LMFDB - Embedded newform 2912.2.k.g.1793.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1793,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.1793"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,44,0,0,0,-4,0,0,0,8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6179217664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13x^{6} + 40x^{4} + 13x^{2} + 1 $ x^8 + 13x^6 + 40x^4 + 13*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.6
Root			$-2.90570i$ of defining polynomial
Character	$\chi$	$=$	2912.1793
Dual form			2912.2.k.g.1793.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53741 q^{3} +3.24985i q^{5} +1.00000i q^{7} +3.43845 q^{9} +5.56155i q^{11} +(1.56155 + 3.24985i) q^{13} +8.24621i q^{15} -3.12311 q^{17} -6.56155i q^{19} +2.53741i q^{21} +0.712445 q^{23} -5.56155 q^{25} +1.11252 q^{27} +5.43845 q^{29} -1.00000i q^{31} +14.1119i q^{33} -3.24985 q^{35} -3.96230i q^{37} +(3.96230 + 8.24621i) q^{39} -2.53741i q^{41} -8.32467 q^{43} +11.1745i q^{45} +12.3693i q^{47} -1.00000 q^{49} -7.92460 q^{51} +8.56155 q^{53} -18.0742 q^{55} -16.6493i q^{57} -15.1231i q^{59} -0.438447 q^{61} +3.43845i q^{63} +(-10.5616 + 5.07482i) q^{65} +0.684658i q^{67} +1.80776 q^{69} -6.24621i q^{71} +4.36237i q^{73} -14.1119 q^{75} -5.56155 q^{77} -5.78726 q^{79} -7.49242 q^{81} +3.43845i q^{83} -10.1496i q^{85} +13.7996 q^{87} +3.24985i q^{89} +(-3.24985 + 1.56155i) q^{91} -2.53741i q^{93} +21.3241 q^{95} +10.8621i q^{97} +19.1231i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{9} - 4 q^{13} + 8 q^{17} - 28 q^{25} + 60 q^{29} - 8 q^{49} + 52 q^{53} - 20 q^{61} - 68 q^{65} - 68 q^{69} - 28 q^{77} + 72 q^{81}+O(q^{100}) $ 8 * q + 44 * q^9 - 4 * q^13 + 8 * q^17 - 28 * q^25 + 60 * q^29 - 8 * q^49 + 52 * q^53 - 20 * q^61 - 68 * q^65 - 68 * q^69 - 28 * q^77 + 72 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	2.53741			1.46497			0.732487	−	0.680781i	$-0.238359\pi$
$3$	2.53741			1.46497			0.732487	+	0.680781i	$0.238359\pi$
$4$		0			0
$5$			3.24985i			1.45338i	0.686966	+	0.726690i	$0.258942\pi$
$5$			3.24985i			1.45338i	−0.686966	+	0.726690i	$0.741058\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	3.43845			1.14615
$10$		0			0
$11$			5.56155i			1.67687i	0.545001	+	0.838436i	$0.316529\pi$
$11$			5.56155i			1.67687i	−0.545001	+	0.838436i	$0.683471\pi$
$12$		0			0
$13$	1.56155	+	3.24985i	0.433097	+	0.901347i
$13$	1.56155	+	3.24985i	0.433097	+	0.901347i
$14$		0			0
$15$			8.24621i			2.12916i
$16$		0			0
$17$	−3.12311			−0.757464			−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311			−0.757464			−0.378732	+	0.925506i	$0.623640\pi$
$18$		0			0
$19$		−	6.56155i		−	1.50532i	−0.658407	−	0.752662i	$-0.728770\pi$
$19$		−	6.56155i		−	1.50532i	0.658407	−	0.752662i	$-0.271230\pi$
$20$		0			0
$21$			2.53741i			0.553708i
$22$		0			0
$23$	0.712445			0.148555			0.0742775	−	0.997238i	$-0.476335\pi$
$23$	0.712445			0.148555			0.0742775	+	0.997238i	$0.476335\pi$
$24$		0			0
$25$	−5.56155			−1.11231
$26$		0			0
$27$	1.11252			0.214105
$28$		0			0
$29$	5.43845			1.00989			0.504947	−	0.863150i	$-0.331512\pi$
$29$	5.43845			1.00989			0.504947	+	0.863150i	$0.331512\pi$
$30$		0			0
$31$		−	1.00000i		−	0.179605i	−0.995960	−	0.0898027i	$-0.971376\pi$
$31$		−	1.00000i		−	0.179605i	0.995960	−	0.0898027i	$-0.0286236\pi$
$32$		0			0
$33$			14.1119i			2.45657i
$34$		0			0
$35$	−3.24985			−0.549326
$36$		0			0
$37$		−	3.96230i		−	0.651398i	−0.945474	−	0.325699i	$-0.894400\pi$
$37$		−	3.96230i		−	0.651398i	0.945474	−	0.325699i	$-0.105600\pi$
$38$		0			0
$39$	3.96230	+	8.24621i	0.634476	+	1.32045i
$40$		0			0
$41$		−	2.53741i		−	0.396277i	−0.980174	−	0.198138i	$-0.936510\pi$
$41$		−	2.53741i		−	0.396277i	0.980174	−	0.198138i	$-0.0634895\pi$
$42$		0			0
$43$	−8.32467			−1.26950			−0.634750	−	0.772717i	$-0.718897\pi$
$43$	−8.32467			−1.26950			−0.634750	+	0.772717i	$0.718897\pi$
$44$		0			0
$45$			11.1745i			1.66579i
$46$		0			0
$47$			12.3693i			1.80425i	0.431474	+	0.902125i	$0.357994\pi$
$47$			12.3693i			1.80425i	−0.431474	+	0.902125i	$0.642006\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.92460			−1.10967
$52$		0			0
$53$	8.56155			1.17602			0.588010	−	0.808854i	$-0.299912\pi$
$53$	8.56155			1.17602			0.588010	+	0.808854i	$0.299912\pi$
$54$		0			0
$55$	−18.0742			−2.43713
$56$		0			0
$57$		−	16.6493i		−	2.20526i
$58$		0			0
$59$		−	15.1231i		−	1.96886i	−0.175775	−	0.984430i	$-0.556243\pi$
$59$		−	15.1231i		−	1.96886i	0.175775	−	0.984430i	$-0.443757\pi$
$60$		0			0
$61$	−0.438447			−0.0561374			−0.0280687	−	0.999606i	$-0.508936\pi$
$61$	−0.438447			−0.0561374			−0.0280687	+	0.999606i	$0.508936\pi$
$62$		0			0
$63$			3.43845i			0.433204i
$64$		0			0
$65$	−10.5616	+	5.07482i	−1.31000	+	0.629454i
$66$		0			0
$67$			0.684658i			0.0836443i	0.999125	+	0.0418222i	$0.0133163\pi$
$67$			0.684658i			0.0836443i	−0.999125	+	0.0418222i	$0.986684\pi$
$68$		0			0
$69$	1.80776			0.217629
$70$		0			0
$71$		−	6.24621i		−	0.741289i	−0.928775	−	0.370644i	$-0.879137\pi$
$71$		−	6.24621i		−	0.741289i	0.928775	−	0.370644i	$-0.120863\pi$
$72$		0			0
$73$			4.36237i			0.510577i	0.966865	+	0.255289i	$0.0821705\pi$
$73$			4.36237i			0.510577i	−0.966865	+	0.255289i	$0.917830\pi$
