LMFDB - Embedded newform 2208.2.j.c.47.42

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2208,2,Mod(47,2208)] 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("2208.47"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2208 = 2^{5} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2208.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [42,0,-2,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$17.6309687663$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.42
Character	$\chi$	$=$	2208.47
Dual form			2208.2.j.c.47.41

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.73179 + 0.0302019i) q^{3} +0.871841 q^{5} +3.09231i q^{7} +(2.99818 + 0.104607i) q^{9} +1.90350i q^{11} +1.23856i q^{13} +(1.50984 + 0.0263313i) q^{15} -1.98300i q^{17} +4.96848 q^{19} +(-0.0933936 + 5.35522i) q^{21} +1.00000 q^{23} -4.23989 q^{25} +(5.18904 + 0.271707i) q^{27} -2.31109 q^{29} +3.48492i q^{31} +(-0.0574892 + 3.29645i) q^{33} +2.69600i q^{35} +4.56655i q^{37} +(-0.0374068 + 2.14492i) q^{39} -8.71457i q^{41} +3.26780 q^{43} +(2.61393 + 0.0912003i) q^{45} -7.68114 q^{47} -2.56238 q^{49} +(0.0598904 - 3.43414i) q^{51} +7.53051 q^{53} +1.65955i q^{55} +(8.60435 + 0.150058i) q^{57} +12.2120i q^{59} +10.7975i q^{61} +(-0.323476 + 9.27129i) q^{63} +1.07983i q^{65} +0.0562647 q^{67} +(1.73179 + 0.0302019i) q^{69} -7.84847 q^{71} -7.04796 q^{73} +(-7.34259 - 0.128053i) q^{75} -5.88620 q^{77} +5.10654i q^{79} +(8.97811 + 0.627258i) q^{81} +0.795490i q^{83} -1.72886i q^{85} +(-4.00232 - 0.0697994i) q^{87} -17.3867i q^{89} -3.83001 q^{91} +(-0.105251 + 6.03513i) q^{93} +4.33173 q^{95} +9.05221 q^{97} +(-0.199118 + 5.70701i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 2 q^{3} - 8 q^{5} + 2 q^{9} - 8 q^{15} - 4 q^{19} - 8 q^{21} + 42 q^{23} + 22 q^{25} + 16 q^{27} + 12 q^{33} + 8 q^{39} - 28 q^{43} + 8 q^{45} - 50 q^{49} - 28 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63}+ \cdots + 20 q^{97}+O(q^{100}) $ 42 * q - 2 * q^3 - 8 * q^5 + 2 * q^9 - 8 * q^15 - 4 * q^19 - 8 * q^21 + 42 * q^23 + 22 * q^25 + 16 * q^27 + 12 * q^33 + 8 * q^39 - 28 * q^43 + 8 * q^45 - 50 * q^49 - 28 * q^51 - 24 * q^53 - 8 * q^57 - 16 * q^63 + 4 * q^67 - 2 * q^69 + 4 * q^73 + 6 * q^75 - 32 * q^77 + 18 * q^81 - 48 * q^87 + 8 * q^91 + 22 * q^93 - 16 * q^95 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2208\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$737$	$1381$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	1.73179	+	0.0302019i	0.999848	+	0.0174371i
$3$	1.73179	+	0.0302019i	0.999848	+	0.0174371i
$4$		0			0
$5$	0.871841			0.389899			0.194950	−	0.980813i	$-0.437546\pi$
$5$	0.871841			0.389899			0.194950	+	0.980813i	$0.437546\pi$
$6$		0			0
$7$			3.09231i			1.16878i	0.811472	+	0.584392i	$0.198667\pi$
$7$			3.09231i			1.16878i	−0.811472	+	0.584392i	$0.801333\pi$
$8$		0			0
$9$	2.99818	+	0.104607i	0.999392	+	0.0348688i
$10$		0			0
$11$			1.90350i			0.573926i	0.957942	+	0.286963i	$0.0926456\pi$
$11$			1.90350i			0.573926i	−0.957942	+	0.286963i	$0.907354\pi$
$12$		0			0
$13$			1.23856i			0.343514i	0.985139	+	0.171757i	$0.0549444\pi$
$13$			1.23856i			0.343514i	−0.985139	+	0.171757i	$0.945056\pi$
$14$		0			0
$15$	1.50984	+	0.0263313i	0.389840	+	0.00679870i
$16$		0			0
$17$		−	1.98300i		−	0.480949i	−0.970655	−	0.240474i	$-0.922697\pi$
$17$		−	1.98300i		−	0.480949i	0.970655	−	0.240474i	$-0.0773030\pi$
$18$		0			0
$19$	4.96848			1.13985			0.569924	−	0.821697i	$-0.306973\pi$
$19$	4.96848			1.13985			0.569924	+	0.821697i	$0.306973\pi$
$20$		0			0
$21$	−0.0933936	+	5.35522i	−0.0203802	+	1.16861i
$22$		0			0
$23$	1.00000			0.208514
$23$	1.00000			0.208514
$24$		0			0
$25$	−4.23989			−0.847979
$26$		0			0
$27$	5.18904	+	0.271707i	0.998632	+	0.0522900i
$28$		0			0
$29$	−2.31109			−0.429159			−0.214580	−	0.976706i	$-0.568838\pi$
$29$	−2.31109			−0.429159			−0.214580	+	0.976706i	$0.568838\pi$
$30$		0			0
$31$			3.48492i			0.625909i	0.949768	+	0.312955i	$0.101319\pi$
$31$			3.48492i			0.625909i	−0.949768	+	0.312955i	$0.898681\pi$
$32$		0			0
$33$	−0.0574892	+	3.29645i	−0.0100076	+	0.573838i
$34$		0			0
$35$			2.69600i			0.455708i
$36$		0			0
$37$			4.56655i			0.750735i	0.926876	+	0.375368i	$0.122484\pi$
$37$			4.56655i			0.750735i	−0.926876	+	0.375368i	$0.877516\pi$
$38$		0			0
$39$	−0.0374068	+	2.14492i	−0.00598989	+	0.343462i
$40$		0			0
$41$		−	8.71457i		−	1.36099i	−0.732754	−	0.680493i	$-0.761766\pi$
$41$		−	8.71457i		−	1.36099i	0.732754	−	0.680493i	$-0.238234\pi$
$42$		0			0
$43$	3.26780			0.498336			0.249168	−	0.968460i	$-0.419843\pi$
$43$	3.26780			0.498336			0.249168	+	0.968460i	$0.419843\pi$
$44$		0			0
$45$	2.61393	+	0.0912003i	0.389662	+	0.0135953i
$46$		0			0
$47$	−7.68114			−1.12041			−0.560205	−	0.828354i	$-0.689277\pi$
$47$	−7.68114			−1.12041			−0.560205	+	0.828354i	$0.689277\pi$
$48$		0			0
$49$	−2.56238			−0.366054
$50$		0			0
$51$	0.0598904	−	3.43414i	0.00838634	−	0.480876i
$52$		0			0
$53$	7.53051			1.03439			0.517197	−	0.855866i	$-0.326975\pi$
$53$	7.53051			1.03439			0.517197	+	0.855866i	$0.326975\pi$
$54$		0			0
$55$			1.65955i			0.223773i
$56$		0			0
$57$	8.60435	+	0.150058i	1.13967	+	0.0198756i
$58$		0			0
$59$			12.2120i			1.58986i	0.606698	+	0.794932i	$0.292494\pi$
$59$			12.2120i			1.58986i	−0.606698	+	0.794932i	$0.707506\pi$
$60$		0			0
$61$			10.7975i			1.38248i	0.722623	+	0.691242i	$0.242936\pi$
$61$			10.7975i			1.38248i	−0.722623	+	0.691242i	$0.757064\pi$
$62$		0			0
$63$	−0.323476	+	9.27129i	−0.0407541	+	1.16807i
$64$		0			0
$65$			1.07983i			0.133936i
$66$		0			0
$67$	0.0562647			0.00687383			0.00343692	−	0.999994i	$-0.498906\pi$
$67$	0.0562647			0.00687383			0.00343692	+	0.999994i	$0.498906\pi$
$68$		0			0
$69$	1.73179	+	0.0302019i	0.208483	+	0.00363588i
$70$		0			0
$71$	−7.84847			−0.931442			−0.465721	−	0.884932i	$-0.654205\pi$
