LMFDB - Embedded newform 6040.2.a.q.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.32609 q^{3} -1.00000 q^{5} +2.45028 q^{7} -1.24148 q^{9} +O(q^{10})$	$q-1.32609 q^{3} -1.00000 q^{5} +2.45028 q^{7} -1.24148 q^{9} -3.05619 q^{11} -6.89432 q^{13} +1.32609 q^{15} -1.76603 q^{17} +0.405973 q^{19} -3.24929 q^{21} +9.08386 q^{23} +1.00000 q^{25} +5.62459 q^{27} +10.5088 q^{29} +8.61528 q^{31} +4.05279 q^{33} -2.45028 q^{35} -3.78048 q^{37} +9.14249 q^{39} -3.40559 q^{41} -1.15467 q^{43} +1.24148 q^{45} +3.98738 q^{47} -0.996141 q^{49} +2.34191 q^{51} +10.8106 q^{53} +3.05619 q^{55} -0.538357 q^{57} -0.272318 q^{59} +3.90654 q^{61} -3.04198 q^{63} +6.89432 q^{65} -7.45708 q^{67} -12.0460 q^{69} -13.0724 q^{71} -11.2825 q^{73} -1.32609 q^{75} -7.48852 q^{77} -4.68384 q^{79} -3.73427 q^{81} -10.5763 q^{83} +1.76603 q^{85} -13.9356 q^{87} +5.06349 q^{89} -16.8930 q^{91} -11.4246 q^{93} -0.405973 q^{95} +4.82364 q^{97} +3.79422 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 4 q^{3} - 23 q^{5} - 9 q^{7} + 37 q^{9}+O(q^{10}) $ 23 * q - 4 * q^3 - 23 * q^5 - 9 * q^7 + 37 * q^9	$ 23 q - 4 q^{3} - 23 q^{5} - 9 q^{7} + 37 q^{9} - 17 q^{11} - 10 q^{13} + 4 q^{15} - 5 q^{21} - 17 q^{23} + 23 q^{25} - 16 q^{27} - 5 q^{29} - 2 q^{31} - 3 q^{33} + 9 q^{35} - 10 q^{37} - 19 q^{39} - 6 q^{41} - 23 q^{43} - 37 q^{45} - 20 q^{47} + 68 q^{49} - 32 q^{51} - 23 q^{53} + 17 q^{55} + 8 q^{57} - 13 q^{59} + 4 q^{61} - 33 q^{63} + 10 q^{65} - 23 q^{67} + 27 q^{69} - 64 q^{71} - 5 q^{73} - 4 q^{75} - 50 q^{77} - 2 q^{79} + 47 q^{81} - 22 q^{83} - 38 q^{87} + q^{89} - 6 q^{91} - 43 q^{93} + 31 q^{97} - 68 q^{99}+O(q^{100}) $ 23 * q - 4 * q^3 - 23 * q^5 - 9 * q^7 + 37 * q^9 - 17 * q^11 - 10 * q^13 + 4 * q^15 - 5 * q^21 - 17 * q^23 + 23 * q^25 - 16 * q^27 - 5 * q^29 - 2 * q^31 - 3 * q^33 + 9 * q^35 - 10 * q^37 - 19 * q^39 - 6 * q^41 - 23 * q^43 - 37 * q^45 - 20 * q^47 + 68 * q^49 - 32 * q^51 - 23 * q^53 + 17 * q^55 + 8 * q^57 - 13 * q^59 + 4 * q^61 - 33 * q^63 + 10 * q^65 - 23 * q^67 + 27 * q^69 - 64 * q^71 - 5 * q^73 - 4 * q^75 - 50 * q^77 - 2 * q^79 + 47 * q^81 - 22 * q^83 - 38 * q^87 + q^89 - 6 * q^91 - 43 * q^93 + 31 * q^97 - 68 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.32609	−0.765619	−0.382809	−	0.923827i	$-0.625043\pi$
$3$	−1.32609	−0.765619	−0.382809	+	0.923827i	$0.625043\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.45028	0.926118	0.463059	−	0.886328i	$-0.346752\pi$
$7$	2.45028	0.926118	0.463059	+	0.886328i	$0.346752\pi$
$8$	0	0
$9$	−1.24148	−0.413828
$10$	0	0
$11$	−3.05619	−0.921477	−0.460739	−	0.887536i	$-0.652415\pi$
$11$	−3.05619	−0.921477	−0.460739	+	0.887536i	$0.652415\pi$
$12$	0	0
$13$	−6.89432	−1.91214	−0.956070	−	0.293138i	$-0.905300\pi$
$13$	−6.89432	−1.91214	−0.956070	+	0.293138i	$0.905300\pi$
$14$	0	0
$15$	1.32609	0.342395
$16$	0	0
$17$	−1.76603	−0.428325	−0.214162	−	0.976798i	$-0.568702\pi$
$17$	−1.76603	−0.428325	−0.214162	+	0.976798i	$0.568702\pi$
$18$	0	0
$19$	0.405973	0.0931366	0.0465683	−	0.998915i	$-0.485171\pi$
$19$	0.405973	0.0931366	0.0465683	+	0.998915i	$0.485171\pi$
$20$	0	0
$21$	−3.24929	−0.709053
$22$	0	0
$23$	9.08386	1.89412	0.947058	−	0.321062i	$-0.104040\pi$
$23$	9.08386	1.89412	0.947058	+	0.321062i	$0.104040\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.62459	1.08245
$28$	0	0
$29$	10.5088	1.95143	0.975716	−	0.219040i	$-0.0702926\pi$
$29$	10.5088	1.95143	0.975716	+	0.219040i	$0.0702926\pi$
$30$	0	0
$31$	8.61528	1.54735	0.773675	−	0.633583i	$-0.218416\pi$
$31$	8.61528	1.54735	0.773675	+	0.633583i	$0.218416\pi$
$32$	0	0
$33$	4.05279	0.705500
$34$	0	0
$35$	−2.45028	−0.414172
$36$	0	0
$37$	−3.78048	−0.621508	−0.310754	−	0.950490i	$-0.600581\pi$
$37$	−3.78048	−0.621508	−0.310754	+	0.950490i	$0.600581\pi$
$38$	0	0
$39$	9.14249	1.46397
$40$	0	0
$41$	−3.40559	−0.531863	−0.265932	−	0.963992i	$-0.585680\pi$
$41$	−3.40559	−0.531863	−0.265932	+	0.963992i	$0.585680\pi$
$42$	0	0
$43$	−1.15467	−0.176085	−0.0880427	−	0.996117i	$-0.528061\pi$
$43$	−1.15467	−0.176085	−0.0880427	+	0.996117i	$0.528061\pi$
$44$	0	0
$45$	1.24148	0.185069
$46$	0	0
$47$	3.98738	0.581620	0.290810	−	0.956781i	$-0.406075\pi$
$47$	3.98738	0.581620	0.290810	+	0.956781i	$0.406075\pi$
$48$	0	0
$49$	−0.996141	−0.142306
$50$	0	0
$51$	2.34191	0.327933
$52$	0	0
$53$	10.8106	1.48495	0.742475	−	0.669874i	$-0.233652\pi$
$53$	10.8106	1.48495	0.742475	+	0.669874i	$0.233652\pi$
$54$	0	0
$55$	3.05619	0.412097
$56$	0	0
$57$	−0.538357	−0.0713071
$58$	0	0
$59$	−0.272318	−0.0354527	−0.0177264	−	0.999843i	$-0.505643\pi$
$59$	−0.272318	−0.0354527	−0.0177264	+	0.999843i	$0.505643\pi$
$60$	0	0
$61$	3.90654	0.500182	0.250091	−	0.968222i	$-0.419540\pi$
$61$	3.90654	0.500182	0.250091	+	0.968222i	$0.419540\pi$
