LMFDB - Embedded newform 6348.2.a.s.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6348,2,Mod(1,6348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6348, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6348.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6348 = 2^{2} \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6348.a (trivial)

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:	yes
Analytic conductor:	$50.6890352031$
Analytic rank:	$1$
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} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 $ x^10 - 4x^9 - 14x^8 + 65x^7 + 57x^6 - 354x^5 - 46x^4 + 714x^3 - 74x^2 - 323*x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0733115$ of defining polynomial
Character	$\chi$	$=$	6348.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.12335 q^{5} +1.05967 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.12335 q^{5} +1.05967 q^{7} +1.00000 q^{9} -2.44394 q^{11} +1.19020 q^{13} -1.12335 q^{15} +0.844588 q^{17} -1.90992 q^{19} +1.05967 q^{21} -3.73809 q^{25} +1.00000 q^{27} +0.574992 q^{29} +0.266531 q^{31} -2.44394 q^{33} -1.19038 q^{35} -8.67745 q^{37} +1.19020 q^{39} +4.87707 q^{41} -3.29738 q^{43} -1.12335 q^{45} +0.561056 q^{47} -5.87710 q^{49} +0.844588 q^{51} -6.19850 q^{53} +2.74540 q^{55} -1.90992 q^{57} -1.47004 q^{59} +8.88933 q^{61} +1.05967 q^{63} -1.33700 q^{65} -4.98206 q^{67} +0.280823 q^{71} +5.14720 q^{73} -3.73809 q^{75} -2.58977 q^{77} -5.85935 q^{79} +1.00000 q^{81} -5.17033 q^{83} -0.948765 q^{85} +0.574992 q^{87} -12.1485 q^{89} +1.26121 q^{91} +0.266531 q^{93} +2.14550 q^{95} -4.03392 q^{97} -2.44394 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	$ 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * q^99

Display 100 coefficients 10 coefficients

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
$5$	−1.12335	−0.502376	−0.251188	−	0.967938i	$-0.580821\pi$
$5$	−1.12335	−0.502376	−0.251188	+	0.967938i	$0.580821\pi$
$6$	0	0
$7$	1.05967	0.400518	0.200259	−	0.979743i	$-0.435822\pi$
$7$	1.05967	0.400518	0.200259	+	0.979743i	$0.435822\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.44394	−0.736876	−0.368438	−	0.929652i	$-0.620107\pi$
$11$	−2.44394	−0.736876	−0.368438	+	0.929652i	$0.620107\pi$
$12$	0	0
$13$	1.19020	0.330101	0.165050	−	0.986285i	$-0.447221\pi$
$13$	1.19020	0.330101	0.165050	+	0.986285i	$0.447221\pi$
$14$	0	0
$15$	−1.12335	−0.290047
$16$	0	0
$17$	0.844588	0.204843	0.102421	−	0.994741i	$-0.467341\pi$
$17$	0.844588	0.204843	0.102421	+	0.994741i	$0.467341\pi$
$18$	0	0
$19$	−1.90992	−0.438166	−0.219083	−	0.975706i	$-0.570307\pi$
$19$	−1.90992	−0.438166	−0.219083	+	0.975706i	$0.570307\pi$
$20$	0	0
$21$	1.05967	0.231239
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.73809	−0.747618
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.574992	0.106773	0.0533867	−	0.998574i	$-0.482998\pi$
$29$	0.574992	0.106773	0.0533867	+	0.998574i	$0.482998\pi$
$30$	0	0
$31$	0.266531	0.0478704	0.0239352	−	0.999714i	$-0.492380\pi$
$31$	0.266531	0.0478704	0.0239352	+	0.999714i	$0.492380\pi$
$32$	0	0
$33$	−2.44394	−0.425436
$34$	0	0
$35$	−1.19038	−0.201210
$36$	0	0
$37$	−8.67745	−1.42656	−0.713282	−	0.700877i	$-0.752792\pi$
$37$	−8.67745	−1.42656	−0.713282	+	0.700877i	$0.752792\pi$
$38$	0	0
$39$	1.19020	0.190584
$40$	0	0
$41$	4.87707	0.761671	0.380835	−	0.924643i	$-0.375636\pi$
$41$	4.87707	0.761671	0.380835	+	0.924643i	$0.375636\pi$
$42$	0	0
$43$	−3.29738	−0.502845	−0.251423	−	0.967877i	$-0.580898\pi$
$43$	−3.29738	−0.502845	−0.251423	+	0.967877i	$0.580898\pi$
$44$	0	0
$45$	−1.12335	−0.167459
$46$	0	0
$47$	0.561056	0.0818384	0.0409192	−	0.999162i	$-0.486971\pi$
$47$	0.561056	0.0818384	0.0409192	+	0.999162i	$0.486971\pi$
$48$	0	0
$49$	−5.87710	−0.839586
$50$	0	0
$51$	0.844588	0.118266
$52$	0	0
$53$	−6.19850	−0.851430	−0.425715	−	0.904857i	$-0.639977\pi$
$53$	−6.19850	−0.851430	−0.425715	+	0.904857i	$0.639977\pi$
$54$	0	0
$55$	2.74540	0.370189
$56$	0	0
$57$	−1.90992	−0.252975
$58$	0	0
$59$	−1.47004	−0.191383	−0.0956916	−	0.995411i	$-0.530506\pi$
$59$	−1.47004	−0.191383	−0.0956916	+	0.995411i	$0.530506\pi$
$60$	0	0
$61$	8.88933	1.13816	0.569081	−	0.822281i	$-0.307299\pi$
$61$	8.88933	1.13816	0.569081	+	0.822281i	$0.307299\pi$
$62$	0	0
$63$	1.05967	0.133506
$64$	0	0
$65$	−1.33700	−0.165835
$66$	0	0
$67$	−4.98206	−0.608656	−0.304328	−	0.952567i	$-0.598432\pi$
$67$	−4.98206	−0.608656	−0.304328	+	0.952567i	$0.598432\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.280823	0.0333275	0.0166638	−	0.999861i	$-0.494696\pi$
$71$	0.280823	0.0333275	0.0166638	+	0.999861i	$0.494696\pi$
$72$	0	0
$73$	5.14720	0.602434	0.301217	−	0.953556i	$-0.402607\pi$
