LMFDB - Embedded newform 4368.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.138892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 2x + 12 $ x^4 - x^3 - 10x^2 + 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.52690$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.152457 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.152457 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.385245 q^{11} +1.00000 q^{13} -0.152457 q^{15} +7.43905 q^{17} +7.20627 q^{19} +1.00000 q^{21} -2.90135 q^{23} -4.97676 q^{25} +1.00000 q^{27} -5.20627 q^{29} -1.76721 q^{31} -0.385245 q^{33} -0.152457 q^{35} +7.43905 q^{37} +1.00000 q^{39} +7.05381 q^{41} -2.90135 q^{43} -0.152457 q^{45} +3.59151 q^{47} +1.00000 q^{49} +7.43905 q^{51} -10.9502 q^{53} +0.0587335 q^{55} +7.20627 q^{57} +5.82430 q^{59} +12.5145 q^{61} +1.00000 q^{63} -0.152457 q^{65} -9.80270 q^{67} -2.90135 q^{69} -5.82430 q^{71} +3.09865 q^{73} -4.97676 q^{75} -0.385245 q^{77} -12.6453 q^{79} +1.00000 q^{81} -11.8964 q^{83} -1.13414 q^{85} -5.20627 q^{87} -5.59151 q^{89} +1.00000 q^{91} -1.76721 q^{93} -1.09865 q^{95} +18.9502 q^{97} -0.385245 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * 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
$5$	−0.152457	−0.0681810	−0.0340905	−	0.999419i	$-0.510853\pi$
$5$	−0.152457	−0.0681810	−0.0340905	+	0.999419i	$0.510853\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.385245	−0.116156	−0.0580779	−	0.998312i	$-0.518497\pi$
$11$	−0.385245	−0.116156	−0.0580779	+	0.998312i	$0.518497\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.152457	−0.0393643
$16$	0	0
$17$	7.43905	1.80424	0.902118	−	0.431490i	$-0.142012\pi$
$17$	7.43905	1.80424	0.902118	+	0.431490i	$0.142012\pi$
$18$	0	0
$19$	7.20627	1.65323	0.826615	−	0.562767i	$-0.190263\pi$
$19$	7.20627	1.65323	0.826615	+	0.562767i	$0.190263\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.90135	−0.604974	−0.302487	−	0.953154i	$-0.597817\pi$
$23$	−2.90135	−0.604974	−0.302487	+	0.953154i	$0.597817\pi$
$24$	0	0
$25$	−4.97676	−0.995351
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.20627	−0.966779	−0.483390	−	0.875405i	$-0.660595\pi$
$29$	−5.20627	−0.966779	−0.483390	+	0.875405i	$0.660595\pi$
$30$	0	0
$31$	−1.76721	−0.317401	−0.158700	−	0.987327i	$-0.550730\pi$
$31$	−1.76721	−0.317401	−0.158700	+	0.987327i	$0.550730\pi$
$32$	0	0
$33$	−0.385245	−0.0670626
$34$	0	0
$35$	−0.152457	−0.0257700
$36$	0	0
$37$	7.43905	1.22297	0.611486	−	0.791255i	$-0.290572\pi$
$37$	7.43905	1.22297	0.611486	+	0.791255i	$0.290572\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.05381	1.10162	0.550810	−	0.834631i	$-0.314319\pi$
$41$	7.05381	1.10162	0.550810	+	0.834631i	$0.314319\pi$
$42$	0	0
$43$	−2.90135	−0.442452	−0.221226	−	0.975223i	$-0.571006\pi$
$43$	−2.90135	−0.442452	−0.221226	+	0.975223i	$0.571006\pi$
$44$	0	0
$45$	−0.152457	−0.0227270
$46$	0	0
$47$	3.59151	0.523876	0.261938	−	0.965085i	$-0.415638\pi$
$47$	3.59151	0.523876	0.261938	+	0.965085i	$0.415638\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.43905	1.04168
$52$	0	0
$53$	−10.9502	−1.50413	−0.752065	−	0.659089i	$-0.770942\pi$
$53$	−10.9502	−1.50413	−0.752065	+	0.659089i	$0.770942\pi$
$54$	0	0
$55$	0.0587335	0.00791963
$56$	0	0
$57$	7.20627	0.954493
$58$	0	0
$59$	5.82430	0.758259	0.379130	−	0.925344i	$-0.376223\pi$
$59$	5.82430	0.758259	0.379130	+	0.925344i	$0.376223\pi$
$60$	0	0
$61$	12.5145	1.60231	0.801156	−	0.598455i	$-0.204219\pi$
$61$	12.5145	1.60231	0.801156	+	0.598455i	$0.204219\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.152457	−0.0189100
$66$	0	0
$67$	−9.80270	−1.19759	−0.598795	−	0.800902i	$-0.704354\pi$
$67$	−9.80270	−1.19759	−0.598795	+	0.800902i	$0.704354\pi$
$68$	0	0
$69$	−2.90135	−0.349282
$70$	0	0
$71$	−5.82430	−0.691217	−0.345609	−	0.938379i	$-0.612328\pi$
$71$	−5.82430	−0.691217	−0.345609	+	0.938379i	$0.612328\pi$
$72$	0	0
$73$	3.09865	0.362669	0.181335	−	0.983421i	$-0.441958\pi$
$73$	3.09865	0.362669	0.181335	+	0.983421i	$0.441958\pi$
$74$	0	0
$75$	−4.97676	−0.574666
$76$	0	0
$77$	−0.385245	−0.0439028
$78$	0	0
$79$	−12.6453	−1.42271	−0.711355	−	0.702833i	$-0.751918\pi$
$79$	−12.6453	−1.42271	−0.711355	+	0.702833i	$0.751918\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.8964	−1.30580	−0.652901	−	0.757443i	$-0.726448\pi$
$83$	−11.8964	−1.30580	−0.652901	+	0.757443i	$0.726448\pi$
$84$	0	0
$85$	−1.13414	−0.123015
$86$	0	0
$87$	−5.20627	−0.558170
