LMFDB - Embedded newform 600.2.r.a.593.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,2,Mod(257,600)] 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("600.257"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.r (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	600.593
Dual form			600.2.r.a.257.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70711 - 0.292893i) q^{3} +(-2.00000 + 2.00000i) q^{7} +(2.82843 + 1.00000i) q^{9} -5.65685i q^{11} +(2.82843 + 2.82843i) q^{17} +4.00000i q^{19} +(4.00000 - 2.82843i) q^{21} +(4.24264 - 4.24264i) q^{23} +(-4.53553 - 2.53553i) q^{27} +5.65685 q^{29} +8.00000 q^{31} +(-1.65685 + 9.65685i) q^{33} +(8.00000 - 8.00000i) q^{37} -5.65685i q^{41} +(-2.00000 - 2.00000i) q^{43} +(-1.41421 - 1.41421i) q^{47} -1.00000i q^{49} +(-4.00000 - 5.65685i) q^{51} +(5.65685 - 5.65685i) q^{53} +(1.17157 - 6.82843i) q^{57} +5.65685 q^{59} -6.00000 q^{61} +(-7.65685 + 3.65685i) q^{63} +(-6.00000 + 6.00000i) q^{67} +(-8.48528 + 6.00000i) q^{69} +11.3137i q^{71} +(8.00000 + 8.00000i) q^{73} +(11.3137 + 11.3137i) q^{77} +(7.00000 + 5.65685i) q^{81} +(9.89949 - 9.89949i) q^{83} +(-9.65685 - 1.65685i) q^{87} -11.3137 q^{89} +(-13.6569 - 2.34315i) q^{93} +(-8.00000 + 8.00000i) q^{97} +(5.65685 - 16.0000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{7} + 16 q^{21} - 4 q^{27} + 32 q^{31} + 16 q^{33} + 32 q^{37} - 8 q^{43} - 16 q^{51} + 16 q^{57} - 24 q^{61} - 8 q^{63} - 24 q^{67} + 32 q^{73} + 28 q^{81} - 16 q^{87} - 32 q^{93} - 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^7 + 16 * q^21 - 4 * q^27 + 32 * q^31 + 16 * q^33 + 32 * q^37 - 8 * q^43 - 16 * q^51 + 16 * q^57 - 24 * q^61 - 8 * q^63 - 24 * q^67 + 32 * q^73 + 28 * q^81 - 16 * q^87 - 32 * q^93 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	−1.70711	−	0.292893i	−0.985599	−	0.169102i
$3$	−1.70711	−	0.292893i	−0.985599	−	0.169102i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.00000	+	2.00000i	−0.755929	+	0.755929i	−0.975579	−	0.219650i	$-0.929509\pi$
$7$	−2.00000	+	2.00000i	−0.755929	+	0.755929i	0.219650	+	0.975579i	$0.429509\pi$
$8$		0			0
$9$	2.82843	+	1.00000i	0.942809	+	0.333333i
$10$		0			0
$11$		−	5.65685i		−	1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$		−	5.65685i		−	1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.82843	+	2.82843i	0.685994	+	0.685994i	0.961344	−	0.275350i	$-0.0887937\pi$
$17$	2.82843	+	2.82843i	0.685994	+	0.685994i	−0.275350	+	0.961344i	$0.588794\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$	4.00000	−	2.82843i	0.872872	−	0.617213i
$22$		0			0
$23$	4.24264	−	4.24264i	0.884652	−	0.884652i	−0.109351	−	0.994003i	$-0.534877\pi$
$23$	4.24264	−	4.24264i	0.884652	−	0.884652i	0.994003	+	0.109351i	$0.0348774\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.53553	−	2.53553i	−0.872864	−	0.487964i
$28$		0			0
$29$	5.65685			1.05045			0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685			1.05045			0.525226	+	0.850963i	$0.323981\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$	−1.65685	+	9.65685i	−0.288421	+	1.68104i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.00000	−	8.00000i	1.31519	−	1.31519i	0.397658	−	0.917534i	$-0.369823\pi$
$37$	8.00000	−	8.00000i	1.31519	−	1.31519i	0.917534	−	0.397658i	$-0.130177\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$	−2.00000	−	2.00000i	−0.304997	−	0.304997i	0.537968	−	0.842965i	$-0.319192\pi$
$43$	−2.00000	−	2.00000i	−0.304997	−	0.304997i	−0.842965	+	0.537968i	$0.819192\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.41421	−	1.41421i	−0.206284	−	0.206284i	0.596402	−	0.802686i	$-0.296597\pi$
$47$	−1.41421	−	1.41421i	−0.206284	−	0.206284i	−0.802686	+	0.596402i	$0.796597\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$	−4.00000	−	5.65685i	−0.560112	−	0.792118i
$52$		0			0
$53$	5.65685	−	5.65685i	0.777029	−	0.777029i	−0.202296	−	0.979324i	$-0.564840\pi$
$53$	5.65685	−	5.65685i	0.777029	−	0.777029i	0.979324	+	0.202296i	$0.0648402\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.17157	−	6.82843i	0.155179	−	0.904447i
$58$		0			0
$59$	5.65685			0.736460			0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685			0.736460			0.368230	+	0.929735i	$0.379964\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$	−7.65685	+	3.65685i	−0.964673	+	0.460720i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.00000	+	6.00000i	−0.733017	+	0.733017i	−0.971216	−	0.238200i	$-0.923443\pi$
$67$	−6.00000	+	6.00000i	−0.733017	+	0.733017i	0.238200	+	0.971216i	$0.423443\pi$
$68$		0			0
$69$	−8.48528	+	6.00000i	−1.02151	+	0.722315i
$70$		0			0
$71$			11.3137i			1.34269i	0.741145	+	0.671345i	$0.234283\pi$
$71$			11.3137i			1.34269i	−0.741145	+	0.671345i	$0.765717\pi$
$72$		0			0
$73$	8.00000	+	8.00000i	0.936329	+	0.936329i	0.998091	−	0.0617617i	$-0.0196719\pi$
$73$	8.00000	+	8.00000i	0.936329	+	0.936329i	−0.0617617	+	0.998091i	$0.519672\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.3137	+	11.3137i	1.28932	+	1.28932i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	7.00000	+	5.65685i	0.777778	+	0.628539i
$82$		0			0
$83$	9.89949	−	9.89949i	1.08661	−	1.08661i	0.0907357	−	0.995875i	$-0.471078\pi$
