LMFDB - Embedded newform 7623.2.a.cn.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.698857$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43091 q^{2} +3.90931 q^{4} -1.43091 q^{5} +1.00000 q^{7} +4.64136 q^{8} +O(q^{10})$	$q+2.43091 q^{2} +3.90931 q^{4} -1.43091 q^{5} +1.00000 q^{7} +4.64136 q^{8} -3.47841 q^{10} -2.30114 q^{13} +2.43091 q^{14} +3.46410 q^{16} +1.14407 q^{17} -5.47841 q^{19} -5.59387 q^{20} -3.00588 q^{23} -2.95250 q^{25} -5.59387 q^{26} +3.90931 q^{28} -3.16727 q^{29} -6.99589 q^{31} -0.861816 q^{32} +2.78112 q^{34} -1.43091 q^{35} -8.16884 q^{37} -13.3175 q^{38} -6.64136 q^{40} +5.59387 q^{41} -10.5939 q^{43} -7.30703 q^{46} +12.9384 q^{47} +1.00000 q^{49} -7.17726 q^{50} -8.99589 q^{52} -9.28273 q^{53} +4.64136 q^{56} -7.69933 q^{58} +6.89501 q^{59} +8.50160 q^{61} -17.0064 q^{62} -9.02320 q^{64} +3.29272 q^{65} +7.61405 q^{67} +4.47252 q^{68} -3.47841 q^{70} +1.92362 q^{71} -4.83704 q^{73} -19.8577 q^{74} -21.4168 q^{76} -13.5348 q^{79} -4.95681 q^{80} +13.5982 q^{82} +9.40661 q^{83} -1.63706 q^{85} -25.7527 q^{86} +4.35911 q^{89} -2.30114 q^{91} -11.7509 q^{92} +31.4520 q^{94} +7.83909 q^{95} -6.56067 q^{97} +2.43091 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7	$ 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} - 10 q^{10} - 10 q^{13} + 2 q^{14} + 6 q^{17} - 18 q^{19} + 2 q^{23} - 8 q^{25} + 4 q^{28} + 6 q^{29} + 12 q^{32} - 2 q^{34} + 2 q^{35} - 4 q^{37} - 8 q^{40} - 20 q^{43} - 16 q^{46} + 6 q^{47} + 4 q^{49} - 24 q^{50} - 8 q^{52} + 24 q^{58} + 6 q^{59} + 10 q^{61} - 16 q^{64} - 10 q^{65} + 4 q^{67} + 28 q^{68} - 10 q^{70} + 6 q^{71} - 34 q^{73} - 36 q^{74} - 36 q^{76} - 24 q^{79} - 12 q^{80} + 28 q^{82} + 6 q^{83} + 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} + 6 q^{94} - 18 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 - 10 * q^10 - 10 * q^13 + 2 * q^14 + 6 * q^17 - 18 * q^19 + 2 * q^23 - 8 * q^25 + 4 * q^28 + 6 * q^29 + 12 * q^32 - 2 * q^34 + 2 * q^35 - 4 * q^37 - 8 * q^40 - 20 * q^43 - 16 * q^46 + 6 * q^47 + 4 * q^49 - 24 * q^50 - 8 * q^52 + 24 * q^58 + 6 * q^59 + 10 * q^61 - 16 * q^64 - 10 * q^65 + 4 * q^67 + 28 * q^68 - 10 * q^70 + 6 * q^71 - 34 * q^73 - 36 * q^74 - 36 * q^76 - 24 * q^79 - 12 * q^80 + 28 * q^82 + 6 * q^83 + 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 + 6 * q^94 - 18 * q^95 - 10 * q^97 + 2 * q^98

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 $

$2$	2.43091	1.71891	0.859456	−	0.511210i	$-0.170803\pi$
