LMFDB - Embedded newform 6028.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00219 q^{3} +0.977497 q^{5} -3.03319 q^{7} +1.00878 q^{9} +O(q^{10})$	$q-2.00219 q^{3} +0.977497 q^{5} -3.03319 q^{7} +1.00878 q^{9} +1.00000 q^{11} -4.12665 q^{13} -1.95714 q^{15} -0.328908 q^{17} -8.36999 q^{19} +6.07303 q^{21} +2.85060 q^{23} -4.04450 q^{25} +3.98681 q^{27} +1.72310 q^{29} +5.02515 q^{31} -2.00219 q^{33} -2.96493 q^{35} -3.87184 q^{37} +8.26235 q^{39} -12.5021 q^{41} -10.5086 q^{43} +0.986076 q^{45} +5.81610 q^{47} +2.20023 q^{49} +0.658537 q^{51} -13.3047 q^{53} +0.977497 q^{55} +16.7583 q^{57} -4.50069 q^{59} +8.17394 q^{61} -3.05981 q^{63} -4.03379 q^{65} +6.37395 q^{67} -5.70746 q^{69} +14.0597 q^{71} -2.78699 q^{73} +8.09787 q^{75} -3.03319 q^{77} -4.40230 q^{79} -11.0087 q^{81} +7.44808 q^{83} -0.321506 q^{85} -3.44997 q^{87} -15.2301 q^{89} +12.5169 q^{91} -10.0613 q^{93} -8.18164 q^{95} +13.5413 q^{97} +1.00878 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	$ 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) $ 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * 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
$3$	−2.00219	−1.15597	−0.577983	−	0.816049i	$-0.696160\pi$
$3$	−2.00219	−1.15597	−0.577983	+	0.816049i	$0.696160\pi$
$4$	0	0
$5$	0.977497	0.437150	0.218575	−	0.975820i	$-0.429859\pi$
$5$	0.977497	0.437150	0.218575	+	0.975820i	$0.429859\pi$
$6$	0	0
$7$	−3.03319	−1.14644	−0.573219	−	0.819402i	$-0.694305\pi$
$7$	−3.03319	−1.14644	−0.573219	+	0.819402i	$0.694305\pi$
$8$	0	0
$9$	1.00878	0.336259
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.12665	−1.14453	−0.572264	−	0.820070i	$-0.693935\pi$
$13$	−4.12665	−1.14453	−0.572264	+	0.820070i	$0.693935\pi$
$14$	0	0
$15$	−1.95714	−0.505331
$16$	0	0
$17$	−0.328908	−0.0797719	−0.0398859	−	0.999204i	$-0.512699\pi$
$17$	−0.328908	−0.0797719	−0.0398859	+	0.999204i	$0.512699\pi$
$18$	0	0
$19$	−8.36999	−1.92021	−0.960104	−	0.279645i	$-0.909783\pi$
$19$	−8.36999	−1.92021	−0.960104	+	0.279645i	$0.909783\pi$
$20$	0	0
$21$	6.07303	1.32524
$22$	0	0
$23$	2.85060	0.594392	0.297196	−	0.954817i	$-0.403949\pi$
$23$	2.85060	0.594392	0.297196	+	0.954817i	$0.403949\pi$
$24$	0	0
$25$	−4.04450	−0.808900
$26$	0	0
$27$	3.98681	0.767263
$28$	0	0
$29$	1.72310	0.319971	0.159986	−	0.987119i	$-0.448855\pi$
$29$	1.72310	0.319971	0.159986	+	0.987119i	$0.448855\pi$
$30$	0	0
$31$	5.02515	0.902544	0.451272	−	0.892386i	$-0.350970\pi$
$31$	5.02515	0.902544	0.451272	+	0.892386i	$0.350970\pi$
$32$	0	0
$33$	−2.00219	−0.348537
$34$	0	0
$35$	−2.96493	−0.501165
$36$	0	0
$37$	−3.87184	−0.636527	−0.318264	−	0.948002i	$-0.603100\pi$
$37$	−3.87184	−0.636527	−0.318264	+	0.948002i	$0.603100\pi$
$38$	0	0
$39$	8.26235	1.32304
$40$	0	0
$41$	−12.5021	−1.95250	−0.976248	−	0.216656i	$-0.930485\pi$
$41$	−12.5021	−1.95250	−0.976248	+	0.216656i	$0.930485\pi$
$42$	0	0
$43$	−10.5086	−1.60254	−0.801272	−	0.598300i	$-0.795843\pi$
$43$	−10.5086	−1.60254	−0.801272	+	0.598300i	$0.795843\pi$
$44$	0	0
$45$	0.986076	0.146996
$46$	0	0
$47$	5.81610	0.848365	0.424183	−	0.905577i	$-0.360561\pi$
$47$	5.81610	0.848365	0.424183	+	0.905577i	$0.360561\pi$
$48$	0	0
$49$	2.20023	0.314319
$50$	0	0
$51$	0.658537	0.0922136
$52$	0	0
$53$	−13.3047	−1.82754	−0.913771	−	0.406230i	$-0.866843\pi$
$53$	−13.3047	−1.82754	−0.913771	+	0.406230i	$0.866843\pi$
$54$	0	0
$55$	0.977497	0.131806
$56$	0	0
$57$	16.7583	2.21970
$58$	0	0
$59$	−4.50069	−0.585940	−0.292970	−	0.956122i	$-0.594644\pi$
$59$	−4.50069	−0.585940	−0.292970	+	0.956122i	$0.594644\pi$
$60$	0	0
$61$	8.17394	1.04657	0.523283	−	0.852159i	$-0.324707\pi$
$61$	8.17394	1.04657	0.523283	+	0.852159i	$0.324707\pi$
$62$	0	0
$63$	−3.05981	−0.385500
$64$	0	0
$65$	−4.03379	−0.500330
$66$	0	0
$67$	6.37395	0.778702	0.389351	−	0.921090i	$-0.372699\pi$
$67$	6.37395	0.778702	0.389351	+	0.921090i	$0.372699\pi$
$68$	0	0
$69$	−5.70746	−0.687097
$70$	0	0
$71$	14.0597	1.66857	0.834287	−	0.551330i	$-0.185880\pi$
$71$	14.0597	1.66857	0.834287	+	0.551330i	$0.185880\pi$
$72$	0	0
$73$	−2.78699	−0.326192	−0.163096	−	0.986610i	$-0.552148\pi$
$73$	−2.78699	−0.326192	−0.163096	+	0.986610i	$0.552148\pi$
$74$	0	0
$75$	8.09787	0.935061
$76$	0	0
$77$	−3.03319	−0.345664
$78$	0	0
$79$	−4.40230	−0.495297	−0.247649	−	0.968850i	$-0.579658\pi$
$79$	−4.40230	−0.495297	−0.247649	+	0.968850i	$0.579658\pi$
$80$	0	0
$81$	−11.0087	−1.22319
$82$	0	0
$83$	7.44808	0.817533	0.408767	−	0.912639i	$-0.365959\pi$
$83$	7.44808	0.817533	0.408767	+	0.912639i	$0.365959\pi$
$84$	0	0
$85$	−0.321506	−0.0348723
$86$	0	0
$87$	−3.44997	−0.369876