$83$	9.89949	−	9.89949i	1.08661	−	1.08661i	0.995875	−	0.0907357i	$-0.0289219\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−9.65685	−	1.65685i	−1.03532	−	0.177633i
$88$		0			0
$89$	−11.3137			−1.19925			−0.599625	−	0.800281i	$-0.704684\pi$
$89$	−11.3137			−1.19925			−0.599625	+	0.800281i	$0.704684\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−13.6569	−	2.34315i	−1.41615	−	0.242973i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.00000	+	8.00000i	−0.812277	+	0.812277i	−0.984975	−	0.172698i	$-0.944752\pi$
$97$	−8.00000	+	8.00000i	−0.812277	+	0.812277i	0.172698	+	0.984975i	$0.444752\pi$
$98$		0			0
$99$	5.65685	−	16.0000i	0.568535	−	1.60806i
$100$		0			0
$101$			5.65685i			0.562878i	0.959579	+	0.281439i	$0.0908117\pi$
$101$			5.65685i			0.562878i	−0.959579	+	0.281439i	$0.909188\pi$
$102$		0			0
$103$	−6.00000	−	6.00000i	−0.591198	−	0.591198i	0.346757	−	0.937955i	$-0.387283\pi$
$103$	−6.00000	−	6.00000i	−0.591198	−	0.591198i	−0.937955	+	0.346757i	$0.887283\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.24264	+	4.24264i	0.410152	+	0.410152i	0.881791	−	0.471640i	$-0.156338\pi$
$107$	4.24264	+	4.24264i	0.410152	+	0.410152i	−0.471640	+	0.881791i	$0.656338\pi$
$108$		0			0
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$	−16.0000	+	11.3137i	−1.51865	+	1.07385i
$112$		0			0
$113$	−2.82843	+	2.82843i	−0.266076	+	0.266076i	−0.827517	−	0.561441i	$-0.810247\pi$
$113$	−2.82843	+	2.82843i	−0.266076	+	0.266076i	0.561441	+	0.827517i	$0.310247\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.3137			−1.03713
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$	−1.65685	+	9.65685i	−0.149394	+	0.870729i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	−0.612781	−	0.790253i	$-0.709949\pi$
$127$	2.00000	−	2.00000i	0.177471	−	0.177471i	0.790253	+	0.612781i	$0.209949\pi$
$128$		0			0
$129$	2.82843	+	4.00000i	0.249029	+	0.352180i
$130$		0			0
$131$			5.65685i			0.494242i	0.968985	+	0.247121i	$0.0794845\pi$
$131$			5.65685i			0.494242i	−0.968985	+	0.247121i	$0.920516\pi$
$132$		0			0
$133$	−8.00000	−	8.00000i	−0.693688	−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.48528	−	8.48528i	−0.724947	−	0.724947i	0.244662	−	0.969608i	$-0.421323\pi$
$137$	−8.48528	−	8.48528i	−0.724947	−	0.724947i	−0.969608	+	0.244662i	$0.921323\pi$
$138$		0			0
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$		0			0
$141$	2.00000	+	2.82843i	0.168430	+	0.238197i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.292893	+	1.70711i	−0.0241574	+	0.140800i
$148$		0			0
$149$	−11.3137			−0.926855			−0.463428	−	0.886135i	$-0.653381\pi$
$149$	−11.3137			−0.926855			−0.463428	+	0.886135i	$0.653381\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	5.17157	+	10.8284i	0.418097	+	0.875426i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	8.00000	−	8.00000i	0.638470	−	0.638470i	−0.311708	−	0.950178i	$-0.600901\pi$
$157$	8.00000	−	8.00000i	0.638470	−	0.638470i	0.950178	+	0.311708i	$0.100901\pi$
$158$		0			0
$159$	−11.3137	+	8.00000i	−0.897235	+	0.634441i
$160$		0			0
$161$			16.9706i			1.33747i
$162$		0			0
$163$	2.00000	+	2.00000i	0.156652	+	0.156652i	0.781081	−	0.624429i	$-0.214668\pi$
$163$	2.00000	+	2.00000i	0.156652	+	0.156652i	−0.624429	+	0.781081i	$0.714668\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.07107	+	7.07107i	0.547176	+	0.547176i	0.925623	−	0.378447i	$-0.123542\pi$
$167$	7.07107	+	7.07107i	0.547176	+	0.547176i	−0.378447	+	0.925623i	$0.623542\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−4.00000	+	11.3137i	−0.305888	+	0.865181i
$172$		0			0
$173$	16.9706	−	16.9706i	1.29025	−	1.29025i	0.355616	−	0.934632i	$-0.384271\pi$
$173$	16.9706	−	16.9706i	1.29025	−	1.29025i	0.934632	−	0.355616i	$-0.115729\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.65685	−	1.65685i	−0.725854	−	0.124537i
$178$		0			0
$179$	−5.65685			−0.422813			−0.211407	−	0.977398i	$-0.567804\pi$
$179$	−5.65685			−0.422813			−0.211407	+	0.977398i	$0.567804\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$	10.2426	+	1.75736i	0.757158	+	0.129908i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.0000	−	16.0000i	1.17004	−	1.17004i
$188$		0			0
$189$	14.1421	−	4.00000i	1.02869	−	0.290957i
$190$		0			0
$191$		−	11.3137i		−	0.818631i	−0.912393	−	0.409316i	$-0.865768\pi$
$191$		−	11.3137i		−	0.818631i	0.912393	−	0.409316i	$-0.134232\pi$
$192$		0			0
$193$	8.00000	+	8.00000i	0.575853	+	0.575853i	0.933758	−	0.357905i	$-0.116509\pi$
$193$	8.00000	+	8.00000i	0.575853	+	0.575853i	−0.357905	+	0.933758i	$0.616509\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	12.0000	−	8.48528i	0.846415	−	0.598506i
$202$		0			0
$203$	−11.3137	+	11.3137i	−0.794067	+	0.794067i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	16.2426	−	7.75736i	1.12894	−	0.539174i
$208$		0			0
$209$	22.6274			1.56517
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	3.31371	−	19.3137i	0.227052	−	1.32335i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−16.0000	+	16.0000i	−1.08615	+	1.08615i
$218$		0			0
$219$	−11.3137	−	16.0000i	−0.764510	−	1.08118i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	18.0000	+	18.0000i	1.20537	+	1.20537i	0.972511	+	0.232859i	$0.0748079\pi$
$223$	18.0000	+	18.0000i	1.20537	+	1.20537i	0.232859	+	0.972511i	$0.425192\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.89949	−	9.89949i	−0.657053	−	0.657053i	0.297629	−	0.954682i	$-0.403804\pi$