$2$	2.43091	1.71891	0.859456	+	0.511210i	$0.170803\pi$
$3$	0	0
$3$	0	0
$4$	3.90931	1.95466
$5$	−1.43091	−0.639921	−0.319961	−	0.947431i	$-0.603670\pi$
$5$	−1.43091	−0.639921	−0.319961	+	0.947431i	$0.603670\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	4.64136	1.64097
$9$	0	0
$10$	−3.47841	−1.09997
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.30114	−0.638222	−0.319111	−	0.947717i	$-0.603384\pi$
$13$	−2.30114	−0.638222	−0.319111	+	0.947717i	$0.603384\pi$
$14$	2.43091	0.649687
$15$	0	0
$16$	3.46410	0.866025
$17$	1.14407	0.277477	0.138739	−	0.990329i	$-0.455695\pi$
$17$	1.14407	0.277477	0.138739	+	0.990329i	$0.455695\pi$
$18$	0	0
$19$	−5.47841	−1.25683	−0.628416	−	0.777877i	$-0.716296\pi$
$19$	−5.47841	−1.25683	−0.628416	+	0.777877i	$0.716296\pi$
$20$	−5.59387	−1.25083
$21$	0	0
$22$	0	0
$23$	−3.00588	−0.626770	−0.313385	−	0.949626i	$-0.601463\pi$
$23$	−3.00588	−0.626770	−0.313385	+	0.949626i	$0.601463\pi$
$24$	0	0
$25$	−2.95250	−0.590501
$26$	−5.59387	−1.09705
$27$	0	0
$28$	3.90931	0.738791
$29$	−3.16727	−0.588147	−0.294073	−	0.955783i	$-0.595011\pi$
$29$	−3.16727	−0.588147	−0.294073	+	0.955783i	$0.595011\pi$
$30$	0	0
$31$	−6.99589	−1.25650	−0.628249	−	0.778012i	$-0.716228\pi$
$31$	−6.99589	−1.25650	−0.628249	+	0.778012i	$0.716228\pi$
$32$	−0.861816	−0.152349
$33$	0	0
$34$	2.78112	0.476959
$35$	−1.43091	−0.241868
$36$	0	0
$37$	−8.16884	−1.34295	−0.671475	−	0.741027i	$-0.734339\pi$
$37$	−8.16884	−1.34295	−0.671475	+	0.741027i	$0.734339\pi$
$38$	−13.3175	−2.16038
$39$	0	0
$40$	−6.64136	−1.05009
$41$	5.59387	0.873615	0.436808	−	0.899555i	$-0.356109\pi$
$41$	5.59387	0.873615	0.436808	+	0.899555i	$0.356109\pi$
$42$	0	0
$43$	−10.5939	−1.61555	−0.807775	−	0.589491i	$-0.799328\pi$
$43$	−10.5939	−1.61555	−0.807775	+	0.589491i	$0.799328\pi$
$44$	0	0
$45$	0	0
$46$	−7.30703	−1.07736
$47$	12.9384	1.88726	0.943629	−	0.331004i	$-0.107387\pi$
$47$	12.9384	1.88726	0.943629	+	0.331004i	$0.107387\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.17726	−1.01502
$51$	0	0
$52$	−8.99589	−1.24751
$53$	−9.28273	−1.27508	−0.637540	−	0.770417i	$-0.720048\pi$
$53$	−9.28273	−1.27508	−0.637540	+	0.770417i	$0.720048\pi$
$54$	0	0
$55$	0	0
$56$	4.64136	0.620228
$57$	0	0
$58$	−7.69933	−1.01097
$59$	6.89501	0.897654	0.448827	−	0.893619i	$-0.351842\pi$
$59$	6.89501	0.897654	0.448827	+	0.893619i	$0.351842\pi$
$60$	0	0
$61$	8.50160	1.08852	0.544259	−	0.838917i	$-0.316811\pi$
$61$	8.50160	1.08852	0.544259	+	0.838917i	$0.316811\pi$