$71$	−7.84847			−0.931442			−0.465721	+	0.884932i	$0.654205\pi$
$72$		0			0
$73$	−7.04796			−0.824902			−0.412451	−	0.910980i	$-0.635327\pi$
$73$	−7.04796			−0.824902			−0.412451	+	0.910980i	$0.635327\pi$
$74$		0			0
$75$	−7.34259	−	0.128053i	−0.847850	−	0.0147863i
$76$		0			0
$77$	−5.88620			−0.670795
$78$		0			0
$79$			5.10654i			0.574531i	0.957851	+	0.287265i	$0.0927462\pi$
$79$			5.10654i			0.574531i	−0.957851	+	0.287265i	$0.907254\pi$
$80$		0			0
$81$	8.97811	+	0.627258i	0.997568	+	0.0696953i
$82$		0			0
$83$			0.795490i			0.0873163i	0.999047	+	0.0436582i	$0.0139012\pi$
$83$			0.795490i			0.0873163i	−0.999047	+	0.0436582i	$0.986099\pi$
$84$		0			0
$85$		−	1.72886i		−	0.187522i
$86$		0			0
$87$	−4.00232	−	0.0697994i	−0.429094	−	0.00748329i
$88$		0			0
$89$		−	17.3867i		−	1.84298i	−0.388399	−	0.921491i	$-0.626972\pi$
$89$		−	17.3867i		−	1.84298i	0.388399	−	0.921491i	$-0.373028\pi$
$90$		0			0
$91$	−3.83001			−0.401494
$92$		0			0
$93$	−0.105251	+	6.03513i	−0.0109140	+	0.625814i
$94$		0			0
$95$	4.33173			0.444426
$96$		0			0
$97$	9.05221			0.919113			0.459557	−	0.888149i	$-0.348008\pi$
$97$	9.05221			0.919113			0.459557	+	0.888149i	$0.348008\pi$
$98$		0			0
$99$	−0.199118	+	5.70701i	−0.0200121	+	0.573577i
$100$		0			0
$101$	19.5806			1.94835			0.974173	−	0.225803i	$-0.0725005\pi$
$101$	19.5806			1.94835			0.974173	+	0.225803i	$0.0725005\pi$
$102$		0			0
$103$			2.96013i			0.291670i	0.989309	+	0.145835i	$0.0465869\pi$
$103$			2.96013i			0.291670i	−0.989309	+	0.145835i	$0.953413\pi$
$104$		0			0
$105$	−0.0814244	+	4.66890i	−0.00794621	+	0.455638i
$106$		0			0
$107$		−	8.95826i		−	0.866028i	−0.901387	−	0.433014i	$-0.857450\pi$
$107$		−	8.95826i		−	0.866028i	0.901387	−	0.433014i	$-0.142550\pi$
$108$		0			0
$109$		−	9.35024i		−	0.895591i	−0.894136	−	0.447795i	$-0.852209\pi$
$109$		−	9.35024i		−	0.895591i	0.894136	−	0.447795i	$-0.147791\pi$
$110$		0			0
$111$	−0.137918	+	7.90829i	−0.0130906	+	0.750621i
$112$		0			0
$113$		−	1.47151i		−	0.138428i	−0.997602	−	0.0692140i	$-0.977951\pi$
$113$		−	1.47151i		−	0.138428i	0.997602	−	0.0692140i	$-0.0220491\pi$
$114$		0			0
$115$	0.871841			0.0812996
$116$		0			0
$117$	−0.129561	+	3.71342i	−0.0119779	+	0.343305i
$118$		0			0
$119$	6.13206			0.562125
$120$		0			0
$121$	7.37670			0.670609
$122$		0			0
$123$	0.263196	−	15.0918i	0.0237316	−	1.36078i
$124$		0			0
$125$	−8.05572			−0.720525
$126$		0			0
$127$		−	6.90921i		−	0.613093i	−0.951856	−	0.306547i	$-0.900826\pi$
$127$		−	6.90921i		−	0.613093i	0.951856	−	0.306547i	$-0.0991735\pi$
$128$		0			0
$129$	5.65914	+	0.0986939i	0.498260	+	0.00868951i
$130$		0			0
$131$			12.5716i			1.09839i	0.835695	+	0.549193i	$0.185065\pi$
$131$			12.5716i			1.09839i	−0.835695	+	0.549193i	$0.814935\pi$
$132$		0			0
$133$			15.3641i			1.33224i
$134$		0			0
$135$	4.52402	+	0.236885i	0.389366	+	0.0203878i
$136$		0			0
$137$			1.78484i			0.152490i	0.997089	+	0.0762448i	$0.0242930\pi$
$137$			1.78484i			0.152490i	−0.997089	+	0.0762448i	$0.975707\pi$
$138$		0			0
$139$	−5.90847			−0.501150			−0.250575	−	0.968097i	$-0.580620\pi$
$139$	−5.90847			−0.501150			−0.250575	+	0.968097i	$0.580620\pi$
$140$		0			0
$141$	−13.3021	−	0.231985i	−1.12024	−	0.0195367i
$142$		0			0
$143$	−2.35759			−0.197152
$144$		0			0
$145$	−2.01491			−0.167329
$146$		0			0
$147$	−4.43750	−	0.0773888i	−0.365999	−	0.00638292i
$148$		0			0
$149$	−18.3640			−1.50444			−0.752219	−	0.658914i	$-0.771016\pi$
$149$	−18.3640			−1.50444			−0.752219	+	0.658914i	$0.771016\pi$
$150$		0			0
$151$		−	5.65286i		−	0.460023i	−0.973188	−	0.230012i	$-0.926124\pi$
$151$		−	5.65286i		−	0.460023i	0.973188	−	0.230012i	$-0.0738764\pi$
$152$		0			0
$153$	0.207435	−	5.94539i	0.0167701	−	0.480656i
$154$		0			0
$155$			3.03829i			0.244041i
$156$		0			0
$157$		−	17.6786i		−	1.41090i	−0.708758	−	0.705451i	$-0.750744\pi$
$157$		−	17.6786i		−	1.41090i	0.708758	−	0.705451i	$-0.249256\pi$
$158$		0			0
$159$	13.0412	+	0.227436i	1.03424	+	0.0180368i
$160$		0			0
$161$			3.09231i			0.243708i
$162$		0			0
$163$	15.4158			1.20746			0.603728	−	0.797191i	$-0.293681\pi$
$163$	15.4158			1.20746			0.603728	+	0.797191i	$0.293681\pi$
$164$		0			0
$165$	−0.0501214	+	2.87398i	−0.00390195	+	0.223739i
$166$		0			0
$167$	9.04899			0.700232			0.350116	−	0.936706i	$-0.386142\pi$
$167$	9.04899			0.700232			0.350116	+	0.936706i	$0.386142\pi$
$168$		0			0
$169$	11.4660			0.881998
$170$		0			0
$171$	14.8964	+	0.519736i	1.13915	+	0.0397452i
$172$		0			0
$173$	−2.83727			−0.215714			−0.107857	−	0.994166i	$-0.534399\pi$
$173$	−2.83727			−0.215714			−0.107857	+	0.994166i	$0.534399\pi$
$174$		0			0
$175$		−	13.1111i		−	0.991103i
$176$		0			0
$177$	−0.368825	+	21.1486i	−0.0277226	+	1.58962i
$178$		0			0
$179$		−	10.5834i		−	0.791043i	−0.918457	−	0.395521i	$-0.870564\pi$
$179$		−	10.5834i		−	0.791043i	0.918457	−	0.395521i	$-0.129436\pi$
$180$		0			0
$181$		−	8.60505i		−	0.639608i	−0.947484	−	0.319804i	$-0.896383\pi$
$181$		−	8.60505i		−	0.639608i	0.947484	−	0.319804i	$-0.103617\pi$
$182$		0			0
$183$	−0.326106	+	18.6991i	−0.0241065	+	1.38227i
$184$		0			0
$185$			3.98130i			0.292711i
$186$		0			0
$187$	3.77464			0.276029
$188$		0			0
$189$	−0.840202	+	16.0461i	−0.0611157	+	1.16718i
$190$		0			0
$191$	−14.6068			−1.05691			−0.528454	−	0.848962i	$-0.677228\pi$
$191$	−14.6068			−1.05691			−0.528454	+	0.848962i	$0.677228\pi$
$192$		0			0
$193$	−5.48507			−0.394824			−0.197412	−	0.980321i	$-0.563254\pi$
$193$	−5.48507			−0.394824			−0.197412	+	0.980321i	$0.563254\pi$
$194$		0			0
$195$	−0.0326128	+	1.87003i	−0.00233545	+	0.133916i
$196$		0			0
$197$	21.8619			1.55760			0.778798	−	0.627274i	$-0.215830\pi$
$197$	21.8619			1.55760			0.778798	+	0.627274i	$0.215830\pi$
$198$		0			0
$199$		−	19.5960i		−	1.38912i	−0.719435	−	0.694560i	$-0.755599\pi$