$227$	−9.89949	−	9.89949i	−0.657053	−	0.657053i	−0.954682	+	0.297629i	$0.903804\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$	−16.0000	−	22.6274i	−1.05272	−	1.48877i
$232$		0			0
$233$	−14.1421	+	14.1421i	−0.926482	+	0.926482i	−0.997477	−	0.0709946i	$-0.977383\pi$
$233$	−14.1421	+	14.1421i	−0.926482	+	0.926482i	0.0709946	+	0.997477i	$0.477383\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.3137			0.731823			0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137			0.731823			0.365911	+	0.930650i	$0.380757\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$	−10.2929	−	11.7071i	−0.660289	−	0.751011i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−19.7990	+	14.0000i	−1.25471	+	0.887214i
$250$		0			0
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		0			0
$253$	−24.0000	−	24.0000i	−1.50887	−	1.50887i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.48528	−	8.48528i	−0.529297	−	0.529297i	0.391066	−	0.920363i	$-0.372107\pi$
$257$	−8.48528	−	8.48528i	−0.529297	−	0.529297i	−0.920363	+	0.391066i	$0.872107\pi$
$258$		0			0
$259$			32.0000i			1.98838i
$260$		0			0
$261$	16.0000	+	5.65685i	0.990375	+	0.350150i
$262$		0			0
$263$	−4.24264	+	4.24264i	−0.261612	+	0.261612i	−0.825709	−	0.564096i	$-0.809225\pi$
$263$	−4.24264	+	4.24264i	−0.261612	+	0.261612i	0.564096	+	0.825709i	$0.309225\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	19.3137	+	3.31371i	1.18198	+	0.202796i
$268$		0			0
$269$	11.3137			0.689809			0.344904	−	0.938638i	$-0.387911\pi$
$269$	11.3137			0.689809			0.344904	+	0.938638i	$0.387911\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−8.00000	+	8.00000i	−0.480673	+	0.480673i	−0.905347	−	0.424673i	$-0.860389\pi$
$277$	−8.00000	+	8.00000i	−0.480673	+	0.480673i	0.424673	+	0.905347i	$0.360389\pi$
$278$		0			0
$279$	22.6274	+	8.00000i	1.35467	+	0.478947i
$280$		0			0
$281$		−	16.9706i		−	1.01238i	−0.862422	−	0.506189i	$-0.831054\pi$
$281$		−	16.9706i		−	1.01238i	0.862422	−	0.506189i	$-0.168946\pi$
$282$		0			0
$283$	6.00000	+	6.00000i	0.356663	+	0.356663i	0.862581	−	0.505918i	$-0.168846\pi$
$283$	6.00000	+	6.00000i	0.356663	+	0.356663i	−0.505918	+	0.862581i	$0.668846\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.3137	+	11.3137i	0.667827	+	0.667827i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$	16.0000	−	11.3137i	0.937937	−	0.663221i
$292$		0			0
$293$	−5.65685	+	5.65685i	−0.330477	+	0.330477i	−0.852768	−	0.522291i	$-0.825078\pi$
$293$	−5.65685	+	5.65685i	−0.330477	+	0.330477i	0.522291	+	0.852768i	$0.325078\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−14.3431	+	25.6569i	−0.832274	+	1.48876i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	8.00000			0.461112
$302$		0			0
$303$	1.65685	−	9.65685i	0.0951838	−	0.554772i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10.0000	+	10.0000i	−0.570730	+	0.570730i	−0.932332	−	0.361602i	$-0.882230\pi$
$307$	−10.0000	+	10.0000i	−0.570730	+	0.570730i	0.361602	+	0.932332i	$0.382230\pi$
$308$		0			0
$309$	8.48528	+	12.0000i	0.482711	+	0.682656i
$310$		0			0
$311$		−	11.3137i		−	0.641542i	−0.947157	−	0.320771i	$-0.896058\pi$
$311$		−	11.3137i		−	0.641542i	0.947157	−	0.320771i	$-0.103942\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−22.6274	−	22.6274i	−1.27088	−	1.27088i	−0.945626	−	0.325257i	$-0.894549\pi$
$317$	−22.6274	−	22.6274i	−1.27088	−	1.27088i	−0.325257	−	0.945626i	$-0.605451\pi$
$318$		0			0
$319$		−	32.0000i		−	1.79166i
$320$		0			0
$321$	−6.00000	−	8.48528i	−0.334887	−	0.473602i
$322$		0			0
$323$	−11.3137	+	11.3137i	−0.629512	+	0.629512i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−2.92893	+	17.0711i	−0.161970	+	0.944032i
$328$		0			0
$329$	5.65685			0.311872
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$	30.6274	−	14.6274i	1.67837	−	0.801578i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$		0			0
$339$	5.65685	−	4.00000i	0.307238	−	0.217250i
$340$		0			0
$341$		−	45.2548i		−	2.45069i
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.5563	+	15.5563i	0.835109	+	0.835109i	0.988210	−	0.153102i	$-0.0489263\pi$
$347$	15.5563	+	15.5563i	0.835109	+	0.835109i	−0.153102	+	0.988210i	$0.548926\pi$
$348$		0			0
$349$			2.00000i			0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$			2.00000i			0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.82843	+	2.82843i	−0.150542	+	0.150542i	−0.778360	−	0.627818i	$-0.783948\pi$
$353$	−2.82843	+	2.82843i	−0.150542	+	0.150542i	0.627818	+	0.778360i	$0.283948\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	19.3137	+	3.31371i	1.02219	+	0.175380i
$358$		0			0
$359$	22.6274			1.19423			0.597115	−	0.802156i	$-0.296314\pi$
$359$	22.6274			1.19423			0.597115	+	0.802156i	$0.296314\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	35.8492	+	6.15076i	1.88160	+	0.322831i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	18.0000	−	18.0000i	0.939592	−	0.939592i	−0.0586842	−	0.998277i	$-0.518691\pi$
$367$	18.0000	−	18.0000i	0.939592	−	0.939592i	0.998277	+	0.0586842i	$0.0186905\pi$
$368$		0			0
$369$	5.65685	−	16.0000i	0.294484	−	0.832927i
$370$		0			0
$371$			22.6274i			1.17476i
$372$		0			0
