LMFDB - Embedded newform 6044.2.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22958 q^{3} -3.12332 q^{5} +0.924225 q^{7} +1.97105 q^{9} +O(q^{10})$	$q-2.22958 q^{3} -3.12332 q^{5} +0.924225 q^{7} +1.97105 q^{9} -2.65938 q^{11} +6.43066 q^{13} +6.96371 q^{15} +1.95143 q^{17} +3.85126 q^{19} -2.06064 q^{21} -1.97512 q^{23} +4.75513 q^{25} +2.29414 q^{27} -5.18431 q^{29} -1.63667 q^{31} +5.92930 q^{33} -2.88665 q^{35} +2.27713 q^{37} -14.3377 q^{39} -8.13524 q^{41} +1.54289 q^{43} -6.15621 q^{45} +8.53117 q^{47} -6.14581 q^{49} -4.35087 q^{51} +11.5930 q^{53} +8.30608 q^{55} -8.58670 q^{57} +2.04749 q^{59} -12.7185 q^{61} +1.82169 q^{63} -20.0850 q^{65} +3.15867 q^{67} +4.40369 q^{69} -3.24373 q^{71} +5.39026 q^{73} -10.6020 q^{75} -2.45786 q^{77} -6.11199 q^{79} -11.0281 q^{81} -1.07683 q^{83} -6.09493 q^{85} +11.5589 q^{87} -7.21940 q^{89} +5.94338 q^{91} +3.64910 q^{93} -12.0287 q^{95} -3.90744 q^{97} -5.24175 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * 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
$3$	−2.22958	−1.28725	−0.643625	−	0.765341i	$-0.722571\pi$
$3$	−2.22958	−1.28725	−0.643625	+	0.765341i	$0.722571\pi$
$4$	0	0
$5$	−3.12332	−1.39679	−0.698396	−	0.715712i	$-0.746102\pi$
$5$	−3.12332	−1.39679	−0.698396	+	0.715712i	$0.746102\pi$
$6$	0	0
$7$	0.924225	0.349324	0.174662	−	0.984628i	$-0.444117\pi$
$7$	0.924225	0.349324	0.174662	+	0.984628i	$0.444117\pi$
$8$	0	0
$9$	1.97105	0.657015
$10$	0	0
$11$	−2.65938	−0.801832	−0.400916	−	0.916115i	$-0.631308\pi$
$11$	−2.65938	−0.801832	−0.400916	+	0.916115i	$0.631308\pi$
$12$	0	0
$13$	6.43066	1.78354	0.891772	−	0.452485i	$-0.149462\pi$
$13$	6.43066	1.78354	0.891772	+	0.452485i	$0.149462\pi$
$14$	0	0
$15$	6.96371	1.79802
$16$	0	0
$17$	1.95143	0.473291	0.236645	−	0.971596i	$-0.423952\pi$
$17$	1.95143	0.473291	0.236645	+	0.971596i	$0.423952\pi$
$18$	0	0
$19$	3.85126	0.883539	0.441770	−	0.897129i	$-0.354351\pi$
$19$	3.85126	0.883539	0.441770	+	0.897129i	$0.354351\pi$
$20$	0	0
$21$	−2.06064	−0.449668
$22$	0	0
$23$	−1.97512	−0.411840	−0.205920	−	0.978569i	$-0.566019\pi$
$23$	−1.97512	−0.411840	−0.205920	+	0.978569i	$0.566019\pi$
$24$	0	0
$25$	4.75513	0.951027
$26$	0	0
$27$	2.29414	0.441508
$28$	0	0
$29$	−5.18431	−0.962703	−0.481352	−	0.876528i	$-0.659854\pi$
$29$	−5.18431	−0.962703	−0.481352	+	0.876528i	$0.659854\pi$
$30$	0	0
$31$	−1.63667	−0.293955	−0.146978	−	0.989140i	$-0.546955\pi$
$31$	−1.63667	−0.293955	−0.146978	+	0.989140i	$0.546955\pi$
$32$	0	0
$33$	5.92930	1.03216
$34$	0	0
$35$	−2.88665	−0.487933
$36$	0	0
$37$	2.27713	0.374357	0.187179	−	0.982326i	$-0.440066\pi$
$37$	2.27713	0.374357	0.187179	+	0.982326i	$0.440066\pi$
$38$	0	0
$39$	−14.3377	−2.29587
$40$	0	0
$41$	−8.13524	−1.27051	−0.635256	−	0.772302i	$-0.719105\pi$
$41$	−8.13524	−1.27051	−0.635256	+	0.772302i	$0.719105\pi$
$42$	0	0
$43$	1.54289	0.235288	0.117644	−	0.993056i	$-0.462466\pi$
$43$	1.54289	0.235288	0.117644	+	0.993056i	$0.462466\pi$
$44$	0	0
$45$	−6.15621	−0.917713
$46$	0	0
$47$	8.53117	1.24440	0.622200	−	0.782859i	$-0.286239\pi$
$47$	8.53117	1.24440	0.622200	+	0.782859i	$0.286239\pi$
$48$	0	0
$49$	−6.14581	−0.877973
$50$	0	0
$51$	−4.35087	−0.609244
$52$	0	0
$53$	11.5930	1.59242	0.796212	−	0.605018i	$-0.206834\pi$
$53$	11.5930	1.59242	0.796212	+	0.605018i	$0.206834\pi$
$54$	0	0
$55$	8.30608	1.11999
$56$	0	0
$57$	−8.58670	−1.13734
$58$	0	0
$59$	2.04749	0.266560	0.133280	−	0.991078i	$-0.457449\pi$
$59$	2.04749	0.266560	0.133280	+	0.991078i	$0.457449\pi$
$60$	0	0
$61$	−12.7185	−1.62843	−0.814217	−	0.580560i	$-0.802834\pi$
$61$	−12.7185	−1.62843	−0.814217	+	0.580560i	$0.802834\pi$
$62$	0	0
$63$	1.82169	0.229511
$64$	0	0
$65$	−20.0850	−2.49124
$66$	0	0
$67$	3.15867	0.385893	0.192947	−	0.981209i	$-0.438196\pi$
$67$	3.15867	0.385893	0.192947	+	0.981209i	$0.438196\pi$
$68$	0	0
$69$	4.40369	0.530142
$70$	0	0
$71$	−3.24373	−0.384960	−0.192480	−	0.981301i	$-0.561653\pi$
$71$	−3.24373	−0.384960	−0.192480	+	0.981301i	$0.561653\pi$
$72$	0	0
$73$	5.39026	0.630882	0.315441	−	0.948945i	$-0.397848\pi$
$73$	5.39026	0.630882	0.315441	+	0.948945i	$0.397848\pi$
$74$	0	0
$75$	−10.6020	−1.22421
$76$	0	0
$77$	−2.45786	−0.280099
$78$	0	0
$79$	−6.11199	−0.687652	−0.343826	−	0.939033i	$-0.611723\pi$
$79$	−6.11199	−0.687652	−0.343826	+	0.939033i	$0.611723\pi$
$80$	0	0
$81$	−11.0281	−1.22535
$82$	0	0
$83$	−1.07683	−0.118197	−0.0590987	−	0.998252i	$-0.518823\pi$
