LMFDB - Embedded newform 2912.2.h.a.2575.42

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(2575,2912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("2912.2575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2575.42
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.a.2575.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44281i q^{3} +3.85137 q^{5} +(-2.49284 - 0.886436i) q^{7} -2.96733 q^{9} +1.81225 q^{11} -1.00000 q^{13} +9.40816i q^{15} +3.46118i q^{17} +4.50271i q^{19} +(2.16540 - 6.08953i) q^{21} +0.813440i q^{23} +9.83303 q^{25} +0.0798187i q^{27} -2.95921i q^{29} -7.74273 q^{31} +4.42699i q^{33} +(-9.60083 - 3.41399i) q^{35} -0.986953i q^{37} -2.44281i q^{39} +3.45219i q^{41} -5.13822 q^{43} -11.4283 q^{45} +12.5979 q^{47} +(5.42846 + 4.41948i) q^{49} -8.45502 q^{51} +1.21088i q^{53} +6.97966 q^{55} -10.9993 q^{57} +12.8822i q^{59} -4.69520 q^{61} +(7.39705 + 2.63034i) q^{63} -3.85137 q^{65} +2.12856 q^{67} -1.98708 q^{69} +3.99821i q^{71} +9.25114i q^{73} +24.0202i q^{75} +(-4.51765 - 1.60645i) q^{77} +11.8410i q^{79} -9.09696 q^{81} -17.6272i q^{83} +13.3303i q^{85} +7.22879 q^{87} +0.651173i q^{89} +(2.49284 + 0.886436i) q^{91} -18.9140i q^{93} +17.3416i q^{95} +10.3289i q^{97} -5.37755 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) $ 48 * q - 48 * q^9 + 4 * q^11 - 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 + 24 * q^45 + 40 * q^51 - 20 * q^63 + 4 * q^67 - 20 * q^77 + 64 * q^81 + 40 * q^87 - 20 * q^99

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
$3$			2.44281i			1.41036i	0.709030	+	0.705179i	$0.249133\pi$
$3$			2.44281i			1.41036i	−0.709030	+	0.705179i	$0.750867\pi$
$4$		0			0
$5$	3.85137			1.72238			0.861192	−	0.508280i	$-0.169718\pi$
$5$	3.85137			1.72238			0.861192	+	0.508280i	$0.169718\pi$
$6$		0			0
$7$	−2.49284	−	0.886436i	−0.942203	−	0.335041i
$7$	−2.49284	−	0.886436i	−0.942203	−	0.335041i
$8$		0			0
$9$	−2.96733			−0.989108
$10$		0			0
$11$	1.81225			0.546415			0.273208	−	0.961955i	$-0.411915\pi$
$11$	1.81225			0.546415			0.273208	+	0.961955i	$0.411915\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$			9.40816i			2.42918i
$16$		0			0
$17$			3.46118i			0.839461i	0.907649	+	0.419730i	$0.137875\pi$
$17$			3.46118i			0.839461i	−0.907649	+	0.419730i	$0.862125\pi$
$18$		0			0
$19$			4.50271i			1.03299i	0.856289	+	0.516496i	$0.172764\pi$
$19$			4.50271i			1.03299i	−0.856289	+	0.516496i	$0.827236\pi$
$20$		0			0
$21$	2.16540	−	6.08953i	0.472528	−	1.32884i
$22$		0			0
$23$			0.813440i			0.169614i	0.996397	+	0.0848070i	$0.0270274\pi$
$23$			0.813440i			0.169614i	−0.996397	+	0.0848070i	$0.972973\pi$
$24$		0			0
$25$	9.83303			1.96661
$26$		0			0
$27$			0.0798187i			0.0153611i
$28$		0			0
$29$		−	2.95921i		−	0.549511i	−0.961514	−	0.274756i	$-0.911403\pi$
$29$		−	2.95921i		−	0.549511i	0.961514	−	0.274756i	$-0.0885970\pi$
$30$		0			0
$31$	−7.74273			−1.39063			−0.695317	−	0.718703i	$-0.744736\pi$
$31$	−7.74273			−1.39063			−0.695317	+	0.718703i	$0.744736\pi$
$32$		0			0
$33$			4.42699i			0.770641i
$34$		0			0
$35$	−9.60083	−	3.41399i	−1.62284	−	0.577070i
$36$		0			0
$37$		−	0.986953i		−	0.162254i	−0.996704	−	0.0811271i	$-0.974148\pi$
$37$		−	0.986953i		−	0.162254i	0.996704	−	0.0811271i	$-0.0258520\pi$
$38$		0			0
$39$		−	2.44281i		−	0.391163i
$40$		0			0
$41$			3.45219i			0.539142i	0.962981	+	0.269571i	$0.0868819\pi$
$41$			3.45219i			0.539142i	−0.962981	+	0.269571i	$0.913118\pi$
$42$		0			0
$43$	−5.13822			−0.783572			−0.391786	−	0.920056i	$-0.628143\pi$
$43$	−5.13822			−0.783572			−0.391786	+	0.920056i	$0.628143\pi$
$44$		0			0
$45$	−11.4283			−1.70362
$46$		0			0
$47$	12.5979			1.83759			0.918797	−	0.394729i	$-0.129162\pi$
$47$	12.5979			1.83759			0.918797	+	0.394729i	$0.129162\pi$
$48$		0			0
$49$	5.42846	+	4.41948i	0.775494	+	0.631354i
$50$		0			0
$51$	−8.45502			−1.18394
$52$		0			0
$53$			1.21088i			0.166328i	0.996536	+	0.0831639i	$0.0265025\pi$
$53$			1.21088i			0.166328i	−0.996536	+	0.0831639i	$0.973498\pi$
$54$		0			0
$55$	6.97966			0.941137
$56$		0			0
$57$	−10.9993			−1.45689
$58$		0			0
$59$			12.8822i			1.67712i	0.544807	+	0.838561i	$0.316603\pi$
$59$			12.8822i			1.67712i	−0.544807	+	0.838561i	$0.683397\pi$
$60$		0			0
$61$	−4.69520			−0.601159			−0.300579	−	0.953757i	$-0.597180\pi$
$61$	−4.69520			−0.601159			−0.300579	+	0.953757i	$0.597180\pi$
$62$		0			0
$63$	7.39705	+	2.63034i	0.931941	+	0.331392i
$64$		0			0
$65$	−3.85137			−0.477703
$66$		0			0
$67$	2.12856			0.260045			0.130023	−	0.991511i	$-0.458495\pi$
$67$	2.12856			0.260045			0.130023	+	0.991511i	$0.458495\pi$
$68$		0			0
$69$	−1.98708			−0.239216
$70$		0			0
$71$			3.99821i			0.474500i	0.971449	+	0.237250i	$0.0762460\pi$
$71$			3.99821i			0.474500i	−0.971449	+	0.237250i	$0.923754\pi$
$72$		0			0
$73$			9.25114i			1.08276i	0.840777	+	0.541382i	$0.182099\pi$
$73$			9.25114i			1.08276i	−0.840777	+	0.541382i	$0.817901\pi$
$74$		0			0
$75$			24.0202i			2.77362i
$76$		0			0
$77$	−4.51765	−	1.60645i	−0.514834	−	0.183072i
$78$		0			0
$79$			11.8410i			1.33222i	0.745854	+	0.666110i	$0.232042\pi$
$79$			11.8410i			1.33222i	−0.745854	+	0.666110i	$0.767958\pi$
$80$		0			0
$81$	−9.09696			−1.01077
$82$		0			0
$83$		−	17.6272i		−	1.93484i	−0.253182	−	0.967419i	$-0.581477\pi$
$83$		−	17.6272i		−	1.93484i	0.253182	−	0.967419i	$-0.418523\pi$
