LMFDB - Embedded newform 2752.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2752,2,Mod(1,2752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2752, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2752 = 2^{6} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2752.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:	$21.9748306363$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.386404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 3x^{2} + 20x + 13 $ x^5 - 9x^3 - 3x^2 + 20*x + 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1376)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.45376$ of defining polynomial
Character	$\chi$	$=$	2752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.45376 q^{3} -0.937992 q^{5} -3.67241 q^{7} -0.886571 q^{9} +O(q^{10})$	$q-1.45376 q^{3} -0.937992 q^{5} -3.67241 q^{7} -0.886571 q^{9} +4.81268 q^{11} -2.23960 q^{13} +1.36362 q^{15} -2.60003 q^{17} -0.317862 q^{19} +5.33882 q^{21} +0.208278 q^{23} -4.12017 q^{25} +5.65016 q^{27} -8.63006 q^{29} -4.60003 q^{31} -6.99650 q^{33} +3.44469 q^{35} +6.70962 q^{37} +3.25585 q^{39} -11.8816 q^{41} -1.00000 q^{43} +0.831597 q^{45} -0.212643 q^{47} +6.48661 q^{49} +3.77984 q^{51} -1.09921 q^{53} -4.51425 q^{55} +0.462097 q^{57} +4.70309 q^{59} -5.56957 q^{61} +3.25585 q^{63} +2.10073 q^{65} +12.9993 q^{67} -0.302788 q^{69} +1.43869 q^{71} +15.4844 q^{73} +5.98975 q^{75} -17.6741 q^{77} -3.55212 q^{79} -5.55428 q^{81} +8.25737 q^{83} +2.43881 q^{85} +12.5461 q^{87} +12.1728 q^{89} +8.22475 q^{91} +6.68736 q^{93} +0.298152 q^{95} +5.20223 q^{97} -4.26678 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - q^5 - 4 * q^7 + 8 * q^9	$ 5 q + 5 q^{3} - q^{5} - 4 q^{7} + 8 q^{9} + 7 q^{11} + 3 q^{13} - 2 q^{17} + 19 q^{19} - 6 q^{21} + 6 q^{23} + 10 q^{25} + 20 q^{27} - 7 q^{29} - 12 q^{31} - 6 q^{33} + 18 q^{35} - 3 q^{37} - 8 q^{39} - 10 q^{41} - 5 q^{43} + 16 q^{45} + 5 q^{47} + 9 q^{49} + 13 q^{51} + 13 q^{53} - 6 q^{55} + 24 q^{57} + 32 q^{59} + 4 q^{61} - 8 q^{63} + 17 q^{67} + 10 q^{69} + 4 q^{71} - 2 q^{73} + 44 q^{75} + 6 q^{77} - 27 q^{79} - 3 q^{81} + 25 q^{83} + 16 q^{85} + 11 q^{87} + 8 q^{89} + 38 q^{91} + 3 q^{93} + 19 q^{95} - 30 q^{97} - q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - q^5 - 4 * q^7 + 8 * q^9 + 7 * q^11 + 3 * q^13 - 2 * q^17 + 19 * q^19 - 6 * q^21 + 6 * q^23 + 10 * q^25 + 20 * q^27 - 7 * q^29 - 12 * q^31 - 6 * q^33 + 18 * q^35 - 3 * q^37 - 8 * q^39 - 10 * q^41 - 5 * q^43 + 16 * q^45 + 5 * q^47 + 9 * q^49 + 13 * q^51 + 13 * q^53 - 6 * q^55 + 24 * q^57 + 32 * q^59 + 4 * q^61 - 8 * q^63 + 17 * q^67 + 10 * q^69 + 4 * q^71 - 2 * q^73 + 44 * q^75 + 6 * q^77 - 27 * q^79 - 3 * q^81 + 25 * q^83 + 16 * q^85 + 11 * q^87 + 8 * q^89 + 38 * q^91 + 3 * q^93 + 19 * q^95 - 30 * q^97 - 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
$3$	−1.45376	−0.839331	−0.419665	−	0.907679i	$-0.637853\pi$
$3$	−1.45376	−0.839331	−0.419665	+	0.907679i	$0.637853\pi$
$4$	0	0
$5$	−0.937992	−0.419483	−0.209741	−	0.977757i	$-0.567262\pi$
$5$	−0.937992	−0.419483	−0.209741	+	0.977757i	$0.567262\pi$
$6$	0	0
$7$	−3.67241	−1.38804	−0.694021	−	0.719955i	$-0.744162\pi$
$7$	−3.67241	−1.38804	−0.694021	+	0.719955i	$0.744162\pi$
$8$	0	0
$9$	−0.886571	−0.295524
$10$	0	0
$11$	4.81268	1.45108	0.725538	−	0.688182i	$-0.241591\pi$
$11$	4.81268	1.45108	0.725538	+	0.688182i	$0.241591\pi$
$12$	0	0
$13$	−2.23960	−0.621154	−0.310577	−	0.950548i	$-0.600522\pi$
$13$	−2.23960	−0.621154	−0.310577	+	0.950548i	$0.600522\pi$
$14$	0	0
$15$	1.36362	0.352085
$16$	0	0
$17$	−2.60003	−0.630601	−0.315300	−	0.948992i	$-0.602105\pi$
$17$	−2.60003	−0.630601	−0.315300	+	0.948992i	$0.602105\pi$
$18$	0	0
$19$	−0.317862	−0.0729226	−0.0364613	−	0.999335i	$-0.511609\pi$
$19$	−0.317862	−0.0729226	−0.0364613	+	0.999335i	$0.511609\pi$
$20$	0	0
$21$	5.33882	1.16503
$22$	0	0
$23$	0.208278	0.0434291	0.0217145	−	0.999764i	$-0.493088\pi$
$23$	0.208278	0.0434291	0.0217145	+	0.999764i	$0.493088\pi$
$24$	0	0
$25$	−4.12017	−0.824034
$26$	0	0
$27$	5.65016	1.08737
$28$	0	0
$29$	−8.63006	−1.60256	−0.801281	−	0.598288i	$-0.795848\pi$
$29$	−8.63006	−1.60256	−0.801281	+	0.598288i	$0.795848\pi$
$30$	0	0
$31$	−4.60003	−0.826191	−0.413095	−	0.910688i	$-0.635552\pi$
$31$	−4.60003	−0.826191	−0.413095	+	0.910688i	$0.635552\pi$
$32$	0	0
$33$	−6.99650	−1.21793
$34$	0	0
$35$	3.44469	0.582260
$36$	0	0
$37$	6.70962	1.10305	0.551527	−	0.834157i	$-0.314045\pi$
$37$	6.70962	1.10305	0.551527	+	0.834157i	$0.314045\pi$
$38$	0	0
$39$	3.25585	0.521354
$40$	0	0
$41$	−11.8816	−1.85559	−0.927793	−	0.373095i	$-0.878297\pi$
$41$	−11.8816	−1.85559	−0.927793	+	0.373095i	$0.878297\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	0.831597	0.123967
$46$	0	0
$47$	−0.212643	−0.0310172	−0.0155086	−	0.999880i	$-0.504937\pi$