$62$	−17.0064	−2.15981
$63$	0	0
$64$	−9.02320	−1.12790
$65$	3.29272	0.408412
$66$	0	0
$67$	7.61405	0.930205	0.465102	−	0.885257i	$-0.346018\pi$
$67$	7.61405	0.930205	0.465102	+	0.885257i	$0.346018\pi$
$68$	4.47252	0.542373
$69$	0	0
$70$	−3.47841	−0.415749
$71$	1.92362	0.228291	0.114146	−	0.993464i	$-0.463587\pi$
$71$	1.92362	0.228291	0.114146	+	0.993464i	$0.463587\pi$
$72$	0	0
$73$	−4.83704	−0.566133	−0.283066	−	0.959100i	$-0.591352\pi$
$73$	−4.83704	−0.566133	−0.283066	+	0.959100i	$0.591352\pi$
$74$	−19.8577	−2.30841
$75$	0	0
$76$	−21.4168	−2.45668
$77$	0	0
$78$	0	0
$79$	−13.5348	−1.52278	−0.761392	−	0.648292i	$-0.775484\pi$
$79$	−13.5348	−1.52278	−0.761392	+	0.648292i	$0.775484\pi$
$80$	−4.95681	−0.554188
$81$	0	0
$82$	13.5982	1.50167
$83$	9.40661	1.03251	0.516255	−	0.856435i	$-0.327326\pi$
$83$	9.40661	1.03251	0.516255	+	0.856435i	$0.327326\pi$
$84$	0	0
$85$	−1.63706	−0.177564
$86$	−25.7527	−2.77699
$87$	0	0
$88$	0	0
$89$	4.35911	0.462065	0.231032	−	0.972946i	$-0.425790\pi$
$89$	4.35911	0.462065	0.231032	+	0.972946i	$0.425790\pi$
$90$	0	0
$91$	−2.30114	−0.241225
$92$	−11.7509	−1.22512
$93$	0	0
$94$	31.4520	3.24403
$95$	7.83909	0.804274
$96$	0	0
$97$	−6.56067	−0.666135	−0.333068	−	0.942903i	$-0.608084\pi$
$97$	−6.56067	−0.666135	−0.333068	+	0.942903i	$0.608084\pi$
$98$	2.43091	0.245559
$99$	0	0
$100$	−11.5423	−1.15423
$101$	10.1239	1.00736	0.503682	−	0.863889i	$-0.331978\pi$
$101$	10.1239	1.00736	0.503682	+	0.863889i	$0.331978\pi$
$102$	0	0
$103$	−5.00158	−0.492820	−0.246410	−	0.969166i	$-0.579251\pi$
$103$	−5.00158	−0.492820	−0.246410	+	0.969166i	$0.579251\pi$
$104$	−10.6804	−1.04730
$105$	0	0
$106$	−22.5655	−2.19175
$107$	−17.6186	−1.70326	−0.851629	−	0.524145i	$-0.824385\pi$
$107$	−17.6186	−1.70326	−0.851629	+	0.524145i	$0.824385\pi$
$108$	0	0
$109$	12.5813	1.20507	0.602537	−	0.798091i	$-0.294157\pi$
$109$	12.5813	1.20507	0.602537	+	0.798091i	$0.294157\pi$
$110$	0	0
$111$	0	0
$112$	3.46410	0.327327
$113$	−3.20457	−0.301461	−0.150730	−	0.988575i	$-0.548163\pi$
$113$	−3.20457	−0.301461	−0.150730	+	0.988575i	$0.548163\pi$
$114$	0	0
$115$	4.30114	0.401084
$116$	−12.3818	−1.14962
$117$	0	0
$118$	16.7611	1.54299
$119$	1.14407	0.104877
$120$	0	0
$121$	0	0
$122$	20.6666	1.87107
$123$	0	0
$124$	−27.3491	−2.45602
$125$	11.3793	1.01780
$126$	0	0
$127$	−4.77524	−0.423734	−0.211867	−	0.977298i	$-0.567954\pi$
$127$	−4.77524	−0.423734	−0.211867	+	0.977298i	$0.567954\pi$
$128$	−20.2109	−1.78641
$129$	0	0