$62$	0	0
$63$	−3.04198	−0.383253
$64$	0	0
$65$	6.89432	0.855135
$66$	0	0
$67$	−7.45708	−0.911027	−0.455514	−	0.890229i	$-0.650544\pi$
$67$	−7.45708	−0.911027	−0.455514	+	0.890229i	$0.650544\pi$
$68$	0	0
$69$	−12.0460	−1.45017
$70$	0	0
$71$	−13.0724	−1.55141	−0.775706	−	0.631095i	$-0.782606\pi$
$71$	−13.0724	−1.55141	−0.775706	+	0.631095i	$0.782606\pi$
$72$	0	0
$73$	−11.2825	−1.32052	−0.660259	−	0.751038i	$-0.729553\pi$
$73$	−11.2825	−1.32052	−0.660259	+	0.751038i	$0.729553\pi$
$74$	0	0
$75$	−1.32609	−0.153124
$76$	0	0
$77$	−7.48852	−0.853396
$78$	0	0
$79$	−4.68384	−0.526973	−0.263486	−	0.964663i	$-0.584872\pi$
$79$	−4.68384	−0.526973	−0.263486	+	0.964663i	$0.584872\pi$
$80$	0	0
$81$	−3.73427	−0.414918
$82$	0	0
$83$	−10.5763	−1.16090	−0.580449	−	0.814297i	$-0.697123\pi$
$83$	−10.5763	−1.16090	−0.580449	+	0.814297i	$0.697123\pi$
$84$	0	0
$85$	1.76603	0.191553
$86$	0	0
$87$	−13.9356	−1.49405
$88$	0	0
$89$	5.06349	0.536729	0.268364	−	0.963317i	$-0.413517\pi$
$89$	5.06349	0.536729	0.268364	+	0.963317i	$0.413517\pi$
$90$	0	0
$91$	−16.8930	−1.77087
$92$	0	0
$93$	−11.4246	−1.18468
$94$	0	0
$95$	−0.405973	−0.0416519
$96$	0	0
$97$	4.82364	0.489767	0.244883	−	0.969553i	$-0.421250\pi$
$97$	4.82364	0.489767	0.244883	+	0.969553i	$0.421250\pi$
$98$	0	0
$99$	3.79422	0.381333
$100$	0	0
$101$	3.46027	0.344310	0.172155	−	0.985070i	$-0.444927\pi$
$101$	3.46027	0.344310	0.172155	+	0.985070i	$0.444927\pi$
$102$	0	0
$103$	10.8209	1.06621	0.533106	−	0.846049i	$-0.321025\pi$
$103$	10.8209	1.06621	0.533106	+	0.846049i	$0.321025\pi$
$104$	0	0
$105$	3.24929	0.317098
$106$	0	0
$107$	−10.8476	−1.04867	−0.524337	−	0.851511i	$-0.675687\pi$
$107$	−10.8476	−1.04867	−0.524337	+	0.851511i	$0.675687\pi$
$108$	0	0
$109$	0.849403	0.0813581	0.0406790	−	0.999172i	$-0.487048\pi$
$109$	0.849403	0.0813581	0.0406790	+	0.999172i	$0.487048\pi$
$110$	0	0
$111$	5.01326	0.475838
$112$	0	0
$113$	9.44620	0.888624	0.444312	−	0.895872i	$-0.353448\pi$
$113$	9.44620	0.888624	0.444312	+	0.895872i	$0.353448\pi$
$114$	0	0
$115$	−9.08386	−0.847075
$116$	0	0
$117$	8.55919	0.791297
$118$	0	0
$119$	−4.32726	−0.396679
$120$	0	0
$121$	−1.65968	−0.150880
$122$	0	0
$123$	4.51612	0.407204
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.31747	0.738056	0.369028	−	0.929418i	$-0.379691\pi$
$127$	8.31747	0.738056	0.369028	+	0.929418i	$0.379691\pi$
$128$	0	0
$129$	1.53120	0.134814
$130$	0	0
$131$	−16.7050	−1.45953	−0.729763	−	0.683701i	$-0.760369\pi$
$131$	−16.7050	−1.45953	−0.729763	+	0.683701i	$0.760369\pi$
$132$	0	0
$133$	0.994746	0.0862554
$134$	0	0
$135$	−5.62459	−0.484088
$136$	0	0
$137$	−18.2628	−1.56030	−0.780150	−	0.625592i	$-0.784857\pi$
$137$	−18.2628	−1.56030	−0.780150	+	0.625592i	$0.784857\pi$
$138$	0	0
$139$	−6.84999	−0.581008	−0.290504	−	0.956874i	$-0.593823\pi$
$139$	−6.84999	−0.581008	−0.290504	+	0.956874i	$0.593823\pi$
$140$	0	0
$141$	−5.28763	−0.445299
$142$	0	0
$143$	21.0704	1.76199
$144$	0	0
$145$	−10.5088	−0.872707
$146$	0	0
$147$	1.32097	0.108952
$148$	0	0
$149$	−18.3934	−1.50685	−0.753424	−	0.657535i	$-0.771599\pi$
$149$	−18.3934	−1.50685	−0.753424	+	0.657535i	$0.771599\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	2.19249	0.177253
$154$	0	0
$155$	−8.61528	−0.691996
$156$	0	0
$157$	−16.9982	−1.35660	−0.678302	−	0.734783i	$-0.737284\pi$
$157$	−16.9982	−1.35660	−0.678302	+	0.734783i	$0.737284\pi$
$158$	0	0
$159$	−14.3358	−1.13690
$160$	0	0
$161$	22.2580	1.75417
$162$	0	0
$163$	−11.3865	−0.891862	−0.445931	−	0.895067i	$-0.647127\pi$
$163$	−11.3865	−0.891862	−0.445931	+	0.895067i	$0.647127\pi$
$164$	0	0
$165$	−4.05279	−0.315509
$166$	0	0
$167$	−12.1368	−0.939177	−0.469589	−	0.882885i	$-0.655598\pi$
$167$	−12.1368	−0.939177	−0.469589	+	0.882885i	$0.655598\pi$
$168$	0	0
$169$	34.5316	2.65628
$170$	0	0
$171$	−0.504009	−0.0385425
$172$	0	0
$173$	20.2781	1.54171	0.770857	−	0.637008i	$-0.219828\pi$
$173$	20.2781	1.54171	0.770857	+	0.637008i	$0.219828\pi$
$174$	0	0
$175$	2.45028	0.185224
$176$	0	0
$177$	0.361118	0.0271433
$178$	0	0
$179$	−13.6104	−1.01729	−0.508644	−	0.860977i	$-0.669853\pi$
$179$	−13.6104	−1.01729	−0.508644	+	0.860977i	$0.669853\pi$
$180$	0	0
$181$	−2.13360	−0.158589	−0.0792947	−	0.996851i	$-0.525267\pi$
$181$	−2.13360	−0.158589	−0.0792947	+	0.996851i	$0.525267\pi$
$182$	0	0
$183$	−5.18043	−0.382948
$184$	0	0
$185$	3.78048	0.277947
$186$	0	0
$187$	5.39732	0.394691
$188$	0	0
$189$	13.7818	1.00248
$190$	0	0
$191$	−7.33504	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$191$	−7.33504	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$192$	0	0
$193$	−20.0612	−1.44403	−0.722017	−	0.691875i	$-0.756785\pi$
$193$	−20.0612	−1.44403	−0.722017	+	0.691875i	$0.756785\pi$