$373$	−8.00000	−	8.00000i	−0.414224	−	0.414224i	0.468983	−	0.883207i	$-0.344621\pi$
$373$	−8.00000	−	8.00000i	−0.414224	−	0.414224i	−0.883207	+	0.468983i	$0.844621\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			36.0000i			1.84920i	0.380945	+	0.924598i	$0.375599\pi$
$379$			36.0000i			1.84920i	−0.380945	+	0.924598i	$0.624401\pi$
$380$		0			0
$381$	−4.00000	+	2.82843i	−0.204926	+	0.144905i
$382$		0			0
$383$	−21.2132	+	21.2132i	−1.08394	+	1.08394i	−0.0878065	+	0.996138i	$0.527986\pi$
$383$	−21.2132	+	21.2132i	−1.08394	+	1.08394i	−0.996138	+	0.0878065i	$0.972014\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.65685	−	7.65685i	−0.185888	−	0.389220i
$388$		0			0
$389$	−33.9411			−1.72088			−0.860442	−	0.509549i	$-0.829812\pi$
$389$	−33.9411			−1.72088			−0.860442	+	0.509549i	$0.829812\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$	1.65685	−	9.65685i	0.0835772	−	0.487124i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$	11.3137	+	16.0000i	0.566394	+	0.801002i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−45.2548	−	45.2548i	−2.24320	−	2.24320i
$408$		0			0
$409$			30.0000i			1.48340i	0.670729	+	0.741702i	$0.265981\pi$
$409$			30.0000i			1.48340i	−0.670729	+	0.741702i	$0.734019\pi$
$410$		0			0
$411$	12.0000	+	16.9706i	0.591916	+	0.837096i
$412$		0			0
$413$	−11.3137	+	11.3137i	−0.556711	+	0.556711i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.17157	−	6.82843i	0.0573722	−	0.334390i
$418$		0			0
$419$	−16.9706			−0.829066			−0.414533	−	0.910034i	$-0.636055\pi$
$419$	−16.9706			−0.829066			−0.414533	+	0.910034i	$0.636055\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		0			0
$423$	−2.58579	−	5.41421i	−0.125725	−	0.263248i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	12.0000	−	12.0000i	0.580721	−	0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	22.6274i		−	1.08992i	−0.838461	−	0.544962i	$-0.816544\pi$
$431$		−	22.6274i		−	1.08992i	0.838461	−	0.544962i	$-0.183456\pi$
$432$		0			0
$433$	24.0000	+	24.0000i	1.15337	+	1.15337i	0.985873	+	0.167493i	$0.0535672\pi$
$433$	24.0000	+	24.0000i	1.15337	+	1.15337i	0.167493	+	0.985873i	$0.446433\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.9706	+	16.9706i	0.811812	+	0.811812i
$438$		0			0
$439$		−	32.0000i		−	1.52728i	−0.645644	−	0.763638i	$-0.723411\pi$
$439$		−	32.0000i		−	1.52728i	0.645644	−	0.763638i	$-0.276589\pi$
$440$		0			0
$441$	1.00000	−	2.82843i	0.0476190	−	0.134687i
$442$		0			0
$443$	7.07107	−	7.07107i	0.335957	−	0.335957i	−0.518887	−	0.854843i	$-0.673653\pi$
$443$	7.07107	−	7.07107i	0.335957	−	0.335957i	0.854843	+	0.518887i	$0.173653\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	19.3137	+	3.31371i	0.913507	+	0.156733i
$448$		0			0
$449$	−5.65685			−0.266963			−0.133482	−	0.991051i	$-0.542616\pi$
$449$	−5.65685			−0.266963			−0.133482	+	0.991051i	$0.542616\pi$
$450$		0			0
$451$	−32.0000			−1.50682
$452$		0			0
$453$	−13.6569	−	2.34315i	−0.641655	−	0.110091i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	16.0000	−	16.0000i	0.748448	−	0.748448i	−0.225739	−	0.974188i	$-0.572480\pi$
$457$	16.0000	−	16.0000i	0.748448	−	0.748448i	0.974188	+	0.225739i	$0.0724798\pi$
$458$		0			0
$459$	−5.65685	−	20.0000i	−0.264039	−	0.933520i
$460$		0			0
$461$			39.5980i			1.84426i	0.386878	+	0.922131i	$0.373553\pi$
$461$			39.5980i			1.84426i	−0.386878	+	0.922131i	$0.626447\pi$
$462$		0			0
$463$	6.00000	+	6.00000i	0.278844	+	0.278844i	0.832647	−	0.553804i	$-0.186824\pi$
$463$	6.00000	+	6.00000i	0.278844	+	0.278844i	−0.553804	+	0.832647i	$0.686824\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.89949	−	9.89949i	−0.458094	−	0.458094i	0.439935	−	0.898029i	$-0.355001\pi$
$467$	−9.89949	−	9.89949i	−0.458094	−	0.458094i	−0.898029	+	0.439935i	$0.855001\pi$
$468$		0			0
$469$		−	24.0000i		−	1.10822i
$470$		0			0
$471$	−16.0000	+	11.3137i	−0.737241	+	0.521308i
$472$		0			0
$473$	−11.3137	+	11.3137i	−0.520205	+	0.520205i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	21.6569	−	10.3431i	0.991599	−	0.473580i
$478$		0			0
$479$	11.3137			0.516937			0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137			0.516937			0.258468	+	0.966020i	$0.416782\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	4.97056	−	28.9706i	0.226168	−	1.31821i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−18.0000	+	18.0000i	−0.815658	+	0.815658i	−0.985476	−	0.169818i	$-0.945682\pi$
$487$	−18.0000	+	18.0000i	−0.815658	+	0.815658i	0.169818	+	0.985476i	$0.445682\pi$
$488$		0			0
$489$	−2.82843	−	4.00000i	−0.127906	−	0.180886i
$490$		0			0
$491$			5.65685i			0.255290i	0.991820	+	0.127645i	$0.0407419\pi$
$491$			5.65685i			0.255290i	−0.991820	+	0.127645i	$0.959258\pi$
$492$		0			0
$493$	16.0000	+	16.0000i	0.720604	+	0.720604i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−22.6274	−	22.6274i	−1.01498	−	1.01498i
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$	−10.0000	−	14.1421i	−0.446767	−	0.631824i
$502$		0			0
$503$	4.24264	−	4.24264i	0.189170	−	0.189170i	−0.606167	−	0.795337i	$-0.707294\pi$
$503$	4.24264	−	4.24264i	0.189170	−	0.189170i	0.795337	+	0.606167i	$0.207294\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.80761	+	22.1924i	−0.169102	+	0.985599i
$508$		0			0
$509$	16.9706			0.752207			0.376103	−	0.926578i	$-0.377264\pi$