$88$	0	0
$89$	−15.2301	−1.61438	−0.807192	−	0.590288i	$-0.799014\pi$
$89$	−15.2301	−1.61438	−0.807192	+	0.590288i	$0.799014\pi$
$90$	0	0
$91$	12.5169	1.31213
$92$	0	0
$93$	−10.0613	−1.04331
$94$	0	0
$95$	−8.18164	−0.839418
$96$	0	0
$97$	13.5413	1.37491	0.687456	−	0.726226i	$-0.258727\pi$
$97$	13.5413	1.37491	0.687456	+	0.726226i	$0.258727\pi$
$98$	0	0
$99$	1.00878	0.101386
$100$	0	0
$101$	−5.59880	−0.557102	−0.278551	−	0.960421i	$-0.589854\pi$
$101$	−5.59880	−0.557102	−0.278551	+	0.960421i	$0.589854\pi$
$102$	0	0
$103$	18.2952	1.80268	0.901338	−	0.433117i	$-0.142586\pi$
$103$	18.2952	1.80268	0.901338	+	0.433117i	$0.142586\pi$
$104$	0	0
$105$	5.93637	0.579330
$106$	0	0
$107$	−3.55388	−0.343567	−0.171783	−	0.985135i	$-0.554953\pi$
$107$	−3.55388	−0.343567	−0.171783	+	0.985135i	$0.554953\pi$
$108$	0	0
$109$	12.8476	1.23058	0.615288	−	0.788302i	$-0.289040\pi$
$109$	12.8476	1.23058	0.615288	+	0.788302i	$0.289040\pi$
$110$	0	0
$111$	7.75218	0.735804
$112$	0	0
$113$	−19.4453	−1.82926	−0.914630	−	0.404293i	$-0.867518\pi$
$113$	−19.4453	−1.82926	−0.914630	+	0.404293i	$0.867518\pi$
$114$	0	0
$115$	2.78645	0.259838
$116$	0	0
$117$	−4.16287	−0.384858
$118$	0	0
$119$	0.997639	0.0914535
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	25.0316	2.25702
$124$	0	0
$125$	−8.84097	−0.790760
$126$	0	0
$127$	16.2060	1.43805	0.719027	−	0.694983i	$-0.244588\pi$
$127$	16.2060	1.43805	0.719027	+	0.694983i	$0.244588\pi$
$128$	0	0
$129$	21.0402	1.85249
$130$	0	0
$131$	−20.9613	−1.83139	−0.915697	−	0.401870i	$-0.868360\pi$
$131$	−20.9613	−1.83139	−0.915697	+	0.401870i	$0.868360\pi$
$132$	0	0
$133$	25.3878	2.20140
$134$	0	0
$135$	3.89710	0.335409
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−18.7907	−1.59381	−0.796904	−	0.604106i	$-0.793530\pi$
$139$	−18.7907	−1.59381	−0.796904	+	0.604106i	$0.793530\pi$
$140$	0	0
$141$	−11.6450	−0.980682
$142$	0	0
$143$	−4.12665	−0.345088
$144$	0	0
$145$	1.68432	0.139875
$146$	0	0
$147$	−4.40529	−0.363343
$148$	0	0
$149$	−9.74776	−0.798567	−0.399284	−	0.916827i	$-0.630741\pi$
$149$	−9.74776	−0.798567	−0.399284	+	0.916827i	$0.630741\pi$
$150$	0	0
$151$	−11.5230	−0.937726	−0.468863	−	0.883271i	$-0.655336\pi$
$151$	−11.5230	−0.937726	−0.468863	+	0.883271i	$0.655336\pi$
$152$	0	0
$153$	−0.331795	−0.0268240
$154$	0	0
$155$	4.91207	0.394547
$156$	0	0
$157$	17.2000	1.37271	0.686356	−	0.727266i	$-0.259209\pi$
$157$	17.2000	1.37271	0.686356	+	0.727266i	$0.259209\pi$
$158$	0	0
$159$	26.6386	2.11258
$160$	0	0
$161$	−8.64641	−0.681433
$162$	0	0
$163$	9.54825	0.747876	0.373938	−	0.927454i	$-0.378007\pi$
$163$	9.54825	0.747876	0.373938	+	0.927454i	$0.378007\pi$
$164$	0	0
$165$	−1.95714	−0.152363
$166$	0	0
$167$	11.0646	0.856207	0.428103	−	0.903730i	$-0.359182\pi$
$167$	11.0646	0.856207	0.428103	+	0.903730i	$0.359182\pi$
$168$	0	0
$169$	4.02925	0.309942
$170$	0	0
$171$	−8.44345	−0.645687
$172$	0	0
$173$	7.86589	0.598033	0.299016	−	0.954248i	$-0.403341\pi$
$173$	7.86589	0.598033	0.299016	+	0.954248i	$0.403341\pi$
$174$	0	0
$175$	12.2677	0.927353
$176$	0	0
$177$	9.01125	0.677327
$178$	0	0
$179$	13.1201	0.980645	0.490322	−	0.871541i	$-0.336879\pi$
$179$	13.1201	0.980645	0.490322	+	0.871541i	$0.336879\pi$
$180$	0	0
$181$	−3.19900	−0.237780	−0.118890	−	0.992907i	$-0.537934\pi$
$181$	−3.19900	−0.237780	−0.118890	+	0.992907i	$0.537934\pi$
$182$	0	0
$183$	−16.3658	−1.20979
$184$	0	0
$185$	−3.78472	−0.278258
$186$	0	0
$187$	−0.328908	−0.0240521
$188$	0	0
$189$	−12.0928	−0.879619
$190$	0	0
$191$	2.31823	0.167741	0.0838706	−	0.996477i	$-0.473272\pi$
$191$	2.31823	0.167741	0.0838706	+	0.996477i	$0.473272\pi$
$192$	0	0
$193$	−19.9716	−1.43759	−0.718793	−	0.695224i	$-0.755305\pi$
$193$	−19.9716	−1.43759	−0.718793	+	0.695224i	$0.755305\pi$
$194$	0	0
$195$	8.07642	0.578365
$196$	0	0
$197$	5.99517	0.427138	0.213569	−	0.976928i	$-0.431491\pi$
$197$	5.99517	0.427138	0.213569	+	0.976928i	$0.431491\pi$
$198$	0	0
$199$	1.48710	0.105418	0.0527090	−	0.998610i	$-0.483214\pi$
$199$	1.48710	0.105418	0.0527090	+	0.998610i	$0.483214\pi$
$200$	0	0
$201$	−12.7619	−0.900153
$202$	0	0
$203$	−5.22648	−0.366827
$204$	0	0
$205$	−12.2207	−0.853533
$206$	0	0
$207$	2.87562	0.199870
$208$	0	0
$209$	−8.36999	−0.578964
$210$	0	0
$211$	−20.8499	−1.43537	−0.717684	−	0.696369i	$-0.754798\pi$
$211$	−20.8499	−1.43537	−0.717684	+	0.696369i	$0.754798\pi$
$212$	0	0
$213$	−28.1501	−1.92882
$214$	0	0
$215$	−10.2721	−0.700552
$216$	0	0
$217$	−15.2422	−1.03471
$218$	0	0
$219$	5.58009	0.377068
$220$	0	0
$221$	1.35729	0.0913011
$222$	0	0
$223$	−17.6100	−1.17925	−0.589625	−	0.807677i	$-0.700725\pi$
