LMFDB - Embedded newform 2752.2.a.ba.1.3

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:	5.5.792644.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 7x^3 + 4x^2 + 4*x - 2
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			1.3
Root			$0.466307$ 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)

$f(q)$	$=$	$q+0.466307 q^{3} +3.41528 q^{5} -0.908821 q^{7} -2.78256 q^{9} +O(q^{10})$	$q+0.466307 q^{3} +3.41528 q^{5} -0.908821 q^{7} -2.78256 q^{9} +6.23799 q^{11} +0.340044 q^{13} +1.59257 q^{15} +0.321320 q^{17} -2.42615 q^{19} -0.423790 q^{21} +5.18999 q^{23} +6.66415 q^{25} -2.69645 q^{27} -3.35133 q^{29} +9.57663 q^{31} +2.90882 q^{33} -3.10388 q^{35} +0.442513 q^{37} +0.158565 q^{39} -7.55931 q^{41} -1.00000 q^{43} -9.50322 q^{45} +5.84650 q^{47} -6.17405 q^{49} +0.149834 q^{51} +3.94897 q^{53} +21.3045 q^{55} -1.13133 q^{57} +9.85413 q^{59} +9.48686 q^{61} +2.52885 q^{63} +1.16135 q^{65} -6.66274 q^{67} +2.42013 q^{69} +3.91969 q^{71} +2.48408 q^{73} +3.10754 q^{75} -5.66922 q^{77} +6.50551 q^{79} +7.09030 q^{81} -10.5270 q^{83} +1.09740 q^{85} -1.56275 q^{87} -13.7721 q^{89} -0.309039 q^{91} +4.46565 q^{93} -8.28600 q^{95} +8.91572 q^{97} -17.3576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + q^{5} + 8 q^{7}+O(q^{10}) $ 5 * q + q^3 + q^5 + 8 * q^7	$ 5 q + q^{3} + q^{5} + 8 q^{7} - q^{11} - q^{13} + 8 q^{15} - 2 q^{17} - 13 q^{19} + 10 q^{21} + 12 q^{23} + 2 q^{25} + 4 q^{27} + 7 q^{29} + 22 q^{31} + 2 q^{33} + 2 q^{35} - 9 q^{37} + 16 q^{39} - 2 q^{41} - 5 q^{43} + 12 q^{45} + 23 q^{47} + 9 q^{49} - 11 q^{51} + 5 q^{53} + 34 q^{55} - 16 q^{57} + 4 q^{59} + 28 q^{63} - 4 q^{65} - 11 q^{67} - 18 q^{69} + 24 q^{71} - 10 q^{73} + 16 q^{75} - 6 q^{77} + 7 q^{79} + 5 q^{81} - 3 q^{83} - 12 q^{85} + 33 q^{87} + 10 q^{91} + q^{93} + 9 q^{95} - 6 q^{97} - 5 q^{99}+O(q^{100}) $ 5 * q + q^3 + q^5 + 8 * q^7 - q^11 - q^13 + 8 * q^15 - 2 * q^17 - 13 * q^19 + 10 * q^21 + 12 * q^23 + 2 * q^25 + 4 * q^27 + 7 * q^29 + 22 * q^31 + 2 * q^33 + 2 * q^35 - 9 * q^37 + 16 * q^39 - 2 * q^41 - 5 * q^43 + 12 * q^45 + 23 * q^47 + 9 * q^49 - 11 * q^51 + 5 * q^53 + 34 * q^55 - 16 * q^57 + 4 * q^59 + 28 * q^63 - 4 * q^65 - 11 * q^67 - 18 * q^69 + 24 * q^71 - 10 * q^73 + 16 * q^75 - 6 * q^77 + 7 * q^79 + 5 * q^81 - 3 * q^83 - 12 * q^85 + 33 * q^87 + 10 * q^91 + q^93 + 9 * q^95 - 6 * 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
$3$	0.466307	0.269223	0.134611	−	0.990898i	$-0.457021\pi$
$3$	0.466307	0.269223	0.134611	+	0.990898i	$0.457021\pi$
$4$	0	0
$5$	3.41528	1.52736	0.763680	−	0.645595i	$-0.223390\pi$
$5$	3.41528	1.52736	0.763680	+	0.645595i	$0.223390\pi$
$6$	0	0
$7$	−0.908821	−0.343502	−0.171751	−	0.985140i	$-0.554942\pi$
$7$	−0.908821	−0.343502	−0.171751	+	0.985140i	$0.554942\pi$
$8$	0	0
$9$	−2.78256	−0.927519
$10$	0	0
$11$	6.23799	1.88083	0.940413	−	0.340035i	$-0.110439\pi$
$11$	6.23799	1.88083	0.940413	+	0.340035i	$0.110439\pi$
$12$	0	0
$13$	0.340044	0.0943113	0.0471556	−	0.998888i	$-0.484984\pi$
$13$	0.340044	0.0943113	0.0471556	+	0.998888i	$0.484984\pi$
$14$	0	0
$15$	1.59257	0.411200
$16$	0	0
$17$	0.321320	0.0779316	0.0389658	−	0.999241i	$-0.487594\pi$
$17$	0.321320	0.0779316	0.0389658	+	0.999241i	$0.487594\pi$
$18$	0	0
$19$	−2.42615	−0.556598	−0.278299	−	0.960495i	$-0.589771\pi$
$19$	−2.42615	−0.556598	−0.278299	+	0.960495i	$0.589771\pi$
$20$	0	0
$21$	−0.423790	−0.0924785
$22$	0	0
$23$	5.18999	1.08219	0.541094	−	0.840962i	$-0.318010\pi$
$23$	5.18999	1.08219	0.541094	+	0.840962i	$0.318010\pi$
$24$	0	0
$25$	6.66415	1.33283
$26$	0	0
$27$	−2.69645	−0.518932
$28$	0	0
$29$	−3.35133	−0.622327	−0.311164	−	0.950356i	$-0.600719\pi$
$29$	−3.35133	−0.622327	−0.311164	+	0.950356i	$0.600719\pi$
$30$	0	0
$31$	9.57663	1.72001	0.860007	−	0.510283i	$-0.170459\pi$
$31$	9.57663	1.72001	0.860007	+	0.510283i	$0.170459\pi$
$32$	0	0
$33$	2.90882	0.506361
$34$	0	0
$35$	−3.10388	−0.524651
$36$	0	0
$37$	0.442513	0.0727488	0.0363744	−	0.999338i	$-0.488419\pi$
$37$	0.442513	0.0727488	0.0363744	+	0.999338i	$0.488419\pi$
$38$	0	0
$39$	0.158565	0.0253907
$40$	0	0
$41$	−7.55931	−1.18057	−0.590283	−	0.807196i	$-0.700984\pi$
$41$	−7.55931	−1.18057	−0.590283	+	0.807196i	$0.700984\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−9.50322	−1.41666
$46$	0	0
$47$	5.84650	0.852800	0.426400	−	0.904535i	$-0.359782\pi$
$47$	5.84650	0.852800	0.426400	+	0.904535i	$0.359782\pi$
$48$	0	0
$49$	−6.17405	−0.882006
$50$	0	0
$51$	0.149834	0.0209810
$52$	0	0
$53$	3.94897	0.542433	0.271217	−	0.962518i	$-0.412574\pi$
$53$	3.94897	0.542433	0.271217	+	0.962518i	$0.412574\pi$
$54$	0	0
$55$	21.3045	2.87270
$56$	0	0
$57$	−1.13133	−0.149849
$58$	0	0
$59$	9.85413	1.28290	0.641449	−	0.767165i	$-0.278333\pi$
$59$	9.85413	1.28290	0.641449	+	0.767165i	$0.278333\pi$