$194$	0	0
$195$	−9.14249	−0.654707
$196$	0	0
$197$	−19.8768	−1.41617	−0.708083	−	0.706129i	$-0.750440\pi$
$197$	−19.8768	−1.41617	−0.708083	+	0.706129i	$0.750440\pi$
$198$	0	0
$199$	15.7828	1.11881	0.559405	−	0.828895i	$-0.311030\pi$
$199$	15.7828	1.11881	0.559405	+	0.828895i	$0.311030\pi$
$200$	0	0
$201$	9.88876	0.697500
$202$	0	0
$203$	25.7494	1.80726
$204$	0	0
$205$	3.40559	0.237856
$206$	0	0
$207$	−11.2775	−0.783838
$208$	0	0
$209$	−1.24073	−0.0858232
$210$	0	0
$211$	−6.81495	−0.469161	−0.234580	−	0.972097i	$-0.575372\pi$
$211$	−6.81495	−0.469161	−0.234580	+	0.972097i	$0.575372\pi$
$212$	0	0
$213$	17.3352	1.18779
$214$	0	0
$215$	1.15467	0.0787478
$216$	0	0
$217$	21.1098	1.43303
$218$	0	0
$219$	14.9616	1.01101
$220$	0	0
$221$	12.1756	0.819017
$222$	0	0
$223$	22.0201	1.47457	0.737286	−	0.675581i	$-0.236107\pi$
$223$	22.0201	1.47457	0.737286	+	0.675581i	$0.236107\pi$
$224$	0	0
$225$	−1.24148	−0.0827656
$226$	0	0
$227$	3.36332	0.223232	0.111616	−	0.993751i	$-0.464397\pi$
$227$	3.36332	0.223232	0.111616	+	0.993751i	$0.464397\pi$
$228$	0	0
$229$	26.0973	1.72456	0.862280	−	0.506432i	$-0.169036\pi$
$229$	26.0973	1.72456	0.862280	+	0.506432i	$0.169036\pi$
$230$	0	0
$231$	9.93046	0.653376
$232$	0	0
$233$	22.3358	1.46327	0.731633	−	0.681699i	$-0.238759\pi$
$233$	22.3358	1.46327	0.731633	+	0.681699i	$0.238759\pi$
$234$	0	0
$235$	−3.98738	−0.260108
$236$	0	0
$237$	6.21119	0.403460
$238$	0	0
$239$	5.63227	0.364321	0.182161	−	0.983269i	$-0.441691\pi$
$239$	5.63227	0.364321	0.182161	+	0.983269i	$0.441691\pi$
$240$	0	0
$241$	14.7347	0.949147	0.474574	−	0.880216i	$-0.342602\pi$
$241$	14.7347	0.949147	0.474574	+	0.880216i	$0.342602\pi$
$242$	0	0
$243$	−11.9218	−0.764784
$244$	0	0
$245$	0.996141	0.0636411
$246$	0	0
$247$	−2.79891	−0.178090
$248$	0	0
$249$	14.0251	0.888805
$250$	0	0
$251$	−14.6707	−0.926007	−0.463004	−	0.886356i	$-0.653228\pi$
$251$	−14.6707	−0.926007	−0.463004	+	0.886356i	$0.653228\pi$
$252$	0	0
$253$	−27.7620	−1.74539
$254$	0	0
$255$	−2.34191	−0.146656
$256$	0	0
$257$	−18.8882	−1.17821	−0.589105	−	0.808056i	$-0.700520\pi$
$257$	−18.8882	−1.17821	−0.589105	+	0.808056i	$0.700520\pi$
$258$	0	0
$259$	−9.26323	−0.575589
$260$	0	0
$261$	−13.0465	−0.807557
$262$	0	0
$263$	−19.3047	−1.19038	−0.595190	−	0.803585i	$-0.702923\pi$
$263$	−19.3047	−1.19038	−0.595190	+	0.803585i	$0.702923\pi$
$264$	0	0
$265$	−10.8106	−0.664089
$266$	0	0
$267$	−6.71465	−0.410930
$268$	0	0
$269$	−18.3329	−1.11777	−0.558887	−	0.829244i	$-0.688772\pi$
$269$	−18.3329	−1.11777	−0.558887	+	0.829244i	$0.688772\pi$
$270$	0	0
$271$	25.3312	1.53876	0.769380	−	0.638791i	$-0.220565\pi$
$271$	25.3312	1.53876	0.769380	+	0.638791i	$0.220565\pi$
$272$	0	0
$273$	22.4016	1.35581
$274$	0	0
$275$	−3.05619	−0.184295
$276$	0	0
$277$	−17.2352	−1.03556	−0.517782	−	0.855513i	$-0.673242\pi$
$277$	−17.2352	−1.03556	−0.517782	+	0.855513i	$0.673242\pi$
$278$	0	0
$279$	−10.6957	−0.640337
$280$	0	0
$281$	13.4379	0.801640	0.400820	−	0.916157i	$-0.368725\pi$
$281$	13.4379	0.801640	0.400820	+	0.916157i	$0.368725\pi$
$282$	0	0
$283$	22.1625	1.31742	0.658712	−	0.752395i	$-0.271101\pi$
$283$	22.1625	1.31742	0.658712	+	0.752395i	$0.271101\pi$
$284$	0	0
$285$	0.538357	0.0318895
$286$	0	0
$287$	−8.34463	−0.492568
$288$	0	0
$289$	−13.8811	−0.816538
$290$	0	0
$291$	−6.39659	−0.374975
$292$	0	0
$293$	−22.0700	−1.28934	−0.644672	−	0.764460i	$-0.723006\pi$
$293$	−22.0700	−1.28934	−0.644672	+	0.764460i	$0.723006\pi$
$294$	0	0
$295$	0.272318	0.0158549
$296$	0	0
$297$	−17.1898	−0.997456
$298$	0	0
$299$	−62.6270	−3.62182
$300$	0	0
$301$	−2.82926	−0.163076
$302$	0	0
$303$	−4.58864	−0.263610
$304$	0	0
$305$	−3.90654	−0.223688
$306$	0	0
$307$	−4.75680	−0.271485	−0.135742	−	0.990744i	$-0.543342\pi$
$307$	−4.75680	−0.271485	−0.135742	+	0.990744i	$0.543342\pi$
$308$	0	0
$309$	−14.3494	−0.816312
$310$	0	0
$311$	19.1842	1.08784	0.543919	−	0.839138i	$-0.316940\pi$
$311$	19.1842	1.08784	0.543919	+	0.839138i	$0.316940\pi$
$312$	0	0
$313$	22.7823	1.28773	0.643867	−	0.765137i	$-0.277329\pi$
$313$	22.7823	1.28773	0.643867	+	0.765137i	$0.277329\pi$
$314$	0	0
$315$	3.04198	0.171396
$316$	0	0
$317$	12.7650	0.716953	0.358477	−	0.933539i	$-0.383296\pi$
$317$	12.7650	0.716953	0.358477	+	0.933539i	$0.383296\pi$
$318$	0	0
$319$	−32.1169	−1.79820
$320$	0	0
$321$	14.3849	0.802884
$322$	0	0
$323$	−0.716959	−0.0398927
$324$	0	0
$325$	−6.89432	−0.382428
$326$	0	0
$327$	−1.12639	−0.0622893
$328$	0	0
$329$	9.77020	0.538648
$330$	0	0
$331$	−0.371726	−0.0204319	−0.0102160	−	0.999948i	$-0.503252\pi$
$331$	−0.371726	−0.0204319	−0.0102160	+	0.999948i	$0.503252\pi$
$332$	0	0
$333$	4.69341	0.257197
$334$	0	0
$335$	7.45708	0.407424
$336$	0	0
$337$	−12.3274	−0.671514	−0.335757	−	0.941949i	$-0.608992\pi$
