LMFDB - Embedded newform 4368.2.h.r.337.6

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 $ x^10 + 20x^8 + 138x^6 + 364x^4 + 249x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$2.69449i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.r.337.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.100328i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.100328i q^{5} +1.00000i q^{7} +1.00000 q^{9} -1.72967i q^{11} +(-2.69449 + 2.39577i) q^{13} -0.100328i q^{15} -2.15994 q^{17} -0.600958i q^{19} -1.00000i q^{21} -4.58736 q^{23} +4.98993 q^{25} -1.00000 q^{27} +5.17825 q^{29} -5.01832i q^{31} +1.72967i q^{33} -0.100328 q^{35} +2.36059i q^{37} +(2.69449 - 2.39577i) q^{39} +5.02116i q^{41} -9.12149 q^{43} +0.100328i q^{45} -9.59248i q^{47} -1.00000 q^{49} +2.15994 q^{51} +3.75964 q^{53} +0.173534 q^{55} +0.600958i q^{57} -7.29218i q^{59} +4.43095 q^{61} +1.00000i q^{63} +(-0.240363 - 0.270332i) q^{65} +11.2985i q^{67} +4.58736 q^{69} -0.0902623i q^{71} -7.78928i q^{73} -4.98993 q^{75} +1.72967 q^{77} +14.6704 q^{79} +1.00000 q^{81} -3.53002i q^{83} -0.216702i q^{85} -5.17825 q^{87} -12.6369i q^{89} +(-2.39577 - 2.69449i) q^{91} +5.01832i q^{93} +0.0602929 q^{95} -0.975148i q^{97} -1.72967i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * 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
$5$			0.100328i			0.0448680i	0.999748	+	0.0224340i	$0.00714157\pi$
$5$			0.100328i			0.0448680i	−0.999748	+	0.0224340i	$0.992858\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.72967i		−	0.521514i	−0.965404	−	0.260757i	$-0.916028\pi$
$11$		−	1.72967i		−	0.521514i	0.965404	−	0.260757i	$-0.0839722\pi$
$12$		0			0
$13$	−2.69449	+	2.39577i	−0.747317	+	0.664468i
$13$	−2.69449	+	2.39577i	−0.747317	+	0.664468i
$14$		0			0
$15$		−	0.100328i		−	0.0259046i
$16$		0			0
$17$	−2.15994			−0.523862			−0.261931	−	0.965087i	$-0.584359\pi$
$17$	−2.15994			−0.523862			−0.261931	+	0.965087i	$0.584359\pi$
$18$		0			0
$19$		−	0.600958i		−	0.137869i	−0.997621	−	0.0689346i	$-0.978040\pi$
$19$		−	0.600958i		−	0.137869i	0.997621	−	0.0689346i	$-0.0219600\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	−4.58736			−0.956531			−0.478266	−	0.878215i	$-0.658734\pi$
$23$	−4.58736			−0.956531			−0.478266	+	0.878215i	$0.658734\pi$
$24$		0			0
$25$	4.98993			0.997987
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	5.17825			0.961578			0.480789	−	0.876836i	$-0.340350\pi$
$29$	5.17825			0.961578			0.480789	+	0.876836i	$0.340350\pi$
$30$		0			0
$31$		−	5.01832i		−	0.901316i	−0.892697	−	0.450658i	$-0.851189\pi$
$31$		−	5.01832i		−	0.901316i	0.892697	−	0.450658i	$-0.148811\pi$
$32$		0			0
$33$			1.72967i			0.301096i
$34$		0			0
$35$	−0.100328			−0.0169585
$36$		0			0
$37$			2.36059i			0.388079i	0.980994	+	0.194040i	$0.0621591\pi$
$37$			2.36059i			0.388079i	−0.980994	+	0.194040i	$0.937841\pi$
$38$		0			0
$39$	2.69449	−	2.39577i	0.431463	−	0.383631i
$40$		0			0
$41$			5.02116i			0.784174i	0.919928	+	0.392087i	$0.128247\pi$
$41$			5.02116i			0.784174i	−0.919928	+	0.392087i	$0.871753\pi$
$42$		0			0
$43$	−9.12149			−1.39101			−0.695507	−	0.718519i	$-0.744820\pi$
$43$	−9.12149			−1.39101			−0.695507	+	0.718519i	$0.744820\pi$
$44$		0			0
$45$			0.100328i			0.0149560i
$46$		0			0
$47$		−	9.59248i		−	1.39921i	−0.714531	−	0.699603i	$-0.753360\pi$
$47$		−	9.59248i		−	1.39921i	0.714531	−	0.699603i	$-0.246640\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.15994			0.302452
$52$		0			0
$53$	3.75964			0.516426			0.258213	−	0.966088i	$-0.416866\pi$
$53$	3.75964			0.516426			0.258213	+	0.966088i	$0.416866\pi$
$54$		0			0
$55$	0.173534			0.0233993
$56$		0			0
$57$			0.600958i			0.0795988i
$58$		0			0
$59$		−	7.29218i		−	0.949361i	−0.880158	−	0.474680i	$-0.842564\pi$
$59$		−	7.29218i		−	0.949361i	0.880158	−	0.474680i	$-0.157436\pi$
$60$		0			0
$61$	4.43095			0.567325			0.283663	−	0.958924i	$-0.408450\pi$
$61$	4.43095			0.567325			0.283663	+	0.958924i	$0.408450\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−0.240363	−	0.270332i	−0.0298134	−	0.0335306i
$66$		0			0
$67$			11.2985i			1.38033i	0.723652	+	0.690165i	$0.242462\pi$
$67$			11.2985i			1.38033i	−0.723652	+	0.690165i	$0.757538\pi$
$68$		0			0
$69$	4.58736			0.552254
$70$		0			0
$71$		−	0.0902623i		−	0.0107122i	−0.999986	−	0.00535608i	$-0.998295\pi$
$71$		−	0.0902623i		−	0.0107122i	0.999986	−	0.00535608i	$-0.00170490\pi$
$72$		0			0
$73$		−	7.78928i		−	0.911666i	−0.890065	−	0.455833i	$-0.849341\pi$
$73$		−	7.78928i		−	0.911666i	0.890065	−	0.455833i	$-0.150659\pi$
$74$		0			0
$75$	−4.98993			−0.576188
$76$		0			0
$77$	1.72967			0.197114
$78$		0			0
$79$	14.6704			1.65055			0.825275	−	0.564731i	$-0.191020\pi$
$79$	14.6704			1.65055			0.825275	+	0.564731i	$0.191020\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	3.53002i		−	0.387470i	−0.981054	−	0.193735i	$-0.937940\pi$
$83$		−	3.53002i		−	0.387470i	0.981054	−	0.193735i	$-0.0620602\pi$
$84$		0			0
$85$		−	0.216702i		−	0.0235047i
$86$		0			0
$87$	−5.17825			−0.555167
$88$		0			0
$89$		−	12.6369i		−	1.33951i	−0.742582	−	0.669755i	$-0.766399\pi$
$89$		−	12.6369i		−	1.33951i	0.742582	−	0.669755i	$-0.233601\pi$