$83$	−1.07683	−0.118197	−0.0590987	+	0.998252i	$0.518823\pi$
$84$	0	0
$85$	−6.09493	−0.661088
$86$	0	0
$87$	11.5589	1.23924
$88$	0	0
$89$	−7.21940	−0.765255	−0.382628	−	0.923903i	$-0.624981\pi$
$89$	−7.21940	−0.765255	−0.382628	+	0.923903i	$0.624981\pi$
$90$	0	0
$91$	5.94338	0.623035
$92$	0	0
$93$	3.64910	0.378394
$94$	0	0
$95$	−12.0287	−1.23412
$96$	0	0
$97$	−3.90744	−0.396741	−0.198370	−	0.980127i	$-0.563565\pi$
$97$	−3.90744	−0.396741	−0.198370	+	0.980127i	$0.563565\pi$
$98$	0	0
$99$	−5.24175	−0.526816
$100$	0	0
$101$	−10.0503	−1.00004	−0.500019	−	0.866015i	$-0.666674\pi$
$101$	−10.0503	−1.00004	−0.500019	+	0.866015i	$0.666674\pi$
$102$	0	0
$103$	−2.25212	−0.221908	−0.110954	−	0.993826i	$-0.535391\pi$
$103$	−2.25212	−0.221908	−0.110954	+	0.993826i	$0.535391\pi$
$104$	0	0
$105$	6.43603	0.628092
$106$	0	0
$107$	−13.5188	−1.30691	−0.653456	−	0.756965i	$-0.726682\pi$
$107$	−13.5188	−1.30691	−0.653456	+	0.756965i	$0.726682\pi$
$108$	0	0
$109$	−14.0824	−1.34885	−0.674424	−	0.738344i	$-0.735608\pi$
$109$	−14.0824	−1.34885	−0.674424	+	0.738344i	$0.735608\pi$
$110$	0	0
$111$	−5.07705	−0.481892
$112$	0	0
$113$	2.75678	0.259336	0.129668	−	0.991557i	$-0.458609\pi$
$113$	2.75678	0.259336	0.129668	+	0.991557i	$0.458609\pi$
$114$	0	0
$115$	6.16892	0.575255
$116$	0	0
$117$	12.6751	1.17182
$118$	0	0
$119$	1.80356	0.165332
$120$	0	0
$121$	−3.92772	−0.357066
$122$	0	0
$123$	18.1382	1.63547
$124$	0	0
$125$	0.764795	0.0684053
$126$	0	0
$127$	−1.67924	−0.149008	−0.0745042	−	0.997221i	$-0.523737\pi$
$127$	−1.67924	−0.149008	−0.0745042	+	0.997221i	$0.523737\pi$
$128$	0	0
$129$	−3.44000	−0.302875
$130$	0	0
$131$	4.10291	0.358473	0.179237	−	0.983806i	$-0.442637\pi$
$131$	4.10291	0.358473	0.179237	+	0.983806i	$0.442637\pi$
$132$	0	0
$133$	3.55943	0.308642
$134$	0	0
$135$	−7.16534	−0.616694
$136$	0	0
$137$	11.4673	0.979714	0.489857	−	0.871803i	$-0.337049\pi$
$137$	11.4673	0.979714	0.489857	+	0.871803i	$0.337049\pi$
$138$	0	0
$139$	19.2586	1.63350	0.816749	−	0.576994i	$-0.195774\pi$
$139$	19.2586	1.63350	0.816749	+	0.576994i	$0.195774\pi$
$140$	0	0
$141$	−19.0210	−1.60185
$142$	0	0
$143$	−17.1015	−1.43010
$144$	0	0
$145$	16.1923	1.34470
$146$	0	0
$147$	13.7026	1.13017
$148$	0	0
$149$	7.61811	0.624100	0.312050	−	0.950066i	$-0.398984\pi$
$149$	7.61811	0.624100	0.312050	+	0.950066i	$0.398984\pi$
$150$	0	0
$151$	9.56011	0.777991	0.388995	−	0.921240i	$-0.372822\pi$
$151$	9.56011	0.777991	0.388995	+	0.921240i	$0.372822\pi$
$152$	0	0
$153$	3.84635	0.310959
$154$	0	0
$155$	5.11186	0.410594
$156$	0	0
$157$	20.5055	1.63652	0.818258	−	0.574851i	$-0.194940\pi$
$157$	20.5055	1.63652	0.818258	+	0.574851i	$0.194940\pi$
$158$	0	0
$159$	−25.8476	−2.04985
$160$	0	0
$161$	−1.82545	−0.143866
$162$	0	0
$163$	8.44850	0.661738	0.330869	−	0.943677i	$-0.392658\pi$
$163$	8.44850	0.661738	0.330869	+	0.943677i	$0.392658\pi$
$164$	0	0
$165$	−18.5191	−1.44171
$166$	0	0
$167$	5.40540	0.418283	0.209141	−	0.977885i	$-0.432933\pi$
$167$	5.40540	0.418283	0.209141	+	0.977885i	$0.432933\pi$
$168$	0	0
$169$	28.3534	2.18103
$170$	0	0
$171$	7.59100	0.580499
$172$	0	0
$173$	−19.5277	−1.48466	−0.742331	−	0.670033i	$-0.766280\pi$
$173$	−19.5277	−1.48466	−0.742331	+	0.670033i	$0.766280\pi$
$174$	0	0
$175$	4.39481	0.332217
$176$	0	0
$177$	−4.56505	−0.343130
$178$	0	0
$179$	−4.84844	−0.362389	−0.181195	−	0.983447i	$-0.557996\pi$
$179$	−4.84844	−0.362389	−0.181195	+	0.983447i	$0.557996\pi$
$180$	0	0
$181$	17.5142	1.30182	0.650911	−	0.759154i	$-0.274387\pi$
$181$	17.5142	1.30182	0.650911	+	0.759154i	$0.274387\pi$
$182$	0	0
$183$	28.3569	2.09620
$184$	0	0
$185$	−7.11220	−0.522899
$186$	0	0
$187$	−5.18958	−0.379500
$188$	0	0
$189$	2.12030	0.154229
$190$	0	0
$191$	−11.0229	−0.797592	−0.398796	−	0.917040i	$-0.630572\pi$
$191$	−11.0229	−0.797592	−0.398796	+	0.917040i	$0.630572\pi$
$192$	0	0
$193$	14.4829	1.04250	0.521250	−	0.853404i	$-0.325466\pi$
$193$	14.4829	1.04250	0.521250	+	0.853404i	$0.325466\pi$
$194$	0	0
$195$	44.7812	3.20685
$196$	0	0
$197$	−2.56156	−0.182503	−0.0912517	−	0.995828i	$-0.529087\pi$
$197$	−2.56156	−0.182503	−0.0912517	+	0.995828i	$0.529087\pi$
$198$	0	0
$199$	12.7518	0.903950	0.451975	−	0.892031i	$-0.350720\pi$
$199$	12.7518	0.903950	0.451975	+	0.892031i	$0.350720\pi$
$200$	0	0
$201$	−7.04252	−0.496741
$202$	0	0
$203$	−4.79147	−0.336295
$204$	0	0
$205$	25.4090	1.77464
$206$	0	0
$207$	−3.89305	−0.270585
$208$	0	0
$209$	−10.2419	−0.708450
$210$	0	0