$199$		−	19.5960i		−	1.38912i	0.719435	−	0.694560i	$-0.244401\pi$
$200$		0			0
$201$	0.0974385	+	0.00169930i	0.00687279	+	0.000119859i
$202$		0			0
$203$		−	7.14662i		−	0.501594i
$204$		0			0
$205$		−	7.59772i		−	0.530647i
$206$		0			0
$207$	2.99818	+	0.104607i	0.208388	+	0.00727066i
$208$		0			0
$209$			9.45748i			0.654188i
$210$		0			0
$211$	−9.82214			−0.676184			−0.338092	−	0.941113i	$-0.609782\pi$
$211$	−9.82214			−0.676184			−0.338092	+	0.941113i	$0.609782\pi$
$212$		0			0
$213$	−13.5919	−	0.237039i	−0.931300	−	0.0162416i
$214$		0			0
$215$	2.84901			0.194301
$216$		0			0
$217$	−10.7764			−0.731552
$218$		0			0
$219$	−12.2056	−	0.212862i	−0.824776	−	0.0143839i
$220$		0			0
$221$	2.45607			0.165213
$222$		0			0
$223$			5.37339i			0.359829i	0.983682	+	0.179914i	$0.0575821\pi$
$223$			5.37339i			0.359829i	−0.983682	+	0.179914i	$0.942418\pi$
$224$		0			0
$225$	−12.7119	−	0.443521i	−0.847463	−	0.0295680i
$226$		0			0
$227$		−	23.0536i		−	1.53012i	−0.643957	−	0.765061i	$-0.722709\pi$
$227$		−	23.0536i		−	1.53012i	0.643957	−	0.765061i	$-0.277291\pi$
$228$		0			0
$229$		−	27.2263i		−	1.79916i	−0.436752	−	0.899582i	$-0.643871\pi$
$229$		−	27.2263i		−	1.79916i	0.436752	−	0.899582i	$-0.356129\pi$
$230$		0			0
$231$	−10.1936	−	0.177774i	−0.670693	−	0.0116967i
$232$		0			0
$233$			21.6173i			1.41620i	0.706114	+	0.708098i	$0.250447\pi$
$233$			21.6173i			1.41620i	−0.706114	+	0.708098i	$0.749553\pi$
$234$		0			0
$235$	−6.69673			−0.436846
$236$		0			0
$237$	−0.154227	+	8.84345i	−0.0100181	+	0.574444i
$238$		0			0
$239$	17.6777			1.14348			0.571738	−	0.820436i	$-0.306270\pi$
$239$	17.6777			1.14348			0.571738	+	0.820436i	$0.306270\pi$
$240$		0			0
$241$	−16.3445			−1.05284			−0.526419	−	0.850225i	$-0.676466\pi$
$241$	−16.3445			−1.05284			−0.526419	+	0.850225i	$0.676466\pi$
$242$		0			0
$243$	15.5292	+	1.35743i	0.996201	+	0.0870794i
$244$		0			0
$245$	−2.23399			−0.142724
$246$		0			0
$247$			6.15376i			0.391554i
$248$		0			0
$249$	−0.0240253	+	1.37762i	−0.00152254	+	0.0873030i
$250$		0			0
$251$			14.1286i			0.891791i	0.895085	+	0.445896i	$0.147115\pi$
$251$			14.1286i			0.891791i	−0.895085	+	0.445896i	$0.852885\pi$
$252$		0			0
$253$			1.90350i			0.119672i
$254$		0			0
$255$	0.0522149	−	2.99402i	0.00326983	−	0.187493i
$256$		0			0
$257$			13.2937i			0.829238i	0.909995	+	0.414619i	$0.136085\pi$
$257$			13.2937i			0.829238i	−0.909995	+	0.414619i	$0.863915\pi$
$258$		0			0
$259$	−14.1212			−0.877447
$260$		0			0
$261$	−6.92907	−	0.241756i	−0.428898	−	0.0149643i
$262$		0			0
$263$	−25.3919			−1.56573			−0.782865	−	0.622192i	$-0.786242\pi$
$263$	−25.3919			−1.56573			−0.782865	+	0.622192i	$0.786242\pi$
$264$		0			0
$265$	6.56541			0.403310
$266$		0			0
$267$	0.525110	−	30.1100i	0.0321362	−	1.84270i
$268$		0			0
$269$	2.58876			0.157840			0.0789198	−	0.996881i	$-0.474853\pi$
$269$	2.58876			0.157840			0.0789198	+	0.996881i	$0.474853\pi$
$270$		0			0
$271$		−	29.5135i		−	1.79282i	−0.443226	−	0.896410i	$-0.646166\pi$
$271$		−	29.5135i		−	1.79282i	0.443226	−	0.896410i	$-0.353834\pi$
$272$		0			0
$273$	−6.63276	−	0.115673i	−0.401433	−	0.00700088i
$274$		0			0
$275$		−	8.07062i		−	0.486677i
$276$		0			0
$277$		−	1.08750i		−	0.0653413i	−0.999466	−	0.0326707i	$-0.989599\pi$
$277$		−	1.08750i		−	0.0653413i	0.999466	−	0.0326707i	$-0.0104012\pi$
$278$		0			0
$279$	−0.364545	+	10.4484i	−0.0218247	+	0.625529i
$280$		0			0
$281$			3.26761i			0.194929i	0.995239	+	0.0974647i	$0.0310733\pi$
$281$			3.26761i			0.194929i	−0.995239	+	0.0974647i	$0.968927\pi$
$282$		0			0
$283$	−6.37119			−0.378728			−0.189364	−	0.981907i	$-0.560643\pi$
$283$	−6.37119			−0.378728			−0.189364	+	0.981907i	$0.560643\pi$
$284$		0			0
$285$	7.50163	+	0.130826i	0.444358	+	0.00774948i
$286$		0			0
$287$	26.9481			1.59070
$288$		0			0
$289$	13.0677			0.768688
$290$		0			0
$291$	15.6765	+	0.273394i	0.918973	+	0.0160266i
$292$		0			0
$293$	24.5319			1.43317			0.716584	−	0.697501i	$-0.245705\pi$
$293$	24.5319			1.43317			0.716584	+	0.697501i	$0.245705\pi$
$294$		0			0
$295$			10.6469i			0.619887i
$296$		0			0
$297$	−0.517193	+	9.87732i	−0.0300106	+	0.573140i
$298$		0			0
$299$			1.23856i			0.0716277i
$300$		0			0
$301$			10.1051i			0.582446i
$302$		0			0
$303$	33.9095	+	0.591372i	1.94805	+	0.0339735i
$304$		0			0
$305$			9.41374i			0.539029i
$306$		0			0
$307$	−6.49267			−0.370556			−0.185278	−	0.982686i	$-0.559319\pi$
$307$	−6.49267			−0.370556			−0.185278	+	0.982686i	$0.559319\pi$
$308$		0			0
$309$	−0.0894015	+	5.12631i	−0.00508587	+	0.291626i
$310$		0			0
$311$	−26.3522			−1.49430			−0.747149	−	0.664657i	$-0.768578\pi$
$311$	−26.3522			−1.49430			−0.747149	+	0.664657i	$0.768578\pi$
$312$		0			0
$313$	−33.2164			−1.87750			−0.938750	−	0.344599i	$-0.888015\pi$
$313$	−33.2164			−1.87750			−0.938750	+	0.344599i	$0.888015\pi$
$314$		0			0
$315$	−0.282019	+	8.08309i	−0.0158900	+	0.455430i
$316$		0			0
$317$	−21.3265			−1.19782			−0.598908	−	0.800818i	$-0.704399\pi$
$317$	−21.3265			−1.19782			−0.598908	+	0.800818i	$0.704399\pi$
$318$		0			0
$319$		−	4.39916i		−	0.246306i
$320$		0			0
$321$	0.270556	−	15.5138i	0.0151010	−	0.865896i
$322$		0			0
$323$		−	9.85251i		−	0.548208i
$324$		0			0
$325$		−	5.25136i		−	0.291293i
$326$		0			0
$327$	0.282395	−	16.1926i	0.0156165	−	0.895454i
$328$		0			0
$329$		−	23.7525i		−	1.30952i
$330$		0			0
$331$	7.14989			0.392994			0.196497	−	0.980504i	$-0.437043\pi$
$331$	7.14989			0.392994			0.196497	+	0.980504i	$0.437043\pi$
$332$		0			0
$333$	−0.477691	+	13.6913i	−0.0261773	+	0.750279i
$334$		0			0
$335$	0.0490539			0.00268010
$336$		0			0
$337$	7.67027			0.417826			0.208913	−	0.977934i	$-0.433007\pi$
$337$	7.67027			0.417826			0.208913	+	0.977934i	$0.433007\pi$
$338$		0			0
$339$	0.0444424	−	2.54834i	0.00241378	−	0.138407i
$340$		0			0
