) q^{7} +5.57646 q^{11} +(1.79226 + 1.79226i) q^{13} +(-5.56203 + 5.56203i) q^{17} +2.61960 q^{19} +(-4.44948 - 4.44948i) q^{23} +(-4.90719 - 0.958877i) q^{25} -2.12491i q^{29} +4.52171 q^{31} +(-5.66484 + 4.66506i) q^{35} +(5.02845 - 5.02845i) q^{37} -1.73215i q^{41} +(-1.19790 - 1.19790i) q^{43} +(-0.849681 + 0.849681i) q^{47} +3.77064i q^{49} +(-4.22906 + 4.22906i) q^{53} +(-1.20124 + 12.4113i) q^{55} +8.08999i q^{59} +3.13786i q^{61} +(-4.37505 + 3.60290i) q^{65} +(1.86836 - 1.86836i) q^{67} +6.95452i q^{71} +(-5.86607 + 5.86607i) q^{73} +(12.9409 + 12.9409i) q^{77} -2.52204i q^{79} +(0.694298 - 0.694298i) q^{83} +(-11.1811 - 13.5774i) q^{85} -12.1475 q^{89} +8.31834i q^{91} +(-0.564297 + 5.83036i) q^{95} +(4.09526 + 4.09526i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) $ 48 * q - 32 * q^31 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$	$1$

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
$5$	−0.215413	+	2.22567i	−0.0963358	+	0.995349i
$5$	−0.215413	+	2.22567i	−0.0963358	+	0.995349i
$6$		0			0
$7$	2.32063	+	2.32063i	0.877115	+	0.877115i	0.993235	−	0.116120i	$-0.0370456\pi$
$7$	2.32063	+	2.32063i	0.877115	+	0.877115i	−0.116120	+	0.993235i	$0.537046\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.57646			1.68137			0.840683	−	0.541528i	$-0.182154\pi$
$11$	5.57646			1.68137			0.840683	+	0.541528i	$0.182154\pi$
$12$		0			0
$13$	1.79226	+	1.79226i	0.497083	+	0.497083i	0.910529	−	0.413446i	$-0.135675\pi$
$13$	1.79226	+	1.79226i	0.497083	+	0.497083i	−0.413446	+	0.910529i	$0.635675\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.56203	+	5.56203i	−1.34899	+	1.34899i	−0.462233	+	0.886758i	$0.652952\pi$
$17$	−5.56203	+	5.56203i	−1.34899	+	1.34899i	−0.886758	+	0.462233i	$0.847048\pi$
$18$		0			0
$19$	2.61960			0.600978			0.300489	−	0.953785i	$-0.402850\pi$
$19$	2.61960			0.600978			0.300489	+	0.953785i	$0.402850\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.44948	−	4.44948i	−0.927781	−	0.927781i	0.0697816	−	0.997562i	$-0.477770\pi$
$23$	−4.44948	−	4.44948i	−0.927781	−	0.927781i	−0.997562	+	0.0697816i	$0.977770\pi$
$24$		0			0
$25$	−4.90719	−	0.958877i	−0.981439	−	0.191775i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.12491i		−	0.394587i	−0.980345	−	0.197293i	$-0.936785\pi$
$29$		−	2.12491i		−	0.394587i	0.980345	−	0.197293i	$-0.0632151\pi$
$30$		0			0
$31$	4.52171			0.812123			0.406061	−	0.913846i	$-0.366902\pi$
$31$	4.52171			0.812123			0.406061	+	0.913846i	$0.366902\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.66484	+	4.66506i	−0.957534	+	0.788538i
$36$		0			0
$37$	5.02845	−	5.02845i	0.826673	−	0.826673i	−0.160382	−	0.987055i	$-0.551273\pi$
$37$	5.02845	−	5.02845i	0.826673	−	0.826673i	0.987055	+	0.160382i	$0.0512727\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.73215i		−	0.270516i	−0.990810	−	0.135258i	$-0.956814\pi$
$41$		−	1.73215i		−	0.270516i	0.990810	−	0.135258i	$-0.0431863\pi$
$42$		0			0
$43$	−1.19790	−	1.19790i	−0.182678	−	0.182678i	0.609844	−	0.792522i	$-0.291232\pi$
$43$	−1.19790	−	1.19790i	−0.182678	−	0.182678i	−0.792522	+	0.609844i	$0.791232\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.849681	+	0.849681i	−0.123939	+	0.123939i	−0.766355	−	0.642417i	$-0.777932\pi$
$47$	−0.849681	+	0.849681i	−0.123939	+	0.123939i	0.642417	+	0.766355i	$0.277932\pi$
$48$		0			0
$49$			3.77064i			0.538663i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.22906	+	4.22906i	−0.580906	+	0.580906i	−0.935152	−	0.354246i	$-0.884737\pi$
$53$	−4.22906	+	4.22906i	−0.580906	+	0.580906i	0.354246	+	0.935152i	$0.384737\pi$
$54$		0			0
$55$	−1.20124	+	12.4113i	−0.161976	+	1.67355i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.08999i			1.05323i	0.850105	+	0.526614i	$0.176539\pi$
$59$			8.08999i			1.05323i	−0.850105	+	0.526614i	$0.823461\pi$
$60$		0			0
$61$			3.13786i			0.401762i	0.979616	+	0.200881i	$0.0643805\pi$
$61$			3.13786i			0.401762i	−0.979616	+	0.200881i	$0.935620\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.37505	+	3.60290i	−0.542658	+	0.446884i
$66$		0			0
$67$	1.86836	−	1.86836i	0.228256	−	0.228256i	−0.583708	−	0.811964i	$-0.698399\pi$
$67$	1.86836	−	1.86836i	0.228256	−	0.228256i	0.811964	+	0.583708i	$0.198399\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.95452i			0.825350i	0.910878	+	0.412675i	$0.135405\pi$
$71$			6.95452i			0.825350i	−0.910878	+	0.412675i	$0.864595\pi$
$72$		0			0
$73$	−5.86607	+	5.86607i	−0.686572	+	0.686572i	−0.961473	−	0.274901i	$-0.911355\pi$
$73$	−5.86607	+	5.86607i	−0.686572	+	0.686572i	0.274901	+	0.961473i	$0.411355\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.9409	+	12.9409i	1.47475	+	1.47475i
$78$		0			0
$79$		−	2.52204i		−	0.283752i	−0.989884	−	0.141876i	$-0.954687\pi$
$79$		−	2.52204i		−	0.283752i	0.989884	−	0.141876i	$-0.0453134\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.694298	−	0.694298i	0.0762091	−	0.0762091i	−0.667975	−	0.744184i	$-0.732839\pi$
$83$	0.694298	−	0.694298i	0.0762091	−	0.0762091i	0.744184	+	0.667975i	$0.232839\pi$
$84$		0			0
$85$	−11.1811	−	13.5774i	−1.21276	−	1.47267i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.1475			−1.28763			−0.643816	−	0.765180i	$-0.722650\pi$
$89$	−12.1475			−1.28763			−0.643816	+	0.765180i	$0.722650\pi$
$90$		0			0
$91$			8.31834i			0.871999i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.564297	+	5.83036i	−0.0578957	+	0.598183i
$96$		0			0
$97$	4.09526	+	4.09526i	0.415811	+	0.415811i	0.883757	−	0.467946i	$-0.155006\pi$
$97$	4.09526	+	4.09526i	0.415811	+	0.415811i	−0.467946	+	0.883757i	$0.655006\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.617937			−0.0614871			−0.0307435	−	0.999527i	$-0.509788\pi$
