LMFDB - Embedded newform 6348.2.a.s.1.9

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			1.9
Root			$-1.92165$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.68761 q^{5} -0.168413 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.68761 q^{5} -0.168413 q^{7} +1.00000 q^{9} -5.32400 q^{11} +3.84191 q^{13} +2.68761 q^{15} -6.65709 q^{17} -6.89290 q^{19} -0.168413 q^{21} +2.22326 q^{25} +1.00000 q^{27} -0.569888 q^{29} -6.95948 q^{31} -5.32400 q^{33} -0.452628 q^{35} +8.85711 q^{37} +3.84191 q^{39} +4.86755 q^{41} -6.75736 q^{43} +2.68761 q^{45} -2.26730 q^{47} -6.97164 q^{49} -6.65709 q^{51} +4.87576 q^{53} -14.3088 q^{55} -6.89290 q^{57} +2.28099 q^{59} -13.9609 q^{61} -0.168413 q^{63} +10.3255 q^{65} -1.61122 q^{67} -9.49666 q^{71} +3.83123 q^{73} +2.22326 q^{75} +0.896630 q^{77} +2.62588 q^{79} +1.00000 q^{81} -8.96954 q^{83} -17.8917 q^{85} -0.569888 q^{87} +6.28643 q^{89} -0.647026 q^{91} -6.95948 q^{93} -18.5254 q^{95} -13.2783 q^{97} -5.32400 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.68761	1.20194	0.600968	−	0.799273i	$-0.294782\pi$
$5$	2.68761	1.20194	0.600968	+	0.799273i	$0.294782\pi$
$6$	0	0
$7$	−0.168413	−0.0636541	−0.0318270	−	0.999493i	$-0.510133\pi$
$7$	−0.168413	−0.0636541	−0.0318270	+	0.999493i	$0.510133\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.32400	−1.60525	−0.802623	−	0.596487i	$-0.796563\pi$
$11$	−5.32400	−1.60525	−0.802623	+	0.596487i	$0.796563\pi$
$12$	0	0
$13$	3.84191	1.06555	0.532776	−	0.846256i	$-0.321149\pi$
$13$	3.84191	1.06555	0.532776	+	0.846256i	$0.321149\pi$
$14$	0	0
$15$	2.68761	0.693938
$16$	0	0
$17$	−6.65709	−1.61458	−0.807291	−	0.590154i	$-0.799067\pi$
$17$	−6.65709	−1.61458	−0.807291	+	0.590154i	$0.799067\pi$
$18$	0	0
$19$	−6.89290	−1.58134	−0.790669	−	0.612243i	$-0.790267\pi$
$19$	−6.89290	−1.58134	−0.790669	+	0.612243i	$0.790267\pi$
$20$	0	0
$21$	−0.168413	−0.0367507
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	2.22326	0.444651
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.569888	−0.105826	−0.0529128	−	0.998599i	$-0.516851\pi$
$29$	−0.569888	−0.105826	−0.0529128	+	0.998599i	$0.516851\pi$
$30$	0	0
$31$	−6.95948	−1.24996	−0.624980	−	0.780641i	$-0.714893\pi$
$31$	−6.95948	−1.24996	−0.624980	+	0.780641i	$0.714893\pi$
$32$	0	0
$33$	−5.32400	−0.926789
$34$	0	0
$35$	−0.452628	−0.0765082
$36$	0	0
$37$	8.85711	1.45610	0.728050	−	0.685524i	$-0.240427\pi$
$37$	8.85711	1.45610	0.728050	+	0.685524i	$0.240427\pi$
$38$	0	0
$39$	3.84191	0.615197
$40$	0	0
$41$	4.86755	0.760183	0.380092	−	0.924949i	$-0.375892\pi$
$41$	4.86755	0.760183	0.380092	+	0.924949i	$0.375892\pi$
$42$	0	0
$43$	−6.75736	−1.03049	−0.515244	−	0.857044i	$-0.672299\pi$
$43$	−6.75736	−1.03049	−0.515244	+	0.857044i	$0.672299\pi$
$44$	0	0
$45$	2.68761	0.400645
$46$	0	0
$47$	−2.26730	−0.330720	−0.165360	−	0.986233i	$-0.552879\pi$
$47$	−2.26730	−0.330720	−0.165360	+	0.986233i	$0.552879\pi$
$48$	0	0
$49$	−6.97164	−0.995948
$50$	0	0
$51$	−6.65709	−0.932179
$52$	0	0
$53$	4.87576	0.669737	0.334868	−	0.942265i	$-0.391308\pi$
$53$	4.87576	0.669737	0.334868	+	0.942265i	$0.391308\pi$
$54$	0	0
$55$	−14.3088	−1.92940
$56$	0	0
$57$	−6.89290	−0.912986
$58$	0	0
$59$	2.28099	0.296959	0.148480	−	0.988915i	$-0.452562\pi$
$59$	2.28099	0.296959	0.148480	+	0.988915i	$0.452562\pi$
$60$	0	0
$61$	−13.9609	−1.78751	−0.893757	−	0.448551i	$-0.851940\pi$
$61$	−13.9609	−1.78751	−0.893757	+	0.448551i	$0.851940\pi$
$62$	0	0
$63$	−0.168413	−0.0212180
$64$	0	0
$65$	10.3255	1.28073
$66$	0	0
$67$	−1.61122	−0.196842	−0.0984211	−	0.995145i	$-0.531379\pi$
$67$	−1.61122	−0.196842	−0.0984211	+	0.995145i	$0.531379\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.49666	−1.12705	−0.563523	−	0.826100i	$-0.690554\pi$
$71$	−9.49666	−1.12705	−0.563523	+	0.826100i	$0.690554\pi$
$72$	0	0
$73$	3.83123	0.448411	0.224206	−	0.974542i	$-0.428021\pi$
$73$	3.83123	0.448411	0.224206	+	0.974542i	$0.428021\pi$
$74$	0	0
$75$	2.22326	0.256719
$76$	0	0
$77$	0.896630	0.102180
$78$	0	0
$79$	2.62588	0.295435	0.147717	−	0.989030i	$-0.452807\pi$
$79$	2.62588	0.295435	0.147717	+	0.989030i	$0.452807\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.96954	−0.984535	−0.492268	−	0.870444i	$-0.663832\pi$
$83$	−8.96954	−0.984535	−0.492268	+	0.870444i	$0.663832\pi$
$84$	0	0
$85$	−17.8917	−1.94062
$86$	0	0
$87$	−0.569888	−0.0610984
$88$	0	0
$89$	6.28643	0.666361	0.333180	−	0.942863i	$-0.391878\pi$
$89$	6.28643	0.666361	0.333180	+	0.942863i	$0.391878\pi$
