LMFDB - Embedded newform 387.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,2,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("387.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	387.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:	$3.09021055822$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 4 $ x^4 - 9*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.684742$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +3.00000 q^{4} -0.684742 q^{5} +3.53113 q^{7} +2.23607 q^{8} +O(q^{10})$	$q+2.23607 q^{2} +3.00000 q^{4} -0.684742 q^{5} +3.53113 q^{7} +2.23607 q^{8} -1.53113 q^{10} -0.684742 q^{11} -1.53113 q^{13} +7.89584 q^{14} -1.00000 q^{16} +1.36948 q^{17} +3.53113 q^{19} -2.05422 q^{20} -1.53113 q^{22} -5.84162 q^{23} -4.53113 q^{25} -3.42371 q^{26} +10.5934 q^{28} +2.05422 q^{29} -6.70820 q^{32} +3.06226 q^{34} -2.41791 q^{35} +6.00000 q^{37} +7.89584 q^{38} -1.53113 q^{40} -11.6832 q^{41} +1.00000 q^{43} -2.05422 q^{44} -13.0623 q^{46} +2.05422 q^{47} +5.46887 q^{49} -10.1319 q^{50} -4.59339 q^{52} -11.6832 q^{53} +0.468871 q^{55} +7.89584 q^{56} +4.59339 q^{58} -5.84162 q^{59} -2.00000 q^{61} -13.0000 q^{64} +1.04843 q^{65} +4.00000 q^{67} +4.10845 q^{68} -5.40661 q^{70} +13.0527 q^{71} +6.00000 q^{73} +13.4164 q^{74} +10.5934 q^{76} -2.41791 q^{77} -15.0623 q^{79} +0.684742 q^{80} -26.1245 q^{82} +10.9985 q^{83} -0.937742 q^{85} +2.23607 q^{86} -1.53113 q^{88} +13.4164 q^{89} -5.40661 q^{91} -17.5249 q^{92} +4.59339 q^{94} -2.41791 q^{95} +13.5311 q^{97} +12.2288 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{4} - 2 q^{7}+O(q^{10}) $ 4 * q + 12 * q^4 - 2 * q^7	$ 4 q + 12 q^{4} - 2 q^{7} + 10 q^{10} + 10 q^{13} - 4 q^{16} - 2 q^{19} + 10 q^{22} - 2 q^{25} - 6 q^{28} - 20 q^{34} + 24 q^{37} + 10 q^{40} + 4 q^{43} - 20 q^{46} + 38 q^{49} + 30 q^{52} + 18 q^{55} - 30 q^{58} - 8 q^{61} - 52 q^{64} + 16 q^{67} - 70 q^{70} + 24 q^{73} - 6 q^{76} - 28 q^{79} - 40 q^{82} - 36 q^{85} + 10 q^{88} - 70 q^{91} - 30 q^{94} + 38 q^{97}+O(q^{100}) $ 4 * q + 12 * q^4 - 2 * q^7 + 10 * q^10 + 10 * q^13 - 4 * q^16 - 2 * q^19 + 10 * q^22 - 2 * q^25 - 6 * q^28 - 20 * q^34 + 24 * q^37 + 10 * q^40 + 4 * q^43 - 20 * q^46 + 38 * q^49 + 30 * q^52 + 18 * q^55 - 30 * q^58 - 8 * q^61 - 52 * q^64 + 16 * q^67 - 70 * q^70 + 24 * q^73 - 6 * q^76 - 28 * q^79 - 40 * q^82 - 36 * q^85 + 10 * q^88 - 70 * q^91 - 30 * q^94 + 38 * q^97

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	−0.684742	−0.306226	−0.153113	−	0.988209i	$-0.548930\pi$
$5$	−0.684742	−0.306226	−0.153113	+	0.988209i	$0.548930\pi$
$6$	0	0
$7$	3.53113	1.33464	0.667321	−	0.744771i	$-0.267441\pi$
$7$	3.53113	1.33464	0.667321	+	0.744771i	$0.267441\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−1.53113	−0.484185
$11$	−0.684742	−0.206457	−0.103229	−	0.994658i	$-0.532917\pi$
$11$	−0.684742	−0.206457	−0.103229	+	0.994658i	$0.532917\pi$
$12$	0	0
$13$	−1.53113	−0.424659	−0.212329	−	0.977198i	$-0.568105\pi$
$13$	−1.53113	−0.424659	−0.212329	+	0.977198i	$0.568105\pi$
$14$	7.89584	2.11025
$15$	0	0
$16$	−1.00000	−0.250000
$17$	1.36948	0.332148	0.166074	−	0.986113i	$-0.446891\pi$
$17$	1.36948	0.332148	0.166074	+	0.986113i	$0.446891\pi$
$18$	0	0
$19$	3.53113	0.810097	0.405048	−	0.914295i	$-0.367255\pi$
$19$	3.53113	0.810097	0.405048	+	0.914295i	$0.367255\pi$
$20$	−2.05422	−0.459339
$21$	0	0
$22$	−1.53113	−0.326438
$23$	−5.84162	−1.21806	−0.609031	−	0.793146i	$-0.708441\pi$
$23$	−5.84162	−1.21806	−0.609031	+	0.793146i	$0.708441\pi$
$24$	0	0
$25$	−4.53113	−0.906226
$26$	−3.42371	−0.671444
$27$	0	0
$28$	10.5934	2.00196
$29$	2.05422	0.381460	0.190730	−	0.981643i	$-0.438915\pi$
$29$	2.05422	0.381460	0.190730	+	0.981643i	$0.438915\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	3.06226	0.525173
$35$	−2.41791	−0.408702
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	7.89584	1.28088
$39$	0	0
$40$	−1.53113	−0.242093
$41$	−11.6832	−1.82462	−0.912308	−	0.409505i	$-0.865701\pi$
$41$	−11.6832	−1.82462	−0.912308	+	0.409505i	$0.865701\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−2.05422	−0.309686
$45$	0	0
$46$	−13.0623	−1.92592
$47$	2.05422	0.299640	0.149820	−	0.988713i	$-0.452131\pi$
$47$	2.05422	0.299640	0.149820	+	0.988713i	$0.452131\pi$
$48$	0	0
$49$	5.46887	0.781267
$50$	−10.1319	−1.43287
$51$	0	0
$52$	−4.59339	−0.636988
$53$	−11.6832	−1.60482	−0.802408	−	0.596776i	$-0.796448\pi$
$53$	−11.6832	−1.60482	−0.802408	+	0.596776i	$0.796448\pi$
$54$	0	0
$55$	0.468871	0.0632226
$56$	7.89584	1.05513
$57$	0	0
$58$	4.59339	0.603141
$59$	−5.84162	−0.760514	−0.380257	−	0.924881i	$-0.624164\pi$
$59$	−5.84162	−0.760514	−0.380257	+	0.924881i	$0.624164\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	−13.0000	−1.62500
$65$	1.04843	0.130041
