LMFDB - Embedded newform 2240.2.h.f.671.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(671,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.h (of order $2$, degree $1$, 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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.6
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.f.671.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59614i q^{3} +1.00000 q^{5} +(-2.28275 + 1.33755i) q^{7} -3.73996 q^{9} +O(q^{10})$	$q+2.59614i q^{3} +1.00000 q^{5} +(-2.28275 + 1.33755i) q^{7} -3.73996 q^{9} -3.05894 q^{11} -6.11716 q^{13} +2.59614i q^{15} -1.25746i q^{17} +3.37720i q^{19} +(-3.47247 - 5.92635i) q^{21} -4.24285i q^{23} +1.00000 q^{25} -1.92104i q^{27} -1.52586i q^{29} +9.46731 q^{31} -7.94145i q^{33} +(-2.28275 + 1.33755i) q^{35} -3.12779i q^{37} -15.8810i q^{39} -3.19570i q^{41} +10.7755 q^{43} -3.73996 q^{45} -7.89284 q^{47} +(3.42191 - 6.10660i) q^{49} +3.26456 q^{51} -12.5951i q^{53} -3.05894 q^{55} -8.76770 q^{57} +14.5828i q^{59} +0.274335 q^{61} +(8.53740 - 5.00239i) q^{63} -6.11716 q^{65} -10.7513 q^{67} +11.0150 q^{69} -7.23258i q^{71} +0.742408i q^{73} +2.59614i q^{75} +(6.98281 - 4.09149i) q^{77} -10.1284i q^{79} -6.23258 q^{81} +6.15502i q^{83} -1.25746i q^{85} +3.96135 q^{87} +5.78158i q^{89} +(13.9640 - 8.18202i) q^{91} +24.5785i q^{93} +3.37720i q^{95} -14.3036i q^{97} +11.4403 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) $ 24 * q + 24 * q^5 - 36 * q^9	$ 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) $ 24 * q + 24 * q^5 - 36 * q^9 + 4 * q^13 + 8 * q^21 + 24 * q^25 - 36 * q^45 + 24 * q^57 + 56 * q^61 + 4 * q^65 + 96 * q^69 - 12 * q^77 + 24 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$			2.59614i			1.49888i	0.662070	+	0.749442i	$0.269678\pi$
$3$			2.59614i			1.49888i	−0.662070	+	0.749442i	$0.730322\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−2.28275	+	1.33755i	−0.862799	+	0.505547i
$7$	−2.28275	+	1.33755i	−0.862799	+	0.505547i
$8$		0			0
$9$	−3.73996			−1.24665
$10$		0			0
$11$	−3.05894			−0.922306			−0.461153	−	0.887321i	$-0.652564\pi$
$11$	−3.05894			−0.922306			−0.461153	+	0.887321i	$0.652564\pi$
$12$		0			0
$13$	−6.11716			−1.69660			−0.848298	−	0.529519i	$-0.822372\pi$
$13$	−6.11716			−1.69660			−0.848298	+	0.529519i	$0.822372\pi$
$14$		0			0
$15$			2.59614i			0.670321i
$16$		0			0
$17$		−	1.25746i		−	0.304980i	−0.988305	−	0.152490i	$-0.951271\pi$
$17$		−	1.25746i		−	0.304980i	0.988305	−	0.152490i	$-0.0487292\pi$
$18$		0			0
$19$			3.37720i			0.774784i	0.921915	+	0.387392i	$0.126624\pi$
$19$			3.37720i			0.774784i	−0.921915	+	0.387392i	$0.873376\pi$
$20$		0			0
$21$	−3.47247	−	5.92635i	−0.757756	−	1.29324i
$22$		0			0
$23$		−	4.24285i		−	0.884695i	−0.896844	−	0.442348i	$-0.854146\pi$
$23$		−	4.24285i		−	0.884695i	0.896844	−	0.442348i	$-0.145854\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	1.92104i		−	0.369704i
$28$		0			0
$29$		−	1.52586i		−	0.283345i	−0.989914	−	0.141673i	$-0.954752\pi$
$29$		−	1.52586i		−	0.283345i	0.989914	−	0.141673i	$-0.0452480\pi$
$30$		0			0
$31$	9.46731			1.70038			0.850190	−	0.526476i	$-0.176487\pi$
$31$	9.46731			1.70038			0.850190	+	0.526476i	$0.176487\pi$
$32$		0			0
$33$		−	7.94145i		−	1.38243i
$34$		0			0
$35$	−2.28275	+	1.33755i	−0.385856	+	0.226087i
$36$		0			0
$37$		−	3.12779i		−	0.514206i	−0.966384	−	0.257103i	$-0.917232\pi$
$37$		−	3.12779i		−	0.514206i	0.966384	−	0.257103i	$-0.0827680\pi$
$38$		0			0
$39$		−	15.8810i		−	2.54300i
$40$		0			0
$41$		−	3.19570i		−	0.499085i	−0.968364	−	0.249543i	$-0.919720\pi$
$41$		−	3.19570i		−	0.499085i	0.968364	−	0.249543i	$-0.0802803\pi$
$42$		0			0
$43$	10.7755			1.64325			0.821623	−	0.570031i	$-0.193069\pi$
$43$	10.7755			1.64325			0.821623	+	0.570031i	$0.193069\pi$
$44$		0			0
$45$	−3.73996			−0.557520
$46$		0			0
$47$	−7.89284			−1.15129			−0.575645	−	0.817700i	$-0.695249\pi$
$47$	−7.89284			−1.15129			−0.575645	+	0.817700i	$0.695249\pi$
$48$		0			0
$49$	3.42191	−	6.10660i	0.488845	−	0.872371i
$50$		0			0
$51$	3.26456			0.457129
$52$		0			0
$53$		−	12.5951i		−	1.73007i	−0.501710	−	0.865036i	$-0.667296\pi$
$53$		−	12.5951i		−	1.73007i	0.501710	−	0.865036i	$-0.332704\pi$
$54$		0			0
$55$	−3.05894			−0.412468
$56$		0			0
$57$	−8.76770			−1.16131
$58$		0			0
$59$			14.5828i			1.89852i	0.314497	+	0.949259i	$0.398164\pi$
$59$			14.5828i			1.89852i	−0.314497	+	0.949259i	$0.601836\pi$
$60$		0			0
$61$	0.274335			0.0351250			0.0175625	−	0.999846i	$-0.494409\pi$
$61$	0.274335			0.0351250			0.0175625	+	0.999846i	$0.494409\pi$
$62$		0			0
$63$	8.53740	−	5.00239i	1.07561	−	0.630242i
$64$		0			0
$65$	−6.11716			−0.758741
$66$		0			0
$67$	−10.7513			−1.31348			−0.656740	−	0.754117i	$-0.728065\pi$
$67$	−10.7513			−1.31348			−0.656740	+	0.754117i	$0.728065\pi$
$68$		0			0
$69$	11.0150			1.32606
$70$		0			0
$71$		−	7.23258i		−	0.858350i	−0.903221	−	0.429175i	$-0.858804\pi$
$71$		−	7.23258i		−	0.858350i	0.903221	−	0.429175i	$-0.141196\pi$
$72$		0			0
$73$			0.742408i			0.0868922i	0.999056	+	0.0434461i	$0.0138337\pi$
$73$			0.742408i			0.0868922i	−0.999056	+	0.0434461i	$0.986166\pi$
$74$		0			0
$75$			2.59614i			0.299777i
$76$		0			0
$77$	6.98281	−	4.09149i	0.795765	−	0.466269i
$78$		0			0
$79$		−	10.1284i		−	1.13953i	−0.821807	−	0.569765i	$-0.807034\pi$
$79$		−	10.1284i		−	1.13953i	0.821807	−	0.569765i	$-0.192966\pi$