$337$	−12.3274	−0.671514	−0.335757	+	0.941949i	$0.608992\pi$
$338$	0	0
$339$	−12.5265	−0.680347
$340$	0	0
$341$	−26.3300	−1.42585
$342$	0	0
$343$	−19.5928	−1.05791
$344$	0	0
$345$	12.0460	0.648536
$346$	0	0
$347$	−9.46698	−0.508214	−0.254107	−	0.967176i	$-0.581782\pi$
$347$	−9.46698	−0.508214	−0.254107	+	0.967176i	$0.581782\pi$
$348$	0	0
$349$	19.6808	1.05349	0.526745	−	0.850024i	$-0.323412\pi$
$349$	19.6808	1.05349	0.526745	+	0.850024i	$0.323412\pi$
$350$	0	0
$351$	−38.7777	−2.06980
$352$	0	0
$353$	−21.4346	−1.14085	−0.570426	−	0.821349i	$-0.693222\pi$
$353$	−21.4346	−1.14085	−0.570426	+	0.821349i	$0.693222\pi$
$354$	0	0
$355$	13.0724	0.693812
$356$	0	0
$357$	5.73833	0.303705
$358$	0	0
$359$	−33.9280	−1.79065	−0.895326	−	0.445412i	$-0.853057\pi$
$359$	−33.9280	−1.79065	−0.895326	+	0.445412i	$0.853057\pi$
$360$	0	0
$361$	−18.8352	−0.991326
$362$	0	0
$363$	2.20088	0.115516
$364$	0	0
$365$	11.2825	0.590553
$366$	0	0
$367$	−20.5267	−1.07149	−0.535743	−	0.844381i	$-0.679968\pi$
$367$	−20.5267	−1.07149	−0.535743	+	0.844381i	$0.679968\pi$
$368$	0	0
$369$	4.22798	0.220100
$370$	0	0
$371$	26.4890	1.37524
$372$	0	0
$373$	−4.43889	−0.229837	−0.114919	−	0.993375i	$-0.536661\pi$
$373$	−4.43889	−0.229837	−0.114919	+	0.993375i	$0.536661\pi$
$374$	0	0
$375$	1.32609	0.0684790
$376$	0	0
$377$	−72.4509	−3.73141
$378$	0	0
$379$	16.9577	0.871059	0.435529	−	0.900174i	$-0.356561\pi$
$379$	16.9577	0.871059	0.435529	+	0.900174i	$0.356561\pi$
$380$	0	0
$381$	−11.0297	−0.565069
$382$	0	0
$383$	−32.2735	−1.64910	−0.824548	−	0.565792i	$-0.808571\pi$
$383$	−32.2735	−1.64910	−0.824548	+	0.565792i	$0.808571\pi$
$384$	0	0
$385$	7.48852	0.381651
$386$	0	0
$387$	1.43350	0.0728691
$388$	0	0
$389$	23.0629	1.16934	0.584668	−	0.811273i	$-0.301225\pi$
$389$	23.0629	1.16934	0.584668	+	0.811273i	$0.301225\pi$
$390$	0	0
$391$	−16.0424	−0.811297
$392$	0	0
$393$	22.1524	1.11744
$394$	0	0
$395$	4.68384	0.235669
$396$	0	0
$397$	15.1738	0.761553	0.380777	−	0.924667i	$-0.375657\pi$
$397$	15.1738	0.761553	0.380777	+	0.924667i	$0.375657\pi$
$398$	0	0
$399$	−1.31912	−0.0660388
$400$	0	0
$401$	−7.28799	−0.363945	−0.181973	−	0.983304i	$-0.558248\pi$
$401$	−7.28799	−0.363945	−0.181973	+	0.983304i	$0.558248\pi$
$402$	0	0
$403$	−59.3965	−2.95875
$404$	0	0
$405$	3.73427	0.185557
$406$	0	0
$407$	11.5539	0.572705
$408$	0	0
$409$	15.7767	0.780109	0.390055	−	0.920792i	$-0.372456\pi$
$409$	15.7767	0.780109	0.390055	+	0.920792i	$0.372456\pi$
$410$	0	0
$411$	24.2182	1.19460
$412$	0	0
$413$	−0.667254	−0.0328334
$414$	0	0
$415$	10.5763	0.519169
$416$	0	0
$417$	9.08370	0.444831
$418$	0	0
$419$	25.5504	1.24822	0.624110	−	0.781336i	$-0.285462\pi$
$419$	25.5504	1.24822	0.624110	+	0.781336i	$0.285462\pi$
$420$	0	0
$421$	−22.5045	−1.09680	−0.548402	−	0.836215i	$-0.684763\pi$
$421$	−22.5045	−1.09680	−0.548402	+	0.836215i	$0.684763\pi$
$422$	0	0
$423$	−4.95027	−0.240691
$424$	0	0
$425$	−1.76603	−0.0856649
$426$	0	0
$427$	9.57212	0.463227
$428$	0	0
$429$	−27.9412	−1.34902
$430$	0	0
$431$	−11.8757	−0.572031	−0.286015	−	0.958225i	$-0.592331\pi$
$431$	−11.8757	−0.572031	−0.286015	+	0.958225i	$0.592331\pi$
$432$	0	0
$433$	21.8654	1.05078	0.525392	−	0.850860i	$-0.323919\pi$
$433$	21.8654	1.05078	0.525392	+	0.850860i	$0.323919\pi$
$434$	0	0
$435$	13.9356	0.668161
$436$	0	0
$437$	3.68780	0.176411
$438$	0	0
$439$	33.4347	1.59575	0.797876	−	0.602822i	$-0.205957\pi$
$439$	33.4347	1.59575	0.797876	+	0.602822i	$0.205957\pi$
$440$	0	0
$441$	1.23669	0.0588901
$442$	0	0
$443$	−14.4779	−0.687867	−0.343934	−	0.938994i	$-0.611760\pi$
$443$	−14.4779	−0.687867	−0.343934	+	0.938994i	$0.611760\pi$
$444$	0	0
$445$	−5.06349	−0.240032
$446$	0	0
$447$	24.3913	1.15367
$448$	0	0
$449$	−26.2506	−1.23884	−0.619421	−	0.785059i	$-0.712633\pi$
$449$	−26.2506	−1.23884	−0.619421	+	0.785059i	$0.712633\pi$
$450$	0	0
$451$	10.4081	0.490100
$452$	0	0
$453$	1.32609	0.0623052
$454$	0	0
$455$	16.8930	0.791956
$456$	0	0
$457$	−28.2433	−1.32116	−0.660582	−	0.750754i	$-0.729690\pi$
$457$	−28.2433	−1.32116	−0.660582	+	0.750754i	$0.729690\pi$
$458$	0	0
$459$	−9.93318	−0.463641
$460$	0	0
$461$	7.72179	0.359640	0.179820	−	0.983700i	$-0.442449\pi$
$461$	7.72179	0.359640	0.179820	+	0.983700i	$0.442449\pi$
$462$	0	0
$463$	−15.6283	−0.726309	−0.363155	−	0.931729i	$-0.618300\pi$
$463$	−15.6283	−0.726309	−0.363155	+	0.931729i	$0.618300\pi$
$464$	0	0
$465$	11.4246	0.529805
$466$	0	0
$467$	−15.5618	−0.720116	−0.360058	−	0.932930i	$-0.617243\pi$
$467$	−15.5618	−0.720116	−0.360058	+	0.932930i	$0.617243\pi$
$468$	0	0
$469$	−18.2719	−0.843719
$470$	0	0
$471$	22.5412	1.03864
$472$	0	0
$473$	3.52889	0.162259
$474$	0	0
$475$	0.405973	0.0186273
$476$	0	0
$477$	−13.4212	−0.614513