$90$	0	0
$91$	−0.647026	−0.0678268
$92$	0	0
$93$	−6.95948	−0.721665
$94$	0	0
$95$	−18.5254	−1.90067
$96$	0	0
$97$	−13.2783	−1.34821	−0.674105	−	0.738635i	$-0.735471\pi$
$97$	−13.2783	−1.34821	−0.674105	+	0.738635i	$0.735471\pi$
$98$	0	0
$99$	−5.32400	−0.535082
$100$	0	0
$101$	−7.26663	−0.723056	−0.361528	−	0.932361i	$-0.617745\pi$
$101$	−7.26663	−0.723056	−0.361528	+	0.932361i	$0.617745\pi$
$102$	0	0
$103$	−4.28252	−0.421969	−0.210984	−	0.977489i	$-0.567667\pi$
$103$	−4.28252	−0.421969	−0.210984	+	0.977489i	$0.567667\pi$
$104$	0	0
$105$	−0.452628	−0.0441720
$106$	0	0
$107$	9.71161	0.938857	0.469428	−	0.882971i	$-0.344460\pi$
$107$	9.71161	0.938857	0.469428	+	0.882971i	$0.344460\pi$
$108$	0	0
$109$	−8.85005	−0.847681	−0.423841	−	0.905737i	$-0.639318\pi$
$109$	−8.85005	−0.847681	−0.423841	+	0.905737i	$0.639318\pi$
$110$	0	0
$111$	8.85711	0.840679
$112$	0	0
$113$	8.54541	0.803884	0.401942	−	0.915665i	$-0.368335\pi$
$113$	8.54541	0.803884	0.401942	+	0.915665i	$0.368335\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.84191	0.355184
$118$	0	0
$119$	1.12114	0.102775
$120$	0	0
$121$	17.3450	1.57681
$122$	0	0
$123$	4.86755	0.438892
$124$	0	0
$125$	−7.46281	−0.667494
$126$	0	0
$127$	−15.1659	−1.34575	−0.672877	−	0.739755i	$-0.734942\pi$
$127$	−15.1659	−1.34575	−0.672877	+	0.739755i	$0.734942\pi$
$128$	0	0
$129$	−6.75736	−0.594952
$130$	0	0
$131$	−1.75962	−0.153739	−0.0768696	−	0.997041i	$-0.524493\pi$
$131$	−1.75962	−0.153739	−0.0768696	+	0.997041i	$0.524493\pi$
$132$	0	0
$133$	1.16085	0.100659
$134$	0	0
$135$	2.68761	0.231313
$136$	0	0
$137$	14.6865	1.25475	0.627375	−	0.778717i	$-0.284129\pi$
$137$	14.6865	1.25475	0.627375	+	0.778717i	$0.284129\pi$
$138$	0	0
$139$	22.3528	1.89594	0.947972	−	0.318355i	$-0.103130\pi$
$139$	22.3528	1.89594	0.947972	+	0.318355i	$0.103130\pi$
$140$	0	0
$141$	−2.26730	−0.190941
$142$	0	0
$143$	−20.4543	−1.71047
$144$	0	0
$145$	−1.53164	−0.127196
$146$	0	0
$147$	−6.97164	−0.575011
$148$	0	0
$149$	−6.86314	−0.562250	−0.281125	−	0.959671i	$-0.590708\pi$
$149$	−6.86314	−0.562250	−0.281125	+	0.959671i	$0.590708\pi$
$150$	0	0
$151$	−1.51838	−0.123564	−0.0617820	−	0.998090i	$-0.519678\pi$
$151$	−1.51838	−0.123564	−0.0617820	+	0.998090i	$0.519678\pi$
$152$	0	0
$153$	−6.65709	−0.538194
$154$	0	0
$155$	−18.7044	−1.50237
$156$	0	0
$157$	−6.59043	−0.525974	−0.262987	−	0.964799i	$-0.584708\pi$
$157$	−6.59043	−0.525974	−0.262987	+	0.964799i	$0.584708\pi$
$158$	0	0
$159$	4.87576	0.386673
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.45066	0.191950	0.0959751	−	0.995384i	$-0.469403\pi$
$163$	2.45066	0.191950	0.0959751	+	0.995384i	$0.469403\pi$
$164$	0	0
$165$	−14.3088	−1.11394
$166$	0	0
$167$	10.3057	0.797479	0.398740	−	0.917064i	$-0.369448\pi$
$167$	10.3057	0.797479	0.398740	+	0.917064i	$0.369448\pi$
$168$	0	0
$169$	1.76024	0.135403
$170$	0	0
$171$	−6.89290	−0.527113
$172$	0	0
$173$	−3.01452	−0.229190	−0.114595	−	0.993412i	$-0.536557\pi$
$173$	−3.01452	−0.229190	−0.114595	+	0.993412i	$0.536557\pi$
$174$	0	0
$175$	−0.374425	−0.0283039
$176$	0	0
$177$	2.28099	0.171449
$178$	0	0
$179$	−13.5122	−1.00995	−0.504973	−	0.863135i	$-0.668498\pi$
$179$	−13.5122	−1.00995	−0.504973	+	0.863135i	$0.668498\pi$
$180$	0	0
$181$	13.8996	1.03315	0.516575	−	0.856242i	$-0.327207\pi$
$181$	13.8996	1.03315	0.516575	+	0.856242i	$0.327207\pi$
$182$	0	0
$183$	−13.9609	−1.03202
$184$	0	0
$185$	23.8045	1.75014
$186$	0	0
$187$	35.4423	2.59180
$188$	0	0
$189$	−0.168413	−0.0122502
$190$	0	0
$191$	12.7981	0.926035	0.463018	−	0.886349i	$-0.346767\pi$
$191$	12.7981	0.926035	0.463018	+	0.886349i	$0.346767\pi$
$192$	0	0
$193$	16.1381	1.16164	0.580822	−	0.814031i	$-0.302731\pi$
$193$	16.1381	1.16164	0.580822	+	0.814031i	$0.302731\pi$
$194$	0	0
$195$	10.3255	0.739428
$196$	0	0
$197$	13.1747	0.938661	0.469330	−	0.883023i	$-0.344495\pi$
$197$	13.1747	0.938661	0.469330	+	0.883023i	$0.344495\pi$
$198$	0	0
$199$	−9.25611	−0.656149	−0.328074	−	0.944652i	$-0.606400\pi$
$199$	−9.25611	−0.656149	−0.328074	+	0.944652i	$0.606400\pi$
$200$	0	0
$201$	−1.61122	−0.113647
$202$	0	0
$203$	0.0959765	0.00673623
$204$	0	0
$205$	13.0821	0.913692
$206$	0	0
$207$	0	0
$208$	0	0
$209$	36.6978	2.53844
$210$	0	0
$211$	−14.7860	−1.01791	−0.508955	−	0.860793i	$-0.669968\pi$
$211$	−14.7860	−1.01791	−0.508955	+	0.860793i	$0.669968\pi$
$212$	0	0
$213$	−9.49666	−0.650700
$214$	0	0
$215$	−18.1612	−1.23858
$216$	0	0
$217$	1.17207	0.0795650
