LMFDB - Embedded newform 1512.2.k.b.377.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(377,1512)] 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("1512.377"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + \cdots + 256 $ x^16 - 6x^15 + 18x^14 - 36x^13 - 45x^12 + 306x^11 - 378x^10 + 1704x^9 + 917x^8 - 12522x^7 + 27090x^6 - 35292x^5 + 30948x^4 - 18000x^3 + 7200x^2 - 1920*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.10
Root			$3.41914 + 0.916156i$ of defining polynomial
Character	$\chi$	$=$	1512.377
Dual form			1512.2.k.b.377.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.933004 q^{5} +(2.45692 + 0.981597i) q^{7} +1.75675i q^{11} +4.20692i q^{13} -0.231144 q^{17} +5.00597i q^{19} -2.61601i q^{23} -4.12950 q^{25} +3.77311i q^{29} +0.840005i q^{31} +(2.29232 + 0.915835i) q^{35} +3.01636 q^{37} -8.98174 q^{41} +2.40035 q^{43} +1.03019 q^{47} +(5.07293 + 4.82342i) q^{49} -9.04335i q^{53} +1.63905i q^{55} +6.77898 q^{59} +8.93236i q^{61} +3.92508i q^{65} +1.10252 q^{67} -7.21168i q^{71} +3.44883i q^{73} +(-1.72442 + 4.31619i) q^{77} +5.29783 q^{79} +14.8713 q^{83} -0.215658 q^{85} -15.0053 q^{89} +(-4.12950 + 10.3361i) q^{91} +4.67059i q^{95} +1.63905i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{7} + 36 q^{25} - 8 q^{37} + 20 q^{43} + 2 q^{49} + 44 q^{67} + 40 q^{79} + 16 q^{85} + 36 q^{91}+O(q^{100}) $ 16 * q - 2 * q^7 + 36 * q^25 - 8 * q^37 + 20 * q^43 + 2 * q^49 + 44 * q^67 + 40 * q^79 + 16 * q^85 + 36 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.933004			0.417252			0.208626	−	0.977995i	$-0.433101\pi$
$5$	0.933004			0.417252			0.208626	+	0.977995i	$0.433101\pi$
$6$		0			0
$7$	2.45692	+	0.981597i	0.928629	+	0.371009i
$7$	2.45692	+	0.981597i	0.928629	+	0.371009i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.75675i			0.529679i	0.964293	+	0.264839i	$0.0853189\pi$
$11$			1.75675i			0.529679i	−0.964293	+	0.264839i	$0.914681\pi$
$12$		0			0
$13$			4.20692i			1.16679i	0.812189	+	0.583395i	$0.198276\pi$
$13$			4.20692i			1.16679i	−0.812189	+	0.583395i	$0.801724\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.231144			−0.0560606			−0.0280303	−	0.999607i	$-0.508923\pi$
$17$	−0.231144			−0.0560606			−0.0280303	+	0.999607i	$0.508923\pi$
$18$		0			0
$19$			5.00597i			1.14845i	0.818698	+	0.574224i	$0.194696\pi$
$19$			5.00597i			1.14845i	−0.818698	+	0.574224i	$0.805304\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.61601i		−	0.545476i	−0.962088	−	0.272738i	$-0.912071\pi$
$23$		−	2.61601i		−	0.545476i	0.962088	−	0.272738i	$-0.0879292\pi$
$24$		0			0
$25$	−4.12950			−0.825901
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.77311i			0.700649i	0.936628	+	0.350324i	$0.113929\pi$
$29$			3.77311i			0.700649i	−0.936628	+	0.350324i	$0.886071\pi$
$30$		0			0
$31$			0.840005i			0.150869i	0.997151	+	0.0754347i	$0.0240344\pi$
$31$			0.840005i			0.150869i	−0.997151	+	0.0754347i	$0.975966\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.29232	+	0.915835i	0.387473	+	0.154804i
$36$		0			0
$37$	3.01636			0.495887			0.247944	−	0.968774i	$-0.420245\pi$
$37$	3.01636			0.495887			0.247944	+	0.968774i	$0.420245\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.98174			−1.40271			−0.701356	−	0.712811i	$-0.747422\pi$
$41$	−8.98174			−1.40271			−0.701356	+	0.712811i	$0.747422\pi$
$42$		0			0
$43$	2.40035			0.366050			0.183025	−	0.983108i	$-0.441411\pi$
$43$	2.40035			0.366050			0.183025	+	0.983108i	$0.441411\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.03019			0.150269			0.0751343	−	0.997173i	$-0.476061\pi$
$47$	1.03019			0.150269			0.0751343	+	0.997173i	$0.476061\pi$
$48$		0			0
$49$	5.07293	+	4.82342i	0.724705	+	0.689060i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.04335i		−	1.24220i	−0.783732	−	0.621100i	$-0.786686\pi$
$53$		−	9.04335i		−	1.24220i	0.783732	−	0.621100i	$-0.213314\pi$
$54$		0			0
$55$			1.63905i			0.221010i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.77898			0.882548			0.441274	−	0.897372i	$-0.354527\pi$
$59$	6.77898			0.882548			0.441274	+	0.897372i	$0.354527\pi$
$60$		0			0
$61$			8.93236i			1.14367i	0.820368	+	0.571836i	$0.193768\pi$
$61$			8.93236i			1.14367i	−0.820368	+	0.571836i	$0.806232\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			3.92508i			0.486846i
$66$		0			0
$67$	1.10252			0.134694			0.0673471	−	0.997730i	$-0.478547\pi$
$67$	1.10252			0.134694			0.0673471	+	0.997730i	$0.478547\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.21168i		−	0.855869i	−0.903810	−	0.427934i	$-0.859241\pi$
$71$		−	7.21168i		−	0.855869i	0.903810	−	0.427934i	$-0.140759\pi$
$72$		0			0
$73$			3.44883i			0.403655i	0.979421	+	0.201828i	$0.0646881\pi$
$73$			3.44883i			0.403655i	−0.979421	+	0.201828i	$0.935312\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.72442	+	4.31619i	−0.196516	+	0.491875i
$78$		0			0
$79$	5.29783			0.596053			0.298026	−	0.954558i	$-0.403672\pi$
$79$	5.29783			0.596053			0.298026	+	0.954558i	$0.403672\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.8713			1.63234			0.816170	−	0.577812i	$-0.196093\pi$
