LMFDB - Embedded newform 4140.2.s.a.737.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("4140.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.s (of order $4$, degree $2$, 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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.16
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.54074 + 1.62053i) q^{5} +(-0.122636 - 0.122636i) q^{7} +O(q^{10})$	$q+(1.54074 + 1.62053i) q^{5} +(-0.122636 - 0.122636i) q^{7} -1.17251i q^{11} +(0.426185 - 0.426185i) q^{13} +(1.86148 - 1.86148i) q^{17} -5.98959i q^{19} +(0.707107 + 0.707107i) q^{23} +(-0.252254 + 4.99363i) q^{25} +4.59971 q^{29} -5.13810 q^{31} +(0.00978578 - 0.387686i) q^{35} +(5.60631 + 5.60631i) q^{37} -11.3295i q^{41} +(1.83217 - 1.83217i) q^{43} +(1.48761 - 1.48761i) q^{47} -6.96992i q^{49} +(1.12963 + 1.12963i) q^{53} +(1.90010 - 1.80654i) q^{55} -2.00765 q^{59} +2.11052 q^{61} +(1.34729 + 0.0340075i) q^{65} +(-3.16400 - 3.16400i) q^{67} -5.05304i q^{71} +(6.29383 - 6.29383i) q^{73} +(-0.143793 + 0.143793i) q^{77} +11.4369i q^{79} +(-8.94646 - 8.94646i) q^{83} +(5.88463 + 0.148537i) q^{85} +6.02046 q^{89} -0.104531 q^{91} +(9.70634 - 9.22839i) q^{95} +(-10.3024 - 10.3024i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 12 q^{7}+O(q^{10}) $ 44 * q - 12 * q^7	$ 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) $ 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.54074	+	1.62053i	0.689039	+	0.724724i
$5$	1.54074	+	1.62053i	0.689039	+	0.724724i
$6$		0			0
$7$	−0.122636	−	0.122636i	−0.0463521	−	0.0463521i	0.683551	−	0.729903i	$-0.260435\pi$
$7$	−0.122636	−	0.122636i	−0.0463521	−	0.0463521i	−0.729903	+	0.683551i	$0.760435\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.17251i		−	0.353526i	−0.984253	−	0.176763i	$-0.943437\pi$
$11$		−	1.17251i		−	0.353526i	0.984253	−	0.176763i	$-0.0565626\pi$
$12$		0			0
$13$	0.426185	−	0.426185i	0.118202	−	0.118202i	−0.645531	−	0.763734i	$-0.723364\pi$
$13$	0.426185	−	0.426185i	0.118202	−	0.118202i	0.763734	+	0.645531i	$0.223364\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.86148	−	1.86148i	0.451474	−	0.451474i	−0.444369	−	0.895844i	$-0.646572\pi$
$17$	1.86148	−	1.86148i	0.451474	−	0.451474i	0.895844	+	0.444369i	$0.146572\pi$
$18$		0			0
$19$		−	5.98959i		−	1.37411i	−0.726607	−	0.687054i	$-0.758904\pi$
$19$		−	5.98959i		−	1.37411i	0.726607	−	0.687054i	$-0.241096\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$24$		0			0
$25$	−0.252254	+	4.99363i	−0.0504508	+	0.998727i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.59971			0.854145			0.427072	−	0.904217i	$-0.359545\pi$
$29$	4.59971			0.854145			0.427072	+	0.904217i	$0.359545\pi$
$30$		0			0
$31$	−5.13810			−0.922829			−0.461415	−	0.887185i	$-0.652658\pi$
$31$	−5.13810			−0.922829			−0.461415	+	0.887185i	$0.652658\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.00978578	−	0.387686i	0.00165410	−	0.0655310i
$36$		0			0
$37$	5.60631	+	5.60631i	0.921672	+	0.921672i	0.997148	−	0.0754756i	$-0.0240475\pi$
$37$	5.60631	+	5.60631i	0.921672	+	0.921672i	−0.0754756	+	0.997148i	$0.524048\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.3295i		−	1.76936i	−0.466195	−	0.884682i	$-0.654376\pi$
$41$		−	11.3295i		−	1.76936i	0.466195	−	0.884682i	$-0.345624\pi$
$42$		0			0
$43$	1.83217	−	1.83217i	0.279403	−	0.279403i	−0.553468	−	0.832871i	$-0.686696\pi$
$43$	1.83217	−	1.83217i	0.279403	−	0.279403i	0.832871	+	0.553468i	$0.186696\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.48761	−	1.48761i	0.216990	−	0.216990i	−0.590239	−	0.807229i	$-0.700966\pi$
$47$	1.48761	−	1.48761i	0.216990	−	0.216990i	0.807229	+	0.590239i	$0.200966\pi$
$48$		0			0
$49$		−	6.96992i		−	0.995703i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.12963	+	1.12963i	0.155167	+	0.155167i	0.780421	−	0.625254i	$-0.215005\pi$
$53$	1.12963	+	1.12963i	0.155167	+	0.155167i	−0.625254	+	0.780421i	$0.715005\pi$
$54$		0			0
$55$	1.90010	−	1.80654i	0.256209	−	0.243593i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00765			−0.261374			−0.130687	−	0.991424i	$-0.541718\pi$
$59$	−2.00765			−0.261374			−0.130687	+	0.991424i	$0.541718\pi$
$60$		0			0
$61$	2.11052			0.270224			0.135112	−	0.990830i	$-0.456861\pi$
$61$	2.11052			0.270224			0.135112	+	0.990830i	$0.456861\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.34729	+	0.0340075i	0.167110	+	0.00421811i
$66$		0			0
$67$	−3.16400	−	3.16400i	−0.386544	−	0.386544i	0.486909	−	0.873453i	$-0.338124\pi$
$67$	−3.16400	−	3.16400i	−0.386544	−	0.386544i	−0.873453	+	0.486909i	$0.838124\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.05304i		−	0.599686i	−0.953989	−	0.299843i	$-0.903066\pi$
$71$		−	5.05304i		−	0.599686i	0.953989	−	0.299843i	$-0.0969342\pi$
$72$		0			0
$73$	6.29383	−	6.29383i	0.736637	−	0.736637i	−0.235288	−	0.971926i	$-0.575603\pi$
$73$	6.29383	−	6.29383i	0.736637	−	0.736637i	0.971926	+	0.235288i	$0.0756035\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.143793	+	0.143793i	−0.0163867	+	0.0163867i
$78$		0			0
$79$			11.4369i			1.28675i	0.765551	+	0.643375i	$0.222466\pi$
$79$			11.4369i			1.28675i	−0.765551	+	0.643375i	$0.777534\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.94646	−	8.94646i	−0.982001	−	0.982001i	0.0178398	−	0.999841i	$-0.494321\pi$
$83$	−8.94646	−	8.94646i	−0.982001	−	0.982001i	−0.999841	+	0.0178398i	$0.994321\pi$
$84$		0			0
$85$	5.88463	+	0.148537i	0.638278	+	0.0161111i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.02046			0.638167			0.319084	−	0.947727i	$-0.396625\pi$