$84$		0			0
$85$			13.3303i			1.44587i
$86$		0			0
$87$	7.22879			0.775007
$88$		0			0
$89$			0.651173i			0.0690242i	0.999404	+	0.0345121i	$0.0109877\pi$
$89$			0.651173i			0.0690242i	−0.999404	+	0.0345121i	$0.989012\pi$
$90$		0			0
$91$	2.49284	+	0.886436i	0.261320	+	0.0929238i
$92$		0			0
$93$		−	18.9140i		−	1.96129i
$94$		0			0
$95$			17.3416i			1.77921i
$96$		0			0
$97$			10.3289i			1.04874i	0.851490	+	0.524371i	$0.175699\pi$
$97$			10.3289i			1.04874i	−0.851490	+	0.524371i	$0.824301\pi$
$98$		0			0
$99$	−5.37755			−0.540464
$100$		0			0
$101$	18.0120			1.79226			0.896132	−	0.443788i	$-0.146366\pi$
$101$	18.0120			1.79226			0.896132	+	0.443788i	$0.146366\pi$
$102$		0			0
$103$	−3.56069			−0.350845			−0.175422	−	0.984493i	$-0.556129\pi$
$103$	−3.56069			−0.350845			−0.175422	+	0.984493i	$0.556129\pi$
$104$		0			0
$105$	8.33974	−	23.4530i	0.813875	−	2.28878i
$106$		0			0
$107$	14.4988			1.40165			0.700824	−	0.713334i	$-0.252816\pi$
$107$	14.4988			1.40165			0.700824	+	0.713334i	$0.252816\pi$
$108$		0			0
$109$			17.5262i			1.67870i	0.543589	+	0.839351i	$0.317065\pi$
$109$			17.5262i			1.67870i	−0.543589	+	0.839351i	$0.682935\pi$
$110$		0			0
$111$	2.41094			0.228836
$112$		0			0
$113$	−9.73680			−0.915962			−0.457981	−	0.888962i	$-0.651427\pi$
$113$	−9.73680			−0.915962			−0.457981	+	0.888962i	$0.651427\pi$
$114$		0			0
$115$			3.13286i			0.292141i
$116$		0			0
$117$	2.96733			0.274329
$118$		0			0
$119$	3.06812	−	8.62817i	0.281254	−	0.790943i
$120$		0			0
$121$	−7.71574			−0.701431
$122$		0			0
$123$	−8.43305			−0.760383
$124$		0			0
$125$	18.6138			1.66487
$126$		0			0
$127$		−	0.775751i		−	0.0688368i	−0.999408	−	0.0344184i	$-0.989042\pi$
$127$		−	0.775751i		−	0.0688368i	0.999408	−	0.0344184i	$-0.0109579\pi$
$128$		0			0
$129$		−	12.5517i		−	1.10512i
$130$		0			0
$131$		−	5.40650i		−	0.472368i	−0.971708	−	0.236184i	$-0.924103\pi$
$131$		−	5.40650i		−	0.472368i	0.971708	−	0.236184i	$-0.0758969\pi$
$132$		0			0
$133$	3.99137	−	11.2245i	0.346095	−	0.973289i
$134$		0			0
$135$			0.307411i			0.0264577i
$136$		0			0
$137$	5.29585			0.452455			0.226227	−	0.974075i	$-0.427361\pi$
$137$	5.29585			0.452455			0.226227	+	0.974075i	$0.427361\pi$
$138$		0			0
$139$		−	15.7618i		−	1.33690i	−0.743758	−	0.668449i	$-0.766959\pi$
$139$		−	15.7618i		−	1.33690i	0.743758	−	0.668449i	$-0.233041\pi$
$140$		0			0
$141$			30.7743i			2.59167i
$142$		0			0
$143$	−1.81225			−0.151548
$144$		0			0
$145$		−	11.3970i		−	0.946470i
$146$		0			0
$147$	−10.7960	+	13.2607i	−0.890435	+	1.09372i
$148$		0			0
$149$		−	20.2534i		−	1.65923i	−0.558339	−	0.829613i	$-0.688561\pi$
$149$		−	20.2534i		−	1.65923i	0.558339	−	0.829613i	$-0.311439\pi$
$150$		0			0
$151$		−	10.6491i		−	0.866609i	−0.901248	−	0.433304i	$-0.857348\pi$
$151$		−	10.6491i		−	0.866609i	0.901248	−	0.433304i	$-0.142652\pi$
$152$		0			0
$153$		−	10.2705i		−	0.830318i
$154$		0			0
$155$	−29.8201			−2.39521
$156$		0			0
$157$	−12.8972			−1.02931			−0.514653	−	0.857398i	$-0.672079\pi$
$157$	−12.8972			−1.02931			−0.514653	+	0.857398i	$0.672079\pi$
$158$		0			0
$159$	−2.95796			−0.234582
$160$		0			0
$161$	0.721063	−	2.02777i	0.0568277	−	0.159811i
$162$		0			0
$163$	−17.2917			−1.35439			−0.677195	−	0.735804i	$-0.736805\pi$
$163$	−17.2917			−1.35439			−0.677195	+	0.735804i	$0.736805\pi$
$164$		0			0
$165$			17.0500i			1.32734i
$166$		0			0
$167$	9.67383			0.748583			0.374292	−	0.927311i	$-0.377886\pi$
$167$	9.67383			0.748583			0.374292	+	0.927311i	$0.377886\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		−	13.3610i		−	1.02174i
$172$		0			0
$173$	−7.65616			−0.582087			−0.291043	−	0.956710i	$-0.594002\pi$
$173$	−7.65616			−0.582087			−0.291043	+	0.956710i	$0.594002\pi$
$174$		0			0
$175$	−24.5121	−	8.71636i	−1.85294	−	0.658895i
$176$		0			0
$177$	−31.4688			−2.36534
$178$		0			0
$179$	22.6980			1.69652			0.848262	−	0.529576i	$-0.177649\pi$
$179$	22.6980			1.69652			0.848262	+	0.529576i	$0.177649\pi$
$180$		0			0
$181$	16.7233			1.24303			0.621517	−	0.783400i	$-0.286517\pi$
$181$	16.7233			1.24303			0.621517	+	0.783400i	$0.286517\pi$
$182$		0			0
$183$		−	11.4695i		−	0.847849i
$184$		0			0
$185$		−	3.80112i		−	0.279464i
$186$		0			0
$187$			6.27255i			0.458694i
$188$		0			0
$189$	0.0707542	−	0.198975i	0.00514661	−	0.0144733i
$190$		0			0
$191$		−	8.61903i		−	0.623651i	−0.950139	−	0.311826i	$-0.899060\pi$
$191$		−	8.61903i		−	0.623651i	0.950139	−	0.311826i	$-0.100940\pi$
$192$		0			0
$193$	−18.9038			−1.36072			−0.680361	−	0.732877i	$-0.738177\pi$
$193$	−18.9038			−1.36072			−0.680361	+	0.732877i	$0.738177\pi$
$194$		0			0
$195$		−	9.40816i		−	0.673733i
$196$		0			0
$197$		−	4.49011i		−	0.319907i	−0.987125	−	0.159954i	$-0.948865\pi$
$197$		−	4.49011i		−	0.319907i	0.987125	−	0.159954i	$-0.0511345\pi$
$198$		0			0
$199$	9.15280			0.648825			0.324413	−	0.945916i	$-0.394833\pi$
$199$	9.15280			0.648825			0.324413	+	0.945916i	$0.394833\pi$
$200$		0			0
$201$			5.19968i			0.366757i
$202$		0			0
$203$	−2.62315	+	7.37682i	−0.184109	+	0.517751i
$204$		0			0
$205$			13.2957i			0.928609i
$206$		0			0
$207$		−	2.41374i		−	0.167767i
$208$		0			0
$209$			8.16005i			0.564443i
$210$		0			0
$211$	−8.77366			−0.604004			−0.302002	−	0.953307i	$-0.597655\pi$
$211$	−8.77366			−0.604004			−0.302002	+	0.953307i	$0.597655\pi$
