LMFDB - Embedded newform 4176.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4176,2,Mod(1,4176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4176 = 2^{4} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4176.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:	$33.3455278841$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{5} -0.236068 q^{7} +O(q^{10})$	$q-3.23607 q^{5} -0.236068 q^{7} -0.236068 q^{11} -5.47214 q^{13} -3.00000 q^{17} +7.23607 q^{19} -7.70820 q^{23} +5.47214 q^{25} +1.00000 q^{29} -3.70820 q^{31} +0.763932 q^{35} +5.23607 q^{37} -2.00000 q^{41} -4.00000 q^{43} +4.70820 q^{47} -6.94427 q^{49} -11.2361 q^{53} +0.763932 q^{55} +4.47214 q^{59} -5.23607 q^{61} +17.7082 q^{65} +13.1803 q^{67} -5.23607 q^{71} +6.76393 q^{73} +0.0557281 q^{77} +12.7639 q^{79} +2.94427 q^{83} +9.70820 q^{85} -5.00000 q^{89} +1.29180 q^{91} -23.4164 q^{95} +18.6525 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^7	$ 2 q - 2 q^{5} + 4 q^{7} + 4 q^{11} - 2 q^{13} - 6 q^{17} + 10 q^{19} - 2 q^{23} + 2 q^{25} + 2 q^{29} + 6 q^{31} + 6 q^{35} + 6 q^{37} - 4 q^{41} - 8 q^{43} - 4 q^{47} + 4 q^{49} - 18 q^{53} + 6 q^{55} - 6 q^{61} + 22 q^{65} + 4 q^{67} - 6 q^{71} + 18 q^{73} + 18 q^{77} + 30 q^{79} - 12 q^{83} + 6 q^{85} - 10 q^{89} + 16 q^{91} - 20 q^{95} + 6 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^7 + 4 * q^11 - 2 * q^13 - 6 * q^17 + 10 * q^19 - 2 * q^23 + 2 * q^25 + 2 * q^29 + 6 * q^31 + 6 * q^35 + 6 * q^37 - 4 * q^41 - 8 * q^43 - 4 * q^47 + 4 * q^49 - 18 * q^53 + 6 * q^55 - 6 * q^61 + 22 * q^65 + 4 * q^67 - 6 * q^71 + 18 * q^73 + 18 * q^77 + 30 * q^79 - 12 * q^83 + 6 * q^85 - 10 * q^89 + 16 * q^91 - 20 * q^95 + 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.70820	−0.666013	−0.333007	−	0.942925i	$-0.608063\pi$
$31$	−3.70820	−0.666013	−0.333007	+	0.942925i	$0.608063\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.763932	0.129128
$36$	0	0
$37$	5.23607	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$37$	5.23607	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.70820	0.686762	0.343381	−	0.939196i	$-0.388428\pi$
$47$	4.70820	0.686762	0.343381	+	0.939196i	$0.388428\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.2361	−1.54339	−0.771696	−	0.635991i	$-0.780591\pi$
$53$	−11.2361	−1.54339	−0.771696	+	0.635991i	$0.780591\pi$
$54$	0	0
$55$	0.763932	0.103009
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	−5.23607	−0.670410	−0.335205	−	0.942145i	$-0.608806\pi$
$61$	−5.23607	−0.670410	−0.335205	+	0.942145i	$0.608806\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.7082	2.19643
$66$	0	0
$67$	13.1803	1.61023	0.805117	−	0.593115i	$-0.202102\pi$
$67$	13.1803	1.61023	0.805117	+	0.593115i	$0.202102\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.23607	−0.621407	−0.310703	−	0.950507i	$-0.600565\pi$
$71$	−5.23607	−0.621407	−0.310703	+	0.950507i	$0.600565\pi$
$72$	0	0
$73$	6.76393	0.791658	0.395829	−	0.918324i	$-0.370457\pi$
$73$	6.76393	0.791658	0.395829	+	0.918324i	$0.370457\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.0557281	0.00635081
$78$	0	0
$79$	12.7639	1.43605	0.718027	−	0.696015i	$-0.245045\pi$
$79$	12.7639	1.43605	0.718027	+	0.696015i	$0.245045\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.94427	0.323176	0.161588	−	0.986858i	$-0.448338\pi$
$83$	2.94427	0.323176	0.161588	+	0.986858i	$0.448338\pi$
$84$	0	0
$85$	9.70820	1.05300
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	0	0
$91$	1.29180	0.135417
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−23.4164	−2.40247
$96$	0	0
$97$	18.6525	1.89387	0.946936	−	0.321422i	$-0.104161\pi$
$97$	18.6525	1.89387	0.946936	+	0.321422i	$0.104161\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.47214	0.743505	0.371753	−	0.928332i	$-0.378757\pi$
$101$	7.47214	0.743505	0.371753	+	0.928332i	$0.378757\pi$
$102$	0	0
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.70820	0.938527	0.469264	−	0.883058i	$-0.344519\pi$
$107$	9.70820	0.938527	0.469264	+	0.883058i	$0.344519\pi$
$108$	0	0
$109$	9.47214	0.907266	0.453633	−	0.891189i	$-0.350128\pi$