$88$	0	0
$89$	−5.59151	−0.592699	−0.296350	−	0.955080i	$-0.595769\pi$
$89$	−5.59151	−0.592699	−0.296350	+	0.955080i	$0.595769\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.76721	−0.183251
$94$	0	0
$95$	−1.09865	−0.112719
$96$	0	0
$97$	18.9502	1.92410	0.962052	−	0.272865i	$-0.0879711\pi$
$97$	18.9502	1.92410	0.962052	+	0.272865i	$0.0879711\pi$
$98$	0	0
$99$	−0.385245	−0.0387186
$100$	0	0
$101$	11.3803	1.13238	0.566192	−	0.824273i	$-0.308416\pi$
$101$	11.3803	1.13238	0.566192	+	0.824273i	$0.308416\pi$
$102$	0	0
$103$	−5.13414	−0.505882	−0.252941	−	0.967482i	$-0.581398\pi$
$103$	−5.13414	−0.505882	−0.252941	+	0.967482i	$0.581398\pi$
$104$	0	0
$105$	−0.152457	−0.0148783
$106$	0	0
$107$	18.8781	1.82502	0.912508	−	0.409059i	$-0.134143\pi$
$107$	18.8781	1.82502	0.912508	+	0.409059i	$0.134143\pi$
$108$	0	0
$109$	19.7440	1.89113	0.945565	−	0.325434i	$-0.105511\pi$
$109$	19.7440	1.89113	0.945565	+	0.325434i	$0.105511\pi$
$110$	0	0
$111$	7.43905	0.706084
$112$	0	0
$113$	0.842618	0.0792668	0.0396334	−	0.999214i	$-0.487381\pi$
$113$	0.842618	0.0792668	0.0396334	+	0.999214i	$0.487381\pi$
$114$	0	0
$115$	0.442333	0.0412477
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	7.43905	0.681937
$120$	0	0
$121$	−10.8516	−0.986508
$122$	0	0
$123$	7.05381	0.636021
$124$	0	0
$125$	1.52103	0.136045
$126$	0	0
$127$	20.9857	1.86218	0.931091	−	0.364787i	$-0.118858\pi$
$127$	20.9857	1.86218	0.931091	+	0.364787i	$0.118858\pi$
$128$	0	0
$129$	−2.90135	−0.255450
$130$	0	0
$131$	−9.13414	−0.798053	−0.399027	−	0.916939i	$-0.630652\pi$
$131$	−9.13414	−0.798053	−0.399027	+	0.916939i	$0.630652\pi$
$132$	0	0
$133$	7.20627	0.624863
$134$	0	0
$135$	−0.152457	−0.0131214
$136$	0	0
$137$	22.6005	1.93089	0.965445	−	0.260608i	$-0.0839231\pi$
$137$	22.6005	1.93089	0.965445	+	0.260608i	$0.0839231\pi$
$138$	0	0
$139$	19.7929	1.67881	0.839404	−	0.543508i	$-0.182904\pi$
$139$	19.7929	1.67881	0.839404	+	0.543508i	$0.182904\pi$
$140$	0	0
$141$	3.59151	0.302460
$142$	0	0
$143$	−0.385245	−0.0322158
$144$	0	0
$145$	0.793734	0.0659160
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−12.5948	−1.03181	−0.515903	−	0.856647i	$-0.672543\pi$
$149$	−12.5948	−1.03181	−0.515903	+	0.856647i	$0.672543\pi$
$150$	0	0
$151$	−21.1341	−1.71987	−0.859936	−	0.510402i	$-0.829497\pi$
$151$	−21.1341	−1.71987	−0.859936	+	0.510402i	$0.829497\pi$
$152$	0	0
$153$	7.43905	0.601412
$154$	0	0
$155$	0.269425	0.0216407
$156$	0	0
$157$	15.4391	1.23217	0.616085	−	0.787679i	$-0.288718\pi$
$157$	15.4391	1.23217	0.616085	+	0.787679i	$0.288718\pi$
$158$	0	0
$159$	−10.9502	−0.868410
$160$	0	0
$161$	−2.90135	−0.228659
$162$	0	0
$163$	3.22951	0.252955	0.126477	−	0.991969i	$-0.459633\pi$
$163$	3.22951	0.252955	0.126477	+	0.991969i	$0.459633\pi$
$164$	0	0
$165$	0.0587335	0.00457240
$166$	0	0
$167$	−17.3355	−1.34146	−0.670730	−	0.741702i	$-0.734019\pi$
$167$	−17.3355	−1.34146	−0.670730	+	0.741702i	$0.734019\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.20627	0.551077
$172$	0	0
$173$	8.57319	0.651808	0.325904	−	0.945403i	$-0.394331\pi$
$173$	8.57319	0.651808	0.325904	+	0.945403i	$0.394331\pi$
$174$	0	0
$175$	−4.97676	−0.376207
$176$	0	0
$177$	5.82430	0.437781
$178$	0	0
$179$	−1.15738	−0.0865068	−0.0432534	−	0.999064i	$-0.513772\pi$
$179$	−1.15738	−0.0865068	−0.0432534	+	0.999064i	$0.513772\pi$
$180$	0	0
$181$	6.60983	0.491305	0.245652	−	0.969358i	$-0.420998\pi$
$181$	6.60983	0.491305	0.245652	+	0.969358i	$0.420998\pi$
$182$	0	0
$183$	12.5145	0.925095
$184$	0	0
$185$	−1.13414	−0.0833836
$186$	0	0
$187$	−2.86586	−0.209573
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	20.0122	1.44804	0.724018	−	0.689781i	$-0.242293\pi$
$191$	20.0122	1.44804	0.724018	+	0.689781i	$0.242293\pi$
$192$	0	0
$193$	−0.878108	−0.0632076	−0.0316038	−	0.999500i	$-0.510061\pi$
$193$	−0.878108	−0.0632076	−0.0316038	+	0.999500i	$0.510061\pi$
$194$	0	0
$195$	−0.152457	−0.0109177
$196$	0	0
$197$	−0.336362	−0.0239648	−0.0119824	−	0.999928i	$-0.503814\pi$
$197$	−0.336362	−0.0239648	−0.0119824	+	0.999928i	$0.503814\pi$
$198$	0	0
$199$	−5.90463	−0.418568	−0.209284	−	0.977855i	$-0.567113\pi$
$199$	−5.90463	−0.418568	−0.209284	+	0.977855i	$0.567113\pi$
$200$	0	0
$201$	−9.80270	−0.691429
$202$	0	0
$203$	−5.20627	−0.365408
$204$	0	0
$205$	−1.07541	−0.0751096
$206$	0	0
$207$	−2.90135	−0.201658
$208$	0	0
$209$	−2.77618	−0.192032
$210$	0	0
$211$	−3.67184	−0.252780	−0.126390	−	0.991981i	$-0.540339\pi$