$89$	6.02046			0.638167			0.319084	+	0.947727i	$0.396625\pi$
$90$		0			0
$91$	−0.104531			−0.0109579
$92$		0			0
$93$		0			0
$94$		0			0
$95$	9.70634	−	9.22839i	0.995849	−	0.946813i
$96$		0			0
$97$	−10.3024	−	10.3024i	−1.04605	−	1.04605i	−0.998887	−	0.0471670i	$-0.984981\pi$
$97$	−10.3024	−	10.3024i	−1.04605	−	1.04605i	−0.0471670	−	0.998887i	$-0.515019\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			0.989072i			0.0984164i	0.998789	+	0.0492082i	$0.0156698\pi$
$101$			0.989072i			0.0984164i	−0.998789	+	0.0492082i	$0.984330\pi$
$102$		0			0
$103$	−2.33159	+	2.33159i	−0.229738	+	0.229738i	−0.812583	−	0.582845i	$-0.801939\pi$
$103$	−2.33159	+	2.33159i	−0.229738	+	0.229738i	0.582845	+	0.812583i	$0.301939\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.88229	−	7.88229i	0.762010	−	0.762010i	−0.214675	−	0.976685i	$-0.568869\pi$
$107$	7.88229	−	7.88229i	0.762010	−	0.762010i	0.976685	+	0.214675i	$0.0688694\pi$
$108$		0			0
$109$			12.4281i			1.19039i	0.803580	+	0.595197i	$0.202926\pi$
$109$			12.4281i			1.19039i	−0.803580	+	0.595197i	$0.797074\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.8551	+	12.8551i	1.20931	+	1.20931i	0.971251	+	0.238058i	$0.0765107\pi$
$113$	12.8551	+	12.8551i	1.20931	+	1.20931i	0.238058	+	0.971251i	$0.423489\pi$
$114$		0			0
$115$	−0.0564237	+	2.23536i	−0.00526154	+	0.208448i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.456569			−0.0418536
$120$		0			0
$121$	9.62521			0.875019
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.48100	+	7.28509i	−0.758564	+	0.651598i
$126$		0			0
$127$	9.45737	+	9.45737i	0.839206	+	0.839206i	0.988754	−	0.149548i	$-0.0477820\pi$
$127$	9.45737	+	9.45737i	0.839206	+	0.839206i	−0.149548	+	0.988754i	$0.547782\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	0.851632i		−	0.0744074i	−0.999308	−	0.0372037i	$-0.988155\pi$
$131$		−	0.851632i		−	0.0744074i	0.999308	−	0.0372037i	$-0.0118450\pi$
$132$		0			0
$133$	−0.734541	+	0.734541i	−0.0636928	+	0.0636928i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.7401	−	13.7401i	1.17390	−	1.17390i	0.192624	−	0.981273i	$-0.438300\pi$
$137$	13.7401	−	13.7401i	1.17390	−	1.17390i	0.981273	−	0.192624i	$-0.0616997\pi$
$138$		0			0
$139$			20.8384i			1.76749i	0.467965	+	0.883747i	$0.344987\pi$
$139$			20.8384i			1.76749i	−0.467965	+	0.883747i	$0.655013\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.499708	−	0.499708i	−0.0417877	−	0.0417877i
$144$		0			0
$145$	7.08695	+	7.45398i	0.588539	+	0.619020i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.97371			−0.325539			−0.162769	−	0.986664i	$-0.552043\pi$
$149$	−3.97371			−0.325539			−0.162769	+	0.986664i	$0.552043\pi$
$150$		0			0
$151$	16.3615			1.33148			0.665740	−	0.746184i	$-0.268116\pi$
$151$	16.3615			1.33148			0.665740	+	0.746184i	$0.268116\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.91646	−	8.32645i	−0.635865	−	0.668797i
$156$		0			0
$157$	7.46362	+	7.46362i	0.595662	+	0.595662i	0.939155	−	0.343493i	$-0.111610\pi$
$157$	7.46362	+	7.46362i	0.595662	+	0.595662i	−0.343493	+	0.939155i	$0.611610\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.173434i		−	0.0136685i
$162$		0			0
$163$	8.48980	−	8.48980i	0.664973	−	0.664973i	−0.291575	−	0.956548i	$-0.594179\pi$
$163$	8.48980	−	8.48980i	0.664973	−	0.664973i	0.956548	+	0.291575i	$0.0941793\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.73164	−	2.73164i	0.211381	−	0.211381i	−0.593473	−	0.804854i	$-0.702244\pi$
$167$	2.73164	−	2.73164i	0.211381	−	0.211381i	0.804854	+	0.593473i	$0.202244\pi$
$168$		0			0
$169$			12.6367i			0.972056i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.3591	−	13.3591i	−1.01567	−	1.01567i	−0.999875	−	0.0157968i	$-0.994972\pi$
$173$	−13.3591	−	13.3591i	−1.01567	−	1.01567i	−0.0157968	−	0.999875i	$-0.505028\pi$
$174$		0			0
$175$	0.643336	−	0.581465i	0.0486316	−	0.0439546i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.478013			−0.0357284			−0.0178642	−	0.999840i	$-0.505687\pi$
$179$	−0.478013			−0.0357284			−0.0178642	+	0.999840i	$0.505687\pi$
$180$		0			0
$181$	−5.48677			−0.407828			−0.203914	−	0.978989i	$-0.565366\pi$
$181$	−5.48677			−0.407828			−0.203914	+	0.978989i	$0.565366\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.447357	+	17.7231i	−0.0328903	+	1.30303i
$186$		0			0
$187$	−2.18261	−	2.18261i	−0.159608	−	0.159608i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	19.8801i		−	1.43847i	−0.694766	−	0.719236i	$-0.744492\pi$
$191$		−	19.8801i		−	1.43847i	0.694766	−	0.719236i	$-0.255508\pi$
$192$		0			0
$193$	−7.20517	+	7.20517i	−0.518640	+	0.518640i	−0.917160	−	0.398520i	$-0.869524\pi$
$193$	−7.20517	+	7.20517i	−0.518640	+	0.518640i	0.398520	+	0.917160i	$0.369524\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.16166	−	7.16166i	0.510247	−	0.510247i	−0.404355	−	0.914602i	$-0.632504\pi$
$197$	7.16166	−	7.16166i	0.510247	−	0.510247i	0.914602	+	0.404355i	$0.132504\pi$
$198$		0			0
$199$		−	4.66521i		−	0.330708i	−0.986234	−	0.165354i	$-0.947123\pi$
$199$		−	4.66521i		−	0.330708i	0.986234	−	0.165354i	$-0.0528767\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.564091	−	0.564091i	−0.0395914	−	0.0395914i
$204$		0			0
$205$	18.3598	−	17.4557i	1.28230	−	1.21916i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.02288			−0.485783
$210$		0			0
$211$	6.00046			0.413088			0.206544	−	0.978437i	$-0.433778\pi$
$211$	6.00046			0.413088			0.206544	+	0.978437i	$0.433778\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.79198	+	0.146198i	0.395010	+	0.00997063i