$80$		0			0
$81$	−6.23258			−0.692509
$82$		0			0
$83$			6.15502i			0.675601i	0.941218	+	0.337801i	$0.109683\pi$
$83$			6.15502i			0.675601i	−0.941218	+	0.337801i	$0.890317\pi$
$84$		0			0
$85$		−	1.25746i		−	0.136391i
$86$		0			0
$87$	3.96135			0.424702
$88$		0			0
$89$			5.78158i			0.612846i	0.951895	+	0.306423i	$0.0991322\pi$
$89$			5.78158i			0.612846i	−0.951895	+	0.306423i	$0.900868\pi$
$90$		0			0
$91$	13.9640	−	8.18202i	1.46382	−	0.857709i
$92$		0			0
$93$			24.5785i			2.54867i
$94$		0			0
$95$			3.37720i			0.346494i
$96$		0			0
$97$		−	14.3036i		−	1.45231i	−0.687533	−	0.726153i	$-0.741306\pi$
$97$		−	14.3036i		−	1.45231i	0.687533	−	0.726153i	$-0.258694\pi$
$98$		0			0
$99$	11.4403			1.14980
$100$		0			0
$101$	−2.31463			−0.230314			−0.115157	−	0.993347i	$-0.536737\pi$
$101$	−2.31463			−0.230314			−0.115157	+	0.993347i	$0.536737\pi$
$102$		0			0
$103$	−13.8344			−1.36315			−0.681573	−	0.731750i	$-0.738704\pi$
$103$	−13.8344			−1.36315			−0.681573	+	0.731750i	$0.738704\pi$
$104$		0			0
$105$	−3.47247	−	5.92635i	−0.338879	−	0.578353i
$106$		0			0
$107$	−13.6267			−1.31734			−0.658672	−	0.752430i	$-0.728882\pi$
$107$	−13.6267			−1.31734			−0.658672	+	0.752430i	$0.728882\pi$
$108$		0			0
$109$			7.26068i			0.695447i	0.937597	+	0.347723i	$0.113045\pi$
$109$			7.26068i			0.695447i	−0.937597	+	0.347723i	$0.886955\pi$
$110$		0			0
$111$	8.12020			0.770736
$112$		0			0
$113$	−9.04766			−0.851133			−0.425566	−	0.904927i	$-0.639925\pi$
$113$	−9.04766			−0.851133			−0.425566	+	0.904927i	$0.639925\pi$
$114$		0			0
$115$		−	4.24285i		−	0.395648i
$116$		0			0
$117$	22.8779			2.11507
$118$		0			0
$119$	1.68192	+	2.87048i	0.154182	+	0.263136i
$120$		0			0
$121$	−1.64287			−0.149352
$122$		0			0
$123$	8.29651			0.748071
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			9.27159i			0.822721i	0.911473	+	0.411360i	$0.134946\pi$
$127$			9.27159i			0.822721i	−0.911473	+	0.411360i	$0.865054\pi$
$128$		0			0
$129$			27.9747i			2.46303i
$130$		0			0
$131$		−	9.14476i		−	0.798981i	−0.916737	−	0.399490i	$-0.869187\pi$
$131$		−	9.14476i		−	0.798981i	0.916737	−	0.399490i	$-0.130813\pi$
$132$		0			0
$133$	−4.51718	−	7.70932i	−0.391689	−	0.668483i
$134$		0			0
$135$		−	1.92104i		−	0.165337i
$136$		0			0
$137$	−11.7481			−1.00371			−0.501853	−	0.864953i	$-0.667348\pi$
$137$	−11.7481			−1.00371			−0.501853	+	0.864953i	$0.667348\pi$
$138$		0			0
$139$		−	11.2894i		−	0.957552i	−0.877937	−	0.478776i	$-0.841081\pi$
$139$		−	11.2894i		−	0.957552i	0.877937	−	0.478776i	$-0.158919\pi$
$140$		0			0
$141$		−	20.4910i		−	1.72565i
$142$		0			0
$143$	18.7121			1.56478
$144$		0			0
$145$		−	1.52586i		−	0.126716i
$146$		0			0
$147$	15.8536	+	8.88378i	1.30758	+	0.732721i
$148$		0			0
$149$			18.6432i			1.52731i	0.645624	+	0.763655i	$0.276597\pi$
$149$			18.6432i			1.52731i	−0.645624	+	0.763655i	$0.723403\pi$
$150$		0			0
$151$		−	3.29824i		−	0.268407i	−0.990954	−	0.134204i	$-0.957152\pi$
$151$		−	3.29824i		−	0.268407i	0.990954	−	0.134204i	$-0.0428476\pi$
$152$		0			0
$153$			4.70286i			0.380204i
$154$		0			0
$155$	9.46731			0.760433
$156$		0			0
$157$	−0.308165			−0.0245943			−0.0122971	−	0.999924i	$-0.503914\pi$
$157$	−0.308165			−0.0245943			−0.0122971	+	0.999924i	$0.503914\pi$
$158$		0			0
$159$	32.6987			2.59318
$160$		0			0
$161$	5.67503	+	9.68537i	0.447255	+	0.763314i
$162$		0			0
$163$	−12.8164			−1.00386			−0.501929	−	0.864909i	$-0.667376\pi$
$163$	−12.8164			−1.00386			−0.501929	+	0.864909i	$0.667376\pi$
$164$		0			0
$165$		−	7.94145i		−	0.618241i
$166$		0			0
$167$	−8.34267			−0.645575			−0.322788	−	0.946471i	$-0.604620\pi$
$167$	−8.34267			−0.645575			−0.322788	+	0.946471i	$0.604620\pi$
$168$		0			0
$169$	24.4197			1.87844
$170$		0			0
$171$		−	12.6306i		−	0.965887i
$172$		0			0
$173$	0.924877			0.0703171			0.0351585	−	0.999382i	$-0.488806\pi$
$173$	0.924877			0.0703171			0.0351585	+	0.999382i	$0.488806\pi$
$174$		0			0
$175$	−2.28275	+	1.33755i	−0.172560	+	0.101109i
$176$		0			0
$177$	−37.8590			−2.84566
$178$		0			0
$179$	2.99825			0.224100			0.112050	−	0.993703i	$-0.464258\pi$
$179$	2.99825			0.224100			0.112050	+	0.993703i	$0.464258\pi$
$180$		0			0
$181$	−1.52021			−0.112997			−0.0564983	−	0.998403i	$-0.517994\pi$
$181$	−1.52021			−0.112997			−0.0564983	+	0.998403i	$0.517994\pi$
$182$		0			0
$183$			0.712214i			0.0526484i
$184$		0			0
$185$		−	3.12779i		−	0.229960i
$186$		0			0
$187$			3.84651i			0.281285i
$188$		0			0
$189$	2.56949	+	4.38526i	0.186903	+	0.318981i
$190$		0			0
$191$		−	4.77816i		−	0.345736i	−0.984945	−	0.172868i	$-0.944697\pi$
$191$		−	4.77816i		−	0.345736i	0.984945	−	0.172868i	$-0.0553034\pi$
$192$		0			0
$193$	14.3979			1.03638			0.518191	−	0.855265i	$-0.326606\pi$
$193$	14.3979			1.03638			0.518191	+	0.855265i	$0.326606\pi$
$194$		0			0
$195$		−	15.8810i		−	1.13726i
$196$		0			0
$197$		−	20.8023i		−	1.48210i	−0.671449	−	0.741050i	$-0.734328\pi$
$197$		−	20.8023i		−	1.48210i	0.671449	−	0.741050i	$-0.265672\pi$
$198$		0			0
$199$	−4.94052			−0.350224			−0.175112	−	0.984549i	$-0.556029\pi$
$199$	−4.94052			−0.350224			−0.175112	+	0.984549i	$0.556029\pi$
$200$		0			0
$201$		−	27.9119i		−	1.96875i
$202$		0			0
$203$	2.04092	+	3.48316i	0.143244	+	0.244470i
$204$		0			0
$205$		−	3.19570i		−	0.223198i
$206$		0			0
$207$			15.8681i			1.10291i
$208$		0			0
$209$		−	10.3307i		−	0.714588i
$210$		0			0