$101$	−0.617937			−0.0614871			−0.0307435	+	0.999527i	$0.509788\pi$
$102$		0			0
$103$	2.59696	−	2.59696i	0.255886	−	0.255886i	−0.567493	−	0.823379i	$-0.692086\pi$
$103$	2.59696	−	2.59696i	0.255886	−	0.255886i	0.823379	+	0.567493i	$0.192086\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.74306	−	1.74306i	−0.168508	−	0.168508i	0.617815	−	0.786323i	$-0.288018\pi$
$107$	−1.74306	−	1.74306i	−0.168508	−	0.168508i	−0.786323	+	0.617815i	$0.788018\pi$
$108$		0			0
$109$	15.1487			1.45098			0.725491	−	0.688231i	$-0.241613\pi$
$109$	15.1487			1.45098			0.725491	+	0.688231i	$0.241613\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.41591	+	2.41591i	0.227270	+	0.227270i	0.811551	−	0.584281i	$-0.198624\pi$
$113$	2.41591	+	2.41591i	0.227270	+	0.227270i	−0.584281	+	0.811551i	$0.698624\pi$
$114$		0			0
$115$	10.8615	−	8.94459i	1.01284	−	0.834087i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−25.8148			−2.36644
$120$		0			0
$121$	20.0969			1.82699
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.19122	−	10.7152i	0.285431	−	0.958399i
$126$		0			0
$127$	1.40304	+	1.40304i	0.124500	+	0.124500i	0.766611	−	0.642112i	$-0.221941\pi$
$127$	1.40304	+	1.40304i	0.124500	+	0.124500i	−0.642112	+	0.766611i	$0.721941\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.14182			0.623984			0.311992	−	0.950085i	$-0.399004\pi$
$131$	7.14182			0.623984			0.311992	+	0.950085i	$0.399004\pi$
$132$		0			0
$133$	6.07912	+	6.07912i	0.527127	+	0.527127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.58546	−	6.58546i	0.562634	−	0.562634i	−0.367421	−	0.930055i	$-0.619759\pi$
$137$	6.58546	−	6.58546i	0.562634	−	0.562634i	0.930055	+	0.367421i	$0.119759\pi$
$138$		0			0
$139$	15.1568			1.28558			0.642791	−	0.766042i	$-0.277776\pi$
$139$	15.1568			1.28558			0.642791	+	0.766042i	$0.277776\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.99446	+	9.99446i	0.835779	+	0.835779i
$144$		0			0
$145$	4.72935	+	0.457735i	0.392751	+	0.0380128i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	5.51376i		−	0.451705i	−0.974162	−	0.225852i	$-0.927483\pi$
$149$		−	5.51376i		−	0.451705i	0.974162	−	0.225852i	$-0.0725167\pi$
$150$		0			0
$151$	−14.7042			−1.19661			−0.598307	−	0.801267i	$-0.704160\pi$
$151$	−14.7042			−1.19661			−0.598307	+	0.801267i	$0.704160\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.974036	+	10.0638i	−0.0782365	+	0.808345i
$156$		0			0
$157$	−11.3534	+	11.3534i	−0.906103	+	0.906103i	−0.995955	−	0.0898521i	$-0.971361\pi$
$157$	−11.3534	+	11.3534i	−0.906103	+	0.906103i	0.0898521	+	0.995955i	$0.471361\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	20.6512i		−	1.62754i
$162$		0			0
$163$	−11.7245	−	11.7245i	−0.918331	−	0.918331i	0.0785768	−	0.996908i	$-0.474962\pi$
$163$	−11.7245	−	11.7245i	−0.918331	−	0.918331i	−0.996908	+	0.0785768i	$0.974962\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.91997	−	7.91997i	0.612865	−	0.612865i	−0.330826	−	0.943692i	$-0.607327\pi$
$167$	7.91997	−	7.91997i	0.612865	−	0.612865i	0.943692	+	0.330826i	$0.107327\pi$
$168$		0			0
$169$		−	6.57562i		−	0.505817i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.22597	+	7.22597i	−0.549380	+	0.549380i	−0.926262	−	0.376881i	$-0.876996\pi$
$173$	−7.22597	+	7.22597i	−0.549380	+	0.549380i	0.376881	+	0.926262i	$0.376996\pi$
$174$		0			0
$175$	−9.16258	−	13.6130i	−0.692626	−	1.02904i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	13.7942i		−	1.03103i	−0.856880	−	0.515515i	$-0.827601\pi$
$179$		−	13.7942i		−	1.03103i	0.856880	−	0.515515i	$-0.172399\pi$
$180$		0			0
$181$			18.3310i			1.36254i	0.732034	+	0.681268i	$0.238571\pi$
$181$			18.3310i			1.36254i	−0.732034	+	0.681268i	$0.761429\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.1085	+	12.2749i	0.743190	+	0.902466i
$186$		0			0
$187$	−31.0165	+	31.0165i	−2.26815	+	2.26815i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.2206i		−	0.884251i	−0.896953	−	0.442125i	$-0.854225\pi$
$191$		−	12.2206i		−	0.884251i	0.896953	−	0.442125i	$-0.145775\pi$
$192$		0			0
$193$	−1.67882	+	1.67882i	−0.120844	+	0.120844i	−0.764943	−	0.644099i	$-0.777233\pi$
$193$	−1.67882	+	1.67882i	−0.120844	+	0.120844i	0.644099	+	0.764943i	$0.277233\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.43727	+	4.43727i	0.316143	+	0.316143i	0.847283	−	0.531141i	$-0.178237\pi$
$197$	4.43727	+	4.43727i	0.316143	+	0.316143i	−0.531141	+	0.847283i	$0.678237\pi$
$198$		0			0
$199$		−	16.2067i		−	1.14886i	−0.818553	−	0.574432i	$-0.805223\pi$
$199$		−	16.2067i		−	1.14886i	0.818553	−	0.574432i	$-0.194777\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.93114	−	4.93114i	0.346098	−	0.346098i
$204$		0			0
$205$	3.85518	+	0.373128i	0.269258	+	0.0260604i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	14.6081			1.01046
$210$		0			0
$211$		−	22.6302i		−	1.55793i	−0.627068	−	0.778964i	$-0.715745\pi$
$211$		−	22.6302i		−	1.55793i	0.627068	−	0.778964i	$-0.284255\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.92417	−	2.40808i	0.199427	−	0.164230i
$216$		0			0
$217$	10.4932	+	10.4932i	0.712325	+	0.712325i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−19.9372			−1.34112
$222$		0			0
$223$	−10.8423	+	10.8423i	−0.726057	+	0.726057i	−0.969832	−	0.243775i	$-0.921614\pi$
$223$	−10.8423	+	10.8423i	−0.726057	+	0.726057i	0.243775	+	0.969832i	$0.421614\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.6177	−	12.6177i	−0.837465	−	0.837465i	0.151059	−	0.988525i	$-0.451732\pi$