$73$	5.14720	0.602434	0.301217	+	0.953556i	$0.402607\pi$
$74$	0	0
$75$	−3.73809	−0.431638
$76$	0	0
$77$	−2.58977	−0.295132
$78$	0	0
$79$	−5.85935	−0.659228	−0.329614	−	0.944116i	$-0.606919\pi$
$79$	−5.85935	−0.659228	−0.329614	+	0.944116i	$0.606919\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.17033	−0.567517	−0.283759	−	0.958896i	$-0.591581\pi$
$83$	−5.17033	−0.567517	−0.283759	+	0.958896i	$0.591581\pi$
$84$	0	0
$85$	−0.948765	−0.102908
$86$	0	0
$87$	0.574992	0.0616456
$88$	0	0
$89$	−12.1485	−1.28774	−0.643869	−	0.765136i	$-0.722672\pi$
$89$	−12.1485	−1.28774	−0.643869	+	0.765136i	$0.722672\pi$
$90$	0	0
$91$	1.26121	0.132211
$92$	0	0
$93$	0.266531	0.0276380
$94$	0	0
$95$	2.14550	0.220124
$96$	0	0
$97$	−4.03392	−0.409582	−0.204791	−	0.978806i	$-0.565652\pi$
$97$	−4.03392	−0.409582	−0.204791	+	0.978806i	$0.565652\pi$
$98$	0	0
$99$	−2.44394	−0.245625
$100$	0	0
$101$	12.0737	1.20138	0.600690	−	0.799482i	$-0.294893\pi$
$101$	12.0737	1.20138	0.600690	+	0.799482i	$0.294893\pi$
$102$	0	0
$103$	−16.9492	−1.67006	−0.835028	−	0.550207i	$-0.814549\pi$
$103$	−16.9492	−1.67006	−0.835028	+	0.550207i	$0.814549\pi$
$104$	0	0
$105$	−1.19038	−0.116169
$106$	0	0
$107$	12.4987	1.20830	0.604150	−	0.796871i	$-0.293513\pi$
$107$	12.4987	1.20830	0.604150	+	0.796871i	$0.293513\pi$
$108$	0	0
$109$	−0.557809	−0.0534284	−0.0267142	−	0.999643i	$-0.508504\pi$
$109$	−0.557809	−0.0534284	−0.0267142	+	0.999643i	$0.508504\pi$
$110$	0	0
$111$	−8.67745	−0.823627
$112$	0	0
$113$	8.42646	0.792695	0.396347	−	0.918101i	$-0.370278\pi$
$113$	8.42646	0.792695	0.396347	+	0.918101i	$0.370278\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.19020	0.110034
$118$	0	0
$119$	0.894984	0.0820431
$120$	0	0
$121$	−5.02715	−0.457013
$122$	0	0
$123$	4.87707	0.439751
$124$	0	0
$125$	9.81591	0.877962
$126$	0	0
$127$	−14.5499	−1.29110	−0.645549	−	0.763719i	$-0.723371\pi$
$127$	−14.5499	−1.29110	−0.645549	+	0.763719i	$0.723371\pi$
$128$	0	0
$129$	−3.29738	−0.290318
$130$	0	0
$131$	10.1604	0.887714	0.443857	−	0.896098i	$-0.353610\pi$
$131$	10.1604	0.887714	0.443857	+	0.896098i	$0.353610\pi$
$132$	0	0
$133$	−2.02389	−0.175493
$134$	0	0
$135$	−1.12335	−0.0966823
$136$	0	0
$137$	10.5603	0.902226	0.451113	−	0.892467i	$-0.351027\pi$
$137$	10.5603	0.902226	0.451113	+	0.892467i	$0.351027\pi$
$138$	0	0
$139$	−6.20115	−0.525975	−0.262987	−	0.964799i	$-0.584708\pi$
$139$	−6.20115	−0.525975	−0.262987	+	0.964799i	$0.584708\pi$
$140$	0	0
$141$	0.561056	0.0472494
$142$	0	0
$143$	−2.90877	−0.243244
$144$	0	0
$145$	−0.645916	−0.0536404
$146$	0	0
$147$	−5.87710	−0.484735
$148$	0	0
$149$	−13.4491	−1.10179	−0.550897	−	0.834573i	$-0.685714\pi$
$149$	−13.4491	−1.10179	−0.550897	+	0.834573i	$0.685714\pi$
$150$	0	0
$151$	−18.2690	−1.48671	−0.743357	−	0.668895i	$-0.766768\pi$
$151$	−18.2690	−1.48671	−0.743357	+	0.668895i	$0.766768\pi$
$152$	0	0
$153$	0.844588	0.0682809
$154$	0	0
$155$	−0.299407	−0.0240489
$156$	0	0
$157$	−18.8900	−1.50758	−0.753791	−	0.657114i	$-0.771777\pi$
$157$	−18.8900	−1.50758	−0.753791	+	0.657114i	$0.771777\pi$
$158$	0	0
$159$	−6.19850	−0.491573
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.103527	−0.00810886	−0.00405443	−	0.999992i	$-0.501291\pi$
$163$	−0.103527	−0.00810886	−0.00405443	+	0.999992i	$0.501291\pi$
$164$	0	0
$165$	2.74540	0.213729
$166$	0	0
$167$	−17.1255	−1.32521	−0.662605	−	0.748969i	$-0.730549\pi$
$167$	−17.1255	−1.32521	−0.662605	+	0.748969i	$0.730549\pi$
$168$	0	0
$169$	−11.5834	−0.891033
$170$	0	0
$171$	−1.90992	−0.146055
$172$	0	0
$173$	−9.98124	−0.758860	−0.379430	−	0.925220i	$-0.623880\pi$
$173$	−9.98124	−0.758860	−0.379430	+	0.925220i	$0.623880\pi$
$174$	0	0
$175$	−3.96114	−0.299434
$176$	0	0
$177$	−1.47004	−0.110495
$178$	0	0
$179$	25.5880	1.91254	0.956270	−	0.292486i	$-0.0944825\pi$
$179$	25.5880	1.91254	0.956270	+	0.292486i	$0.0944825\pi$
$180$	0	0
$181$	0.984696	0.0731918	0.0365959	−	0.999330i	$-0.488349\pi$
$181$	0.984696	0.0731918	0.0365959	+	0.999330i	$0.488349\pi$
$182$	0	0
$183$	8.88933	0.657118
$184$	0	0
$185$	9.74779	0.716672
$186$	0	0
$187$	−2.06412	−0.150944
$188$	0	0
$189$	1.05967	0.0770796
$190$	0	0
$191$	−4.63831	−0.335617	−0.167808	−	0.985820i	$-0.553669\pi$
$191$	−4.63831	−0.335617	−0.167808	+	0.985820i	$0.553669\pi$
$192$	0	0
$193$	17.2519	1.24182	0.620910	−	0.783882i	$-0.286763\pi$
$193$	17.2519	1.24182	0.620910	+	0.783882i	$0.286763\pi$
$194$	0	0
$195$	−1.33700	−0.0957448
$196$	0	0
$197$	−1.45685	−0.103796	−0.0518980	−	0.998652i	$-0.516527\pi$
$197$	−1.45685	−0.103796	−0.0518980	+	0.998652i	$0.516527\pi$
$198$	0	0
$199$	−7.29300	−0.516987	−0.258493	−	0.966013i	$-0.583226\pi$
$199$	−7.29300	−0.516987	−0.258493	+	0.966013i	$0.583226\pi$
$200$	0	0
$201$	−4.98206	−0.351408