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	4.10845	0.498223
$69$	0	0
$70$	−5.40661	−0.646214
$71$	13.0527	1.54907	0.774537	−	0.632529i	$-0.217983\pi$
$71$	13.0527	1.54907	0.774537	+	0.632529i	$0.217983\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	13.4164	1.55963
$75$	0	0
$76$	10.5934	1.21514
$77$	−2.41791	−0.275547
$78$	0	0
$79$	−15.0623	−1.69464	−0.847318	−	0.531086i	$-0.821784\pi$
$79$	−15.0623	−1.69464	−0.847318	+	0.531086i	$0.821784\pi$
$80$	0.684742	0.0765564
$81$	0	0
$82$	−26.1245	−2.88497
$83$	10.9985	1.20724	0.603621	−	0.797271i	$-0.293724\pi$
$83$	10.9985	1.20724	0.603621	+	0.797271i	$0.293724\pi$
$84$	0	0
$85$	−0.937742	−0.101712
$86$	2.23607	0.241121
$87$	0	0
$88$	−1.53113	−0.163219
$89$	13.4164	1.42214	0.711068	−	0.703123i	$-0.248212\pi$
$89$	13.4164	1.42214	0.711068	+	0.703123i	$0.248212\pi$
$90$	0	0
$91$	−5.40661	−0.566767
$92$	−17.5249	−1.82709
$93$	0	0
$94$	4.59339	0.473772
$95$	−2.41791	−0.248072
$96$	0	0
$97$	13.5311	1.37388	0.686939	−	0.726715i	$-0.258954\pi$
$97$	13.5311	1.37388	0.686939	+	0.726715i	$0.258954\pi$
$98$	12.2288	1.23529
$99$	0	0
$100$	−13.5934	−1.35934
$101$	8.94427	0.889988	0.444994	−	0.895533i	$-0.353206\pi$
$101$	8.94427	0.889988	0.444994	+	0.895533i	$0.353206\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−3.42371	−0.335722
$105$	0	0
$106$	−26.1245	−2.53744
$107$	−10.9985	−1.06326	−0.531632	−	0.846975i	$-0.678421\pi$
$107$	−10.9985	−1.06326	−0.531632	+	0.846975i	$0.678421\pi$
$108$	0	0
$109$	16.5934	1.58936	0.794679	−	0.607030i	$-0.207639\pi$
$109$	16.5934	1.58936	0.794679	+	0.607030i	$0.207639\pi$
$110$	1.04843	0.0999637
$111$	0	0
$112$	−3.53113	−0.333660
$113$	−0.684742	−0.0644151	−0.0322075	−	0.999481i	$-0.510254\pi$
$113$	−0.684742	−0.0644151	−0.0322075	+	0.999481i	$0.510254\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	6.16267	0.572190
$117$	0	0
$118$	−13.0623	−1.20248
$119$	4.83582	0.443299
$120$	0	0
$121$	−10.5311	−0.957375
$122$	−4.47214	−0.404888
$123$	0	0
$124$	0	0
$125$	6.52636	0.583735
$126$	0	0
$127$	−7.06226	−0.626674	−0.313337	−	0.949642i	$-0.601447\pi$
$127$	−7.06226	−0.626674	−0.313337	+	0.949642i	$0.601447\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	2.34436	0.205614
$131$	11.6832	1.02077	0.510385	−	0.859946i	$-0.329503\pi$
$131$	11.6832	1.02077	0.510385	+	0.859946i	$0.329503\pi$
$132$	0	0
$133$	12.4689	1.08119
$134$	8.94427	0.772667
$135$	0	0
$136$	3.06226	0.262586
$137$	−2.05422	−0.175504	−0.0877521	−	0.996142i	$-0.527968\pi$
$137$	−2.05422	−0.175504	−0.0877521	+	0.996142i	$0.527968\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−7.25373	−0.613052
$141$	0	0
$142$	29.1868	2.44930
$143$	1.04843	0.0876739
$144$	0	0
$145$	−1.40661	−0.116813
$146$	13.4164	1.11035
$147$	0	0
$148$	18.0000	1.47959
$149$	21.6333	1.77227	0.886135	−	0.463428i	$-0.153381\pi$
$149$	21.6333	1.77227	0.886135	+	0.463428i	$0.153381\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	7.89584	0.640438
$153$	0	0
$154$	−5.40661	−0.435677
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−33.6802	−2.67946
$159$	0	0
$160$	4.59339	0.363139
$161$	−20.6275	−1.62568
$162$	0	0
$163$	−14.5934	−1.14304	−0.571521	−	0.820587i	$-0.693646\pi$
$163$	−14.5934	−1.14304	−0.571521	+	0.820587i	$0.693646\pi$
$164$	−35.0497	−2.73692
$165$	0	0
$166$	24.5934	1.90882
$167$	−15.1069	−1.16901	−0.584505	−	0.811390i	$-0.698711\pi$
$167$	−15.1069	−1.16901	−0.584505	+	0.811390i	$0.698711\pi$
$168$	0	0
$169$	−10.6556	−0.819665
$170$	−2.09686	−0.160821
$171$	0	0
$172$	3.00000	0.228748
$173$	17.8885	1.36004	0.680020	−	0.733193i	$-0.261971\pi$
$173$	17.8885	1.36004	0.680020	+	0.733193i	$0.261971\pi$
$174$	0	0
$175$	−16.0000	−1.20949
$176$	0.684742	0.0516143
$177$	0	0
$178$	30.0000	2.24860
$179$	−4.83582	−0.361446	−0.180723	−	0.983534i	$-0.557844\pi$
$179$	−4.83582	−0.361446	−0.180723	+	0.983534i	$0.557844\pi$
$180$	0	0
$181$	13.5311	1.00576	0.502880	−	0.864356i	$-0.332274\pi$
$181$	13.5311	1.00576	0.502880	+	0.864356i	$0.332274\pi$
$182$	−12.0896	−0.896137
$183$	0	0
$184$	−13.0623	−0.962962
$185$	−4.10845	−0.302059
$186$	0	0
$187$	−0.937742	−0.0685745
$188$	6.16267	0.449459
$189$	0	0
$190$	−5.40661	−0.392237
$191$	−11.6832	−0.845369	−0.422685	−	0.906277i	$-0.638912\pi$
$191$	−11.6832	−0.845369	−0.422685	+	0.906277i	$0.638912\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	30.2565	2.17229
$195$	0	0
$196$	16.4066	1.17190
$197$	−21.9970	−1.56722	−0.783610	−	0.621252i	$-0.786624\pi$
$197$	−21.9970	−1.56722	−0.783610	+	0.621252i	$0.786624\pi$
$198$	0	0
$199$	−26.1245	−1.85192	−0.925959	−	0.377624i	$-0.876741\pi$
$199$	−26.1245	−1.85192	−0.925959	+	0.377624i	$0.876741\pi$
$200$	−10.1319	−0.716434
$201$	0	0
$202$	20.0000	1.40720
$203$	7.25373	0.509112
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	0	0
