LMFDB - Embedded newform 7800.2.a.bz.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.504568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 2 $ x^5 - x^4 - 6x^3 + 3x^2 + 7*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.28447$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.71215 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.71215 q^{7} +1.00000 q^{9} +3.31842 q^{11} -1.00000 q^{13} -1.55706 q^{17} -5.33709 q^{19} -3.71215 q^{21} -0.442940 q^{23} -1.00000 q^{27} +2.56894 q^{29} +0.613062 q^{31} -3.31842 q^{33} +0.257322 q^{37} +1.00000 q^{39} -10.6114 q^{41} -12.6935 q^{43} -7.44442 q^{47} +6.78003 q^{49} +1.55706 q^{51} +5.39521 q^{53} +5.33709 q^{57} -13.1358 q^{59} -5.27452 q^{61} +3.71215 q^{63} -10.5384 q^{67} +0.442940 q^{69} +0.311845 q^{71} -9.46841 q^{73} +12.3184 q^{77} +16.1871 q^{79} +1.00000 q^{81} -11.7546 q^{83} -2.56894 q^{87} -4.58762 q^{89} -3.71215 q^{91} -0.613062 q^{93} -10.8500 q^{97} +3.31842 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - q^7 + 5 * q^9	$ 5 q - 5 q^{3} - q^{7} + 5 q^{9} + 5 q^{11} - 5 q^{13} - 7 q^{17} + 6 q^{19} + q^{21} - 3 q^{23} - 5 q^{27} + 2 q^{29} - 5 q^{33} - 9 q^{37} + 5 q^{39} - q^{41} - 4 q^{43} - 14 q^{47} + 2 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{57} + 20 q^{59} - 19 q^{61} - q^{63} - 12 q^{67} + 3 q^{69} + 13 q^{71} - 16 q^{73} - 11 q^{77} + 7 q^{79} + 5 q^{81} + 2 q^{83} - 2 q^{87} + 9 q^{89} + q^{91} - 13 q^{97} + 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - q^7 + 5 * q^9 + 5 * q^11 - 5 * q^13 - 7 * q^17 + 6 * q^19 + q^21 - 3 * q^23 - 5 * q^27 + 2 * q^29 - 5 * q^33 - 9 * q^37 + 5 * q^39 - q^41 - 4 * q^43 - 14 * q^47 + 2 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^57 + 20 * q^59 - 19 * q^61 - q^63 - 12 * q^67 + 3 * q^69 + 13 * q^71 - 16 * q^73 - 11 * q^77 + 7 * q^79 + 5 * q^81 + 2 * q^83 - 2 * q^87 + 9 * q^89 + q^91 - 13 * q^97 + 5 * 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	0
$5$	0	0
$6$	0	0
$7$	3.71215	1.40306	0.701530	−	0.712640i	$-0.252501\pi$
$7$	3.71215	1.40306	0.701530	+	0.712640i	$0.252501\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.31842	1.00054	0.500270	−	0.865869i	$-0.333234\pi$
$11$	3.31842	1.00054	0.500270	+	0.865869i	$0.333234\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.55706	−0.377642	−0.188821	−	0.982011i	$-0.560467\pi$
$17$	−1.55706	−0.377642	−0.188821	+	0.982011i	$0.560467\pi$
$18$	0	0
$19$	−5.33709	−1.22441	−0.612207	−	0.790698i	$-0.709718\pi$
$19$	−5.33709	−1.22441	−0.612207	+	0.790698i	$0.709718\pi$
$20$	0	0
$21$	−3.71215	−0.810057
$22$	0	0
$23$	−0.442940	−0.0923594	−0.0461797	−	0.998933i	$-0.514705\pi$
$23$	−0.442940	−0.0923594	−0.0461797	+	0.998933i	$0.514705\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.56894	0.477041	0.238521	−	0.971137i	$-0.423338\pi$
$29$	2.56894	0.477041	0.238521	+	0.971137i	$0.423338\pi$
$30$	0	0
$31$	0.613062	0.110109	0.0550546	−	0.998483i	$-0.482467\pi$
$31$	0.613062	0.110109	0.0550546	+	0.998483i	$0.482467\pi$
$32$	0	0
$33$	−3.31842	−0.577662
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.257322	0.0423034	0.0211517	−	0.999776i	$-0.493267\pi$
$37$	0.257322	0.0423034	0.0211517	+	0.999776i	$0.493267\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−10.6114	−1.65722	−0.828611	−	0.559826i	$-0.810868\pi$
$41$	−10.6114	−1.65722	−0.828611	+	0.559826i	$0.810868\pi$
$42$	0	0
$43$	−12.6935	−1.93574	−0.967870	−	0.251450i	$-0.919093\pi$
$43$	−12.6935	−1.93574	−0.967870	+	0.251450i	$0.919093\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.44442	−1.08588	−0.542940	−	0.839771i	$-0.682689\pi$
$47$	−7.44442	−1.08588	−0.542940	+	0.839771i	$0.682689\pi$
$48$	0	0
$49$	6.78003	0.968576
$50$	0	0
$51$	1.55706	0.218032
$52$	0	0
$53$	5.39521	0.741089	0.370545	−	0.928815i	$-0.379171\pi$
$53$	5.39521	0.741089	0.370545	+	0.928815i	$0.379171\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.33709	0.706915
$58$	0	0
$59$	−13.1358	−1.71014	−0.855068	−	0.518516i	$-0.826485\pi$
$59$	−13.1358	−1.71014	−0.855068	+	0.518516i	$0.826485\pi$
$60$	0	0
$61$	−5.27452	−0.675333	−0.337667	−	0.941266i	$-0.609638\pi$
$61$	−5.27452	−0.675333	−0.337667	+	0.941266i	$0.609638\pi$
$62$	0	0
$63$	3.71215	0.467687
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.5384	−1.28747	−0.643736	−	0.765248i	$-0.722617\pi$
$67$	−10.5384	−1.28747	−0.643736	+	0.765248i	$0.722617\pi$
$68$	0	0
$69$	0.442940	0.0533237
$70$	0	0
$71$	0.311845	0.0370092	0.0185046	−	0.999829i	$-0.494109\pi$
$71$	0.311845	0.0370092	0.0185046	+	0.999829i	$0.494109\pi$
$72$	0	0
$73$	−9.46841	−1.10819	−0.554097	−	0.832452i	$-0.686936\pi$