$212$		0			0
$213$	−9.76686			−0.669214
$214$		0			0
$215$	−19.7892			−1.34961
$216$		0			0
$217$	19.3013	+	6.86343i	1.31026	+	0.465920i
$218$		0			0
$219$	−22.5988			−1.52708
$220$		0			0
$221$		−	3.46118i		−	0.232824i
$222$		0			0
$223$	1.21343			0.0812571			0.0406286	−	0.999174i	$-0.487064\pi$
$223$	1.21343			0.0812571			0.0406286	+	0.999174i	$0.487064\pi$
$224$		0			0
$225$	−29.1778			−1.94519
$226$		0			0
$227$		−	27.1333i		−	1.80090i	−0.434958	−	0.900451i	$-0.643237\pi$
$227$		−	27.1333i		−	1.80090i	0.434958	−	0.900451i	$-0.356763\pi$
$228$		0			0
$229$	−0.338616			−0.0223764			−0.0111882	−	0.999937i	$-0.503561\pi$
$229$	−0.338616			−0.0223764			−0.0111882	+	0.999937i	$0.503561\pi$
$230$		0			0
$231$	3.92425	−	11.0358i	0.258197	−	0.726100i
$232$		0			0
$233$	−2.38874			−0.156492			−0.0782459	−	0.996934i	$-0.524932\pi$
$233$	−2.38874			−0.156492			−0.0782459	+	0.996934i	$0.524932\pi$
$234$		0			0
$235$	48.5192			3.16504
$236$		0			0
$237$	−28.9254			−1.87891
$238$		0			0
$239$		−	22.6748i		−	1.46671i	−0.679846	−	0.733355i	$-0.737953\pi$
$239$		−	22.6748i		−	1.46671i	0.679846	−	0.733355i	$-0.262047\pi$
$240$		0			0
$241$			24.4172i			1.57285i	0.617685	+	0.786426i	$0.288071\pi$
$241$			24.4172i			1.57285i	−0.617685	+	0.786426i	$0.711929\pi$
$242$		0			0
$243$		−	21.9827i		−	1.41019i
$244$		0			0
$245$	20.9070	+	17.0210i	1.33570	+	1.08743i
$246$		0			0
$247$		−	4.50271i		−	0.286501i
$248$		0			0
$249$	43.0599			2.72881
$250$		0			0
$251$			10.9009i			0.688057i	0.938959	+	0.344029i	$0.111792\pi$
$251$			10.9009i			0.688057i	−0.938959	+	0.344029i	$0.888208\pi$
$252$		0			0
$253$			1.47416i			0.0926797i
$254$		0			0
$255$	−32.5634			−2.03920
$256$		0			0
$257$		−	27.1459i		−	1.69331i	−0.532140	−	0.846657i	$-0.678612\pi$
$257$		−	27.1459i		−	1.69331i	0.532140	−	0.846657i	$-0.321388\pi$
$258$		0			0
$259$	−0.874871	+	2.46031i	−0.0543619	+	0.152876i
$260$		0			0
$261$			8.78094i			0.543526i
$262$		0			0
$263$			27.3617i			1.68720i	0.536975	+	0.843598i	$0.319567\pi$
$263$			27.3617i			1.68720i	−0.536975	+	0.843598i	$0.680433\pi$
$264$		0			0
$265$			4.66356i			0.286480i
$266$		0			0
$267$	−1.59069			−0.0973488
$268$		0			0
$269$	29.3298			1.78827			0.894134	−	0.447799i	$-0.147792\pi$
$269$	29.3298			1.78827			0.894134	+	0.447799i	$0.147792\pi$
$270$		0			0
$271$	−4.63752			−0.281709			−0.140855	−	0.990030i	$-0.544985\pi$
$271$	−4.63752			−0.281709			−0.140855	+	0.990030i	$0.544985\pi$
$272$		0			0
$273$	−2.16540	+	6.08953i	−0.131056	+	0.368555i
$274$		0			0
$275$	17.8200			1.07458
$276$		0			0
$277$			2.25309i			0.135375i	0.997707	+	0.0676876i	$0.0215621\pi$
$277$			2.25309i			0.135375i	−0.997707	+	0.0676876i	$0.978438\pi$
$278$		0			0
$279$	22.9752			1.37549
$280$		0			0
$281$	−7.46920			−0.445575			−0.222788	−	0.974867i	$-0.571516\pi$
$281$	−7.46920			−0.445575			−0.222788	+	0.974867i	$0.571516\pi$
$282$		0			0
$283$		−	0.265875i		−	0.0158047i	−0.999969	−	0.00790233i	$-0.997485\pi$
$283$		−	0.265875i		−	0.0158047i	0.999969	−	0.00790233i	$-0.00251541\pi$
$284$		0			0
$285$	−42.3622			−2.50932
$286$		0			0
$287$	3.06015	−	8.60575i	0.180635	−	0.507981i
$288$		0			0
$289$	5.02020			0.295306
$290$		0			0
$291$	−25.2316			−1.47910
$292$		0			0
$293$	4.28179			0.250145			0.125072	−	0.992148i	$-0.460084\pi$
$293$	4.28179			0.250145			0.125072	+	0.992148i	$0.460084\pi$
$294$		0			0
$295$			49.6142i			2.88865i
$296$		0			0
$297$			0.144652i			0.00839354i
$298$		0			0
$299$		−	0.813440i		−	0.0470425i
$300$		0			0
$301$	12.8088	+	4.55471i	0.738284	+	0.262529i
$302$		0			0
$303$			44.0000i			2.52773i
$304$		0			0
$305$	−18.0829			−1.03543
$306$		0			0
$307$		−	2.40502i		−	0.137262i	−0.997642	−	0.0686309i	$-0.978137\pi$
$307$		−	2.40502i		−	0.137262i	0.997642	−	0.0686309i	$-0.0218631\pi$
$308$		0			0
$309$		−	8.69808i		−	0.494817i
$310$		0			0
$311$	18.9445			1.07424			0.537121	−	0.843505i	$-0.319512\pi$
$311$	18.9445			1.07424			0.537121	+	0.843505i	$0.319512\pi$
$312$		0			0
$313$			2.60232i			0.147092i	0.997292	+	0.0735460i	$0.0234316\pi$
$313$			2.60232i			0.147092i	−0.997292	+	0.0735460i	$0.976568\pi$
$314$		0			0
$315$	28.4888	+	10.1304i	1.60516	+	0.570785i
$316$		0			0
$317$			2.67627i			0.150314i	0.997172	+	0.0751571i	$0.0239458\pi$
$317$			2.67627i			0.150314i	−0.997172	+	0.0751571i	$0.976054\pi$
$318$		0			0
$319$		−	5.36284i		−	0.300261i
$320$		0			0
$321$			35.4177i			1.97682i
$322$		0			0
$323$	−15.5847			−0.867157
$324$		0			0
$325$	−9.83303			−0.545439
$326$		0			0
$327$	−42.8131			−2.36757
$328$		0			0
$329$	−31.4045	−	11.1672i	−1.73139	−	0.615670i
$330$		0			0
$331$	−15.6728			−0.861453			−0.430726	−	0.902483i	$-0.641743\pi$
$331$	−15.6728			−0.861453			−0.430726	+	0.902483i	$0.641743\pi$
$332$		0			0
$333$			2.92861i			0.160487i
$334$		0			0
$335$	8.19788			0.447898
$336$		0			0
$337$	27.0995			1.47620			0.738102	−	0.674690i	$-0.235723\pi$
$337$	27.0995			1.47620			0.738102	+	0.674690i	$0.235723\pi$
$338$		0			0
$339$		−	23.7852i		−	1.29183i
$340$		0			0
$341$	−14.0318			−0.759864
$342$		0			0
$343$	−9.61467	−	15.8290i	−0.519144	−	0.854687i
$344$		0			0
$345$	−7.65298			−0.412023
$346$		0			0
$347$	2.59125			0.139106			0.0695528	−	0.997578i	$-0.477843\pi$
$347$	2.59125			0.139106			0.0695528	+	0.997578i	$0.477843\pi$
