LMFDB - Embedded newform 675.2.e.c.226.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(226,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("675.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.1223810289.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 $ x^8 - 2x^7 + 8x^6 - 2x^5 + 23x^4 - 8x^3 + 37x^2 + 15*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			226.3
Root			$-0.236627 - 0.409850i$ of defining polynomial
Character	$\chi$	$=$	675.226
Dual form			675.2.e.c.451.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.236627 + 0.409850i) q^{2} +(0.888015 - 1.53809i) q^{4} +(-1.28153 - 2.21967i) q^{7} +1.78702 q^{8} +O(q^{10})$	\(q+(0.236627 + 0.409850i) q^{2} +(0.888015 - 1.53809i) q^{4} +(-1.28153 - 2.21967i) q^{7} +1.78702 q^{8} +(-3.08430 - 5.34217i) q^{11} +(-1.06615 + 1.84662i) q^{13} +(0.606488 - 1.05047i) q^{14} +(-1.35317 - 2.34376i) q^{16} -3.16860 q^{17} +0.356267 q^{19} +(1.45966 - 2.52821i) q^{22} +(-2.10649 + 3.64854i) q^{23} -1.00912 q^{26} -4.55206 q^{28} +(0.843116 + 1.46032i) q^{29} +(4.12920 - 7.15199i) q^{31} +(2.42742 - 4.20441i) q^{32} +(-0.749778 - 1.29865i) q^{34} +3.63274 q^{37} +(0.0843024 + 0.146016i) q^{38} +(-1.36677 + 2.36731i) q^{41} +(3.83908 + 6.64949i) q^{43} -10.9556 q^{44} -1.99381 q^{46} +(-5.71444 - 9.89770i) q^{47} +(0.215378 - 0.373046i) q^{49} +(1.89351 + 3.27966i) q^{52} +9.43507 q^{53} +(-2.29012 - 3.96660i) q^{56} +(-0.399008 + 0.691103i) q^{58} +(5.10795 - 8.84723i) q^{59} +(0.00549659 + 0.00952038i) q^{61} +3.90833 q^{62} -3.11511 q^{64} +(-0.491409 + 0.851145i) q^{67} +(-2.81377 + 4.87359i) q^{68} +6.43507 q^{71} +6.61467 q^{73} +(0.859605 + 1.48888i) q^{74} +(0.316370 - 0.547969i) q^{76} +(-7.90523 + 13.6923i) q^{77} +(4.73569 + 8.20246i) q^{79} -1.29366 q^{82} +(-5.20988 - 9.02378i) q^{83} +(-1.81686 + 3.14690i) q^{86} +(-5.51172 - 9.54658i) q^{88} +6.26940 q^{89} +5.46519 q^{91} +(3.74119 + 6.47993i) q^{92} +(2.70439 - 4.68413i) q^{94} +(-3.60339 - 6.24126i) q^{97} +0.203858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8	$ 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} - 4 q^{16} + 22 q^{17} + 4 q^{19} + 3 q^{22} - 15 q^{23} + 20 q^{26} + 8 q^{28} + q^{29} + 4 q^{31} - 10 q^{32} - 9 q^{34} + 2 q^{37} - 23 q^{38} - 5 q^{41} - 10 q^{43} - 44 q^{44} - 20 q^{47} + 3 q^{49} + 17 q^{52} + 40 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} + 13 q^{61} - 12 q^{62} + 38 q^{64} + 17 q^{67} - 34 q^{68} + 16 q^{71} - 4 q^{73} + 40 q^{74} - 11 q^{76} - 12 q^{77} + 7 q^{79} - 24 q^{82} - 30 q^{83} - 34 q^{86} + 9 q^{88} + 18 q^{89} - 34 q^{91} + 12 q^{92} - 3 q^{94} - 19 q^{97} + 26 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8 - q^11 + 2 * q^13 + 3 * q^14 - 4 * q^16 + 22 * q^17 + 4 * q^19 + 3 * q^22 - 15 * q^23 + 20 * q^26 + 8 * q^28 + q^29 + 4 * q^31 - 10 * q^32 - 9 * q^34 + 2 * q^37 - 23 * q^38 - 5 * q^41 - 10 * q^43 - 44 * q^44 - 20 * q^47 + 3 * q^49 + 17 * q^52 + 40 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 + 13 * q^61 - 12 * q^62 + 38 * q^64 + 17 * q^67 - 34 * q^68 + 16 * q^71 - 4 * q^73 + 40 * q^74 - 11 * q^76 - 12 * q^77 + 7 * q^79 - 24 * q^82 - 30 * q^83 - 34 * q^86 + 9 * q^88 + 18 * q^89 - 34 * q^91 + 12 * q^92 - 3 * q^94 - 19 * q^97 + 26 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	0.236627	+	0.409850i	0.167321	+	0.289808i	0.937477	−	0.348047i	$-0.113155\pi$
$2$	0.236627	+	0.409850i	0.167321	+	0.289808i	−0.770156	+	0.637855i	$0.779822\pi$
$3$		0			0
$3$		0			0
$4$	0.888015	−	1.53809i	0.444008	−	0.769044i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.28153	−	2.21967i	−0.484372	−	0.838956i	0.515467	−	0.856909i	$-0.327618\pi$
$7$	−1.28153	−	2.21967i	−0.484372	−	0.838956i	−0.999839	+	0.0179531i	$0.994285\pi$
$8$	1.78702			0.631808
$9$		0			0
$10$		0			0
$11$	−3.08430	−	5.34217i	−0.929952	−	1.61072i	−0.783397	−	0.621522i	$-0.786515\pi$
$11$	−3.08430	−	5.34217i	−0.929952	−	1.61072i	−0.146555	−	0.989202i	$-0.546819\pi$
$12$		0			0
$13$	−1.06615	+	1.84662i	−0.295696	+	0.512161i	−0.975147	−	0.221560i	$-0.928885\pi$
$13$	−1.06615	+	1.84662i	−0.295696	+	0.512161i	0.679450	+	0.733722i	$0.262218\pi$
$14$	0.606488	−	1.05047i	0.162091	−	0.280750i
$15$		0			0
$16$	−1.35317	−	2.34376i	−0.338293	−	0.585941i
$17$	−3.16860			−0.768500			−0.384250	−	0.923229i	$-0.625540\pi$
$17$	−3.16860			−0.768500			−0.384250	+	0.923229i	$0.625540\pi$
$18$		0			0
$19$	0.356267			0.0817332			0.0408666	−	0.999165i	$-0.486988\pi$
$19$	0.356267			0.0817332			0.0408666	+	0.999165i	$0.486988\pi$
$20$		0			0
$21$		0			0
$22$	1.45966	−	2.52821i	0.311201	−	0.539015i
$23$	−2.10649	+	3.64854i	−0.439233	+	0.760774i	−0.997631	−	0.0687995i	$-0.978083\pi$
$23$	−2.10649	+	3.64854i	−0.439233	+	0.760774i	0.558397	+	0.829574i	$0.311416\pi$
$24$		0			0
$25$		0			0
$26$	−1.00912			−0.197905
$27$		0			0
$28$	−4.55206			−0.860259
$29$	0.843116	+	1.46032i	0.156563	+	0.271174i	0.933627	−	0.358247i	$-0.116625\pi$
$29$	0.843116	+	1.46032i	0.156563	+	0.271174i	−0.777064	+	0.629421i	$0.783292\pi$
$30$		0			0
$31$	4.12920	−	7.15199i	0.741627	−	1.28453i	−0.210128	−	0.977674i	$-0.567388\pi$
$31$	4.12920	−	7.15199i	0.741627	−	1.28453i	0.951754	−	0.306861i	$-0.0992787\pi$
$32$	2.42742	−	4.20441i	0.429111	−	0.743242i
$33$		0			0
$34$	−0.749778	−	1.29865i	−0.128586	−	0.222717i
$35$		0			0
$36$		0			0
$37$	3.63274			0.597219			0.298609	−	0.954375i	$-0.403477\pi$
$37$	3.63274			0.597219			0.298609	+	0.954375i	$0.403477\pi$
