LMFDB - Embedded newform 4368.2.a.bd.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:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.37228 q^{11} -1.00000 q^{13} -3.37228 q^{15} -1.37228 q^{17} -3.37228 q^{19} -1.00000 q^{21} -0.627719 q^{23} +6.37228 q^{25} -1.00000 q^{27} -1.37228 q^{29} -4.00000 q^{31} +1.37228 q^{33} +3.37228 q^{35} +9.37228 q^{37} +1.00000 q^{39} +8.00000 q^{41} +7.37228 q^{43} +3.37228 q^{45} +4.74456 q^{47} +1.00000 q^{49} +1.37228 q^{51} +6.00000 q^{53} -4.62772 q^{55} +3.37228 q^{57} +4.74456 q^{59} +12.1168 q^{61} +1.00000 q^{63} -3.37228 q^{65} -2.74456 q^{67} +0.627719 q^{69} +4.74456 q^{71} -5.37228 q^{73} -6.37228 q^{75} -1.37228 q^{77} +6.74456 q^{79} +1.00000 q^{81} +2.00000 q^{83} -4.62772 q^{85} +1.37228 q^{87} +2.74456 q^{89} -1.00000 q^{91} +4.00000 q^{93} -11.3723 q^{95} -2.00000 q^{97} -1.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} - q^{15} + 3 q^{17} - q^{19} - 2 q^{21} - 7 q^{23} + 7 q^{25} - 2 q^{27} + 3 q^{29} - 8 q^{31} - 3 q^{33} + q^{35} + 13 q^{37} + 2 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 2 q^{47} + 2 q^{49} - 3 q^{51} + 12 q^{53} - 15 q^{55} + q^{57} - 2 q^{59} + 7 q^{61} + 2 q^{63} - q^{65} + 6 q^{67} + 7 q^{69} - 2 q^{71} - 5 q^{73} - 7 q^{75} + 3 q^{77} + 2 q^{79} + 2 q^{81} + 4 q^{83} - 15 q^{85} - 3 q^{87} - 6 q^{89} - 2 q^{91} + 8 q^{93} - 17 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 3 * q^11 - 2 * q^13 - q^15 + 3 * q^17 - q^19 - 2 * q^21 - 7 * q^23 + 7 * q^25 - 2 * q^27 + 3 * q^29 - 8 * q^31 - 3 * q^33 + q^35 + 13 * q^37 + 2 * q^39 + 16 * q^41 + 9 * q^43 + q^45 - 2 * q^47 + 2 * q^49 - 3 * q^51 + 12 * q^53 - 15 * q^55 + q^57 - 2 * q^59 + 7 * q^61 + 2 * q^63 - q^65 + 6 * q^67 + 7 * q^69 - 2 * q^71 - 5 * q^73 - 7 * q^75 + 3 * q^77 + 2 * q^79 + 2 * q^81 + 4 * q^83 - 15 * q^85 - 3 * q^87 - 6 * q^89 - 2 * q^91 + 8 * q^93 - 17 * q^95 - 4 * 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
$5$	3.37228	1.50813	0.754065	−	0.656800i	$-0.228090\pi$
$5$	3.37228	1.50813	0.754065	+	0.656800i	$0.228090\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.37228	−0.413758	−0.206879	−	0.978366i	$-0.566331\pi$
$11$	−1.37228	−0.413758	−0.206879	+	0.978366i	$0.566331\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.37228	−0.870719
$16$	0	0
$17$	−1.37228	−0.332827	−0.166414	−	0.986056i	$-0.553219\pi$
$17$	−1.37228	−0.332827	−0.166414	+	0.986056i	$0.553219\pi$
$18$	0	0
$19$	−3.37228	−0.773654	−0.386827	−	0.922152i	$-0.626429\pi$
$19$	−3.37228	−0.773654	−0.386827	+	0.922152i	$0.626429\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−0.627719	−0.130888	−0.0654442	−	0.997856i	$-0.520846\pi$
$23$	−0.627719	−0.130888	−0.0654442	+	0.997856i	$0.520846\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	1.37228	0.238884
$34$	0	0
$35$	3.37228	0.570020
$36$	0	0
$37$	9.37228	1.54079	0.770397	−	0.637565i	$-0.220058\pi$
$37$	9.37228	1.54079	0.770397	+	0.637565i	$0.220058\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	7.37228	1.12426	0.562131	−	0.827048i	$-0.309982\pi$
$43$	7.37228	1.12426	0.562131	+	0.827048i	$0.309982\pi$
$44$	0	0
$45$	3.37228	0.502710
$46$	0	0
$47$	4.74456	0.692066	0.346033	−	0.938222i	$-0.387529\pi$
$47$	4.74456	0.692066	0.346033	+	0.938222i	$0.387529\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.37228	0.192158
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−4.62772	−0.624001
$56$	0	0
$57$	3.37228	0.446670
$58$	0	0
$59$	4.74456	0.617689	0.308845	−	0.951112i	$-0.400058\pi$
$59$	4.74456	0.617689	0.308845	+	0.951112i	$0.400058\pi$
$60$	0	0
$61$	12.1168	1.55140	0.775701	−	0.631100i	$-0.217396\pi$
$61$	12.1168	1.55140	0.775701	+	0.631100i	$0.217396\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.37228	−0.418280
$66$	0	0
$67$	−2.74456	−0.335302	−0.167651	−	0.985846i	$-0.553618\pi$
$67$	−2.74456	−0.335302	−0.167651	+	0.985846i	$0.553618\pi$
$68$	0	0
$69$	0.627719	0.0755684
$70$	0	0
$71$	4.74456	0.563076	0.281538	−	0.959550i	$-0.409155\pi$
$71$	4.74456	0.563076	0.281538	+	0.959550i	$0.409155\pi$
$72$	0	0
$73$	−5.37228	−0.628778	−0.314389	−	0.949294i	$-0.601800\pi$
$73$	−5.37228	−0.628778	−0.314389	+	0.949294i	$0.601800\pi$
$74$	0	0
$75$	−6.37228	−0.735808
$76$	0	0
$77$	−1.37228	−0.156386
$78$	0	0
$79$	6.74456	0.758823	0.379411	−	0.925228i	$-0.376127\pi$
