LMFDB - Embedded newform 6348.2.a.r.1.3

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:	$0$
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} - 2x^{9} - 15x^{8} + 29x^{7} + 74x^{6} - 143x^{5} - 126x^{4} + 259x^{3} + 21x^{2} - 98x + 23 $ x^10 - 2x^9 - 15x^8 + 29x^7 + 74x^6 - 143x^5 - 126x^4 + 259x^3 + 21x^2 - 98*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			1.3
Root			$-2.46955$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.50596 q^{5} -3.28346 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.50596 q^{5} -3.28346 q^{7} +1.00000 q^{9} +0.390966 q^{11} -2.76618 q^{13} +1.50596 q^{15} -3.19006 q^{17} +0.507939 q^{19} +3.28346 q^{21} -2.73208 q^{25} -1.00000 q^{27} -6.31611 q^{29} -9.57602 q^{31} -0.390966 q^{33} +4.94477 q^{35} +1.19306 q^{37} +2.76618 q^{39} +3.24670 q^{41} +7.97880 q^{43} -1.50596 q^{45} -11.8320 q^{47} +3.78114 q^{49} +3.19006 q^{51} -0.700042 q^{53} -0.588780 q^{55} -0.507939 q^{57} -0.0143753 q^{59} +0.745517 q^{61} -3.28346 q^{63} +4.16576 q^{65} +4.06026 q^{67} -10.0709 q^{71} -8.38306 q^{73} +2.73208 q^{75} -1.28372 q^{77} +0.102997 q^{79} +1.00000 q^{81} +6.12300 q^{83} +4.80412 q^{85} +6.31611 q^{87} +0.138727 q^{89} +9.08266 q^{91} +9.57602 q^{93} -0.764937 q^{95} +6.92787 q^{97} +0.390966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 9 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 9 * q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 9 q^{7} + 10 q^{9} + 11 q^{11} - 4 q^{13} + 7 q^{17} - 2 q^{19} - 9 q^{21} + 6 q^{25} - 10 q^{27} + 9 q^{29} + 11 q^{31} - 11 q^{33} + 43 q^{35} - 9 q^{37} + 4 q^{39} + 2 q^{41} + 24 q^{43} - 27 q^{47} + 37 q^{49} - 7 q^{51} + 4 q^{53} - 22 q^{55} + 2 q^{57} + 2 q^{59} - 27 q^{61} + 9 q^{63} + 22 q^{65} + 24 q^{67} + 11 q^{71} + 2 q^{73} - 6 q^{75} - 11 q^{77} + 23 q^{79} + 10 q^{81} + 37 q^{83} - q^{85} - 9 q^{87} + 11 q^{89} - 8 q^{91} - 11 q^{93} - 10 q^{95} + 10 q^{97} + 11 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 9 * q^7 + 10 * q^9 + 11 * q^11 - 4 * q^13 + 7 * q^17 - 2 * q^19 - 9 * q^21 + 6 * q^25 - 10 * q^27 + 9 * q^29 + 11 * q^31 - 11 * q^33 + 43 * q^35 - 9 * q^37 + 4 * q^39 + 2 * q^41 + 24 * q^43 - 27 * q^47 + 37 * q^49 - 7 * q^51 + 4 * q^53 - 22 * q^55 + 2 * q^57 + 2 * q^59 - 27 * q^61 + 9 * q^63 + 22 * q^65 + 24 * q^67 + 11 * q^71 + 2 * q^73 - 6 * q^75 - 11 * q^77 + 23 * q^79 + 10 * q^81 + 37 * q^83 - q^85 - 9 * q^87 + 11 * q^89 - 8 * q^91 - 11 * q^93 - 10 * q^95 + 10 * 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
$5$	−1.50596	−0.673487	−0.336743	−	0.941596i	$-0.609325\pi$
$5$	−1.50596	−0.673487	−0.336743	+	0.941596i	$0.609325\pi$
$6$	0	0
$7$	−3.28346	−1.24103	−0.620516	−	0.784193i	$-0.713077\pi$
$7$	−3.28346	−1.24103	−0.620516	+	0.784193i	$0.713077\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.390966	0.117881	0.0589403	−	0.998262i	$-0.481228\pi$
$11$	0.390966	0.117881	0.0589403	+	0.998262i	$0.481228\pi$
$12$	0	0
$13$	−2.76618	−0.767201	−0.383600	−	0.923499i	$-0.625316\pi$
$13$	−2.76618	−0.767201	−0.383600	+	0.923499i	$0.625316\pi$
$14$	0	0
$15$	1.50596	0.388838
$16$	0	0
$17$	−3.19006	−0.773704	−0.386852	−	0.922142i	$-0.626438\pi$
$17$	−3.19006	−0.773704	−0.386852	+	0.922142i	$0.626438\pi$
$18$	0	0
$19$	0.507939	0.116529	0.0582646	−	0.998301i	$-0.481443\pi$
$19$	0.507939	0.116529	0.0582646	+	0.998301i	$0.481443\pi$
$20$	0	0
$21$	3.28346	0.716511
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.73208	−0.546416
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.31611	−1.17287	−0.586436	−	0.809996i	$-0.699469\pi$
$29$	−6.31611	−1.17287	−0.586436	+	0.809996i	$0.699469\pi$
$30$	0	0
$31$	−9.57602	−1.71990	−0.859952	−	0.510375i	$-0.829507\pi$
$31$	−9.57602	−1.71990	−0.859952	+	0.510375i	$0.829507\pi$
$32$	0	0
$33$	−0.390966	−0.0680584
$34$	0	0
$35$	4.94477	0.835819
$36$	0	0
$37$	1.19306	0.196138	0.0980691	−	0.995180i	$-0.468733\pi$
$37$	1.19306	0.196138	0.0980691	+	0.995180i	$0.468733\pi$
$38$	0	0
$39$	2.76618	0.442943
$40$	0	0
$41$	3.24670	0.507049	0.253525	−	0.967329i	$-0.418410\pi$
$41$	3.24670	0.507049	0.253525	+	0.967329i	$0.418410\pi$
$42$	0	0
$43$	7.97880	1.21676	0.608378	−	0.793648i	$-0.291821\pi$
$43$	7.97880	1.21676	0.608378	+	0.793648i	$0.291821\pi$
$44$	0	0
$45$	−1.50596	−0.224496
$46$	0	0
$47$	−11.8320	−1.72588	−0.862939	−	0.505309i	$-0.831379\pi$
$47$	−11.8320	−1.72588	−0.862939	+	0.505309i	$0.831379\pi$
$48$	0	0
$49$	3.78114	0.540163
$50$	0	0
$51$	3.19006	0.446698
$52$	0	0
$53$	−0.700042	−0.0961581	−0.0480791	−	0.998844i	$-0.515310\pi$
$53$	−0.700042	−0.0961581	−0.0480791	+	0.998844i	$0.515310\pi$
$54$	0	0
$55$	−0.588780	−0.0793910
$56$	0	0
$57$	−0.507939	−0.0672782
$58$	0	0
$59$	−0.0143753	−0.00187151	−0.000935754	−	1.00000i	$-0.500298\pi$
$59$	−0.0143753	−0.00187151	−0.000935754		1.00000i	$0.500298\pi$