$348$		0			0
$349$	−12.4308			−0.665408			−0.332704	−	0.943031i	$-0.607961\pi$
$349$	−12.4308			−0.665408			−0.332704	+	0.943031i	$0.607961\pi$
$350$		0			0
$351$		−	0.0798187i		−	0.00426041i
$352$		0			0
$353$			12.9113i			0.687198i	0.939116	+	0.343599i	$0.111646\pi$
$353$			12.9113i			0.687198i	−0.939116	+	0.343599i	$0.888354\pi$
$354$		0			0
$355$			15.3986i			0.817271i
$356$		0			0
$357$	21.0770	+	7.49484i	1.11551	+	0.396669i
$358$		0			0
$359$		−	15.4518i		−	0.815517i	−0.913090	−	0.407759i	$-0.866310\pi$
$359$		−	15.4518i		−	0.815517i	0.913090	−	0.407759i	$-0.133690\pi$
$360$		0			0
$361$	−1.27441			−0.0670740
$362$		0			0
$363$		−	18.8481i		−	0.989268i
$364$		0			0
$365$			35.6295i			1.86494i
$366$		0			0
$367$	−31.0339			−1.61995			−0.809977	−	0.586461i	$-0.800521\pi$
$367$	−31.0339			−1.61995			−0.809977	+	0.586461i	$0.800521\pi$
$368$		0			0
$369$		−	10.2438i		−	0.533270i
$370$		0			0
$371$	1.07337	−	3.01854i	0.0557267	−	0.156715i
$372$		0			0
$373$		−	23.1844i		−	1.20044i	−0.799835	−	0.600220i	$-0.795080\pi$
$373$		−	23.1844i		−	1.20044i	0.799835	−	0.600220i	$-0.204920\pi$
$374$		0			0
$375$			45.4700i			2.34806i
$376$		0			0
$377$			2.95921i			0.152407i
$378$		0			0
$379$	9.21018			0.473095			0.236547	−	0.971620i	$-0.423984\pi$
$379$	9.21018			0.473095			0.236547	+	0.971620i	$0.423984\pi$
$380$		0			0
$381$	1.89501			0.0970845
$382$		0			0
$383$	−1.04939			−0.0536214			−0.0268107	−	0.999641i	$-0.508535\pi$
$383$	−1.04939			−0.0536214			−0.0268107	+	0.999641i	$0.508535\pi$
$384$		0			0
$385$	−17.3991	−	6.18702i	−0.886742	−	0.315320i
$386$		0			0
$387$	15.2468			0.775038
$388$		0			0
$389$		−	33.8614i		−	1.71684i	−0.512948	−	0.858419i	$-0.671447\pi$
$389$		−	33.8614i		−	1.71684i	0.512948	−	0.858419i	$-0.328553\pi$
$390$		0			0
$391$	−2.81547			−0.142384
$392$		0			0
$393$	13.2071			0.666208
$394$		0			0
$395$			45.6041i			2.29459i
$396$		0			0
$397$	−15.8664			−0.796314			−0.398157	−	0.917317i	$-0.630350\pi$
$397$	−15.8664			−0.796314			−0.398157	+	0.917317i	$0.630350\pi$
$398$		0			0
$399$	27.4194	+	9.75015i	1.37269	+	0.488118i
$400$		0			0
$401$	−2.08850			−0.104295			−0.0521473	−	0.998639i	$-0.516607\pi$
$401$	−2.08850			−0.104295			−0.0521473	+	0.998639i	$0.516607\pi$
$402$		0			0
$403$	7.74273			0.385693
$404$		0			0
$405$	−35.0357			−1.74094
$406$		0			0
$407$		−	1.78861i		−	0.0886581i
$408$		0			0
$409$			1.03994i			0.0514215i	0.999669	+	0.0257107i	$0.00818488\pi$
$409$			1.03994i			0.0514215i	−0.999669	+	0.0257107i	$0.991815\pi$
$410$		0			0
$411$			12.9368i			0.638123i
$412$		0			0
$413$	11.4193	−	32.1133i	0.561906	−	1.58019i
$414$		0			0
$415$		−	67.8889i		−	3.33253i
$416$		0			0
$417$	38.5031			1.88550
$418$		0			0
$419$			17.2962i			0.844973i	0.906369	+	0.422486i	$0.138843\pi$
$419$			17.2962i			0.844973i	−0.906369	+	0.422486i	$0.861157\pi$
$420$		0			0
$421$		−	21.9062i		−	1.06764i	−0.845597	−	0.533821i	$-0.820755\pi$
$421$		−	21.9062i		−	1.06764i	0.845597	−	0.533821i	$-0.179245\pi$
$422$		0			0
$423$	−37.3821			−1.81758
$424$		0			0
$425$			34.0339i			1.65089i
$426$		0			0
$427$	11.7044	+	4.16200i	0.566414	+	0.201413i
$428$		0			0
$429$		−	4.42699i		−	0.213737i
$430$		0			0
$431$		−	23.8919i		−	1.15083i	−0.817860	−	0.575417i	$-0.804840\pi$
$431$		−	23.8919i		−	1.15083i	0.817860	−	0.575417i	$-0.195160\pi$
$432$		0			0
$433$			2.82638i			0.135827i	0.997691	+	0.0679135i	$0.0216342\pi$
$433$			2.82638i			0.135827i	−0.997691	+	0.0679135i	$0.978366\pi$
$434$		0			0
$435$	27.8407			1.33486
$436$		0			0
$437$	−3.66269			−0.175210
$438$		0			0
$439$	1.33988			0.0639489			0.0319745	−	0.999489i	$-0.489820\pi$
$439$	1.33988			0.0639489			0.0319745	+	0.999489i	$0.489820\pi$
$440$		0			0
$441$	−16.1080	−	13.1140i	−0.767048	−	0.624478i
$442$		0			0
$443$	7.54624			0.358533			0.179266	−	0.983801i	$-0.442628\pi$
$443$	7.54624			0.358533			0.179266	+	0.983801i	$0.442628\pi$
$444$		0			0
$445$			2.50791i			0.118886i
$446$		0			0
$447$	49.4753			2.34010
$448$		0			0
$449$	17.7511			0.837725			0.418862	−	0.908050i	$-0.362429\pi$
$449$	17.7511			0.837725			0.418862	+	0.908050i	$0.362429\pi$
$450$		0			0
$451$			6.25625i			0.294595i
$452$		0			0
$453$	26.0137			1.22223
$454$		0			0
$455$	9.60083	+	3.41399i	0.450094	+	0.160050i
$456$		0			0
$457$	1.09635			0.0512851			0.0256425	−	0.999671i	$-0.491837\pi$
$457$	1.09635			0.0512851			0.0256425	+	0.999671i	$0.491837\pi$
$458$		0			0
$459$	−0.276267			−0.0128950
$460$		0			0
$461$	3.61769			0.168493			0.0842463	−	0.996445i	$-0.473152\pi$
$461$	3.61769			0.168493			0.0842463	+	0.996445i	$0.473152\pi$
$462$		0			0
$463$		−	3.14597i		−	0.146206i	−0.997324	−	0.0731029i	$-0.976710\pi$
$463$		−	3.14597i		−	0.146206i	0.997324	−	0.0731029i	$-0.0232901\pi$
$464$		0			0
$465$		−	72.8448i		−	3.37810i
$466$		0			0
$467$			22.1648i			1.02567i	0.858488	+	0.512833i	$0.171404\pi$
$467$			22.1648i			1.02567i	−0.858488	+	0.512833i	$0.828596\pi$
$468$		0			0
$469$	−5.30616	−	1.88684i	−0.245016	−	0.0871260i
$470$		0			0
$471$		−	31.5054i		−	1.45169i
$472$		0			0
$473$	−9.31177			−0.428155
$474$		0			0
$475$			44.2753i			2.03149i
$476$		0			0
$477$		−	3.59309i		−	0.164516i
$478$		0			0
$479$	22.6530			1.03504			0.517521	−	0.855671i	$-0.326855\pi$
$479$	22.6530			1.03504			0.517521	+	0.855671i	$0.326855\pi$
$480$		0			0