$109$	9.47214	0.907266	0.453633	+	0.891189i	$0.350128\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.0000	−1.78737	−0.893685	−	0.448695i	$-0.851889\pi$
$113$	−19.0000	−1.78737	−0.893685	+	0.448695i	$0.851889\pi$
$114$	0	0
$115$	24.9443	2.32607
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.708204	0.0649209
$120$	0	0
$121$	−10.9443	−0.994934
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−12.4721	−1.10672	−0.553362	−	0.832941i	$-0.686655\pi$
$127$	−12.4721	−1.10672	−0.553362	+	0.832941i	$0.686655\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.70820	0.760839	0.380420	−	0.924814i	$-0.375780\pi$
$131$	8.70820	0.760839	0.380420	+	0.924814i	$0.375780\pi$
$132$	0	0
$133$	−1.70820	−0.148120
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.52786	−0.301406	−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786	−0.301406	−0.150703	+	0.988579i	$0.548154\pi$
$138$	0	0
$139$	−1.18034	−0.100115	−0.0500576	−	0.998746i	$-0.515940\pi$
$139$	−1.18034	−0.100115	−0.0500576	+	0.998746i	$0.515940\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.29180	0.108025
$144$	0	0
$145$	−3.23607	−0.268741
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.4164	1.91835	0.959173	−	0.282819i	$-0.0912694\pi$
$149$	23.4164	1.91835	0.959173	+	0.282819i	$0.0912694\pi$
$150$	0	0
$151$	−3.05573	−0.248672	−0.124336	−	0.992240i	$-0.539680\pi$
$151$	−3.05573	−0.248672	−0.124336	+	0.992240i	$0.539680\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.81966	0.143409
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.47214	−0.500829	−0.250414	−	0.968139i	$-0.580567\pi$
$167$	−6.47214	−0.500829	−0.250414	+	0.968139i	$0.580567\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.05573	0.536437	0.268219	−	0.963358i	$-0.413565\pi$
$173$	7.05573	0.536437	0.268219	+	0.963358i	$0.413565\pi$
$174$	0	0
$175$	−1.29180	−0.0976506
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.2361	1.28828	0.644142	−	0.764906i	$-0.277215\pi$
$179$	17.2361	1.28828	0.644142	+	0.764906i	$0.277215\pi$
$180$	0	0
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−16.9443	−1.24577
$186$	0	0
$187$	0.708204	0.0517890
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	8.47214	0.609838	0.304919	−	0.952378i	$-0.401371\pi$
$193$	8.47214	0.609838	0.304919	+	0.952378i	$0.401371\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.2361	1.37051	0.685257	−	0.728302i	$-0.259690\pi$
$197$	19.2361	1.37051	0.685257	+	0.728302i	$0.259690\pi$
$198$	0	0
$199$	15.6525	1.10957	0.554787	−	0.831992i	$-0.312800\pi$
$199$	15.6525	1.10957	0.554787	+	0.831992i	$0.312800\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.236068	−0.0165687
$204$	0	0
$205$	6.47214	0.452034
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.70820	−0.118159
$210$	0	0
$211$	−10.9443	−0.753435	−0.376717	−	0.926328i	$-0.622947\pi$
$211$	−10.9443	−0.753435	−0.376717	+	0.926328i	$0.622947\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.9443	0.882792
$216$	0	0
$217$	0.875388	0.0594252
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.4164	1.10429
$222$	0	0
$223$	−15.1803	−1.01655	−0.508275	−	0.861195i	$-0.669717\pi$
$223$	−15.1803	−1.01655	−0.508275	+	0.861195i	$0.669717\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.9443	−0.726397	−0.363198	−	0.931712i	$-0.618315\pi$
$227$	−10.9443	−0.726397	−0.363198	+	0.931712i	$0.618315\pi$
$228$	0	0
$229$	−16.1803	−1.06923	−0.534613	−	0.845097i	$-0.679543\pi$
$229$	−16.1803	−1.06923	−0.534613	+	0.845097i	$0.679543\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.4164	−1.14099	−0.570493	−	0.821302i	$-0.693248\pi$
$233$	−17.4164	−1.14099	−0.570493	+	0.821302i	$0.693248\pi$
$234$	0	0
$235$	−15.2361	−0.993891
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.5967	1.26761	0.633804	−	0.773494i	$-0.281493\pi$
$239$	19.5967	1.26761	0.633804	+	0.773494i	$0.281493\pi$
$240$	0	0
$241$	10.4164	0.670980	0.335490	−	0.942044i	$-0.391098\pi$
$241$	10.4164	0.670980	0.335490	+	0.942044i	$0.391098\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	22.4721	1.43569
$246$	0	0
$247$	−39.5967	−2.51948
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.18034	−0.579458	−0.289729	−	0.957109i	$-0.593565\pi$