$341$	−6.63352			−0.359225
$342$		0			0
$343$			13.7225i			0.740945i
$344$		0			0
$345$	1.50984	+	0.0263313i	0.0812872	+	0.00141763i
$346$		0			0
$347$			27.3593i			1.46872i	0.678759	+	0.734361i	$0.262518\pi$
$347$			27.3593i			1.46872i	−0.678759	+	0.734361i	$0.737482\pi$
$348$		0			0
$349$			6.07938i			0.325422i	0.986674	+	0.162711i	$0.0520238\pi$
$349$			6.07938i			0.325422i	−0.986674	+	0.162711i	$0.947976\pi$
$350$		0			0
$351$	−0.336525	+	6.42693i	−0.0179624	+	0.343044i
$352$		0			0
$353$		−	25.6872i		−	1.36719i	−0.729861	−	0.683595i	$-0.760415\pi$
$353$		−	25.6872i		−	1.36719i	0.729861	−	0.683595i	$-0.239585\pi$
$354$		0			0
$355$	−6.84262			−0.363168
$356$		0			0
$357$	10.6194	+	0.185200i	0.562039	+	0.00980181i
$358$		0			0
$359$	−17.9590			−0.947838			−0.473919	−	0.880568i	$-0.657161\pi$
$359$	−17.9590			−0.947838			−0.473919	+	0.880568i	$0.657161\pi$
$360$		0			0
$361$	5.68581			0.299253
$362$		0			0
$363$	12.7749	+	0.222790i	0.670508	+	0.0116935i
$364$		0			0
$365$	−6.14470			−0.321628
$366$		0			0
$367$			22.5647i			1.17787i	0.808180	+	0.588935i	$0.200453\pi$
$367$			22.5647i			1.17787i	−0.808180	+	0.588935i	$0.799547\pi$
$368$		0			0
$369$	0.911601	−	26.1278i	0.0474560	−	1.36016i
$370$		0			0
$371$			23.2867i			1.20898i
$372$		0			0
$373$			19.6359i			1.01671i	0.861147	+	0.508355i	$0.169746\pi$
$373$			19.6359i			1.01671i	−0.861147	+	0.508355i	$0.830254\pi$
$374$		0			0
$375$	−13.9508	−	0.243298i	−0.720416	−	0.0125639i
$376$		0			0
$377$		−	2.86243i		−	0.147422i
$378$		0			0
$379$	13.9167			0.714855			0.357428	−	0.933941i	$-0.383654\pi$
$379$	13.9167			0.714855			0.357428	+	0.933941i	$0.383654\pi$
$380$		0			0
$381$	0.208671	−	11.9653i	0.0106905	−	0.613000i
$382$		0			0
$383$	27.2328			1.39153			0.695766	−	0.718269i	$-0.255065\pi$
$383$	27.2328			1.39153			0.695766	+	0.718269i	$0.255065\pi$
$384$		0			0
$385$	−5.13183			−0.261542
$386$		0			0
$387$	9.79745	+	0.341834i	0.498032	+	0.0173764i
$388$		0			0
$389$	−0.198260			−0.0100522			−0.00502608	−	0.999987i	$-0.501600\pi$
$389$	−0.198260			−0.0100522			−0.00502608	+	0.999987i	$0.501600\pi$
$390$		0			0
$391$		−	1.98300i		−	0.100285i
$392$		0			0
$393$	−0.379687	+	21.7714i	−0.0191527	+	1.09822i
$394$		0			0
$395$			4.45209i			0.224009i
$396$		0			0
$397$		−	12.8510i		−	0.644971i	−0.946574	−	0.322485i	$-0.895482\pi$
$397$		−	12.8510i		−	0.644971i	0.946574	−	0.322485i	$-0.104518\pi$
$398$		0			0
$399$	−0.464025	+	26.6073i	−0.0232303	+	1.33203i
$400$		0			0
$401$		−	23.3200i		−	1.16454i	−0.812994	−	0.582272i	$-0.802164\pi$
$401$		−	23.3200i		−	1.16454i	0.812994	−	0.582272i	$-0.197836\pi$
$402$		0			0
$403$	−4.31627			−0.215009
$404$		0			0
$405$	7.82749	+	0.546869i	0.388951	+	0.0271741i
$406$		0			0
$407$	−8.69240			−0.430866
$408$		0			0
$409$	2.90989			0.143885			0.0719425	−	0.997409i	$-0.477080\pi$
$409$	2.90989			0.143885			0.0719425	+	0.997409i	$0.477080\pi$
$410$		0			0
$411$	−0.0539057	+	3.09097i	−0.00265897	+	0.152466i
$412$		0			0
$413$	−37.7632			−1.85821
$414$		0			0
$415$			0.693540i			0.0340446i
$416$		0			0
$417$	−10.2322	−	0.178447i	−0.501074	−	0.00873859i
$418$		0			0
$419$			15.9445i			0.778941i	0.921039	+	0.389470i	$0.127342\pi$
$419$			15.9445i			0.778941i	−0.921039	+	0.389470i	$0.872658\pi$
$420$		0			0
$421$		−	30.6791i		−	1.49521i	−0.664144	−	0.747604i	$-0.731204\pi$
$421$		−	30.6791i		−	1.49521i	0.664144	−	0.747604i	$-0.268796\pi$
$422$		0			0
$423$	−23.0294	−	0.803497i	−1.11973	−	0.0390674i
$424$		0			0
$425$			8.40772i			0.407834i
$426$		0			0
$427$	−33.3894			−1.61582
$428$		0			0
$429$	−4.08285	−	0.0712037i	−0.197122	−	0.00343775i
$430$		0			0
$431$	35.6205			1.71578			0.857890	−	0.513833i	$-0.171775\pi$
$431$	35.6205			1.71578			0.857890	+	0.513833i	$0.171775\pi$
$432$		0			0
$433$	3.15606			0.151670			0.0758352	−	0.997120i	$-0.475838\pi$
$433$	3.15606			0.151670			0.0758352	+	0.997120i	$0.475838\pi$
$434$		0			0
$435$	−3.48939	−	0.0608540i	−0.167303	−	0.00291773i
$436$		0			0
$437$	4.96848			0.237675
$438$		0			0
$439$		−	27.8197i		−	1.32776i	−0.747839	−	0.663880i	$-0.768908\pi$
$439$		−	27.8197i		−	1.32776i	0.747839	−	0.663880i	$-0.231092\pi$
$440$		0			0
$441$	−7.68247	−	0.268042i	−0.365832	−	0.0127639i
$442$		0			0
$443$			6.53287i			0.310386i	0.987884	+	0.155193i	$0.0496000\pi$
$443$			6.53287i			0.310386i	−0.987884	+	0.155193i	$0.950400\pi$
$444$		0			0
$445$		−	15.1584i		−	0.718577i
$446$		0			0
$447$	−31.8025	−	0.554628i	−1.50421	−	0.0262330i
$448$		0			0
$449$			25.4871i			1.20281i	0.798943	+	0.601406i	$0.205393\pi$
$449$			25.4871i			1.20281i	−0.798943	+	0.601406i	$0.794607\pi$
$450$		0			0
$451$	16.5881			0.781105
$452$		0			0
$453$	0.170727	−	9.78955i	0.00802146	−	0.459953i
$454$		0			0
$455$	−3.33916			−0.156542
$456$		0			0
$457$	2.40286			0.112401			0.0562004	−	0.998420i	$-0.482101\pi$
$457$	2.40286			0.112401			0.0562004	+	0.998420i	$0.482101\pi$
$458$		0			0
$459$	0.538796	−	10.2899i	0.0251488	−	0.480291i
$460$		0			0
$461$	14.0637			0.655013			0.327506	−	0.944849i	$-0.393792\pi$
$461$	14.0637			0.655013			0.327506	+	0.944849i	$0.393792\pi$
$462$		0			0
$463$		−	35.7689i		−	1.66232i	−0.556031	−	0.831161i	$-0.687677\pi$
$463$		−	35.7689i		−	1.66232i	0.556031	−	0.831161i	$-0.312323\pi$
$464$		0			0
$465$	−0.0917622	+	5.26168i	−0.00425537	+	0.244004i
$466$		0			0
$467$		−	39.3360i		−	1.82025i	−0.414331	−	0.910126i	$-0.635985\pi$
$467$		−	39.3360i		−	1.82025i	0.414331	−	0.910126i	$-0.364015\pi$
$468$		0			0
$469$			0.173988i			0.00803402i
$470$		0			0
$471$	0.533926	−	30.6155i	0.0246020	−	1.41069i
$472$		0			0
$473$			6.22025i			0.286007i
$474$		0			0
$475$	−21.0658			−0.966567
$476$		0			0
$477$	22.5778	+	0.787740i	1.03377	+	0.0360682i
$478$		0			0
$479$	2.16149			0.0987611			0.0493805	−	0.998780i	$-0.484275\pi$