$38$	0.0843024	+	0.146016i	0.0136757	+	0.0236869i
$39$		0			0
$40$		0			0
$41$	−1.36677	+	2.36731i	−0.213453	+	0.369711i	−0.952793	−	0.303621i	$-0.901804\pi$
$41$	−1.36677	+	2.36731i	−0.213453	+	0.369711i	0.739340	+	0.673332i	$0.235138\pi$
$42$		0			0
$43$	3.83908	+	6.64949i	0.585455	+	1.01404i	0.994819	+	0.101666i	$0.0324174\pi$
$43$	3.83908	+	6.64949i	0.585455	+	1.01404i	−0.409364	+	0.912371i	$0.634249\pi$
$44$	−10.9556			−1.65162
$45$		0			0
$46$	−1.99381			−0.293971
$47$	−5.71444	−	9.89770i	−0.833537	−	1.44373i	−0.895216	−	0.445632i	$-0.852979\pi$
$47$	−5.71444	−	9.89770i	−0.833537	−	1.44373i	0.0616792	−	0.998096i	$-0.480354\pi$
$48$		0			0
$49$	0.215378	−	0.373046i	0.0307683	−	0.0532923i
$50$		0			0
$51$		0			0
$52$	1.89351	+	3.27966i	0.262583	+	0.454807i
$53$	9.43507			1.29601			0.648003	−	0.761637i	$-0.275604\pi$
$53$	9.43507			1.29601			0.648003	+	0.761637i	$0.275604\pi$
$54$		0			0
$55$		0			0
$56$	−2.29012	−	3.96660i	−0.306030	−	0.530059i
$57$		0			0
$58$	−0.399008	+	0.691103i	−0.0523924	+	0.0907462i
$59$	5.10795	−	8.84723i	0.664999	−	1.15181i	−0.314287	−	0.949328i	$-0.601765\pi$
$59$	5.10795	−	8.84723i	0.664999	−	1.15181i	0.979286	−	0.202484i	$-0.0649013\pi$
$60$		0			0
$61$	0.00549659	+	0.00952038i	0.000703767	+	0.00121896i	0.866377	−	0.499390i	$-0.166443\pi$
$61$	0.00549659	+	0.00952038i	0.000703767	+	0.00121896i	−0.865673	+	0.500609i	$0.833109\pi$
$62$	3.90833			0.496358
$63$		0			0
$64$	−3.11511			−0.389389
$65$		0			0
$66$		0			0
$67$	−0.491409	+	0.851145i	−0.0600351	+	0.103984i	−0.894481	−	0.447106i	$-0.852455\pi$
$67$	−0.491409	+	0.851145i	−0.0600351	+	0.103984i	0.834446	+	0.551090i	$0.185788\pi$
$68$	−2.81377	+	4.87359i	−0.341220	+	0.591010i
$69$		0			0
$70$		0			0
$71$	6.43507			0.763703			0.381851	−	0.924224i	$-0.375287\pi$
$71$	6.43507			0.763703			0.381851	+	0.924224i	$0.375287\pi$
$72$		0			0
$73$	6.61467			0.774189			0.387094	−	0.922040i	$-0.373479\pi$
$73$	6.61467			0.774189			0.387094	+	0.922040i	$0.373479\pi$
$74$	0.859605	+	1.48888i	0.0999271	+	0.173079i
$75$		0			0
$76$	0.316370	−	0.547969i	0.0362901	−	0.0628564i
$77$	−7.90523	+	13.6923i	−0.900885	+	1.56038i
$78$		0			0
$79$	4.73569	+	8.20246i	0.532807	+	0.922848i	0.999266	+	0.0383057i	$0.0121961\pi$
$79$	4.73569	+	8.20246i	0.532807	+	0.922848i	−0.466459	+	0.884543i	$0.654471\pi$
$80$		0			0
$81$		0			0
$82$	−1.29366			−0.142860
$83$	−5.20988	−	9.02378i	−0.571859	−	0.990489i	−0.996375	−	0.0850682i	$-0.972889\pi$
$83$	−5.20988	−	9.02378i	−0.571859	−	0.990489i	0.424516	−	0.905420i	$-0.360444\pi$
$84$		0			0
$85$		0			0
$86$	−1.81686	+	3.14690i	−0.195917	+	0.339339i
$87$		0			0
$88$	−5.51172	−	9.54658i	−0.587551	−	1.01767i
$89$	6.26940			0.664555			0.332277	−	0.943182i	$-0.392183\pi$
$89$	6.26940			0.664555			0.332277	+	0.943182i	$0.392183\pi$
$90$		0			0
$91$	5.46519			0.572908
$92$	3.74119	+	6.47993i	0.390046	+	0.675579i
$93$		0			0
$94$	2.70439	−	4.68413i	0.278936	−	0.483131i
$95$		0			0
$96$		0			0
$97$	−3.60339	−	6.24126i	−0.365869	−	0.633704i	0.623046	−	0.782185i	$-0.285895\pi$
$97$	−3.60339	−	6.24126i	−0.365869	−	0.633704i	−0.988915	+	0.148481i	$0.952562\pi$
$98$	0.203858			0.0205927
$99$		0			0
$100$		0			0
$101$	3.48547	+	6.03701i	0.346817	+	0.600705i	0.985682	−	0.168614i	$-0.0539291\pi$
$101$	3.48547	+	6.03701i	0.346817	+	0.600705i	−0.638865	+	0.769319i	$0.720596\pi$
$102$		0			0
$103$	3.05756	−	5.29584i	0.301270	−	0.521815i	−0.675154	−	0.737677i	$-0.735923\pi$
$103$	3.05756	−	5.29584i	0.301270	−	0.521815i	0.976424	+	0.215862i	$0.0692561\pi$
$104$	−1.90523	+	3.29996i	−0.186823	+	0.323588i
$105$		0			0
$106$	2.23260	+	3.86697i	0.216849	+	0.375593i
$107$	14.5349			1.40514			0.702570	−	0.711615i	$-0.252036\pi$
$107$	14.5349			1.40514			0.702570	+	0.711615i	$0.252036\pi$
$108$		0			0
$109$	1.90214			0.182192			0.0910958	−	0.995842i	$-0.470963\pi$
$109$	1.90214			0.182192			0.0910958	+	0.995842i	$0.470963\pi$
$110$		0			0
$111$		0			0
$112$	−3.46825	+	6.00719i	−0.327719	+	0.567626i
$113$	−3.28962	+	5.69780i	−0.309462	+	0.536004i	−0.978245	−	0.207454i	$-0.933482\pi$
$113$	−3.28962	+	5.69780i	−0.309462	+	0.536004i	0.668783	+	0.743458i	$0.266816\pi$
$114$		0			0
$115$		0			0
$116$	2.99480			0.278060
$117$		0			0
$118$	4.83472			0.445072
$119$	4.06065	+	7.03326i	0.372239	+	0.644737i
$120$		0			0
$121$	−13.5258	+	23.4274i	−1.22962	+	2.12977i
$122$	−0.00260129	+	0.00450556i	−0.000235509	+	0.000407914i
$123$		0			0
$124$	−7.33359	−	12.7021i	−0.658576	−	1.14069i
$125$		0			0
$126$		0			0
$127$	−9.25840			−0.821550			−0.410775	−	0.911737i	$-0.634742\pi$
$127$	−9.25840			−0.821550			−0.410775	+	0.911737i	$0.634742\pi$
$128$	−5.59196	−	9.68555i	−0.494264	−	0.856090i
$129$		0			0
$130$		0			0
$131$	0.134698	−	0.233305i	0.0117687	−	0.0203839i	−0.860081	−	0.510157i	$-0.829587\pi$
$131$	0.134698	−	0.233305i	0.0117687	−	0.0203839i	0.871850	+	0.489773i	$0.162920\pi$
$132$		0			0
$133$	−0.456565	−	0.790794i	−0.0395892	−	0.0685705i
$134$	−0.465123			−0.0401805
$135$		0			0
$136$	−5.66237			−0.485544
$137$	1.73809	+	3.01046i	0.148495	+	0.257201i	0.930671	−	0.365856i	$-0.119224\pi$
$137$	1.73809	+	3.01046i	0.148495	+	0.257201i	−0.782176	+	0.623057i	$0.785890\pi$
$138$		0			0
$139$	−7.37393	+	12.7720i	−0.625448	+	1.08331i	0.363006	+	0.931787i	$0.381751\pi$
$139$	−7.37393	+	12.7720i	−0.625448	+	1.08331i	−0.988454	+	0.151521i	$0.951583\pi$
$140$		0			0
$141$		0			0
$142$	1.52271	+	2.63742i	0.127783	+	0.221327i
$143$	13.1533			1.09993
$144$		0			0
$145$		0			0
