LMFDB - Embedded newform 4368.2.h.q.337.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 273)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.7
Root			$-2.54814i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.q.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.49301i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.49301i q^{5} -1.00000i q^{7} +1.00000 q^{9} +0.708126i q^{11} +(-3.25627 - 1.54814i) q^{13} +3.49301i q^{15} +7.09628 q^{17} +0.311392i q^{19} -1.00000i q^{21} +7.88116 q^{23} -7.20114 q^{25} +1.00000 q^{27} +5.29742 q^{29} -7.29742i q^{31} +0.708126i q^{33} +3.49301 q^{35} +1.41625i q^{37} +(-3.25627 - 1.54814i) q^{39} +11.8044i q^{41} +3.29742 q^{43} +3.49301i q^{45} -6.11580i q^{47} -1.00000 q^{49} +7.09628 q^{51} -3.72764 q^{53} -2.47349 q^{55} +0.311392i q^{57} +2.19560i q^{59} +2.51253 q^{61} -1.00000i q^{63} +(5.40767 - 11.3742i) q^{65} +9.17858i q^{67} +7.88116 q^{69} -0.708126i q^{71} +5.21511i q^{73} -7.20114 q^{75} +0.708126 q^{77} +2.78489 q^{79} +1.00000 q^{81} +6.11580i q^{83} +24.7874i q^{85} +5.29742 q^{87} +11.1018i q^{89} +(-1.54814 + 3.25627i) q^{91} -7.29742i q^{93} -1.08770 q^{95} -7.79886i q^{97} +0.708126i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 8 * q^9	$ 8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 6 * q^13 + 20 * q^17 + 6 * q^23 - 34 * q^25 + 8 * q^27 - 18 * q^29 + 6 * q^35 - 6 * q^39 - 34 * q^43 - 8 * q^49 + 20 * q^51 - 10 * q^53 - 16 * q^55 - 20 * q^61 - 10 * q^65 + 6 * q^69 - 34 * q^75 + 4 * q^77 + 2 * q^79 + 8 * q^81 - 18 * q^87 + 6 * q^91 + 78 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			3.49301i			1.56212i	0.624454	+	0.781061i	$0.285321\pi$
$5$			3.49301i			1.56212i	−0.624454	+	0.781061i	$0.714679\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			0.708126i			0.213508i	0.994285	+	0.106754i	$0.0340458\pi$
$11$			0.708126i			0.213508i	−0.994285	+	0.106754i	$0.965954\pi$
$12$		0			0
$13$	−3.25627	−	1.54814i	−0.903126	−	0.429377i
$13$	−3.25627	−	1.54814i	−0.903126	−	0.429377i
$14$		0			0
$15$			3.49301i			0.901892i
$16$		0			0
$17$	7.09628			1.72110			0.860550	−	0.509366i	$-0.170120\pi$
$17$	7.09628			1.72110			0.860550	+	0.509366i	$0.170120\pi$
$18$		0			0
$19$			0.311392i			0.0714382i	0.999362	+	0.0357191i	$0.0113722\pi$
$19$			0.311392i			0.0714382i	−0.999362	+	0.0357191i	$0.988628\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	7.88116			1.64334			0.821668	−	0.569966i	$-0.193044\pi$
$23$	7.88116			1.64334			0.821668	+	0.569966i	$0.193044\pi$
$24$		0			0
$25$	−7.20114			−1.44023
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	5.29742			0.983706			0.491853	−	0.870678i	$-0.336320\pi$
$29$	5.29742			0.983706			0.491853	+	0.870678i	$0.336320\pi$
$30$		0			0
$31$		−	7.29742i		−	1.31065i	−0.755345	−	0.655327i	$-0.772531\pi$
$31$		−	7.29742i		−	1.31065i	0.755345	−	0.655327i	$-0.227469\pi$
$32$		0			0
$33$			0.708126i			0.123269i
$34$		0			0
$35$	3.49301			0.590427
$36$		0			0
$37$			1.41625i			0.232831i	0.993201	+	0.116415i	$0.0371403\pi$
$37$			1.41625i			0.232831i	−0.993201	+	0.116415i	$0.962860\pi$
$38$		0			0
$39$	−3.25627	−	1.54814i	−0.521420	−	0.247901i
$40$		0			0
$41$			11.8044i			1.84354i	0.387739	+	0.921769i	$0.373256\pi$
$41$			11.8044i			1.84354i	−0.387739	+	0.921769i	$0.626744\pi$
$42$		0			0
$43$	3.29742			0.502851			0.251426	−	0.967877i	$-0.419101\pi$
$43$	3.29742			0.502851			0.251426	+	0.967877i	$0.419101\pi$
$44$		0			0
$45$			3.49301i			0.520708i
$46$		0			0
$47$		−	6.11580i		−	0.892081i	−0.895013	−	0.446040i	$-0.852834\pi$
$47$		−	6.11580i		−	0.892081i	0.895013	−	0.446040i	$-0.147166\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	7.09628			0.993678
$52$		0			0
$53$	−3.72764			−0.512031			−0.256016	−	0.966673i	$-0.582410\pi$
$53$	−3.72764			−0.512031			−0.256016	+	0.966673i	$0.582410\pi$
$54$		0			0
$55$	−2.47349			−0.333526
$56$		0			0
$57$			0.311392i			0.0412448i
$58$		0			0
$59$			2.19560i			0.285842i	0.989734	+	0.142921i	$0.0456495\pi$
$59$			2.19560i			0.285842i	−0.989734	+	0.142921i	$0.954350\pi$
$60$		0			0
$61$	2.51253			0.321697			0.160848	−	0.986979i	$-0.448577\pi$
$61$	2.51253			0.321697			0.160848	+	0.986979i	$0.448577\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	5.40767	−	11.3742i	0.670739	−	1.41079i
$66$		0			0
$67$			9.17858i			1.12134i	0.828039	+	0.560671i	$0.189457\pi$
$67$			9.17858i			1.12134i	−0.828039	+	0.560671i	$0.810543\pi$
$68$		0			0
$69$	7.88116			0.948781
$70$		0			0
$71$		−	0.708126i		−	0.0840391i	−0.999117	−	0.0420196i	$-0.986621\pi$
$71$		−	0.708126i		−	0.0840391i	0.999117	−	0.0420196i	$-0.0133792\pi$
$72$		0			0
$73$			5.21511i			0.610383i	0.952291	+	0.305191i	$0.0987205\pi$
$73$			5.21511i			0.610383i	−0.952291	+	0.305191i	$0.901280\pi$
$74$		0			0
$75$	−7.20114			−0.831516
$76$		0			0
$77$	0.708126			0.0806985
$78$		0			0
$79$	2.78489			0.313324			0.156662	−	0.987652i	$-0.449927\pi$
$79$	2.78489			0.313324			0.156662	+	0.987652i	$0.449927\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			6.11580i			0.671296i	0.941988	+	0.335648i	$0.108955\pi$
$83$			6.11580i			0.671296i	−0.941988	+	0.335648i	$0.891045\pi$
$84$		0			0
$85$			24.7874i			2.68857i
$86$		0			0
$87$	5.29742			0.567943
$88$		0			0