$47$	−0.212643	−0.0310172	−0.0155086	+	0.999880i	$0.504937\pi$
$48$	0	0
$49$	6.48661	0.926658
$50$	0	0
$51$	3.77984	0.529283
$52$	0	0
$53$	−1.09921	−0.150989	−0.0754944	−	0.997146i	$-0.524053\pi$
$53$	−1.09921	−0.150989	−0.0754944	+	0.997146i	$0.524053\pi$
$54$	0	0
$55$	−4.51425	−0.608702
$56$	0	0
$57$	0.462097	0.0612062
$58$	0	0
$59$	4.70309	0.612291	0.306145	−	0.951985i	$-0.400961\pi$
$59$	4.70309	0.612291	0.306145	+	0.951985i	$0.400961\pi$
$60$	0	0
$61$	−5.56957	−0.713110	−0.356555	−	0.934274i	$-0.616049\pi$
$61$	−5.56957	−0.713110	−0.356555	+	0.934274i	$0.616049\pi$
$62$	0	0
$63$	3.25585	0.410199
$64$	0	0
$65$	2.10073	0.260564
$66$	0	0
$67$	12.9993	1.58812	0.794061	−	0.607838i	$-0.207963\pi$
$67$	12.9993	1.58812	0.794061	+	0.607838i	$0.207963\pi$
$68$	0	0
$69$	−0.302788	−0.0364513
$70$	0	0
$71$	1.43869	0.170741	0.0853705	−	0.996349i	$-0.472793\pi$
$71$	1.43869	0.170741	0.0853705	+	0.996349i	$0.472793\pi$
$72$	0	0
$73$	15.4844	1.81232	0.906158	−	0.422940i	$-0.139002\pi$
$73$	15.4844	1.81232	0.906158	+	0.422940i	$0.139002\pi$
$74$	0	0
$75$	5.98975	0.691637
$76$	0	0
$77$	−17.6741	−2.01415
$78$	0	0
$79$	−3.55212	−0.399644	−0.199822	−	0.979832i	$-0.564036\pi$
$79$	−3.55212	−0.399644	−0.199822	+	0.979832i	$0.564036\pi$
$80$	0	0
$81$	−5.55428	−0.617142
$82$	0	0
$83$	8.25737	0.906364	0.453182	−	0.891418i	$-0.350289\pi$
$83$	8.25737	0.906364	0.453182	+	0.891418i	$0.350289\pi$
$84$	0	0
$85$	2.43881	0.264526
$86$	0	0
$87$	12.5461	1.34508
$88$	0	0
$89$	12.1728	1.29031	0.645157	−	0.764050i	$-0.276792\pi$
$89$	12.1728	1.29031	0.645157	+	0.764050i	$0.276792\pi$
$90$	0	0
$91$	8.22475	0.862188
$92$	0	0
$93$	6.68736	0.693447
$94$	0	0
$95$	0.298152	0.0305898
$96$	0	0
$97$	5.20223	0.528206	0.264103	−	0.964494i	$-0.414924\pi$
$97$	5.20223	0.528206	0.264103	+	0.964494i	$0.414924\pi$
$98$	0	0
$99$	−4.26678	−0.428828
$100$	0	0
$101$	11.3097	1.12535	0.562676	−	0.826677i	$-0.309772\pi$
$101$	11.3097	1.12535	0.562676	+	0.826677i	$0.309772\pi$
$102$	0	0
$103$	13.7338	1.35323	0.676614	−	0.736338i	$-0.263447\pi$
$103$	13.7338	1.35323	0.676614	+	0.736338i	$0.263447\pi$
$104$	0	0
$105$	−5.00777	−0.488708
$106$	0	0
$107$	−18.3184	−1.77091	−0.885454	−	0.464728i	$-0.846152\pi$
$107$	−18.3184	−1.77091	−0.885454	+	0.464728i	$0.846152\pi$
$108$	0	0
$109$	−6.16053	−0.590072	−0.295036	−	0.955486i	$-0.595332\pi$
$109$	−6.16053	−0.590072	−0.295036	+	0.955486i	$0.595332\pi$
$110$	0	0
$111$	−9.75420	−0.925828
$112$	0	0
$113$	7.93907	0.746845	0.373422	−	0.927661i	$-0.378184\pi$
$113$	7.93907	0.746845	0.373422	+	0.927661i	$0.378184\pi$
$114$	0	0
$115$	−0.195364	−0.0182177
$116$	0	0
$117$	1.98557	0.183566
$118$	0	0
$119$	9.54840	0.875300
$120$	0	0
$121$	12.1619	1.10562
$122$	0	0
$123$	17.2730	1.55745
$124$	0	0
$125$	8.55465	0.765151
$126$	0	0
$127$	−15.1571	−1.34497	−0.672486	−	0.740110i	$-0.734773\pi$
$127$	−15.1571	−1.34497	−0.672486	+	0.740110i	$0.734773\pi$
$128$	0	0
$129$	1.45376	0.127997
$130$	0	0
$131$	4.98099	0.435191	0.217595	−	0.976039i	$-0.430179\pi$
$131$	4.98099	0.435191	0.217595	+	0.976039i	$0.430179\pi$
$132$	0	0
$133$	1.16732	0.101220
$134$	0	0
$135$	−5.29980	−0.456134
$136$	0	0
$137$	−9.65550	−0.824925	−0.412463	−	0.910974i	$-0.635331\pi$
$137$	−9.65550	−0.824925	−0.412463	+	0.910974i	$0.635331\pi$
$138$	0	0
$139$	18.2993	1.55212	0.776062	−	0.630656i	$-0.217214\pi$
$139$	18.2993	1.55212	0.776062	+	0.630656i	$0.217214\pi$
$140$	0	0
$141$	0.309133	0.0260337
$142$	0	0
$143$	−10.7785	−0.901343
$144$	0	0
$145$	8.09493	0.672247
$146$	0	0
$147$	−9.42999	−0.777773
$148$	0	0
$149$	−11.1888	−0.916624	−0.458312	−	0.888791i	$-0.651546\pi$
$149$	−11.1888	−0.916624	−0.458312	+	0.888791i	$0.651546\pi$
$150$	0	0
$151$	7.83834	0.637875	0.318937	−	0.947776i	$-0.396674\pi$
$151$	7.83834	0.637875	0.318937	+	0.947776i	$0.396674\pi$
$152$	0	0
$153$	2.30512	0.186358
$154$	0	0
$155$	4.31480	0.346573
$156$	0	0
$157$	15.1835	1.21177	0.605887	−	0.795551i	$-0.292818\pi$
$157$	15.1835	1.21177	0.605887	+	0.795551i	$0.292818\pi$
$158$	0	0
$159$	1.59800	0.126729
$160$	0	0
$161$	−0.764884	−0.0602813
$162$	0	0
$163$	8.14747	0.638159	0.319080	−	0.947728i	$-0.396626\pi$
$163$	8.14747	0.638159	0.319080	+	0.947728i	$0.396626\pi$
$164$	0	0
$165$	6.56266	0.510902
$166$	0	0
$167$	7.72146	0.597504	0.298752	−	0.954331i	$-0.403430\pi$
$167$	7.72146	0.597504	0.298752	+	0.954331i	$0.403430\pi$
$168$	0	0
$169$	−7.98417	−0.614167
$170$	0	0
$171$	0.281808	0.0215504
$172$	0	0
$173$	8.17025	0.621172	0.310586	−	0.950545i	$-0.399475\pi$
$173$	8.17025	0.621172	0.310586	+	0.950545i	$0.399475\pi$
$174$	0	0
$175$	15.1310	1.14379
$176$	0	0
$177$	−6.83719	−0.513914
$178$	0	0
$179$	−5.84552	−0.436915	−0.218457	−	0.975847i	$-0.570102\pi$
$179$	−5.84552	−0.436915	−0.218457	+	0.975847i	$0.570102\pi$
$180$	0	0
$181$	−5.55930	−0.413219	−0.206610	−	0.978423i	$-0.566243\pi$