$202$	0	0
$203$	0.609302	0.0427646
$204$	0	0
$205$	−5.47865	−0.382645
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.66774	0.322874
$210$	0	0
$211$	24.9741	1.71929	0.859645	−	0.510892i	$-0.170685\pi$
$211$	24.9741	1.71929	0.859645	+	0.510892i	$0.170685\pi$
$212$	0	0
$213$	0.280823	0.0192417
$214$	0	0
$215$	3.70410	0.252617
$216$	0	0
$217$	0.282435	0.0191729
$218$	0	0
$219$	5.14720	0.347815
$220$	0	0
$221$	1.00522	0.0676187
$222$	0	0
$223$	−6.32348	−0.423451	−0.211726	−	0.977329i	$-0.567908\pi$
$223$	−6.32348	−0.423451	−0.211726	+	0.977329i	$0.567908\pi$
$224$	0	0
$225$	−3.73809	−0.249206
$226$	0	0
$227$	−24.8259	−1.64775	−0.823877	−	0.566769i	$-0.808193\pi$
$227$	−24.8259	−1.64775	−0.823877	+	0.566769i	$0.808193\pi$
$228$	0	0
$229$	8.33079	0.550514	0.275257	−	0.961371i	$-0.411237\pi$
$229$	8.33079	0.550514	0.275257	+	0.961371i	$0.411237\pi$
$230$	0	0
$231$	−2.58977	−0.170394
$232$	0	0
$233$	−23.7405	−1.55529	−0.777647	−	0.628701i	$-0.783587\pi$
$233$	−23.7405	−1.55529	−0.777647	+	0.628701i	$0.783587\pi$
$234$	0	0
$235$	−0.630261	−0.0411137
$236$	0	0
$237$	−5.85935	−0.380605
$238$	0	0
$239$	24.4018	1.57842	0.789212	−	0.614121i	$-0.210489\pi$
$239$	24.4018	1.57842	0.789212	+	0.614121i	$0.210489\pi$
$240$	0	0
$241$	−12.2133	−0.786729	−0.393365	−	0.919383i	$-0.628689\pi$
$241$	−12.2133	−0.786729	−0.393365	+	0.919383i	$0.628689\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.60202	0.421788
$246$	0	0
$247$	−2.27318	−0.144639
$248$	0	0
$249$	−5.17033	−0.327656
$250$	0	0
$251$	−24.9817	−1.57683	−0.788417	−	0.615141i	$-0.789099\pi$
$251$	−24.9817	−1.57683	−0.788417	+	0.615141i	$0.789099\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.948765	−0.0594140
$256$	0	0
$257$	−11.8628	−0.739980	−0.369990	−	0.929036i	$-0.620639\pi$
$257$	−11.8628	−0.739980	−0.369990	+	0.929036i	$0.620639\pi$
$258$	0	0
$259$	−9.19523	−0.571364
$260$	0	0
$261$	0.574992	0.0355911
$262$	0	0
$263$	9.30561	0.573808	0.286904	−	0.957959i	$-0.407374\pi$
$263$	9.30561	0.573808	0.286904	+	0.957959i	$0.407374\pi$
$264$	0	0
$265$	6.96307	0.427738
$266$	0	0
$267$	−12.1485	−0.743476
$268$	0	0
$269$	−12.3771	−0.754646	−0.377323	−	0.926082i	$-0.623155\pi$
$269$	−12.3771	−0.754646	−0.377323	+	0.926082i	$0.623155\pi$
$270$	0	0
$271$	12.7080	0.771956	0.385978	−	0.922508i	$-0.373864\pi$
$271$	12.7080	0.771956	0.385978	+	0.922508i	$0.373864\pi$
$272$	0	0
$273$	1.26121	0.0763322
$274$	0	0
$275$	9.13568	0.550902
$276$	0	0
$277$	19.8326	1.19163	0.595813	−	0.803123i	$-0.296830\pi$
$277$	19.8326	1.19163	0.595813	+	0.803123i	$0.296830\pi$
$278$	0	0
$279$	0.266531	0.0159568
$280$	0	0
$281$	−13.8243	−0.824689	−0.412344	−	0.911028i	$-0.635290\pi$
$281$	−13.8243	−0.824689	−0.412344	+	0.911028i	$0.635290\pi$
$282$	0	0
$283$	−7.16308	−0.425801	−0.212901	−	0.977074i	$-0.568291\pi$
$283$	−7.16308	−0.425801	−0.212901	+	0.977074i	$0.568291\pi$
$284$	0	0
$285$	2.14550	0.127089
$286$	0	0
$287$	5.16809	0.305062
$288$	0	0
$289$	−16.2867	−0.958039
$290$	0	0
$291$	−4.03392	−0.236472
$292$	0	0
$293$	18.2242	1.06467	0.532333	−	0.846535i	$-0.321315\pi$
$293$	18.2242	1.06467	0.532333	+	0.846535i	$0.321315\pi$
$294$	0	0
$295$	1.65137	0.0961464
$296$	0	0
$297$	−2.44394	−0.141812
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.49413	−0.201398
$302$	0	0
$303$	12.0737	0.693617
$304$	0	0
$305$	−9.98581	−0.571786
$306$	0	0
$307$	15.9314	0.909252	0.454626	−	0.890682i	$-0.349773\pi$
$307$	15.9314	0.909252	0.454626	+	0.890682i	$0.349773\pi$
$308$	0	0
$309$	−16.9492	−0.964207
$310$	0	0
$311$	−29.1322	−1.65194	−0.825968	−	0.563717i	$-0.809371\pi$
$311$	−29.1322	−1.65194	−0.825968	+	0.563717i	$0.809371\pi$
$312$	0	0
$313$	−17.8209	−1.00730	−0.503649	−	0.863908i	$-0.668010\pi$
$313$	−17.8209	−1.00730	−0.503649	+	0.863908i	$0.668010\pi$
$314$	0	0
$315$	−1.19038	−0.0670701
$316$	0	0
$317$	23.9318	1.34414	0.672072	−	0.740486i	$-0.265405\pi$
$317$	23.9318	1.34414	0.672072	+	0.740486i	$0.265405\pi$
$318$	0	0
$319$	−1.40525	−0.0786788
$320$	0	0
$321$	12.4987	0.697612
$322$	0	0
$323$	−1.61310	−0.0897551
$324$	0	0
$325$	−4.44906	−0.246789
$326$	0	0
$327$	−0.557809	−0.0308469
$328$	0	0
$329$	0.594534	0.0327777
$330$	0	0
$331$	−6.46271	−0.355223	−0.177611	−	0.984101i	$-0.556837\pi$
$331$	−6.46271	−0.355223	−0.177611	+	0.984101i	$0.556837\pi$
$332$	0	0
$333$	−8.67745	−0.475521
$334$	0	0
$335$	5.59659	0.305774
$336$	0	0
$337$	−8.39310	−0.457201	−0.228601	−	0.973520i	$-0.573415\pi$
$337$	−8.39310	−0.457201	−0.228601	+	0.973520i	$0.573415\pi$
$338$	0	0
$339$	8.42646	0.457663
$340$	0	0
$341$	−0.651387	−0.0352746
$342$	0	0
$343$	−13.6455	−0.736786
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.1733	0.814544	0.407272	−	0.913307i	$-0.366480\pi$
