LMFDB - Embedded newform 6028.2.a.c.1.2

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:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-3.29075 q^{3} +0.224551 q^{5} +1.38599 q^{7} +7.82903 q^{9} +O(q^{10})$	$q-3.29075 q^{3} +0.224551 q^{5} +1.38599 q^{7} +7.82903 q^{9} +1.00000 q^{11} +4.56916 q^{13} -0.738940 q^{15} +6.79020 q^{17} -5.08808 q^{19} -4.56096 q^{21} -2.53376 q^{23} -4.94958 q^{25} -15.8911 q^{27} +1.59262 q^{29} -10.8772 q^{31} -3.29075 q^{33} +0.311226 q^{35} +0.774631 q^{37} -15.0360 q^{39} -5.97631 q^{41} +0.900590 q^{43} +1.75802 q^{45} -1.69126 q^{47} -5.07902 q^{49} -22.3448 q^{51} -0.188215 q^{53} +0.224551 q^{55} +16.7436 q^{57} -3.39900 q^{59} -0.834329 q^{61} +10.8510 q^{63} +1.02601 q^{65} +10.7924 q^{67} +8.33798 q^{69} -0.605771 q^{71} -2.02309 q^{73} +16.2878 q^{75} +1.38599 q^{77} +8.16382 q^{79} +28.8066 q^{81} -5.40726 q^{83} +1.52474 q^{85} -5.24092 q^{87} +6.67430 q^{89} +6.33283 q^{91} +35.7941 q^{93} -1.14253 q^{95} -13.5712 q^{97} +7.82903 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * 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
$3$	−3.29075	−1.89992	−0.949958	−	0.312379i	$-0.898874\pi$
$3$	−3.29075	−1.89992	−0.949958	+	0.312379i	$0.898874\pi$
$4$	0	0
$5$	0.224551	0.100422	0.0502111	−	0.998739i	$-0.484011\pi$
$5$	0.224551	0.100422	0.0502111	+	0.998739i	$0.484011\pi$
$6$	0	0
$7$	1.38599	0.523857	0.261928	−	0.965087i	$-0.415642\pi$
$7$	1.38599	0.523857	0.261928	+	0.965087i	$0.415642\pi$
$8$	0	0
$9$	7.82903	2.60968
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.56916	1.26726	0.633629	−	0.773637i	$-0.281565\pi$
$13$	4.56916	1.26726	0.633629	+	0.773637i	$0.281565\pi$
$14$	0	0
$15$	−0.738940	−0.190794
$16$	0	0
$17$	6.79020	1.64686	0.823432	−	0.567415i	$-0.192056\pi$
$17$	6.79020	1.64686	0.823432	+	0.567415i	$0.192056\pi$
$18$	0	0
$19$	−5.08808	−1.16728	−0.583642	−	0.812011i	$-0.698373\pi$
$19$	−5.08808	−1.16728	−0.583642	+	0.812011i	$0.698373\pi$
$20$	0	0
$21$	−4.56096	−0.995283
$22$	0	0
$23$	−2.53376	−0.528326	−0.264163	−	0.964478i	$-0.585096\pi$
$23$	−2.53376	−0.528326	−0.264163	+	0.964478i	$0.585096\pi$
$24$	0	0
$25$	−4.94958	−0.989915
$26$	0	0
$27$	−15.8911	−3.05825
$28$	0	0
$29$	1.59262	0.295742	0.147871	−	0.989007i	$-0.452758\pi$
$29$	1.59262	0.295742	0.147871	+	0.989007i	$0.452758\pi$
$30$	0	0
$31$	−10.8772	−1.95360	−0.976801	−	0.214147i	$-0.931303\pi$
$31$	−10.8772	−1.95360	−0.976801	+	0.214147i	$0.931303\pi$
$32$	0	0
$33$	−3.29075	−0.572846
$34$	0	0
$35$	0.311226	0.0526068
$36$	0	0
$37$	0.774631	0.127349	0.0636743	−	0.997971i	$-0.479718\pi$
$37$	0.774631	0.127349	0.0636743	+	0.997971i	$0.479718\pi$
$38$	0	0
$39$	−15.0360	−2.40768
$40$	0	0
$41$	−5.97631	−0.933343	−0.466671	−	0.884431i	$-0.654547\pi$
$41$	−5.97631	−0.933343	−0.466671	+	0.884431i	$0.654547\pi$
$42$	0	0
$43$	0.900590	0.137339	0.0686694	−	0.997639i	$-0.478125\pi$
$43$	0.900590	0.137339	0.0686694	+	0.997639i	$0.478125\pi$
$44$	0	0
$45$	1.75802	0.262069
$46$	0	0
$47$	−1.69126	−0.246695	−0.123347	−	0.992364i	$-0.539363\pi$
$47$	−1.69126	−0.246695	−0.123347	+	0.992364i	$0.539363\pi$
$48$	0	0
$49$	−5.07902	−0.725574
$50$	0	0
$51$	−22.3448	−3.12890
$52$	0	0
$53$	−0.188215	−0.0258533	−0.0129266	−	0.999916i	$-0.504115\pi$
$53$	−0.188215	−0.0258533	−0.0129266	+	0.999916i	$0.504115\pi$
$54$	0	0
$55$	0.224551	0.0302784
$56$	0	0
$57$	16.7436	2.21774
$58$	0	0
$59$	−3.39900	−0.442512	−0.221256	−	0.975216i	$-0.571016\pi$
$59$	−3.39900	−0.442512	−0.221256	+	0.975216i	$0.571016\pi$
$60$	0	0
$61$	−0.834329	−0.106825	−0.0534124	−	0.998573i	$-0.517010\pi$
$61$	−0.834329	−0.106825	−0.0534124	+	0.998573i	$0.517010\pi$
$62$	0	0
$63$	10.8510	1.36710
$64$	0	0
$65$	1.02601	0.127261
$66$	0	0
$67$	10.7924	1.31850	0.659249	−	0.751924i	$-0.270874\pi$
$67$	10.7924	1.31850	0.659249	+	0.751924i	$0.270874\pi$
$68$	0	0
$69$	8.33798	1.00377
$70$	0	0
$71$	−0.605771	−0.0718918	−0.0359459	−	0.999354i	$-0.511444\pi$
$71$	−0.605771	−0.0718918	−0.0359459	+	0.999354i	$0.511444\pi$
$72$	0	0
$73$	−2.02309	−0.236784	−0.118392	−	0.992967i	$-0.537774\pi$
$73$	−2.02309	−0.236784	−0.118392	+	0.992967i	$0.537774\pi$
$74$	0	0
$75$	16.2878	1.88076
$76$	0	0
$77$	1.38599	0.157949
$78$	0	0
$79$	8.16382	0.918501	0.459251	−	0.888307i	$-0.348118\pi$
$79$	8.16382	0.918501	0.459251	+	0.888307i	$0.348118\pi$
$80$	0	0
$81$	28.8066	3.20074
$82$	0	0
$83$	−5.40726	−0.593524	−0.296762	−	0.954951i	$-0.595907\pi$