$223$	−17.6100	−1.17925	−0.589625	+	0.807677i	$0.700725\pi$
$224$	0	0
$225$	−4.08000	−0.272000
$226$	0	0
$227$	−7.20216	−0.478024	−0.239012	−	0.971017i	$-0.576824\pi$
$227$	−7.20216	−0.478024	−0.239012	+	0.971017i	$0.576824\pi$
$228$	0	0
$229$	16.2665	1.07492	0.537459	−	0.843290i	$-0.319384\pi$
$229$	16.2665	1.07492	0.537459	+	0.843290i	$0.319384\pi$
$230$	0	0
$231$	6.07303	0.399576
$232$	0	0
$233$	−6.64502	−0.435330	−0.217665	−	0.976024i	$-0.569844\pi$
$233$	−6.64502	−0.435330	−0.217665	+	0.976024i	$0.569844\pi$
$234$	0	0
$235$	5.68522	0.370863
$236$	0	0
$237$	8.81425	0.572547
$238$	0	0
$239$	22.7141	1.46925	0.734627	−	0.678471i	$-0.237357\pi$
$239$	22.7141	1.46925	0.734627	+	0.678471i	$0.237357\pi$
$240$	0	0
$241$	18.8268	1.21274	0.606371	−	0.795182i	$-0.292625\pi$
$241$	18.8268	1.21274	0.606371	+	0.795182i	$0.292625\pi$
$242$	0	0
$243$	10.0811	0.646703
$244$	0	0
$245$	2.15072	0.137405
$246$	0	0
$247$	34.5400	2.19773
$248$	0	0
$249$	−14.9125	−0.945041
$250$	0	0
$251$	3.39230	0.214120	0.107060	−	0.994253i	$-0.465856\pi$
$251$	3.39230	0.214120	0.107060	+	0.994253i	$0.465856\pi$
$252$	0	0
$253$	2.85060	0.179216
$254$	0	0
$255$	0.643718	0.0403112
$256$	0	0
$257$	24.7136	1.54159	0.770796	−	0.637083i	$-0.219859\pi$
$257$	24.7136	1.54159	0.770796	+	0.637083i	$0.219859\pi$
$258$	0	0
$259$	11.7440	0.729739
$260$	0	0
$261$	1.73822	0.107593
$262$	0	0
$263$	15.9557	0.983871	0.491936	−	0.870632i	$-0.336290\pi$
$263$	15.9557	0.983871	0.491936	+	0.870632i	$0.336290\pi$
$264$	0	0
$265$	−13.0053	−0.798910
$266$	0	0
$267$	30.4936	1.86618
$268$	0	0
$269$	−5.78458	−0.352692	−0.176346	−	0.984328i	$-0.556428\pi$
$269$	−5.78458	−0.352692	−0.176346	+	0.984328i	$0.556428\pi$
$270$	0	0
$271$	−6.88186	−0.418043	−0.209022	−	0.977911i	$-0.567028\pi$
$271$	−6.88186	−0.418043	−0.209022	+	0.977911i	$0.567028\pi$
$272$	0	0
$273$	−25.0613	−1.51678
$274$	0	0
$275$	−4.04450	−0.243893
$276$	0	0
$277$	−13.0005	−0.781125	−0.390563	−	0.920576i	$-0.627720\pi$
$277$	−13.0005	−0.781125	−0.390563	+	0.920576i	$0.627720\pi$
$278$	0	0
$279$	5.06926	0.303489
$280$	0	0
$281$	16.3084	0.972878	0.486439	−	0.873715i	$-0.338296\pi$
$281$	16.3084	0.972878	0.486439	+	0.873715i	$0.338296\pi$
$282$	0	0
$283$	15.1041	0.897847	0.448923	−	0.893570i	$-0.351808\pi$
$283$	15.1041	0.897847	0.448923	+	0.893570i	$0.351808\pi$
$284$	0	0
$285$	16.3812	0.970340
$286$	0	0
$287$	37.9212	2.23841
$288$	0	0
$289$	−16.8918	−0.993636
$290$	0	0
$291$	−27.1123	−1.58935
$292$	0	0
$293$	15.6206	0.912568	0.456284	−	0.889834i	$-0.349180\pi$
$293$	15.6206	0.912568	0.456284	+	0.889834i	$0.349180\pi$
$294$	0	0
$295$	−4.39941	−0.256144
$296$	0	0
$297$	3.98681	0.231338
$298$	0	0
$299$	−11.7634	−0.680297
$300$	0	0
$301$	31.8745	1.83722
$302$	0	0
$303$	11.2099	0.643991
$304$	0	0
$305$	7.99000	0.457506
$306$	0	0
$307$	8.85495	0.505378	0.252689	−	0.967547i	$-0.418685\pi$
$307$	8.85495	0.505378	0.252689	+	0.967547i	$0.418685\pi$
$308$	0	0
$309$	−36.6304	−2.08383
$310$	0	0
$311$	−15.2638	−0.865533	−0.432766	−	0.901506i	$-0.642462\pi$
$311$	−15.2638	−0.865533	−0.432766	+	0.901506i	$0.642462\pi$
$312$	0	0
$313$	23.5499	1.33112	0.665559	−	0.746345i	$-0.268193\pi$
$313$	23.5499	1.33112	0.665559	+	0.746345i	$0.268193\pi$
$314$	0	0
$315$	−2.99096	−0.168521
$316$	0	0
$317$	18.4581	1.03671	0.518354	−	0.855166i	$-0.326545\pi$
$317$	18.4581	1.03671	0.518354	+	0.855166i	$0.326545\pi$
$318$	0	0
$319$	1.72310	0.0964749
$320$	0	0
$321$	7.11556	0.397152
$322$	0	0
$323$	2.75295	0.153179
$324$	0	0
$325$	16.6902	0.925808
$326$	0	0
$327$	−25.7234	−1.42251
$328$	0	0
$329$	−17.6413	−0.972598
$330$	0	0
$331$	−20.2499	−1.11303	−0.556517	−	0.830836i	$-0.687862\pi$
$331$	−20.2499	−1.11303	−0.556517	+	0.830836i	$0.687862\pi$
$332$	0	0
$333$	−3.90583	−0.214038
$334$	0	0
$335$	6.23052	0.340409
$336$	0	0
$337$	8.87065	0.483215	0.241608	−	0.970374i	$-0.422325\pi$
$337$	8.87065	0.483215	0.241608	+	0.970374i	$0.422325\pi$
$338$	0	0
$339$	38.9332	2.11456
$340$	0	0
$341$	5.02515	0.272127
$342$	0	0
$343$	14.5586	0.786090
$344$	0	0
$345$	−5.57902	−0.300364
$346$	0	0
$347$	3.02668	0.162481	0.0812404	−	0.996695i	$-0.474112\pi$
$347$	3.02668	0.162481	0.0812404	+	0.996695i	$0.474112\pi$
$348$	0	0
$349$	14.9226	0.798791	0.399395	−	0.916779i	$-0.369220\pi$
$349$	14.9226	0.798791	0.399395	+	0.916779i	$0.369220\pi$
$350$	0	0
$351$	−16.4522	−0.878153
$352$	0	0
$353$	20.2247	1.07645	0.538227	−	0.842800i	$-0.319094\pi$
$353$	20.2247	1.07645	0.538227	+	0.842800i	$0.319094\pi$
$354$	0	0
$355$	13.7433	0.729417
$356$	0	0
$357$	−1.99747	−0.105717
$358$	0	0