$73$	−9.46841	−1.10819	−0.554097	+	0.832452i	$0.686936\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.3184	1.40382
$78$	0	0
$79$	16.1871	1.82119	0.910597	−	0.413295i	$-0.135622\pi$
$79$	16.1871	1.82119	0.910597	+	0.413295i	$0.135622\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.7546	−1.29023	−0.645117	−	0.764084i	$-0.723191\pi$
$83$	−11.7546	−1.29023	−0.645117	+	0.764084i	$0.723191\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.56894	−0.275420
$88$	0	0
$89$	−4.58762	−0.486287	−0.243144	−	0.969990i	$-0.578179\pi$
$89$	−4.58762	−0.486287	−0.243144	+	0.969990i	$0.578179\pi$
$90$	0	0
$91$	−3.71215	−0.389139
$92$	0	0
$93$	−0.613062	−0.0635716
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.8500	−1.10165	−0.550827	−	0.834619i	$-0.685688\pi$
$97$	−10.8500	−1.10165	−0.550827	+	0.834619i	$0.685688\pi$
$98$	0	0
$99$	3.31842	0.333513
$100$	0	0
$101$	1.68306	0.167471	0.0837356	−	0.996488i	$-0.473315\pi$
$101$	1.68306	0.167471	0.0837356	+	0.996488i	$0.473315\pi$
$102$	0	0
$103$	−1.78535	−0.175916	−0.0879578	−	0.996124i	$-0.528034\pi$
$103$	−1.78535	−0.175916	−0.0879578	+	0.996124i	$0.528034\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.11943	−0.108220	−0.0541098	−	0.998535i	$-0.517232\pi$
$107$	−1.11943	−0.108220	−0.0541098	+	0.998535i	$0.517232\pi$
$108$	0	0
$109$	−0.914914	−0.0876329	−0.0438164	−	0.999040i	$-0.513952\pi$
$109$	−0.914914	−0.0876329	−0.0438164	+	0.999040i	$0.513952\pi$
$110$	0	0
$111$	−0.257322	−0.0244239
$112$	0	0
$113$	−2.41386	−0.227077	−0.113538	−	0.993534i	$-0.536218\pi$
$113$	−2.41386	−0.227077	−0.113538	+	0.993534i	$0.536218\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−5.78003	−0.529855
$120$	0	0
$121$	0.0118848	0.00108043
$122$	0	0
$123$	10.6114	0.956797
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.0418	1.24601	0.623005	−	0.782218i	$-0.285912\pi$
$127$	14.0418	1.24601	0.623005	+	0.782218i	$0.285912\pi$
$128$	0	0
$129$	12.6935	1.11760
$130$	0	0
$131$	13.7650	1.20265	0.601327	−	0.799003i	$-0.294639\pi$
$131$	13.7650	1.20265	0.601327	+	0.799003i	$0.294639\pi$
$132$	0	0
$133$	−19.8121	−1.71792
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.50470	0.555734	0.277867	−	0.960620i	$-0.410373\pi$
$137$	6.50470	0.555734	0.277867	+	0.960620i	$0.410373\pi$
$138$	0	0
$139$	9.78003	0.829532	0.414766	−	0.909928i	$-0.363863\pi$
$139$	9.78003	0.829532	0.414766	+	0.909928i	$0.363863\pi$
$140$	0	0
$141$	7.44442	0.626933
$142$	0	0
$143$	−3.31842	−0.277500
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.78003	−0.559208
$148$	0	0
$149$	−5.93148	−0.485926	−0.242963	−	0.970036i	$-0.578119\pi$
$149$	−5.93148	−0.485926	−0.242963	+	0.970036i	$0.578119\pi$
$150$	0	0
$151$	−0.272818	−0.0222016	−0.0111008	−	0.999938i	$-0.503534\pi$
$151$	−0.272818	−0.0222016	−0.0111008	+	0.999938i	$0.503534\pi$
$152$	0	0
$153$	−1.55706	−0.125881
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.6561	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$157$	−14.6561	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$158$	0	0
$159$	−5.39521	−0.427868
$160$	0	0
$161$	−1.64426	−0.129586
$162$	0	0
$163$	3.48345	0.272845	0.136422	−	0.990651i	$-0.456440\pi$
$163$	3.48345	0.272845	0.136422	+	0.990651i	$0.456440\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.69135	−0.363028	−0.181514	−	0.983388i	$-0.558100\pi$
$167$	−4.69135	−0.363028	−0.181514	+	0.983388i	$0.558100\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.33709	−0.408138
$172$	0	0
$173$	−3.26475	−0.248214	−0.124107	−	0.992269i	$-0.539607\pi$
$173$	−3.26475	−0.248214	−0.124107	+	0.992269i	$0.539607\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.1358	0.987348
$178$	0	0
$179$	19.7204	1.47397	0.736987	−	0.675907i	$-0.236248\pi$
$179$	19.7204	1.47397	0.736987	+	0.675907i	$0.236248\pi$
$180$	0	0
$181$	−22.0558	−1.63940	−0.819698	−	0.572796i	$-0.805859\pi$
$181$	−22.0558	−1.63940	−0.819698	+	0.572796i	$0.805859\pi$
$182$	0	0
$183$	5.27452	0.389904
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.16697	−0.377846
$188$	0	0
$189$	−3.71215	−0.270019
$190$	0	0
$191$	22.2729	1.61161	0.805805	−	0.592182i	$-0.201733\pi$
$191$	22.2729	1.61161	0.805805	+	0.592182i	$0.201733\pi$
$192$	0	0
$193$	15.7733	1.13539	0.567693	−	0.823241i	$-0.307836\pi$
$193$	15.7733	1.13539	0.567693	+	0.823241i	$0.307836\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.72871	0.194413	0.0972063	−	0.995264i	$-0.469009\pi$
$197$	2.72871	0.194413	0.0972063	+	0.995264i	$0.469009\pi$
$198$	0	0
$199$	−22.1029	−1.56684	−0.783418	−	0.621495i	$-0.786526\pi$
$199$	−22.1029	−1.56684	−0.783418	+	0.621495i	$0.786526\pi$
$200$	0	0