$89$			11.1018i			1.17679i	0.808573	+	0.588395i	$0.200240\pi$
$89$			11.1018i			1.17679i	−0.808573	+	0.588395i	$0.799760\pi$
$90$		0			0
$91$	−1.54814	+	3.25627i	−0.162289	+	0.341349i
$92$		0			0
$93$		−	7.29742i		−	0.756707i
$94$		0			0
$95$	−1.08770			−0.111595
$96$		0			0
$97$		−	7.79886i		−	0.791854i	−0.918282	−	0.395927i	$-0.870423\pi$
$97$		−	7.79886i		−	0.791854i	0.918282	−	0.395927i	$-0.129577\pi$
$98$		0			0
$99$			0.708126i			0.0711694i
$100$		0			0
$101$	17.6911			1.76033			0.880166	−	0.474666i	$-0.157431\pi$
$101$	17.6911			1.76033			0.880166	+	0.474666i	$0.157431\pi$
$102$		0			0
$103$	−5.52651			−0.544543			−0.272271	−	0.962221i	$-0.587775\pi$
$103$	−5.52651			−0.544543			−0.272271	+	0.962221i	$0.587775\pi$
$104$		0			0
$105$	3.49301			0.340883
$106$		0			0
$107$	−19.1786			−1.85406			−0.927032	−	0.374983i	$-0.877649\pi$
$107$	−19.1786			−1.85406			−0.927032	+	0.374983i	$0.877649\pi$
$108$		0			0
$109$			3.20653i			0.307130i	0.988139	+	0.153565i	$0.0490754\pi$
$109$			3.20653i			0.307130i	−0.988139	+	0.153565i	$0.950925\pi$
$110$		0			0
$111$			1.41625i			0.134425i
$112$		0			0
$113$	−5.29742			−0.498339			−0.249170	−	0.968460i	$-0.580158\pi$
$113$	−5.29742			−0.498339			−0.249170	+	0.968460i	$0.580158\pi$
$114$		0			0
$115$			27.5290i			2.56709i
$116$		0			0
$117$	−3.25627	−	1.54814i	−0.301042	−	0.143126i
$118$		0			0
$119$		−	7.09628i		−	0.650515i
$120$		0			0
$121$	10.4986			0.954414
$122$		0			0
$123$			11.8044i			1.06437i
$124$		0			0
$125$		−	7.68861i		−	0.687690i
$126$		0			0
$127$	14.3028			1.26917			0.634585	−	0.772853i	$-0.281171\pi$
$127$	14.3028			1.26917			0.634585	+	0.772853i	$0.281171\pi$
$128$		0			0
$129$	3.29742			0.290321
$130$		0			0
$131$	−12.4023			−1.08359			−0.541796	−	0.840510i	$-0.682255\pi$
$131$	−12.4023			−1.08359			−0.541796	+	0.840510i	$0.682255\pi$
$132$		0			0
$133$	0.311392			0.0270011
$134$		0			0
$135$			3.49301i			0.300631i
$136$		0			0
$137$			5.58390i			0.477065i	0.971135	+	0.238532i	$0.0766663\pi$
$137$			5.58390i			0.477065i	−0.971135	+	0.238532i	$0.923334\pi$
$138$		0			0
$139$	1.52651			0.129476			0.0647382	−	0.997902i	$-0.479379\pi$
$139$	1.52651			0.129476			0.0647382	+	0.997902i	$0.479379\pi$
$140$		0			0
$141$		−	6.11580i		−	0.515043i
$142$		0			0
$143$	1.09628	−	2.30585i	0.0916754	−	0.192825i
$144$		0			0
$145$			18.5039i			1.53667i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$		−	15.7765i		−	1.29246i	−0.763144	−	0.646229i	$-0.776345\pi$
$149$		−	15.7765i		−	1.29246i	0.763144	−	0.646229i	$-0.223655\pi$
$150$		0			0
$151$			6.83251i			0.556021i	0.960578	+	0.278011i	$0.0896751\pi$
$151$			6.83251i			0.556021i	−0.960578	+	0.278011i	$0.910325\pi$
$152$		0			0
$153$	7.09628			0.573700
$154$		0			0
$155$	25.4900			2.04740
$156$		0			0
$157$	−12.1926			−0.973072			−0.486536	−	0.873661i	$-0.661740\pi$
$157$	−12.1926			−0.973072			−0.486536	+	0.873661i	$0.661740\pi$
$158$		0			0
$159$	−3.72764			−0.295621
$160$		0			0
$161$		−	7.88116i		−	0.621123i
$162$		0			0
$163$			10.0390i			0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$163$			10.0390i			0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$164$		0			0
$165$	−2.47349			−0.192561
$166$		0			0
$167$		−	24.7106i		−	1.91217i	−0.293096	−	0.956083i	$-0.594686\pi$
$167$		−	24.7106i		−	1.91217i	0.293096	−	0.956083i	$-0.405314\pi$
$168$		0			0
$169$	8.20653	+	10.0823i	0.631272	+	0.775562i
$170$		0			0
$171$			0.311392i			0.0238127i
$172$		0			0
$173$	10.4735			0.796285			0.398143	−	0.917324i	$-0.369655\pi$
$173$	10.4735			0.796285			0.398143	+	0.917324i	$0.369655\pi$
$174$		0			0
$175$			7.20114i			0.544355i
$176$		0			0
$177$			2.19560i			0.165031i
$178$		0			0
$179$	−11.2584			−0.841491			−0.420745	−	0.907179i	$-0.638231\pi$
$179$	−11.2584			−0.841491			−0.420745	+	0.907179i	$0.638231\pi$
$180$		0			0
$181$	−8.59484			−0.638849			−0.319425	−	0.947612i	$-0.603490\pi$
$181$	−8.59484			−0.638849			−0.319425	+	0.947612i	$0.603490\pi$
$182$		0			0
$183$	2.51253			0.185732
$184$		0			0
$185$	−4.94699			−0.363710
$186$		0			0
$187$			5.02506i			0.367469i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	−4.98603			−0.360776			−0.180388	−	0.983596i	$-0.557735\pi$
$191$	−4.98603			−0.360776			−0.180388	+	0.983596i	$0.557735\pi$
$192$		0			0
$193$			6.98603i			0.502865i	0.967875	+	0.251433i	$0.0809017\pi$
$193$			6.98603i			0.502865i	−0.967875	+	0.251433i	$0.919098\pi$
$194$		0			0
$195$	5.40767	−	11.3742i	0.387251	−	0.814522i
$196$		0			0
$197$			21.8044i			1.55350i	0.629809	+	0.776750i	$0.283133\pi$
$197$			21.8044i			1.55350i	−0.629809	+	0.776750i	$0.716867\pi$
$198$		0			0
$199$	−0.622783			−0.0441479			−0.0220740	−	0.999756i	$-0.507027\pi$
$199$	−0.622783			−0.0441479			−0.0220740	+	0.999756i	$0.507027\pi$
$200$		0			0
$201$			9.17858i			0.647407i
$202$		0			0
$203$		−	5.29742i		−	0.371806i
$204$		0			0
$205$	−41.2329			−2.87983
$206$		0			0
$207$	7.88116			0.547779
$208$		0			0
$209$	−0.220505			−0.0152526
$210$		0			0
$211$	−16.3547			−1.12590			−0.562951	−	0.826491i	$-0.690334\pi$
$211$	−16.3547			−1.12590			−0.562951	+	0.826491i	$0.690334\pi$
$212$		0			0
$213$		−	0.708126i		−	0.0485200i
$214$		0			0
$215$			11.5179i			0.785516i
$216$		0			0