$181$	−5.55930	−0.413219	−0.206610	+	0.978423i	$0.566243\pi$
$182$	0	0
$183$	8.09684	0.598535
$184$	0	0
$185$	−6.29357	−0.462712
$186$	0	0
$187$	−12.5131	−0.915050
$188$	0	0
$189$	−20.7497	−1.50932
$190$	0	0
$191$	26.0792	1.88702	0.943511	−	0.331341i	$-0.107501\pi$
$191$	26.0792	1.88702	0.943511	+	0.331341i	$0.107501\pi$
$192$	0	0
$193$	2.51258	0.180860	0.0904298	−	0.995903i	$-0.471176\pi$
$193$	2.51258	0.180860	0.0904298	+	0.995903i	$0.471176\pi$
$194$	0	0
$195$	−3.05397	−0.218699
$196$	0	0
$197$	2.20487	0.157090	0.0785452	−	0.996911i	$-0.474973\pi$
$197$	2.20487	0.157090	0.0785452	+	0.996911i	$0.474973\pi$
$198$	0	0
$199$	−10.0419	−0.711852	−0.355926	−	0.934514i	$-0.615835\pi$
$199$	−10.0419	−0.711852	−0.355926	+	0.934514i	$0.615835\pi$
$200$	0	0
$201$	−18.8980	−1.33296
$202$	0	0
$203$	31.6931	2.22442
$204$	0	0
$205$	11.1448	0.778387
$206$	0	0
$207$	−0.184654	−0.0128343
$208$	0	0
$209$	−1.52977	−0.105816
$210$	0	0
$211$	−13.5842	−0.935176	−0.467588	−	0.883947i	$-0.654877\pi$
$211$	−13.5842	−0.935176	−0.467588	+	0.883947i	$0.654877\pi$
$212$	0	0
$213$	−2.09151	−0.143308
$214$	0	0
$215$	0.937992	0.0639705
$216$	0	0
$217$	16.8932	1.14679
$218$	0	0
$219$	−22.5107	−1.52113
$220$	0	0
$221$	5.82305	0.391701
$222$	0	0
$223$	19.4143	1.30008	0.650040	−	0.759900i	$-0.274752\pi$
$223$	19.4143	1.30008	0.650040	+	0.759900i	$0.274752\pi$
$224$	0	0
$225$	3.65282	0.243522
$226$	0	0
$227$	18.3403	1.21729	0.608645	−	0.793443i	$-0.291713\pi$
$227$	18.3403	1.21729	0.608645	+	0.793443i	$0.291713\pi$
$228$	0	0
$229$	−3.19605	−0.211201	−0.105601	−	0.994409i	$-0.533676\pi$
$229$	−3.19605	−0.211201	−0.105601	+	0.994409i	$0.533676\pi$
$230$	0	0
$231$	25.6940	1.69054
$232$	0	0
$233$	−23.0121	−1.50757	−0.753786	−	0.657120i	$-0.771775\pi$
$233$	−23.0121	−1.50757	−0.753786	+	0.657120i	$0.771775\pi$
$234$	0	0
$235$	0.199458	0.0130112
$236$	0	0
$237$	5.16394	0.335434
$238$	0	0
$239$	−11.9706	−0.774316	−0.387158	−	0.922013i	$-0.626543\pi$
$239$	−11.9706	−0.774316	−0.387158	+	0.922013i	$0.626543\pi$
$240$	0	0
$241$	−15.3837	−0.990951	−0.495476	−	0.868622i	$-0.665006\pi$
$241$	−15.3837	−0.990951	−0.495476	+	0.868622i	$0.665006\pi$
$242$	0	0
$243$	−8.87586	−0.569387
$244$	0	0
$245$	−6.08439	−0.388717
$246$	0	0
$247$	0.711885	0.0452962
$248$	0	0
$249$	−12.0043	−0.760739
$250$	0	0
$251$	−8.02172	−0.506327	−0.253163	−	0.967424i	$-0.581471\pi$
$251$	−8.02172	−0.506327	−0.253163	+	0.967424i	$0.581471\pi$
$252$	0	0
$253$	1.00238	0.0630189
$254$	0	0
$255$	−3.54546	−0.222025
$256$	0	0
$257$	16.5518	1.03247	0.516235	−	0.856447i	$-0.327333\pi$
$257$	16.5518	1.03247	0.516235	+	0.856447i	$0.327333\pi$
$258$	0	0
$259$	−24.6405	−1.53108
$260$	0	0
$261$	7.65116	0.473595
$262$	0	0
$263$	−10.1914	−0.628430	−0.314215	−	0.949352i	$-0.601741\pi$
$263$	−10.1914	−0.628430	−0.314215	+	0.949352i	$0.601741\pi$
$264$	0	0
$265$	1.03105	0.0633372
$266$	0	0
$267$	−17.6964	−1.08300
$268$	0	0
$269$	−17.9933	−1.09707	−0.548535	−	0.836127i	$-0.684814\pi$
$269$	−17.9933	−1.09707	−0.548535	+	0.836127i	$0.684814\pi$
$270$	0	0
$271$	−16.4918	−1.00180	−0.500902	−	0.865504i	$-0.666998\pi$
$271$	−16.4918	−1.00180	−0.500902	+	0.865504i	$0.666998\pi$
$272$	0	0
$273$	−11.9568	−0.723661
$274$	0	0
$275$	−19.8291	−1.19574
$276$	0	0
$277$	−4.14982	−0.249339	−0.124669	−	0.992198i	$-0.539787\pi$
$277$	−4.14982	−0.249339	−0.124669	+	0.992198i	$0.539787\pi$
$278$	0	0
$279$	4.07826	0.244159
$280$	0	0
$281$	−8.68349	−0.518013	−0.259007	−	0.965876i	$-0.583395\pi$
$281$	−8.68349	−0.518013	−0.259007	+	0.965876i	$0.583395\pi$
$282$	0	0
$283$	14.7604	0.877412	0.438706	−	0.898631i	$-0.355437\pi$
$283$	14.7604	0.877412	0.438706	+	0.898631i	$0.355437\pi$
$284$	0	0
$285$	−0.433443	−0.0256749
$286$	0	0
$287$	43.6339	2.57563
$288$	0	0
$289$	−10.2398	−0.602342
$290$	0	0
$291$	−7.56281	−0.443340
$292$	0	0
$293$	29.3246	1.71316	0.856579	−	0.516015i	$-0.172585\pi$
$293$	29.3246	1.71316	0.856579	+	0.516015i	$0.172585\pi$
$294$	0	0
$295$	−4.41147	−0.256845
$296$	0	0
$297$	27.1924	1.57786
$298$	0	0
$299$	−0.466461	−0.0269761
$300$	0	0
$301$	3.67241	0.211674
$302$	0	0
$303$	−16.4416	−0.944543
$304$	0	0
$305$	5.22421	0.299138
$306$	0	0
$307$	−17.9480	−1.02435	−0.512173	−	0.858883i	$-0.671159\pi$
$307$	−17.9480	−1.02435	−0.512173	+	0.858883i	$0.671159\pi$
$308$	0	0
$309$	−19.9656	−1.13581
$310$	0	0
$311$	28.8087	1.63359	0.816796	−	0.576927i	$-0.195748\pi$
$311$	28.8087	1.63359	0.816796	+	0.576927i	$0.195748\pi$
$312$	0	0
$313$	21.2488	1.20106	0.600528	−	0.799604i	$-0.294957\pi$
$313$	21.2488	1.20106	0.600528	+	0.799604i	$0.294957\pi$
$314$	0	0
$315$	−3.05397	−0.172072
$316$	0	0
$317$	33.9903	1.90908	0.954542	−	0.298078i	$-0.0963455\pi$
$317$	33.9903	1.90908	0.954542	+	0.298078i	$0.0963455\pi$
$318$	0	0
$319$	−41.5337	−2.32544
$320$	0	0
$321$	26.6306	1.48638
$322$	0	0
$323$	0.826453	0.0459851
$324$	0	0