$146$	1.56521	+	2.71103i	0.129538	+	0.224366i
$147$		0			0
$148$	3.22593	−	5.58747i	0.265170	−	0.459287i
$149$	−5.07665	+	8.79301i	−0.415895	+	0.720352i	−0.995522	−	0.0945305i	$-0.969865\pi$
$149$	−5.07665	+	8.79301i	−0.415895	+	0.720352i	0.579627	+	0.814882i	$0.303198\pi$
$150$		0			0
$151$	5.15811	+	8.93410i	0.419761	+	0.727047i	0.995915	−	0.0902940i	$-0.0287807\pi$
$151$	5.15811	+	8.93410i	0.419761	+	0.727047i	−0.576155	+	0.817341i	$0.695447\pi$
$152$	0.636657			0.0516397
$153$		0			0
$154$	−7.48237			−0.602947
$155$		0			0
$156$		0			0
$157$	−0.531305	+	0.920247i	−0.0424028	+	0.0734437i	−0.886448	−	0.462828i	$-0.846835\pi$
$157$	−0.531305	+	0.920247i	−0.0424028	+	0.0734437i	0.844045	+	0.536272i	$0.180168\pi$
$158$	−2.24119	+	3.88185i	−0.178299	+	0.308823i
$159$		0			0
$160$		0			0
$161$	10.7981			0.851008
$162$		0			0
$163$	−17.1386			−1.34240			−0.671198	−	0.741278i	$-0.734220\pi$
$163$	−17.1386			−1.34240			−0.671198	+	0.741278i	$0.734220\pi$
$164$	2.42742	+	4.20441i	0.189549	+	0.328309i
$165$		0			0
$166$	2.46560	−	4.27054i	0.191368	−	0.331459i
$167$	−2.18672	+	3.78752i	−0.169214	+	0.293087i	−0.938144	−	0.346246i	$-0.887456\pi$
$167$	−2.18672	+	3.78752i	−0.169214	+	0.293087i	0.768930	+	0.639333i	$0.220790\pi$
$168$		0			0
$169$	4.22666	+	7.32078i	0.325127	+	0.563137i
$170$		0			0
$171$		0			0
$172$	13.6367			1.03979
$173$	−7.33005	−	12.6960i	−0.557293	−	0.965260i	−0.997721	−	0.0674723i	$-0.978507\pi$
$173$	−7.33005	−	12.6960i	−0.557293	−	0.965260i	0.440428	−	0.897788i	$-0.354827\pi$
$174$		0			0
$175$		0			0
$176$	−8.34718	+	14.4577i	−0.629192	+	1.08979i
$177$		0			0
$178$	1.48351	+	2.56952i	0.111194	+	0.192593i
$179$	6.87014			0.513499			0.256749	−	0.966478i	$-0.417349\pi$
$179$	6.87014			0.513499			0.256749	+	0.966478i	$0.417349\pi$
$180$		0			0
$181$	−10.9709			−0.815463			−0.407732	−	0.913102i	$-0.633680\pi$
$181$	−10.9709			−0.815463			−0.407732	+	0.913102i	$0.633680\pi$
$182$	1.29321	+	2.23991i	0.0958593	+	0.166033i
$183$		0			0
$184$	−3.76434	+	6.52003i	−0.277511	+	0.480663i
$185$		0			0
$186$		0			0
$187$	9.77294	+	16.9272i	0.714668	+	1.23784i
$188$	−20.2980			−1.48039
$189$		0			0
$190$		0			0
$191$	6.86627	+	11.8927i	0.496826	+	0.860528i	0.999993	−	0.00366109i	$-0.00116536\pi$
$191$	6.86627	+	11.8927i	0.496826	+	0.860528i	−0.503167	+	0.864189i	$0.667832\pi$
$192$		0			0
$193$	0.241187	−	0.417748i	0.0173610	−	0.0300701i	−0.857214	−	0.514960i	$-0.827807\pi$
$193$	0.241187	−	0.417748i	0.0173610	−	0.0300701i	0.874575	+	0.484890i	$0.161140\pi$
$194$	1.70532	−	2.95370i	0.122435	−	0.212064i
$195$		0			0
$196$	−0.382518	−	0.662541i	−0.0273227	−	0.0473244i
$197$	−5.53488			−0.394344			−0.197172	−	0.980369i	$-0.563176\pi$
$197$	−5.53488			−0.394344			−0.197172	+	0.980369i	$0.563176\pi$
$198$		0			0
$199$	17.4590			1.23764			0.618818	−	0.785534i	$-0.287612\pi$
$199$	17.4590			1.23764			0.618818	+	0.785534i	$0.287612\pi$
$200$		0			0
$201$		0			0
$202$	−1.64951	+	2.85704i	−0.116059	+	0.201021i
$203$	2.16095	−	3.74288i	0.151669	−	0.262698i
$204$		0			0
$205$		0			0
$206$	2.89401			0.201635
$207$		0			0
$208$	5.77073			0.400128
$209$	−1.09883	−	1.90324i	−0.0760079	−	0.131650i
$210$		0			0
$211$	0.818328	−	1.41739i	0.0563360	−	0.0975769i	−0.836482	−	0.547994i	$-0.815392\pi$
$211$	0.818328	−	1.41739i	0.0563360	−	0.0975769i	0.892818	+	0.450417i	$0.148725\pi$
$212$	8.37849	−	14.5120i	0.575437	−	0.996686i
$213$		0			0
$214$	3.43935	+	5.95713i	0.235109	+	0.407221i
$215$		0			0
$216$		0			0
$217$	−21.1667			−1.43689
$218$	0.450098	+	0.779592i	0.0304844	+	0.0528006i
$219$		0			0
$220$		0			0
$221$	3.37820	−	5.85122i	0.227243	−	0.393596i
$222$		0			0
$223$	−3.87393	−	6.70984i	−0.259417	−	0.449324i	0.706669	−	0.707545i	$-0.250197\pi$
$223$	−3.87393	−	6.70984i	−0.259417	−	0.449324i	−0.966086	+	0.258221i	$0.916864\pi$
$224$	−12.4432			−0.831397
$225$		0			0
$226$	−3.11366			−0.207118
$227$	−5.63014	−	9.75169i	−0.373685	−	0.647242i	0.616444	−	0.787399i	$-0.288573\pi$
$227$	−5.63014	−	9.75169i	−0.373685	−	0.647242i	−0.990129	+	0.140157i	$0.955239\pi$
$228$		0			0
$229$	5.23879	−	9.07384i	0.346189	−	0.599616i	−0.639380	−	0.768891i	$-0.720809\pi$
$229$	5.23879	−	9.07384i	0.346189	−	0.599616i	0.985569	+	0.169274i	$0.0541424\pi$
$230$		0			0
$231$		0			0
$232$	1.50667	+	2.60962i	0.0989176	+	0.171330i
$233$	2.90214			0.190125			0.0950627	−	0.995471i	$-0.469695\pi$
$233$	2.90214			0.190125			0.0950627	+	0.995471i	$0.469695\pi$
$234$		0			0
$235$		0			0
$236$	−9.07188	−	15.7130i	−0.590529	−	1.02283i
$237$		0			0
$238$	−1.92172	+	3.32852i	−0.124567	+	0.215756i
$239$	8.17723	−	14.1634i	0.528941	−	0.916153i	−0.470489	−	0.882406i	$-0.655923\pi$
$239$	8.17723	−	14.1634i	0.528941	−	0.916153i	0.999430	−	0.0337471i	$-0.0107441\pi$
$240$		0			0
$241$	−8.76194	−	15.1761i	−0.564406	−	0.977580i	−0.997105	−	0.0760416i	$-0.975772\pi$
$241$	−8.76194	−	15.1761i	−0.564406	−	0.977580i	0.432698	−	0.901539i	$-0.357562\pi$
$242$	−12.8023			−0.822965
$243$		0			0
$244$	0.0195242			0.00124991
$245$		0			0
$246$		0			0
$247$	−0.379833	+	0.657890i	−0.0241682	+	0.0418605i
$248$	7.37898	−	12.7808i	0.468566	−	0.811580i
$249$		0			0
$250$		0			0
$251$	−8.46999			−0.534621			−0.267311	−	0.963610i	$-0.586135\pi$
$251$	−8.46999			−0.534621			−0.267311	+	0.963610i	$0.586135\pi$
$252$		0			0
$253$	25.9882			1.63386
$254$	−2.19079	−	3.79456i	−0.137462	−	0.238092i
$255$		0			0
$256$	−0.468695	+	0.811804i	−0.0292934	+	0.0507377i
$257$	−1.43625	+	2.48766i	−0.0895910	+	0.155176i	−0.907338	−	0.420401i	$-0.861889\pi$