$83$	14.8713			1.63234			0.816170	+	0.577812i	$0.196093\pi$
$84$		0			0
$85$	−0.215658			−0.0233914
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.0053			−1.59056			−0.795279	−	0.606244i	$-0.792675\pi$
$89$	−15.0053			−1.59056			−0.795279	+	0.606244i	$0.792675\pi$
$90$		0			0
$91$	−4.12950	+	10.3361i	−0.432890	+	1.08352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			4.67059i			0.479192i
$96$		0			0
$97$			1.63905i			0.166420i	0.996532	+	0.0832102i	$0.0265173\pi$
$97$			1.63905i			0.166420i	−0.996532	+	0.0832102i	$0.973483\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.58464			−0.456189			−0.228094	−	0.973639i	$-0.573249\pi$
$101$	−4.58464			−0.456189			−0.228094	+	0.973639i	$0.573249\pi$
$102$		0			0
$103$			16.1411i			1.59043i	0.606328	+	0.795215i	$0.292642\pi$
$103$			16.1411i			1.59043i	−0.606328	+	0.795215i	$0.707358\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	18.4601i		−	1.78460i	−0.451441	−	0.892301i	$-0.649090\pi$
$107$		−	18.4601i		−	1.78460i	0.451441	−	0.892301i	$-0.350910\pi$
$108$		0			0
$109$	8.81132			0.843972			0.421986	−	0.906602i	$-0.361333\pi$
$109$	8.81132			0.843972			0.421986	+	0.906602i	$0.361333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.9026i			1.30785i	0.756560	+	0.653924i	$0.226878\pi$
$113$			13.9026i			1.30785i	−0.756560	+	0.653924i	$0.773122\pi$
$114$		0			0
$115$		−	2.44075i		−	0.227601i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.567902	−	0.226890i	−0.0520595	−	0.0207990i
$120$		0			0
$121$	7.91384			0.719440
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.51787			−0.761861
$126$		0			0
$127$	1.61601			0.143398			0.0716989	−	0.997426i	$-0.477158\pi$
$127$	1.61601			0.143398			0.0716989	+	0.997426i	$0.477158\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	17.7239			1.54855			0.774273	−	0.632852i	$-0.218116\pi$
$131$	17.7239			1.54855			0.774273	+	0.632852i	$0.218116\pi$
$132$		0			0
$133$	−4.91384	+	12.2993i	−0.426084	+	1.06648i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.9848i			1.28024i	0.768276	+	0.640118i	$0.221115\pi$
$137$			14.9848i			1.28024i	−0.768276	+	0.640118i	$0.778885\pi$
$138$		0			0
$139$			1.05435i			0.0894285i	0.999000	+	0.0447143i	$0.0142377\pi$
$139$			1.05435i			0.0894285i	−0.999000	+	0.0447143i	$0.985762\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.39049			−0.618024
$144$		0			0
$145$			3.52033i			0.292347i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			1.45493i			0.119193i	0.998223	+	0.0595963i	$0.0189813\pi$
$149$			1.45493i			0.119193i	−0.998223	+	0.0595963i	$0.981019\pi$
$150$		0			0
$151$	12.6757			1.03154			0.515768	−	0.856728i	$-0.327507\pi$
$151$	12.6757			1.03154			0.515768	+	0.856728i	$0.327507\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			0.783729i			0.0629506i
$156$		0			0
$157$		−	21.7953i		−	1.73946i	−0.493530	−	0.869729i	$-0.664294\pi$
$157$		−	21.7953i		−	1.73946i	0.493530	−	0.869729i	$-0.335706\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.56787	−	6.42734i	0.202376	−	0.506545i
$162$		0			0
$163$	−8.65936			−0.678253			−0.339127	−	0.940741i	$-0.610132\pi$
$163$	−8.65936			−0.678253			−0.339127	+	0.940741i	$0.610132\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.49611			0.580066			0.290033	−	0.957017i	$-0.406334\pi$
$167$	7.49611			0.580066			0.290033	+	0.957017i	$0.406334\pi$
$168$		0			0
$169$	−4.69819			−0.361399
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.71863			−0.206694			−0.103347	−	0.994645i	$-0.532955\pi$
$173$	−2.71863			−0.206694			−0.103347	+	0.994645i	$0.532955\pi$
$174$		0			0
$175$	−10.1459	−	4.05351i	−0.766955	−	0.306416i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	18.3851i		−	1.37417i	−0.726577	−	0.687085i	$-0.758890\pi$
$179$		−	18.3851i		−	1.37417i	0.726577	−	0.687085i	$-0.241110\pi$
$180$		0			0
$181$		−	8.45745i		−	0.628638i	−0.949317	−	0.314319i	$-0.898224\pi$
$181$		−	8.45745i		−	0.628638i	0.949317	−	0.314319i	$-0.101776\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.81428			0.206910
$186$		0			0
$187$		−	0.406061i		−	0.0296941i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			25.3412i			1.83362i	0.399320	+	0.916812i	$0.369246\pi$
$191$			25.3412i			1.83362i	−0.399320	+	0.916812i	$0.630754\pi$
$192$		0			0
$193$	−0.404334			−0.0291046			−0.0145523	−	0.999894i	$-0.504632\pi$
$193$	−0.404334			−0.0291046			−0.0145523	+	0.999894i	$0.504632\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	7.58443i		−	0.540369i	−0.962809	−	0.270184i	$-0.912915\pi$
$197$		−	7.58443i		−	0.540369i	0.962809	−	0.270184i	$-0.0870846\pi$
$198$		0			0
$199$		−	25.7879i		−	1.82806i	−0.405649	−	0.914029i	$-0.632955\pi$
$199$		−	25.7879i		−	1.82806i	0.405649	−	0.914029i	$-0.367045\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.70367	+	9.27024i	−0.259947	+	0.650643i
$204$		0			0
$205$	−8.38001			−0.585285
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.79421			−0.608308