$211$	−9.55483	−0.657781	−0.328891	−	0.944368i	$-0.606675\pi$
$211$	−9.55483	−0.657781	−0.328891	+	0.944368i	$0.606675\pi$
$212$	0	0
$213$	7.23216	0.495540
$214$	0	0
$215$	−4.81894	−0.328649
$216$	0	0
$217$	−1.51266	−0.102686
$218$	0	0
$219$	−12.0180	−0.812103
$220$	0	0
$221$	12.5490	0.844135
$222$	0	0
$223$	−15.5739	−1.04291	−0.521453	−	0.853280i	$-0.674610\pi$
$223$	−15.5739	−1.04291	−0.521453	+	0.853280i	$0.674610\pi$
$224$	0	0
$225$	9.37259	0.624839
$226$	0	0
$227$	−0.916895	−0.0608565	−0.0304282	−	0.999537i	$-0.509687\pi$
$227$	−0.916895	−0.0608565	−0.0304282	+	0.999537i	$0.509687\pi$
$228$	0	0
$229$	20.9598	1.38506	0.692530	−	0.721389i	$-0.256496\pi$
$229$	20.9598	1.38506	0.692530	+	0.721389i	$0.256496\pi$
$230$	0	0
$231$	5.48001	0.360558
$232$	0	0
$233$	10.4289	0.683221	0.341610	−	0.939842i	$-0.389028\pi$
$233$	10.4289	0.683221	0.341610	+	0.939842i	$0.389028\pi$
$234$	0	0
$235$	−26.6456	−1.73817
$236$	0	0
$237$	13.6272	0.885181
$238$	0	0
$239$	−1.23488	−0.0798778	−0.0399389	−	0.999202i	$-0.512716\pi$
$239$	−1.23488	−0.0798778	−0.0399389	+	0.999202i	$0.512716\pi$
$240$	0	0
$241$	1.79934	0.115905	0.0579527	−	0.998319i	$-0.481543\pi$
$241$	1.79934	0.115905	0.0579527	+	0.998319i	$0.481543\pi$
$242$	0	0
$243$	17.7057	1.13582
$244$	0	0
$245$	19.1953	1.22634
$246$	0	0
$247$	24.7661	1.57583
$248$	0	0
$249$	2.40088	0.152150
$250$	0	0
$251$	5.67158	0.357987	0.178993	−	0.983850i	$-0.442716\pi$
$251$	5.67158	0.357987	0.178993	+	0.983850i	$0.442716\pi$
$252$	0	0
$253$	5.25258	0.330227
$254$	0	0
$255$	13.5892	0.850987
$256$	0	0
$257$	−7.59209	−0.473581	−0.236791	−	0.971561i	$-0.576096\pi$
$257$	−7.59209	−0.473581	−0.236791	+	0.971561i	$0.576096\pi$
$258$	0	0
$259$	2.10458	0.130772
$260$	0	0
$261$	−10.2185	−0.632510
$262$	0	0
$263$	16.3628	1.00898	0.504488	−	0.863419i	$-0.331681\pi$
$263$	16.3628	1.00898	0.504488	+	0.863419i	$0.331681\pi$
$264$	0	0
$265$	−36.2087	−2.22428
$266$	0	0
$267$	16.0963	0.985075
$268$	0	0
$269$	19.5569	1.19241	0.596203	−	0.802834i	$-0.296675\pi$
$269$	19.5569	1.19241	0.596203	+	0.802834i	$0.296675\pi$
$270$	0	0
$271$	8.60796	0.522896	0.261448	−	0.965218i	$-0.415800\pi$
$271$	8.60796	0.522896	0.261448	+	0.965218i	$0.415800\pi$
$272$	0	0
$273$	−13.2513	−0.802003
$274$	0	0
$275$	−12.6457	−0.762564
$276$	0	0
$277$	−15.6219	−0.938628	−0.469314	−	0.883031i	$-0.655499\pi$
$277$	−15.6219	−0.938628	−0.469314	+	0.883031i	$0.655499\pi$
$278$	0	0
$279$	−3.22596	−0.193133
$280$	0	0
$281$	−16.8199	−1.00339	−0.501696	−	0.865044i	$-0.667290\pi$
$281$	−16.8199	−1.00339	−0.501696	+	0.865044i	$0.667290\pi$
$282$	0	0
$283$	26.6629	1.58494	0.792471	−	0.609909i	$-0.208794\pi$
$283$	26.6629	1.58494	0.792471	+	0.609909i	$0.208794\pi$
$284$	0	0
$285$	26.8190	1.58862
$286$	0	0
$287$	−7.51880	−0.443820
$288$	0	0
$289$	−13.1919	−0.775996
$290$	0	0
$291$	8.71198	0.510705
$292$	0	0
$293$	−18.9143	−1.10499	−0.552493	−	0.833517i	$-0.686323\pi$
$293$	−18.9143	−1.10499	−0.552493	+	0.833517i	$0.686323\pi$
$294$	0	0
$295$	−6.39497	−0.372329
$296$	0	0
$297$	−6.10098	−0.354015
$298$	0	0
$299$	−12.7013	−0.734536
$300$	0	0
$301$	1.42598	0.0821919
$302$	0	0
$303$	22.4079	1.28730
$304$	0	0
$305$	39.7239	2.27458
$306$	0	0
$307$	11.2918	0.644458	0.322229	−	0.946662i	$-0.395568\pi$
$307$	11.2918	0.644458	0.322229	+	0.946662i	$0.395568\pi$
$308$	0	0
$309$	5.02128	0.285651
$310$	0	0
$311$	−21.1671	−1.20028	−0.600138	−	0.799896i	$-0.704888\pi$
$311$	−21.1671	−1.20028	−0.600138	+	0.799896i	$0.704888\pi$
$312$	0	0
$313$	−16.4400	−0.929244	−0.464622	−	0.885509i	$-0.653810\pi$
$313$	−16.4400	−0.929244	−0.464622	+	0.885509i	$0.653810\pi$
$314$	0	0
$315$	−5.68972	−0.320579
$316$	0	0
$317$	−1.87265	−0.105179	−0.0525893	−	0.998616i	$-0.516747\pi$
$317$	−1.87265	−0.105179	−0.0525893	+	0.998616i	$0.516747\pi$
$318$	0	0
$319$	13.7870	0.771926
$320$	0	0
$321$	30.1413	1.68232
$322$	0	0
$323$	7.51545	0.418171
$324$	0	0
$325$	30.5787	1.69620
$326$	0	0
$327$	31.3979	1.73631
$328$	0	0
$329$	7.88472	0.434699
$330$	0	0
$331$	35.9477	1.97586	0.987932	−	0.154890i	$-0.0495023\pi$
$331$	35.9477	1.97586	0.987932	+	0.154890i	$0.0495023\pi$
$332$	0	0
$333$	4.48832	0.245958
$334$	0	0
$335$	−9.86554	−0.539012
$336$	0	0
$337$	−29.8012	−1.62338	−0.811688	−	0.584091i	$-0.801451\pi$
$337$	−29.8012	−1.62338	−0.811688	+	0.584091i	$0.801451\pi$
$338$	0	0
$339$	−6.14647	−0.333830
$340$	0	0
$341$	4.35253	0.235703
$342$	0	0
$343$	−12.1497	−0.656021
$344$	0	0
$345$	−13.7541	−0.740498