$257$	−1.43625	+	2.48766i	−0.0895910	+	0.155176i	0.817747	+	0.575577i	$0.195223\pi$
$258$		0			0
$259$	−4.65545	−	8.06348i	−0.289276	−	0.501040i
$260$		0			0
$261$		0			0
$262$	0.127493			0.00787656
$263$	12.8119	+	22.1909i	0.790017	+	1.36835i	0.925955	+	0.377633i	$0.123262\pi$
$263$	12.8119	+	22.1909i	0.790017	+	1.36835i	−0.135938	+	0.990717i	$0.543405\pi$
$264$		0			0
$265$		0			0
$266$	0.216072	−	0.374247i	0.0132482	−	0.0229465i
$267$		0			0
$268$	0.872756	+	1.51166i	0.0533121	+	0.0923392i
$269$	−0.337210			−0.0205600			−0.0102800	−	0.999947i	$-0.503272\pi$
$269$	−0.337210			−0.0205600			−0.0102800	+	0.999947i	$0.503272\pi$
$270$		0			0
$271$	21.5927			1.31166			0.655831	−	0.754908i	$-0.272318\pi$
$271$	21.5927			1.31166			0.655831	+	0.754908i	$0.272318\pi$
$272$	4.28767	+	7.42646i	0.259978	+	0.450295i
$273$		0			0
$274$	−0.822560	+	1.42472i	−0.0496927	+	0.0860702i
$275$		0			0
$276$		0			0
$277$	−12.0669	−	20.9004i	−0.725028	−	1.25579i	−0.958962	−	0.283533i	$-0.908493\pi$
$277$	−12.0669	−	20.9004i	−0.725028	−	1.25579i	0.233934	−	0.972252i	$-0.424840\pi$
$278$	−6.97949			−0.418602
$279$		0			0
$280$		0			0
$281$	1.68363	+	2.91613i	0.100437	+	0.173962i	0.911865	−	0.410491i	$-0.134643\pi$
$281$	1.68363	+	2.91613i	0.100437	+	0.173962i	−0.811428	+	0.584453i	$0.801309\pi$
$282$		0			0
$283$	−10.9249	+	18.9224i	−0.649415	+	1.12482i	0.333848	+	0.942627i	$0.391653\pi$
$283$	−10.9249	+	18.9224i	−0.649415	+	1.12482i	−0.983263	+	0.182193i	$0.941681\pi$
$284$	5.71444	−	9.89770i	0.339090	−	0.587321i
$285$		0			0
$286$	3.11243	+	5.39088i	0.184042	+	0.318770i
$287$	7.00619			0.413562
$288$		0			0
$289$	−6.95994			−0.409408
$290$		0			0
$291$		0			0
$292$	5.87393	−	10.1739i	0.343746	−	0.595385i
$293$	6.87702	−	11.9114i	0.401760	−	0.695869i	−0.592179	−	0.805807i	$-0.701732\pi$
$293$	6.87702	−	11.9114i	0.401760	−	0.695869i	0.993938	+	0.109938i	$0.0350653\pi$
$294$		0			0
$295$		0			0
$296$	6.49179			0.377328
$297$		0			0
$298$	−4.80509			−0.278352
$299$	−4.49166	−	7.77978i	−0.259759	−	0.449916i
$300$		0			0
$301$	9.83978	−	17.0430i	0.567155	−	0.982342i
$302$	−2.44110	+	4.22810i	−0.140469	+	0.243300i
$303$		0			0
$304$	−0.482090	−	0.835004i	−0.0276498	−	0.0478908i
$305$		0			0
$306$		0			0
$307$	34.2183			1.95294			0.976472	−	0.215644i	$-0.0691850\pi$
$307$	34.2183			1.95294			0.976472	+	0.215644i	$0.0691850\pi$
$308$	14.0399	+	24.3179i	0.799999	+	1.38564i
$309$		0			0
$310$		0			0
$311$	11.5199	−	19.9530i	0.653232	−	1.13143i	−0.329102	−	0.944294i	$-0.606746\pi$
$311$	11.5199	−	19.9530i	0.653232	−	1.13143i	0.982334	−	0.187136i	$-0.0599206\pi$
$312$		0			0
$313$	1.79565	+	3.11016i	0.101496	+	0.175796i	0.912301	−	0.409520i	$-0.134304\pi$
$313$	1.79565	+	3.11016i	0.101496	+	0.175796i	−0.810805	+	0.585316i	$0.800970\pi$
$314$	−0.502885			−0.0283794
$315$		0			0
$316$	16.8215			0.946281
$317$	6.59033	+	11.4148i	0.370150	+	0.641118i	0.989588	−	0.143927i	$-0.0459731\pi$
$317$	6.59033	+	11.4148i	0.370150	+	0.641118i	−0.619439	+	0.785045i	$0.712640\pi$
$318$		0			0
$319$	5.20085	−	9.00813i	0.291192	−	0.504359i
$320$		0			0
$321$		0			0
$322$	2.55512	+	4.42560i	0.142391	+	0.246629i
$323$	−1.12887			−0.0628119
$324$		0			0
$325$		0			0
$326$	−4.05545	−	7.02424i	−0.224611	−	0.389037i
$327$		0			0
$328$	−2.44244	+	4.23044i	−0.134861	+	0.233587i
$329$	−14.6464	+	25.3683i	−0.807483	+	1.39860i
$330$		0			0
$331$	−0.591264	−	1.02410i	−0.0324988	−	0.0562896i	0.849319	−	0.527881i	$-0.177013\pi$
$331$	−0.591264	−	1.02410i	−0.0324988	−	0.0562896i	−0.881817	+	0.471591i	$0.843680\pi$
$332$	−18.5058			−1.01564
$333$		0			0
$334$	−2.06975			−0.113252
$335$		0			0
$336$		0			0
$337$	12.3997	−	21.4770i	0.675457	−	1.16993i	−0.300879	−	0.953662i	$-0.597280\pi$
$337$	12.3997	−	21.4770i	0.675457	−	1.16993i	0.976335	−	0.216263i	$-0.0693868\pi$
$338$	−2.00028	+	3.46459i	−0.108801	+	0.188449i
$339$		0			0
$340$		0			0
$341$	−50.9428			−2.75871
$342$		0			0
$343$	−19.0454			−1.02836
$344$	6.86053	+	11.8828i	0.369895	+	0.640677i
$345$		0			0
$346$	3.46898	−	6.00845i	0.186493	−	0.323016i
$347$	11.0846	−	19.1991i	0.595052	−	1.03066i	−0.398488	−	0.917174i	$-0.630465\pi$
$347$	11.0846	−	19.1991i	0.595052	−	1.03066i	0.993540	−	0.113486i	$-0.0362018\pi$
$348$		0			0
$349$	−7.45925	−	12.9198i	−0.399285	−	0.691581i	0.594353	−	0.804204i	$-0.297408\pi$
$349$	−7.45925	−	12.9198i	−0.399285	−	0.691581i	−0.993638	+	0.112623i	$0.964075\pi$
$350$		0			0
$351$		0			0
$352$	−29.9476			−1.59621
$353$	8.45726	+	14.6484i	0.450134	+	0.779656i	0.998394	−	0.0566525i	$-0.0180427\pi$
$353$	8.45726	+	14.6484i	0.450134	+	0.779656i	−0.548260	+	0.836308i	$0.684709\pi$
$354$		0			0
$355$		0			0
$356$	5.56732	−	9.64288i	0.295067	−	0.511072i
$357$		0			0
$358$	1.62566	+	2.81573i	0.0859190	+	0.148816i
$359$	−0.636657			−0.0336015			−0.0168007	−	0.999859i	$-0.505348\pi$
$359$	−0.636657			−0.0336015			−0.0168007	+	0.999859i	$0.505348\pi$
$360$		0			0
$361$	−18.8731			−0.993320
$362$	−2.59602	−	4.49644i	−0.136444	−	0.236328i
$363$		0			0
$364$	4.85317	−	8.40594i	0.254375	−	0.440591i
$365$		0			0
$366$		0			0
$367$	−10.0490	−	17.4053i	−0.524552	−	0.908550i	−0.999591	−	0.0285858i	$-0.990900\pi$
$367$	−10.0490	−	17.4053i	−0.524552	−	0.908550i	0.475040	−	0.879964i	$-0.342434\pi$
$368$	11.4018			0.594358
$369$		0			0
$370$		0			0
$371$	−12.0913	−	20.9427i	−0.627749	−	1.08729i
$372$		0			0
$373$	9.82146	−	17.0113i	0.508536	−	0.880810i	−0.491415	−	0.870925i	$-0.663520\pi$