$509$	16.9706			0.752207			0.376103	+	0.926578i	$0.377264\pi$
$510$		0			0
$511$	−32.0000			−1.41560
$512$		0			0
$513$	10.1421	−	18.1421i	0.447786	−	0.800995i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.00000	+	8.00000i	−0.351840	+	0.351840i
$518$		0			0
$519$	−33.9411	+	24.0000i	−1.48985	+	1.05348i
$520$		0			0
$521$			33.9411i			1.48699i	0.668743	+	0.743494i	$0.266833\pi$
$521$			33.9411i			1.48699i	−0.668743	+	0.743494i	$0.733167\pi$
$522$		0			0
$523$	−6.00000	−	6.00000i	−0.262362	−	0.262362i	0.563651	−	0.826013i	$-0.309396\pi$
$523$	−6.00000	−	6.00000i	−0.262362	−	0.262362i	−0.826013	+	0.563651i	$0.809396\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	22.6274	+	22.6274i	0.985666	+	0.985666i
$528$		0			0
$529$		−	13.0000i		−	0.565217i
$530$		0			0
$531$	16.0000	+	5.65685i	0.694341	+	0.245487i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	9.65685	+	1.65685i	0.416724	+	0.0714985i
$538$		0			0
$539$	−5.65685			−0.243658
$540$		0			0
$541$	34.0000			1.46177			0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000			1.46177			0.730887	+	0.682498i	$0.239107\pi$
$542$		0			0
$543$	17.0711	+	2.92893i	0.732590	+	0.125693i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−30.0000	+	30.0000i	−1.28271	+	1.28271i	−0.343586	+	0.939121i	$0.611642\pi$
$547$	−30.0000	+	30.0000i	−1.28271	+	1.28271i	−0.939121	+	0.343586i	$0.888358\pi$
$548$		0			0
$549$	−16.9706	−	6.00000i	−0.724286	−	0.256074i
$550$		0			0
$551$			22.6274i			0.963960i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.9706	+	16.9706i	0.719066	+	0.719066i	0.968414	−	0.249348i	$-0.0802163\pi$
$557$	16.9706	+	16.9706i	0.719066	+	0.719066i	−0.249348	+	0.968414i	$0.580216\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−32.0000	+	22.6274i	−1.35104	+	0.955330i
$562$		0			0
$563$	−9.89949	+	9.89949i	−0.417214	+	0.417214i	−0.884242	−	0.467028i	$-0.845325\pi$
$563$	−9.89949	+	9.89949i	−0.417214	+	0.417214i	0.467028	+	0.884242i	$0.345325\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−25.3137	+	2.68629i	−1.06308	+	0.112814i
$568$		0			0
$569$	−28.2843			−1.18574			−0.592869	−	0.805299i	$-0.702005\pi$
$569$	−28.2843			−1.18574			−0.592869	+	0.805299i	$0.702005\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$	−3.31371	+	19.3137i	−0.138432	+	0.806842i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	16.0000	−	16.0000i	0.666089	−	0.666089i	−0.290720	−	0.956808i	$-0.593895\pi$
$577$	16.0000	−	16.0000i	0.666089	−	0.666089i	0.956808	+	0.290720i	$0.0938947\pi$
$578$		0			0
$579$	−11.3137	−	16.0000i	−0.470182	−	0.664937i
$580$		0			0
$581$			39.5980i			1.64280i
$582$		0			0
$583$	−32.0000	−	32.0000i	−1.32530	−	1.32530i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.5563	−	15.5563i	−0.642079	−	0.642079i	0.308987	−	0.951066i	$-0.400010\pi$
$587$	−15.5563	−	15.5563i	−0.642079	−	0.642079i	−0.951066	+	0.308987i	$0.900010\pi$
$588$		0			0
$589$			32.0000i			1.31854i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.48528	+	8.48528i	−0.348449	+	0.348449i	−0.859532	−	0.511083i	$-0.829245\pi$
$593$	−8.48528	+	8.48528i	−0.348449	+	0.348449i	0.511083	+	0.859532i	$0.329245\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−45.2548			−1.84906			−0.924531	−	0.381106i	$-0.875543\pi$
$599$	−45.2548			−1.84906			−0.924531	+	0.381106i	$0.875543\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$	−22.9706	+	10.9706i	−0.935434	+	0.446756i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−10.0000	+	10.0000i	−0.405887	+	0.405887i	−0.880302	−	0.474414i	$-0.842660\pi$
$607$	−10.0000	+	10.0000i	−0.405887	+	0.405887i	0.474414	+	0.880302i	$0.342660\pi$
$608$		0			0
$609$	22.6274	−	16.0000i	0.916909	−	0.648353i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.4558	+	25.4558i	1.02481	+	1.02481i	0.999684	+	0.0251295i	$0.00799981\pi$
$617$	25.4558	+	25.4558i	1.02481	+	1.02481i	0.0251295	+	0.999684i	$0.492000\pi$
$618$		0			0
$619$			12.0000i			0.482321i	0.970485	+	0.241160i	$0.0775280\pi$
$619$			12.0000i			0.482321i	−0.970485	+	0.241160i	$0.922472\pi$
$620$		0			0
$621$	−30.0000	+	8.48528i	−1.20386	+	0.340503i
$622$		0			0
$623$	22.6274	−	22.6274i	0.906548	−	0.906548i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−38.6274	−	6.62742i	−1.54263	−	0.264674i
$628$		0			0
$629$	45.2548			1.80443
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$	−6.82843	−	1.17157i	−0.271406	−	0.0465658i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−11.3137	+	32.0000i	−0.447563	+	1.26590i
$640$		0			0
$641$		−	28.2843i		−	1.11716i	−0.829450	−	0.558581i	$-0.811346\pi$
$641$		−	28.2843i		−	1.11716i	0.829450	−	0.558581i	$-0.188654\pi$
$642$		0			0
$643$	−22.0000	−	22.0000i	−0.867595	−	0.867595i	0.124610	−	0.992206i	$-0.460232\pi$
$643$	−22.0000	−	22.0000i	−0.867595	−	0.867595i	−0.992206	+	0.124610i	$0.960232\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.24264	−	4.24264i	−0.166795	−	0.166795i	0.618774	−	0.785569i	$-0.287630\pi$
$647$	−4.24264	−	4.24264i	−0.166795	−	0.166795i	−0.785569	+	0.618774i	$0.787630\pi$
$648$		0			0
$649$		−	32.0000i		−	1.25611i
$650$		0			0
$651$	32.0000	−	22.6274i	1.25418	−	0.886838i
$652$		0			0
$653$	−16.9706	+	16.9706i	−0.664109	+	0.664109i	−0.956346	−	0.292237i	$-0.905601\pi$