$211$	−3.67184	−0.252780	−0.126390	+	0.991981i	$0.540339\pi$
$212$	0	0
$213$	−5.82430	−0.399074
$214$	0	0
$215$	0.442333	0.0301668
$216$	0	0
$217$	−1.76721	−0.119966
$218$	0	0
$219$	3.09865	0.209387
$220$	0	0
$221$	7.43905	0.500405
$222$	0	0
$223$	12.6453	0.846793	0.423397	−	0.905944i	$-0.360838\pi$
$223$	12.6453	0.846793	0.423397	+	0.905944i	$0.360838\pi$
$224$	0	0
$225$	−4.97676	−0.331784
$226$	0	0
$227$	−6.59479	−0.437712	−0.218856	−	0.975757i	$-0.570232\pi$
$227$	−6.59479	−0.437712	−0.218856	+	0.975757i	$0.570232\pi$
$228$	0	0
$229$	−2.60983	−0.172462	−0.0862312	−	0.996275i	$-0.527482\pi$
$229$	−2.60983	−0.172462	−0.0862312	+	0.996275i	$0.527482\pi$
$230$	0	0
$231$	−0.385245	−0.0253473
$232$	0	0
$233$	−1.35468	−0.0887480	−0.0443740	−	0.999015i	$-0.514129\pi$
$233$	−1.35468	−0.0887480	−0.0443740	+	0.999015i	$0.514129\pi$
$234$	0	0
$235$	−0.547553	−0.0357184
$236$	0	0
$237$	−12.6453	−0.821402
$238$	0	0
$239$	−14.4341	−0.933666	−0.466833	−	0.884345i	$-0.654605\pi$
$239$	−14.4341	−0.933666	−0.466833	+	0.884345i	$0.654605\pi$
$240$	0	0
$241$	17.7306	1.14213	0.571063	−	0.820906i	$-0.306531\pi$
$241$	17.7306	1.14213	0.571063	+	0.820906i	$0.306531\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.152457	−0.00974015
$246$	0	0
$247$	7.20627	0.458524
$248$	0	0
$249$	−11.8964	−0.753905
$250$	0	0
$251$	−4.05873	−0.256185	−0.128092	−	0.991762i	$-0.540885\pi$
$251$	−4.05873	−0.256185	−0.128092	+	0.991762i	$0.540885\pi$
$252$	0	0
$253$	1.11773	0.0702712
$254$	0	0
$255$	−1.13414	−0.0710225
$256$	0	0
$257$	−30.2584	−1.88747	−0.943734	−	0.330704i	$-0.892714\pi$
$257$	−30.2584	−1.88747	−0.943734	+	0.330704i	$0.892714\pi$
$258$	0	0
$259$	7.43905	0.462240
$260$	0	0
$261$	−5.20627	−0.322260
$262$	0	0
$263$	−12.9502	−0.798546	−0.399273	−	0.916832i	$-0.630737\pi$
$263$	−12.9502	−0.798546	−0.399273	+	0.916832i	$0.630737\pi$
$264$	0	0
$265$	1.66944	0.102553
$266$	0	0
$267$	−5.59151	−0.342195
$268$	0	0
$269$	17.7929	1.08485	0.542425	−	0.840104i	$-0.317506\pi$
$269$	17.7929	1.08485	0.542425	+	0.840104i	$0.317506\pi$
$270$	0	0
$271$	−17.9470	−1.09020	−0.545100	−	0.838371i	$-0.683508\pi$
$271$	−17.9470	−1.09020	−0.545100	+	0.838371i	$0.683508\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	1.91727	0.115616
$276$	0	0
$277$	−5.87483	−0.352984	−0.176492	−	0.984302i	$-0.556475\pi$
$277$	−5.87483	−0.352984	−0.176492	+	0.984302i	$0.556475\pi$
$278$	0	0
$279$	−1.76721	−0.105800
$280$	0	0
$281$	−26.5417	−1.58335	−0.791674	−	0.610944i	$-0.790790\pi$
$281$	−26.5417	−1.58335	−0.791674	+	0.610944i	$0.790790\pi$
$282$	0	0
$283$	−17.7562	−1.05550	−0.527749	−	0.849401i	$-0.676964\pi$
$283$	−17.7562	−1.05550	−0.527749	+	0.849401i	$0.676964\pi$
$284$	0	0
$285$	−1.09865	−0.0650783
$286$	0	0
$287$	7.05381	0.416373
$288$	0	0
$289$	38.3395	2.25527
$290$	0	0
$291$	18.9502	1.11088
$292$	0	0
$293$	−14.0139	−0.818700	−0.409350	−	0.912377i	$-0.634245\pi$
$293$	−14.0139	−0.818700	−0.409350	+	0.912377i	$0.634245\pi$
$294$	0	0
$295$	−0.887958	−0.0516989
$296$	0	0
$297$	−0.385245	−0.0223542
$298$	0	0
$299$	−2.90135	−0.167789
$300$	0	0
$301$	−2.90135	−0.167231
$302$	0	0
$303$	11.3803	0.653782
$304$	0	0
$305$	−1.90792	−0.109247
$306$	0	0
$307$	−22.4912	−1.28364	−0.641821	−	0.766855i	$-0.721821\pi$
$307$	−22.4912	−1.28364	−0.641821	+	0.766855i	$0.721821\pi$
$308$	0	0
$309$	−5.13414	−0.292071
$310$	0	0
$311$	33.1300	1.87863	0.939314	−	0.343058i	$-0.111463\pi$
$311$	33.1300	1.87863	0.939314	+	0.343058i	$0.111463\pi$
$312$	0	0
$313$	19.3371	1.09300	0.546500	−	0.837459i	$-0.315960\pi$
$313$	19.3371	1.09300	0.546500	+	0.837459i	$0.315960\pi$
$314$	0	0
$315$	−0.152457	−0.00859000
$316$	0	0
$317$	−16.0860	−0.903481	−0.451740	−	0.892149i	$-0.649197\pi$
$317$	−16.0860	−0.903481	−0.451740	+	0.892149i	$0.649197\pi$
$318$	0	0
$319$	2.00569	0.112297
$320$	0	0
$321$	18.8781	1.05367
$322$	0	0
$323$	53.6078	2.98282
$324$	0	0
$325$	−4.97676	−0.276061
$326$	0	0
$327$	19.7440	1.09184
$328$	0	0
$329$	3.59151	0.198006
$330$	0	0
$331$	27.9535	1.53646	0.768232	−	0.640172i	$-0.221137\pi$
$331$	27.9535	1.53646	0.768232	+	0.640172i	$0.221137\pi$
$332$	0	0
$333$	7.43905	0.407658
$334$	0	0
$335$	1.49449	0.0816530
$336$	0	0
$337$	7.31388	0.398413	0.199206	−	0.979958i	$-0.436164\pi$
$337$	7.31388	0.398413	0.199206	+	0.979958i	$0.436164\pi$
$338$	0	0
$339$	0.842618	0.0457647
$340$	0	0
$341$	0.680810	0.0368679
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.442333	0.0238144
$346$	0	0