$201$	10.5384	0.743322
$202$	0	0
$203$	9.53630	0.669317
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.442940	−0.0307865
$208$	0	0
$209$	−17.7107	−1.22507
$210$	0	0
$211$	1.01720	0.0700268	0.0350134	−	0.999387i	$-0.488853\pi$
$211$	1.01720	0.0700268	0.0350134	+	0.999387i	$0.488853\pi$
$212$	0	0
$213$	−0.311845	−0.0213672
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.27578	0.154490
$218$	0	0
$219$	9.46841	0.639816
$220$	0	0
$221$	1.55706	0.104739
$222$	0	0
$223$	6.85535	0.459068	0.229534	−	0.973301i	$-0.426280\pi$
$223$	6.85535	0.459068	0.229534	+	0.973301i	$0.426280\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.88224	−0.191301	−0.0956504	−	0.995415i	$-0.530493\pi$
$227$	−2.88224	−0.191301	−0.0956504	+	0.995415i	$0.530493\pi$
$228$	0	0
$229$	14.1857	0.937416	0.468708	−	0.883353i	$-0.344720\pi$
$229$	14.1857	0.937416	0.468708	+	0.883353i	$0.344720\pi$
$230$	0	0
$231$	−12.3184	−0.810494
$232$	0	0
$233$	−0.429354	−0.0281279	−0.0140639	−	0.999901i	$-0.504477\pi$
$233$	−0.429354	−0.0281279	−0.0140639	+	0.999901i	$0.504477\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.1871	−1.05147
$238$	0	0
$239$	19.7255	1.27594	0.637969	−	0.770062i	$-0.279775\pi$
$239$	19.7255	1.27594	0.637969	+	0.770062i	$0.279775\pi$
$240$	0	0
$241$	19.6598	1.26640	0.633200	−	0.773988i	$-0.281741\pi$
$241$	19.6598	1.26640	0.633200	+	0.773988i	$0.281741\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.33709	0.339591
$248$	0	0
$249$	11.7546	0.744917
$250$	0	0
$251$	23.4175	1.47810	0.739051	−	0.673650i	$-0.235274\pi$
$251$	23.4175	1.47810	0.739051	+	0.673650i	$0.235274\pi$
$252$	0	0
$253$	−1.46986	−0.0924093
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−30.7987	−1.92117	−0.960586	−	0.277981i	$-0.910335\pi$
$257$	−30.7987	−1.92117	−0.960586	+	0.277981i	$0.910335\pi$
$258$	0	0
$259$	0.955216	0.0593543
$260$	0	0
$261$	2.56894	0.159014
$262$	0	0
$263$	24.1471	1.48897	0.744486	−	0.667638i	$-0.232695\pi$
$263$	24.1471	1.48897	0.744486	+	0.667638i	$0.232695\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.58762	0.280758
$268$	0	0
$269$	12.7557	0.777726	0.388863	−	0.921295i	$-0.372868\pi$
$269$	12.7557	0.777726	0.388863	+	0.921295i	$0.372868\pi$
$270$	0	0
$271$	−10.4222	−0.633102	−0.316551	−	0.948575i	$-0.602525\pi$
$271$	−10.4222	−0.633102	−0.316551	+	0.948575i	$0.602525\pi$
$272$	0	0
$273$	3.71215	0.224669
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.9404	−1.19810	−0.599052	−	0.800710i	$-0.704456\pi$
$277$	−19.9404	−1.19810	−0.599052	+	0.800710i	$0.704456\pi$
$278$	0	0
$279$	0.613062	0.0367031
$280$	0	0
$281$	28.5272	1.70179	0.850895	−	0.525336i	$-0.176060\pi$
$281$	28.5272	1.70179	0.850895	+	0.525336i	$0.176060\pi$
$282$	0	0
$283$	−19.1888	−1.14066	−0.570329	−	0.821416i	$-0.693184\pi$
$283$	−19.1888	−1.14066	−0.570329	+	0.821416i	$0.693184\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−39.3910	−2.32518
$288$	0	0
$289$	−14.5756	−0.857386
$290$	0	0
$291$	10.8500	0.636040
$292$	0	0
$293$	11.9828	0.700045	0.350022	−	0.936741i	$-0.386174\pi$
$293$	11.9828	0.700045	0.350022	+	0.936741i	$0.386174\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.31842	−0.192554
$298$	0	0
$299$	0.442940	0.0256159
$300$	0	0
$301$	−47.1201	−2.71596
$302$	0	0
$303$	−1.68306	−0.0966895
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.0299	−1.42853	−0.714267	−	0.699874i	$-0.753240\pi$
$307$	−25.0299	−1.42853	−0.714267	+	0.699874i	$0.753240\pi$
$308$	0	0
$309$	1.78535	0.101565
$310$	0	0
$311$	27.6360	1.56710	0.783548	−	0.621331i	$-0.213408\pi$
$311$	27.6360	1.56710	0.783548	+	0.621331i	$0.213408\pi$
$312$	0	0
$313$	−14.5441	−0.822083	−0.411042	−	0.911617i	$-0.634835\pi$
$313$	−14.5441	−0.822083	−0.411042	+	0.911617i	$0.634835\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.00364	−0.449529	−0.224765	−	0.974413i	$-0.572161\pi$
$317$	−8.00364	−0.449529	−0.224765	+	0.974413i	$0.572161\pi$
$318$	0	0
$319$	8.52483	0.477299
$320$	0	0
$321$	1.11943	0.0624806
$322$	0	0
$323$	8.31017	0.462390
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.914914	0.0505949
$328$	0	0
$329$	−27.6348	−1.52355
$330$	0	0
$331$	5.59929	0.307765	0.153882	−	0.988089i	$-0.450822\pi$
$331$	5.59929	0.307765	0.153882	+	0.988089i	$0.450822\pi$
$332$	0	0
$333$	0.257322	0.0141011
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−34.5802	−1.88370	−0.941852	−	0.336027i	$-0.890917\pi$
$337$	−34.5802	−1.88370	−0.941852	+	0.336027i	$0.890917\pi$
$338$	0	0
$339$	2.41386	0.131103
$340$	0	0
$341$	2.03440	0.110169
$342$	0	0
$343$	−0.816545	−0.0440893
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.0537	−1.66705	−0.833525	−	0.552482i	$-0.813681\pi$