$83$	−5.40726	−0.593524	−0.296762	+	0.954951i	$0.595907\pi$
$84$	0	0
$85$	1.52474	0.165382
$86$	0	0
$87$	−5.24092	−0.561885
$88$	0	0
$89$	6.67430	0.707474	0.353737	−	0.935345i	$-0.384911\pi$
$89$	6.67430	0.707474	0.353737	+	0.935345i	$0.384911\pi$
$90$	0	0
$91$	6.33283	0.663861
$92$	0	0
$93$	35.7941	3.71168
$94$	0	0
$95$	−1.14253	−0.117221
$96$	0	0
$97$	−13.5712	−1.37795	−0.688976	−	0.724785i	$-0.741939\pi$
$97$	−13.5712	−1.37795	−0.688976	+	0.724785i	$0.741939\pi$
$98$	0	0
$99$	7.82903	0.786847
$100$	0	0
$101$	3.17255	0.315680	0.157840	−	0.987465i	$-0.449547\pi$
$101$	3.17255	0.315680	0.157840	+	0.987465i	$0.449547\pi$
$102$	0	0
$103$	−15.5465	−1.53185	−0.765923	−	0.642933i	$-0.777717\pi$
$103$	−15.5465	−1.53185	−0.765923	+	0.642933i	$0.777717\pi$
$104$	0	0
$105$	−1.02417	−0.0999485
$106$	0	0
$107$	11.8789	1.14838	0.574190	−	0.818722i	$-0.305317\pi$
$107$	11.8789	1.14838	0.574190	+	0.818722i	$0.305317\pi$
$108$	0	0
$109$	−7.92685	−0.759254	−0.379627	−	0.925140i	$-0.623948\pi$
$109$	−7.92685	−0.759254	−0.379627	+	0.925140i	$0.623948\pi$
$110$	0	0
$111$	−2.54912	−0.241952
$112$	0	0
$113$	−13.0033	−1.22325	−0.611626	−	0.791147i	$-0.709484\pi$
$113$	−13.0033	−1.22325	−0.611626	+	0.791147i	$0.709484\pi$
$114$	0	0
$115$	−0.568958	−0.0530556
$116$	0	0
$117$	35.7721	3.30713
$118$	0	0
$119$	9.41117	0.862721
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	19.6665	1.77327
$124$	0	0
$125$	−2.23419	−0.199832
$126$	0	0
$127$	−7.83625	−0.695355	−0.347677	−	0.937614i	$-0.613030\pi$
$127$	−7.83625	−0.695355	−0.347677	+	0.937614i	$0.613030\pi$
$128$	0	0
$129$	−2.96362	−0.260932
$130$	0	0
$131$	−11.2710	−0.984752	−0.492376	−	0.870383i	$-0.663872\pi$
$131$	−11.2710	−0.984752	−0.492376	+	0.870383i	$0.663872\pi$
$132$	0	0
$133$	−7.05205	−0.611490
$134$	0	0
$135$	−3.56837	−0.307116
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	−4.63125	−0.392817	−0.196409	−	0.980522i	$-0.562928\pi$
$139$	−4.63125	−0.392817	−0.196409	+	0.980522i	$0.562928\pi$
$140$	0	0
$141$	5.56550	0.468699
$142$	0	0
$143$	4.56916	0.382092
$144$	0	0
$145$	0.357624	0.0296991
$146$	0	0
$147$	16.7138	1.37853
$148$	0	0
$149$	−16.3136	−1.33646	−0.668230	−	0.743955i	$-0.732948\pi$
$149$	−16.3136	−1.33646	−0.668230	+	0.743955i	$0.732948\pi$
$150$	0	0
$151$	−17.7554	−1.44491	−0.722456	−	0.691417i	$-0.756987\pi$
$151$	−17.7554	−1.44491	−0.722456	+	0.691417i	$0.756987\pi$
$152$	0	0
$153$	53.1607	4.29778
$154$	0	0
$155$	−2.44248	−0.196185
$156$	0	0
$157$	19.4344	1.55104	0.775518	−	0.631325i	$-0.217489\pi$
$157$	19.4344	1.55104	0.775518	+	0.631325i	$0.217489\pi$
$158$	0	0
$159$	0.619368	0.0491191
$160$	0	0
$161$	−3.51178	−0.276767
$162$	0	0
$163$	13.3100	1.04252	0.521260	−	0.853398i	$-0.325462\pi$
$163$	13.3100	1.04252	0.521260	+	0.853398i	$0.325462\pi$
$164$	0	0
$165$	−0.738940	−0.0575264
$166$	0	0
$167$	7.41466	0.573764	0.286882	−	0.957966i	$-0.407381\pi$
$167$	7.41466	0.573764	0.286882	+	0.957966i	$0.407381\pi$
$168$	0	0
$169$	7.87723	0.605941
$170$	0	0
$171$	−39.8347	−3.04624
$172$	0	0
$173$	−14.4508	−1.09867	−0.549337	−	0.835601i	$-0.685120\pi$
$173$	−14.4508	−1.09867	−0.549337	+	0.835601i	$0.685120\pi$
$174$	0	0
$175$	−6.86009	−0.518574
$176$	0	0
$177$	11.1852	0.840735
$178$	0	0
$179$	15.1928	1.13556	0.567782	−	0.823179i	$-0.307802\pi$
$179$	15.1928	1.13556	0.567782	+	0.823179i	$0.307802\pi$
$180$	0	0
$181$	−3.03007	−0.225223	−0.112612	−	0.993639i	$-0.535922\pi$
$181$	−3.03007	−0.225223	−0.112612	+	0.993639i	$0.535922\pi$
$182$	0	0
$183$	2.74557	0.202958
$184$	0	0
$185$	0.173944	0.0127886
$186$	0	0
$187$	6.79020	0.496548
$188$	0	0
$189$	−22.0250	−1.60209
$190$	0	0
$191$	11.5054	0.832499	0.416250	−	0.909250i	$-0.363344\pi$
$191$	11.5054	0.832499	0.416250	+	0.909250i	$0.363344\pi$
$192$	0	0
$193$	−23.3331	−1.67956	−0.839778	−	0.542930i	$-0.817315\pi$
$193$	−23.3331	−1.67956	−0.839778	+	0.542930i	$0.817315\pi$
$194$	0	0
$195$	−3.37634	−0.241785
$196$	0	0
$197$	3.68272	0.262383	0.131191	−	0.991357i	$-0.458120\pi$
$197$	3.68272	0.262383	0.131191	+	0.991357i	$0.458120\pi$
$198$	0	0
$199$	7.25744	0.514466	0.257233	−	0.966349i	$-0.417189\pi$
$199$	7.25744	0.514466	0.257233	+	0.966349i	$0.417189\pi$
$200$	0	0
$201$	−35.5150	−2.50504
$202$	0	0
$203$	2.20737	0.154927
$204$	0	0
$205$	−1.34199	−0.0937283
$206$	0	0
$207$	−19.8369	−1.37876
$208$	0	0
$209$	−5.08808	−0.351950
$210$	0	0
$211$	1.56621	0.107822	0.0539111	−	0.998546i	$-0.482831\pi$