$346$	0	0
$347$	23.8159	1.27851	0.639253	−	0.768996i	$-0.279243\pi$
$347$	23.8159	1.27851	0.639253	+	0.768996i	$0.279243\pi$
$348$	0	0
$349$	33.9230	1.81586	0.907929	−	0.419124i	$-0.137663\pi$
$349$	33.9230	1.81586	0.907929	+	0.419124i	$0.137663\pi$
$350$	0	0
$351$	14.7528	0.787448
$352$	0	0
$353$	7.61846	0.405490	0.202745	−	0.979232i	$-0.435014\pi$
$353$	7.61846	0.405490	0.202745	+	0.979232i	$0.435014\pi$
$354$	0	0
$355$	10.1312	0.537709
$356$	0	0
$357$	−4.02118	−0.212824
$358$	0	0
$359$	12.2902	0.648650	0.324325	−	0.945946i	$-0.394863\pi$
$359$	12.2902	0.648650	0.324325	+	0.945946i	$0.394863\pi$
$360$	0	0
$361$	−4.16781	−0.219359
$362$	0	0
$363$	8.75719	0.459633
$364$	0	0
$365$	−16.8355	−0.881211
$366$	0	0
$367$	25.9693	1.35559	0.677793	−	0.735253i	$-0.262936\pi$
$367$	25.9693	1.35559	0.677793	+	0.735253i	$0.262936\pi$
$368$	0	0
$369$	−16.0349	−0.834745
$370$	0	0
$371$	10.7146	0.556272
$372$	0	0
$373$	−35.1978	−1.82247	−0.911236	−	0.411885i	$-0.864870\pi$
$373$	−35.1978	−1.82247	−0.911236	+	0.411885i	$0.864870\pi$
$374$	0	0
$375$	−1.70517	−0.0880548
$376$	0	0
$377$	−33.3386	−1.71702
$378$	0	0
$379$	−13.1797	−0.676995	−0.338497	−	0.940967i	$-0.609919\pi$
$379$	−13.1797	−0.676995	−0.338497	+	0.940967i	$0.609919\pi$
$380$	0	0
$381$	3.74401	0.191811
$382$	0	0
$383$	1.19949	0.0612913	0.0306457	−	0.999530i	$-0.490244\pi$
$383$	1.19949	0.0612913	0.0306457	+	0.999530i	$0.490244\pi$
$384$	0	0
$385$	7.67669	0.391240
$386$	0	0
$387$	3.04111	0.154588
$388$	0	0
$389$	−3.86961	−0.196197	−0.0980986	−	0.995177i	$-0.531276\pi$
$389$	−3.86961	−0.196197	−0.0980986	+	0.995177i	$0.531276\pi$
$390$	0	0
$391$	−3.85430	−0.194920
$392$	0	0
$393$	−9.14779	−0.461445
$394$	0	0
$395$	19.0897	0.960507
$396$	0	0
$397$	18.6831	0.937678	0.468839	−	0.883284i	$-0.344673\pi$
$397$	18.6831	0.937678	0.468839	+	0.883284i	$0.344673\pi$
$398$	0	0
$399$	−7.93605	−0.397299
$400$	0	0
$401$	2.51575	0.125630	0.0628152	−	0.998025i	$-0.479992\pi$
$401$	2.51575	0.125630	0.0628152	+	0.998025i	$0.479992\pi$
$402$	0	0
$403$	−10.5249	−0.524282
$404$	0	0
$405$	34.4443	1.71155
$406$	0	0
$407$	−6.05574	−0.300172
$408$	0	0
$409$	−4.31996	−0.213608	−0.106804	−	0.994280i	$-0.534062\pi$
$409$	−4.31996	−0.213608	−0.106804	+	0.994280i	$0.534062\pi$
$410$	0	0
$411$	−25.5672	−1.26114
$412$	0	0
$413$	1.89234	0.0931160
$414$	0	0
$415$	3.36328	0.165097
$416$	0	0
$417$	−42.9388	−2.10272
$418$	0	0
$419$	−14.7764	−0.721874	−0.360937	−	0.932590i	$-0.617543\pi$
$419$	−14.7764	−0.721874	−0.360937	+	0.932590i	$0.617543\pi$
$420$	0	0
$421$	−3.22129	−0.156996	−0.0784981	−	0.996914i	$-0.525012\pi$
$421$	−3.22129	−0.156996	−0.0784981	+	0.996914i	$0.525012\pi$
$422$	0	0
$423$	16.8153	0.817589
$424$	0	0
$425$	9.27930	0.450112
$426$	0	0
$427$	−11.7547	−0.568852
$428$	0	0
$429$	38.1293	1.84090
$430$	0	0
$431$	18.2633	0.879713	0.439857	−	0.898068i	$-0.355029\pi$
$431$	18.2633	0.879713	0.439857	+	0.898068i	$0.355029\pi$
$432$	0	0
$433$	16.2513	0.780986	0.390493	−	0.920606i	$-0.372304\pi$
$433$	16.2513	0.780986	0.390493	+	0.920606i	$0.372304\pi$
$434$	0	0
$435$	−36.1020	−1.73096
$436$	0	0
$437$	−7.60669	−0.363877
$438$	0	0
$439$	−1.22233	−0.0583388	−0.0291694	−	0.999574i	$-0.509286\pi$
$439$	−1.22233	−0.0583388	−0.0291694	+	0.999574i	$0.509286\pi$
$440$	0	0
$441$	−12.1137	−0.576841
$442$	0	0
$443$	30.5080	1.44948	0.724740	−	0.689023i	$-0.241960\pi$
$443$	30.5080	1.44948	0.724740	+	0.689023i	$0.241960\pi$
$444$	0	0
$445$	22.5485	1.06890
$446$	0	0
$447$	−16.9852	−0.803373
$448$	0	0
$449$	−12.1323	−0.572561	−0.286280	−	0.958146i	$-0.592419\pi$
$449$	−12.1323	−0.572561	−0.286280	+	0.958146i	$0.592419\pi$
$450$	0	0
$451$	21.6347	1.01874
$452$	0	0
$453$	−21.3151	−1.00147
$454$	0	0
$455$	−18.5631	−0.870250
$456$	0	0
$457$	−19.7866	−0.925578	−0.462789	−	0.886468i	$-0.653151\pi$
$457$	−19.7866	−0.925578	−0.462789	+	0.886468i	$0.653151\pi$
$458$	0	0
$459$	4.47685	0.208961
$460$	0	0
$461$	−33.5824	−1.56409	−0.782043	−	0.623224i	$-0.785822\pi$
$461$	−33.5824	−1.56409	−0.782043	+	0.623224i	$0.785822\pi$
$462$	0	0
$463$	26.0241	1.20944	0.604721	−	0.796437i	$-0.293284\pi$
$463$	26.0241	1.20944	0.604721	+	0.796437i	$0.293284\pi$
$464$	0	0
$465$	−11.3973	−0.528538
$466$	0	0
$467$	4.30938	0.199414	0.0997071	−	0.995017i	$-0.468209\pi$
$467$	4.30938	0.199414	0.0997071	+	0.995017i	$0.468209\pi$
$468$	0	0
$469$	2.91932	0.134802
$470$	0	0
$471$	−45.7187	−2.10661
$472$	0	0
$473$	−4.10312	−0.188662