$60$	0	0
$61$	9.48686	1.21467	0.607334	−	0.794447i	$-0.292239\pi$
$61$	9.48686	1.21467	0.607334	+	0.794447i	$0.292239\pi$
$62$	0	0
$63$	2.52885	0.318605
$64$	0	0
$65$	1.16135	0.144047
$66$	0	0
$67$	−6.66274	−0.813983	−0.406991	−	0.913432i	$-0.633422\pi$
$67$	−6.66274	−0.813983	−0.406991	+	0.913432i	$0.633422\pi$
$68$	0	0
$69$	2.42013	0.291349
$70$	0	0
$71$	3.91969	0.465182	0.232591	−	0.972575i	$-0.425280\pi$
$71$	3.91969	0.465182	0.232591	+	0.972575i	$0.425280\pi$
$72$	0	0
$73$	2.48408	0.290739	0.145370	−	0.989377i	$-0.453563\pi$
$73$	2.48408	0.290739	0.145370	+	0.989377i	$0.453563\pi$
$74$	0	0
$75$	3.10754	0.358828
$76$	0	0
$77$	−5.66922	−0.646067
$78$	0	0
$79$	6.50551	0.731927	0.365963	−	0.930629i	$-0.380740\pi$
$79$	6.50551	0.731927	0.365963	+	0.930629i	$0.380740\pi$
$80$	0	0
$81$	7.09030	0.787811
$82$	0	0
$83$	−10.5270	−1.15549	−0.577745	−	0.816217i	$-0.696067\pi$
$83$	−10.5270	−1.15549	−0.577745	+	0.816217i	$0.696067\pi$
$84$	0	0
$85$	1.09740	0.119030
$86$	0	0
$87$	−1.56275	−0.167545
$88$	0	0
$89$	−13.7721	−1.45984	−0.729920	−	0.683533i	$-0.760443\pi$
$89$	−13.7721	−1.45984	−0.729920	+	0.683533i	$0.760443\pi$
$90$	0	0
$91$	−0.309039	−0.0323961
$92$	0	0
$93$	4.46565	0.463066
$94$	0	0
$95$	−8.28600	−0.850125
$96$	0	0
$97$	8.91572	0.905254	0.452627	−	0.891700i	$-0.350487\pi$
$97$	8.91572	0.905254	0.452627	+	0.891700i	$0.350487\pi$
$98$	0	0
$99$	−17.3576	−1.74450
$100$	0	0
$101$	−6.83873	−0.680479	−0.340239	−	0.940339i	$-0.610508\pi$
$101$	−6.83873	−0.680479	−0.340239	+	0.940339i	$0.610508\pi$
$102$	0	0
$103$	−5.49471	−0.541410	−0.270705	−	0.962662i	$-0.587257\pi$
$103$	−5.49471	−0.541410	−0.270705	+	0.962662i	$0.587257\pi$
$104$	0	0
$105$	−1.44736	−0.141248
$106$	0	0
$107$	14.3592	1.38815	0.694077	−	0.719900i	$-0.255813\pi$
$107$	14.3592	1.38815	0.694077	+	0.719900i	$0.255813\pi$
$108$	0	0
$109$	−4.20527	−0.402792	−0.201396	−	0.979510i	$-0.564548\pi$
$109$	−4.20527	−0.402792	−0.201396	+	0.979510i	$0.564548\pi$
$110$	0	0
$111$	0.206347	0.0195856
$112$	0	0
$113$	−6.16490	−0.579945	−0.289972	−	0.957035i	$-0.593646\pi$
$113$	−6.16490	−0.579945	−0.289972	+	0.957035i	$0.593646\pi$
$114$	0	0
$115$	17.7253	1.65289
$116$	0	0
$117$	−0.946193	−0.0874755
$118$	0	0
$119$	−0.292023	−0.0267697
$120$	0	0
$121$	27.9125	2.53750
$122$	0	0
$123$	−3.52496	−0.317835
$124$	0	0
$125$	5.68353	0.508350
$126$	0	0
$127$	4.86286	0.431509	0.215755	−	0.976448i	$-0.430779\pi$
$127$	4.86286	0.431509	0.215755	+	0.976448i	$0.430779\pi$
$128$	0	0
$129$	−0.466307	−0.0410561
$130$	0	0
$131$	−4.45822	−0.389516	−0.194758	−	0.980851i	$-0.562392\pi$
$131$	−4.45822	−0.389516	−0.194758	+	0.980851i	$0.562392\pi$
$132$	0	0
$133$	2.20494	0.191192
$134$	0	0
$135$	−9.20913	−0.792596
$136$	0	0
$137$	19.2738	1.64667	0.823337	−	0.567553i	$-0.192110\pi$
$137$	19.2738	1.64667	0.823337	+	0.567553i	$0.192110\pi$
$138$	0	0
$139$	−6.29623	−0.534039	−0.267020	−	0.963691i	$-0.586039\pi$
$139$	−6.29623	−0.534039	−0.267020	+	0.963691i	$0.586039\pi$
$140$	0	0
$141$	2.72627	0.229593
$142$	0	0
$143$	2.12119	0.177383
$144$	0	0
$145$	−11.4457	−0.950518
$146$	0	0
$147$	−2.87900	−0.237456
$148$	0	0
$149$	−11.2888	−0.924813	−0.462406	−	0.886668i	$-0.653014\pi$
$149$	−11.2888	−0.924813	−0.462406	+	0.886668i	$0.653014\pi$
$150$	0	0
$151$	12.0137	0.977657	0.488829	−	0.872380i	$-0.337424\pi$
$151$	12.0137	0.977657	0.488829	+	0.872380i	$0.337424\pi$
$152$	0	0
$153$	−0.894092	−0.0722831
$154$	0	0
$155$	32.7069	2.62708
$156$	0	0
$157$	15.3624	1.22605	0.613027	−	0.790062i	$-0.289952\pi$
$157$	15.3624	1.22605	0.613027	+	0.790062i	$0.289952\pi$
$158$	0	0
$159$	1.84143	0.146035
$160$	0	0
$161$	−4.71677	−0.371733
$162$	0	0
$163$	−15.7735	−1.23548	−0.617738	−	0.786384i	$-0.711951\pi$
$163$	−15.7735	−1.23548	−0.617738	+	0.786384i	$0.711951\pi$
$164$	0	0
$165$	9.93444	0.773395
$166$	0	0
$167$	−21.4850	−1.66256	−0.831280	−	0.555853i	$-0.812392\pi$
$167$	−21.4850	−1.66256	−0.831280	+	0.555853i	$0.812392\pi$
$168$	0	0
$169$	−12.8844	−0.991105
$170$	0	0
$171$	6.75091	0.516255
$172$	0	0
$173$	−1.48481	−0.112888	−0.0564439	−	0.998406i	$-0.517976\pi$
$173$	−1.48481	−0.112888	−0.0564439	+	0.998406i	$0.517976\pi$
$174$	0	0
$175$	−6.05651	−0.457829
$176$	0	0
$177$	4.59505	0.345385
$178$	0	0
$179$	−4.00492	−0.299342	−0.149671	−	0.988736i	$-0.547821\pi$
$179$	−4.00492	−0.299342	−0.149671	+	0.988736i	$0.547821\pi$
$180$	0	0
$181$	3.20593	0.238295	0.119147	−	0.992877i	$-0.461984\pi$
$181$	3.20593	0.238295	0.119147	+	0.992877i	$0.461984\pi$
$182$	0	0
$183$	4.42379	0.327016
$184$	0	0
$185$	1.51131	0.111114
$186$	0	0
$187$	2.00439	0.146576
$188$	0	0
$189$	2.45059	0.178254
$190$	0	0
$191$	−21.1787	−1.53243	−0.766217	−	0.642582i	$-0.777863\pi$
$191$	−21.1787	−1.53243	−0.766217	+	0.642582i	$0.777863\pi$
$192$	0	0