$60$	0	0
$61$	0.745517	0.0954536	0.0477268	−	0.998860i	$-0.484802\pi$
$61$	0.745517	0.0954536	0.0477268	+	0.998860i	$0.484802\pi$
$62$	0	0
$63$	−3.28346	−0.413678
$64$	0	0
$65$	4.16576	0.516699
$66$	0	0
$67$	4.06026	0.496039	0.248020	−	0.968755i	$-0.420220\pi$
$67$	4.06026	0.496039	0.248020	+	0.968755i	$0.420220\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.0709	−1.19519	−0.597597	−	0.801797i	$-0.703878\pi$
$71$	−10.0709	−1.19519	−0.597597	+	0.801797i	$0.703878\pi$
$72$	0	0
$73$	−8.38306	−0.981163	−0.490582	−	0.871395i	$-0.663216\pi$
$73$	−8.38306	−0.981163	−0.490582	+	0.871395i	$0.663216\pi$
$74$	0	0
$75$	2.73208	0.315473
$76$	0	0
$77$	−1.28372	−0.146294
$78$	0	0
$79$	0.102997	0.0115880	0.00579401	−	0.999983i	$-0.498156\pi$
$79$	0.102997	0.0115880	0.00579401	+	0.999983i	$0.498156\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.12300	0.672086	0.336043	−	0.941847i	$-0.390911\pi$
$83$	6.12300	0.672086	0.336043	+	0.941847i	$0.390911\pi$
$84$	0	0
$85$	4.80412	0.521080
$86$	0	0
$87$	6.31611	0.677158
$88$	0	0
$89$	0.138727	0.0147050	0.00735249	−	0.999973i	$-0.497660\pi$
$89$	0.138727	0.0147050	0.00735249	+	0.999973i	$0.497660\pi$
$90$	0	0
$91$	9.08266	0.952121
$92$	0	0
$93$	9.57602	0.992987
$94$	0	0
$95$	−0.764937	−0.0784809
$96$	0	0
$97$	6.92787	0.703419	0.351709	−	0.936109i	$-0.385601\pi$
$97$	6.92787	0.703419	0.351709	+	0.936109i	$0.385601\pi$
$98$	0	0
$99$	0.390966	0.0392935
$100$	0	0
$101$	−13.4937	−1.34267	−0.671334	−	0.741155i	$-0.734278\pi$
$101$	−13.4937	−1.34267	−0.671334	+	0.741155i	$0.734278\pi$
$102$	0	0
$103$	−14.2848	−1.40752	−0.703762	−	0.710435i	$-0.748498\pi$
$103$	−14.2848	−1.40752	−0.703762	+	0.710435i	$0.748498\pi$
$104$	0	0
$105$	−4.94477	−0.482561
$106$	0	0
$107$	−15.4607	−1.49464	−0.747320	−	0.664464i	$-0.768660\pi$
$107$	−15.4607	−1.49464	−0.747320	+	0.664464i	$0.768660\pi$
$108$	0	0
$109$	−13.9386	−1.33508	−0.667540	−	0.744574i	$-0.732653\pi$
$109$	−13.9386	−1.33508	−0.667540	+	0.744574i	$0.732653\pi$
$110$	0	0
$111$	−1.19306	−0.113240
$112$	0	0
$113$	17.1220	1.61070	0.805350	−	0.592800i	$-0.201978\pi$
$113$	17.1220	1.61070	0.805350	+	0.592800i	$0.201978\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.76618	−0.255734
$118$	0	0
$119$	10.4745	0.960193
$120$	0	0
$121$	−10.8471	−0.986104
$122$	0	0
$123$	−3.24670	−0.292745
$124$	0	0
$125$	11.6442	1.04149
$126$	0	0
$127$	12.8324	1.13869	0.569344	−	0.822099i	$-0.307197\pi$
$127$	12.8324	1.13869	0.569344	+	0.822099i	$0.307197\pi$
$128$	0	0
$129$	−7.97880	−0.702494
$130$	0	0
$131$	6.04869	0.528476	0.264238	−	0.964457i	$-0.414880\pi$
$131$	6.04869	0.528476	0.264238	+	0.964457i	$0.414880\pi$
$132$	0	0
$133$	−1.66780	−0.144617
$134$	0	0
$135$	1.50596	0.129613
$136$	0	0
$137$	9.86419	0.842754	0.421377	−	0.906885i	$-0.361547\pi$
$137$	9.86419	0.842754	0.421377	+	0.906885i	$0.361547\pi$
$138$	0	0
$139$	18.2084	1.54441	0.772206	−	0.635372i	$-0.219153\pi$
$139$	18.2084	1.54441	0.772206	+	0.635372i	$0.219153\pi$
$140$	0	0
$141$	11.8320	0.996436
$142$	0	0
$143$	−1.08148	−0.0904381
$144$	0	0
$145$	9.51182	0.789914
$146$	0	0
$147$	−3.78114	−0.311863
$148$	0	0
$149$	−14.4546	−1.18417	−0.592083	−	0.805877i	$-0.701694\pi$
$149$	−14.4546	−1.18417	−0.592083	+	0.805877i	$0.701694\pi$
$150$	0	0
$151$	−10.3011	−0.838292	−0.419146	−	0.907919i	$-0.637671\pi$
$151$	−10.3011	−0.838292	−0.419146	+	0.907919i	$0.637671\pi$
$152$	0	0
$153$	−3.19006	−0.257901
$154$	0	0
$155$	14.4211	1.15833
$156$	0	0
$157$	−8.37041	−0.668032	−0.334016	−	0.942567i	$-0.608404\pi$
$157$	−8.37041	−0.668032	−0.334016	+	0.942567i	$0.608404\pi$
$158$	0	0
$159$	0.700042	0.0555169
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.79796	0.767435	0.383718	−	0.923450i	$-0.374644\pi$
$163$	9.79796	0.767435	0.383718	+	0.923450i	$0.374644\pi$
$164$	0	0
$165$	0.588780	0.0458364
$166$	0	0
$167$	−11.5088	−0.890577	−0.445288	−	0.895387i	$-0.646899\pi$
$167$	−11.5088	−0.890577	−0.445288	+	0.895387i	$0.646899\pi$
$168$	0	0
$169$	−5.34824	−0.411403
$170$	0	0
$171$	0.507939	0.0388431
$172$	0	0
$173$	13.9292	1.05902	0.529509	−	0.848304i	$-0.322376\pi$
$173$	13.9292	1.05902	0.529509	+	0.848304i	$0.322376\pi$
$174$	0	0
$175$	8.97068	0.678120
$176$	0	0
$177$	0.0143753	0.00108052
$178$	0	0
$179$	−10.7942	−0.806796	−0.403398	−	0.915025i	$-0.632171\pi$
$179$	−10.7942	−0.806796	−0.403398	+	0.915025i	$0.632171\pi$
$180$	0	0
$181$	−21.5016	−1.59820	−0.799100	−	0.601198i	$-0.794690\pi$
$181$	−21.5016	−1.59820	−0.799100	+	0.601198i	$0.794690\pi$
$182$	0	0
$183$	−0.745517	−0.0551102
$184$	0	0
$185$	−1.79671	−0.132096
$186$	0	0
$187$	−1.24721	−0.0912047
$188$	0	0
$189$	3.28346	0.238837
$190$	0	0
$191$	24.8000	1.79446	0.897231	−	0.441561i	$-0.145575\pi$
$191$	24.8000	1.79446	0.897231	+	0.441561i	$0.145575\pi$