$347$	15.1733	0.814544	0.407272	+	0.913307i	$0.366480\pi$
$348$	0	0
$349$	6.05659	0.324202	0.162101	−	0.986774i	$-0.448173\pi$
$349$	6.05659	0.324202	0.162101	+	0.986774i	$0.448173\pi$
$350$	0	0
$351$	1.19020	0.0635280
$352$	0	0
$353$	−32.8618	−1.74906	−0.874529	−	0.484974i	$-0.838829\pi$
$353$	−32.8618	−1.74906	−0.874529	+	0.484974i	$0.838829\pi$
$354$	0	0
$355$	−0.315462	−0.0167430
$356$	0	0
$357$	0.894984	0.0473676
$358$	0	0
$359$	4.53967	0.239595	0.119797	−	0.992798i	$-0.461776\pi$
$359$	4.53967	0.239595	0.119797	+	0.992798i	$0.461776\pi$
$360$	0	0
$361$	−15.3522	−0.808010
$362$	0	0
$363$	−5.02715	−0.263857
$364$	0	0
$365$	−5.78209	−0.302648
$366$	0	0
$367$	31.9179	1.66610	0.833050	−	0.553198i	$-0.186593\pi$
$367$	31.9179	1.66610	0.833050	+	0.553198i	$0.186593\pi$
$368$	0	0
$369$	4.87707	0.253890
$370$	0	0
$371$	−6.56836	−0.341012
$372$	0	0
$373$	−2.25521	−0.116770	−0.0583852	−	0.998294i	$-0.518595\pi$
$373$	−2.25521	−0.116770	−0.0583852	+	0.998294i	$0.518595\pi$
$374$	0	0
$375$	9.81591	0.506891
$376$	0	0
$377$	0.684353	0.0352460
$378$	0	0
$379$	−25.7863	−1.32455	−0.662277	−	0.749259i	$-0.730410\pi$
$379$	−25.7863	−1.32455	−0.662277	+	0.749259i	$0.730410\pi$
$380$	0	0
$381$	−14.5499	−0.745416
$382$	0	0
$383$	−11.2687	−0.575805	−0.287903	−	0.957660i	$-0.592958\pi$
$383$	−11.2687	−0.575805	−0.287903	+	0.957660i	$0.592958\pi$
$384$	0	0
$385$	2.90921	0.148267
$386$	0	0
$387$	−3.29738	−0.167615
$388$	0	0
$389$	28.7820	1.45931	0.729654	−	0.683817i	$-0.239681\pi$
$389$	28.7820	1.45931	0.729654	+	0.683817i	$0.239681\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	10.1604	0.512522
$394$	0	0
$395$	6.58208	0.331180
$396$	0	0
$397$	−34.2399	−1.71845	−0.859225	−	0.511598i	$-0.829054\pi$
$397$	−34.2399	−1.71845	−0.859225	+	0.511598i	$0.829054\pi$
$398$	0	0
$399$	−2.02389	−0.101321
$400$	0	0
$401$	28.2860	1.41254	0.706268	−	0.707945i	$-0.250378\pi$
$401$	28.2860	1.41254	0.706268	+	0.707945i	$0.250378\pi$
$402$	0	0
$403$	0.317224	0.0158021
$404$	0	0
$405$	−1.12335	−0.0558196
$406$	0	0
$407$	21.2072	1.05120
$408$	0	0
$409$	2.23897	0.110710	0.0553550	−	0.998467i	$-0.482371\pi$
$409$	2.23897	0.110710	0.0553550	+	0.998467i	$0.482371\pi$
$410$	0	0
$411$	10.5603	0.520900
$412$	0	0
$413$	−1.55776	−0.0766523
$414$	0	0
$415$	5.80807	0.285107
$416$	0	0
$417$	−6.20115	−0.303672
$418$	0	0
$419$	−22.1369	−1.08146	−0.540729	−	0.841197i	$-0.681852\pi$
$419$	−22.1369	−1.08146	−0.540729	+	0.841197i	$0.681852\pi$
$420$	0	0
$421$	−7.20472	−0.351137	−0.175568	−	0.984467i	$-0.556176\pi$
$421$	−7.20472	−0.351137	−0.175568	+	0.984467i	$0.556176\pi$
$422$	0	0
$423$	0.561056	0.0272795
$424$	0	0
$425$	−3.15715	−0.153144
$426$	0	0
$427$	9.41976	0.455854
$428$	0	0
$429$	−2.90877	−0.140437
$430$	0	0
$431$	12.9708	0.624783	0.312391	−	0.949953i	$-0.398870\pi$
$431$	12.9708	0.624783	0.312391	+	0.949953i	$0.398870\pi$
$432$	0	0
$433$	−14.9212	−0.717065	−0.358532	−	0.933517i	$-0.616723\pi$
$433$	−14.9212	−0.717065	−0.358532	+	0.933517i	$0.616723\pi$
$434$	0	0
$435$	−0.645916	−0.0309693
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0.533264	0.0254513	0.0127256	−	0.999919i	$-0.495949\pi$
$439$	0.533264	0.0254513	0.0127256	+	0.999919i	$0.495949\pi$
$440$	0	0
$441$	−5.87710	−0.279862
$442$	0	0
$443$	−19.2096	−0.912675	−0.456338	−	0.889807i	$-0.650839\pi$
$443$	−19.2096	−0.912675	−0.456338	+	0.889807i	$0.650839\pi$
$444$	0	0
$445$	13.6470	0.646929
$446$	0	0
$447$	−13.4491	−0.636121
$448$	0	0
$449$	−5.76230	−0.271940	−0.135970	−	0.990713i	$-0.543415\pi$
$449$	−5.76230	−0.271940	−0.135970	+	0.990713i	$0.543415\pi$
$450$	0	0
$451$	−11.9193	−0.561257
$452$	0	0
$453$	−18.2690	−0.858355
$454$	0	0
$455$	−1.41678	−0.0664197
$456$	0	0
$457$	13.3639	0.625135	0.312568	−	0.949896i	$-0.398811\pi$
$457$	13.3639	0.625135	0.312568	+	0.949896i	$0.398811\pi$
$458$	0	0
$459$	0.844588	0.0394220
$460$	0	0
$461$	22.9200	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$461$	22.9200	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$462$	0	0
$463$	12.0624	0.560589	0.280295	−	0.959914i	$-0.409568\pi$
$463$	12.0624	0.560589	0.280295	+	0.959914i	$0.409568\pi$
$464$	0	0
$465$	−0.299407	−0.0138847
$466$	0	0
$467$	−7.32562	−0.338989	−0.169495	−	0.985531i	$-0.554214\pi$
$467$	−7.32562	−0.338989	−0.169495	+	0.985531i	$0.554214\pi$
$468$	0	0
$469$	−5.27934	−0.243777
$470$	0	0
$471$	−18.8900	−0.870403
$472$	0	0
$473$	8.05860	0.370535
$474$	0	0
$475$	7.13946	0.327581
$476$	0	0
$477$	−6.19850	−0.283810
$478$	0	0
$479$	34.3758	1.57067	0.785335	−	0.619071i	$-0.212491\pi$
$479$	34.3758	1.57067	0.785335	+	0.619071i	$0.212491\pi$
$480$	0	0
$481$	−10.3279	−0.470910
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.53149	0.205764