$130$	8.00431	0.702024
$131$	−16.5713	−1.44784	−0.723922	−	0.689882i	$-0.757663\pi$
$131$	−16.5713	−1.44784	−0.723922	+	0.689882i	$0.757663\pi$
$132$	0	0
$133$	−5.47841	−0.475038
$134$	18.5091	1.59894
$135$	0	0
$136$	5.31004	0.455332
$137$	−9.64725	−0.824220	−0.412110	−	0.911134i	$-0.635208\pi$
$137$	−9.64725	−0.824220	−0.412110	+	0.911134i	$0.635208\pi$
$138$	0	0
$139$	20.9647	1.77821	0.889103	−	0.457707i	$-0.151329\pi$
$139$	20.9647	1.77821	0.889103	+	0.457707i	$0.151329\pi$
$140$	−5.59387	−0.472768
$141$	0	0
$142$	4.67613	0.392412
$143$	0	0
$144$	0	0
$145$	4.53207	0.376368
$146$	−11.7584	−0.973132
$147$	0	0
$148$	−31.9346	−2.62500
$149$	12.3823	1.01440	0.507199	−	0.861829i	$-0.330681\pi$
$149$	12.3823	1.01440	0.507199	+	0.861829i	$0.330681\pi$
$150$	0	0
$151$	19.6952	1.60277	0.801387	−	0.598146i	$-0.204096\pi$
$151$	19.6952	1.60277	0.801387	+	0.598146i	$0.204096\pi$
$152$	−25.4273	−2.06242
$153$	0	0
$154$	0	0
$155$	10.0105	0.804060
$156$	0	0
$157$	−1.54001	−0.122906	−0.0614531	−	0.998110i	$-0.519573\pi$
$157$	−1.54001	−0.122906	−0.0614531	+	0.998110i	$0.519573\pi$
$158$	−32.9018	−2.61753
$159$	0	0
$160$	1.23318	0.0974913
$161$	−3.00588	−0.236897
$162$	0	0
$163$	24.7343	1.93734	0.968670	−	0.248352i	$-0.0798890\pi$
$163$	24.7343	1.93734	0.968670	+	0.248352i	$0.0798890\pi$
$164$	21.8682	1.70762
$165$	0	0
$166$	22.8666	1.77479
$167$	−12.2020	−0.944222	−0.472111	−	0.881539i	$-0.656508\pi$
$167$	−12.2020	−0.944222	−0.472111	+	0.881539i	$0.656508\pi$
$168$	0	0
$169$	−7.70474	−0.592672
$170$	−3.97953	−0.305216
$171$	0	0
$172$	−41.4147	−3.15784
$173$	−12.3486	−0.938850	−0.469425	−	0.882972i	$-0.655539\pi$
$173$	−12.3486	−0.938850	−0.469425	+	0.882972i	$0.655539\pi$
$174$	0	0
$175$	−2.95250	−0.223188
$176$	0	0
$177$	0	0
$178$	10.5966	0.794249
$179$	−17.7036	−1.32323	−0.661616	−	0.749843i	$-0.730129\pi$
$179$	−17.7036	−1.32323	−0.661616	+	0.749843i	$0.730129\pi$
$180$	0	0
$181$	−5.64677	−0.419721	−0.209861	−	0.977731i	$-0.567301\pi$
$181$	−5.64677	−0.419721	−0.209861	+	0.977731i	$0.567301\pi$
$182$	−5.59387	−0.414645
$183$	0	0
$184$	−13.9514	−1.02851
$185$	11.6889	0.859382
$186$	0	0
$187$	0	0
$188$	50.5802	3.68894
$189$	0	0
$190$	19.0561	1.38248
$191$	6.86229	0.496538	0.248269	−	0.968691i	$-0.420138\pi$
$191$	6.86229	0.496538	0.248269	+	0.968691i	$0.420138\pi$
$192$	0	0
$193$	−0.803848	−0.0578622	−0.0289311	−	0.999581i	$-0.509210\pi$
$193$	−0.803848	−0.0578622	−0.0289311	+	0.999581i	$0.509210\pi$
$194$	−15.9484	−1.14503