$79$	6.74456	0.758823	0.379411	+	0.925228i	$0.376127\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−4.62772	−0.501947
$86$	0	0
$87$	1.37228	0.147124
$88$	0	0
$89$	2.74456	0.290923	0.145462	−	0.989364i	$-0.453533\pi$
$89$	2.74456	0.290923	0.145462	+	0.989364i	$0.453533\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−11.3723	−1.16677
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−1.37228	−0.137919
$100$	0	0
$101$	−4.74456	−0.472102	−0.236051	−	0.971741i	$-0.575853\pi$
$101$	−4.74456	−0.472102	−0.236051	+	0.971741i	$0.575853\pi$
$102$	0	0
$103$	10.1168	0.996842	0.498421	−	0.866935i	$-0.333913\pi$
$103$	10.1168	0.996842	0.498421	+	0.866935i	$0.333913\pi$
$104$	0	0
$105$	−3.37228	−0.329101
$106$	0	0
$107$	−6.74456	−0.652021	−0.326011	−	0.945366i	$-0.605705\pi$
$107$	−6.74456	−0.652021	−0.326011	+	0.945366i	$0.605705\pi$
$108$	0	0
$109$	12.1168	1.16058	0.580292	−	0.814409i	$-0.302939\pi$
$109$	12.1168	1.16058	0.580292	+	0.814409i	$0.302939\pi$
$110$	0	0
$111$	−9.37228	−0.889578
$112$	0	0
$113$	−7.48913	−0.704518	−0.352259	−	0.935903i	$-0.614586\pi$
$113$	−7.48913	−0.704518	−0.352259	+	0.935903i	$0.614586\pi$
$114$	0	0
$115$	−2.11684	−0.197397
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−1.37228	−0.125797
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	−8.00000	−0.721336
$124$	0	0
$125$	4.62772	0.413916
$126$	0	0
$127$	6.74456	0.598483	0.299242	−	0.954177i	$-0.403266\pi$
$127$	6.74456	0.598483	0.299242	+	0.954177i	$0.403266\pi$
$128$	0	0
$129$	−7.37228	−0.649093
$130$	0	0
$131$	4.62772	0.404326	0.202163	−	0.979352i	$-0.435203\pi$
$131$	4.62772	0.404326	0.202163	+	0.979352i	$0.435203\pi$
$132$	0	0
$133$	−3.37228	−0.292414
$134$	0	0
$135$	−3.37228	−0.290240
$136$	0	0
$137$	11.3723	0.971600	0.485800	−	0.874070i	$-0.338528\pi$
$137$	11.3723	0.971600	0.485800	+	0.874070i	$0.338528\pi$
$138$	0	0
$139$	9.48913	0.804857	0.402429	−	0.915451i	$-0.368166\pi$
$139$	9.48913	0.804857	0.402429	+	0.915451i	$0.368166\pi$
$140$	0	0
$141$	−4.74456	−0.399564
$142$	0	0
$143$	1.37228	0.114756
$144$	0	0
$145$	−4.62772	−0.384311
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	17.4891	1.43276	0.716382	−	0.697708i	$-0.245797\pi$
$149$	17.4891	1.43276	0.716382	+	0.697708i	$0.245797\pi$
$150$	0	0
$151$	15.3723	1.25098	0.625489	−	0.780233i	$-0.284899\pi$
$151$	15.3723	1.25098	0.625489	+	0.780233i	$0.284899\pi$
$152$	0	0
$153$	−1.37228	−0.110942
$154$	0	0
$155$	−13.4891	−1.08347
$156$	0	0
$157$	−10.8614	−0.866835	−0.433417	−	0.901193i	$-0.642692\pi$
$157$	−10.8614	−0.866835	−0.433417	+	0.901193i	$0.642692\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−0.627719	−0.0494712
$162$	0	0
$163$	1.25544	0.0983334	0.0491667	−	0.998791i	$-0.484343\pi$
$163$	1.25544	0.0983334	0.0491667	+	0.998791i	$0.484343\pi$
$164$	0	0
$165$	4.62772	0.360267
$166$	0	0
$167$	3.88316	0.300488	0.150244	−	0.988649i	$-0.451994\pi$
$167$	3.88316	0.300488	0.150244	+	0.988649i	$0.451994\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.37228	−0.257885
$172$	0	0
$173$	−3.48913	−0.265273	−0.132637	−	0.991165i	$-0.542344\pi$
$173$	−3.48913	−0.265273	−0.132637	+	0.991165i	$0.542344\pi$
$174$	0	0
$175$	6.37228	0.481699
$176$	0	0
$177$	−4.74456	−0.356623
$178$	0	0
$179$	−17.4891	−1.30720	−0.653599	−	0.756841i	$-0.726742\pi$
$179$	−17.4891	−1.30720	−0.653599	+	0.756841i	$0.726742\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−12.1168	−0.895703
$184$	0	0
$185$	31.6060	2.32372
$186$	0	0
$187$	1.88316	0.137710
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	5.88316	0.425690	0.212845	−	0.977086i	$-0.431727\pi$
$191$	5.88316	0.425690	0.212845	+	0.977086i	$0.431727\pi$
$192$	0	0
$193$	−26.2337	−1.88834	−0.944171	−	0.329456i	$-0.893135\pi$
$193$	−26.2337	−1.88834	−0.944171	+	0.329456i	$0.893135\pi$
$194$	0	0
$195$	3.37228	0.241494
$196$	0	0
$197$	−18.9783	−1.35214	−0.676072	−	0.736835i	$-0.736319\pi$
$197$	−18.9783	−1.35214	−0.676072	+	0.736835i	$0.736319\pi$
$198$	0	0
$199$	−4.62772	−0.328050	−0.164025	−	0.986456i	$-0.552448\pi$
$199$	−4.62772	−0.328050	−0.164025	+	0.986456i	$0.552448\pi$
$200$	0	0
$201$	2.74456	0.193587
$202$	0	0
$203$	−1.37228	−0.0963153
$204$	0	0
$205$	26.9783	1.88424
$206$	0	0
$207$	−0.627719	−0.0436295
$208$	0	0