$218$	0	0
$219$	3.83123	0.258890
$220$	0	0
$221$	−25.5759	−1.72042
$222$	0	0
$223$	14.9643	1.00209	0.501043	−	0.865422i	$-0.332950\pi$
$223$	14.9643	1.00209	0.501043	+	0.865422i	$0.332950\pi$
$224$	0	0
$225$	2.22326	0.148217
$226$	0	0
$227$	19.3027	1.28117	0.640583	−	0.767889i	$-0.278693\pi$
$227$	19.3027	1.28117	0.640583	+	0.767889i	$0.278693\pi$
$228$	0	0
$229$	−26.6442	−1.76070	−0.880349	−	0.474326i	$-0.842692\pi$
$229$	−26.6442	−1.76070	−0.880349	+	0.474326i	$0.842692\pi$
$230$	0	0
$231$	0.896630	0.0589939
$232$	0	0
$233$	−14.4481	−0.946527	−0.473263	−	0.880921i	$-0.656924\pi$
$233$	−14.4481	−0.946527	−0.473263	+	0.880921i	$0.656924\pi$
$234$	0	0
$235$	−6.09362	−0.397504
$236$	0	0
$237$	2.62588	0.170569
$238$	0	0
$239$	−24.1116	−1.55965	−0.779823	−	0.626000i	$-0.784691\pi$
$239$	−24.1116	−1.55965	−0.779823	+	0.626000i	$0.784691\pi$
$240$	0	0
$241$	−11.2125	−0.722259	−0.361129	−	0.932516i	$-0.617609\pi$
$241$	−11.2125	−0.722259	−0.361129	+	0.932516i	$0.617609\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−18.7371	−1.19707
$246$	0	0
$247$	−26.4819	−1.68500
$248$	0	0
$249$	−8.96954	−0.568422
$250$	0	0
$251$	−28.6735	−1.80986	−0.904929	−	0.425563i	$-0.860076\pi$
$251$	−28.6735	−1.80986	−0.904929	+	0.425563i	$0.860076\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−17.8917	−1.12042
$256$	0	0
$257$	18.6188	1.16141	0.580705	−	0.814114i	$-0.302777\pi$
$257$	18.6188	1.16141	0.580705	+	0.814114i	$0.302777\pi$
$258$	0	0
$259$	−1.49165	−0.0926867
$260$	0	0
$261$	−0.569888	−0.0352752
$262$	0	0
$263$	−14.3106	−0.882427	−0.441214	−	0.897402i	$-0.645452\pi$
$263$	−14.3106	−0.882427	−0.441214	+	0.897402i	$0.645452\pi$
$264$	0	0
$265$	13.1041	0.804981
$266$	0	0
$267$	6.28643	0.384724
$268$	0	0
$269$	−7.01605	−0.427776	−0.213888	−	0.976858i	$-0.568613\pi$
$269$	−7.01605	−0.427776	−0.213888	+	0.976858i	$0.568613\pi$
$270$	0	0
$271$	10.6915	0.649462	0.324731	−	0.945806i	$-0.394726\pi$
$271$	10.6915	0.649462	0.324731	+	0.945806i	$0.394726\pi$
$272$	0	0
$273$	−0.647026	−0.0391598
$274$	0	0
$275$	−11.8366	−0.713774
$276$	0	0
$277$	22.1540	1.33111	0.665553	−	0.746351i	$-0.268196\pi$
$277$	22.1540	1.33111	0.665553	+	0.746351i	$0.268196\pi$
$278$	0	0
$279$	−6.95948	−0.416653
$280$	0	0
$281$	7.06273	0.421327	0.210664	−	0.977559i	$-0.432438\pi$
$281$	7.06273	0.421327	0.210664	+	0.977559i	$0.432438\pi$
$282$	0	0
$283$	26.9335	1.60103	0.800515	−	0.599313i	$-0.204559\pi$
$283$	26.9335	1.60103	0.800515	+	0.599313i	$0.204559\pi$
$284$	0	0
$285$	−18.5254	−1.09735
$286$	0	0
$287$	−0.819758	−0.0483888
$288$	0	0
$289$	27.3168	1.60687
$290$	0	0
$291$	−13.2783	−0.778390
$292$	0	0
$293$	28.5308	1.66679	0.833393	−	0.552681i	$-0.186395\pi$
$293$	28.5308	1.66679	0.833393	+	0.552681i	$0.186395\pi$
$294$	0	0
$295$	6.13041	0.356926
$296$	0	0
$297$	−5.32400	−0.308930
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.13803	0.0655948
$302$	0	0
$303$	−7.26663	−0.417457
$304$	0	0
$305$	−37.5216	−2.14848
$306$	0	0
$307$	−11.6303	−0.663774	−0.331887	−	0.943319i	$-0.607685\pi$
$307$	−11.6303	−0.663774	−0.331887	+	0.943319i	$0.607685\pi$
$308$	0	0
$309$	−4.28252	−0.243624
$310$	0	0
$311$	26.2524	1.48864	0.744319	−	0.667825i	$-0.232774\pi$
$311$	26.2524	1.48864	0.744319	+	0.667825i	$0.232774\pi$
$312$	0	0
$313$	−25.6165	−1.44793	−0.723964	−	0.689837i	$-0.757682\pi$
$313$	−25.6165	−1.44793	−0.723964	+	0.689837i	$0.757682\pi$
$314$	0	0
$315$	−0.452628	−0.0255027
$316$	0	0
$317$	20.1963	1.13434	0.567169	−	0.823602i	$-0.308039\pi$
$317$	20.1963	1.13434	0.567169	+	0.823602i	$0.308039\pi$
$318$	0	0
$319$	3.03408	0.169876
$320$	0	0
$321$	9.71161	0.542049
$322$	0	0
$323$	45.8866	2.55320
$324$	0	0
$325$	8.54154	0.473799
$326$	0	0
$327$	−8.85005	−0.489409
$328$	0	0
$329$	0.381843	0.0210517
$330$	0	0
$331$	−21.5420	−1.18406	−0.592028	−	0.805917i	$-0.701673\pi$
$331$	−21.5420	−1.18406	−0.592028	+	0.805917i	$0.701673\pi$
$332$	0	0
$333$	8.85711	0.485366
$334$	0	0
$335$	−4.33034	−0.236592
$336$	0	0
$337$	1.95917	0.106723	0.0533615	−	0.998575i	$-0.483006\pi$
$337$	1.95917	0.106723	0.0533615	+	0.998575i	$0.483006\pi$
$338$	0	0
$339$	8.54541	0.464123
$340$	0	0
$341$	37.0523	2.00649
$342$	0	0
$343$	2.35300	0.127050
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.75901	0.523891	0.261946	−	0.965083i	$-0.415636\pi$
$347$	9.75901	0.523891	0.261946	+	0.965083i	$0.415636\pi$
$348$	0	0
$349$	−16.3641	−0.875951	−0.437975	−	0.898987i	$-0.644304\pi$