$479$	2.16149			0.0987611			0.0493805	+	0.998780i	$0.484275\pi$
$480$		0			0
$481$	−5.65593			−0.257888
$482$		0			0
$483$	−0.0933936	+	5.35522i	−0.00424956	+	0.243671i
$484$		0			0
$485$	7.89209			0.358361
$486$		0			0
$487$		−	28.2115i		−	1.27839i	−0.769046	−	0.639193i	$-0.779268\pi$
$487$		−	28.2115i		−	1.27839i	0.769046	−	0.639193i	$-0.220732\pi$
$488$		0			0
$489$	26.6968	+	0.465585i	1.20727	+	0.0210545i
$490$		0			0
$491$		−	5.65453i		−	0.255185i	−0.991827	−	0.127593i	$-0.959275\pi$
$491$		−	5.65453i		−	0.255185i	0.991827	−	0.127593i	$-0.0407250\pi$
$492$		0			0
$493$			4.58291i			0.206404i
$494$		0			0
$495$	−0.173599	+	4.97561i	−0.00780271	+	0.223637i
$496$		0			0
$497$		−	24.2699i		−	1.08865i
$498$		0			0
$499$	−31.1605			−1.39494			−0.697468	−	0.716616i	$-0.745690\pi$
$499$	−31.1605			−1.39494			−0.697468	+	0.716616i	$0.745690\pi$
$500$		0			0
$501$	15.6709	+	0.273297i	0.700125	+	0.0122100i
$502$		0			0
$503$	−0.784770			−0.0349912			−0.0174956	−	0.999847i	$-0.505569\pi$
$503$	−0.784770			−0.0349912			−0.0174956	+	0.999847i	$0.505569\pi$
$504$		0			0
$505$	17.0712			0.759658
$506$		0			0
$507$	19.8566	+	0.346294i	0.881864	+	0.0153795i
$508$		0			0
$509$	−1.62618			−0.0720790			−0.0360395	−	0.999350i	$-0.511474\pi$
$509$	−1.62618			−0.0720790			−0.0360395	+	0.999350i	$0.511474\pi$
$510$		0			0
$511$		−	21.7945i		−	0.964131i
$512$		0			0
$513$	25.7817	+	1.34997i	1.13829	+	0.0596027i
$514$		0			0
$515$			2.58076i			0.113722i
$516$		0			0
$517$		−	14.6210i		−	0.643031i
$518$		0			0
$519$	−4.91355	−	0.0856910i	−0.215681	−	0.00376142i
$520$		0			0
$521$		−	22.0363i		−	0.965429i	−0.875778	−	0.482715i	$-0.839651\pi$
$521$		−	22.0363i		−	0.965429i	0.875778	−	0.482715i	$-0.160349\pi$
$522$		0			0
$523$	0.873026			0.0381748			0.0190874	−	0.999818i	$-0.493924\pi$
$523$	0.873026			0.0381748			0.0190874	+	0.999818i	$0.493924\pi$
$524$		0			0
$525$	0.395979	−	22.7056i	0.0172819	−	0.990953i
$526$		0			0
$527$	6.91060			0.301030
$528$		0			0
$529$	1.00000			0.0434783
$530$		0			0
$531$	−1.27745	+	36.6137i	−0.0554368	+	1.58890i
$532$		0			0
$533$	10.7935			0.467518
$534$		0			0
$535$		−	7.81018i		−	0.337663i
$536$		0			0
$537$	0.319640	−	18.3282i	0.0137935	−	0.790922i
$538$		0			0
$539$		−	4.87748i		−	0.210088i
$540$		0			0
$541$		−	40.4127i		−	1.73748i	−0.495269	−	0.868739i	$-0.664931\pi$
$541$		−	40.4127i		−	1.73748i	0.495269	−	0.868739i	$-0.335069\pi$
$542$		0			0
$543$	0.259889	−	14.9021i	0.0111529	−	0.639511i
$544$		0			0
$545$		−	8.15192i		−	0.349190i
$546$		0			0
$547$	−26.3268			−1.12565			−0.562825	−	0.826576i	$-0.690286\pi$
$547$	−26.3268			−1.12565			−0.562825	+	0.826576i	$0.690286\pi$
$548$		0			0
$549$	−1.12949	+	32.3729i	−0.0482056	+	1.38164i
$550$		0			0
$551$	−11.4826			−0.489176
$552$		0			0
$553$	−15.7910			−0.671502
$554$		0			0
$555$	−0.120243	+	6.89477i	−0.00510402	+	0.292667i
$556$		0			0
$557$	−26.7166			−1.13202			−0.566009	−	0.824399i	$-0.691513\pi$
$557$	−26.7166			−1.13202			−0.566009	+	0.824399i	$0.691513\pi$
$558$		0			0
$559$			4.04737i			0.171185i
$560$		0			0
$561$	6.53687	+	0.114001i	0.275987	+	0.00481313i
$562$		0			0
$563$		−	13.0343i		−	0.549330i	−0.961540	−	0.274665i	$-0.911433\pi$
$563$		−	13.0343i		−	0.549330i	0.961540	−	0.274665i	$-0.0885669\pi$
$564$		0			0
$565$		−	1.28292i		−	0.0539729i
$566$		0			0
$567$	−1.93967	+	27.7631i	−0.0814587	+	1.16594i
$568$		0			0
$569$			38.2253i			1.60249i	0.598338	+	0.801244i	$0.295828\pi$
$569$			38.2253i			1.60249i	−0.598338	+	0.801244i	$0.704172\pi$
$570$		0			0
$571$	−25.1413			−1.05213			−0.526066	−	0.850444i	$-0.676333\pi$
$571$	−25.1413			−1.05213			−0.526066	+	0.850444i	$0.676333\pi$
$572$		0			0
$573$	−25.2958	−	0.441152i	−1.05675	−	0.0184294i
$574$		0			0
$575$	−4.23989			−0.176816
$576$		0			0
$577$	−20.3082			−0.845442			−0.422721	−	0.906260i	$-0.638925\pi$
$577$	−20.3082			−0.845442			−0.422721	+	0.906260i	$0.638925\pi$
$578$		0			0
$579$	−9.49898	−	0.165660i	−0.394764	−	0.00688458i
$580$		0			0
$581$	−2.45990			−0.102054
$582$		0			0
$583$			14.3343i			0.593666i
$584$		0			0
$585$	−0.112957	+	3.23751i	−0.00467019	+	0.133854i
$586$		0			0
$587$			15.5366i			0.641263i	0.947204	+	0.320631i	$0.103895\pi$
$587$			15.5366i			0.641263i	−0.947204	+	0.320631i	$0.896105\pi$
$588$		0			0
$589$			17.3147i			0.713441i
$590$		0			0
$591$	37.8602	+	0.660271i	1.55736	+	0.0271599i
$592$		0			0
$593$			6.77501i			0.278216i	0.990277	+	0.139108i	$0.0444236\pi$
$593$			6.77501i			0.278216i	−0.990277	+	0.139108i	$0.955576\pi$
$594$		0			0
$595$	5.34618			0.219172
$596$		0			0
$597$	0.591835	−	33.9360i	0.0242222	−	1.38891i
$598$		0			0
$599$	−32.7522			−1.33822			−0.669109	−	0.743164i	$-0.733324\pi$
$599$	−32.7522			−1.33822			−0.669109	+	0.743164i	$0.733324\pi$
$600$		0			0
$601$	−7.83996			−0.319799			−0.159899	−	0.987133i	$-0.551117\pi$
$601$	−7.83996			−0.319799			−0.159899	+	0.987133i	$0.551117\pi$
$602$		0			0
$603$	0.168692	+	0.00588566i	0.00686965	+	0.000239683i
$604$		0			0
$605$	6.43131			0.261470
$606$		0			0
$607$		−	11.1151i		−	0.451147i	−0.974226	−	0.225574i	$-0.927574\pi$
$607$		−	11.1151i		−	0.451147i	0.974226	−	0.225574i	$-0.0724256\pi$
$608$		0			0
$609$	0.215841	−	12.3764i	0.00874634	−	0.501518i
$610$		0			0
$611$		−	9.51354i		−	0.384877i
$612$		0			0
$613$		−	2.83979i		−	0.114698i	−0.998354	−	0.0573490i	$-0.981735\pi$
$613$		−	2.83979i		−	0.114698i	0.998354	−	0.0573490i	$-0.0182648\pi$
$614$		0			0
$615$	0.229465	−	13.1576i	0.00925294	−	0.530567i
$616$		0			0
$617$			2.11914i			0.0853133i	0.999090	+	0.0426566i	$0.0135822\pi$
$617$			2.11914i			0.0853133i	−0.999090	+	0.0426566i	$0.986418\pi$
$618$		0			0
$619$	19.2856			0.775154			0.387577	−	0.921837i	$-0.373312\pi$
$619$	19.2856			0.775154			0.387577	+	0.921837i	$0.373312\pi$
$620$		0			0