$74$		0			0
$75$	−14.1119			−1.62951
$76$		0			0
$77$	−5.56155			−0.633798
$78$		0			0
$79$	−5.78726			−0.651118			−0.325559	−	0.945522i	$-0.605552\pi$
$79$	−5.78726			−0.651118			−0.325559	+	0.945522i	$0.605552\pi$
$80$		0			0
$81$	−7.49242			−0.832491
$82$		0			0
$83$			3.43845i			0.377419i	0.982033	+	0.188709i	$0.0604304\pi$
$83$			3.43845i			0.377419i	−0.982033	+	0.188709i	$0.939570\pi$
$84$		0			0
$85$		−	10.1496i		−	1.10088i
$86$		0			0
$87$	13.7996			1.47947
$88$		0			0
$89$			3.24985i			0.344484i	0.985055	+	0.172242i	$0.0551011\pi$
$89$			3.24985i			0.344484i	−0.985055	+	0.172242i	$0.944899\pi$
$90$		0			0
$91$	−3.24985	+	1.56155i	−0.340677	+	0.163695i
$92$		0			0
$93$		−	2.53741i		−	0.263117i
$94$		0			0
$95$	21.3241			2.18781
$96$		0			0
$97$			10.8621i			1.10288i	0.834215	+	0.551439i	$0.185921\pi$
$97$			10.8621i			1.10288i	−0.834215	+	0.551439i	$0.814079\pi$
$98$		0			0
$99$			19.1231i			1.92194i
$100$		0			0
$101$	−8.93087			−0.888655			−0.444327	−	0.895864i	$-0.646557\pi$
$101$	−8.93087			−0.888655			−0.444327	+	0.895864i	$0.646557\pi$
$102$		0			0
$103$	6.49971			0.640435			0.320218	−	0.947344i	$-0.396244\pi$
$103$	6.49971			0.640435			0.320218	+	0.947344i	$0.396244\pi$
$104$		0			0
$105$	−8.24621			−0.804748
$106$		0			0
$107$	18.0742			1.74730			0.873651	−	0.486553i	$-0.161746\pi$
$107$	18.0742			1.74730			0.873651	+	0.486553i	$0.161746\pi$
$108$		0			0
$109$			11.5745i			1.10864i	0.832304	+	0.554319i	$0.187021\pi$
$109$			11.5745i			1.10864i	−0.832304	+	0.554319i	$0.812979\pi$
$110$		0			0
$111$		−	10.0540i		−	0.954281i
$112$		0			0
$113$	17.4924			1.64555			0.822774	−	0.568368i	$-0.192425\pi$
$113$	17.4924			1.64555			0.822774	+	0.568368i	$0.192425\pi$
$114$		0			0
$115$			2.31534i			0.215907i
$116$		0			0
$117$	5.36932	+	11.1745i	0.496394	+	1.03308i
$118$		0			0
$119$		−	3.12311i		−	0.286295i
$120$		0			0
$121$	−19.9309			−1.81190
$122$		0			0
$123$		−	6.43845i		−	0.580535i
$124$		0			0
$125$		−	1.82496i		−	0.163230i
$126$		0			0
$127$	−1.11252			−0.0987202			−0.0493601	−	0.998781i	$-0.515718\pi$
$127$	−1.11252			−0.0987202			−0.0493601	+	0.998781i	$0.515718\pi$
$128$		0			0
$129$	−21.1231			−1.85979
$130$		0			0
$131$	−3.64993			−0.318896			−0.159448	−	0.987206i	$-0.550971\pi$
$131$	−3.64993			−0.318896			−0.159448	+	0.987206i	$0.550971\pi$
$132$		0			0
$133$	6.56155			0.568959
$134$		0			0
$135$			3.61553i			0.311175i
$136$		0			0
$137$		−	19.4991i		−	1.66592i	−0.553331	−	0.832961i	$-0.686644\pi$
$137$		−	19.4991i		−	1.66592i	0.553331	−	0.832961i	$-0.313356\pi$
$138$		0			0
$139$	23.1491			1.96348			0.981739	−	0.190235i	$-0.0609249\pi$
$139$	23.1491			1.96348			0.981739	+	0.190235i	$0.0609249\pi$
$140$		0			0
$141$			31.3860i			2.64318i
$142$		0			0
$143$	−18.0742	+	8.68466i	−1.51144	+	0.726248i
$144$		0			0
$145$			17.6742i			1.46776i
$146$		0			0
$147$	−2.53741			−0.209282
$148$		0			0
$149$			14.1119i			1.15609i	0.816003	+	0.578047i	$0.196185\pi$
$149$			14.1119i			1.15609i	−0.816003	+	0.578047i	$0.803815\pi$
$150$		0			0
$151$			4.87689i			0.396876i	0.980113	+	0.198438i	$0.0635869\pi$
$151$			4.87689i			0.396876i	−0.980113	+	0.198438i	$0.936413\pi$
$152$		0			0
$153$	−10.7386			−0.868167
$154$		0			0
$155$	3.24985			0.261035
$156$		0			0
$157$	1.31534			0.104976			0.0524878	−	0.998622i	$-0.483285\pi$
$157$	1.31534			0.104976			0.0524878	+	0.998622i	$0.483285\pi$
$158$		0			0
$159$	21.7242			1.72284
$160$		0			0
$161$			0.712445i			0.0561485i
$162$		0			0
$163$		−	19.3693i		−	1.51712i	−0.651602	−	0.758561i	$-0.725903\pi$
$163$		−	19.3693i		−	1.51712i	0.651602	−	0.758561i	$-0.274097\pi$
$164$		0			0
$165$	−45.8617			−3.57033
$166$		0			0
$167$		−	9.43845i		−	0.730369i	−0.930935	−	0.365184i	$-0.881006\pi$
$167$		−	9.43845i		−	0.730369i	0.930935	−	0.365184i	$-0.118994\pi$
$168$		0			0
$169$	−8.12311	+	10.1496i	−0.624854	+	0.780741i
$170$		0			0
$171$		−	22.5616i		−	1.72533i
$172$		0			0
$173$	17.3693			1.32056			0.660282	−	0.751017i	$-0.270437\pi$
$173$	17.3693			1.32056			0.660282	+	0.751017i	$0.270437\pi$
$174$		0			0
$175$		−	5.56155i		−	0.420414i
$176$		0			0
$177$		−	38.3735i		−	2.88433i
$178$		0			0
$179$	−8.32467			−0.622215			−0.311108	−	0.950375i	$-0.600700\pi$
$179$	−8.32467			−0.622215			−0.311108	+	0.950375i	$0.600700\pi$
$180$		0			0
$181$	−15.8078			−1.17498			−0.587491	−	0.809231i	$-0.699884\pi$
$181$	−15.8078			−1.17498			−0.587491	+	0.809231i	$0.699884\pi$
$182$		0			0
$183$	−1.11252			−0.0822399
$184$		0			0
$185$	12.8769			0.946728
$186$		0			0
$187$		−	17.3693i		−	1.27017i
$188$		0			0
$189$			1.11252i			0.0809239i
$190$		0			0
$191$	2.84978			0.206203			0.103101	−	0.994671i	$-0.467123\pi$
$191$	2.84978			0.206203			0.103101	+	0.994671i	$0.467123\pi$
$192$		0			0
$193$			25.9988i			1.87144i	0.352748	+	0.935719i	$0.385247\pi$
$193$			25.9988i			1.87144i	−0.352748	+	0.935719i	$0.614753\pi$
$194$		0			0
$195$	−26.7990	+	12.8769i	−1.91912	+	0.922134i
$196$		0			0
$197$			24.2616i			1.72857i	0.503006	+	0.864283i	$0.332227\pi$
$197$			24.2616i			1.72857i	−0.503006	+	0.864283i	$0.667773\pi$
$198$		0			0
$199$	12.9994			0.921504			0.460752	−	0.887529i	$-0.347580\pi$
$199$	12.9994			0.921504			0.460752	+	0.887529i	$0.347580\pi$