$359$	13.0082	0.686548	0.343274	−	0.939235i	$-0.388464\pi$
$359$	13.0082	0.686548	0.343274	+	0.939235i	$0.388464\pi$
$360$	0	0
$361$	51.0567	2.68720
$362$	0	0
$363$	−2.00219	−0.105088
$364$	0	0
$365$	−2.72427	−0.142595
$366$	0	0
$367$	8.55269	0.446447	0.223223	−	0.974767i	$-0.428342\pi$
$367$	8.55269	0.446447	0.223223	+	0.974767i	$0.428342\pi$
$368$	0	0
$369$	−12.6118	−0.656544
$370$	0	0
$371$	40.3557	2.09516
$372$	0	0
$373$	−11.7305	−0.607382	−0.303691	−	0.952771i	$-0.598219\pi$
$373$	−11.7305	−0.607382	−0.303691	+	0.952771i	$0.598219\pi$
$374$	0	0
$375$	17.7013	0.914093
$376$	0	0
$377$	−7.11062	−0.366216
$378$	0	0
$379$	20.9245	1.07482	0.537411	−	0.843321i	$-0.319403\pi$
$379$	20.9245	1.07482	0.537411	+	0.843321i	$0.319403\pi$
$380$	0	0
$381$	−32.4476	−1.66234
$382$	0	0
$383$	−6.27724	−0.320752	−0.160376	−	0.987056i	$-0.551271\pi$
$383$	−6.27724	−0.320752	−0.160376	+	0.987056i	$0.551271\pi$
$384$	0	0
$385$	−2.96493	−0.151107
$386$	0	0
$387$	−10.6008	−0.538870
$388$	0	0
$389$	−19.3789	−0.982551	−0.491276	−	0.871004i	$-0.663469\pi$
$389$	−19.3789	−0.982551	−0.491276	+	0.871004i	$0.663469\pi$
$390$	0	0
$391$	−0.937585	−0.0474157
$392$	0	0
$393$	41.9685	2.11703
$394$	0	0
$395$	−4.30323	−0.216519
$396$	0	0
$397$	20.1112	1.00935	0.504677	−	0.863308i	$-0.331612\pi$
$397$	20.1112	1.00935	0.504677	+	0.863308i	$0.331612\pi$
$398$	0	0
$399$	−50.8312	−2.54474
$400$	0	0
$401$	−15.4657	−0.772321	−0.386160	−	0.922432i	$-0.626199\pi$
$401$	−15.4657	−0.772321	−0.386160	+	0.922432i	$0.626199\pi$
$402$	0	0
$403$	−20.7371	−1.03299
$404$	0	0
$405$	−10.7610	−0.534717
$406$	0	0
$407$	−3.87184	−0.191920
$408$	0	0
$409$	−2.32781	−0.115103	−0.0575514	−	0.998343i	$-0.518329\pi$
$409$	−2.32781	−0.115103	−0.0575514	+	0.998343i	$0.518329\pi$
$410$	0	0
$411$	−2.00219	−0.0987609
$412$	0	0
$413$	13.6514	0.671743
$414$	0	0
$415$	7.28048	0.357385
$416$	0	0
$417$	37.6227	1.84239
$418$	0	0
$419$	−3.20899	−0.156769	−0.0783847	−	0.996923i	$-0.524976\pi$
$419$	−3.20899	−0.156769	−0.0783847	+	0.996923i	$0.524976\pi$
$420$	0	0
$421$	−4.39568	−0.214232	−0.107116	−	0.994247i	$-0.534162\pi$
$421$	−4.39568	−0.214232	−0.107116	+	0.994247i	$0.534162\pi$
$422$	0	0
$423$	5.86715	0.285270
$424$	0	0
$425$	1.33027	0.0645275
$426$	0	0
$427$	−24.7931	−1.19982
$428$	0	0
$429$	8.26235	0.398910
$430$	0	0
$431$	−8.09241	−0.389798	−0.194899	−	0.980823i	$-0.562438\pi$
$431$	−8.09241	−0.389798	−0.194899	+	0.980823i	$0.562438\pi$
$432$	0	0
$433$	−6.07296	−0.291848	−0.145924	−	0.989296i	$-0.546615\pi$
$433$	−6.07296	−0.291848	−0.145924	+	0.989296i	$0.546615\pi$
$434$	0	0
$435$	−3.37234	−0.161691
$436$	0	0
$437$	−23.8595	−1.14136
$438$	0	0
$439$	28.4709	1.35884	0.679422	−	0.733748i	$-0.262231\pi$
$439$	28.4709	1.35884	0.679422	+	0.733748i	$0.262231\pi$
$440$	0	0
$441$	2.21955	0.105693
$442$	0	0
$443$	16.1690	0.768213	0.384107	−	0.923289i	$-0.374510\pi$
$443$	16.1690	0.768213	0.384107	+	0.923289i	$0.374510\pi$
$444$	0	0
$445$	−14.8874	−0.705728
$446$	0	0
$447$	19.5169	0.923117
$448$	0	0
$449$	15.0987	0.712551	0.356276	−	0.934381i	$-0.384046\pi$
$449$	15.0987	0.712551	0.356276	+	0.934381i	$0.384046\pi$
$450$	0	0
$451$	−12.5021	−0.588700
$452$	0	0
$453$	23.0712	1.08398
$454$	0	0
$455$	12.2352	0.573597
$456$	0	0
$457$	1.75977	0.0823188	0.0411594	−	0.999153i	$-0.486895\pi$
$457$	1.75977	0.0823188	0.0411594	+	0.999153i	$0.486895\pi$
$458$	0	0
$459$	−1.31129	−0.0612060
$460$	0	0
$461$	29.4532	1.37177	0.685887	−	0.727708i	$-0.259414\pi$
$461$	29.4532	1.37177	0.685887	+	0.727708i	$0.259414\pi$
$462$	0	0
$463$	−11.1996	−0.520491	−0.260245	−	0.965543i	$-0.583803\pi$
$463$	−11.1996	−0.520491	−0.260245	+	0.965543i	$0.583803\pi$
$464$	0	0
$465$	−9.83492	−0.456083
$466$	0	0
$467$	4.64000	0.214713	0.107357	−	0.994221i	$-0.465761\pi$
$467$	4.64000	0.214713	0.107357	+	0.994221i	$0.465761\pi$
$468$	0	0
$469$	−19.3334	−0.892733
$470$	0	0
$471$	−34.4378	−1.58681
$472$	0	0
$473$	−10.5086	−0.483185
$474$	0	0
$475$	33.8524	1.55326
$476$	0	0
$477$	−13.4215	−0.614527
$478$	0	0
$479$	−22.7125	−1.03776	−0.518880	−	0.854847i	$-0.673651\pi$
$479$	−22.7125	−1.03776	−0.518880	+	0.854847i	$0.673651\pi$
$480$	0	0
$481$	15.9778	0.728523
$482$	0	0
$483$	17.3118	0.787714
$484$	0	0
$485$	13.2366	0.601043
$486$	0	0
$487$	17.5797	0.796614	0.398307	−	0.917252i	$-0.369598\pi$
$487$	17.5797	0.796614	0.398307	+	0.917252i	$0.369598\pi$
$488$	0	0
$489$	−19.1174	−0.864520
$490$	0	0
$491$	21.0696	0.950860	0.475430	−	0.879754i	$-0.342293\pi$
$491$	21.0696	0.950860	0.475430	+	0.879754i	$0.342293\pi$
$492$	0	0
$493$	−0.566740	−0.0255247
$494$	0	0