$621$	5.18904	+	0.271707i	0.208229	+	0.0109032i
$622$		0			0
$623$	53.7649			2.15405
$624$		0			0
$625$	14.1762			0.567047
$626$		0			0
$627$	−0.285634	+	16.3784i	−0.0114071	+	0.654088i
$628$		0			0
$629$	9.05547			0.361065
$630$		0			0
$631$		−	12.4518i		−	0.495696i	−0.968799	−	0.247848i	$-0.920277\pi$
$631$		−	12.4518i		−	0.495696i	0.968799	−	0.247848i	$-0.0797234\pi$
$632$		0			0
$633$	−17.0099	−	0.296647i	−0.676081	−	0.0117907i
$634$		0			0
$635$		−	6.02373i		−	0.239044i
$636$		0			0
$637$		−	3.17366i		−	0.125745i
$638$		0			0
$639$	−23.5311	−	0.821001i	−0.930875	−	0.0324783i
$640$		0			0
$641$		−	18.3406i		−	0.724411i	−0.932098	−	0.362205i	$-0.882024\pi$
$641$		−	18.3406i		−	0.724411i	0.932098	−	0.362205i	$-0.117976\pi$
$642$		0			0
$643$	−42.8115			−1.68832			−0.844160	−	0.536091i	$-0.819900\pi$
$643$	−42.8115			−1.68832			−0.844160	+	0.536091i	$0.819900\pi$
$644$		0			0
$645$	4.93387	+	0.0860454i	0.194271	+	0.00338803i
$646$		0			0
$647$	−24.2136			−0.951935			−0.475968	−	0.879463i	$-0.657902\pi$
$647$	−24.2136			−0.951935			−0.475968	+	0.879463i	$0.657902\pi$
$648$		0			0
$649$	−23.2455			−0.912464
$650$		0			0
$651$	−18.6625	−	0.325469i	−0.731441	−	0.0127561i
$652$		0			0
$653$	37.6716			1.47420			0.737101	−	0.675782i	$-0.236194\pi$
$653$	37.6716			1.47420			0.737101	+	0.675782i	$0.236194\pi$
$654$		0			0
$655$			10.9604i			0.428260i
$656$		0			0
$657$	−21.1310	−	0.737263i	−0.824400	−	0.0287634i
$658$		0			0
$659$		−	24.6271i		−	0.959337i	−0.877450	−	0.479669i	$-0.840757\pi$
$659$		−	24.6271i		−	0.959337i	0.877450	−	0.479669i	$-0.159243\pi$
$660$		0			0
$661$		−	43.8276i		−	1.70470i	−0.522976	−	0.852348i	$-0.675178\pi$
$661$		−	43.8276i		−	1.70470i	0.522976	−	0.852348i	$-0.324822\pi$
$662$		0			0
$663$	4.25338	+	0.0741778i	0.165188	+	0.00288083i
$664$		0			0
$665$			13.3950i			0.519437i
$666$		0			0
$667$	−2.31109			−0.0894859
$668$		0			0
$669$	−0.162287	+	9.30557i	−0.00627436	+	0.359774i
$670$		0			0
$671$	−20.5531			−0.793443
$672$		0			0
$673$	−33.5022			−1.29142			−0.645708	−	0.763584i	$-0.723438\pi$
$673$	−33.5022			−1.29142			−0.645708	+	0.763584i	$0.723438\pi$
$674$		0			0
$675$	−22.0010	−	1.15201i	−0.846819	−	0.0443408i
$676$		0			0
$677$	14.9572			0.574850			0.287425	−	0.957803i	$-0.407201\pi$
$677$	14.9572			0.574850			0.287425	+	0.957803i	$0.407201\pi$
$678$		0			0
$679$			27.9923i			1.07424i
$680$		0			0
$681$	0.696263	−	39.9240i	0.0266809	−	1.52989i
$682$		0			0
$683$		−	4.05767i		−	0.155262i	−0.996982	−	0.0776312i	$-0.975264\pi$
$683$		−	4.05767i		−	0.155262i	0.996982	−	0.0776312i	$-0.0247357\pi$
$684$		0			0
$685$			1.55610i			0.0594555i
$686$		0			0
$687$	0.822286	−	47.1502i	0.0313722	−	1.79889i
$688$		0			0
$689$			9.32698i			0.355329i
$690$		0			0
$691$	23.0400			0.876485			0.438242	−	0.898857i	$-0.355601\pi$
$691$	23.0400			0.876485			0.438242	+	0.898857i	$0.355601\pi$
$692$		0			0
$693$	−17.6479	−	0.615735i	−0.670387	−	0.0233898i
$694$		0			0
$695$	−5.15125			−0.195398
$696$		0			0
$697$	−17.2810			−0.654565
$698$		0			0
$699$	−0.652884	+	37.4366i	−0.0246943	+	1.41598i
$700$		0			0
$701$	−22.3112			−0.842683			−0.421342	−	0.906902i	$-0.638441\pi$
$701$	−22.3112			−0.842683			−0.421342	+	0.906902i	$0.638441\pi$
$702$		0			0
$703$			22.6888i			0.855724i
$704$		0			0
$705$	−11.5973	−	0.202254i	−0.436780	−	0.00761732i
$706$		0			0
$707$			60.5494i			2.27719i
$708$		0			0
$709$			17.9668i			0.674758i	0.941369	+	0.337379i	$0.109540\pi$
$709$			17.9668i			0.674758i	−0.941369	+	0.337379i	$0.890460\pi$
$710$		0			0
$711$	−0.534178	+	15.3103i	−0.0200332	+	0.574182i
$712$		0			0
$713$			3.48492i			0.130511i
$714$		0			0
$715$	−2.05544			−0.0768693
$716$		0			0
$717$	30.6140	+	0.533900i	1.14330	+	0.0199389i
$718$		0			0
$719$	34.2271			1.27646			0.638228	−	0.769848i	$-0.279668\pi$
$719$	34.2271			1.27646			0.638228	+	0.769848i	$0.279668\pi$
$720$		0			0
$721$	−9.15363			−0.340899
$722$		0			0
$723$	−28.3051	−	0.493634i	−1.05268	−	0.0183584i
$724$		0			0
$725$	9.79879			0.363918
$726$		0			0
$727$		−	19.2864i		−	0.715293i	−0.933857	−	0.357646i	$-0.883579\pi$
$727$		−	19.2864i		−	0.715293i	0.933857	−	0.357646i	$-0.116421\pi$
$728$		0			0
$729$	26.8524	+	2.81980i	0.994532	+	0.104437i
$730$		0			0
$731$		−	6.48007i		−	0.239674i
$732$		0			0
$733$			13.7227i			0.506860i	0.967354	+	0.253430i	$0.0815587\pi$
$733$			13.7227i			0.506860i	−0.967354	+	0.253430i	$0.918441\pi$
$734$		0			0
$735$	−3.86879	−	0.0674707i	−0.142703	−	0.00248869i
$736$		0			0
$737$			0.107100i			0.00394507i
$738$		0			0
$739$	21.1541			0.778167			0.389083	−	0.921203i	$-0.372792\pi$
$739$	21.1541			0.778167			0.389083	+	0.921203i	$0.372792\pi$
$740$		0			0
$741$	−0.185855	+	10.6570i	−0.00682756	+	0.391495i
$742$		0			0
$743$	−26.8675			−0.985673			−0.492837	−	0.870122i	$-0.664040\pi$
$743$	−26.8675			−0.985673			−0.492837	+	0.870122i	$0.664040\pi$
$744$		0			0
$745$	−16.0105			−0.586579
$746$		0			0
$747$	−0.0832134	+	2.38502i	−0.00304462	+	0.0872632i
$748$		0			0
$749$	27.7017			1.01220
$750$		0			0
$751$		−	6.80908i		−	0.248467i	−0.992253	−	0.124233i	$-0.960353\pi$
$751$		−	6.80908i		−	0.248467i	0.992253	−	0.124233i	$-0.0396472\pi$
$752$		0			0
$753$	−0.426711	+	24.4678i	−0.0155502	+	0.891656i
$754$		0			0
$755$		−	4.92839i		−	0.179363i
$756$		0			0
$757$		−	5.76958i		−	0.209699i	−0.994488	−	0.104849i	$-0.966564\pi$
$757$		−	5.76958i		−	0.209699i	0.994488	−	0.104849i	$-0.0334361\pi$
$758$		0			0
$759$	−0.0574892	+	3.29645i	−0.00208673	+	0.119654i
$760$		0			0
$761$			39.7502i			1.44095i	0.693483	+	0.720473i	$0.256075\pi$
$761$			39.7502i			1.44095i	−0.693483	+	0.720473i	$0.743925\pi$
$762$		0			0
$763$	28.9138			1.04675
$764$		0			0
$765$	0.180850	−	5.18343i	0.00653866	−	0.187407i
$766$		0			0
$767$	−15.1253			−0.546141
$768$		0			0