$209$	4.62772	0.320106
$210$	0	0
$211$	−12.8614	−0.885416	−0.442708	−	0.896666i	$-0.645982\pi$
$211$	−12.8614	−0.885416	−0.442708	+	0.896666i	$0.645982\pi$
$212$	0	0
$213$	−4.74456	−0.325092
$214$	0	0
$215$	24.8614	1.69553
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	5.37228	0.363025
$220$	0	0
$221$	1.37228	0.0923096
$222$	0	0
$223$	−13.2554	−0.887650	−0.443825	−	0.896114i	$-0.646379\pi$
$223$	−13.2554	−0.887650	−0.443825	+	0.896114i	$0.646379\pi$
$224$	0	0
$225$	6.37228	0.424819
$226$	0	0
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	1.37228	0.0902895
$232$	0	0
$233$	26.2337	1.71863	0.859313	−	0.511450i	$-0.170891\pi$
$233$	26.2337	1.71863	0.859313	+	0.511450i	$0.170891\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	−6.74456	−0.438106
$238$	0	0
$239$	−8.74456	−0.565639	−0.282819	−	0.959173i	$-0.591270\pi$
$239$	−8.74456	−0.565639	−0.282819	+	0.959173i	$0.591270\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.37228	0.215447
$246$	0	0
$247$	3.37228	0.214573
$248$	0	0
$249$	−2.00000	−0.126745
$250$	0	0
$251$	2.11684	0.133614	0.0668070	−	0.997766i	$-0.478719\pi$
$251$	2.11684	0.133614	0.0668070	+	0.997766i	$0.478719\pi$
$252$	0	0
$253$	0.861407	0.0541562
$254$	0	0
$255$	4.62772	0.289799
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	9.37228	0.582365
$260$	0	0
$261$	−1.37228	−0.0849421
$262$	0	0
$263$	−14.9783	−0.923598	−0.461799	−	0.886984i	$-0.652796\pi$
$263$	−14.9783	−0.923598	−0.461799	+	0.886984i	$0.652796\pi$
$264$	0	0
$265$	20.2337	1.24295
$266$	0	0
$267$	−2.74456	−0.167965
$268$	0	0
$269$	15.4891	0.944389	0.472194	−	0.881494i	$-0.343462\pi$
$269$	15.4891	0.944389	0.472194	+	0.881494i	$0.343462\pi$
$270$	0	0
$271$	−6.74456	−0.409703	−0.204852	−	0.978793i	$-0.565671\pi$
$271$	−6.74456	−0.409703	−0.204852	+	0.978793i	$0.565671\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−8.74456	−0.527317
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−20.2337	−1.20704	−0.603520	−	0.797348i	$-0.706236\pi$
$281$	−20.2337	−1.20704	−0.603520	+	0.797348i	$0.706236\pi$
$282$	0	0
$283$	17.4891	1.03962	0.519810	−	0.854282i	$-0.326003\pi$
$283$	17.4891	1.03962	0.519810	+	0.854282i	$0.326003\pi$
$284$	0	0
$285$	11.3723	0.673636
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	5.25544	0.307026	0.153513	−	0.988147i	$-0.450941\pi$
$293$	5.25544	0.307026	0.153513	+	0.988147i	$0.450941\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	1.37228	0.0796278
$298$	0	0
$299$	0.627719	0.0363019
$300$	0	0
$301$	7.37228	0.424931
$302$	0	0
$303$	4.74456	0.272568
$304$	0	0
$305$	40.8614	2.33972
$306$	0	0
$307$	9.48913	0.541573	0.270786	−	0.962639i	$-0.412716\pi$
$307$	9.48913	0.541573	0.270786	+	0.962639i	$0.412716\pi$
$308$	0	0
$309$	−10.1168	−0.575527
$310$	0	0
$311$	2.74456	0.155630	0.0778149	−	0.996968i	$-0.475206\pi$
$311$	2.74456	0.155630	0.0778149	+	0.996968i	$0.475206\pi$
$312$	0	0
$313$	−32.7446	−1.85083	−0.925416	−	0.378953i	$-0.876284\pi$
$313$	−32.7446	−1.85083	−0.925416	+	0.378953i	$0.876284\pi$
$314$	0	0
$315$	3.37228	0.190007
$316$	0	0
$317$	16.0000	0.898650	0.449325	−	0.893368i	$-0.351665\pi$
$317$	16.0000	0.898650	0.449325	+	0.893368i	$0.351665\pi$
$318$	0	0
$319$	1.88316	0.105436
$320$	0	0
$321$	6.74456	0.376445
$322$	0	0
$323$	4.62772	0.257493
$324$	0	0
$325$	−6.37228	−0.353471
$326$	0	0
$327$	−12.1168	−0.670063
$328$	0	0
$329$	4.74456	0.261576
$330$	0	0
$331$	34.7446	1.90973	0.954867	−	0.297034i	$-0.0959974\pi$
$331$	34.7446	1.90973	0.954867	+	0.297034i	$0.0959974\pi$
$332$	0	0
$333$	9.37228	0.513598
$334$	0	0
$335$	−9.25544	−0.505679
$336$	0	0
$337$	−4.11684	−0.224259	−0.112129	−	0.993694i	$-0.535767\pi$
$337$	−4.11684	−0.224259	−0.112129	+	0.993694i	$0.535767\pi$
$338$	0	0
$339$	7.48913	0.406753
$340$	0	0
$341$	5.48913	0.297253
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.11684	0.113967
$346$	0	0
$347$	25.4891	1.36833	0.684164	−	0.729328i	$-0.260167\pi$
$347$	25.4891	1.36833	0.684164	+	0.729328i	$0.260167\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−12.2337	−0.651134	−0.325567	−	0.945519i	$-0.605555\pi$
$353$	−12.2337	−0.651134	−0.325567	+	0.945519i	$0.605555\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	1.37228	0.0726288
$358$	0	0
$359$	20.7446	1.09486	0.547428	−	0.836853i	$-0.315607\pi$