$495$	0.986076	0.0443208
$496$	0	0
$497$	−42.6456	−1.91292
$498$	0	0
$499$	37.3214	1.67074	0.835369	−	0.549690i	$-0.185254\pi$
$499$	37.3214	1.67074	0.835369	+	0.549690i	$0.185254\pi$
$500$	0	0
$501$	−22.1535	−0.989747
$502$	0	0
$503$	−23.5940	−1.05200	−0.526002	−	0.850483i	$-0.676310\pi$
$503$	−23.5940	−1.05200	−0.526002	+	0.850483i	$0.676310\pi$
$504$	0	0
$505$	−5.47281	−0.243537
$506$	0	0
$507$	−8.06734	−0.358283
$508$	0	0
$509$	−0.134884	−0.00597864	−0.00298932	−	0.999996i	$-0.500952\pi$
$509$	−0.134884	−0.00597864	−0.00298932	+	0.999996i	$0.500952\pi$
$510$	0	0
$511$	8.45347	0.373959
$512$	0	0
$513$	−33.3696	−1.47330
$514$	0	0
$515$	17.8835	0.788039
$516$	0	0
$517$	5.81610	0.255792
$518$	0	0
$519$	−15.7490	−0.691306
$520$	0	0
$521$	38.7979	1.69977	0.849883	−	0.526972i	$-0.176673\pi$
$521$	38.7979	1.69977	0.849883	+	0.526972i	$0.176673\pi$
$522$	0	0
$523$	2.45738	0.107454	0.0537270	−	0.998556i	$-0.482890\pi$
$523$	2.45738	0.107454	0.0537270	+	0.998556i	$0.482890\pi$
$524$	0	0
$525$	−24.5624	−1.07199
$526$	0	0
$527$	−1.65281	−0.0719976
$528$	0	0
$529$	−14.8741	−0.646699
$530$	0	0
$531$	−4.54019	−0.197027
$532$	0	0
$533$	51.5917	2.23468
$534$	0	0
$535$	−3.47391	−0.150190
$536$	0	0
$537$	−26.2690	−1.13359
$538$	0	0
$539$	2.20023	0.0947708
$540$	0	0
$541$	−29.7999	−1.28120	−0.640599	−	0.767875i	$-0.721314\pi$
$541$	−29.7999	−1.28120	−0.640599	+	0.767875i	$0.721314\pi$
$542$	0	0
$543$	6.40502	0.274866
$544$	0	0
$545$	12.5585	0.537946
$546$	0	0
$547$	−30.5484	−1.30615	−0.653077	−	0.757292i	$-0.726522\pi$
$547$	−30.5484	−1.30615	−0.653077	+	0.757292i	$0.726522\pi$
$548$	0	0
$549$	8.24568	0.351917
$550$	0	0
$551$	−14.4223	−0.614411
$552$	0	0
$553$	13.3530	0.567827
$554$	0	0
$555$	7.57773	0.321657
$556$	0	0
$557$	14.3934	0.609866	0.304933	−	0.952374i	$-0.401366\pi$
$557$	14.3934	0.609866	0.304933	+	0.952374i	$0.401366\pi$
$558$	0	0
$559$	43.3653	1.83416
$560$	0	0
$561$	0.658537	0.0278035
$562$	0	0
$563$	33.9249	1.42976	0.714882	−	0.699245i	$-0.246480\pi$
$563$	33.9249	1.42976	0.714882	+	0.699245i	$0.246480\pi$
$564$	0	0
$565$	−19.0077	−0.799660
$566$	0	0
$567$	33.3915	1.40231
$568$	0	0
$569$	−19.1419	−0.802470	−0.401235	−	0.915975i	$-0.631419\pi$
$569$	−19.1419	−0.802470	−0.401235	+	0.915975i	$0.631419\pi$
$570$	0	0
$571$	16.5075	0.690816	0.345408	−	0.938453i	$-0.387741\pi$
$571$	16.5075	0.690816	0.345408	+	0.938453i	$0.387741\pi$
$572$	0	0
$573$	−4.64154	−0.193903
$574$	0	0
$575$	−11.5293	−0.480803
$576$	0	0
$577$	−17.0245	−0.708740	−0.354370	−	0.935105i	$-0.615305\pi$
$577$	−17.0245	−0.708740	−0.354370	+	0.935105i	$0.615305\pi$
$578$	0	0
$579$	39.9870	1.66180
$580$	0	0
$581$	−22.5914	−0.937251
$582$	0	0
$583$	−13.3047	−0.551025
$584$	0	0
$585$	−4.06919	−0.168240
$586$	0	0
$587$	9.77851	0.403602	0.201801	−	0.979427i	$-0.435321\pi$
$587$	9.77851	0.403602	0.201801	+	0.979427i	$0.435321\pi$
$588$	0	0
$589$	−42.0605	−1.73307
$590$	0	0
$591$	−12.0035	−0.493758
$592$	0	0
$593$	−1.49585	−0.0614274	−0.0307137	−	0.999528i	$-0.509778\pi$
$593$	−1.49585	−0.0614274	−0.0307137	+	0.999528i	$0.509778\pi$
$594$	0	0
$595$	0.975190	0.0399789
$596$	0	0
$597$	−2.97747	−0.121860
$598$	0	0
$599$	−4.61882	−0.188720	−0.0943599	−	0.995538i	$-0.530080\pi$
$599$	−4.61882	−0.188720	−0.0943599	+	0.995538i	$0.530080\pi$
$600$	0	0
$601$	−42.5865	−1.73714	−0.868570	−	0.495567i	$-0.834960\pi$
$601$	−42.5865	−1.73714	−0.868570	+	0.495567i	$0.834960\pi$
$602$	0	0
$603$	6.42989	0.261845
$604$	0	0
$605$	0.977497	0.0397409
$606$	0	0
$607$	−27.4872	−1.11567	−0.557836	−	0.829951i	$-0.688368\pi$
$607$	−27.4872	−1.11567	−0.557836	+	0.829951i	$0.688368\pi$
$608$	0	0
$609$	10.4644	0.424040
$610$	0	0
$611$	−24.0010	−0.970977
$612$	0	0
$613$	−18.2387	−0.736652	−0.368326	−	0.929697i	$-0.620069\pi$
$613$	−18.2387	−0.736652	−0.368326	+	0.929697i	$0.620069\pi$
$614$	0	0
$615$	24.4683	0.986656
$616$	0	0
$617$	−40.9697	−1.64938	−0.824688	−	0.565587i	$-0.808650\pi$
$617$	−40.9697	−1.64938	−0.824688	+	0.565587i	$0.808650\pi$
$618$	0	0
$619$	10.8199	0.434887	0.217443	−	0.976073i	$-0.430228\pi$
$619$	10.8199	0.434887	0.217443	+	0.976073i	$0.430228\pi$
$620$	0	0
$621$	11.3648	0.456054
$622$	0	0
$623$	46.1957	1.85079
$624$	0	0
$625$	11.5805	0.463219
$626$	0	0
$627$	16.7583	0.669263
$628$	0	0
$629$	1.27348	0.0507770
$630$	0	0
$631$	−25.5163	−1.01579	−0.507894	−	0.861420i	$-0.669576\pi$
$631$	−25.5163	−1.01579	−0.507894	+	0.861420i	$0.669576\pi$
$632$	0	0
$633$	41.7456	1.65924
$634$	0	0
$635$	15.8414	0.628645
$636$	0	0
$637$	−9.07960	−0.359747
$638$	0	0
$639$	14.1831	0.561073
$640$	0	0