$474$	0	0
$475$	18.3132	0.840269
$476$	0	0
$477$	22.8504	1.04625
$478$	0	0
$479$	−28.6535	−1.30921	−0.654605	−	0.755971i	$-0.727165\pi$
$479$	−28.6535	−1.30921	−0.654605	+	0.755971i	$0.727165\pi$
$480$	0	0
$481$	14.6434	0.667683
$482$	0	0
$483$	4.07000	0.185191
$484$	0	0
$485$	12.2042	0.554164
$486$	0	0
$487$	1.66047	0.0752433	0.0376216	−	0.999292i	$-0.488022\pi$
$487$	1.66047	0.0752433	0.0376216	+	0.999292i	$0.488022\pi$
$488$	0	0
$489$	−18.8366	−0.851822
$490$	0	0
$491$	28.3304	1.27853	0.639267	−	0.768985i	$-0.279238\pi$
$491$	28.3304	1.27853	0.639267	+	0.768985i	$0.279238\pi$
$492$	0	0
$493$	−10.1168	−0.455638
$494$	0	0
$495$	16.3717	0.735852
$496$	0	0
$497$	−2.99793	−0.134476
$498$	0	0
$499$	7.56475	0.338645	0.169322	−	0.985561i	$-0.445842\pi$
$499$	7.56475	0.338645	0.169322	+	0.985561i	$0.445842\pi$
$500$	0	0
$501$	−12.0518	−0.538435
$502$	0	0
$503$	−0.889661	−0.0396680	−0.0198340	−	0.999803i	$-0.506314\pi$
$503$	−0.889661	−0.0396680	−0.0198340	+	0.999803i	$0.506314\pi$
$504$	0	0
$505$	31.3902	1.39684
$506$	0	0
$507$	−63.2163	−2.80753
$508$	0	0
$509$	27.6798	1.22689	0.613443	−	0.789739i	$-0.289784\pi$
$509$	27.6798	1.22689	0.613443	+	0.789739i	$0.289784\pi$
$510$	0	0
$511$	4.98181	0.220382
$512$	0	0
$513$	8.83533	0.390089
$514$	0	0
$515$	7.03408	0.309959
$516$	0	0
$517$	−22.6876	−0.997799
$518$	0	0
$519$	43.5386	1.91113
$520$	0	0
$521$	15.6904	0.687407	0.343704	−	0.939078i	$-0.388318\pi$
$521$	15.6904	0.687407	0.343704	+	0.939078i	$0.388318\pi$
$522$	0	0
$523$	19.9238	0.871208	0.435604	−	0.900138i	$-0.356535\pi$
$523$	19.9238	0.871208	0.435604	+	0.900138i	$0.356535\pi$
$524$	0	0
$525$	−9.79861	−0.427646
$526$	0	0
$527$	−3.19385	−0.139126
$528$	0	0
$529$	−19.0989	−0.830388
$530$	0	0
$531$	4.03569	0.175134
$532$	0	0
$533$	−52.3150	−2.26601
$534$	0	0
$535$	42.2236	1.82548
$536$	0	0
$537$	10.8100	0.466486
$538$	0	0
$539$	16.3440	0.703986
$540$	0	0
$541$	31.3312	1.34703	0.673516	−	0.739173i	$-0.264783\pi$
$541$	31.3312	1.34703	0.673516	+	0.739173i	$0.264783\pi$
$542$	0	0
$543$	−39.0494	−1.67577
$544$	0	0
$545$	43.9838	1.88406
$546$	0	0
$547$	43.7356	1.87000	0.935000	−	0.354649i	$-0.115399\pi$
$547$	43.7356	1.87000	0.935000	+	0.354649i	$0.115399\pi$
$548$	0	0
$549$	−25.0687	−1.06991
$550$	0	0
$551$	−19.9661	−0.850586
$552$	0	0
$553$	−5.64885	−0.240214
$554$	0	0
$555$	15.8572	0.673103
$556$	0	0
$557$	44.5322	1.88689	0.943445	−	0.331530i	$-0.107565\pi$
$557$	44.5322	1.88689	0.943445	+	0.331530i	$0.107565\pi$
$558$	0	0
$559$	9.92180	0.419647
$560$	0	0
$561$	11.5706	0.488511
$562$	0	0
$563$	34.5317	1.45534	0.727670	−	0.685928i	$-0.240603\pi$
$563$	34.5317	1.45534	0.727670	+	0.685928i	$0.240603\pi$
$564$	0	0
$565$	−8.61030	−0.362238
$566$	0	0
$567$	−10.1925	−0.428043
$568$	0	0
$569$	30.6699	1.28575	0.642874	−	0.765972i	$-0.277742\pi$
$569$	30.6699	1.28575	0.642874	+	0.765972i	$0.277742\pi$
$570$	0	0
$571$	40.9073	1.71192	0.855959	−	0.517044i	$-0.172968\pi$
$571$	40.9073	1.71192	0.855959	+	0.517044i	$0.172968\pi$
$572$	0	0
$573$	24.5766	1.02670
$574$	0	0
$575$	−9.39195	−0.391671
$576$	0	0
$577$	21.8264	0.908646	0.454323	−	0.890837i	$-0.349881\pi$
$577$	21.8264	0.908646	0.454323	+	0.890837i	$0.349881\pi$
$578$	0	0
$579$	−32.2908	−1.34196
$580$	0	0
$581$	−0.995232	−0.0412892
$582$	0	0
$583$	−30.8302	−1.27686
$584$	0	0
$585$	−39.5885	−1.63678
$586$	0	0
$587$	11.8400	0.488687	0.244344	−	0.969689i	$-0.421427\pi$
$587$	11.8400	0.488687	0.244344	+	0.969689i	$0.421427\pi$
$588$	0	0
$589$	−6.30325	−0.259721
$590$	0	0
$591$	5.71121	0.234928
$592$	0	0
$593$	−7.04852	−0.289448	−0.144724	−	0.989472i	$-0.546229\pi$
$593$	−7.04852	−0.289448	−0.144724	+	0.989472i	$0.546229\pi$
$594$	0	0
$595$	−5.63309	−0.230934
$596$	0	0
$597$	−28.4312	−1.16361
$598$	0	0
$599$	−0.847547	−0.0346298	−0.0173149	−	0.999850i	$-0.505512\pi$
$599$	−0.847547	−0.0346298	−0.0173149	+	0.999850i	$0.505512\pi$
$600$	0	0
$601$	−17.1433	−0.699290	−0.349645	−	0.936882i	$-0.613698\pi$
$601$	−17.1433	−0.699290	−0.349645	+	0.936882i	$0.613698\pi$
$602$	0	0
$603$	6.22588	0.253538
$604$	0	0
$605$	12.2675	0.498746
$606$	0	0
$607$	−20.8526	−0.846379	−0.423190	−	0.906041i	$-0.639090\pi$
$607$	−20.8526	−0.846379	−0.423190	+	0.906041i	$0.639090\pi$
$608$	0	0
$609$	10.6830	0.432897
$610$	0	0
$611$	54.8611	2.21944
$612$	0	0
$613$	23.4367	0.946600	0.473300	−	0.880901i	$-0.343063\pi$
$613$	23.4367	0.946600	0.473300	+	0.880901i	$0.343063\pi$
$614$	0	0