$208$	1.53113	0.106165
$209$	−2.41791	−0.167250
$210$	0	0
$211$	−16.4689	−1.13376	−0.566882	−	0.823799i	$-0.691850\pi$
$211$	−16.4689	−1.13376	−0.566882	+	0.823799i	$0.691850\pi$
$212$	−35.0497	−2.40722
$213$	0	0
$214$	−24.5934	−1.68117
$215$	−0.684742	−0.0466990
$216$	0	0
$217$	0	0
$218$	37.1039	2.51300
$219$	0	0
$220$	1.40661	0.0948339
$221$	−2.09686	−0.141050
$222$	0	0
$223$	−13.6556	−0.914450	−0.457225	−	0.889351i	$-0.651157\pi$
$223$	−13.6556	−0.914450	−0.457225	+	0.889351i	$0.651157\pi$
$224$	−23.6875	−1.58269
$225$	0	0
$226$	−1.53113	−0.101849
$227$	26.1054	1.73268	0.866340	−	0.499455i	$-0.166467\pi$
$227$	26.1054	1.73268	0.866340	+	0.499455i	$0.166467\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	8.94427	0.589768
$231$	0	0
$232$	4.59339	0.301571
$233$	21.3123	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$233$	21.3123	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$234$	0	0
$235$	−1.40661	−0.0917573
$236$	−17.5249	−1.14077
$237$	0	0
$238$	10.8132	0.700917
$239$	28.8870	1.86855	0.934274	−	0.356557i	$-0.116049\pi$
$239$	28.8870	1.86855	0.934274	+	0.356557i	$0.116049\pi$
$240$	0	0
$241$	19.1868	1.23593	0.617964	−	0.786206i	$-0.287958\pi$
$241$	19.1868	1.23593	0.617964	+	0.786206i	$0.287958\pi$
$242$	−23.5483	−1.51374
$243$	0	0
$244$	−6.00000	−0.384111
$245$	−3.74476	−0.239244
$246$	0	0
$247$	−5.40661	−0.344015
$248$	0	0
$249$	0	0
$250$	14.5934	0.922967
$251$	−10.9985	−0.694219	−0.347109	−	0.937825i	$-0.612837\pi$
$251$	−10.9985	−0.694219	−0.347109	+	0.937825i	$0.612837\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−15.7917	−0.990859
$255$	0	0
$256$	−9.00000	−0.562500
$257$	2.05422	0.128139	0.0640695	−	0.997945i	$-0.479592\pi$
$257$	2.05422	0.128139	0.0640695	+	0.997945i	$0.479592\pi$
$258$	0	0
$259$	21.1868	1.31648
$260$	3.14528	0.195062
$261$	0	0
$262$	26.1245	1.61398
$263$	15.7917	0.973757	0.486879	−	0.873470i	$-0.338135\pi$
$263$	15.7917	0.973757	0.486879	+	0.873470i	$0.338135\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	27.8812	1.70951
$267$	0	0
$268$	12.0000	0.733017
$269$	4.10845	0.250497	0.125248	−	0.992125i	$-0.460027\pi$
$269$	4.10845	0.250497	0.125248	+	0.992125i	$0.460027\pi$
$270$	0	0
$271$	−21.1868	−1.28700	−0.643502	−	0.765444i	$-0.722519\pi$
$271$	−21.1868	−1.28700	−0.643502	+	0.765444i	$0.722519\pi$
$272$	−1.36948	−0.0830371
$273$	0	0
$274$	−4.59339	−0.277497
$275$	3.10265	0.187097
$276$	0	0
$277$	9.06226	0.544498	0.272249	−	0.962227i	$-0.412233\pi$
$277$	9.06226	0.544498	0.272249	+	0.962227i	$0.412233\pi$
$278$	26.8328	1.60933
$279$	0	0
$280$	−5.40661	−0.323107
$281$	13.0527	0.778660	0.389330	−	0.921098i	$-0.372707\pi$
$281$	13.0527	0.778660	0.389330	+	0.921098i	$0.372707\pi$
$282$	0	0
$283$	−19.0623	−1.13313	−0.566567	−	0.824016i	$-0.691729\pi$
$283$	−19.0623	−1.13313	−0.566567	+	0.824016i	$0.691729\pi$
$284$	39.1582	2.32361
$285$	0	0
$286$	2.34436	0.138625
$287$	−41.2550	−2.43521
$288$	0	0
$289$	−15.1245	−0.889677
$290$	−3.14528	−0.184697
$291$	0	0
$292$	18.0000	1.05337
$293$	6.84742	0.400030	0.200015	−	0.979793i	$-0.435901\pi$
$293$	6.84742	0.400030	0.200015	+	0.979793i	$0.435901\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	13.4164	0.779813
$297$	0	0
$298$	48.3735	2.80220
$299$	8.94427	0.517261
$300$	0	0
$301$	3.53113	0.203531
$302$	17.8885	1.02937
$303$	0	0
$304$	−3.53113	−0.202524
$305$	1.36948	0.0784164
$306$	0	0
$307$	4.93774	0.281812	0.140906	−	0.990023i	$-0.454998\pi$
$307$	4.93774	0.281812	0.140906	+	0.990023i	$0.454998\pi$
$308$	−7.25373	−0.413320
$309$	0	0
$310$	0	0
$311$	−28.8870	−1.63803	−0.819017	−	0.573769i	$-0.805481\pi$
$311$	−28.8870	−1.63803	−0.819017	+	0.573769i	$0.805481\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	31.3050	1.76664
$315$	0	0
$316$	−45.1868	−2.54195
$317$	−7.57479	−0.425443	−0.212721	−	0.977113i	$-0.568233\pi$
$317$	−7.57479	−0.425443	−0.212721	+	0.977113i	$0.568233\pi$
$318$	0	0
$319$	−1.40661	−0.0787552
$320$	8.90164	0.497617
$321$	0	0
$322$	−46.1245	−2.57042
$323$	4.83582	0.269072
$324$	0	0
$325$	6.93774	0.384837
$326$	−32.6318	−1.80731
$327$	0	0
$328$	−26.1245	−1.44249
$329$	7.25373	0.399911
$330$	0	0
$331$	−3.53113	−0.194088	−0.0970442	−	0.995280i	$-0.530939\pi$
$331$	−3.53113	−0.194088	−0.0970442	+	0.995280i	$0.530939\pi$
$332$	32.9955	1.81086
$333$	0	0
$334$	−33.7802	−1.84837
$335$	−2.73897	−0.149646
$336$	0	0
$337$	−3.40661	−0.185570	−0.0927850	−	0.995686i	$-0.529577\pi$
$337$	−3.40661	−0.185570	−0.0927850	+	0.995686i	$0.529577\pi$
$338$	−23.8267	−1.29600
$339$	0	0
$340$	−2.81323	−0.152569
$341$	0	0
$342$	0	0
$343$	−5.40661	−0.291930
$344$	2.23607	0.120561
$345$	0	0
$346$	40.0000	2.15041
$347$	−13.0527	−0.700707	−0.350353	−	0.936618i	$-0.613939\pi$
$347$	−13.0527	−0.700707	−0.350353	+	0.936618i	$0.613939\pi$