$478$	0	0
$479$	−29.4481	−1.34552	−0.672760	−	0.739861i	$-0.734891\pi$
$479$	−29.4481	−1.34552	−0.672760	+	0.739861i	$0.734891\pi$
$480$	0	0
$481$	26.0639	1.18841
$482$	0	0
$483$	−29.5161	−1.34303
$484$	0	0
$485$	−4.82364	−0.219030
$486$	0	0
$487$	13.0907	0.593198	0.296599	−	0.955002i	$-0.404147\pi$
$487$	13.0907	0.593198	0.296599	+	0.955002i	$0.404147\pi$
$488$	0	0
$489$	15.0996	0.682826
$490$	0	0
$491$	−34.2538	−1.54585	−0.772927	−	0.634495i	$-0.781208\pi$
$491$	−34.2538	−1.54585	−0.772927	+	0.634495i	$0.781208\pi$
$492$	0	0
$493$	−18.5588	−0.835846
$494$	0	0
$495$	−3.79422	−0.170537
$496$	0	0
$497$	−32.0311	−1.43679
$498$	0	0
$499$	8.67268	0.388242	0.194121	−	0.980978i	$-0.437814\pi$
$499$	8.67268	0.388242	0.194121	+	0.980978i	$0.437814\pi$
$500$	0	0
$501$	16.0946	0.719052
$502$	0	0
$503$	30.4959	1.35974	0.679872	−	0.733331i	$-0.262035\pi$
$503$	30.4959	1.35974	0.679872	+	0.733331i	$0.262035\pi$
$504$	0	0
$505$	−3.46027	−0.153980
$506$	0	0
$507$	−45.7921	−2.03370
$508$	0	0
$509$	−12.9237	−0.572833	−0.286416	−	0.958105i	$-0.592464\pi$
$509$	−12.9237	−0.572833	−0.286416	+	0.958105i	$0.592464\pi$
$510$	0	0
$511$	−27.6453	−1.22295
$512$	0	0
$513$	2.28343	0.100816
$514$	0	0
$515$	−10.8209	−0.476824
$516$	0	0
$517$	−12.1862	−0.535949
$518$	0	0
$519$	−26.8906	−1.18037
$520$	0	0
$521$	43.1909	1.89223	0.946115	−	0.323831i	$-0.104971\pi$
$521$	43.1909	1.89223	0.946115	+	0.323831i	$0.104971\pi$
$522$	0	0
$523$	−26.4894	−1.15830	−0.579149	−	0.815222i	$-0.696615\pi$
$523$	−26.4894	−1.15830	−0.579149	+	0.815222i	$0.696615\pi$
$524$	0	0
$525$	−3.24929	−0.141811
$526$	0	0
$527$	−15.2148	−0.662768
$528$	0	0
$529$	59.5166	2.58768
$530$	0	0
$531$	0.338078	0.0146713
$532$	0	0
$533$	23.4792	1.01700
$534$	0	0
$535$	10.8476	0.468981
$536$	0	0
$537$	18.0486	0.778855
$538$	0	0
$539$	3.04440	0.131132
$540$	0	0
$541$	33.1698	1.42608	0.713040	−	0.701123i	$-0.247318\pi$
$541$	33.1698	1.42608	0.713040	+	0.701123i	$0.247318\pi$
$542$	0	0
$543$	2.82935	0.121419
$544$	0	0
$545$	−0.849403	−0.0363844
$546$	0	0
$547$	−1.76044	−0.0752711	−0.0376356	−	0.999292i	$-0.511983\pi$
$547$	−1.76044	−0.0752711	−0.0376356	+	0.999292i	$0.511983\pi$
$548$	0	0
$549$	−4.84991	−0.206989
$550$	0	0
$551$	4.26628	0.181750
$552$	0	0
$553$	−11.4767	−0.488039
$554$	0	0
$555$	−5.01326	−0.212801
$556$	0	0
$557$	43.1224	1.82716	0.913578	−	0.406663i	$-0.133308\pi$
$557$	43.1224	1.82716	0.913578	+	0.406663i	$0.133308\pi$
$558$	0	0
$559$	7.96066	0.336700
$560$	0	0
$561$	−7.15734	−0.302183
$562$	0	0
$563$	−42.4808	−1.79035	−0.895175	−	0.445715i	$-0.852950\pi$
$563$	−42.4808	−1.79035	−0.895175	+	0.445715i	$0.852950\pi$
$564$	0	0
$565$	−9.44620	−0.397405
$566$	0	0
$567$	−9.14999	−0.384263
$568$	0	0
$569$	−23.9706	−1.00490	−0.502451	−	0.864606i	$-0.667568\pi$
$569$	−23.9706	−1.00490	−0.502451	+	0.864606i	$0.667568\pi$
$570$	0	0
$571$	−1.76806	−0.0739909	−0.0369954	−	0.999315i	$-0.511779\pi$
$571$	−1.76806	−0.0739909	−0.0369954	+	0.999315i	$0.511779\pi$
$572$	0	0
$573$	9.72692	0.406348
$574$	0	0
$575$	9.08386	0.378823
$576$	0	0
$577$	−30.8760	−1.28539	−0.642693	−	0.766124i	$-0.722183\pi$
$577$	−30.8760	−1.28539	−0.642693	+	0.766124i	$0.722183\pi$
$578$	0	0
$579$	26.6029	1.10558
$580$	0	0
$581$	−25.9148	−1.07513
$582$	0	0
$583$	−33.0393	−1.36835
$584$	0	0
$585$	−8.55919	−0.353879
$586$	0	0
$587$	−19.6353	−0.810435	−0.405217	−	0.914220i	$-0.632804\pi$
$587$	−19.6353	−0.810435	−0.405217	+	0.914220i	$0.632804\pi$
$588$	0	0
$589$	3.49757	0.144115
$590$	0	0
$591$	26.3585	1.08424
$592$	0	0
$593$	−21.0275	−0.863495	−0.431747	−	0.901995i	$-0.642103\pi$
$593$	−21.0275	−0.863495	−0.431747	+	0.901995i	$0.642103\pi$
$594$	0	0
$595$	4.32726	0.177400
$596$	0	0
$597$	−20.9294	−0.856582
$598$	0	0
$599$	−13.8388	−0.565440	−0.282720	−	0.959203i	$-0.591237\pi$
$599$	−13.8388	−0.565440	−0.282720	+	0.959203i	$0.591237\pi$
$600$	0	0
$601$	2.52586	0.103032	0.0515159	−	0.998672i	$-0.483595\pi$
$601$	2.52586	0.103032	0.0515159	+	0.998672i	$0.483595\pi$
$602$	0	0
$603$	9.25785	0.377009
$604$	0	0
$605$	1.65968	0.0674754
$606$	0	0
$607$	37.4721	1.52095	0.760473	−	0.649369i	$-0.224967\pi$
$607$	37.4721	1.52095	0.760473	+	0.649369i	$0.224967\pi$
$608$	0	0
$609$	−34.1461	−1.38367
$610$	0	0
$611$	−27.4903	−1.11214
$612$	0	0
$613$	−13.7626	−0.555867	−0.277933	−	0.960600i	$-0.589649\pi$
$613$	−13.7626	−0.555867	−0.277933	+	0.960600i	$0.589649\pi$
$614$	0	0
$615$	−4.51612	−0.182107
$616$	0	0
$617$	−33.8750	−1.36376	−0.681879	−	0.731465i	$-0.738837\pi$
$617$	−33.8750	−1.36376	−0.681879	+	0.731465i	$0.738837\pi$
$618$	0	0
$619$	−2.28327	−0.0917725	−0.0458863	−	0.998947i	$-0.514611\pi$
$619$	−2.28327	−0.0917725	−0.0458863	+	0.998947i	$0.514611\pi$
$620$	0	0