$481$			0.986953i			0.0450012i
$482$		0			0
$483$	4.95347	+	1.76142i	0.225391	+	0.0801474i
$484$		0			0
$485$			39.7804i			1.80634i
$486$		0			0
$487$			0.888954i			0.0402824i	0.999797	+	0.0201412i	$0.00641157\pi$
$487$			0.888954i			0.0402824i	−0.999797	+	0.0201412i	$0.993588\pi$
$488$		0			0
$489$		−	42.2403i		−	1.91017i
$490$		0			0
$491$	−1.30699			−0.0589837			−0.0294918	−	0.999565i	$-0.509389\pi$
$491$	−1.30699			−0.0589837			−0.0294918	+	0.999565i	$0.509389\pi$
$492$		0			0
$493$	10.2424			0.461293
$494$		0			0
$495$	−20.7109			−0.930886
$496$		0			0
$497$	3.54415	−	9.96687i	0.158977	−	0.447075i
$498$		0			0
$499$	28.7592			1.28744			0.643720	−	0.765262i	$-0.277390\pi$
$499$	28.7592			1.28744			0.643720	+	0.765262i	$0.277390\pi$
$500$		0			0
$501$			23.6313i			1.05577i
$502$		0			0
$503$	11.8526			0.528479			0.264240	−	0.964457i	$-0.414879\pi$
$503$	11.8526			0.528479			0.264240	+	0.964457i	$0.414879\pi$
$504$		0			0
$505$	69.3710			3.08697
$506$		0			0
$507$			2.44281i			0.108489i
$508$		0			0
$509$	15.4601			0.685259			0.342629	−	0.939471i	$-0.388682\pi$
$509$	15.4601			0.685259			0.342629	+	0.939471i	$0.388682\pi$
$510$		0			0
$511$	8.20055	−	23.0616i	0.362771	−	1.02018i
$512$		0			0
$513$	−0.359400			−0.0158679
$514$		0			0
$515$	−13.7135			−0.604289
$516$		0			0
$517$	22.8306			1.00409
$518$		0			0
$519$		−	18.7025i		−	0.820950i
$520$		0			0
$521$			34.6336i			1.51733i	0.651484	+	0.758663i	$0.274147\pi$
$521$			34.6336i			1.51733i	−0.651484	+	0.758663i	$0.725853\pi$
$522$		0			0
$523$		−	25.9793i		−	1.13600i	−0.823030	−	0.567998i	$-0.807718\pi$
$523$		−	25.9793i		−	1.13600i	0.823030	−	0.567998i	$-0.192282\pi$
$524$		0			0
$525$	21.2924	−	59.8785i	0.929277	−	2.61331i
$526$		0			0
$527$		−	26.7990i		−	1.16738i
$528$		0			0
$529$	22.3383			0.971231
$530$		0			0
$531$		−	38.2258i		−	1.65886i
$532$		0			0
$533$		−	3.45219i		−	0.149531i
$534$		0			0
$535$	55.8401			2.41418
$536$		0			0
$537$			55.4468i			2.39271i
$538$		0			0
$539$	9.83775	+	8.00922i	0.423742	+	0.344982i
$540$		0			0
$541$		−	12.5538i		−	0.539732i	−0.962898	−	0.269866i	$-0.913021\pi$
$541$		−	12.5538i		−	0.539732i	0.962898	−	0.269866i	$-0.0869794\pi$
$542$		0			0
$543$			40.8519i			1.75312i
$544$		0			0
$545$			67.4997i			2.89137i
$546$		0			0
$547$	19.8173			0.847327			0.423663	−	0.905820i	$-0.360744\pi$
$547$	19.8173			0.847327			0.423663	+	0.905820i	$0.360744\pi$
$548$		0			0
$549$	13.9322			0.594611
$550$		0			0
$551$	13.3245			0.567641
$552$		0			0
$553$	10.4963	−	29.5177i	0.446349	−	1.25522i
$554$		0			0
$555$	9.28542			0.394144
$556$		0			0
$557$			22.7061i			0.962089i	0.876696	+	0.481044i	$0.159742\pi$
$557$			22.7061i			0.962089i	−0.876696	+	0.481044i	$0.840258\pi$
$558$		0			0
$559$	5.13822			0.217324
$560$		0			0
$561$	−15.3226			−0.646922
$562$		0			0
$563$			6.86719i			0.289417i	0.989474	+	0.144709i	$0.0462245\pi$
$563$			6.86719i			0.289417i	−0.989474	+	0.144709i	$0.953775\pi$
$564$		0			0
$565$	−37.5000			−1.57764
$566$		0			0
$567$	22.6772	+	8.06387i	0.952354	+	0.338651i
$568$		0			0
$569$	0.587660			0.0246360			0.0123180	−	0.999924i	$-0.496079\pi$
$569$	0.587660			0.0246360			0.0123180	+	0.999924i	$0.496079\pi$
$570$		0			0
$571$	11.7541			0.491894			0.245947	−	0.969283i	$-0.420901\pi$
$571$	11.7541			0.491894			0.245947	+	0.969283i	$0.420901\pi$
$572$		0			0
$573$	21.0547			0.879571
$574$		0			0
$575$			7.99859i			0.333564i
$576$		0			0
$577$			19.3871i			0.807095i	0.914959	+	0.403547i	$0.132223\pi$
$577$			19.3871i			0.807095i	−0.914959	+	0.403547i	$0.867777\pi$
$578$		0			0
$579$		−	46.1783i		−	1.91910i
$580$		0			0
$581$	−15.6254	+	43.9417i	−0.648251	+	1.82301i
$582$		0			0
$583$			2.19443i			0.0908840i
$584$		0			0
$585$	11.4283			0.472500
$586$		0			0
$587$		−	41.4962i		−	1.71273i	−0.516368	−	0.856367i	$-0.672716\pi$
$587$		−	41.4962i		−	1.71273i	0.516368	−	0.856367i	$-0.327284\pi$
$588$		0			0
$589$		−	34.8633i		−	1.43652i
$590$		0			0
$591$	10.9685			0.451184
$592$		0			0
$593$		−	23.4256i		−	0.961976i	−0.876727	−	0.480988i	$-0.840278\pi$
$593$		−	23.4256i		−	0.961976i	0.876727	−	0.480988i	$-0.159722\pi$
$594$		0			0
$595$	11.8165	−	33.2302i	0.484428	−	1.36231i
$596$		0			0
$597$			22.3586i			0.915075i
$598$		0			0
$599$			16.8992i			0.690485i	0.938514	+	0.345242i	$0.112203\pi$
$599$			16.8992i			0.690485i	−0.938514	+	0.345242i	$0.887797\pi$
$600$		0			0
$601$		−	41.1108i		−	1.67695i	−0.544943	−	0.838473i	$-0.683449\pi$
$601$		−	41.1108i		−	1.67695i	0.544943	−	0.838473i	$-0.316551\pi$
$602$		0			0
$603$	−6.31614			−0.257213
$604$		0			0
$605$	−29.7161			−1.20813
$606$		0			0
$607$	35.6288			1.44613			0.723064	−	0.690781i	$-0.242733\pi$
$607$	35.6288			1.44613			0.723064	+	0.690781i	$0.242733\pi$
$608$		0			0
$609$	−18.0202	−	6.40786i	−0.730215	−	0.259660i
$610$		0			0
$611$	−12.5979			−0.509657
$612$		0			0
$613$		−	15.0402i		−	0.607466i	−0.952757	−	0.303733i	$-0.901767\pi$
$613$		−	15.0402i		−	0.607466i	0.952757	−	0.303733i	$-0.0982331\pi$
$614$		0			0
$615$	−32.4788			−1.30967
$616$		0			0
$617$	−20.9019			−0.841479			−0.420739	−	0.907182i	$-0.638229\pi$
$617$	−20.9019			−0.841479			−0.420739	+	0.907182i	$0.638229\pi$
$618$		0			0
$619$			32.1484i			1.29215i	0.763272	+	0.646077i	$0.223591\pi$
$619$			32.1484i			1.29215i	−0.763272	+	0.646077i	$0.776409\pi$