$90$		0			0
$91$	−2.39577	−	2.69449i	−0.251145	−	0.282459i
$92$		0			0
$93$			5.01832i			0.520375i
$94$		0			0
$95$	0.0602929			0.00618592
$96$		0			0
$97$		−	0.975148i		−	0.0990113i	−0.998774	−	0.0495056i	$-0.984235\pi$
$97$		−	0.975148i		−	0.0990113i	0.998774	−	0.0495056i	$-0.0157646\pi$
$98$		0			0
$99$		−	1.72967i		−	0.173838i
$100$		0			0
$101$	12.7212			1.26581			0.632903	−	0.774231i	$-0.281863\pi$
$101$	12.7212			1.26581			0.632903	+	0.774231i	$0.281863\pi$
$102$		0			0
$103$	4.28706			0.422417			0.211208	−	0.977441i	$-0.432260\pi$
$103$	4.28706			0.422417			0.211208	+	0.977441i	$0.432260\pi$
$104$		0			0
$105$	0.100328			0.00979101
$106$		0			0
$107$	−7.65209			−0.739756			−0.369878	−	0.929080i	$-0.620600\pi$
$107$	−7.65209			−0.739756			−0.369878	+	0.929080i	$0.620600\pi$
$108$		0			0
$109$			5.61927i			0.538229i	0.963108	+	0.269114i	$0.0867310\pi$
$109$			5.61927i			0.538229i	−0.963108	+	0.269114i	$0.913269\pi$
$110$		0			0
$111$		−	2.36059i		−	0.224058i
$112$		0			0
$113$	−2.95701			−0.278172			−0.139086	−	0.990280i	$-0.544417\pi$
$113$	−2.95701			−0.278172			−0.139086	+	0.990280i	$0.544417\pi$
$114$		0			0
$115$		−	0.460241i		−	0.0429177i
$116$		0			0
$117$	−2.69449	+	2.39577i	−0.249106	+	0.221489i
$118$		0			0
$119$		−	2.15994i		−	0.198001i
$120$		0			0
$121$	8.00825			0.728023
$122$		0			0
$123$		−	5.02116i		−	0.452743i
$124$		0			0
$125$			1.00227i			0.0896457i
$126$		0			0
$127$	−3.86970			−0.343381			−0.171690	−	0.985151i	$-0.554923\pi$
$127$	−3.86970			−0.343381			−0.171690	+	0.985151i	$0.554923\pi$
$128$		0			0
$129$	9.12149			0.803103
$130$		0			0
$131$	−10.9947			−0.960607			−0.480304	−	0.877102i	$-0.659474\pi$
$131$	−10.9947			−0.960607			−0.480304	+	0.877102i	$0.659474\pi$
$132$		0			0
$133$	0.600958			0.0521097
$134$		0			0
$135$		−	0.100328i		−	0.00863486i
$136$		0			0
$137$			15.3082i			1.30787i	0.756551	+	0.653935i	$0.226883\pi$
$137$			15.3082i			1.30787i	−0.756551	+	0.653935i	$0.773117\pi$
$138$		0			0
$139$	8.45154			0.716850			0.358425	−	0.933558i	$-0.383314\pi$
$139$	8.45154			0.716850			0.358425	+	0.933558i	$0.383314\pi$
$140$		0			0
$141$			9.59248i			0.807833i
$142$		0			0
$143$	4.14389	+	4.66057i	0.346530	+	0.389736i
$144$		0			0
$145$			0.519524i			0.0431441i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			0.514225i			0.0421270i	0.999778	+	0.0210635i	$0.00670521\pi$
$149$			0.514225i			0.0421270i	−0.999778	+	0.0210635i	$0.993295\pi$
$150$		0			0
$151$		−	7.01479i		−	0.570855i	−0.958400	−	0.285428i	$-0.907864\pi$
$151$		−	7.01479i		−	0.570855i	0.958400	−	0.285428i	$-0.0921356\pi$
$152$		0			0
$153$	−2.15994			−0.174621
$154$		0			0
$155$	0.503477			0.0404403
$156$		0			0
$157$	−13.3619			−1.06639			−0.533196	−	0.845992i	$-0.679009\pi$
$157$	−13.3619			−1.06639			−0.533196	+	0.845992i	$0.679009\pi$
$158$		0			0
$159$	−3.75964			−0.298159
$160$		0			0
$161$		−	4.58736i		−	0.361535i
$162$		0			0
$163$		−	1.85611i		−	0.145382i	−0.997355	−	0.0726908i	$-0.976841\pi$
$163$		−	1.85611i		−	0.145382i	0.997355	−	0.0726908i	$-0.0231586\pi$
$164$		0			0
$165$	−0.173534			−0.0135096
$166$		0			0
$167$		−	20.9884i		−	1.62413i	−0.583564	−	0.812067i	$-0.698342\pi$
$167$		−	20.9884i		−	1.62413i	0.583564	−	0.812067i	$-0.301658\pi$
$168$		0			0
$169$	1.52053	−	12.9108i	0.116964	−	0.993136i
$170$		0			0
$171$		−	0.600958i		−	0.0459564i
$172$		0			0
$173$	15.1250			1.14993			0.574967	−	0.818177i	$-0.305015\pi$
$173$	15.1250			1.14993			0.574967	+	0.818177i	$0.305015\pi$
$174$		0			0
$175$			4.98993i			0.377204i
$176$		0			0
$177$			7.29218i			0.548114i
$178$		0			0
$179$	8.33040			0.622643			0.311322	−	0.950305i	$-0.399228\pi$
$179$	8.33040			0.622643			0.311322	+	0.950305i	$0.399228\pi$
$180$		0			0
$181$	−15.1024			−1.12255			−0.561275	−	0.827629i	$-0.689689\pi$
$181$	−15.1024			−1.12255			−0.561275	+	0.827629i	$0.689689\pi$
$182$		0			0
$183$	−4.43095			−0.327545
$184$		0			0
$185$	−0.236834			−0.0174124
$186$		0			0
$187$			3.73598i			0.273202i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	15.9301			1.15266			0.576331	−	0.817217i	$-0.304484\pi$
$191$	15.9301			1.15266			0.576331	+	0.817217i	$0.304484\pi$
$192$		0			0
$193$		−	12.6341i		−	0.909420i	−0.890640	−	0.454710i	$-0.849743\pi$
$193$		−	12.6341i		−	0.909420i	0.890640	−	0.454710i	$-0.150257\pi$
$194$		0			0
$195$	0.240363	+	0.270332i	0.0172128	+	0.0193589i
$196$		0			0
$197$		−	11.6935i		−	0.833127i	−0.909107	−	0.416563i	$-0.863234\pi$
$197$		−	11.6935i		−	0.833127i	0.909107	−	0.416563i	$-0.136766\pi$
$198$		0			0
$199$	11.2824			0.799791			0.399895	−	0.916561i	$-0.369046\pi$
$199$	11.2824			0.799791			0.399895	+	0.916561i	$0.369046\pi$
$200$		0			0
$201$		−	11.2985i		−	0.796934i
$202$		0			0
$203$			5.17825i			0.363442i
$204$		0			0
$205$	−0.503763			−0.0351843
$206$		0			0
$207$	−4.58736			−0.318844
$208$		0			0
$209$	−1.03946			−0.0719008
$210$		0			0
$211$	15.3086			1.05388			0.526942	−	0.849901i	$-0.323338\pi$
$211$	15.3086			1.05388			0.526942	+	0.849901i	$0.323338\pi$
$212$		0			0
$213$			0.0902623i			0.00618467i
$214$		0			0
$215$		−	0.915141i		−	0.0624121i
$216$		0			0
$217$	5.01832			0.340665