$193$	−21.3964	−1.54015	−0.770074	−	0.637954i	$-0.779781\pi$
$193$	−21.3964	−1.54015	−0.770074	+	0.637954i	$0.779781\pi$
$194$	0	0
$195$	0.541544	0.0387808
$196$	0	0
$197$	−18.7221	−1.33390	−0.666949	−	0.745104i	$-0.732400\pi$
$197$	−18.7221	−1.33390	−0.666949	+	0.745104i	$0.732400\pi$
$198$	0	0
$199$	21.8919	1.55188	0.775938	−	0.630809i	$-0.217277\pi$
$199$	21.8919	1.55188	0.775938	+	0.630809i	$0.217277\pi$
$200$	0	0
$201$	−3.10688	−0.219143
$202$	0	0
$203$	3.04576	0.213771
$204$	0	0
$205$	−25.8172	−1.80315
$206$	0	0
$207$	−14.4414	−1.00375
$208$	0	0
$209$	−15.1343	−1.04686
$210$	0	0
$211$	23.2339	1.59949	0.799743	−	0.600343i	$-0.204969\pi$
$211$	23.2339	1.59949	0.799743	+	0.600343i	$0.204969\pi$
$212$	0	0
$213$	1.82778	0.125237
$214$	0	0
$215$	−3.41528	−0.232920
$216$	0	0
$217$	−8.70344	−0.590828
$218$	0	0
$219$	1.15834	0.0782735
$220$	0	0
$221$	0.109263	0.00734983
$222$	0	0
$223$	27.8281	1.86351	0.931753	−	0.363093i	$-0.118279\pi$
$223$	27.8281	1.86351	0.931753	+	0.363093i	$0.118279\pi$
$224$	0	0
$225$	−18.5434	−1.23622
$226$	0	0
$227$	5.29548	0.351473	0.175737	−	0.984437i	$-0.443769\pi$
$227$	5.29548	0.351473	0.175737	+	0.984437i	$0.443769\pi$
$228$	0	0
$229$	−1.70022	−0.112354	−0.0561769	−	0.998421i	$-0.517891\pi$
$229$	−1.70022	−0.112354	−0.0561769	+	0.998421i	$0.517891\pi$
$230$	0	0
$231$	−2.64360	−0.173936
$232$	0	0
$233$	9.22800	0.604546	0.302273	−	0.953221i	$-0.402255\pi$
$233$	9.22800	0.604546	0.302273	+	0.953221i	$0.402255\pi$
$234$	0	0
$235$	19.9675	1.30253
$236$	0	0
$237$	3.03356	0.197051
$238$	0	0
$239$	13.9414	0.901795	0.450897	−	0.892576i	$-0.351104\pi$
$239$	13.9414	0.901795	0.450897	+	0.892576i	$0.351104\pi$
$240$	0	0
$241$	19.2590	1.24058	0.620289	−	0.784373i	$-0.287015\pi$
$241$	19.2590	1.24058	0.620289	+	0.784373i	$0.287015\pi$
$242$	0	0
$243$	11.3956	0.731028
$244$	0	0
$245$	−21.0861	−1.34714
$246$	0	0
$247$	−0.824999	−0.0524935
$248$	0	0
$249$	−4.90882	−0.311084
$250$	0	0
$251$	−19.5445	−1.23364	−0.616820	−	0.787104i	$-0.711579\pi$
$251$	−19.5445	−1.23364	−0.616820	+	0.787104i	$0.711579\pi$
$252$	0	0
$253$	32.3751	2.03541
$254$	0	0
$255$	0.511725	0.0320455
$256$	0	0
$257$	13.8640	0.864813	0.432407	−	0.901679i	$-0.357664\pi$
$257$	13.8640	0.864813	0.432407	+	0.901679i	$0.357664\pi$
$258$	0	0
$259$	−0.402165	−0.0249893
$260$	0	0
$261$	9.32528	0.577220
$262$	0	0
$263$	5.87064	0.361999	0.181000	−	0.983483i	$-0.442067\pi$
$263$	5.87064	0.361999	0.181000	+	0.983483i	$0.442067\pi$
$264$	0	0
$265$	13.4869	0.828491
$266$	0	0
$267$	−6.42203	−0.393022
$268$	0	0
$269$	3.34970	0.204235	0.102117	−	0.994772i	$-0.467438\pi$
$269$	3.34970	0.204235	0.102117	+	0.994772i	$0.467438\pi$
$270$	0	0
$271$	0.506795	0.0307856	0.0153928	−	0.999882i	$-0.495100\pi$
$271$	0.506795	0.0307856	0.0153928	+	0.999882i	$0.495100\pi$
$272$	0	0
$273$	−0.144107	−0.00872176
$274$	0	0
$275$	41.5709	2.50682
$276$	0	0
$277$	−31.6872	−1.90390	−0.951949	−	0.306258i	$-0.900923\pi$
$277$	−31.6872	−1.90390	−0.951949	+	0.306258i	$0.900923\pi$
$278$	0	0
$279$	−26.6475	−1.59535
$280$	0	0
$281$	−12.2674	−0.731810	−0.365905	−	0.930652i	$-0.619240\pi$
$281$	−12.2674	−0.731810	−0.365905	+	0.930652i	$0.619240\pi$
$282$	0	0
$283$	−3.97628	−0.236365	−0.118183	−	0.992992i	$-0.537707\pi$
$283$	−3.97628	−0.236365	−0.118183	+	0.992992i	$0.537707\pi$
$284$	0	0
$285$	−3.86382	−0.228873
$286$	0	0
$287$	6.87006	0.405527
$288$	0	0
$289$	−16.8968	−0.993927
$290$	0	0
$291$	4.15746	0.243715
$292$	0	0
$293$	−15.6602	−0.914877	−0.457438	−	0.889241i	$-0.651233\pi$
$293$	−15.6602	−0.914877	−0.457438	+	0.889241i	$0.651233\pi$
$294$	0	0
$295$	33.6546	1.95945
$296$	0	0
$297$	−16.8204	−0.976020
$298$	0	0
$299$	1.76483	0.102062
$300$	0	0
$301$	0.908821	0.0523835
$302$	0	0
$303$	−3.18895	−0.183200
$304$	0	0
$305$	32.4003	1.85523
$306$	0	0
$307$	−21.7993	−1.24415	−0.622077	−	0.782956i	$-0.713711\pi$
$307$	−21.7993	−1.24415	−0.622077	+	0.782956i	$0.713711\pi$
$308$	0	0
$309$	−2.56222	−0.145760
$310$	0	0
$311$	19.5616	1.10923	0.554617	−	0.832106i	$-0.312865\pi$
$311$	19.5616	1.10923	0.554617	+	0.832106i	$0.312865\pi$
$312$	0	0
$313$	5.70666	0.322560	0.161280	−	0.986909i	$-0.448438\pi$
$313$	5.70666	0.322560	0.161280	+	0.986909i	$0.448438\pi$
$314$	0	0
$315$	8.63672	0.486624
$316$	0	0
$317$	22.0067	1.23602	0.618011	−	0.786170i	$-0.287939\pi$
$317$	22.0067	1.23602	0.618011	+	0.786170i	$0.287939\pi$
$318$	0	0
$319$	−20.9056	−1.17049
$320$	0	0
$321$	6.69579	0.373723
$322$	0	0
$323$	−0.779573	−0.0433766
$324$	0	0
$325$	2.26610	0.125701
$326$	0	0
$327$	−1.96095	−0.108441
$328$	0	0
$329$	−5.31342	−0.292939
$330$	0	0
$331$	22.2201	1.22133	0.610663	−	0.791890i	$-0.290903\pi$
$331$	22.2201	1.22133	0.610663	+	0.791890i	$0.290903\pi$
$332$	0	0
$333$	−1.23132	−0.0674759
$334$	0	0
$335$	−22.7551	−1.24325