$348$	0	0
$349$	−31.1868	−1.66939	−0.834695	−	0.550713i	$-0.814356\pi$
$349$	−31.1868	−1.66939	−0.834695	+	0.550713i	$0.814356\pi$
$350$	−35.7771	−1.91237
$351$	0	0
$352$	4.59339	0.244828
$353$	−21.9970	−1.17078	−0.585391	−	0.810751i	$-0.699059\pi$
$353$	−21.9970	−1.17078	−0.585391	+	0.810751i	$0.699059\pi$
$354$	0	0
$355$	−8.93774	−0.474366
$356$	40.2492	2.13320
$357$	0	0
$358$	−10.8132	−0.571497
$359$	8.58059	0.452866	0.226433	−	0.974027i	$-0.427294\pi$
$359$	8.58059	0.452866	0.226433	+	0.974027i	$0.427294\pi$
$360$	0	0
$361$	−6.53113	−0.343744
$362$	30.2565	1.59025
$363$	0	0
$364$	−16.2198	−0.850151
$365$	−4.10845	−0.215046
$366$	0	0
$367$	−15.0623	−0.786243	−0.393122	−	0.919486i	$-0.628605\pi$
$367$	−15.0623	−0.786243	−0.393122	+	0.919486i	$0.628605\pi$
$368$	5.84162	0.304515
$369$	0	0
$370$	−9.18677	−0.477598
$371$	−41.2550	−2.14185
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−2.09686	−0.108426
$375$	0	0
$376$	4.59339	0.236886
$377$	−3.14528	−0.161990
$378$	0	0
$379$	2.12452	0.109129	0.0545645	−	0.998510i	$-0.482623\pi$
$379$	2.12452	0.109129	0.0545645	+	0.998510i	$0.482623\pi$
$380$	−7.25373	−0.372109
$381$	0	0
$382$	−26.1245	−1.33665
$383$	30.9413	1.58102	0.790512	−	0.612446i	$-0.209814\pi$
$383$	30.9413	1.58102	0.790512	+	0.612446i	$0.209814\pi$
$384$	0	0
$385$	1.65564	0.0843795
$386$	−22.3607	−1.13813
$387$	0	0
$388$	40.5934	2.06082
$389$	−19.9428	−1.01114	−0.505569	−	0.862786i	$-0.668717\pi$
$389$	−19.9428	−1.01114	−0.505569	+	0.862786i	$0.668717\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	12.2288	0.617646
$393$	0	0
$394$	−49.1868	−2.47799
$395$	10.3138	0.518941
$396$	0	0
$397$	−0.593387	−0.0297812	−0.0148906	−	0.999889i	$-0.504740\pi$
$397$	−0.593387	−0.0297812	−0.0148906	+	0.999889i	$0.504740\pi$
$398$	−58.4162	−2.92814
$399$	0	0
$400$	4.53113	0.226556
$401$	4.10845	0.205166	0.102583	−	0.994724i	$-0.467289\pi$
$401$	4.10845	0.205166	0.102583	+	0.994724i	$0.467289\pi$
$402$	0	0
$403$	0	0
$404$	26.8328	1.33498
$405$	0	0
$406$	16.2198	0.804977
$407$	−4.10845	−0.203648
$408$	0	0
$409$	−25.0623	−1.23925	−0.619624	−	0.784898i	$-0.712715\pi$
$409$	−25.0623	−1.23925	−0.619624	+	0.784898i	$0.712715\pi$
$410$	17.8885	0.883452
$411$	0	0
$412$	0	0
$413$	−20.6275	−1.01501
$414$	0	0
$415$	−7.53113	−0.369689
$416$	10.2711	0.503583
$417$	0	0
$418$	−5.40661	−0.264446
$419$	−32.3107	−1.57848	−0.789242	−	0.614083i	$-0.789526\pi$
$419$	−32.3107	−1.57848	−0.789242	+	0.614083i	$0.789526\pi$
$420$	0	0
$421$	2.93774	0.143177	0.0715884	−	0.997434i	$-0.477193\pi$
$421$	2.93774	0.143177	0.0715884	+	0.997434i	$0.477193\pi$
$422$	−36.8255	−1.79264
$423$	0	0
$424$	−26.1245	−1.26872
$425$	−6.20531	−0.301002
$426$	0	0
$427$	−7.06226	−0.341767
$428$	−32.9955	−1.59490
$429$	0	0
$430$	−1.53113	−0.0738376
$431$	4.79319	0.230880	0.115440	−	0.993314i	$-0.463172\pi$
$431$	4.79319	0.230880	0.115440	+	0.993314i	$0.463172\pi$
$432$	0	0
$433$	23.1868	1.11429	0.557143	−	0.830417i	$-0.311898\pi$
$433$	23.1868	1.11429	0.557143	+	0.830417i	$0.311898\pi$
$434$	0	0
$435$	0	0
$436$	49.7802	2.38404
$437$	−20.6275	−0.986748
$438$	0	0
$439$	21.1868	1.01119	0.505595	−	0.862771i	$-0.331273\pi$
$439$	21.1868	1.01119	0.505595	+	0.862771i	$0.331273\pi$
$440$	1.04843	0.0499818
$441$	0	0
$442$	−4.68871	−0.223019
$443$	4.79319	0.227731	0.113866	−	0.993496i	$-0.463677\pi$
$443$	4.79319	0.227731	0.113866	+	0.993496i	$0.463677\pi$
$444$	0	0
$445$	−9.18677	−0.435495
$446$	−30.5349	−1.44587
$447$	0	0
$448$	−45.9047	−2.16879
$449$	−6.16267	−0.290835	−0.145417	−	0.989370i	$-0.546452\pi$
$449$	−6.16267	−0.290835	−0.145417	+	0.989370i	$0.546452\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	−2.05422	−0.0966226
$453$	0	0
$454$	58.3735	2.73961
$455$	3.70213	0.173559
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	13.4164	0.626908
$459$	0	0
$460$	12.0000	0.559503
$461$	28.2023	1.31351	0.656756	−	0.754103i	$-0.271928\pi$
$461$	28.2023	1.31351	0.656756	+	0.754103i	$0.271928\pi$
$462$	0	0
$463$	22.5934	1.05000	0.525002	−	0.851101i	$-0.324065\pi$
$463$	22.5934	1.05000	0.525002	+	0.851101i	$0.324065\pi$
$464$	−2.05422	−0.0953650
$465$	0	0
$466$	47.6556	2.20761
$467$	−4.83582	−0.223775	−0.111888	−	0.993721i	$-0.535690\pi$
$467$	−4.83582	−0.223775	−0.111888	+	0.993721i	$0.535690\pi$
$468$	0	0
$469$	14.1245	0.652210
$470$	−3.14528	−0.145081
$471$	0	0
$472$	−13.0623	−0.601239
$473$	−0.684742	−0.0314845
$474$	0	0
$475$	−16.0000	−0.734130
$476$	14.5075	0.664949
$477$	0	0
$478$	64.5934	2.95443
$479$	−36.4618	−1.66598	−0.832992	−	0.553285i	$-0.813374\pi$
$479$	−36.4618	−1.66598	−0.832992	+	0.553285i	$0.813374\pi$
$480$	0	0
$481$	−9.18677	−0.418881
$482$	42.9029	1.95417
$483$	0	0
$484$	−31.5934	−1.43606
$485$	−9.26533	−0.420717
$486$	0	0
$487$	−7.06226	−0.320021	−0.160011	−	0.987115i	$-0.551153\pi$