$218$		0			0
$219$			7.78928i			0.526351i
$220$		0			0
$221$	5.81993	−	5.17473i	0.391491	−	0.348090i
$222$		0			0
$223$		−	20.3168i		−	1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$223$		−	20.3168i		−	1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$224$		0			0
$225$	4.98993			0.332662
$226$		0			0
$227$		−	6.32301i		−	0.419673i	−0.977736	−	0.209836i	$-0.932707\pi$
$227$		−	6.32301i		−	0.419673i	0.977736	−	0.209836i	$-0.0672932\pi$
$228$		0			0
$229$		−	10.9153i		−	0.721304i	−0.932700	−	0.360652i	$-0.882554\pi$
$229$		−	10.9153i		−	0.721304i	0.932700	−	0.360652i	$-0.117446\pi$
$230$		0			0
$231$	−1.72967			−0.113804
$232$		0			0
$233$	28.2001			1.84745			0.923725	−	0.383056i	$-0.125128\pi$
$233$	28.2001			1.84745			0.923725	+	0.383056i	$0.125128\pi$
$234$		0			0
$235$	0.962394			0.0627797
$236$		0			0
$237$	−14.6704			−0.952945
$238$		0			0
$239$		−	18.8481i		−	1.21918i	−0.792716	−	0.609591i	$-0.791334\pi$
$239$		−	18.8481i		−	1.21918i	0.792716	−	0.609591i	$-0.208666\pi$
$240$		0			0
$241$		−	9.60921i		−	0.618984i	−0.950902	−	0.309492i	$-0.899841\pi$
$241$		−	9.60921i		−	0.618984i	0.950902	−	0.309492i	$-0.100159\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	0.100328i		−	0.00640972i
$246$		0			0
$247$	1.43976	+	1.61927i	0.0916097	+	0.103032i
$248$		0			0
$249$			3.53002i			0.223706i
$250$		0			0
$251$	8.40292			0.530388			0.265194	−	0.964195i	$-0.414564\pi$
$251$	8.40292			0.530388			0.265194	+	0.964195i	$0.414564\pi$
$252$		0			0
$253$			7.93461i			0.498845i
$254$		0			0
$255$			0.216702i			0.0135704i
$256$		0			0
$257$	19.6399			1.22510			0.612550	−	0.790432i	$-0.290144\pi$
$257$	19.6399			1.22510			0.612550	+	0.790432i	$0.290144\pi$
$258$		0			0
$259$	−2.36059			−0.146680
$260$		0			0
$261$	5.17825			0.320526
$262$		0			0
$263$	25.0516			1.54475			0.772374	−	0.635169i	$-0.219069\pi$
$263$	25.0516			1.54475			0.772374	+	0.635169i	$0.219069\pi$
$264$		0			0
$265$			0.377197i			0.0231710i
$266$		0			0
$267$			12.6369i			0.773366i
$268$		0			0
$269$	16.8418			1.02686			0.513431	−	0.858131i	$-0.328374\pi$
$269$	16.8418			1.02686			0.513431	+	0.858131i	$0.328374\pi$
$270$		0			0
$271$			28.8203i			1.75071i	0.483483	+	0.875354i	$0.339371\pi$
$271$			28.8203i			1.75071i	−0.483483	+	0.875354i	$0.660629\pi$
$272$		0			0
$273$	2.39577	+	2.69449i	0.144999	+	0.163078i
$274$		0			0
$275$		−	8.63093i		−	0.520464i
$276$		0			0
$277$	−4.22023			−0.253569			−0.126785	−	0.991930i	$-0.540466\pi$
$277$	−4.22023			−0.253569			−0.126785	+	0.991930i	$0.540466\pi$
$278$		0			0
$279$		−	5.01832i		−	0.300439i
$280$		0			0
$281$		−	20.1470i		−	1.20187i	−0.799298	−	0.600935i	$-0.794795\pi$
$281$		−	20.1470i		−	1.20187i	0.799298	−	0.600935i	$-0.205205\pi$
$282$		0			0
$283$	−14.4581			−0.859443			−0.429722	−	0.902961i	$-0.641388\pi$
$283$	−14.4581			−0.859443			−0.429722	+	0.902961i	$0.641388\pi$
$284$		0			0
$285$	−0.0602929			−0.00357144
$286$		0			0
$287$	−5.02116			−0.296390
$288$		0			0
$289$	−12.3347			−0.725568
$290$		0			0
$291$			0.975148i			0.0571642i
$292$		0			0
$293$		−	3.54225i		−	0.206941i	−0.994633	−	0.103470i	$-0.967005\pi$
$293$		−	3.54225i		−	0.206941i	0.994633	−	0.103470i	$-0.0329947\pi$
$294$		0			0
$295$	0.731609			0.0425959
$296$		0			0
$297$			1.72967i			0.100365i
$298$		0			0
$299$	12.3606	−	10.9903i	0.714832	−	0.635585i
$300$		0			0
$301$		−	9.12149i		−	0.525754i
$302$		0			0
$303$	−12.7212			−0.730813
$304$		0			0
$305$			0.444548i			0.0254548i
$306$		0			0
$307$		−	15.2814i		−	0.872157i	−0.899909	−	0.436079i	$-0.856367\pi$
$307$		−	15.2814i		−	0.872157i	0.899909	−	0.436079i	$-0.143633\pi$
$308$		0			0
$309$	−4.28706			−0.243882
$310$		0			0
$311$	8.84178			0.501371			0.250686	−	0.968069i	$-0.419344\pi$
$311$	8.84178			0.501371			0.250686	+	0.968069i	$0.419344\pi$
$312$		0			0
$313$	−10.5838			−0.598234			−0.299117	−	0.954217i	$-0.596692\pi$
$313$	−10.5838			−0.598234			−0.299117	+	0.954217i	$0.596692\pi$
$314$		0			0
$315$	−0.100328			−0.00565284
$316$		0			0
$317$		−	22.8435i		−	1.28302i	−0.767114	−	0.641511i	$-0.778308\pi$
$317$		−	22.8435i		−	1.28302i	0.767114	−	0.641511i	$-0.221692\pi$
$318$		0			0
$319$		−	8.95666i		−	0.501477i
$320$		0			0
$321$	7.65209			0.427098
$322$		0			0
$323$			1.29803i			0.0722244i
$324$		0			0
$325$	−13.4453	+	11.9548i	−0.745812	+	0.663130i
$326$		0			0
$327$		−	5.61927i		−	0.310747i
$328$		0			0
$329$	9.59248			0.528851
$330$		0			0
$331$			24.3610i			1.33900i	0.742810	+	0.669502i	$0.233493\pi$
$331$			24.3610i			1.33900i	−0.742810	+	0.669502i	$0.766507\pi$
$332$		0			0
$333$			2.36059i			0.129360i
$334$		0			0
$335$	−1.13355			−0.0619327
$336$		0			0
$337$	−15.5047			−0.844593			−0.422297	−	0.906458i	$-0.638776\pi$
$337$	−15.5047			−0.844593			−0.422297	+	0.906458i	$0.638776\pi$
$338$		0			0
$339$	2.95701			0.160603
$340$		0			0
$341$	−8.68002			−0.470049
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			0.460241i			0.0247785i
$346$		0			0
$347$	−9.30754			−0.499655			−0.249828	−	0.968290i	$-0.580374\pi$
$347$	−9.30754			−0.499655			−0.249828	+	0.968290i	$0.580374\pi$
$348$		0			0
$349$		−	3.43892i		−	0.184081i	−0.995755	−	0.0920405i	$-0.970661\pi$