$210$		0			0
$211$	−3.69819			−0.254594			−0.127297	−	0.991865i	$-0.540630\pi$
$211$	−3.69819			−0.254594			−0.127297	+	0.991865i	$0.540630\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.23954			0.152735
$216$		0			0
$217$	−0.824547	+	2.06383i	−0.0559739	+	0.140102i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.972404i		−	0.0654110i
$222$		0			0
$223$		−	8.17692i		−	0.547567i	−0.961791	−	0.273784i	$-0.911725\pi$
$223$		−	8.17692i		−	0.547567i	0.961791	−	0.273784i	$-0.0882752\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.80735			0.319075			0.159538	−	0.987192i	$-0.449000\pi$
$227$	4.80735			0.319075			0.159538	+	0.987192i	$0.449000\pi$
$228$		0			0
$229$			3.65010i			0.241205i	0.992701	+	0.120603i	$0.0384827\pi$
$229$			3.65010i			0.241205i	−0.992701	+	0.120603i	$0.961517\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	22.0118i		−	1.44204i	−0.692915	−	0.721020i	$-0.743674\pi$
$233$		−	22.0118i		−	1.44204i	0.692915	−	0.721020i	$-0.256326\pi$
$234$		0			0
$235$	0.961172			0.0627000
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5.64361i			0.365055i	0.983201	+	0.182527i	$0.0584278\pi$
$239$			5.64361i			0.365055i	−0.983201	+	0.182527i	$0.941572\pi$
$240$		0			0
$241$		−	24.9479i		−	1.60704i	−0.595280	−	0.803518i	$-0.702959\pi$
$241$		−	24.9479i		−	1.60704i	0.595280	−	0.803518i	$-0.297041\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.73307	+	4.50027i	0.302385	+	0.287512i
$246$		0			0
$247$	−21.0597			−1.34000
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.8598			1.06418			0.532089	−	0.846688i	$-0.321407\pi$
$251$	16.8598			1.06418			0.532089	+	0.846688i	$0.321407\pi$
$252$		0			0
$253$	4.59567			0.288927
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.13580			−0.0708496			−0.0354248	−	0.999372i	$-0.511278\pi$
$257$	−1.13580			−0.0708496			−0.0354248	+	0.999372i	$0.511278\pi$
$258$		0			0
$259$	7.41097	+	2.96085i	0.460495	+	0.183979i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	0.485895i		−	0.0299616i	−0.999888	−	0.0149808i	$-0.995231\pi$
$263$		−	0.485895i		−	0.0299616i	0.999888	−	0.0149808i	$-0.00476871\pi$
$264$		0			0
$265$		−	8.43748i		−	0.518310i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.3100			−0.689585			−0.344793	−	0.938679i	$-0.612051\pi$
$269$	−11.3100			−0.689585			−0.344793	+	0.938679i	$0.612051\pi$
$270$		0			0
$271$			5.00862i			0.304252i	0.988361	+	0.152126i	$0.0486119\pi$
$271$			5.00862i			0.304252i	−0.988361	+	0.152126i	$0.951388\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	7.25449i		−	0.437462i
$276$		0			0
$277$	−20.6717			−1.24204			−0.621022	−	0.783793i	$-0.713282\pi$
$277$	−20.6717			−1.24204			−0.621022	+	0.783793i	$0.713282\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	13.9448i		−	0.831878i	−0.909393	−	0.415939i	$-0.863453\pi$
$281$		−	13.9448i		−	0.831878i	0.909393	−	0.415939i	$-0.136547\pi$
$282$		0			0
$283$		−	22.5944i		−	1.34310i	−0.740961	−	0.671549i	$-0.765629\pi$
$283$		−	22.5944i		−	1.34310i	0.740961	−	0.671549i	$-0.234371\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−22.0674	−	8.81646i	−1.30260	−	0.520419i
$288$		0			0
$289$	−16.9466			−0.996857
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.27838			−0.366787			−0.183394	−	0.983040i	$-0.558708\pi$
$293$	−6.27838			−0.366787			−0.183394	+	0.983040i	$0.558708\pi$
$294$		0			0
$295$	6.32482			0.368245
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.0054			0.636456
$300$		0			0
$301$	5.89748	+	2.35618i	0.339925	+	0.135808i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			8.33393i			0.477199i
$306$		0			0
$307$		−	22.1578i		−	1.26461i	−0.774719	−	0.632306i	$-0.782109\pi$
$307$		−	22.1578i		−	1.26461i	0.774719	−	0.632306i	$-0.217891\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.1295			−1.14144			−0.570719	−	0.821145i	$-0.693336\pi$
$311$	−20.1295			−1.14144			−0.570719	+	0.821145i	$0.693336\pi$
$312$		0			0
$313$			17.2233i			0.973520i	0.873536	+	0.486760i	$0.161821\pi$
$313$			17.2233i			0.973520i	−0.873536	+	0.486760i	$0.838179\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			29.3779i			1.65003i	0.565113	+	0.825013i	$0.308832\pi$
$317$			29.3779i			1.65003i	−0.565113	+	0.825013i	$0.691168\pi$
$318$		0			0
$319$	−6.62839			−0.371119
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.15710i		−	0.0643827i
$324$		0			0
$325$		−	17.3725i		−	0.963652i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.53110	+	1.01123i	0.139544	+	0.0557510i
$330$		0			0
$331$	13.2550			0.728562			0.364281	−	0.931289i	$-0.381315\pi$
$331$	13.2550			0.728562			0.364281	+	0.931289i	$0.381315\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.02866			0.0562015
$336$		0			0
$337$	12.1728			0.663097			0.331549	−	0.943438i	$-0.392429\pi$
$337$	12.1728			0.663097			0.331549	+	0.943438i	$0.392429\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.47568			−0.0799123
$342$		0			0
$343$	7.72915	+	16.8303i	0.417335	+	0.908753i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	8.55745i		−	0.459388i	−0.973263	−	0.229694i	$-0.926227\pi$