$349$	−16.3641	−0.875951	−0.437975	+	0.898987i	$0.644304\pi$
$350$	0	0
$351$	3.84191	0.205066
$352$	0	0
$353$	−18.8249	−1.00195	−0.500975	−	0.865462i	$-0.667025\pi$
$353$	−18.8249	−1.00195	−0.500975	+	0.865462i	$0.667025\pi$
$354$	0	0
$355$	−25.5233	−1.35464
$356$	0	0
$357$	1.12114	0.0593370
$358$	0	0
$359$	−13.1714	−0.695158	−0.347579	−	0.937651i	$-0.612996\pi$
$359$	−13.1714	−0.695158	−0.347579	+	0.937651i	$0.612996\pi$
$360$	0	0
$361$	28.5120	1.50063
$362$	0	0
$363$	17.3450	0.910374
$364$	0	0
$365$	10.2968	0.538962
$366$	0	0
$367$	12.2223	0.638000	0.319000	−	0.947755i	$-0.396653\pi$
$367$	12.2223	0.638000	0.319000	+	0.947755i	$0.396653\pi$
$368$	0	0
$369$	4.86755	0.253394
$370$	0	0
$371$	−0.821140	−0.0426315
$372$	0	0
$373$	−25.7142	−1.33143	−0.665715	−	0.746206i	$-0.731873\pi$
$373$	−25.7142	−1.33143	−0.665715	+	0.746206i	$0.731873\pi$
$374$	0	0
$375$	−7.46281	−0.385378
$376$	0	0
$377$	−2.18946	−0.112763
$378$	0	0
$379$	19.3168	0.992239	0.496120	−	0.868254i	$-0.334758\pi$
$379$	19.3168	0.992239	0.496120	+	0.868254i	$0.334758\pi$
$380$	0	0
$381$	−15.1659	−0.776971
$382$	0	0
$383$	26.7155	1.36510	0.682548	−	0.730840i	$-0.260872\pi$
$383$	26.7155	1.36510	0.682548	+	0.730840i	$0.260872\pi$
$384$	0	0
$385$	2.40979	0.122814
$386$	0	0
$387$	−6.75736	−0.343496
$388$	0	0
$389$	−5.66024	−0.286986	−0.143493	−	0.989651i	$-0.545833\pi$
$389$	−5.66024	−0.286986	−0.143493	+	0.989651i	$0.545833\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1.75962	−0.0887613
$394$	0	0
$395$	7.05735	0.355094
$396$	0	0
$397$	7.82688	0.392820	0.196410	−	0.980522i	$-0.437072\pi$
$397$	7.82688	0.392820	0.196410	+	0.980522i	$0.437072\pi$
$398$	0	0
$399$	1.16085	0.0581153
$400$	0	0
$401$	−8.86169	−0.442532	−0.221266	−	0.975214i	$-0.571019\pi$
$401$	−8.86169	−0.442532	−0.221266	+	0.975214i	$0.571019\pi$
$402$	0	0
$403$	−26.7377	−1.33190
$404$	0	0
$405$	2.68761	0.133548
$406$	0	0
$407$	−47.1552	−2.33740
$408$	0	0
$409$	13.5450	0.669755	0.334878	−	0.942262i	$-0.391305\pi$
$409$	13.5450	0.669755	0.334878	+	0.942262i	$0.391305\pi$
$410$	0	0
$411$	14.6865	0.724430
$412$	0	0
$413$	−0.384147	−0.0189027
$414$	0	0
$415$	−24.1066	−1.18335
$416$	0	0
$417$	22.3528	1.09462
$418$	0	0
$419$	9.44850	0.461590	0.230795	−	0.973002i	$-0.425867\pi$
$419$	9.44850	0.461590	0.230795	+	0.973002i	$0.425867\pi$
$420$	0	0
$421$	−31.1359	−1.51747	−0.758735	−	0.651399i	$-0.774182\pi$
$421$	−31.1359	−1.51747	−0.758735	+	0.651399i	$0.774182\pi$
$422$	0	0
$423$	−2.26730	−0.110240
$424$	0	0
$425$	−14.8004	−0.717925
$426$	0	0
$427$	2.35120	0.113783
$428$	0	0
$429$	−20.4543	−0.987543
$430$	0	0
$431$	−21.5752	−1.03924	−0.519621	−	0.854397i	$-0.673927\pi$
$431$	−21.5752	−1.03924	−0.519621	+	0.854397i	$0.673927\pi$
$432$	0	0
$433$	−0.528096	−0.0253787	−0.0126893	−	0.999919i	$-0.504039\pi$
$433$	−0.528096	−0.0253787	−0.0126893	+	0.999919i	$0.504039\pi$
$434$	0	0
$435$	−1.53164	−0.0734364
$436$	0	0
$437$	0	0
$438$	0	0
$439$	21.6220	1.03196	0.515980	−	0.856601i	$-0.327428\pi$
$439$	21.6220	1.03196	0.515980	+	0.856601i	$0.327428\pi$
$440$	0	0
$441$	−6.97164	−0.331983
$442$	0	0
$443$	22.2377	1.05654	0.528272	−	0.849075i	$-0.322840\pi$
$443$	22.2377	1.05654	0.528272	+	0.849075i	$0.322840\pi$
$444$	0	0
$445$	16.8955	0.800923
$446$	0	0
$447$	−6.86314	−0.324615
$448$	0	0
$449$	−41.7152	−1.96866	−0.984332	−	0.176325i	$-0.943579\pi$
$449$	−41.7152	−1.96866	−0.984332	+	0.176325i	$0.943579\pi$
$450$	0	0
$451$	−25.9148	−1.22028
$452$	0	0
$453$	−1.51838	−0.0713397
$454$	0	0
$455$	−1.73896	−0.0815235
$456$	0	0
$457$	19.6407	0.918751	0.459376	−	0.888242i	$-0.348073\pi$
$457$	19.6407	0.918751	0.459376	+	0.888242i	$0.348073\pi$
$458$	0	0
$459$	−6.65709	−0.310726
$460$	0	0
$461$	−14.8017	−0.689383	−0.344692	−	0.938716i	$-0.612017\pi$
$461$	−14.8017	−0.689383	−0.344692	+	0.938716i	$0.612017\pi$
$462$	0	0
$463$	15.2602	0.709200	0.354600	−	0.935018i	$-0.384617\pi$
$463$	15.2602	0.709200	0.354600	+	0.935018i	$0.384617\pi$
$464$	0	0
$465$	−18.7044	−0.867395
$466$	0	0
$467$	15.4487	0.714881	0.357440	−	0.933936i	$-0.383650\pi$
$467$	15.4487	0.714881	0.357440	+	0.933936i	$0.383650\pi$
$468$	0	0
$469$	0.271351	0.0125298
$470$	0	0
$471$	−6.59043	−0.303671
$472$	0	0
$473$	35.9762	1.65419
$474$	0	0
$475$	−15.3247	−0.703144
$476$	0	0
$477$	4.87576	0.223246
$478$	0	0
$479$	18.9013	0.863622	0.431811	−	0.901964i	$-0.357875\pi$
$479$	18.9013	0.863622	0.431811	+	0.901964i	$0.357875\pi$
$480$	0	0
$481$	34.0282	1.55155