$621$	51.0930	2.05029
$622$	0	0
$623$	12.4070	0.497074
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.64532	0.0657079
$628$	0	0
$629$	6.67644	0.266207
$630$	0	0
$631$	9.10654	0.362526	0.181263	−	0.983435i	$-0.441982\pi$
$631$	9.10654	0.362526	0.181263	+	0.983435i	$0.441982\pi$
$632$	0	0
$633$	9.03724	0.359198
$634$	0	0
$635$	−8.31747	−0.330069
$636$	0	0
$637$	6.86771	0.272109
$638$	0	0
$639$	16.2292	0.642017
$640$	0	0
$641$	5.75646	0.227367	0.113683	−	0.993517i	$-0.463735\pi$
$641$	5.75646	0.227367	0.113683	+	0.993517i	$0.463735\pi$
$642$	0	0
$643$	−31.5278	−1.24334	−0.621668	−	0.783281i	$-0.713545\pi$
$643$	−31.5278	−1.24334	−0.621668	+	0.783281i	$0.713545\pi$
$644$	0	0
$645$	−1.53120	−0.0602908
$646$	0	0
$647$	−12.4565	−0.489715	−0.244858	−	0.969559i	$-0.578741\pi$
$647$	−12.4565	−0.489715	−0.244858	+	0.969559i	$0.578741\pi$
$648$	0	0
$649$	0.832256	0.0326689
$650$	0	0
$651$	−27.9935	−1.09715
$652$	0	0
$653$	37.5751	1.47043	0.735215	−	0.677834i	$-0.237081\pi$
$653$	37.5751	1.47043	0.735215	+	0.677834i	$0.237081\pi$
$654$	0	0
$655$	16.7050	0.652719
$656$	0	0
$657$	14.0070	0.546467
$658$	0	0
$659$	26.9362	1.04928	0.524642	−	0.851323i	$-0.324199\pi$
$659$	26.9362	1.04928	0.524642	+	0.851323i	$0.324199\pi$
$660$	0	0
$661$	30.3558	1.18070	0.590352	−	0.807146i	$-0.298989\pi$
$661$	30.3558	1.18070	0.590352	+	0.807146i	$0.298989\pi$
$662$	0	0
$663$	−16.1459	−0.627054
$664$	0	0
$665$	−0.994746	−0.0385746
$666$	0	0
$667$	95.4603	3.69624
$668$	0	0
$669$	−29.2006	−1.12896
$670$	0	0
$671$	−11.9392	−0.460906
$672$	0	0
$673$	39.2554	1.51318	0.756592	−	0.653887i	$-0.226863\pi$
$673$	39.2554	1.51318	0.756592	+	0.653887i	$0.226863\pi$
$674$	0	0
$675$	5.62459	0.216491
$676$	0	0
$677$	33.6900	1.29481	0.647406	−	0.762145i	$-0.275854\pi$
$677$	33.6900	1.29481	0.647406	+	0.762145i	$0.275854\pi$
$678$	0	0
$679$	11.8193	0.453582
$680$	0	0
$681$	−4.46007	−0.170910
$682$	0	0
$683$	−3.80962	−0.145771	−0.0728855	−	0.997340i	$-0.523221\pi$
$683$	−3.80962	−0.145771	−0.0728855	+	0.997340i	$0.523221\pi$
$684$	0	0
$685$	18.2628	0.697787
$686$	0	0
$687$	−34.6074	−1.32035
$688$	0	0
$689$	−74.5317	−2.83943
$690$	0	0
$691$	30.0689	1.14388	0.571938	−	0.820297i	$-0.306192\pi$
$691$	30.0689	1.14388	0.571938	+	0.820297i	$0.306192\pi$
$692$	0	0
$693$	9.29688	0.353159
$694$	0	0
$695$	6.84999	0.259835
$696$	0	0
$697$	6.01436	0.227810
$698$	0	0
$699$	−29.6193	−1.12030
$700$	0	0
$701$	4.49238	0.169675	0.0848374	−	0.996395i	$-0.472963\pi$
$701$	4.49238	0.169675	0.0848374	+	0.996395i	$0.472963\pi$
$702$	0	0
$703$	−1.53477	−0.0578851
$704$	0	0
$705$	5.28763	0.199144
$706$	0	0
$707$	8.47863	0.318872
$708$	0	0
$709$	−31.5582	−1.18519	−0.592596	−	0.805500i	$-0.701897\pi$
$709$	−31.5582	−1.18519	−0.592596	+	0.805500i	$0.701897\pi$
$710$	0	0
$711$	5.81491	0.218076
$712$	0	0
$713$	78.2600	2.93086
$714$	0	0
$715$	−21.0704	−0.787987
$716$	0	0
$717$	−7.46890	−0.278931
$718$	0	0
$719$	38.4278	1.43312	0.716558	−	0.697528i	$-0.245717\pi$
$719$	38.4278	1.43312	0.716558	+	0.697528i	$0.245717\pi$
$720$	0	0
$721$	26.5141	0.987438
$722$	0	0
$723$	−19.5396	−0.726685
$724$	0	0
$725$	10.5088	0.390286
$726$	0	0
$727$	−29.5079	−1.09439	−0.547194	−	0.837005i	$-0.684304\pi$
$727$	−29.5079	−1.09439	−0.547194	+	0.837005i	$0.684304\pi$
$728$	0	0
$729$	27.0122	1.00045
$730$	0	0
$731$	2.03918	0.0754217
$732$	0	0
$733$	3.47781	0.128456	0.0642279	−	0.997935i	$-0.479542\pi$
$733$	3.47781	0.128456	0.0642279	+	0.997935i	$0.479542\pi$
$734$	0	0
$735$	−1.32097	−0.0487248
$736$	0	0
$737$	22.7903	0.839491
$738$	0	0
$739$	−42.2800	−1.55529	−0.777647	−	0.628702i	$-0.783587\pi$
$739$	−42.2800	−1.55529	−0.777647	+	0.628702i	$0.783587\pi$
$740$	0	0
$741$	3.71160	0.136349
$742$	0	0
$743$	−31.1119	−1.14138	−0.570692	−	0.821164i	$-0.693325\pi$
$743$	−31.1119	−1.14138	−0.570692	+	0.821164i	$0.693325\pi$
$744$	0	0
$745$	18.3934	0.673883
$746$	0	0
$747$	13.1303	0.480412
$748$	0	0
$749$	−26.5795	−0.971195
$750$	0	0
$751$	−0.476533	−0.0173889	−0.00869447	−	0.999962i	$-0.502768\pi$
$751$	−0.476533	−0.0173889	−0.00869447	+	0.999962i	$0.502768\pi$
$752$	0	0
$753$	19.4547	0.708969
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	17.6390	0.641099	0.320549	−	0.947232i	$-0.396132\pi$
$757$	17.6390	0.641099	0.320549	+	0.947232i	$0.396132\pi$
$758$	0	0
$759$	36.8150	1.33630
$760$	0	0
$761$	−40.2857	−1.46036	−0.730178	−	0.683257i	$-0.760563\pi$
$761$	−40.2857	−1.46036	−0.730178	+	0.683257i	$0.760563\pi$
$762$	0	0
$763$	2.08127	0.0753472
$764$	0	0
$765$	−2.19249	−0.0792698
$766$	0	0
$767$	1.87744	0.0677906
$768$	0	0
$769$	−12.6025	−0.454457	−0.227228	−	0.973842i	$-0.572966\pi$
$769$	−12.6025	−0.454457	−0.227228	+	0.973842i	$0.572966\pi$
$770$	0	0
$771$	25.0474	0.902060