$192$	0	0
$193$	−3.28393	−0.236382	−0.118191	−	0.992991i	$-0.537710\pi$
$193$	−3.28393	−0.236382	−0.118191	+	0.992991i	$0.537710\pi$
$194$	0	0
$195$	−4.16576	−0.298317
$196$	0	0
$197$	27.3897	1.95143	0.975717	−	0.219033i	$-0.0702902\pi$
$197$	27.3897	1.95143	0.975717	+	0.219033i	$0.0702902\pi$
$198$	0	0
$199$	11.8414	0.839416	0.419708	−	0.907659i	$-0.362133\pi$
$199$	11.8414	0.839416	0.419708	+	0.907659i	$0.362133\pi$
$200$	0	0
$201$	−4.06026	−0.286388
$202$	0	0
$203$	20.7387	1.45557
$204$	0	0
$205$	−4.88940	−0.341491
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.198587	0.0137365
$210$	0	0
$211$	12.5281	0.862468	0.431234	−	0.902240i	$-0.358078\pi$
$211$	12.5281	0.862468	0.431234	+	0.902240i	$0.358078\pi$
$212$	0	0
$213$	10.0709	0.690046
$214$	0	0
$215$	−12.0158	−0.819469
$216$	0	0
$217$	31.4425	2.13446
$218$	0	0
$219$	8.38306	0.566475
$220$	0	0
$221$	8.82430	0.593586
$222$	0	0
$223$	15.3136	1.02547	0.512736	−	0.858546i	$-0.328632\pi$
$223$	15.3136	1.02547	0.512736	+	0.858546i	$0.328632\pi$
$224$	0	0
$225$	−2.73208	−0.182139
$226$	0	0
$227$	16.4454	1.09152	0.545761	−	0.837941i	$-0.316241\pi$
$227$	16.4454	1.09152	0.545761	+	0.837941i	$0.316241\pi$
$228$	0	0
$229$	6.90633	0.456383	0.228192	−	0.973616i	$-0.426719\pi$
$229$	6.90633	0.456383	0.228192	+	0.973616i	$0.426719\pi$
$230$	0	0
$231$	1.28372	0.0844627
$232$	0	0
$233$	24.8931	1.63080	0.815400	−	0.578897i	$-0.196517\pi$
$233$	24.8931	1.63080	0.815400	+	0.578897i	$0.196517\pi$
$234$	0	0
$235$	17.8186	1.16236
$236$	0	0
$237$	−0.102997	−0.00669034
$238$	0	0
$239$	8.41669	0.544430	0.272215	−	0.962236i	$-0.412244\pi$
$239$	8.41669	0.544430	0.272215	+	0.962236i	$0.412244\pi$
$240$	0	0
$241$	−21.4774	−1.38348	−0.691740	−	0.722147i	$-0.743155\pi$
$241$	−21.4774	−1.38348	−0.691740	+	0.722147i	$0.743155\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.69425	−0.363793
$246$	0	0
$247$	−1.40505	−0.0894013
$248$	0	0
$249$	−6.12300	−0.388029
$250$	0	0
$251$	9.57312	0.604250	0.302125	−	0.953268i	$-0.402304\pi$
$251$	9.57312	0.604250	0.302125	+	0.953268i	$0.402304\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.80412	−0.300845
$256$	0	0
$257$	0.630921	0.0393558	0.0196779	−	0.999806i	$-0.493736\pi$
$257$	0.630921	0.0393558	0.0196779	+	0.999806i	$0.493736\pi$
$258$	0	0
$259$	−3.91738	−0.243414
$260$	0	0
$261$	−6.31611	−0.390957
$262$	0	0
$263$	22.2949	1.37476	0.687382	−	0.726296i	$-0.258760\pi$
$263$	22.2949	1.37476	0.687382	+	0.726296i	$0.258760\pi$
$264$	0	0
$265$	1.05424	0.0647612
$266$	0	0
$267$	−0.138727	−0.00848992
$268$	0	0
$269$	−25.2879	−1.54183	−0.770915	−	0.636938i	$-0.780201\pi$
$269$	−25.2879	−1.54183	−0.770915	+	0.636938i	$0.780201\pi$
$270$	0	0
$271$	10.8247	0.657555	0.328778	−	0.944407i	$-0.393363\pi$
$271$	10.8247	0.657555	0.328778	+	0.944407i	$0.393363\pi$
$272$	0	0
$273$	−9.08266	−0.549707
$274$	0	0
$275$	−1.06815	−0.0644118
$276$	0	0
$277$	24.7512	1.48716	0.743579	−	0.668648i	$-0.233127\pi$
$277$	24.7512	1.48716	0.743579	+	0.668648i	$0.233127\pi$
$278$	0	0
$279$	−9.57602	−0.573301
$280$	0	0
$281$	−1.90741	−0.113786	−0.0568932	−	0.998380i	$-0.518119\pi$
$281$	−1.90741	−0.113786	−0.0568932	+	0.998380i	$0.518119\pi$
$282$	0	0
$283$	0.935738	0.0556238	0.0278119	−	0.999613i	$-0.491146\pi$
$283$	0.935738	0.0556238	0.0278119	+	0.999613i	$0.491146\pi$
$284$	0	0
$285$	0.764937	0.0453110
$286$	0	0
$287$	−10.6604	−0.629265
$288$	0	0
$289$	−6.82349	−0.401382
$290$	0	0
$291$	−6.92787	−0.406119
$292$	0	0
$293$	−23.4028	−1.36721	−0.683604	−	0.729853i	$-0.739589\pi$
$293$	−23.4028	−1.36721	−0.683604	+	0.729853i	$0.739589\pi$
$294$	0	0
$295$	0.0216487	0.00126044
$296$	0	0
$297$	−0.390966	−0.0226861
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−26.1981	−1.51003
$302$	0	0
$303$	13.4937	0.775190
$304$	0	0
$305$	−1.12272	−0.0642868
$306$	0	0
$307$	−28.3393	−1.61741	−0.808706	−	0.588213i	$-0.799832\pi$
$307$	−28.3393	−1.61741	−0.808706	+	0.588213i	$0.799832\pi$
$308$	0	0
$309$	14.2848	0.812635
$310$	0	0
$311$	18.1727	1.03048	0.515240	−	0.857046i	$-0.327703\pi$
$311$	18.1727	1.03048	0.515240	+	0.857046i	$0.327703\pi$
$312$	0	0
$313$	4.52686	0.255873	0.127937	−	0.991782i	$-0.459165\pi$
$313$	4.52686	0.255873	0.127937	+	0.991782i	$0.459165\pi$
$314$	0	0
$315$	4.94477	0.278606
$316$	0	0
$317$	−2.72256	−0.152914	−0.0764571	−	0.997073i	$-0.524361\pi$
$317$	−2.72256	−0.152914	−0.0764571	+	0.997073i	$0.524361\pi$
$318$	0	0
$319$	−2.46938	−0.138259
$320$	0	0
$321$	15.4607	0.862931
$322$	0	0
$323$	−1.62036	−0.0901592
$324$	0	0
$325$	7.55742	0.419210
$326$	0	0
$327$	13.9386	0.770808
$328$	0	0
$329$	38.8500	2.14187
$330$	0	0
$331$	3.23888	0.178025	0.0890125	−	0.996031i	$-0.471629\pi$
$331$	3.23888	0.178025	0.0890125	+	0.996031i	$0.471629\pi$
$332$	0	0
$333$	1.19306	0.0653794
$334$	0	0