$349$		−	3.43892i		−	0.184081i	0.995755	−	0.0920405i	$-0.0293389\pi$
$350$		0			0
$351$	2.69449	−	2.39577i	0.143821	−	0.127877i
$352$		0			0
$353$		−	22.3563i		−	1.18990i	−0.803761	−	0.594952i	$-0.797171\pi$
$353$		−	22.3563i		−	1.18990i	0.803761	−	0.594952i	$-0.202829\pi$
$354$		0			0
$355$	0.00905583			0.000480633
$356$		0			0
$357$			2.15994i			0.114316i
$358$		0			0
$359$		−	21.9464i		−	1.15829i	−0.815226	−	0.579143i	$-0.803387\pi$
$359$		−	21.9464i		−	1.15829i	0.815226	−	0.579143i	$-0.196613\pi$
$360$		0			0
$361$	18.6388			0.980992
$362$		0			0
$363$	−8.00825			−0.420324
$364$		0			0
$365$	0.781482			0.0409047
$366$		0			0
$367$	19.2817			1.00650			0.503249	−	0.864142i	$-0.332138\pi$
$367$	19.2817			1.00650			0.503249	+	0.864142i	$0.332138\pi$
$368$		0			0
$369$			5.02116i			0.261391i
$370$		0			0
$371$			3.75964i			0.195191i
$372$		0			0
$373$	13.9095			0.720207			0.360104	−	0.932912i	$-0.382741\pi$
$373$	13.9095			0.720207			0.360104	+	0.932912i	$0.382741\pi$
$374$		0			0
$375$		−	1.00227i		−	0.0517570i
$376$		0			0
$377$	−13.9527	+	12.4059i	−0.718603	+	0.638938i
$378$		0			0
$379$			7.95974i			0.408864i	0.978881	+	0.204432i	$0.0655348\pi$
$379$			7.95974i			0.408864i	−0.978881	+	0.204432i	$0.934465\pi$
$380$		0			0
$381$	3.86970			0.198251
$382$		0			0
$383$			5.90565i			0.301765i	0.988552	+	0.150882i	$0.0482115\pi$
$383$			5.90565i			0.301765i	−0.988552	+	0.150882i	$0.951789\pi$
$384$		0			0
$385$			0.173534i			0.00884411i
$386$		0			0
$387$	−9.12149			−0.463671
$388$		0			0
$389$	−6.14072			−0.311347			−0.155673	−	0.987809i	$-0.549755\pi$
$389$	−6.14072			−0.311347			−0.155673	+	0.987809i	$0.549755\pi$
$390$		0			0
$391$	9.90842			0.501090
$392$		0			0
$393$	10.9947			0.554607
$394$		0			0
$395$			1.47185i			0.0740569i
$396$		0			0
$397$		−	23.7330i		−	1.19112i	−0.803309	−	0.595562i	$-0.796929\pi$
$397$		−	23.7330i		−	1.19112i	0.803309	−	0.595562i	$-0.203071\pi$
$398$		0			0
$399$	−0.600958			−0.0300855
$400$		0			0
$401$			17.3288i			0.865359i	0.901548	+	0.432680i	$0.142432\pi$
$401$			17.3288i			0.865359i	−0.901548	+	0.432680i	$0.857568\pi$
$402$		0			0
$403$	12.0228	+	13.5218i	0.598896	+	0.673569i
$404$		0			0
$405$			0.100328i			0.00498534i
$406$		0			0
$407$	4.08304			0.202389
$408$		0			0
$409$			12.9697i			0.641310i	0.947196	+	0.320655i	$0.103903\pi$
$409$			12.9697i			0.641310i	−0.947196	+	0.320655i	$0.896097\pi$
$410$		0			0
$411$		−	15.3082i		−	0.755099i
$412$		0			0
$413$	7.29218			0.358825
$414$		0			0
$415$	0.354160			0.0173850
$416$		0			0
$417$	−8.45154			−0.413874
$418$		0			0
$419$	15.0831			0.736860			0.368430	−	0.929655i	$-0.379895\pi$
$419$	15.0831			0.736860			0.368430	+	0.929655i	$0.379895\pi$
$420$		0			0
$421$		−	8.38118i		−	0.408474i	−0.978921	−	0.204237i	$-0.934529\pi$
$421$		−	8.38118i		−	0.408474i	0.978921	−	0.204237i	$-0.0654713\pi$
$422$		0			0
$423$		−	9.59248i		−	0.466402i
$424$		0			0
$425$	−10.7780			−0.522808
$426$		0			0
$427$			4.43095i			0.214429i
$428$		0			0
$429$	−4.14389	−	4.66057i	−0.200069	−	0.225014i
$430$		0			0
$431$		−	20.6630i		−	0.995303i	−0.867377	−	0.497651i	$-0.834196\pi$
$431$		−	20.6630i		−	0.995303i	0.867377	−	0.497651i	$-0.165804\pi$
$432$		0			0
$433$	−25.3286			−1.21721			−0.608607	−	0.793472i	$-0.708271\pi$
$433$	−25.3286			−1.21721			−0.608607	+	0.793472i	$0.708271\pi$
$434$		0			0
$435$		−	0.519524i		−	0.0249093i
$436$		0			0
$437$			2.75681i			0.131876i
$438$		0			0
$439$	−11.8537			−0.565744			−0.282872	−	0.959158i	$-0.591287\pi$
$439$	−11.8537			−0.565744			−0.282872	+	0.959158i	$0.591287\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	9.60467			0.456332			0.228166	−	0.973622i	$-0.426727\pi$
$443$	9.60467			0.456332			0.228166	+	0.973622i	$0.426727\pi$
$444$		0			0
$445$	1.26784			0.0601011
$446$		0			0
$447$		−	0.514225i		−	0.0243220i
$448$		0			0
$449$		−	20.0907i		−	0.948140i	−0.880487	−	0.474070i	$-0.842784\pi$
$449$		−	20.0907i		−	0.948140i	0.880487	−	0.474070i	$-0.157216\pi$
$450$		0			0
$451$	8.68494			0.408958
$452$		0			0
$453$			7.01479i			0.329583i
$454$		0			0
$455$	0.270332	−	0.240363i	0.0126734	−	0.0112684i
$456$		0			0
$457$		−	23.2985i		−	1.08986i	−0.838482	−	0.544929i	$-0.816557\pi$
$457$		−	23.2985i		−	1.08986i	0.838482	−	0.544929i	$-0.183443\pi$
$458$		0			0
$459$	2.15994			0.100817
$460$		0			0
$461$			12.6580i			0.589544i	0.955568	+	0.294772i	$0.0952437\pi$
$461$			12.6580i			0.589544i	−0.955568	+	0.294772i	$0.904756\pi$
$462$		0			0
$463$			0.838801i			0.0389824i	0.999810	+	0.0194912i	$0.00620464\pi$
$463$			0.838801i			0.0389824i	−0.999810	+	0.0194912i	$0.993795\pi$
$464$		0			0
$465$	−0.503477			−0.0233482
$466$		0			0
$467$	5.81287			0.268988			0.134494	−	0.990914i	$-0.457059\pi$
$467$	5.81287			0.268988			0.134494	+	0.990914i	$0.457059\pi$
$468$		0			0
$469$	−11.2985			−0.521716
$470$		0			0
$471$	13.3619			0.615682
$472$		0			0
$473$			15.7771i			0.725434i
$474$		0			0
$475$		−	2.99874i		−	0.137592i
$476$		0			0
$477$	3.75964			0.172142
$478$		0			0
$479$			7.73229i			0.353297i	0.984274	+	0.176649i	$0.0565256\pi$
$479$			7.73229i			0.353297i	−0.984274	+	0.176649i	$0.943474\pi$