$615$	−56.6514	−2.28441
$616$	0	0
$617$	−5.31002	−0.213773	−0.106887	−	0.994271i	$-0.534088\pi$
$617$	−5.31002	−0.213773	−0.106887	+	0.994271i	$0.534088\pi$
$618$	0	0
$619$	−31.5861	−1.26955	−0.634776	−	0.772696i	$-0.718908\pi$
$619$	−31.5861	−1.26955	−0.634776	+	0.772696i	$0.718908\pi$
$620$	0	0
$621$	−4.53120	−0.181831
$622$	0	0
$623$	−6.67235	−0.267322
$624$	0	0
$625$	−26.1644	−1.04657
$626$	0	0
$627$	22.8353	0.911953
$628$	0	0
$629$	4.44365	0.177180
$630$	0	0
$631$	6.73995	0.268313	0.134157	−	0.990960i	$-0.457167\pi$
$631$	6.73995	0.268313	0.134157	+	0.990960i	$0.457167\pi$
$632$	0	0
$633$	21.3033	0.846730
$634$	0	0
$635$	5.24480	0.208134
$636$	0	0
$637$	−39.5216	−1.56590
$638$	0	0
$639$	−6.39354	−0.252924
$640$	0	0
$641$	44.1076	1.74215	0.871073	−	0.491153i	$-0.163425\pi$
$641$	44.1076	1.74215	0.871073	+	0.491153i	$0.163425\pi$
$642$	0	0
$643$	38.8226	1.53101	0.765507	−	0.643427i	$-0.222488\pi$
$643$	38.8226	1.53101	0.765507	+	0.643427i	$0.222488\pi$
$644$	0	0
$645$	10.7442	0.423054
$646$	0	0
$647$	−45.4490	−1.78679	−0.893393	−	0.449276i	$-0.851682\pi$
$647$	−45.4490	−1.78679	−0.893393	+	0.449276i	$0.851682\pi$
$648$	0	0
$649$	−5.44504	−0.213737
$650$	0	0
$651$	3.37259	0.132182
$652$	0	0
$653$	−2.07672	−0.0812683	−0.0406341	−	0.999174i	$-0.512938\pi$
$653$	−2.07672	−0.0812683	−0.0406341	+	0.999174i	$0.512938\pi$
$654$	0	0
$655$	−12.8147	−0.500712
$656$	0	0
$657$	10.6244	0.414499
$658$	0	0
$659$	−23.4328	−0.912811	−0.456405	−	0.889772i	$-0.650863\pi$
$659$	−23.4328	−0.912811	−0.456405	+	0.889772i	$0.650863\pi$
$660$	0	0
$661$	−1.14268	−0.0444451	−0.0222226	−	0.999753i	$-0.507074\pi$
$661$	−1.14268	−0.0444451	−0.0222226	+	0.999753i	$0.507074\pi$
$662$	0	0
$663$	−27.9790	−1.08661
$664$	0	0
$665$	−11.1172	−0.431108
$666$	0	0
$667$	10.2396	0.396480
$668$	0	0
$669$	34.7233	1.34248
$670$	0	0
$671$	33.8232	1.30573
$672$	0	0
$673$	−12.5724	−0.484630	−0.242315	−	0.970198i	$-0.577907\pi$
$673$	−12.5724	−0.484630	−0.242315	+	0.970198i	$0.577907\pi$
$674$	0	0
$675$	10.9089	0.419886
$676$	0	0
$677$	22.1422	0.850992	0.425496	−	0.904960i	$-0.360100\pi$
$677$	22.1422	0.850992	0.425496	+	0.904960i	$0.360100\pi$
$678$	0	0
$679$	−3.61136	−0.138591
$680$	0	0
$681$	2.04429	0.0783375
$682$	0	0
$683$	−2.20276	−0.0842862	−0.0421431	−	0.999112i	$-0.513419\pi$
$683$	−2.20276	−0.0842862	−0.0421431	+	0.999112i	$0.513419\pi$
$684$	0	0
$685$	−35.8159	−1.36846
$686$	0	0
$687$	−46.7316	−1.78292
$688$	0	0
$689$	74.5508	2.84016
$690$	0	0
$691$	−44.8271	−1.70530	−0.852651	−	0.522481i	$-0.825007\pi$
$691$	−44.8271	−1.70530	−0.852651	+	0.522481i	$0.825007\pi$
$692$	0	0
$693$	−4.84456	−0.184029
$694$	0	0
$695$	−60.1509	−2.28166
$696$	0	0
$697$	−15.8753	−0.601321
$698$	0	0
$699$	−23.2521	−0.879477
$700$	0	0
$701$	−6.33268	−0.239182	−0.119591	−	0.992823i	$-0.538158\pi$
$701$	−6.33268	−0.239182	−0.119591	+	0.992823i	$0.538158\pi$
$702$	0	0
$703$	8.76980	0.330759
$704$	0	0
$705$	59.4086	2.23746
$706$	0	0
$707$	−9.28869	−0.349337
$708$	0	0
$709$	4.96193	0.186349	0.0931745	−	0.995650i	$-0.470299\pi$
$709$	4.96193	0.186349	0.0931745	+	0.995650i	$0.470299\pi$
$710$	0	0
$711$	−12.0470	−0.451798
$712$	0	0
$713$	3.23262	0.121063
$714$	0	0
$715$	53.4136	1.99756
$716$	0	0
$717$	2.75327	0.102823
$718$	0	0
$719$	37.6713	1.40490	0.702452	−	0.711731i	$-0.252089\pi$
$719$	37.6713	1.40490	0.702452	+	0.711731i	$0.252089\pi$
$720$	0	0
$721$	−2.08146	−0.0775177
$722$	0	0
$723$	−4.01177	−0.149199
$724$	0	0
$725$	−24.6521	−0.915556
$726$	0	0
$727$	−34.4148	−1.27637	−0.638187	−	0.769881i	$-0.720315\pi$
$727$	−34.4148	−1.27637	−0.638187	+	0.769881i	$0.720315\pi$
$728$	0	0
$729$	−6.39198	−0.236740
$730$	0	0
$731$	3.01084	0.111360
$732$	0	0
$733$	−13.2150	−0.488107	−0.244054	−	0.969762i	$-0.578477\pi$
$733$	−13.2150	−0.488107	−0.244054	+	0.969762i	$0.578477\pi$
$734$	0	0
$735$	−42.7976	−1.57861
$736$	0	0
$737$	−8.40009	−0.309421
$738$	0	0
$739$	12.8665	0.473302	0.236651	−	0.971595i	$-0.423950\pi$
$739$	12.8665	0.473302	0.236651	+	0.971595i	$0.423950\pi$
$740$	0	0
$741$	−55.2182	−2.02849
$742$	0	0
$743$	25.1035	0.920958	0.460479	−	0.887671i	$-0.347678\pi$
$743$	25.1035	0.920958	0.460479	+	0.887671i	$0.347678\pi$
$744$	0	0
$745$	−23.7938	−0.871738
$746$	0	0
$747$	−2.12248	−0.0776575
$748$	0	0
$749$	−12.4944	−0.456536
$750$	0	0
$751$	47.8949	1.74771	0.873855	−	0.486186i	$-0.161612\pi$
$751$	47.8949	1.74771	0.873855	+	0.486186i	$0.161612\pi$
$752$	0	0
$753$	−12.6453	−0.460819