$336$	0	0
$337$	−12.8887	−0.702093	−0.351046	−	0.936358i	$-0.614174\pi$
$337$	−12.8887	−0.702093	−0.351046	+	0.936358i	$0.614174\pi$
$338$	0	0
$339$	−2.87474	−0.156134
$340$	0	0
$341$	59.7389	3.23504
$342$	0	0
$343$	11.9728	0.646473
$344$	0	0
$345$	8.26542	0.444995
$346$	0	0
$347$	18.5154	0.993958	0.496979	−	0.867763i	$-0.334443\pi$
$347$	18.5154	0.993958	0.496979	+	0.867763i	$0.334443\pi$
$348$	0	0
$349$	−25.6049	−1.37060	−0.685300	−	0.728261i	$-0.740329\pi$
$349$	−25.6049	−1.37060	−0.685300	+	0.728261i	$0.740329\pi$
$350$	0	0
$351$	−0.916912	−0.0489411
$352$	0	0
$353$	25.4323	1.35363	0.676814	−	0.736154i	$-0.263360\pi$
$353$	25.4323	1.35363	0.676814	+	0.736154i	$0.263360\pi$
$354$	0	0
$355$	13.3869	0.710500
$356$	0	0
$357$	−0.136172	−0.00720700
$358$	0	0
$359$	−5.75426	−0.303698	−0.151849	−	0.988404i	$-0.548523\pi$
$359$	−5.75426	−0.303698	−0.151849	+	0.988404i	$0.548523\pi$
$360$	0	0
$361$	−13.1138	−0.690199
$362$	0	0
$363$	13.0158	0.683153
$364$	0	0
$365$	8.48382	0.444063
$366$	0	0
$367$	−33.5548	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$367$	−33.5548	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$368$	0	0
$369$	21.0342	1.09500
$370$	0	0
$371$	−3.58891	−0.186327
$372$	0	0
$373$	7.16480	0.370979	0.185490	−	0.982646i	$-0.440613\pi$
$373$	7.16480	0.370979	0.185490	+	0.982646i	$0.440613\pi$
$374$	0	0
$375$	2.65027	0.136859
$376$	0	0
$377$	−1.13960	−0.0586925
$378$	0	0
$379$	−26.2537	−1.34856	−0.674281	−	0.738475i	$-0.735546\pi$
$379$	−26.2537	−1.34856	−0.674281	+	0.738475i	$0.735546\pi$
$380$	0	0
$381$	2.26759	0.116172
$382$	0	0
$383$	−7.23829	−0.369859	−0.184930	−	0.982752i	$-0.559206\pi$
$383$	−7.23829	−0.369859	−0.184930	+	0.982752i	$0.559206\pi$
$384$	0	0
$385$	−19.3620	−0.986777
$386$	0	0
$387$	2.78256	0.141445
$388$	0	0
$389$	2.37159	0.120245	0.0601223	−	0.998191i	$-0.480851\pi$
$389$	2.37159	0.120245	0.0601223	+	0.998191i	$0.480851\pi$
$390$	0	0
$391$	1.66765	0.0843366
$392$	0	0
$393$	−2.07890	−0.104867
$394$	0	0
$395$	22.2181	1.11792
$396$	0	0
$397$	34.1381	1.71334	0.856670	−	0.515865i	$-0.172529\pi$
$397$	34.1381	1.71334	0.856670	+	0.515865i	$0.172529\pi$
$398$	0	0
$399$	1.02818	0.0514733
$400$	0	0
$401$	−17.0351	−0.850691	−0.425346	−	0.905031i	$-0.639847\pi$
$401$	−17.0351	−0.850691	−0.425346	+	0.905031i	$0.639847\pi$
$402$	0	0
$403$	3.25648	0.162217
$404$	0	0
$405$	24.2154	1.20327
$406$	0	0
$407$	2.76040	0.136828
$408$	0	0
$409$	34.2939	1.69572	0.847862	−	0.530216i	$-0.177889\pi$
$409$	34.2939	1.69572	0.847862	+	0.530216i	$0.177889\pi$
$410$	0	0
$411$	8.98752	0.443322
$412$	0	0
$413$	−8.95564	−0.440678
$414$	0	0
$415$	−35.9527	−1.76485
$416$	0	0
$417$	−2.93598	−0.143775
$418$	0	0
$419$	−31.9357	−1.56016	−0.780080	−	0.625680i	$-0.784822\pi$
$419$	−31.9357	−1.56016	−0.780080	+	0.625680i	$0.784822\pi$
$420$	0	0
$421$	−24.6177	−1.19979	−0.599897	−	0.800077i	$-0.704792\pi$
$421$	−24.6177	−1.19979	−0.599897	+	0.800077i	$0.704792\pi$
$422$	0	0
$423$	−16.2682	−0.790989
$424$	0	0
$425$	2.14133	0.103870
$426$	0	0
$427$	−8.62185	−0.417241
$428$	0	0
$429$	0.989128	0.0477555
$430$	0	0
$431$	−28.6010	−1.37766	−0.688832	−	0.724921i	$-0.741876\pi$
$431$	−28.6010	−1.37766	−0.688832	+	0.724921i	$0.741876\pi$
$432$	0	0
$433$	−16.9347	−0.813828	−0.406914	−	0.913466i	$-0.633395\pi$
$433$	−16.9347	−0.813828	−0.406914	+	0.913466i	$0.633395\pi$
$434$	0	0
$435$	−5.33723	−0.255901
$436$	0	0
$437$	−12.5917	−0.602343
$438$	0	0
$439$	−24.6233	−1.17521	−0.587604	−	0.809149i	$-0.699929\pi$
$439$	−24.6233	−1.17521	−0.587604	+	0.809149i	$0.699929\pi$
$440$	0	0
$441$	17.1796	0.818078
$442$	0	0
$443$	−19.6330	−0.932793	−0.466396	−	0.884576i	$-0.654448\pi$
$443$	−19.6330	−0.932793	−0.466396	+	0.884576i	$0.654448\pi$
$444$	0	0
$445$	−47.0356	−2.22970
$446$	0	0
$447$	−5.26404	−0.248980
$448$	0	0
$449$	25.5510	1.20583	0.602913	−	0.797807i	$-0.294007\pi$
$449$	25.5510	1.20583	0.602913	+	0.797807i	$0.294007\pi$
$450$	0	0
$451$	−47.1549	−2.22044
$452$	0	0
$453$	5.60205	0.263207
$454$	0	0
$455$	−1.05546	−0.0494805
$456$	0	0
$457$	8.09806	0.378811	0.189406	−	0.981899i	$-0.439344\pi$
$457$	8.09806	0.378811	0.189406	+	0.981899i	$0.439344\pi$
$458$	0	0
$459$	−0.866424	−0.0404412
$460$	0	0
$461$	−36.5951	−1.70440	−0.852201	−	0.523214i	$-0.824733\pi$
$461$	−36.5951	−1.70440	−0.852201	+	0.523214i	$0.824733\pi$
$462$	0	0
$463$	−37.4335	−1.73968	−0.869840	−	0.493334i	$-0.835778\pi$
$463$	−37.4335	−1.73968	−0.869840	+	0.493334i	$0.835778\pi$
$464$	0	0
$465$	15.2515	0.707269
$466$	0	0
$467$	36.4666	1.68747	0.843736	−	0.536758i	$-0.180351\pi$
$467$	36.4666	1.68747	0.843736	+	0.536758i	$0.180351\pi$
$468$	0	0
$469$	6.05523	0.279605
$470$	0	0
$471$	7.16361	0.330082
$472$	0	0
$473$	−6.23799	−0.286823
$474$	0	0
$475$	−16.1682	−0.741850