$480$		0			0
$481$	−5.65545	−	6.36059i	−0.257866	−	0.290018i
$482$		0			0
$483$			4.58736i			0.208732i
$484$		0			0
$485$	0.0978346			0.00444244
$486$		0			0
$487$			41.4021i			1.87611i	0.346488	+	0.938054i	$0.387374\pi$
$487$			41.4021i			1.87611i	−0.346488	+	0.938054i	$0.612626\pi$
$488$		0			0
$489$			1.85611i			0.0839361i
$490$		0			0
$491$	−29.6958			−1.34015			−0.670076	−	0.742293i	$-0.733738\pi$
$491$	−29.6958			−1.34015			−0.670076	+	0.742293i	$0.733738\pi$
$492$		0			0
$493$	−11.1847			−0.503734
$494$		0			0
$495$	0.173534			0.00779977
$496$		0			0
$497$	0.0902623			0.00404882
$498$		0			0
$499$			32.4816i			1.45408i	0.686596	+	0.727039i	$0.259104\pi$
$499$			32.4816i			1.45408i	−0.686596	+	0.727039i	$0.740896\pi$
$500$		0			0
$501$			20.9884i			0.937695i
$502$		0			0
$503$	14.0437			0.626177			0.313089	−	0.949724i	$-0.398636\pi$
$503$	14.0437			0.626177			0.313089	+	0.949724i	$0.398636\pi$
$504$		0			0
$505$			1.27629i			0.0567942i
$506$		0			0
$507$	−1.52053	+	12.9108i	−0.0675293	+	0.573387i
$508$		0			0
$509$		−	18.4985i		−	0.819930i	−0.912101	−	0.409965i	$-0.865541\pi$
$509$		−	18.4985i		−	0.819930i	0.912101	−	0.409965i	$-0.134459\pi$
$510$		0			0
$511$	7.78928			0.344577
$512$		0			0
$513$			0.600958i			0.0265329i
$514$		0			0
$515$			0.430112i			0.0189530i
$516$		0			0
$517$	−16.5918			−0.729707
$518$		0			0
$519$	−15.1250			−0.663915
$520$		0			0
$521$	−5.96065			−0.261141			−0.130570	−	0.991439i	$-0.541681\pi$
$521$	−5.96065			−0.261141			−0.130570	+	0.991439i	$0.541681\pi$
$522$		0			0
$523$	−30.7879			−1.34626			−0.673131	−	0.739523i	$-0.735051\pi$
$523$	−30.7879			−1.34626			−0.673131	+	0.739523i	$0.735051\pi$
$524$		0			0
$525$		−	4.98993i		−	0.217779i
$526$		0			0
$527$			10.8393i			0.472165i
$528$		0			0
$529$	−1.95610			−0.0850480
$530$		0			0
$531$		−	7.29218i		−	0.316454i
$532$		0			0
$533$	−12.0296	−	13.5295i	−0.521059	−	0.586026i
$534$		0			0
$535$		−	0.767719i		−	0.0331914i
$536$		0			0
$537$	−8.33040			−0.359483
$538$		0			0
$539$			1.72967i			0.0745021i
$540$		0			0
$541$		−	31.3216i		−	1.34662i	−0.739360	−	0.673310i	$-0.764872\pi$
$541$		−	31.3216i		−	1.34662i	0.739360	−	0.673310i	$-0.235128\pi$
$542$		0			0
$543$	15.1024			0.648105
$544$		0			0
$545$	−0.563770			−0.0241493
$546$		0			0
$547$	9.69833			0.414671			0.207335	−	0.978270i	$-0.433521\pi$
$547$	9.69833			0.414671			0.207335	+	0.978270i	$0.433521\pi$
$548$		0			0
$549$	4.43095			0.189108
$550$		0			0
$551$		−	3.11191i		−	0.132572i
$552$		0			0
$553$			14.6704i			0.623849i
$554$		0			0
$555$	0.236834			0.0100530
$556$		0			0
$557$			16.5268i			0.700265i	0.936700	+	0.350132i	$0.113863\pi$
$557$			16.5268i			0.700265i	−0.936700	+	0.350132i	$0.886137\pi$
$558$		0			0
$559$	24.5778	−	21.8530i	1.03953	−	0.924285i
$560$		0			0
$561$		−	3.73598i		−	0.157733i
$562$		0			0
$563$	−40.2421			−1.69600			−0.848001	−	0.529995i	$-0.822194\pi$
$563$	−40.2421			−1.69600			−0.848001	+	0.529995i	$0.822194\pi$
$564$		0			0
$565$		−	0.296671i		−	0.0124810i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	2.77987			0.116538			0.0582692	−	0.998301i	$-0.481442\pi$
$569$	2.77987			0.116538			0.0582692	+	0.998301i	$0.481442\pi$
$570$		0			0
$571$	33.0993			1.38516			0.692581	−	0.721340i	$-0.256474\pi$
$571$	33.0993			1.38516			0.692581	+	0.721340i	$0.256474\pi$
$572$		0			0
$573$	−15.9301			−0.665489
$574$		0			0
$575$	−22.8906			−0.954606
$576$		0			0
$577$			12.0432i			0.501364i	0.968070	+	0.250682i	$0.0806548\pi$
$577$			12.0432i			0.501364i	−0.968070	+	0.250682i	$0.919345\pi$
$578$		0			0
$579$			12.6341i			0.525054i
$580$		0			0
$581$	3.53002			0.146450
$582$		0			0
$583$		−	6.50292i		−	0.269324i
$584$		0			0
$585$	−0.240363	−	0.270332i	−0.00993779	−	0.0111769i
$586$		0			0
$587$			16.7989i			0.693366i	0.937982	+	0.346683i	$0.112692\pi$
$587$			16.7989i			0.693366i	−0.937982	+	0.346683i	$0.887308\pi$
$588$		0			0
$589$	−3.01580			−0.124264
$590$		0			0
$591$			11.6935i			0.481006i
$592$		0			0
$593$		−	24.2969i		−	0.997754i	−0.866673	−	0.498877i	$-0.833746\pi$
$593$		−	24.2969i		−	0.997754i	0.866673	−	0.498877i	$-0.166254\pi$
$594$		0			0
$595$	0.216702			0.00888393
$596$		0			0
$597$	−11.2824			−0.461760
$598$		0			0
$599$	42.7379			1.74622			0.873111	−	0.487522i	$-0.162099\pi$
$599$	42.7379			1.74622			0.873111	+	0.487522i	$0.162099\pi$
$600$		0			0
$601$	−23.7948			−0.970611			−0.485306	−	0.874345i	$-0.661292\pi$
$601$	−23.7948			−0.970611			−0.485306	+	0.874345i	$0.661292\pi$
$602$		0			0
$603$			11.2985i			0.460110i
$604$		0			0
$605$			0.803451i			0.0326649i
$606$		0			0
$607$	−24.2740			−0.985251			−0.492625	−	0.870241i	$-0.663963\pi$
$607$	−24.2740			−0.985251			−0.492625	+	0.870241i	$0.663963\pi$
$608$		0			0
$609$		−	5.17825i		−	0.209833i
$610$		0			0
$611$	22.9814	+	25.8468i	0.929728	+	1.04565i
$612$		0			0
$613$			22.0876i			0.892109i	0.895006	+	0.446055i	$0.147171\pi$
$613$			22.0876i			0.892109i	−0.895006	+	0.446055i	$0.852829\pi$
$614$		0			0
$615$	0.503763			0.0203137
$616$		0			0
$617$		−	9.89122i		−	0.398205i	−0.979979	−	0.199103i	$-0.936197\pi$