$251$	−9.18034	−0.579458	−0.289729	+	0.957109i	$0.593565\pi$
$252$	0	0
$253$	1.81966	0.114401
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.4164	0.961649	0.480825	−	0.876817i	$-0.340337\pi$
$257$	15.4164	0.961649	0.480825	+	0.876817i	$0.340337\pi$
$258$	0	0
$259$	−1.23607	−0.0768055
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.8328	−1.40793	−0.703966	−	0.710234i	$-0.748589\pi$
$263$	−22.8328	−1.40793	−0.703966	+	0.710234i	$0.748589\pi$
$264$	0	0
$265$	36.3607	2.23362
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.00000	0.304855	0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000	0.304855	0.152428	+	0.988315i	$0.451291\pi$
$270$	0	0
$271$	26.9443	1.63675	0.818374	−	0.574686i	$-0.194876\pi$
$271$	26.9443	1.63675	0.818374	+	0.574686i	$0.194876\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.29180	−0.0778982
$276$	0	0
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.4164	1.27760	0.638798	−	0.769375i	$-0.279432\pi$
$281$	21.4164	1.27760	0.638798	+	0.769375i	$0.279432\pi$
$282$	0	0
$283$	−7.41641	−0.440860	−0.220430	−	0.975403i	$-0.570746\pi$
$283$	−7.41641	−0.440860	−0.220430	+	0.975403i	$0.570746\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.472136	0.0278693
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.9443	1.16516	0.582578	−	0.812775i	$-0.302044\pi$
$293$	19.9443	1.16516	0.582578	+	0.812775i	$0.302044\pi$
$294$	0	0
$295$	−14.4721	−0.842600
$296$	0	0
$297$	0	0
$298$	0	0
$299$	42.1803	2.43935
$300$	0	0
$301$	0.944272	0.0544269
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.9443	0.970226
$306$	0	0
$307$	12.6525	0.722115	0.361057	−	0.932544i	$-0.382416\pi$
$307$	12.6525	0.722115	0.361057	+	0.932544i	$0.382416\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.7082	1.62789	0.813946	−	0.580940i	$-0.197315\pi$
$311$	28.7082	1.62789	0.813946	+	0.580940i	$0.197315\pi$
$312$	0	0
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.47214	0.0826834	0.0413417	−	0.999145i	$-0.486837\pi$
$317$	1.47214	0.0826834	0.0413417	+	0.999145i	$0.486837\pi$
$318$	0	0
$319$	−0.236068	−0.0132173
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−21.7082	−1.20788
$324$	0	0
$325$	−29.9443	−1.66101
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.11146	−0.0612766
$330$	0	0
$331$	20.7639	1.14129	0.570644	−	0.821197i	$-0.306693\pi$
$331$	20.7639	1.14129	0.570644	+	0.821197i	$0.306693\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−42.6525	−2.33035
$336$	0	0
$337$	−8.18034	−0.445612	−0.222806	−	0.974863i	$-0.571522\pi$
$337$	−8.18034	−0.445612	−0.222806	+	0.974863i	$0.571522\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.875388	0.0474049
$342$	0	0
$343$	3.29180	0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.2361	1.35474	0.677372	−	0.735641i	$-0.263119\pi$
$347$	25.2361	1.35474	0.677372	+	0.735641i	$0.263119\pi$
$348$	0	0
$349$	−13.4164	−0.718164	−0.359082	−	0.933306i	$-0.616910\pi$
$349$	−13.4164	−0.718164	−0.359082	+	0.933306i	$0.616910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.23607	0.172239	0.0861193	−	0.996285i	$-0.472553\pi$
$353$	3.23607	0.172239	0.0861193	+	0.996285i	$0.472553\pi$
$354$	0	0
$355$	16.9443	0.899309
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.4721	−0.763810	−0.381905	−	0.924202i	$-0.624732\pi$
$359$	−14.4721	−0.763810	−0.381905	+	0.924202i	$0.624732\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−21.8885	−1.14570
$366$	0	0
$367$	−13.5279	−0.706149	−0.353074	−	0.935595i	$-0.614864\pi$
$367$	−13.5279	−0.706149	−0.353074	+	0.935595i	$0.614864\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.65248	0.137710
$372$	0	0
$373$	−11.5279	−0.596890	−0.298445	−	0.954427i	$-0.596468\pi$
$373$	−11.5279	−0.596890	−0.298445	+	0.954427i	$0.596468\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.47214	−0.281829
$378$	0	0
$379$	−1.70820	−0.0877445	−0.0438723	−	0.999037i	$-0.513969\pi$
$379$	−1.70820	−0.0877445	−0.0438723	+	0.999037i	$0.513969\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.2361	0.574136	0.287068	−	0.957910i	$-0.407319\pi$