$769$	44.3637			1.59980			0.799898	−	0.600136i	$-0.204887\pi$
$769$	44.3637			1.59980			0.799898	+	0.600136i	$0.204887\pi$
$770$		0			0
$771$	−0.401495	+	23.0219i	−0.0144595	+	0.829112i
$772$		0			0
$773$	−20.9167			−0.752321			−0.376161	−	0.926554i	$-0.622756\pi$
$773$	−20.9167			−0.752321			−0.376161	+	0.926554i	$0.622756\pi$
$774$		0			0
$775$		−	14.7757i		−	0.530758i
$776$		0			0
$777$	−24.4549	−	0.426486i	−0.877314	−	0.0153001i
$778$		0			0
$779$		−	43.2982i		−	1.55132i
$780$		0			0
$781$		−	14.9395i		−	0.534578i
$782$		0			0
$783$	−11.9924	−	0.627940i	−0.428572	−	0.0224408i
$784$		0			0
$785$		−	15.4129i		−	0.550110i
$786$		0			0
$787$	52.1207			1.85790			0.928951	−	0.370202i	$-0.120711\pi$
$787$	52.1207			1.85790			0.928951	+	0.370202i	$0.120711\pi$
$788$		0			0
$789$	−43.9733	−	0.766883i	−1.56549	−	0.0273017i
$790$		0			0
$791$	4.55036			0.161792
$792$		0			0
$793$	−13.3734			−0.474903
$794$		0			0
$795$	11.3699	+	0.198288i	0.403248	+	0.00703254i
$796$		0			0
$797$	−35.1231			−1.24412			−0.622062	−	0.782968i	$-0.713705\pi$
$797$	−35.1231			−1.24412			−0.622062	+	0.782968i	$0.713705\pi$
$798$		0			0
$799$			15.2317i			0.538859i
$800$		0			0
$801$	1.81876	−	52.1283i	0.0642627	−	1.84186i
$802$		0			0
$803$		−	13.4158i		−	0.473432i
$804$		0			0
$805$			2.69600i			0.0950216i
$806$		0			0
$807$	4.48318	+	0.0781855i	0.157816	+	0.00275226i
$808$		0			0
$809$			11.2110i			0.394157i	0.980388	+	0.197078i	$0.0631454\pi$
$809$			11.2110i			0.394157i	−0.980388	+	0.197078i	$0.936855\pi$
$810$		0			0
$811$	23.2309			0.815746			0.407873	−	0.913039i	$-0.366271\pi$
$811$	23.2309			0.815746			0.407873	+	0.913039i	$0.366271\pi$
$812$		0			0
$813$	0.891365	−	51.1112i	0.0312615	−	1.79255i
$814$		0			0
$815$	13.4401			0.470786
$816$		0			0
$817$	16.2360			0.568027
$818$		0			0
$819$	−11.4830	−	0.400644i	−0.401250	−	0.0139996i
$820$		0			0
$821$	8.28219			0.289050			0.144525	−	0.989501i	$-0.453835\pi$
$821$	8.28219			0.289050			0.144525	+	0.989501i	$0.453835\pi$
$822$		0			0
$823$		−	17.0583i		−	0.594616i	−0.954782	−	0.297308i	$-0.903911\pi$
$823$		−	17.0583i		−	0.594616i	0.954782	−	0.297308i	$-0.0960889\pi$
$824$		0			0
$825$	0.243748	−	13.9766i	0.00848622	−	0.486603i
$826$		0			0
$827$			11.3622i			0.395101i	0.980293	+	0.197551i	$0.0632987\pi$
$827$			11.3622i			0.395101i	−0.980293	+	0.197551i	$0.936701\pi$
$828$		0			0
$829$			33.5403i			1.16490i	0.812865	+	0.582452i	$0.197906\pi$
$829$			33.5403i			1.16490i	−0.812865	+	0.582452i	$0.802094\pi$
$830$		0			0
$831$	0.0328444	−	1.88331i	0.00113936	−	0.0653314i
$832$		0			0
$833$			5.08121i			0.176053i
$834$		0			0
$835$	7.88928			0.273020
$836$		0			0
$837$	−0.946876	+	18.0834i	−0.0327288	+	0.625053i
$838$		0			0
$839$	−0.880528			−0.0303992			−0.0151996	−	0.999884i	$-0.504838\pi$
$839$	−0.880528			−0.0303992			−0.0151996	+	0.999884i	$0.504838\pi$
$840$		0			0
$841$	−23.6588			−0.815822
$842$		0			0
$843$	−0.0986881	+	5.65881i	−0.00339900	+	0.194900i
$844$		0			0
$845$	9.99650			0.343890
$846$		0			0
$847$			22.8111i			0.783797i
$848$		0			0
$849$	−11.0336	−	0.192422i	−0.378671	−	0.00660391i
$850$		0			0
$851$			4.56655i			0.156539i
$852$		0			0
$853$			51.3054i			1.75666i	0.478052	+	0.878331i	$0.341343\pi$
$853$			51.3054i			1.75666i	−0.478052	+	0.878331i	$0.658657\pi$
$854$		0			0
$855$	12.9873	+	0.453127i	0.444155	+	0.0154966i
$856$		0			0
$857$		−	20.2353i		−	0.691224i	−0.938378	−	0.345612i	$-0.887671\pi$
$857$		−	20.2353i		−	0.691224i	0.938378	−	0.345612i	$-0.112329\pi$
$858$		0			0
$859$	23.4886			0.801421			0.400710	−	0.916205i	$-0.368763\pi$
$859$	23.4886			0.801421			0.400710	+	0.916205i	$0.368763\pi$
$860$		0			0
$861$	46.6685	+	0.813885i	1.59046	+	0.0277371i
$862$		0			0
$863$	4.02794			0.137113			0.0685563	−	0.997647i	$-0.478161\pi$
$863$	4.02794			0.137113			0.0685563	+	0.997647i	$0.478161\pi$
$864$		0			0
$865$	−2.47365			−0.0841066
$866$		0			0
$867$	22.6305	+	0.394669i	0.768571	+	0.0134037i
$868$		0			0
$869$	−9.72028			−0.329738
$870$		0			0
$871$			0.0696872i			0.00236126i
$872$		0			0
$873$	27.1401	+	0.946921i	0.918554	+	0.0320484i
$874$		0			0
$875$		−	24.9108i		−	0.842138i
$876$		0			0
$877$			37.8678i			1.27870i	0.768914	+	0.639352i	$0.220798\pi$
$877$			37.8678i			1.27870i	−0.768914	+	0.639352i	$0.779202\pi$
$878$		0			0
$879$	42.4840	+	0.740910i	1.43295	+	0.0249903i
$880$		0			0
$881$		−	48.6628i		−	1.63949i	−0.572727	−	0.819746i	$-0.694115\pi$
$881$		−	48.6628i		−	1.63949i	0.572727	−	0.819746i	$-0.305885\pi$
$882$		0			0
$883$	6.80087			0.228868			0.114434	−	0.993431i	$-0.463495\pi$
$883$	6.80087			0.228868			0.114434	+	0.993431i	$0.463495\pi$
$884$		0			0
$885$	−0.321557	+	18.4382i	−0.0108090	+	0.619793i
$886$		0			0
$887$	−2.86430			−0.0961737			−0.0480869	−	0.998843i	$-0.515312\pi$
$887$	−2.86430			−0.0961737			−0.0480869	+	0.998843i	$0.515312\pi$
$888$		0			0
$889$	21.3654			0.716573
$890$		0			0
$891$	−1.19398	+	17.0898i	−0.0399999	+	0.572530i
$892$		0			0
$893$	−38.1636			−1.27710
$894$		0			0
$895$		−	9.22707i		−	0.308427i
$896$		0			0
$897$	−0.0374068	+	2.14492i	−0.00124898	+	0.0716168i
$898$		0			0
$899$		−	8.05397i		−	0.268615i
$900$		0			0
$901$		−	14.9330i		−	0.497491i
$902$		0			0
$903$	−0.305192	+	17.4998i	−0.0101562	+	0.582358i
$904$		0			0
$905$		−	7.50224i		−	0.249383i
$906$		0			0
$907$	19.3695			0.643153			0.321576	−	0.946884i	$-0.395787\pi$
$907$	19.3695			0.643153			0.321576	+	0.946884i	$0.395787\pi$
$908$		0			0
$909$	58.7062	+	2.04826i	1.94716	+	0.0679366i
$910$		0			0
$911$	−22.5365			−0.746668			−0.373334	−	0.927697i	$-0.621785\pi$
$911$	−22.5365			−0.746668			−0.373334	+	0.927697i	$0.621785\pi$
$912$		0			0
$913$	−1.51421			−0.0501131
$914$		0			0
$915$	−0.284313	+	16.3026i	−0.00939909	+	0.538947i
$916$		0			0
$917$	−38.8753			−1.28378
$918$		0			0