$217$	−7.29742			−0.495381
$218$		0			0
$219$			5.21511i			0.352405i
$220$		0			0
$221$	−23.1074	−	10.9860i	−1.55437	−	0.739000i
$222$		0			0
$223$			8.71367i			0.583511i	0.956493	+	0.291755i	$0.0942393\pi$
$223$			8.71367i			0.583511i	−0.956493	+	0.291755i	$0.905761\pi$
$224$		0			0
$225$	−7.20114			−0.480076
$226$		0			0
$227$			12.9440i			0.859120i	0.903038	+	0.429560i	$0.141331\pi$
$227$			12.9440i			0.859120i	−0.903038	+	0.429560i	$0.858669\pi$
$228$		0			0
$229$		−	19.6521i		−	1.29865i	−0.760513	−	0.649323i	$-0.775052\pi$
$229$		−	19.6521i		−	1.29865i	0.760513	−	0.649323i	$-0.224948\pi$
$230$		0			0
$231$	0.708126			0.0465913
$232$		0			0
$233$	3.72764			0.244206			0.122103	−	0.992517i	$-0.461036\pi$
$233$	3.72764			0.244206			0.122103	+	0.992517i	$0.461036\pi$
$234$		0			0
$235$	21.3626			1.39354
$236$		0			0
$237$	2.78489			0.180898
$238$		0			0
$239$			8.86165i			0.573212i	0.958048	+	0.286606i	$0.0925271\pi$
$239$			8.86165i			0.573212i	−0.958048	+	0.286606i	$0.907473\pi$
$240$		0			0
$241$		−	10.0025i		−	0.644318i	−0.946686	−	0.322159i	$-0.895591\pi$
$241$		−	10.0025i		−	0.644318i	0.946686	−	0.322159i	$-0.104409\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	3.49301i		−	0.223160i
$246$		0			0
$247$	0.482078	−	1.01397i	0.0306739	−	0.0645176i
$248$		0			0
$249$			6.11580i			0.387573i
$250$		0			0
$251$	−9.24557			−0.583575			−0.291788	−	0.956483i	$-0.594250\pi$
$251$	−9.24557			−0.583575			−0.291788	+	0.956483i	$0.594250\pi$
$252$		0			0
$253$			5.58086i			0.350866i
$254$		0			0
$255$			24.7874i			1.55225i
$256$		0			0
$257$	−14.5516			−0.907702			−0.453851	−	0.891078i	$-0.649950\pi$
$257$	−14.5516			−0.907702			−0.453851	+	0.891078i	$0.649950\pi$
$258$		0			0
$259$	1.41625			0.0880017
$260$		0			0
$261$	5.29742			0.327902
$262$		0			0
$263$	14.7137			0.907284			0.453642	−	0.891184i	$-0.350125\pi$
$263$	14.7137			0.907284			0.453642	+	0.891184i	$0.350125\pi$
$264$		0			0
$265$		−	13.0207i		−	0.799856i
$266$		0			0
$267$			11.1018i			0.679420i
$268$		0			0
$269$	−12.0433			−0.734291			−0.367145	−	0.930164i	$-0.619665\pi$
$269$	−12.0433			−0.734291			−0.367145	+	0.930164i	$0.619665\pi$
$270$		0			0
$271$		−	7.38830i		−	0.448808i	−0.974496	−	0.224404i	$-0.927957\pi$
$271$		−	7.38830i		−	0.448808i	0.974496	−	0.224404i	$-0.0720434\pi$
$272$		0			0
$273$	−1.54814	+	3.25627i	−0.0936976	+	0.197078i
$274$		0			0
$275$		−	5.09932i		−	0.307500i
$276$		0			0
$277$	13.6997			0.823135			0.411567	−	0.911379i	$-0.364981\pi$
$277$	13.6997			0.823135			0.411567	+	0.911379i	$0.364981\pi$
$278$		0			0
$279$		−	7.29742i		−	0.436885i
$280$		0			0
$281$		−	4.62582i		−	0.275953i	−0.990435	−	0.137977i	$-0.955940\pi$
$281$		−	4.62582i		−	0.275953i	0.990435	−	0.137977i	$-0.0440599\pi$
$282$		0			0
$283$	−17.9721			−1.06833			−0.534164	−	0.845381i	$-0.679373\pi$
$283$	−17.9721			−1.06833			−0.534164	+	0.845381i	$0.679373\pi$
$284$		0			0
$285$	−1.08770			−0.0644295
$286$		0			0
$287$	11.8044			0.696792
$288$		0			0
$289$	33.3572			1.96219
$290$		0			0
$291$		−	7.79886i		−	0.457177i
$292$		0			0
$293$			6.87023i			0.401363i	0.979657	+	0.200682i	$0.0643156\pi$
$293$			6.87023i			0.401363i	−0.979657	+	0.200682i	$0.935684\pi$
$294$		0			0
$295$	−7.66924			−0.446521
$296$		0			0
$297$			0.708126i			0.0410897i
$298$		0			0
$299$	−25.6632	−	12.2011i	−1.48414	−	0.705610i
$300$		0			0
$301$		−	3.29742i		−	0.190060i
$302$		0			0
$303$	17.6911			1.01633
$304$		0			0
$305$			8.77630i			0.502530i
$306$		0			0
$307$			11.9202i			0.680322i	0.940367	+	0.340161i	$0.110482\pi$
$307$			11.9202i			0.680322i	−0.940367	+	0.340161i	$0.889518\pi$
$308$		0			0
$309$	−5.52651			−0.314392
$310$		0			0
$311$	24.7874			1.40556			0.702782	−	0.711405i	$-0.251941\pi$
$311$	24.7874			1.40556			0.702782	+	0.711405i	$0.251941\pi$
$312$		0			0
$313$	−6.73304			−0.380574			−0.190287	−	0.981729i	$-0.560942\pi$
$313$	−6.73304			−0.380574			−0.190287	+	0.981729i	$0.560942\pi$
$314$		0			0
$315$	3.49301			0.196809
$316$		0			0
$317$			19.1537i			1.07578i	0.843016	+	0.537889i	$0.180778\pi$
$317$			19.1537i			1.07578i	−0.843016	+	0.537889i	$0.819222\pi$
$318$		0			0
$319$			3.75124i			0.210029i
$320$		0			0
$321$	−19.1786			−1.07044
$322$		0			0
$323$			2.20972i			0.122952i
$324$		0			0
$325$	23.4488	+	11.1484i	1.30071	+	0.618400i
$326$		0			0
$327$			3.20653i			0.177322i
$328$		0			0
$329$	−6.11580			−0.337175
$330$		0			0
$331$		−	23.5247i		−	1.29303i	−0.762900	−	0.646516i	$-0.776225\pi$
$331$		−	23.5247i		−	1.29303i	0.762900	−	0.646516i	$-0.223775\pi$
$332$		0			0
$333$			1.41625i			0.0776102i
$334$		0			0
$335$	−32.0609			−1.75167
$336$		0			0
$337$	29.7258			1.61927			0.809634	−	0.586935i	$-0.199666\pi$
$337$	29.7258			1.61927			0.809634	+	0.586935i	$0.199666\pi$
$338$		0			0
$339$	−5.29742			−0.287716
$340$		0			0
$341$	5.16749			0.279836
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			27.5290i			1.48211i
$346$		0			0
$347$	32.2036			1.72878			0.864391	−	0.502820i	$-0.167704\pi$
$347$	32.2036			1.72878			0.864391	+	0.502820i	$0.167704\pi$
$348$		0			0
$349$			12.5752i			0.673133i	0.941660	+	0.336567i	$0.109266\pi$
$349$			12.5752i			0.673133i	−0.941660	+	0.336567i	$0.890734\pi$
$350$		0			0
$351$	−3.25627	−	1.54814i	−0.173807	−	0.0826336i