$617$		−	9.89122i		−	0.398205i	0.979979	−	0.199103i	$-0.0638027\pi$
$618$		0			0
$619$			34.8194i			1.39951i	0.714384	+	0.699754i	$0.246707\pi$
$619$			34.8194i			1.39951i	−0.714384	+	0.699754i	$0.753293\pi$
$620$		0			0
$621$	4.58736			0.184085
$622$		0			0
$623$	12.6369			0.506287
$624$		0			0
$625$	24.8491			0.993965
$626$		0			0
$627$	1.03946			0.0415119
$628$		0			0
$629$		−	5.09874i		−	0.203300i
$630$		0			0
$631$			12.5965i			0.501459i	0.968057	+	0.250730i	$0.0806705\pi$
$631$			12.5965i			0.501459i	−0.968057	+	0.250730i	$0.919329\pi$
$632$		0			0
$633$	−15.3086			−0.608460
$634$		0			0
$635$		−	0.388239i		−	0.0154068i
$636$		0			0
$637$	2.69449	−	2.39577i	0.106760	−	0.0949240i
$638$		0			0
$639$		−	0.0902623i		−	0.00357072i
$640$		0			0
$641$	23.2769			0.919382			0.459691	−	0.888079i	$-0.347960\pi$
$641$	23.2769			0.919382			0.459691	+	0.888079i	$0.347960\pi$
$642$		0			0
$643$		−	19.6817i		−	0.776172i	−0.921623	−	0.388086i	$-0.873136\pi$
$643$		−	19.6817i		−	0.776172i	0.921623	−	0.388086i	$-0.126864\pi$
$644$		0			0
$645$			0.915141i			0.0360336i
$646$		0			0
$647$	18.7578			0.737446			0.368723	−	0.929539i	$-0.379795\pi$
$647$	18.7578			0.737446			0.368723	+	0.929539i	$0.379795\pi$
$648$		0			0
$649$	−12.6130			−0.495105
$650$		0			0
$651$	−5.01832			−0.196683
$652$		0			0
$653$	−6.61599			−0.258904			−0.129452	−	0.991586i	$-0.541322\pi$
$653$	−6.61599			−0.258904			−0.129452	+	0.991586i	$0.541322\pi$
$654$		0			0
$655$		−	1.10307i		−	0.0431006i
$656$		0			0
$657$		−	7.78928i		−	0.303889i
$658$		0			0
$659$	−30.3670			−1.18293			−0.591466	−	0.806330i	$-0.701451\pi$
$659$	−30.3670			−1.18293			−0.591466	+	0.806330i	$0.701451\pi$
$660$		0			0
$661$			43.6458i			1.69763i	0.528693	+	0.848813i	$0.322682\pi$
$661$			43.6458i			1.69763i	−0.528693	+	0.848813i	$0.677318\pi$
$662$		0			0
$663$	−5.81993	+	5.17473i	−0.226027	+	0.200970i
$664$		0			0
$665$			0.0602929i			0.00233806i
$666$		0			0
$667$	−23.7545			−0.919779
$668$		0			0
$669$			20.3168i			0.785493i
$670$		0			0
$671$		−	7.66408i		−	0.295868i
$672$		0			0
$673$	33.9517			1.30874			0.654370	−	0.756174i	$-0.272934\pi$
$673$	33.9517			1.30874			0.654370	+	0.756174i	$0.272934\pi$
$674$		0			0
$675$	−4.98993			−0.192063
$676$		0			0
$677$	−33.8557			−1.30118			−0.650589	−	0.759430i	$-0.725478\pi$
$677$	−33.8557			−1.30118			−0.650589	+	0.759430i	$0.725478\pi$
$678$		0			0
$679$	0.975148			0.0374227
$680$		0			0
$681$			6.32301i			0.242298i
$682$		0			0
$683$		−	7.24440i		−	0.277199i	−0.990348	−	0.138600i	$-0.955740\pi$
$683$		−	7.24440i		−	0.277199i	0.990348	−	0.138600i	$-0.0442601\pi$
$684$		0			0
$685$	−1.53584			−0.0586815
$686$		0			0
$687$			10.9153i			0.416445i
$688$		0			0
$689$	−10.1303	+	9.00724i	−0.385934	+	0.343149i
$690$		0			0
$691$			6.57468i			0.250113i	0.992150	+	0.125056i	$0.0399111\pi$
$691$			6.57468i			0.250113i	−0.992150	+	0.125056i	$0.960089\pi$
$692$		0			0
$693$	1.72967			0.0657046
$694$		0			0
$695$			0.847926i			0.0321637i
$696$		0			0
$697$		−	10.8454i		−	0.410799i
$698$		0			0
$699$	−28.2001			−1.06663
$700$		0			0
$701$	−5.00672			−0.189101			−0.0945506	−	0.995520i	$-0.530141\pi$
$701$	−5.00672			−0.189101			−0.0945506	+	0.995520i	$0.530141\pi$
$702$		0			0
$703$	1.41862			0.0535042
$704$		0			0
$705$	−0.962394			−0.0362459
$706$		0			0
$707$			12.7212i			0.478430i
$708$		0			0
$709$			9.72245i			0.365134i	0.983193	+	0.182567i	$0.0584407\pi$
$709$			9.72245i			0.365134i	−0.983193	+	0.182567i	$0.941559\pi$
$710$		0			0
$711$	14.6704			0.550183
$712$		0			0
$713$			23.0208i			0.862137i
$714$		0			0
$715$	−0.467585	+	0.415748i	−0.0174867	+	0.0155481i
$716$		0			0
$717$			18.8481i			0.703895i
$718$		0			0
$719$	−11.3786			−0.424349			−0.212174	−	0.977232i	$-0.568054\pi$
$719$	−11.3786			−0.424349			−0.212174	+	0.977232i	$0.568054\pi$
$720$		0			0
$721$			4.28706i			0.159658i
$722$		0			0
$723$			9.60921i			0.357370i
$724$		0			0
$725$	25.8392			0.959642
$726$		0			0
$727$	39.9815			1.48283			0.741415	−	0.671046i	$-0.234155\pi$
$727$	39.9815			1.48283			0.741415	+	0.671046i	$0.234155\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	19.7019			0.728700
$732$		0			0
$733$		−	16.4073i		−	0.606017i	−0.952988	−	0.303009i	$-0.902009\pi$
$733$		−	16.4073i		−	0.606017i	0.952988	−	0.303009i	$-0.0979911\pi$
$734$		0			0
$735$			0.100328i			0.00370065i
$736$		0			0
$737$	19.5426			0.719862
$738$		0			0
$739$			40.5818i			1.49283i	0.665482	+	0.746413i	$0.268226\pi$
$739$			40.5818i			1.49283i	−0.665482	+	0.746413i	$0.731774\pi$
$740$		0			0
$741$	−1.43976	−	1.61927i	−0.0528909	−	0.0594855i
$742$		0			0
$743$		−	27.9435i		−	1.02515i	−0.858643	−	0.512574i	$-0.828692\pi$
$743$		−	27.9435i		−	1.02515i	0.858643	−	0.512574i	$-0.171308\pi$
$744$		0			0
$745$	−0.0515911			−0.00189015
$746$		0			0
$747$		−	3.53002i		−	0.129157i
$748$		0			0
$749$		−	7.65209i		−	0.279601i
$750$		0			0
$751$	−24.0228			−0.876603			−0.438301	−	0.898828i	$-0.644420\pi$
$751$	−24.0228			−0.876603			−0.438301	+	0.898828i	$0.644420\pi$
$752$		0			0
$753$	−8.40292			−0.306219
$754$		0			0
$755$	0.703779			0.0256131
$756$		0			0
$757$	−35.8309			−1.30230			−0.651148	−	0.758951i	$-0.725712\pi$