$347$	26.1076	1.40153	0.700765	−	0.713392i	$-0.252842\pi$
$347$	26.1076	1.40153	0.700765	+	0.713392i	$0.252842\pi$
$348$	0	0
$349$	−3.76721	−0.201654	−0.100827	−	0.994904i	$-0.532149\pi$
$349$	−3.76721	−0.201654	−0.100827	+	0.994904i	$0.532149\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−11.8243	−0.629344	−0.314672	−	0.949200i	$-0.601895\pi$
$353$	−11.8243	−0.629344	−0.314672	+	0.949200i	$0.601895\pi$
$354$	0	0
$355$	0.887958	0.0471279
$356$	0	0
$357$	7.43905	0.393716
$358$	0	0
$359$	−14.4341	−0.761804	−0.380902	−	0.924615i	$-0.624387\pi$
$359$	−14.4341	−0.761804	−0.380902	+	0.924615i	$0.624387\pi$
$360$	0	0
$361$	32.9303	1.73317
$362$	0	0
$363$	−10.8516	−0.569561
$364$	0	0
$365$	−0.472412	−0.0247272
$366$	0	0
$367$	13.9634	0.728882	0.364441	−	0.931227i	$-0.381260\pi$
$367$	13.9634	0.728882	0.364441	+	0.931227i	$0.381260\pi$
$368$	0	0
$369$	7.05381	0.367207
$370$	0	0
$371$	−10.9502	−0.568508
$372$	0	0
$373$	−31.0289	−1.60662	−0.803308	−	0.595563i	$-0.796929\pi$
$373$	−31.0289	−1.60662	−0.803308	+	0.595563i	$0.796929\pi$
$374$	0	0
$375$	1.52103	0.0785457
$376$	0	0
$377$	−5.20627	−0.268136
$378$	0	0
$379$	25.9634	1.33365	0.666824	−	0.745215i	$-0.267653\pi$
$379$	25.9634	1.33365	0.666824	+	0.745215i	$0.267653\pi$
$380$	0	0
$381$	20.9857	1.07513
$382$	0	0
$383$	−13.4175	−0.685600	−0.342800	−	0.939408i	$-0.611375\pi$
$383$	−13.4175	−0.685600	−0.342800	+	0.939408i	$0.611375\pi$
$384$	0	0
$385$	0.0587335	0.00299334
$386$	0	0
$387$	−2.90135	−0.147484
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	−21.5833	−1.09151
$392$	0	0
$393$	−9.13414	−0.460756
$394$	0	0
$395$	1.92787	0.0970018
$396$	0	0
$397$	−25.5234	−1.28098	−0.640492	−	0.767965i	$-0.721270\pi$
$397$	−25.5234	−1.28098	−0.640492	+	0.767965i	$0.721270\pi$
$398$	0	0
$399$	7.20627	0.360765
$400$	0	0
$401$	3.00732	0.150178	0.0750892	−	0.997177i	$-0.476076\pi$
$401$	3.00732	0.150178	0.0750892	+	0.997177i	$0.476076\pi$
$402$	0	0
$403$	−1.76721	−0.0880311
$404$	0	0
$405$	−0.152457	−0.00757567
$406$	0	0
$407$	−2.86586	−0.142055
$408$	0	0
$409$	−22.8483	−1.12978	−0.564888	−	0.825168i	$-0.691081\pi$
$409$	−22.8483	−1.12978	−0.564888	+	0.825168i	$0.691081\pi$
$410$	0	0
$411$	22.6005	1.11480
$412$	0	0
$413$	5.82430	0.286595
$414$	0	0
$415$	1.81370	0.0890310
$416$	0	0
$417$	19.7929	0.969260
$418$	0	0
$419$	−16.1565	−0.789297	−0.394648	−	0.918832i	$-0.629134\pi$
$419$	−16.1565	−0.789297	−0.394648	+	0.918832i	$0.629134\pi$
$420$	0	0
$421$	21.7929	1.06212	0.531059	−	0.847335i	$-0.321794\pi$
$421$	21.7929	1.06212	0.531059	+	0.847335i	$0.321794\pi$
$422$	0	0
$423$	3.59151	0.174625
$424$	0	0
$425$	−37.0224	−1.79585
$426$	0	0
$427$	12.5145	0.605617
$428$	0	0
$429$	−0.385245	−0.0185998
$430$	0	0
$431$	6.80194	0.327638	0.163819	−	0.986490i	$-0.447619\pi$
$431$	6.80194	0.327638	0.163819	+	0.986490i	$0.447619\pi$
$432$	0	0
$433$	6.26172	0.300919	0.150460	−	0.988616i	$-0.451925\pi$
$433$	6.26172	0.300919	0.150460	+	0.988616i	$0.451925\pi$
$434$	0	0
$435$	0.793734	0.0380566
$436$	0	0
$437$	−20.9079	−1.00016
$438$	0	0
$439$	21.0877	1.00646	0.503229	−	0.864153i	$-0.332145\pi$
$439$	21.0877	1.00646	0.503229	+	0.864153i	$0.332145\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.39345	0.303762	0.151881	−	0.988399i	$-0.451467\pi$
$443$	6.39345	0.303762	0.151881	+	0.988399i	$0.451467\pi$
$444$	0	0
$445$	0.852467	0.0404108
$446$	0	0
$447$	−12.5948	−0.595713
$448$	0	0
$449$	4.49286	0.212031	0.106016	−	0.994364i	$-0.466191\pi$
$449$	4.49286	0.212031	0.106016	+	0.994364i	$0.466191\pi$
$450$	0	0
$451$	−2.71745	−0.127960
$452$	0	0
$453$	−21.1341	−0.992968
$454$	0	0
$455$	−0.152457	−0.00714731
$456$	0	0
$457$	−31.7398	−1.48473	−0.742363	−	0.669998i	$-0.766295\pi$
$457$	−31.7398	−1.48473	−0.742363	+	0.669998i	$0.766295\pi$
$458$	0	0
$459$	7.43905	0.347225
$460$	0	0
$461$	3.76886	0.175533	0.0877666	−	0.996141i	$-0.472027\pi$
$461$	3.76886	0.175533	0.0877666	+	0.996141i	$0.472027\pi$
$462$	0	0
$463$	−18.1664	−0.844262	−0.422131	−	0.906535i	$-0.638718\pi$
$463$	−18.1664	−0.844262	−0.422131	+	0.906535i	$0.638718\pi$
$464$	0	0
$465$	0.269425	0.0124943
$466$	0	0
$467$	14.0057	0.648106	0.324053	−	0.946039i	$-0.394954\pi$
$467$	14.0057	0.648106	0.324053	+	0.946039i	$0.394954\pi$
$468$	0	0
$469$	−9.80270	−0.452647
$470$	0	0
$471$	15.4391	0.711394
$472$	0	0
$473$	1.11773	0.0513934
$474$	0	0
$475$	−35.8638	−1.64555
$476$	0	0
$477$	−10.9502	−0.501377
$478$	0	0