$383$	11.2361	0.574136	0.287068	+	0.957910i	$0.407319\pi$
$384$	0	0
$385$	−0.180340	−0.00919097
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.3607	−0.880221	−0.440111	−	0.897944i	$-0.645061\pi$
$389$	−17.3607	−0.880221	−0.440111	+	0.897944i	$0.645061\pi$
$390$	0	0
$391$	23.1246	1.16946
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−41.3050	−2.07828
$396$	0	0
$397$	6.94427	0.348523	0.174262	−	0.984699i	$-0.444246\pi$
$397$	6.94427	0.348523	0.174262	+	0.984699i	$0.444246\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−28.1803	−1.40726	−0.703630	−	0.710567i	$-0.748439\pi$
$401$	−28.1803	−1.40726	−0.703630	+	0.710567i	$0.748439\pi$
$402$	0	0
$403$	20.2918	1.01081
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.23607	−0.0612696
$408$	0	0
$409$	−4.47214	−0.221133	−0.110566	−	0.993869i	$-0.535266\pi$
$409$	−4.47214	−0.221133	−0.110566	+	0.993869i	$0.535266\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.05573	−0.0519490
$414$	0	0
$415$	−9.52786	−0.467704
$416$	0	0
$417$	0	0
$418$	0	0
$419$	27.2361	1.33057	0.665284	−	0.746590i	$-0.268310\pi$
$419$	27.2361	1.33057	0.665284	+	0.746590i	$0.268310\pi$
$420$	0	0
$421$	−35.8885	−1.74910	−0.874550	−	0.484935i	$-0.838843\pi$
$421$	−35.8885	−1.74910	−0.874550	+	0.484935i	$0.838843\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.4164	−0.796313
$426$	0	0
$427$	1.23607	0.0598175
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	−6.65248	−0.319698	−0.159849	−	0.987142i	$-0.551101\pi$
$433$	−6.65248	−0.319698	−0.159849	+	0.987142i	$0.551101\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−55.7771	−2.66818
$438$	0	0
$439$	−13.2918	−0.634383	−0.317191	−	0.948362i	$-0.602740\pi$
$439$	−13.2918	−0.634383	−0.317191	+	0.948362i	$0.602740\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.76393	−0.178830	−0.0894149	−	0.995994i	$-0.528500\pi$
$443$	−3.76393	−0.178830	−0.0894149	+	0.995994i	$0.528500\pi$
$444$	0	0
$445$	16.1803	0.767022
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.4721	0.918947	0.459473	−	0.888192i	$-0.348038\pi$
$449$	19.4721	0.918947	0.459473	+	0.888192i	$0.348038\pi$
$450$	0	0
$451$	0.472136	0.0222320
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.18034	−0.195977
$456$	0	0
$457$	−1.47214	−0.0688636	−0.0344318	−	0.999407i	$-0.510962\pi$
$457$	−1.47214	−0.0688636	−0.0344318	+	0.999407i	$0.510962\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.8885	0.740003	0.370002	−	0.929031i	$-0.379357\pi$
$461$	15.8885	0.740003	0.370002	+	0.929031i	$0.379357\pi$
$462$	0	0
$463$	−15.1803	−0.705490	−0.352745	−	0.935719i	$-0.614752\pi$
$463$	−15.1803	−0.705490	−0.352745	+	0.935719i	$0.614752\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−3.11146	−0.143674
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.944272	0.0434177
$474$	0	0
$475$	39.5967	1.81682
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.4721	0.661249	0.330624	−	0.943762i	$-0.392741\pi$
$479$	14.4721	0.661249	0.330624	+	0.943762i	$0.392741\pi$
$480$	0	0
$481$	−28.6525	−1.30644
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−60.3607	−2.74084
$486$	0	0
$487$	−36.9443	−1.67410	−0.837052	−	0.547123i	$-0.815723\pi$
$487$	−36.9443	−1.67410	−0.837052	+	0.547123i	$0.815723\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.23607	0.0554452
$498$	0	0
$499$	−3.29180	−0.147361	−0.0736805	−	0.997282i	$-0.523475\pi$
$499$	−3.29180	−0.147361	−0.0736805	+	0.997282i	$0.523475\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.5410	1.22799	0.613997	−	0.789309i	$-0.289561\pi$
$503$	27.5410	1.22799	0.613997	+	0.789309i	$0.289561\pi$
$504$	0	0
$505$	−24.1803	−1.07601
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	−1.59675	−0.0706360
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−1.11146	−0.0488818
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.4164	1.37638	0.688189	−	0.725532i	$-0.258406\pi$
$521$	31.4164	1.37638	0.688189	+	0.725532i	$0.258406\pi$
$522$	0	0
$523$	−14.1246	−0.617626	−0.308813	−	0.951123i	$-0.599932\pi$
$523$	−14.1246	−0.617626	−0.308813	+	0.951123i	$0.599932\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.1246	0.484596