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−35.6870	−1.62046
$486$	0	0
$487$	10.3167	0.467494	0.233747	−	0.972297i	$-0.424901\pi$
$487$	10.3167	0.467494	0.233747	+	0.972297i	$0.424901\pi$
$488$	0	0
$489$	2.45066	0.110822
$490$	0	0
$491$	6.96153	0.314170	0.157085	−	0.987585i	$-0.449790\pi$
$491$	6.96153	0.314170	0.157085	+	0.987585i	$0.449790\pi$
$492$	0	0
$493$	3.79380	0.170864
$494$	0	0
$495$	−14.3088	−0.643135
$496$	0	0
$497$	1.59936	0.0717411
$498$	0	0
$499$	−1.02074	−0.0456945	−0.0228472	−	0.999739i	$-0.507273\pi$
$499$	−1.02074	−0.0456945	−0.0228472	+	0.999739i	$0.507273\pi$
$500$	0	0
$501$	10.3057	0.460425
$502$	0	0
$503$	−14.4196	−0.642937	−0.321468	−	0.946920i	$-0.604176\pi$
$503$	−14.4196	−0.642937	−0.321468	+	0.946920i	$0.604176\pi$
$504$	0	0
$505$	−19.5299	−0.869068
$506$	0	0
$507$	1.76024	0.0781750
$508$	0	0
$509$	−25.0238	−1.10916	−0.554580	−	0.832131i	$-0.687121\pi$
$509$	−25.0238	−1.10916	−0.554580	+	0.832131i	$0.687121\pi$
$510$	0	0
$511$	−0.645228	−0.0285432
$512$	0	0
$513$	−6.89290	−0.304329
$514$	0	0
$515$	−11.5097	−0.507180
$516$	0	0
$517$	12.0711	0.530887
$518$	0	0
$519$	−3.01452	−0.132323
$520$	0	0
$521$	34.4792	1.51056	0.755280	−	0.655402i	$-0.227501\pi$
$521$	34.4792	1.51056	0.755280	+	0.655402i	$0.227501\pi$
$522$	0	0
$523$	−12.5663	−0.549485	−0.274742	−	0.961518i	$-0.588593\pi$
$523$	−12.5663	−0.549485	−0.274742	+	0.961518i	$0.588593\pi$
$524$	0	0
$525$	−0.374425	−0.0163412
$526$	0	0
$527$	46.3299	2.01816
$528$	0	0
$529$	0	0
$530$	0	0
$531$	2.28099	0.0989864
$532$	0	0
$533$	18.7007	0.810016
$534$	0	0
$535$	26.1010	1.12845
$536$	0	0
$537$	−13.5122	−0.583093
$538$	0	0
$539$	37.1170	1.59874
$540$	0	0
$541$	−13.1406	−0.564960	−0.282480	−	0.959273i	$-0.591157\pi$
$541$	−13.1406	−0.564960	−0.282480	+	0.959273i	$0.591157\pi$
$542$	0	0
$543$	13.8996	0.596489
$544$	0	0
$545$	−23.7855	−1.01886
$546$	0	0
$547$	−35.0838	−1.50008	−0.750038	−	0.661395i	$-0.769965\pi$
$547$	−35.0838	−1.50008	−0.750038	+	0.661395i	$0.769965\pi$
$548$	0	0
$549$	−13.9609	−0.595838
$550$	0	0
$551$	3.92818	0.167346
$552$	0	0
$553$	−0.442233	−0.0188056
$554$	0	0
$555$	23.8045	1.01044
$556$	0	0
$557$	−0.643547	−0.0272680	−0.0136340	−	0.999907i	$-0.504340\pi$
$557$	−0.643547	−0.0272680	−0.0136340	+	0.999907i	$0.504340\pi$
$558$	0	0
$559$	−25.9611	−1.09804
$560$	0	0
$561$	35.4423	1.49638
$562$	0	0
$563$	−11.6418	−0.490644	−0.245322	−	0.969442i	$-0.578894\pi$
$563$	−11.6418	−0.490644	−0.245322	+	0.969442i	$0.578894\pi$
$564$	0	0
$565$	22.9667	0.966218
$566$	0	0
$567$	−0.168413	−0.00707268
$568$	0	0
$569$	−35.5814	−1.49165	−0.745825	−	0.666141i	$-0.767945\pi$
$569$	−35.5814	−1.49165	−0.745825	+	0.666141i	$0.767945\pi$
$570$	0	0
$571$	4.13581	0.173078	0.0865392	−	0.996248i	$-0.472419\pi$
$571$	4.13581	0.173078	0.0865392	+	0.996248i	$0.472419\pi$
$572$	0	0
$573$	12.7981	0.534647
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.33356	−0.138778	−0.0693890	−	0.997590i	$-0.522105\pi$
$577$	−3.33356	−0.138778	−0.0693890	+	0.997590i	$0.522105\pi$
$578$	0	0
$579$	16.1381	0.670676
$580$	0	0
$581$	1.51059	0.0626697
$582$	0	0
$583$	−25.9585	−1.07509
$584$	0	0
$585$	10.3255	0.426909
$586$	0	0
$587$	13.8017	0.569656	0.284828	−	0.958579i	$-0.408063\pi$
$587$	13.8017	0.569656	0.284828	+	0.958579i	$0.408063\pi$
$588$	0	0
$589$	47.9710	1.97661
$590$	0	0
$591$	13.1747	0.541936
$592$	0	0
$593$	−36.1689	−1.48528	−0.742639	−	0.669692i	$-0.766426\pi$
$593$	−36.1689	−1.48528	−0.742639	+	0.669692i	$0.766426\pi$
$594$	0	0
$595$	3.01319	0.123529
$596$	0	0
$597$	−9.25611	−0.378828
$598$	0	0
$599$	18.8219	0.769041	0.384520	−	0.923117i	$-0.374367\pi$
$599$	18.8219	0.769041	0.384520	+	0.923117i	$0.374367\pi$
$600$	0	0
$601$	−44.4453	−1.81296	−0.906481	−	0.422247i	$-0.861241\pi$
$601$	−44.4453	−1.81296	−0.906481	+	0.422247i	$0.861241\pi$
$602$	0	0
$603$	−1.61122	−0.0656140
$604$	0	0
$605$	46.6165	1.89523
$606$	0	0
$607$	−24.5329	−0.995760	−0.497880	−	0.867246i	$-0.665888\pi$
$607$	−24.5329	−0.995760	−0.497880	+	0.867246i	$0.665888\pi$
$608$	0	0
$609$	0.0959765	0.00388917
$610$	0	0
$611$	−8.71075	−0.352399
$612$	0	0
$613$	−2.27696	−0.0919657	−0.0459829	−	0.998942i	$-0.514642\pi$
$613$	−2.27696	−0.0919657	−0.0459829	+	0.998942i	$0.514642\pi$
$614$	0	0
$615$	13.0821	0.527520
$616$	0	0
$617$	−12.3066	−0.495445	−0.247722	−	0.968831i	$-0.579682\pi$
$617$	−12.3066	−0.495445	−0.247722	+	0.968831i	$0.579682\pi$
$618$	0	0