$200$		0			0
$201$			1.73726i			0.122537i
$202$		0			0
$203$			5.43845i			0.381704i
$204$		0			0
$205$	8.24621			0.575940
$206$		0			0
$207$	2.44970			0.170266
$208$		0			0
$209$	36.4924			2.52423
$210$		0			0
$211$	18.4743			1.27182			0.635912	−	0.771762i	$-0.280624\pi$
$211$	18.4743			1.27182			0.635912	+	0.771762i	$0.280624\pi$
$212$		0			0
$213$		−	15.8492i		−	1.08597i
$214$		0			0
$215$		−	27.0540i		−	1.84507i
$216$		0			0
$217$	1.00000			0.0678844
$218$		0			0
$219$			11.0691i			0.747983i
$220$		0			0
$221$	−4.87689	−	10.1496i	−0.328055	−	0.682739i
$222$		0			0
$223$			5.00000i			0.334825i	0.985887	+	0.167412i	$0.0535411\pi$
$223$			5.00000i			0.334825i	−0.985887	+	0.167412i	$0.946459\pi$
$224$		0			0
$225$	−19.1231			−1.27487
$226$		0			0
$227$		−	15.6155i		−	1.03644i	−0.855248	−	0.518220i	$-0.826595\pi$
$227$		−	15.6155i		−	1.03644i	0.855248	−	0.518220i	$-0.173405\pi$
$228$		0			0
$229$		−	16.6493i		−	1.10022i	−0.835092	−	0.550110i	$-0.814586\pi$
$229$		−	16.6493i		−	1.10022i	0.835092	−	0.550110i	$-0.185414\pi$
$230$		0			0
$231$	−14.1119			−0.928497
$232$		0			0
$233$	−7.00000			−0.458585			−0.229293	−	0.973358i	$-0.573641\pi$
$233$	−7.00000			−0.458585			−0.229293	+	0.973358i	$0.573641\pi$
$234$		0			0
$235$	−40.1985			−2.62226
$236$		0			0
$237$	−14.6847			−0.953871
$238$		0			0
$239$			4.87689i			0.315460i	0.987482	+	0.157730i	$0.0504176\pi$
$239$			4.87689i			0.315460i	−0.987482	+	0.157730i	$0.949582\pi$
$240$		0			0
$241$		−	6.89978i		−	0.444454i	−0.974995	−	0.222227i	$-0.928667\pi$
$241$		−	6.89978i		−	0.444454i	0.974995	−	0.222227i	$-0.0713326\pi$
$242$		0			0
$243$	−22.3489			−1.43368
$244$		0			0
$245$		−	3.24985i		−	0.207626i
$246$		0			0
$247$	21.3241	−	10.2462i	1.35682	−	0.651951i
$248$		0			0
$249$			8.72475i			0.552908i
$250$		0			0
$251$	−5.38719			−0.340036			−0.170018	−	0.985441i	$-0.554383\pi$
$251$	−5.38719			−0.340036			−0.170018	+	0.985441i	$0.554383\pi$
$252$		0			0
$253$			3.96230i			0.249108i
$254$		0			0
$255$		−	25.7538i		−	1.61276i
$256$		0			0
$257$	23.6155			1.47310			0.736548	−	0.676385i	$-0.236455\pi$
$257$	23.6155			1.47310			0.736548	+	0.676385i	$0.236455\pi$
$258$		0			0
$259$	3.96230			0.246205
$260$		0			0
$261$	18.6998			1.15749
$262$		0			0
$263$	24.9740			1.53996			0.769982	−	0.638066i	$-0.220265\pi$
$263$	24.9740			1.53996			0.769982	+	0.638066i	$0.220265\pi$
$264$		0			0
$265$			27.8238i			1.70920i
$266$		0			0
$267$			8.24621i			0.504660i
$268$		0			0
$269$	−3.31534			−0.202140			−0.101070	−	0.994879i	$-0.532227\pi$
$269$	−3.31534			−0.202140			−0.101070	+	0.994879i	$0.532227\pi$
$270$		0			0
$271$		−	16.6847i		−	1.01352i	−0.862087	−	0.506760i	$-0.830843\pi$
$271$		−	16.6847i		−	1.01352i	0.862087	−	0.506760i	$-0.169157\pi$
$272$		0			0
$273$	−8.24621	+	3.96230i	−0.499083	+	0.239809i
$274$		0			0
$275$		−	30.9309i		−	1.86520i
$276$		0			0
$277$	−1.68466			−0.101221			−0.0506107	−	0.998718i	$-0.516117\pi$
$277$	−1.68466			−0.101221			−0.0506107	+	0.998718i	$0.516117\pi$
$278$		0			0
$279$		−	3.43845i		−	0.205854i
$280$		0			0
$281$			20.2993i			1.21095i	0.795863	+	0.605477i	$0.207017\pi$
$281$			20.2993i			1.21095i	−0.795863	+	0.605477i	$0.792983\pi$
$282$		0			0
$283$	10.4620			0.621902			0.310951	−	0.950426i	$-0.399353\pi$
$283$	10.4620			0.621902			0.310951	+	0.950426i	$0.399353\pi$
$284$		0			0
$285$	54.1080			3.20508
$286$		0			0
$287$	2.53741			0.149779
$288$		0			0
$289$	−7.24621			−0.426248
$290$		0			0
$291$			27.5616i			1.61569i
$292$		0			0
$293$		−	16.2493i		−	0.949293i	−0.880177	−	0.474646i	$-0.842576\pi$
$293$		−	16.2493i		−	0.949293i	0.880177	−	0.474646i	$-0.157424\pi$
$294$		0			0
$295$	49.1479			2.86150
$296$		0			0
$297$			6.18734i			0.359026i
$298$		0			0
$299$	1.11252	+	2.31534i	0.0643387	+	0.133900i
$300$		0			0
$301$		−	8.32467i		−	0.479826i
$302$		0			0
$303$	−22.6613			−1.30186
$304$		0			0
$305$		−	1.42489i		−	0.0815889i
$306$		0			0
$307$			4.56155i			0.260342i	0.991492	+	0.130171i	$0.0415526\pi$
$307$			4.56155i			0.260342i	−0.991492	+	0.130171i	$0.958447\pi$
$308$		0			0
$309$	16.4924			0.938221
$310$		0			0
$311$	16.6493			0.944098			0.472049	−	0.881572i	$-0.343515\pi$
$311$	16.6493			0.944098			0.472049	+	0.881572i	$0.343515\pi$
$312$		0			0
$313$	1.12311			0.0634817			0.0317408	−	0.999496i	$-0.489895\pi$
$313$	1.12311			0.0634817			0.0317408	+	0.999496i	$0.489895\pi$
$314$		0			0
$315$	−11.1745			−0.629609
$316$		0			0
$317$			11.2622i			0.632546i	0.948668	+	0.316273i	$0.102432\pi$
$317$			11.2622i			0.632546i	−0.948668	+	0.316273i	$0.897568\pi$
$318$		0			0
$319$			30.2462i			1.69346i
$320$		0			0
$321$	45.8617			2.55975
$322$		0			0
$323$			20.4924i			1.14023i
$324$		0			0
$325$	−8.68466	−	18.0742i	−0.481738	−	1.00258i
$326$		0			0
$327$			29.3693i			1.62413i
$328$		0			0
$329$	−12.3693			−0.681943
$330$		0			0
$331$		−	14.6847i		−	0.807142i	−0.914948	−	0.403571i	$-0.867769\pi$
$331$		−	14.6847i		−	0.807142i	0.914948	−	0.403571i	$-0.132231\pi$
$332$		0			0
$333$		−	13.6242i		−	0.746599i
$334$		0			0
$335$	−2.22504			−0.121567
$336$		0			0
$337$	25.7386			1.40207			0.701036	−	0.713126i	$-0.252721\pi$
$337$	25.7386			1.40207			0.701036	+	0.713126i	$0.252721\pi$