$347$		−	8.55745i		−	0.459388i	0.973263	−	0.229694i	$-0.0737725\pi$
$348$		0			0
$349$		−	21.3840i		−	1.14466i	−0.820024	−	0.572329i	$-0.806040\pi$
$349$		−	21.3840i		−	1.14466i	0.820024	−	0.572329i	$-0.193960\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	35.9774			1.91489			0.957443	−	0.288623i	$-0.0931976\pi$
$353$	35.9774			1.91489			0.957443	+	0.288623i	$0.0931976\pi$
$354$		0			0
$355$		−	6.72853i		−	0.357113i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	0.864752i		−	0.0456399i	−0.999740	−	0.0228199i	$-0.992736\pi$
$359$		−	0.864752i		−	0.0456399i	0.999740	−	0.0228199i	$-0.00726444\pi$
$360$		0			0
$361$	−6.05971			−0.318932
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3.21778i			0.168426i
$366$		0			0
$367$		−	28.2287i		−	1.47352i	−0.676152	−	0.736762i	$-0.736354\pi$
$367$		−	28.2287i		−	1.47352i	0.676152	−	0.736762i	$-0.263646\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.87693	−	22.2188i	0.460867	−	1.15354i
$372$		0			0
$373$	22.5074			1.16539			0.582694	−	0.812691i	$-0.301999\pi$
$373$	22.5074			1.16539			0.582694	+	0.812691i	$0.301999\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.8732			−0.817510
$378$		0			0
$379$	−15.3306			−0.787478			−0.393739	−	0.919222i	$-0.628819\pi$
$379$	−15.3306			−0.787478			−0.393739	+	0.919222i	$0.628819\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−31.2215			−1.59534			−0.797671	−	0.603093i	$-0.793935\pi$
$383$	−31.2215			−1.59534			−0.797671	+	0.603093i	$0.793935\pi$
$384$		0			0
$385$	−1.60889	+	4.02702i	−0.0819966	+	0.205236i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			10.4695i			0.530826i	0.964135	+	0.265413i	$0.0855084\pi$
$389$			10.4695i			0.530826i	−0.964135	+	0.265413i	$0.914492\pi$
$390$		0			0
$391$			0.604675i			0.0305797i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.94290			0.248704
$396$		0			0
$397$			2.27161i			0.114009i	0.998374	+	0.0570044i	$0.0181549\pi$
$397$			2.27161i			0.114009i	−0.998374	+	0.0570044i	$0.981845\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	17.0822i		−	0.853043i	−0.904477	−	0.426522i	$-0.859739\pi$
$401$		−	17.0822i		−	0.853043i	0.904477	−	0.426522i	$-0.140261\pi$
$402$		0			0
$403$	−3.53384			−0.176033
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5.29898i			0.262661i
$408$		0			0
$409$		−	34.4145i		−	1.70169i	−0.525418	−	0.850844i	$-0.676091\pi$
$409$		−	34.4145i		−	1.70169i	0.525418	−	0.850844i	$-0.323909\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16.6554	+	6.65423i	0.819560	+	0.327433i
$414$		0			0
$415$	13.8750			0.681098
$416$		0			0
$417$		0			0
$418$		0			0
$419$	35.4899			1.73380			0.866898	−	0.498486i	$-0.166110\pi$
$419$	35.4899			1.73380			0.866898	+	0.498486i	$0.166110\pi$
$420$		0			0
$421$	−33.1464			−1.61546			−0.807728	−	0.589555i	$-0.799303\pi$
$421$	−33.1464			−1.61546			−0.807728	+	0.589555i	$0.799303\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.954509			0.0463005
$426$		0			0
$427$	−8.76798	+	21.9461i	−0.424312	+	1.06205i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			6.80071i			0.327578i	0.986495	+	0.163789i	$0.0523717\pi$
$431$			6.80071i			0.327578i	−0.986495	+	0.163789i	$0.947628\pi$
$432$		0			0
$433$		−	15.1067i		−	0.725982i	−0.931793	−	0.362991i	$-0.881755\pi$
$433$		−	15.1067i		−	0.725982i	0.931793	−	0.362991i	$-0.118245\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13.0957			0.626451
$438$		0			0
$439$		−	8.25619i		−	0.394046i	−0.980399	−	0.197023i	$-0.936873\pi$
$439$		−	8.25619i		−	0.394046i	0.980399	−	0.197023i	$-0.0631274\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			20.0749i			0.953788i	0.878961	+	0.476894i	$0.158238\pi$
$443$			20.0749i			0.953788i	−0.878961	+	0.476894i	$0.841762\pi$
$444$		0			0
$445$	−14.0000			−0.663664
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.18983i		−	0.244923i	−0.992473	−	0.122462i	$-0.960921\pi$
$449$		−	5.18983i		−	0.244923i	0.992473	−	0.122462i	$-0.0390788\pi$
$450$		0			0
$451$		−	15.7786i		−	0.742987i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.85284	+	9.64361i	−0.180624	+	0.452099i
$456$		0			0
$457$	−19.5074			−0.912517			−0.456259	−	0.889847i	$-0.650811\pi$
$457$	−19.5074			−0.912517			−0.456259	+	0.889847i	$0.650811\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−34.1566			−1.59083			−0.795416	−	0.606064i	$-0.792748\pi$
$461$	−34.1566			−1.59083			−0.795416	+	0.606064i	$0.792748\pi$
$462$		0			0
$463$	−9.23866			−0.429357			−0.214678	−	0.976685i	$-0.568870\pi$
$463$	−9.23866			−0.429357			−0.214678	+	0.976685i	$0.568870\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.4374			0.760630			0.380315	−	0.924857i	$-0.375816\pi$
$467$	16.4374			0.760630			0.380315	+	0.924857i	$0.375816\pi$
$468$		0			0
$469$	2.70881	+	1.08223i	0.125081	+	0.0499728i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.21681i			0.193889i
$474$		0			0