$227$	−12.6177	−	12.6177i	−0.837465	−	0.837465i	−0.988525	+	0.151059i	$0.951732\pi$
$228$		0			0
$229$	−0.594060			−0.0392566			−0.0196283	−	0.999807i	$-0.506248\pi$
$229$	−0.594060			−0.0392566			−0.0196283	+	0.999807i	$0.506248\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.12779	−	5.12779i	−0.335933	−	0.335933i	0.518901	−	0.854834i	$-0.326341\pi$
$233$	−5.12779	−	5.12779i	−0.335933	−	0.335933i	−0.854834	+	0.518901i	$0.826341\pi$
$234$		0			0
$235$	−1.70807	−	2.07414i	−0.111423	−	0.135302i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.1582			0.721763			0.360881	−	0.932612i	$-0.382476\pi$
$239$	11.1582			0.721763			0.360881	+	0.932612i	$0.382476\pi$
$240$		0			0
$241$	−19.9706			−1.28642			−0.643209	−	0.765691i	$-0.722397\pi$
$241$	−19.9706			−1.28642			−0.643209	+	0.765691i	$0.722397\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.39219	−	0.812247i	−0.536158	−	0.0518925i
$246$		0			0
$247$	4.69500	+	4.69500i	0.298736	+	0.298736i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.4134			1.28848			0.644240	−	0.764823i	$-0.277174\pi$
$251$	20.4134			1.28848			0.644240	+	0.764823i	$0.277174\pi$
$252$		0			0
$253$	−24.8123	−	24.8123i	−1.55994	−	1.55994i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.9742	−	13.9742i	0.871688	−	0.871688i	−0.120968	−	0.992656i	$-0.538600\pi$
$257$	13.9742	−	13.9742i	0.871688	−	0.871688i	0.992656	+	0.120968i	$0.0385999\pi$
$258$		0			0
$259$	23.3384			1.45017
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.4243	+	10.4243i	0.642788	+	0.642788i	0.951240	−	0.308452i	$-0.0998109\pi$
$263$	10.4243	+	10.4243i	0.642788	+	0.642788i	−0.308452	+	0.951240i	$0.599811\pi$
$264$		0			0
$265$	−8.50148	−	10.3235i	−0.522242	−	0.634166i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	29.5007i		−	1.79869i	−0.437241	−	0.899345i	$-0.644044\pi$
$269$		−	29.5007i		−	1.79869i	0.437241	−	0.899345i	$-0.355956\pi$
$270$		0			0
$271$	19.3447			1.17511			0.587554	−	0.809185i	$-0.300091\pi$
$271$	19.3447			1.17511			0.587554	+	0.809185i	$0.300091\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−27.3648	−	5.34714i	−1.65016	−	0.322445i
$276$		0			0
$277$	1.44346	−	1.44346i	0.0867293	−	0.0867293i	−0.662411	−	0.749140i	$-0.730467\pi$
$277$	1.44346	−	1.44346i	0.0867293	−	0.0867293i	0.749140	+	0.662411i	$0.230467\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			23.9276i			1.42740i	0.700450	+	0.713702i	$0.252983\pi$
$281$			23.9276i			1.42740i	−0.700450	+	0.713702i	$0.747017\pi$
$282$		0			0
$283$	−7.90740	−	7.90740i	−0.470046	−	0.470046i	0.431883	−	0.901930i	$-0.357849\pi$
$283$	−7.90740	−	7.90740i	−0.470046	−	0.470046i	−0.901930	+	0.431883i	$0.857849\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.01967	−	4.01967i	0.237274	−	0.237274i
$288$		0			0
$289$		−	44.8725i		−	2.63956i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.38544	−	5.38544i	0.314621	−	0.314621i	−0.532076	−	0.846697i	$-0.678588\pi$
$293$	5.38544	−	5.38544i	0.314621	−	0.314621i	0.846697	+	0.532076i	$0.178588\pi$
$294$		0			0
$295$	−18.0056	−	1.74269i	−1.04833	−	0.101463i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	15.9492i		−	0.922368i
$300$		0			0
$301$		−	5.55977i		−	0.320460i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.98384	−	0.675938i	−0.399893	−	0.0387041i
$306$		0			0
$307$	−0.116076	+	0.116076i	−0.00662479	+	0.00662479i	−0.710411	−	0.703787i	$-0.751491\pi$
$307$	−0.116076	+	0.116076i	−0.00662479	+	0.00662479i	0.703787	+	0.710411i	$0.251491\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.42835i			0.0809944i	0.999180	+	0.0404972i	$0.0128942\pi$
$311$			1.42835i			0.0809944i	−0.999180	+	0.0404972i	$0.987106\pi$
$312$		0			0
$313$	−6.59396	+	6.59396i	−0.372712	+	0.372712i	−0.868464	−	0.495752i	$-0.834893\pi$
$313$	−6.59396	+	6.59396i	−0.372712	+	0.372712i	0.495752	+	0.868464i	$0.334893\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.2572	+	11.2572i	0.632269	+	0.632269i	0.948637	−	0.316367i	$-0.102463\pi$
$317$	11.2572	+	11.2572i	0.632269	+	0.632269i	−0.316367	+	0.948637i	$0.602463\pi$
$318$		0			0
$319$		−	11.8495i		−	0.663444i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−14.5703	+	14.5703i	−0.810714	+	0.810714i
$324$		0			0
$325$	−7.07640	−	10.5135i	−0.392528	−	0.583185i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.94359			−0.217417
$330$		0			0
$331$			35.0445i			1.92622i	0.269113	+	0.963109i	$0.413270\pi$
$331$			35.0445i			1.92622i	−0.269113	+	0.963109i	$0.586730\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.75587	+	4.56081i	0.205205	+	0.249184i
$336$		0			0
$337$	1.94865	+	1.94865i	0.106150	+	0.106150i	0.758187	−	0.652037i	$-0.226085\pi$
$337$	1.94865	+	1.94865i	0.106150	+	0.106150i	−0.652037	+	0.758187i	$0.726085\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	25.2151			1.36548
$342$		0			0
$343$	7.49415	−	7.49415i	0.404646	−	0.404646i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.0347	+	18.0347i	0.968151	+	0.968151i	0.999508	−	0.0313569i	$-0.00998286\pi$
$347$	18.0347	+	18.0347i	0.968151	+	0.968151i	−0.0313569	+	0.999508i	$0.509983\pi$
$348$		0			0
$349$	1.54696			0.0828069			0.0414035	−	0.999143i	$-0.486817\pi$
$349$	1.54696			0.0828069			0.0414035	+	0.999143i	$0.486817\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.4669	+	13.4669i	0.716770	+	0.716770i	0.967942	−	0.251172i	$-0.0808161\pi$
$353$	13.4669	+	13.4669i	0.716770	+	0.716770i	−0.251172	+	0.967942i	$0.580816\pi$
$354$		0			0
$355$	−15.4785	−	1.49810i	−0.821511	−	0.0795107i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	32.2230			1.70067			0.850334	−	0.526244i	$-0.176400\pi$