$486$	0	0
$487$	11.4197	0.517474	0.258737	−	0.965948i	$-0.416694\pi$
$487$	11.4197	0.517474	0.258737	+	0.965948i	$0.416694\pi$
$488$	0	0
$489$	−0.103527	−0.00468165
$490$	0	0
$491$	−39.5529	−1.78500	−0.892498	−	0.451052i	$-0.851049\pi$
$491$	−39.5529	−1.78500	−0.892498	+	0.451052i	$0.851049\pi$
$492$	0	0
$493$	0.485631	0.0218717
$494$	0	0
$495$	2.74540	0.123396
$496$	0	0
$497$	0.297579	0.0133483
$498$	0	0
$499$	12.2792	0.549694	0.274847	−	0.961488i	$-0.411373\pi$
$499$	12.2792	0.549694	0.274847	+	0.961488i	$0.411373\pi$
$500$	0	0
$501$	−17.1255	−0.765110
$502$	0	0
$503$	−25.4872	−1.13642	−0.568208	−	0.822885i	$-0.692363\pi$
$503$	−25.4872	−1.13642	−0.568208	+	0.822885i	$0.692363\pi$
$504$	0	0
$505$	−13.5630	−0.603544
$506$	0	0
$507$	−11.5834	−0.514438
$508$	0	0
$509$	−23.6734	−1.04930	−0.524652	−	0.851317i	$-0.675804\pi$
$509$	−23.6734	−1.04930	−0.524652	+	0.851317i	$0.675804\pi$
$510$	0	0
$511$	5.45433	0.241285
$512$	0	0
$513$	−1.90992	−0.0843251
$514$	0	0
$515$	19.0399	0.838996
$516$	0	0
$517$	−1.37119	−0.0603048
$518$	0	0
$519$	−9.98124	−0.438128
$520$	0	0
$521$	22.6423	0.991978	0.495989	−	0.868329i	$-0.334806\pi$
$521$	22.6423	0.991978	0.495989	+	0.868329i	$0.334806\pi$
$522$	0	0
$523$	−9.13792	−0.399573	−0.199787	−	0.979839i	$-0.564025\pi$
$523$	−9.13792	−0.399573	−0.199787	+	0.979839i	$0.564025\pi$
$524$	0	0
$525$	−3.96114	−0.172878
$526$	0	0
$527$	0.225109	0.00980590
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−1.47004	−0.0637944
$532$	0	0
$533$	5.80467	0.251428
$534$	0	0
$535$	−14.0404	−0.607021
$536$	0	0
$537$	25.5880	1.10421
$538$	0	0
$539$	14.3633	0.618671
$540$	0	0
$541$	33.8581	1.45567	0.727837	−	0.685750i	$-0.240526\pi$
$541$	33.8581	1.45567	0.727837	+	0.685750i	$0.240526\pi$
$542$	0	0
$543$	0.984696	0.0422573
$544$	0	0
$545$	0.626614	0.0268412
$546$	0	0
$547$	−11.0700	−0.473321	−0.236660	−	0.971592i	$-0.576053\pi$
$547$	−11.0700	−0.473321	−0.236660	+	0.971592i	$0.576053\pi$
$548$	0	0
$549$	8.88933	0.379388
$550$	0	0
$551$	−1.09819	−0.0467845
$552$	0	0
$553$	−6.20897	−0.264032
$554$	0	0
$555$	9.74779	0.413771
$556$	0	0
$557$	15.3225	0.649233	0.324617	−	0.945846i	$-0.394765\pi$
$557$	15.3225	0.649233	0.324617	+	0.945846i	$0.394765\pi$
$558$	0	0
$559$	−3.92452	−0.165990
$560$	0	0
$561$	−2.06412	−0.0871474
$562$	0	0
$563$	0.969804	0.0408724	0.0204362	−	0.999791i	$-0.493495\pi$
$563$	0.969804	0.0408724	0.0204362	+	0.999791i	$0.493495\pi$
$564$	0	0
$565$	−9.46584	−0.398231
$566$	0	0
$567$	1.05967	0.0445019
$568$	0	0
$569$	2.64569	0.110913	0.0554564	−	0.998461i	$-0.482339\pi$
$569$	2.64569	0.110913	0.0554564	+	0.998461i	$0.482339\pi$
$570$	0	0
$571$	26.0441	1.08991	0.544955	−	0.838465i	$-0.316547\pi$
$571$	26.0441	1.08991	0.544955	+	0.838465i	$0.316547\pi$
$572$	0	0
$573$	−4.63831	−0.193768
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.6943	0.986406	0.493203	−	0.869914i	$-0.335826\pi$
$577$	23.6943	0.986406	0.493203	+	0.869914i	$0.335826\pi$
$578$	0	0
$579$	17.2519	0.716966
$580$	0	0
$581$	−5.47884	−0.227301
$582$	0	0
$583$	15.1488	0.627398
$584$	0	0
$585$	−1.33700	−0.0552783
$586$	0	0
$587$	−16.3627	−0.675361	−0.337681	−	0.941261i	$-0.609642\pi$
$587$	−16.3627	−0.675361	−0.337681	+	0.941261i	$0.609642\pi$
$588$	0	0
$589$	−0.509053	−0.0209752
$590$	0	0
$591$	−1.45685	−0.0599266
$592$	0	0
$593$	45.1123	1.85254	0.926269	−	0.376863i	$-0.122997\pi$
$593$	45.1123	1.85254	0.926269	+	0.376863i	$0.122997\pi$
$594$	0	0
$595$	−1.00538	−0.0412165
$596$	0	0
$597$	−7.29300	−0.298483
$598$	0	0
$599$	−12.5788	−0.513957	−0.256978	−	0.966417i	$-0.582727\pi$
$599$	−12.5788	−0.513957	−0.256978	+	0.966417i	$0.582727\pi$
$600$	0	0
$601$	−31.8652	−1.29981	−0.649904	−	0.760017i	$-0.725191\pi$
$601$	−31.8652	−1.29981	−0.649904	+	0.760017i	$0.725191\pi$
$602$	0	0
$603$	−4.98206	−0.202885
$604$	0	0
$605$	5.64723	0.229593
$606$	0	0
$607$	−21.0762	−0.855457	−0.427729	−	0.903907i	$-0.640686\pi$
$607$	−21.0762	−0.855457	−0.427729	+	0.903907i	$0.640686\pi$
$608$	0	0
$609$	0.609302	0.0246901
$610$	0	0
$611$	0.667767	0.0270149
$612$	0	0
$613$	−4.97023	−0.200746	−0.100373	−	0.994950i	$-0.532004\pi$
$613$	−4.97023	−0.200746	−0.100373	+	0.994950i	$0.532004\pi$
$614$	0	0
$615$	−5.47865	−0.220920
$616$	0	0
$617$	5.64543	0.227276	0.113638	−	0.993522i	$-0.463750\pi$
$617$	5.64543	0.227276	0.113638	+	0.993522i	$0.463750\pi$
$618$	0	0
$619$	2.09404	0.0841668	0.0420834	−	0.999114i	$-0.486600\pi$
$619$	2.09404	0.0841668	0.0420834	+	0.999114i	$0.486600\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.8734	−0.515762
$624$	0	0
$625$	7.66378	0.306551
$626$	0	0
$627$	4.66774	0.186412
$628$	0	0
$629$	−7.32887	−0.292221
$630$	0	0
$631$	−21.9835	−0.875150	−0.437575	−	0.899182i	$-0.644162\pi$