$347$	−31.0537	−1.66705	−0.833525	+	0.552482i	$0.813681\pi$
$348$	0	0
$349$	2.94804	0.157805	0.0789026	−	0.996882i	$-0.474858\pi$
$349$	2.94804	0.157805	0.0789026	+	0.996882i	$0.474858\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−1.80547	−0.0960957	−0.0480478	−	0.998845i	$-0.515300\pi$
$353$	−1.80547	−0.0960957	−0.0480478	+	0.998845i	$0.515300\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.78003	0.305912
$358$	0	0
$359$	−9.29216	−0.490422	−0.245211	−	0.969470i	$-0.578857\pi$
$359$	−9.29216	−0.490422	−0.245211	+	0.969470i	$0.578857\pi$
$360$	0	0
$361$	9.48457	0.499188
$362$	0	0
$363$	−0.0118848	−0.000623788 0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.0955	0.944579	0.472289	−	0.881444i	$-0.343428\pi$
$367$	18.0955	0.944579	0.472289	+	0.881444i	$0.343428\pi$
$368$	0	0
$369$	−10.6114	−0.552407
$370$	0	0
$371$	20.0278	1.03979
$372$	0	0
$373$	29.4689	1.52584	0.762922	−	0.646491i	$-0.223764\pi$
$373$	29.4689	1.52584	0.762922	+	0.646491i	$0.223764\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.56894	−0.132307
$378$	0	0
$379$	−21.2489	−1.09148	−0.545740	−	0.837954i	$-0.683751\pi$
$379$	−21.2489	−1.09148	−0.545740	+	0.837954i	$0.683751\pi$
$380$	0	0
$381$	−14.0418	−0.719384
$382$	0	0
$383$	8.19163	0.418573	0.209286	−	0.977854i	$-0.432886\pi$
$383$	8.19163	0.418573	0.209286	+	0.977854i	$0.432886\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.6935	−0.645247
$388$	0	0
$389$	−24.4557	−1.23995	−0.619976	−	0.784621i	$-0.712858\pi$
$389$	−24.4557	−1.23995	−0.619976	+	0.784621i	$0.712858\pi$
$390$	0	0
$391$	0.689684	0.0348788
$392$	0	0
$393$	−13.7650	−0.694352
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−34.4916	−1.73108	−0.865541	−	0.500837i	$-0.833025\pi$
$397$	−34.4916	−1.73108	−0.865541	+	0.500837i	$0.833025\pi$
$398$	0	0
$399$	19.8121	0.991844
$400$	0	0
$401$	−16.9434	−0.846114	−0.423057	−	0.906103i	$-0.639043\pi$
$401$	−16.9434	−0.846114	−0.423057	+	0.906103i	$0.639043\pi$
$402$	0	0
$403$	−0.613062	−0.0305388
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.853901	0.0423263
$408$	0	0
$409$	−3.05299	−0.150961	−0.0754804	−	0.997147i	$-0.524049\pi$
$409$	−3.05299	−0.150961	−0.0754804	+	0.997147i	$0.524049\pi$
$410$	0	0
$411$	−6.50470	−0.320853
$412$	0	0
$413$	−48.7620	−2.39942
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.78003	−0.478930
$418$	0	0
$419$	20.9567	1.02380	0.511902	−	0.859044i	$-0.328941\pi$
$419$	20.9567	1.02380	0.511902	+	0.859044i	$0.328941\pi$
$420$	0	0
$421$	−37.4462	−1.82502	−0.912508	−	0.409059i	$-0.865857\pi$
$421$	−37.4462	−1.82502	−0.912508	+	0.409059i	$0.865857\pi$
$422$	0	0
$423$	−7.44442	−0.361960
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19.5798	−0.947533
$428$	0	0
$429$	3.31842	0.160215
$430$	0	0
$431$	−27.2671	−1.31341	−0.656706	−	0.754147i	$-0.728051\pi$
$431$	−27.2671	−1.31341	−0.656706	+	0.754147i	$0.728051\pi$
$432$	0	0
$433$	−12.5679	−0.603975	−0.301988	−	0.953312i	$-0.597650\pi$
$433$	−12.5679	−0.603975	−0.301988	+	0.953312i	$0.597650\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.36401	0.113086
$438$	0	0
$439$	14.7241	0.702742	0.351371	−	0.936236i	$-0.385716\pi$
$439$	14.7241	0.702742	0.351371	+	0.936236i	$0.385716\pi$
$440$	0	0
$441$	6.78003	0.322859
$442$	0	0
$443$	−8.65996	−0.411447	−0.205724	−	0.978610i	$-0.565955\pi$
$443$	−8.65996	−0.411447	−0.205724	+	0.978610i	$0.565955\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.93148	0.280549
$448$	0	0
$449$	−21.5690	−1.01791	−0.508953	−	0.860795i	$-0.669967\pi$
$449$	−21.5690	−1.01791	−0.508953	+	0.860795i	$0.669967\pi$
$450$	0	0
$451$	−35.2130	−1.65812
$452$	0	0
$453$	0.272818	0.0128181
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.192394	0.00899982	0.00449991	−	0.999990i	$-0.498568\pi$
$457$	0.192394	0.00899982	0.00449991	+	0.999990i	$0.498568\pi$
$458$	0	0
$459$	1.55706	0.0726773
$460$	0	0
$461$	−12.0821	−0.562720	−0.281360	−	0.959602i	$-0.590785\pi$
$461$	−12.0821	−0.562720	−0.281360	+	0.959602i	$0.590785\pi$
$462$	0	0
$463$	−25.7245	−1.19552	−0.597759	−	0.801676i	$-0.703942\pi$
$463$	−25.7245	−1.19552	−0.597759	+	0.801676i	$0.703942\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.4317	−0.852918	−0.426459	−	0.904507i	$-0.640239\pi$
$467$	−18.4317	−0.852918	−0.426459	+	0.904507i	$0.640239\pi$
$468$	0	0
$469$	−39.1201	−1.80640
$470$	0	0
$471$	14.6561	0.675319
$472$	0	0
$473$	−42.1223	−1.93679
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.39521	0.247030
$478$	0	0
$479$	25.5826	1.16890	0.584450	−	0.811430i	$-0.301310\pi$
$479$	25.5826	1.16890	0.584450	+	0.811430i	$0.301310\pi$
$480$	0	0