$195$	0	0
$196$	3.90931	0.279237
$197$	−16.4482	−1.17189	−0.585944	−	0.810352i	$-0.699276\pi$
$197$	−16.4482	−1.17189	−0.585944	+	0.810352i	$0.699276\pi$
$198$	0	0
$199$	−3.65156	−0.258852	−0.129426	−	0.991589i	$-0.541313\pi$
$199$	−3.65156	−0.258852	−0.129426	+	0.991589i	$0.541313\pi$
$200$	−13.7036	−0.968994
$201$	0	0
$202$	24.6102	1.73157
$203$	−3.16727	−0.222298
$204$	0	0
$205$	−8.00431	−0.559045
$206$	−12.1584	−0.847114
$207$	0	0
$208$	−7.97139	−0.552717
$209$	0	0
$210$	0	0
$211$	−18.2468	−1.25616	−0.628081	−	0.778148i	$-0.716159\pi$
$211$	−18.2468	−1.25616	−0.628081	+	0.778148i	$0.716159\pi$
$212$	−36.2891	−2.49234
$213$	0	0
$214$	−42.8293	−2.92775
$215$	15.1588	1.03382
$216$	0	0
$217$	−6.99589	−0.474912
$218$	30.5841	2.07141
$219$	0	0
$220$	0	0
$221$	−2.63266	−0.177092
$222$	0	0
$223$	−24.3164	−1.62835	−0.814173	−	0.580622i	$-0.802809\pi$
$223$	−24.3164	−1.62835	−0.814173	+	0.580622i	$0.802809\pi$
$224$	−0.861816	−0.0575825
$225$	0	0
$226$	−7.79002	−0.518184
$227$	9.08069	0.602707	0.301353	−	0.953513i	$-0.402562\pi$
$227$	9.08069	0.602707	0.301353	+	0.953513i	$0.402562\pi$
$228$	0	0
$229$	1.23318	0.0814907	0.0407454	−	0.999170i	$-0.487027\pi$
$229$	1.23318	0.0814907	0.0407454	+	0.999170i	$0.487027\pi$
$230$	10.4557	0.689427
$231$	0	0
$232$	−14.7004	−0.965131
$233$	−9.13996	−0.598778	−0.299389	−	0.954131i	$-0.596783\pi$
$233$	−9.13996	−0.598778	−0.299389	+	0.954131i	$0.596783\pi$
$234$	0	0
$235$	−18.5137	−1.20770
$236$	26.9547	1.75460
$237$	0	0
$238$	2.78112	0.180274
$239$	25.7745	1.66721	0.833606	−	0.552359i	$-0.186272\pi$
$239$	25.7745	1.66721	0.833606	+	0.552359i	$0.186272\pi$
$240$	0	0
$241$	−3.70617	−0.238736	−0.119368	−	0.992850i	$-0.538087\pi$
$241$	−3.70617	−0.238736	−0.119368	+	0.992850i	$0.538087\pi$
$242$	0	0
$243$	0	0
$244$	33.2354	2.12768
$245$	−1.43091	−0.0914173
$246$	0	0
$247$	12.6066	0.802138
$248$	−32.4705	−2.06188
$249$	0	0
$250$	27.6620	1.74950
$251$	17.7354	1.11945	0.559724	−	0.828679i	$-0.310907\pi$
$251$	17.7354	1.11945	0.559724	+	0.828679i	$0.310907\pi$
$252$	0	0
$253$	0	0
$254$	−11.6082	−0.728361
$255$	0	0
$256$	−31.0845	−1.94278
$257$	−8.43426	−0.526114	−0.263057	−	0.964780i	$-0.584731\pi$
$257$	−8.43426	−0.526114	−0.263057	+	0.964780i	$0.584731\pi$
$258$	0	0
$259$	−8.16884	−0.507587
$260$	12.8723	0.798305
$261$	0	0
$262$	−40.2834	−2.48872
$263$	−3.09226	−0.190677	−0.0953386	−	0.995445i	$-0.530393\pi$
$263$	−3.09226	−0.190677	−0.0953386	+	0.995445i	$0.530393\pi$
$264$	0	0