$631$	−21.9835	−0.875150	−0.437575	+	0.899182i	$0.644162\pi$
$632$	0	0
$633$	24.9741	0.992633
$634$	0	0
$635$	16.3446	0.648617
$636$	0	0
$637$	−6.99490	−0.277148
$638$	0	0
$639$	0.280823	0.0111092
$640$	0	0
$641$	−4.47430	−0.176724	−0.0883620	−	0.996088i	$-0.528163\pi$
$641$	−4.47430	−0.176724	−0.0883620	+	0.996088i	$0.528163\pi$
$642$	0	0
$643$	15.7233	0.620067	0.310034	−	0.950726i	$-0.399660\pi$
$643$	15.7233	0.620067	0.310034	+	0.950726i	$0.399660\pi$
$644$	0	0
$645$	3.70410	0.145849
$646$	0	0
$647$	−22.0204	−0.865711	−0.432856	−	0.901463i	$-0.642494\pi$
$647$	−22.0204	−0.865711	−0.432856	+	0.901463i	$0.642494\pi$
$648$	0	0
$649$	3.59270	0.141026
$650$	0	0
$651$	0.282435	0.0110695
$652$	0	0
$653$	−6.33003	−0.247713	−0.123857	−	0.992300i	$-0.539526\pi$
$653$	−6.33003	−0.247713	−0.123857	+	0.992300i	$0.539526\pi$
$654$	0	0
$655$	−11.4136	−0.445966
$656$	0	0
$657$	5.14720	0.200811
$658$	0	0
$659$	−4.83254	−0.188249	−0.0941245	−	0.995560i	$-0.530005\pi$
$659$	−4.83254	−0.188249	−0.0941245	+	0.995560i	$0.530005\pi$
$660$	0	0
$661$	5.92542	0.230472	0.115236	−	0.993338i	$-0.463238\pi$
$661$	5.92542	0.230472	0.115236	+	0.993338i	$0.463238\pi$
$662$	0	0
$663$	1.00522	0.0390397
$664$	0	0
$665$	2.27353	0.0881636
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−6.32348	−0.244480
$670$	0	0
$671$	−21.7250	−0.838685
$672$	0	0
$673$	50.4050	1.94297	0.971484	−	0.237104i	$-0.0761982\pi$
$673$	50.4050	1.94297	0.971484	+	0.237104i	$0.0761982\pi$
$674$	0	0
$675$	−3.73809	−0.143879
$676$	0	0
$677$	41.5453	1.59672	0.798359	−	0.602182i	$-0.205702\pi$
$677$	41.5453	1.59672	0.798359	+	0.602182i	$0.205702\pi$
$678$	0	0
$679$	−4.27462	−0.164045
$680$	0	0
$681$	−24.8259	−0.951331
$682$	0	0
$683$	43.4156	1.66125	0.830625	−	0.556832i	$-0.187983\pi$
$683$	43.4156	1.66125	0.830625	+	0.556832i	$0.187983\pi$
$684$	0	0
$685$	−11.8629	−0.453257
$686$	0	0
$687$	8.33079	0.317839
$688$	0	0
$689$	−7.37743	−0.281058
$690$	0	0
$691$	−16.4600	−0.626168	−0.313084	−	0.949725i	$-0.601362\pi$
$691$	−16.4600	−0.626168	−0.313084	+	0.949725i	$0.601362\pi$
$692$	0	0
$693$	−2.58977	−0.0983773
$694$	0	0
$695$	6.96605	0.264237
$696$	0	0
$697$	4.11912	0.156023
$698$	0	0
$699$	−23.7405	−0.897949
$700$	0	0
$701$	10.0634	0.380091	0.190045	−	0.981775i	$-0.439136\pi$
$701$	10.0634	0.380091	0.190045	+	0.981775i	$0.439136\pi$
$702$	0	0
$703$	16.5732	0.625072
$704$	0	0
$705$	−0.630261	−0.0237370
$706$	0	0
$707$	12.7941	0.481174
$708$	0	0
$709$	−34.4009	−1.29195	−0.645977	−	0.763357i	$-0.723550\pi$
$709$	−34.4009	−1.29195	−0.645977	+	0.763357i	$0.723550\pi$
$710$	0	0
$711$	−5.85935	−0.219743
$712$	0	0
$713$	0	0
$714$	0	0
$715$	3.26756	0.122200
$716$	0	0
$717$	24.4018	0.911304
$718$	0	0
$719$	−14.3969	−0.536914	−0.268457	−	0.963292i	$-0.586514\pi$
$719$	−14.3969	−0.536914	−0.268457	+	0.963292i	$0.586514\pi$
$720$	0	0
$721$	−17.9606	−0.668887
$722$	0	0
$723$	−12.2133	−0.454218
$724$	0	0
$725$	−2.14937	−0.0798257
$726$	0	0
$727$	−3.37140	−0.125038	−0.0625192	−	0.998044i	$-0.519913\pi$
$727$	−3.37140	−0.125038	−0.0625192	+	0.998044i	$0.519913\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.78492	−0.103004
$732$	0	0
$733$	17.7744	0.656514	0.328257	−	0.944588i	$-0.393539\pi$
$733$	17.7744	0.656514	0.328257	+	0.944588i	$0.393539\pi$
$734$	0	0
$735$	6.60202	0.243519
$736$	0	0
$737$	12.1759	0.448504
$738$	0	0
$739$	32.3268	1.18916	0.594580	−	0.804036i	$-0.297318\pi$
$739$	32.3268	1.18916	0.594580	+	0.804036i	$0.297318\pi$
$740$	0	0
$741$	−2.27318	−0.0835074
$742$	0	0
$743$	7.00810	0.257102	0.128551	−	0.991703i	$-0.458967\pi$
$743$	7.00810	0.257102	0.128551	+	0.991703i	$0.458967\pi$
$744$	0	0
$745$	15.1080	0.553515
$746$	0	0
$747$	−5.17033	−0.189172
$748$	0	0
$749$	13.2445	0.483945
$750$	0	0
$751$	−3.65592	−0.133406	−0.0667031	−	0.997773i	$-0.521248\pi$
$751$	−3.65592	−0.133406	−0.0667031	+	0.997773i	$0.521248\pi$
$752$	0	0
$753$	−24.9817	−0.910385
$754$	0	0
$755$	20.5225	0.746890
$756$	0	0
$757$	42.1696	1.53268	0.766340	−	0.642435i	$-0.222076\pi$
$757$	42.1696	1.53268	0.766340	+	0.642435i	$0.222076\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.4691	−1.10450	−0.552252	−	0.833677i	$-0.686231\pi$
$761$	−30.4691	−1.10450	−0.552252	+	0.833677i	$0.686231\pi$
$762$	0	0
$763$	−0.591094	−0.0213990
$764$	0	0
$765$	−0.948765	−0.0343027
$766$	0	0
$767$	−1.74964	−0.0631758
$768$	0	0
$769$	−43.1158	−1.55480	−0.777398	−	0.629009i	$-0.783461\pi$
$769$	−43.1158	−1.55480	−0.777398	+	0.629009i	$0.783461\pi$
$770$	0	0
$771$	−11.8628	−0.427228
$772$	0	0
$773$	41.9850	1.51010	0.755049	−	0.655669i	$-0.227613\pi$
$773$	41.9850	1.51010	0.755049	+	0.655669i	$0.227613\pi$
$774$	0	0
$775$	−0.996317	−0.0357888