$325$	9.22755	0.511852
$326$	0	0
$327$	8.95596	0.495266
$328$	0	0
$329$	0.780913	0.0430531
$330$	0	0
$331$	19.8007	1.08835	0.544174	−	0.838973i	$-0.316843\pi$
$331$	19.8007	1.08835	0.544174	+	0.838973i	$0.316843\pi$
$332$	0	0
$333$	−5.94855	−0.325979
$334$	0	0
$335$	−12.1933	−0.666190
$336$	0	0
$337$	4.07488	0.221973	0.110986	−	0.993822i	$-0.464599\pi$
$337$	4.07488	0.221973	0.110986	+	0.993822i	$0.464599\pi$
$338$	0	0
$339$	−11.5415	−0.626850
$340$	0	0
$341$	−22.1385	−1.19887
$342$	0	0
$343$	1.88540	0.101802
$344$	0	0
$345$	0.284012	0.0152907
$346$	0	0
$347$	−16.6382	−0.893185	−0.446593	−	0.894737i	$-0.647363\pi$
$347$	−16.6382	−0.893185	−0.446593	+	0.894737i	$0.647363\pi$
$348$	0	0
$349$	5.00730	0.268035	0.134017	−	0.990979i	$-0.457212\pi$
$349$	5.00730	0.268035	0.134017	+	0.990979i	$0.457212\pi$
$350$	0	0
$351$	−12.6541	−0.675426
$352$	0	0
$353$	11.3274	0.602898	0.301449	−	0.953482i	$-0.402530\pi$
$353$	11.3274	0.602898	0.301449	+	0.953482i	$0.402530\pi$
$354$	0	0
$355$	−1.34948	−0.0716229
$356$	0	0
$357$	−13.8811	−0.734666
$358$	0	0
$359$	12.5719	0.663517	0.331758	−	0.943364i	$-0.392358\pi$
$359$	12.5719	0.663517	0.331758	+	0.943364i	$0.392358\pi$
$360$	0	0
$361$	−18.8990	−0.994682
$362$	0	0
$363$	−17.6805	−0.927984
$364$	0	0
$365$	−14.5243	−0.760235
$366$	0	0
$367$	5.49610	0.286894	0.143447	−	0.989658i	$-0.454181\pi$
$367$	5.49610	0.286894	0.143447	+	0.989658i	$0.454181\pi$
$368$	0	0
$369$	10.5338	0.548370
$370$	0	0
$371$	4.03677	0.209579
$372$	0	0
$373$	19.6138	1.01556	0.507781	−	0.861486i	$-0.330466\pi$
$373$	19.6138	1.01556	0.507781	+	0.861486i	$0.330466\pi$
$374$	0	0
$375$	−12.4364	−0.642215
$376$	0	0
$377$	19.3279	0.995438
$378$	0	0
$379$	19.4414	0.998637	0.499319	−	0.866418i	$-0.333584\pi$
$379$	19.4414	0.998637	0.499319	+	0.866418i	$0.333584\pi$
$380$	0	0
$381$	22.0348	1.12888
$382$	0	0
$383$	28.7221	1.46763	0.733817	−	0.679348i	$-0.237737\pi$
$383$	28.7221	1.46763	0.733817	+	0.679348i	$0.237737\pi$
$384$	0	0
$385$	16.5782	0.844903
$386$	0	0
$387$	0.886571	0.0450670
$388$	0	0
$389$	−18.8941	−0.957971	−0.478986	−	0.877823i	$-0.658995\pi$
$389$	−18.8941	−0.957971	−0.478986	+	0.877823i	$0.658995\pi$
$390$	0	0
$391$	−0.541531	−0.0273864
$392$	0	0
$393$	−7.24118	−0.365269
$394$	0	0
$395$	3.33186	0.167644
$396$	0	0
$397$	−18.7260	−0.939830	−0.469915	−	0.882712i	$-0.655716\pi$
$397$	−18.7260	−0.939830	−0.469915	+	0.882712i	$0.655716\pi$
$398$	0	0
$399$	−1.69701	−0.0849567
$400$	0	0
$401$	11.5832	0.578439	0.289219	−	0.957263i	$-0.406604\pi$
$401$	11.5832	0.578439	0.289219	+	0.957263i	$0.406604\pi$
$402$	0	0
$403$	10.3023	0.513192
$404$	0	0
$405$	5.20987	0.258881
$406$	0	0
$407$	32.2912	1.60062
$408$	0	0
$409$	−12.0829	−0.597461	−0.298731	−	0.954338i	$-0.596563\pi$
$409$	−12.0829	−0.597461	−0.298731	+	0.954338i	$0.596563\pi$
$410$	0	0
$411$	14.0368	0.692385
$412$	0	0
$413$	−17.2717	−0.849885
$414$	0	0
$415$	−7.74535	−0.380204
$416$	0	0
$417$	−26.6028	−1.30275
$418$	0	0
$419$	−8.34411	−0.407636	−0.203818	−	0.979009i	$-0.565335\pi$
$419$	−8.34411	−0.407636	−0.203818	+	0.979009i	$0.565335\pi$
$420$	0	0
$421$	28.3196	1.38021	0.690106	−	0.723708i	$-0.257564\pi$
$421$	28.3196	1.38021	0.690106	+	0.723708i	$0.257564\pi$
$422$	0	0
$423$	0.188523	0.00916632
$424$	0	0
$425$	10.7126	0.519637
$426$	0	0
$427$	20.4538	0.989826
$428$	0	0
$429$	15.6694	0.756525
$430$	0	0
$431$	16.8363	0.810978	0.405489	−	0.914100i	$-0.367101\pi$
$431$	16.8363	0.810978	0.405489	+	0.914100i	$0.367101\pi$
$432$	0	0
$433$	−11.2159	−0.539002	−0.269501	−	0.963000i	$-0.586859\pi$
$433$	−11.2159	−0.539002	−0.269501	+	0.963000i	$0.586859\pi$
$434$	0	0
$435$	−11.7681	−0.564238
$436$	0	0
$437$	−0.0662038	−0.00316696
$438$	0	0
$439$	−26.8581	−1.28187	−0.640933	−	0.767597i	$-0.721452\pi$
$439$	−26.8581	−1.28187	−0.640933	+	0.767597i	$0.721452\pi$
$440$	0	0
$441$	−5.75084	−0.273849
$442$	0	0
$443$	−34.9914	−1.66249	−0.831245	−	0.555906i	$-0.812372\pi$
$443$	−34.9914	−1.66249	−0.831245	+	0.555906i	$0.812372\pi$
$444$	0	0
$445$	−11.4180	−0.541264
$446$	0	0
$447$	16.2659	0.769351
$448$	0	0
$449$	13.1543	0.620790	0.310395	−	0.950608i	$-0.399539\pi$
$449$	13.1543	0.620790	0.310395	+	0.950608i	$0.399539\pi$
$450$	0	0
$451$	−57.1821	−2.69260
$452$	0	0
$453$	−11.3951	−0.535388
$454$	0	0
$455$	−7.71475	−0.361673
$456$	0	0
$457$	−28.5941	−1.33758	−0.668788	−	0.743453i	$-0.733187\pi$
$457$	−28.5941	−1.33758	−0.668788	+	0.743453i	$0.733187\pi$
$458$	0	0
$459$	−14.6906	−0.685698
$460$	0	0
$461$	28.5107	1.32788	0.663938	−	0.747788i	$-0.268884\pi$
$461$	28.5107	1.32788	0.663938	+	0.747788i	$0.268884\pi$
$462$	0	0
$463$	−23.9711	−1.11403	−0.557015	−	0.830502i	$-0.688054\pi$
$463$	−23.9711	−1.11403	−0.557015	+	0.830502i	$0.688054\pi$
$464$	0	0
$465$	−6.27269	−0.290889
$466$	0	0
$467$	−1.17510	−0.0543770	−0.0271885	−	0.999630i	$-0.508655\pi$