$338$		0			0
$339$	44.3854			2.41069
$340$		0			0
$341$	5.56155			0.301175
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			5.87497i			0.316298i
$346$		0			0
$347$	−15.8492			−0.850829			−0.425415	−	0.904999i	$-0.639872\pi$
$347$	−15.8492			−0.850829			−0.425415	+	0.904999i	$0.639872\pi$
$348$		0			0
$349$		−	6.89978i		−	0.369337i	−0.982801	−	0.184668i	$-0.940879\pi$
$349$		−	6.89978i		−	0.369337i	0.982801	−	0.184668i	$-0.0591211\pi$
$350$		0			0
$351$	1.73726	+	3.61553i	0.0927280	+	0.192983i
$352$		0			0
$353$			4.76245i			0.253480i	0.991936	+	0.126740i	$0.0404513\pi$
$353$			4.76245i			0.253480i	−0.991936	+	0.126740i	$0.959549\pi$
$354$		0			0
$355$	20.2993			1.07737
$356$		0			0
$357$		−	7.92460i		−	0.419414i
$358$		0			0
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$		0			0
$361$	−24.0540			−1.26600
$362$		0			0
$363$	−50.5728			−2.65438
$364$		0			0
$365$	−14.1771			−0.742062
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		−	8.72475i		−	0.454192i
$370$		0			0
$371$			8.56155i			0.444494i
$372$		0			0
$373$	−30.4924			−1.57884			−0.789419	−	0.613855i	$-0.789618\pi$
$373$	−30.4924			−1.57884			−0.789419	+	0.613855i	$0.789618\pi$
$374$		0			0
$375$		−	4.63068i		−	0.239127i
$376$		0			0
$377$	8.49242	+	17.6742i	0.437382	+	0.910266i
$378$		0			0
$379$		−	5.12311i		−	0.263156i	−0.991306	−	0.131578i	$-0.957996\pi$
$379$		−	5.12311i		−	0.263156i	0.991306	−	0.131578i	$-0.0420044\pi$
$380$		0			0
$381$	−2.82292			−0.144623
$382$		0			0
$383$			30.4384i			1.55533i	0.628678	+	0.777666i	$0.283597\pi$
$383$			30.4384i			1.55533i	−0.628678	+	0.777666i	$0.716403\pi$
$384$		0			0
$385$		−	18.0742i		−	0.921148i
$386$		0			0
$387$	−28.6239			−1.45504
$388$		0			0
$389$	3.75379			0.190325			0.0951623	−	0.995462i	$-0.469663\pi$
$389$	3.75379			0.190325			0.0951623	+	0.995462i	$0.469663\pi$
$390$		0			0
$391$	−2.22504			−0.112525
$392$		0			0
$393$	−9.26137			−0.467174
$394$		0			0
$395$		−	18.8078i		−	0.946321i
$396$		0			0
$397$		−	19.8992i		−	0.998712i	−0.866397	−	0.499356i	$-0.833570\pi$
$397$		−	19.8992i		−	0.998712i	0.866397	−	0.499356i	$-0.166430\pi$
$398$		0			0
$399$	16.6493			0.833510
$400$		0			0
$401$		−	12.9994i		−	0.649160i	−0.945858	−	0.324580i	$-0.894777\pi$
$401$		−	12.9994i		−	0.649160i	0.945858	−	0.324580i	$-0.105223\pi$
$402$		0			0
$403$	3.24985	−	1.56155i	0.161887	−	0.0777865i
$404$		0			0
$405$		−	24.3493i		−	1.20993i
$406$		0			0
$407$	22.0365			1.09231
$408$		0			0
$409$			12.5993i			0.622997i	0.950247	+	0.311499i	$0.100831\pi$
$409$			12.5993i			0.622997i	−0.950247	+	0.311499i	$0.899169\pi$
$410$		0			0
$411$		−	49.4773i		−	2.44053i
$412$		0			0
$413$	15.1231			0.744159
$414$		0			0
$415$	−11.1745			−0.548532
$416$		0			0
$417$	58.7386			2.87644
$418$		0			0
$419$	−14.1119			−0.689413			−0.344707	−	0.938710i	$-0.612022\pi$
$419$	−14.1119			−0.689413			−0.344707	+	0.938710i	$0.612022\pi$
$420$		0			0
$421$			19.1868i			0.935105i	0.883965	+	0.467553i	$0.154864\pi$
$421$			19.1868i			0.935105i	−0.883965	+	0.467553i	$0.845136\pi$
$422$		0			0
$423$			42.5312i			2.06794i
$424$		0			0
$425$	17.3693			0.842536
$426$		0			0
$427$		−	0.438447i		−	0.0212179i
$428$		0			0
$429$	−45.8617	+	22.0365i	−2.21423	+	1.06393i
$430$		0			0
$431$		−	38.7386i		−	1.86597i	−0.359910	−	0.932987i	$-0.617192\pi$
$431$		−	38.7386i		−	1.86597i	0.359910	−	0.932987i	$-0.382808\pi$
$432$		0			0
$433$	7.61553			0.365979			0.182989	−	0.983115i	$-0.441423\pi$
$433$	7.61553			0.365979			0.182989	+	0.983115i	$0.441423\pi$
$434$		0			0
$435$			44.8466i			2.15023i
$436$		0			0
$437$		−	4.67474i		−	0.223623i
$438$		0			0
$439$	−36.1485			−1.72527			−0.862636	−	0.505825i	$-0.831188\pi$
$439$	−36.1485			−1.72527			−0.862636	+	0.505825i	$0.831188\pi$
$440$		0			0
$441$	−3.43845			−0.163736
$442$		0			0
$443$	−27.8238			−1.32195			−0.660974	−	0.750409i	$-0.729857\pi$
$443$	−27.8238			−1.32195			−0.660974	+	0.750409i	$0.729857\pi$
$444$		0			0
$445$	−10.5616			−0.500666
$446$		0			0
$447$			35.8078i			1.69365i
$448$		0			0
$449$			14.4243i			0.680725i	0.940294	+	0.340363i	$0.110550\pi$
$449$			14.4243i			0.680725i	−0.940294	+	0.340363i	$0.889450\pi$
$450$		0			0
$451$	14.1119			0.664505
$452$		0			0
$453$			12.3747i			0.581413i
$454$		0			0
$455$	−5.07482	−	10.5616i	−0.237911	−	0.495133i
$456$		0			0
$457$			6.49971i			0.304044i	0.988377	+	0.152022i	$0.0485784\pi$
$457$			6.49971i			0.304044i	−0.988377	+	0.152022i	$0.951422\pi$
$458$		0			0
$459$	−3.47452			−0.162177
$460$		0			0
$461$			39.7984i			1.85360i	0.375560	+	0.926798i	$0.377450\pi$
$461$			39.7984i			1.85360i	−0.375560	+	0.926798i	$0.622550\pi$
$462$		0			0
$463$			0.876894i			0.0407527i	0.999792	+	0.0203764i	$0.00648645\pi$
$463$			0.876894i			0.0407527i	−0.999792	+	0.0203764i	$0.993514\pi$
$464$		0			0
$465$	8.24621			0.382409
$466$		0			0
$467$	−26.7990			−1.24011			−0.620055	−	0.784559i	$-0.712890\pi$
$467$	−26.7990			−1.24011			−0.620055	+	0.784559i	$0.712890\pi$
$468$		0			0
$469$	−0.684658			−0.0316146
$470$		0			0
$471$	3.33756			0.153787
$472$		0			0
$473$		−	46.2981i		−	2.12879i
$474$		0			0
$475$			36.4924i			1.67439i
$476$		0			0
$477$	29.4384			1.34789
$478$		0			0