$211$	1.56621	0.107822	0.0539111	+	0.998546i	$0.482831\pi$
$212$	0	0
$213$	1.99344	0.136588
$214$	0	0
$215$	0.202228	0.0137919
$216$	0	0
$217$	−15.0757	−1.02341
$218$	0	0
$219$	6.65747	0.449870
$220$	0	0
$221$	31.0255	2.08700
$222$	0	0
$223$	−9.55400	−0.639783	−0.319891	−	0.947454i	$-0.603646\pi$
$223$	−9.55400	−0.639783	−0.319891	+	0.947454i	$0.603646\pi$
$224$	0	0
$225$	−38.7504	−2.58336
$226$	0	0
$227$	−17.9965	−1.19447	−0.597235	−	0.802066i	$-0.703734\pi$
$227$	−17.9965	−1.19447	−0.597235	+	0.802066i	$0.703734\pi$
$228$	0	0
$229$	21.8938	1.44679	0.723393	−	0.690437i	$-0.242582\pi$
$229$	21.8938	1.44679	0.723393	+	0.690437i	$0.242582\pi$
$230$	0	0
$231$	−4.56096	−0.300089
$232$	0	0
$233$	6.73150	0.440995	0.220498	−	0.975388i	$-0.429232\pi$
$233$	6.73150	0.440995	0.220498	+	0.975388i	$0.429232\pi$
$234$	0	0
$235$	−0.379773	−0.0247736
$236$	0	0
$237$	−26.8651	−1.74507
$238$	0	0
$239$	1.39387	0.0901619	0.0450809	−	0.998983i	$-0.485645\pi$
$239$	1.39387	0.0901619	0.0450809	+	0.998983i	$0.485645\pi$
$240$	0	0
$241$	9.80535	0.631618	0.315809	−	0.948823i	$-0.397724\pi$
$241$	9.80535	0.631618	0.315809	+	0.948823i	$0.397724\pi$
$242$	0	0
$243$	−47.1220	−3.02288
$244$	0	0
$245$	−1.14050	−0.0728637
$246$	0	0
$247$	−23.2482	−1.47925
$248$	0	0
$249$	17.7939	1.12765
$250$	0	0
$251$	10.8229	0.683137	0.341568	−	0.939857i	$-0.389042\pi$
$251$	10.8229	0.683137	0.341568	+	0.939857i	$0.389042\pi$
$252$	0	0
$253$	−2.53376	−0.159296
$254$	0	0
$255$	−5.01755	−0.314211
$256$	0	0
$257$	−9.61862	−0.599993	−0.299997	−	0.953940i	$-0.596986\pi$
$257$	−9.61862	−0.599993	−0.299997	+	0.953940i	$0.596986\pi$
$258$	0	0
$259$	1.07364	0.0667124
$260$	0	0
$261$	12.4687	0.771792
$262$	0	0
$263$	−21.2541	−1.31059	−0.655293	−	0.755375i	$-0.727455\pi$
$263$	−21.2541	−1.31059	−0.655293	+	0.755375i	$0.727455\pi$
$264$	0	0
$265$	−0.0422638	−0.00259624
$266$	0	0
$267$	−21.9634	−1.34414
$268$	0	0
$269$	0.539063	0.0328673	0.0164336	−	0.999865i	$-0.494769\pi$
$269$	0.539063	0.0328673	0.0164336	+	0.999865i	$0.494769\pi$
$270$	0	0
$271$	6.64394	0.403591	0.201795	−	0.979428i	$-0.435322\pi$
$271$	6.64394	0.403591	0.201795	+	0.979428i	$0.435322\pi$
$272$	0	0
$273$	−20.8398	−1.26128
$274$	0	0
$275$	−4.94958	−0.298471
$276$	0	0
$277$	−14.4545	−0.868487	−0.434244	−	0.900795i	$-0.642984\pi$
$277$	−14.4545	−0.868487	−0.434244	+	0.900795i	$0.642984\pi$
$278$	0	0
$279$	−85.1579	−5.09827
$280$	0	0
$281$	−26.3634	−1.57271	−0.786355	−	0.617774i	$-0.788035\pi$
$281$	−26.3634	−1.57271	−0.786355	+	0.617774i	$0.788035\pi$
$282$	0	0
$283$	−4.93558	−0.293390	−0.146695	−	0.989182i	$-0.546864\pi$
$283$	−4.93558	−0.293390	−0.146695	+	0.989182i	$0.546864\pi$
$284$	0	0
$285$	3.75978	0.222710
$286$	0	0
$287$	−8.28313	−0.488938
$288$	0	0
$289$	29.1068	1.71216
$290$	0	0
$291$	44.6596	2.61799
$292$	0	0
$293$	13.6724	0.798747	0.399374	−	0.916788i	$-0.369228\pi$
$293$	13.6724	0.798747	0.399374	+	0.916788i	$0.369228\pi$
$294$	0	0
$295$	−0.763247	−0.0444380
$296$	0	0
$297$	−15.8911	−0.922097
$298$	0	0
$299$	−11.5772	−0.669525
$300$	0	0
$301$	1.24821	0.0719458
$302$	0	0
$303$	−10.4401	−0.599766
$304$	0	0
$305$	−0.187349	−0.0107276
$306$	0	0
$307$	24.2366	1.38325	0.691627	−	0.722255i	$-0.256894\pi$
$307$	24.2366	1.38325	0.691627	+	0.722255i	$0.256894\pi$
$308$	0	0
$309$	51.1597	2.91038
$310$	0	0
$311$	2.17097	0.123104	0.0615522	−	0.998104i	$-0.480395\pi$
$311$	2.17097	0.123104	0.0615522	+	0.998104i	$0.480395\pi$
$312$	0	0
$313$	−9.33593	−0.527698	−0.263849	−	0.964564i	$-0.584992\pi$
$313$	−9.33593	−0.527698	−0.263849	+	0.964564i	$0.584992\pi$
$314$	0	0
$315$	2.43660	0.137287
$316$	0	0
$317$	−28.5607	−1.60413	−0.802064	−	0.597237i	$-0.796265\pi$
$317$	−28.5607	−1.60413	−0.802064	+	0.597237i	$0.796265\pi$
$318$	0	0
$319$	1.59262	0.0891697
$320$	0	0
$321$	−39.0906	−2.18182
$322$	0	0
$323$	−34.5490	−1.92236
$324$	0	0
$325$	−22.6154	−1.25448
$326$	0	0
$327$	26.0853	1.44252
$328$	0	0
$329$	−2.34407	−0.129233
$330$	0	0
$331$	24.9418	1.37093	0.685463	−	0.728107i	$-0.259600\pi$
$331$	24.9418	1.37093	0.685463	+	0.728107i	$0.259600\pi$
$332$	0	0
$333$	6.06461	0.332339
$334$	0	0
$335$	2.42344	0.132406
$336$	0	0
$337$	−2.44100	−0.132970	−0.0664848	−	0.997787i	$-0.521178\pi$
$337$	−2.44100	−0.132970	−0.0664848	+	0.997787i	$0.521178\pi$
$338$	0	0
$339$	42.7907	2.32407
$340$	0	0
$341$	−10.8772	−0.589033
$342$	0	0
$343$	−16.7415	−0.903954
$344$	0	0
$345$	1.87230	0.100801
$346$	0	0