$481$	−0.257322	−0.0117329
$482$	0	0
$483$	1.64426	0.0748164
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.1997	1.36848	0.684239	−	0.729258i	$-0.260134\pi$
$487$	30.1997	1.36848	0.684239	+	0.729258i	$0.260134\pi$
$488$	0	0
$489$	−3.48345	−0.157527
$490$	0	0
$491$	−10.7256	−0.484039	−0.242020	−	0.970271i	$-0.577810\pi$
$491$	−10.7256	−0.484039	−0.242020	+	0.970271i	$0.577810\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.15761	0.0519260
$498$	0	0
$499$	7.01044	0.313830	0.156915	−	0.987612i	$-0.449845\pi$
$499$	7.01044	0.313830	0.156915	+	0.987612i	$0.449845\pi$
$500$	0	0
$501$	4.69135	0.209594
$502$	0	0
$503$	32.2604	1.43842	0.719210	−	0.694793i	$-0.244504\pi$
$503$	32.2604	1.43842	0.719210	+	0.694793i	$0.244504\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	17.3631	0.769604	0.384802	−	0.922999i	$-0.374270\pi$
$509$	17.3631	0.769604	0.384802	+	0.922999i	$0.374270\pi$
$510$	0	0
$511$	−35.1481	−1.55486
$512$	0	0
$513$	5.33709	0.235638
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−24.7037	−1.08647
$518$	0	0
$519$	3.26475	0.143307
$520$	0	0
$521$	27.6513	1.21142	0.605712	−	0.795684i	$-0.292888\pi$
$521$	27.6513	1.21142	0.605712	+	0.795684i	$0.292888\pi$
$522$	0	0
$523$	7.49745	0.327840	0.163920	−	0.986474i	$-0.447586\pi$
$523$	7.49745	0.327840	0.163920	+	0.986474i	$0.447586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.954575	−0.0415819
$528$	0	0
$529$	−22.8038	−0.991470
$530$	0	0
$531$	−13.1358	−0.570045
$532$	0	0
$533$	10.6114	0.459630
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−19.7204	−0.850999
$538$	0	0
$539$	22.4990	0.969099
$540$	0	0
$541$	18.8836	0.811871	0.405936	−	0.913902i	$-0.366946\pi$
$541$	18.8836	0.811871	0.405936	+	0.913902i	$0.366946\pi$
$542$	0	0
$543$	22.0558	0.946506
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.3808	0.914178	0.457089	−	0.889421i	$-0.348892\pi$
$547$	21.3808	0.914178	0.457089	+	0.889421i	$0.348892\pi$
$548$	0	0
$549$	−5.27452	−0.225111
$550$	0	0
$551$	−13.7107	−0.584095
$552$	0	0
$553$	60.0890	2.55524
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.80681	−0.161300	−0.0806498	−	0.996743i	$-0.525700\pi$
$557$	−3.80681	−0.161300	−0.0806498	+	0.996743i	$0.525700\pi$
$558$	0	0
$559$	12.6935	0.536878
$560$	0	0
$561$	5.16697	0.218150
$562$	0	0
$563$	−29.3983	−1.23899	−0.619495	−	0.785001i	$-0.712662\pi$
$563$	−29.3983	−1.23899	−0.619495	+	0.785001i	$0.712662\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.71215	0.155896
$568$	0	0
$569$	12.4014	0.519892	0.259946	−	0.965623i	$-0.416295\pi$
$569$	12.4014	0.519892	0.259946	+	0.965623i	$0.416295\pi$
$570$	0	0
$571$	−4.56899	−0.191206	−0.0956032	−	0.995420i	$-0.530478\pi$
$571$	−4.56899	−0.191206	−0.0956032	+	0.995420i	$0.530478\pi$
$572$	0	0
$573$	−22.2729	−0.930463
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27.4496	−1.14274	−0.571370	−	0.820692i	$-0.693588\pi$
$577$	−27.4496	−1.14274	−0.571370	+	0.820692i	$0.693588\pi$
$578$	0	0
$579$	−15.7733	−0.655515
$580$	0	0
$581$	−43.6348	−1.81028
$582$	0	0
$583$	17.9036	0.741489
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.5191	0.805641	0.402820	−	0.915279i	$-0.368030\pi$
$587$	19.5191	0.805641	0.402820	+	0.915279i	$0.368030\pi$
$588$	0	0
$589$	−3.27197	−0.134819
$590$	0	0
$591$	−2.72871	−0.112244
$592$	0	0
$593$	14.8539	0.609975	0.304987	−	0.952356i	$-0.401348\pi$
$593$	14.8539	0.609975	0.304987	+	0.952356i	$0.401348\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.1029	0.904613
$598$	0	0
$599$	−42.6098	−1.74099	−0.870494	−	0.492178i	$-0.836201\pi$
$599$	−42.6098	−1.74099	−0.870494	+	0.492178i	$0.836201\pi$
$600$	0	0
$601$	34.4960	1.40712	0.703561	−	0.710635i	$-0.251592\pi$
$601$	34.4960	1.40712	0.703561	+	0.710635i	$0.251592\pi$
$602$	0	0
$603$	−10.5384	−0.429157
$604$	0	0
$605$	0	0
$606$	0	0
$607$	44.0061	1.78615	0.893076	−	0.449906i	$-0.148543\pi$
$607$	44.0061	1.78615	0.893076	+	0.449906i	$0.148543\pi$
$608$	0	0
$609$	−9.53630	−0.386430
$610$	0	0
$611$	7.44442	0.301169
$612$	0	0
$613$	−11.4907	−0.464104	−0.232052	−	0.972703i	$-0.574544\pi$
$613$	−11.4907	−0.464104	−0.232052	+	0.972703i	$0.574544\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	47.2506	1.90224	0.951120	−	0.308823i	$-0.0999350\pi$
$617$	47.2506	1.90224	0.951120	+	0.308823i	$0.0999350\pi$
$618$	0	0
$619$	25.5964	1.02881	0.514403	−	0.857549i	$-0.328014\pi$
$619$	25.5964	1.02881	0.514403	+	0.857549i	$0.328014\pi$
$620$	0	0
$621$	0.442940	0.0177746
$622$	0	0
$623$	−17.0299	−0.682290
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17.7107	0.707297
$628$	0	0
$629$	−0.400665	−0.0159756
$630$	0	0