$653$	−16.9706	+	16.9706i	−0.664109	+	0.664109i	0.292237	+	0.956346i	$0.405601\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	14.6274	+	30.6274i	0.570670	+	1.19489i
$658$		0			0
$659$	16.9706			0.661079			0.330540	−	0.943792i	$-0.392769\pi$
$659$	16.9706			0.661079			0.330540	+	0.943792i	$0.392769\pi$
$660$		0			0
$661$	−34.0000			−1.32245			−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000			−1.32245			−0.661223	+	0.750189i	$0.729962\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	24.0000	−	24.0000i	0.929284	−	0.929284i
$668$		0			0
$669$	−25.4558	−	36.0000i	−0.984180	−	1.39184i
$670$		0			0
$671$			33.9411i			1.31028i
$672$		0			0
$673$	−24.0000	−	24.0000i	−0.925132	−	0.925132i	0.0722542	−	0.997386i	$-0.476981\pi$
$673$	−24.0000	−	24.0000i	−0.925132	−	0.925132i	−0.997386	+	0.0722542i	$0.976981\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.9706	−	16.9706i	−0.652232	−	0.652232i	0.301298	−	0.953530i	$-0.402580\pi$
$677$	−16.9706	−	16.9706i	−0.652232	−	0.652232i	−0.953530	+	0.301298i	$0.902580\pi$
$678$		0			0
$679$		−	32.0000i		−	1.22805i
$680$		0			0
$681$	14.0000	+	19.7990i	0.536481	+	0.758699i
$682$		0			0
$683$	−4.24264	+	4.24264i	−0.162340	+	0.162340i	−0.783603	−	0.621262i	$-0.786620\pi$
$683$	−4.24264	+	4.24264i	−0.162340	+	0.162340i	0.621262	+	0.783603i	$0.286620\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.75736	−	10.2426i	0.0670474	−	0.390781i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$	20.6863	+	43.3137i	0.785807	+	1.64535i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	16.0000	−	16.0000i	0.606043	−	0.606043i
$698$		0			0
$699$	28.2843	−	20.0000i	1.06981	−	0.756469i
$700$		0			0
$701$		−	22.6274i		−	0.854626i	−0.904104	−	0.427313i	$-0.859460\pi$
$701$		−	22.6274i		−	0.854626i	0.904104	−	0.427313i	$-0.140540\pi$
$702$		0			0
$703$	32.0000	+	32.0000i	1.20690	+	1.20690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.3137	−	11.3137i	−0.425496	−	0.425496i
$708$		0			0
$709$			6.00000i			0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	33.9411	−	33.9411i	1.27111	−	1.27111i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−19.3137	−	3.31371i	−0.721284	−	0.123753i
$718$		0			0
$719$	−45.2548			−1.68772			−0.843860	−	0.536563i	$-0.819722\pi$
$719$	−45.2548			−1.68772			−0.843860	+	0.536563i	$0.819722\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$		0			0
$723$	−17.0711	−	2.92893i	−0.634880	−	0.108928i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−30.0000	+	30.0000i	−1.11264	+	1.11264i	−0.119846	+	0.992793i	$0.538240\pi$
$727$	−30.0000	+	30.0000i	−1.11264	+	1.11264i	−0.992793	+	0.119846i	$0.961760\pi$
$728$		0			0
$729$	14.1421	+	23.0000i	0.523783	+	0.851852i
$730$		0			0
$731$		−	11.3137i		−	0.418453i
$732$		0			0
$733$	24.0000	+	24.0000i	0.886460	+	0.886460i	0.994181	−	0.107721i	$-0.0343553\pi$
$733$	24.0000	+	24.0000i	0.886460	+	0.886460i	−0.107721	+	0.994181i	$0.534355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	33.9411	+	33.9411i	1.25024	+	1.25024i
$738$		0			0
$739$			4.00000i			0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$			4.00000i			0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.3848	−	18.3848i	0.674472	−	0.674472i	−0.284272	−	0.958744i	$-0.591752\pi$
$743$	18.3848	−	18.3848i	0.674472	−	0.674472i	0.958744	+	0.284272i	$0.0917518\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	37.8995	−	18.1005i	1.38667	−	0.662263i
$748$		0			0
$749$	−16.9706			−0.620091
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$		0			0
$753$	−1.65685	+	9.65685i	−0.0603791	+	0.351915i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	24.0000	−	24.0000i	0.872295	−	0.872295i	−0.120427	−	0.992722i	$-0.538426\pi$
$757$	24.0000	−	24.0000i	0.872295	−	0.872295i	0.992722	+	0.120427i	$0.0384265\pi$
$758$		0			0
$759$	33.9411	+	48.0000i	1.23198	+	1.74229i
$760$		0			0
$761$		−	22.6274i		−	0.820243i	−0.912031	−	0.410122i	$-0.865486\pi$
$761$		−	22.6274i		−	0.820243i	0.912031	−	0.410122i	$-0.134514\pi$
$762$		0			0
$763$	20.0000	+	20.0000i	0.724049	+	0.724049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			50.0000i			1.80305i	0.432731	+	0.901523i	$0.357550\pi$
$769$			50.0000i			1.80305i	−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$	12.0000	+	16.9706i	0.432169	+	0.611180i
$772$		0			0
$773$	11.3137	−	11.3137i	0.406926	−	0.406926i	−0.473739	−	0.880665i	$-0.657096\pi$
$773$	11.3137	−	11.3137i	0.406926	−	0.406926i	0.880665	+	0.473739i	$0.157096\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.37258	−	54.6274i	0.336240	−	1.95975i
$778$		0			0
$779$	22.6274			0.810711
$780$		0			0
$781$	64.0000			2.29010
$782$		0			0
$783$	−25.6569	−	14.3431i	−0.916901	−	0.512582i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−22.0000	+	22.0000i	−0.784215	+	0.784215i	−0.980539	−	0.196324i	$-0.937100\pi$
$787$	−22.0000	+	22.0000i	−0.784215	+	0.784215i	0.196324	+	0.980539i	$0.437100\pi$
$788$		0			0
$789$	8.48528	−	6.00000i	0.302084	−	0.213606i
$790$		0			0
$791$		−	11.3137i		−	0.402269i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		−	8.00000i		−	0.283020i
$800$		0			0
$801$	−32.0000	−	11.3137i	−1.13066	−	0.399750i
$802$		0			0