$352$		0			0
$353$		−	15.4132i		−	0.820363i	−0.912004	−	0.410181i	$-0.865465\pi$
$353$		−	15.4132i		−	0.820363i	0.912004	−	0.410181i	$-0.134535\pi$
$354$		0			0
$355$	2.47349			0.131279
$356$		0			0
$357$		−	7.09628i		−	0.375575i
$358$		0			0
$359$			24.6910i			1.30314i	0.758589	+	0.651570i	$0.225889\pi$
$359$			24.6910i			1.30314i	−0.758589	+	0.651570i	$0.774111\pi$
$360$		0			0
$361$	18.9030			0.994897
$362$		0			0
$363$	10.4986			0.551031
$364$		0			0
$365$	−18.2165			−0.953493
$366$		0			0
$367$	−10.8758			−0.567711			−0.283855	−	0.958867i	$-0.591614\pi$
$367$	−10.8758			−0.567711			−0.283855	+	0.958867i	$0.591614\pi$
$368$		0			0
$369$			11.8044i			0.614513i
$370$		0			0
$371$			3.72764i			0.193530i
$372$		0			0
$373$	−1.71906			−0.0890096			−0.0445048	−	0.999009i	$-0.514171\pi$
$373$	−1.71906			−0.0890096			−0.0445048	+	0.999009i	$0.514171\pi$
$374$		0			0
$375$		−	7.68861i		−	0.397038i
$376$		0			0
$377$	−17.2498	−	8.20114i	−0.888410	−	0.422380i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$	14.3028			0.732755
$382$		0			0
$383$		−	36.5528i		−	1.86776i	−0.357588	−	0.933879i	$-0.616401\pi$
$383$		−	36.5528i		−	1.86776i	0.357588	−	0.933879i	$-0.383599\pi$
$384$		0			0
$385$			2.47349i			0.126061i
$386$		0			0
$387$	3.29742			0.167617
$388$		0			0
$389$	36.1367			1.83220			0.916101	−	0.400948i	$-0.131319\pi$
$389$	36.1367			1.83220			0.916101	+	0.400948i	$0.131319\pi$
$390$		0			0
$391$	55.9269			2.82835
$392$		0			0
$393$	−12.4023			−0.625612
$394$		0			0
$395$			9.72764i			0.489451i
$396$		0			0
$397$		−	24.3657i		−	1.22288i	−0.791290	−	0.611441i	$-0.790590\pi$
$397$		−	24.3657i		−	1.22288i	0.791290	−	0.611441i	$-0.209410\pi$
$398$		0			0
$399$	0.311392			0.0155891
$400$		0			0
$401$			3.76537i			0.188034i	0.995571	+	0.0940168i	$0.0299707\pi$
$401$			3.76537i			0.188034i	−0.995571	+	0.0940168i	$0.970029\pi$
$402$		0			0
$403$	−11.2974	+	23.7623i	−0.562764	+	1.18369i
$404$		0			0
$405$			3.49301i			0.173569i
$406$		0			0
$407$	−1.00289			−0.0497112
$408$		0			0
$409$		−	31.3235i		−	1.54885i	−0.632667	−	0.774424i	$-0.718040\pi$
$409$		−	31.3235i		−	1.54885i	0.632667	−	0.774424i	$-0.281960\pi$
$410$		0			0
$411$			5.58390i			0.275433i
$412$		0			0
$413$	2.19560			0.108038
$414$		0			0
$415$	−21.3626			−1.04865
$416$		0			0
$417$	1.52651			0.0747533
$418$		0			0
$419$	26.3572			1.28763			0.643816	−	0.765180i	$-0.277350\pi$
$419$	26.3572			1.28763			0.643816	+	0.765180i	$0.277350\pi$
$420$		0			0
$421$			22.5168i			1.09740i	0.836019	+	0.548700i	$0.184877\pi$
$421$			22.5168i			1.09740i	−0.836019	+	0.548700i	$0.815123\pi$
$422$		0			0
$423$		−	6.11580i		−	0.297360i
$424$		0			0
$425$	−51.1013			−2.47878
$426$		0			0
$427$		−	2.51253i		−	0.121590i
$428$		0			0
$429$	1.09628	−	2.30585i	0.0529288	−	0.111327i
$430$		0			0
$431$		−	16.5374i		−	0.796580i	−0.917259	−	0.398290i	$-0.869604\pi$
$431$		−	16.5374i		−	0.796580i	0.917259	−	0.398290i	$-0.130396\pi$
$432$		0			0
$433$	25.2720			1.21449			0.607247	−	0.794513i	$-0.292274\pi$
$433$	25.2720			1.21449			0.607247	+	0.794513i	$0.292274\pi$
$434$		0			0
$435$			18.5039i			0.887196i
$436$		0			0
$437$			2.45413i			0.117397i
$438$		0			0
$439$	−5.52651			−0.263766			−0.131883	−	0.991265i	$-0.542102\pi$
$439$	−5.52651			−0.263766			−0.131883	+	0.991265i	$0.542102\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	7.25838			0.344856			0.172428	−	0.985022i	$-0.444839\pi$
$443$	7.25838			0.344856			0.172428	+	0.985022i	$0.444839\pi$
$444$		0			0
$445$	−38.7788			−1.83829
$446$		0			0
$447$		−	15.7765i		−	0.746201i
$448$		0			0
$449$		−	24.1787i		−	1.14107i	−0.821275	−	0.570533i	$-0.806737\pi$
$449$		−	24.1787i		−	1.14107i	0.821275	−	0.570533i	$-0.193263\pi$
$450$		0			0
$451$	−8.35901			−0.393610
$452$		0			0
$453$			6.83251i			0.321019i
$454$		0			0
$455$	−11.3742	−	5.40767i	−0.533230	−	0.253515i
$456$		0			0
$457$		−	33.3991i		−	1.56234i	−0.624316	−	0.781172i	$-0.714622\pi$
$457$		−	33.3991i		−	1.56234i	0.624316	−	0.781172i	$-0.285378\pi$
$458$		0			0
$459$	7.09628			0.331226
$460$		0			0
$461$			10.3882i			0.483824i	0.970298	+	0.241912i	$0.0777746\pi$
$461$			10.3882i			0.483824i	−0.970298	+	0.241912i	$0.922225\pi$
$462$		0			0
$463$			23.9330i			1.11226i	0.831095	+	0.556131i	$0.187715\pi$
$463$			23.9330i			1.11226i	−0.831095	+	0.556131i	$0.812285\pi$
$464$		0			0
$465$	25.4900			1.18207
$466$		0			0
$467$	−8.84329			−0.409219			−0.204609	−	0.978844i	$-0.565592\pi$
$467$	−8.84329			−0.409219			−0.204609	+	0.978844i	$0.565592\pi$
$468$		0			0
$469$	9.17858			0.423828
$470$		0			0
$471$	−12.1926			−0.561803
$472$		0			0
$473$			2.33499i			0.107363i
$474$		0			0
$475$		−	2.24238i		−	0.102887i
$476$		0			0
$477$	−3.72764			−0.170677
$478$		0			0
$479$		−	13.9623i		−	0.637953i	−0.947763	−	0.318976i	$-0.896661\pi$
$479$		−	13.9623i		−	0.637953i	0.947763	−	0.318976i	$-0.103339\pi$
$480$		0			0
$481$	2.19256	−	4.61170i	0.0999720	−	0.210275i
$482$		0			0
$483$		−	7.88116i		−	0.358605i
$484$		0			0
$485$	27.2415			1.23697
$486$		0			0
$487$		−	2.41306i		−	0.109346i	−0.998504	−	0.0546731i	$-0.982588\pi$
$487$		−	2.41306i		−	0.109346i	0.998504	−	0.0546731i	$-0.0174117\pi$
$488$		0			0
$489$			10.0390i			0.453981i