$479$			13.3002i			0.607701i	0.952720	+	0.303851i	$0.0982723\pi$
$479$			13.3002i			0.607701i	−0.952720	+	0.303851i	$0.901728\pi$
$480$		0			0
$481$	12.8769	−	6.18734i	0.587136	−	0.282118i
$482$		0			0
$483$			1.80776i			0.0822561i
$484$		0			0
$485$	−35.3002			−1.60290
$486$		0			0
$487$		−	23.3693i		−	1.05896i	−0.848321	−	0.529482i	$-0.822386\pi$
$487$		−	23.3693i		−	1.05896i	0.848321	−	0.529482i	$-0.177614\pi$
$488$		0			0
$489$		−	49.1479i		−	2.22254i
$490$		0			0
$491$	−2.22504			−0.100415			−0.0502073	−	0.998739i	$-0.515988\pi$
$491$	−2.22504			−0.100415			−0.0502073	+	0.998739i	$0.515988\pi$
$492$		0			0
$493$	−16.9848			−0.764959
$494$		0			0
$495$	−62.1473			−2.79331
$496$		0			0
$497$	6.24621			0.280181
$498$		0			0
$499$		−	3.56155i		−	0.159437i	−0.996817	−	0.0797185i	$-0.974598\pi$
$499$		−	3.56155i		−	0.159437i	0.996817	−	0.0797185i	$-0.0254021\pi$
$500$		0			0
$501$		−	23.9492i		−	1.06997i
$502$		0			0
$503$	31.8738			1.42118			0.710591	−	0.703605i	$-0.248428\pi$
$503$	31.8738			1.42118			0.710591	+	0.703605i	$0.248428\pi$
$504$		0			0
$505$		−	29.0240i		−	1.29155i
$506$		0			0
$507$	−20.6116	+	25.7538i	−0.915395	+	1.14377i
$508$		0			0
$509$		−	30.0488i		−	1.33189i	−0.746000	−	0.665946i	$-0.768028\pi$
$509$		−	30.0488i		−	1.33189i	0.746000	−	0.665946i	$-0.231972\pi$
$510$		0			0
$511$	−4.36237			−0.192980
$512$		0			0
$513$		−	7.29986i		−	0.322297i
$514$		0			0
$515$			21.1231i			0.930795i
$516$		0			0
$517$	−68.7926			−3.02550
$518$		0			0
$519$	44.0731			1.93459
$520$		0			0
$521$	−7.36932			−0.322856			−0.161428	−	0.986885i	$-0.551610\pi$
$521$	−7.36932			−0.322856			−0.161428	+	0.986885i	$0.551610\pi$
$522$		0			0
$523$	6.18734			0.270553			0.135277	−	0.990808i	$-0.456808\pi$
$523$	6.18734			0.270553			0.135277	+	0.990808i	$0.456808\pi$
$524$		0			0
$525$		−	14.1119i		−	0.615895i
$526$		0			0
$527$			3.12311i			0.136045i
$528$		0			0
$529$	−22.4924			−0.977931
$530$		0			0
$531$		−	52.0000i		−	2.25661i
$532$		0			0
$533$	8.24621	−	3.96230i	0.357183	−	0.171626i
$534$		0			0
$535$			58.7386i			2.53949i
$536$		0			0
$537$	−21.1231			−0.911529
$538$		0			0
$539$		−	5.56155i		−	0.239553i
$540$		0			0
$541$			21.7242i			0.933995i	0.884259	+	0.466997i	$0.154664\pi$
$541$			21.7242i			0.933995i	−0.884259	+	0.466997i	$0.845336\pi$
$542$		0			0
$543$	−40.1108			−1.72132
$544$		0			0
$545$	−37.6155			−1.61127
$546$		0			0
$547$	19.8992			0.850828			0.425414	−	0.904999i	$-0.360128\pi$
$547$	19.8992			0.850828			0.425414	+	0.904999i	$0.360128\pi$
$548$		0			0
$549$	−1.50758			−0.0643418
$550$		0			0
$551$		−	35.6847i		−	1.52022i
$552$		0			0
$553$		−	5.78726i		−	0.246099i
$554$		0			0
$555$	32.6740			1.38693
$556$		0			0
$557$		−	36.4608i		−	1.54490i	−0.635078	−	0.772448i	$-0.719032\pi$
$557$		−	36.4608i		−	1.54490i	0.635078	−	0.772448i	$-0.280968\pi$
$558$		0			0
$559$	−12.9994	−	27.0540i	−0.549817	−	1.14426i
$560$		0			0
$561$		−	44.0731i		−	1.86077i
$562$		0			0
$563$	16.9617			0.714851			0.357426	−	0.933942i	$-0.383655\pi$
$563$	16.9617			0.714851			0.357426	+	0.933942i	$0.383655\pi$
$564$		0			0
$565$			56.8478i			2.39161i
$566$		0			0
$567$		−	7.49242i		−	0.314652i
$568$		0			0
$569$	33.1080			1.38796			0.693979	−	0.719995i	$-0.255856\pi$
$569$	33.1080			1.38796			0.693979	+	0.719995i	$0.255856\pi$
$570$		0			0
$571$	6.89978			0.288747			0.144373	−	0.989523i	$-0.453883\pi$
$571$	6.89978			0.288747			0.144373	+	0.989523i	$0.453883\pi$
$572$		0			0
$573$	7.23106			0.302082
$574$		0			0
$575$	−3.96230			−0.165239
$576$		0			0
$577$		−	44.8732i		−	1.86810i	−0.357147	−	0.934048i	$-0.616250\pi$
$577$		−	44.8732i		−	1.86810i	0.357147	−	0.934048i	$-0.383750\pi$
$578$		0			0
$579$			65.9697i			2.74161i
$580$		0			0
$581$	−3.43845			−0.142651
$582$		0			0
$583$			47.6155i			1.97203i
$584$		0			0
$585$	−36.3153	+	17.4495i	−1.50145	+	0.721448i
$586$		0			0
$587$		−	37.0540i		−	1.52938i	−0.644398	−	0.764691i	$-0.722892\pi$
$587$		−	37.0540i		−	1.52938i	0.644398	−	0.764691i	$-0.277108\pi$
$588$		0			0
$589$	−6.56155			−0.270364
$590$		0			0
$591$			61.5616i			2.53230i
$592$		0			0
$593$			12.5993i			0.517393i	0.965959	+	0.258696i	$0.0832929\pi$
$593$			12.5993i			0.517393i	−0.965959	+	0.258696i	$0.916707\pi$
$594$		0			0
$595$	10.1496			0.416095
$596$		0			0
$597$	32.9848			1.34998
$598$		0			0
$599$	−18.7867			−0.767603			−0.383801	−	0.923416i	$-0.625385\pi$
$599$	−18.7867			−0.767603			−0.383801	+	0.923416i	$0.625385\pi$
$600$		0			0
$601$	5.75379			0.234702			0.117351	−	0.993091i	$-0.462560\pi$
$601$	5.75379			0.234702			0.117351	+	0.993091i	$0.462560\pi$
$602$		0			0
$603$			2.35416i			0.0958689i
$604$		0			0
$605$		−	64.7724i		−	2.63337i
$606$		0			0
$607$	5.87497			0.238458			0.119229	−	0.992867i	$-0.461958\pi$
$607$	5.87497			0.238458			0.119229	+	0.992867i	$0.461958\pi$
$608$		0			0
$609$			13.7996i			0.559187i
$610$		0			0
$611$	−40.1985	+	19.3153i	−1.62626	+	0.781415i
$612$		0			0
$613$		−	30.7613i		−	1.24244i	−0.783638	−	0.621218i	$-0.786638\pi$
$613$		−	30.7613i		−	1.24244i	0.783638	−	0.621218i	$-0.213362\pi$
$614$		0			0
$615$	20.9240			0.843738
$616$		0			0
$617$		−	13.7996i		−	0.555550i	−0.960646	−	0.277775i	$-0.910403\pi$