$467$	−1.17510	−0.0543770	−0.0271885	+	0.999630i	$0.508655\pi$
$468$	0	0
$469$	−47.7389	−2.20438
$470$	0	0
$471$	−22.0732	−1.01708
$472$	0	0
$473$	−4.81268	−0.221287
$474$	0	0
$475$	1.30965	0.0600907
$476$	0	0
$477$	0.974532	0.0446208
$478$	0	0
$479$	−25.5908	−1.16927	−0.584637	−	0.811295i	$-0.698763\pi$
$479$	−25.5908	−1.16927	−0.584637	+	0.811295i	$0.698763\pi$
$480$	0	0
$481$	−15.0269	−0.685167
$482$	0	0
$483$	1.11196	0.0505960
$484$	0	0
$485$	−4.87965	−0.221574
$486$	0	0
$487$	−38.1353	−1.72808	−0.864038	−	0.503426i	$-0.832073\pi$
$487$	−38.1353	−1.72808	−0.864038	+	0.503426i	$0.832073\pi$
$488$	0	0
$489$	−11.8445	−0.535627
$490$	0	0
$491$	16.8788	0.761729	0.380864	−	0.924631i	$-0.375626\pi$
$491$	16.8788	0.761729	0.380864	+	0.924631i	$0.375626\pi$
$492$	0	0
$493$	22.4385	1.01058
$494$	0	0
$495$	4.00221	0.179886
$496$	0	0
$497$	−5.28346	−0.236995
$498$	0	0
$499$	22.0583	0.987464	0.493732	−	0.869614i	$-0.335632\pi$
$499$	22.0583	0.987464	0.493732	+	0.869614i	$0.335632\pi$
$500$	0	0
$501$	−11.2252	−0.501504
$502$	0	0
$503$	9.12832	0.407012	0.203506	−	0.979074i	$-0.434766\pi$
$503$	9.12832	0.407012	0.203506	+	0.979074i	$0.434766\pi$
$504$	0	0
$505$	−10.6084	−0.472066
$506$	0	0
$507$	11.6071	0.515490
$508$	0	0
$509$	−14.1100	−0.625416	−0.312708	−	0.949849i	$-0.601236\pi$
$509$	−14.1100	−0.625416	−0.312708	+	0.949849i	$0.601236\pi$
$510$	0	0
$511$	−56.8652	−2.51557
$512$	0	0
$513$	−1.79597	−0.0792941
$514$	0	0
$515$	−12.8822	−0.567656
$516$	0	0
$517$	−1.02338	−0.0450083
$518$	0	0
$519$	−11.8776	−0.521369
$520$	0	0
$521$	15.3980	0.674599	0.337300	−	0.941397i	$-0.390486\pi$
$521$	15.3980	0.674599	0.337300	+	0.941397i	$0.390486\pi$
$522$	0	0
$523$	−25.3214	−1.10723	−0.553614	−	0.832773i	$-0.686752\pi$
$523$	−25.3214	−1.10723	−0.553614	+	0.832773i	$0.686752\pi$
$524$	0	0
$525$	−21.9968	−0.960021
$526$	0	0
$527$	11.9602	0.520997
$528$	0	0
$529$	−22.9566	−0.998114
$530$	0	0
$531$	−4.16963	−0.180946
$532$	0	0
$533$	26.6100	1.15261
$534$	0	0
$535$	17.1825	0.742865
$536$	0	0
$537$	8.49800	0.366716
$538$	0	0
$539$	31.2179	1.34465
$540$	0	0
$541$	−30.6816	−1.31910	−0.659552	−	0.751659i	$-0.729254\pi$
$541$	−30.6816	−1.31910	−0.659552	+	0.751659i	$0.729254\pi$
$542$	0	0
$543$	8.08191	0.346828
$544$	0	0
$545$	5.77853	0.247525
$546$	0	0
$547$	6.87711	0.294044	0.147022	−	0.989133i	$-0.453031\pi$
$547$	6.87711	0.294044	0.147022	+	0.989133i	$0.453031\pi$
$548$	0	0
$549$	4.93782	0.210741
$550$	0	0
$551$	2.74317	0.116863
$552$	0	0
$553$	13.0448	0.554723
$554$	0	0
$555$	9.14936	0.388369
$556$	0	0
$557$	−11.5604	−0.489830	−0.244915	−	0.969545i	$-0.578760\pi$
$557$	−11.5604	−0.489830	−0.244915	+	0.969545i	$0.578760\pi$
$558$	0	0
$559$	2.23960	0.0947251
$560$	0	0
$561$	18.1911	0.768030
$562$	0	0
$563$	−29.0711	−1.22520	−0.612600	−	0.790393i	$-0.709876\pi$
$563$	−29.0711	−1.22520	−0.612600	+	0.790393i	$0.709876\pi$
$564$	0	0
$565$	−7.44679	−0.313289
$566$	0	0
$567$	20.3976	0.856618
$568$	0	0
$569$	−11.3911	−0.477541	−0.238770	−	0.971076i	$-0.576744\pi$
$569$	−11.3911	−0.477541	−0.238770	+	0.971076i	$0.576744\pi$
$570$	0	0
$571$	21.1474	0.884991	0.442496	−	0.896771i	$-0.354093\pi$
$571$	21.1474	0.884991	0.442496	+	0.896771i	$0.354093\pi$
$572$	0	0
$573$	−37.9129	−1.58384
$574$	0	0
$575$	−0.858143	−0.0357870
$576$	0	0
$577$	23.5687	0.981177	0.490589	−	0.871391i	$-0.336782\pi$
$577$	23.5687	0.981177	0.490589	+	0.871391i	$0.336782\pi$
$578$	0	0
$579$	−3.65270	−0.151801
$580$	0	0
$581$	−30.3245	−1.25807
$582$	0	0
$583$	−5.29016	−0.219096
$584$	0	0
$585$	−1.86245	−0.0770027
$586$	0	0
$587$	0.395137	0.0163091	0.00815453	−	0.999967i	$-0.497404\pi$
$587$	0.395137	0.0163091	0.00815453	+	0.999967i	$0.497404\pi$
$588$	0	0
$589$	1.46218	0.0602480
$590$	0	0
$591$	−3.20536	−0.131851
$592$	0	0
$593$	14.6741	0.602591	0.301296	−	0.953531i	$-0.402581\pi$
$593$	14.6741	0.602591	0.301296	+	0.953531i	$0.402581\pi$
$594$	0	0
$595$	−8.95632	−0.367173
$596$	0	0
$597$	14.5986	0.597480
$598$	0	0
$599$	4.34297	0.177449	0.0887244	−	0.996056i	$-0.471721\pi$
$599$	4.34297	0.177449	0.0887244	+	0.996056i	$0.471721\pi$
$600$	0	0
$601$	−40.3289	−1.64505	−0.822525	−	0.568728i	$-0.807436\pi$
$601$	−40.3289	−1.64505	−0.822525	+	0.568728i	$0.807436\pi$
$602$	0	0
$603$	−11.5248	−0.469328
$604$	0	0
$605$	−11.4077	−0.463790
$606$	0	0
$607$	−25.2675	−1.02558	−0.512788	−	0.858516i	$-0.671387\pi$
$607$	−25.2675	−1.02558	−0.512788	+	0.858516i	$0.671387\pi$
$608$	0	0
$609$	−46.0743	−1.86703
$610$	0	0
$611$	0.476236	0.0192665
$612$	0	0
$613$	33.7920	1.36485	0.682423	−	0.730957i	$-0.260926\pi$
$613$	33.7920	1.36485	0.682423	+	0.730957i	$0.260926\pi$
$614$	0	0
$615$	−16.2019	−0.653324
$616$	0	0
$617$	−16.9245	−0.681354	−0.340677	−	0.940180i	$-0.610656\pi$
$617$	−16.9245	−0.681354	−0.340677	+	0.940180i	$0.610656\pi$
$618$	0	0