$487$	−7.06226	−0.320021	−0.160011	+	0.987115i	$0.551153\pi$
$488$	−4.47214	−0.202444
$489$	0	0
$490$	−8.37355	−0.378278
$491$	13.7801	0.621887	0.310943	−	0.950428i	$-0.399355\pi$
$491$	13.7801	0.621887	0.310943	+	0.950428i	$0.399355\pi$
$492$	0	0
$493$	2.81323	0.126701
$494$	−12.0896	−0.543935
$495$	0	0
$496$	0	0
$497$	46.0908	2.06746
$498$	0	0
$499$	−10.5934	−0.474225	−0.237113	−	0.971482i	$-0.576201\pi$
$499$	−10.5934	−0.474225	−0.237113	+	0.971482i	$0.576201\pi$
$500$	19.5791	0.875603
$501$	0	0
$502$	−24.5934	−1.09766
$503$	13.7801	0.614424	0.307212	−	0.951641i	$-0.400604\pi$
$503$	13.7801	0.614424	0.307212	+	0.951641i	$0.400604\pi$
$504$	0	0
$505$	−6.12452	−0.272537
$506$	8.94427	0.397621
$507$	0	0
$508$	−21.1868	−0.940011
$509$	−8.21690	−0.364208	−0.182104	−	0.983279i	$-0.558291\pi$
$509$	−8.21690	−0.364208	−0.182104	+	0.983279i	$0.558291\pi$
$510$	0	0
$511$	21.1868	0.937248
$512$	11.1803	0.494106
$513$	0	0
$514$	4.59339	0.202606
$515$	0	0
$516$	0	0
$517$	−1.40661	−0.0618628
$518$	47.3751	2.08154
$519$	0	0
$520$	2.34436	0.102807
$521$	−41.9398	−1.83741	−0.918707	−	0.394939i	$-0.870766\pi$
$521$	−41.9398	−1.83741	−0.918707	+	0.394939i	$0.870766\pi$
$522$	0	0
$523$	−10.5934	−0.463216	−0.231608	−	0.972809i	$-0.574399\pi$
$523$	−10.5934	−0.463216	−0.231608	+	0.972809i	$0.574399\pi$
$524$	35.0497	1.53115
$525$	0	0
$526$	35.3113	1.53965
$527$	0	0
$528$	0	0
$529$	11.1245	0.483675
$530$	17.8885	0.777029
$531$	0	0
$532$	37.4066	1.62178
$533$	17.8885	0.774839
$534$	0	0
$535$	7.53113	0.325599
$536$	8.94427	0.386334
$537$	0	0
$538$	9.18677	0.396070
$539$	−3.74476	−0.161298
$540$	0	0
$541$	32.1245	1.38114	0.690570	−	0.723265i	$-0.257360\pi$
$541$	32.1245	1.38114	0.690570	+	0.723265i	$0.257360\pi$
$542$	−47.3751	−2.03493
$543$	0	0
$544$	−9.18677	−0.393880
$545$	−11.3622	−0.486702
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−6.16267	−0.263256
$549$	0	0
$550$	6.93774	0.295826
$551$	7.25373	0.309019
$552$	0	0
$553$	−53.1868	−2.26173
$554$	20.2638	0.860927
$555$	0	0
$556$	36.0000	1.52674
$557$	−8.21690	−0.348161	−0.174081	−	0.984731i	$-0.555695\pi$
$557$	−8.21690	−0.348161	−0.174081	+	0.984731i	$0.555695\pi$
$558$	0	0
$559$	−1.53113	−0.0647599
$560$	2.41791	0.102175
$561$	0	0
$562$	29.1868	1.23117
$563$	−20.2638	−0.854018	−0.427009	−	0.904247i	$-0.640433\pi$
$563$	−20.2638	−0.854018	−0.427009	+	0.904247i	$0.640433\pi$
$564$	0	0
$565$	0.468871	0.0197256
$566$	−42.6245	−1.79164
$567$	0	0
$568$	29.1868	1.22465
$569$	−28.2023	−1.18230	−0.591151	−	0.806561i	$-0.701326\pi$
$569$	−28.2023	−1.18230	−0.591151	+	0.806561i	$0.701326\pi$
$570$	0	0
$571$	−31.5311	−1.31954	−0.659768	−	0.751469i	$-0.729346\pi$
$571$	−31.5311	−1.31954	−0.659768	+	0.751469i	$0.729346\pi$
$572$	3.14528	0.131511
$573$	0	0
$574$	−92.2490	−3.85040
$575$	26.4691	1.10384
$576$	0	0
$577$	24.1245	1.00432	0.502158	−	0.864776i	$-0.332539\pi$
$577$	24.1245	1.00432	0.502158	+	0.864776i	$0.332539\pi$
$578$	−33.8194	−1.40670
$579$	0	0
$580$	−4.21984	−0.175219
$581$	38.8371	1.61123
$582$	0	0
$583$	8.00000	0.331326
$584$	13.4164	0.555175
$585$	0	0
$586$	15.3113	0.632504
$587$	27.4749	1.13401	0.567006	−	0.823714i	$-0.308102\pi$
$587$	27.4749	1.13401	0.567006	+	0.823714i	$0.308102\pi$
$588$	0	0
$589$	0	0
$590$	8.94427	0.368230
$591$	0	0
$592$	−6.00000	−0.246598
$593$	33.3165	1.36815	0.684073	−	0.729413i	$-0.260207\pi$
$593$	33.3165	1.36815	0.684073	+	0.729413i	$0.260207\pi$
$594$	0	0
$595$	−3.31129	−0.135750
$596$	64.8999	2.65840
$597$	0	0
$598$	20.0000	0.817861
$599$	5.52056	0.225564	0.112782	−	0.993620i	$-0.464024\pi$
$599$	5.52056	0.225564	0.112782	+	0.993620i	$0.464024\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	7.89584	0.321811
$603$	0	0
$604$	24.0000	0.976546
$605$	7.21110	0.293173
$606$	0	0
$607$	30.5934	1.24175	0.620874	−	0.783911i	$-0.286778\pi$
$607$	30.5934	1.24175	0.620874	+	0.783911i	$0.286778\pi$
$608$	−23.6875	−0.960656
$609$	0	0
$610$	3.06226	0.123987
$611$	−3.14528	−0.127245
$612$	0	0
$613$	8.12452	0.328146	0.164073	−	0.986448i	$-0.447537\pi$
$613$	8.12452	0.328146	0.164073	+	0.986448i	$0.447537\pi$
$614$	11.0411	0.445584
$615$	0	0
$616$	−5.40661	−0.217839
$617$	−8.21690	−0.330800	−0.165400	−	0.986227i	$-0.552892\pi$
$617$	−8.21690	−0.330800	−0.165400	+	0.986227i	$0.552892\pi$
$618$	0	0
$619$	9.18677	0.369248	0.184624	−	0.982809i	$-0.440893\pi$
$619$	9.18677	0.369248	0.184624	+	0.982809i	$0.440893\pi$
$620$	0	0
$621$	0	0
$622$	−64.5934	−2.58996
$623$	47.3751	1.89804
$624$	0	0
$625$	18.1868	0.727471
$626$	4.47214	0.178743
$627$	0	0
$628$	42.0000	1.67598
$629$	8.21690	0.327629
$630$	0	0
$631$	−9.87548	−0.393137	−0.196568	−	0.980490i	$-0.562980\pi$
$631$	−9.87548	−0.393137	−0.196568	+	0.980490i	$0.562980\pi$
$632$	−33.6802	−1.33973