$776$	0	0
$777$	−9.19523	−0.329877
$778$	0	0
$779$	−9.31483	−0.333738
$780$	0	0
$781$	−0.686315	−0.0245583
$782$	0	0
$783$	0.574992	0.0205485
$784$	0	0
$785$	21.2200	0.757374
$786$	0	0
$787$	−41.6587	−1.48497	−0.742487	−	0.669860i	$-0.766354\pi$
$787$	−41.6587	−1.48497	−0.742487	+	0.669860i	$0.766354\pi$
$788$	0	0
$789$	9.30561	0.331288
$790$	0	0
$791$	8.92927	0.317488
$792$	0	0
$793$	10.5800	0.375709
$794$	0	0
$795$	6.96307	0.246955
$796$	0	0
$797$	30.9676	1.09693	0.548464	−	0.836174i	$-0.315213\pi$
$797$	30.9676	1.09693	0.548464	+	0.836174i	$0.315213\pi$
$798$	0	0
$799$	0.473861	0.0167640
$800$	0	0
$801$	−12.1485	−0.429246
$802$	0	0
$803$	−12.5795	−0.443919
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.3771	−0.435695
$808$	0	0
$809$	−55.2835	−1.94367	−0.971833	−	0.235672i	$-0.924271\pi$
$809$	−55.2835	−1.94367	−0.971833	+	0.235672i	$0.924271\pi$
$810$	0	0
$811$	30.8655	1.08383	0.541917	−	0.840432i	$-0.317699\pi$
$811$	30.8655	1.08383	0.541917	+	0.840432i	$0.317699\pi$
$812$	0	0
$813$	12.7080	0.445689
$814$	0	0
$815$	0.116297	0.00407370
$816$	0	0
$817$	6.29773	0.220330
$818$	0	0
$819$	1.26121	0.0440704
$820$	0	0
$821$	−0.377087	−0.0131604	−0.00658021	−	0.999978i	$-0.502095\pi$
$821$	−0.377087	−0.0131604	−0.00658021	+	0.999978i	$0.502095\pi$
$822$	0	0
$823$	−4.38556	−0.152871	−0.0764354	−	0.997075i	$-0.524354\pi$
$823$	−4.38556	−0.152871	−0.0764354	+	0.997075i	$0.524354\pi$
$824$	0	0
$825$	9.13568	0.318064
$826$	0	0
$827$	−37.5607	−1.30611	−0.653056	−	0.757309i	$-0.726513\pi$
$827$	−37.5607	−1.30611	−0.653056	+	0.757309i	$0.726513\pi$
$828$	0	0
$829$	18.6526	0.647832	0.323916	−	0.946086i	$-0.395000\pi$
$829$	18.6526	0.647832	0.323916	+	0.946086i	$0.395000\pi$
$830$	0	0
$831$	19.8326	0.687986
$832$	0	0
$833$	−4.96373	−0.171983
$834$	0	0
$835$	19.2379	0.665753
$836$	0	0
$837$	0.266531	0.00921266
$838$	0	0
$839$	35.0156	1.20887	0.604436	−	0.796654i	$-0.293399\pi$
$839$	35.0156	1.20887	0.604436	+	0.796654i	$0.293399\pi$
$840$	0	0
$841$	−28.6694	−0.988599
$842$	0	0
$843$	−13.8243	−0.476134
$844$	0	0
$845$	13.0122	0.447634
$846$	0	0
$847$	−5.32711	−0.183042
$848$	0	0
$849$	−7.16308	−0.245836
$850$	0	0
$851$	0	0
$852$	0	0
$853$	36.7819	1.25939	0.629693	−	0.776844i	$-0.283181\pi$
$853$	36.7819	1.25939	0.629693	+	0.776844i	$0.283181\pi$
$854$	0	0
$855$	2.14550	0.0733747
$856$	0	0
$857$	30.1316	1.02928	0.514638	−	0.857408i	$-0.327926\pi$
$857$	30.1316	1.02928	0.514638	+	0.857408i	$0.327926\pi$
$858$	0	0
$859$	14.2005	0.484515	0.242258	−	0.970212i	$-0.422112\pi$
$859$	14.2005	0.484515	0.242258	+	0.970212i	$0.422112\pi$
$860$	0	0
$861$	5.16809	0.176128
$862$	0	0
$863$	−11.3456	−0.386210	−0.193105	−	0.981178i	$-0.561856\pi$
$863$	−11.3456	−0.386210	−0.193105	+	0.981178i	$0.561856\pi$
$864$	0	0
$865$	11.2124	0.381233
$866$	0	0
$867$	−16.2867	−0.553124
$868$	0	0
$869$	14.3199	0.485770
$870$	0	0
$871$	−5.92963	−0.200918
$872$	0	0
$873$	−4.03392	−0.136527
$874$	0	0
$875$	10.4016	0.351639
$876$	0	0
$877$	24.1765	0.816383	0.408191	−	0.912896i	$-0.366160\pi$
$877$	24.1765	0.816383	0.408191	+	0.912896i	$0.366160\pi$
$878$	0	0
$879$	18.2242	0.614685
$880$	0	0
$881$	34.3697	1.15795	0.578973	−	0.815347i	$-0.303454\pi$
$881$	34.3697	1.15795	0.578973	+	0.815347i	$0.303454\pi$
$882$	0	0
$883$	−17.4668	−0.587804	−0.293902	−	0.955835i	$-0.594954\pi$
$883$	−17.4668	−0.587804	−0.293902	+	0.955835i	$0.594954\pi$
$884$	0	0
$885$	1.65137	0.0555101
$886$	0	0
$887$	31.1198	1.04490	0.522450	−	0.852670i	$-0.325018\pi$
$887$	31.1198	1.04490	0.522450	+	0.852670i	$0.325018\pi$
$888$	0	0
$889$	−15.4181	−0.517107
$890$	0	0
$891$	−2.44394	−0.0818752
$892$	0	0
$893$	−1.07157	−0.0358588
$894$	0	0
$895$	−28.7442	−0.960814
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.153253	0.00511128
$900$	0	0
$901$	−5.23518	−0.174409
$902$	0	0
$903$	−3.49413	−0.116277
$904$	0	0
$905$	−1.10615	−0.0367698
$906$	0	0
$907$	−36.1749	−1.20117	−0.600583	−	0.799562i	$-0.705065\pi$
$907$	−36.1749	−1.20117	−0.600583	+	0.799562i	$0.705065\pi$
$908$	0	0
$909$	12.0737	0.400460
$910$	0	0
$911$	11.8624	0.393020	0.196510	−	0.980502i	$-0.437039\pi$
$911$	11.8624	0.393020	0.196510	+	0.980502i	$0.437039\pi$
$912$	0	0
$913$	12.6360	0.418190
$914$	0	0
$915$	−9.98581	−0.330121
$916$	0	0
$917$	10.7666	0.355545
$918$	0	0
$919$	8.62999	0.284677	0.142338	−	0.989818i	$-0.454538\pi$
$919$	8.62999	0.284677	0.142338	+	0.989818i	$0.454538\pi$
$920$	0	0
$921$	15.9314	0.524957
$922$	0	0
$923$	0.334234	0.0110015
$924$	0	0
$925$	32.4371	1.06653
$926$	0	0
$927$	−16.9492	−0.556685
$928$	0	0
$929$	12.0951	0.396828	0.198414	−	0.980118i	$-0.436421\pi$
$929$	12.0951	0.396828	0.198414	+	0.980118i	$0.436421\pi$
$930$	0	0
$931$	11.2248	0.367878
$932$	0	0