$359$	32.2230			1.70067			0.850334	+	0.526244i	$0.176400\pi$
$360$		0			0
$361$	−12.1377			−0.638826
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.7923	−	14.3196i	−0.617237	−	0.749520i
$366$		0			0
$367$	7.07186	+	7.07186i	0.369148	+	0.369148i	0.867167	−	0.498018i	$-0.165939\pi$
$367$	7.07186	+	7.07186i	0.369148	+	0.369148i	−0.498018	+	0.867167i	$0.665939\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−19.6282			−1.01904
$372$		0			0
$373$	5.32291	+	5.32291i	0.275610	+	0.275610i	0.831354	−	0.555744i	$-0.187566\pi$
$373$	5.32291	+	5.32291i	0.275610	+	0.275610i	−0.555744	+	0.831354i	$0.687566\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.80839	−	3.80839i	0.196142	−	0.196142i
$378$		0			0
$379$	−19.7611			−1.01506			−0.507529	−	0.861635i	$-0.669441\pi$
$379$	−19.7611			−1.01506			−0.507529	+	0.861635i	$0.669441\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.2191	−	13.2191i	−0.675466	−	0.675466i	0.283505	−	0.958971i	$-0.408503\pi$
$383$	−13.2191	−	13.2191i	−0.675466	−	0.675466i	−0.958971	+	0.283505i	$0.908503\pi$
$384$		0			0
$385$	−31.5898	+	26.0145i	−1.60996	+	1.32582i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	27.1107i		−	1.37457i	−0.726389	−	0.687284i	$-0.758803\pi$
$389$		−	27.1107i		−	1.37457i	0.726389	−	0.687284i	$-0.241197\pi$
$390$		0			0
$391$	49.4963			2.50314
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.61322	+	0.543281i	0.282432	+	0.0273354i
$396$		0			0
$397$	−8.68567	+	8.68567i	−0.435921	+	0.435921i	−0.890637	−	0.454715i	$-0.849741\pi$
$397$	−8.68567	+	8.68567i	−0.435921	+	0.435921i	0.454715	+	0.890637i	$0.349741\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	3.09967i		−	0.154790i	−0.997000	−	0.0773951i	$-0.975340\pi$
$401$		−	3.09967i		−	0.154790i	0.997000	−	0.0773951i	$-0.0246603\pi$
$402$		0			0
$403$	8.10407	+	8.10407i	0.403692	+	0.403692i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	28.0410	−	28.0410i	1.38994	−	1.38994i
$408$		0			0
$409$		−	14.1083i		−	0.697610i	−0.937195	−	0.348805i	$-0.886588\pi$
$409$		−	14.1083i		−	0.697610i	0.937195	−	0.348805i	$-0.113412\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.7739	+	18.7739i	−0.923802	+	0.923802i
$414$		0			0
$415$	1.39571	+	1.69484i	0.0685129	+	0.0831963i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	30.4747i		−	1.48878i	−0.667743	−	0.744392i	$-0.732739\pi$
$419$		−	30.4747i		−	1.48878i	0.667743	−	0.744392i	$-0.267261\pi$
$420$		0			0
$421$			38.0605i			1.85496i	0.373878	+	0.927478i	$0.378028\pi$
$421$			38.0605i			1.85496i	−0.373878	+	0.927478i	$0.621972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	32.6273	−	21.9607i	1.58266	−	1.06525i
$426$		0			0
$427$	−7.28181	+	7.28181i	−0.352392	+	0.352392i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	30.0050i		−	1.44529i	−0.691220	−	0.722644i	$-0.742927\pi$
$431$		−	30.0050i		−	1.44529i	0.691220	−	0.722644i	$-0.257073\pi$
$432$		0			0
$433$	20.6496	−	20.6496i	0.992355	−	0.992355i	−0.00761624	−	0.999971i	$-0.502424\pi$
$433$	20.6496	−	20.6496i	0.992355	−	0.992355i	0.999971	+	0.00761624i	$0.00242435\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.6559	−	11.6559i	−0.557576	−	0.557576i
$438$		0			0
$439$		−	0.227040i		−	0.0108360i	−0.999985	−	0.00541802i	$-0.998275\pi$
$439$		−	0.227040i		−	0.0108360i	0.999985	−	0.00541802i	$-0.00172462\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.3406	−	16.3406i	0.776364	−	0.776364i	−0.202846	−	0.979211i	$-0.565019\pi$
$443$	16.3406	−	16.3406i	0.776364	−	0.776364i	0.979211	+	0.202846i	$0.0650192\pi$
$444$		0			0
$445$	2.61673	−	27.0363i	0.124045	−	1.28164i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.4698			−1.01322			−0.506611	−	0.862175i	$-0.669102\pi$
$449$	−21.4698			−1.01322			−0.506611	+	0.862175i	$0.669102\pi$
$450$		0			0
$451$		−	9.65925i		−	0.454836i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−18.5139	−	1.79188i	−0.867943	−	0.0840047i
$456$		0			0
$457$	−19.4509	−	19.4509i	−0.909876	−	0.909876i	0.0863855	−	0.996262i	$-0.472468\pi$
$457$	−19.4509	−	19.4509i	−0.909876	−	0.909876i	−0.996262	+	0.0863855i	$0.972468\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.2557			0.896828			0.448414	−	0.893826i	$-0.351989\pi$
$461$	19.2557			0.896828			0.448414	+	0.893826i	$0.351989\pi$
$462$		0			0
$463$	−10.6135	+	10.6135i	−0.493250	+	0.493250i	−0.909329	−	0.416079i	$-0.863404\pi$
$463$	−10.6135	+	10.6135i	−0.493250	+	0.493250i	0.416079	+	0.909329i	$0.363404\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.94796	−	1.94796i	−0.0901407	−	0.0901407i	0.660599	−	0.750739i	$-0.270303\pi$
$467$	−1.94796	−	1.94796i	−0.0901407	−	0.0901407i	−0.750739	+	0.660599i	$0.770303\pi$
$468$		0			0
$469$	8.67152			0.400414
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.68004	−	6.68004i	−0.307149	−	0.307149i
$474$		0			0
$475$	−12.8549	−	2.51188i	−0.589823	−	0.115253i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.9363			−0.910911			−0.455456	−	0.890258i	$-0.650524\pi$
$479$	−19.9363			−0.910911			−0.455456	+	0.890258i	$0.650524\pi$
$480$		0			0
$481$	18.0246			0.821850
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.99687	+	8.23252i	−0.453935	+	0.373820i
$486$		0			0
$487$	−18.5407	−	18.5407i	−0.840161	−	0.840161i	0.148719	−	0.988880i	$-0.452485\pi$
$487$	−18.5407	−	18.5407i	−0.840161	−	0.840161i	−0.988880	+	0.148719i	$0.952485\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.08218			−0.0488379			−0.0244189	−	0.999702i	$-0.507774\pi$
$491$	−1.08218			−0.0488379			−0.0244189	+	0.999702i	$0.507774\pi$
$492$		0			0