$475$		−	20.6722i		−	0.948504i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−33.1011			−1.51243			−0.756215	−	0.654323i	$-0.772954\pi$
$479$	−33.1011			−1.51243			−0.756215	+	0.654323i	$0.772954\pi$
$480$		0			0
$481$			12.6896i			0.578596i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			1.52924i			0.0694393i
$486$		0			0
$487$	−35.1137			−1.59115			−0.795576	−	0.605853i	$-0.792832\pi$
$487$	−35.1137			−1.59115			−0.795576	+	0.605853i	$0.792832\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			10.9560i			0.494439i	0.968959	+	0.247220i	$0.0795169\pi$
$491$			10.9560i			0.494439i	−0.968959	+	0.247220i	$0.920483\pi$
$492$		0			0
$493$		−	0.872131i		−	0.0392788i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.07896	−	17.7185i	0.317535	−	0.794785i
$498$		0			0
$499$	−12.0985			−0.541605			−0.270802	−	0.962635i	$-0.587289\pi$
$499$	−12.0985			−0.541605			−0.270802	+	0.962635i	$0.587289\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17.5594			−0.782936			−0.391468	−	0.920192i	$-0.628033\pi$
$503$	−17.5594			−0.782936			−0.391468	+	0.920192i	$0.628033\pi$
$504$		0			0
$505$	−4.27749			−0.190346
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.06881			0.0916984			0.0458492	−	0.998948i	$-0.485401\pi$
$509$	2.06881			0.0916984			0.0458492	+	0.998948i	$0.485401\pi$
$510$		0			0
$511$	−3.38537	+	8.47352i	−0.149760	+	0.374846i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			15.0597i			0.663610i
$516$		0			0
$517$			1.80978i			0.0795941i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.9440			1.22425			0.612123	−	0.790762i	$-0.290316\pi$
$521$	27.9440			1.22425			0.612123	+	0.790762i	$0.290316\pi$
$522$		0			0
$523$			31.2026i			1.36439i	0.731168	+	0.682197i	$0.238976\pi$
$523$			31.2026i			1.36439i	−0.731168	+	0.682197i	$0.761024\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	0.194162i		−	0.00845783i
$528$		0			0
$529$	16.1565			0.702456
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	37.7855i		−	1.63667i
$534$		0			0
$535$		−	17.2233i		−	0.744629i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.47352	+	8.91185i	−0.364980	+	0.383861i
$540$		0			0
$541$	37.9905			1.63334			0.816669	−	0.577107i	$-0.195818\pi$
$541$	37.9905			1.63334			0.816669	+	0.577107i	$0.195818\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.22101			0.352149
$546$		0			0
$547$	31.4173			1.34330			0.671652	−	0.740866i	$-0.265585\pi$
$547$	31.4173			1.34330			0.671652	+	0.740866i	$0.265585\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.8881			−0.804659
$552$		0			0
$553$	13.0164	+	5.20034i	0.553512	+	0.221141i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	32.8158i		−	1.39045i	−0.718791	−	0.695226i	$-0.755304\pi$
$557$		−	32.8158i		−	1.39045i	0.718791	−	0.695226i	$-0.244696\pi$
$558$		0			0
$559$			10.0981i			0.427104i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−23.4092			−0.986579			−0.493289	−	0.869865i	$-0.664206\pi$
$563$	−23.4092			−0.986579			−0.493289	+	0.869865i	$0.664206\pi$
$564$		0			0
$565$			12.9712i			0.545702i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			33.2320i			1.39316i	0.717480	+	0.696579i	$0.245296\pi$
$569$			33.2320i			1.39316i	−0.717480	+	0.696579i	$0.754704\pi$
$570$		0			0
$571$	19.5996			0.820220			0.410110	−	0.912036i	$-0.365490\pi$
$571$	19.5996			0.820220			0.410110	+	0.912036i	$0.365490\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			10.8028i			0.450509i
$576$		0			0
$577$		−	11.3337i		−	0.471830i	−0.971774	−	0.235915i	$-0.924191\pi$
$577$		−	11.3337i		−	0.471830i	0.971774	−	0.235915i	$-0.0758086\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	36.5377	+	14.5977i	1.51584	+	0.605613i
$582$		0			0
$583$	15.8869			0.657967
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.6923			0.523866			0.261933	−	0.965086i	$-0.415640\pi$
$587$	12.6923			0.523866			0.261933	+	0.965086i	$0.415640\pi$
$588$		0			0
$589$	−4.20504			−0.173266
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13.2663			0.544783			0.272392	−	0.962186i	$-0.412185\pi$
$593$	13.2663			0.544783			0.272392	+	0.962186i	$0.412185\pi$
$594$		0			0
$595$	−0.529856	−	0.211690i	−0.0217220	−	0.00867843i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			4.74613i			0.193921i	0.995288	+	0.0969607i	$0.0309121\pi$
$599$			4.74613i			0.193921i	−0.995288	+	0.0969607i	$0.969088\pi$
$600$		0			0
$601$			23.0666i			0.940908i	0.882425	+	0.470454i	$0.155910\pi$
$601$			23.0666i			0.940908i	−0.882425	+	0.470454i	$0.844090\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.38365			0.300188
$606$		0			0
$607$			35.6212i			1.44582i	0.690942	+	0.722910i	$0.257196\pi$
$607$			35.6212i			1.44582i	−0.690942	+	0.722910i	$0.742804\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4.33393i			0.175332i
$612$		0			0
$613$	33.8931			1.36893			0.684466	−	0.729045i	$-0.260036\pi$
$613$	33.8931			1.36893			0.684466	+	0.729045i	$0.260036\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.09014i			0.0438872i	0.999759	+	0.0219436i	$0.00698543\pi$
$617$			1.09014i			0.0438872i	−0.999759	+	0.0219436i	$0.993015\pi$