$211$	4.89937			0.337287			0.168643	−	0.985677i	$-0.446061\pi$
$211$	4.89937			0.337287			0.168643	+	0.985677i	$0.446061\pi$
$212$		0			0
$213$	18.7768			1.28657
$214$		0			0
$215$	10.7755			0.734882
$216$		0			0
$217$	−21.6115	+	12.6630i	−1.46709	+	0.859622i
$218$		0			0
$219$	−1.92740			−0.130241
$220$		0			0
$221$			7.69211i			0.517428i
$222$		0			0
$223$	−3.08814			−0.206797			−0.103399	−	0.994640i	$-0.532972\pi$
$223$	−3.08814			−0.206797			−0.103399	+	0.994640i	$0.532972\pi$
$224$		0			0
$225$	−3.73996			−0.249331
$226$		0			0
$227$			2.04747i			0.135896i	0.997689	+	0.0679478i	$0.0216451\pi$
$227$			2.04747i			0.135896i	−0.997689	+	0.0679478i	$0.978355\pi$
$228$		0			0
$229$	−9.06904			−0.599299			−0.299649	−	0.954049i	$-0.596870\pi$
$229$	−9.06904			−0.599299			−0.299649	+	0.954049i	$0.596870\pi$
$230$		0			0
$231$	10.6221	+	18.1284i	0.698883	+	1.19276i
$232$		0			0
$233$	−12.8301			−0.840529			−0.420265	−	0.907402i	$-0.638063\pi$
$233$	−12.8301			−0.840529			−0.420265	+	0.907402i	$0.638063\pi$
$234$		0			0
$235$	−7.89284			−0.514872
$236$		0			0
$237$	26.2947			1.70802
$238$		0			0
$239$		−	14.4982i		−	0.937811i	−0.883248	−	0.468905i	$-0.844649\pi$
$239$		−	14.4982i		−	0.937811i	0.883248	−	0.468905i	$-0.155351\pi$
$240$		0			0
$241$			0.632410i			0.0407371i	0.999793	+	0.0203686i	$0.00648396\pi$
$241$			0.632410i			0.0407371i	−0.999793	+	0.0203686i	$0.993516\pi$
$242$		0			0
$243$		−	21.9438i		−	1.40770i
$244$		0			0
$245$	3.42191	−	6.10660i	0.218618	−	0.390136i
$246$		0			0
$247$		−	20.6589i		−	1.31449i
$248$		0			0
$249$	−15.9793			−1.01265
$250$		0			0
$251$		−	9.77158i		−	0.616777i	−0.951261	−	0.308388i	$-0.900210\pi$
$251$		−	9.77158i		−	0.616777i	0.951261	−	0.308388i	$-0.0997897\pi$
$252$		0			0
$253$			12.9786i			0.815960i
$254$		0			0
$255$	3.26456			0.204434
$256$		0			0
$257$		−	3.31038i		−	0.206496i	−0.994656	−	0.103248i	$-0.967077\pi$
$257$		−	3.31038i		−	0.206496i	0.994656	−	0.103248i	$-0.0329235\pi$
$258$		0			0
$259$	4.18359	+	7.13998i	0.259955	+	0.443657i
$260$		0			0
$261$			5.70666i			0.353233i
$262$		0			0
$263$		−	28.2314i		−	1.74082i	−0.492324	−	0.870412i	$-0.663852\pi$
$263$		−	28.2314i		−	1.74082i	0.492324	−	0.870412i	$-0.336148\pi$
$264$		0			0
$265$		−	12.5951i		−	0.773711i
$266$		0			0
$267$	−15.0098			−0.918585
$268$		0			0
$269$	−28.0436			−1.70985			−0.854925	−	0.518751i	$-0.826397\pi$
$269$	−28.0436			−1.70985			−0.854925	+	0.518751i	$0.826397\pi$
$270$		0			0
$271$	−8.79282			−0.534126			−0.267063	−	0.963679i	$-0.586053\pi$
$271$	−8.79282			−0.534126			−0.267063	+	0.963679i	$0.586053\pi$
$272$		0			0
$273$	21.2417	+	36.2525i	1.28561	+	2.19410i
$274$		0			0
$275$	−3.05894			−0.184461
$276$		0			0
$277$			18.3990i			1.10549i	0.833351	+	0.552744i	$0.186419\pi$
$277$			18.3990i			1.10549i	−0.833351	+	0.552744i	$0.813581\pi$
$278$		0			0
$279$	−35.4074			−2.11978
$280$		0			0
$281$	31.1469			1.85807			0.929035	−	0.369991i	$-0.120639\pi$
$281$	31.1469			1.85807			0.929035	+	0.369991i	$0.120639\pi$
$282$		0			0
$283$			29.8362i			1.77358i	0.462172	+	0.886790i	$0.347070\pi$
$283$			29.8362i			1.77358i	−0.462172	+	0.886790i	$0.652930\pi$
$284$		0			0
$285$	−8.76770			−0.519354
$286$		0			0
$287$	4.27442	+	7.29500i	0.252311	+	0.430610i
$288$		0			0
$289$	15.4188			0.906987
$290$		0			0
$291$	37.1341			2.17684
$292$		0			0
$293$	16.5550			0.967156			0.483578	−	0.875301i	$-0.339337\pi$
$293$	16.5550			0.967156			0.483578	+	0.875301i	$0.339337\pi$
$294$		0			0
$295$			14.5828i			0.849043i
$296$		0			0
$297$			5.87635i			0.340981i
$298$		0			0
$299$			25.9542i			1.50097i
$300$		0			0
$301$	−24.5978	+	14.4128i	−1.41779	+	0.830738i
$302$		0			0
$303$		−	6.00911i		−	0.345215i
$304$		0			0
$305$	0.274335			0.0157084
$306$		0			0
$307$			22.4879i			1.28345i	0.766933	+	0.641727i	$0.221782\pi$
$307$			22.4879i			1.28345i	−0.766933	+	0.641727i	$0.778218\pi$
$308$		0			0
$309$		−	35.9161i		−	2.04320i
$310$		0			0
$311$	3.02754			0.171676			0.0858380	−	0.996309i	$-0.472643\pi$
$311$	3.02754			0.171676			0.0858380	+	0.996309i	$0.472643\pi$
$312$		0			0
$313$			23.0675i			1.30385i	0.758282	+	0.651926i	$0.226039\pi$
$313$			23.0675i			1.30385i	−0.758282	+	0.651926i	$0.773961\pi$
$314$		0			0
$315$	8.53740	−	5.00239i	0.481028	−	0.281853i
$316$		0			0
$317$			10.8809i			0.611135i	0.952170	+	0.305567i	$0.0988462\pi$
$317$			10.8809i			0.611135i	−0.952170	+	0.305567i	$0.901154\pi$
$318$		0			0
$319$			4.66752i			0.261331i
$320$		0			0
$321$		−	35.3769i		−	1.97455i
$322$		0			0
$323$	4.24671			0.236293
$324$		0			0
$325$	−6.11716			−0.339319
$326$		0			0
$327$	−18.8498			−1.04239
$328$		0			0
$329$	18.0174	−	10.5571i	0.993332	−	0.582031i
$330$		0			0
$331$	−33.3053			−1.83062			−0.915312	−	0.402745i	$-0.868056\pi$
$331$	−33.3053			−1.83062			−0.915312	+	0.402745i	$0.868056\pi$
$332$		0			0
$333$			11.6978i			0.641037i
$334$		0			0
$335$	−10.7513			−0.587406
$336$		0			0
$337$	14.3149			0.779783			0.389892	−	0.920861i	$-0.372512\pi$
$337$	14.3149			0.779783			0.389892	+	0.920861i	$0.372512\pi$
$338$		0			0
$339$		−	23.4890i		−	1.27575i
$340$		0			0
$341$	−28.9600			−1.56827
$342$		0			0
$343$	0.356507	+	18.5168i	0.0192496	+	0.999815i
$344$		0			0
$345$	11.0150			0.593030
$346$		0			0
$347$	−19.7204			−1.05865			−0.529324	−	0.848420i	$-0.677554\pi$
$347$	−19.7204			−1.05865			−0.529324	+	0.848420i	$0.677554\pi$
$348$		0			0