$772$	0	0
$773$	−11.3568	−0.408476	−0.204238	−	0.978921i	$-0.565472\pi$
$773$	−11.3568	−0.408476	−0.204238	+	0.978921i	$0.565472\pi$
$774$	0	0
$775$	8.61528	0.309470
$776$	0	0
$777$	12.2839	0.440682
$778$	0	0
$779$	−1.38258	−0.0495359
$780$	0	0
$781$	39.9519	1.42959
$782$	0	0
$783$	59.1076	2.11233
$784$	0	0
$785$	16.9982	0.606692
$786$	0	0
$787$	−3.51083	−0.125148	−0.0625738	−	0.998040i	$-0.519931\pi$
$787$	−3.51083	−0.125148	−0.0625738	+	0.998040i	$0.519931\pi$
$788$	0	0
$789$	25.5998	0.911378
$790$	0	0
$791$	23.1458	0.822971
$792$	0	0
$793$	−26.9330	−0.956417
$794$	0	0
$795$	14.3358	0.508439
$796$	0	0
$797$	−25.0890	−0.888699	−0.444350	−	0.895854i	$-0.646565\pi$
$797$	−25.0890	−0.888699	−0.444350	+	0.895854i	$0.646565\pi$
$798$	0	0
$799$	−7.04183	−0.249122
$800$	0	0
$801$	−6.28624	−0.222113
$802$	0	0
$803$	34.4815	1.21683
$804$	0	0
$805$	−22.2580	−0.784491
$806$	0	0
$807$	24.3110	0.855789
$808$	0	0
$809$	18.1441	0.637914	0.318957	−	0.947769i	$-0.396668\pi$
$809$	18.1441	0.637914	0.318957	+	0.947769i	$0.396668\pi$
$810$	0	0
$811$	44.0988	1.54852	0.774259	−	0.632869i	$-0.218123\pi$
$811$	44.0988	1.54852	0.774259	+	0.632869i	$0.218123\pi$
$812$	0	0
$813$	−33.5915	−1.17810
$814$	0	0
$815$	11.3865	0.398853
$816$	0	0
$817$	−0.468765	−0.0164000
$818$	0	0
$819$	20.9724	0.732834
$820$	0	0
$821$	−30.6464	−1.06957	−0.534784	−	0.844989i	$-0.679607\pi$
$821$	−30.6464	−1.06957	−0.534784	+	0.844989i	$0.679607\pi$
$822$	0	0
$823$	1.25552	0.0437648	0.0218824	−	0.999761i	$-0.493034\pi$
$823$	1.25552	0.0437648	0.0218824	+	0.999761i	$0.493034\pi$
$824$	0	0
$825$	4.05279	0.141100
$826$	0	0
$827$	−32.0263	−1.11366	−0.556832	−	0.830625i	$-0.687983\pi$
$827$	−32.0263	−1.11366	−0.556832	+	0.830625i	$0.687983\pi$
$828$	0	0
$829$	−16.3683	−0.568495	−0.284248	−	0.958751i	$-0.591744\pi$
$829$	−16.3683	−0.568495	−0.284248	+	0.958751i	$0.591744\pi$
$830$	0	0
$831$	22.8555	0.792847
$832$	0	0
$833$	1.75921	0.0609531
$834$	0	0
$835$	12.1368	0.420013
$836$	0	0
$837$	48.4574	1.67493
$838$	0	0
$839$	6.38404	0.220402	0.110201	−	0.993909i	$-0.464851\pi$
$839$	6.38404	0.220402	0.110201	+	0.993909i	$0.464851\pi$
$840$	0	0
$841$	81.4345	2.80809
$842$	0	0
$843$	−17.8199	−0.613751
$844$	0	0
$845$	−34.5316	−1.18792
$846$	0	0
$847$	−4.06667	−0.139732
$848$	0	0
$849$	−29.3895	−1.00865
$850$	0	0
$851$	−34.3414	−1.17721
$852$	0	0
$853$	−30.6379	−1.04902	−0.524512	−	0.851403i	$-0.675752\pi$
$853$	−30.6379	−1.04902	−0.524512	+	0.851403i	$0.675752\pi$
$854$	0	0
$855$	0.504009	0.0172367
$856$	0	0
$857$	−36.0473	−1.23135	−0.615675	−	0.788000i	$-0.711117\pi$
$857$	−36.0473	−1.23135	−0.615675	+	0.788000i	$0.711117\pi$
$858$	0	0
$859$	20.7085	0.706566	0.353283	−	0.935517i	$-0.385065\pi$
$859$	20.7085	0.706566	0.353283	+	0.935517i	$0.385065\pi$
$860$	0	0
$861$	11.0657	0.377119
$862$	0	0
$863$	11.8016	0.401731	0.200866	−	0.979619i	$-0.435625\pi$
$863$	11.8016	0.401731	0.200866	+	0.979619i	$0.435625\pi$
$864$	0	0
$865$	−20.2781	−0.689476
$866$	0	0
$867$	18.4077	0.625157
$868$	0	0
$869$	14.3147	0.485594
$870$	0	0
$871$	51.4115	1.74201
$872$	0	0
$873$	−5.98847	−0.202679
$874$	0	0
$875$	−2.45028	−0.0828345
$876$	0	0
$877$	−30.2809	−1.02251	−0.511257	−	0.859428i	$-0.670820\pi$
$877$	−30.2809	−1.02251	−0.511257	+	0.859428i	$0.670820\pi$
$878$	0	0
$879$	29.2668	0.987145
$880$	0	0
$881$	5.98679	0.201700	0.100850	−	0.994902i	$-0.467844\pi$
$881$	5.98679	0.201700	0.100850	+	0.994902i	$0.467844\pi$
$882$	0	0
$883$	−25.4093	−0.855091	−0.427545	−	0.903994i	$-0.640622\pi$
$883$	−25.4093	−0.855091	−0.427545	+	0.903994i	$0.640622\pi$
$884$	0	0
$885$	−0.361118	−0.0121388
$886$	0	0
$887$	−22.5196	−0.756135	−0.378067	−	0.925778i	$-0.623411\pi$
$887$	−22.5196	−0.756135	−0.378067	+	0.925778i	$0.623411\pi$
$888$	0	0
$889$	20.3801	0.683527
$890$	0	0
$891$	11.4126	0.382338
$892$	0	0
$893$	1.61877	0.0541701
$894$	0	0
$895$	13.6104	0.454945
$896$	0	0
$897$	83.0491	2.77293
$898$	0	0
$899$	90.5361	3.01955
$900$	0	0
$901$	−19.0918	−0.636040
$902$	0	0
$903$	3.75186	0.124854
$904$	0	0
$905$	2.13360	0.0709234
$906$	0	0
$907$	55.7517	1.85121	0.925603	−	0.378496i	$-0.123559\pi$
$907$	55.7517	1.85121	0.925603	+	0.378496i	$0.123559\pi$
$908$	0	0
$909$	−4.29587	−0.142485
$910$	0	0
$911$	42.7042	1.41485	0.707426	−	0.706787i	$-0.249856\pi$
$911$	42.7042	1.41485	0.707426	+	0.706787i	$0.249856\pi$
$912$	0	0
$913$	32.3232	1.06974
$914$	0	0
$915$	5.18043	0.171260
$916$	0	0
$917$	−40.9320	−1.35169
$918$	0	0
$919$	−10.5678	−0.348599	−0.174299	−	0.984693i	$-0.555766\pi$
$919$	−10.5678	−0.348599	−0.174299	+	0.984693i	$0.555766\pi$
$920$	0	0
$921$	6.30795	0.207854
$922$	0	0
$923$	90.1255	2.96652
$924$	0	0
$925$	−3.78048	−0.124302
$926$	0	0
$927$	−13.4339	−0.441228