$618$		0			0
$619$		−	21.1907i		−	0.851725i	−0.904788	−	0.425862i	$-0.859971\pi$
$619$		−	21.1907i		−	0.851725i	0.904788	−	0.425862i	$-0.140029\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−36.8668	−	14.7291i	−1.47704	−	0.590111i
$624$		0			0
$625$	12.7003			0.508012
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.697214			−0.0277997
$630$		0			0
$631$	−14.9020			−0.593239			−0.296620	−	0.954996i	$-0.595859\pi$
$631$	−14.9020			−0.593239			−0.296620	+	0.954996i	$0.595859\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.50775			0.0598331
$636$		0			0
$637$	−20.2917	+	21.3414i	−0.803988	+	0.845578i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	24.3260i		−	0.960818i	−0.877045	−	0.480409i	$-0.840488\pi$
$641$		−	24.3260i		−	0.960818i	0.877045	−	0.480409i	$-0.159512\pi$
$642$		0			0
$643$			22.4341i			0.884714i	0.896839	+	0.442357i	$0.145858\pi$
$643$			22.4341i			0.884714i	−0.896839	+	0.442357i	$0.854142\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−37.5251			−1.47526			−0.737631	−	0.675204i	$-0.764056\pi$
$647$	−37.5251			−1.47526			−0.737631	+	0.675204i	$0.764056\pi$
$648$		0			0
$649$			11.9089i			0.467467i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	29.4549i		−	1.15266i	−0.817217	−	0.576330i	$-0.804484\pi$
$653$		−	29.4549i		−	1.15266i	0.817217	−	0.576330i	$-0.195516\pi$
$654$		0			0
$655$	16.5365			0.646134
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	43.8012i		−	1.70625i	−0.521703	−	0.853127i	$-0.674703\pi$
$659$		−	43.8012i		−	1.70625i	0.521703	−	0.853127i	$-0.325297\pi$
$660$		0			0
$661$			8.90182i			0.346241i	0.984901	+	0.173120i	$0.0553850\pi$
$661$			8.90182i			0.346241i	−0.984901	+	0.173120i	$0.944615\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.58464	+	11.4753i	−0.177785	+	0.444992i
$666$		0			0
$667$	9.87050			0.382187
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15.6919			−0.605778
$672$		0			0
$673$	21.0761			0.812423			0.406211	−	0.913779i	$-0.366850\pi$
$673$	21.0761			0.812423			0.406211	+	0.913779i	$0.366850\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.1464			1.19705			0.598526	−	0.801103i	$-0.295753\pi$
$677$	31.1464			1.19705			0.598526	+	0.801103i	$0.295753\pi$
$678$		0			0
$679$	−1.60889	+	4.02702i	−0.0617435	+	0.154543i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			19.5805i			0.749225i	0.927181	+	0.374613i	$0.122224\pi$
$683$			19.5805i			0.749225i	−0.927181	+	0.374613i	$0.877776\pi$
$684$		0			0
$685$			13.9809i			0.534182i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	38.0446			1.44939
$690$		0			0
$691$			34.1739i			1.30004i	0.759919	+	0.650018i	$0.225239\pi$
$691$			34.1739i			1.30004i	−0.759919	+	0.650018i	$0.774761\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			0.983710i			0.0373143i
$696$		0			0
$697$	2.07608			0.0786370
$698$		0			0
$699$		0			0
$700$		0			0
$701$			42.1904i			1.59351i	0.604304	+	0.796754i	$0.293451\pi$
$701$			42.1904i			1.59351i	−0.604304	+	0.796754i	$0.706549\pi$
$702$		0			0
$703$			15.0998i			0.569501i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−11.2641	−	4.50027i	−0.423630	−	0.169250i
$708$		0			0
$709$	−37.4937			−1.40810			−0.704052	−	0.710148i	$-0.748628\pi$
$709$	−37.4937			−1.40810			−0.704052	+	0.710148i	$0.748628\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.19746			0.0822956
$714$		0			0
$715$	−6.89536			−0.257872
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.26820			0.121883			0.0609416	−	0.998141i	$-0.480590\pi$
$719$	3.26820			0.121883			0.0609416	+	0.998141i	$0.480590\pi$
$720$		0			0
$721$	−15.8441	+	39.6574i	−0.590063	+	1.47692i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	15.5811i		−	0.578666i
$726$		0			0
$727$			21.3997i			0.793671i	0.917890	+	0.396836i	$0.129892\pi$
$727$			21.3997i			0.793671i	−0.917890	+	0.396836i	$0.870108\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.554827			−0.0205210
$732$		0			0
$733$			39.3611i			1.45384i	0.686724	+	0.726919i	$0.259048\pi$
$733$			39.3611i			1.45384i	−0.686724	+	0.726919i	$0.740952\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.93685i			0.0713447i
$738$		0			0
$739$	−9.40433			−0.345944			−0.172972	−	0.984927i	$-0.555337\pi$
$739$	−9.40433			−0.345944			−0.172972	+	0.984927i	$0.555337\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			32.8880i			1.20654i	0.797535	+	0.603272i	$0.206137\pi$
$743$			32.8880i			1.20654i	−0.797535	+	0.603272i	$0.793863\pi$
$744$		0			0
$745$			1.35746i			0.0497334i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.1203	−	45.3549i	0.662103	−	1.65723i
$750$		0			0
$751$	−23.9081			−0.872419			−0.436209	−	0.899845i	$-0.643679\pi$
$751$	−23.9081			−0.872419			−0.436209	+	0.899845i	$0.643679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.8265			0.430411
$756$		0			0
$757$	48.2944			1.75529			0.877645	−	0.479312i	$-0.159114\pi$
$757$	48.2944			1.75529			0.877645	+	0.479312i	$0.159114\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−34.5737			−1.25330			−0.626648	−	0.779302i	$-0.715574\pi$