$349$	9.99879			0.535223			0.267612	−	0.963527i	$-0.413766\pi$
$349$	9.99879			0.535223			0.267612	+	0.963527i	$0.413766\pi$
$350$		0			0
$351$			11.7513i			0.627239i
$352$		0			0
$353$		−	37.0563i		−	1.97231i	−0.165839	−	0.986153i	$-0.553033\pi$
$353$		−	37.0563i		−	1.97231i	0.165839	−	0.986153i	$-0.446967\pi$
$354$		0			0
$355$		−	7.23258i		−	0.383866i
$356$		0			0
$357$	−7.45217	+	4.36651i	−0.394411	+	0.231100i
$358$		0			0
$359$			14.8233i			0.782343i	0.920318	+	0.391172i	$0.127930\pi$
$359$			14.8233i			0.782343i	−0.920318	+	0.391172i	$0.872070\pi$
$360$		0			0
$361$	7.59449			0.399710
$362$		0			0
$363$		−	4.26512i		−	0.223861i
$364$		0			0
$365$			0.742408i			0.0388594i
$366$		0			0
$367$	−28.3172			−1.47814			−0.739072	−	0.673626i	$-0.764736\pi$
$367$	−28.3172			−1.47814			−0.739072	+	0.673626i	$0.764736\pi$
$368$		0			0
$369$			11.9518i			0.622186i
$370$		0			0
$371$	16.8466	+	28.7515i	0.874632	+	1.49270i
$372$		0			0
$373$		−	8.43536i		−	0.436767i	−0.975863	−	0.218383i	$-0.929922\pi$
$373$		−	8.43536i		−	0.436767i	0.975863	−	0.218383i	$-0.0700783\pi$
$374$		0			0
$375$			2.59614i			0.134064i
$376$		0			0
$377$			9.33394i			0.480722i
$378$		0			0
$379$	−10.1855			−0.523196			−0.261598	−	0.965177i	$-0.584249\pi$
$379$	−10.1855			−0.523196			−0.261598	+	0.965177i	$0.584249\pi$
$380$		0			0
$381$	−24.0704			−1.23316
$382$		0			0
$383$	−37.5763			−1.92006			−0.960031	−	0.279895i	$-0.909700\pi$
$383$	−37.5763			−1.92006			−0.960031	+	0.279895i	$0.909700\pi$
$384$		0			0
$385$	6.98281	−	4.09149i	0.355877	−	0.208522i
$386$		0			0
$387$	−40.2999			−2.04856
$388$		0			0
$389$			4.88157i			0.247505i	0.992313	+	0.123753i	$0.0394929\pi$
$389$			4.88157i			0.247505i	−0.992313	+	0.123753i	$0.960507\pi$
$390$		0			0
$391$	−5.33523			−0.269814
$392$		0			0
$393$	23.7411			1.19758
$394$		0			0
$395$		−	10.1284i		−	0.509614i
$396$		0			0
$397$	−22.4288			−1.12567			−0.562835	−	0.826569i	$-0.690289\pi$
$397$	−22.4288			−1.12567			−0.562835	+	0.826569i	$0.690289\pi$
$398$		0			0
$399$	20.0145	−	11.7273i	1.00198	−	0.587097i
$400$		0			0
$401$	4.25212			0.212341			0.106170	−	0.994348i	$-0.466141\pi$
$401$	4.25212			0.212341			0.106170	+	0.994348i	$0.466141\pi$
$402$		0			0
$403$	−57.9131			−2.88486
$404$		0			0
$405$	−6.23258			−0.309699
$406$		0			0
$407$			9.56775i			0.474256i
$408$		0			0
$409$			27.7723i			1.37325i	0.727010	+	0.686627i	$0.240909\pi$
$409$			27.7723i			1.37325i	−0.727010	+	0.686627i	$0.759091\pi$
$410$		0			0
$411$		−	30.4997i		−	1.50444i
$412$		0			0
$413$	−19.5052	−	33.2889i	−0.959789	−	1.63804i
$414$		0			0
$415$			6.15502i			0.302138i
$416$		0			0
$417$	29.3088			1.43526
$418$		0			0
$419$			9.91246i			0.484255i	0.970244	+	0.242128i	$0.0778453\pi$
$419$			9.91246i			0.484255i	−0.970244	+	0.242128i	$0.922155\pi$
$420$		0			0
$421$		−	33.1235i		−	1.61434i	−0.590318	−	0.807171i	$-0.700998\pi$
$421$		−	33.1235i		−	1.61434i	0.590318	−	0.807171i	$-0.299002\pi$
$422$		0			0
$423$	29.5189			1.43526
$424$		0			0
$425$		−	1.25746i		−	0.0609960i
$426$		0			0
$427$	−0.626240	+	0.366938i	−0.0303059	+	0.0177574i
$428$		0			0
$429$			48.5792i			2.34542i
$430$		0			0
$431$			24.9585i			1.20221i	0.799171	+	0.601104i	$0.205272\pi$
$431$			24.9585i			1.20221i	−0.799171	+	0.601104i	$0.794728\pi$
$432$		0			0
$433$		−	4.83379i		−	0.232297i	−0.993232	−	0.116149i	$-0.962945\pi$
$433$		−	4.83379i		−	0.232297i	0.993232	−	0.116149i	$-0.0370549\pi$
$434$		0			0
$435$	3.96135			0.189932
$436$		0			0
$437$	14.3290			0.685447
$438$		0			0
$439$	18.2643			0.871708			0.435854	−	0.900017i	$-0.356446\pi$
$439$	18.2643			0.871708			0.435854	+	0.900017i	$0.356446\pi$
$440$		0			0
$441$	−12.7978	+	22.8384i	−0.609420	+	1.08754i
$442$		0			0
$443$	26.2979			1.24945			0.624725	−	0.780844i	$-0.285211\pi$
$443$	26.2979			1.24945			0.624725	+	0.780844i	$0.285211\pi$
$444$		0			0
$445$			5.78158i			0.274073i
$446$		0			0
$447$	−48.4004			−2.28926
$448$		0			0
$449$	−38.9639			−1.83882			−0.919411	−	0.393299i	$-0.871334\pi$
$449$	−38.9639			−1.83882			−0.919411	+	0.393299i	$0.871334\pi$
$450$		0			0
$451$			9.77548i			0.460309i
$452$		0			0
$453$	8.56271			0.402311
$454$		0			0
$455$	13.9640	−	8.18202i	0.654641	−	0.383579i
$456$		0			0
$457$	35.8688			1.67787			0.838936	−	0.544230i	$-0.183178\pi$
$457$	35.8688			1.67787			0.838936	+	0.544230i	$0.183178\pi$
$458$		0			0
$459$	−2.41564			−0.112752
$460$		0			0
$461$	−33.9750			−1.58237			−0.791187	−	0.611574i	$-0.790537\pi$
$461$	−33.9750			−1.58237			−0.791187	+	0.611574i	$0.790537\pi$
$462$		0			0
$463$			15.8826i			0.738125i	0.929405	+	0.369063i	$0.120321\pi$
$463$			15.8826i			0.738125i	−0.929405	+	0.369063i	$0.879679\pi$
$464$		0			0
$465$			24.5785i			1.13980i
$466$		0			0
$467$			13.1435i			0.608209i	0.952639	+	0.304105i	$0.0983572\pi$
$467$			13.1435i			0.608209i	−0.952639	+	0.304105i	$0.901643\pi$
$468$		0			0
$469$	24.5425	−	14.3804i	1.13327	−	0.664026i
$470$		0			0
$471$		−	0.800041i		−	0.0368639i
$472$		0			0
$473$	−32.9616			−1.51558
$474$		0			0
$475$			3.37720i			0.154957i
$476$		0			0
$477$			47.1052i			2.15680i
$478$		0			0
$479$	−22.5908			−1.03220			−0.516100	−	0.856528i	$-0.672617\pi$
$479$	−22.5908			−1.03220			−0.516100	+	0.856528i	$0.672617\pi$
$480$		0			0
$481$			19.1332i			0.872400i
$482$		0			0
$483$	−25.1446	+	14.7332i	−1.14412	+	0.670383i
$484$		0			0