$617$		−	13.7996i		−	0.555550i	0.960646	−	0.277775i	$-0.0895969\pi$
$618$		0			0
$619$		−	7.50758i		−	0.301755i	−0.988552	−	0.150878i	$-0.951790\pi$
$619$		−	7.50758i		−	0.301755i	0.988552	−	0.150878i	$-0.0482099\pi$
$620$		0			0
$621$	0.792609			0.0318063
$622$		0			0
$623$	−3.24985			−0.130203
$624$		0			0
$625$	−21.8769			−0.875076
$626$		0			0
$627$	92.5962			3.69794
$628$		0			0
$629$			12.3747i			0.493411i
$630$		0			0
$631$			26.8769i			1.06995i	0.844867	+	0.534976i	$0.179679\pi$
$631$			26.8769i			1.06995i	−0.844867	+	0.534976i	$0.820321\pi$
$632$		0			0
$633$	46.8769			1.86319
$634$		0			0
$635$		−	3.61553i		−	0.143478i
$636$		0			0
$637$	−1.56155	−	3.24985i	−0.0618710	−	0.128764i
$638$		0			0
$639$		−	21.4773i		−	0.849628i
$640$		0			0
$641$	30.6155			1.20924			0.604620	−	0.796514i	$-0.293325\pi$
$641$	30.6155			1.20924			0.604620	+	0.796514i	$0.293325\pi$
$642$		0			0
$643$			11.6155i			0.458072i	0.973418	+	0.229036i	$0.0735573\pi$
$643$			11.6155i			0.458072i	−0.973418	+	0.229036i	$0.926443\pi$
$644$		0			0
$645$		−	68.6470i		−	2.70297i
$646$		0			0
$647$	−35.5237			−1.39658			−0.698291	−	0.715814i	$-0.746056\pi$
$647$	−35.5237			−1.39658			−0.698291	+	0.715814i	$0.746056\pi$
$648$		0			0
$649$	84.1080			3.30153
$650$		0			0
$651$	2.53741			0.0994489
$652$		0			0
$653$	19.3693			0.757980			0.378990	−	0.925401i	$-0.376271\pi$
$653$	19.3693			0.757980			0.378990	+	0.925401i	$0.376271\pi$
$654$		0			0
$655$		−	11.8617i		−	0.463477i
$656$		0			0
$657$			14.9998i			0.585198i
$658$		0			0
$659$	35.1237			1.36822			0.684112	−	0.729377i	$-0.260190\pi$
$659$	35.1237			1.36822			0.684112	+	0.729377i	$0.260190\pi$
$660$		0			0
$661$			31.4737i			1.22419i	0.790786	+	0.612093i	$0.209672\pi$
$661$			31.4737i			1.22419i	−0.790786	+	0.612093i	$0.790328\pi$
$662$		0			0
$663$	−12.3747	−	25.7538i	−0.480593	−	1.00019i
$664$		0			0
$665$			21.3241i			0.826913i
$666$		0			0
$667$	3.87459			0.150025
$668$		0			0
$669$			12.6870i			0.490510i
$670$		0			0
$671$		−	2.43845i		−	0.0941352i
$672$		0			0
$673$	47.1080			1.81588			0.907939	−	0.419102i	$-0.137655\pi$
$673$	47.1080			1.81588			0.907939	+	0.419102i	$0.137655\pi$
$674$		0			0
$675$	−6.18734			−0.238151
$676$		0			0
$677$	−17.8078			−0.684408			−0.342204	−	0.939626i	$-0.611173\pi$
$677$	−17.8078			−0.684408			−0.342204	+	0.939626i	$0.611173\pi$
$678$		0			0
$679$	−10.8621			−0.416848
$680$		0			0
$681$		−	39.6230i		−	1.51836i
$682$		0			0
$683$		−	12.0540i		−	0.461233i	−0.973045	−	0.230616i	$-0.925926\pi$
$683$		−	12.0540i		−	0.461233i	0.973045	−	0.230616i	$-0.0740742\pi$
$684$		0			0
$685$	63.3693			2.42122
$686$		0			0
$687$		−	42.2462i		−	1.61179i
$688$		0			0
$689$	13.3693	+	27.8238i	0.509330	+	1.06000i
$690$		0			0
$691$		−	37.5464i		−	1.42833i	−0.699976	−	0.714166i	$-0.746806\pi$
$691$		−	37.5464i		−	1.42833i	0.699976	−	0.714166i	$-0.253194\pi$
$692$		0			0
$693$	−19.1231			−0.726427
$694$		0			0
$695$			75.2311i			2.85368i
$696$		0			0
$697$			7.92460i			0.300166i
$698$		0			0
$699$	−17.7619			−0.671816
$700$		0			0
$701$	33.5464			1.26703			0.633515	−	0.773730i	$-0.281611\pi$
$701$	33.5464			1.26703			0.633515	+	0.773730i	$0.281611\pi$
$702$		0			0
$703$	−25.9988			−0.980565
$704$		0			0
$705$	−102.000			−3.84154
$706$		0			0
$707$		−	8.93087i		−	0.335880i
$708$		0			0
$709$			34.4112i			1.29234i	0.763193	+	0.646170i	$0.223630\pi$
$709$			34.4112i			1.29234i	−0.763193	+	0.646170i	$0.776370\pi$
$710$		0			0
$711$	−19.8992			−0.746278
$712$		0			0
$713$		−	0.712445i		−	0.0266813i
$714$		0			0
$715$	−28.2239	−	58.7386i	−1.05551	−	2.19670i
$716$		0			0
$717$			12.3747i			0.462141i
$718$		0			0
$719$	−42.6482			−1.59051			−0.795254	−	0.606276i	$-0.792663\pi$
$719$	−42.6482			−1.59051			−0.795254	+	0.606276i	$0.792663\pi$
$720$		0			0
$721$			6.49971i			0.242062i
$722$		0			0
$723$		−	17.5076i		−	0.651114i
$724$		0			0
$725$	−30.2462			−1.12332
$726$		0			0
$727$	27.4237			1.01709			0.508545	−	0.861036i	$-0.330184\pi$
$727$	27.4237			1.01709			0.508545	+	0.861036i	$0.330184\pi$
$728$		0			0
$729$	−34.2311			−1.26782
$730$		0			0
$731$	25.9988			0.961602
$732$		0			0
$733$		−	18.4743i		−	0.682364i	−0.939997	−	0.341182i	$-0.889173\pi$
$733$		−	18.4743i		−	0.682364i	0.939997	−	0.341182i	$-0.110827\pi$
$734$		0			0
$735$		−	8.24621i		−	0.304166i
$736$		0			0
$737$	−3.80776			−0.140261
$738$		0			0
$739$		−	28.6307i		−	1.05320i	−0.850114	−	0.526598i	$-0.823467\pi$
$739$		−	28.6307i		−	1.05320i	0.850114	−	0.526598i	$-0.176533\pi$
$740$		0			0
$741$	54.1080	−	25.9988i	1.98771	−	0.955091i
$742$		0			0
$743$			12.2462i			0.449270i	0.974443	+	0.224635i	$0.0721190\pi$
$743$			12.2462i			0.449270i	−0.974443	+	0.224635i	$0.927881\pi$
$744$		0			0
$745$	−45.8617			−1.68024
$746$		0			0
$747$			11.8229i			0.432578i
$748$		0			0
$749$			18.0742i			0.660418i
$750$		0			0
$751$	−22.4366			−0.818724			−0.409362	−	0.912372i	$-0.634249\pi$
$751$	−22.4366			−0.818724			−0.409362	+	0.912372i	$0.634249\pi$
$752$		0			0
$753$	−13.6695			−0.498144
$754$		0			0
$755$	−15.8492			−0.576811
$756$		0			0
$757$	−2.31534			−0.0841525			−0.0420763	−	0.999114i	$-0.513397\pi$
$757$	−2.31534			−0.0841525			−0.0420763	+	0.999114i	$0.513397\pi$
$758$		0			0