$347$	−13.3483	−0.716575	−0.358287	−	0.933611i	$-0.616639\pi$
$347$	−13.3483	−0.716575	−0.358287	+	0.933611i	$0.616639\pi$
$348$	0	0
$349$	34.5768	1.85085	0.925426	−	0.378928i	$-0.123707\pi$
$349$	34.5768	1.85085	0.925426	+	0.378928i	$0.123707\pi$
$350$	0	0
$351$	−72.6091	−3.87559
$352$	0	0
$353$	9.02812	0.480518	0.240259	−	0.970709i	$-0.422768\pi$
$353$	9.02812	0.480518	0.240259	+	0.970709i	$0.422768\pi$
$354$	0	0
$355$	−0.136026	−0.00721953
$356$	0	0
$357$	−30.9698	−1.63910
$358$	0	0
$359$	29.6193	1.56324	0.781622	−	0.623752i	$-0.214392\pi$
$359$	29.6193	1.56324	0.781622	+	0.623752i	$0.214392\pi$
$360$	0	0
$361$	6.88851	0.362553
$362$	0	0
$363$	−3.29075	−0.172720
$364$	0	0
$365$	−0.454285	−0.0237784
$366$	0	0
$367$	8.11045	0.423362	0.211681	−	0.977339i	$-0.432106\pi$
$367$	8.11045	0.423362	0.211681	+	0.977339i	$0.432106\pi$
$368$	0	0
$369$	−46.7887	−2.43572
$370$	0	0
$371$	−0.260865	−0.0135434
$372$	0	0
$373$	−3.88170	−0.200987	−0.100494	−	0.994938i	$-0.532042\pi$
$373$	−3.88170	−0.200987	−0.100494	+	0.994938i	$0.532042\pi$
$374$	0	0
$375$	7.35214	0.379663
$376$	0	0
$377$	7.27694	0.374782
$378$	0	0
$379$	−24.9198	−1.28004	−0.640022	−	0.768357i	$-0.721075\pi$
$379$	−24.9198	−1.28004	−0.640022	+	0.768357i	$0.721075\pi$
$380$	0	0
$381$	25.7871	1.32112
$382$	0	0
$383$	−26.8445	−1.37169	−0.685845	−	0.727748i	$-0.740567\pi$
$383$	−26.8445	−1.37169	−0.685845	+	0.727748i	$0.740567\pi$
$384$	0	0
$385$	0.311226	0.0158616
$386$	0	0
$387$	7.05075	0.358410
$388$	0	0
$389$	−18.8791	−0.957209	−0.478604	−	0.878031i	$-0.658857\pi$
$389$	−18.8791	−0.957209	−0.478604	+	0.878031i	$0.658857\pi$
$390$	0	0
$391$	−17.2047	−0.870081
$392$	0	0
$393$	37.0901	1.87095
$394$	0	0
$395$	1.83319	0.0922379
$396$	0	0
$397$	−13.3701	−0.671029	−0.335514	−	0.942035i	$-0.608910\pi$
$397$	−13.3701	−0.671029	−0.335514	+	0.942035i	$0.608910\pi$
$398$	0	0
$399$	23.2065	1.16178
$400$	0	0
$401$	−2.99932	−0.149779	−0.0748894	−	0.997192i	$-0.523860\pi$
$401$	−2.99932	−0.149779	−0.0748894	+	0.997192i	$0.523860\pi$
$402$	0	0
$403$	−49.6997	−2.47572
$404$	0	0
$405$	6.46856	0.321425
$406$	0	0
$407$	0.774631	0.0383971
$408$	0	0
$409$	−4.98633	−0.246558	−0.123279	−	0.992372i	$-0.539341\pi$
$409$	−4.98633	−0.246558	−0.123279	+	0.992372i	$0.539341\pi$
$410$	0	0
$411$	3.29075	0.162321
$412$	0	0
$413$	−4.71099	−0.231813
$414$	0	0
$415$	−1.21421	−0.0596030
$416$	0	0
$417$	15.2403	0.746319
$418$	0	0
$419$	11.9908	0.585787	0.292893	−	0.956145i	$-0.405382\pi$
$419$	11.9908	0.585787	0.292893	+	0.956145i	$0.405382\pi$
$420$	0	0
$421$	11.3784	0.554551	0.277276	−	0.960790i	$-0.410569\pi$
$421$	11.3784	0.554551	0.277276	+	0.960790i	$0.410569\pi$
$422$	0	0
$423$	−13.2409	−0.643794
$424$	0	0
$425$	−33.6086	−1.63026
$426$	0	0
$427$	−1.15638	−0.0559609
$428$	0	0
$429$	−15.0360	−0.725943
$430$	0	0
$431$	9.36409	0.451052	0.225526	−	0.974237i	$-0.427590\pi$
$431$	9.36409	0.451052	0.225526	+	0.974237i	$0.427590\pi$
$432$	0	0
$433$	−31.7615	−1.52636	−0.763180	−	0.646186i	$-0.776363\pi$
$433$	−31.7615	−1.52636	−0.763180	+	0.646186i	$0.776363\pi$
$434$	0	0
$435$	−1.17685	−0.0564258
$436$	0	0
$437$	12.8920	0.616707
$438$	0	0
$439$	−9.32975	−0.445285	−0.222642	−	0.974900i	$-0.571468\pi$
$439$	−9.32975	−0.445285	−0.222642	+	0.974900i	$0.571468\pi$
$440$	0	0
$441$	−39.7638	−1.89351
$442$	0	0
$443$	−35.6628	−1.69439	−0.847195	−	0.531282i	$-0.821710\pi$
$443$	−35.6628	−1.69439	−0.847195	+	0.531282i	$0.821710\pi$
$444$	0	0
$445$	1.49872	0.0710461
$446$	0	0
$447$	53.6839	2.53916
$448$	0	0
$449$	14.1096	0.665871	0.332935	−	0.942950i	$-0.391961\pi$
$449$	14.1096	0.665871	0.332935	+	0.942950i	$0.391961\pi$
$450$	0	0
$451$	−5.97631	−0.281413
$452$	0	0
$453$	58.4285	2.74521
$454$	0	0
$455$	1.42204	0.0666664
$456$	0	0
$457$	42.6912	1.99701	0.998505	−	0.0546557i	$-0.0174061\pi$
$457$	42.6912	1.99701	0.998505	+	0.0546557i	$0.0174061\pi$
$458$	0	0
$459$	−107.904	−5.03652
$460$	0	0
$461$	−3.53003	−0.164410	−0.0822048	−	0.996615i	$-0.526196\pi$
$461$	−3.53003	−0.164410	−0.0822048	+	0.996615i	$0.526196\pi$
$462$	0	0
$463$	30.7068	1.42707	0.713534	−	0.700621i	$-0.247094\pi$
$463$	30.7068	1.42707	0.713534	+	0.700621i	$0.247094\pi$
$464$	0	0
$465$	8.03760	0.372735
$466$	0	0
$467$	−18.2711	−0.845486	−0.422743	−	0.906250i	$-0.638933\pi$
$467$	−18.2711	−0.845486	−0.422743	+	0.906250i	$0.638933\pi$
$468$	0	0
$469$	14.9582	0.690704
$470$	0	0
$471$	−63.9538	−2.94684
$472$	0	0
$473$	0.900590	0.0414092
$474$	0	0