$754$	0	0
$755$	−29.8593	−1.08669
$756$	0	0
$757$	−44.7254	−1.62557	−0.812786	−	0.582562i	$-0.802050\pi$
$757$	−44.7254	−1.62557	−0.812786	+	0.582562i	$0.802050\pi$
$758$	0	0
$759$	−11.7111	−0.425085
$760$	0	0
$761$	−36.5827	−1.32612	−0.663061	−	0.748566i	$-0.730743\pi$
$761$	−36.5827	−1.32612	−0.663061	+	0.748566i	$0.730743\pi$
$762$	0	0
$763$	−13.0153	−0.471185
$764$	0	0
$765$	−12.0134	−0.434345
$766$	0	0
$767$	13.1667	0.475422
$768$	0	0
$769$	−19.9914	−0.720909	−0.360454	−	0.932777i	$-0.617378\pi$
$769$	−19.9914	−0.720909	−0.360454	+	0.932777i	$0.617378\pi$
$770$	0	0
$771$	16.9272	0.609618
$772$	0	0
$773$	−41.3344	−1.48669	−0.743347	−	0.668906i	$-0.766763\pi$
$773$	−41.3344	−1.48669	−0.743347	+	0.668906i	$0.766763\pi$
$774$	0	0
$775$	−7.78261	−0.279559
$776$	0	0
$777$	−4.69233	−0.168337
$778$	0	0
$779$	−31.3309	−1.12255
$780$	0	0
$781$	8.62629	0.308673
$782$	0	0
$783$	−11.8935	−0.425041
$784$	0	0
$785$	−64.0452	−2.28587
$786$	0	0
$787$	50.5238	1.80098	0.900489	−	0.434878i	$-0.143209\pi$
$787$	50.5238	1.80098	0.900489	+	0.434878i	$0.143209\pi$
$788$	0	0
$789$	−36.4823	−1.29881
$790$	0	0
$791$	2.54788	0.0905923
$792$	0	0
$793$	−81.7882	−2.90439
$794$	0	0
$795$	80.7304	2.86321
$796$	0	0
$797$	15.8959	0.563060	0.281530	−	0.959552i	$-0.409158\pi$
$797$	15.8959	0.563060	0.281530	+	0.959552i	$0.409158\pi$
$798$	0	0
$799$	16.6480	0.588963
$800$	0	0
$801$	−14.2298	−0.502784
$802$	0	0
$803$	−14.3347	−0.505861
$804$	0	0
$805$	5.70147	0.200951
$806$	0	0
$807$	−43.6038	−1.53493
$808$	0	0
$809$	1.83889	0.0646518	0.0323259	−	0.999477i	$-0.489709\pi$
$809$	1.83889	0.0646518	0.0323259	+	0.999477i	$0.489709\pi$
$810$	0	0
$811$	−26.5177	−0.931161	−0.465580	−	0.885006i	$-0.654154\pi$
$811$	−26.5177	−0.931161	−0.465580	+	0.885006i	$0.654154\pi$
$812$	0	0
$813$	−19.1922	−0.673099
$814$	0	0
$815$	−26.3874	−0.924309
$816$	0	0
$817$	5.94207	0.207887
$818$	0	0
$819$	11.7147	0.409344
$820$	0	0
$821$	27.2828	0.952177	0.476088	−	0.879398i	$-0.342054\pi$
$821$	27.2828	0.952177	0.476088	+	0.879398i	$0.342054\pi$
$822$	0	0
$823$	33.8050	1.17837	0.589185	−	0.807998i	$-0.299449\pi$
$823$	33.8050	1.17837	0.589185	+	0.807998i	$0.299449\pi$
$824$	0	0
$825$	28.1946	0.981611
$826$	0	0
$827$	6.77654	0.235643	0.117822	−	0.993035i	$-0.462409\pi$
$827$	6.77654	0.235643	0.117822	+	0.993035i	$0.462409\pi$
$828$	0	0
$829$	8.15715	0.283309	0.141655	−	0.989916i	$-0.454758\pi$
$829$	8.15715	0.283309	0.141655	+	0.989916i	$0.454758\pi$
$830$	0	0
$831$	34.8303	1.20825
$832$	0	0
$833$	−11.9931	−0.415536
$834$	0	0
$835$	−16.8828	−0.584254
$836$	0	0
$837$	−3.75476	−0.129784
$838$	0	0
$839$	−10.1109	−0.349067	−0.174533	−	0.984651i	$-0.555842\pi$
$839$	−10.1109	−0.349067	−0.174533	+	0.984651i	$0.555842\pi$
$840$	0	0
$841$	−2.12288	−0.0732029
$842$	0	0
$843$	37.5014	1.29162
$844$	0	0
$845$	−88.5568	−3.04644
$846$	0	0
$847$	−3.63010	−0.124732
$848$	0	0
$849$	−59.4471	−2.04022
$850$	0	0
$851$	−4.49759	−0.154176
$852$	0	0
$853$	13.5690	0.464595	0.232298	−	0.972645i	$-0.425376\pi$
$853$	13.5690	0.464595	0.232298	+	0.972645i	$0.425376\pi$
$854$	0	0
$855$	−23.7091	−0.810836
$856$	0	0
$857$	−18.0261	−0.615758	−0.307879	−	0.951425i	$-0.599619\pi$
$857$	−18.0261	−0.615758	−0.307879	+	0.951425i	$0.599619\pi$
$858$	0	0
$859$	−47.7315	−1.62858	−0.814288	−	0.580461i	$-0.802872\pi$
$859$	−47.7315	−1.62858	−0.814288	+	0.580461i	$0.802872\pi$
$860$	0	0
$861$	16.7638	0.571308
$862$	0	0
$863$	−5.16838	−0.175934	−0.0879668	−	0.996123i	$-0.528037\pi$
$863$	−5.16838	−0.175934	−0.0879668	+	0.996123i	$0.528037\pi$
$864$	0	0
$865$	60.9912	2.07376
$866$	0	0
$867$	29.4125	0.998902
$868$	0	0
$869$	16.2541	0.551382
$870$	0	0
$871$	20.3123	0.688257
$872$	0	0
$873$	−7.70175	−0.260665
$874$	0	0
$875$	0.706842	0.0238956
$876$	0	0
$877$	3.13247	0.105776	0.0528880	−	0.998600i	$-0.483157\pi$
$877$	3.13247	0.105776	0.0528880	+	0.998600i	$0.483157\pi$
$878$	0	0
$879$	42.1711	1.42239
$880$	0	0
$881$	−42.6209	−1.43593	−0.717967	−	0.696077i	$-0.754927\pi$
$881$	−42.6209	−1.43593	−0.717967	+	0.696077i	$0.754927\pi$
$882$	0	0
$883$	27.9559	0.940789	0.470395	−	0.882456i	$-0.344112\pi$
$883$	27.9559	0.940789	0.470395	+	0.882456i	$0.344112\pi$
$884$	0	0
$885$	14.2581	0.479281
$886$	0	0
$887$	−41.0574	−1.37857	−0.689286	−	0.724489i	$-0.742076\pi$
$887$	−41.0574	−1.37857	−0.689286	+	0.724489i	$0.742076\pi$
$888$	0	0
$889$	−1.55200	−0.0520522
$890$	0	0
$891$	29.3279	0.982522
$892$	0	0