$490$		0			0
$491$	−9.15063			−0.412962			−0.206481	−	0.978451i	$-0.566201\pi$
$491$	−9.15063			−0.412962			−0.206481	+	0.978451i	$0.566201\pi$
$492$		0			0
$493$	37.5919			1.69306
$494$		0			0
$495$	−2.47349			−0.111175
$496$		0			0
$497$	−0.708126			−0.0317638
$498$		0			0
$499$			5.34927i			0.239466i	0.992806	+	0.119733i	$0.0382039\pi$
$499$			5.34927i			0.239466i	−0.992806	+	0.119733i	$0.961796\pi$
$500$		0			0
$501$		−	24.7106i		−	1.10399i
$502$		0			0
$503$	−35.1617			−1.56778			−0.783892	−	0.620897i	$-0.786768\pi$
$503$	−35.1617			−1.56778			−0.783892	+	0.620897i	$0.786768\pi$
$504$		0			0
$505$			61.7953i			2.74985i
$506$		0			0
$507$	8.20653	+	10.0823i	0.364465	+	0.447771i
$508$		0			0
$509$		−	18.4790i		−	0.819069i	−0.912295	−	0.409534i	$-0.865691\pi$
$509$		−	18.4790i		−	0.819069i	0.912295	−	0.409534i	$-0.134309\pi$
$510$		0			0
$511$	5.21511			0.230703
$512$		0			0
$513$			0.311392i			0.0137483i
$514$		0			0
$515$		−	19.3042i		−	0.850643i
$516$		0			0
$517$	4.33076			0.190466
$518$		0			0
$519$	10.4735			0.459735
$520$		0			0
$521$	12.1385			0.531798			0.265899	−	0.964001i	$-0.414331\pi$
$521$	12.1385			0.531798			0.265899	+	0.964001i	$0.414331\pi$
$522$		0			0
$523$	11.4205			0.499383			0.249691	−	0.968325i	$-0.419671\pi$
$523$	11.4205			0.499383			0.249691	+	0.968325i	$0.419671\pi$
$524$		0			0
$525$			7.20114i			0.314283i
$526$		0			0
$527$		−	51.7845i		−	2.25577i
$528$		0			0
$529$	39.1128			1.70055
$530$		0			0
$531$			2.19560i			0.0952807i
$532$		0			0
$533$	18.2749	−	38.4383i	0.791572	−	1.66495i
$534$		0			0
$535$		−	66.9910i		−	2.89628i
$536$		0			0
$537$	−11.2584			−0.485835
$538$		0			0
$539$		−	0.708126i		−	0.0305012i
$540$		0			0
$541$		−	16.3851i		−	0.704451i	−0.935915	−	0.352226i	$-0.885425\pi$
$541$		−	16.3851i		−	0.704451i	0.935915	−	0.352226i	$-0.114575\pi$
$542$		0			0
$543$	−8.59484			−0.368840
$544$		0			0
$545$	−11.2005			−0.479775
$546$		0			0
$547$	19.4620			0.832136			0.416068	−	0.909333i	$-0.363408\pi$
$547$	19.4620			0.832136			0.416068	+	0.909333i	$0.363408\pi$
$548$		0			0
$549$	2.51253			0.107232
$550$		0			0
$551$			1.64957i			0.0702741i
$552$		0			0
$553$		−	2.78489i		−	0.118425i
$554$		0			0
$555$	−4.94699			−0.209988
$556$		0			0
$557$		−	38.5248i		−	1.63235i	−0.577806	−	0.816174i	$-0.696091\pi$
$557$		−	38.5248i		−	1.63235i	0.577806	−	0.816174i	$-0.303909\pi$
$558$		0			0
$559$	−10.7373	−	5.10486i	−0.454138	−	0.215913i
$560$		0			0
$561$			5.02506i			0.212158i
$562$		0			0
$563$	13.7903			0.581191			0.290595	−	0.956846i	$-0.406147\pi$
$563$	13.7903			0.581191			0.290595	+	0.956846i	$0.406147\pi$
$564$		0			0
$565$		−	18.5039i		−	0.778467i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−5.29742			−0.222079			−0.111040	−	0.993816i	$-0.535418\pi$
$569$	−5.29742			−0.222079			−0.111040	+	0.993816i	$0.535418\pi$
$570$		0			0
$571$	27.7928			1.16309			0.581546	−	0.813514i	$-0.302448\pi$
$571$	27.7928			1.16309			0.581546	+	0.813514i	$0.302448\pi$
$572$		0			0
$573$	−4.98603			−0.208294
$574$		0			0
$575$	−56.7534			−2.36678
$576$		0			0
$577$			18.7333i			0.779879i	0.920840	+	0.389940i	$0.127504\pi$
$577$			18.7333i			0.779879i	−0.920840	+	0.389940i	$0.872496\pi$
$578$		0			0
$579$			6.98603i			0.290329i
$580$		0			0
$581$	6.11580			0.253726
$582$		0			0
$583$		−	2.63964i		−	0.109323i
$584$		0			0
$585$	5.40767	−	11.3742i	0.223580	−	0.470264i
$586$		0			0
$587$		−	8.93721i		−	0.368878i	−0.982844	−	0.184439i	$-0.940953\pi$
$587$		−	8.93721i		−	0.368878i	0.982844	−	0.184439i	$-0.0590469\pi$
$588$		0			0
$589$	2.27236			0.0936308
$590$		0			0
$591$			21.8044i			0.896913i
$592$		0			0
$593$		−	10.9655i		−	0.450298i	−0.974324	−	0.225149i	$-0.927713\pi$
$593$		−	10.9655i		−	0.450298i	0.974324	−	0.225149i	$-0.0722869\pi$
$594$		0			0
$595$	24.7874			1.01618
$596$		0			0
$597$	−0.622783			−0.0254888
$598$		0			0
$599$	−32.2663			−1.31836			−0.659182	−	0.751983i	$-0.729097\pi$
$599$	−32.2663			−1.31836			−0.659182	+	0.751983i	$0.729097\pi$
$600$		0			0
$601$	−19.2999			−0.787260			−0.393630	−	0.919269i	$-0.628781\pi$
$601$	−19.2999			−0.787260			−0.393630	+	0.919269i	$0.628781\pi$
$602$		0			0
$603$			9.17858i			0.373781i
$604$		0			0
$605$			36.6716i			1.49091i
$606$		0			0
$607$	−31.6632			−1.28517			−0.642584	−	0.766215i	$-0.722138\pi$
$607$	−31.6632			−1.28517			−0.642584	+	0.766215i	$0.722138\pi$
$608$		0			0
$609$		−	5.29742i		−	0.214662i
$610$		0			0
$611$	−9.46810	+	19.9147i	−0.383038	+	0.805661i
$612$		0			0
$613$			3.23478i			0.130652i	0.997864	+	0.0653259i	$0.0208087\pi$
$613$			3.23478i			0.130652i	−0.997864	+	0.0653259i	$0.979191\pi$
$614$		0			0
$615$	−41.2329			−1.66267
$616$		0			0
$617$		−	11.1365i		−	0.448339i	−0.974550	−	0.224169i	$-0.928033\pi$
$617$		−	11.1365i		−	0.448339i	0.974550	−	0.224169i	$-0.0719669\pi$
$618$		0			0
$619$		−	6.82643i		−	0.274377i	−0.990545	−	0.137189i	$-0.956193\pi$
$619$		−	6.82643i		−	0.274377i	0.990545	−	0.137189i	$-0.0438067\pi$
$620$		0			0
$621$	7.88116			0.316260
$622$		0			0
$623$	11.1018			0.444785
$624$		0			0
$625$	−9.14929			−0.365972
$626$		0			0
$627$	−0.220505			−0.00880611
$628$		0			0
$629$			10.0501i			0.400725i
$630$		0			0
$631$			18.9689i			0.755138i	0.925981	+	0.377569i	$0.123240\pi$