$265$	13.2827	0.815951
$266$	−13.3175	−0.816548
$267$	0	0
$268$	29.7657	1.81823
$269$	12.7468	0.777188	0.388594	−	0.921409i	$-0.372961\pi$
$269$	12.7468	0.777188	0.388594	+	0.921409i	$0.372961\pi$
$270$	0	0
$271$	−22.6995	−1.37890	−0.689449	−	0.724334i	$-0.742147\pi$
$271$	−22.6995	−1.37890	−0.689449	+	0.724334i	$0.742147\pi$
$272$	3.96317	0.240302
$273$	0	0
$274$	−23.4516	−1.41676
$275$	0	0
$276$	0	0
$277$	−6.43679	−0.386749	−0.193375	−	0.981125i	$-0.561943\pi$
$277$	−6.43679	−0.386749	−0.193375	+	0.981125i	$0.561943\pi$
$278$	50.9634	3.05658
$279$	0	0
$280$	−6.64136	−0.396897
$281$	−1.70455	−0.101685	−0.0508423	−	0.998707i	$-0.516191\pi$
$281$	−1.70455	−0.101685	−0.0508423	+	0.998707i	$0.516191\pi$
$282$	0	0
$283$	0.00861477	0.000512095 0	0.000256048	−	1.00000i	$-0.499918\pi$
$283$	0.00861477	0.000512095 0	0.000256048		1.00000i	$0.499918\pi$
$284$	7.52002	0.446231
$285$	0	0
$286$	0	0
$287$	5.59387	0.330195
$288$	0	0
$289$	−15.6911	−0.923006
$290$	11.0170	0.646943
$291$	0	0
$292$	−18.9095	−1.10660
$293$	2.99130	0.174754	0.0873768	−	0.996175i	$-0.472152\pi$
$293$	2.99130	0.174754	0.0873768	+	0.996175i	$0.472152\pi$
$294$	0	0
$295$	−9.86612	−0.574428
$296$	−37.9146	−2.20374
$297$	0	0
$298$	30.1003	1.74366
$299$	6.91697	0.400019
$300$	0	0
$301$	−10.5939	−0.610620
$302$	47.8773	2.75503
$303$	0	0
$304$	−18.9778	−1.08845
$305$	−12.1650	−0.696566
$306$	0	0
$307$	−16.6115	−0.948069	−0.474035	−	0.880506i	$-0.657203\pi$
$307$	−16.6115	−0.948069	−0.474035	+	0.880506i	$0.657203\pi$
$308$	0	0
$309$	0	0
$310$	24.3345	1.38211
$311$	−18.0280	−1.02227	−0.511136	−	0.859500i	$-0.670775\pi$
$311$	−18.0280	−1.02227	−0.511136	+	0.859500i	$0.670775\pi$
$312$	0	0
$313$	32.1350	1.81638	0.908189	−	0.418559i	$-0.137465\pi$
$313$	32.1350	1.81638	0.908189	+	0.418559i	$0.137465\pi$
$314$	−3.74362	−0.211265
$315$	0	0
$316$	−52.9118	−2.97652
$317$	25.9290	1.45632	0.728159	−	0.685408i	$-0.240376\pi$
$317$	25.9290	1.45632	0.728159	+	0.685408i	$0.240376\pi$
$318$	0	0
$319$	0	0
$320$	12.9114	0.721767
$321$	0	0
$322$	−7.30703	−0.407205
$323$	−6.26767	−0.348742
$324$	0	0
$325$	6.79413	0.376871
$326$	60.1268	3.33012
$327$	0	0
$328$	25.9632	1.43358
$329$	12.9384	0.713317
$330$	0	0
$331$	−10.6498	−0.585365	−0.292683	−	0.956210i	$-0.594548\pi$
$331$	−10.6498	−0.585365	−0.292683	+	0.956210i	$0.594548\pi$
$332$	36.7734	2.01820
$333$	0	0
$334$	−29.6620	−1.62303
$335$	−10.8950	−0.595258
$336$	0	0
$337$	−14.5148	−0.790669	−0.395334	−	0.918537i	$-0.629371\pi$