$479$	−36.1704	−1.65267	−0.826334	−	0.563181i	$-0.809577\pi$
$479$	−36.1704	−1.65267	−0.826334	+	0.563181i	$0.809577\pi$
$480$	0	0
$481$	7.43905	0.339192
$482$	0	0
$483$	−2.90135	−0.132016
$484$	0	0
$485$	−2.88910	−0.131187
$486$	0	0
$487$	−5.03221	−0.228031	−0.114016	−	0.993479i	$-0.536371\pi$
$487$	−5.03221	−0.228031	−0.114016	+	0.993479i	$0.536371\pi$
$488$	0	0
$489$	3.22951	0.146043
$490$	0	0
$491$	−10.8781	−0.490922	−0.245461	−	0.969406i	$-0.578939\pi$
$491$	−10.8781	−0.490922	−0.245461	+	0.969406i	$0.578939\pi$
$492$	0	0
$493$	−38.7297	−1.74430
$494$	0	0
$495$	0.0587335	0.00263988
$496$	0	0
$497$	−5.82430	−0.261256
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	−17.3355	−0.774492
$502$	0	0
$503$	−13.8459	−0.617358	−0.308679	−	0.951166i	$-0.599887\pi$
$503$	−13.8459	−0.617358	−0.308679	+	0.951166i	$0.599887\pi$
$504$	0	0
$505$	−1.73501	−0.0772071
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−37.1814	−1.64804	−0.824018	−	0.566564i	$-0.808272\pi$
$509$	−37.1814	−1.64804	−0.824018	+	0.566564i	$0.808272\pi$
$510$	0	0
$511$	3.09865	0.137076
$512$	0	0
$513$	7.20627	0.318164
$514$	0	0
$515$	0.782738	0.0344915
$516$	0	0
$517$	−1.38361	−0.0608512
$518$	0	0
$519$	8.57319	0.376321
$520$	0	0
$521$	43.4936	1.90549	0.952745	−	0.303771i	$-0.0982456\pi$
$521$	43.4936	1.90549	0.952745	+	0.303771i	$0.0982456\pi$
$522$	0	0
$523$	25.9634	1.13530	0.567649	−	0.823270i	$-0.307853\pi$
$523$	25.9634	1.13530	0.567649	+	0.823270i	$0.307853\pi$
$524$	0	0
$525$	−4.97676	−0.217203
$526$	0	0
$527$	−13.1464	−0.572666
$528$	0	0
$529$	−14.5822	−0.634007
$530$	0	0
$531$	5.82430	0.252753
$532$	0	0
$533$	7.05381	0.305534
$534$	0	0
$535$	−2.87811	−0.124431
$536$	0	0
$537$	−1.15738	−0.0499447
$538$	0	0
$539$	−0.385245	−0.0165937
$540$	0	0
$541$	−14.1565	−0.608636	−0.304318	−	0.952571i	$-0.598428\pi$
$541$	−14.1565	−0.608636	−0.304318	+	0.952571i	$0.598428\pi$
$542$	0	0
$543$	6.60983	0.283655
$544$	0	0
$545$	−3.01011	−0.128939
$546$	0	0
$547$	14.3338	0.612871	0.306436	−	0.951891i	$-0.400864\pi$
$547$	14.3338	0.612871	0.306436	+	0.951891i	$0.400864\pi$
$548$	0	0
$549$	12.5145	0.534104
$550$	0	0
$551$	−37.5177	−1.59831
$552$	0	0
$553$	−12.6453	−0.537734
$554$	0	0
$555$	−1.13414	−0.0481415
$556$	0	0
$557$	−38.4953	−1.63110	−0.815548	−	0.578689i	$-0.803564\pi$
$557$	−38.4953	−1.63110	−0.815548	+	0.578689i	$0.803564\pi$
$558$	0	0
$559$	−2.90135	−0.122714
$560$	0	0
$561$	−2.86586	−0.120997
$562$	0	0
$563$	39.9494	1.68366	0.841832	−	0.539739i	$-0.181477\pi$
$563$	39.9494	1.68366	0.841832	+	0.539739i	$0.181477\pi$
$564$	0	0
$565$	−0.128463	−0.00540449
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	2.83606	0.118894	0.0594469	−	0.998231i	$-0.481066\pi$
$569$	2.83606	0.118894	0.0594469	+	0.998231i	$0.481066\pi$
$570$	0	0
$571$	15.9890	0.669119	0.334559	−	0.942375i	$-0.391412\pi$
$571$	15.9890	0.669119	0.334559	+	0.942375i	$0.391412\pi$
$572$	0	0
$573$	20.0122	0.836024
$574$	0	0
$575$	14.4393	0.602161
$576$	0	0
$577$	15.2826	0.636221	0.318111	−	0.948054i	$-0.396952\pi$
$577$	15.2826	0.636221	0.318111	+	0.948054i	$0.396952\pi$
$578$	0	0
$579$	−0.878108	−0.0364929
$580$	0	0
$581$	−11.8964	−0.493547
$582$	0	0
$583$	4.21853	0.174714
$584$	0	0
$585$	−0.152457	−0.00630334
$586$	0	0
$587$	44.5340	1.83812	0.919058	−	0.394122i	$-0.128951\pi$
$587$	44.5340	1.83812	0.919058	+	0.394122i	$0.128951\pi$
$588$	0	0
$589$	−12.7350	−0.524737
$590$	0	0
$591$	−0.336362	−0.0138361
$592$	0	0
$593$	−23.9063	−0.981713	−0.490857	−	0.871240i	$-0.663316\pi$
$593$	−23.9063	−0.981713	−0.490857	+	0.871240i	$0.663316\pi$
$594$	0	0
$595$	−1.13414	−0.0464952
$596$	0	0
$597$	−5.90463	−0.241660
$598$	0	0
$599$	−23.9303	−0.977764	−0.488882	−	0.872350i	$-0.662595\pi$
$599$	−23.9303	−0.977764	−0.488882	+	0.872350i	$0.662595\pi$
$600$	0	0
$601$	6.51206	0.265633	0.132816	−	0.991141i	$-0.457598\pi$
$601$	6.51206	0.265633	0.132816	+	0.991141i	$0.457598\pi$
$602$	0	0
$603$	−9.80270	−0.399197
$604$	0	0
$605$	1.65440	0.0672611
$606$	0	0
$607$	−40.5756	−1.64691	−0.823456	−	0.567380i	$-0.807957\pi$
$607$	−40.5756	−1.64691	−0.823456	+	0.567380i	$0.807957\pi$
$608$	0	0
$609$	−5.20627	−0.210969
$610$	0	0
$611$	3.59151	0.145297
$612$	0	0
$613$	39.1422	1.58094	0.790470	−	0.612501i	$-0.209836\pi$
$613$	39.1422	1.58094	0.790470	+	0.612501i	$0.209836\pi$
$614$	0	0
$615$	−1.07541	−0.0433645
$616$	0	0
$617$	−31.7191	−1.27696	−0.638481	−	0.769638i	$-0.720437\pi$
$617$	−31.7191	−1.27696	−0.638481	+	0.769638i	$0.720437\pi$