$373$	9.82146	−	17.0113i	0.508536	−	0.880810i	0.999951	−	0.00988448i	$-0.00314638\pi$
$374$	−4.62509	+	8.01088i	−0.239157	+	0.414233i
$375$		0			0
$376$	−10.2118	−	17.6874i	−0.526635	−	0.912159i
$377$	−3.59555			−0.185180
$378$		0			0
$379$	−7.94219			−0.407963			−0.203982	−	0.978975i	$-0.565388\pi$
$379$	−7.94219			−0.407963			−0.203982	+	0.978975i	$0.565388\pi$
$380$		0			0
$381$		0			0
$382$	−3.24949	+	5.62829i	−0.166259	+	0.287968i
$383$	−15.5944	+	27.0103i	−0.796836	+	1.38016i	0.124830	+	0.992178i	$0.460161\pi$
$383$	−15.5944	+	27.0103i	−0.796836	+	1.38016i	−0.921667	+	0.387983i	$0.873172\pi$
$384$		0			0
$385$		0			0
$386$	0.228285			0.0116194
$387$		0			0
$388$	−12.7995			−0.649795
$389$	−15.7247	−	27.2360i	−0.797274	−	1.38092i	−0.921385	−	0.388650i	$-0.872941\pi$
$389$	−15.7247	−	27.2360i	−0.797274	−	1.38092i	0.124111	−	0.992268i	$-0.460392\pi$
$390$		0			0
$391$	6.67463	−	11.5608i	0.337550	−	0.584655i
$392$	0.384886	−	0.666642i	0.0194397	−	0.0336705i
$393$		0			0
$394$	−1.30970	−	2.26847i	−0.0659819	−	0.114284i
$395$		0			0
$396$		0			0
$397$	17.7174			0.889211			0.444606	−	0.895726i	$-0.353344\pi$
$397$	17.7174			0.889211			0.444606	+	0.895726i	$0.353344\pi$
$398$	4.13128	+	7.15558i	0.207082	+	0.358677i
$399$		0			0
$400$		0			0
$401$	−3.57124	+	6.18556i	−0.178339	+	0.308892i	−0.941312	−	0.337538i	$-0.890406\pi$
$401$	−3.57124	+	6.18556i	−0.178339	+	0.308892i	0.762973	+	0.646431i	$0.223739\pi$
$402$		0			0
$403$	8.80468	+	15.2502i	0.438593	+	0.759665i
$404$	12.3806			0.615958
$405$		0			0
$406$	2.04536			0.101509
$407$	−11.2045	−	19.4067i	−0.555385	−	0.961955i
$408$		0			0
$409$	−12.3759	+	21.4357i	−0.611948	+	1.05993i	0.378964	+	0.925412i	$0.376281\pi$
$409$	−12.3759	+	21.4357i	−0.611948	+	1.05993i	−0.990912	+	0.134514i	$0.957053\pi$
$410$		0			0
$411$		0			0
$412$	−5.43031	−	9.40558i	−0.267532	−	0.463380i
$413$	−26.1839			−1.28843
$414$		0			0
$415$		0			0
$416$	5.17598	+	8.96505i	0.253773	+	0.439548i
$417$		0			0
$418$	0.520028	−	0.900715i	0.0254354	−	0.0440554i
$419$	−5.32956	+	9.23106i	−0.260366	+	0.450967i	−0.966339	−	0.257272i	$-0.917176\pi$
$419$	−5.32956	+	9.23106i	−0.260366	+	0.450967i	0.705973	+	0.708238i	$0.250510\pi$
$420$		0			0
$421$	4.08931	+	7.08288i	0.199301	+	0.345199i	0.948302	−	0.317370i	$-0.102800\pi$
$421$	4.08931	+	7.08288i	0.199301	+	0.345199i	−0.749001	+	0.662569i	$0.769466\pi$
$422$	0.774555			0.0377048
$423$		0			0
$424$	16.8607			0.818828
$425$		0			0
$426$		0			0
$427$	0.0140881	−	0.0244012i	0.000681769	−	0.00118086i
$428$	12.9072	−	22.3559i	0.623893	−	1.08061i
$429$		0			0
$430$		0			0
$431$	−1.67248			−0.0805604			−0.0402802	−	0.999188i	$-0.512825\pi$
$431$	−1.67248			−0.0805604			−0.0402802	+	0.999188i	$0.512825\pi$
$432$		0			0
$433$	−9.95994			−0.478644			−0.239322	−	0.970940i	$-0.576925\pi$
$433$	−9.95994			−0.478644			−0.239322	+	0.970940i	$0.576925\pi$
$434$	−5.00863	−	8.67519i	−0.240422	−	0.416423i
$435$		0			0
$436$	1.68913	−	2.92565i	0.0808945	−	0.140113i
$437$	−0.750471	+	1.29985i	−0.0358999	+	0.0621805i
$438$		0			0
$439$	−6.40788	−	11.0988i	−0.305832	−	0.529716i	0.671615	−	0.740901i	$-0.265601\pi$
$439$	−6.40788	−	11.0988i	−0.305832	−	0.529716i	−0.977446	+	0.211185i	$0.932268\pi$
$440$		0			0
$441$		0			0
$442$	3.19750			0.152090
$443$	−3.87702	−	6.71520i	−0.184203	−	0.319049i	0.759105	−	0.650968i	$-0.225637\pi$
$443$	−3.87702	−	6.71520i	−0.184203	−	0.319049i	−0.943308	+	0.331920i	$0.892304\pi$
$444$		0			0
$445$		0			0
$446$	1.83335	−	3.17546i	0.0868118	−	0.150362i
$447$		0			0
$448$	3.99210	+	6.91452i	0.188609	+	0.326681i
$449$	33.3401			1.57342			0.786709	−	0.617324i	$-0.211783\pi$
$449$	33.3401			1.57342			0.786709	+	0.617324i	$0.211783\pi$
$450$		0			0
$451$	16.8621			0.794004
$452$	5.84247	+	10.1195i	0.274807	+	0.475979i
$453$		0			0
$454$	2.66449	−	4.61503i	0.125051	−	0.216594i
$455$		0			0
$456$		0			0
$457$	19.1096	+	33.0988i	0.893910	+	1.54830i	0.835148	+	0.550026i	$0.185382\pi$
$457$	19.1096	+	33.0988i	0.893910	+	1.54830i	0.0587626	+	0.998272i	$0.481285\pi$
$458$	4.95856			0.231698
$459$		0			0
$460$		0			0
$461$	15.6517	+	27.1095i	0.728971	+	1.26261i	0.957319	+	0.289035i	$0.0933343\pi$
$461$	15.6517	+	27.1095i	0.728971	+	1.26261i	−0.228348	+	0.973580i	$0.573332\pi$
$462$		0			0
$463$	−6.04258	+	10.4661i	−0.280823	+	0.486399i	−0.971588	−	0.236680i	$-0.923941\pi$
$463$	−6.04258	+	10.4661i	−0.280823	+	0.486399i	0.690765	+	0.723079i	$0.257274\pi$
$464$	2.28176	−	3.95212i	0.105928	−	0.183473i
$465$		0			0
$466$	0.686725	+	1.18944i	0.0318119	+	0.0550998i
$467$	7.60466			0.351902			0.175951	−	0.984399i	$-0.443700\pi$
$467$	7.60466			0.351902			0.175951	+	0.984399i	$0.443700\pi$
$468$		0			0
$469$	2.51901			0.116317
$470$		0			0
$471$		0			0
$472$	9.12803	−	15.8102i	0.420152	−	0.727724i
$473$	23.6818	−	41.0181i	1.08889	−	1.88601i
$474$		0			0
$475$		0			0
$476$	14.4237			0.661108
$477$		0			0
$478$	7.73982			0.354011
$479$	−16.2417	−	28.1314i	−0.742101	−	1.28536i	−0.951537	−	0.307534i	$-0.900496\pi$
$479$	−16.2417	−	28.1314i	−0.742101	−	1.28536i	0.209437	−	0.977822i	$-0.432837\pi$
$480$		0			0
$481$	−3.87304	+	6.70830i	−0.176595	+	0.305872i
$482$	4.14663	−	7.18217i	0.188874	−	0.327139i
$483$		0			0
$484$	24.0223	+	41.6079i	1.09192	+	1.89127i
$485$		0			0
$486$		0			0
$487$	−4.46121			−0.202157			−0.101078	−	0.994878i	$-0.532229\pi$
$487$	−4.46121			−0.202157			−0.101078	+	0.994878i	$0.532229\pi$
$488$	0.00982254	+	0.0170131i	0.000444645	+	0.000770148i