$216$		0			0
$217$	0.630117	+	0.630117i	0.0427751	+	0.0427751i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	1.58667i		−	0.106731i
$222$		0			0
$223$	−6.15097	+	6.15097i	−0.411899	+	0.411899i	−0.882400	−	0.470500i	$-0.844073\pi$
$223$	−6.15097	+	6.15097i	−0.411899	+	0.411899i	0.470500	+	0.882400i	$0.344073\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.96865	+	7.96865i	−0.528898	+	0.528898i	−0.920244	−	0.391346i	$-0.872010\pi$
$227$	−7.96865	+	7.96865i	−0.528898	+	0.528898i	0.391346	+	0.920244i	$0.372010\pi$
$228$		0			0
$229$			7.81865i			0.516671i	0.966055	+	0.258336i	$0.0831740\pi$
$229$			7.81865i			0.516671i	−0.966055	+	0.258336i	$0.916826\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.79852	+	7.79852i	0.510898	+	0.510898i	0.914802	−	0.403904i	$-0.132347\pi$
$233$	7.79852	+	7.79852i	0.510898	+	0.510898i	−0.403904	+	0.914802i	$0.632347\pi$
$234$		0			0
$235$	4.70274	+	0.118704i	0.306773	+	0.00774340i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.8518			−1.41348			−0.706739	−	0.707475i	$-0.749834\pi$
$239$	−21.8518			−1.41348			−0.706739	+	0.707475i	$0.749834\pi$
$240$		0			0
$241$	−16.7323			−1.07782			−0.538912	−	0.842362i	$-0.681165\pi$
$241$	−16.7323			−1.07782			−0.538912	+	0.842362i	$0.681165\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.2950	−	10.7388i	0.721610	−	0.686078i
$246$		0			0
$247$	−2.55268	−	2.55268i	−0.162423	−	0.162423i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	29.3265i		−	1.85107i	−0.378661	−	0.925535i	$-0.623615\pi$
$251$		−	29.3265i		−	1.85107i	0.378661	−	0.925535i	$-0.376385\pi$
$252$		0			0
$253$	0.829092	−	0.829092i	0.0521246	−	0.0521246i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.97186	−	8.97186i	0.559649	−	0.559649i	−0.369558	−	0.929208i	$-0.620491\pi$
$257$	8.97186	−	8.97186i	0.559649	−	0.559649i	0.929208	+	0.369558i	$0.120491\pi$
$258$		0			0
$259$		−	1.37507i		−	0.0854429i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.90412	−	1.90412i	−0.117413	−	0.117413i	0.645959	−	0.763372i	$-0.276458\pi$
$263$	−1.90412	−	1.90412i	−0.117413	−	0.117413i	−0.763372	+	0.645959i	$0.776458\pi$
$264$		0			0
$265$	−0.0901392	+	3.57107i	−0.00553721	+	0.219369i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.88030			0.297557			0.148779	−	0.988871i	$-0.452466\pi$
$269$	4.88030			0.297557			0.148779	+	0.988871i	$0.452466\pi$
$270$		0			0
$271$	18.0053			1.09374			0.546871	−	0.837217i	$-0.315819\pi$
$271$	18.0053			1.09374			0.546871	+	0.837217i	$0.315819\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.85510	+	0.295772i	0.353076	+	0.0178357i
$276$		0			0
$277$	17.8461	+	17.8461i	1.07227	+	1.07227i	0.997177	+	0.0750920i	$0.0239250\pi$
$277$	17.8461	+	17.8461i	1.07227	+	1.07227i	0.0750920	+	0.997177i	$0.476075\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.20890i		−	0.191427i	−0.995409	−	0.0957135i	$-0.969487\pi$
$281$		−	3.20890i		−	0.191427i	0.995409	−	0.0957135i	$-0.0305133\pi$
$282$		0			0
$283$	−2.31588	+	2.31588i	−0.137665	+	0.137665i	−0.772581	−	0.634916i	$-0.781035\pi$
$283$	−2.31588	+	2.31588i	−0.137665	+	0.137665i	0.634916	+	0.772581i	$0.281035\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.38940	+	1.38940i	−0.0820138	+	0.0820138i
$288$		0			0
$289$			10.0698i			0.592342i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.5143	+	12.5143i	0.731094	+	0.731094i	0.970836	−	0.239743i	$-0.0770630\pi$
$293$	12.5143	+	12.5143i	0.731094	+	0.731094i	−0.239743	+	0.970836i	$0.577063\pi$
$294$		0			0
$295$	−3.09326	−	3.25346i	−0.180097	−	0.189424i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.602717			0.0348560
$300$		0			0
$301$	−0.449380			−0.0259019
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.25176	+	3.42016i	0.186195	+	0.195838i
$306$		0			0
$307$	−4.20663	−	4.20663i	−0.240085	−	0.240085i	0.576800	−	0.816885i	$-0.304301\pi$
$307$	−4.20663	−	4.20663i	−0.240085	−	0.240085i	−0.816885	+	0.576800i	$0.804301\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			27.5792i			1.56387i	0.623359	+	0.781936i	$0.285767\pi$
$311$			27.5792i			1.56387i	−0.623359	+	0.781936i	$0.714233\pi$
$312$		0			0
$313$	9.97172	−	9.97172i	0.563635	−	0.563635i	−0.366703	−	0.930338i	$-0.619513\pi$
$313$	9.97172	−	9.97172i	0.563635	−	0.563635i	0.930338	+	0.366703i	$0.119513\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.02731	−	8.02731i	0.450859	−	0.450859i	−0.444781	−	0.895639i	$-0.646718\pi$
$317$	8.02731	−	8.02731i	0.450859	−	0.450859i	0.895639	+	0.444781i	$0.146718\pi$
$318$		0			0
$319$		−	5.39322i		−	0.301963i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−11.1495	−	11.1495i	−0.620374	−	0.620374i
$324$		0			0
$325$	2.02070	+	2.23572i	0.112089	+	0.124015i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.364870			−0.0201159
$330$		0			0
$331$	−11.3607			−0.624442			−0.312221	−	0.950010i	$-0.601073\pi$
$331$	−11.3607			−0.624442			−0.312221	+	0.950010i	$0.601073\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.252472	−	10.0022i	0.0137940	−	0.546481i
$336$		0			0
$337$	1.71325	+	1.71325i	0.0933267	+	0.0933267i	0.752229	−	0.658902i	$-0.228979\pi$
$337$	1.71325	+	1.71325i	0.0933267	+	0.0933267i	−0.658902	+	0.752229i	$0.728979\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			6.02449i			0.326244i
$342$		0			0
$343$	−1.71322	+	1.71322i	−0.0925051	+	0.0925051i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	23.3151	−	23.3151i	1.25162	−	1.25162i	0.296625	−	0.954994i	$-0.404139\pi$
$347$	23.3151	−	23.3151i	1.25162	−	1.25162i	0.954994	−	0.296625i	$-0.0958611\pi$