$618$	0	0
$619$	−0.246181	−0.00989486	−0.00494743	−	0.999988i	$-0.501575\pi$
$619$	−0.246181	−0.00989486	−0.00494743	+	0.999988i	$0.501575\pi$
$620$	0	0
$621$	−2.90135	−0.116427
$622$	0	0
$623$	−5.59151	−0.224019
$624$	0	0
$625$	24.6519	0.986076
$626$	0	0
$627$	−2.77618	−0.110870
$628$	0	0
$629$	55.3395	2.20653
$630$	0	0
$631$	−24.0587	−0.957763	−0.478882	−	0.877880i	$-0.658958\pi$
$631$	−24.0587	−0.957763	−0.478882	+	0.877880i	$0.658958\pi$
$632$	0	0
$633$	−3.67184	−0.145943
$634$	0	0
$635$	−3.19943	−0.126965
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−5.82430	−0.230406
$640$	0	0
$641$	−18.1962	−0.718705	−0.359352	−	0.933202i	$-0.617002\pi$
$641$	−18.1962	−0.718705	−0.359352	+	0.933202i	$0.617002\pi$
$642$	0	0
$643$	21.3926	0.843641	0.421820	−	0.906679i	$-0.361391\pi$
$643$	21.3926	0.843641	0.421820	+	0.906679i	$0.361391\pi$
$644$	0	0
$645$	0.442333	0.0174168
$646$	0	0
$647$	−19.8557	−0.780610	−0.390305	−	0.920686i	$-0.627630\pi$
$647$	−19.8557	−0.780610	−0.390305	+	0.920686i	$0.627630\pi$
$648$	0	0
$649$	−2.24378	−0.0880762
$650$	0	0
$651$	−1.76721	−0.0692625
$652$	0	0
$653$	−34.4182	−1.34689	−0.673445	−	0.739238i	$-0.735186\pi$
$653$	−34.4182	−1.34689	−0.673445	+	0.739238i	$0.735186\pi$
$654$	0	0
$655$	1.39257	0.0544121
$656$	0	0
$657$	3.09865	0.120890
$658$	0	0
$659$	−35.7851	−1.39399	−0.696996	−	0.717075i	$-0.745480\pi$
$659$	−35.7851	−1.39399	−0.696996	+	0.717075i	$0.745480\pi$
$660$	0	0
$661$	18.5988	0.723411	0.361705	−	0.932292i	$-0.382195\pi$
$661$	18.5988	0.723411	0.361705	+	0.932292i	$0.382195\pi$
$662$	0	0
$663$	7.43905	0.288909
$664$	0	0
$665$	−1.09865	−0.0426038
$666$	0	0
$667$	15.1052	0.584876
$668$	0	0
$669$	12.6453	0.488896
$670$	0	0
$671$	−4.82114	−0.186118
$672$	0	0
$673$	−42.6576	−1.64433	−0.822164	−	0.569250i	$-0.807233\pi$
$673$	−42.6576	−1.64433	−0.822164	+	0.569250i	$0.807233\pi$
$674$	0	0
$675$	−4.97676	−0.191555
$676$	0	0
$677$	32.3660	1.24393	0.621964	−	0.783046i	$-0.286335\pi$
$677$	32.3660	1.24393	0.621964	+	0.783046i	$0.286335\pi$
$678$	0	0
$679$	18.9502	0.727243
$680$	0	0
$681$	−6.59479	−0.252713
$682$	0	0
$683$	−7.40761	−0.283444	−0.141722	−	0.989906i	$-0.545264\pi$
$683$	−7.40761	−0.283444	−0.141722	+	0.989906i	$0.545264\pi$
$684$	0	0
$685$	−3.44561	−0.131650
$686$	0	0
$687$	−2.60983	−0.0995712
$688$	0	0
$689$	−10.9502	−0.417171
$690$	0	0
$691$	−39.0147	−1.48419	−0.742094	−	0.670296i	$-0.766167\pi$
$691$	−39.0147	−1.48419	−0.742094	+	0.670296i	$0.766167\pi$
$692$	0	0
$693$	−0.385245	−0.0146343
$694$	0	0
$695$	−3.01757	−0.114463
$696$	0	0
$697$	52.4737	1.98758
$698$	0	0
$699$	−1.35468	−0.0512387
$700$	0	0
$701$	4.27598	0.161502	0.0807508	−	0.996734i	$-0.474268\pi$
$701$	4.27598	0.161502	0.0807508	+	0.996734i	$0.474268\pi$
$702$	0	0
$703$	53.6078	2.02186
$704$	0	0
$705$	−0.547553	−0.0206220
$706$	0	0
$707$	11.3803	0.428001
$708$	0	0
$709$	45.5425	1.71038	0.855192	−	0.518311i	$-0.173439\pi$
$709$	45.5425	1.71038	0.855192	+	0.518311i	$0.173439\pi$
$710$	0	0
$711$	−12.6453	−0.474237
$712$	0	0
$713$	5.12730	0.192019
$714$	0	0
$715$	0.0587335	0.00219651
$716$	0	0
$717$	−14.4341	−0.539052
$718$	0	0
$719$	−23.3005	−0.868962	−0.434481	−	0.900681i	$-0.643068\pi$
$719$	−23.3005	−0.868962	−0.434481	+	0.900681i	$0.643068\pi$
$720$	0	0
$721$	−5.13414	−0.191205
$722$	0	0
$723$	17.7306	0.659407
$724$	0	0
$725$	25.9103	0.962285
$726$	0	0
$727$	−1.55652	−0.0577282	−0.0288641	−	0.999583i	$-0.509189\pi$
$727$	−1.55652	−0.0577282	−0.0288641	+	0.999583i	$0.509189\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.5833	−0.798287
$732$	0	0
$733$	32.6087	1.20443	0.602215	−	0.798334i	$-0.294285\pi$
$733$	32.6087	1.20443	0.602215	+	0.798334i	$0.294285\pi$
$734$	0	0
$735$	−0.152457	−0.00562348
$736$	0	0
$737$	3.77645	0.139107
$738$	0	0
$739$	−28.5633	−1.05072	−0.525360	−	0.850880i	$-0.676069\pi$
$739$	−28.5633	−1.05072	−0.525360	+	0.850880i	$0.676069\pi$
$740$	0	0
$741$	7.20627	0.264729
$742$	0	0
$743$	20.4440	0.750017	0.375008	−	0.927021i	$-0.377640\pi$
$743$	20.4440	0.750017	0.375008	+	0.927021i	$0.377640\pi$
$744$	0	0
$745$	1.92017	0.0703496
$746$	0	0
$747$	−11.8964	−0.435267
$748$	0	0
$749$	18.8781	0.689791
$750$	0	0
$751$	−11.8748	−0.433319	−0.216659	−	0.976247i	$-0.569516\pi$
$751$	−11.8748	−0.433319	−0.216659	+	0.976247i	$0.569516\pi$
$752$	0	0
$753$	−4.05873	−0.147908
$754$	0	0
$755$	3.22206	0.117263
$756$	0	0
$757$	−37.1720	−1.35104	−0.675520	−	0.737342i	$-0.736081\pi$