$335$	−6.11459	−0.334076
$336$	0	0
$337$	−27.1932	−1.48131	−0.740654	−	0.671887i	$-0.765484\pi$
$337$	−27.1932	−1.48131	−0.740654	+	0.671887i	$0.765484\pi$
$338$	0	0
$339$	−17.1220	−0.929938
$340$	0	0
$341$	−3.74390	−0.202743
$342$	0	0
$343$	10.5690	0.570673
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.0473955	−0.00254432	−0.00127216	−	0.999999i	$-0.500405\pi$
$347$	−0.0473955	−0.00254432	−0.00127216	+	0.999999i	$0.500405\pi$
$348$	0	0
$349$	19.3503	1.03580	0.517898	−	0.855442i	$-0.326715\pi$
$349$	19.3503	1.03580	0.517898	+	0.855442i	$0.326715\pi$
$350$	0	0
$351$	2.76618	0.147648
$352$	0	0
$353$	7.92211	0.421651	0.210826	−	0.977524i	$-0.432385\pi$
$353$	7.92211	0.421651	0.210826	+	0.977524i	$0.432385\pi$
$354$	0	0
$355$	15.1664	0.804947
$356$	0	0
$357$	−10.4745	−0.554367
$358$	0	0
$359$	0.233786	0.0123388	0.00616938	−	0.999981i	$-0.498036\pi$
$359$	0.233786	0.0123388	0.00616938	+	0.999981i	$0.498036\pi$
$360$	0	0
$361$	−18.7420	−0.986421
$362$	0	0
$363$	10.8471	0.569328
$364$	0	0
$365$	12.6246	0.660800
$366$	0	0
$367$	15.0127	0.783656	0.391828	−	0.920039i	$-0.371843\pi$
$367$	15.0127	0.783656	0.391828	+	0.920039i	$0.371843\pi$
$368$	0	0
$369$	3.24670	0.169016
$370$	0	0
$371$	2.29856	0.119335
$372$	0	0
$373$	−32.6123	−1.68860	−0.844302	−	0.535868i	$-0.819984\pi$
$373$	−32.6123	−1.68860	−0.844302	+	0.535868i	$0.819984\pi$
$374$	0	0
$375$	−11.6442	−0.601305
$376$	0	0
$377$	17.4715	0.899828
$378$	0	0
$379$	31.7114	1.62891	0.814453	−	0.580230i	$-0.197037\pi$
$379$	31.7114	1.62891	0.814453	+	0.580230i	$0.197037\pi$
$380$	0	0
$381$	−12.8324	−0.657422
$382$	0	0
$383$	30.3718	1.55193	0.775963	−	0.630778i	$-0.217264\pi$
$383$	30.3718	1.55193	0.775963	+	0.630778i	$0.217264\pi$
$384$	0	0
$385$	1.93324	0.0985269
$386$	0	0
$387$	7.97880	0.405585
$388$	0	0
$389$	28.1479	1.42716	0.713579	−	0.700575i	$-0.247073\pi$
$389$	28.1479	1.42716	0.713579	+	0.700575i	$0.247073\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.04869	−0.305116
$394$	0	0
$395$	−0.155109	−0.00780438
$396$	0	0
$397$	−12.2913	−0.616883	−0.308441	−	0.951243i	$-0.599807\pi$
$397$	−12.2913	−0.616883	−0.308441	+	0.951243i	$0.599807\pi$
$398$	0	0
$399$	1.66780	0.0834945
$400$	0	0
$401$	−34.7918	−1.73742	−0.868709	−	0.495322i	$-0.835050\pi$
$401$	−34.7918	−1.73742	−0.868709	+	0.495322i	$0.835050\pi$
$402$	0	0
$403$	26.4890	1.31951
$404$	0	0
$405$	−1.50596	−0.0748319
$406$	0	0
$407$	0.466446	0.0231209
$408$	0	0
$409$	−1.73698	−0.0858883	−0.0429442	−	0.999077i	$-0.513674\pi$
$409$	−1.73698	−0.0858883	−0.0429442	+	0.999077i	$0.513674\pi$
$410$	0	0
$411$	−9.86419	−0.486564
$412$	0	0
$413$	0.0472009	0.00232260
$414$	0	0
$415$	−9.22100	−0.452641
$416$	0	0
$417$	−18.2084	−0.891667
$418$	0	0
$419$	−14.2139	−0.694392	−0.347196	−	0.937793i	$-0.612866\pi$
$419$	−14.2139	−0.694392	−0.347196	+	0.937793i	$0.612866\pi$
$420$	0	0
$421$	−1.71488	−0.0835779	−0.0417890	−	0.999126i	$-0.513306\pi$
$421$	−1.71488	−0.0835779	−0.0417890	+	0.999126i	$0.513306\pi$
$422$	0	0
$423$	−11.8320	−0.575292
$424$	0	0
$425$	8.71550	0.422764
$426$	0	0
$427$	−2.44788	−0.118461
$428$	0	0
$429$	1.08148	0.0522144
$430$	0	0
$431$	39.5165	1.90344	0.951721	−	0.306966i	$-0.0993137\pi$
$431$	39.5165	1.90344	0.951721	+	0.306966i	$0.0993137\pi$
$432$	0	0
$433$	−5.60498	−0.269358	−0.134679	−	0.990889i	$-0.543000\pi$
$433$	−5.60498	−0.269358	−0.134679	+	0.990889i	$0.543000\pi$
$434$	0	0
$435$	−9.51182	−0.456057
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−5.43798	−0.259541	−0.129770	−	0.991544i	$-0.541424\pi$
$439$	−5.43798	−0.259541	−0.129770	+	0.991544i	$0.541424\pi$
$440$	0	0
$441$	3.78114	0.180054
$442$	0	0
$443$	−29.7584	−1.41386	−0.706932	−	0.707282i	$-0.749921\pi$
$443$	−29.7584	−1.41386	−0.706932	+	0.707282i	$0.749921\pi$
$444$	0	0
$445$	−0.208917	−0.00990361
$446$	0	0
$447$	14.4546	0.683679
$448$	0	0
$449$	32.1684	1.51812	0.759061	−	0.651020i	$-0.225658\pi$
$449$	32.1684	1.51812	0.759061	+	0.651020i	$0.225658\pi$
$450$	0	0
$451$	1.26935	0.0597713
$452$	0	0
$453$	10.3011	0.483988
$454$	0	0
$455$	−13.6781	−0.641241
$456$	0	0
$457$	34.3821	1.60833	0.804164	−	0.594407i	$-0.202613\pi$
$457$	34.3821	1.60833	0.804164	+	0.594407i	$0.202613\pi$
$458$	0	0
$459$	3.19006	0.148899
$460$	0	0
$461$	0.317693	0.0147964	0.00739822	−	0.999973i	$-0.497645\pi$
$461$	0.317693	0.0147964	0.00739822	+	0.999973i	$0.497645\pi$
$462$	0	0
$463$	−7.12574	−0.331161	−0.165581	−	0.986196i	$-0.552950\pi$
$463$	−7.12574	−0.331161	−0.165581	+	0.986196i	$0.552950\pi$
$464$	0	0
$465$	−14.4211	−0.668764
$466$	0	0
$467$	25.7928	1.19355	0.596774	−	0.802409i	$-0.296449\pi$
$467$	25.7928	1.19355	0.596774	+	0.802409i	$0.296449\pi$
$468$	0	0
$469$	−13.3317	−0.615601
$470$	0	0
$471$	8.37041	0.385688
$472$	0	0
$473$	3.11944	0.143432
$474$	0	0
$475$	−1.38773	−0.0636734