$641$	40.5303	1.60085	0.800424	−	0.599434i	$-0.204608\pi$
$641$	40.5303	1.60085	0.800424	+	0.599434i	$0.204608\pi$
$642$	0	0
$643$	37.0951	1.46289	0.731444	−	0.681901i	$-0.238847\pi$
$643$	37.0951	1.46289	0.731444	+	0.681901i	$0.238847\pi$
$644$	0	0
$645$	20.5667	0.809815
$646$	0	0
$647$	19.4603	0.765065	0.382533	−	0.923942i	$-0.375052\pi$
$647$	19.4603	0.765065	0.382533	+	0.923942i	$0.375052\pi$
$648$	0	0
$649$	−4.50069	−0.176667
$650$	0	0
$651$	30.5179	1.19609
$652$	0	0
$653$	−22.2969	−0.872544	−0.436272	−	0.899815i	$-0.643702\pi$
$653$	−22.2969	−0.872544	−0.436272	+	0.899815i	$0.643702\pi$
$654$	0	0
$655$	−20.4896	−0.800594
$656$	0	0
$657$	−2.81145	−0.109685
$658$	0	0
$659$	−5.38133	−0.209627	−0.104814	−	0.994492i	$-0.533425\pi$
$659$	−5.38133	−0.209627	−0.104814	+	0.994492i	$0.533425\pi$
$660$	0	0
$661$	−16.1394	−0.627751	−0.313875	−	0.949464i	$-0.601627\pi$
$661$	−16.1394	−0.627751	−0.313875	+	0.949464i	$0.601627\pi$
$662$	0	0
$663$	−2.71755	−0.105541
$664$	0	0
$665$	24.8165	0.962341
$666$	0	0
$667$	4.91186	0.190188
$668$	0	0
$669$	35.2585	1.36317
$670$	0	0
$671$	8.17394	0.315551
$672$	0	0
$673$	−9.52981	−0.367347	−0.183674	−	0.982987i	$-0.558799\pi$
$673$	−9.52981	−0.367347	−0.183674	+	0.982987i	$0.558799\pi$
$674$	0	0
$675$	−16.1247	−0.620639
$676$	0	0
$677$	−26.6949	−1.02597	−0.512984	−	0.858398i	$-0.671460\pi$
$677$	−26.6949	−1.02597	−0.512984	+	0.858398i	$0.671460\pi$
$678$	0	0
$679$	−41.0734	−1.57625
$680$	0	0
$681$	14.4201	0.552580
$682$	0	0
$683$	−13.2958	−0.508749	−0.254374	−	0.967106i	$-0.581870\pi$
$683$	−13.2958	−0.508749	−0.254374	+	0.967106i	$0.581870\pi$
$684$	0	0
$685$	0.977497	0.0373482
$686$	0	0
$687$	−32.5686	−1.24257
$688$	0	0
$689$	54.9039	2.09167
$690$	0	0
$691$	−17.0819	−0.649825	−0.324913	−	0.945744i	$-0.605335\pi$
$691$	−17.0819	−0.649825	−0.324913	+	0.945744i	$0.605335\pi$
$692$	0	0
$693$	−3.05981	−0.116233
$694$	0	0
$695$	−18.3679	−0.696733
$696$	0	0
$697$	4.11203	0.155754
$698$	0	0
$699$	13.3046	0.503227
$700$	0	0
$701$	4.13965	0.156353	0.0781763	−	0.996940i	$-0.475090\pi$
$701$	4.13965	0.156353	0.0781763	+	0.996940i	$0.475090\pi$
$702$	0	0
$703$	32.4073	1.22226
$704$	0	0
$705$	−11.3829	−0.428705
$706$	0	0
$707$	16.9822	0.638683
$708$	0	0
$709$	7.99415	0.300226	0.150113	−	0.988669i	$-0.452036\pi$
$709$	7.99415	0.300226	0.150113	+	0.988669i	$0.452036\pi$
$710$	0	0
$711$	−4.44094	−0.166548
$712$	0	0
$713$	14.3247	0.536465
$714$	0	0
$715$	−4.03379	−0.150855
$716$	0	0
$717$	−45.4780	−1.69841
$718$	0	0
$719$	−23.7584	−0.886040	−0.443020	−	0.896512i	$-0.646093\pi$
$719$	−23.7584	−0.886040	−0.443020	+	0.896512i	$0.646093\pi$
$720$	0	0
$721$	−55.4927	−2.06665
$722$	0	0
$723$	−37.6949	−1.40189
$724$	0	0
$725$	−6.96906	−0.258825
$726$	0	0
$727$	−33.3570	−1.23714	−0.618572	−	0.785728i	$-0.712289\pi$
$727$	−33.3570	−1.23714	−0.618572	+	0.785728i	$0.712289\pi$
$728$	0	0
$729$	12.8418	0.475622
$730$	0	0
$731$	3.45636	0.127838
$732$	0	0
$733$	13.2988	0.491203	0.245601	−	0.969371i	$-0.421015\pi$
$733$	13.2988	0.491203	0.245601	+	0.969371i	$0.421015\pi$
$734$	0	0
$735$	−4.30616	−0.158835
$736$	0	0
$737$	6.37395	0.234787
$738$	0	0
$739$	39.2207	1.44276	0.721379	−	0.692541i	$-0.243509\pi$
$739$	39.2207	1.44276	0.721379	+	0.692541i	$0.243509\pi$
$740$	0	0
$741$	−69.1558	−2.54050
$742$	0	0
$743$	40.2666	1.47724	0.738619	−	0.674123i	$-0.235478\pi$
$743$	40.2666	1.47724	0.738619	+	0.674123i	$0.235478\pi$
$744$	0	0
$745$	−9.52840	−0.349094
$746$	0	0
$747$	7.51346	0.274903
$748$	0	0
$749$	10.7796	0.393878
$750$	0	0
$751$	−29.8856	−1.09054	−0.545271	−	0.838260i	$-0.683573\pi$
$751$	−29.8856	−1.09054	−0.545271	+	0.838260i	$0.683573\pi$
$752$	0	0
$753$	−6.79204	−0.247516
$754$	0	0
$755$	−11.2637	−0.409927
$756$	0	0
$757$	−39.3272	−1.42937	−0.714686	−	0.699445i	$-0.753431\pi$
$757$	−39.3272	−1.42937	−0.714686	+	0.699445i	$0.753431\pi$
$758$	0	0
$759$	−5.70746	−0.207168
$760$	0	0
$761$	15.2195	0.551706	0.275853	−	0.961200i	$-0.411040\pi$
$761$	15.2195	0.551706	0.275853	+	0.961200i	$0.411040\pi$
$762$	0	0
$763$	−38.9692	−1.41078
$764$	0	0
$765$	−0.324328	−0.0117261
$766$	0	0
$767$	18.5728	0.670624
$768$	0	0
$769$	35.2618	1.27157	0.635787	−	0.771865i	$-0.280676\pi$
$769$	35.2618	1.27157	0.635787	+	0.771865i	$0.280676\pi$
$770$	0	0
$771$	−49.4814	−1.78203
$772$	0	0
$773$	−48.1191	−1.73072	−0.865361	−	0.501149i	$-0.832911\pi$
$773$	−48.1191	−1.73072	−0.865361	+	0.501149i	$0.832911\pi$
$774$	0	0
$775$	−20.3242	−0.730068
$776$	0	0
$777$	−23.5138	−0.843554
$778$	0	0
$779$	104.642	3.74920
$780$	0	0
$781$	14.0597	0.503094
$782$	0	0
$783$	6.86966	0.245502