$619$	8.92236	0.358620	0.179310	−	0.983793i	$-0.442613\pi$
$619$	8.92236	0.358620	0.179310	+	0.983793i	$0.442613\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.05872	−0.0424166
$624$	0	0
$625$	−31.1734	−1.24694
$626$	0	0
$627$	36.6978	1.46557
$628$	0	0
$629$	−58.9625	−2.35099
$630$	0	0
$631$	−33.0833	−1.31702	−0.658512	−	0.752570i	$-0.728814\pi$
$631$	−33.0833	−1.31702	−0.658512	+	0.752570i	$0.728814\pi$
$632$	0	0
$633$	−14.7860	−0.587691
$634$	0	0
$635$	−40.7600	−1.61751
$636$	0	0
$637$	−26.7844	−1.06124
$638$	0	0
$639$	−9.49666	−0.375682
$640$	0	0
$641$	45.5559	1.79935	0.899675	−	0.436560i	$-0.143803\pi$
$641$	45.5559	1.79935	0.899675	+	0.436560i	$0.143803\pi$
$642$	0	0
$643$	−6.72806	−0.265329	−0.132664	−	0.991161i	$-0.542353\pi$
$643$	−6.72806	−0.265329	−0.132664	+	0.991161i	$0.542353\pi$
$644$	0	0
$645$	−18.1612	−0.715095
$646$	0	0
$647$	−4.78380	−0.188070	−0.0940352	−	0.995569i	$-0.529977\pi$
$647$	−4.78380	−0.188070	−0.0940352	+	0.995569i	$0.529977\pi$
$648$	0	0
$649$	−12.1440	−0.476692
$650$	0	0
$651$	1.17207	0.0459369
$652$	0	0
$653$	31.3121	1.22534	0.612668	−	0.790340i	$-0.290096\pi$
$653$	31.3121	1.22534	0.612668	+	0.790340i	$0.290096\pi$
$654$	0	0
$655$	−4.72919	−0.184785
$656$	0	0
$657$	3.83123	0.149470
$658$	0	0
$659$	−7.80466	−0.304026	−0.152013	−	0.988378i	$-0.548576\pi$
$659$	−7.80466	−0.304026	−0.152013	+	0.988378i	$0.548576\pi$
$660$	0	0
$661$	46.7053	1.81663	0.908313	−	0.418291i	$-0.137371\pi$
$661$	46.7053	1.81663	0.908313	+	0.418291i	$0.137371\pi$
$662$	0	0
$663$	−25.5759	−0.993286
$664$	0	0
$665$	3.11992	0.120985
$666$	0	0
$667$	0	0
$668$	0	0
$669$	14.9643	0.578555
$670$	0	0
$671$	74.3280	2.86940
$672$	0	0
$673$	−43.8070	−1.68864	−0.844318	−	0.535842i	$-0.819994\pi$
$673$	−43.8070	−1.68864	−0.844318	+	0.535842i	$0.819994\pi$
$674$	0	0
$675$	2.22326	0.0855731
$676$	0	0
$677$	26.0495	1.00116	0.500582	−	0.865689i	$-0.333119\pi$
$677$	26.0495	1.00116	0.500582	+	0.865689i	$0.333119\pi$
$678$	0	0
$679$	2.23624	0.0858191
$680$	0	0
$681$	19.3027	0.739682
$682$	0	0
$683$	12.0695	0.461826	0.230913	−	0.972974i	$-0.425829\pi$
$683$	12.0695	0.461826	0.230913	+	0.972974i	$0.425829\pi$
$684$	0	0
$685$	39.4715	1.50813
$686$	0	0
$687$	−26.6442	−1.01654
$688$	0	0
$689$	18.7322	0.713640
$690$	0	0
$691$	−41.2368	−1.56872	−0.784360	−	0.620305i	$-0.787009\pi$
$691$	−41.2368	−1.56872	−0.784360	+	0.620305i	$0.787009\pi$
$692$	0	0
$693$	0.896630	0.0340602
$694$	0	0
$695$	60.0758	2.27880
$696$	0	0
$697$	−32.4037	−1.22738
$698$	0	0
$699$	−14.4481	−0.546477
$700$	0	0
$701$	−15.9467	−0.602299	−0.301149	−	0.953577i	$-0.597370\pi$
$701$	−15.9467	−0.602299	−0.301149	+	0.953577i	$0.597370\pi$
$702$	0	0
$703$	−61.0511	−2.30259
$704$	0	0
$705$	−6.09362	−0.229499
$706$	0	0
$707$	1.22379	0.0460255
$708$	0	0
$709$	−5.43912	−0.204270	−0.102135	−	0.994771i	$-0.532567\pi$
$709$	−5.43912	−0.204270	−0.102135	+	0.994771i	$0.532567\pi$
$710$	0	0
$711$	2.62588	0.0984783
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−54.9732	−2.05588
$716$	0	0
$717$	−24.1116	−0.900463
$718$	0	0
$719$	−5.60236	−0.208933	−0.104466	−	0.994528i	$-0.533313\pi$
$719$	−5.60236	−0.208933	−0.104466	+	0.994528i	$0.533313\pi$
$720$	0	0
$721$	0.721231	0.0268600
$722$	0	0
$723$	−11.2125	−0.416996
$724$	0	0
$725$	−1.26701	−0.0470555
$726$	0	0
$727$	39.2481	1.45563	0.727815	−	0.685773i	$-0.240536\pi$
$727$	39.2481	1.45563	0.727815	+	0.685773i	$0.240536\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.9844	1.66381
$732$	0	0
$733$	−30.2116	−1.11589	−0.557946	−	0.829877i	$-0.688410\pi$
$733$	−30.2116	−1.11589	−0.557946	+	0.829877i	$0.688410\pi$
$734$	0	0
$735$	−18.7371	−0.691127
$736$	0	0
$737$	8.57814	0.315980
$738$	0	0
$739$	−13.6068	−0.500534	−0.250267	−	0.968177i	$-0.580518\pi$
$739$	−13.6068	−0.500534	−0.250267	+	0.968177i	$0.580518\pi$
$740$	0	0
$741$	−26.4819	−0.972835
$742$	0	0
$743$	−3.28957	−0.120683	−0.0603413	−	0.998178i	$-0.519219\pi$
$743$	−3.28957	−0.120683	−0.0603413	+	0.998178i	$0.519219\pi$
$744$	0	0
$745$	−18.4454	−0.675789
$746$	0	0
$747$	−8.96954	−0.328178
$748$	0	0
$749$	−1.63556	−0.0597621
$750$	0	0
$751$	−20.4896	−0.747675	−0.373837	−	0.927494i	$-0.621958\pi$
$751$	−20.4896	−0.747675	−0.373837	+	0.927494i	$0.621958\pi$
$752$	0	0
$753$	−28.6735	−1.04492
$754$	0	0
$755$	−4.08082	−0.148516
$756$	0	0
$757$	−47.7612	−1.73591	−0.867955	−	0.496643i	$-0.834566\pi$