$757$	−37.1720	−1.35104	−0.675520	+	0.737342i	$0.736081\pi$
$758$	0	0
$759$	1.11773	0.0405711
$760$	0	0
$761$	−44.8920	−1.62733	−0.813667	−	0.581331i	$-0.802532\pi$
$761$	−44.8920	−1.62733	−0.813667	+	0.581331i	$0.802532\pi$
$762$	0	0
$763$	19.7440	0.714780
$764$	0	0
$765$	−1.13414	−0.0410049
$766$	0	0
$767$	5.82430	0.210303
$768$	0	0
$769$	46.7008	1.68407	0.842036	−	0.539421i	$-0.181357\pi$
$769$	46.7008	1.68407	0.842036	+	0.539421i	$0.181357\pi$
$770$	0	0
$771$	−30.2584	−1.08973
$772$	0	0
$773$	9.56827	0.344147	0.172073	−	0.985084i	$-0.444953\pi$
$773$	9.56827	0.344147	0.172073	+	0.985084i	$0.444953\pi$
$774$	0	0
$775$	8.79498	0.315925
$776$	0	0
$777$	7.43905	0.266875
$778$	0	0
$779$	50.8316	1.82123
$780$	0	0
$781$	2.24378	0.0802889
$782$	0	0
$783$	−5.20627	−0.186057
$784$	0	0
$785$	−2.35380	−0.0840107
$786$	0	0
$787$	40.2898	1.43617	0.718087	−	0.695953i	$-0.245018\pi$
$787$	40.2898	1.43617	0.718087	+	0.695953i	$0.245018\pi$
$788$	0	0
$789$	−12.9502	−0.461041
$790$	0	0
$791$	0.842618	0.0299600
$792$	0	0
$793$	12.5145	0.444401
$794$	0	0
$795$	1.66944	0.0592091
$796$	0	0
$797$	−23.6584	−0.838025	−0.419013	−	0.907980i	$-0.637624\pi$
$797$	−23.6584	−0.838025	−0.419013	+	0.907980i	$0.637624\pi$
$798$	0	0
$799$	26.7174	0.945195
$800$	0	0
$801$	−5.59151	−0.197566
$802$	0	0
$803$	−1.19374	−0.0421262
$804$	0	0
$805$	0.442333	0.0155902
$806$	0	0
$807$	17.7929	0.626338
$808$	0	0
$809$	−38.7431	−1.36213	−0.681067	−	0.732221i	$-0.738484\pi$
$809$	−38.7431	−1.36213	−0.681067	+	0.732221i	$0.738484\pi$
$810$	0	0
$811$	1.86144	0.0653639	0.0326819	−	0.999466i	$-0.489595\pi$
$811$	1.86144	0.0653639	0.0326819	+	0.999466i	$0.489595\pi$
$812$	0	0
$813$	−17.9470	−0.629427
$814$	0	0
$815$	−0.492363	−0.0172467
$816$	0	0
$817$	−20.9079	−0.731475
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	10.7024	0.373517	0.186758	−	0.982406i	$-0.440202\pi$
$821$	10.7024	0.373517	0.186758	+	0.982406i	$0.440202\pi$
$822$	0	0
$823$	−38.9759	−1.35861	−0.679307	−	0.733854i	$-0.737719\pi$
$823$	−38.9759	−1.35861	−0.679307	+	0.733854i	$0.737719\pi$
$824$	0	0
$825$	1.91727	0.0667509
$826$	0	0
$827$	−24.1349	−0.839253	−0.419626	−	0.907697i	$-0.637839\pi$
$827$	−24.1349	−0.839253	−0.419626	+	0.907697i	$0.637839\pi$
$828$	0	0
$829$	10.3636	0.359944	0.179972	−	0.983672i	$-0.442399\pi$
$829$	10.3636	0.359944	0.179972	+	0.983672i	$0.442399\pi$
$830$	0	0
$831$	−5.87483	−0.203796
$832$	0	0
$833$	7.43905	0.257748
$834$	0	0
$835$	2.64292	0.0914621
$836$	0	0
$837$	−1.76721	−0.0610838
$838$	0	0
$839$	−22.7855	−0.786644	−0.393322	−	0.919401i	$-0.628674\pi$
$839$	−22.7855	−0.786644	−0.393322	+	0.919401i	$0.628674\pi$
$840$	0	0
$841$	−1.89479	−0.0653377
$842$	0	0
$843$	−26.5417	−0.914146
$844$	0	0
$845$	−0.152457	−0.00524470
$846$	0	0
$847$	−10.8516	−0.372865
$848$	0	0
$849$	−17.7562	−0.609392
$850$	0	0
$851$	−21.5833	−0.739866
$852$	0	0
$853$	−24.0289	−0.822735	−0.411367	−	0.911470i	$-0.634949\pi$
$853$	−24.0289	−0.822735	−0.411367	+	0.911470i	$0.634949\pi$
$854$	0	0
$855$	−1.09865	−0.0375730
$856$	0	0
$857$	26.8683	0.917802	0.458901	−	0.888487i	$-0.348243\pi$
$857$	26.8683	0.917802	0.458901	+	0.888487i	$0.348243\pi$
$858$	0	0
$859$	−18.6164	−0.635183	−0.317591	−	0.948228i	$-0.602874\pi$
$859$	−18.6164	−0.635183	−0.317591	+	0.948228i	$0.602874\pi$
$860$	0	0
$861$	7.05381	0.240393
$862$	0	0
$863$	−19.8341	−0.675162	−0.337581	−	0.941296i	$-0.609609\pi$
$863$	−19.8341	−0.675162	−0.337581	+	0.941296i	$0.609609\pi$
$864$	0	0
$865$	−1.30705	−0.0444409
$866$	0	0
$867$	38.3395	1.30208
$868$	0	0
$869$	4.87155	0.165256
$870$	0	0
$871$	−9.80270	−0.332152
$872$	0	0
$873$	18.9502	0.641368
$874$	0	0
$875$	1.52103	0.0514202
$876$	0	0
$877$	−7.07541	−0.238919	−0.119460	−	0.992839i	$-0.538116\pi$
$877$	−7.07541	−0.238919	−0.119460	+	0.992839i	$0.538116\pi$
$878$	0	0
$879$	−14.0139	−0.472677
$880$	0	0
$881$	−23.0412	−0.776277	−0.388138	−	0.921601i	$-0.626882\pi$
$881$	−23.0412	−0.776277	−0.388138	+	0.921601i	$0.626882\pi$
$882$	0	0
$883$	21.6462	0.728453	0.364226	−	0.931310i	$-0.381333\pi$
$883$	21.6462	0.728453	0.364226	+	0.931310i	$0.381333\pi$
$884$	0	0
$885$	−0.887958	−0.0298484
$886$	0	0
$887$	52.1223	1.75009	0.875047	−	0.484038i	$-0.160830\pi$
$887$	52.1223	1.75009	0.875047	+	0.484038i	$0.160830\pi$
$888$	0	0
$889$	20.9857	0.703839
$890$	0	0
$891$	−0.385245	−0.0129062
$892$	0	0
$893$	25.8814	0.866088
$894$	0	0
$895$	0.176452	0.00589812
$896$	0	0
$897$	−2.90135	−0.0968733