$348$		0			0
$349$			30.0879i			1.61057i	0.592890	+	0.805284i	$0.297987\pi$
$349$			30.0879i			1.61057i	−0.592890	+	0.805284i	$0.702013\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.0872711	+	0.0872711i	0.00464497	+	0.00464497i	0.709425	−	0.704780i	$-0.248955\pi$
$353$	0.0872711	+	0.0872711i	0.00464497	+	0.00464497i	−0.704780	+	0.709425i	$0.748955\pi$
$354$		0			0
$355$	8.18862	−	7.78541i	0.434607	−	0.413207i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.5942			0.559138			0.279569	−	0.960126i	$-0.409808\pi$
$359$	10.5942			0.559138			0.279569	+	0.960126i	$0.409808\pi$
$360$		0			0
$361$	−16.8752			−0.888171
$362$		0			0
$363$		0			0
$364$		0			0
$365$	19.8965	+	0.502217i	1.04143	+	0.0262873i
$366$		0			0
$367$	−1.99228	−	1.99228i	−0.103996	−	0.103996i	0.653194	−	0.757190i	$-0.273428\pi$
$367$	−1.99228	−	1.99228i	−0.103996	−	0.103996i	−0.757190	+	0.653194i	$0.773428\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	0.277068i		−	0.0143846i
$372$		0			0
$373$	6.00511	−	6.00511i	0.310933	−	0.310933i	−0.534338	−	0.845271i	$-0.679439\pi$
$373$	6.00511	−	6.00511i	0.310933	−	0.310933i	0.845271	+	0.534338i	$0.179439\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.96033	−	1.96033i	0.100962	−	0.100962i
$378$		0			0
$379$		−	29.1506i		−	1.49736i	−0.662930	−	0.748682i	$-0.730687\pi$
$379$		−	29.1506i		−	1.49736i	0.662930	−	0.748682i	$-0.269313\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.4510	−	13.4510i	−0.687312	−	0.687312i	0.274325	−	0.961637i	$-0.411546\pi$
$383$	−13.4510	−	13.4510i	−0.687312	−	0.687312i	−0.961637	+	0.274325i	$0.911546\pi$
$384$		0			0
$385$	−0.454568	−	0.0114740i	−0.0231669	−	0.000584767i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.52175			0.432070			0.216035	−	0.976386i	$-0.430687\pi$
$389$	8.52175			0.432070			0.216035	+	0.976386i	$0.430687\pi$
$390$		0			0
$391$	2.63252			0.133132
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−18.5338	+	17.6212i	−0.932539	+	0.886620i
$396$		0			0
$397$	−24.2287	−	24.2287i	−1.21600	−	1.21600i	−0.969020	−	0.246983i	$-0.920561\pi$
$397$	−24.2287	−	24.2287i	−1.21600	−	1.21600i	−0.246983	−	0.969020i	$-0.579439\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.63372i			0.181459i	0.995876	+	0.0907297i	$0.0289199\pi$
$401$			3.63372i			0.181459i	−0.995876	+	0.0907297i	$0.971080\pi$
$402$		0			0
$403$	−2.18978	+	2.18978i	−0.109081	+	0.109081i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.57348	−	6.57348i	0.325835	−	0.325835i
$408$		0			0
$409$			2.40032i			0.118688i	0.998238	+	0.0593440i	$0.0189009\pi$
$409$			2.40032i			0.118688i	−0.998238	+	0.0593440i	$0.981099\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.246210	+	0.246210i	0.0121152	+	0.0121152i
$414$		0			0
$415$	0.713884	−	28.2822i	0.0350432	−	1.38832i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.15036			−0.251612			−0.125806	−	0.992055i	$-0.540152\pi$
$419$	−5.15036			−0.251612			−0.125806	+	0.992055i	$0.540152\pi$
$420$		0			0
$421$	−11.5520			−0.563010			−0.281505	−	0.959560i	$-0.590834\pi$
$421$	−11.5520			−0.563010			−0.281505	+	0.959560i	$0.590834\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8.82596	+	9.76509i	0.428122	+	0.473677i
$426$		0			0
$427$	−0.258826	−	0.258826i	−0.0125255	−	0.0125255i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			21.5028i			1.03575i	0.855455	+	0.517877i	$0.173278\pi$
$431$			21.5028i			1.03575i	−0.855455	+	0.517877i	$0.826722\pi$
$432$		0			0
$433$	17.4438	−	17.4438i	0.838294	−	0.838294i	−0.150341	−	0.988634i	$-0.548037\pi$
$433$	17.4438	−	17.4438i	0.838294	−	0.838294i	0.988634	+	0.150341i	$0.0480371\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.23528	−	4.23528i	0.202601	−	0.202601i
$438$		0			0
$439$			17.3115i			0.826233i	0.910678	+	0.413117i	$0.135560\pi$
$439$			17.3115i			0.826233i	−0.910678	+	0.413117i	$0.864440\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	22.9656	+	22.9656i	1.09113	+	1.09113i	0.995408	+	0.0957193i	$0.0305151\pi$
$443$	22.9656	+	22.9656i	1.09113	+	1.09113i	0.0957193	+	0.995408i	$0.469485\pi$
$444$		0			0
$445$	9.27595	+	9.75635i	0.439722	+	0.462495i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.80908			0.462919			0.231460	−	0.972845i	$-0.425650\pi$
$449$	9.80908			0.462919			0.231460	+	0.972845i	$0.425650\pi$
$450$		0			0
$451$	−13.2839			−0.625516
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.161056	−	0.169397i	−0.00755040	−	0.00794144i
$456$		0			0
$457$	−9.54719	−	9.54719i	−0.446599	−	0.446599i	0.447623	−	0.894222i	$-0.352271\pi$
$457$	−9.54719	−	9.54719i	−0.446599	−	0.446599i	−0.894222	+	0.447623i	$0.852271\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	11.5663i		−	0.538698i	−0.963043	−	0.269349i	$-0.913191\pi$
$461$		−	11.5663i		−	0.538698i	0.963043	−	0.269349i	$-0.0868085\pi$
$462$		0			0
$463$	−5.41187	+	5.41187i	−0.251511	+	0.251511i	−0.821590	−	0.570079i	$-0.806913\pi$
$463$	−5.41187	+	5.41187i	−0.251511	+	0.251511i	0.570079	+	0.821590i	$0.306913\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.3418	+	24.3418i	−1.12640	+	1.12640i	−0.135647	+	0.990757i	$0.543311\pi$
$467$	−24.3418	+	24.3418i	−1.12640	+	1.12640i	−0.990757	+	0.135647i	$0.956689\pi$
$468$		0			0
$469$			0.776041i			0.0358342i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.14824	−	2.14824i	−0.0987763	−	0.0987763i
$474$		0			0
$475$	29.9098	+	1.51090i	1.37236	+	0.0693249i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.02822			−0.321128			−0.160564	−	0.987025i	$-0.551331\pi$