$493$	11.8188	+	11.8188i	0.532294	+	0.532294i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.1389	+	16.1389i	−0.723927	+	0.723927i
$498$		0			0
$499$	16.6371			0.744779			0.372390	−	0.928076i	$-0.378539\pi$
$499$	16.6371			0.744779			0.372390	+	0.928076i	$0.378539\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	23.0316	+	23.0316i	1.02693	+	1.02693i	0.999627	+	0.0273024i	$0.00869169\pi$
$503$	23.0316	+	23.0316i	1.02693	+	1.02693i	0.0273024	+	0.999627i	$0.491308\pi$
$504$		0			0
$505$	0.133112	−	1.37532i	0.00592341	−	0.0612011i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			25.0494i			1.11030i	0.831751	+	0.555148i	$0.187338\pi$
$509$			25.0494i			1.11030i	−0.831751	+	0.555148i	$0.812662\pi$
$510$		0			0
$511$	−27.2260			−1.20441
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.22055	+	6.33939i	0.230045	+	0.279347i
$516$		0			0
$517$	−4.73821	+	4.73821i	−0.208386	+	0.208386i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12.1278i		−	0.531327i	−0.964066	−	0.265663i	$-0.914409\pi$
$521$		−	12.1278i		−	0.531327i	0.964066	−	0.265663i	$-0.0855909\pi$
$522$		0			0
$523$	15.8272	+	15.8272i	0.692077	+	0.692077i	0.962689	−	0.270611i	$-0.0872259\pi$
$523$	15.8272	+	15.8272i	0.692077	+	0.692077i	−0.270611	+	0.962689i	$0.587226\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−25.1499	+	25.1499i	−1.09555	+	1.09555i
$528$		0			0
$529$			16.5957i			0.721554i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.10446	−	3.10446i	0.134469	−	0.134469i
$534$		0			0
$535$	4.25496	−	3.50400i	0.183958	−	0.151491i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			21.0268i			0.905690i
$540$		0			0
$541$		−	9.09722i		−	0.391120i	−0.980692	−	0.195560i	$-0.937348\pi$
$541$		−	9.09722i		−	0.391120i	0.980692	−	0.195560i	$-0.0626524\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.26323	+	33.7160i	−0.139782	+	1.44423i
$546$		0			0
$547$	28.5935	−	28.5935i	1.22257	−	1.22257i	0.255856	−	0.966715i	$-0.417643\pi$
$547$	28.5935	−	28.5935i	1.22257	−	1.22257i	0.966715	−	0.255856i	$-0.0823573\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	5.56643i		−	0.237138i
$552$		0			0
$553$	5.85272	−	5.85272i	0.248883	−	0.248883i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−27.5733	−	27.5733i	−1.16832	−	1.16832i	−0.982604	−	0.185716i	$-0.940540\pi$
$557$	−27.5733	−	27.5733i	−1.16832	−	1.16832i	−0.185716	−	0.982604i	$-0.559460\pi$
$558$		0			0
$559$		−	4.29389i		−	0.181612i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.6071	−	10.6071i	0.447036	−	0.447036i	−0.447332	−	0.894368i	$-0.647626\pi$
$563$	10.6071	−	10.6071i	0.447036	−	0.447036i	0.894368	+	0.447332i	$0.147626\pi$
$564$		0			0
$565$	−5.89743	+	4.85659i	−0.248107	+	0.204318i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.8127			−0.579060			−0.289530	−	0.957169i	$-0.593499\pi$
$569$	−13.8127			−0.579060			−0.289530	+	0.957169i	$0.593499\pi$
$570$		0			0
$571$			27.2118i			1.13878i	0.822068	+	0.569390i	$0.192821\pi$
$571$			27.2118i			1.13878i	−0.822068	+	0.569390i	$0.807179\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	17.5680	+	26.1010i	0.732634	+	1.08849i
$576$		0			0
$577$	23.7851	+	23.7851i	0.990188	+	0.990188i	0.999952	−	0.00976420i	$-0.00310809\pi$
$577$	23.7851	+	23.7851i	0.990188	+	0.990188i	−0.00976420	+	0.999952i	$0.503108\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.22241			0.133688
$582$		0			0
$583$	−23.5832	+	23.5832i	−0.976715	+	0.976715i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.8981	+	20.8981i	0.862557	+	0.862557i	0.991635	−	0.129077i	$-0.0412016\pi$
$587$	20.8981	+	20.8981i	0.862557	+	0.862557i	−0.129077	+	0.991635i	$0.541202\pi$
$588$		0			0
$589$	11.8451			0.488068
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.5965	+	15.5965i	0.640471	+	0.640471i	0.950671	−	0.310200i	$-0.100396\pi$
$593$	15.5965	+	15.5965i	0.640471	+	0.640471i	−0.310200	+	0.950671i	$0.600396\pi$
$594$		0			0
$595$	5.56086	−	57.4553i	0.227973	−	2.35544i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.7577			−0.725558			−0.362779	−	0.931875i	$-0.618172\pi$
$599$	−17.7577			−0.725558			−0.362779	+	0.931875i	$0.618172\pi$
$600$		0			0
$601$	10.1083			0.412325			0.206163	−	0.978518i	$-0.433902\pi$
$601$	10.1083			0.412325			0.206163	+	0.978518i	$0.433902\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.32914	+	44.7290i	−0.176005	+	1.81849i
$606$		0			0
$607$	15.9642	+	15.9642i	0.647966	+	0.647966i	0.952501	−	0.304535i	$-0.0985011\pi$
$607$	15.9642	+	15.9642i	0.647966	+	0.647966i	−0.304535	+	0.952501i	$0.598501\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.04570			−0.123216
$612$		0			0
$613$	−16.9357	−	16.9357i	−0.684026	−	0.684026i	0.276878	−	0.960905i	$-0.410700\pi$
$613$	−16.9357	−	16.9357i	−0.684026	−	0.684026i	−0.960905	+	0.276878i	$0.910700\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.84335	−	8.84335i	0.356020	−	0.356020i	−0.506324	−	0.862344i	$-0.668996\pi$
$617$	8.84335	−	8.84335i	0.356020	−	0.356020i	0.862344	+	0.506324i	$0.168996\pi$
$618$		0			0
$619$	7.27180			0.292278			0.146139	−	0.989264i	$-0.453315\pi$
$619$	7.27180			0.292278			0.146139	+	0.989264i	$0.453315\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.1898	−	28.1898i	−1.12940	−	1.12940i
$624$		0			0
$625$	23.1611	+	9.41079i	0.926444	+	0.376432i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			55.9369i			2.23035i
$630$		0			0
$631$	13.2389			0.527031			0.263515	−	0.964655i	$-0.415118\pi$
$631$	13.2389			0.527031			0.263515	+	0.964655i	$0.415118\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.42494	+	2.82047i	−0.135914	+	0.111927i