$359$	20.7446	1.09486	0.547428	+	0.836853i	$0.315607\pi$
$360$	0	0
$361$	−7.62772	−0.401459
$362$	0	0
$363$	9.11684	0.478510
$364$	0	0
$365$	−18.1168	−0.948279
$366$	0	0
$367$	−18.9783	−0.990657	−0.495328	−	0.868706i	$-0.664952\pi$
$367$	−18.9783	−0.990657	−0.495328	+	0.868706i	$0.664952\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	−28.7446	−1.48834	−0.744169	−	0.667992i	$-0.767154\pi$
$373$	−28.7446	−1.48834	−0.744169	+	0.667992i	$0.767154\pi$
$374$	0	0
$375$	−4.62772	−0.238974
$376$	0	0
$377$	1.37228	0.0706761
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	−6.74456	−0.345534
$382$	0	0
$383$	16.3505	0.835473	0.417737	−	0.908568i	$-0.362823\pi$
$383$	16.3505	0.835473	0.417737	+	0.908568i	$0.362823\pi$
$384$	0	0
$385$	−4.62772	−0.235850
$386$	0	0
$387$	7.37228	0.374754
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	0.861407	0.0435632
$392$	0	0
$393$	−4.62772	−0.233438
$394$	0	0
$395$	22.7446	1.14440
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	3.37228	0.168825
$400$	0	0
$401$	13.2554	0.661945	0.330972	−	0.943640i	$-0.392623\pi$
$401$	13.2554	0.661945	0.330972	+	0.943640i	$0.392623\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	3.37228	0.167570
$406$	0	0
$407$	−12.8614	−0.637516
$408$	0	0
$409$	−17.6060	−0.870559	−0.435280	−	0.900295i	$-0.643350\pi$
$409$	−17.6060	−0.870559	−0.435280	+	0.900295i	$0.643350\pi$
$410$	0	0
$411$	−11.3723	−0.560953
$412$	0	0
$413$	4.74456	0.233465
$414$	0	0
$415$	6.74456	0.331078
$416$	0	0
$417$	−9.48913	−0.464684
$418$	0	0
$419$	−34.1168	−1.66672	−0.833358	−	0.552733i	$-0.813585\pi$
$419$	−34.1168	−1.66672	−0.833358	+	0.552733i	$0.813585\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	4.74456	0.230689
$424$	0	0
$425$	−8.74456	−0.424174
$426$	0	0
$427$	12.1168	0.586375
$428$	0	0
$429$	−1.37228	−0.0662544
$430$	0	0
$431$	−0.510875	−0.0246080	−0.0123040	−	0.999924i	$-0.503917\pi$
$431$	−0.510875	−0.0246080	−0.0123040	+	0.999924i	$0.503917\pi$
$432$	0	0
$433$	18.2337	0.876255	0.438128	−	0.898913i	$-0.355642\pi$
$433$	18.2337	0.876255	0.438128	+	0.898913i	$0.355642\pi$
$434$	0	0
$435$	4.62772	0.221882
$436$	0	0
$437$	2.11684	0.101262
$438$	0	0
$439$	15.6060	0.744832	0.372416	−	0.928066i	$-0.378529\pi$
$439$	15.6060	0.744832	0.372416	+	0.928066i	$0.378529\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	30.7446	1.46072	0.730359	−	0.683063i	$-0.239353\pi$
$443$	30.7446	1.46072	0.730359	+	0.683063i	$0.239353\pi$
$444$	0	0
$445$	9.25544	0.438750
$446$	0	0
$447$	−17.4891	−0.827207
$448$	0	0
$449$	−24.6277	−1.16225	−0.581127	−	0.813813i	$-0.697388\pi$
$449$	−24.6277	−1.16225	−0.581127	+	0.813813i	$0.697388\pi$
$450$	0	0
$451$	−10.9783	−0.516946
$452$	0	0
$453$	−15.3723	−0.722253
$454$	0	0
$455$	−3.37228	−0.158095
$456$	0	0
$457$	−22.2337	−1.04005	−0.520024	−	0.854152i	$-0.674077\pi$
$457$	−22.2337	−1.04005	−0.520024	+	0.854152i	$0.674077\pi$
$458$	0	0
$459$	1.37228	0.0640526
$460$	0	0
$461$	−13.8832	−0.646603	−0.323302	−	0.946296i	$-0.604793\pi$
$461$	−13.8832	−0.646603	−0.323302	+	0.946296i	$0.604793\pi$
$462$	0	0
$463$	16.8614	0.783616	0.391808	−	0.920047i	$-0.371850\pi$
$463$	16.8614	0.783616	0.391808	+	0.920047i	$0.371850\pi$
$464$	0	0
$465$	13.4891	0.625543
$466$	0	0
$467$	26.1168	1.20854	0.604272	−	0.796778i	$-0.293464\pi$
$467$	26.1168	1.20854	0.604272	+	0.796778i	$0.293464\pi$
$468$	0	0
$469$	−2.74456	−0.126732
$470$	0	0
$471$	10.8614	0.500467
$472$	0	0
$473$	−10.1168	−0.465173
$474$	0	0
$475$	−21.4891	−0.985989
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−12.3505	−0.564310	−0.282155	−	0.959369i	$-0.591049\pi$
$479$	−12.3505	−0.564310	−0.282155	+	0.959369i	$0.591049\pi$
$480$	0	0
$481$	−9.37228	−0.427339
$482$	0	0
$483$	0.627719	0.0285622
$484$	0	0
$485$	−6.74456	−0.306255
$486$	0	0
$487$	14.5109	0.657550	0.328775	−	0.944408i	$-0.393364\pi$
$487$	14.5109	0.657550	0.328775	+	0.944408i	$0.393364\pi$
$488$	0	0
$489$	−1.25544	−0.0567728
$490$	0	0
$491$	20.2337	0.913134	0.456567	−	0.889689i	$-0.349079\pi$
$491$	20.2337	0.913134	0.456567	+	0.889689i	$0.349079\pi$
$492$	0	0
$493$	1.88316	0.0848131
$494$	0	0
$495$	−4.62772	−0.208000
$496$	0	0
$497$	4.74456	0.212823
$498$	0	0
$499$	25.4891	1.14105	0.570525	−	0.821280i	$-0.306740\pi$
$499$	25.4891	1.14105	0.570525	+	0.821280i	$0.306740\pi$