$476$	0	0
$477$	−10.9882	−0.503117
$478$	0	0
$479$	−6.11528	−0.279414	−0.139707	−	0.990193i	$-0.544616\pi$
$479$	−6.11528	−0.279414	−0.139707	+	0.990193i	$0.544616\pi$
$480$	0	0
$481$	0.150474	0.00686103
$482$	0	0
$483$	−2.19946	−0.100079
$484$	0	0
$485$	30.4497	1.38265
$486$	0	0
$487$	−10.6944	−0.484609	−0.242304	−	0.970200i	$-0.577903\pi$
$487$	−10.6944	−0.484609	−0.242304	+	0.970200i	$0.577903\pi$
$488$	0	0
$489$	−7.35530	−0.332618
$490$	0	0
$491$	12.6577	0.571234	0.285617	−	0.958344i	$-0.407801\pi$
$491$	12.6577	0.571234	0.285617	+	0.958344i	$0.407801\pi$
$492$	0	0
$493$	−1.07685	−0.0484990
$494$	0	0
$495$	−59.2810	−2.66448
$496$	0	0
$497$	−3.56230	−0.159791
$498$	0	0
$499$	−11.3901	−0.509891	−0.254945	−	0.966955i	$-0.582057\pi$
$499$	−11.3901	−0.509891	−0.254945	+	0.966955i	$0.582057\pi$
$500$	0	0
$501$	−10.0186	−0.447599
$502$	0	0
$503$	30.2936	1.35073	0.675363	−	0.737485i	$-0.263987\pi$
$503$	30.2936	1.35073	0.675363	+	0.737485i	$0.263987\pi$
$504$	0	0
$505$	−23.3562	−1.03934
$506$	0	0
$507$	−6.00807	−0.266828
$508$	0	0
$509$	−17.1050	−0.758168	−0.379084	−	0.925362i	$-0.623761\pi$
$509$	−17.1050	−0.758168	−0.379084	+	0.925362i	$0.623761\pi$
$510$	0	0
$511$	−2.25758	−0.0998694
$512$	0	0
$513$	6.54200	0.288836
$514$	0	0
$515$	−18.7660	−0.826928
$516$	0	0
$517$	36.4705	1.60397
$518$	0	0
$519$	−0.692377	−0.0303920
$520$	0	0
$521$	−4.08191	−0.178832	−0.0894159	−	0.995994i	$-0.528500\pi$
$521$	−4.08191	−0.178832	−0.0894159	+	0.995994i	$0.528500\pi$
$522$	0	0
$523$	−17.3929	−0.760538	−0.380269	−	0.924876i	$-0.624169\pi$
$523$	−17.3929	−0.760538	−0.380269	+	0.924876i	$0.624169\pi$
$524$	0	0
$525$	−2.82420	−0.123258
$526$	0	0
$527$	3.07717	0.134043
$528$	0	0
$529$	3.93597	0.171129
$530$	0	0
$531$	−27.4197	−1.18991
$532$	0	0
$533$	−2.57050	−0.111341
$534$	0	0
$535$	49.0407	2.12021
$536$	0	0
$537$	−1.86752	−0.0805896
$538$	0	0
$539$	−38.5136	−1.65890
$540$	0	0
$541$	−25.2133	−1.08401	−0.542003	−	0.840376i	$-0.682334\pi$
$541$	−25.2133	−1.08401	−0.542003	+	0.840376i	$0.682334\pi$
$542$	0	0
$543$	1.49495	0.0641544
$544$	0	0
$545$	−14.3622	−0.615209
$546$	0	0
$547$	−6.94293	−0.296858	−0.148429	−	0.988923i	$-0.547422\pi$
$547$	−6.94293	−0.296858	−0.148429	+	0.988923i	$0.547422\pi$
$548$	0	0
$549$	−26.3977	−1.12663
$550$	0	0
$551$	8.13085	0.346386
$552$	0	0
$553$	−5.91234	−0.251418
$554$	0	0
$555$	0.704734	0.0299143
$556$	0	0
$557$	33.6896	1.42747	0.713737	−	0.700414i	$-0.247001\pi$
$557$	33.6896	1.42747	0.713737	+	0.700414i	$0.247001\pi$
$558$	0	0
$559$	−0.340044	−0.0143823
$560$	0	0
$561$	0.934663	0.0394615
$562$	0	0
$563$	−18.7684	−0.790995	−0.395498	−	0.918467i	$-0.629428\pi$
$563$	−18.7684	−0.790995	−0.395498	+	0.918467i	$0.629428\pi$
$564$	0	0
$565$	−21.0549	−0.885784
$566$	0	0
$567$	−6.44381	−0.270615
$568$	0	0
$569$	−35.8226	−1.50176	−0.750880	−	0.660438i	$-0.770371\pi$
$569$	−35.8226	−1.50176	−0.750880	+	0.660438i	$0.770371\pi$
$570$	0	0
$571$	−15.0116	−0.628217	−0.314108	−	0.949387i	$-0.601706\pi$
$571$	−15.0116	−0.628217	−0.314108	+	0.949387i	$0.601706\pi$
$572$	0	0
$573$	−9.87576	−0.412566
$574$	0	0
$575$	34.5868	1.44237
$576$	0	0
$577$	28.6291	1.19185	0.595923	−	0.803041i	$-0.296786\pi$
$577$	28.6291	1.19185	0.595923	+	0.803041i	$0.296786\pi$
$578$	0	0
$579$	−9.97731	−0.414643
$580$	0	0
$581$	9.56716	0.396913
$582$	0	0
$583$	24.6337	1.02022
$584$	0	0
$585$	−3.23151	−0.133607
$586$	0	0
$587$	−38.4779	−1.58815	−0.794076	−	0.607818i	$-0.792045\pi$
$587$	−38.4779	−1.58815	−0.794076	+	0.607818i	$0.792045\pi$
$588$	0	0
$589$	−23.2344	−0.957356
$590$	0	0
$591$	−8.73027	−0.359115
$592$	0	0
$593$	−8.97167	−0.368422	−0.184211	−	0.982887i	$-0.558973\pi$
$593$	−8.97167	−0.368422	−0.184211	+	0.982887i	$0.558973\pi$
$594$	0	0
$595$	−0.997339	−0.0408869
$596$	0	0
$597$	10.2084	0.417800
$598$	0	0
$599$	19.3980	0.792583	0.396291	−	0.918125i	$-0.370297\pi$
$599$	19.3980	0.792583	0.396291	+	0.918125i	$0.370297\pi$
$600$	0	0
$601$	−21.3048	−0.869040	−0.434520	−	0.900662i	$-0.643082\pi$
$601$	−21.3048	−0.869040	−0.434520	+	0.900662i	$0.643082\pi$
$602$	0	0
$603$	18.5395	0.754985
$604$	0	0
$605$	95.3292	3.87568
$606$	0	0
$607$	−13.2895	−0.539405	−0.269703	−	0.962944i	$-0.586925\pi$
$607$	−13.2895	−0.539405	−0.269703	+	0.962944i	$0.586925\pi$
$608$	0	0
$609$	1.42026	0.0575519
$610$	0	0
$611$	1.98807	0.0804287
$612$	0	0
$613$	−3.90623	−0.157771	−0.0788855	−	0.996884i	$-0.525136\pi$
$613$	−3.90623	−0.157771	−0.0788855	+	0.996884i	$0.525136\pi$
$614$	0	0
$615$	−12.0387	−0.485449
$616$	0	0
$617$	−37.9611	−1.52826	−0.764128	−	0.645064i	$-0.776831\pi$
$617$	−37.9611	−1.52826	−0.764128	+	0.645064i	$0.776831\pi$
$618$	0	0
$619$	−29.6338	−1.19108	−0.595542	−	0.803324i	$-0.703063\pi$
$619$	−29.6338	−1.19108	−0.595542	+	0.803324i	$0.703063\pi$
$620$	0	0