$636$		0			0
$637$	−6.75796	+	6.75796i	−0.267760	+	0.267760i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	16.7648i		−	0.662171i	−0.943601	−	0.331086i	$-0.892585\pi$
$641$		−	16.7648i		−	0.662171i	0.943601	−	0.331086i	$-0.107415\pi$
$642$		0			0
$643$	−35.3834	−	35.3834i	−1.39538	−	1.39538i	−0.812695	−	0.582690i	$-0.802000\pi$
$643$	−35.3834	−	35.3834i	−1.39538	−	1.39538i	−0.582690	−	0.812695i	$-0.698000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−23.0431	+	23.0431i	−0.905919	+	0.905919i	−0.995940	−	0.0900206i	$-0.971307\pi$
$647$	−23.0431	+	23.0431i	−0.905919	+	0.905919i	0.0900206	+	0.995940i	$0.471307\pi$
$648$		0			0
$649$			45.1135i			1.77086i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	31.6390	−	31.6390i	1.23813	−	1.23813i	0.277365	−	0.960765i	$-0.410539\pi$
$653$	31.6390	−	31.6390i	1.23813	−	1.23813i	0.960765	−	0.277365i	$-0.0894612\pi$
$654$		0			0
$655$	−1.53844	+	15.8953i	−0.0601120	+	0.621082i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			20.2093i			0.787242i	0.919273	+	0.393621i	$0.128778\pi$
$659$			20.2093i			0.787242i	−0.919273	+	0.393621i	$0.871222\pi$
$660$		0			0
$661$		−	44.1882i		−	1.71872i	−0.511369	−	0.859361i	$-0.670861\pi$
$661$		−	44.1882i		−	1.71872i	0.511369	−	0.859361i	$-0.329139\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−14.8396	+	12.2206i	−0.575456	+	0.473894i
$666$		0			0
$667$	−9.45476	+	9.45476i	−0.366090	+	0.366090i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			17.4982i			0.675509i
$672$		0			0
$673$	−23.8238	+	23.8238i	−0.918340	+	0.918340i	−0.996909	−	0.0785687i	$-0.974965\pi$
$673$	−23.8238	+	23.8238i	−0.918340	+	0.918340i	0.0785687	+	0.996909i	$0.474965\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.72866	+	5.72866i	0.220170	+	0.220170i	0.808570	−	0.588400i	$-0.200242\pi$
$677$	5.72866	+	5.72866i	0.220170	+	0.220170i	−0.588400	+	0.808570i	$0.700242\pi$
$678$		0			0
$679$			19.0072i			0.729429i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−33.0998	+	33.0998i	−1.26653	+	1.26653i	−0.318659	+	0.947869i	$0.603232\pi$
$683$	−33.0998	+	33.0998i	−1.26653	+	1.26653i	−0.947869	+	0.318659i	$0.896768\pi$
$684$		0			0
$685$	13.2384	+	16.0756i	0.505815	+	0.614219i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.1591			−0.577517
$690$		0			0
$691$			0.570988i			0.0217214i	0.999941	+	0.0108607i	$0.00345714\pi$
$691$			0.570988i			0.0217214i	−0.999941	+	0.0108607i	$0.996543\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.26498	+	33.7340i	−0.123848	+	1.27960i
$696$		0			0
$697$	9.63426	+	9.63426i	0.364924	+	0.364924i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	25.9738			0.981018			0.490509	−	0.871436i	$-0.336811\pi$
$701$	25.9738			0.981018			0.490509	+	0.871436i	$0.336811\pi$
$702$		0			0
$703$	13.1725	−	13.1725i	0.496812	−	0.496812i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.43400	−	1.43400i	−0.0539313	−	0.0539313i
$708$		0			0
$709$	35.8807			1.34753			0.673764	−	0.738947i	$-0.264677\pi$
$709$	35.8807			1.34753			0.673764	+	0.738947i	$0.264677\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−20.1192	−	20.1192i	−0.753472	−	0.753472i
$714$		0			0
$715$	−24.3973	+	20.0914i	−0.912407	+	0.751376i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.2588			−0.904699			−0.452350	−	0.891841i	$-0.649414\pi$
$719$	−24.2588			−0.904699			−0.452350	+	0.891841i	$0.649414\pi$
$720$		0			0
$721$	12.0532			0.448883
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.03753	+	10.4274i	−0.0756720	+	0.387263i
$726$		0			0
$727$	−23.7764	−	23.7764i	−0.881819	−	0.881819i	0.111900	−	0.993719i	$-0.464306\pi$
$727$	−23.7764	−	23.7764i	−0.881819	−	0.881819i	−0.993719	+	0.111900i	$0.964306\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13.3255			0.492862
$732$		0			0
$733$	1.58667	+	1.58667i	0.0586051	+	0.0586051i	0.735802	−	0.677197i	$-0.236805\pi$
$733$	1.58667	+	1.58667i	0.0586051	+	0.0586051i	−0.677197	+	0.735802i	$0.736805\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.4188	−	10.4188i	0.383782	−	0.383782i
$738$		0			0
$739$	25.6836			0.944786			0.472393	−	0.881388i	$-0.343390\pi$
$739$	25.6836			0.944786			0.472393	+	0.881388i	$0.343390\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.375129	−	0.375129i	−0.0137622	−	0.0137622i	0.700192	−	0.713954i	$-0.253098\pi$
$743$	−0.375129	−	0.375129i	−0.0137622	−	0.0137622i	−0.713954	+	0.700192i	$0.753098\pi$
$744$		0			0
$745$	12.2718	+	1.18774i	0.449604	+	0.0435153i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	8.09000i		−	0.295602i
$750$		0			0
$751$	−50.4813			−1.84209			−0.921045	−	0.389457i	$-0.872663\pi$
$751$	−50.4813			−1.84209			−0.921045	+	0.389457i	$0.872663\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.16749	−	32.7268i	0.115277	−	1.19105i
$756$		0			0
$757$	−7.30248	+	7.30248i	−0.265413	+	0.265413i	−0.827249	−	0.561836i	$-0.810095\pi$
$757$	−7.30248	+	7.30248i	−0.265413	+	0.265413i	0.561836	+	0.827249i	$0.310095\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			13.1137i			0.475371i	0.971342	+	0.237685i	$0.0763887\pi$
$761$			13.1137i			0.475371i	−0.971342	+	0.237685i	$0.923611\pi$
$762$		0			0
$763$	35.1545	+	35.1545i	1.27268	+	1.27268i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.4994	+	14.4994i	−0.523541	+	0.523541i
$768$		0			0
$769$		−	16.1276i		−	0.581577i	−0.956787	−	0.290788i	$-0.906082\pi$
$769$		−	16.1276i		−	0.581577i	0.956787	−	0.290788i	$-0.0939175\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.60986	−	7.60986i	0.273708	−	0.273708i	−0.556883	−	0.830591i	$-0.688003\pi$