$761$	−34.5737			−1.25330			−0.626648	+	0.779302i	$0.715574\pi$
$762$		0			0
$763$	21.6487	+	8.64917i	0.783737	+	0.313121i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			28.5186i			1.02975i
$768$		0			0
$769$		−	33.1831i		−	1.19661i	−0.801267	−	0.598307i	$-0.795840\pi$
$769$		−	33.1831i		−	1.19661i	0.801267	−	0.598307i	$-0.204160\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	37.5014			1.34883			0.674415	−	0.738352i	$-0.264396\pi$
$773$	37.5014			1.34883			0.674415	+	0.738352i	$0.264396\pi$
$774$		0			0
$775$		−	3.46880i		−	0.124603i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	44.9623i		−	1.61094i
$780$		0			0
$781$	12.6691			0.453335
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	20.3352i		−	0.725793i
$786$		0			0
$787$			9.33842i			0.332879i	0.986052	+	0.166439i	$0.0532270\pi$
$787$			9.33842i			0.332879i	−0.986052	+	0.166439i	$0.946773\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−13.6468	+	34.1576i	−0.485223	+	1.21451i
$792$		0			0
$793$	−37.5777			−1.33442
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−31.6239			−1.12018			−0.560089	−	0.828433i	$-0.689233\pi$
$797$	−31.6239			−1.12018			−0.560089	+	0.828433i	$0.689233\pi$
$798$		0			0
$799$	−0.238122			−0.00842416
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.05872			−0.213808
$804$		0			0
$805$	2.39583	−	5.99673i	0.0844420	−	0.211357i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			27.7318i			0.974998i	0.873123	+	0.487499i	$0.162091\pi$
$809$			27.7318i			0.974998i	−0.873123	+	0.487499i	$0.837909\pi$
$810$		0			0
$811$			8.45321i			0.296832i	0.988925	+	0.148416i	$0.0474175\pi$
$811$			8.45321i			0.296832i	−0.988925	+	0.148416i	$0.952582\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8.07922			−0.283003
$816$		0			0
$817$			12.0161i			0.420390i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			0.496589i			0.0173311i	0.999962	+	0.00866554i	$0.00275836\pi$
$821$			0.496589i			0.0173311i	−0.999962	+	0.00866554i	$0.997242\pi$
$822$		0			0
$823$	48.0721			1.67569			0.837844	−	0.545910i	$-0.183816\pi$
$823$	48.0721			1.67569			0.837844	+	0.545910i	$0.183816\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			11.6211i			0.404107i	0.979375	+	0.202053i	$0.0647614\pi$
$827$			11.6211i			0.404107i	−0.979375	+	0.202053i	$0.935239\pi$
$828$		0			0
$829$		−	39.4221i		−	1.36919i	−0.728926	−	0.684593i	$-0.759980\pi$
$829$		−	39.4221i		−	1.36919i	0.728926	−	0.684593i	$-0.240020\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.17258	−	1.11490i	−0.0406274	−	0.0386291i
$834$		0			0
$835$	6.99390			0.242034
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.9656			0.827384			0.413692	−	0.910417i	$-0.364239\pi$
$839$	23.9656			0.827384			0.413692	+	0.910417i	$0.364239\pi$
$840$		0			0
$841$	14.7636			0.509091
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.38343			−0.150795
$846$		0			0
$847$	19.4437	+	7.76821i	0.668093	+	0.266919i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	7.89084i		−	0.270495i
$852$		0			0
$853$			17.9030i			0.612988i	0.951873	+	0.306494i	$0.0991559\pi$
$853$			17.9030i			0.612988i	−0.951873	+	0.306494i	$0.900844\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.1623			−0.517934			−0.258967	−	0.965886i	$-0.583382\pi$
$857$	−15.1623			−0.517934			−0.258967	+	0.965886i	$0.583382\pi$
$858$		0			0
$859$		−	19.8862i		−	0.678508i	−0.940695	−	0.339254i	$-0.889825\pi$
$859$		−	19.8862i		−	0.678508i	0.940695	−	0.339254i	$-0.110175\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			21.1813i			0.721018i	0.932756	+	0.360509i	$0.117397\pi$
$863$			21.1813i			0.721018i	−0.932756	+	0.360509i	$0.882603\pi$
$864$		0			0
$865$	−2.53649			−0.0862434
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9.30695i			0.315717i
$870$		0			0
$871$			4.63821i			0.157160i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20.9277	−	8.36112i	−0.707487	−	0.282657i
$876$		0			0
$877$	−4.89802			−0.165394			−0.0826972	−	0.996575i	$-0.526353\pi$
$877$	−4.89802			−0.165394			−0.0826972	+	0.996575i	$0.526353\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14.0502			0.473362			0.236681	−	0.971587i	$-0.423940\pi$
$881$	14.0502			0.473362			0.236681	+	0.971587i	$0.423940\pi$
$882$		0			0
$883$	17.4928			0.588679			0.294340	−	0.955701i	$-0.404900\pi$
$883$	17.4928			0.588679			0.294340	+	0.955701i	$0.404900\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.286309			0.00961331			0.00480665	−	0.999988i	$-0.498470\pi$
$887$	0.286309			0.00961331			0.00480665	+	0.999988i	$0.498470\pi$
$888$		0			0
$889$	3.97041	+	1.58627i	0.133163	+	0.0532019i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			5.15710i			0.172576i
$894$		0			0
$895$		−	17.1534i		−	0.573376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.16943			−0.105706
$900$		0			0
$901$			2.09031i			0.0696385i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	7.89084i		−	0.262300i
$906$		0			0