$759$			10.0540i			0.364936i
$760$		0			0
$761$		−	41.1356i		−	1.49116i	−0.666414	−	0.745582i	$-0.732172\pi$
$761$		−	41.1356i		−	1.49116i	0.666414	−	0.745582i	$-0.267828\pi$
$762$		0			0
$763$	−11.5745			−0.419026
$764$		0			0
$765$		−	34.8990i		−	1.26178i
$766$		0			0
$767$	49.1479	−	23.6155i	1.77463	−	0.852707i
$768$		0			0
$769$			19.5868i			0.706319i	0.935563	+	0.353160i	$0.114893\pi$
$769$			19.5868i			0.706319i	−0.935563	+	0.353160i	$0.885107\pi$
$770$		0			0
$771$	59.9223			2.15805
$772$		0			0
$773$		−	40.4231i		−	1.45392i	−0.686680	−	0.726960i	$-0.740933\pi$
$773$		−	40.4231i		−	1.45392i	0.686680	−	0.726960i	$-0.259067\pi$
$774$		0			0
$775$			5.56155i			0.199777i
$776$		0			0
$777$	10.0540			0.360684
$778$		0			0
$779$	−16.6493			−0.596525
$780$		0			0
$781$	34.7386			1.24305
$782$		0			0
$783$	6.05038			0.216223
$784$		0			0
$785$			4.27467i			0.152569i
$786$		0			0
$787$		−	54.4233i		−	1.93998i	−0.243143	−	0.969990i	$-0.578178\pi$
$787$		−	54.4233i		−	1.93998i	0.243143	−	0.969990i	$-0.421822\pi$
$788$		0			0
$789$	63.3693			2.25601
$790$		0			0
$791$			17.4924i			0.621959i
$792$		0			0
$793$	−0.684658	−	1.42489i	−0.0243129	−	0.0505993i
$794$		0			0
$795$			70.6004i			2.50394i
$796$		0			0
$797$	−3.31534			−0.117435			−0.0587177	−	0.998275i	$-0.518701\pi$
$797$	−3.31534			−0.117435			−0.0587177	+	0.998275i	$0.518701\pi$
$798$		0			0
$799$		−	38.6307i		−	1.36666i
$800$		0			0
$801$			11.1745i			0.394830i
$802$		0			0
$803$	−24.2616			−0.856172
$804$		0			0
$805$	−2.31534			−0.0816051
$806$		0			0
$807$	−8.41238			−0.296130
$808$		0			0
$809$	27.9309			0.981997			0.490999	−	0.871160i	$-0.336632\pi$
$809$	27.9309			0.981997			0.490999	+	0.871160i	$0.336632\pi$
$810$		0			0
$811$		−	14.7386i		−	0.517543i	−0.965939	−	0.258772i	$-0.916682\pi$
$811$		−	14.7386i		−	0.517543i	0.965939	−	0.258772i	$-0.0833177\pi$
$812$		0			0
$813$		−	42.3358i		−	1.48478i
$814$		0			0
$815$	62.9475			2.20495
$816$		0			0
$817$			54.6228i			1.91101i
$818$		0			0
$819$	−11.1745	+	5.36932i	−0.390467	+	0.187619i
$820$		0			0
$821$		−	31.8738i		−	1.11240i	−0.831047	−	0.556202i	$-0.812258\pi$
$821$		−	31.8738i		−	1.11240i	0.831047	−	0.556202i	$-0.187742\pi$
$822$		0			0
$823$	−24.2616			−0.845705			−0.422853	−	0.906198i	$-0.638971\pi$
$823$	−24.2616			−0.845705			−0.422853	+	0.906198i	$0.638971\pi$
$824$		0			0
$825$		−	78.4843i		−	2.73247i
$826$		0			0
$827$			49.1231i			1.70818i	0.520127	+	0.854089i	$0.325885\pi$
$827$			49.1231i			1.70818i	−0.520127	+	0.854089i	$0.674115\pi$
$828$		0			0
$829$	−25.3693			−0.881113			−0.440556	−	0.897725i	$-0.645219\pi$
$829$	−25.3693			−0.881113			−0.440556	+	0.897725i	$0.645219\pi$
$830$		0			0
$831$	−4.27467			−0.148287
$832$		0			0
$833$	3.12311			0.108209
$834$		0			0
$835$	30.6736			1.06150
$836$		0			0
$837$		−	1.11252i		−	0.0384543i
$838$		0			0
$839$		−	51.4233i		−	1.77533i	−0.460491	−	0.887665i	$-0.652326\pi$
$839$		−	51.4233i		−	1.77533i	0.460491	−	0.887665i	$-0.347674\pi$
$840$		0			0
$841$	0.576708			0.0198865
$842$		0			0
$843$			51.5076i			1.77402i
$844$		0			0
$845$	−32.9848	−	26.3989i	−1.13471	−	0.908150i
$846$		0			0
$847$		−	19.9309i		−	0.684833i
$848$		0			0
$849$	26.5464			0.911070
$850$		0			0
$851$		−	2.82292i		−	0.0967684i
$852$		0			0
$853$			53.8226i			1.84285i	0.388554	+	0.921426i	$0.372975\pi$
$853$			53.8226i			1.84285i	−0.388554	+	0.921426i	$0.627025\pi$
$854$		0			0
$855$	73.3218			2.50755
$856$		0			0
$857$	−40.4924			−1.38319			−0.691597	−	0.722283i	$-0.743093\pi$
$857$	−40.4924			−1.38319			−0.691597	+	0.722283i	$0.743093\pi$
$858$		0			0
$859$	19.8115			0.675959			0.337980	−	0.941153i	$-0.390256\pi$
$859$	19.8115			0.675959			0.337980	+	0.941153i	$0.390256\pi$
$860$		0			0
$861$	6.43845			0.219422
$862$		0			0
$863$		−	10.4924i		−	0.357166i	−0.983925	−	0.178583i	$-0.942849\pi$
$863$		−	10.4924i		−	0.357166i	0.983925	−	0.178583i	$-0.0571513\pi$
$864$		0			0
$865$			56.4477i			1.91928i
$866$		0			0
$867$	−18.3866			−0.624442
$868$		0			0
$869$		−	32.1862i		−	1.09184i
$870$		0			0
$871$	−2.22504	+	1.06913i	−0.0753926	+	0.0362261i
$872$		0			0
$873$			37.3487i			1.26406i
$874$		0			0
$875$	1.82496			0.0616951
$876$		0			0
$877$		−	23.4614i		−	0.792236i	−0.918200	−	0.396118i	$-0.870357\pi$
$877$		−	23.4614i		−	0.792236i	0.918200	−	0.396118i	$-0.129643\pi$
$878$		0			0
$879$		−	41.2311i		−	1.39069i
$880$		0			0
$881$	15.3693			0.517805			0.258903	−	0.965903i	$-0.416639\pi$
$881$	15.3693			0.517805			0.258903	+	0.965903i	$0.416639\pi$
$882$		0			0
$883$	−2.84978			−0.0959027			−0.0479513	−	0.998850i	$-0.515269\pi$
$883$	−2.84978			−0.0959027			−0.0479513	+	0.998850i	$0.515269\pi$
$884$		0			0
$885$	124.708			4.19202
$886$		0			0
$887$	−30.4489			−1.02237			−0.511187	−	0.859469i	$-0.670794\pi$
$887$	−30.4489			−1.02237			−0.511187	+	0.859469i	$0.670794\pi$
$888$		0			0
$889$		−	1.11252i		−	0.0373127i
$890$		0			0
$891$		−	41.6695i		−	1.39598i
$892$		0			0
$893$	81.1619			2.71598
$894$		0			0
$895$		−	27.0540i		−	0.904315i
$896$		0			0
$897$	2.82292	+	5.87497i	0.0942545	+	0.196160i
$898$		0			0
$899$		−	5.43845i		−	0.181382i
$900$		0			0
$901$	−26.7386			−0.890793
$902$		0			0
$903$		−	21.1231i		−	0.702933i
$904$		0			0