$475$	25.1838	1.15551
$476$	0	0
$477$	−1.47354	−0.0674688
$478$	0	0
$479$	−39.6377	−1.81109	−0.905547	−	0.424245i	$-0.860539\pi$
$479$	−39.6377	−1.81109	−0.905547	+	0.424245i	$0.860539\pi$
$480$	0	0
$481$	3.53942	0.161383
$482$	0	0
$483$	11.5564	0.525834
$484$	0	0
$485$	−3.04743	−0.138377
$486$	0	0
$487$	41.4570	1.87860	0.939298	−	0.343103i	$-0.111478\pi$
$487$	41.4570	1.87860	0.939298	+	0.343103i	$0.111478\pi$
$488$	0	0
$489$	−43.7999	−1.98070
$490$	0	0
$491$	−15.6182	−0.704839	−0.352419	−	0.935842i	$-0.614641\pi$
$491$	−15.6182	−0.704839	−0.352419	+	0.935842i	$0.614641\pi$
$492$	0	0
$493$	10.8142	0.487048
$494$	0	0
$495$	1.75802	0.0790169
$496$	0	0
$497$	−0.839595	−0.0376610
$498$	0	0
$499$	5.11809	0.229117	0.114559	−	0.993417i	$-0.463455\pi$
$499$	5.11809	0.229117	0.114559	+	0.993417i	$0.463455\pi$
$500$	0	0
$501$	−24.3998	−1.09010
$502$	0	0
$503$	−21.4678	−0.957202	−0.478601	−	0.878032i	$-0.658856\pi$
$503$	−21.4678	−0.957202	−0.478601	+	0.878032i	$0.658856\pi$
$504$	0	0
$505$	0.712399	0.0317013
$506$	0	0
$507$	−25.9220	−1.15124
$508$	0	0
$509$	−36.0545	−1.59809	−0.799045	−	0.601272i	$-0.794661\pi$
$509$	−36.0545	−1.59809	−0.799045	+	0.601272i	$0.794661\pi$
$510$	0	0
$511$	−2.80399	−0.124041
$512$	0	0
$513$	80.8553	3.56985
$514$	0	0
$515$	−3.49099	−0.153831
$516$	0	0
$517$	−1.69126	−0.0743813
$518$	0	0
$519$	47.5540	2.08739
$520$	0	0
$521$	−36.8245	−1.61331	−0.806655	−	0.591022i	$-0.798725\pi$
$521$	−36.8245	−1.61331	−0.806655	+	0.591022i	$0.798725\pi$
$522$	0	0
$523$	12.4290	0.543483	0.271742	−	0.962370i	$-0.412400\pi$
$523$	12.4290	0.543483	0.271742	+	0.962370i	$0.412400\pi$
$524$	0	0
$525$	22.5748	0.985246
$526$	0	0
$527$	−73.8583	−3.21732
$528$	0	0
$529$	−16.5801	−0.720872
$530$	0	0
$531$	−26.6109	−1.15481
$532$	0	0
$533$	−27.3067	−1.18279
$534$	0	0
$535$	2.66742	0.115323
$536$	0	0
$537$	−49.9957	−2.15747
$538$	0	0
$539$	−5.07902	−0.218769
$540$	0	0
$541$	2.35938	0.101438	0.0507188	−	0.998713i	$-0.483849\pi$
$541$	2.35938	0.101438	0.0507188	+	0.998713i	$0.483849\pi$
$542$	0	0
$543$	9.97119	0.427905
$544$	0	0
$545$	−1.77998	−0.0762460
$546$	0	0
$547$	1.63272	0.0698101	0.0349050	−	0.999391i	$-0.488887\pi$
$547$	1.63272	0.0698101	0.0349050	+	0.999391i	$0.488887\pi$
$548$	0	0
$549$	−6.53199	−0.278778
$550$	0	0
$551$	−8.10338	−0.345216
$552$	0	0
$553$	11.3150	0.481163
$554$	0	0
$555$	−0.572407	−0.0242973
$556$	0	0
$557$	−11.0177	−0.466833	−0.233417	−	0.972377i	$-0.574991\pi$
$557$	−11.0177	−0.466833	−0.233417	+	0.972377i	$0.574991\pi$
$558$	0	0
$559$	4.11494	0.174043
$560$	0	0
$561$	−22.3448	−0.943399
$562$	0	0
$563$	24.9774	1.05267	0.526336	−	0.850277i	$-0.323565\pi$
$563$	24.9774	1.05267	0.526336	+	0.850277i	$0.323565\pi$
$564$	0	0
$565$	−2.91991	−0.122842
$566$	0	0
$567$	39.9259	1.67673
$568$	0	0
$569$	39.7669	1.66712	0.833558	−	0.552432i	$-0.186300\pi$
$569$	39.7669	1.66712	0.833558	+	0.552432i	$0.186300\pi$
$570$	0	0
$571$	−4.45018	−0.186234	−0.0931171	−	0.995655i	$-0.529683\pi$
$571$	−4.45018	−0.186234	−0.0931171	+	0.995655i	$0.529683\pi$
$572$	0	0
$573$	−37.8613	−1.58168
$574$	0	0
$575$	12.5410	0.522998
$576$	0	0
$577$	−24.2946	−1.01140	−0.505699	−	0.862710i	$-0.668766\pi$
$577$	−24.2946	−1.01140	−0.505699	+	0.862710i	$0.668766\pi$
$578$	0	0
$579$	76.7835	3.19101
$580$	0	0
$581$	−7.49444	−0.310922
$582$	0	0
$583$	−0.188215	−0.00779506
$584$	0	0
$585$	8.03265	0.332109
$586$	0	0
$587$	−15.6625	−0.646460	−0.323230	−	0.946320i	$-0.604769\pi$
$587$	−15.6625	−0.646460	−0.323230	+	0.946320i	$0.604769\pi$
$588$	0	0
$589$	55.3440	2.28041
$590$	0	0
$591$	−12.1189	−0.498505
$592$	0	0
$593$	−2.63496	−0.108205	−0.0541023	−	0.998535i	$-0.517230\pi$
$593$	−2.63496	−0.108205	−0.0541023	+	0.998535i	$0.517230\pi$
$594$	0	0
$595$	2.11329	0.0866363
$596$	0	0
$597$	−23.8824	−0.977442
$598$	0	0
$599$	13.2955	0.543241	0.271621	−	0.962404i	$-0.412440\pi$
$599$	13.2955	0.543241	0.271621	+	0.962404i	$0.412440\pi$
$600$	0	0
$601$	9.32689	0.380452	0.190226	−	0.981740i	$-0.439078\pi$
$601$	9.32689	0.380452	0.190226	+	0.981740i	$0.439078\pi$
$602$	0	0
$603$	84.4938	3.44086
$604$	0	0
$605$	0.224551	0.00912929
$606$	0	0
$607$	−11.6824	−0.474175	−0.237088	−	0.971488i	$-0.576193\pi$
$607$	−11.6824	−0.474175	−0.237088	+	0.971488i	$0.576193\pi$
$608$	0	0
$609$	−7.26389	−0.294347
$610$	0	0
$611$	−7.72762	−0.312626
$612$	0	0
$613$	−15.7497	−0.636126	−0.318063	−	0.948070i	$-0.603032\pi$