$784$	0	0
$785$	16.8130	0.600081
$786$	0	0
$787$	13.6998	0.488347	0.244173	−	0.969732i	$-0.421483\pi$
$787$	13.6998	0.488347	0.244173	+	0.969732i	$0.421483\pi$
$788$	0	0
$789$	−31.9464	−1.13732
$790$	0	0
$791$	58.9812	2.09713
$792$	0	0
$793$	−33.7310	−1.19782
$794$	0	0
$795$	26.0391	0.923513
$796$	0	0
$797$	−5.92621	−0.209917	−0.104959	−	0.994477i	$-0.533471\pi$
$797$	−5.92621	−0.209917	−0.104959	+	0.994477i	$0.533471\pi$
$798$	0	0
$799$	−1.91296	−0.0676757
$800$	0	0
$801$	−15.3638	−0.542851
$802$	0	0
$803$	−2.78699	−0.0983507
$804$	0	0
$805$	−8.45184	−0.297888
$806$	0	0
$807$	11.5818	0.407700
$808$	0	0
$809$	−17.0552	−0.599629	−0.299814	−	0.953998i	$-0.596925\pi$
$809$	−17.0552	−0.599629	−0.299814	+	0.953998i	$0.596925\pi$
$810$	0	0
$811$	22.9900	0.807289	0.403644	−	0.914916i	$-0.367743\pi$
$811$	22.9900	0.807289	0.403644	+	0.914916i	$0.367743\pi$
$812$	0	0
$813$	13.7788	0.483244
$814$	0	0
$815$	9.33338	0.326934
$816$	0	0
$817$	87.9568	3.07722
$818$	0	0
$819$	12.6268	0.441215
$820$	0	0
$821$	−0.539841	−0.0188406	−0.00942029	−	0.999956i	$-0.502999\pi$
$821$	−0.539841	−0.0188406	−0.00942029	+	0.999956i	$0.502999\pi$
$822$	0	0
$823$	17.0813	0.595418	0.297709	−	0.954657i	$-0.403777\pi$
$823$	17.0813	0.595418	0.297709	+	0.954657i	$0.403777\pi$
$824$	0	0
$825$	8.09787	0.281932
$826$	0	0
$827$	−16.2959	−0.566666	−0.283333	−	0.959022i	$-0.591440\pi$
$827$	−16.2959	−0.566666	−0.283333	+	0.959022i	$0.591440\pi$
$828$	0	0
$829$	42.4427	1.47410	0.737048	−	0.675840i	$-0.236219\pi$
$829$	42.4427	1.47410	0.737048	+	0.675840i	$0.236219\pi$
$830$	0	0
$831$	26.0295	0.902955
$832$	0	0
$833$	−0.723674	−0.0250738
$834$	0	0
$835$	10.8156	0.374291
$836$	0	0
$837$	20.0343	0.692488
$838$	0	0
$839$	−25.8648	−0.892950	−0.446475	−	0.894796i	$-0.647321\pi$
$839$	−25.8648	−0.892950	−0.446475	+	0.894796i	$0.647321\pi$
$840$	0	0
$841$	−26.0309	−0.897619
$842$	0	0
$843$	−32.6526	−1.12461
$844$	0	0
$845$	3.93858	0.135491
$846$	0	0
$847$	−3.03319	−0.104222
$848$	0	0
$849$	−30.2414	−1.03788
$850$	0	0
$851$	−11.0371	−0.378347
$852$	0	0
$853$	41.1039	1.40737	0.703685	−	0.710512i	$-0.251537\pi$
$853$	41.1039	1.40737	0.703685	+	0.710512i	$0.251537\pi$
$854$	0	0
$855$	−8.25345	−0.282262
$856$	0	0
$857$	40.5227	1.38423	0.692115	−	0.721787i	$-0.256679\pi$
$857$	40.5227	1.38423	0.692115	+	0.721787i	$0.256679\pi$
$858$	0	0
$859$	−25.0346	−0.854168	−0.427084	−	0.904212i	$-0.640459\pi$
$859$	−25.0346	−0.854168	−0.427084	+	0.904212i	$0.640459\pi$
$860$	0	0
$861$	−75.9255	−2.58753
$862$	0	0
$863$	26.7230	0.909662	0.454831	−	0.890578i	$-0.349700\pi$
$863$	26.7230	0.909662	0.454831	+	0.890578i	$0.349700\pi$
$864$	0	0
$865$	7.68888	0.261430
$866$	0	0
$867$	33.8207	1.14861
$868$	0	0
$869$	−4.40230	−0.149338
$870$	0	0
$871$	−26.3031	−0.891245
$872$	0	0
$873$	13.6602	0.462326
$874$	0	0
$875$	26.8163	0.906558
$876$	0	0
$877$	33.5316	1.13228	0.566140	−	0.824309i	$-0.308436\pi$
$877$	33.5316	1.13228	0.566140	+	0.824309i	$0.308436\pi$
$878$	0	0
$879$	−31.2756	−1.05490
$880$	0	0
$881$	40.2863	1.35728	0.678641	−	0.734470i	$-0.262569\pi$
$881$	40.2863	1.35728	0.678641	+	0.734470i	$0.262569\pi$
$882$	0	0
$883$	0.202900	0.00682812	0.00341406	−	0.999994i	$-0.498913\pi$
$883$	0.202900	0.00682812	0.00341406	+	0.999994i	$0.498913\pi$
$884$	0	0
$885$	8.80847	0.296093
$886$	0	0
$887$	−16.5107	−0.554376	−0.277188	−	0.960816i	$-0.589402\pi$
$887$	−16.5107	−0.554376	−0.277188	+	0.960816i	$0.589402\pi$
$888$	0	0
$889$	−49.1560	−1.64864
$890$	0	0
$891$	−11.0087	−0.368805
$892$	0	0
$893$	−48.6807	−1.62904
$894$	0	0
$895$	12.8249	0.428689
$896$	0	0
$897$	23.5527	0.786401
$898$	0	0
$899$	8.65883	0.288788
$900$	0	0
$901$	4.37602	0.145786
$902$	0	0
$903$	−63.8190	−2.12376
$904$	0	0
$905$	−3.12701	−0.103945
$906$	0	0
$907$	−46.2763	−1.53658	−0.768290	−	0.640102i	$-0.778892\pi$
$907$	−46.2763	−1.53658	−0.768290	+	0.640102i	$0.778892\pi$
$908$	0	0
$909$	−5.64794	−0.187331
$910$	0	0
$911$	−22.1774	−0.734770	−0.367385	−	0.930069i	$-0.619747\pi$
$911$	−22.1774	−0.734770	−0.367385	+	0.930069i	$0.619747\pi$
$912$	0	0
$913$	7.44808	0.246496
$914$	0	0
$915$	−15.9975	−0.528862
$916$	0	0
$917$	63.5795	2.09958
$918$	0	0
$919$	26.2421	0.865646	0.432823	−	0.901479i	$-0.357518\pi$
$919$	26.2421	0.865646	0.432823	+	0.901479i	$0.357518\pi$
$920$	0	0
$921$	−17.7293	−0.584201
$922$	0	0
$923$	−58.0193	−1.90973
$924$	0	0
$925$	15.6597	0.514887
$926$	0	0
$927$	18.4557	0.606166
$928$	0	0
$929$	46.7248	1.53299	0.766495	−	0.642250i	$-0.221999\pi$
$929$	46.7248	1.53299	0.766495	+	0.642250i	$0.221999\pi$
$930$	0	0