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.9443	0.474049
$534$	0	0
$535$	−31.4164	−1.35825
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.63932	0.0706105
$540$	0	0
$541$	8.18034	0.351700	0.175850	−	0.984417i	$-0.443733\pi$
$541$	8.18034	0.351700	0.175850	+	0.984417i	$0.443733\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−30.6525	−1.31301
$546$	0	0
$547$	−3.65248	−0.156169	−0.0780843	−	0.996947i	$-0.524880\pi$
$547$	−3.65248	−0.156169	−0.0780843	+	0.996947i	$0.524880\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.23607	0.308267
$552$	0	0
$553$	−3.01316	−0.128132
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.4164	1.07693	0.538464	−	0.842649i	$-0.319005\pi$
$557$	25.4164	1.07693	0.538464	+	0.842649i	$0.319005\pi$
$558$	0	0
$559$	21.8885	0.925787
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−45.0689	−1.89943	−0.949713	−	0.313120i	$-0.898626\pi$
$563$	−45.0689	−1.89943	−0.949713	+	0.313120i	$0.898626\pi$
$564$	0	0
$565$	61.4853	2.58671
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.2492	−1.47772	−0.738862	−	0.673857i	$-0.764637\pi$
$569$	−35.2492	−1.47772	−0.738862	+	0.673857i	$0.764637\pi$
$570$	0	0
$571$	−15.4164	−0.645157	−0.322578	−	0.946543i	$-0.604550\pi$
$571$	−15.4164	−0.645157	−0.322578	+	0.946543i	$0.604550\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−42.1803	−1.75904
$576$	0	0
$577$	−40.9443	−1.70453	−0.852266	−	0.523108i	$-0.824772\pi$
$577$	−40.9443	−1.70453	−0.852266	+	0.523108i	$0.824772\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.695048	−0.0288355
$582$	0	0
$583$	2.65248	0.109854
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.41641	−0.223559	−0.111780	−	0.993733i	$-0.535655\pi$
$587$	−5.41641	−0.223559	−0.111780	+	0.993733i	$0.535655\pi$
$588$	0	0
$589$	−26.8328	−1.10563
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.5967	0.640482	0.320241	−	0.947336i	$-0.396236\pi$
$593$	15.5967	0.640482	0.320241	+	0.947336i	$0.396236\pi$
$594$	0	0
$595$	−2.29180	−0.0939545
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.2918	1.36027	0.680133	−	0.733089i	$-0.261922\pi$
$599$	33.2918	1.36027	0.680133	+	0.733089i	$0.261922\pi$
$600$	0	0
$601$	8.18034	0.333683	0.166842	−	0.985984i	$-0.446643\pi$
$601$	8.18034	0.333683	0.166842	+	0.985984i	$0.446643\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	35.4164	1.43988
$606$	0	0
$607$	−1.41641	−0.0574902	−0.0287451	−	0.999587i	$-0.509151\pi$
$607$	−1.41641	−0.0574902	−0.0287451	+	0.999587i	$0.509151\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25.7639	−1.04230
$612$	0	0
$613$	5.58359	0.225519	0.112760	−	0.993622i	$-0.464031\pi$
$613$	5.58359	0.225519	0.112760	+	0.993622i	$0.464031\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.4721	−0.502109	−0.251055	−	0.967973i	$-0.580777\pi$
$617$	−12.4721	−0.502109	−0.251055	+	0.967973i	$0.580777\pi$
$618$	0	0
$619$	1.70820	0.0686585	0.0343293	−	0.999411i	$-0.489071\pi$
$619$	1.70820	0.0686585	0.0343293	+	0.999411i	$0.489071\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.18034	0.0472893
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−15.7082	−0.626327
$630$	0	0
$631$	23.6525	0.941590	0.470795	−	0.882243i	$-0.343967\pi$
$631$	23.6525	0.941590	0.470795	+	0.882243i	$0.343967\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	40.3607	1.60166
$636$	0	0
$637$	38.0000	1.50561
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−38.3050	−1.51295	−0.756477	−	0.654020i	$-0.773081\pi$
$641$	−38.3050	−1.51295	−0.756477	+	0.654020i	$0.773081\pi$
$642$	0	0
$643$	−0.708204	−0.0279288	−0.0139644	−	0.999902i	$-0.504445\pi$
$643$	−0.708204	−0.0279288	−0.0139644	+	0.999902i	$0.504445\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.1803	0.557487	0.278743	−	0.960366i	$-0.410082\pi$
$647$	14.1803	0.557487	0.278743	+	0.960366i	$0.410082\pi$
$648$	0	0
$649$	−1.05573	−0.0414410
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	−28.1803	−1.10110
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.5967	0.958153	0.479077	−	0.877773i	$-0.340972\pi$
$659$	24.5967	0.958153	0.479077	+	0.877773i	$0.340972\pi$