$489$		0			0
$490$		0			0
$491$	16.4210	−	28.4420i	0.741070	−	1.28357i	−0.210938	−	0.977499i	$-0.567652\pi$
$491$	16.4210	−	28.4420i	0.741070	−	1.28357i	0.952008	−	0.306072i	$-0.0990148\pi$
$492$		0			0
$493$	−2.67150	−	4.62717i	−0.120318	−	0.208397i
$494$	−0.359515			−0.0161754
$495$		0			0
$496$	−22.3501			−1.00355
$497$	−8.24672	−	14.2837i	−0.369916	−	0.640713i
$498$		0			0
$499$	17.1010	−	29.6198i	0.765547	−	1.32597i	−0.174410	−	0.984673i	$-0.555802\pi$
$499$	17.1010	−	29.6198i	0.765547	−	1.32597i	0.939957	−	0.341293i	$-0.110865\pi$
$500$		0			0
$501$		0			0
$502$	−2.00423	−	3.47143i	−0.0894532	−	0.154938i
$503$	−22.1773			−0.988837			−0.494419	−	0.869224i	$-0.664619\pi$
$503$	−22.1773			−0.988837			−0.494419	+	0.869224i	$0.664619\pi$
$504$		0			0
$505$		0			0
$506$	6.14951	+	10.6513i	0.273379	+	0.473507i
$507$		0			0
$508$	−8.22160	+	14.2402i	−0.364775	+	0.631808i
$509$	−10.7816	+	18.6743i	−0.477887	+	0.827724i	−0.999679	−	0.0253489i	$-0.991930\pi$
$509$	−10.7816	+	18.6743i	−0.477887	+	0.827724i	0.521792	+	0.853073i	$0.325264\pi$
$510$		0			0
$511$	−8.47688	−	14.6824i	−0.374995	−	0.649510i
$512$	−22.8115			−1.00813
$513$		0			0
$514$	−1.35943			−0.0599617
$515$		0			0
$516$		0			0
$517$	−35.2501	+	61.0550i	−1.55030	+	2.68520i
$518$	2.20321	−	3.81608i	0.0968037	−	0.167669i
$519$		0			0
$520$		0			0
$521$	−20.2626			−0.887718			−0.443859	−	0.896097i	$-0.646391\pi$
$521$	−20.2626			−0.887718			−0.443859	+	0.896097i	$0.646391\pi$
$522$		0			0
$523$	31.8114			1.39101			0.695507	−	0.718520i	$-0.255180\pi$
$523$	31.8114			1.39101			0.695507	+	0.718520i	$0.255180\pi$
$524$	−0.239229	−	0.414356i	−0.0104507	−	0.0181012i
$525$		0			0
$526$	−6.06330	+	10.5019i	−0.264373	+	0.457907i
$527$	−13.0838	+	22.6618i	−0.569940	+	0.987164i
$528$		0			0
$529$	2.62541	+	4.54735i	0.114148	+	0.197711i
$530$		0			0
$531$		0			0
$532$	−1.62175			−0.0703117
$533$	−2.91435	−	5.04780i	−0.126235	−	0.218645i
$534$		0			0
$535$		0			0
$536$	−0.878159	+	1.52102i	−0.0379307	+	0.0656978i
$537$		0			0
$538$	−0.0797930	−	0.138205i	−0.00344012	−	0.00595846i
$539$	−2.65717			−0.114452
$540$		0			0
$541$	−15.1315			−0.650553			−0.325277	−	0.945619i	$-0.605457\pi$
$541$	−15.1315			−0.650553			−0.325277	+	0.945619i	$0.605457\pi$
$542$	5.10942	+	8.84977i	0.219468	+	0.380130i
$543$		0			0
$544$	−7.69153	+	13.3221i	−0.329772	+	0.571181i
$545$		0			0
$546$		0			0
$547$	2.04372	+	3.53982i	0.0873831	+	0.151352i	0.906404	−	0.422411i	$-0.138816\pi$
$547$	2.04372	+	3.53982i	0.0873831	+	0.151352i	−0.819021	+	0.573763i	$0.805483\pi$
$548$	6.17381			0.263732
$549$		0			0
$550$		0			0
$551$	0.300374	+	0.520263i	0.0127964	+	0.0221639i
$552$		0			0
$553$	12.1378	−	21.0233i	0.516153	−	0.894003i
$554$	5.71070	−	9.89123i	0.242625	−	0.420238i
$555$		0			0
$556$	13.0963	+	22.6835i	0.555408	+	0.961994i
$557$	13.1425			0.556864			0.278432	−	0.960456i	$-0.410185\pi$
$557$	13.1425			0.556864			0.278432	+	0.960456i	$0.410185\pi$
$558$		0			0
$559$	−16.3721			−0.692467
$560$		0			0
$561$		0			0
$562$	−0.796785	+	1.38007i	−0.0336104	+	0.0582149i
$563$	−12.2611	+	21.2368i	−0.516742	+	0.895023i	0.483069	+	0.875582i	$0.339522\pi$
$563$	−12.2611	+	21.2368i	−0.516742	+	0.895023i	−0.999811	+	0.0194410i	$0.993811\pi$
$564$		0			0
$565$		0			0
$566$	−10.3405			−0.434642
$567$		0			0
$568$	11.4996			0.482514
$569$	11.3649	+	19.6846i	0.476442	+	0.825223i	0.999636	−	0.0269915i	$-0.00859271\pi$
$569$	11.3649	+	19.6846i	0.476442	+	0.825223i	−0.523193	+	0.852214i	$0.675259\pi$
$570$		0			0
$571$	0.247093	−	0.427977i	0.0103405	−	0.0179103i	−0.860809	−	0.508928i	$-0.830042\pi$
$571$	0.247093	−	0.427977i	0.0103405	−	0.0179103i	0.871149	+	0.491018i	$0.163375\pi$
$572$	11.6803	−	20.2309i	0.488379	−	0.845897i
$573$		0			0
$574$	1.65786	+	2.87149i	0.0691976	+	0.119854i
$575$		0			0
$576$		0			0
$577$	−9.41187			−0.391821			−0.195911	−	0.980622i	$-0.562766\pi$
$577$	−9.41187			−0.391821			−0.195911	+	0.980622i	$0.562766\pi$
$578$	−1.64691	−	2.85254i	−0.0685025	−	0.118650i
$579$		0			0
$580$		0			0
$581$	−13.3532	+	23.1284i	−0.553984	+	0.959529i
$582$		0			0
$583$	−29.1006	−	50.4037i	−1.20522	−	2.08751i
$584$	11.8206			0.489139
$585$		0			0
$586$	6.50916			0.268891
$587$	4.98661	+	8.63705i	0.205819	+	0.356489i	0.950393	−	0.311050i	$-0.100681\pi$
$587$	4.98661	+	8.63705i	0.205819	+	0.356489i	−0.744574	+	0.667540i	$0.767347\pi$
$588$		0			0
$589$	1.47110	−	2.54801i	0.0606155	−	0.104989i
$590$		0			0
$591$		0			0
$592$	−4.91572	−	8.51428i	−0.202035	−	0.349935i
$593$	−38.3421			−1.57452			−0.787260	−	0.616621i	$-0.788501\pi$
$593$	−38.3421			−1.57452			−0.787260	+	0.616621i	$0.788501\pi$
$594$		0			0
$595$		0			0
$596$	9.01628	+	15.6167i	0.369321	+	0.639683i
$597$		0			0
$598$	2.12570	−	3.68182i	0.0869262	−	0.150561i
$599$	−5.07665	+	8.79301i	−0.207426	+	0.359273i	−0.950903	−	0.309489i	$-0.899842\pi$
$599$	−5.07665	+	8.79301i	−0.207426	+	0.359273i	0.743477	+	0.668762i	$0.233175\pi$
$600$		0			0
$601$	10.6371	+	18.4241i	0.433898	+	0.751533i	0.997205	−	0.0747146i	$-0.0238046\pi$
$601$	10.6371	+	18.4241i	0.433898	+	0.751533i	−0.563307	+	0.826248i	$0.690471\pi$
$602$	9.31344			0.379587
$603$		0			0
$604$	18.3219			0.745508
$605$		0			0
$606$		0			0
$607$	18.8678	−	32.6799i	0.765819	−	1.32644i	−0.173993	−	0.984747i	$-0.555667\pi$
$607$	18.8678	−	32.6799i	0.765819	−	1.32644i	0.939812	−	0.341691i	$-0.111000\pi$
$608$	0.864808	−	1.49789i	0.0350726	−	0.0607475i
$609$		0			0
$610$		0			0