$803$	45.2548	−	45.2548i	1.59701	−	1.59701i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−19.3137	−	3.31371i	−0.679875	−	0.116648i
$808$		0			0
$809$	−11.3137			−0.397769			−0.198884	−	0.980023i	$-0.563732\pi$
$809$	−11.3137			−0.397769			−0.198884	+	0.980023i	$0.563732\pi$
$810$		0			0
$811$	−44.0000			−1.54505			−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000			−1.54505			−0.772524	+	0.634985i	$0.781006\pi$
$812$		0			0
$813$	−27.3137	−	4.68629i	−0.957934	−	0.164355i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	8.00000	−	8.00000i	0.279885	−	0.279885i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	22.6274i		−	0.789702i	−0.918745	−	0.394851i	$-0.870796\pi$
$821$		−	22.6274i		−	0.789702i	0.918745	−	0.394851i	$-0.129204\pi$
$822$		0			0
$823$	6.00000	+	6.00000i	0.209147	+	0.209147i	0.803905	−	0.594758i	$-0.202752\pi$
$823$	6.00000	+	6.00000i	0.209147	+	0.209147i	−0.594758	+	0.803905i	$0.702752\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29.6985	+	29.6985i	1.03272	+	1.03272i	0.999446	+	0.0332711i	$0.0105925\pi$
$827$	29.6985	+	29.6985i	1.03272	+	1.03272i	0.0332711	+	0.999446i	$0.489408\pi$
$828$		0			0
$829$			26.0000i			0.903017i	0.892267	+	0.451509i	$0.149114\pi$
$829$			26.0000i			0.903017i	−0.892267	+	0.451509i	$0.850886\pi$
$830$		0			0
$831$	16.0000	−	11.3137i	0.555034	−	0.392468i
$832$		0			0
$833$	2.82843	−	2.82843i	0.0979992	−	0.0979992i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−36.2843	−	20.2843i	−1.25417	−	0.701127i
$838$		0			0
$839$	45.2548			1.56237			0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548			1.56237			0.781185	+	0.624299i	$0.214615\pi$
$840$		0			0
$841$	3.00000			0.103448
$842$		0			0
$843$	−4.97056	+	28.9706i	−0.171195	+	0.997799i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	42.0000	−	42.0000i	1.44314	−	1.44314i
$848$		0			0
$849$	−8.48528	−	12.0000i	−0.291214	−	0.411839i
$850$		0			0
$851$		−	67.8823i		−	2.32697i
$852$		0			0
$853$	32.0000	+	32.0000i	1.09566	+	1.09566i	0.994912	+	0.100747i	$0.0321233\pi$
$853$	32.0000	+	32.0000i	1.09566	+	1.09566i	0.100747	+	0.994912i	$0.467877\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.48528	−	8.48528i	−0.289852	−	0.289852i	0.547170	−	0.837022i	$-0.315705\pi$
$857$	−8.48528	−	8.48528i	−0.289852	−	0.289852i	−0.837022	+	0.547170i	$0.815705\pi$
$858$		0			0
$859$		−	20.0000i		−	0.682391i	−0.939992	−	0.341196i	$-0.889168\pi$
$859$		−	20.0000i		−	0.682391i	0.939992	−	0.341196i	$-0.110832\pi$
$860$		0			0
$861$	−16.0000	−	22.6274i	−0.545279	−	0.771140i
$862$		0			0
$863$	−32.5269	+	32.5269i	−1.10723	+	1.10723i	−0.113716	+	0.993513i	$0.536275\pi$
$863$	−32.5269	+	32.5269i	−1.10723	+	1.10723i	−0.993513	+	0.113716i	$0.963725\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.292893	+	1.70711i	−0.00994718	+	0.0579764i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−30.6274	+	14.6274i	−1.03658	+	0.495063i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−24.0000	+	24.0000i	−0.810422	+	0.810422i	−0.984697	−	0.174275i	$-0.944242\pi$
$877$	−24.0000	+	24.0000i	−0.810422	+	0.810422i	0.174275	+	0.984697i	$0.444242\pi$
$878$		0			0
$879$	11.3137	−	8.00000i	0.381602	−	0.269833i
$880$		0			0
$881$		−	5.65685i		−	0.190584i	−0.995449	−	0.0952921i	$-0.969621\pi$
$881$		−	5.65685i		−	0.190584i	0.995449	−	0.0952921i	$-0.0303785\pi$
$882$		0			0
$883$	−2.00000	−	2.00000i	−0.0673054	−	0.0673054i	0.672653	−	0.739958i	$-0.265155\pi$
$883$	−2.00000	−	2.00000i	−0.0673054	−	0.0673054i	−0.739958	+	0.672653i	$0.765155\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	15.5563	+	15.5563i	0.522331	+	0.522331i	0.918275	−	0.395944i	$-0.129582\pi$
$887$	15.5563	+	15.5563i	0.522331	+	0.522331i	−0.395944	+	0.918275i	$0.629582\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$	32.0000	−	39.5980i	1.07204	−	1.32658i
$892$		0			0
$893$	5.65685	−	5.65685i	0.189299	−	0.189299i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	45.2548			1.50933
$900$		0			0
$901$	32.0000			1.06607
$902$		0			0
$903$	−13.6569	−	2.34315i	−0.454472	−	0.0779750i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.00000	+	6.00000i	−0.199227	+	0.199227i	−0.799668	−	0.600442i	$-0.794991\pi$
$907$	−6.00000	+	6.00000i	−0.199227	+	0.199227i	0.600442	+	0.799668i	$0.294991\pi$
$908$		0			0
$909$	−5.65685	+	16.0000i	−0.187626	+	0.530687i
$910$		0			0
$911$			22.6274i			0.749680i	0.927090	+	0.374840i	$0.122302\pi$
$911$			22.6274i			0.749680i	−0.927090	+	0.374840i	$0.877698\pi$
$912$		0			0
$913$	−56.0000	−	56.0000i	−1.85333	−	1.85333i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11.3137	−	11.3137i	−0.373612	−	0.373612i
$918$		0			0
$919$		−	8.00000i		−	0.263896i	−0.991257	−	0.131948i	$-0.957877\pi$
$919$		−	8.00000i		−	0.263896i	0.991257	−	0.131948i	$-0.0421231\pi$
$920$		0			0
$921$	20.0000	−	14.1421i	0.659022	−	0.465999i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−10.9706	−	22.9706i	−0.360321	−	0.754452i
$928$		0			0
$929$	5.65685			0.185595			0.0927977	−	0.995685i	$-0.470419\pi$
$929$	5.65685			0.185595			0.0927977	+	0.995685i	$0.470419\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$	−3.31371	+	19.3137i	−0.108486	+	0.632302i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24.0000	+	24.0000i	−0.784046	+	0.784046i	−0.980511	−	0.196465i	$-0.937054\pi$