$905$		−	51.3729i		−	1.70769i
$906$		0			0
$907$	−31.4737			−1.04507			−0.522534	−	0.852618i	$-0.675013\pi$
$907$	−31.4737			−1.04507			−0.522534	+	0.852618i	$0.675013\pi$
$908$		0			0
$909$	−30.7083			−1.01853
$910$		0			0
$911$	20.5239			0.679988			0.339994	−	0.940428i	$-0.389575\pi$
$911$	20.5239			0.679988			0.339994	+	0.940428i	$0.389575\pi$
$912$		0			0
$913$	−19.1231			−0.632882
$914$		0			0
$915$		−	3.61553i		−	0.119526i
$916$		0			0
$917$		−	3.64993i		−	0.120531i
$918$		0			0
$919$	−32.8109			−1.08233			−0.541166	−	0.840916i	$-0.682017\pi$
$919$	−32.8109			−1.08233			−0.541166	+	0.840916i	$0.682017\pi$
$920$		0			0
$921$			11.5745i			0.381394i
$922$		0			0
$923$	20.2993	−	9.75379i	0.668159	−	0.321050i
$924$		0			0
$925$			22.0365i			0.724557i
$926$		0			0
$927$	22.3489			0.734034
$928$		0			0
$929$		−	47.0106i		−	1.54237i	−0.636613	−	0.771183i	$-0.719665\pi$
$929$		−	47.0106i		−	1.54237i	0.636613	−	0.771183i	$-0.280335\pi$
$930$		0			0
$931$			6.56155i			0.215046i
$932$		0			0
$933$	42.2462			1.38308
$934$		0			0
$935$	56.4477			1.84604
$936$		0			0
$937$	30.4924			0.996144			0.498072	−	0.867136i	$-0.334042\pi$
$937$	30.4924			0.996144			0.498072	+	0.867136i	$0.334042\pi$
$938$		0			0
$939$	2.84978			0.0929990
$940$		0			0
$941$			27.8238i			0.907030i	0.891249	+	0.453515i	$0.149830\pi$
$941$			27.8238i			0.907030i	−0.891249	+	0.453515i	$0.850170\pi$
$942$		0			0
$943$		−	1.80776i		−	0.0588689i
$944$		0			0
$945$	−3.61553			−0.117613
$946$		0			0
$947$		−	27.8617i		−	0.905385i	−0.891667	−	0.452692i	$-0.850464\pi$
$947$		−	27.8617i		−	0.905385i	0.891667	−	0.452692i	$-0.149536\pi$
$948$		0			0
$949$	−14.1771	+	6.81208i	−0.460208	+	0.221129i
$950$		0			0
$951$			28.5767i			0.926663i
$952$		0			0
$953$	−27.9309			−0.904770			−0.452385	−	0.891823i	$-0.649427\pi$
$953$	−27.9309			−0.904770			−0.452385	+	0.891823i	$0.649427\pi$
$954$		0			0
$955$			9.26137i			0.299691i
$956$		0			0
$957$			76.7470i			2.48088i
$958$		0			0
$959$	19.4991			0.629660
$960$		0			0
$961$	30.0000			0.967742
$962$		0			0
$963$	62.1473			2.00267
$964$		0			0
$965$	−84.4924			−2.71991
$966$		0			0
$967$			25.6155i			0.823740i	0.911243	+	0.411870i	$0.135124\pi$
$967$			25.6155i			0.823740i	−0.911243	+	0.411870i	$0.864876\pi$
$968$		0			0
$969$			51.9977i			1.67041i
$970$		0			0
$971$	26.3112			0.844367			0.422183	−	0.906510i	$-0.361264\pi$
$971$	26.3112			0.844367			0.422183	+	0.906510i	$0.361264\pi$
$972$		0			0
$973$			23.1491i			0.742125i
$974$		0			0
$975$	−22.0365	−	45.8617i	−0.705734	−	1.46875i
$976$		0			0
$977$			17.2741i			0.552647i	0.961065	+	0.276323i	$0.0891161\pi$
$977$			17.2741i			0.552647i	−0.961065	+	0.276323i	$0.910884\pi$
$978$		0			0
$979$	−18.0742			−0.577655
$980$		0			0
$981$			39.7984i			1.27067i
$982$		0			0
$983$		−	24.9460i		−	0.795655i	−0.917460	−	0.397827i	$-0.869764\pi$
$983$		−	24.9460i		−	0.795655i	0.917460	−	0.397827i	$-0.130236\pi$
$984$		0			0
$985$	−78.8466			−2.51226
$986$		0			0
$987$	−31.3860			−0.999028
$988$		0			0
$989$	−5.93087			−0.188591
$990$		0			0
$991$	4.58704			0.145712			0.0728560	−	0.997342i	$-0.476789\pi$
$991$	4.58704			0.145712			0.0728560	+	0.997342i	$0.476789\pi$
$992$		0			0
$993$		−	37.2610i		−	1.18244i
$994$		0			0
$995$			42.2462i			1.33929i
$996$		0			0
$997$	39.5616			1.25293			0.626463	−	0.779451i	$-0.284502\pi$
$997$	39.5616			1.25293			0.626463	+	0.779451i	$0.284502\pi$
$998$		0			0
$999$		−	4.40814i		−	0.139467i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.k.g.1793.6	yes	8
4.3	odd	2	inner	2912.2.k.g.1793.4	yes	8
13.12	even	2	inner	2912.2.k.g.1793.5	yes	8
52.51	odd	2	inner	2912.2.k.g.1793.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.k.g.1793.3	✓	8	52.51	odd	2	inner
2912.2.k.g.1793.4	yes	8	4.3	odd	2	inner
2912.2.k.g.1793.5	yes	8	13.12	even	2	inner
2912.2.k.g.1793.6	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.53741 q^{3} +3.24985i q^{5} +1.00000i q^{7} +3.43845 q^{9} +5.56155i q^{11} +(1.56155 + 3.24985i) q^{13} +8.24621i q^{15} -3.12311 q^{17} -6.56155i q^{19} +2.53741i q^{21} +0.712445 q^{23} -5.56155 q^{25} +1.11252 q^{27} +5.43845 q^{29} -1.00000i q^{31} +14.1119i q^{33} -3.24985 q^{35} -3.96230i q^{37} +(3.96230 + 8.24621i) q^{39} -2.53741i q^{41} -8.32467 q^{43} +11.1745i q^{45} +12.3693i q^{47} -1.00000 q^{49} -7.92460 q^{51} +8.56155 q^{53} -18.0742 q^{55} -16.6493i q^{57} -15.1231i q^{59} -0.438447 q^{61} +3.43845i q^{63} +(-10.5616 + 5.07482i) q^{65} +0.684658i q^{67} +1.80776 q^{69} -6.24621i q^{71} +4.36237i q^{73} -14.1119 q^{75} -5.56155 q^{77} -5.78726 q^{79} -7.49242 q^{81} +3.43845i q^{83} -10.1496i q^{85} +13.7996 q^{87} +3.24985i q^{89} +(-3.24985 + 1.56155i) q^{91} -2.53741i q^{93} +21.3241 q^{95} +10.8621i q^{97} +19.1231i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{9} - 4 q^{13} + 8 q^{17} - 28 q^{25} + 60 q^{29} - 8 q^{49} + 52 q^{53} - 20 q^{61} - 68 q^{65} - 68 q^{69} - 28 q^{77} + 72 q^{81}+O(q^{100}) \) 8 * q + 44 * q^9 - 4 * q^13 + 8 * q^17 - 28 * q^25 + 60 * q^29 - 8 * q^49 + 52 * q^53 - 20 * q^61 - 68 * q^65 - 68 * q^69 - 28 * q^77 + 72 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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