$660$	0	0
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.52786	0.214361
$666$	0	0
$667$	−7.70820	−0.298463
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.23607	0.0477179
$672$	0	0
$673$	30.3050	1.16817	0.584085	−	0.811692i	$-0.301453\pi$
$673$	30.3050	1.16817	0.584085	+	0.811692i	$0.301453\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.416408	0.0160039	0.00800193	−	0.999968i	$-0.497453\pi$
$677$	0.416408	0.0160039	0.00800193	+	0.999968i	$0.497453\pi$
$678$	0	0
$679$	−4.40325	−0.168981
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.8328	0.414506	0.207253	−	0.978287i	$-0.433548\pi$
$683$	10.8328	0.414506	0.207253	+	0.978287i	$0.433548\pi$
$684$	0	0
$685$	11.4164	0.436199
$686$	0	0
$687$	0	0
$688$	0	0
$689$	61.4853	2.34240
$690$	0	0
$691$	32.5967	1.24004	0.620019	−	0.784587i	$-0.287125\pi$
$691$	32.5967	1.24004	0.620019	+	0.784587i	$0.287125\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.81966	0.144888
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.5410	1.00244	0.501220	−	0.865320i	$-0.332885\pi$
$701$	26.5410	1.00244	0.501220	+	0.865320i	$0.332885\pi$
$702$	0	0
$703$	37.8885	1.42899
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.76393	−0.0663395
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	28.5836	1.07046
$714$	0	0
$715$	−4.18034	−0.156336
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−50.2492	−1.87398	−0.936990	−	0.349356i	$-0.886400\pi$
$719$	−50.2492	−1.87398	−0.936990	+	0.349356i	$0.886400\pi$
$720$	0	0
$721$	−1.16718	−0.0434682
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.47214	0.203230
$726$	0	0
$727$	14.7639	0.547564	0.273782	−	0.961792i	$-0.411725\pi$
$727$	14.7639	0.547564	0.273782	+	0.961792i	$0.411725\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	18.4721	0.682284	0.341142	−	0.940012i	$-0.389186\pi$
$733$	18.4721	0.682284	0.341142	+	0.940012i	$0.389186\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.11146	−0.114612
$738$	0	0
$739$	−6.18034	−0.227347	−0.113674	−	0.993518i	$-0.536262\pi$
$739$	−6.18034	−0.227347	−0.113674	+	0.993518i	$0.536262\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.3475	−0.379614	−0.189807	−	0.981821i	$-0.560786\pi$
$743$	−10.3475	−0.379614	−0.189807	+	0.981821i	$0.560786\pi$
$744$	0	0
$745$	−75.7771	−2.77626
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.29180	−0.0837404
$750$	0	0
$751$	43.7771	1.59745	0.798724	−	0.601697i	$-0.205509\pi$
$751$	43.7771	1.59745	0.798724	+	0.601697i	$0.205509\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.88854	0.359881
$756$	0	0
$757$	−20.9443	−0.761233	−0.380616	−	0.924733i	$-0.624288\pi$
$757$	−20.9443	−0.761233	−0.380616	+	0.924733i	$0.624288\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.1803	−1.02154	−0.510768	−	0.859718i	$-0.670639\pi$
$761$	−28.1803	−1.02154	−0.510768	+	0.859718i	$0.670639\pi$
$762$	0	0
$763$	−2.23607	−0.0809511
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.4721	−0.883638
$768$	0	0
$769$	17.2361	0.621549	0.310774	−	0.950484i	$-0.399412\pi$
$769$	17.2361	0.621549	0.310774	+	0.950484i	$0.399412\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.4721	−1.02407	−0.512036	−	0.858964i	$-0.671108\pi$
$773$	−28.4721	−1.02407	−0.512036	+	0.858964i	$0.671108\pi$
$774$	0	0
$775$	−20.2918	−0.728903
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.4721	−0.518518
$780$	0	0
$781$	1.23607	0.0442300
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.47214	0.231000
$786$	0	0
$787$	46.4721	1.65655	0.828276	−	0.560320i	$-0.189322\pi$
$787$	46.4721	1.65655	0.828276	+	0.560320i	$0.189322\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.48529	0.159479
$792$	0	0
$793$	28.6525	1.01748
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.05573	−0.320770	−0.160385	−	0.987055i	$-0.551274\pi$
$797$	−9.05573	−0.320770	−0.160385	+	0.987055i	$0.551274\pi$
$798$	0	0
$799$	−14.1246	−0.499693
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.59675	−0.0563480
$804$	0	0
$805$	−5.88854	−0.207544
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.3050	−1.27641	−0.638207	−	0.769865i	$-0.720324\pi$