$757$	−35.8309			−1.30230			−0.651148	+	0.758951i	$0.725712\pi$
$758$		0			0
$759$		−	7.93461i		−	0.288008i
$760$		0			0
$761$			1.91320i			0.0693534i	0.999399	+	0.0346767i	$0.0110402\pi$
$761$			1.91320i			0.0693534i	−0.999399	+	0.0346767i	$0.988960\pi$
$762$		0			0
$763$	−5.61927			−0.203431
$764$		0			0
$765$		−	0.216702i		−	0.00783489i
$766$		0			0
$767$	17.4704	+	19.6487i	0.630820	+	0.709473i
$768$		0			0
$769$			17.8802i			0.644775i	0.946608	+	0.322387i	$0.104485\pi$
$769$			17.8802i			0.644775i	−0.946608	+	0.322387i	$0.895515\pi$
$770$		0			0
$771$	−19.6399			−0.707312
$772$		0			0
$773$		−	10.5791i		−	0.380505i	−0.981735	−	0.190253i	$-0.939069\pi$
$773$		−	10.5791i		−	0.380505i	0.981735	−	0.190253i	$-0.0609307\pi$
$774$		0			0
$775$		−	25.0411i		−	0.899502i
$776$		0			0
$777$	2.36059			0.0846859
$778$		0			0
$779$	3.01751			0.108113
$780$		0			0
$781$	−0.156124			−0.00558654
$782$		0			0
$783$	−5.17825			−0.185056
$784$		0			0
$785$		−	1.34057i		−	0.0478469i
$786$		0			0
$787$			20.0459i			0.714558i	0.933998	+	0.357279i	$0.116295\pi$
$787$			20.0459i			0.714558i	−0.933998	+	0.357279i	$0.883705\pi$
$788$		0			0
$789$	−25.0516			−0.891860
$790$		0			0
$791$		−	2.95701i		−	0.105139i
$792$		0			0
$793$	−11.9392	+	10.6156i	−0.423972	+	0.376970i
$794$		0			0
$795$		−	0.377197i		−	0.0133778i
$796$		0			0
$797$	35.2505			1.24864			0.624319	−	0.781170i	$-0.285377\pi$
$797$	35.2505			1.24864			0.624319	+	0.781170i	$0.285377\pi$
$798$		0			0
$799$			20.7192i			0.732992i
$800$		0			0
$801$		−	12.6369i		−	0.446503i
$802$		0			0
$803$	−13.4729			−0.475447
$804$		0			0
$805$	0.460241			0.0162214
$806$		0			0
$807$	−16.8418			−0.592859
$808$		0			0
$809$	−31.2729			−1.09950			−0.549748	−	0.835330i	$-0.685276\pi$
$809$	−31.2729			−1.09950			−0.549748	+	0.835330i	$0.685276\pi$
$810$		0			0
$811$		−	14.6818i		−	0.515549i	−0.966205	−	0.257775i	$-0.917011\pi$
$811$		−	14.6818i		−	0.515549i	0.966205	−	0.257775i	$-0.0829892\pi$
$812$		0			0
$813$		−	28.8203i		−	1.01077i
$814$		0			0
$815$	0.186219			0.00652298
$816$		0			0
$817$			5.48163i			0.191778i
$818$		0			0
$819$	−2.39577	−	2.69449i	−0.0837151	−	0.0941530i
$820$		0			0
$821$		−	18.5314i		−	0.646750i	−0.946271	−	0.323375i	$-0.895183\pi$
$821$		−	18.5314i		−	0.646750i	0.946271	−	0.323375i	$-0.104817\pi$
$822$		0			0
$823$	1.08133			0.0376928			0.0188464	−	0.999822i	$-0.494001\pi$
$823$	1.08133			0.0376928			0.0188464	+	0.999822i	$0.494001\pi$
$824$		0			0
$825$			8.63093i			0.300490i
$826$		0			0
$827$		−	15.5392i		−	0.540353i	−0.962811	−	0.270176i	$-0.912918\pi$
$827$		−	15.5392i		−	0.540353i	0.962811	−	0.270176i	$-0.0870820\pi$
$828$		0			0
$829$	10.1608			0.352898			0.176449	−	0.984310i	$-0.443539\pi$
$829$	10.1608			0.352898			0.176449	+	0.984310i	$0.443539\pi$
$830$		0			0
$831$	4.22023			0.146398
$832$		0			0
$833$	2.15994			0.0748374
$834$		0			0
$835$	2.10573			0.0728717
$836$		0			0
$837$			5.01832i			0.173458i
$838$		0			0
$839$		−	13.2082i		−	0.455999i	−0.973661	−	0.227999i	$-0.926782\pi$
$839$		−	13.2082i		−	0.455999i	0.973661	−	0.227999i	$-0.0732184\pi$
$840$		0			0
$841$	−2.18568			−0.0753682
$842$		0			0
$843$			20.1470i			0.693900i
$844$		0			0
$845$	1.29531	+	0.152552i	0.0445601	+	0.00524795i
$846$		0			0
$847$			8.00825i			0.275167i
$848$		0			0
$849$	14.4581			0.496200
$850$		0			0
$851$		−	10.8289i		−	0.371210i
$852$		0			0
$853$			52.8473i			1.80946i	0.425989	+	0.904729i	$0.359926\pi$
$853$			52.8473i			1.80946i	−0.425989	+	0.904729i	$0.640074\pi$
$854$		0			0
$855$	0.0602929			0.00206197
$856$		0			0
$857$	−46.5246			−1.58925			−0.794625	−	0.607100i	$-0.792333\pi$
$857$	−46.5246			−1.58925			−0.794625	+	0.607100i	$0.792333\pi$
$858$		0			0
$859$	−30.4633			−1.03939			−0.519697	−	0.854350i	$-0.673955\pi$
$859$	−30.4633			−1.03939			−0.519697	+	0.854350i	$0.673955\pi$
$860$		0			0
$861$	5.02116			0.171121
$862$		0			0
$863$			30.6461i			1.04320i	0.853189	+	0.521602i	$0.174665\pi$
$863$			30.6461i			1.04320i	−0.853189	+	0.521602i	$0.825335\pi$
$864$		0			0
$865$			1.51746i			0.0515953i
$866$		0			0
$867$	12.3347			0.418907
$868$		0			0
$869$		−	25.3749i		−	0.860785i
$870$		0			0
$871$	−27.0686	−	30.4436i	−0.917185	−	1.03154i
$872$		0			0
$873$		−	0.975148i		−	0.0330038i
$874$		0			0
$875$	−1.00227			−0.0338829
$876$		0			0
$877$			19.8663i			0.670839i	0.942069	+	0.335419i	$0.108878\pi$
$877$			19.8663i			0.670839i	−0.942069	+	0.335419i	$0.891122\pi$
$878$		0			0
$879$			3.54225i			0.119477i
$880$		0			0
$881$	−19.5688			−0.659289			−0.329644	−	0.944105i	$-0.606929\pi$
$881$	−19.5688			−0.659289			−0.329644	+	0.944105i	$0.606929\pi$
$882$		0			0
$883$	−40.0463			−1.34767			−0.673833	−	0.738883i	$-0.735353\pi$
$883$	−40.0463			−1.34767			−0.673833	+	0.738883i	$0.735353\pi$
$884$		0			0
$885$	−0.731609			−0.0245928
$886$		0			0
$887$	13.8417			0.464758			0.232379	−	0.972625i	$-0.425349\pi$
$887$	13.8417			0.464758			0.232379	+	0.972625i	$0.425349\pi$
$888$		0			0
$889$		−	3.86970i		−	0.129786i
$890$		0			0
$891$		−	1.72967i		−	0.0579460i
$892$		0			0
$893$	−5.76468			−0.192908
$894$		0			0