$773$	7.60986	−	7.60986i	0.273708	−	0.273708i	0.830591	+	0.556883i	$0.188003\pi$
$774$		0			0
$775$	−22.1889	−	4.33576i	−0.797049	−	0.155745i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	4.53753i		−	0.162574i
$780$		0			0
$781$			38.7816i			1.38771i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.8233	−	27.7147i	−0.814598	−	0.989179i
$786$		0			0
$787$	24.1198	−	24.1198i	0.859779	−	0.859779i	−0.131533	−	0.991312i	$-0.541990\pi$
$787$	24.1198	−	24.1198i	0.859779	−	0.859779i	0.991312	+	0.131533i	$0.0419899\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			11.2129i			0.398683i
$792$		0			0
$793$	−5.62386	+	5.62386i	−0.199709	+	0.199709i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.34128	+	9.34128i	0.330885	+	0.330885i	0.852923	−	0.522037i	$-0.174828\pi$
$797$	9.34128	+	9.34128i	0.330885	+	0.330885i	−0.522037	+	0.852923i	$0.674828\pi$
$798$		0			0
$799$		−	9.45191i		−	0.334385i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−32.7119	+	32.7119i	−1.15438	+	1.15438i
$804$		0			0
$805$	45.9627	+	4.44854i	1.61997	+	0.156791i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.47429			0.297940			0.148970	−	0.988842i	$-0.452404\pi$
$809$	8.47429			0.297940			0.148970	+	0.988842i	$0.452404\pi$
$810$		0			0
$811$		−	2.25683i		−	0.0792481i	−0.999215	−	0.0396240i	$-0.987384\pi$
$811$		−	2.25683i		−	0.0792481i	0.999215	−	0.0396240i	$-0.0126160\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	28.6204	−	23.5692i	1.00253	−	0.825592i
$816$		0			0
$817$	−3.13802	−	3.13802i	−0.109785	−	0.109785i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25.8885			−0.903515			−0.451757	−	0.892141i	$-0.649203\pi$
$821$	−25.8885			−0.903515			−0.451757	+	0.892141i	$0.649203\pi$
$822$		0			0
$823$	32.4100	−	32.4100i	1.12974	−	1.12974i	0.139522	−	0.990219i	$-0.455443\pi$
$823$	32.4100	−	32.4100i	1.12974	−	1.12974i	0.990219	−	0.139522i	$-0.0445567\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.2295	+	18.2295i	0.633901	+	0.633901i	0.949044	−	0.315143i	$-0.102053\pi$
$827$	18.2295	+	18.2295i	0.633901	+	0.633901i	−0.315143	+	0.949044i	$0.602053\pi$
$828$		0			0
$829$	−14.3355			−0.497894			−0.248947	−	0.968517i	$-0.580085\pi$
$829$	−14.3355			−0.497894			−0.248947	+	0.968517i	$0.580085\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20.9724	−	20.9724i	−0.726652	−	0.726652i
$834$		0			0
$835$	15.9211	+	19.3333i	0.550974	+	0.669056i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−29.2314			−1.00918			−0.504590	−	0.863359i	$-0.668356\pi$
$839$	−29.2314			−1.00918			−0.504590	+	0.863359i	$0.668356\pi$
$840$		0			0
$841$	24.4847			0.844301
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.6351	+	1.41648i	0.503464	+	0.0487283i
$846$		0			0
$847$	46.6375	+	46.6375i	1.60248	+	1.60248i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−44.7480			−1.53394
$852$		0			0
$853$	−28.2054	−	28.2054i	−0.965735	−	0.965735i	0.0336967	−	0.999432i	$-0.489272\pi$
$853$	−28.2054	−	28.2054i	−0.965735	−	0.965735i	−0.999432	+	0.0336967i	$0.989272\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.8982	+	12.8982i	−0.440593	+	0.440593i	−0.892211	−	0.451619i	$-0.850847\pi$
$857$	−12.8982	+	12.8982i	−0.440593	+	0.440593i	0.451619	+	0.892211i	$0.350847\pi$
$858$		0			0
$859$	−14.2270			−0.485421			−0.242710	−	0.970099i	$-0.578036\pi$
$859$	−14.2270			−0.485421			−0.242710	+	0.970099i	$0.578036\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.78453	−	3.78453i	−0.128827	−	0.128827i	0.639753	−	0.768580i	$-0.279036\pi$
$863$	−3.78453	−	3.78453i	−0.128827	−	0.128827i	−0.768580	+	0.639753i	$0.779036\pi$
$864$		0			0
$865$	−14.5260	−	17.6392i	−0.493900	−	0.599750i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	14.0641i		−	0.477090i
$870$		0			0
$871$	6.69716			0.226924
$872$		0			0
$873$		0			0
$874$		0			0
$875$	32.2717	−	17.4604i	1.09098	−	0.590271i
$876$		0			0
$877$	18.4253	−	18.4253i	0.622177	−	0.622177i	−0.323910	−	0.946088i	$-0.604998\pi$
$877$	18.4253	−	18.4253i	0.622177	−	0.622177i	0.946088	+	0.323910i	$0.104998\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			2.06935i			0.0697181i	0.999392	+	0.0348591i	$0.0110982\pi$
$881$			2.06935i			0.0697181i	−0.999392	+	0.0348591i	$0.988902\pi$
$882$		0			0
$883$	37.3458	+	37.3458i	1.25679	+	1.25679i	0.952618	+	0.304170i	$0.0983790\pi$
$883$	37.3458	+	37.3458i	1.25679	+	1.25679i	0.304170	+	0.952618i	$0.401621\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.7113	−	20.7113i	0.695418	−	0.695418i	−0.268001	−	0.963419i	$-0.586363\pi$
$887$	20.7113	−	20.7113i	0.695418	−	0.695418i	0.963419	+	0.268001i	$0.0863629\pi$
$888$		0			0
$889$			6.51188i			0.218401i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.22583	+	2.22583i	−0.0744844	+	0.0744844i
$894$		0			0
$895$	30.7014	+	2.97147i	1.02623	+	0.0993251i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	9.60824i		−	0.320453i
$900$		0			0
$901$		−	47.0443i		−	1.56727i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−40.7988	−	3.94875i	−1.35620	−	0.131261i
$906$		0			0
$907$	10.5449	−	10.5449i	0.350139	−	0.350139i	−0.510022	−	0.860161i	$-0.670363\pi$
$907$	10.5449	−	10.5449i	0.350139	−	0.350139i	0.860161	+	0.510022i	$0.170363\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			26.8997i			0.891226i	0.895226	+	0.445613i	$0.147014\pi$
$911$			26.8997i			0.891226i	−0.895226	+	0.445613i	$0.852986\pi$
$912$		0			0
$913$	3.87172	−	3.87172i	0.128135	−	0.128135i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.5735	+	16.5735i	0.547306	+	0.547306i
$918$		0			0
$919$			30.3696i			1.00180i	0.865505	+	0.500900i	$0.166998\pi$