$809$	−36.3050	−1.27641	−0.638207	+	0.769865i	$0.720324\pi$
$810$	0	0
$811$	23.6525	0.830551	0.415275	−	0.909696i	$-0.363685\pi$
$811$	23.6525	0.830551	0.415275	+	0.909696i	$0.363685\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.4164	−0.680127
$816$	0	0
$817$	−28.9443	−1.01263
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.4164	−0.887039	−0.443519	−	0.896265i	$-0.646270\pi$
$821$	−25.4164	−0.887039	−0.443519	+	0.896265i	$0.646270\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.16718	0.0405870	0.0202935	−	0.999794i	$-0.493540\pi$
$827$	1.16718	0.0405870	0.0202935	+	0.999794i	$0.493540\pi$
$828$	0	0
$829$	30.2492	1.05060	0.525299	−	0.850917i	$-0.323953\pi$
$829$	30.2492	1.05060	0.525299	+	0.850917i	$0.323953\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20.8328	0.721814
$834$	0	0
$835$	20.9443	0.724806
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−35.6525	−1.23086	−0.615430	−	0.788191i	$-0.711018\pi$
$839$	−35.6525	−1.23086	−0.615430	+	0.788191i	$0.711018\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−54.8328	−1.88631
$846$	0	0
$847$	2.58359	0.0887733
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−40.3607	−1.38355
$852$	0	0
$853$	7.41641	0.253933	0.126966	−	0.991907i	$-0.459476\pi$
$853$	7.41641	0.253933	0.126966	+	0.991907i	$0.459476\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.5967	1.07932	0.539662	−	0.841882i	$-0.318552\pi$
$857$	31.5967	1.07932	0.539662	+	0.841882i	$0.318552\pi$
$858$	0	0
$859$	36.8328	1.25672	0.628360	−	0.777923i	$-0.283727\pi$
$859$	36.8328	1.25672	0.628360	+	0.777923i	$0.283727\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.0132	1.25994	0.629971	−	0.776618i	$-0.283067\pi$
$863$	37.0132	1.25994	0.629971	+	0.776618i	$0.283067\pi$
$864$	0	0
$865$	−22.8328	−0.776339
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.01316	−0.102214
$870$	0	0
$871$	−72.1246	−2.44385
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.360680	0.0121932
$876$	0	0
$877$	21.4164	0.723181	0.361590	−	0.932337i	$-0.382234\pi$
$877$	21.4164	0.723181	0.361590	+	0.932337i	$0.382234\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−22.5279	−0.758983	−0.379492	−	0.925195i	$-0.623901\pi$
$881$	−22.5279	−0.758983	−0.379492	+	0.925195i	$0.623901\pi$
$882$	0	0
$883$	−27.4164	−0.922636	−0.461318	−	0.887235i	$-0.652623\pi$
$883$	−27.4164	−0.922636	−0.461318	+	0.887235i	$0.652623\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.8197	0.564749	0.282374	−	0.959304i	$-0.408878\pi$
$887$	16.8197	0.564749	0.282374	+	0.959304i	$0.408878\pi$
$888$	0	0
$889$	2.94427	0.0987477
$890$	0	0
$891$	0	0
$892$	0	0
$893$	34.0689	1.14007
$894$	0	0
$895$	−55.7771	−1.86442
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.70820	−0.123676
$900$	0	0
$901$	33.7082	1.12298
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.70820	0.322712
$906$	0	0
$907$	17.7771	0.590279	0.295139	−	0.955454i	$-0.404634\pi$
$907$	17.7771	0.590279	0.295139	+	0.955454i	$0.404634\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.8197	0.358471	0.179236	−	0.983806i	$-0.442638\pi$
$911$	10.8197	0.358471	0.179236	+	0.983806i	$0.442638\pi$
$912$	0	0
$913$	−0.695048	−0.0230027
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.05573	−0.0678861
$918$	0	0
$919$	−58.0132	−1.91368	−0.956839	−	0.290619i	$-0.906139\pi$
$919$	−58.0132	−1.91368	−0.956839	+	0.290619i	$0.906139\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.6525	0.943108
$924$	0	0
$925$	28.6525	0.942088
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.5279	1.16563	0.582816	−	0.812604i	$-0.301951\pi$
$929$	35.5279	1.16563	0.582816	+	0.812604i	$0.301951\pi$
$930$	0	0
$931$	−50.2492	−1.64685
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.29180	−0.0749497
$936$	0	0
$937$	35.3607	1.15518	0.577592	−	0.816326i	$-0.303993\pi$
$937$	35.3607	1.15518	0.577592	+	0.816326i	$0.303993\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.1115	0.329624	0.164812	−	0.986325i	$-0.447298\pi$
$941$	10.1115	0.329624	0.164812	+	0.986325i	$0.447298\pi$
$942$	0	0
$943$	15.4164	0.502027