$500$	0	0
$501$	−3.88316	−0.173487
$502$	0	0
$503$	2.74456	0.122374	0.0611870	−	0.998126i	$-0.480511\pi$
$503$	2.74456	0.122374	0.0611870	+	0.998126i	$0.480511\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−24.8614	−1.10196	−0.550981	−	0.834518i	$-0.685746\pi$
$509$	−24.8614	−1.10196	−0.550981	+	0.834518i	$0.685746\pi$
$510$	0	0
$511$	−5.37228	−0.237656
$512$	0	0
$513$	3.37228	0.148890
$514$	0	0
$515$	34.1168	1.50337
$516$	0	0
$517$	−6.51087	−0.286348
$518$	0	0
$519$	3.48913	0.153156
$520$	0	0
$521$	41.8397	1.83303	0.916514	−	0.400002i	$-0.130991\pi$
$521$	41.8397	1.83303	0.916514	+	0.400002i	$0.130991\pi$
$522$	0	0
$523$	6.51087	0.284701	0.142350	−	0.989816i	$-0.454534\pi$
$523$	6.51087	0.284701	0.142350	+	0.989816i	$0.454534\pi$
$524$	0	0
$525$	−6.37228	−0.278109
$526$	0	0
$527$	5.48913	0.239110
$528$	0	0
$529$	−22.6060	−0.982868
$530$	0	0
$531$	4.74456	0.205896
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	−22.7446	−0.983333
$536$	0	0
$537$	17.4891	0.754711
$538$	0	0
$539$	−1.37228	−0.0591083
$540$	0	0
$541$	−3.88316	−0.166950	−0.0834750	−	0.996510i	$-0.526602\pi$
$541$	−3.88316	−0.166950	−0.0834750	+	0.996510i	$0.526602\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	40.8614	1.75031
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	12.1168	0.517134
$550$	0	0
$551$	4.62772	0.197147
$552$	0	0
$553$	6.74456	0.286808
$554$	0	0
$555$	−31.6060	−1.34160
$556$	0	0
$557$	41.4891	1.75795	0.878975	−	0.476867i	$-0.158228\pi$
$557$	41.4891	1.75795	0.878975	+	0.476867i	$0.158228\pi$
$558$	0	0
$559$	−7.37228	−0.311814
$560$	0	0
$561$	−1.88316	−0.0795069
$562$	0	0
$563$	22.1168	0.932114	0.466057	−	0.884755i	$-0.345674\pi$
$563$	22.1168	0.932114	0.466057	+	0.884755i	$0.345674\pi$
$564$	0	0
$565$	−25.2554	−1.06250
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−12.5109	−0.524483	−0.262242	−	0.965002i	$-0.584462\pi$
$569$	−12.5109	−0.524483	−0.262242	+	0.965002i	$0.584462\pi$
$570$	0	0
$571$	36.4674	1.52611	0.763056	−	0.646332i	$-0.223698\pi$
$571$	36.4674	1.52611	0.763056	+	0.646332i	$0.223698\pi$
$572$	0	0
$573$	−5.88316	−0.245772
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−11.4891	−0.478299	−0.239149	−	0.970983i	$-0.576869\pi$
$577$	−11.4891	−0.478299	−0.239149	+	0.970983i	$0.576869\pi$
$578$	0	0
$579$	26.2337	1.09023
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	−8.23369	−0.341005
$584$	0	0
$585$	−3.37228	−0.139427
$586$	0	0
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	13.4891	0.555810
$590$	0	0
$591$	18.9783	0.780661
$592$	0	0
$593$	−32.2337	−1.32368	−0.661839	−	0.749646i	$-0.730224\pi$
$593$	−32.2337	−1.32368	−0.661839	+	0.749646i	$0.730224\pi$
$594$	0	0
$595$	−4.62772	−0.189718
$596$	0	0
$597$	4.62772	0.189400
$598$	0	0
$599$	−7.37228	−0.301223	−0.150612	−	0.988593i	$-0.548124\pi$
$599$	−7.37228	−0.301223	−0.150612	+	0.988593i	$0.548124\pi$
$600$	0	0
$601$	−11.7228	−0.478184	−0.239092	−	0.970997i	$-0.576850\pi$
$601$	−11.7228	−0.478184	−0.239092	+	0.970997i	$0.576850\pi$
$602$	0	0
$603$	−2.74456	−0.111767
$604$	0	0
$605$	−30.7446	−1.24994
$606$	0	0
$607$	−46.3505	−1.88131	−0.940655	−	0.339364i	$-0.889788\pi$
$607$	−46.3505	−1.88131	−0.940655	+	0.339364i	$0.889788\pi$
$608$	0	0
$609$	1.37228	0.0556076
$610$	0	0
$611$	−4.74456	−0.191944
$612$	0	0
$613$	5.60597	0.226423	0.113211	−	0.993571i	$-0.463886\pi$
$613$	5.60597	0.226423	0.113211	+	0.993571i	$0.463886\pi$
$614$	0	0
$615$	−26.9783	−1.08787
$616$	0	0
$617$	21.0951	0.849257	0.424628	−	0.905368i	$-0.360405\pi$
$617$	21.0951	0.849257	0.424628	+	0.905368i	$0.360405\pi$
$618$	0	0
$619$	−47.8397	−1.92284	−0.961419	−	0.275088i	$-0.911293\pi$
$619$	−47.8397	−1.92284	−0.961419	+	0.275088i	$0.911293\pi$
$620$	0	0
$621$	0.627719	0.0251895
$622$	0	0
$623$	2.74456	0.109959
$624$	0	0
$625$	−16.2554	−0.650217
$626$	0	0
$627$	−4.62772	−0.184813
$628$	0	0
$629$	−12.8614	−0.512818
$630$	0	0
$631$	−49.0951	−1.95444	−0.977222	−	0.212218i	$-0.931931\pi$
$631$	−49.0951	−1.95444	−0.977222	+	0.212218i	$0.931931\pi$
$632$	0	0
$633$	12.8614	0.511195
$634$	0	0
$635$	22.7446	0.902590
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	4.74456	0.187692
$640$	0	0
$641$	20.5109	0.810131	0.405065	−	0.914288i	$-0.367249\pi$
$641$	20.5109	0.810131	0.405065	+	0.914288i	$0.367249\pi$