$620$		0			0
$621$	−0.0649277			−0.00260546
$622$		0			0
$623$	0.577223	−	1.62327i	0.0231260	−	0.0650348i
$624$		0			0
$625$	22.5234			0.900935
$626$		0			0
$627$	−19.9335			−0.796066
$628$		0			0
$629$	3.41603			0.136206
$630$		0			0
$631$			20.4474i			0.813998i	0.913429	+	0.406999i	$0.133425\pi$
$631$			20.4474i			0.813998i	−0.913429	+	0.406999i	$0.866575\pi$
$632$		0			0
$633$		−	21.4324i		−	0.851861i
$634$		0			0
$635$		−	2.98770i		−	0.118563i
$636$		0			0
$637$	−5.42846	−	4.41948i	−0.215083	−	0.175106i
$638$		0			0
$639$		−	11.8640i		−	0.469332i
$640$		0			0
$641$	19.9013			0.786056			0.393028	−	0.919527i	$-0.371428\pi$
$641$	19.9013			0.786056			0.393028	+	0.919527i	$0.371428\pi$
$642$		0			0
$643$		−	36.6222i		−	1.44424i	−0.691768	−	0.722120i	$-0.743168\pi$
$643$		−	36.6222i		−	1.44424i	0.691768	−	0.722120i	$-0.256832\pi$
$644$		0			0
$645$		−	48.3413i		−	1.90343i
$646$		0			0
$647$	−33.4255			−1.31409			−0.657045	−	0.753851i	$-0.728194\pi$
$647$	−33.4255			−1.31409			−0.657045	+	0.753851i	$0.728194\pi$
$648$		0			0
$649$			23.3459i			0.916405i
$650$		0			0
$651$	−16.7661	+	47.1495i	−0.657114	+	1.84794i
$652$		0			0
$653$			15.8386i			0.619811i	0.950767	+	0.309905i	$0.100297\pi$
$653$			15.8386i			0.619811i	−0.950767	+	0.309905i	$0.899703\pi$
$654$		0			0
$655$		−	20.8224i		−	0.813600i
$656$		0			0
$657$		−	27.4511i		−	1.07097i
$658$		0			0
$659$	35.1659			1.36987			0.684935	−	0.728604i	$-0.259831\pi$
$659$	35.1659			1.36987			0.684935	+	0.728604i	$0.259831\pi$
$660$		0			0
$661$	48.9746			1.90489			0.952446	−	0.304707i	$-0.0985586\pi$
$661$	48.9746			1.90489			0.952446	+	0.304707i	$0.0985586\pi$
$662$		0			0
$663$	8.45502			0.328366
$664$		0			0
$665$	15.3722	−	43.2298i	0.596109	−	1.67638i
$666$		0			0
$667$	2.40714			0.0932049
$668$		0			0
$669$			2.96417i			0.114602i
$670$		0			0
$671$	−8.50890			−0.328482
$672$		0			0
$673$	−12.2361			−0.471666			−0.235833	−	0.971794i	$-0.575782\pi$
$673$	−12.2361			−0.471666			−0.235833	+	0.971794i	$0.575782\pi$
$674$		0			0
$675$			0.784860i			0.0302093i
$676$		0			0
$677$	12.6987			0.488051			0.244025	−	0.969769i	$-0.421532\pi$
$677$	12.6987			0.488051			0.244025	+	0.969769i	$0.421532\pi$
$678$		0			0
$679$	9.15592	−	25.7483i	0.351372	−	0.988128i
$680$		0			0
$681$	66.2816			2.53992
$682$		0			0
$683$	14.5785			0.557831			0.278916	−	0.960316i	$-0.410025\pi$
$683$	14.5785			0.557831			0.278916	+	0.960316i	$0.410025\pi$
$684$		0			0
$685$	20.3963			0.779301
$686$		0			0
$687$		−	0.827175i		−	0.0315587i
$688$		0			0
$689$		−	1.21088i		−	0.0461310i
$690$		0			0
$691$			23.6667i			0.900323i	0.892947	+	0.450162i	$0.148634\pi$
$691$			23.6667i			0.900323i	−0.892947	+	0.450162i	$0.851366\pi$
$692$		0			0
$693$	13.4053	+	4.76685i	0.509227	+	0.181078i
$694$		0			0
$695$		−	60.7045i		−	2.30265i
$696$		0			0
$697$	−11.9487			−0.452588
$698$		0			0
$699$		−	5.83525i		−	0.220709i
$700$		0			0
$701$			7.44505i			0.281196i	0.990067	+	0.140598i	$0.0449025\pi$
$701$			7.44505i			0.281196i	−0.990067	+	0.140598i	$0.955098\pi$
$702$		0			0
$703$	4.44397			0.167607
$704$		0			0
$705$			118.523i			4.46384i
$706$		0			0
$707$	−44.9010	−	15.9665i	−1.68868	−	0.600483i
$708$		0			0
$709$		−	44.0918i		−	1.65590i	−0.560800	−	0.827951i	$-0.689506\pi$
$709$		−	44.0918i		−	1.65590i	0.560800	−	0.827951i	$-0.310494\pi$
$710$		0			0
$711$		−	35.1362i		−	1.31771i
$712$		0			0
$713$		−	6.29825i		−	0.235871i
$714$		0			0
$715$	−6.97966			−0.261024
$716$		0			0
$717$	55.3902			2.06859
$718$		0			0
$719$	−24.0504			−0.896929			−0.448464	−	0.893801i	$-0.648029\pi$
$719$	−24.0504			−0.896929			−0.448464	+	0.893801i	$0.648029\pi$
$720$		0			0
$721$	8.87621	+	3.15632i	0.330567	+	0.117548i
$722$		0			0
$723$	−59.6467			−2.21828
$724$		0			0
$725$		−	29.0980i		−	1.08067i
$726$		0			0
$727$	−36.9514			−1.37045			−0.685226	−	0.728330i	$-0.740297\pi$
$727$	−36.9514			−1.37045			−0.685226	+	0.728330i	$0.740297\pi$
$728$		0			0
$729$	26.4087			0.978099
$730$		0			0
$731$		−	17.7843i		−	0.657778i
$732$		0			0
$733$	−22.0569			−0.814692			−0.407346	−	0.913274i	$-0.633546\pi$
$733$	−22.0569			−0.814692			−0.407346	+	0.913274i	$0.633546\pi$
$734$		0			0
$735$	−41.5792	+	51.0718i	−1.53367	+	1.88381i
$736$		0			0
$737$	3.85750			0.142093
$738$		0			0
$739$	−36.0166			−1.32489			−0.662445	−	0.749110i	$-0.730481\pi$
$739$	−36.0166			−1.32489			−0.662445	+	0.749110i	$0.730481\pi$
$740$		0			0
$741$	10.9993			0.404068
$742$		0			0
$743$		−	10.2860i		−	0.377358i	−0.982039	−	0.188679i	$-0.939579\pi$
$743$		−	10.2860i		−	0.377358i	0.982039	−	0.188679i	$-0.0604206\pi$
$744$		0			0
$745$		−	78.0034i		−	2.85782i
$746$		0			0
$747$			52.3057i			1.91376i
$748$		0			0
$749$	−36.1430	−	12.8522i	−1.32064	−	0.469610i
$750$		0			0
$751$			34.8371i			1.27122i	0.772009	+	0.635612i	$0.219252\pi$
$751$			34.8371i			1.27122i	−0.772009	+	0.635612i	$0.780748\pi$
$752$		0			0
$753$	−26.6288			−0.970407
$754$		0			0
$755$		−	41.0135i		−	1.49263i
$756$		0			0
$757$			6.93156i			0.251932i	0.992035	+	0.125966i	$0.0402030\pi$
$757$			6.93156i			0.251932i	−0.992035	+	0.125966i	$0.959797\pi$
$758$		0			0
$759$	−3.60110			−0.130711
$760$		0			0
$761$			32.8554i			1.19101i	0.803352	+	0.595504i	$0.203048\pi$
$761$			32.8554i			1.19101i	−0.803352	+	0.595504i	$0.796952\pi$