$485$		−	14.3036i		−	0.649491i
$486$		0			0
$487$			25.3997i			1.15097i	0.817812	+	0.575485i	$0.195187\pi$
$487$			25.3997i			1.15097i	−0.817812	+	0.575485i	$0.804813\pi$
$488$		0			0
$489$		−	33.2732i		−	1.50467i
$490$		0			0
$491$	−12.4146			−0.560264			−0.280132	−	0.959961i	$-0.590378\pi$
$491$	−12.4146			−0.560264			−0.280132	+	0.959961i	$0.590378\pi$
$492$		0			0
$493$	−1.91872			−0.0864146
$494$		0			0
$495$	11.4403			0.514204
$496$		0			0
$497$	9.67395	+	16.5102i	0.433936	+	0.740583i
$498$		0			0
$499$	−20.3343			−0.910290			−0.455145	−	0.890417i	$-0.650413\pi$
$499$	−20.3343			−0.910290			−0.455145	+	0.890417i	$0.650413\pi$
$500$		0			0
$501$		−	21.6588i		−	0.967642i
$502$		0			0
$503$	−3.07370			−0.137050			−0.0685248	−	0.997649i	$-0.521829\pi$
$503$	−3.07370			−0.137050			−0.0685248	+	0.997649i	$0.521829\pi$
$504$		0			0
$505$	−2.31463			−0.103000
$506$		0			0
$507$			63.3970i			2.81556i
$508$		0			0
$509$	32.3148			1.43233			0.716163	−	0.697933i	$-0.245897\pi$
$509$	32.3148			1.43233			0.716163	+	0.697933i	$0.245897\pi$
$510$		0			0
$511$	−0.993008	−	1.69473i	−0.0439281	−	0.0749706i
$512$		0			0
$513$	6.48775			0.286441
$514$		0			0
$515$	−13.8344			−0.609618
$516$		0			0
$517$	24.1438			1.06184
$518$		0			0
$519$			2.40111i			0.105397i
$520$		0			0
$521$		−	25.7947i		−	1.13009i	−0.825062	−	0.565043i	$-0.808860\pi$
$521$		−	25.7947i		−	1.13009i	0.825062	−	0.565043i	$-0.191140\pi$
$522$		0			0
$523$		−	11.7945i		−	0.515736i	−0.966180	−	0.257868i	$-0.916980\pi$
$523$		−	11.7945i		−	0.515736i	0.966180	−	0.257868i	$-0.0830200\pi$
$524$		0			0
$525$	−3.47247	−	5.92635i	−0.151551	−	0.258647i
$526$		0			0
$527$		−	11.9048i		−	0.518582i
$528$		0			0
$529$	4.99824			0.217315
$530$		0			0
$531$		−	54.5390i		−	2.36679i
$532$		0			0
$533$			19.5486i			0.846746i
$534$		0			0
$535$	−13.6267			−0.589134
$536$		0			0
$537$			7.78390i			0.335900i
$538$		0			0
$539$	−10.4674	+	18.6797i	−0.450864	+	0.804593i
$540$		0			0
$541$		−	2.07521i		−	0.0892204i	−0.999004	−	0.0446102i	$-0.985795\pi$
$541$		−	2.07521i		−	0.0892204i	0.999004	−	0.0446102i	$-0.0142046\pi$
$542$		0			0
$543$		−	3.94669i		−	0.169369i
$544$		0			0
$545$			7.26068i			0.311013i
$546$		0			0
$547$	17.5339			0.749697			0.374849	−	0.927086i	$-0.377695\pi$
$547$	17.5339			0.749697			0.374849	+	0.927086i	$0.377695\pi$
$548$		0			0
$549$	−1.02600			−0.0437887
$550$		0			0
$551$	5.15314			0.219531
$552$		0			0
$553$	13.5472	+	23.1206i	0.576086	+	0.983186i
$554$		0			0
$555$	8.12020			0.344683
$556$		0			0
$557$			17.9855i			0.762068i	0.924561	+	0.381034i	$0.124432\pi$
$557$			17.9855i			0.762068i	−0.924561	+	0.381034i	$0.875568\pi$
$558$		0			0
$559$	−65.9154			−2.78792
$560$		0			0
$561$	−9.98609			−0.421613
$562$		0			0
$563$		−	26.5159i		−	1.11751i	−0.829332	−	0.558756i	$-0.811279\pi$
$563$		−	26.5159i		−	1.11751i	0.829332	−	0.558756i	$-0.188721\pi$
$564$		0			0
$565$	−9.04766			−0.380638
$566$		0			0
$567$	14.2274	−	8.33640i	0.597496	−	0.350096i
$568$		0			0
$569$	−9.60242			−0.402554			−0.201277	−	0.979534i	$-0.564509\pi$
$569$	−9.60242			−0.402554			−0.201277	+	0.979534i	$0.564509\pi$
$570$		0			0
$571$	14.4225			0.603563			0.301782	−	0.953377i	$-0.402419\pi$
$571$	14.4225			0.603563			0.301782	+	0.953377i	$0.402419\pi$
$572$		0			0
$573$	12.4048			0.518218
$574$		0			0
$575$		−	4.24285i		−	0.176939i
$576$		0			0
$577$			33.6880i			1.40245i	0.712940	+	0.701225i	$0.247363\pi$
$577$			33.6880i			1.40245i	−0.712940	+	0.701225i	$0.752637\pi$
$578$		0			0
$579$			37.3789i			1.55342i
$580$		0			0
$581$	−8.23266	−	14.0504i	−0.341548	−	0.582908i
$582$		0			0
$583$			38.5277i			1.59566i
$584$		0			0
$585$	22.8779			0.945887
$586$		0			0
$587$		−	33.5627i		−	1.38528i	−0.721283	−	0.692641i	$-0.756447\pi$
$587$		−	33.5627i		−	1.38528i	0.721283	−	0.692641i	$-0.243553\pi$
$588$		0			0
$589$			31.9731i			1.31743i
$590$		0			0
$591$	54.0057			2.22150
$592$		0			0
$593$		−	24.5203i		−	1.00693i	−0.864016	−	0.503464i	$-0.832059\pi$
$593$		−	24.5203i		−	1.00693i	0.864016	−	0.503464i	$-0.167941\pi$
$594$		0			0
$595$	1.68192	+	2.87048i	0.0689521	+	0.117678i
$596$		0			0
$597$		−	12.8263i		−	0.524945i
$598$		0			0
$599$		−	3.17590i		−	0.129764i	−0.997893	−	0.0648818i	$-0.979333\pi$
$599$		−	3.17590i		−	0.129764i	0.997893	−	0.0648818i	$-0.0206670\pi$
$600$		0			0
$601$			0.334920i			0.0136617i	0.999977	+	0.00683083i	$0.00217434\pi$
$601$			0.334920i			0.0136617i	−0.999977	+	0.00683083i	$0.997826\pi$
$602$		0			0
$603$	40.2094			1.63745
$604$		0			0
$605$	−1.64287			−0.0667921
$606$		0			0
$607$	5.55422			0.225439			0.112719	−	0.993627i	$-0.464044\pi$
$607$	5.55422			0.225439			0.112719	+	0.993627i	$0.464044\pi$
$608$		0			0
$609$	−9.04279	+	5.29851i	−0.366432	+	0.214707i
$610$		0			0
$611$	48.2818			1.95327
$612$		0			0
$613$		−	0.100256i		−	0.00404931i	−0.999998	−	0.00202465i	$-0.999356\pi$
$613$		−	0.100256i		−	0.00404931i	0.999998	−	0.00202465i	$-0.000644468\pi$
$614$		0			0
$615$	8.29651			0.334547
$616$		0			0
$617$	13.6533			0.549660			0.274830	−	0.961493i	$-0.411378\pi$
$617$	13.6533			0.549660			0.274830	+	0.961493i	$0.411378\pi$
$618$		0			0
$619$		−	21.7703i		−	0.875024i	−0.899213	−	0.437512i	$-0.855860\pi$
$619$		−	21.7703i		−	0.875024i	0.899213	−	0.437512i	$-0.144140\pi$
$620$		0			0
$621$	−8.15068			−0.327076
$622$		0			0