$619$	26.0353	1.04645	0.523224	−	0.852195i	$-0.324729\pi$
$619$	26.0353	1.04645	0.523224	+	0.852195i	$0.324729\pi$
$620$	0	0
$621$	1.17681	0.0472236
$622$	0	0
$623$	−44.7035	−1.79101
$624$	0	0
$625$	12.5767	0.503066
$626$	0	0
$627$	2.22392	0.0888149
$628$	0	0
$629$	−17.4452	−0.695587
$630$	0	0
$631$	40.4126	1.60880	0.804399	−	0.594089i	$-0.202487\pi$
$631$	40.4126	1.60880	0.804399	+	0.594089i	$0.202487\pi$
$632$	0	0
$633$	19.7482	0.784922
$634$	0	0
$635$	14.2172	0.564193
$636$	0	0
$637$	−14.5274	−0.575598
$638$	0	0
$639$	−1.27550	−0.0504580
$640$	0	0
$641$	9.31678	0.367991	0.183995	−	0.982927i	$-0.441097\pi$
$641$	9.31678	0.367991	0.183995	+	0.982927i	$0.441097\pi$
$642$	0	0
$643$	−0.751383	−0.0296316	−0.0148158	−	0.999890i	$-0.504716\pi$
$643$	−0.751383	−0.0296316	−0.0148158	+	0.999890i	$0.504716\pi$
$644$	0	0
$645$	−1.36362	−0.0536925
$646$	0	0
$647$	−22.3687	−0.879403	−0.439702	−	0.898144i	$-0.644916\pi$
$647$	−22.3687	−0.879403	−0.439702	+	0.898144i	$0.644916\pi$
$648$	0	0
$649$	22.6345	0.888481
$650$	0	0
$651$	−24.5587	−0.962533
$652$	0	0
$653$	19.1981	0.751279	0.375639	−	0.926766i	$-0.377423\pi$
$653$	19.1981	0.751279	0.375639	+	0.926766i	$0.377423\pi$
$654$	0	0
$655$	−4.67213	−0.182555
$656$	0	0
$657$	−13.7281	−0.535582
$658$	0	0
$659$	17.5429	0.683376	0.341688	−	0.939813i	$-0.389001\pi$
$659$	17.5429	0.683376	0.341688	+	0.939813i	$0.389001\pi$
$660$	0	0
$661$	−39.1053	−1.52102	−0.760509	−	0.649327i	$-0.775051\pi$
$661$	−39.1053	−1.52102	−0.760509	+	0.649327i	$0.775051\pi$
$662$	0	0
$663$	−8.46533	−0.328766
$664$	0	0
$665$	−1.09494	−0.0424599
$666$	0	0
$667$	−1.79746	−0.0695978
$668$	0	0
$669$	−28.2239	−1.09120
$670$	0	0
$671$	−26.8045	−1.03478
$672$	0	0
$673$	−49.5981	−1.91187	−0.955933	−	0.293586i	$-0.905151\pi$
$673$	−49.5981	−1.91187	−0.955933	+	0.293586i	$0.905151\pi$
$674$	0	0
$675$	−23.2796	−0.896032
$676$	0	0
$677$	15.9508	0.613040	0.306520	−	0.951864i	$-0.400835\pi$
$677$	15.9508	0.613040	0.306520	+	0.951864i	$0.400835\pi$
$678$	0	0
$679$	−19.1047	−0.733172
$680$	0	0
$681$	−26.6625	−1.02171
$682$	0	0
$683$	−4.04859	−0.154915	−0.0774575	−	0.996996i	$-0.524680\pi$
$683$	−4.04859	−0.154915	−0.0774575	+	0.996996i	$0.524680\pi$
$684$	0	0
$685$	9.05679	0.346042
$686$	0	0
$687$	4.64630	0.177268
$688$	0	0
$689$	2.46180	0.0937873
$690$	0	0
$691$	18.1691	0.691184	0.345592	−	0.938385i	$-0.387678\pi$
$691$	18.1691	0.691184	0.345592	+	0.938385i	$0.387678\pi$
$692$	0	0
$693$	15.6694	0.595230
$694$	0	0
$695$	−17.1646	−0.651090
$696$	0	0
$697$	30.8924	1.17013
$698$	0	0
$699$	33.4541	1.26535
$700$	0	0
$701$	39.5708	1.49457	0.747285	−	0.664504i	$-0.231357\pi$
$701$	39.5708	1.49457	0.747285	+	0.664504i	$0.231357\pi$
$702$	0	0
$703$	−2.13273	−0.0804376
$704$	0	0
$705$	−0.289964	−0.0109207
$706$	0	0
$707$	−41.5337	−1.56204
$708$	0	0
$709$	17.7125	0.665207	0.332604	−	0.943067i	$-0.392073\pi$
$709$	17.7125	0.665207	0.332604	+	0.943067i	$0.392073\pi$
$710$	0	0
$711$	3.14921	0.118104
$712$	0	0
$713$	−0.958088	−0.0358807
$714$	0	0
$715$	10.1101	0.378098
$716$	0	0
$717$	17.4025	0.649908
$718$	0	0
$719$	−28.9335	−1.07904	−0.539519	−	0.841973i	$-0.681394\pi$
$719$	−28.9335	−1.07904	−0.539519	+	0.841973i	$0.681394\pi$
$720$	0	0
$721$	−50.4360	−1.87834
$722$	0	0
$723$	22.3643	0.831736
$724$	0	0
$725$	35.5573	1.32057
$726$	0	0
$727$	−12.1860	−0.451953	−0.225977	−	0.974133i	$-0.572557\pi$
$727$	−12.1860	−0.451953	−0.225977	+	0.974133i	$0.572557\pi$
$728$	0	0
$729$	29.5662	1.09505
$730$	0	0
$731$	2.60003	0.0961657
$732$	0	0
$733$	19.3847	0.715989	0.357994	−	0.933724i	$-0.383461\pi$
$733$	19.3847	0.715989	0.357994	+	0.933724i	$0.383461\pi$
$734$	0	0
$735$	8.84526	0.326262
$736$	0	0
$737$	62.5616	2.30449
$738$	0	0
$739$	−16.9316	−0.622840	−0.311420	−	0.950272i	$-0.600805\pi$
$739$	−16.9316	−0.622840	−0.311420	+	0.950272i	$0.600805\pi$
$740$	0	0
$741$	−1.03491	−0.0380185
$742$	0	0
$743$	47.4170	1.73956	0.869780	−	0.493439i	$-0.164261\pi$
$743$	47.4170	1.73956	0.869780	+	0.493439i	$0.164261\pi$
$744$	0	0
$745$	10.4950	0.384508
$746$	0	0
$747$	−7.32075	−0.267852
$748$	0	0
$749$	67.2727	2.45809
$750$	0	0
$751$	−38.5960	−1.40839	−0.704194	−	0.710007i	$-0.748692\pi$
$751$	−38.5960	−1.40839	−0.704194	+	0.710007i	$0.748692\pi$
$752$	0	0
$753$	11.6617	0.424976
$754$	0	0
$755$	−7.35230	−0.267578
$756$	0	0
$757$	−1.32592	−0.0481913	−0.0240956	−	0.999710i	$-0.507671\pi$
$757$	−1.32592	−0.0481913	−0.0240956	+	0.999710i	$0.507671\pi$
$758$	0	0
$759$	−1.45722	−0.0528937
$760$	0	0
$761$	−26.0851	−0.945584	−0.472792	−	0.881174i	$-0.656754\pi$
$761$	−26.0851	−0.945584	−0.472792	+	0.881174i	$0.656754\pi$
$762$	0	0
$763$	22.6240	0.819044
$764$	0	0
$765$	−2.16218	−0.0781738
$766$	0	0
$767$	−10.5331	−0.380327
$768$	0	0
$769$	−36.1945	−1.30521	−0.652603	−	0.757700i	$-0.726323\pi$
$769$	−36.1945	−1.30521	−0.652603	+	0.757700i	$0.726323\pi$