$907$	−16.5299			−0.548865			−0.274432	−	0.961606i	$-0.588490\pi$
$907$	−16.5299			−0.548865			−0.274432	+	0.961606i	$0.588490\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	48.8889i		−	1.61976i	−0.586594	−	0.809881i	$-0.699532\pi$
$911$		−	48.8889i		−	1.61976i	0.586594	−	0.809881i	$-0.300468\pi$
$912$		0			0
$913$			26.1251i			0.864616i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	43.5463	+	17.3977i	1.43803	+	0.574524i
$918$		0			0
$919$	−8.24441			−0.271958			−0.135979	−	0.990712i	$-0.543418\pi$
$919$	−8.24441			−0.271958			−0.135979	+	0.990712i	$0.543418\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	30.3390			0.998619
$924$		0			0
$925$	−12.4561			−0.409553
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.4613			−0.638504			−0.319252	−	0.947670i	$-0.603432\pi$
$929$	−19.4613			−0.638504			−0.319252	+	0.947670i	$0.603432\pi$
$930$		0			0
$931$	−24.1459	+	25.3949i	−0.791349	+	0.832286i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	0.378857i		−	0.0123899i
$936$		0			0
$937$			19.2022i			0.627310i	0.949537	+	0.313655i	$0.101554\pi$
$937$			19.2022i			0.627310i	−0.949537	+	0.313655i	$0.898446\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−32.3809			−1.05559			−0.527794	−	0.849373i	$-0.676981\pi$
$941$	−32.3809			−1.05559			−0.527794	+	0.849373i	$0.676981\pi$
$942$		0			0
$943$			23.4963i			0.765146i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			54.5365i			1.77220i	0.463496	+	0.886099i	$0.346595\pi$
$947$			54.5365i			1.77220i	−0.463496	+	0.886099i	$0.653405\pi$
$948$		0			0
$949$	−14.5090			−0.470981
$950$		0			0
$951$		0			0
$952$		0			0
$953$			41.5158i			1.34483i	0.740175	+	0.672414i	$0.234743\pi$
$953$			41.5158i			1.34483i	−0.740175	+	0.672414i	$0.765257\pi$
$954$		0			0
$955$			23.6434i			0.765084i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.7090	+	36.8165i	−0.474979	+	1.18887i
$960$		0			0
$961$	30.2944			0.977238
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.377245			−0.0121439
$966$		0			0
$967$	33.7358			1.08487			0.542435	−	0.840098i	$-0.317503\pi$
$967$	33.7358			1.08487			0.542435	+	0.840098i	$0.317503\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.5369			1.17253			0.586263	−	0.810121i	$-0.300599\pi$
$971$	36.5369			1.17253			0.586263	+	0.810121i	$0.300599\pi$
$972$		0			0
$973$	−1.03494	+	2.59045i	−0.0331788	+	0.0830460i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	44.7254i		−	1.43089i	−0.698669	−	0.715445i	$-0.746224\pi$
$977$		−	44.7254i		−	1.43089i	0.698669	−	0.715445i	$-0.253776\pi$
$978$		0			0
$979$		−	26.3605i		−	0.842484i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−43.7614			−1.39577			−0.697885	−	0.716209i	$-0.745876\pi$
$983$	−43.7614			−1.39577			−0.697885	+	0.716209i	$0.745876\pi$
$984$		0			0
$985$		−	7.07631i		−	0.225470i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	6.27935i		−	0.199672i
$990$		0			0
$991$	−17.4844			−0.555410			−0.277705	−	0.960666i	$-0.589574\pi$
$991$	−17.4844			−0.555410			−0.277705	+	0.960666i	$0.589574\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	24.0602i		−	0.762761i
$996$		0			0
$997$			23.3062i			0.738115i	0.929407	+	0.369058i	$0.120320\pi$
$997$			23.3062i			0.738115i	−0.929407	+	0.369058i	$0.879680\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.k.b.377.10	yes	16
3.2	odd	2	inner	1512.2.k.b.377.8	yes	16
4.3	odd	2		3024.2.k.l.1889.9		16
7.6	odd	2	inner	1512.2.k.b.377.7	✓	16
12.11	even	2		3024.2.k.l.1889.7		16
21.20	even	2	inner	1512.2.k.b.377.9	yes	16
28.27	even	2		3024.2.k.l.1889.8		16
84.83	odd	2		3024.2.k.l.1889.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.k.b.377.7	✓	16	7.6	odd	2	inner
1512.2.k.b.377.8	yes	16	3.2	odd	2	inner
1512.2.k.b.377.9	yes	16	21.20	even	2	inner
1512.2.k.b.377.10	yes	16	1.1	even	1	trivial
3024.2.k.l.1889.7		16	12.11	even	2
3024.2.k.l.1889.8		16	28.27	even	2
3024.2.k.l.1889.9		16	4.3	odd	2
3024.2.k.l.1889.10		16	84.83	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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.933004 q^{5} +(2.45692 + 0.981597i) q^{7} +1.75675i q^{11} +4.20692i q^{13} -0.231144 q^{17} +5.00597i q^{19} -2.61601i q^{23} -4.12950 q^{25} +3.77311i q^{29} +0.840005i q^{31} +(2.29232 + 0.915835i) q^{35} +3.01636 q^{37} -8.98174 q^{41} +2.40035 q^{43} +1.03019 q^{47} +(5.07293 + 4.82342i) q^{49} -9.04335i q^{53} +1.63905i q^{55} +6.77898 q^{59} +8.93236i q^{61} +3.92508i q^{65} +1.10252 q^{67} -7.21168i q^{71} +3.44883i q^{73} +(-1.72442 + 4.31619i) q^{77} +5.29783 q^{79} +14.8713 q^{83} -0.215658 q^{85} -15.0053 q^{89} +(-4.12950 + 10.3361i) q^{91} +4.67059i q^{95} +1.63905i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{7} + 36 q^{25} - 8 q^{37} + 20 q^{43} + 2 q^{49} + 44 q^{67} + 40 q^{79} + 16 q^{85} + 36 q^{91}+O(q^{100}) \) 16 * q - 2 * q^7 + 36 * q^25 - 8 * q^37 + 20 * q^43 + 2 * q^49 + 44 * q^67 + 40 * q^79 + 16 * q^85 + 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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