$479$	−7.02822			−0.321128			−0.160564	+	0.987025i	$0.551331\pi$
$480$		0			0
$481$	4.77865			0.217888
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.822085	−	32.5688i	0.0373290	−	1.47887i
$486$		0			0
$487$	−25.2253	−	25.2253i	−1.14307	−	1.14307i	−0.987886	−	0.155183i	$-0.950403\pi$
$487$	−25.2253	−	25.2253i	−1.14307	−	1.14307i	−0.155183	−	0.987886i	$-0.549597\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	32.7771i		−	1.47921i	−0.673041	−	0.739605i	$-0.735012\pi$
$491$		−	32.7771i		−	1.47921i	0.673041	−	0.739605i	$-0.264988\pi$
$492$		0			0
$493$	8.56225	−	8.56225i	0.385624	−	0.385624i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.619686	+	0.619686i	−0.0277967	+	0.0277967i
$498$		0			0
$499$			11.9751i			0.536081i	0.963408	+	0.268040i	$0.0863761\pi$
$499$			11.9751i			0.536081i	−0.963408	+	0.268040i	$0.913624\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.70157	−	1.70157i	−0.0758692	−	0.0758692i	0.668154	−	0.744023i	$-0.267085\pi$
$503$	−1.70157	−	1.70157i	−0.0758692	−	0.0758692i	−0.744023	+	0.668154i	$0.767085\pi$
$504$		0			0
$505$	−1.60282	+	1.52390i	−0.0713247	+	0.0678127i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−30.0703			−1.33284			−0.666420	−	0.745576i	$-0.732174\pi$
$509$	−30.0703			−1.33284			−0.666420	+	0.745576i	$0.732174\pi$
$510$		0			0
$511$	−1.54370			−0.0682894
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.37078	−	0.186049i	−0.324795	−	0.00819832i
$516$		0			0
$517$	−1.74424	−	1.74424i	−0.0767117	−	0.0767117i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	18.0902i		−	0.792547i	−0.918133	−	0.396273i	$-0.870303\pi$
$521$		−	18.0902i		−	0.792547i	0.918133	−	0.396273i	$-0.129697\pi$
$522$		0			0
$523$	30.3824	−	30.3824i	1.32853	−	1.32853i	0.421876	−	0.906653i	$-0.361372\pi$
$523$	30.3824	−	30.3824i	1.32853	−	1.32853i	0.906653	−	0.421876i	$-0.138628\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.56444	+	9.56444i	−0.416634	+	0.416634i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.82844	−	4.82844i	−0.209143	−	0.209143i
$534$		0			0
$535$	24.9181	+	0.628969i	1.07730	+	0.0271927i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.17233			−0.352007
$540$		0			0
$541$	−31.6275			−1.35977			−0.679886	−	0.733318i	$-0.737971\pi$
$541$	−31.6275			−1.35977			−0.679886	+	0.733318i	$0.737971\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.1401	+	19.1484i	−0.862708	+	0.820228i
$546$		0			0
$547$	3.14771	+	3.14771i	0.134586	+	0.134586i	0.771191	−	0.636604i	$-0.219662\pi$
$547$	3.14771	+	3.14771i	0.134586	+	0.134586i	−0.636604	+	0.771191i	$0.719662\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	27.5504i		−	1.17369i
$552$		0			0
$553$	1.40258	−	1.40258i	0.0596436	−	0.0596436i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.15640	+	8.15640i	−0.345598	+	0.345598i	−0.858467	−	0.512869i	$-0.828583\pi$
$557$	−8.15640	+	8.15640i	−0.345598	+	0.345598i	0.512869	+	0.858467i	$0.328583\pi$
$558$		0			0
$559$		−	1.56169i		−	0.0660523i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.3555	+	16.3555i	0.689302	+	0.689302i	0.962078	−	0.272775i	$-0.0879416\pi$
$563$	16.3555	+	16.3555i	0.689302	+	0.689302i	−0.272775	+	0.962078i	$0.587942\pi$
$564$		0			0
$565$	−1.02578	+	40.6385i	−0.0431548	+	1.70968i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.4314			−0.479231			−0.239616	−	0.970868i	$-0.577021\pi$
$569$	−11.4314			−0.479231			−0.239616	+	0.970868i	$0.577021\pi$
$570$		0			0
$571$	−5.29909			−0.221760			−0.110880	−	0.993834i	$-0.535367\pi$
$571$	−5.29909			−0.221760			−0.110880	+	0.993834i	$0.535367\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.70940	+	3.35266i	−0.154693	+	0.139816i
$576$		0			0
$577$	−13.3542	−	13.3542i	−0.555941	−	0.555941i	0.372208	−	0.928149i	$-0.378601\pi$
$577$	−13.3542	−	13.3542i	−0.555941	−	0.555941i	−0.928149	+	0.372208i	$0.878601\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			2.19432i			0.0910357i
$582$		0			0
$583$	1.32451	−	1.32451i	0.0548556	−	0.0548556i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.7762	−	19.7762i	0.816252	−	0.816252i	−0.169311	−	0.985563i	$-0.554154\pi$
$587$	19.7762	−	19.7762i	0.816252	−	0.816252i	0.985563	+	0.169311i	$0.0541541\pi$
$588$		0			0
$589$			30.7751i			1.26807i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	27.7985	+	27.7985i	1.14155	+	1.14155i	0.988167	+	0.153379i	$0.0490155\pi$
$593$	27.7985	+	27.7985i	1.14155	+	1.14155i	0.153379	+	0.988167i	$0.450985\pi$
$594$		0			0
$595$	−0.703453	−	0.739885i	−0.0288388	−	0.0303323i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.37124			0.0968864			0.0484432	−	0.998826i	$-0.484574\pi$
$599$	2.37124			0.0968864			0.0484432	+	0.998826i	$0.484574\pi$
$600$		0			0
$601$	−46.8084			−1.90935			−0.954677	−	0.297645i	$-0.903799\pi$
$601$	−46.8084			−1.90935			−0.954677	+	0.297645i	$0.903799\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.8299	+	15.5980i	0.602922	+	0.634148i
$606$		0			0
$607$	−21.2692	−	21.2692i	−0.863290	−	0.863290i	0.128428	−	0.991719i	$-0.459007\pi$
$607$	−21.2692	−	21.2692i	−0.863290	−	0.863290i	−0.991719	+	0.128428i	$0.959007\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1.26799i		−	0.0512976i
$612$		0			0
$613$	−5.99228	+	5.99228i	−0.242026	+	0.242026i	−0.817688	−	0.575662i	$-0.804744\pi$
$613$	−5.99228	+	5.99228i	−0.242026	+	0.242026i	0.575662	+	0.817688i	$0.304744\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.95730	+	6.95730i	−0.280090	+	0.280090i	−0.833145	−	0.553055i	$-0.813462\pi$