$933$	−29.1322	−0.953745
$934$	0	0
$935$	2.31873	0.0758305
$936$	0	0
$937$	−16.3790	−0.535080	−0.267540	−	0.963547i	$-0.586211\pi$
$937$	−16.3790	−0.535080	−0.267540	+	0.963547i	$0.586211\pi$
$938$	0	0
$939$	−17.8209	−0.581564
$940$	0	0
$941$	28.5387	0.930336	0.465168	−	0.885222i	$-0.345994\pi$
$941$	28.5387	0.930336	0.465168	+	0.885222i	$0.345994\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.19038	−0.0387230
$946$	0	0
$947$	3.49771	0.113660	0.0568301	−	0.998384i	$-0.481901\pi$
$947$	3.49771	0.113660	0.0568301	+	0.998384i	$0.481901\pi$
$948$	0	0
$949$	6.12617	0.198864
$950$	0	0
$951$	23.9318	0.776042
$952$	0	0
$953$	38.3548	1.24243	0.621217	−	0.783639i	$-0.286639\pi$
$953$	38.3548	1.24243	0.621217	+	0.783639i	$0.286639\pi$
$954$	0	0
$955$	5.21044	0.168606
$956$	0	0
$957$	−1.40525	−0.0454252
$958$	0	0
$959$	11.1904	0.361357
$960$	0	0
$961$	−30.9290	−0.997708
$962$	0	0
$963$	12.4987	0.402767
$964$	0	0
$965$	−19.3799	−0.623861
$966$	0	0
$967$	−55.1290	−1.77283	−0.886415	−	0.462892i	$-0.846812\pi$
$967$	−55.1290	−1.77283	−0.886415	+	0.462892i	$0.846812\pi$
$968$	0	0
$969$	−1.61310	−0.0518201
$970$	0	0
$971$	2.74726	0.0881637	0.0440819	−	0.999028i	$-0.485964\pi$
$971$	2.74726	0.0881637	0.0440819	+	0.999028i	$0.485964\pi$
$972$	0	0
$973$	−6.57117	−0.210662
$974$	0	0
$975$	−4.44906	−0.142484
$976$	0	0
$977$	47.4319	1.51748	0.758740	−	0.651394i	$-0.225815\pi$
$977$	47.4319	1.51748	0.758740	+	0.651394i	$0.225815\pi$
$978$	0	0
$979$	29.6902	0.948904
$980$	0	0
$981$	−0.557809	−0.0178095
$982$	0	0
$983$	−23.0938	−0.736578	−0.368289	−	0.929711i	$-0.620056\pi$
$983$	−23.0938	−0.736578	−0.368289	+	0.929711i	$0.620056\pi$
$984$	0	0
$985$	1.63654	0.0521446
$986$	0	0
$987$	0.594534	0.0189242
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−40.9799	−1.30177	−0.650885	−	0.759176i	$-0.725602\pi$
$991$	−40.9799	−1.30177	−0.650885	+	0.759176i	$0.725602\pi$
$992$	0	0
$993$	−6.46271	−0.205088
$994$	0	0
$995$	8.19257	0.259722
$996$	0	0
$997$	13.8248	0.437836	0.218918	−	0.975743i	$-0.429747\pi$
$997$	13.8248	0.437836	0.218918	+	0.975743i	$0.429747\pi$
$998$	0	0
$999$	−8.67745	−0.274542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.s.1.5		10
23.13	even	11		276.2.i.a.169.2	yes	20
23.16	even	11		276.2.i.a.49.2	✓	20
23.22	odd	2		6348.2.a.t.1.6		10
69.59	odd	22		828.2.q.c.721.1		20
69.62	odd	22		828.2.q.c.325.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.49.2	✓	20	23.16	even	11
276.2.i.a.169.2	yes	20	23.13	even	11
828.2.q.c.325.1		20	69.62	odd	22
828.2.q.c.721.1		20	69.59	odd	22
6348.2.a.s.1.5		10	1.1	even	1	trivial
6348.2.a.t.1.6		10	23.22	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:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.12335 q^{5} +1.05967 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.12335 q^{5} +1.05967 q^{7} +1.00000 q^{9} -2.44394 q^{11} +1.19020 q^{13} -1.12335 q^{15} +0.844588 q^{17} -1.90992 q^{19} +1.05967 q^{21} -3.73809 q^{25} +1.00000 q^{27} +0.574992 q^{29} +0.266531 q^{31} -2.44394 q^{33} -1.19038 q^{35} -8.67745 q^{37} +1.19020 q^{39} +4.87707 q^{41} -3.29738 q^{43} -1.12335 q^{45} +0.561056 q^{47} -5.87710 q^{49} +0.844588 q^{51} -6.19850 q^{53} +2.74540 q^{55} -1.90992 q^{57} -1.47004 q^{59} +8.88933 q^{61} +1.05967 q^{63} -1.33700 q^{65} -4.98206 q^{67} +0.280823 q^{71} +5.14720 q^{73} -3.73809 q^{75} -2.58977 q^{77} -5.85935 q^{79} +1.00000 q^{81} -5.17033 q^{83} -0.948765 q^{85} +0.574992 q^{87} -12.1485 q^{89} +1.26121 q^{91} +0.266531 q^{93} +2.14550 q^{95} -4.03392 q^{97} -2.44394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9} - 11 q^{11} - 2 q^{15} - 13 q^{17} - 18 q^{19} - 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} - 11 q^{33} - 13 q^{35} - 5 q^{37} + 24 q^{41} - 40 q^{43} - 2 q^{45} - 9 q^{47} + 5 q^{49} - 13 q^{51} - 6 q^{53} - 14 q^{55} - 18 q^{57} + 28 q^{59} - 39 q^{61} - 11 q^{63} - 14 q^{65} - 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} - 33 q^{79} + 10 q^{81} - 29 q^{83} - 21 q^{85} + 5 q^{87} - 17 q^{89} - 36 q^{91} + 15 q^{93} - 42 q^{95} - 46 q^{97} - 11 q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 - 2 * q^5 - 11 * q^7 + 10 * q^9 - 11 * q^11 - 2 * q^15 - 13 * q^17 - 18 * q^19 - 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 - 11 * q^33 - 13 * q^35 - 5 * q^37 + 24 * q^41 - 40 * q^43 - 2 * q^45 - 9 * q^47 + 5 * q^49 - 13 * q^51 - 6 * q^53 - 14 * q^55 - 18 * q^57 + 28 * q^59 - 39 * q^61 - 11 * q^63 - 14 * q^65 - 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 - 33 * q^79 + 10 * q^81 - 29 * q^83 - 21 * q^85 + 5 * q^87 - 17 * q^89 - 36 * q^91 + 15 * q^93 - 42 * q^95 - 46 * q^97 - 11 * q^99

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