$476$	0	0
$477$	−0.700042	−0.0320527
$478$	0	0
$479$	3.45526	0.157875	0.0789374	−	0.996880i	$-0.474847\pi$
$479$	3.45526	0.157875	0.0789374	+	0.996880i	$0.474847\pi$
$480$	0	0
$481$	−3.30022	−0.150477
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.4331	−0.473743
$486$	0	0
$487$	−3.11298	−0.141062	−0.0705312	−	0.997510i	$-0.522469\pi$
$487$	−3.11298	−0.141062	−0.0705312	+	0.997510i	$0.522469\pi$
$488$	0	0
$489$	−9.79796	−0.443079
$490$	0	0
$491$	−40.4737	−1.82655	−0.913277	−	0.407339i	$-0.866457\pi$
$491$	−40.4737	−1.82655	−0.913277	+	0.407339i	$0.866457\pi$
$492$	0	0
$493$	20.1488	0.907456
$494$	0	0
$495$	−0.588780	−0.0264637
$496$	0	0
$497$	33.0674	1.48328
$498$	0	0
$499$	−3.79660	−0.169959	−0.0849796	−	0.996383i	$-0.527083\pi$
$499$	−3.79660	−0.169959	−0.0849796	+	0.996383i	$0.527083\pi$
$500$	0	0
$501$	11.5088	0.514175
$502$	0	0
$503$	30.0364	1.33926	0.669629	−	0.742696i	$-0.266453\pi$
$503$	30.0364	1.33926	0.669629	+	0.742696i	$0.266453\pi$
$504$	0	0
$505$	20.3209	0.904269
$506$	0	0
$507$	5.34824	0.237524
$508$	0	0
$509$	7.91819	0.350967	0.175484	−	0.984482i	$-0.443851\pi$
$509$	7.91819	0.350967	0.175484	+	0.984482i	$0.443851\pi$
$510$	0	0
$511$	27.5255	1.21766
$512$	0	0
$513$	−0.507939	−0.0224261
$514$	0	0
$515$	21.5124	0.947949
$516$	0	0
$517$	−4.62591	−0.203447
$518$	0	0
$519$	−13.9292	−0.611424
$520$	0	0
$521$	25.6327	1.12299	0.561495	−	0.827480i	$-0.310226\pi$
$521$	25.6327	1.12299	0.561495	+	0.827480i	$0.310226\pi$
$522$	0	0
$523$	−29.3297	−1.28250	−0.641250	−	0.767332i	$-0.721584\pi$
$523$	−29.3297	−1.28250	−0.641250	+	0.767332i	$0.721584\pi$
$524$	0	0
$525$	−8.97068	−0.391513
$526$	0	0
$527$	30.5481	1.33070
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−0.0143753	−0.000623836 0
$532$	0	0
$533$	−8.98095	−0.389008
$534$	0	0
$535$	23.2832	1.00662
$536$	0	0
$537$	10.7942	0.465804
$538$	0	0
$539$	1.47830	0.0636747
$540$	0	0
$541$	−22.0787	−0.949237	−0.474619	−	0.880192i	$-0.657414\pi$
$541$	−22.0787	−0.949237	−0.474619	+	0.880192i	$0.657414\pi$
$542$	0	0
$543$	21.5016	0.922721
$544$	0	0
$545$	20.9911	0.899158
$546$	0	0
$547$	9.86543	0.421816	0.210908	−	0.977506i	$-0.432358\pi$
$547$	9.86543	0.421816	0.210908	+	0.977506i	$0.432358\pi$
$548$	0	0
$549$	0.745517	0.0318179
$550$	0	0
$551$	−3.20820	−0.136674
$552$	0	0
$553$	−0.338185	−0.0143811
$554$	0	0
$555$	1.79671	0.0762659
$556$	0	0
$557$	12.7728	0.541202	0.270601	−	0.962692i	$-0.412778\pi$
$557$	12.7728	0.541202	0.270601	+	0.962692i	$0.412778\pi$
$558$	0	0
$559$	−22.0708	−0.933495
$560$	0	0
$561$	1.24721	0.0526571
$562$	0	0
$563$	23.1637	0.976233	0.488116	−	0.872779i	$-0.337684\pi$
$563$	23.1637	0.976233	0.488116	+	0.872779i	$0.337684\pi$
$564$	0	0
$565$	−25.7850	−1.08478
$566$	0	0
$567$	−3.28346	−0.137893
$568$	0	0
$569$	−14.6959	−0.616082	−0.308041	−	0.951373i	$-0.599673\pi$
$569$	−14.6959	−0.616082	−0.308041	+	0.951373i	$0.599673\pi$
$570$	0	0
$571$	−16.1324	−0.675119	−0.337560	−	0.941304i	$-0.609601\pi$
$571$	−16.1324	−0.675119	−0.337560	+	0.941304i	$0.609601\pi$
$572$	0	0
$573$	−24.8000	−1.03603
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−26.6042	−1.10755	−0.553774	−	0.832667i	$-0.686813\pi$
$577$	−26.6042	−1.10755	−0.553774	+	0.832667i	$0.686813\pi$
$578$	0	0
$579$	3.28393	0.136475
$580$	0	0
$581$	−20.1046	−0.834081
$582$	0	0
$583$	−0.273692	−0.0113352
$584$	0	0
$585$	4.16576	0.172233
$586$	0	0
$587$	−9.56562	−0.394816	−0.197408	−	0.980321i	$-0.563252\pi$
$587$	−9.56562	−0.394816	−0.197408	+	0.980321i	$0.563252\pi$
$588$	0	0
$589$	−4.86404	−0.200419
$590$	0	0
$591$	−27.3897	−1.12666
$592$	0	0
$593$	−22.2609	−0.914147	−0.457074	−	0.889429i	$-0.651102\pi$
$593$	−22.2609	−0.914147	−0.457074	+	0.889429i	$0.651102\pi$
$594$	0	0
$595$	−15.7741	−0.646677
$596$	0	0
$597$	−11.8414	−0.484637
$598$	0	0
$599$	−8.47121	−0.346124	−0.173062	−	0.984911i	$-0.555366\pi$
$599$	−8.47121	−0.346124	−0.173062	+	0.984911i	$0.555366\pi$
$600$	0	0
$601$	−29.8764	−1.21868	−0.609342	−	0.792907i	$-0.708566\pi$
$601$	−29.8764	−1.21868	−0.609342	+	0.792907i	$0.708566\pi$
$602$	0	0
$603$	4.06026	0.165346
$604$	0	0
$605$	16.3354	0.664128
$606$	0	0
$607$	−33.0548	−1.34165	−0.670827	−	0.741614i	$-0.734061\pi$
$607$	−33.0548	−1.34165	−0.670827	+	0.741614i	$0.734061\pi$
$608$	0	0
$609$	−20.7387	−0.840375
$610$	0	0
$611$	32.7295	1.32409
$612$	0	0
$613$	−7.51577	−0.303559	−0.151780	−	0.988414i	$-0.548500\pi$
$613$	−7.51577	−0.303559	−0.151780	+	0.988414i	$0.548500\pi$
$614$	0	0
$615$	4.88940	0.197160
$616$	0	0
$617$	−30.7579	−1.23827	−0.619134	−	0.785285i	$-0.712516\pi$
$617$	−30.7579	−1.23827	−0.619134	+	0.785285i	$0.712516\pi$
$618$	0	0
$619$	−42.8508	−1.72232	−0.861159	−	0.508335i	$-0.830261\pi$
$619$	−42.8508	−1.72232	−0.861159	+	0.508335i	$0.830261\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.455504	−0.0182494