$762$		0			0
$763$	15.5358	−	43.6899i	0.562435	−	1.58168i
$764$		0			0
$765$		−	39.5553i		−	1.43013i
$766$		0			0
$767$		−	12.8822i		−	0.465150i
$768$		0			0
$769$		−	29.5660i		−	1.06618i	−0.846059	−	0.533089i	$-0.821031\pi$
$769$		−	29.5660i		−	1.06618i	0.846059	−	0.533089i	$-0.178969\pi$
$770$		0			0
$771$	66.3122			2.38818
$772$		0			0
$773$	−28.4754			−1.02419			−0.512094	−	0.858929i	$-0.671130\pi$
$773$	−28.4754			−1.02419			−0.512094	+	0.858929i	$0.671130\pi$
$774$		0			0
$775$	−76.1345			−2.73483
$776$		0			0
$777$	−6.01008	−	2.13715i	−0.215610	−	0.0766697i
$778$		0			0
$779$	−15.5442			−0.556930
$780$		0			0
$781$			7.24576i			0.259274i
$782$		0			0
$783$	0.236200			0.00844110
$784$		0			0
$785$	−49.6717			−1.77286
$786$		0			0
$787$			39.7645i			1.41745i	0.705484	+	0.708726i	$0.250730\pi$
$787$			39.7645i			1.41745i	−0.705484	+	0.708726i	$0.749270\pi$
$788$		0			0
$789$	−66.8395			−2.37955
$790$		0			0
$791$	24.2723	+	8.63106i	0.863022	+	0.306885i
$792$		0			0
$793$	4.69520			0.166731
$794$		0			0
$795$	−11.3922			−0.404039
$796$		0			0
$797$	1.40341			0.0497114			0.0248557	−	0.999691i	$-0.492087\pi$
$797$	1.40341			0.0497114			0.0248557	+	0.999691i	$0.492087\pi$
$798$		0			0
$799$			43.6037i			1.54259i
$800$		0			0
$801$		−	1.93224i		−	0.0682724i
$802$		0			0
$803$			16.7654i			0.591639i
$804$		0			0
$805$	2.77708	−	7.80970i	0.0978792	−	0.275256i
$806$		0			0
$807$			71.6471i			2.52210i
$808$		0			0
$809$	14.9618			0.526027			0.263014	−	0.964792i	$-0.415284\pi$
$809$	14.9618			0.526027			0.263014	+	0.964792i	$0.415284\pi$
$810$		0			0
$811$		−	3.70960i		−	0.130262i	−0.997877	−	0.0651308i	$-0.979254\pi$
$811$		−	3.70960i		−	0.130262i	0.997877	−	0.0651308i	$-0.0207464\pi$
$812$		0			0
$813$		−	11.3286i		−	0.397311i
$814$		0			0
$815$	−66.5967			−2.33278
$816$		0			0
$817$		−	23.1359i		−	0.809424i
$818$		0			0
$819$	−7.39705	−	2.63034i	−0.258474	−	0.0919117i
$820$		0			0
$821$			36.0973i			1.25981i	0.776674	+	0.629903i	$0.216905\pi$
$821$			36.0973i			1.25981i	−0.776674	+	0.629903i	$0.783095\pi$
$822$		0			0
$823$		−	7.14831i		−	0.249175i	−0.992209	−	0.124587i	$-0.960239\pi$
$823$		−	7.14831i		−	0.249175i	0.992209	−	0.124587i	$-0.0397607\pi$
$824$		0			0
$825$			43.5308i			1.51555i
$826$		0			0
$827$	45.0128			1.56525			0.782625	−	0.622493i	$-0.213880\pi$
$827$	45.0128			1.56525			0.782625	+	0.622493i	$0.213880\pi$
$828$		0			0
$829$	−1.70523			−0.0592250			−0.0296125	−	0.999561i	$-0.509427\pi$
$829$	−1.70523			−0.0592250			−0.0296125	+	0.999561i	$0.509427\pi$
$830$		0			0
$831$	−5.50388			−0.190927
$832$		0			0
$833$	−15.2966	+	18.7889i	−0.529997	+	0.650997i
$834$		0			0
$835$	37.2575			1.28935
$836$		0			0
$837$		−	0.618014i		−	0.0213617i
$838$		0			0
$839$	9.92632			0.342694			0.171347	−	0.985211i	$-0.445188\pi$
$839$	9.92632			0.342694			0.171347	+	0.985211i	$0.445188\pi$
$840$		0			0
$841$	20.2431			0.698037
$842$		0			0
$843$		−	18.2458i		−	0.628420i
$844$		0			0
$845$	3.85137			0.132491
$846$		0			0
$847$	19.2341	+	6.83951i	0.660890	+	0.235008i
$848$		0			0
$849$	0.649483			0.0222902
$850$		0			0
$851$	0.802828			0.0275206
$852$		0			0
$853$	7.29950			0.249930			0.124965	−	0.992161i	$-0.460118\pi$
$853$	7.29950			0.249930			0.124965	+	0.992161i	$0.460118\pi$
$854$		0			0
$855$		−	51.4582i		−	1.75983i
$856$		0			0
$857$			42.5363i			1.45301i	0.687160	+	0.726506i	$0.258857\pi$
$857$			42.5363i			1.45301i	−0.687160	+	0.726506i	$0.741143\pi$
$858$		0			0
$859$			18.3243i			0.625217i	0.949882	+	0.312609i	$0.101203\pi$
$859$			18.3243i			0.625217i	−0.949882	+	0.312609i	$0.898797\pi$
$860$		0			0
$861$	21.0222	+	7.47536i	0.716435	+	0.254760i
$862$		0			0
$863$			31.0819i			1.05804i	0.848610	+	0.529019i	$0.177440\pi$
$863$			31.0819i			1.05804i	−0.848610	+	0.529019i	$0.822560\pi$
$864$		0			0
$865$	−29.4867			−1.00258
$866$		0			0
$867$			12.2634i			0.416487i
$868$		0			0
$869$			21.4589i			0.727945i
$870$		0			0
$871$	−2.12856			−0.0721236
$872$		0			0
$873$		−	30.6492i		−	1.03732i
$874$		0			0
$875$	−46.4011	−	16.4999i	−1.56864	−	0.557800i
$876$		0			0
$877$		−	50.5638i		−	1.70742i	−0.520752	−	0.853708i	$-0.674348\pi$
$877$		−	50.5638i		−	1.70742i	0.520752	−	0.853708i	$-0.325652\pi$
$878$		0			0
$879$			10.4596i			0.352794i
$880$		0			0
$881$		−	0.189942i		−	0.00639930i	−0.999995	−	0.00319965i	$-0.998982\pi$
$881$		−	0.189942i		−	0.00639930i	0.999995	−	0.00319965i	$-0.00101848\pi$
$882$		0			0
$883$	32.9338			1.10831			0.554156	−	0.832413i	$-0.313041\pi$
$883$	32.9338			1.10831			0.554156	+	0.832413i	$0.313041\pi$
$884$		0			0
$885$	−121.198			−4.07403
$886$		0			0
$887$	−36.6972			−1.23217			−0.616086	−	0.787679i	$-0.711283\pi$
$887$	−36.6972			−1.23217			−0.616086	+	0.787679i	$0.711283\pi$
$888$		0			0
$889$	−0.687654	+	1.93382i	−0.0230632	+	0.0648583i
$890$		0			0
$891$	−16.4860			−0.552302
$892$		0			0
$893$			56.7248i			1.89822i
$894$		0			0
$895$	87.4182			2.92207
$896$		0			0
$897$	1.98708			0.0663467
$898$		0			0
$899$			22.9123i			0.764170i
$900$		0			0
$901$	−4.19109			−0.139626
$902$		0			0
$903$	−11.1263	+	31.2894i	−0.370260	+	1.04124i
$904$		0			0
$905$	64.4077			2.14098
$906$		0			0
$907$	30.6366			1.01727			0.508636	−	0.860982i	$-0.330150\pi$
$907$	30.6366			1.01727			0.508636	+	0.860982i	$0.330150\pi$