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.12461	−0.0690406	−0.0345203	−	0.999404i	$-0.510990\pi$
$947$	−2.12461	−0.0690406	−0.0345203	+	0.999404i	$0.510990\pi$
$948$	0	0
$949$	−37.0132	−1.20150
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−41.8885	−1.35690	−0.678452	−	0.734645i	$-0.737349\pi$
$953$	−41.8885	−1.35690	−0.678452	+	0.734645i	$0.737349\pi$
$954$	0	0
$955$	−38.8328	−1.25660
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.832816	0.0268930
$960$	0	0
$961$	−17.2492	−0.556427
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27.4164	−0.882565
$966$	0	0
$967$	−0.763932	−0.0245664	−0.0122832	−	0.999925i	$-0.503910\pi$
$967$	−0.763932	−0.0245664	−0.0122832	+	0.999925i	$0.503910\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.360680	−0.0115748	−0.00578738	−	0.999983i	$-0.501842\pi$
$971$	−0.360680	−0.0115748	−0.00578738	+	0.999983i	$0.501842\pi$
$972$	0	0
$973$	0.278640	0.00893280
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.7082	−1.27038	−0.635189	−	0.772357i	$-0.719078\pi$
$977$	−39.7082	−1.27038	−0.635189	+	0.772357i	$0.719078\pi$
$978$	0	0
$979$	1.18034	0.0377238
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.94427	−0.157698	−0.0788489	−	0.996887i	$-0.525124\pi$
$983$	−4.94427	−0.157698	−0.0788489	+	0.996887i	$0.525124\pi$
$984$	0	0
$985$	−62.2492	−1.98343
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30.8328	0.980427
$990$	0	0
$991$	26.0132	0.826335	0.413168	−	0.910655i	$-0.364422\pi$
$991$	26.0132	0.826335	0.413168	+	0.910655i	$0.364422\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−50.6525	−1.60579
$996$	0	0
$997$	−60.1378	−1.90458	−0.952291	−	0.305191i	$-0.901280\pi$
$997$	−60.1378	−1.90458	−0.952291	+	0.305191i	$0.901280\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4176.2.a.bn.1.1		2
3.2	odd	2		1392.2.a.q.1.2		2
4.3	odd	2		261.2.a.b.1.2		2
12.11	even	2		87.2.a.a.1.1	✓	2
20.19	odd	2		6525.2.a.ba.1.1		2
24.5	odd	2		5568.2.a.bs.1.1		2
24.11	even	2		5568.2.a.bl.1.1		2
60.23	odd	4		2175.2.c.k.349.3		4
60.47	odd	4		2175.2.c.k.349.2		4
60.59	even	2		2175.2.a.l.1.2		2
84.83	odd	2		4263.2.a.j.1.1		2
116.115	odd	2		7569.2.a.k.1.1		2
348.347	even	2		2523.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.a.a.1.1	✓	2	12.11	even	2
261.2.a.b.1.2		2	4.3	odd	2
1392.2.a.q.1.2		2	3.2	odd	2
2175.2.a.l.1.2		2	60.59	even	2
2175.2.c.k.349.2		4	60.47	odd	4
2175.2.c.k.349.3		4	60.23	odd	4
2523.2.a.c.1.2		2	348.347	even	2
4176.2.a.bn.1.1		2	1.1	even	1	trivial
4263.2.a.j.1.1		2	84.83	odd	2
5568.2.a.bl.1.1		2	24.11	even	2
5568.2.a.bs.1.1		2	24.5	odd	2
6525.2.a.ba.1.1		2	20.19	odd	2
7569.2.a.k.1.1		2	116.115	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4176.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{5} -0.236068 q^{7} +O(q^{10})\)	\(q-3.23607 q^{5} -0.236068 q^{7} -0.236068 q^{11} -5.47214 q^{13} -3.00000 q^{17} +7.23607 q^{19} -7.70820 q^{23} +5.47214 q^{25} +1.00000 q^{29} -3.70820 q^{31} +0.763932 q^{35} +5.23607 q^{37} -2.00000 q^{41} -4.00000 q^{43} +4.70820 q^{47} -6.94427 q^{49} -11.2361 q^{53} +0.763932 q^{55} +4.47214 q^{59} -5.23607 q^{61} +17.7082 q^{65} +13.1803 q^{67} -5.23607 q^{71} +6.76393 q^{73} +0.0557281 q^{77} +12.7639 q^{79} +2.94427 q^{83} +9.70820 q^{85} -5.00000 q^{89} +1.29180 q^{91} -23.4164 q^{95} +18.6525 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^7	\( 2 q - 2 q^{5} + 4 q^{7} + 4 q^{11} - 2 q^{13} - 6 q^{17} + 10 q^{19} - 2 q^{23} + 2 q^{25} + 2 q^{29} + 6 q^{31} + 6 q^{35} + 6 q^{37} - 4 q^{41} - 8 q^{43} - 4 q^{47} + 4 q^{49} - 18 q^{53} + 6 q^{55} - 6 q^{61} + 22 q^{65} + 4 q^{67} - 6 q^{71} + 18 q^{73} + 18 q^{77} + 30 q^{79} - 12 q^{83} + 6 q^{85} - 10 q^{89} + 16 q^{91} - 20 q^{95} + 6 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^7 + 4 * q^11 - 2 * q^13 - 6 * q^17 + 10 * q^19 - 2 * q^23 + 2 * q^25 + 2 * q^29 + 6 * q^31 + 6 * q^35 + 6 * q^37 - 4 * q^41 - 8 * q^43 - 4 * q^47 + 4 * q^49 - 18 * q^53 + 6 * q^55 - 6 * q^61 + 22 * q^65 + 4 * q^67 - 6 * q^71 + 18 * q^73 + 18 * q^77 + 30 * q^79 - 12 * q^83 + 6 * q^85 - 10 * q^89 + 16 * q^91 - 20 * q^95 + 6 * q^97

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