$928$	0	0
$929$	−8.64583	−0.283660	−0.141830	−	0.989891i	$-0.545299\pi$
$929$	−8.64583	−0.283660	−0.141830	+	0.989891i	$0.545299\pi$
$930$	0	0
$931$	−0.404406	−0.0132539
$932$	0	0
$933$	−25.4400	−0.832869
$934$	0	0
$935$	−5.39732	−0.176511
$936$	0	0
$937$	−58.5895	−1.91404	−0.957018	−	0.290029i	$-0.906335\pi$
$937$	−58.5895	−1.91404	−0.957018	+	0.290029i	$0.906335\pi$
$938$	0	0
$939$	−30.2114	−0.985913
$940$	0	0
$941$	−10.2522	−0.334212	−0.167106	−	0.985939i	$-0.553442\pi$
$941$	−10.2522	−0.334212	−0.167106	+	0.985939i	$0.553442\pi$
$942$	0	0
$943$	−30.9359	−1.00741
$944$	0	0
$945$	−13.7818	−0.448322
$946$	0	0
$947$	−23.0782	−0.749941	−0.374970	−	0.927037i	$-0.622347\pi$
$947$	−23.0782	−0.749941	−0.374970	+	0.927037i	$0.622347\pi$
$948$	0	0
$949$	77.7852	2.52501
$950$	0	0
$951$	−16.9275	−0.548913
$952$	0	0
$953$	9.19116	0.297731	0.148865	−	0.988857i	$-0.452438\pi$
$953$	9.19116	0.297731	0.148865	+	0.988857i	$0.452438\pi$
$954$	0	0
$955$	7.33504	0.237356
$956$	0	0
$957$	42.5899	1.37674
$958$	0	0
$959$	−44.7490	−1.44502
$960$	0	0
$961$	43.2230	1.39429
$962$	0	0
$963$	13.4671	0.433970
$964$	0	0
$965$	20.0612	0.645792
$966$	0	0
$967$	−33.7595	−1.08563	−0.542816	−	0.839852i	$-0.682642\pi$
$967$	−33.7595	−1.08563	−0.542816	+	0.839852i	$0.682642\pi$
$968$	0	0
$969$	0.950753	0.0305426
$970$	0	0
$971$	−41.9156	−1.34514	−0.672568	−	0.740035i	$-0.734809\pi$
$971$	−41.9156	−1.34514	−0.672568	+	0.740035i	$0.734809\pi$
$972$	0	0
$973$	−16.7844	−0.538082
$974$	0	0
$975$	9.14249	0.292794
$976$	0	0
$977$	−11.0950	−0.354959	−0.177479	−	0.984125i	$-0.556794\pi$
$977$	−11.0950	−0.354959	−0.177479	+	0.984125i	$0.556794\pi$
$978$	0	0
$979$	−15.4750	−0.494584
$980$	0	0
$981$	−1.05452	−0.0336682
$982$	0	0
$983$	−37.2295	−1.18744	−0.593719	−	0.804673i	$-0.702341\pi$
$983$	−37.2295	−1.18744	−0.593719	+	0.804673i	$0.702341\pi$
$984$	0	0
$985$	19.8768	0.633328
$986$	0	0
$987$	−12.9562	−0.412399
$988$	0	0
$989$	−10.4889	−0.333526
$990$	0	0
$991$	−19.1788	−0.609236	−0.304618	−	0.952475i	$-0.598529\pi$
$991$	−19.1788	−0.609236	−0.304618	+	0.952475i	$0.598529\pi$
$992$	0	0
$993$	0.492942	0.0156430
$994$	0	0
$995$	−15.7828	−0.500347
$996$	0	0
$997$	10.8795	0.344557	0.172278	−	0.985048i	$-0.444887\pi$
$997$	10.8795	0.344557	0.172278	+	0.985048i	$0.444887\pi$
$998$	0	0
$999$	−21.2637	−0.672753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.q.1.9	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.q.1.9	✓	23	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.32609 q^{3} -1.00000 q^{5} +2.45028 q^{7} -1.24148 q^{9} +O(q^{10})\)	\(q-1.32609 q^{3} -1.00000 q^{5} +2.45028 q^{7} -1.24148 q^{9} -3.05619 q^{11} -6.89432 q^{13} +1.32609 q^{15} -1.76603 q^{17} +0.405973 q^{19} -3.24929 q^{21} +9.08386 q^{23} +1.00000 q^{25} +5.62459 q^{27} +10.5088 q^{29} +8.61528 q^{31} +4.05279 q^{33} -2.45028 q^{35} -3.78048 q^{37} +9.14249 q^{39} -3.40559 q^{41} -1.15467 q^{43} +1.24148 q^{45} +3.98738 q^{47} -0.996141 q^{49} +2.34191 q^{51} +10.8106 q^{53} +3.05619 q^{55} -0.538357 q^{57} -0.272318 q^{59} +3.90654 q^{61} -3.04198 q^{63} +6.89432 q^{65} -7.45708 q^{67} -12.0460 q^{69} -13.0724 q^{71} -11.2825 q^{73} -1.32609 q^{75} -7.48852 q^{77} -4.68384 q^{79} -3.73427 q^{81} -10.5763 q^{83} +1.76603 q^{85} -13.9356 q^{87} +5.06349 q^{89} -16.8930 q^{91} -11.4246 q^{93} -0.405973 q^{95} +4.82364 q^{97} +3.79422 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 4 q^{3} - 23 q^{5} - 9 q^{7} + 37 q^{9}+O(q^{10}) \) 23 * q - 4 * q^3 - 23 * q^5 - 9 * q^7 + 37 * q^9	\( 23 q - 4 q^{3} - 23 q^{5} - 9 q^{7} + 37 q^{9} - 17 q^{11} - 10 q^{13} + 4 q^{15} - 5 q^{21} - 17 q^{23} + 23 q^{25} - 16 q^{27} - 5 q^{29} - 2 q^{31} - 3 q^{33} + 9 q^{35} - 10 q^{37} - 19 q^{39} - 6 q^{41} - 23 q^{43} - 37 q^{45} - 20 q^{47} + 68 q^{49} - 32 q^{51} - 23 q^{53} + 17 q^{55} + 8 q^{57} - 13 q^{59} + 4 q^{61} - 33 q^{63} + 10 q^{65} - 23 q^{67} + 27 q^{69} - 64 q^{71} - 5 q^{73} - 4 q^{75} - 50 q^{77} - 2 q^{79} + 47 q^{81} - 22 q^{83} - 38 q^{87} + q^{89} - 6 q^{91} - 43 q^{93} + 31 q^{97} - 68 q^{99}+O(q^{100}) \) 23 * q - 4 * q^3 - 23 * q^5 - 9 * q^7 + 37 * q^9 - 17 * q^11 - 10 * q^13 + 4 * q^15 - 5 * q^21 - 17 * q^23 + 23 * q^25 - 16 * q^27 - 5 * q^29 - 2 * q^31 - 3 * q^33 + 9 * q^35 - 10 * q^37 - 19 * q^39 - 6 * q^41 - 23 * q^43 - 37 * q^45 - 20 * q^47 + 68 * q^49 - 32 * q^51 - 23 * q^53 + 17 * q^55 + 8 * q^57 - 13 * q^59 + 4 * q^61 - 33 * q^63 + 10 * q^65 - 23 * q^67 + 27 * q^69 - 64 * q^71 - 5 * q^73 - 4 * q^75 - 50 * q^77 - 2 * q^79 + 47 * q^81 - 22 * q^83 - 38 * q^87 + q^89 - 6 * q^91 - 43 * q^93 + 31 * q^97 - 68 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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