$642$	0	0
$643$	−25.0951	−0.989654	−0.494827	−	0.868992i	$-0.664769\pi$
$643$	−25.0951	−0.989654	−0.494827	+	0.868992i	$0.664769\pi$
$644$	0	0
$645$	−24.8614	−0.978917
$646$	0	0
$647$	−45.4891	−1.78836	−0.894181	−	0.447706i	$-0.852241\pi$
$647$	−45.4891	−1.78836	−0.894181	+	0.447706i	$0.852241\pi$
$648$	0	0
$649$	−6.51087	−0.255574
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	18.8614	0.738104	0.369052	−	0.929409i	$-0.379682\pi$
$653$	18.8614	0.738104	0.369052	+	0.929409i	$0.379682\pi$
$654$	0	0
$655$	15.6060	0.609776
$656$	0	0
$657$	−5.37228	−0.209593
$658$	0	0
$659$	32.2337	1.25565	0.627823	−	0.778356i	$-0.283946\pi$
$659$	32.2337	1.25565	0.627823	+	0.778356i	$0.283946\pi$
$660$	0	0
$661$	−26.4674	−1.02946	−0.514731	−	0.857352i	$-0.672108\pi$
$661$	−26.4674	−1.02946	−0.514731	+	0.857352i	$0.672108\pi$
$662$	0	0
$663$	−1.37228	−0.0532950
$664$	0	0
$665$	−11.3723	−0.440998
$666$	0	0
$667$	0.861407	0.0333538
$668$	0	0
$669$	13.2554	0.512485
$670$	0	0
$671$	−16.6277	−0.641906
$672$	0	0
$673$	−21.1386	−0.814833	−0.407416	−	0.913242i	$-0.633570\pi$
$673$	−21.1386	−0.814833	−0.407416	+	0.913242i	$0.633570\pi$
$674$	0	0
$675$	−6.37228	−0.245269
$676$	0	0
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	5.37228	0.205565	0.102782	−	0.994704i	$-0.467225\pi$
$683$	5.37228	0.205565	0.102782	+	0.994704i	$0.467225\pi$
$684$	0	0
$685$	38.3505	1.46530
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−46.9783	−1.78714	−0.893568	−	0.448927i	$-0.851806\pi$
$691$	−46.9783	−1.78714	−0.893568	+	0.448927i	$0.851806\pi$
$692$	0	0
$693$	−1.37228	−0.0521287
$694$	0	0
$695$	32.0000	1.21383
$696$	0	0
$697$	−10.9783	−0.415831
$698$	0	0
$699$	−26.2337	−0.992249
$700$	0	0
$701$	−4.51087	−0.170373	−0.0851867	−	0.996365i	$-0.527149\pi$
$701$	−4.51087	−0.170373	−0.0851867	+	0.996365i	$0.527149\pi$
$702$	0	0
$703$	−31.6060	−1.19204
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	−4.74456	−0.178438
$708$	0	0
$709$	−51.9565	−1.95127	−0.975634	−	0.219406i	$-0.929588\pi$
$709$	−51.9565	−1.95127	−0.975634	+	0.219406i	$0.929588\pi$
$710$	0	0
$711$	6.74456	0.252941
$712$	0	0
$713$	2.51087	0.0940330
$714$	0	0
$715$	4.62772	0.173067
$716$	0	0
$717$	8.74456	0.326572
$718$	0	0
$719$	−34.7446	−1.29575	−0.647877	−	0.761745i	$-0.724343\pi$
$719$	−34.7446	−1.29575	−0.647877	+	0.761745i	$0.724343\pi$
$720$	0	0
$721$	10.1168	0.376771
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	−8.74456	−0.324765
$726$	0	0
$727$	−27.3723	−1.01518	−0.507591	−	0.861598i	$-0.669464\pi$
$727$	−27.3723	−1.01518	−0.507591	+	0.861598i	$0.669464\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.1168	−0.374185
$732$	0	0
$733$	−7.25544	−0.267986	−0.133993	−	0.990982i	$-0.542780\pi$
$733$	−7.25544	−0.267986	−0.133993	+	0.990982i	$0.542780\pi$
$734$	0	0
$735$	−3.37228	−0.124388
$736$	0	0
$737$	3.76631	0.138734
$738$	0	0
$739$	16.2337	0.597166	0.298583	−	0.954384i	$-0.403486\pi$
$739$	16.2337	0.597166	0.298583	+	0.954384i	$0.403486\pi$
$740$	0	0
$741$	−3.37228	−0.123884
$742$	0	0
$743$	−4.51087	−0.165488	−0.0827440	−	0.996571i	$-0.526368\pi$
$743$	−4.51087	−0.165488	−0.0827440	+	0.996571i	$0.526368\pi$
$744$	0	0
$745$	58.9783	2.16080
$746$	0	0
$747$	2.00000	0.0731762
$748$	0	0
$749$	−6.74456	−0.246441
$750$	0	0
$751$	−10.9783	−0.400602	−0.200301	−	0.979734i	$-0.564192\pi$
$751$	−10.9783	−0.400602	−0.200301	+	0.979734i	$0.564192\pi$
$752$	0	0
$753$	−2.11684	−0.0771421
$754$	0	0
$755$	51.8397	1.88664
$756$	0	0
$757$	−3.48913	−0.126814	−0.0634072	−	0.997988i	$-0.520197\pi$
$757$	−3.48913	−0.126814	−0.0634072	+	0.997988i	$0.520197\pi$
$758$	0	0
$759$	−0.861407	−0.0312671
$760$	0	0
$761$	−34.9783	−1.26796	−0.633980	−	0.773349i	$-0.718580\pi$
$761$	−34.9783	−1.26796	−0.633980	+	0.773349i	$0.718580\pi$
$762$	0	0
$763$	12.1168	0.438659
$764$	0	0
$765$	−4.62772	−0.167316
$766$	0	0
$767$	−4.74456	−0.171316
$768$	0	0
$769$	34.8614	1.25713	0.628567	−	0.777755i	$-0.283642\pi$
$769$	34.8614	1.25713	0.628567	+	0.777755i	$0.283642\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−15.3723	−0.552903	−0.276451	−	0.961028i	$-0.589158\pi$
$773$	−15.3723	−0.552903	−0.276451	+	0.961028i	$0.589158\pi$
$774$	0	0
$775$	−25.4891	−0.915596
$776$	0	0
$777$	−9.37228	−0.336229
$778$	0	0
$779$	−26.9783	−0.966596
$780$	0	0