$617$	−6.95730	+	6.95730i	−0.280090	+	0.280090i	0.553055	+	0.833145i	$0.313462\pi$
$618$		0			0
$619$			9.65634i			0.388121i	0.980990	+	0.194061i	$0.0621658\pi$
$619$			9.65634i			0.388121i	−0.980990	+	0.194061i	$0.937834\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.738326	−	0.738326i	−0.0295804	−	0.0295804i
$624$		0			0
$625$	−24.8727	−	2.51933i	−0.994909	−	0.100773i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	20.8720			0.832222
$630$		0			0
$631$	23.3908			0.931172			0.465586	−	0.885003i	$-0.345844\pi$
$631$	23.3908			0.931172			0.465586	+	0.885003i	$0.345844\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.754653	+	29.8973i	−0.0299475	+	1.18644i
$636$		0			0
$637$	−2.97048	−	2.97048i	−0.117695	−	0.117695i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			34.7879i			1.37404i	0.726639	+	0.687020i	$0.241081\pi$
$641$			34.7879i			1.37404i	−0.726639	+	0.687020i	$0.758919\pi$
$642$		0			0
$643$	2.80230	−	2.80230i	0.110512	−	0.110512i	−0.649689	−	0.760200i	$-0.725101\pi$
$643$	2.80230	−	2.80230i	0.110512	−	0.110512i	0.760200	+	0.649689i	$0.225101\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.0564	+	21.0564i	−0.827814	+	0.827814i	−0.987214	−	0.159400i	$-0.949044\pi$
$647$	−21.0564	+	21.0564i	−0.827814	+	0.827814i	0.159400	+	0.987214i	$0.449044\pi$
$648$		0			0
$649$			2.35399i			0.0924024i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.3052	+	22.3052i	0.872872	+	0.872872i	0.992784	−	0.119913i	$-0.0382615\pi$
$653$	22.3052	+	22.3052i	0.872872	+	0.872872i	−0.119913	+	0.992784i	$0.538262\pi$
$654$		0			0
$655$	1.38010	−	1.31214i	0.0539249	−	0.0512696i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.33016			−0.246588			−0.123294	−	0.992370i	$-0.539346\pi$
$659$	−6.33016			−0.246588			−0.123294	+	0.992370i	$0.539346\pi$
$660$		0			0
$661$	−9.61901			−0.374136			−0.187068	−	0.982347i	$-0.559898\pi$
$661$	−9.61901			−0.374136			−0.187068	+	0.982347i	$0.559898\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.32208	−	0.0586129i	−0.0900466	−	0.00227291i
$666$		0			0
$667$	3.25249	+	3.25249i	0.125937	+	0.125937i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.47461i		−	0.0955313i
$672$		0			0
$673$	−15.4144	+	15.4144i	−0.594182	+	0.594182i	−0.938758	−	0.344577i	$-0.888023\pi$
$673$	−15.4144	+	15.4144i	−0.594182	+	0.594182i	0.344577	+	0.938758i	$0.388023\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.27453	+	8.27453i	−0.318016	+	0.318016i	−0.848005	−	0.529989i	$-0.822196\pi$
$677$	−8.27453	+	8.27453i	−0.318016	+	0.318016i	0.529989	+	0.848005i	$0.322196\pi$
$678$		0			0
$679$			2.52690i			0.0969737i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.90153	−	3.90153i	−0.149288	−	0.149288i	0.628512	−	0.777800i	$-0.283664\pi$
$683$	−3.90153	−	3.90153i	−0.149288	−	0.149288i	−0.777800	+	0.628512i	$0.783664\pi$
$684$		0			0
$685$	43.4362	+	1.09639i	1.65961	+	0.0418911i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.962865			0.0366822
$690$		0			0
$691$	−29.9199			−1.13821			−0.569104	−	0.822266i	$-0.692710\pi$
$691$	−29.9199			−1.13821			−0.569104	+	0.822266i	$0.692710\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−33.7694	+	32.1066i	−1.28095	+	1.21787i
$696$		0			0
$697$	−21.0895	−	21.0895i	−0.798822	−	0.798822i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			17.3738i			0.656200i	0.944643	+	0.328100i	$0.106408\pi$
$701$			17.3738i			0.656200i	−0.944643	+	0.328100i	$0.893592\pi$
$702$		0			0
$703$	33.5795	−	33.5795i	1.26648	−	1.26648i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.121296	−	0.121296i	0.00456181	−	0.00456181i
$708$		0			0
$709$			18.9856i			0.713018i	0.934292	+	0.356509i	$0.116033\pi$
$709$			18.9856i			0.713018i	−0.934292	+	0.356509i	$0.883967\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.63318	−	3.63318i	−0.136064	−	0.136064i
$714$		0			0
$715$	0.0398743	−	1.57971i	0.00149121	−	0.0590779i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−43.8982			−1.63713			−0.818564	−	0.574416i	$-0.805229\pi$
$719$	−43.8982			−1.63713			−0.818564	+	0.574416i	$0.805229\pi$
$720$		0			0
$721$	0.571874			0.0212977
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.16030	+	22.9693i	−0.0430923	+	0.853057i
$726$		0			0
$727$	3.26413	+	3.26413i	0.121060	+	0.121060i	0.765041	−	0.643981i	$-0.222719\pi$
$727$	3.26413	+	3.26413i	0.121060	+	0.121060i	−0.643981	+	0.765041i	$0.722719\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	6.82107i		−	0.252287i
$732$		0			0
$733$	−23.3414	+	23.3414i	−0.862136	+	0.862136i	−0.991586	−	0.129450i	$-0.958679\pi$
$733$	−23.3414	+	23.3414i	−0.862136	+	0.862136i	0.129450	+	0.991586i	$0.458679\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.70983	+	3.70983i	−0.136653	+	0.136653i
$738$		0			0
$739$			31.9197i			1.17418i	0.809520	+	0.587092i	$0.199727\pi$
$739$			31.9197i			1.17418i	−0.809520	+	0.587092i	$0.800273\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.2855	−	13.2855i	−0.487398	−	0.487398i	0.420086	−	0.907484i	$-0.362000\pi$
$743$	−13.2855	−	13.2855i	−0.487398	−	0.487398i	−0.907484	+	0.420086i	$0.862000\pi$
$744$		0			0
$745$	−6.12244	−	6.43952i	−0.224309	−	0.235926i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.93331			−0.0706416
$750$		0			0
$751$	−10.1881			−0.371771			−0.185885	−	0.982571i	$-0.559515\pi$
$751$	−10.1881			−0.371771			−0.185885	+	0.982571i	$0.559515\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25.2088	+	26.5143i	0.917441	+	0.964956i
$756$		0			0