$895$			0.835772i			0.0279368i
$896$		0			0
$897$	−12.3606	+	10.9903i	−0.412708	+	0.366955i
$898$		0			0
$899$		−	25.9861i		−	0.866686i
$900$		0			0
$901$	−8.12059			−0.270536
$902$		0			0
$903$			9.12149i			0.303544i
$904$		0			0
$905$		−	1.51519i		−	0.0503666i
$906$		0			0
$907$	−34.2782			−1.13819			−0.569094	−	0.822272i	$-0.692706\pi$
$907$	−34.2782			−1.13819			−0.569094	+	0.822272i	$0.692706\pi$
$908$		0			0
$909$	12.7212			0.421935
$910$		0			0
$911$	−8.32005			−0.275655			−0.137828	−	0.990456i	$-0.544012\pi$
$911$	−8.32005			−0.275655			−0.137828	+	0.990456i	$0.544012\pi$
$912$		0			0
$913$	−6.10576			−0.202071
$914$		0			0
$915$		−	0.444548i		−	0.0146963i
$916$		0			0
$917$		−	10.9947i		−	0.363075i
$918$		0			0
$919$	−34.3211			−1.13215			−0.566074	−	0.824355i	$-0.691538\pi$
$919$	−34.3211			−1.13215			−0.566074	+	0.824355i	$0.691538\pi$
$920$		0			0
$921$			15.2814i			0.503540i
$922$		0			0
$923$	0.216248	+	0.243211i	0.00711789	+	0.00800537i
$924$		0			0
$925$			11.7792i			0.387298i
$926$		0			0
$927$	4.28706			0.140806
$928$		0			0
$929$			41.9922i			1.37772i	0.724895	+	0.688859i	$0.241888\pi$
$929$			41.9922i			1.37772i	−0.724895	+	0.688859i	$0.758112\pi$
$930$		0			0
$931$			0.600958i			0.0196956i
$932$		0			0
$933$	−8.84178			−0.289467
$934$		0			0
$935$	−0.374823			−0.0122580
$936$		0			0
$937$	−50.0568			−1.63529			−0.817643	−	0.575726i	$-0.804719\pi$
$937$	−50.0568			−1.63529			−0.817643	+	0.575726i	$0.804719\pi$
$938$		0			0
$939$	10.5838			0.345390
$940$		0			0
$941$		−	6.63237i		−	0.216209i	−0.994140	−	0.108105i	$-0.965522\pi$
$941$		−	6.63237i		−	0.216209i	0.994140	−	0.108105i	$-0.0344781\pi$
$942$		0			0
$943$		−	23.0339i		−	0.750087i
$944$		0			0
$945$	0.100328			0.00326367
$946$		0			0
$947$			48.0550i			1.56158i	0.624795	+	0.780789i	$0.285182\pi$
$947$			48.0550i			1.56158i	−0.624795	+	0.780789i	$0.714818\pi$
$948$		0			0
$949$	18.6614	+	20.9881i	0.605773	+	0.681303i
$950$		0			0
$951$			22.8435i			0.740753i
$952$		0			0
$953$	24.5129			0.794051			0.397026	−	0.917808i	$-0.370042\pi$
$953$	24.5129			0.794051			0.397026	+	0.917808i	$0.370042\pi$
$954$		0			0
$955$			1.59823i			0.0517176i
$956$		0			0
$957$			8.95666i			0.289528i
$958$		0			0
$959$	−15.3082			−0.494328
$960$		0			0
$961$	5.81650			0.187629
$962$		0			0
$963$	−7.65209			−0.246585
$964$		0			0
$965$	1.26755			0.0408039
$966$		0			0
$967$		−	34.0407i		−	1.09468i	−0.836912	−	0.547338i	$-0.815641\pi$
$967$		−	34.0407i		−	1.09468i	0.836912	−	0.547338i	$-0.184359\pi$
$968$		0			0
$969$		−	1.29803i		−	0.0416988i
$970$		0			0
$971$	−32.0733			−1.02928			−0.514640	−	0.857406i	$-0.672074\pi$
$971$	−32.0733			−1.02928			−0.514640	+	0.857406i	$0.672074\pi$
$972$		0			0
$973$			8.45154i			0.270944i
$974$		0			0
$975$	13.4453	−	11.9548i	0.430595	−	0.382859i
$976$		0			0
$977$			15.4439i			0.494094i	0.969003	+	0.247047i	$0.0794602\pi$
$977$			15.4439i			0.494094i	−0.969003	+	0.247047i	$0.920540\pi$
$978$		0			0
$979$	−21.8576			−0.698573
$980$		0			0
$981$			5.61927i			0.179410i
$982$		0			0
$983$		−	43.2099i		−	1.37818i	−0.724674	−	0.689091i	$-0.758010\pi$
$983$		−	43.2099i		−	1.37818i	0.724674	−	0.689091i	$-0.241990\pi$
$984$		0			0
$985$	1.17318			0.0373807
$986$		0			0
$987$	−9.59248			−0.305332
$988$		0			0
$989$	41.8436			1.33055
$990$		0			0
$991$	0.153790			0.00488529			0.00244264	−	0.999997i	$-0.499222\pi$
$991$	0.153790			0.00488529			0.00244264	+	0.999997i	$0.499222\pi$
$992$		0			0
$993$		−	24.3610i		−	0.773075i
$994$		0			0
$995$			1.13194i			0.0358850i
$996$		0			0
$997$	38.2907			1.21268			0.606339	−	0.795206i	$-0.292638\pi$
$997$	38.2907			1.21268			0.606339	+	0.795206i	$0.292638\pi$
$998$		0			0
$999$		−	2.36059i		−	0.0746859i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.r.337.6		10
4.3	odd	2		2184.2.h.f.337.6	yes	10
13.12	even	2	inner	4368.2.h.r.337.5		10
52.51	odd	2		2184.2.h.f.337.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.f.337.5	✓	10	52.51	odd	2
2184.2.h.f.337.6	yes	10	4.3	odd	2
4368.2.h.r.337.5		10	13.12	even	2	inner
4368.2.h.r.337.6		10	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.100328i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.100328i q^{5} +1.00000i q^{7} +1.00000 q^{9} -1.72967i q^{11} +(-2.69449 + 2.39577i) q^{13} -0.100328i q^{15} -2.15994 q^{17} -0.600958i q^{19} -1.00000i q^{21} -4.58736 q^{23} +4.98993 q^{25} -1.00000 q^{27} +5.17825 q^{29} -5.01832i q^{31} +1.72967i q^{33} -0.100328 q^{35} +2.36059i q^{37} +(2.69449 - 2.39577i) q^{39} +5.02116i q^{41} -9.12149 q^{43} +0.100328i q^{45} -9.59248i q^{47} -1.00000 q^{49} +2.15994 q^{51} +3.75964 q^{53} +0.173534 q^{55} +0.600958i q^{57} -7.29218i q^{59} +4.43095 q^{61} +1.00000i q^{63} +(-0.240363 - 0.270332i) q^{65} +11.2985i q^{67} +4.58736 q^{69} -0.0902623i q^{71} -7.78928i q^{73} -4.98993 q^{75} +1.72967 q^{77} +14.6704 q^{79} +1.00000 q^{81} -3.53002i q^{83} -0.216702i q^{85} -5.17825 q^{87} -12.6369i q^{89} +(-2.39577 - 2.69449i) q^{91} +5.01832i q^{93} +0.0602929 q^{95} -0.975148i q^{97} -1.72967i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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