$631$			18.9689i			0.755138i	−0.925981	+	0.377569i	$0.876760\pi$
$632$		0			0
$633$	−16.3547			−0.650039
$634$		0			0
$635$			49.9599i			1.98260i
$636$		0			0
$637$	3.25627	+	1.54814i	0.129018	+	0.0613395i
$638$		0			0
$639$		−	0.708126i		−	0.0280130i
$640$		0			0
$641$	−8.45413			−0.333918			−0.166959	−	0.985964i	$-0.553395\pi$
$641$	−8.45413			−0.333918			−0.166959	+	0.985964i	$0.553395\pi$
$642$		0			0
$643$			37.4102i			1.47531i	0.675176	+	0.737657i	$0.264068\pi$
$643$			37.4102i			1.47531i	−0.675176	+	0.737657i	$0.735932\pi$
$644$		0			0
$645$			11.5179i			0.453518i
$646$		0			0
$647$	17.2627			0.678668			0.339334	−	0.940666i	$-0.389798\pi$
$647$	17.2627			0.678668			0.339334	+	0.940666i	$0.389798\pi$
$648$		0			0
$649$	−1.55476			−0.0610296
$650$		0			0
$651$	−7.29742			−0.286008
$652$		0			0
$653$	27.2456			1.06620			0.533101	−	0.846052i	$-0.321027\pi$
$653$	27.2456			1.06620			0.533101	+	0.846052i	$0.321027\pi$
$654$		0			0
$655$		−	43.3213i		−	1.69270i
$656$		0			0
$657$			5.21511i			0.203461i
$658$		0			0
$659$	36.0458			1.40414			0.702072	−	0.712106i	$-0.252258\pi$
$659$	36.0458			1.40414			0.702072	+	0.712106i	$0.252258\pi$
$660$		0			0
$661$		−	25.3626i		−	0.986489i	−0.869891	−	0.493245i	$-0.835811\pi$
$661$		−	25.3626i		−	0.986489i	0.869891	−	0.493245i	$-0.164189\pi$
$662$		0			0
$663$	−23.1074	−	10.9860i	−0.897416	−	0.426662i
$664$		0			0
$665$			1.08770i			0.0421790i
$666$		0			0
$667$	41.7498			1.61656
$668$		0			0
$669$			8.71367i			0.336890i
$670$		0			0
$671$			1.77919i			0.0686849i
$672$		0			0
$673$	−40.7355			−1.57024			−0.785120	−	0.619344i	$-0.787399\pi$
$673$	−40.7355			−1.57024			−0.785120	+	0.619344i	$0.787399\pi$
$674$		0			0
$675$	−7.20114			−0.277172
$676$		0			0
$677$	−38.7333			−1.48864			−0.744322	−	0.667821i	$-0.767227\pi$
$677$	−38.7333			−1.48864			−0.744322	+	0.667821i	$0.767227\pi$
$678$		0			0
$679$	−7.79886			−0.299293
$680$		0			0
$681$			12.9440i			0.496013i
$682$		0			0
$683$			32.0682i			1.22705i	0.789673	+	0.613527i	$0.210250\pi$
$683$			32.0682i			1.22705i	−0.789673	+	0.613527i	$0.789750\pi$
$684$		0			0
$685$	−19.5046			−0.745234
$686$		0			0
$687$		−	19.6521i		−	0.749774i
$688$		0			0
$689$	12.1382	+	5.77091i	0.462429	+	0.219854i
$690$		0			0
$691$		−	14.4259i		−	0.548786i	−0.961618	−	0.274393i	$-0.911523\pi$
$691$		−	14.4259i		−	0.548786i	0.961618	−	0.274393i	$-0.0884769\pi$
$692$		0			0
$693$	0.708126			0.0268995
$694$		0			0
$695$			5.33210i			0.202258i
$696$		0			0
$697$			83.7673i			3.17291i
$698$		0			0
$699$	3.72764			0.140992
$700$		0			0
$701$	11.0490			0.417314			0.208657	−	0.977989i	$-0.433091\pi$
$701$	11.0490			0.417314			0.208657	+	0.977989i	$0.433091\pi$
$702$		0			0
$703$	−0.441009			−0.0166330
$704$		0			0
$705$	21.3626			0.804560
$706$		0			0
$707$		−	17.6911i		−	0.665343i
$708$		0			0
$709$			19.8293i			0.744706i	0.928091	+	0.372353i	$0.121449\pi$
$709$			19.8293i			0.744706i	−0.928091	+	0.372353i	$0.878551\pi$
$710$		0			0
$711$	2.78489			0.104441
$712$		0			0
$713$		−	57.5122i		−	2.15385i
$714$		0			0
$715$	8.05436	+	3.82931i	0.301216	+	0.143208i
$716$		0			0
$717$			8.86165i			0.330944i
$718$		0			0
$719$	3.21261			0.119810			0.0599050	−	0.998204i	$-0.480920\pi$
$719$	3.21261			0.119810			0.0599050	+	0.998204i	$0.480920\pi$
$720$		0			0
$721$			5.52651i			0.205818i
$722$		0			0
$723$		−	10.0025i		−	0.371997i
$724$		0			0
$725$	−38.1474			−1.41676
$726$		0			0
$727$	13.3493			0.495097			0.247548	−	0.968876i	$-0.420375\pi$
$727$	13.3493			0.495097			0.247548	+	0.968876i	$0.420375\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	23.3994			0.865458
$732$		0			0
$733$			17.4185i			0.643365i	0.946848	+	0.321683i	$0.104248\pi$
$733$			17.4185i			0.643365i	−0.946848	+	0.321683i	$0.895752\pi$
$734$		0			0
$735$		−	3.49301i		−	0.128842i
$736$		0			0
$737$	−6.49960			−0.239416
$738$		0			0
$739$		−	1.80137i		−	0.0662643i	−0.999451	−	0.0331322i	$-0.989452\pi$
$739$		−	1.80137i		−	0.0662643i	0.999451	−	0.0331322i	$-0.0105482\pi$
$740$		0			0
$741$	0.482078	−	1.01397i	0.0177096	−	0.0372493i
$742$		0			0
$743$		−	3.86484i		−	0.141787i	−0.997484	−	0.0708936i	$-0.977415\pi$
$743$		−	3.86484i		−	0.141787i	0.997484	−	0.0708936i	$-0.0225851\pi$
$744$		0			0
$745$	55.1074			2.01898
$746$		0			0
$747$			6.11580i			0.223765i
$748$		0			0
$749$			19.1786i			0.700770i
$750$		0			0
$751$	−29.8923			−1.09078			−0.545392	−	0.838181i	$-0.683619\pi$
$751$	−29.8923			−1.09078			−0.545392	+	0.838181i	$0.683619\pi$
$752$		0			0
$753$	−9.24557			−0.336927
$754$		0			0
$755$	−23.8660			−0.868574
$756$		0			0
$757$	−38.8933			−1.41360			−0.706800	−	0.707413i	$-0.749862\pi$
$757$	−38.8933			−1.41360			−0.706800	+	0.707413i	$0.749862\pi$
$758$		0			0
$759$			5.58086i			0.202572i
$760$		0			0
$761$			8.33630i			0.302191i	0.988519	+	0.151095i	$0.0482800\pi$
$761$			8.33630i			0.302191i	−0.988519	+	0.151095i	$0.951720\pi$
$762$		0			0
$763$	3.20653			0.116084
$764$		0			0
$765$			24.7874i			0.896190i
$766$		0			0
$767$	3.39909	−	7.14944i	0.122734	−	0.258151i
$768$		0			0
$769$			17.0811i			0.615962i	0.951393	+	0.307981i	$0.0996533\pi$
$769$			17.0811i			0.615962i	−0.951393	+	0.307981i	$0.900347\pi$
$770$		0			0