$613$	−15.7497	−0.636126	−0.318063	+	0.948070i	$0.603032\pi$
$614$	0	0
$615$	4.41614	0.178076
$616$	0	0
$617$	21.9516	0.883738	0.441869	−	0.897080i	$-0.354316\pi$
$617$	21.9516	0.883738	0.441869	+	0.897080i	$0.354316\pi$
$618$	0	0
$619$	30.4160	1.22252	0.611261	−	0.791429i	$-0.290662\pi$
$619$	30.4160	1.22252	0.611261	+	0.791429i	$0.290662\pi$
$620$	0	0
$621$	40.2644	1.61575
$622$	0	0
$623$	9.25054	0.370615
$624$	0	0
$625$	24.2462	0.969848
$626$	0	0
$627$	16.7436	0.668674
$628$	0	0
$629$	5.25990	0.209726
$630$	0	0
$631$	−24.4938	−0.975083	−0.487541	−	0.873100i	$-0.662106\pi$
$631$	−24.4938	−0.975083	−0.487541	+	0.873100i	$0.662106\pi$
$632$	0	0
$633$	−5.15400	−0.204853
$634$	0	0
$635$	−1.75964	−0.0698291
$636$	0	0
$637$	−23.2069	−0.919489
$638$	0	0
$639$	−4.74260	−0.187614
$640$	0	0
$641$	24.7428	0.977284	0.488642	−	0.872484i	$-0.337492\pi$
$641$	24.7428	0.977284	0.488642	+	0.872484i	$0.337492\pi$
$642$	0	0
$643$	21.6607	0.854216	0.427108	−	0.904201i	$-0.359532\pi$
$643$	21.6607	0.854216	0.427108	+	0.904201i	$0.359532\pi$
$644$	0	0
$645$	−0.665482	−0.0262033
$646$	0	0
$647$	−16.9430	−0.666097	−0.333049	−	0.942910i	$-0.608077\pi$
$647$	−16.9430	−0.666097	−0.333049	+	0.942910i	$0.608077\pi$
$648$	0	0
$649$	−3.39900	−0.133422
$650$	0	0
$651$	49.6105	1.94439
$652$	0	0
$653$	9.33307	0.365231	0.182616	−	0.983184i	$-0.441544\pi$
$653$	9.33307	0.365231	0.182616	+	0.983184i	$0.441544\pi$
$654$	0	0
$655$	−2.53091	−0.0988910
$656$	0	0
$657$	−15.8388	−0.617930
$658$	0	0
$659$	−21.9292	−0.854241	−0.427120	−	0.904195i	$-0.640472\pi$
$659$	−21.9292	−0.854241	−0.427120	+	0.904195i	$0.640472\pi$
$660$	0	0
$661$	16.0710	0.625090	0.312545	−	0.949903i	$-0.398818\pi$
$661$	16.0710	0.625090	0.312545	+	0.949903i	$0.398818\pi$
$662$	0	0
$663$	−102.097	−3.96512
$664$	0	0
$665$	−1.58354	−0.0614071
$666$	0	0
$667$	−4.03532	−0.156248
$668$	0	0
$669$	31.4398	1.21553
$670$	0	0
$671$	−0.834329	−0.0322089
$672$	0	0
$673$	−40.4526	−1.55933	−0.779667	−	0.626195i	$-0.784611\pi$
$673$	−40.4526	−1.55933	−0.779667	+	0.626195i	$0.784611\pi$
$674$	0	0
$675$	78.6544	3.02741
$676$	0	0
$677$	50.5777	1.94386	0.971930	−	0.235271i	$-0.0755977\pi$
$677$	50.5777	1.94386	0.971930	+	0.235271i	$0.0755977\pi$
$678$	0	0
$679$	−18.8097	−0.721849
$680$	0	0
$681$	59.2220	2.26939
$682$	0	0
$683$	−30.7544	−1.17678	−0.588391	−	0.808576i	$-0.700239\pi$
$683$	−30.7544	−1.17678	−0.588391	+	0.808576i	$0.700239\pi$
$684$	0	0
$685$	−0.224551	−0.00857965
$686$	0	0
$687$	−72.0471	−2.74877
$688$	0	0
$689$	−0.859984	−0.0327628
$690$	0	0
$691$	−23.5974	−0.897686	−0.448843	−	0.893611i	$-0.648164\pi$
$691$	−23.5974	−0.897686	−0.448843	+	0.893611i	$0.648164\pi$
$692$	0	0
$693$	10.8510	0.412195
$694$	0	0
$695$	−1.03995	−0.0394475
$696$	0	0
$697$	−40.5803	−1.53709
$698$	0	0
$699$	−22.1517	−0.837853
$700$	0	0
$701$	0.527257	0.0199142	0.00995711	−	0.999950i	$-0.496831\pi$
$701$	0.527257	0.0199142	0.00995711	+	0.999950i	$0.496831\pi$
$702$	0	0
$703$	−3.94138	−0.148652
$704$	0	0
$705$	1.24974	0.0470678
$706$	0	0
$707$	4.39714	0.165371
$708$	0	0
$709$	−33.8604	−1.27166	−0.635828	−	0.771831i	$-0.719341\pi$
$709$	−33.8604	−1.27166	−0.635828	+	0.771831i	$0.719341\pi$
$710$	0	0
$711$	63.9148	2.39699
$712$	0	0
$713$	27.5602	1.03214
$714$	0	0
$715$	1.02601	0.0383706
$716$	0	0
$717$	−4.58687	−0.171300
$718$	0	0
$719$	29.5125	1.10063	0.550316	−	0.834957i	$-0.314507\pi$
$719$	29.5125	1.10063	0.550316	+	0.834957i	$0.314507\pi$
$720$	0	0
$721$	−21.5474	−0.802467
$722$	0	0
$723$	−32.2669	−1.20002
$724$	0	0
$725$	−7.88280	−0.292760
$726$	0	0
$727$	−44.1906	−1.63894	−0.819469	−	0.573123i	$-0.805732\pi$
$727$	−44.1906	−1.63894	−0.819469	+	0.573123i	$0.805732\pi$
$728$	0	0
$729$	68.6469	2.54248
$730$	0	0
$731$	6.11518	0.226178
$732$	0	0
$733$	23.9591	0.884951	0.442475	−	0.896781i	$-0.354100\pi$
$733$	23.9591	0.884951	0.442475	+	0.896781i	$0.354100\pi$
$734$	0	0
$735$	3.75309	0.138435
$736$	0	0
$737$	10.7924	0.397542
$738$	0	0
$739$	2.48594	0.0914469	0.0457234	−	0.998954i	$-0.485441\pi$
$739$	2.48594	0.0914469	0.0457234	+	0.998954i	$0.485441\pi$
$740$	0	0
$741$	76.5041	2.81045
$742$	0	0
$743$	−40.1084	−1.47143	−0.735717	−	0.677289i	$-0.763155\pi$
$743$	−40.1084	−1.47143	−0.735717	+	0.677289i	$0.763155\pi$
$744$	0	0
$745$	−3.66322	−0.134210
$746$	0	0
$747$	−42.3336	−1.54891
$748$	0	0
$749$	16.4641	0.601587
$750$	0	0
$751$	37.8475	1.38108	0.690538	−	0.723296i	$-0.257374\pi$
$751$	37.8475	1.38108	0.690538	+	0.723296i	$0.257374\pi$