$611$	24.3698			0.985895
$612$		0			0
$613$	−32.2633			−1.30310			−0.651551	−	0.758605i	$-0.725881\pi$
$613$	−32.2633			−1.30310			−0.651551	+	0.758605i	$0.725881\pi$
$614$	8.09699	+	14.0244i	0.326768	+	0.565979i
$615$		0			0
$616$	−14.1268	+	24.4684i	−0.569186	+	0.985860i
$617$	−13.0089	+	22.5321i	−0.523719	+	0.907108i	0.475900	+	0.879499i	$0.342122\pi$
$617$	−13.0089	+	22.5321i	−0.523719	+	0.907108i	−0.999619	+	0.0276084i	$0.991211\pi$
$618$		0			0
$619$	5.94077	+	10.2897i	0.238780	+	0.413578i	0.960364	−	0.278748i	$-0.0899193\pi$
$619$	5.94077	+	10.2897i	0.238780	+	0.413578i	−0.721585	+	0.692326i	$0.756586\pi$
$620$		0			0
$621$		0			0
$622$	10.9037			0.437197
$623$	−8.03440	−	13.9160i	−0.321891	−	0.557532i
$624$		0			0
$625$		0			0
$626$	−0.849799	+	1.47189i	−0.0339648	+	0.0588288i
$627$		0			0
$628$	0.943614	+	1.63439i	0.0376543	+	0.0652191i
$629$	−11.5107			−0.458962
$630$		0			0
$631$	−13.2726			−0.528372			−0.264186	−	0.964472i	$-0.585103\pi$
$631$	−13.2726			−0.528372			−0.264186	+	0.964472i	$0.585103\pi$
$632$	8.46279	+	14.6580i	0.336632	+	0.583063i
$633$		0			0
$634$	−3.11890	+	5.40210i	−0.123867	+	0.214545i
$635$		0			0
$636$		0			0
$637$	0.459251	+	0.795445i	0.0181962	+	0.0315167i
$638$	4.92265			0.194890
$639$		0			0
$640$		0			0
$641$	22.4075	+	38.8109i	0.885042	+	1.53294i	0.845665	+	0.533713i	$0.179204\pi$
$641$	22.4075	+	38.8109i	0.885042	+	1.53294i	0.0393765	+	0.999224i	$0.487463\pi$
$642$		0			0
$643$	7.46275	−	12.9259i	0.294302	−	0.509747i	−0.680520	−	0.732729i	$-0.738246\pi$
$643$	7.46275	−	12.9259i	0.294302	−	0.509747i	0.974822	+	0.222983i	$0.0715794\pi$
$644$	9.58886	−	16.6084i	0.377854	−	0.654462i
$645$		0			0
$646$	−0.267121	−	0.462667i	−0.0105097	−	0.0182034i
$647$	41.2684			1.62243			0.811214	−	0.584749i	$-0.198807\pi$
$647$	41.2684			1.62243			0.811214	+	0.584749i	$0.198807\pi$
$648$		0			0
$649$	−63.0179			−2.47367
$650$		0			0
$651$		0			0
$652$	−15.2193	+	26.3606i	−0.596034	+	1.03236i
$653$	13.8126	−	23.9241i	0.540528	−	0.936223i	−0.458345	−	0.888774i	$-0.651558\pi$
$653$	13.8126	−	23.9241i	0.540528	−	0.936223i	0.998874	−	0.0474484i	$-0.0151090\pi$
$654$		0			0
$655$		0			0
$656$	7.39788			0.288839
$657$		0			0
$658$	−13.8630			−0.540435
$659$	20.0112	+	34.6605i	0.779527	+	1.35018i	0.932215	+	0.361905i	$0.117874\pi$
$659$	20.0112	+	34.6605i	0.779527	+	1.35018i	−0.152688	+	0.988274i	$0.548793\pi$
$660$		0			0
$661$	−12.4965	+	21.6445i	−0.486056	+	0.841874i	−0.999872	−	0.0160270i	$-0.994898\pi$
$661$	−12.4965	+	21.6445i	−0.486056	+	0.841874i	0.513816	+	0.857901i	$0.328232\pi$
$662$	0.279818	−	0.484659i	0.0108754	−	0.0188368i
$663$		0			0
$664$	−9.31018	−	16.1257i	−0.361305	−	0.625799i
$665$		0			0
$666$		0			0
$667$	−7.10405			−0.275070
$668$	3.88369	+	6.72675i	0.150264	+	0.260266i
$669$		0			0
$670$		0			0
$671$	0.0339063	−	0.0587274i	0.00130894	−	0.00226715i
$672$		0			0
$673$	20.4024	+	35.3380i	0.786454	+	1.36218i	0.928126	+	0.372265i	$0.121419\pi$
$673$	20.4024	+	35.3380i	0.786454	+	1.36218i	−0.141672	+	0.989914i	$0.545248\pi$
$674$	11.7365			0.452072
$675$		0			0
$676$	15.0133			0.577436
$677$	20.5947	+	35.6710i	0.791518	+	1.37095i	0.925027	+	0.379901i	$0.124042\pi$
$677$	20.5947	+	35.6710i	0.791518	+	1.37095i	−0.133509	+	0.991048i	$0.542625\pi$
$678$		0			0
$679$	−9.23569	+	15.9967i	−0.354433	+	0.613896i
$680$		0			0
$681$		0			0
$682$	−12.0545	−	20.8789i	−0.461589	−	0.799496i
$683$	−1.33820			−0.0512047			−0.0256023	−	0.999672i	$-0.508150\pi$
$683$	−1.33820			−0.0512047			−0.0256023	+	0.999672i	$0.508150\pi$
$684$		0			0
$685$		0			0
$686$	−4.50667	−	7.80578i	−0.172065	−	0.298026i
$687$		0			0
$688$	10.3899	−	17.9958i	0.396110	−	0.686083i
$689$	−10.0592	+	17.4230i	−0.383225	+	0.663764i
$690$		0			0
$691$	12.6407	+	21.8943i	0.480874	+	0.832898i	0.999759	−	0.0219459i	$-0.00698617\pi$
$691$	12.6407	+	21.8943i	0.480874	+	0.832898i	−0.518885	+	0.854844i	$0.673653\pi$
$692$	−26.0368			−0.989770
$693$		0			0
$694$	10.4917			0.398258
$695$		0			0
$696$		0			0
$697$	4.33074	−	7.50106i	0.164039	−	0.284123i
$698$	3.53012	−	6.11435i	0.133617	−	0.231432i
$699$		0			0
$700$		0			0
$701$	18.2064			0.687645			0.343822	−	0.939035i	$-0.388278\pi$
$701$	18.2064			0.687645			0.343822	+	0.939035i	$0.388278\pi$
$702$		0			0
$703$	1.29422			0.0488126
$704$	9.60795	+	16.6415i	0.362113	+	0.627199i
$705$		0			0
$706$	−4.00244	+	6.93242i	−0.150634	+	0.260905i
$707$	8.93344	−	15.4732i	0.335977	−	0.581929i
$708$		0			0
$709$	−20.9103	−	36.2177i	−0.785304	−	1.36019i	−0.928818	−	0.370537i	$-0.879174\pi$
$709$	−20.9103	−	36.2177i	−0.785304	−	1.36019i	0.143514	−	0.989648i	$-0.454160\pi$
$710$		0			0
$711$		0			0
$712$	11.2036			0.419871
$713$	17.3962	+	30.1312i	0.651494	+	1.12842i
$714$		0			0
$715$		0			0
$716$	6.10079	−	10.5669i	0.227997	−	0.394903i
$717$		0			0
$718$	−0.150650	−	0.260934i	−0.00562222	−	0.00973797i
$719$	−48.9786			−1.82660			−0.913298	−	0.407293i	$-0.866473\pi$
$719$	−48.9786			−1.82660			−0.913298	+	0.407293i	$0.866473\pi$
$720$		0			0
$721$	−15.6734			−0.583707
$722$	−4.46588	−	7.73514i	−0.166203	−	0.287872i
$723$		0			0
$724$	−9.74236	+	16.8743i	−0.362072	+	0.627127i
$725$		0			0
$726$		0			0
$727$	21.9005	+	37.9327i	0.812243	+	1.40685i	0.911291	+	0.411764i	$0.135087\pi$
$727$	21.9005	+	37.9327i	0.812243	+	1.40685i	−0.0990474	+	0.995083i	$0.531580\pi$
$728$	9.76642			0.361968
$729$		0			0
$730$		0			0
$731$	−12.1645	−	21.0696i	−0.449922	−	0.779287i
$732$		0			0