$757$	−47.7612	−1.73591	−0.867955	+	0.496643i	$0.834566\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.1657	0.513505	0.256753	−	0.966477i	$-0.417347\pi$
$761$	14.1657	0.513505	0.256753	+	0.966477i	$0.417347\pi$
$762$	0	0
$763$	1.49046	0.0539584
$764$	0	0
$765$	−17.8917	−0.646875
$766$	0	0
$767$	8.76333	0.316426
$768$	0	0
$769$	−44.9704	−1.62167	−0.810836	−	0.585273i	$-0.800987\pi$
$769$	−44.9704	−1.62167	−0.810836	+	0.585273i	$0.800987\pi$
$770$	0	0
$771$	18.6188	0.670540
$772$	0	0
$773$	31.2256	1.12311	0.561553	−	0.827441i	$-0.310204\pi$
$773$	31.2256	1.12311	0.561553	+	0.827441i	$0.310204\pi$
$774$	0	0
$775$	−15.4727	−0.555796
$776$	0	0
$777$	−1.49165	−0.0535127
$778$	0	0
$779$	−33.5515	−1.20211
$780$	0	0
$781$	50.5602	1.80919
$782$	0	0
$783$	−0.569888	−0.0203661
$784$	0	0
$785$	−17.7125	−0.632187
$786$	0	0
$787$	−6.98462	−0.248975	−0.124487	−	0.992221i	$-0.539729\pi$
$787$	−6.98462	−0.248975	−0.124487	+	0.992221i	$0.539729\pi$
$788$	0	0
$789$	−14.3106	−0.509470
$790$	0	0
$791$	−1.43916	−0.0511705
$792$	0	0
$793$	−53.6366	−1.90469
$794$	0	0
$795$	13.1041	0.464756
$796$	0	0
$797$	3.28738	0.116445	0.0582225	−	0.998304i	$-0.481457\pi$
$797$	3.28738	0.116445	0.0582225	+	0.998304i	$0.481457\pi$
$798$	0	0
$799$	15.0936	0.533974
$800$	0	0
$801$	6.28643	0.222120
$802$	0	0
$803$	−20.3974	−0.719810
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.01605	−0.246977
$808$	0	0
$809$	−36.3906	−1.27942	−0.639712	−	0.768614i	$-0.720946\pi$
$809$	−36.3906	−1.27942	−0.639712	+	0.768614i	$0.720946\pi$
$810$	0	0
$811$	−50.9938	−1.79063	−0.895317	−	0.445430i	$-0.853051\pi$
$811$	−50.9938	−1.79063	−0.895317	+	0.445430i	$0.853051\pi$
$812$	0	0
$813$	10.6915	0.374967
$814$	0	0
$815$	6.58641	0.230712
$816$	0	0
$817$	46.5778	1.62955
$818$	0	0
$819$	−0.647026	−0.0226089
$820$	0	0
$821$	16.5463	0.577469	0.288734	−	0.957409i	$-0.406766\pi$
$821$	16.5463	0.577469	0.288734	+	0.957409i	$0.406766\pi$
$822$	0	0
$823$	8.22811	0.286814	0.143407	−	0.989664i	$-0.454194\pi$
$823$	8.22811	0.286814	0.143407	+	0.989664i	$0.454194\pi$
$824$	0	0
$825$	−11.8366	−0.412098
$826$	0	0
$827$	22.7771	0.792036	0.396018	−	0.918243i	$-0.370392\pi$
$827$	22.7771	0.792036	0.396018	+	0.918243i	$0.370392\pi$
$828$	0	0
$829$	−5.13824	−0.178459	−0.0892293	−	0.996011i	$-0.528440\pi$
$829$	−5.13824	−0.178459	−0.0892293	+	0.996011i	$0.528440\pi$
$830$	0	0
$831$	22.1540	0.768514
$832$	0	0
$833$	46.4108	1.60804
$834$	0	0
$835$	27.6977	0.958519
$836$	0	0
$837$	−6.95948	−0.240555
$838$	0	0
$839$	−10.2732	−0.354671	−0.177336	−	0.984150i	$-0.556748\pi$
$839$	−10.2732	−0.354671	−0.177336	+	0.984150i	$0.556748\pi$
$840$	0	0
$841$	−28.6752	−0.988801
$842$	0	0
$843$	7.06273	0.243253
$844$	0	0
$845$	4.73084	0.162746
$846$	0	0
$847$	−2.92111	−0.100371
$848$	0	0
$849$	26.9335	0.924355
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−31.2764	−1.07089	−0.535443	−	0.844572i	$-0.679855\pi$
$853$	−31.2764	−1.07089	−0.535443	+	0.844572i	$0.679855\pi$
$854$	0	0
$855$	−18.5254	−0.633556
$856$	0	0
$857$	18.7082	0.639060	0.319530	−	0.947576i	$-0.396475\pi$
$857$	18.7082	0.639060	0.319530	+	0.947576i	$0.396475\pi$
$858$	0	0
$859$	41.3968	1.41244	0.706221	−	0.707991i	$-0.250398\pi$
$859$	41.3968	1.41244	0.706221	+	0.707991i	$0.250398\pi$
$860$	0	0
$861$	−0.819758	−0.0279373
$862$	0	0
$863$	−7.83474	−0.266698	−0.133349	−	0.991069i	$-0.542573\pi$
$863$	−7.83474	−0.266698	−0.133349	+	0.991069i	$0.542573\pi$
$864$	0	0
$865$	−8.10187	−0.275472
$866$	0	0
$867$	27.3168	0.927729
$868$	0	0
$869$	−13.9802	−0.474246
$870$	0	0
$871$	−6.19016	−0.209746
$872$	0	0
$873$	−13.2783	−0.449404
$874$	0	0
$875$	1.25683	0.0424887
$876$	0	0
$877$	38.0372	1.28443	0.642213	−	0.766526i	$-0.278016\pi$
$877$	38.0372	1.28443	0.642213	+	0.766526i	$0.278016\pi$
$878$	0	0
$879$	28.5308	0.962320
$880$	0	0
$881$	12.6784	0.427147	0.213573	−	0.976927i	$-0.431490\pi$
$881$	12.6784	0.427147	0.213573	+	0.976927i	$0.431490\pi$
$882$	0	0
$883$	3.53058	0.118813	0.0594067	−	0.998234i	$-0.481079\pi$
$883$	3.53058	0.118813	0.0594067	+	0.998234i	$0.481079\pi$
$884$	0	0
$885$	6.13041	0.206071
$886$	0	0
$887$	40.2404	1.35114	0.675569	−	0.737296i	$-0.263898\pi$
$887$	40.2404	1.35114	0.675569	+	0.737296i	$0.263898\pi$
$888$	0	0
$889$	2.55413	0.0856627
$890$	0	0
$891$	−5.32400	−0.178361
$892$	0	0
$893$	15.6283	0.522980
$894$	0	0
$895$	−36.3154	−1.21389
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.96613	0.132278
$900$	0	0
$901$	−32.4584	−1.08134
$902$	0	0
$903$	1.13803	0.0378712
$904$	0	0