$770$	0	0
$771$	−24.0623	−0.866584
$772$	0	0
$773$	39.7648	1.43024	0.715120	−	0.699002i	$-0.246372\pi$
$773$	39.7648	1.43024	0.715120	+	0.699002i	$0.246372\pi$
$774$	0	0
$775$	18.9529	0.680809
$776$	0	0
$777$	35.8214	1.28509
$778$	0	0
$779$	3.77670	0.135314
$780$	0	0
$781$	6.92395	0.247758
$782$	0	0
$783$	−48.7612	−1.74258
$784$	0	0
$785$	−14.2420	−0.508318
$786$	0	0
$787$	−43.8812	−1.56420	−0.782098	−	0.623155i	$-0.785850\pi$
$787$	−43.8812	−1.56420	−0.782098	+	0.623155i	$0.785850\pi$
$788$	0	0
$789$	14.8159	0.527461
$790$	0	0
$791$	−29.1555	−1.03665
$792$	0	0
$793$	12.4736	0.442952
$794$	0	0
$795$	−1.49891	−0.0531609
$796$	0	0
$797$	3.35980	0.119010	0.0595052	−	0.998228i	$-0.481048\pi$
$797$	3.35980	0.119010	0.0595052	+	0.998228i	$0.481048\pi$
$798$	0	0
$799$	0.552879	0.0195595
$800$	0	0
$801$	−10.7920	−0.381318
$802$	0	0
$803$	74.5216	2.62981
$804$	0	0
$805$	0.717455	0.0252870
$806$	0	0
$807$	26.1580	0.920805
$808$	0	0
$809$	13.1158	0.461126	0.230563	−	0.973057i	$-0.425943\pi$
$809$	13.1158	0.461126	0.230563	+	0.973057i	$0.425943\pi$
$810$	0	0
$811$	39.5403	1.38845	0.694225	−	0.719759i	$-0.255747\pi$
$811$	39.5403	1.38845	0.694225	+	0.719759i	$0.255747\pi$
$812$	0	0
$813$	23.9752	0.840845
$814$	0	0
$815$	−7.64227	−0.267697
$816$	0	0
$817$	0.317862	0.0111206
$818$	0	0
$819$	−7.29182	−0.254797
$820$	0	0
$821$	19.9431	0.696019	0.348009	−	0.937491i	$-0.386858\pi$
$821$	19.9431	0.696019	0.348009	+	0.937491i	$0.386858\pi$
$822$	0	0
$823$	−48.1567	−1.67864	−0.839319	−	0.543639i	$-0.817046\pi$
$823$	−48.1567	−1.67864	−0.839319	+	0.543639i	$0.817046\pi$
$824$	0	0
$825$	28.8268	1.00362
$826$	0	0
$827$	5.36340	0.186504	0.0932518	−	0.995643i	$-0.470274\pi$
$827$	5.36340	0.186504	0.0932518	+	0.995643i	$0.470274\pi$
$828$	0	0
$829$	19.9640	0.693380	0.346690	−	0.937980i	$-0.387306\pi$
$829$	19.9640	0.693380	0.346690	+	0.937980i	$0.387306\pi$
$830$	0	0
$831$	6.03286	0.209278
$832$	0	0
$833$	−16.8654	−0.584351
$834$	0	0
$835$	−7.24267	−0.250643
$836$	0	0
$837$	−25.9909	−0.898377
$838$	0	0
$839$	−5.12883	−0.177067	−0.0885335	−	0.996073i	$-0.528218\pi$
$839$	−5.12883	−0.177067	−0.0885335	+	0.996073i	$0.528218\pi$
$840$	0	0
$841$	45.4779	1.56820
$842$	0	0
$843$	12.6237	0.434785
$844$	0	0
$845$	7.48909	0.257633
$846$	0	0
$847$	−44.6634	−1.53465
$848$	0	0
$849$	−21.4581	−0.736439
$850$	0	0
$851$	1.39747	0.0479046
$852$	0	0
$853$	−12.9218	−0.442433	−0.221217	−	0.975225i	$-0.571003\pi$
$853$	−12.9218	−0.442433	−0.221217	+	0.975225i	$0.571003\pi$
$854$	0	0
$855$	−0.264333	−0.00904001
$856$	0	0
$857$	1.43126	0.0488910	0.0244455	−	0.999701i	$-0.492218\pi$
$857$	1.43126	0.0488910	0.0244455	+	0.999701i	$0.492218\pi$
$858$	0	0
$859$	6.53487	0.222967	0.111484	−	0.993766i	$-0.464440\pi$
$859$	6.53487	0.222967	0.111484	+	0.993766i	$0.464440\pi$
$860$	0	0
$861$	−63.4334	−2.16181
$862$	0	0
$863$	54.4516	1.85355	0.926776	−	0.375614i	$-0.122568\pi$
$863$	54.4516	1.85355	0.926776	+	0.375614i	$0.122568\pi$
$864$	0	0
$865$	−7.66363	−0.260571
$866$	0	0
$867$	14.8863	0.505565
$868$	0	0
$869$	−17.0952	−0.579915
$870$	0	0
$871$	−29.1134	−0.986469
$872$	0	0
$873$	−4.61215	−0.156098
$874$	0	0
$875$	−31.4162	−1.06206
$876$	0	0
$877$	48.2597	1.62962	0.814808	−	0.579731i	$-0.196842\pi$
$877$	48.2597	1.62962	0.814808	+	0.579731i	$0.196842\pi$
$878$	0	0
$879$	−42.6310	−1.43791
$880$	0	0
$881$	11.2062	0.377547	0.188774	−	0.982021i	$-0.439549\pi$
$881$	11.2062	0.377547	0.188774	+	0.982021i	$0.439549\pi$
$882$	0	0
$883$	−1.56257	−0.0525847	−0.0262924	−	0.999654i	$-0.508370\pi$
$883$	−1.56257	−0.0525847	−0.0262924	+	0.999654i	$0.508370\pi$
$884$	0	0
$885$	6.41323	0.215578
$886$	0	0
$887$	49.2011	1.65201	0.826006	−	0.563662i	$-0.190608\pi$
$887$	49.2011	1.65201	0.826006	+	0.563662i	$0.190608\pi$
$888$	0	0
$889$	55.6630	1.86688
$890$	0	0
$891$	−26.7309	−0.895520
$892$	0	0
$893$	0.0675912	0.00226185
$894$	0	0
$895$	5.48305	0.183278
$896$	0	0
$897$	0.678124	0.0226419
$898$	0	0
$899$	39.6986	1.32402
$900$	0	0
$901$	2.85800	0.0952136
$902$	0	0
$903$	−5.33882	−0.177665
$904$	0	0
$905$	5.21458	0.173338
$906$	0	0
$907$	−40.5983	−1.34804	−0.674022	−	0.738711i	$-0.735435\pi$
$907$	−40.5983	−1.34804	−0.674022	+	0.738711i	$0.735435\pi$
$908$	0	0
$909$	−10.0268	−0.332568
$910$	0	0
$911$	37.6775	1.24831	0.624155	−	0.781300i	$-0.285443\pi$
$911$	37.6775	1.24831	0.624155	+	0.781300i	$0.285443\pi$
$912$	0	0
$913$	39.7401	1.31520
$914$	0	0
$915$	−7.59477	−0.251075
$916$	0	0
$917$	−18.2922	−0.604063
$918$	0	0
$919$	−3.95874	−0.130587	−0.0652934	−	0.997866i	$-0.520798\pi$
$919$	−3.95874	−0.130587	−0.0652934	+	0.997866i	$0.520798\pi$
$920$	0	0
$921$	26.0921	0.859764
$922$	0	0
$923$	−3.22209	−0.106056
$924$	0	0
$925$	−27.6448	−0.908954
$926$	0	0
$927$	−12.1760	−0.399911
$928$	0	0
$929$	1.85350	0.0608113	0.0304057	−	0.999538i	$-0.490320\pi$