$733$	3.39332	−	5.87740i	0.125335	−	0.217087i	−0.796529	−	0.604601i	$-0.793333\pi$
$733$	3.39332	−	5.87740i	0.125335	−	0.217087i	0.921864	+	0.387514i	$0.126666\pi$
$734$	4.75572	−	8.23715i	0.175537	−	0.304039i
$735$		0			0
$736$	10.2267	+	17.7131i	0.376960	+	0.652913i
$737$	6.06261			0.223319
$738$		0			0
$739$	28.7245			1.05665			0.528324	−	0.849043i	$-0.322821\pi$
$739$	28.7245			1.05665			0.528324	+	0.849043i	$0.322821\pi$
$740$		0			0
$741$		0			0
$742$	5.72226	−	9.91125i	0.210071	−	0.363853i
$743$	−15.7262	+	27.2385i	−0.576937	+	0.999284i	0.418891	+	0.908036i	$0.362419\pi$
$743$	−15.7262	+	27.2385i	−0.576937	+	0.999284i	−0.995828	+	0.0912477i	$0.970915\pi$
$744$		0			0
$745$		0			0
$746$	9.29610			0.340354
$747$		0			0
$748$	34.7141			1.26927
$749$	−18.6268	−	32.2626i	−0.680610	−	1.17885i
$750$		0			0
$751$	−5.47659	+	9.48574i	−0.199844	+	0.346139i	−0.948478	−	0.316844i	$-0.897377\pi$
$751$	−5.47659	+	9.48574i	−0.199844	+	0.346139i	0.748634	+	0.662984i	$0.230710\pi$
$752$	−15.4652	+	26.7866i	−0.563959	+	0.976806i
$753$		0			0
$754$	−0.850804	−	1.47364i	−0.0309845	−	0.0536667i
$755$		0			0
$756$		0			0
$757$	−45.7942			−1.66442			−0.832210	−	0.554461i	$-0.812925\pi$
$757$	−45.7942			−1.66442			−0.832210	+	0.554461i	$0.812925\pi$
$758$	−1.87934	−	3.25511i	−0.0682607	−	0.118231i
$759$		0			0
$760$		0			0
$761$	−16.9569	+	29.3702i	−0.614687	+	1.06467i	0.375753	+	0.926720i	$0.377384\pi$
$761$	−16.9569	+	29.3702i	−0.614687	+	1.06467i	−0.990439	+	0.137948i	$0.955949\pi$
$762$		0			0
$763$	−2.43764	−	4.22212i	−0.0882485	−	0.152851i
$764$	24.3894			0.882378
$765$		0			0
$766$	−14.7602			−0.533309
$767$	10.8917	+	18.8649i	0.393275	+	0.681173i
$768$		0			0
$769$	−3.57986	+	6.20050i	−0.129093	+	0.223596i	−0.923325	−	0.384018i	$-0.874540\pi$
$769$	−3.57986	+	6.20050i	−0.129093	+	0.223596i	0.794232	+	0.607614i	$0.207873\pi$
$770$		0			0
$771$		0			0
$772$	−0.428355	−	0.741933i	−0.0154168	−	0.0267027i
$773$	14.5998			0.525117			0.262558	−	0.964916i	$-0.415434\pi$
$773$	14.5998			0.525117			0.262558	+	0.964916i	$0.415434\pi$
$774$		0			0
$775$		0			0
$776$	−6.43935	−	11.1533i	−0.231159	−	0.400379i
$777$		0			0
$778$	7.44178	−	12.8895i	0.266801	−	0.462113i
$779$	−0.486933	+	0.843393i	−0.0174462	+	0.0302177i
$780$		0			0
$781$	−19.8477	−	34.3772i	−0.710207	−	1.23011i
$782$	6.31760			0.225917
$783$		0			0
$784$	−1.16578			−0.0416348
$785$		0			0
$786$		0			0
$787$	−9.23638	+	15.9979i	−0.329242	+	0.570263i	−0.982362	−	0.186991i	$-0.940126\pi$
$787$	−9.23638	+	15.9979i	−0.329242	+	0.570263i	0.653120	+	0.757254i	$0.273460\pi$
$788$	−4.91505	+	8.51312i	−0.175092	+	0.303267i
$789$		0			0
$790$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.236627 + 0.409850i) q^{2} +(0.888015 - 1.53809i) q^{4} +(-1.28153 - 2.21967i) q^{7} +1.78702 q^{8} +O(q^{10})\)	\(q+(0.236627 + 0.409850i) q^{2} +(0.888015 - 1.53809i) q^{4} +(-1.28153 - 2.21967i) q^{7} +1.78702 q^{8} +(-3.08430 - 5.34217i) q^{11} +(-1.06615 + 1.84662i) q^{13} +(0.606488 - 1.05047i) q^{14} +(-1.35317 - 2.34376i) q^{16} -3.16860 q^{17} +0.356267 q^{19} +(1.45966 - 2.52821i) q^{22} +(-2.10649 + 3.64854i) q^{23} -1.00912 q^{26} -4.55206 q^{28} +(0.843116 + 1.46032i) q^{29} +(4.12920 - 7.15199i) q^{31} +(2.42742 - 4.20441i) q^{32} +(-0.749778 - 1.29865i) q^{34} +3.63274 q^{37} +(0.0843024 + 0.146016i) q^{38} +(-1.36677 + 2.36731i) q^{41} +(3.83908 + 6.64949i) q^{43} -10.9556 q^{44} -1.99381 q^{46} +(-5.71444 - 9.89770i) q^{47} +(0.215378 - 0.373046i) q^{49} +(1.89351 + 3.27966i) q^{52} +9.43507 q^{53} +(-2.29012 - 3.96660i) q^{56} +(-0.399008 + 0.691103i) q^{58} +(5.10795 - 8.84723i) q^{59} +(0.00549659 + 0.00952038i) q^{61} +3.90833 q^{62} -3.11511 q^{64} +(-0.491409 + 0.851145i) q^{67} +(-2.81377 + 4.87359i) q^{68} +6.43507 q^{71} +6.61467 q^{73} +(0.859605 + 1.48888i) q^{74} +(0.316370 - 0.547969i) q^{76} +(-7.90523 + 13.6923i) q^{77} +(4.73569 + 8.20246i) q^{79} -1.29366 q^{82} +(-5.20988 - 9.02378i) q^{83} +(-1.81686 + 3.14690i) q^{86} +(-5.51172 - 9.54658i) q^{88} +6.26940 q^{89} +5.46519 q^{91} +(3.74119 + 6.47993i) q^{92} +(2.70439 - 4.68413i) q^{94} +(-3.60339 - 6.24126i) q^{97} +0.203858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8	\( 8 q - 2 q^{2} - 4 q^{4} - q^{7} + 18 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} - 4 q^{16} + 22 q^{17} + 4 q^{19} + 3 q^{22} - 15 q^{23} + 20 q^{26} + 8 q^{28} + q^{29} + 4 q^{31} - 10 q^{32} - 9 q^{34} + 2 q^{37} - 23 q^{38} - 5 q^{41} - 10 q^{43} - 44 q^{44} - 20 q^{47} + 3 q^{49} + 17 q^{52} + 40 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} + 13 q^{61} - 12 q^{62} + 38 q^{64} + 17 q^{67} - 34 q^{68} + 16 q^{71} - 4 q^{73} + 40 q^{74} - 11 q^{76} - 12 q^{77} + 7 q^{79} - 24 q^{82} - 30 q^{83} - 34 q^{86} + 9 q^{88} + 18 q^{89} - 34 q^{91} + 12 q^{92} - 3 q^{94} - 19 q^{97} + 26 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 - 4 * q^4 - q^7 + 18 * q^8 - q^11 + 2 * q^13 + 3 * q^14 - 4 * q^16 + 22 * q^17 + 4 * q^19 + 3 * q^22 - 15 * q^23 + 20 * q^26 + 8 * q^28 + q^29 + 4 * q^31 - 10 * q^32 - 9 * q^34 + 2 * q^37 - 23 * q^38 - 5 * q^41 - 10 * q^43 - 44 * q^44 - 20 * q^47 + 3 * q^49 + 17 * q^52 + 40 * q^53 - 30 * q^56 - 18 * q^58 + 17 * q^59 + 13 * q^61 - 12 * q^62 + 38 * q^64 + 17 * q^67 - 34 * q^68 + 16 * q^71 - 4 * q^73 + 40 * q^74 - 11 * q^76 - 12 * q^77 + 7 * q^79 - 24 * q^82 - 30 * q^83 - 34 * q^86 + 9 * q^88 + 18 * q^89 - 34 * q^91 + 12 * q^92 - 3 * q^94 - 19 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more