$937$	−24.0000	+	24.0000i	−0.784046	+	0.784046i	0.196465	+	0.980511i	$0.437054\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	16.9706i		−	0.553225i	−0.960982	−	0.276612i	$-0.910788\pi$
$941$		−	16.9706i		−	0.553225i	0.960982	−	0.276612i	$-0.0892118\pi$
$942$		0			0
$943$	−24.0000	−	24.0000i	−0.781548	−	0.781548i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.7279	−	12.7279i	−0.413602	−	0.413602i	0.469389	−	0.882991i	$-0.344474\pi$
$947$	−12.7279	−	12.7279i	−0.413602	−	0.413602i	−0.882991	+	0.469389i	$0.844474\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	32.0000	+	45.2548i	1.03767	+	1.46749i
$952$		0			0
$953$	42.4264	−	42.4264i	1.37433	−	1.37433i	0.520409	−	0.853917i	$-0.325780\pi$
$953$	42.4264	−	42.4264i	1.37433	−	1.37433i	0.853917	−	0.520409i	$-0.174220\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−9.37258	+	54.6274i	−0.302973	+	1.76585i
$958$		0			0
$959$	33.9411			1.09602
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	7.75736	+	16.2426i	0.249977	+	0.523412i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.0000	−	10.0000i	0.321578	−	0.321578i	−0.527794	−	0.849372i	$-0.676981\pi$
$967$	10.0000	−	10.0000i	0.321578	−	0.321578i	0.849372	+	0.527794i	$0.176981\pi$
$968$		0			0
$969$	22.6274	−	16.0000i	0.726897	−	0.513994i
$970$		0			0
$971$			5.65685i			0.181537i	0.995872	+	0.0907685i	$0.0289323\pi$
$971$			5.65685i			0.181537i	−0.995872	+	0.0907685i	$0.971068\pi$
$972$		0			0
$973$	−8.00000	−	8.00000i	−0.256468	−	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	42.4264	+	42.4264i	1.35734	+	1.35734i	0.877181	+	0.480160i	$0.159421\pi$
$977$	42.4264	+	42.4264i	1.35734	+	1.35734i	0.480160	+	0.877181i	$0.340579\pi$
$978$		0			0
$979$			64.0000i			2.04545i
$980$		0			0
$981$	10.0000	−	28.2843i	0.319275	−	0.903047i
$982$		0			0
$983$	15.5563	−	15.5563i	0.496170	−	0.496170i	−0.414073	−	0.910244i	$-0.635894\pi$
$983$	15.5563	−	15.5563i	0.496170	−	0.496170i	0.910244	+	0.414073i	$0.135894\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−9.65685	−	1.65685i	−0.307381	−	0.0527383i
$988$		0			0
$989$	−16.9706			−0.539633
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$	−6.82843	−	1.17157i	−0.216694	−	0.0371787i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−16.0000	+	16.0000i	−0.506725	+	0.506725i	−0.913520	−	0.406795i	$-0.866646\pi$
$997$	−16.0000	+	16.0000i	−0.506725	+	0.506725i	0.406795	+	0.913520i	$0.366646\pi$
$998$		0			0
$999$	−56.5685	+	16.0000i	−1.78975	+	0.506218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.2.r.a.593.1	yes	4
3.2	odd	2	inner	600.2.r.a.593.2	yes	4
4.3	odd	2		1200.2.v.k.593.2		4
5.2	odd	4	inner	600.2.r.a.257.2	yes	4
5.3	odd	4		600.2.r.e.257.1	yes	4
5.4	even	2		600.2.r.e.593.2	yes	4
12.11	even	2		1200.2.v.k.593.1		4
15.2	even	4	inner	600.2.r.a.257.1	✓	4
15.8	even	4		600.2.r.e.257.2	yes	4
15.14	odd	2		600.2.r.e.593.1	yes	4
20.3	even	4		1200.2.v.a.257.2		4
20.7	even	4		1200.2.v.k.257.1		4
20.19	odd	2		1200.2.v.a.593.1		4
60.23	odd	4		1200.2.v.a.257.1		4
60.47	odd	4		1200.2.v.k.257.2		4
60.59	even	2		1200.2.v.a.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.r.a.257.1	✓	4	15.2	even	4	inner
600.2.r.a.257.2	yes	4	5.2	odd	4	inner
600.2.r.a.593.1	yes	4	1.1	even	1	trivial
600.2.r.a.593.2	yes	4	3.2	odd	2	inner
600.2.r.e.257.1	yes	4	5.3	odd	4
600.2.r.e.257.2	yes	4	15.8	even	4
600.2.r.e.593.1	yes	4	15.14	odd	2
600.2.r.e.593.2	yes	4	5.4	even	2
1200.2.v.a.257.1		4	60.23	odd	4
1200.2.v.a.257.2		4	20.3	even	4
1200.2.v.a.593.1		4	20.19	odd	2
1200.2.v.a.593.2		4	60.59	even	2
1200.2.v.k.257.1		4	20.7	even	4
1200.2.v.k.257.2		4	60.47	odd	4
1200.2.v.k.593.1		4	12.11	even	2
1200.2.v.k.593.2		4	4.3	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.r (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.70711 - 0.292893i) q^{3} +(-2.00000 + 2.00000i) q^{7} +(2.82843 + 1.00000i) q^{9} -5.65685i q^{11} +(2.82843 + 2.82843i) q^{17} +4.00000i q^{19} +(4.00000 - 2.82843i) q^{21} +(4.24264 - 4.24264i) q^{23} +(-4.53553 - 2.53553i) q^{27} +5.65685 q^{29} +8.00000 q^{31} +(-1.65685 + 9.65685i) q^{33} +(8.00000 - 8.00000i) q^{37} -5.65685i q^{41} +(-2.00000 - 2.00000i) q^{43} +(-1.41421 - 1.41421i) q^{47} -1.00000i q^{49} +(-4.00000 - 5.65685i) q^{51} +(5.65685 - 5.65685i) q^{53} +(1.17157 - 6.82843i) q^{57} +5.65685 q^{59} -6.00000 q^{61} +(-7.65685 + 3.65685i) q^{63} +(-6.00000 + 6.00000i) q^{67} +(-8.48528 + 6.00000i) q^{69} +11.3137i q^{71} +(8.00000 + 8.00000i) q^{73} +(11.3137 + 11.3137i) q^{77} +(7.00000 + 5.65685i) q^{81} +(9.89949 - 9.89949i) q^{83} +(-9.65685 - 1.65685i) q^{87} -11.3137 q^{89} +(-13.6569 - 2.34315i) q^{93} +(-8.00000 + 8.00000i) q^{97} +(5.65685 - 16.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{7} + 16 q^{21} - 4 q^{27} + 32 q^{31} + 16 q^{33} + 32 q^{37} - 8 q^{43} - 16 q^{51} + 16 q^{57} - 24 q^{61} - 8 q^{63} - 24 q^{67} + 32 q^{73} + 28 q^{81} - 16 q^{87} - 32 q^{93} - 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^7 + 16 * q^21 - 4 * q^27 + 32 * q^31 + 16 * q^33 + 32 * q^37 - 8 * q^43 - 16 * q^51 + 16 * q^57 - 24 * q^61 - 8 * q^63 - 24 * q^67 + 32 * q^73 + 28 * q^81 - 16 * q^87 - 32 * q^93 - 32 * q^97

Introduction

L-functions

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