$771$	−14.5516			−0.524062
$772$		0			0
$773$		−	16.0532i		−	0.577392i	−0.957421	−	0.288696i	$-0.906778\pi$
$773$		−	16.0532i		−	0.577392i	0.957421	−	0.288696i	$-0.0932217\pi$
$774$		0			0
$775$			52.5497i			1.88764i
$776$		0			0
$777$	1.41625			0.0508078
$778$		0			0
$779$	−3.67579			−0.131699
$780$		0			0
$781$	0.501443			0.0179430
$782$		0			0
$783$	5.29742			0.189314
$784$		0			0
$785$		−	42.5888i		−	1.52006i
$786$		0			0
$787$		−	17.5654i		−	0.626140i	−0.949730	−	0.313070i	$-0.898643\pi$
$787$		−	17.5654i		−	0.626140i	0.949730	−	0.313070i	$-0.101357\pi$
$788$		0			0
$789$	14.7137			0.523821
$790$		0			0
$791$			5.29742i			0.188354i
$792$		0			0
$793$	−8.18147	−	3.88975i	−0.290532	−	0.138129i
$794$		0			0
$795$		−	13.0207i		−	0.461797i
$796$		0			0
$797$	−26.0934			−0.924275			−0.462138	−	0.886808i	$-0.652917\pi$
$797$	−26.0934			−0.924275			−0.462138	+	0.886808i	$0.652917\pi$
$798$		0			0
$799$		−	43.3994i		−	1.53536i
$800$		0			0
$801$			11.1018i			0.392264i
$802$		0			0
$803$	−3.69296			−0.130322
$804$		0			0
$805$	27.5290			0.970270
$806$		0			0
$807$	−12.0433			−0.423943
$808$		0			0
$809$	−6.30030			−0.221507			−0.110753	−	0.993848i	$-0.535326\pi$
$809$	−6.30030			−0.221507			−0.110753	+	0.993848i	$0.535326\pi$
$810$		0			0
$811$		−	1.16749i		−	0.0409963i	−0.999790	−	0.0204981i	$-0.993475\pi$
$811$		−	1.16749i		−	0.0409963i	0.999790	−	0.0204981i	$-0.00652522\pi$
$812$		0			0
$813$		−	7.38830i		−	0.259119i
$814$		0			0
$815$	−35.0665			−1.22833
$816$		0			0
$817$			1.02679i			0.0359228i
$818$		0			0
$819$	−1.54814	+	3.25627i	−0.0540964	+	0.113783i
$820$		0			0
$821$		−	17.9518i		−	0.626524i	−0.949667	−	0.313262i	$-0.898578\pi$
$821$		−	17.9518i		−	0.626524i	0.949667	−	0.313262i	$-0.101422\pi$
$822$		0			0
$823$	5.66501			0.197470			0.0987349	−	0.995114i	$-0.468520\pi$
$823$	5.66501			0.197470			0.0987349	+	0.995114i	$0.468520\pi$
$824$		0			0
$825$		−	5.09932i		−	0.177535i
$826$		0			0
$827$		−	11.6942i		−	0.406646i	−0.979112	−	0.203323i	$-0.934826\pi$
$827$		−	11.6942i		−	0.406646i	0.979112	−	0.203323i	$-0.0651741\pi$
$828$		0			0
$829$	−30.8024			−1.06981			−0.534906	−	0.844912i	$-0.679653\pi$
$829$	−30.8024			−1.06981			−0.534906	+	0.844912i	$0.679653\pi$
$830$		0			0
$831$	13.6997			0.475237
$832$		0			0
$833$	−7.09628			−0.245871
$834$		0			0
$835$	86.3146			2.98704
$836$		0			0
$837$		−	7.29742i		−	0.252236i
$838$		0			0
$839$			8.80760i			0.304072i	0.988375	+	0.152036i	$0.0485830\pi$
$839$			8.80760i			0.304072i	−0.988375	+	0.152036i	$0.951417\pi$
$840$		0			0
$841$	−0.937367			−0.0323230
$842$		0			0
$843$		−	4.62582i		−	0.159322i
$844$		0			0
$845$	−35.2176	+	28.6655i	−1.21152	+	0.986124i
$846$		0			0
$847$		−	10.4986i		−	0.360735i
$848$		0			0
$849$	−17.9721			−0.616799
$850$		0			0
$851$			11.1617i			0.382619i
$852$		0			0
$853$		−	14.4109i		−	0.493419i	−0.969090	−	0.246709i	$-0.920651\pi$
$853$		−	14.4109i		−	0.493419i	0.969090	−	0.246709i	$-0.0793493\pi$
$854$		0			0
$855$	−1.08770			−0.0371984
$856$		0			0
$857$	−28.9646			−0.989413			−0.494706	−	0.869060i	$-0.664724\pi$
$857$	−28.9646			−0.989413			−0.494706	+	0.869060i	$0.664724\pi$
$858$		0			0
$859$	−6.67291			−0.227677			−0.113838	−	0.993499i	$-0.536315\pi$
$859$	−6.67291			−0.227677			−0.113838	+	0.993499i	$0.536315\pi$
$860$		0			0
$861$	11.8044			0.402293
$862$		0			0
$863$		−	18.7300i		−	0.637577i	−0.947826	−	0.318788i	$-0.896724\pi$
$863$		−	18.7300i		−	0.637577i	0.947826	−	0.318788i	$-0.103276\pi$
$864$		0			0
$865$			36.5841i			1.24390i
$866$		0			0
$867$	33.3572			1.13287
$868$		0			0
$869$			1.97205i			0.0668973i
$870$		0			0
$871$	14.2097	−	29.8879i	0.481478	−	1.01271i
$872$		0			0
$873$		−	7.79886i		−	0.263951i
$874$		0			0
$875$	−7.68861			−0.259922
$876$		0			0
$877$		−	46.3743i		−	1.56595i	−0.622053	−	0.782975i	$-0.713701\pi$
$877$		−	46.3743i		−	1.56595i	0.622053	−	0.782975i	$-0.286299\pi$
$878$		0			0
$879$			6.87023i			0.231727i
$880$		0			0
$881$	31.0963			1.04766			0.523830	−	0.851823i	$-0.324503\pi$
$881$	31.0963			1.04766			0.523830	+	0.851823i	$0.324503\pi$
$882$		0			0
$883$	−31.9119			−1.07392			−0.536961	−	0.843607i	$-0.680428\pi$
$883$	−31.9119			−1.07392			−0.536961	+	0.843607i	$0.680428\pi$
$884$		0			0
$885$	−7.66924			−0.257799
$886$		0			0
$887$	−6.43023			−0.215906			−0.107953	−	0.994156i	$-0.534430\pi$
$887$	−6.43023			−0.215906			−0.107953	+	0.994156i	$0.534430\pi$
$888$		0			0
$889$		−	14.3028i		−	0.479701i
$890$		0			0
$891$			0.708126i			0.0237231i
$892$		0			0
$893$	1.90441			0.0637286
$894$		0			0
$895$		−	39.3257i		−	1.31451i
$896$		0			0
$897$	−25.6632	−	12.2011i	−0.856868	−	0.407384i
$898$		0			0
$899$		−	38.6575i		−	1.28930i
$900$		0			0
$901$	−26.4524			−0.881257
$902$		0			0
$903$		−	3.29742i		−	0.109731i
$904$		0			0
$905$		−	30.0219i		−	0.997961i
$906$		0			0
$907$	−39.7541			−1.32001			−0.660006	−	0.751260i	$-0.729446\pi$
$907$	−39.7541			−1.32001			−0.660006	+	0.751260i	$0.729446\pi$
$908$		0			0
$909$	17.6911			0.586777
$910$		0			0
$911$	20.4652			0.678043			0.339021	−	0.940779i	$-0.389904\pi$
$911$	20.4652			0.678043			0.339021	+	0.940779i	$0.389904\pi$
$912$		0			0
$913$	−4.33076			−0.143327
$914$		0			0
$915$			8.77630i			0.290136i
$916$		0			0