$623$	−7.73316	−	13.1979i	−0.309822	−	0.528763i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	26.8199			1.07108
$628$		0			0
$629$	−3.93309			−0.156823
$630$		0			0
$631$			22.1194i			0.880559i	0.897861	+	0.440280i	$0.145121\pi$
$631$			22.1194i			0.880559i	−0.897861	+	0.440280i	$0.854879\pi$
$632$		0			0
$633$			12.7195i			0.505554i
$634$		0			0
$635$			9.27159i			0.367932i
$636$		0			0
$637$	−20.9324	+	37.3550i	−0.829372	+	1.48006i
$638$		0			0
$639$			27.0496i			1.07006i
$640$		0			0
$641$	26.4852			1.04610			0.523050	−	0.852302i	$-0.324794\pi$
$641$	26.4852			1.04610			0.523050	+	0.852302i	$0.324794\pi$
$642$		0			0
$643$		−	39.9374i		−	1.57498i	−0.616330	−	0.787488i	$-0.711381\pi$
$643$		−	39.9374i		−	1.57498i	0.616330	−	0.787488i	$-0.288619\pi$
$644$		0			0
$645$			27.9747i			1.10150i
$646$		0			0
$647$	45.2321			1.77826			0.889128	−	0.457658i	$-0.151312\pi$
$647$	45.2321			1.77826			0.889128	+	0.457658i	$0.151312\pi$
$648$		0			0
$649$		−	44.6079i		−	1.75101i
$650$		0			0
$651$	−32.8750	−	56.1066i	−1.28847	−	2.19899i
$652$		0			0
$653$			9.97322i			0.390282i	0.980775	+	0.195141i	$0.0625165\pi$
$653$			9.97322i			0.390282i	−0.980775	+	0.195141i	$0.937484\pi$
$654$		0			0
$655$		−	9.14476i		−	0.357315i
$656$		0			0
$657$		−	2.77657i		−	0.108324i
$658$		0			0
$659$	−33.7753			−1.31570			−0.657850	−	0.753149i	$-0.728534\pi$
$659$	−33.7753			−1.31570			−0.657850	+	0.753149i	$0.728534\pi$
$660$		0			0
$661$	−18.0667			−0.702715			−0.351357	−	0.936241i	$-0.614280\pi$
$661$	−18.0667			−0.702715			−0.351357	+	0.936241i	$0.614280\pi$
$662$		0			0
$663$	−19.9698			−0.775564
$664$		0			0
$665$	−4.51718	−	7.70932i	−0.175169	−	0.298955i
$666$		0			0
$667$	−6.47400			−0.250674
$668$		0			0
$669$		−	8.01727i		−	0.309965i
$670$		0			0
$671$	−0.839176			−0.0323960
$672$		0			0
$673$	−37.7732			−1.45605			−0.728024	−	0.685551i	$-0.759561\pi$
$673$	−37.7732			−1.45605			−0.728024	+	0.685551i	$0.759561\pi$
$674$		0			0
$675$		−	1.92104i		−	0.0739409i
$676$		0			0
$677$	36.0391			1.38510			0.692548	−	0.721371i	$-0.256488\pi$
$677$	36.0391			1.38510			0.692548	+	0.721371i	$0.256488\pi$
$678$		0			0
$679$	19.1317	+	32.6515i	0.734209	+	1.25305i
$680$		0			0
$681$	−5.31553			−0.203692
$682$		0			0
$683$	11.2784			0.431554			0.215777	−	0.976443i	$-0.430772\pi$
$683$	11.2784			0.431554			0.215777	+	0.976443i	$0.430772\pi$
$684$		0			0
$685$	−11.7481			−0.448871
$686$		0			0
$687$		−	23.5445i		−	0.898280i
$688$		0			0
$689$			77.0463i			2.93523i
$690$		0			0
$691$		−	21.4353i		−	0.815438i	−0.913107	−	0.407719i	$-0.866324\pi$
$691$		−	21.4353i		−	0.815438i	0.913107	−	0.407719i	$-0.133676\pi$
$692$		0			0
$693$	−26.1154	+	15.3020i	−0.992043	+	0.581276i
$694$		0			0
$695$		−	11.2894i		−	0.428230i
$696$		0			0
$697$	−4.01848			−0.152211
$698$		0			0
$699$		−	33.3088i		−	1.25986i
$700$		0			0
$701$			1.09861i			0.0414940i	0.999785	+	0.0207470i	$0.00660446\pi$
$701$			1.09861i			0.0414940i	−0.999785	+	0.0207470i	$0.993396\pi$
$702$		0			0
$703$	10.5632			0.398399
$704$		0			0
$705$		−	20.4910i		−	0.771734i
$706$		0			0
$707$	5.28373	−	3.09594i	0.198715	−	0.116435i
$708$		0			0
$709$		−	20.7287i		−	0.778481i	−0.921136	−	0.389240i	$-0.872738\pi$
$709$		−	20.7287i		−	0.778481i	0.921136	−	0.389240i	$-0.127262\pi$
$710$		0			0
$711$			37.8797i			1.42060i
$712$		0			0
$713$		−	40.1684i		−	1.50432i
$714$		0			0
$715$	18.7121			0.699791
$716$		0			0
$717$	37.6394			1.40567
$718$		0			0
$719$	−26.5825			−0.991361			−0.495680	−	0.868505i	$-0.665081\pi$
$719$	−26.5825			−0.991361			−0.495680	+	0.868505i	$0.665081\pi$
$720$		0			0
$721$	31.5806	−	18.5043i	1.17612	−	0.689134i
$722$		0			0
$723$	−1.64183			−0.0610602
$724$		0			0
$725$		−	1.52586i		−	0.0566691i
$726$		0			0
$727$	2.08151			0.0771989			0.0385995	−	0.999255i	$-0.487710\pi$
$727$	2.08151			0.0771989			0.0385995	+	0.999255i	$0.487710\pi$
$728$		0			0
$729$	38.2715			1.41746
$730$		0			0
$731$		−	13.5498i		−	0.501157i
$732$		0			0
$733$	−2.11229			−0.0780191			−0.0390096	−	0.999239i	$-0.512420\pi$
$733$	−2.11229			−0.0780191			−0.0390096	+	0.999239i	$0.512420\pi$
$734$		0			0
$735$	15.8536	+	8.88378i	0.584769	+	0.327683i
$736$		0			0
$737$	32.8876			1.21143
$738$		0			0
$739$	13.6264			0.501255			0.250628	−	0.968084i	$-0.419363\pi$
$739$	13.6264			0.501255			0.250628	+	0.968084i	$0.419363\pi$
$740$		0			0
$741$	53.6335			1.97028
$742$		0			0
$743$			42.8070i			1.57044i	0.619219	+	0.785218i	$0.287449\pi$
$743$			42.8070i			1.57044i	−0.619219	+	0.785218i	$0.712551\pi$
$744$		0			0
$745$			18.6432i			0.683034i
$746$		0			0
$747$		−	23.0195i		−	0.842241i
$748$		0			0
$749$	31.1064	−	18.2264i	1.13660	−	0.665979i
$750$		0			0
$751$			25.3283i			0.924244i	0.886816	+	0.462122i	$0.152912\pi$
$751$			25.3283i			0.924244i	−0.886816	+	0.462122i	$0.847088\pi$
$752$		0			0
$753$	25.3684			0.924477
$754$		0			0
$755$		−	3.29824i		−	0.120035i
$756$		0			0
$757$			37.2852i			1.35516i	0.735451	+	0.677578i	$0.236970\pi$
$757$			37.2852i			1.35516i	−0.735451	+	0.677578i	$0.763030\pi$
$758$		0			0
$759$	−33.6944			−1.22303
$760$		0			0
$761$			28.5390i			1.03454i	0.855823	+	0.517268i	$0.173051\pi$
$761$			28.5390i			1.03454i	−0.855823	+	0.517268i	$0.826949\pi$
$762$		0			0
$763$	−9.71153	−	16.5743i	−0.351581	−	0.600031i
$764$		0			0
$765$			4.70286i			0.170032i
$766$		0			0
$767$		−	89.2053i		−	3.22102i
$768$		0			0