$624$	0	0
$625$	−3.87536	−0.155015
$626$	0	0
$627$	−0.198587	−0.00793080
$628$	0	0
$629$	−3.80594	−0.151753
$630$	0	0
$631$	22.7175	0.904369	0.452184	−	0.891925i	$-0.350645\pi$
$631$	22.7175	0.904369	0.452184	+	0.891925i	$0.350645\pi$
$632$	0	0
$633$	−12.5281	−0.497946
$634$	0	0
$635$	−19.3251	−0.766891
$636$	0	0
$637$	−10.4593	−0.414413
$638$	0	0
$639$	−10.0709	−0.398398
$640$	0	0
$641$	19.3734	0.765202	0.382601	−	0.923914i	$-0.375028\pi$
$641$	19.3734	0.765202	0.382601	+	0.923914i	$0.375028\pi$
$642$	0	0
$643$	−21.9932	−0.867327	−0.433663	−	0.901075i	$-0.642779\pi$
$643$	−21.9932	−0.867327	−0.433663	+	0.901075i	$0.642779\pi$
$644$	0	0
$645$	12.0158	0.473120
$646$	0	0
$647$	28.0626	1.10326	0.551628	−	0.834090i	$-0.314007\pi$
$647$	28.0626	1.10326	0.551628	+	0.834090i	$0.314007\pi$
$648$	0	0
$649$	−0.00562026	−0.000220615 0
$650$	0	0
$651$	−31.4425	−1.23233
$652$	0	0
$653$	40.4936	1.58464	0.792319	−	0.610107i	$-0.208874\pi$
$653$	40.4936	1.58464	0.792319	+	0.610107i	$0.208874\pi$
$654$	0	0
$655$	−9.10909	−0.355922
$656$	0	0
$657$	−8.38306	−0.327054
$658$	0	0
$659$	3.37680	0.131542	0.0657708	−	0.997835i	$-0.479049\pi$
$659$	3.37680	0.131542	0.0657708	+	0.997835i	$0.479049\pi$
$660$	0	0
$661$	−39.8841	−1.55131	−0.775656	−	0.631156i	$-0.782581\pi$
$661$	−39.8841	−1.55131	−0.775656	+	0.631156i	$0.782581\pi$
$662$	0	0
$663$	−8.82430	−0.342707
$664$	0	0
$665$	2.51164	0.0973974
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−15.3136	−0.592057
$670$	0	0
$671$	0.291471	0.0112521
$672$	0	0
$673$	43.5630	1.67923	0.839615	−	0.543182i	$-0.182781\pi$
$673$	43.5630	1.67923	0.839615	+	0.543182i	$0.182781\pi$
$674$	0	0
$675$	2.73208	0.105158
$676$	0	0
$677$	−51.1071	−1.96420	−0.982102	−	0.188347i	$-0.939687\pi$
$677$	−51.1071	−1.96420	−0.982102	+	0.188347i	$0.939687\pi$
$678$	0	0
$679$	−22.7474	−0.872966
$680$	0	0
$681$	−16.4454	−0.630191
$682$	0	0
$683$	37.7470	1.44435	0.722175	−	0.691711i	$-0.243143\pi$
$683$	37.7470	1.44435	0.722175	+	0.691711i	$0.243143\pi$
$684$	0	0
$685$	−14.8551	−0.567584
$686$	0	0
$687$	−6.90633	−0.263493
$688$	0	0
$689$	1.93644	0.0737726
$690$	0	0
$691$	−15.1130	−0.574927	−0.287463	−	0.957792i	$-0.592812\pi$
$691$	−15.1130	−0.574927	−0.287463	+	0.957792i	$0.592812\pi$
$692$	0	0
$693$	−1.28372	−0.0487646
$694$	0	0
$695$	−27.4211	−1.04014
$696$	0	0
$697$	−10.3572	−0.392306
$698$	0	0
$699$	−24.8931	−0.941543
$700$	0	0
$701$	−1.46695	−0.0554060	−0.0277030	−	0.999616i	$-0.508819\pi$
$701$	−1.46695	−0.0554060	−0.0277030	+	0.999616i	$0.508819\pi$
$702$	0	0
$703$	0.606003	0.0228558
$704$	0	0
$705$	−17.8186	−0.671086
$706$	0	0
$707$	44.3059	1.66630
$708$	0	0
$709$	27.8379	1.04547	0.522737	−	0.852494i	$-0.324911\pi$
$709$	27.8379	1.04547	0.522737	+	0.852494i	$0.324911\pi$
$710$	0	0
$711$	0.102997	0.00386267
$712$	0	0
$713$	0	0
$714$	0	0
$715$	1.62867	0.0609088
$716$	0	0
$717$	−8.41669	−0.314327
$718$	0	0
$719$	−17.1116	−0.638154	−0.319077	−	0.947729i	$-0.603373\pi$
$719$	−17.1116	−0.638154	−0.319077	+	0.947729i	$0.603373\pi$
$720$	0	0
$721$	46.9037	1.74678
$722$	0	0
$723$	21.4774	0.798752
$724$	0	0
$725$	17.2561	0.640875
$726$	0	0
$727$	−8.84961	−0.328214	−0.164107	−	0.986443i	$-0.552474\pi$
$727$	−8.84961	−0.328214	−0.164107	+	0.986443i	$0.552474\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−25.4529	−0.941409
$732$	0	0
$733$	−28.1809	−1.04088	−0.520442	−	0.853897i	$-0.674233\pi$
$733$	−28.1809	−1.04088	−0.520442	+	0.853897i	$0.674233\pi$
$734$	0	0
$735$	5.69425	0.210036
$736$	0	0
$737$	1.58742	0.0584734
$738$	0	0
$739$	22.1296	0.814049	0.407025	−	0.913417i	$-0.366566\pi$
$739$	22.1296	0.814049	0.407025	+	0.913417i	$0.366566\pi$
$740$	0	0
$741$	1.40505	0.0516159
$742$	0	0
$743$	6.64578	0.243810	0.121905	−	0.992542i	$-0.461100\pi$
$743$	6.64578	0.243810	0.121905	+	0.992542i	$0.461100\pi$
$744$	0	0
$745$	21.7681	0.797520
$746$	0	0
$747$	6.12300	0.224029
$748$	0	0
$749$	50.7646	1.85490
$750$	0	0
$751$	17.4830	0.637963	0.318982	−	0.947761i	$-0.396659\pi$
$751$	17.4830	0.637963	0.318982	+	0.947761i	$0.396659\pi$
$752$	0	0
$753$	−9.57312	−0.348864
$754$	0	0
$755$	15.5131	0.564579
$756$	0	0
$757$	37.0936	1.34819	0.674095	−	0.738645i	$-0.264534\pi$
$757$	37.0936	1.34819	0.674095	+	0.738645i	$0.264534\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.494338	0.0179197	0.00895986	−	0.999960i	$-0.497148\pi$
$761$	0.494338	0.0179197	0.00895986	+	0.999960i	$0.497148\pi$
$762$	0	0
$763$	45.7670	1.65688
$764$	0	0
$765$	4.80412	0.173693
$766$	0	0
$767$	0.0397648	0.00143582
$768$	0	0
$769$	−32.2653	−1.16352	−0.581758	−	0.813362i	$-0.697635\pi$
$769$	−32.2653	−1.16352	−0.581758	+	0.813362i	$0.697635\pi$
$770$	0	0
$771$	−0.630921	−0.0227221
$772$	0	0
$773$	−29.2168	−1.05085	−0.525427	−	0.850839i	$-0.676094\pi$