$631$	6.54777	0.260663	0.130331	−	0.991470i	$-0.458396\pi$
$631$	6.54777	0.260663	0.130331	+	0.991470i	$0.458396\pi$
$632$	0	0
$633$	−1.01720	−0.0404300
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.78003	−0.268635
$638$	0	0
$639$	0.311845	0.0123364
$640$	0	0
$641$	−0.899023	−0.0355093	−0.0177546	−	0.999842i	$-0.505652\pi$
$641$	−0.899023	−0.0355093	−0.0177546	+	0.999842i	$0.505652\pi$
$642$	0	0
$643$	2.48862	0.0981416	0.0490708	−	0.998795i	$-0.484374\pi$
$643$	2.48862	0.0981416	0.0490708	+	0.998795i	$0.484374\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.773002	0.0303898	0.0151949	−	0.999885i	$-0.495163\pi$
$647$	0.773002	0.0303898	0.0151949	+	0.999885i	$0.495163\pi$
$648$	0	0
$649$	−43.5901	−1.71106
$650$	0	0
$651$	−2.27578	−0.0891948
$652$	0	0
$653$	43.1740	1.68953	0.844764	−	0.535139i	$-0.179741\pi$
$653$	43.1740	1.68953	0.844764	+	0.535139i	$0.179741\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.46841	−0.369398
$658$	0	0
$659$	−39.3552	−1.53306	−0.766530	−	0.642208i	$-0.778019\pi$
$659$	−39.3552	−1.53306	−0.766530	+	0.642208i	$0.778019\pi$
$660$	0	0
$661$	−20.7673	−0.807756	−0.403878	−	0.914813i	$-0.632338\pi$
$661$	−20.7673	−0.807756	−0.403878	+	0.914813i	$0.632338\pi$
$662$	0	0
$663$	−1.55706	−0.0604712
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.13789	−0.0440592
$668$	0	0
$669$	−6.85535	−0.265043
$670$	0	0
$671$	−17.5031	−0.675698
$672$	0	0
$673$	49.6479	1.91379	0.956893	−	0.290439i	$-0.0938014\pi$
$673$	49.6479	1.91379	0.956893	+	0.290439i	$0.0938014\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.5884	1.17561	0.587803	−	0.809004i	$-0.299993\pi$
$677$	30.5884	1.17561	0.587803	+	0.809004i	$0.299993\pi$
$678$	0	0
$679$	−40.2769	−1.54569
$680$	0	0
$681$	2.88224	0.110448
$682$	0	0
$683$	−28.9405	−1.10738	−0.553688	−	0.832724i	$-0.686780\pi$
$683$	−28.9405	−1.10738	−0.553688	+	0.832724i	$0.686780\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−14.1857	−0.541218
$688$	0	0
$689$	−5.39521	−0.205541
$690$	0	0
$691$	−31.9739	−1.21635	−0.608173	−	0.793805i	$-0.708097\pi$
$691$	−31.9739	−1.21635	−0.608173	+	0.793805i	$0.708097\pi$
$692$	0	0
$693$	12.3184	0.467939
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.5226	0.625837
$698$	0	0
$699$	0.429354	0.0162396
$700$	0	0
$701$	4.51673	0.170595	0.0852973	−	0.996356i	$-0.472816\pi$
$701$	4.51673	0.170595	0.0852973	+	0.996356i	$0.472816\pi$
$702$	0	0
$703$	−1.37335	−0.0517969
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.24778	0.234972
$708$	0	0
$709$	−29.7979	−1.11908	−0.559542	−	0.828802i	$-0.689023\pi$
$709$	−29.7979	−1.11908	−0.559542	+	0.828802i	$0.689023\pi$
$710$	0	0
$711$	16.1871	0.607065
$712$	0	0
$713$	−0.271550	−0.0101696
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.7255	−0.736663
$718$	0	0
$719$	41.8431	1.56049	0.780243	−	0.625477i	$-0.215096\pi$
$719$	41.8431	1.56049	0.780243	+	0.625477i	$0.215096\pi$
$720$	0	0
$721$	−6.62747	−0.246820
$722$	0	0
$723$	−19.6598	−0.731157
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.4317	−0.572330	−0.286165	−	0.958180i	$-0.592381\pi$
$727$	−15.4317	−0.572330	−0.286165	+	0.958180i	$0.592381\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.7645	0.731018
$732$	0	0
$733$	−39.1794	−1.44712	−0.723561	−	0.690260i	$-0.757496\pi$
$733$	−39.1794	−1.44712	−0.723561	+	0.690260i	$0.757496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−34.9708	−1.28817
$738$	0	0
$739$	32.5361	1.19686	0.598430	−	0.801175i	$-0.295791\pi$
$739$	32.5361	1.19686	0.598430	+	0.801175i	$0.295791\pi$
$740$	0	0
$741$	−5.33709	−0.196063
$742$	0	0
$743$	−8.13346	−0.298388	−0.149194	−	0.988808i	$-0.547668\pi$
$743$	−8.13346	−0.298388	−0.149194	+	0.988808i	$0.547668\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.7546	−0.430078
$748$	0	0
$749$	−4.15550	−0.151839
$750$	0	0
$751$	−48.9359	−1.78570	−0.892848	−	0.450358i	$-0.851296\pi$
$751$	−48.9359	−1.78570	−0.892848	+	0.450358i	$0.851296\pi$
$752$	0	0
$753$	−23.4175	−0.853382
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.54775	0.128945	0.0644726	−	0.997919i	$-0.479463\pi$
$757$	3.54775	0.128945	0.0644726	+	0.997919i	$0.479463\pi$
$758$	0	0
$759$	1.46986	0.0533525
$760$	0	0
$761$	40.8645	1.48134	0.740669	−	0.671870i	$-0.234509\pi$
$761$	40.8645	1.48134	0.740669	+	0.671870i	$0.234509\pi$
$762$	0	0
$763$	−3.39630	−0.122954
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.1358	0.474306
$768$	0	0
$769$	−42.6263	−1.53714	−0.768571	−	0.639764i	$-0.779032\pi$
$769$	−42.6263	−1.53714	−0.768571	+	0.639764i	$0.779032\pi$
$770$	0	0
$771$	30.7987	1.10919
$772$	0	0
$773$	−36.3873	−1.30876	−0.654379	−	0.756166i	$-0.727070\pi$