$769$			25.7803i			0.929660i	0.885400	+	0.464830i	$0.153885\pi$
$769$			25.7803i			0.929660i	−0.885400	+	0.464830i	$0.846115\pi$
$770$		0			0
$771$	8.59421			0.309513
$772$		0			0
$773$	−29.8642			−1.07414			−0.537070	−	0.843538i	$-0.680469\pi$
$773$	−29.8642			−1.07414			−0.537070	+	0.843538i	$0.680469\pi$
$774$		0			0
$775$	9.46731			0.340076
$776$		0			0
$777$	−18.5364	+	10.8612i	−0.664990	+	0.389643i
$778$		0			0
$779$	10.7925			0.386683
$780$		0			0
$781$			22.1241i			0.791661i
$782$		0			0
$783$	−2.93124			−0.104754
$784$		0			0
$785$	−0.308165			−0.0109989
$786$		0			0
$787$		−	0.839063i		−	0.0299094i	−0.999888	−	0.0149547i	$-0.995240\pi$
$787$		−	0.839063i		−	0.0299094i	0.999888	−	0.0149547i	$-0.00476040\pi$
$788$		0			0
$789$	73.2928			2.60929
$790$		0			0
$791$	20.6536	−	12.1017i	0.734357	−	0.430287i
$792$		0			0
$793$	−1.67815			−0.0595930
$794$		0			0
$795$	32.6987			1.15970
$796$		0			0
$797$	2.39992			0.0850096			0.0425048	−	0.999096i	$-0.486466\pi$
$797$	2.39992			0.0850096			0.0425048	+	0.999096i	$0.486466\pi$
$798$		0			0
$799$			9.92497i			0.351120i
$800$		0			0
$801$		−	21.6229i		−	0.764007i
$802$		0			0
$803$		−	2.27098i		−	0.0801412i
$804$		0			0
$805$	5.67503	+	9.68537i	0.200018	+	0.341364i
$806$		0			0
$807$		−	72.8053i		−	2.56287i
$808$		0			0
$809$	−31.4961			−1.10734			−0.553672	−	0.832735i	$-0.686774\pi$
$809$	−31.4961			−1.10734			−0.553672	+	0.832735i	$0.686774\pi$
$810$		0			0
$811$			8.89901i			0.312486i	0.987719	+	0.156243i	$0.0499384\pi$
$811$			8.89901i			0.312486i	−0.987719	+	0.156243i	$0.950062\pi$
$812$		0			0
$813$		−	22.8274i		−	0.800592i
$814$		0			0
$815$	−12.8164			−0.448939
$816$		0			0
$817$			36.3910i			1.27316i
$818$		0			0
$819$	−52.2247	+	30.6004i	−1.82488	+	1.06927i
$820$		0			0
$821$		−	24.8802i		−	0.868326i	−0.900834	−	0.434163i	$-0.857044\pi$
$821$		−	24.8802i		−	0.868326i	0.900834	−	0.434163i	$-0.142956\pi$
$822$		0			0
$823$			38.0624i			1.32677i	0.748277	+	0.663386i	$0.230881\pi$
$823$			38.0624i			1.32677i	−0.748277	+	0.663386i	$0.769119\pi$
$824$		0			0
$825$		−	7.94145i		−	0.276486i
$826$		0			0
$827$	28.3847			0.987031			0.493516	−	0.869737i	$-0.335712\pi$
$827$	28.3847			0.987031			0.493516	+	0.869737i	$0.335712\pi$
$828$		0			0
$829$	−31.3879			−1.09015			−0.545074	−	0.838388i	$-0.683498\pi$
$829$	−31.3879			−1.09015			−0.545074	+	0.838388i	$0.683498\pi$
$830$		0			0
$831$	−47.7664			−1.65700
$832$		0			0
$833$	−7.67882	−	4.30293i	−0.266056	−	0.149088i
$834$		0			0
$835$	−8.34267			−0.288710
$836$		0			0
$837$		−	18.1871i		−	0.628638i
$838$		0			0
$839$	10.4588			0.361079			0.180539	−	0.983568i	$-0.442216\pi$
$839$	10.4588			0.361079			0.180539	+	0.983568i	$0.442216\pi$
$840$		0			0
$841$	26.6717			0.919715
$842$		0			0
$843$			80.8619i			2.78503i
$844$		0			0
$845$	24.4197			0.840063
$846$		0			0
$847$	3.75026	−	2.19742i	0.128861	−	0.0755043i
$848$		0			0
$849$	−77.4592			−2.65839
$850$		0			0
$851$	−13.2708			−0.454916
$852$		0			0
$853$	17.0347			0.583256			0.291628	−	0.956532i	$-0.405803\pi$
$853$	17.0347			0.583256			0.291628	+	0.956532i	$0.405803\pi$
$854$		0			0
$855$		−	12.6306i		−	0.431958i
$856$		0			0
$857$			43.2852i			1.47859i	0.673379	+	0.739297i	$0.264842\pi$
$857$			43.2852i			1.47859i	−0.673379	+	0.739297i	$0.735158\pi$
$858$		0			0
$859$		−	23.0541i		−	0.786597i	−0.919411	−	0.393298i	$-0.871334\pi$
$859$		−	23.0541i		−	0.786597i	0.919411	−	0.393298i	$-0.128666\pi$
$860$		0			0
$861$	−18.9389	+	11.0970i	−0.645435	+	0.378185i
$862$		0			0
$863$			24.3924i			0.830327i	0.909747	+	0.415163i	$0.136276\pi$
$863$			24.3924i			0.830327i	−0.909747	+	0.415163i	$0.863724\pi$
$864$		0			0
$865$	0.924877			0.0314468
$866$		0			0
$867$			40.0294i			1.35947i
$868$		0			0
$869$			30.9821i			1.05100i
$870$		0			0
$871$	65.7675			2.22845
$872$		0			0
$873$			53.4947i			1.81052i
$874$		0			0
$875$	−2.28275	+	1.33755i	−0.0771711	+	0.0452175i
$876$		0			0
$877$		−	57.3223i		−	1.93564i	−0.251649	−	0.967819i	$-0.580973\pi$
$877$		−	57.3223i		−	1.93564i	0.251649	−	0.967819i	$-0.419027\pi$
$878$		0			0
$879$			42.9793i			1.44965i
$880$		0			0
$881$		−	44.0539i		−	1.48421i	−0.670282	−	0.742107i	$-0.733827\pi$
$881$		−	44.0539i		−	1.48421i	0.670282	−	0.742107i	$-0.266173\pi$
$882$		0			0
$883$	−44.5580			−1.49950			−0.749749	−	0.661723i	$-0.769826\pi$
$883$	−44.5580			−1.49950			−0.749749	+	0.661723i	$0.769826\pi$
$884$		0			0
$885$	−37.8590			−1.27262
$886$		0			0
$887$	2.89650			0.0972550			0.0486275	−	0.998817i	$-0.484515\pi$
$887$	2.89650			0.0972550			0.0486275	+	0.998817i	$0.484515\pi$
$888$		0			0
$889$	−12.4012	−	21.1647i	−0.415924	−	0.709843i
$890$		0			0
$891$	19.0651			0.638705
$892$		0			0
$893$		−	26.6557i		−	0.892000i
$894$		0			0
$895$	2.99825			0.100221
$896$		0			0
$897$	−67.3808			−2.24978
$898$		0			0
$899$		−	14.4458i		−	0.481795i
$900$		0			0
$901$	−15.8379			−0.527637
$902$		0			0
$903$	−37.4176	−	63.8593i	−1.24518	−	2.12510i
$904$		0			0
$905$	−1.52021			−0.0505336
$906$		0			0
$907$	−26.2835			−0.872728			−0.436364	−	0.899770i	$-0.643734\pi$
$907$	−26.2835			−0.872728			−0.436364	+	0.899770i	$0.643734\pi$
$908$		0			0
$909$	8.65663			0.287122
$910$		0			0
$911$		−	33.5911i		−	1.11292i	−0.830873	−	0.556462i	$-0.812158\pi$
$911$		−	33.5911i		−	1.11292i	0.830873	−	0.556462i	$-0.187842\pi$
$912$		0			0
$913$		−	18.8279i		−	0.623111i
$914$		0			0