$752$	0	0
$753$	−35.6155	−1.29790
$754$	0	0
$755$	−3.98698	−0.145101
$756$	0	0
$757$	37.8570	1.37594	0.687969	−	0.725740i	$-0.258503\pi$
$757$	37.8570	1.37594	0.687969	+	0.725740i	$0.258503\pi$
$758$	0	0
$759$	8.33798	0.302649
$760$	0	0
$761$	−28.0206	−1.01575	−0.507873	−	0.861432i	$-0.669568\pi$
$761$	−28.0206	−1.01575	−0.507873	+	0.861432i	$0.669568\pi$
$762$	0	0
$763$	−10.9866	−0.397740
$764$	0	0
$765$	11.9373	0.431593
$766$	0	0
$767$	−15.5306	−0.560776
$768$	0	0
$769$	−18.5338	−0.668347	−0.334173	−	0.942512i	$-0.608457\pi$
$769$	−18.5338	−0.668347	−0.334173	+	0.942512i	$0.608457\pi$
$770$	0	0
$771$	31.6525	1.13994
$772$	0	0
$773$	27.6554	0.994697	0.497349	−	0.867551i	$-0.334307\pi$
$773$	27.6554	0.994697	0.497349	+	0.867551i	$0.334307\pi$
$774$	0	0
$775$	53.8375	1.93390
$776$	0	0
$777$	−3.53306	−0.126748
$778$	0	0
$779$	30.4079	1.08948
$780$	0	0
$781$	−0.605771	−0.0216762
$782$	0	0
$783$	−25.3086	−0.904454
$784$	0	0
$785$	4.36402	0.155758
$786$	0	0
$787$	−45.7918	−1.63230	−0.816151	−	0.577839i	$-0.803896\pi$
$787$	−45.7918	−1.63230	−0.816151	+	0.577839i	$0.803896\pi$
$788$	0	0
$789$	69.9420	2.49000
$790$	0	0
$791$	−18.0226	−0.640808
$792$	0	0
$793$	−3.81218	−0.135375
$794$	0	0
$795$	0.139080	0.00493264
$796$	0	0
$797$	−8.31770	−0.294628	−0.147314	−	0.989090i	$-0.547063\pi$
$797$	−8.31770	−0.294628	−0.147314	+	0.989090i	$0.547063\pi$
$798$	0	0
$799$	−11.4840	−0.406273
$800$	0	0
$801$	52.2533	1.84628
$802$	0	0
$803$	−2.02309	−0.0713931
$804$	0	0
$805$	−0.788573	−0.0277936
$806$	0	0
$807$	−1.77392	−0.0624450
$808$	0	0
$809$	21.4921	0.755620	0.377810	−	0.925883i	$-0.376677\pi$
$809$	21.4921	0.755620	0.377810	+	0.925883i	$0.376677\pi$
$810$	0	0
$811$	−2.90964	−0.102171	−0.0510856	−	0.998694i	$-0.516268\pi$
$811$	−2.90964	−0.102171	−0.0510856	+	0.998694i	$0.516268\pi$
$812$	0	0
$813$	−21.8636	−0.766788
$814$	0	0
$815$	2.98877	0.104692
$816$	0	0
$817$	−4.58227	−0.160313
$818$	0	0
$819$	49.5799	1.73246
$820$	0	0
$821$	−33.5560	−1.17111	−0.585557	−	0.810632i	$-0.699124\pi$
$821$	−33.5560	−1.17111	−0.585557	+	0.810632i	$0.699124\pi$
$822$	0	0
$823$	16.3594	0.570252	0.285126	−	0.958490i	$-0.407965\pi$
$823$	16.3594	0.570252	0.285126	+	0.958490i	$0.407965\pi$
$824$	0	0
$825$	16.2878	0.567069
$826$	0	0
$827$	18.3440	0.637884	0.318942	−	0.947774i	$-0.396672\pi$
$827$	18.3440	0.637884	0.318942	+	0.947774i	$0.396672\pi$
$828$	0	0
$829$	12.0814	0.419605	0.209803	−	0.977744i	$-0.432718\pi$
$829$	12.0814	0.419605	0.209803	+	0.977744i	$0.432718\pi$
$830$	0	0
$831$	47.5662	1.65005
$832$	0	0
$833$	−34.4875	−1.19492
$834$	0	0
$835$	1.66497	0.0576186
$836$	0	0
$837$	172.851	5.97461
$838$	0	0
$839$	−37.1459	−1.28242	−0.641209	−	0.767366i	$-0.721567\pi$
$839$	−37.1459	−1.28242	−0.641209	+	0.767366i	$0.721567\pi$
$840$	0	0
$841$	−26.4636	−0.912536
$842$	0	0
$843$	86.7555	2.98802
$844$	0	0
$845$	1.76884	0.0608499
$846$	0	0
$847$	1.38599	0.0476233
$848$	0	0
$849$	16.2418	0.557416
$850$	0	0
$851$	−1.96273	−0.0672816
$852$	0	0
$853$	−42.2447	−1.44643	−0.723216	−	0.690622i	$-0.757337\pi$
$853$	−42.2447	−1.44643	−0.723216	+	0.690622i	$0.757337\pi$
$854$	0	0
$855$	−8.94492	−0.305910
$856$	0	0
$857$	−8.96064	−0.306089	−0.153045	−	0.988219i	$-0.548908\pi$
$857$	−8.96064	−0.306089	−0.153045	+	0.988219i	$0.548908\pi$
$858$	0	0
$859$	20.7847	0.709165	0.354583	−	0.935025i	$-0.384623\pi$
$859$	20.7847	0.709165	0.354583	+	0.935025i	$0.384623\pi$
$860$	0	0
$861$	27.2577	0.928940
$862$	0	0
$863$	14.7613	0.502481	0.251240	−	0.967925i	$-0.419162\pi$
$863$	14.7613	0.502481	0.251240	+	0.967925i	$0.419162\pi$
$864$	0	0
$865$	−3.24494	−0.110331
$866$	0	0
$867$	−95.7830	−3.25296
$868$	0	0
$869$	8.16382	0.276939
$870$	0	0
$871$	49.3121	1.67088
$872$	0	0
$873$	−106.250	−3.59601
$874$	0	0
$875$	−3.09657	−0.104683
$876$	0	0
$877$	14.6830	0.495808	0.247904	−	0.968785i	$-0.420258\pi$
$877$	14.6830	0.495808	0.247904	+	0.968785i	$0.420258\pi$
$878$	0	0
$879$	−44.9923	−1.51755
$880$	0	0
$881$	5.29654	0.178445	0.0892224	−	0.996012i	$-0.471562\pi$
$881$	5.29654	0.178445	0.0892224	+	0.996012i	$0.471562\pi$
$882$	0	0
$883$	−0.353432	−0.0118939	−0.00594696	−	0.999982i	$-0.501893\pi$
$883$	−0.353432	−0.0118939	−0.00594696	+	0.999982i	$0.501893\pi$
$884$	0	0
$885$	2.51166	0.0844284
$886$	0	0
$887$	−34.0653	−1.14380	−0.571900	−	0.820323i	$-0.693794\pi$
$887$	−34.0653	−1.14380	−0.571900	+	0.820323i	$0.693794\pi$
$888$	0	0
$889$	−10.8610	−0.364266