$893$	32.8557	1.09948
$894$	0	0
$895$	15.1432	0.506182
$896$	0	0
$897$	28.3186	0.945532
$898$	0	0
$899$	8.48503	0.282992
$900$	0	0
$901$	22.6229	0.753679
$902$	0	0
$903$	−3.17934	−0.105802
$904$	0	0
$905$	−54.7026	−1.81837
$906$	0	0
$907$	−44.2282	−1.46857	−0.734286	−	0.678840i	$-0.762483\pi$
$907$	−44.2282	−1.46857	−0.734286	+	0.678840i	$0.762483\pi$
$908$	0	0
$909$	−19.8095	−0.657040
$910$	0	0
$911$	24.2469	0.803335	0.401667	−	0.915786i	$-0.368431\pi$
$911$	24.2469	0.803335	0.401667	+	0.915786i	$0.368431\pi$
$912$	0	0
$913$	2.86369	0.0947744
$914$	0	0
$915$	−88.5678	−2.92796
$916$	0	0
$917$	3.79201	0.125223
$918$	0	0
$919$	−5.31529	−0.175335	−0.0876677	−	0.996150i	$-0.527941\pi$
$919$	−5.31529	−0.175335	−0.0876677	+	0.996150i	$0.527941\pi$
$920$	0	0
$921$	−25.1761	−0.829580
$922$	0	0
$923$	−20.8593	−0.686593
$924$	0	0
$925$	10.8280	0.356024
$926$	0	0
$927$	−4.43902	−0.145797
$928$	0	0
$929$	26.6315	0.873751	0.436875	−	0.899522i	$-0.356085\pi$
$929$	26.6315	0.873751	0.436875	+	0.899522i	$0.356085\pi$
$930$	0	0
$931$	−23.6691	−0.775723
$932$	0	0
$933$	47.1939	1.54506
$934$	0	0
$935$	16.2087	0.530082
$936$	0	0
$937$	13.4108	0.438112	0.219056	−	0.975712i	$-0.429702\pi$
$937$	13.4108	0.438112	0.219056	+	0.975712i	$0.429702\pi$
$938$	0	0
$939$	36.6544	1.19617
$940$	0	0
$941$	−31.2688	−1.01933	−0.509666	−	0.860372i	$-0.670231\pi$
$941$	−31.2688	−1.01933	−0.509666	+	0.860372i	$0.670231\pi$
$942$	0	0
$943$	16.0681	0.523248
$944$	0	0
$945$	−6.62238	−0.215426
$946$	0	0
$947$	32.9546	1.07088	0.535440	−	0.844573i	$-0.320146\pi$
$947$	32.9546	1.07088	0.535440	+	0.844573i	$0.320146\pi$
$948$	0	0
$949$	34.6629	1.12521
$950$	0	0
$951$	4.17523	0.135391
$952$	0	0
$953$	25.1250	0.813878	0.406939	−	0.913455i	$-0.366596\pi$
$953$	25.1250	0.813878	0.406939	+	0.913455i	$0.366596\pi$
$954$	0	0
$955$	34.4282	1.11407
$956$	0	0
$957$	−30.7394	−0.993662
$958$	0	0
$959$	10.5983	0.342238
$960$	0	0
$961$	−28.3213	−0.913590
$962$	0	0
$963$	−26.6462	−0.858661
$964$	0	0
$965$	−45.2347	−1.45616
$966$	0	0
$967$	12.5694	0.404204	0.202102	−	0.979365i	$-0.435223\pi$
$967$	12.5694	0.404204	0.202102	+	0.979365i	$0.435223\pi$
$968$	0	0
$969$	−16.7563	−0.538291
$970$	0	0
$971$	−56.3517	−1.80841	−0.904207	−	0.427095i	$-0.859537\pi$
$971$	−56.3517	−1.80841	−0.904207	+	0.427095i	$0.859537\pi$
$972$	0	0
$973$	17.7993	0.570620
$974$	0	0
$975$	−68.1777	−2.18343
$976$	0	0
$977$	9.81957	0.314156	0.157078	−	0.987586i	$-0.449793\pi$
$977$	9.81957	0.314156	0.157078	+	0.987586i	$0.449793\pi$
$978$	0	0
$979$	19.1991	0.613606
$980$	0	0
$981$	−27.7570	−0.886213
$982$	0	0
$983$	−28.7958	−0.918443	−0.459221	−	0.888322i	$-0.651872\pi$
$983$	−28.7958	−0.918443	−0.459221	+	0.888322i	$0.651872\pi$
$984$	0	0
$985$	8.00056	0.254919
$986$	0	0
$987$	−17.5797	−0.559567
$988$	0	0
$989$	−3.04739	−0.0969013
$990$	0	0
$991$	6.38780	0.202915	0.101458	−	0.994840i	$-0.467649\pi$
$991$	6.38780	0.202915	0.101458	+	0.994840i	$0.467649\pi$
$992$	0	0
$993$	−80.1484	−2.54343
$994$	0	0
$995$	−39.8279	−1.26263
$996$	0	0
$997$	−47.7982	−1.51378	−0.756892	−	0.653540i	$-0.773283\pi$
$997$	−47.7982	−1.51378	−0.756892	+	0.653540i	$0.773283\pi$
$998$	0	0
$999$	5.22405	0.165282

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.13	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.13	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22958 q^{3} -3.12332 q^{5} +0.924225 q^{7} +1.97105 q^{9} +O(q^{10})\)	\(q-2.22958 q^{3} -3.12332 q^{5} +0.924225 q^{7} +1.97105 q^{9} -2.65938 q^{11} +6.43066 q^{13} +6.96371 q^{15} +1.95143 q^{17} +3.85126 q^{19} -2.06064 q^{21} -1.97512 q^{23} +4.75513 q^{25} +2.29414 q^{27} -5.18431 q^{29} -1.63667 q^{31} +5.92930 q^{33} -2.88665 q^{35} +2.27713 q^{37} -14.3377 q^{39} -8.13524 q^{41} +1.54289 q^{43} -6.15621 q^{45} +8.53117 q^{47} -6.14581 q^{49} -4.35087 q^{51} +11.5930 q^{53} +8.30608 q^{55} -8.58670 q^{57} +2.04749 q^{59} -12.7185 q^{61} +1.82169 q^{63} -20.0850 q^{65} +3.15867 q^{67} +4.40369 q^{69} -3.24373 q^{71} +5.39026 q^{73} -10.6020 q^{75} -2.45786 q^{77} -6.11199 q^{79} -11.0281 q^{81} -1.07683 q^{83} -6.09493 q^{85} +11.5589 q^{87} -7.21940 q^{89} +5.94338 q^{91} +3.64910 q^{93} -12.0287 q^{95} -3.90744 q^{97} -5.24175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Introduction

L-functions

Modular forms

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Database

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