$773$	−29.2168	−1.05085	−0.525427	+	0.850839i	$0.676094\pi$
$774$	0	0
$775$	26.1624	0.939782
$776$	0	0
$777$	3.91738	0.140535
$778$	0	0
$779$	1.64913	0.0590860
$780$	0	0
$781$	−3.93737	−0.140890
$782$	0	0
$783$	6.31611	0.225719
$784$	0	0
$785$	12.6055	0.449910
$786$	0	0
$787$	39.1296	1.39482	0.697410	−	0.716672i	$-0.254335\pi$
$787$	39.1296	1.39482	0.697410	+	0.716672i	$0.254335\pi$
$788$	0	0
$789$	−22.2949	−0.793721
$790$	0	0
$791$	−56.2194	−1.99893
$792$	0	0
$793$	−2.06223	−0.0732321
$794$	0	0
$795$	−1.05424	−0.0373899
$796$	0	0
$797$	31.1794	1.10443	0.552215	−	0.833702i	$-0.313783\pi$
$797$	31.1794	1.10443	0.552215	+	0.833702i	$0.313783\pi$
$798$	0	0
$799$	37.7449	1.33532
$800$	0	0
$801$	0.138727	0.00490166
$802$	0	0
$803$	−3.27749	−0.115660
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.2879	0.890176
$808$	0	0
$809$	−20.4213	−0.717974	−0.358987	−	0.933343i	$-0.616878\pi$
$809$	−20.4213	−0.717974	−0.358987	+	0.933343i	$0.616878\pi$
$810$	0	0
$811$	−32.5888	−1.14435	−0.572173	−	0.820133i	$-0.693899\pi$
$811$	−32.5888	−1.14435	−0.572173	+	0.820133i	$0.693899\pi$
$812$	0	0
$813$	−10.8247	−0.379640
$814$	0	0
$815$	−14.7554	−0.516858
$816$	0	0
$817$	4.05274	0.141788
$818$	0	0
$819$	9.08266	0.317374
$820$	0	0
$821$	−28.8184	−1.00577	−0.502885	−	0.864354i	$-0.667728\pi$
$821$	−28.8184	−1.00577	−0.502885	+	0.864354i	$0.667728\pi$
$822$	0	0
$823$	−2.47289	−0.0861996	−0.0430998	−	0.999071i	$-0.513723\pi$
$823$	−2.47289	−0.0861996	−0.0430998	+	0.999071i	$0.513723\pi$
$824$	0	0
$825$	1.06815	0.0371882
$826$	0	0
$827$	−12.3320	−0.428827	−0.214414	−	0.976743i	$-0.568784\pi$
$827$	−12.3320	−0.428827	−0.214414	+	0.976743i	$0.568784\pi$
$828$	0	0
$829$	14.1559	0.491654	0.245827	−	0.969314i	$-0.420940\pi$
$829$	14.1559	0.491654	0.245827	+	0.969314i	$0.420940\pi$
$830$	0	0
$831$	−24.7512	−0.858611
$832$	0	0
$833$	−12.0621	−0.417926
$834$	0	0
$835$	17.3318	0.599792
$836$	0	0
$837$	9.57602	0.330996
$838$	0	0
$839$	−27.6153	−0.953387	−0.476693	−	0.879070i	$-0.658165\pi$
$839$	−27.6153	−0.953387	−0.476693	+	0.879070i	$0.658165\pi$
$840$	0	0
$841$	10.8932	0.375628
$842$	0	0
$843$	1.90741	0.0656946
$844$	0	0
$845$	8.05425	0.277075
$846$	0	0
$847$	35.6162	1.22379
$848$	0	0
$849$	−0.935738	−0.0321144
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.1725	0.348301	0.174150	−	0.984719i	$-0.444282\pi$
$853$	10.1725	0.348301	0.174150	+	0.984719i	$0.444282\pi$
$854$	0	0
$855$	−0.764937	−0.0261603
$856$	0	0
$857$	15.9562	0.545055	0.272527	−	0.962148i	$-0.412141\pi$
$857$	15.9562	0.545055	0.272527	+	0.962148i	$0.412141\pi$
$858$	0	0
$859$	29.8403	1.01814	0.509069	−	0.860726i	$-0.329990\pi$
$859$	29.8403	1.01814	0.509069	+	0.860726i	$0.329990\pi$
$860$	0	0
$861$	10.6604	0.363306
$862$	0	0
$863$	−7.96875	−0.271259	−0.135630	−	0.990760i	$-0.543306\pi$
$863$	−7.96875	−0.271259	−0.135630	+	0.990760i	$0.543306\pi$
$864$	0	0
$865$	−20.9769	−0.713235
$866$	0	0
$867$	6.82349	0.231738
$868$	0	0
$869$	0.0402681	0.00136600
$870$	0	0
$871$	−11.2314	−0.380562
$872$	0	0
$873$	6.92787	0.234473
$874$	0	0
$875$	−38.2334	−1.29252
$876$	0	0
$877$	−6.40930	−0.216427	−0.108213	−	0.994128i	$-0.534513\pi$
$877$	−6.40930	−0.216427	−0.108213	+	0.994128i	$0.534513\pi$
$878$	0	0
$879$	23.4028	0.789358
$880$	0	0
$881$	28.3974	0.956734	0.478367	−	0.878160i	$-0.341229\pi$
$881$	28.3974	0.956734	0.478367	+	0.878160i	$0.341229\pi$
$882$	0	0
$883$	−36.4959	−1.22818	−0.614092	−	0.789234i	$-0.710478\pi$
$883$	−36.4959	−1.22818	−0.614092	+	0.789234i	$0.710478\pi$
$884$	0	0
$885$	−0.0216487	−0.000727713 0
$886$	0	0
$887$	0.360845	0.0121160	0.00605799	−	0.999982i	$-0.498072\pi$
$887$	0.360845	0.0121160	0.00605799	+	0.999982i	$0.498072\pi$
$888$	0	0
$889$	−42.1346	−1.41315
$890$	0	0
$891$	0.390966	0.0130978
$892$	0	0
$893$	−6.00995	−0.201115
$894$	0	0
$895$	16.2556	0.543366
$896$	0	0
$897$	0	0
$898$	0	0
$899$	60.4832	2.01723
$900$	0	0
$901$	2.23318	0.0743980
$902$	0	0
$903$	26.1981	0.871818
$904$	0	0
$905$	32.3806	1.07637
$906$	0	0
$907$	−23.7931	−0.790036	−0.395018	−	0.918673i	$-0.629262\pi$
$907$	−23.7931	−0.790036	−0.395018	+	0.918673i	$0.629262\pi$
$908$	0	0
$909$	−13.4937	−0.447556
$910$	0	0
$911$	−34.5837	−1.14581	−0.572905	−	0.819622i	$-0.694184\pi$
$911$	−34.5837	−1.14581	−0.572905	+	0.819622i	$0.694184\pi$
$912$	0	0
$913$	2.39388	0.0792259
$914$	0	0
$915$	1.12272	0.0371160
$916$	0	0
$917$	−19.8606	−0.655856
$918$	0	0
$919$	−48.9646	−1.61519	−0.807596	−	0.589736i	$-0.799232\pi$
$919$	−48.9646	−1.61519	−0.807596	+	0.589736i	$0.799232\pi$
$920$	0	0
$921$	28.3393	0.933813
$922$	0	0
$923$	27.8579	0.916953
$924$	0	0
$925$	−3.25954	−0.107173
$926$	0	0
$927$	−14.2848	−0.469175
$928$	0	0
$929$	30.8197	1.01116	0.505581	−	0.862779i	$-0.331278\pi$
$929$	30.8197	1.01116	0.505581	+	0.862779i	$0.331278\pi$