$919$			30.3696i			1.00180i	−0.865505	+	0.500900i	$0.833002\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12.4643	+	12.4643i	−0.410267	+	0.410267i
$924$		0			0
$925$	−29.4973	+	19.8539i	−0.969864	+	0.652793i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	31.6162			1.03729			0.518646	−	0.854989i	$-0.326436\pi$
$929$	31.6162			1.03729			0.518646	+	0.854989i	$0.326436\pi$
$930$		0			0
$931$			9.87758i			0.323724i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−62.3510	−	75.7137i	−2.03910	−	2.47610i
$936$		0			0
$937$	−3.51188	−	3.51188i	−0.114728	−	0.114728i	0.647412	−	0.762140i	$-0.275851\pi$
$937$	−3.51188	−	3.51188i	−0.114728	−	0.114728i	−0.762140	+	0.647412i	$0.775851\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	24.7342			0.806312			0.403156	−	0.915131i	$-0.367913\pi$
$941$	24.7342			0.806312			0.403156	+	0.915131i	$0.367913\pi$
$942$		0			0
$943$	−7.70715	+	7.70715i	−0.250979	+	0.250979i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.0920	+	23.0920i	0.750389	+	0.750389i	0.974552	−	0.224163i	$-0.0719648\pi$
$947$	23.0920	+	23.0920i	0.750389	+	0.750389i	−0.224163	+	0.974552i	$0.571965\pi$
$948$		0			0
$949$	−21.0270			−0.682566
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32.1149	+	32.1149i	1.04030	+	1.04030i	0.999153	+	0.0411513i	$0.0131026\pi$
$953$	32.1149	+	32.1149i	1.04030	+	1.04030i	0.0411513	+	0.999153i	$0.486897\pi$
$954$		0			0
$955$	27.1990	+	2.63248i	0.880138	+	0.0851850i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	30.5648			0.986990
$960$		0			0
$961$	−10.5542			−0.340457
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.37485	−	4.09813i	−0.108640	−	0.131924i
$966$		0			0
$967$	29.3972	+	29.3972i	0.945349	+	0.945349i	0.998582	−	0.0532327i	$-0.0169525\pi$
$967$	29.3972	+	29.3972i	0.945349	+	0.945349i	−0.0532327	+	0.998582i	$0.516953\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	40.7956			1.30919			0.654596	−	0.755979i	$-0.272839\pi$
$971$	40.7956			1.30919			0.654596	+	0.755979i	$0.272839\pi$
$972$		0			0
$973$	35.1733	+	35.1733i	1.12760	+	1.12760i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.32380	+	8.32380i	−0.266302	+	0.266302i	−0.827608	−	0.561306i	$-0.810299\pi$
$977$	−8.32380	+	8.32380i	−0.266302	+	0.266302i	0.561306	+	0.827608i	$0.310299\pi$
$978$		0			0
$979$	−67.7400			−2.16498
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.0382	+	16.0382i	0.511539	+	0.511539i	0.914998	−	0.403458i	$-0.132192\pi$
$983$	16.0382	+	16.0382i	0.511539	+	0.511539i	−0.403458	+	0.914998i	$0.632192\pi$
$984$		0			0
$985$	−10.8317	+	8.92005i	−0.345128	+	0.284216i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			10.6601i			0.338970i
$990$		0			0
$991$	−23.7160			−0.753362			−0.376681	−	0.926343i	$-0.622935\pi$
$991$	−23.7160			−0.753362			−0.376681	+	0.926343i	$0.622935\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	36.0708	+	3.49114i	1.14352	+	0.110677i
$996$		0			0
$997$	−22.4793	+	22.4793i	−0.711926	+	0.711926i	−0.966938	−	0.255012i	$-0.917921\pi$
$997$	−22.4793	+	22.4793i	−0.711926	+	0.711926i	0.255012	+	0.966938i	$0.417921\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bj.a.17.12		48
3.2	odd	2	inner	1440.2.bj.a.17.14		48
4.3	odd	2		360.2.x.a.197.12	yes	48
5.3	odd	4	inner	1440.2.bj.a.593.11		48
8.3	odd	2		360.2.x.a.197.24	yes	48
8.5	even	2	inner	1440.2.bj.a.17.13		48
12.11	even	2		360.2.x.a.197.13	yes	48
15.8	even	4	inner	1440.2.bj.a.593.13		48
20.3	even	4		360.2.x.a.53.1	✓	48
24.5	odd	2	inner	1440.2.bj.a.17.11		48
24.11	even	2		360.2.x.a.197.1	yes	48
40.3	even	4		360.2.x.a.53.13	yes	48
40.13	odd	4	inner	1440.2.bj.a.593.14		48
60.23	odd	4		360.2.x.a.53.24	yes	48
120.53	even	4	inner	1440.2.bj.a.593.12		48
120.83	odd	4		360.2.x.a.53.12	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.x.a.53.1	✓	48	20.3	even	4
360.2.x.a.53.12	yes	48	120.83	odd	4
360.2.x.a.53.13	yes	48	40.3	even	4
360.2.x.a.53.24	yes	48	60.23	odd	4
360.2.x.a.197.1	yes	48	24.11	even	2
360.2.x.a.197.12	yes	48	4.3	odd	2
360.2.x.a.197.13	yes	48	12.11	even	2
360.2.x.a.197.24	yes	48	8.3	odd	2
1440.2.bj.a.17.11		48	24.5	odd	2	inner
1440.2.bj.a.17.12		48	1.1	even	1	trivial
1440.2.bj.a.17.13		48	8.5	even	2	inner
1440.2.bj.a.17.14		48	3.2	odd	2	inner
1440.2.bj.a.593.11		48	5.3	odd	4	inner
1440.2.bj.a.593.12		48	120.53	even	4	inner
1440.2.bj.a.593.13		48	15.8	even	4	inner
1440.2.bj.a.593.14		48	40.13	odd	4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bj (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.215413 + 2.22567i) q^{5} +(2.32063 + 2.32063i) q^{7} +O(q^{10})\)	\(q+(-0.215413 + 2.22567i) q^{5} +(2.32063 + 2.32063i) q^{7} +5.57646 q^{11} +(1.79226 + 1.79226i) q^{13} +(-5.56203 + 5.56203i) q^{17} +2.61960 q^{19} +(-4.44948 - 4.44948i) q^{23} +(-4.90719 - 0.958877i) q^{25} -2.12491i q^{29} +4.52171 q^{31} +(-5.66484 + 4.66506i) q^{35} +(5.02845 - 5.02845i) q^{37} -1.73215i q^{41} +(-1.19790 - 1.19790i) q^{43} +(-0.849681 + 0.849681i) q^{47} +3.77064i q^{49} +(-4.22906 + 4.22906i) q^{53} +(-1.20124 + 12.4113i) q^{55} +8.08999i q^{59} +3.13786i q^{61} +(-4.37505 + 3.60290i) q^{65} +(1.86836 - 1.86836i) q^{67} +6.95452i q^{71} +(-5.86607 + 5.86607i) q^{73} +(12.9409 + 12.9409i) q^{77} -2.52204i q^{79} +(0.694298 - 0.694298i) q^{83} +(-11.1811 - 13.5774i) q^{85} -12.1475 q^{89} +8.31834i q^{91} +(-0.564297 + 5.83036i) q^{95} +(4.09526 + 4.09526i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 32 q^{31} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^31 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more