$781$	−6.51087	−0.232977
$782$	0	0
$783$	1.37228	0.0490413
$784$	0	0
$785$	−36.6277	−1.30730
$786$	0	0
$787$	14.1168	0.503211	0.251606	−	0.967830i	$-0.419041\pi$
$787$	14.1168	0.503211	0.251606	+	0.967830i	$0.419041\pi$
$788$	0	0
$789$	14.9783	0.533240
$790$	0	0
$791$	−7.48913	−0.266283
$792$	0	0
$793$	−12.1168	−0.430282
$794$	0	0
$795$	−20.2337	−0.717615
$796$	0	0
$797$	39.2554	1.39050	0.695249	−	0.718769i	$-0.255294\pi$
$797$	39.2554	1.39050	0.695249	+	0.718769i	$0.255294\pi$
$798$	0	0
$799$	−6.51087	−0.230338
$800$	0	0
$801$	2.74456	0.0969744
$802$	0	0
$803$	7.37228	0.260162
$804$	0	0
$805$	−2.11684	−0.0746089
$806$	0	0
$807$	−15.4891	−0.545243
$808$	0	0
$809$	−26.4674	−0.930543	−0.465272	−	0.885168i	$-0.654043\pi$
$809$	−26.4674	−0.930543	−0.465272	+	0.885168i	$0.654043\pi$
$810$	0	0
$811$	−32.8614	−1.15392	−0.576960	−	0.816772i	$-0.695761\pi$
$811$	−32.8614	−1.15392	−0.576960	+	0.816772i	$0.695761\pi$
$812$	0	0
$813$	6.74456	0.236542
$814$	0	0
$815$	4.23369	0.148300
$816$	0	0
$817$	−24.8614	−0.869791
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−17.4891	−0.610375	−0.305187	−	0.952292i	$-0.598719\pi$
$821$	−17.4891	−0.610375	−0.305187	+	0.952292i	$0.598719\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	8.74456	0.304447
$826$	0	0
$827$	6.62772	0.230468	0.115234	−	0.993338i	$-0.463238\pi$
$827$	6.62772	0.230468	0.115234	+	0.993338i	$0.463238\pi$
$828$	0	0
$829$	−22.8614	−0.794009	−0.397005	−	0.917817i	$-0.629950\pi$
$829$	−22.8614	−0.794009	−0.397005	+	0.917817i	$0.629950\pi$
$830$	0	0
$831$	−14.0000	−0.485655
$832$	0	0
$833$	−1.37228	−0.0475467
$834$	0	0
$835$	13.0951	0.453174
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	21.7663	0.751457	0.375728	−	0.926730i	$-0.377393\pi$
$839$	21.7663	0.751457	0.375728	+	0.926730i	$0.377393\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	20.2337	0.696885
$844$	0	0
$845$	3.37228	0.116010
$846$	0	0
$847$	−9.11684	−0.313258
$848$	0	0
$849$	−17.4891	−0.600225
$850$	0	0
$851$	−5.88316	−0.201672
$852$	0	0
$853$	−24.7446	−0.847238	−0.423619	−	0.905841i	$-0.639240\pi$
$853$	−24.7446	−0.847238	−0.423619	+	0.905841i	$0.639240\pi$
$854$	0	0
$855$	−11.3723	−0.388924
$856$	0	0
$857$	32.5109	1.11055	0.555275	−	0.831667i	$-0.312613\pi$
$857$	32.5109	1.11055	0.555275	+	0.831667i	$0.312613\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	50.0000	1.70202	0.851010	−	0.525150i	$-0.175991\pi$
$863$	50.0000	1.70202	0.851010	+	0.525150i	$0.175991\pi$
$864$	0	0
$865$	−11.7663	−0.400067
$866$	0	0
$867$	15.1168	0.513395
$868$	0	0
$869$	−9.25544	−0.313969
$870$	0	0
$871$	2.74456	0.0929960
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	4.62772	0.156445
$876$	0	0
$877$	−20.5109	−0.692603	−0.346302	−	0.938123i	$-0.612563\pi$
$877$	−20.5109	−0.692603	−0.346302	+	0.938123i	$0.612563\pi$
$878$	0	0
$879$	−5.25544	−0.177262
$880$	0	0
$881$	40.1168	1.35157	0.675785	−	0.737098i	$-0.263805\pi$
$881$	40.1168	1.35157	0.675785	+	0.737098i	$0.263805\pi$
$882$	0	0
$883$	−19.6060	−0.659793	−0.329897	−	0.944017i	$-0.607014\pi$
$883$	−19.6060	−0.659793	−0.329897	+	0.944017i	$0.607014\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	−18.5109	−0.621534	−0.310767	−	0.950486i	$-0.600586\pi$
$887$	−18.5109	−0.621534	−0.310767	+	0.950486i	$0.600586\pi$
$888$	0	0
$889$	6.74456	0.226205
$890$	0	0
$891$	−1.37228	−0.0459732
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	−58.9783	−1.97143
$896$	0	0
$897$	−0.627719	−0.0209589
$898$	0	0
$899$	5.48913	0.183073
$900$	0	0
$901$	−8.23369	−0.274304
$902$	0	0
$903$	−7.37228	−0.245334
$904$	0	0
$905$	20.2337	0.672591
$906$	0	0
$907$	49.9565	1.65878	0.829389	−	0.558671i	$-0.188689\pi$
$907$	49.9565	1.65878	0.829389	+	0.558671i	$0.188689\pi$
$908$	0	0
$909$	−4.74456	−0.157367
$910$	0	0
$911$	−48.8614	−1.61885	−0.809425	−	0.587223i	$-0.800221\pi$
$911$	−48.8614	−1.61885	−0.809425	+	0.587223i	$0.800221\pi$
$912$	0	0
$913$	−2.74456	−0.0908318
$914$	0	0
$915$	−40.8614	−1.35084
$916$	0	0
$917$	4.62772	0.152821
$918$	0	0
$919$	−39.2119	−1.29348	−0.646741	−	0.762709i	$-0.723869\pi$
$919$	−39.2119	−1.29348	−0.646741	+	0.762709i	$0.723869\pi$
$920$	0	0
$921$	−9.48913	−0.312677
$922$	0	0
$923$	−4.74456	−0.156169
$924$	0	0
$925$	59.7228	1.96367
$926$	0	0
$927$	10.1168	0.332281
$928$	0	0