$890$	0	0
$891$	28.8066	0.965059
$892$	0	0
$893$	8.60523	0.287963
$894$	0	0
$895$	3.41156	0.114036
$896$	0	0
$897$	38.0975	1.27204
$898$	0	0
$899$	−17.3233	−0.577763
$900$	0	0
$901$	−1.27802	−0.0425769
$902$	0	0
$903$	−4.10756	−0.136691
$904$	0	0
$905$	−0.680404	−0.0226174
$906$	0	0
$907$	−7.14171	−0.237137	−0.118568	−	0.992946i	$-0.537830\pi$
$907$	−7.14171	−0.237137	−0.118568	+	0.992946i	$0.537830\pi$
$908$	0	0
$909$	24.8380	0.823824
$910$	0	0
$911$	−11.2549	−0.372891	−0.186446	−	0.982465i	$-0.559697\pi$
$911$	−11.2549	−0.372891	−0.186446	+	0.982465i	$0.559697\pi$
$912$	0	0
$913$	−5.40726	−0.178954
$914$	0	0
$915$	0.616519	0.0203815
$916$	0	0
$917$	−15.6216	−0.515869
$918$	0	0
$919$	−52.5765	−1.73434	−0.867170	−	0.498013i	$-0.834063\pi$
$919$	−52.5765	−1.73434	−0.867170	+	0.498013i	$0.834063\pi$
$920$	0	0
$921$	−79.7565	−2.62807
$922$	0	0
$923$	−2.76786	−0.0911054
$924$	0	0
$925$	−3.83410	−0.126064
$926$	0	0
$927$	−121.714	−3.99762
$928$	0	0
$929$	54.3969	1.78470	0.892352	−	0.451341i	$-0.149054\pi$
$929$	54.3969	1.78470	0.892352	+	0.451341i	$0.149054\pi$
$930$	0	0
$931$	25.8424	0.846951
$932$	0	0
$933$	−7.14412	−0.233888
$934$	0	0
$935$	1.52474	0.0498645
$936$	0	0
$937$	−54.3237	−1.77468	−0.887338	−	0.461119i	$-0.847448\pi$
$937$	−54.3237	−1.77468	−0.887338	+	0.461119i	$0.847448\pi$
$938$	0	0
$939$	30.7222	1.00258
$940$	0	0
$941$	4.00618	0.130598	0.0652988	−	0.997866i	$-0.479200\pi$
$941$	4.00618	0.130598	0.0652988	+	0.997866i	$0.479200\pi$
$942$	0	0
$943$	15.1425	0.493109
$944$	0	0
$945$	−4.94574	−0.160885
$946$	0	0
$947$	12.4185	0.403546	0.201773	−	0.979432i	$-0.435330\pi$
$947$	12.4185	0.403546	0.201773	+	0.979432i	$0.435330\pi$
$948$	0	0
$949$	−9.24380	−0.300067
$950$	0	0
$951$	93.9861	3.04771
$952$	0	0
$953$	30.6471	0.992755	0.496378	−	0.868107i	$-0.334663\pi$
$953$	30.6471	0.992755	0.496378	+	0.868107i	$0.334663\pi$
$954$	0	0
$955$	2.58354	0.0836014
$956$	0	0
$957$	−5.24092	−0.169415
$958$	0	0
$959$	−1.38599	−0.0447561
$960$	0	0
$961$	87.3135	2.81656
$962$	0	0
$963$	93.0006	2.99690
$964$	0	0
$965$	−5.23947	−0.168665
$966$	0	0
$967$	−6.72578	−0.216286	−0.108143	−	0.994135i	$-0.534491\pi$
$967$	−6.72578	−0.216286	−0.108143	+	0.994135i	$0.534491\pi$
$968$	0	0
$969$	113.692	3.65232
$970$	0	0
$971$	26.5924	0.853391	0.426695	−	0.904395i	$-0.359678\pi$
$971$	26.5924	0.853391	0.426695	+	0.904395i	$0.359678\pi$
$972$	0	0
$973$	−6.41888	−0.205780
$974$	0	0
$975$	74.4217	2.38340
$976$	0	0
$977$	−59.9969	−1.91947	−0.959736	−	0.280905i	$-0.909366\pi$
$977$	−59.9969	−1.91947	−0.959736	+	0.280905i	$0.909366\pi$
$978$	0	0
$979$	6.67430	0.213312
$980$	0	0
$981$	−62.0595	−1.98141
$982$	0	0
$983$	41.9044	1.33654	0.668271	−	0.743918i	$-0.267035\pi$
$983$	41.9044	1.33654	0.668271	+	0.743918i	$0.267035\pi$
$984$	0	0
$985$	0.826957	0.0263490
$986$	0	0
$987$	7.71375	0.245531
$988$	0	0
$989$	−2.28188	−0.0725596
$990$	0	0
$991$	46.2139	1.46803	0.734016	−	0.679132i	$-0.237644\pi$
$991$	46.2139	1.46803	0.734016	+	0.679132i	$0.237644\pi$
$992$	0	0
$993$	−82.0773	−2.60464
$994$	0	0
$995$	1.62966	0.0516638
$996$	0	0
$997$	45.6587	1.44603	0.723013	−	0.690834i	$-0.242757\pi$
$997$	45.6587	1.44603	0.723013	+	0.690834i	$0.242757\pi$
$998$	0	0
$999$	−12.3098	−0.389464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.c.1.2	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.c.1.2	✓	25	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)

\(f(q)\)	\(=\)	\(q-3.29075 q^{3} +0.224551 q^{5} +1.38599 q^{7} +7.82903 q^{9} +O(q^{10})\)	\(q-3.29075 q^{3} +0.224551 q^{5} +1.38599 q^{7} +7.82903 q^{9} +1.00000 q^{11} +4.56916 q^{13} -0.738940 q^{15} +6.79020 q^{17} -5.08808 q^{19} -4.56096 q^{21} -2.53376 q^{23} -4.94958 q^{25} -15.8911 q^{27} +1.59262 q^{29} -10.8772 q^{31} -3.29075 q^{33} +0.311226 q^{35} +0.774631 q^{37} -15.0360 q^{39} -5.97631 q^{41} +0.900590 q^{43} +1.75802 q^{45} -1.69126 q^{47} -5.07902 q^{49} -22.3448 q^{51} -0.188215 q^{53} +0.224551 q^{55} +16.7436 q^{57} -3.39900 q^{59} -0.834329 q^{61} +10.8510 q^{63} +1.02601 q^{65} +10.7924 q^{67} +8.33798 q^{69} -0.605771 q^{71} -2.02309 q^{73} +16.2878 q^{75} +1.38599 q^{77} +8.16382 q^{79} +28.8066 q^{81} -5.40726 q^{83} +1.52474 q^{85} -5.24092 q^{87} +6.67430 q^{89} +6.33283 q^{91} +35.7941 q^{93} -1.14253 q^{95} -13.5712 q^{97} +7.82903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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