$633$	0	0
$634$	−16.9377	−0.672684
$635$	4.83582	0.191904
$636$	0	0
$637$	−8.37355	−0.331772
$638$	−3.14528	−0.124523
$639$	0	0
$640$	10.7179	0.423662
$641$	−9.95007	−0.393004	−0.196502	−	0.980503i	$-0.562958\pi$
$641$	−9.95007	−0.393004	−0.196502	+	0.980503i	$0.562958\pi$
$642$	0	0
$643$	−33.1868	−1.30876	−0.654379	−	0.756166i	$-0.727070\pi$
$643$	−33.1868	−1.30876	−0.654379	+	0.756166i	$0.727070\pi$
$644$	−61.8825	−2.43851
$645$	0	0
$646$	10.8132	0.425441
$647$	−15.7917	−0.620835	−0.310418	−	0.950600i	$-0.600469\pi$
$647$	−15.7917	−0.620835	−0.310418	+	0.950600i	$0.600469\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	15.5133	0.608480
$651$	0	0
$652$	−43.7802	−1.71456
$653$	24.0512	0.941197	0.470598	−	0.882347i	$-0.344038\pi$
$653$	24.0512	0.941197	0.470598	+	0.882347i	$0.344038\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	11.6832	0.456154
$657$	0	0
$658$	16.2198	0.632315
$659$	32.6744	1.27282	0.636408	−	0.771353i	$-0.280420\pi$
$659$	32.6744	1.27282	0.636408	+	0.771353i	$0.280420\pi$
$660$	0	0
$661$	23.4066	0.910412	0.455206	−	0.890386i	$-0.349566\pi$
$661$	23.4066	0.910412	0.455206	+	0.890386i	$0.349566\pi$
$662$	−7.89584	−0.306881
$663$	0	0
$664$	24.5934	0.954408
$665$	−8.53796	−0.331088
$666$	0	0
$667$	−12.0000	−0.464642
$668$	−45.3208	−1.75352
$669$	0	0
$670$	−6.12452	−0.236611
$671$	1.36948	0.0528683
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	−7.61742	−0.293412
$675$	0	0
$676$	−31.9669	−1.22950
$677$	−16.1554	−0.620901	−0.310451	−	0.950589i	$-0.600480\pi$
$677$	−16.1554	−0.620901	−0.310451	+	0.950589i	$0.600480\pi$
$678$	0	0
$679$	47.7802	1.83363
$680$	−2.09686	−0.0804107
$681$	0	0
$682$	0	0
$683$	−9.30796	−0.356159	−0.178080	−	0.984016i	$-0.556988\pi$
$683$	−9.30796	−0.356159	−0.178080	+	0.984016i	$0.556988\pi$
$684$	0	0
$685$	1.40661	0.0537439
$686$	−12.0896	−0.461581
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	17.8885	0.681499
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	53.6656	2.04006
$693$	0	0
$694$	−29.1868	−1.10791
$695$	−8.21690	−0.311685
$696$	0	0
$697$	−16.0000	−0.606043
$698$	−69.7357	−2.63954
$699$	0	0
$700$	−48.0000	−1.81423
$701$	17.1612	0.648169	0.324084	−	0.946028i	$-0.394944\pi$
$701$	17.1612	0.648169	0.324084	+	0.946028i	$0.394944\pi$
$702$	0	0
$703$	21.1868	0.799074
$704$	8.90164	0.335493
$705$	0	0
$706$	−49.1868	−1.85117
$707$	31.5834	1.18782
$708$	0	0
$709$	34.7179	1.30386	0.651929	−	0.758280i	$-0.273960\pi$
$709$	34.7179	1.30386	0.651929	+	0.758280i	$0.273960\pi$
$710$	−19.9854	−0.750039
$711$	0	0
$712$	30.0000	1.12430
$713$	0	0
$714$	0	0
$715$	−0.717902	−0.0268480
$716$	−14.5075	−0.542169
$717$	0	0
$718$	19.1868	0.716044
$719$	−0.363686	−0.0135632	−0.00678160	−	0.999977i	$-0.502159\pi$
$719$	−0.363686	−0.0135632	−0.00678160	+	0.999977i	$0.502159\pi$
$720$	0	0
$721$	0	0
$722$	−14.6040	−0.543506
$723$	0	0
$724$	40.5934	1.50864
$725$	−9.30796	−0.345689
$726$	0	0
$727$	5.40661	0.200520	0.100260	−	0.994961i	$-0.468033\pi$
$727$	5.40661	0.200520	0.100260	+	0.994961i	$0.468033\pi$
$728$	−12.0896	−0.448069
$729$	0	0
$730$	−9.18677	−0.340018
$731$	1.36948	0.0506522
$732$	0	0
$733$	39.1868	1.44740	0.723698	−	0.690117i	$-0.242441\pi$
$733$	39.1868	1.44740	0.723698	+	0.690117i	$0.242441\pi$
$734$	−33.6802	−1.24316
$735$	0	0
$736$	39.1868	1.44444
$737$	−2.73897	−0.100891
$738$	0	0
$739$	14.5934	0.536826	0.268413	−	0.963304i	$-0.413501\pi$
$739$	14.5934	0.536826	0.268413	+	0.963304i	$0.413501\pi$
$740$	−12.3253	−0.453089
$741$	0	0
$742$	−92.2490	−3.38657
$743$	−33.6802	−1.23561	−0.617804	−	0.786332i	$-0.711977\pi$
$743$	−33.6802	−1.23561	−0.617804	+	0.786332i	$0.711977\pi$
$744$	0	0
$745$	−14.8132	−0.542715
$746$	13.4164	0.491210
$747$	0	0
$748$	−2.81323	−0.102862
$749$	−38.8371	−1.41908
$750$	0	0
$751$	6.59339	0.240596	0.120298	−	0.992738i	$-0.461615\pi$
$751$	6.59339	0.240596	0.120298	+	0.992738i	$0.461615\pi$
$752$	−2.05422	−0.0749099
$753$	0	0
$754$	−7.03307	−0.256129
$755$	−5.47793	−0.199362
$756$	0	0
$757$	39.1868	1.42427	0.712134	−	0.702044i	$-0.247729\pi$
$757$	39.1868	1.42427	0.712134	+	0.702044i	$0.247729\pi$
$758$	4.75056	0.172548
$759$	0	0
$760$	−5.40661	−0.196118
$761$	39.5219	1.43267	0.716333	−	0.697759i	$-0.245819\pi$
$761$	39.5219	1.43267	0.716333	+	0.697759i	$0.245819\pi$
$762$	0	0
$763$	58.5934	2.12122
$764$	−35.0497	−1.26805
$765$	0	0
$766$	69.1868	2.49982
$767$	8.94427	0.322959
$768$	0	0
$769$	−43.6556	−1.57426	−0.787131	−	0.616785i	$-0.788435\pi$
$769$	−43.6556	−1.57426	−0.787131	+	0.616785i	$0.788435\pi$
$770$	3.70213	0.133416
$771$	0	0
$772$	−30.0000	−1.07972
$773$	37.8313	1.36070	0.680349	−	0.732888i	$-0.261828\pi$
$773$	37.8313	1.36070	0.680349	+	0.732888i	$0.261828\pi$
$774$	0	0
$775$	0	0
$776$	30.2565	1.08615
$777$	0	0
$778$	−44.5934	−1.59875