$917$			12.4023i			0.409559i
$918$		0			0
$919$	−40.2749			−1.32855			−0.664273	−	0.747490i	$-0.731259\pi$
$919$	−40.2749			−1.32855			−0.664273	+	0.747490i	$0.731259\pi$
$920$		0			0
$921$			11.9202i			0.392784i
$922$		0			0
$923$	−1.09628	+	2.30585i	−0.0360844	+	0.0758979i
$924$		0			0
$925$		−	10.1986i		−	0.335329i
$926$		0			0
$927$	−5.52651			−0.181514
$928$		0			0
$929$			4.19124i			0.137510i	0.997634	+	0.0687551i	$0.0219027\pi$
$929$			4.19124i			0.137510i	−0.997634	+	0.0687551i	$0.978097\pi$
$930$		0			0
$931$		−	0.311392i		−	0.0102055i
$932$		0			0
$933$	24.7874			0.811503
$934$		0			0
$935$	−17.5526			−0.574032
$936$		0			0
$937$	23.8446			0.778970			0.389485	−	0.921033i	$-0.372653\pi$
$937$	23.8446			0.778970			0.389485	+	0.921033i	$0.372653\pi$
$938$		0			0
$939$	−6.73304			−0.219724
$940$		0			0
$941$		−	5.24456i		−	0.170968i	−0.996340	−	0.0854839i	$-0.972756\pi$
$941$		−	5.24456i		−	0.170968i	0.996340	−	0.0854839i	$-0.0272436\pi$
$942$		0			0
$943$			93.0325i			3.02955i
$944$		0			0
$945$	3.49301			0.113628
$946$		0			0
$947$			10.4486i			0.339533i	0.985484	+	0.169767i	$0.0543014\pi$
$947$			10.4486i			0.339533i	−0.985484	+	0.169767i	$0.945699\pi$
$948$		0			0
$949$	8.07372	−	16.9818i	0.262084	−	0.551252i
$950$		0			0
$951$			19.1537i			0.621100i
$952$		0			0
$953$	−39.4169			−1.27684			−0.638420	−	0.769689i	$-0.720411\pi$
$953$	−39.4169			−1.27684			−0.638420	+	0.769689i	$0.720411\pi$
$954$		0			0
$955$		−	17.4163i		−	0.563577i
$956$		0			0
$957$			3.75124i			0.121260i
$958$		0			0
$959$	5.58390			0.180314
$960$		0			0
$961$	−22.2523			−0.717816
$962$		0			0
$963$	−19.1786			−0.618021
$964$		0			0
$965$	−24.4023			−0.785537
$966$		0			0
$967$			20.5558i			0.661030i	0.943801	+	0.330515i	$0.107222\pi$
$967$			20.5558i			0.661030i	−0.943801	+	0.330515i	$0.892778\pi$
$968$		0			0
$969$			2.20972i			0.0709865i
$970$		0			0
$971$	−22.8325			−0.732730			−0.366365	−	0.930471i	$-0.619398\pi$
$971$	−22.8325			−0.732730			−0.366365	+	0.930471i	$0.619398\pi$
$972$		0			0
$973$		−	1.52651i		−	0.0489375i
$974$		0			0
$975$	23.4488	+	11.1484i	0.750963	+	0.357033i
$976$		0			0
$977$		−	2.49417i		−	0.0797957i	−0.999204	−	0.0398978i	$-0.987297\pi$
$977$		−	2.49417i		−	0.0797957i	0.999204	−	0.0398978i	$-0.0127032\pi$
$978$		0			0
$979$	−7.86149			−0.251254
$980$		0			0
$981$			3.20653i			0.102377i
$982$		0			0
$983$			9.95119i			0.317394i	0.987327	+	0.158697i	$0.0507292\pi$
$983$			9.95119i			0.317394i	−0.987327	+	0.158697i	$0.949271\pi$
$984$		0			0
$985$	−76.1631			−2.42676
$986$		0			0
$987$	−6.11580			−0.194668
$988$		0			0
$989$	25.9875			0.826354
$990$		0			0
$991$	1.08096			0.0343378			0.0171689	−	0.999853i	$-0.494535\pi$
$991$	1.08096			0.0343378			0.0171689	+	0.999853i	$0.494535\pi$
$992$		0			0
$993$		−	23.5247i		−	0.746532i
$994$		0			0
$995$		−	2.17539i		−	0.0689645i
$996$		0			0
$997$	−25.3129			−0.801666			−0.400833	−	0.916151i	$-0.631279\pi$
$997$	−25.3129			−0.801666			−0.400833	+	0.916151i	$0.631279\pi$
$998$		0			0
$999$			1.41625i			0.0448083i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.q.337.7		8
4.3	odd	2		273.2.c.c.64.7	yes	8
12.11	even	2		819.2.c.d.64.2		8
13.12	even	2	inner	4368.2.h.q.337.2		8
28.27	even	2		1911.2.c.l.883.7		8
52.31	even	4		3549.2.a.x.1.4		4
52.47	even	4		3549.2.a.v.1.1		4
52.51	odd	2		273.2.c.c.64.2	✓	8
156.155	even	2		819.2.c.d.64.7		8
364.363	even	2		1911.2.c.l.883.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.c.64.2	✓	8	52.51	odd	2
273.2.c.c.64.7	yes	8	4.3	odd	2
819.2.c.d.64.2		8	12.11	even	2
819.2.c.d.64.7		8	156.155	even	2
1911.2.c.l.883.2		8	364.363	even	2
1911.2.c.l.883.7		8	28.27	even	2
3549.2.a.v.1.1		4	52.47	even	4
3549.2.a.x.1.4		4	52.31	even	4
4368.2.h.q.337.2		8	13.12	even	2	inner
4368.2.h.q.337.7		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.49301i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.49301i q^{5} -1.00000i q^{7} +1.00000 q^{9} +0.708126i q^{11} +(-3.25627 - 1.54814i) q^{13} +3.49301i q^{15} +7.09628 q^{17} +0.311392i q^{19} -1.00000i q^{21} +7.88116 q^{23} -7.20114 q^{25} +1.00000 q^{27} +5.29742 q^{29} -7.29742i q^{31} +0.708126i q^{33} +3.49301 q^{35} +1.41625i q^{37} +(-3.25627 - 1.54814i) q^{39} +11.8044i q^{41} +3.29742 q^{43} +3.49301i q^{45} -6.11580i q^{47} -1.00000 q^{49} +7.09628 q^{51} -3.72764 q^{53} -2.47349 q^{55} +0.311392i q^{57} +2.19560i q^{59} +2.51253 q^{61} -1.00000i q^{63} +(5.40767 - 11.3742i) q^{65} +9.17858i q^{67} +7.88116 q^{69} -0.708126i q^{71} +5.21511i q^{73} -7.20114 q^{75} +0.708126 q^{77} +2.78489 q^{79} +1.00000 q^{81} +6.11580i q^{83} +24.7874i q^{85} +5.29742 q^{87} +11.1018i q^{89} +(-1.54814 + 3.25627i) q^{91} -7.29742i q^{93} -1.08770 q^{95} -7.79886i q^{97} +0.708126i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 8 * q^9	\( 8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 6 * q^13 + 20 * q^17 + 6 * q^23 - 34 * q^25 + 8 * q^27 - 18 * q^29 + 6 * q^35 - 6 * q^39 - 34 * q^43 - 8 * q^49 + 20 * q^51 - 10 * q^53 - 16 * q^55 - 20 * q^61 - 10 * q^65 + 6 * q^69 - 34 * q^75 + 4 * q^77 + 2 * q^79 + 8 * q^81 - 18 * q^87 + 6 * q^91 + 78 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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