$621$	−13.9945	−0.561581
$622$	0	0
$623$	12.5164	0.501458
$624$	0	0
$625$	−13.9099	−0.556396
$626$	0	0
$627$	−7.05725	−0.281839
$628$	0	0
$629$	0.142189	0.00566943
$630$	0	0
$631$	25.9492	1.03302	0.516510	−	0.856281i	$-0.327231\pi$
$631$	25.9492	1.03302	0.516510	+	0.856281i	$0.327231\pi$
$632$	0	0
$633$	10.8341	0.430618
$634$	0	0
$635$	16.6081	0.659070
$636$	0	0
$637$	−2.09945	−0.0831832
$638$	0	0
$639$	−10.9068	−0.431465
$640$	0	0
$641$	−3.37559	−0.133328	−0.0666639	−	0.997775i	$-0.521236\pi$
$641$	−3.37559	−0.133328	−0.0666639	+	0.997775i	$0.521236\pi$
$642$	0	0
$643$	−22.8309	−0.900364	−0.450182	−	0.892937i	$-0.648641\pi$
$643$	−22.8309	−0.900364	−0.450182	+	0.892937i	$0.648641\pi$
$644$	0	0
$645$	−1.59257	−0.0627074
$646$	0	0
$647$	0.192650	0.00757386	0.00378693	−	0.999993i	$-0.498795\pi$
$647$	0.192650	0.00757386	0.00378693	+	0.999993i	$0.498795\pi$
$648$	0	0
$649$	61.4700	2.41291
$650$	0	0
$651$	−4.05847	−0.159064
$652$	0	0
$653$	−19.4072	−0.759462	−0.379731	−	0.925097i	$-0.623983\pi$
$653$	−19.4072	−0.759462	−0.379731	+	0.925097i	$0.623983\pi$
$654$	0	0
$655$	−15.2261	−0.594931
$656$	0	0
$657$	−6.91208	−0.269666
$658$	0	0
$659$	−26.4586	−1.03068	−0.515341	−	0.856985i	$-0.672335\pi$
$659$	−26.4586	−1.03068	−0.515341	+	0.856985i	$0.672335\pi$
$660$	0	0
$661$	11.6258	0.452191	0.226096	−	0.974105i	$-0.427404\pi$
$661$	11.6258	0.452191	0.226096	+	0.974105i	$0.427404\pi$
$662$	0	0
$663$	0.0509502	0.00197874
$664$	0	0
$665$	7.53048	0.292020
$666$	0	0
$667$	−17.3934	−0.673474
$668$	0	0
$669$	12.9764	0.501698
$670$	0	0
$671$	59.1789	2.28458
$672$	0	0
$673$	9.34502	0.360224	0.180112	−	0.983646i	$-0.442354\pi$
$673$	9.34502	0.360224	0.180112	+	0.983646i	$0.442354\pi$
$674$	0	0
$675$	−17.9695	−0.691647
$676$	0	0
$677$	18.2950	0.703135	0.351568	−	0.936162i	$-0.385649\pi$
$677$	18.2950	0.703135	0.351568	+	0.936162i	$0.385649\pi$
$678$	0	0
$679$	−8.10279	−0.310956
$680$	0	0
$681$	2.46932	0.0946245
$682$	0	0
$683$	3.45652	0.132260	0.0661301	−	0.997811i	$-0.478935\pi$
$683$	3.45652	0.132260	0.0661301	+	0.997811i	$0.478935\pi$
$684$	0	0
$685$	65.8255	2.51506
$686$	0	0
$687$	−0.792825	−0.0302482
$688$	0	0
$689$	1.34283	0.0511576
$690$	0	0
$691$	−32.3624	−1.23112	−0.615562	−	0.788088i	$-0.711071\pi$
$691$	−32.3624	−1.23112	−0.615562	+	0.788088i	$0.711071\pi$
$692$	0	0
$693$	15.7749	0.599240
$694$	0	0
$695$	−21.5034	−0.815670
$696$	0	0
$697$	−2.42896	−0.0920035
$698$	0	0
$699$	4.30308	0.162758
$700$	0	0
$701$	−52.4363	−1.98049	−0.990245	−	0.139335i	$-0.955503\pi$
$701$	−52.4363	−1.98049	−0.990245	+	0.139335i	$0.955503\pi$
$702$	0	0
$703$	−1.07361	−0.0404918
$704$	0	0
$705$	9.31097	0.350671
$706$	0	0
$707$	6.21518	0.233746
$708$	0	0
$709$	−15.9333	−0.598388	−0.299194	−	0.954192i	$-0.596718\pi$
$709$	−15.9333	−0.598388	−0.299194	+	0.954192i	$0.596718\pi$
$710$	0	0
$711$	−18.1019	−0.678876
$712$	0	0
$713$	49.7026	1.86138
$714$	0	0
$715$	7.24447	0.270928
$716$	0	0
$717$	6.50098	0.242784
$718$	0	0
$719$	12.7060	0.473854	0.236927	−	0.971527i	$-0.423860\pi$
$719$	12.7060	0.473854	0.236927	+	0.971527i	$0.423860\pi$
$720$	0	0
$721$	4.99370	0.185975
$722$	0	0
$723$	8.98059	0.333992
$724$	0	0
$725$	−22.3338	−0.829456
$726$	0	0
$727$	−3.37543	−0.125188	−0.0625940	−	0.998039i	$-0.519937\pi$
$727$	−3.37543	−0.125188	−0.0625940	+	0.998039i	$0.519937\pi$
$728$	0	0
$729$	−15.9570	−0.591002
$730$	0	0
$731$	−0.321320	−0.0118845
$732$	0	0
$733$	−13.4421	−0.496494	−0.248247	−	0.968697i	$-0.579854\pi$
$733$	−13.4421	−0.496494	−0.248247	+	0.968697i	$0.579854\pi$
$734$	0	0
$735$	−9.83260	−0.362681
$736$	0	0
$737$	−41.5621	−1.53096
$738$	0	0
$739$	−9.45310	−0.347738	−0.173869	−	0.984769i	$-0.555627\pi$
$739$	−9.45310	−0.347738	−0.173869	+	0.984769i	$0.555627\pi$
$740$	0	0
$741$	−0.384703	−0.0141324
$742$	0	0
$743$	−25.1132	−0.921314	−0.460657	−	0.887578i	$-0.652386\pi$
$743$	−25.1132	−0.921314	−0.460657	+	0.887578i	$0.652386\pi$
$744$	0	0
$745$	−38.5544	−1.41252
$746$	0	0
$747$	29.2920	1.07174
$748$	0	0
$749$	−13.0499	−0.476834
$750$	0	0
$751$	4.39645	0.160429	0.0802143	−	0.996778i	$-0.474440\pi$
$751$	4.39645	0.160429	0.0802143	+	0.996778i	$0.474440\pi$
$752$	0	0
$753$	−9.11376	−0.332124
$754$	0	0
$755$	41.0300	1.49323
$756$	0	0
$757$	−27.1909	−0.988269	−0.494134	−	0.869386i	$-0.664515\pi$
$757$	−27.1909	−0.988269	−0.494134	+	0.869386i	$0.664515\pi$
$758$	0	0
$759$	15.0967	0.547977
$760$	0	0
$761$	−37.5961	−1.36286	−0.681428	−	0.731885i	$-0.738641\pi$
$761$	−37.5961	−1.36286	−0.681428	+	0.731885i	$0.738641\pi$
$762$	0	0
$763$	3.82184	0.138360
$764$	0	0
$765$	−3.05358	−0.110402
$766$	0	0
$767$	3.35084	0.120992
$768$	0	0
$769$	6.58730	0.237544	0.118772	−	0.992922i	$-0.462104\pi$
$769$	6.58730	0.237544	0.118772	+	0.992922i	$0.462104\pi$
$770$	0	0