$773$	−36.3873	−1.30876	−0.654379	+	0.756166i	$0.727070\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.955216	−0.0342682
$778$	0	0
$779$	56.6340	2.02912
$780$	0	0
$781$	1.03483	0.0370291
$782$	0	0
$783$	−2.56894	−0.0918066
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12.7280	−0.453705	−0.226853	−	0.973929i	$-0.572844\pi$
$787$	−12.7280	−0.453705	−0.226853	+	0.973929i	$0.572844\pi$
$788$	0	0
$789$	−24.1471	−0.859658
$790$	0	0
$791$	−8.96059	−0.318602
$792$	0	0
$793$	5.27452	0.187304
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20.3707	−0.721566	−0.360783	−	0.932650i	$-0.617490\pi$
$797$	−20.3707	−0.721566	−0.360783	+	0.932650i	$0.617490\pi$
$798$	0	0
$799$	11.5914	0.410074
$800$	0	0
$801$	−4.58762	−0.162096
$802$	0	0
$803$	−31.4201	−1.10879
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.7557	−0.449021
$808$	0	0
$809$	−36.4656	−1.28206	−0.641030	−	0.767516i	$-0.721493\pi$
$809$	−36.4656	−1.28206	−0.641030	+	0.767516i	$0.721493\pi$
$810$	0	0
$811$	32.4823	1.14061	0.570304	−	0.821433i	$-0.306825\pi$
$811$	32.4823	1.14061	0.570304	+	0.821433i	$0.306825\pi$
$812$	0	0
$813$	10.4222	0.365522
$814$	0	0
$815$	0	0
$816$	0	0
$817$	67.7464	2.37015
$818$	0	0
$819$	−3.71215	−0.129713
$820$	0	0
$821$	15.7750	0.550550	0.275275	−	0.961366i	$-0.411231\pi$
$821$	15.7750	0.550550	0.275275	+	0.961366i	$0.411231\pi$
$822$	0	0
$823$	−33.4570	−1.16624	−0.583120	−	0.812386i	$-0.698168\pi$
$823$	−33.4570	−1.16624	−0.583120	+	0.812386i	$0.698168\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.0516	−1.04500	−0.522499	−	0.852640i	$-0.675000\pi$
$827$	−30.0516	−1.04500	−0.522499	+	0.852640i	$0.675000\pi$
$828$	0	0
$829$	3.52850	0.122550	0.0612750	−	0.998121i	$-0.480483\pi$
$829$	3.52850	0.122550	0.0612750	+	0.998121i	$0.480483\pi$
$830$	0	0
$831$	19.9404	0.691726
$832$	0	0
$833$	−10.5569	−0.365776
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.613062	−0.0211905
$838$	0	0
$839$	7.19133	0.248272	0.124136	−	0.992265i	$-0.460384\pi$
$839$	7.19133	0.248272	0.124136	+	0.992265i	$0.460384\pi$
$840$	0	0
$841$	−22.4005	−0.772432
$842$	0	0
$843$	−28.5272	−0.982529
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.0441180	0.00151591
$848$	0	0
$849$	19.1888	0.658559
$850$	0	0
$851$	−0.113978	−0.00390712
$852$	0	0
$853$	−16.1780	−0.553924	−0.276962	−	0.960881i	$-0.589328\pi$
$853$	−16.1780	−0.553924	−0.276962	+	0.960881i	$0.589328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.79125	−0.0611881	−0.0305940	−	0.999532i	$-0.509740\pi$
$857$	−1.79125	−0.0611881	−0.0305940	+	0.999532i	$0.509740\pi$
$858$	0	0
$859$	17.7341	0.605078	0.302539	−	0.953137i	$-0.402166\pi$
$859$	17.7341	0.605078	0.302539	+	0.953137i	$0.402166\pi$
$860$	0	0
$861$	39.3910	1.34244
$862$	0	0
$863$	2.45249	0.0834837	0.0417419	−	0.999128i	$-0.486709\pi$
$863$	2.45249	0.0834837	0.0417419	+	0.999128i	$0.486709\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.5756	0.495012
$868$	0	0
$869$	53.7156	1.82218
$870$	0	0
$871$	10.5384	0.357081
$872$	0	0
$873$	−10.8500	−0.367218
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.8367	1.64910	0.824549	−	0.565791i	$-0.191429\pi$
$877$	48.8367	1.64910	0.824549	+	0.565791i	$0.191429\pi$
$878$	0	0
$879$	−11.9828	−0.404171
$880$	0	0
$881$	6.25328	0.210678	0.105339	−	0.994436i	$-0.466407\pi$
$881$	6.25328	0.210678	0.105339	+	0.994436i	$0.466407\pi$
$882$	0	0
$883$	25.0964	0.844560	0.422280	−	0.906465i	$-0.361230\pi$
$883$	25.0964	0.844560	0.422280	+	0.906465i	$0.361230\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.57354	0.0528344	0.0264172	−	0.999651i	$-0.491590\pi$
$887$	1.57354	0.0528344	0.0264172	+	0.999651i	$0.491590\pi$
$888$	0	0
$889$	52.1253	1.74823
$890$	0	0
$891$	3.31842	0.111171
$892$	0	0
$893$	39.7316	1.32957
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.442940	−0.0147893
$898$	0	0
$899$	1.57492	0.0525266
$900$	0	0
$901$	−8.40067	−0.279867
$902$	0	0
$903$	47.1201	1.56806
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.6504	0.685686	0.342843	−	0.939393i	$-0.388610\pi$
$907$	20.6504	0.685686	0.342843	+	0.939393i	$0.388610\pi$
$908$	0	0
$909$	1.68306	0.0558237
$910$	0	0
$911$	39.4108	1.30574	0.652869	−	0.757471i	$-0.273565\pi$
$911$	39.4108	1.30574	0.652869	+	0.757471i	$0.273565\pi$
$912$	0	0
$913$	−39.0066	−1.29093
$914$	0	0
$915$	0	0
$916$	0	0
$917$	51.0977	1.68739
$918$	0	0
$919$	−7.84896	−0.258913	−0.129457	−	0.991585i	$-0.541323\pi$
$919$	−7.84896	−0.258913	−0.129457	+	0.991585i	$0.541323\pi$
$920$	0	0
$921$	25.0299	0.824764
$922$	0	0
$923$	−0.311845	−0.0102645
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.78535	−0.0586385
$928$	0	0