$779$	−41.2550	−1.47811
$780$	0	0
$781$	−8.93774	−0.319818
$782$	−17.8885	−0.639693
$783$	0	0
$784$	−5.46887	−0.195317
$785$	−9.58638	−0.342153
$786$	0	0
$787$	−48.2490	−1.71989	−0.859946	−	0.510385i	$-0.829503\pi$
$787$	−48.2490	−1.71989	−0.859946	+	0.510385i	$0.829503\pi$
$788$	−65.9910	−2.35083
$789$	0	0
$790$	23.0623	0.820518
$791$	−2.41791	−0.0859710
$792$	0	0
$793$	3.06226	0.108744
$794$	−1.32685	−0.0470882
$795$	0	0
$796$	−78.3735	−2.77788
$797$	−9.67164	−0.342587	−0.171294	−	0.985220i	$-0.554795\pi$
$797$	−9.67164	−0.342587	−0.171294	+	0.985220i	$0.554795\pi$
$798$	0	0
$799$	2.81323	0.0995248
$800$	30.3957	1.07465
$801$	0	0
$802$	9.18677	0.324396
$803$	−4.10845	−0.144984
$804$	0	0
$805$	14.1245	0.497824
$806$	0	0
$807$	0	0
$808$	20.0000	0.703598
$809$	−30.2139	−1.06226	−0.531132	−	0.847289i	$-0.678233\pi$
$809$	−30.2139	−1.06226	−0.531132	+	0.847289i	$0.678233\pi$
$810$	0	0
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	21.7612	0.763668
$813$	0	0
$814$	−9.18677	−0.321996
$815$	9.99270	0.350029
$816$	0	0
$817$	3.53113	0.123539
$818$	−56.0409	−1.95942
$819$	0	0
$820$	24.0000	0.838116
$821$	−26.1054	−0.911086	−0.455543	−	0.890214i	$-0.650555\pi$
$821$	−26.1054	−0.911086	−0.455543	+	0.890214i	$0.650555\pi$
$822$	0	0
$823$	−17.8755	−0.623100	−0.311550	−	0.950230i	$-0.600848\pi$
$823$	−17.8755	−0.623100	−0.311550	+	0.950230i	$0.600848\pi$
$824$	0	0
$825$	0	0
$826$	−46.1245	−1.60488
$827$	8.90164	0.309540	0.154770	−	0.987950i	$-0.450536\pi$
$827$	8.90164	0.309540	0.154770	+	0.987950i	$0.450536\pi$
$828$	0	0
$829$	−13.0623	−0.453671	−0.226835	−	0.973933i	$-0.572838\pi$
$829$	−13.0623	−0.453671	−0.226835	+	0.973933i	$0.572838\pi$
$830$	−16.8401	−0.584529
$831$	0	0
$832$	19.9047	0.690070
$833$	7.48953	0.259497
$834$	0	0
$835$	10.3444	0.357981
$836$	−7.25373	−0.250876
$837$	0	0
$838$	−72.2490	−2.49580
$839$	19.2580	0.664861	0.332430	−	0.943128i	$-0.392131\pi$
$839$	19.2580	0.664861	0.332430	+	0.943128i	$0.392131\pi$
$840$	0	0
$841$	−24.7802	−0.854488
$842$	6.56899	0.226382
$843$	0	0
$844$	−49.4066	−1.70065
$845$	7.29636	0.251003
$846$	0	0
$847$	−37.1868	−1.27775
$848$	11.6832	0.401204
$849$	0	0
$850$	−13.8755	−0.475925
$851$	−35.0497	−1.20149
$852$	0	0
$853$	55.6556	1.90561	0.952806	−	0.303578i	$-0.0981814\pi$
$853$	55.6556	1.90561	0.952806	+	0.303578i	$0.0981814\pi$
$854$	−15.7917	−0.540380
$855$	0	0
$856$	−24.5934	−0.840585
$857$	36.4192	1.24406	0.622028	−	0.782995i	$-0.286309\pi$
$857$	36.4192	1.24406	0.622028	+	0.782995i	$0.286309\pi$
$858$	0	0
$859$	−19.7802	−0.674890	−0.337445	−	0.941345i	$-0.609563\pi$
$859$	−19.7802	−0.674890	−0.337445	+	0.941345i	$0.609563\pi$
$860$	−2.05422	−0.0700485
$861$	0	0
$862$	10.7179	0.365053
$863$	34.4076	1.17125	0.585624	−	0.810583i	$-0.300849\pi$
$863$	34.4076	1.17125	0.585624	+	0.810583i	$0.300849\pi$
$864$	0	0
$865$	−12.2490	−0.416480
$866$	51.8472	1.76184
$867$	0	0
$868$	0	0
$869$	10.3138	0.349870
$870$	0	0
$871$	−6.12452	−0.207521
$872$	37.1039	1.25650
$873$	0	0
$874$	−46.1245	−1.56019
$875$	23.0454	0.779077
$876$	0	0
$877$	12.1245	0.409416	0.204708	−	0.978823i	$-0.434376\pi$
$877$	12.1245	0.409416	0.204708	+	0.978823i	$0.434376\pi$
$878$	47.3751	1.59883
$879$	0	0
$880$	−0.468871	−0.0158056
$881$	−30.2139	−1.01793	−0.508966	−	0.860787i	$-0.669972\pi$
$881$	−30.2139	−1.01793	−0.508966	+	0.860787i	$0.669972\pi$
$882$	0	0
$883$	−46.3735	−1.56059	−0.780297	−	0.625409i	$-0.784932\pi$
$883$	−46.3735	−1.56059	−0.780297	+	0.625409i	$0.784932\pi$
$884$	−6.29057	−0.211575
$885$	0	0
$886$	10.7179	0.360075
$887$	30.9413	1.03891	0.519453	−	0.854499i	$-0.326136\pi$
$887$	30.9413	1.03891	0.519453	+	0.854499i	$0.326136\pi$
$888$	0	0
$889$	−24.9377	−0.836385
$890$	−20.5422	−0.688578
$891$	0	0
$892$	−40.9669	−1.37167
$893$	7.25373	0.242737
$894$	0	0
$895$	3.31129	0.110684
$896$	−55.2709	−1.84647
$897$	0	0
$898$	−13.7802	−0.459850
$899$	0	0
$900$	0	0
$901$	−16.0000	−0.533037
$902$	17.8885	0.595623
$903$	0	0
$904$	−1.53113	−0.0509246
$905$	−9.26533	−0.307990
$906$	0	0
$907$	−9.18677	−0.305042	−0.152521	−	0.988300i	$-0.548739\pi$
$907$	−9.18677	−0.305042	−0.152521	+	0.988300i	$0.548739\pi$
$908$	78.3163	2.59902
$909$	0	0
$910$	8.27822	0.274420
$911$	−39.1582	−1.29737	−0.648684	−	0.761058i	$-0.724680\pi$
$911$	−39.1582	−1.29737	−0.648684	+	0.761058i	$0.724680\pi$
$912$	0	0
$913$	−7.53113	−0.249244
$914$	−40.2492	−1.33133
$915$	0	0
$916$	18.0000	0.594737
$917$	41.2550	1.36236
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	8.94427	0.294884
$921$	0	0
$922$	63.0623	2.07684
$923$	−19.9854	−0.657827
$924$	0	0
$925$	−27.1868	−0.893896
$926$	50.5203	1.66020
$927$	0	0
$928$	−13.7802	−0.452356
$929$	32.2681	1.05868	0.529341	−	0.848409i	$-0.322439\pi$
$929$	32.2681	1.05868	0.529341	+	0.848409i	$0.322439\pi$