$915$			0.712214i			0.0235451i
$916$		0			0
$917$	12.2316	+	20.8752i	0.403922	+	0.689360i
$918$		0			0
$919$			30.6535i			1.01117i	0.862778	+	0.505583i	$0.168723\pi$
$919$			30.6535i			1.01117i	−0.862778	+	0.505583i	$0.831277\pi$
$920$		0			0
$921$	−58.3819			−1.92375
$922$		0			0
$923$			44.2429i			1.45627i
$924$		0			0
$925$		−	3.12779i		−	0.102841i
$926$		0			0
$927$	51.7402			1.69937
$928$		0			0
$929$		−	59.1269i		−	1.93989i	−0.243322	−	0.969946i	$-0.578237\pi$
$929$		−	59.1269i		−	1.93989i	0.243322	−	0.969946i	$-0.421763\pi$
$930$		0			0
$931$	20.6232	+	11.5565i	0.675899	+	0.378749i
$932$		0			0
$933$			7.85992i			0.257322i
$934$		0			0
$935$			3.84651i			0.125794i
$936$		0			0
$937$		−	16.3377i		−	0.533730i	−0.963734	−	0.266865i	$-0.914012\pi$
$937$		−	16.3377i		−	0.533730i	0.963734	−	0.266865i	$-0.0859878\pi$
$938$		0			0
$939$	−59.8866			−1.95432
$940$		0			0
$941$	24.6773			0.804457			0.402228	−	0.915539i	$-0.368236\pi$
$941$	24.6773			0.804457			0.402228	+	0.915539i	$0.368236\pi$
$942$		0			0
$943$	−13.5589			−0.441538
$944$		0			0
$945$	2.56949	+	4.38526i	0.0835855	+	0.142652i
$946$		0			0
$947$	28.5781			0.928664			0.464332	−	0.885661i	$-0.346294\pi$
$947$	28.5781			0.928664			0.464332	+	0.885661i	$0.346294\pi$
$948$		0			0
$949$		−	4.54143i		−	0.147421i
$950$		0			0
$951$	−28.2485			−0.916020
$952$		0			0
$953$	−20.9275			−0.677908			−0.338954	−	0.940803i	$-0.610073\pi$
$953$	−20.9275			−0.677908			−0.338954	+	0.940803i	$0.610073\pi$
$954$		0			0
$955$		−	4.77816i		−	0.154618i
$956$		0			0
$957$	−12.1176			−0.391705
$958$		0			0
$959$	26.8179	−	15.7137i	0.865996	−	0.507420i
$960$		0			0
$961$	58.6300			1.89129
$962$		0			0
$963$	50.9634			1.64227
$964$		0			0
$965$	14.3979			0.463484
$966$		0			0
$967$		−	21.8079i		−	0.701295i	−0.936508	−	0.350647i	$-0.885962\pi$
$967$		−	21.8079i		−	0.701295i	0.936508	−	0.350647i	$-0.114038\pi$
$968$		0			0
$969$			11.0251i			0.354176i
$970$		0			0
$971$		−	32.1900i		−	1.03303i	−0.856279	−	0.516513i	$-0.827230\pi$
$971$		−	32.1900i		−	1.03303i	0.856279	−	0.516513i	$-0.172770\pi$
$972$		0			0
$973$	15.1001	+	25.7708i	0.484088	+	0.826175i
$974$		0			0
$975$		−	15.8810i		−	0.508600i
$976$		0			0
$977$	19.4242			0.621435			0.310717	−	0.950502i	$-0.399431\pi$
$977$	19.4242			0.621435			0.310717	+	0.950502i	$0.399431\pi$
$978$		0			0
$979$		−	17.6855i		−	0.565232i
$980$		0			0
$981$		−	27.1546i		−	0.866981i
$982$		0			0
$983$	57.5903			1.83685			0.918423	−	0.395599i	$-0.129463\pi$
$983$	57.5903			1.83685			0.918423	+	0.395599i	$0.129463\pi$
$984$		0			0
$985$		−	20.8023i		−	0.662816i
$986$		0			0
$987$	27.4077	+	46.7758i	0.872397	+	1.48889i
$988$		0			0
$989$		−	45.7187i		−	1.45377i
$990$		0			0
$991$		−	61.5330i		−	1.95466i	−0.211725	−	0.977329i	$-0.567908\pi$
$991$		−	61.5330i		−	1.95466i	0.211725	−	0.977329i	$-0.432092\pi$
$992$		0			0
$993$		−	86.4653i		−	2.74389i
$994$		0			0
$995$	−4.94052			−0.156625
$996$		0			0
$997$	−29.2889			−0.927589			−0.463795	−	0.885943i	$-0.653512\pi$
$997$	−29.2889			−0.927589			−0.463795	+	0.885943i	$0.653512\pi$
$998$		0			0
$999$	−6.00862			−0.190104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.f.671.6	yes	24
4.3	odd	2	inner	2240.2.h.f.671.19	yes	24
7.6	odd	2		2240.2.h.e.671.19	yes	24
8.3	odd	2		2240.2.h.e.671.20	yes	24
8.5	even	2		2240.2.h.e.671.5	✓	24
28.27	even	2		2240.2.h.e.671.6	yes	24
56.13	odd	2	inner	2240.2.h.f.671.20	yes	24
56.27	even	2	inner	2240.2.h.f.671.5	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.e.671.5	✓	24	8.5	even	2
2240.2.h.e.671.6	yes	24	28.27	even	2
2240.2.h.e.671.19	yes	24	7.6	odd	2
2240.2.h.e.671.20	yes	24	8.3	odd	2
2240.2.h.f.671.5	yes	24	56.27	even	2	inner
2240.2.h.f.671.6	yes	24	1.1	even	1	trivial
2240.2.h.f.671.19	yes	24	4.3	odd	2	inner
2240.2.h.f.671.20	yes	24	56.13	odd	2	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.59614i q^{3} +1.00000 q^{5} +(-2.28275 + 1.33755i) q^{7} -3.73996 q^{9} +O(q^{10})\)	\(q+2.59614i q^{3} +1.00000 q^{5} +(-2.28275 + 1.33755i) q^{7} -3.73996 q^{9} -3.05894 q^{11} -6.11716 q^{13} +2.59614i q^{15} -1.25746i q^{17} +3.37720i q^{19} +(-3.47247 - 5.92635i) q^{21} -4.24285i q^{23} +1.00000 q^{25} -1.92104i q^{27} -1.52586i q^{29} +9.46731 q^{31} -7.94145i q^{33} +(-2.28275 + 1.33755i) q^{35} -3.12779i q^{37} -15.8810i q^{39} -3.19570i q^{41} +10.7755 q^{43} -3.73996 q^{45} -7.89284 q^{47} +(3.42191 - 6.10660i) q^{49} +3.26456 q^{51} -12.5951i q^{53} -3.05894 q^{55} -8.76770 q^{57} +14.5828i q^{59} +0.274335 q^{61} +(8.53740 - 5.00239i) q^{63} -6.11716 q^{65} -10.7513 q^{67} +11.0150 q^{69} -7.23258i q^{71} +0.742408i q^{73} +2.59614i q^{75} +(6.98281 - 4.09149i) q^{77} -10.1284i q^{79} -6.23258 q^{81} +6.15502i q^{83} -1.25746i q^{85} +3.96135 q^{87} +5.78158i q^{89} +(13.9640 - 8.18202i) q^{91} +24.5785i q^{93} +3.37720i q^{95} -14.3036i q^{97} +11.4403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{5} - 36 q^{9}+O(q^{10}) \) 24 * q + 24 * q^5 - 36 * q^9	\( 24 q + 24 q^{5} - 36 q^{9} + 4 q^{13} + 8 q^{21} + 24 q^{25} - 36 q^{45} + 24 q^{57} + 56 q^{61} + 4 q^{65} + 96 q^{69} - 12 q^{77} + 24 q^{81}+O(q^{100}) \) 24 * q + 24 * q^5 - 36 * q^9 + 4 * q^13 + 8 * q^21 + 24 * q^25 - 36 * q^45 + 24 * q^57 + 56 * q^61 + 4 * q^65 + 96 * q^69 - 12 * q^77 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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