$929$	−16.4674	−0.540277	−0.270139	−	0.962821i	$-0.587070\pi$
$929$	−16.4674	−0.540277	−0.270139	+	0.962821i	$0.587070\pi$
$930$	0	0
$931$	−3.37228	−0.110522
$932$	0	0
$933$	−2.74456	−0.0898529
$934$	0	0
$935$	6.35053	0.207685
$936$	0	0
$937$	38.4674	1.25667	0.628337	−	0.777941i	$-0.283736\pi$
$937$	38.4674	1.25667	0.628337	+	0.777941i	$0.283736\pi$
$938$	0	0
$939$	32.7446	1.06858
$940$	0	0
$941$	−35.2119	−1.14788	−0.573938	−	0.818899i	$-0.694585\pi$
$941$	−35.2119	−1.14788	−0.573938	+	0.818899i	$0.694585\pi$
$942$	0	0
$943$	−5.02175	−0.163531
$944$	0	0
$945$	−3.37228	−0.109700
$946$	0	0
$947$	56.5842	1.83874	0.919370	−	0.393394i	$-0.128699\pi$
$947$	56.5842	1.83874	0.919370	+	0.393394i	$0.128699\pi$
$948$	0	0
$949$	5.37228	0.174392
$950$	0	0
$951$	−16.0000	−0.518836
$952$	0	0
$953$	29.2119	0.946268	0.473134	−	0.880991i	$-0.343123\pi$
$953$	29.2119	0.946268	0.473134	+	0.880991i	$0.343123\pi$
$954$	0	0
$955$	19.8397	0.641996
$956$	0	0
$957$	−1.88316	−0.0608738
$958$	0	0
$959$	11.3723	0.367230
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−6.74456	−0.217340
$964$	0	0
$965$	−88.4674	−2.84787
$966$	0	0
$967$	33.0951	1.06427	0.532133	−	0.846661i	$-0.321391\pi$
$967$	33.0951	1.06427	0.532133	+	0.846661i	$0.321391\pi$
$968$	0	0
$969$	−4.62772	−0.148664
$970$	0	0
$971$	12.4674	0.400097	0.200049	−	0.979786i	$-0.435890\pi$
$971$	12.4674	0.400097	0.200049	+	0.979786i	$0.435890\pi$
$972$	0	0
$973$	9.48913	0.304207
$974$	0	0
$975$	6.37228	0.204076
$976$	0	0
$977$	54.1168	1.73135	0.865676	−	0.500605i	$-0.166889\pi$
$977$	54.1168	1.73135	0.865676	+	0.500605i	$0.166889\pi$
$978$	0	0
$979$	−3.76631	−0.120372
$980$	0	0
$981$	12.1168	0.386861
$982$	0	0
$983$	−55.3288	−1.76471	−0.882357	−	0.470581i	$-0.844045\pi$
$983$	−55.3288	−1.76471	−0.882357	+	0.470581i	$0.844045\pi$
$984$	0	0
$985$	−64.0000	−2.03921
$986$	0	0
$987$	−4.74456	−0.151021
$988$	0	0
$989$	−4.62772	−0.147153
$990$	0	0
$991$	−38.7446	−1.23076	−0.615381	−	0.788230i	$-0.710998\pi$
$991$	−38.7446	−1.23076	−0.615381	+	0.788230i	$0.710998\pi$
$992$	0	0
$993$	−34.7446	−1.10259
$994$	0	0
$995$	−15.6060	−0.494742
$996$	0	0
$997$	32.9783	1.04443	0.522216	−	0.852813i	$-0.325106\pi$
$997$	32.9783	1.04443	0.522216	+	0.852813i	$0.325106\pi$
$998$	0	0
$999$	−9.37228	−0.296526

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.r.1.2	✓	2	4.3	odd	2
4368.2.a.bd.1.2		2	1.1	even	1	trivial
6552.2.a.bd.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.37228 q^{11} -1.00000 q^{13} -3.37228 q^{15} -1.37228 q^{17} -3.37228 q^{19} -1.00000 q^{21} -0.627719 q^{23} +6.37228 q^{25} -1.00000 q^{27} -1.37228 q^{29} -4.00000 q^{31} +1.37228 q^{33} +3.37228 q^{35} +9.37228 q^{37} +1.00000 q^{39} +8.00000 q^{41} +7.37228 q^{43} +3.37228 q^{45} +4.74456 q^{47} +1.00000 q^{49} +1.37228 q^{51} +6.00000 q^{53} -4.62772 q^{55} +3.37228 q^{57} +4.74456 q^{59} +12.1168 q^{61} +1.00000 q^{63} -3.37228 q^{65} -2.74456 q^{67} +0.627719 q^{69} +4.74456 q^{71} -5.37228 q^{73} -6.37228 q^{75} -1.37228 q^{77} +6.74456 q^{79} +1.00000 q^{81} +2.00000 q^{83} -4.62772 q^{85} +1.37228 q^{87} +2.74456 q^{89} -1.00000 q^{91} +4.00000 q^{93} -11.3723 q^{95} -2.00000 q^{97} -1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} - q^{15} + 3 q^{17} - q^{19} - 2 q^{21} - 7 q^{23} + 7 q^{25} - 2 q^{27} + 3 q^{29} - 8 q^{31} - 3 q^{33} + q^{35} + 13 q^{37} + 2 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 2 q^{47} + 2 q^{49} - 3 q^{51} + 12 q^{53} - 15 q^{55} + q^{57} - 2 q^{59} + 7 q^{61} + 2 q^{63} - q^{65} + 6 q^{67} + 7 q^{69} - 2 q^{71} - 5 q^{73} - 7 q^{75} + 3 q^{77} + 2 q^{79} + 2 q^{81} + 4 q^{83} - 15 q^{85} - 3 q^{87} - 6 q^{89} - 2 q^{91} + 8 q^{93} - 17 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 3 * q^11 - 2 * q^13 - q^15 + 3 * q^17 - q^19 - 2 * q^21 - 7 * q^23 + 7 * q^25 - 2 * q^27 + 3 * q^29 - 8 * q^31 - 3 * q^33 + q^35 + 13 * q^37 + 2 * q^39 + 16 * q^41 + 9 * q^43 + q^45 - 2 * q^47 + 2 * q^49 - 3 * q^51 + 12 * q^53 - 15 * q^55 + q^57 - 2 * q^59 + 7 * q^61 + 2 * q^63 - q^65 + 6 * q^67 + 7 * q^69 - 2 * q^71 - 5 * q^73 - 7 * q^75 + 3 * q^77 + 2 * q^79 + 2 * q^81 + 4 * q^83 - 15 * q^85 - 3 * q^87 - 6 * q^89 - 2 * q^91 + 8 * q^93 - 17 * q^95 - 4 * q^97 + 3 * q^99

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