$930$	0	0
$931$	19.3113	0.632902
$932$	63.9368	2.09432
$933$	0	0
$934$	−10.8132	−0.353820
$935$	0.642111	0.0209993
$936$	0	0
$937$	25.0623	0.818748	0.409374	−	0.912367i	$-0.365747\pi$
$937$	25.0623	0.818748	0.409374	+	0.912367i	$0.365747\pi$
$938$	31.5834	1.03123
$939$	0	0
$940$	−4.21984	−0.137636
$941$	−48.8298	−1.59181	−0.795903	−	0.605424i	$-0.793004\pi$
$941$	−48.8298	−1.59181	−0.795903	+	0.605424i	$0.793004\pi$
$942$	0	0
$943$	68.2490	2.22249
$944$	5.84162	0.190129
$945$	0	0
$946$	−1.53113	−0.0497813
$947$	50.8840	1.65351	0.826754	−	0.562563i	$-0.190185\pi$
$947$	50.8840	1.65351	0.826754	+	0.562563i	$0.190185\pi$
$948$	0	0
$949$	−9.18677	−0.298215
$950$	−35.7771	−1.16076
$951$	0	0
$952$	10.8132	0.350459
$953$	19.9428	0.646010	0.323005	−	0.946397i	$-0.395307\pi$
$953$	19.9428	0.646010	0.323005	+	0.946397i	$0.395307\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	86.6611	2.80282
$957$	0	0
$958$	−81.5311	−2.63415
$959$	−7.25373	−0.234235
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−20.5422	−0.662309
$963$	0	0
$964$	57.5603	1.85389
$965$	6.84742	0.220426
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−23.5483	−0.756872
$969$	0	0
$970$	−20.7179	−0.665212
$971$	−37.4250	−1.20103	−0.600513	−	0.799615i	$-0.705037\pi$
$971$	−37.4250	−1.20103	−0.600513	+	0.799615i	$0.705037\pi$
$972$	0	0
$973$	42.3735	1.35843
$974$	−15.7917	−0.505998
$975$	0	0
$976$	2.00000	0.0640184
$977$	17.8885	0.572305	0.286153	−	0.958184i	$-0.407624\pi$
$977$	17.8885	0.572305	0.286153	+	0.958184i	$0.407624\pi$
$978$	0	0
$979$	−9.18677	−0.293611
$980$	−11.2343	−0.358866
$981$	0	0
$982$	30.8132	0.983290
$983$	−52.2109	−1.66527	−0.832634	−	0.553823i	$-0.813168\pi$
$983$	−52.2109	−1.66527	−0.832634	+	0.553823i	$0.813168\pi$
$984$	0	0
$985$	15.0623	0.479923
$986$	6.29057	0.200332
$987$	0	0
$988$	−16.2198	−0.516022
$989$	−5.84162	−0.185753
$990$	0	0
$991$	−16.4689	−0.523151	−0.261575	−	0.965183i	$-0.584242\pi$
$991$	−16.4689	−0.523151	−0.261575	+	0.965183i	$0.584242\pi$
$992$	0	0
$993$	0	0
$994$	103.062	3.26894
$995$	17.8885	0.567105
$996$	0	0
$997$	42.2490	1.33804	0.669020	−	0.743244i	$-0.266714\pi$
$997$	42.2490	1.33804	0.669020	+	0.743244i	$0.266714\pi$
$998$	−23.6875	−0.749816
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.2.a.j.1.3	yes	4
3.2	odd	2	inner	387.2.a.j.1.2	✓	4
4.3	odd	2		6192.2.a.bz.1.2		4
5.4	even	2		9675.2.a.bv.1.1		4
12.11	even	2		6192.2.a.bz.1.3		4
15.14	odd	2		9675.2.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
387.2.a.j.1.2	✓	4	3.2	odd	2	inner
387.2.a.j.1.3	yes	4	1.1	even	1	trivial
6192.2.a.bz.1.2		4	4.3	odd	2
6192.2.a.bz.1.3		4	12.11	even	2
9675.2.a.bv.1.1		4	5.4	even	2
9675.2.a.bv.1.3		4	15.14	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} -0.684742 q^{5} +3.53113 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q+2.23607 q^{2} +3.00000 q^{4} -0.684742 q^{5} +3.53113 q^{7} +2.23607 q^{8} -1.53113 q^{10} -0.684742 q^{11} -1.53113 q^{13} +7.89584 q^{14} -1.00000 q^{16} +1.36948 q^{17} +3.53113 q^{19} -2.05422 q^{20} -1.53113 q^{22} -5.84162 q^{23} -4.53113 q^{25} -3.42371 q^{26} +10.5934 q^{28} +2.05422 q^{29} -6.70820 q^{32} +3.06226 q^{34} -2.41791 q^{35} +6.00000 q^{37} +7.89584 q^{38} -1.53113 q^{40} -11.6832 q^{41} +1.00000 q^{43} -2.05422 q^{44} -13.0623 q^{46} +2.05422 q^{47} +5.46887 q^{49} -10.1319 q^{50} -4.59339 q^{52} -11.6832 q^{53} +0.468871 q^{55} +7.89584 q^{56} +4.59339 q^{58} -5.84162 q^{59} -2.00000 q^{61} -13.0000 q^{64} +1.04843 q^{65} +4.00000 q^{67} +4.10845 q^{68} -5.40661 q^{70} +13.0527 q^{71} +6.00000 q^{73} +13.4164 q^{74} +10.5934 q^{76} -2.41791 q^{77} -15.0623 q^{79} +0.684742 q^{80} -26.1245 q^{82} +10.9985 q^{83} -0.937742 q^{85} +2.23607 q^{86} -1.53113 q^{88} +13.4164 q^{89} -5.40661 q^{91} -17.5249 q^{92} +4.59339 q^{94} -2.41791 q^{95} +13.5311 q^{97} +12.2288 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{4} - 2 q^{7}+O(q^{10}) \) 4 * q + 12 * q^4 - 2 * q^7	\( 4 q + 12 q^{4} - 2 q^{7} + 10 q^{10} + 10 q^{13} - 4 q^{16} - 2 q^{19} + 10 q^{22} - 2 q^{25} - 6 q^{28} - 20 q^{34} + 24 q^{37} + 10 q^{40} + 4 q^{43} - 20 q^{46} + 38 q^{49} + 30 q^{52} + 18 q^{55} - 30 q^{58} - 8 q^{61} - 52 q^{64} + 16 q^{67} - 70 q^{70} + 24 q^{73} - 6 q^{76} - 28 q^{79} - 40 q^{82} - 36 q^{85} + 10 q^{88} - 70 q^{91} - 30 q^{94} + 38 q^{97}+O(q^{100}) \) 4 * q + 12 * q^4 - 2 * q^7 + 10 * q^10 + 10 * q^13 - 4 * q^16 - 2 * q^19 + 10 * q^22 - 2 * q^25 - 6 * q^28 - 20 * q^34 + 24 * q^37 + 10 * q^40 + 4 * q^43 - 20 * q^46 + 38 * q^49 + 30 * q^52 + 18 * q^55 - 30 * q^58 - 8 * q^61 - 52 * q^64 + 16 * q^67 - 70 * q^70 + 24 * q^73 - 6 * q^76 - 28 * q^79 - 40 * q^82 - 36 * q^85 + 10 * q^88 - 70 * q^91 - 30 * q^94 + 38 * q^97

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