$771$	6.46489	0.232827
$772$	0	0
$773$	−39.2516	−1.41178	−0.705891	−	0.708320i	$-0.749453\pi$
$773$	−39.2516	−1.41178	−0.705891	+	0.708320i	$0.749453\pi$
$774$	0	0
$775$	63.8200	2.29248
$776$	0	0
$777$	−0.187533	−0.00672769
$778$	0	0
$779$	18.3401	0.657101
$780$	0	0
$781$	24.4510	0.874926
$782$	0	0
$783$	9.03670	0.322945
$784$	0	0
$785$	52.4670	1.87263
$786$	0	0
$787$	8.60698	0.306806	0.153403	−	0.988164i	$-0.450977\pi$
$787$	8.60698	0.306806	0.153403	+	0.988164i	$0.450977\pi$
$788$	0	0
$789$	2.73752	0.0974584
$790$	0	0
$791$	5.60279	0.199212
$792$	0	0
$793$	3.22595	0.114557
$794$	0	0
$795$	6.28902	0.223048
$796$	0	0
$797$	−25.7092	−0.910667	−0.455334	−	0.890321i	$-0.650480\pi$
$797$	−25.7092	−0.910667	−0.455334	+	0.890321i	$0.650480\pi$
$798$	0	0
$799$	1.87860	0.0664601
$800$	0	0
$801$	38.3217	1.35403
$802$	0	0
$803$	15.4956	0.546829
$804$	0	0
$805$	−16.1091	−0.567771
$806$	0	0
$807$	1.56199	0.0549847
$808$	0	0
$809$	44.7509	1.57336	0.786679	−	0.617363i	$-0.211799\pi$
$809$	44.7509	1.57336	0.786679	+	0.617363i	$0.211799\pi$
$810$	0	0
$811$	−20.3825	−0.715727	−0.357864	−	0.933774i	$-0.616495\pi$
$811$	−20.3825	−0.715727	−0.357864	+	0.933774i	$0.616495\pi$
$812$	0	0
$813$	0.236322	0.00828818
$814$	0	0
$815$	−53.8710	−1.88702
$816$	0	0
$817$	2.42615	0.0848804
$818$	0	0
$819$	0.859919	0.0300480
$820$	0	0
$821$	52.1712	1.82079	0.910394	−	0.413742i	$-0.135779\pi$
$821$	52.1712	1.82079	0.910394	+	0.413742i	$0.135779\pi$
$822$	0	0
$823$	−12.0395	−0.419669	−0.209834	−	0.977737i	$-0.567293\pi$
$823$	−12.0395	−0.419669	−0.209834	+	0.977737i	$0.567293\pi$
$824$	0	0
$825$	19.3848	0.674892
$826$	0	0
$827$	49.7907	1.73139	0.865696	−	0.500570i	$-0.166876\pi$
$827$	49.7907	1.73139	0.865696	+	0.500570i	$0.166876\pi$
$828$	0	0
$829$	31.8634	1.10666	0.553331	−	0.832962i	$-0.313357\pi$
$829$	31.8634	1.10666	0.553331	+	0.832962i	$0.313357\pi$
$830$	0	0
$831$	−14.7760	−0.512572
$832$	0	0
$833$	−1.98385	−0.0687362
$834$	0	0
$835$	−73.3774	−2.53933
$836$	0	0
$837$	−25.8229	−0.892569
$838$	0	0
$839$	45.4586	1.56941	0.784703	−	0.619872i	$-0.212815\pi$
$839$	45.4586	1.56941	0.784703	+	0.619872i	$0.212815\pi$
$840$	0	0
$841$	−17.7686	−0.612709
$842$	0	0
$843$	−5.72036	−0.197020
$844$	0	0
$845$	−44.0037	−1.51377
$846$	0	0
$847$	−25.3675	−0.871638
$848$	0	0
$849$	−1.85417	−0.0636349
$850$	0	0
$851$	2.29664	0.0787278
$852$	0	0
$853$	21.6171	0.740156	0.370078	−	0.929001i	$-0.379331\pi$
$853$	21.6171	0.740156	0.370078	+	0.929001i	$0.379331\pi$
$854$	0	0
$855$	23.0563	0.788508
$856$	0	0
$857$	22.6450	0.773540	0.386770	−	0.922176i	$-0.373591\pi$
$857$	22.6450	0.773540	0.386770	+	0.922176i	$0.373591\pi$
$858$	0	0
$859$	−7.58208	−0.258697	−0.129349	−	0.991599i	$-0.541289\pi$
$859$	−7.58208	−0.258697	−0.129349	+	0.991599i	$0.541289\pi$
$860$	0	0
$861$	3.20356	0.109177
$862$	0	0
$863$	43.3666	1.47622	0.738108	−	0.674683i	$-0.235720\pi$
$863$	43.3666	1.47622	0.738108	+	0.674683i	$0.235720\pi$
$864$	0	0
$865$	−5.07104	−0.172420
$866$	0	0
$867$	−7.87908	−0.267587
$868$	0	0
$869$	40.5813	1.37663
$870$	0	0
$871$	−2.26563	−0.0767678
$872$	0	0
$873$	−24.8085	−0.839640
$874$	0	0
$875$	−5.16531	−0.174619
$876$	0	0
$877$	47.0415	1.58848	0.794239	−	0.607605i	$-0.207870\pi$
$877$	47.0415	1.58848	0.794239	+	0.607605i	$0.207870\pi$
$878$	0	0
$879$	−7.30245	−0.246306
$880$	0	0
$881$	−34.0264	−1.14638	−0.573189	−	0.819423i	$-0.694294\pi$
$881$	−34.0264	−1.14638	−0.573189	+	0.819423i	$0.694294\pi$
$882$	0	0
$883$	49.6342	1.67033	0.835163	−	0.550003i	$-0.185373\pi$
$883$	49.6342	1.67033	0.835163	+	0.550003i	$0.185373\pi$
$884$	0	0
$885$	15.6934	0.527528
$886$	0	0
$887$	−44.8813	−1.50697	−0.753484	−	0.657467i	$-0.771628\pi$
$887$	−44.8813	−1.50697	−0.753484	+	0.657467i	$0.771628\pi$
$888$	0	0
$889$	−4.41947	−0.148224
$890$	0	0
$891$	44.2292	1.48174
$892$	0	0
$893$	−14.1845	−0.474667
$894$	0	0
$895$	−13.6779	−0.457203
$896$	0	0
$897$	0.822951	0.0274775
$898$	0	0
$899$	−32.0945	−1.07041
$900$	0	0
$901$	1.26889	0.0422727
$902$	0	0
$903$	0.423790	0.0141028
$904$	0	0
$905$	10.9492	0.363962
$906$	0	0
$907$	27.7937	0.922876	0.461438	−	0.887172i	$-0.347334\pi$
$907$	27.7937	0.922876	0.461438	+	0.887172i	$0.347334\pi$
$908$	0	0
$909$	19.0292	0.631157
$910$	0	0
$911$	26.3176	0.871940	0.435970	−	0.899961i	$-0.356405\pi$
$911$	26.3176	0.871940	0.435970	+	0.899961i	$0.356405\pi$
$912$	0	0
$913$	−65.6674	−2.17327
$914$	0	0
$915$	15.1085	0.499471
$916$	0	0
$917$	4.05172	0.133800
$918$	0	0
$919$	10.6058	0.349852	0.174926	−	0.984582i	$-0.444031\pi$
$919$	10.6058	0.349852	0.174926	+	0.984582i	$0.444031\pi$
$920$	0	0
$921$	−10.1652	−0.334954
$922$	0	0
$923$	1.33287	0.0438719
$924$	0	0
$925$	2.94897	0.0969617
$926$	0	0
$927$	15.2893	0.502168
$928$	0	0
$929$	2.53112	0.0830433	0.0415216	−	0.999138i	$-0.486779\pi$