$757$	35.7721	+	35.7721i	1.30016	+	1.30016i	0.928282	+	0.371877i	$0.121286\pi$
$757$	35.7721	+	35.7721i	1.30016	+	1.30016i	0.371877	+	0.928282i	$0.378714\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	20.3897i		−	0.739126i	−0.929206	−	0.369563i	$-0.879507\pi$
$761$		−	20.3897i		−	0.739126i	0.929206	−	0.369563i	$-0.120493\pi$
$762$		0			0
$763$	1.52413	−	1.52413i	0.0551773	−	0.0551773i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.855630	+	0.855630i	−0.0308950	+	0.0308950i
$768$		0			0
$769$		−	11.6519i		−	0.420180i	−0.977682	−	0.210090i	$-0.932624\pi$
$769$		−	11.6519i		−	0.420180i	0.977682	−	0.210090i	$-0.0673756\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.61936	−	1.61936i	−0.0582445	−	0.0582445i	0.677385	−	0.735629i	$-0.263113\pi$
$773$	−1.61936	−	1.61936i	−0.0582445	−	0.0582445i	−0.735629	+	0.677385i	$0.763113\pi$
$774$		0			0
$775$	1.29611	−	25.6578i	0.0465575	−	0.921654i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−67.8588			−2.43130
$780$		0			0
$781$	−5.92476			−0.212005
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.595561	+	23.5945i	−0.0212565	+	0.842125i
$786$		0			0
$787$	−7.97772	−	7.97772i	−0.284375	−	0.284375i	0.550476	−	0.834851i	$-0.314446\pi$
$787$	−7.97772	−	7.97772i	−0.284375	−	0.284375i	−0.834851	+	0.550476i	$0.814446\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	3.15301i		−	0.112108i
$792$		0			0
$793$	0.899471	−	0.899471i	0.0319412	−	0.0319412i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.0792	−	11.0792i	0.392446	−	0.392446i	−0.483112	−	0.875558i	$-0.660494\pi$
$797$	11.0792	−	11.0792i	0.392446	−	0.392446i	0.875558	+	0.483112i	$0.160494\pi$
$798$		0			0
$799$		−	5.53830i		−	0.195931i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.37960	−	7.37960i	−0.260421	−	0.260421i
$804$		0			0
$805$	0.281055	−	0.267216i	0.00990590	−	0.00941813i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14.1242			0.496581			0.248291	−	0.968686i	$-0.420131\pi$
$809$	14.1242			0.496581			0.248291	+	0.968686i	$0.420131\pi$
$810$		0			0
$811$	30.2642			1.06272			0.531361	−	0.847146i	$-0.321681\pi$
$811$	30.2642			1.06272			0.531361	+	0.847146i	$0.321681\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	26.8386	+	0.677446i	0.940114	+	0.0237299i
$816$		0			0
$817$	−10.9739	−	10.9739i	−0.383930	−	0.383930i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1.62715i		−	0.0567880i	−0.999597	−	0.0283940i	$-0.990961\pi$
$821$		−	1.62715i		−	0.0567880i	0.999597	−	0.0283940i	$-0.00903930\pi$
$822$		0			0
$823$	12.8774	−	12.8774i	0.448878	−	0.448878i	−0.446104	−	0.894981i	$-0.647189\pi$
$823$	12.8774	−	12.8774i	0.448878	−	0.448878i	0.894981	+	0.446104i	$0.147189\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.64136	+	7.64136i	−0.265716	+	0.265716i	−0.827371	−	0.561655i	$-0.810165\pi$
$827$	−7.64136	+	7.64136i	−0.265716	+	0.265716i	0.561655	+	0.827371i	$0.310165\pi$
$828$		0			0
$829$		−	42.5875i		−	1.47912i	−0.673088	−	0.739562i	$-0.735033\pi$
$829$		−	42.5875i		−	1.47912i	0.673088	−	0.739562i	$-0.264967\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.9743	−	12.9743i	−0.449534	−	0.449534i
$834$		0			0
$835$	8.63546	+	0.217972i	0.298842	+	0.00754323i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−48.0541			−1.65901			−0.829505	−	0.558499i	$-0.811378\pi$
$839$	−48.0541			−1.65901			−0.829505	+	0.558499i	$0.811378\pi$
$840$		0			0
$841$	−7.84266			−0.270437
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−20.4782	+	19.4699i	−0.704473	+	0.669785i
$846$		0			0
$847$	−1.18040	−	1.18040i	−0.0405590	−	0.0405590i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			7.92852i			0.271786i
$852$		0			0
$853$	−32.9634	+	32.9634i	−1.12865	+	1.12865i	−0.138249	+	0.990398i	$0.544147\pi$
$853$	−32.9634	+	32.9634i	−1.12865	+	1.12865i	−0.990398	+	0.138249i	$0.955853\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24.6331	+	24.6331i	−0.841451	+	0.841451i	−0.989048	−	0.147596i	$-0.952846\pi$
$857$	−24.6331	+	24.6331i	−0.841451	+	0.841451i	0.147596	+	0.989048i	$0.452846\pi$
$858$		0			0
$859$			5.21251i			0.177849i	0.996038	+	0.0889243i	$0.0283429\pi$
$859$			5.21251i			0.177849i	−0.996038	+	0.0889243i	$0.971657\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17.2870	−	17.2870i	−0.588458	−	0.588458i	0.348756	−	0.937214i	$-0.386604\pi$
$863$	−17.2870	−	17.2870i	−0.588458	−	0.588458i	−0.937214	+	0.348756i	$0.886604\pi$
$864$		0			0
$865$	1.06599	−	42.2317i	0.0362448	−	1.43592i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	13.4099			0.454900
$870$		0			0
$871$	−2.69689			−0.0913808
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.93349	+	0.146662i	0.0653641	+	0.00495808i
$876$		0			0
$877$	−0.501372	−	0.501372i	−0.0169301	−	0.0169301i	0.698591	−	0.715521i	$-0.253811\pi$
$877$	−0.501372	−	0.501372i	−0.0169301	−	0.0169301i	−0.715521	+	0.698591i	$0.753811\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	4.90108i		−	0.165122i	−0.996586	−	0.0825608i	$-0.973690\pi$
$881$		−	4.90108i		−	0.165122i	0.996586	−	0.0825608i	$-0.0263099\pi$
$882$		0			0
$883$	−35.5931	+	35.5931i	−1.19780	+	1.19780i	−0.222982	+	0.974823i	$0.571579\pi$
$883$	−35.5931	+	35.5931i	−1.19780	+	1.19780i	−0.974823	+	0.222982i	$0.928421\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.9679	−	24.9679i	0.838340	−	0.838340i	−0.150300	−	0.988640i	$-0.548024\pi$
$887$	24.9679	−	24.9679i	0.838340	−	0.838340i	0.988640	+	0.150300i	$0.0480241\pi$
$888$		0			0
$889$		−	2.31963i		−	0.0777980i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.91018	−	8.91018i	−0.298168	−	0.298168i