$929$	−29.2279	−0.958935	−0.479467	−	0.877560i	$-0.659170\pi$
$929$	−29.2279	−0.958935	−0.479467	+	0.877560i	$0.659170\pi$
$930$	0	0
$931$	−36.1857	−1.18594
$932$	0	0
$933$	−27.6360	−0.904764
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.62318	0.281707	0.140853	−	0.990030i	$-0.455015\pi$
$937$	8.62318	0.281707	0.140853	+	0.990030i	$0.455015\pi$
$938$	0	0
$939$	14.5441	0.474630
$940$	0	0
$941$	−31.5064	−1.02708	−0.513540	−	0.858066i	$-0.671666\pi$
$941$	−31.5064	−1.02708	−0.513540	+	0.858066i	$0.671666\pi$
$942$	0	0
$943$	4.70021	0.153060
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.8565	1.03520	0.517598	−	0.855624i	$-0.326826\pi$
$947$	31.8565	1.03520	0.517598	+	0.855624i	$0.326826\pi$
$948$	0	0
$949$	9.46841	0.307358
$950$	0	0
$951$	8.00364	0.259536
$952$	0	0
$953$	−32.9471	−1.06726	−0.533631	−	0.845717i	$-0.679173\pi$
$953$	−32.9471	−1.06726	−0.533631	+	0.845717i	$0.679173\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.52483	−0.275569
$958$	0	0
$959$	24.1464	0.779728
$960$	0	0
$961$	−30.6242	−0.987876
$962$	0	0
$963$	−1.11943	−0.0360732
$964$	0	0
$965$	0	0
$966$	0	0
$967$	17.3801	0.558907	0.279453	−	0.960159i	$-0.409847\pi$
$967$	17.3801	0.558907	0.279453	+	0.960159i	$0.409847\pi$
$968$	0	0
$969$	−8.31017	−0.266961
$970$	0	0
$971$	−29.4311	−0.944490	−0.472245	−	0.881467i	$-0.656556\pi$
$971$	−29.4311	−0.944490	−0.472245	+	0.881467i	$0.656556\pi$
$972$	0	0
$973$	36.3049	1.16388
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.49492	−0.111812	−0.0559062	−	0.998436i	$-0.517805\pi$
$977$	−3.49492	−0.111812	−0.0559062	+	0.998436i	$0.517805\pi$
$978$	0	0
$979$	−15.2236	−0.486550
$980$	0	0
$981$	−0.914914	−0.0292110
$982$	0	0
$983$	12.1407	0.387230	0.193615	−	0.981078i	$-0.437979\pi$
$983$	12.1407	0.387230	0.193615	+	0.981078i	$0.437979\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	27.6348	0.879625
$988$	0	0
$989$	5.62246	0.178784
$990$	0	0
$991$	46.5633	1.47913	0.739566	−	0.673084i	$-0.235031\pi$
$991$	46.5633	1.47913	0.739566	+	0.673084i	$0.235031\pi$
$992$	0	0
$993$	−5.59929	−0.177688
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.8908	0.344915	0.172457	−	0.985017i	$-0.444829\pi$
$997$	10.8908	0.344915	0.172457	+	0.985017i	$0.444829\pi$
$998$	0	0
$999$	−0.257322	−0.00814130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bz.1.5		5
5.2	odd	4		1560.2.l.f.1249.10	yes	10
5.3	odd	4		1560.2.l.f.1249.5	✓	10
5.4	even	2		7800.2.a.ca.1.1		5
15.2	even	4		4680.2.l.h.2809.1		10
15.8	even	4		4680.2.l.h.2809.2		10
20.3	even	4		3120.2.l.q.1249.10		10
20.7	even	4		3120.2.l.q.1249.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.f.1249.5	✓	10	5.3	odd	4
1560.2.l.f.1249.10	yes	10	5.2	odd	4
3120.2.l.q.1249.5		10	20.7	even	4
3120.2.l.q.1249.10		10	20.3	even	4
4680.2.l.h.2809.1		10	15.2	even	4
4680.2.l.h.2809.2		10	15.8	even	4
7800.2.a.bz.1.5		5	1.1	even	1	trivial
7800.2.a.ca.1.1		5	5.4	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.71215 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.71215 q^{7} +1.00000 q^{9} +3.31842 q^{11} -1.00000 q^{13} -1.55706 q^{17} -5.33709 q^{19} -3.71215 q^{21} -0.442940 q^{23} -1.00000 q^{27} +2.56894 q^{29} +0.613062 q^{31} -3.31842 q^{33} +0.257322 q^{37} +1.00000 q^{39} -10.6114 q^{41} -12.6935 q^{43} -7.44442 q^{47} +6.78003 q^{49} +1.55706 q^{51} +5.39521 q^{53} +5.33709 q^{57} -13.1358 q^{59} -5.27452 q^{61} +3.71215 q^{63} -10.5384 q^{67} +0.442940 q^{69} +0.311845 q^{71} -9.46841 q^{73} +12.3184 q^{77} +16.1871 q^{79} +1.00000 q^{81} -11.7546 q^{83} -2.56894 q^{87} -4.58762 q^{89} -3.71215 q^{91} -0.613062 q^{93} -10.8500 q^{97} +3.31842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - q^7 + 5 * q^9	\( 5 q - 5 q^{3} - q^{7} + 5 q^{9} + 5 q^{11} - 5 q^{13} - 7 q^{17} + 6 q^{19} + q^{21} - 3 q^{23} - 5 q^{27} + 2 q^{29} - 5 q^{33} - 9 q^{37} + 5 q^{39} - q^{41} - 4 q^{43} - 14 q^{47} + 2 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{57} + 20 q^{59} - 19 q^{61} - q^{63} - 12 q^{67} + 3 q^{69} + 13 q^{71} - 16 q^{73} - 11 q^{77} + 7 q^{79} + 5 q^{81} + 2 q^{83} - 2 q^{87} + 9 q^{89} + q^{91} - 13 q^{97} + 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - q^7 + 5 * q^9 + 5 * q^11 - 5 * q^13 - 7 * q^17 + 6 * q^19 + q^21 - 3 * q^23 - 5 * q^27 + 2 * q^29 - 5 * q^33 - 9 * q^37 + 5 * q^39 - q^41 - 4 * q^43 - 14 * q^47 + 2 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^57 + 20 * q^59 - 19 * q^61 - q^63 - 12 * q^67 + 3 * q^69 + 13 * q^71 - 16 * q^73 - 11 * q^77 + 7 * q^79 + 5 * q^81 + 2 * q^83 - 2 * q^87 + 9 * q^89 + q^91 - 13 * q^97 + 5 * q^99

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