$929$	2.53112	0.0830433	0.0415216	+	0.999138i	$0.486779\pi$
$930$	0	0
$931$	14.9792	0.490923
$932$	0	0
$933$	9.12169	0.298631
$934$	0	0
$935$	6.84557	0.223874
$936$	0	0
$937$	−3.92715	−0.128295	−0.0641473	−	0.997940i	$-0.520433\pi$
$937$	−3.92715	−0.128295	−0.0641473	+	0.997940i	$0.520433\pi$
$938$	0	0
$939$	2.66106	0.0868404
$940$	0	0
$941$	9.84680	0.320997	0.160498	−	0.987036i	$-0.448690\pi$
$941$	9.84680	0.320997	0.160498	+	0.987036i	$0.448690\pi$
$942$	0	0
$943$	−39.2327	−1.27759
$944$	0	0
$945$	8.36945	0.272258
$946$	0	0
$947$	−41.7556	−1.35687	−0.678437	−	0.734658i	$-0.737342\pi$
$947$	−41.7556	−1.35687	−0.678437	+	0.734658i	$0.737342\pi$
$948$	0	0
$949$	0.844695	0.0274200
$950$	0	0
$951$	10.2619	0.332765
$952$	0	0
$953$	38.5402	1.24844	0.624220	−	0.781249i	$-0.285417\pi$
$953$	38.5402	1.24844	0.624220	+	0.781249i	$0.285417\pi$
$954$	0	0
$955$	−72.3311	−2.34058
$956$	0	0
$957$	−9.74843	−0.315122
$958$	0	0
$959$	−17.5165	−0.565636
$960$	0	0
$961$	60.7118	1.95845
$962$	0	0
$963$	−39.9553	−1.28754
$964$	0	0
$965$	−73.0748	−2.35236
$966$	0	0
$967$	19.4260	0.624699	0.312350	−	0.949967i	$-0.398884\pi$
$967$	19.4260	0.624699	0.312350	+	0.949967i	$0.398884\pi$
$968$	0	0
$969$	−0.363520	−0.0116780
$970$	0	0
$971$	2.31871	0.0744108	0.0372054	−	0.999308i	$-0.488154\pi$
$971$	2.31871	0.0744108	0.0372054	+	0.999308i	$0.488154\pi$
$972$	0	0
$973$	5.72214	0.183443
$974$	0	0
$975$	1.05670	0.0338415
$976$	0	0
$977$	−40.4719	−1.29481	−0.647405	−	0.762146i	$-0.724146\pi$
$977$	−40.4719	−1.29481	−0.647405	+	0.762146i	$0.724146\pi$
$978$	0	0
$979$	−85.9103	−2.74570
$980$	0	0
$981$	11.7014	0.373597
$982$	0	0
$983$	0.497440	0.0158659	0.00793294	−	0.999969i	$-0.497475\pi$
$983$	0.497440	0.0158659	0.00793294	+	0.999969i	$0.497475\pi$
$984$	0	0
$985$	−63.9414	−2.03734
$986$	0	0
$987$	−2.47769	−0.0788657
$988$	0	0
$989$	−5.18999	−0.165032
$990$	0	0
$991$	30.1439	0.957552	0.478776	−	0.877937i	$-0.341081\pi$
$991$	30.1439	0.957552	0.478776	+	0.877937i	$0.341081\pi$
$992$	0	0
$993$	10.3614	0.328809
$994$	0	0
$995$	74.7670	2.37027
$996$	0	0
$997$	−32.3068	−1.02317	−0.511583	−	0.859234i	$-0.670941\pi$
$997$	−32.3068	−1.02317	−0.511583	+	0.859234i	$0.670941\pi$
$998$	0	0
$999$	−1.19321	−0.0377516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2752.2.a.ba.1.3		5
4.3	odd	2		2752.2.a.y.1.3		5
8.3	odd	2		1376.2.a.g.1.3	yes	5
8.5	even	2		1376.2.a.f.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1376.2.a.f.1.3	✓	5	8.5	even	2
1376.2.a.g.1.3	yes	5	8.3	odd	2
2752.2.a.y.1.3		5	4.3	odd	2
2752.2.a.ba.1.3		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)

\(f(q)\)	\(=\)	\(q+0.466307 q^{3} +3.41528 q^{5} -0.908821 q^{7} -2.78256 q^{9} +O(q^{10})\)	\(q+0.466307 q^{3} +3.41528 q^{5} -0.908821 q^{7} -2.78256 q^{9} +6.23799 q^{11} +0.340044 q^{13} +1.59257 q^{15} +0.321320 q^{17} -2.42615 q^{19} -0.423790 q^{21} +5.18999 q^{23} +6.66415 q^{25} -2.69645 q^{27} -3.35133 q^{29} +9.57663 q^{31} +2.90882 q^{33} -3.10388 q^{35} +0.442513 q^{37} +0.158565 q^{39} -7.55931 q^{41} -1.00000 q^{43} -9.50322 q^{45} +5.84650 q^{47} -6.17405 q^{49} +0.149834 q^{51} +3.94897 q^{53} +21.3045 q^{55} -1.13133 q^{57} +9.85413 q^{59} +9.48686 q^{61} +2.52885 q^{63} +1.16135 q^{65} -6.66274 q^{67} +2.42013 q^{69} +3.91969 q^{71} +2.48408 q^{73} +3.10754 q^{75} -5.66922 q^{77} +6.50551 q^{79} +7.09030 q^{81} -10.5270 q^{83} +1.09740 q^{85} -1.56275 q^{87} -13.7721 q^{89} -0.309039 q^{91} +4.46565 q^{93} -8.28600 q^{95} +8.91572 q^{97} -17.3576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + q^{5} + 8 q^{7}+O(q^{10}) \) 5 * q + q^3 + q^5 + 8 * q^7	\( 5 q + q^{3} + q^{5} + 8 q^{7} - q^{11} - q^{13} + 8 q^{15} - 2 q^{17} - 13 q^{19} + 10 q^{21} + 12 q^{23} + 2 q^{25} + 4 q^{27} + 7 q^{29} + 22 q^{31} + 2 q^{33} + 2 q^{35} - 9 q^{37} + 16 q^{39} - 2 q^{41} - 5 q^{43} + 12 q^{45} + 23 q^{47} + 9 q^{49} - 11 q^{51} + 5 q^{53} + 34 q^{55} - 16 q^{57} + 4 q^{59} + 28 q^{63} - 4 q^{65} - 11 q^{67} - 18 q^{69} + 24 q^{71} - 10 q^{73} + 16 q^{75} - 6 q^{77} + 7 q^{79} + 5 q^{81} - 3 q^{83} - 12 q^{85} + 33 q^{87} + 10 q^{91} + q^{93} + 9 q^{95} - 6 q^{97} - 5 q^{99}+O(q^{100}) \) 5 * q + q^3 + q^5 + 8 * q^7 - q^11 - q^13 + 8 * q^15 - 2 * q^17 - 13 * q^19 + 10 * q^21 + 12 * q^23 + 2 * q^25 + 4 * q^27 + 7 * q^29 + 22 * q^31 + 2 * q^33 + 2 * q^35 - 9 * q^37 + 16 * q^39 - 2 * q^41 - 5 * q^43 + 12 * q^45 + 23 * q^47 + 9 * q^49 - 11 * q^51 + 5 * q^53 + 34 * q^55 - 16 * q^57 + 4 * q^59 + 28 * q^63 - 4 * q^65 - 11 * q^67 - 18 * q^69 + 24 * q^71 - 10 * q^73 + 16 * q^75 - 6 * q^77 + 7 * q^79 + 5 * q^81 - 3 * q^83 - 12 * q^85 + 33 * q^87 + 10 * q^91 + q^93 + 9 * q^95 - 6 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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