$894$		0			0
$895$	−0.736493	−	0.774636i	−0.0246182	−	0.0258932i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−23.6338			−0.788230
$900$		0			0
$901$	4.20557			0.140108
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.45368	−	8.89149i	−0.281010	−	0.295563i
$906$		0			0
$907$	−20.7600	−	20.7600i	−0.689326	−	0.689326i	0.272757	−	0.962083i	$-0.412065\pi$
$907$	−20.7600	−	20.7600i	−0.689326	−	0.689326i	−0.962083	+	0.272757i	$0.912065\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	24.8240i		−	0.822456i	−0.911533	−	0.411228i	$-0.865100\pi$
$911$		−	24.8240i		−	0.822456i	0.911533	−	0.411228i	$-0.134900\pi$
$912$		0			0
$913$	−10.4898	+	10.4898i	−0.347163	+	0.347163i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.104441	+	0.104441i	−0.00344894	+	0.00344894i
$918$		0			0
$919$			54.3852i			1.79400i	0.442028	+	0.897001i	$0.354259\pi$
$919$			54.3852i			1.79400i	−0.442028	+	0.897001i	$0.645741\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.15353	−	2.15353i	−0.0708843	−	0.0708843i
$924$		0			0
$925$	−29.4101	+	26.5816i	−0.966997	+	0.873999i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−28.3576			−0.930383			−0.465191	−	0.885210i	$-0.654014\pi$
$929$	−28.3576			−0.930383			−0.465191	+	0.885210i	$0.654014\pi$
$930$		0			0
$931$	−41.7470			−1.36820
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.174161	−	6.89981i	0.00569569	−	0.225648i
$936$		0			0
$937$	−21.4758	−	21.4758i	−0.701585	−	0.701585i	0.263165	−	0.964751i	$-0.415233\pi$
$937$	−21.4758	−	21.4758i	−0.701585	−	0.701585i	−0.964751	+	0.263165i	$0.915233\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			58.0427i			1.89214i	0.323965	+	0.946069i	$0.394984\pi$
$941$			58.0427i			1.89214i	−0.323965	+	0.946069i	$0.605016\pi$
$942$		0			0
$943$	8.01113	−	8.01113i	0.260878	−	0.260878i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.41500	−	1.41500i	0.0459814	−	0.0459814i	−0.683742	−	0.729724i	$-0.739649\pi$
$947$	1.41500	−	1.41500i	0.0459814	−	0.0459814i	0.729724	+	0.683742i	$0.239649\pi$
$948$		0			0
$949$		−	5.36467i		−	0.174145i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	39.0662	+	39.0662i	1.26548	+	1.26548i	0.948400	+	0.317077i	$0.102701\pi$
$953$	39.0662	+	39.0662i	1.26548	+	1.26548i	0.317077	+	0.948400i	$0.397299\pi$
$954$		0			0
$955$	32.2163	−	30.6300i	1.04250	−	0.991163i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.37007			−0.108825
$960$		0			0
$961$	−4.59997			−0.148386
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.7775	−	0.574938i	−0.733234	−	0.0185079i
$966$		0			0
$967$	−26.0312	−	26.0312i	−0.837105	−	0.837105i	0.151372	−	0.988477i	$-0.451631\pi$
$967$	−26.0312	−	26.0312i	−0.837105	−	0.837105i	−0.988477	+	0.151372i	$0.951631\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			55.8073i			1.79094i	0.445120	+	0.895471i	$0.353161\pi$
$971$			55.8073i			1.79094i	−0.445120	+	0.895471i	$0.646839\pi$
$972$		0			0
$973$	2.55555	−	2.55555i	0.0819271	−	0.0819271i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.5948	+	20.5948i	−0.658886	+	0.658886i	−0.955116	−	0.296231i	$-0.904270\pi$
$977$	−20.5948	+	20.5948i	−0.658886	+	0.658886i	0.296231	+	0.955116i	$0.404270\pi$
$978$		0			0
$979$		−	7.05907i		−	0.225609i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27.2373	+	27.2373i	0.868737	+	0.868737i	0.992333	−	0.123596i	$-0.0394427\pi$
$983$	27.2373	+	27.2373i	0.868737	+	0.868737i	−0.123596	+	0.992333i	$0.539443\pi$
$984$		0			0
$985$	22.6399	+	0.571466i	0.721369	+	0.0182084i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.59108			0.0823915
$990$		0			0
$991$	8.77379			0.278709			0.139354	−	0.990243i	$-0.455497\pi$
$991$	8.77379			0.278709			0.139354	+	0.990243i	$0.455497\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.56013	−	7.18787i	0.239672	−	0.227871i
$996$		0			0
$997$	−9.86580	−	9.86580i	−0.312453	−	0.312453i	0.533406	−	0.845859i	$-0.320912\pi$
$997$	−9.86580	−	9.86580i	−0.312453	−	0.312453i	−0.845859	+	0.533406i	$0.820912\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.a.737.16	yes	44
3.2	odd	2	inner	4140.2.s.a.737.7	✓	44
5.3	odd	4	inner	4140.2.s.a.2393.7	yes	44
15.8	even	4	inner	4140.2.s.a.2393.16	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.7	✓	44	3.2	odd	2	inner
4140.2.s.a.737.16	yes	44	1.1	even	1	trivial
4140.2.s.a.2393.7	yes	44	5.3	odd	4	inner
4140.2.s.a.2393.16	yes	44	15.8	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.54074 + 1.62053i) q^{5} +(-0.122636 - 0.122636i) q^{7} +O(q^{10})\)	\(q+(1.54074 + 1.62053i) q^{5} +(-0.122636 - 0.122636i) q^{7} -1.17251i q^{11} +(0.426185 - 0.426185i) q^{13} +(1.86148 - 1.86148i) q^{17} -5.98959i q^{19} +(0.707107 + 0.707107i) q^{23} +(-0.252254 + 4.99363i) q^{25} +4.59971 q^{29} -5.13810 q^{31} +(0.00978578 - 0.387686i) q^{35} +(5.60631 + 5.60631i) q^{37} -11.3295i q^{41} +(1.83217 - 1.83217i) q^{43} +(1.48761 - 1.48761i) q^{47} -6.96992i q^{49} +(1.12963 + 1.12963i) q^{53} +(1.90010 - 1.80654i) q^{55} -2.00765 q^{59} +2.11052 q^{61} +(1.34729 + 0.0340075i) q^{65} +(-3.16400 - 3.16400i) q^{67} -5.05304i q^{71} +(6.29383 - 6.29383i) q^{73} +(-0.143793 + 0.143793i) q^{77} +11.4369i q^{79} +(-8.94646 - 8.94646i) q^{83} +(5.88463 + 0.148537i) q^{85} +6.02046 q^{89} -0.104531 q^{91} +(9.70634 - 9.22839i) q^{95} +(-10.3024 - 10.3024i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 44 * q - 12 * q^7	\( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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