$930$	0	0
$931$	1.92059	0.0629448
$932$	0	0
$933$	−18.1727	−0.594948
$934$	0	0
$935$	1.87825	0.0614252
$936$	0	0
$937$	23.7170	0.774802	0.387401	−	0.921911i	$-0.373373\pi$
$937$	23.7170	0.774802	0.387401	+	0.921911i	$0.373373\pi$
$938$	0	0
$939$	−4.52686	−0.147728
$940$	0	0
$941$	−17.5411	−0.571824	−0.285912	−	0.958256i	$-0.592297\pi$
$941$	−17.5411	−0.571824	−0.285912	+	0.958256i	$0.592297\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.94477	−0.160854
$946$	0	0
$947$	−24.6474	−0.800931	−0.400466	−	0.916312i	$-0.631152\pi$
$947$	−24.6474	−0.800931	−0.400466	+	0.916312i	$0.631152\pi$
$948$	0	0
$949$	23.1891	0.752749
$950$	0	0
$951$	2.72256	0.0882851
$952$	0	0
$953$	−8.40890	−0.272391	−0.136196	−	0.990682i	$-0.543488\pi$
$953$	−8.40890	−0.272391	−0.136196	+	0.990682i	$0.543488\pi$
$954$	0	0
$955$	−37.3478	−1.20855
$956$	0	0
$957$	2.46938	0.0798238
$958$	0	0
$959$	−32.3887	−1.04589
$960$	0	0
$961$	60.7002	1.95807
$962$	0	0
$963$	−15.4607	−0.498214
$964$	0	0
$965$	4.94547	0.159200
$966$	0	0
$967$	29.6971	0.954993	0.477497	−	0.878634i	$-0.341544\pi$
$967$	29.6971	0.954993	0.477497	+	0.878634i	$0.341544\pi$
$968$	0	0
$969$	1.62036	0.0520534
$970$	0	0
$971$	11.4460	0.367321	0.183661	−	0.982990i	$-0.441205\pi$
$971$	11.4460	0.367321	0.183661	+	0.982990i	$0.441205\pi$
$972$	0	0
$973$	−59.7865	−1.91667
$974$	0	0
$975$	−7.55742	−0.242031
$976$	0	0
$977$	22.2744	0.712620	0.356310	−	0.934368i	$-0.384035\pi$
$977$	22.2744	0.712620	0.356310	+	0.934368i	$0.384035\pi$
$978$	0	0
$979$	0.0542373	0.00173343
$980$	0	0
$981$	−13.9386	−0.445026
$982$	0	0
$983$	23.8386	0.760334	0.380167	−	0.924918i	$-0.375867\pi$
$983$	23.8386	0.760334	0.380167	+	0.924918i	$0.375867\pi$
$984$	0	0
$985$	−41.2478	−1.31427
$986$	0	0
$987$	−38.8500	−1.23661
$988$	0	0
$989$	0	0
$990$	0	0
$991$	23.4218	0.744019	0.372010	−	0.928229i	$-0.378669\pi$
$991$	23.4218	0.744019	0.372010	+	0.928229i	$0.378669\pi$
$992$	0	0
$993$	−3.23888	−0.102783
$994$	0	0
$995$	−17.8327	−0.565335
$996$	0	0
$997$	21.4747	0.680111	0.340055	−	0.940405i	$-0.389554\pi$
$997$	21.4747	0.680111	0.340055	+	0.940405i	$0.389554\pi$
$998$	0	0
$999$	−1.19306	−0.0377468

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.r.1.3		10
23.9	even	11		276.2.i.b.265.2	yes	20
23.18	even	11		276.2.i.b.25.2	✓	20
23.22	odd	2		6348.2.a.q.1.8		10
69.32	odd	22		828.2.q.b.541.1		20
69.41	odd	22		828.2.q.b.577.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.b.25.2	✓	20	23.18	even	11
276.2.i.b.265.2	yes	20	23.9	even	11
828.2.q.b.541.1		20	69.32	odd	22
828.2.q.b.577.1		20	69.41	odd	22
6348.2.a.q.1.8		10	23.22	odd	2
6348.2.a.r.1.3		10	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.50596 q^{5} -3.28346 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.50596 q^{5} -3.28346 q^{7} +1.00000 q^{9} +0.390966 q^{11} -2.76618 q^{13} +1.50596 q^{15} -3.19006 q^{17} +0.507939 q^{19} +3.28346 q^{21} -2.73208 q^{25} -1.00000 q^{27} -6.31611 q^{29} -9.57602 q^{31} -0.390966 q^{33} +4.94477 q^{35} +1.19306 q^{37} +2.76618 q^{39} +3.24670 q^{41} +7.97880 q^{43} -1.50596 q^{45} -11.8320 q^{47} +3.78114 q^{49} +3.19006 q^{51} -0.700042 q^{53} -0.588780 q^{55} -0.507939 q^{57} -0.0143753 q^{59} +0.745517 q^{61} -3.28346 q^{63} +4.16576 q^{65} +4.06026 q^{67} -10.0709 q^{71} -8.38306 q^{73} +2.73208 q^{75} -1.28372 q^{77} +0.102997 q^{79} +1.00000 q^{81} +6.12300 q^{83} +4.80412 q^{85} +6.31611 q^{87} +0.138727 q^{89} +9.08266 q^{91} +9.57602 q^{93} -0.764937 q^{95} +6.92787 q^{97} +0.390966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 9 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 9 * q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 9 q^{7} + 10 q^{9} + 11 q^{11} - 4 q^{13} + 7 q^{17} - 2 q^{19} - 9 q^{21} + 6 q^{25} - 10 q^{27} + 9 q^{29} + 11 q^{31} - 11 q^{33} + 43 q^{35} - 9 q^{37} + 4 q^{39} + 2 q^{41} + 24 q^{43} - 27 q^{47} + 37 q^{49} - 7 q^{51} + 4 q^{53} - 22 q^{55} + 2 q^{57} + 2 q^{59} - 27 q^{61} + 9 q^{63} + 22 q^{65} + 24 q^{67} + 11 q^{71} + 2 q^{73} - 6 q^{75} - 11 q^{77} + 23 q^{79} + 10 q^{81} + 37 q^{83} - q^{85} - 9 q^{87} + 11 q^{89} - 8 q^{91} - 11 q^{93} - 10 q^{95} + 10 q^{97} + 11 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 9 * q^7 + 10 * q^9 + 11 * q^11 - 4 * q^13 + 7 * q^17 - 2 * q^19 - 9 * q^21 + 6 * q^25 - 10 * q^27 + 9 * q^29 + 11 * q^31 - 11 * q^33 + 43 * q^35 - 9 * q^37 + 4 * q^39 + 2 * q^41 + 24 * q^43 - 27 * q^47 + 37 * q^49 - 7 * q^51 + 4 * q^53 - 22 * q^55 + 2 * q^57 + 2 * q^59 - 27 * q^61 + 9 * q^63 + 22 * q^65 + 24 * q^67 + 11 * q^71 + 2 * q^73 - 6 * q^75 - 11 * q^77 + 23 * q^79 + 10 * q^81 + 37 * q^83 - q^85 - 9 * q^87 + 11 * q^89 - 8 * q^91 - 11 * q^93 - 10 * q^95 + 10 * q^97 + 11 * q^99

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