$929$	1.85350	0.0608113	0.0304057	+	0.999538i	$0.490320\pi$
$930$	0	0
$931$	−2.06185	−0.0675743
$932$	0	0
$933$	−41.8810	−1.37112
$934$	0	0
$935$	11.7372	0.383848
$936$	0	0
$937$	28.6950	0.937423	0.468712	−	0.883351i	$-0.344718\pi$
$937$	28.6950	0.937423	0.468712	+	0.883351i	$0.344718\pi$
$938$	0	0
$939$	−30.8908	−1.00808
$940$	0	0
$941$	−47.9769	−1.56400	−0.782001	−	0.623278i	$-0.785801\pi$
$941$	−47.9769	−1.56400	−0.782001	+	0.623278i	$0.785801\pi$
$942$	0	0
$943$	−2.47467	−0.0805864
$944$	0	0
$945$	19.4631	0.633133
$946$	0	0
$947$	−36.6466	−1.19085	−0.595427	−	0.803409i	$-0.703017\pi$
$947$	−36.6466	−1.19085	−0.595427	+	0.803409i	$0.703017\pi$
$948$	0	0
$949$	−34.6790	−1.12573
$950$	0	0
$951$	−49.4138	−1.60235
$952$	0	0
$953$	−45.0978	−1.46086	−0.730430	−	0.682987i	$-0.760680\pi$
$953$	−45.0978	−1.46086	−0.730430	+	0.682987i	$0.760680\pi$
$954$	0	0
$955$	−24.4621	−0.791574
$956$	0	0
$957$	60.3802	1.95181
$958$	0	0
$959$	35.4590	1.14503
$960$	0	0
$961$	−9.83968	−0.317409
$962$	0	0
$963$	16.2406	0.523345
$964$	0	0
$965$	−2.35678	−0.0758675
$966$	0	0
$967$	−14.2370	−0.457831	−0.228915	−	0.973446i	$-0.573518\pi$
$967$	−14.2370	−0.457831	−0.228915	+	0.973446i	$0.573518\pi$
$968$	0	0
$969$	−1.20147	−0.0385967
$970$	0	0
$971$	12.0708	0.387371	0.193686	−	0.981064i	$-0.437956\pi$
$971$	12.0708	0.387371	0.193686	+	0.981064i	$0.437956\pi$
$972$	0	0
$973$	−67.2025	−2.15441
$974$	0	0
$975$	−13.4147	−0.429613
$976$	0	0
$977$	−44.5670	−1.42582	−0.712912	−	0.701254i	$-0.752624\pi$
$977$	−44.5670	−1.42582	−0.712912	+	0.701254i	$0.752624\pi$
$978$	0	0
$979$	58.5837	1.87234
$980$	0	0
$981$	5.46175	0.174380
$982$	0	0
$983$	31.9215	1.01814	0.509069	−	0.860726i	$-0.329990\pi$
$983$	31.9215	1.01814	0.509069	+	0.860726i	$0.329990\pi$
$984$	0	0
$985$	−2.06815	−0.0658967
$986$	0	0
$987$	−1.13526	−0.0361358
$988$	0	0
$989$	−0.208278	−0.00662287
$990$	0	0
$991$	45.3785	1.44149	0.720747	−	0.693198i	$-0.243799\pi$
$991$	45.3785	1.44149	0.720747	+	0.693198i	$0.243799\pi$
$992$	0	0
$993$	−28.7856	−0.913484
$994$	0	0
$995$	9.41924	0.298610
$996$	0	0
$997$	−3.98689	−0.126266	−0.0631331	−	0.998005i	$-0.520109\pi$
$997$	−3.98689	−0.126266	−0.0631331	+	0.998005i	$0.520109\pi$
$998$	0	0
$999$	37.9104	1.19943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2752.2.a.bb.1.1		5
4.3	odd	2		2752.2.a.w.1.5		5
8.3	odd	2		1376.2.a.h.1.1	yes	5
8.5	even	2		1376.2.a.e.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1376.2.a.e.1.5	✓	5	8.5	even	2
1376.2.a.h.1.1	yes	5	8.3	odd	2
2752.2.a.w.1.5		5	4.3	odd	2
2752.2.a.bb.1.1		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2752 = 2^{6} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2752.a (trivial)

\(f(q)\)	\(=\)	\(q-1.45376 q^{3} -0.937992 q^{5} -3.67241 q^{7} -0.886571 q^{9} +O(q^{10})\)	\(q-1.45376 q^{3} -0.937992 q^{5} -3.67241 q^{7} -0.886571 q^{9} +4.81268 q^{11} -2.23960 q^{13} +1.36362 q^{15} -2.60003 q^{17} -0.317862 q^{19} +5.33882 q^{21} +0.208278 q^{23} -4.12017 q^{25} +5.65016 q^{27} -8.63006 q^{29} -4.60003 q^{31} -6.99650 q^{33} +3.44469 q^{35} +6.70962 q^{37} +3.25585 q^{39} -11.8816 q^{41} -1.00000 q^{43} +0.831597 q^{45} -0.212643 q^{47} +6.48661 q^{49} +3.77984 q^{51} -1.09921 q^{53} -4.51425 q^{55} +0.462097 q^{57} +4.70309 q^{59} -5.56957 q^{61} +3.25585 q^{63} +2.10073 q^{65} +12.9993 q^{67} -0.302788 q^{69} +1.43869 q^{71} +15.4844 q^{73} +5.98975 q^{75} -17.6741 q^{77} -3.55212 q^{79} -5.55428 q^{81} +8.25737 q^{83} +2.43881 q^{85} +12.5461 q^{87} +12.1728 q^{89} +8.22475 q^{91} +6.68736 q^{93} +0.298152 q^{95} +5.20223 q^{97} -4.26678 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - q^5 - 4 * q^7 + 8 * q^9	\( 5 q + 5 q^{3} - q^{5} - 4 q^{7} + 8 q^{9} + 7 q^{11} + 3 q^{13} - 2 q^{17} + 19 q^{19} - 6 q^{21} + 6 q^{23} + 10 q^{25} + 20 q^{27} - 7 q^{29} - 12 q^{31} - 6 q^{33} + 18 q^{35} - 3 q^{37} - 8 q^{39} - 10 q^{41} - 5 q^{43} + 16 q^{45} + 5 q^{47} + 9 q^{49} + 13 q^{51} + 13 q^{53} - 6 q^{55} + 24 q^{57} + 32 q^{59} + 4 q^{61} - 8 q^{63} + 17 q^{67} + 10 q^{69} + 4 q^{71} - 2 q^{73} + 44 q^{75} + 6 q^{77} - 27 q^{79} - 3 q^{81} + 25 q^{83} + 16 q^{85} + 11 q^{87} + 8 q^{89} + 38 q^{91} + 3 q^{93} + 19 q^{95} - 30 q^{97} - q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - q^5 - 4 * q^7 + 8 * q^9 + 7 * q^11 + 3 * q^13 - 2 * q^17 + 19 * q^19 - 6 * q^21 + 6 * q^23 + 10 * q^25 + 20 * q^27 - 7 * q^29 - 12 * q^31 - 6 * q^33 + 18 * q^35 - 3 * q^37 - 8 * q^39 - 10 * q^41 - 5 * q^43 + 16 * q^45 + 5 * q^47 + 9 * q^49 + 13 * q^51 + 13 * q^53 - 6 * q^55 + 24 * q^57 + 32 * q^59 + 4 * q^61 - 8 * q^63 + 17 * q^67 + 10 * q^69 + 4 * q^71 - 2 * q^73 + 44 * q^75 + 6 * q^77 - 27 * q^79 - 3 * q^81 + 25 * q^83 + 16 * q^85 + 11 * q^87 + 8 * q^89 + 38 * q^91 + 3 * q^93 + 19 * q^95 - 30 * q^97 - q^99

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