$898$	0	0
$899$	9.20058	0.306856
$900$	0	0
$901$	−81.4594	−2.71381
$902$	0	0
$903$	−2.90135	−0.0965509
$904$	0	0
$905$	−1.00772	−0.0334977
$906$	0	0
$907$	−38.2841	−1.27120	−0.635601	−	0.772018i	$-0.719248\pi$
$907$	−38.2841	−1.27120	−0.635601	+	0.772018i	$0.719248\pi$
$908$	0	0
$909$	11.3803	0.377461
$910$	0	0
$911$	−15.8161	−0.524011	−0.262005	−	0.965066i	$-0.584384\pi$
$911$	−15.8161	−0.524011	−0.262005	+	0.965066i	$0.584384\pi$
$912$	0	0
$913$	4.58304	0.151677
$914$	0	0
$915$	−1.90792	−0.0630740
$916$	0	0
$917$	−9.13414	−0.301636
$918$	0	0
$919$	−12.0546	−0.397644	−0.198822	−	0.980036i	$-0.563712\pi$
$919$	−12.0546	−0.397644	−0.198822	+	0.980036i	$0.563712\pi$
$920$	0	0
$921$	−22.4912	−0.741111
$922$	0	0
$923$	−5.82430	−0.191709
$924$	0	0
$925$	−37.0224	−1.21729
$926$	0	0
$927$	−5.13414	−0.168627
$928$	0	0
$929$	−1.98940	−0.0652701	−0.0326350	−	0.999467i	$-0.510390\pi$
$929$	−1.98940	−0.0652701	−0.0326350	+	0.999467i	$0.510390\pi$
$930$	0	0
$931$	7.20627	0.236176
$932$	0	0
$933$	33.1300	1.08463
$934$	0	0
$935$	0.436922	0.0142889
$936$	0	0
$937$	1.31919	0.0430961	0.0215480	−	0.999768i	$-0.493141\pi$
$937$	1.31919	0.0430961	0.0215480	+	0.999768i	$0.493141\pi$
$938$	0	0
$939$	19.3371	0.631043
$940$	0	0
$941$	12.9107	0.420877	0.210438	−	0.977607i	$-0.432511\pi$
$941$	12.9107	0.420877	0.210438	+	0.977607i	$0.432511\pi$
$942$	0	0
$943$	−20.4656	−0.666451
$944$	0	0
$945$	−0.152457	−0.00495944
$946$	0	0
$947$	−46.7081	−1.51781	−0.758905	−	0.651202i	$-0.774265\pi$
$947$	−46.7081	−1.51781	−0.758905	+	0.651202i	$0.774265\pi$
$948$	0	0
$949$	3.09865	0.100586
$950$	0	0
$951$	−16.0860	−0.521625
$952$	0	0
$953$	−58.0355	−1.87995	−0.939977	−	0.341238i	$-0.889154\pi$
$953$	−58.0355	−1.87995	−0.939977	+	0.341238i	$0.889154\pi$
$954$	0	0
$955$	−3.05102	−0.0987286
$956$	0	0
$957$	2.00569	0.0648347
$958$	0	0
$959$	22.6005	0.729808
$960$	0	0
$961$	−27.8770	−0.899257
$962$	0	0
$963$	18.8781	0.608339
$964$	0	0
$965$	0.133874	0.00430956
$966$	0	0
$967$	−50.0057	−1.60807	−0.804037	−	0.594579i	$-0.797319\pi$
$967$	−50.0057	−1.60807	−0.804037	+	0.594579i	$0.797319\pi$
$968$	0	0
$969$	53.6078	1.72213
$970$	0	0
$971$	−52.3595	−1.68030	−0.840148	−	0.542357i	$-0.817532\pi$
$971$	−52.3595	−1.68030	−0.840148	+	0.542357i	$0.817532\pi$
$972$	0	0
$973$	19.7929	0.634530
$974$	0	0
$975$	−4.97676	−0.159384
$976$	0	0
$977$	−17.3098	−0.553791	−0.276895	−	0.960900i	$-0.589306\pi$
$977$	−17.3098	−0.553791	−0.276895	+	0.960900i	$0.589306\pi$
$978$	0	0
$979$	2.15410	0.0688455
$980$	0	0
$981$	19.7440	0.630376
$982$	0	0
$983$	−47.5409	−1.51632	−0.758159	−	0.652070i	$-0.773901\pi$
$983$	−47.5409	−1.51632	−0.758159	+	0.652070i	$0.773901\pi$
$984$	0	0
$985$	0.0512808	0.00163394
$986$	0	0
$987$	3.59151	0.114319
$988$	0	0
$989$	8.41784	0.267672
$990$	0	0
$991$	−11.5809	−0.367880	−0.183940	−	0.982937i	$-0.558885\pi$
$991$	−11.5809	−0.367880	−0.183940	+	0.982937i	$0.558885\pi$
$992$	0	0
$993$	27.9535	0.887078
$994$	0	0
$995$	0.900205	0.0285384
$996$	0	0
$997$	16.5103	0.522886	0.261443	−	0.965219i	$-0.415802\pi$
$997$	16.5103	0.522886	0.261443	+	0.965219i	$0.415802\pi$
$998$	0	0
$999$	7.43905	0.235361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bs.1.2		4
4.3	odd	2		2184.2.a.v.1.2	✓	4
12.11	even	2		6552.2.a.bs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.v.1.2	✓	4	4.3	odd	2
4368.2.a.bs.1.2		4	1.1	even	1	trivial
6552.2.a.bs.1.3		4	12.11	even	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.152457 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.152457 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.385245 q^{11} +1.00000 q^{13} -0.152457 q^{15} +7.43905 q^{17} +7.20627 q^{19} +1.00000 q^{21} -2.90135 q^{23} -4.97676 q^{25} +1.00000 q^{27} -5.20627 q^{29} -1.76721 q^{31} -0.385245 q^{33} -0.152457 q^{35} +7.43905 q^{37} +1.00000 q^{39} +7.05381 q^{41} -2.90135 q^{43} -0.152457 q^{45} +3.59151 q^{47} +1.00000 q^{49} +7.43905 q^{51} -10.9502 q^{53} +0.0587335 q^{55} +7.20627 q^{57} +5.82430 q^{59} +12.5145 q^{61} +1.00000 q^{63} -0.152457 q^{65} -9.80270 q^{67} -2.90135 q^{69} -5.82430 q^{71} +3.09865 q^{73} -4.97676 q^{75} -0.385245 q^{77} -12.6453 q^{79} +1.00000 q^{81} -11.8964 q^{83} -1.13414 q^{85} -5.20627 q^{87} -5.59151 q^{89} +1.00000 q^{91} -1.76721 q^{93} -1.09865 q^{95} +18.9502 q^{97} -0.385245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * q^99

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