$931$	−18.4159	−0.603558
$932$	0	0
$933$	30.5611	1.00053
$934$	0	0
$935$	−0.321506	−0.0105144
$936$	0	0
$937$	−14.4290	−0.471373	−0.235687	−	0.971829i	$-0.575734\pi$
$937$	−14.4290	−0.471373	−0.235687	+	0.971829i	$0.575734\pi$
$938$	0	0
$939$	−47.1514	−1.53873
$940$	0	0
$941$	−42.4566	−1.38404	−0.692022	−	0.721876i	$-0.743280\pi$
$941$	−42.4566	−1.38404	−0.692022	+	0.721876i	$0.743280\pi$
$942$	0	0
$943$	−35.6384	−1.16055
$944$	0	0
$945$	−11.8206	−0.384525
$946$	0	0
$947$	−59.4759	−1.93271	−0.966354	−	0.257215i	$-0.917195\pi$
$947$	−59.4759	−1.93271	−0.966354	+	0.257215i	$0.917195\pi$
$948$	0	0
$949$	11.5009	0.373336
$950$	0	0
$951$	−36.9566	−1.19840
$952$	0	0
$953$	−22.7903	−0.738250	−0.369125	−	0.929380i	$-0.620343\pi$
$953$	−22.7903	−0.738250	−0.369125	+	0.929380i	$0.620343\pi$
$954$	0	0
$955$	2.26606	0.0733281
$956$	0	0
$957$	−3.44997	−0.111522
$958$	0	0
$959$	−3.03319	−0.0979468
$960$	0	0
$961$	−5.74784	−0.185414
$962$	0	0
$963$	−3.58507	−0.115527
$964$	0	0
$965$	−19.5222	−0.628441
$966$	0	0
$967$	8.35057	0.268536	0.134268	−	0.990945i	$-0.457132\pi$
$967$	8.35057	0.268536	0.134268	+	0.990945i	$0.457132\pi$
$968$	0	0
$969$	−5.51195	−0.177069
$970$	0	0
$971$	−31.0833	−0.997512	−0.498756	−	0.866742i	$-0.666210\pi$
$971$	−31.0833	−0.997512	−0.498756	+	0.866742i	$0.666210\pi$
$972$	0	0
$973$	56.9958	1.82720
$974$	0	0
$975$	−33.4171	−1.07020
$976$	0	0
$977$	−16.6117	−0.531455	−0.265727	−	0.964048i	$-0.585612\pi$
$977$	−16.6117	−0.531455	−0.265727	+	0.964048i	$0.585612\pi$
$978$	0	0
$979$	−15.2301	−0.486755
$980$	0	0
$981$	12.9604	0.413792
$982$	0	0
$983$	28.2096	0.899748	0.449874	−	0.893092i	$-0.351469\pi$
$983$	28.2096	0.899748	0.449874	+	0.893092i	$0.351469\pi$
$984$	0	0
$985$	5.86026	0.186724
$986$	0	0
$987$	35.3213	1.12429
$988$	0	0
$989$	−29.9558	−0.952539
$990$	0	0
$991$	58.7449	1.86609	0.933047	−	0.359755i	$-0.117140\pi$
$991$	58.7449	1.86609	0.933047	+	0.359755i	$0.117140\pi$
$992$	0	0
$993$	40.5441	1.28663
$994$	0	0
$995$	1.45364	0.0460835
$996$	0	0
$997$	2.42598	0.0768317	0.0384158	−	0.999262i	$-0.487769\pi$
$997$	2.42598	0.0768317	0.0384158	+	0.999262i	$0.487769\pi$
$998$	0	0
$999$	−15.4363	−0.488384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.f.1.5	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.f.1.5	✓	29	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00219 q^{3} +0.977497 q^{5} -3.03319 q^{7} +1.00878 q^{9} +O(q^{10})\)	\(q-2.00219 q^{3} +0.977497 q^{5} -3.03319 q^{7} +1.00878 q^{9} +1.00000 q^{11} -4.12665 q^{13} -1.95714 q^{15} -0.328908 q^{17} -8.36999 q^{19} +6.07303 q^{21} +2.85060 q^{23} -4.04450 q^{25} +3.98681 q^{27} +1.72310 q^{29} +5.02515 q^{31} -2.00219 q^{33} -2.96493 q^{35} -3.87184 q^{37} +8.26235 q^{39} -12.5021 q^{41} -10.5086 q^{43} +0.986076 q^{45} +5.81610 q^{47} +2.20023 q^{49} +0.658537 q^{51} -13.3047 q^{53} +0.977497 q^{55} +16.7583 q^{57} -4.50069 q^{59} +8.17394 q^{61} -3.05981 q^{63} -4.03379 q^{65} +6.37395 q^{67} -5.70746 q^{69} +14.0597 q^{71} -2.78699 q^{73} +8.09787 q^{75} -3.03319 q^{77} -4.40230 q^{79} -11.0087 q^{81} +7.44808 q^{83} -0.321506 q^{85} -3.44997 q^{87} -15.2301 q^{89} +12.5169 q^{91} -10.0613 q^{93} -8.18164 q^{95} +13.5413 q^{97} +1.00878 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9}+O(q^{10}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9	\( 29 q + 14 q^{3} + 9 q^{5} + 14 q^{7} + 43 q^{9} + 29 q^{11} + 10 q^{15} + 29 q^{17} + 7 q^{19} + 2 q^{21} + 36 q^{23} + 36 q^{25} + 50 q^{27} + 9 q^{29} + 28 q^{31} + 14 q^{33} + 15 q^{35} + 25 q^{37} + 9 q^{39} + 19 q^{41} + 23 q^{43} + 5 q^{45} + 27 q^{47} + 27 q^{49} + 13 q^{51} + 4 q^{53} + 9 q^{55} + 14 q^{57} + 40 q^{59} + 20 q^{61} - 17 q^{63} + 9 q^{65} + 59 q^{67} + 30 q^{69} + 29 q^{71} - 5 q^{73} + 46 q^{75} + 14 q^{77} + 29 q^{79} + 61 q^{81} + 35 q^{83} - 57 q^{85} + 45 q^{87} + 39 q^{89} + 45 q^{91} - 8 q^{93} + q^{95} + 55 q^{97} + 43 q^{99}+O(q^{100}) \) 29 * q + 14 * q^3 + 9 * q^5 + 14 * q^7 + 43 * q^9 + 29 * q^11 + 10 * q^15 + 29 * q^17 + 7 * q^19 + 2 * q^21 + 36 * q^23 + 36 * q^25 + 50 * q^27 + 9 * q^29 + 28 * q^31 + 14 * q^33 + 15 * q^35 + 25 * q^37 + 9 * q^39 + 19 * q^41 + 23 * q^43 + 5 * q^45 + 27 * q^47 + 27 * q^49 + 13 * q^51 + 4 * q^53 + 9 * q^55 + 14 * q^57 + 40 * q^59 + 20 * q^61 - 17 * q^63 + 9 * q^65 + 59 * q^67 + 30 * q^69 + 29 * q^71 - 5 * q^73 + 46 * q^75 + 14 * q^77 + 29 * q^79 + 61 * q^81 + 35 * q^83 - 57 * q^85 + 45 * q^87 + 39 * q^89 + 45 * q^91 - 8 * q^93 + q^95 + 55 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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