$919$			47.7774i			1.57603i	0.615656	+	0.788015i	$0.288891\pi$
$919$			47.7774i			1.57603i	−0.615656	+	0.788015i	$0.711109\pi$
$920$		0			0
$921$	−11.2439	−	0.196091i	−0.370500	−	0.00646142i
$922$		0			0
$923$		−	9.72079i		−	0.319964i
$924$		0			0
$925$		−	19.3617i		−	0.636608i
$926$		0			0
$927$	−0.309649	+	8.87498i	−0.0101702	+	0.291493i
$928$		0			0
$929$			40.9144i			1.34236i	0.741295	+	0.671179i	$0.234212\pi$
$929$			40.9144i			1.34236i	−0.741295	+	0.671179i	$0.765788\pi$
$930$		0			0
$931$	−12.7311			−0.417246
$932$		0			0
$933$	−45.6365	−	0.795887i	−1.49407	−	0.0260562i
$934$		0			0
$935$	3.29088			0.107623
$936$		0			0
$937$	27.6662			0.903814			0.451907	−	0.892065i	$-0.350744\pi$
$937$	27.6662			0.903814			0.451907	+	0.892065i	$0.350744\pi$
$938$		0			0
$939$	−57.5237	−	1.00320i	−1.87721	−	0.0327381i
$940$		0			0
$941$	6.59325			0.214934			0.107467	−	0.994209i	$-0.465726\pi$
$941$	6.59325			0.214934			0.107467	+	0.994209i	$0.465726\pi$
$942$		0			0
$943$		−	8.71457i		−	0.283785i
$944$		0			0
$945$	−0.732522	+	13.9897i	−0.0238290	+	0.455084i
$946$		0			0
$947$			48.2638i			1.56836i	0.620531	+	0.784182i	$0.286917\pi$
$947$			48.2638i			1.56836i	−0.620531	+	0.784182i	$0.713083\pi$
$948$		0			0
$949$		−	8.72931i		−	0.283366i
$950$		0			0
$951$	−36.9330	−	0.644101i	−1.19763	−	0.0208864i
$952$		0			0
$953$			48.2640i			1.56342i	0.623640	+	0.781712i	$0.285653\pi$
$953$			48.2640i			1.56342i	−0.623640	+	0.781712i	$0.714347\pi$
$954$		0			0
$955$	−12.7348			−0.412087
$956$		0			0
$957$	0.132863	−	7.61841i	0.00429485	−	0.246268i
$958$		0			0
$959$	−5.51929			−0.178227
$960$		0			0
$961$	18.8554			0.608237
$962$		0			0
$963$	0.937092	−	26.8584i	0.0301974	−	0.865501i
$964$		0			0
$965$	−4.78211			−0.153942
$966$		0			0
$967$		−	51.1796i		−	1.64583i	−0.568168	−	0.822913i	$-0.692348\pi$
$967$		−	51.1796i		−	1.64583i	0.568168	−	0.822913i	$-0.307652\pi$
$968$		0			0
$969$	0.297565	−	17.0625i	0.00955915	−	0.548125i
$970$		0			0
$971$		−	31.6471i		−	1.01560i	−0.861474	−	0.507802i	$-0.830458\pi$
$971$		−	31.6471i		−	1.01560i	0.861474	−	0.507802i	$-0.169542\pi$
$972$		0			0
$973$		−	18.2708i		−	0.585736i
$974$		0			0
$975$	0.158601	−	9.09423i	0.00507929	−	0.291249i
$976$		0			0
$977$			25.9886i			0.831448i	0.909491	+	0.415724i	$0.136472\pi$
$977$			25.9886i			0.831448i	−0.909491	+	0.415724i	$0.863528\pi$
$978$		0			0
$979$	33.0954			1.05773
$980$		0			0
$981$	0.978096	−	28.0337i	0.0312282	−	0.895046i
$982$		0			0
$983$	39.6637			1.26507			0.632537	−	0.774530i	$-0.282014\pi$
$983$	39.6637			1.26507			0.632537	+	0.774530i	$0.282014\pi$
$984$		0			0
$985$	19.0601			0.607305
$986$		0			0
$987$	0.717369	−	41.1342i	0.0228341	−	1.30932i
$988$		0			0
$989$	3.26780			0.103910
$990$		0			0
$991$		−	13.4539i		−	0.427378i	−0.976902	−	0.213689i	$-0.931452\pi$
$991$		−	13.4539i		−	0.427378i	0.976902	−	0.213689i	$-0.0685480\pi$
$992$		0			0
$993$	12.3821	+	0.215940i	0.392934	+	0.00685266i
$994$		0			0
$995$		−	17.0846i		−	0.541617i
$996$		0			0
$997$			57.2452i			1.81297i	0.422234	+	0.906487i	$0.361246\pi$
$997$			57.2452i			1.81297i	−0.422234	+	0.906487i	$0.638754\pi$
$998$		0			0
$999$	−1.24076	+	23.6960i	−0.0392560	+	0.749708i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.j.c.47.42		42
3.2	odd	2		2208.2.j.d.47.41		42
4.3	odd	2		552.2.j.c.323.27	✓	42
8.3	odd	2		2208.2.j.d.47.42		42
8.5	even	2		552.2.j.d.323.15	yes	42
12.11	even	2		552.2.j.d.323.16	yes	42
24.5	odd	2		552.2.j.c.323.28	yes	42
24.11	even	2	inner	2208.2.j.c.47.41		42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.j.c.323.27	✓	42	4.3	odd	2
552.2.j.c.323.28	yes	42	24.5	odd	2
552.2.j.d.323.15	yes	42	8.5	even	2
552.2.j.d.323.16	yes	42	12.11	even	2
2208.2.j.c.47.41		42	24.11	even	2	inner
2208.2.j.c.47.42		42	1.1	even	1	trivial
2208.2.j.d.47.41		42	3.2	odd	2
2208.2.j.d.47.42		42	8.3	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:	\( N \)	\(=\)	\( 2208 = 2^{5} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2208.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.73179 + 0.0302019i) q^{3} +0.871841 q^{5} +3.09231i q^{7} +(2.99818 + 0.104607i) q^{9} +1.90350i q^{11} +1.23856i q^{13} +(1.50984 + 0.0263313i) q^{15} -1.98300i q^{17} +4.96848 q^{19} +(-0.0933936 + 5.35522i) q^{21} +1.00000 q^{23} -4.23989 q^{25} +(5.18904 + 0.271707i) q^{27} -2.31109 q^{29} +3.48492i q^{31} +(-0.0574892 + 3.29645i) q^{33} +2.69600i q^{35} +4.56655i q^{37} +(-0.0374068 + 2.14492i) q^{39} -8.71457i q^{41} +3.26780 q^{43} +(2.61393 + 0.0912003i) q^{45} -7.68114 q^{47} -2.56238 q^{49} +(0.0598904 - 3.43414i) q^{51} +7.53051 q^{53} +1.65955i q^{55} +(8.60435 + 0.150058i) q^{57} +12.2120i q^{59} +10.7975i q^{61} +(-0.323476 + 9.27129i) q^{63} +1.07983i q^{65} +0.0562647 q^{67} +(1.73179 + 0.0302019i) q^{69} -7.84847 q^{71} -7.04796 q^{73} +(-7.34259 - 0.128053i) q^{75} -5.88620 q^{77} +5.10654i q^{79} +(8.97811 + 0.627258i) q^{81} +0.795490i q^{83} -1.72886i q^{85} +(-4.00232 - 0.0697994i) q^{87} -17.3867i q^{89} -3.83001 q^{91} +(-0.105251 + 6.03513i) q^{93} +4.33173 q^{95} +9.05221 q^{97} +(-0.199118 + 5.70701i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 2 q^{3} - 8 q^{5} + 2 q^{9} - 8 q^{15} - 4 q^{19} - 8 q^{21} + 42 q^{23} + 22 q^{25} + 16 q^{27} + 12 q^{33} + 8 q^{39} - 28 q^{43} + 8 q^{45} - 50 q^{49} - 28 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63}+ \cdots + 20 q^{97}+O(q^{100}) \) 42 * q - 2 * q^3 - 8 * q^5 + 2 * q^9 - 8 * q^15 - 4 * q^19 - 8 * q^21 + 42 * q^23 + 22 * q^25 + 16 * q^27 + 12 * q^33 + 8 * q^39 - 28 * q^43 + 8 * q^45 - 50 * q^49 - 28 * q^51 - 24 * q^53 - 8 * q^57 - 16 * q^63 + 4 * q^67 - 2 * q^69 + 4 * q^73 + 6 * q^75 - 32 * q^77 + 18 * q^81 - 48 * q^87 + 8 * q^91 + 22 * q^93 - 16 * q^95 + 20 * q^97

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