$908$		0			0
$909$	−53.4476			−1.77274
$910$		0			0
$911$			32.3852i			1.07297i	0.843910	+	0.536484i	$0.180248\pi$
$911$			32.3852i			1.07297i	−0.843910	+	0.536484i	$0.819752\pi$
$912$		0			0
$913$		−	31.9450i		−	1.05722i
$914$		0			0
$915$		−	44.1732i		−	1.46032i
$916$		0			0
$917$	−4.79252	+	13.4775i	−0.158263	+	0.445067i
$918$		0			0
$919$			34.3537i			1.13322i	0.823985	+	0.566612i	$0.191746\pi$
$919$			34.3537i			1.13322i	−0.823985	+	0.566612i	$0.808254\pi$
$920$		0			0
$921$	5.87502			0.193588
$922$		0			0
$923$		−	3.99821i		−	0.131603i
$924$		0			0
$925$		−	9.70475i		−	0.319090i
$926$		0			0
$927$	10.5657			0.347024
$928$		0			0
$929$		−	41.2449i		−	1.35320i	−0.736350	−	0.676601i	$-0.763452\pi$
$929$		−	41.2449i		−	1.35320i	0.736350	−	0.676601i	$-0.236548\pi$
$930$		0			0
$931$	−19.8996	+	24.4428i	−0.652184	+	0.801080i
$932$		0			0
$933$			46.2778i			1.51507i
$934$		0			0
$935$			24.1579i			0.790047i
$936$		0			0
$937$		−	24.8231i		−	0.810936i	−0.914109	−	0.405468i	$-0.867109\pi$
$937$		−	24.8231i		−	0.810936i	0.914109	−	0.405468i	$-0.132891\pi$
$938$		0			0
$939$	−6.35698			−0.207452
$940$		0			0
$941$	22.0956			0.720295			0.360147	−	0.932895i	$-0.382726\pi$
$941$	22.0956			0.720295			0.360147	+	0.932895i	$0.382726\pi$
$942$		0			0
$943$	−2.80815			−0.0914460
$944$		0			0
$945$	0.272500	−	0.766325i	0.00886444	−	0.0249286i
$946$		0			0
$947$	49.5764			1.61102			0.805508	−	0.592585i	$-0.201893\pi$
$947$	49.5764			1.61102			0.805508	+	0.592585i	$0.201893\pi$
$948$		0			0
$949$		−	9.25114i		−	0.300305i
$950$		0			0
$951$	−6.53761			−0.211997
$952$		0			0
$953$	46.4982			1.50622			0.753112	−	0.657892i	$-0.228552\pi$
$953$	46.4982			1.50622			0.753112	+	0.657892i	$0.228552\pi$
$954$		0			0
$955$		−	33.1951i		−	1.07417i
$956$		0			0
$957$	13.1004			0.423476
$958$		0			0
$959$	−13.2017	−	4.69443i	−0.426305	−	0.151591i
$960$		0			0
$961$	28.9498			0.933865
$962$		0			0
$963$	−43.0225			−1.38638
$964$		0			0
$965$	−72.8053			−2.34369
$966$		0			0
$967$		−	31.7620i		−	1.02140i	−0.859760	−	0.510699i	$-0.829387\pi$
$967$		−	31.7620i		−	1.02140i	0.859760	−	0.510699i	$-0.170613\pi$
$968$		0			0
$969$		−	38.0705i		−	1.22300i
$970$		0			0
$971$		−	1.85166i		−	0.0594226i	−0.999559	−	0.0297113i	$-0.990541\pi$
$971$		−	1.85166i		−	0.0594226i	0.999559	−	0.0297113i	$-0.00945879\pi$
$972$		0			0
$973$	−13.9718	+	39.2916i	−0.447916	+	1.25963i
$974$		0			0
$975$		−	24.0202i		−	0.769263i
$976$		0			0
$977$	−40.6090			−1.29920			−0.649598	−	0.760278i	$-0.725063\pi$
$977$	−40.6090			−1.29920			−0.649598	+	0.760278i	$0.725063\pi$
$978$		0			0
$979$			1.18009i			0.0377159i
$980$		0			0
$981$		−	52.0058i		−	1.66042i
$982$		0			0
$983$	23.1405			0.738066			0.369033	−	0.929416i	$-0.379689\pi$
$983$	23.1405			0.738066			0.369033	+	0.929416i	$0.379689\pi$
$984$		0			0
$985$		−	17.2931i		−	0.551003i
$986$		0			0
$987$	27.2795	−	76.7153i	0.868315	−	2.44188i
$988$		0			0
$989$		−	4.17964i		−	0.132905i
$990$		0			0
$991$			3.02612i			0.0961279i	0.998844	+	0.0480640i	$0.0153051\pi$
$991$			3.02612i			0.0961279i	−0.998844	+	0.0480640i	$0.984695\pi$
$992$		0			0
$993$		−	38.2856i		−	1.21496i
$994$		0			0
$995$	35.2508			1.11753
$996$		0			0
$997$	−25.0174			−0.792308			−0.396154	−	0.918184i	$-0.629655\pi$
$997$	−25.0174			−0.792308			−0.396154	+	0.918184i	$0.629655\pi$
$998$		0			0
$999$	0.0787773			0.00249240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.a.2575.42		48
4.3	odd	2		728.2.h.a.27.9	✓	48
7.6	odd	2		2912.2.h.b.2575.7		48
8.3	odd	2		2912.2.h.b.2575.42		48
8.5	even	2		728.2.h.b.27.10	yes	48
28.27	even	2		728.2.h.b.27.9	yes	48
56.13	odd	2		728.2.h.a.27.10	yes	48
56.27	even	2	inner	2912.2.h.a.2575.7		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.9	✓	48	4.3	odd	2
728.2.h.a.27.10	yes	48	56.13	odd	2
728.2.h.b.27.9	yes	48	28.27	even	2
728.2.h.b.27.10	yes	48	8.5	even	2
2912.2.h.a.2575.7		48	56.27	even	2	inner
2912.2.h.a.2575.42		48	1.1	even	1	trivial
2912.2.h.b.2575.7		48	7.6	odd	2
2912.2.h.b.2575.42		48	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44281i q^{3} +3.85137 q^{5} +(-2.49284 - 0.886436i) q^{7} -2.96733 q^{9} +1.81225 q^{11} -1.00000 q^{13} +9.40816i q^{15} +3.46118i q^{17} +4.50271i q^{19} +(2.16540 - 6.08953i) q^{21} +0.813440i q^{23} +9.83303 q^{25} +0.0798187i q^{27} -2.95921i q^{29} -7.74273 q^{31} +4.42699i q^{33} +(-9.60083 - 3.41399i) q^{35} -0.986953i q^{37} -2.44281i q^{39} +3.45219i q^{41} -5.13822 q^{43} -11.4283 q^{45} +12.5979 q^{47} +(5.42846 + 4.41948i) q^{49} -8.45502 q^{51} +1.21088i q^{53} +6.97966 q^{55} -10.9993 q^{57} +12.8822i q^{59} -4.69520 q^{61} +(7.39705 + 2.63034i) q^{63} -3.85137 q^{65} +2.12856 q^{67} -1.98708 q^{69} +3.99821i q^{71} +9.25114i q^{73} +24.0202i q^{75} +(-4.51765 - 1.60645i) q^{77} +11.8410i q^{79} -9.09696 q^{81} -17.6272i q^{83} +13.3303i q^{85} +7.22879 q^{87} +0.651173i q^{89} +(2.49284 + 0.886436i) q^{91} -18.9140i q^{93} +17.3416i q^{95} +10.3289i q^{97} -5.37755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) \) 48 * q - 48 * q^9 + 4 * q^11 - 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 + 24 * q^45 + 40 * q^51 - 20 * q^63 + 4 * q^67 - 20 * q^77 + 64 * q^81 + 40 * q^87 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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