$905$	37.3568	1.24178
$906$	0	0
$907$	−25.8219	−0.857403	−0.428701	−	0.903446i	$-0.641029\pi$
$907$	−25.8219	−0.857403	−0.428701	+	0.903446i	$0.641029\pi$
$908$	0	0
$909$	−7.26663	−0.241019
$910$	0	0
$911$	−14.8460	−0.491870	−0.245935	−	0.969286i	$-0.579095\pi$
$911$	−14.8460	−0.491870	−0.245935	+	0.969286i	$0.579095\pi$
$912$	0	0
$913$	47.7538	1.58042
$914$	0	0
$915$	−37.5216	−1.24043
$916$	0	0
$917$	0.296343	0.00978612
$918$	0	0
$919$	−25.3485	−0.836171	−0.418085	−	0.908408i	$-0.637299\pi$
$919$	−25.3485	−0.836171	−0.418085	+	0.908408i	$0.637299\pi$
$920$	0	0
$921$	−11.6303	−0.383230
$922$	0	0
$923$	−36.4853	−1.20093
$924$	0	0
$925$	19.6916	0.647456
$926$	0	0
$927$	−4.28252	−0.140656
$928$	0	0
$929$	45.2024	1.48304	0.741521	−	0.670929i	$-0.234105\pi$
$929$	45.2024	1.48304	0.741521	+	0.670929i	$0.234105\pi$
$930$	0	0
$931$	48.0548	1.57493
$932$	0	0
$933$	26.2524	0.859465
$934$	0	0
$935$	95.2552	3.11518
$936$	0	0
$937$	52.7763	1.72413	0.862064	−	0.506800i	$-0.169172\pi$
$937$	52.7763	1.72413	0.862064	+	0.506800i	$0.169172\pi$
$938$	0	0
$939$	−25.6165	−0.835962
$940$	0	0
$941$	−35.3014	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$941$	−35.3014	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−0.452628	−0.0147240
$946$	0	0
$947$	59.7208	1.94066	0.970332	−	0.241775i	$-0.0777296\pi$
$947$	59.7208	1.94066	0.970332	+	0.241775i	$0.0777296\pi$
$948$	0	0
$949$	14.7192	0.477806
$950$	0	0
$951$	20.1963	0.654910
$952$	0	0
$953$	54.1571	1.75432	0.877161	−	0.480197i	$-0.159435\pi$
$953$	54.1571	1.75432	0.877161	+	0.480197i	$0.159435\pi$
$954$	0	0
$955$	34.3962	1.11304
$956$	0	0
$957$	3.03408	0.0980780
$958$	0	0
$959$	−2.47339	−0.0798700
$960$	0	0
$961$	17.4344	0.562399
$962$	0	0
$963$	9.71161	0.312952
$964$	0	0
$965$	43.3729	1.39622
$966$	0	0
$967$	−13.5723	−0.436457	−0.218229	−	0.975898i	$-0.570028\pi$
$967$	−13.5723	−0.436457	−0.218229	+	0.975898i	$0.570028\pi$
$968$	0	0
$969$	45.8866	1.47409
$970$	0	0
$971$	−3.68048	−0.118112	−0.0590561	−	0.998255i	$-0.518809\pi$
$971$	−3.68048	−0.118112	−0.0590561	+	0.998255i	$0.518809\pi$
$972$	0	0
$973$	−3.76451	−0.120685
$974$	0	0
$975$	8.54154	0.273548
$976$	0	0
$977$	22.0511	0.705477	0.352738	−	0.935722i	$-0.385251\pi$
$977$	22.0511	0.705477	0.352738	+	0.935722i	$0.385251\pi$
$978$	0	0
$979$	−33.4690	−1.06967
$980$	0	0
$981$	−8.85005	−0.282560
$982$	0	0
$983$	−27.2107	−0.867887	−0.433944	−	0.900940i	$-0.642878\pi$
$983$	−27.2107	−0.867887	−0.433944	+	0.900940i	$0.642878\pi$
$984$	0	0
$985$	35.4086	1.12821
$986$	0	0
$987$	0.381843	0.0121542
$988$	0	0
$989$	0	0
$990$	0	0
$991$	14.5200	0.461244	0.230622	−	0.973043i	$-0.425924\pi$
$991$	14.5200	0.461244	0.230622	+	0.973043i	$0.425924\pi$
$992$	0	0
$993$	−21.5420	−0.683616
$994$	0	0
$995$	−24.8768	−0.788649
$996$	0	0
$997$	−5.08045	−0.160899	−0.0804497	−	0.996759i	$-0.525636\pi$
$997$	−5.08045	−0.160899	−0.0804497	+	0.996759i	$0.525636\pi$
$998$	0	0
$999$	8.85711	0.280226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.s.1.9		10
23.4	even	11		276.2.i.a.85.2	yes	20
23.6	even	11		276.2.i.a.13.2	✓	20
23.22	odd	2		6348.2.a.t.1.2		10
69.29	odd	22		828.2.q.c.289.1		20
69.50	odd	22		828.2.q.c.361.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.13.2	✓	20	23.6	even	11
276.2.i.a.85.2	yes	20	23.4	even	11
828.2.q.c.289.1		20	69.29	odd	22
828.2.q.c.361.1		20	69.50	odd	22
6348.2.a.s.1.9		10	1.1	even	1	trivial
6348.2.a.t.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.68761 q^{5} -0.168413 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.68761 q^{5} -0.168413 q^{7} +1.00000 q^{9} -5.32400 q^{11} +3.84191 q^{13} +2.68761 q^{15} -6.65709 q^{17} -6.89290 q^{19} -0.168413 q^{21} +2.22326 q^{25} +1.00000 q^{27} -0.569888 q^{29} -6.95948 q^{31} -5.32400 q^{33} -0.452628 q^{35} +8.85711 q^{37} +3.84191 q^{39} +4.86755 q^{41} -6.75736 q^{43} +2.68761 q^{45} -2.26730 q^{47} -6.97164 q^{49} -6.65709 q^{51} +4.87576 q^{53} -14.3088 q^{55} -6.89290 q^{57} +2.28099 q^{59} -13.9609 q^{61} -0.168413 q^{63} +10.3255 q^{65} -1.61122 q^{67} -9.49666 q^{71} +3.83123 q^{73} +2.22326 q^{75} +0.896630 q^{77} +2.62588 q^{79} +1.00000 q^{81} -8.96954 q^{83} -17.8917 q^{85} -0.569888 q^{87} +6.28643 q^{89} -0.647026 q^{91} -6.95948 q^{93} -18.5254 q^{95} -13.2783 q^{97} -5.32400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) \) 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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