$337$	−14.5148	−0.790669	−0.395334	+	0.918537i	$0.629371\pi$
$338$	−18.7295	−1.01875
$339$	0	0
$340$	−6.39977	−0.347076
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−49.1700	−2.65107
$345$	0	0
$346$	−30.0184	−1.61380
$347$	14.7064	0.789479	0.394740	−	0.918793i	$-0.370835\pi$
$347$	14.7064	0.789479	0.394740	+	0.918793i	$0.370835\pi$
$348$	0	0
$349$	1.89548	0.101463	0.0507315	−	0.998712i	$-0.483845\pi$
$349$	1.89548	0.101463	0.0507315	+	0.998712i	$0.483845\pi$
$350$	−7.17726	−0.383641
$351$	0	0
$352$	0	0
$353$	−23.1014	−1.22956	−0.614780	−	0.788698i	$-0.710755\pi$
$353$	−23.1014	−1.22956	−0.614780	+	0.788698i	$0.710755\pi$
$354$	0	0
$355$	−2.75252	−0.146088
$356$	17.0411	0.903178
$357$	0	0
$358$	−43.0359	−2.27452
$359$	−4.19869	−0.221598	−0.110799	−	0.993843i	$-0.535341\pi$
$359$	−4.19869	−0.221598	−0.110799	+	0.993843i	$0.535341\pi$
$360$	0	0
$361$	11.0129	0.579627
$362$	−13.7268	−0.721464
$363$	0	0
$364$	−8.99589	−0.471513
$365$	6.92136	0.362281
$366$	0	0
$367$	−2.95073	−0.154027	−0.0770134	−	0.997030i	$-0.524538\pi$
$367$	−2.95073	−0.154027	−0.0770134	+	0.997030i	$0.524538\pi$
$368$	−10.4127	−0.542799
$369$	0	0
$370$	28.4145	1.47720
$371$	−9.28273	−0.481935
$372$	0	0
$373$	21.6666	1.12185	0.560927	−	0.827865i	$-0.310445\pi$
$373$	21.6666	1.12185	0.560927	+	0.827865i	$0.310445\pi$
$374$	0	0
$375$	0	0
$376$	60.0518	3.09693
$377$	7.28833	0.375368
$378$	0	0
$379$	29.3165	1.50589	0.752945	−	0.658084i	$-0.228633\pi$
$379$	29.3165	1.50589	0.752945	+	0.658084i	$0.228633\pi$
$380$	30.6455	1.57208
$381$	0	0
$382$	16.6816	0.853505
$383$	−37.2326	−1.90249	−0.951247	−	0.308429i	$-0.900197\pi$
$383$	−37.2326	−1.90249	−0.951247	+	0.308429i	$0.900197\pi$
$384$	0	0
$385$	0	0
$386$	−1.95408	−0.0994600
$387$	0	0
$388$	−25.6477	−1.30207
$389$	29.3869	1.48997	0.744987	−	0.667079i	$-0.232455\pi$
$389$	29.3869	1.48997	0.744987	+	0.667079i	$0.232455\pi$
$390$	0	0
$391$	−3.43894	−0.173915
$392$	4.64136	0.234424
$393$	0	0
$394$	−39.9841	−2.01437
$395$	19.3670	0.974462
$396$	0	0
$397$	−18.5537	−0.931183	−0.465591	−	0.885000i	$-0.654158\pi$
$397$	−18.5537	−0.931183	−0.465591	+	0.885000i	$0.654158\pi$
$398$	−8.87659	−0.444943
$399$	0	0
$400$	−10.2278	−0.511388
$401$	−3.09911	−0.154762	−0.0773810	−	0.997002i	$-0.524656\pi$
$401$	−3.09911	−0.154762	−0.0773810	+	0.997002i	$0.524656\pi$
$402$	0	0
$403$	16.0985	0.801925
$404$	39.5774	1.96905
$405$	0	0
$406$	−7.69933	−0.382111
$407$	0	0
$408$	0	0
$409$	−13.5912	−0.672041	−0.336020	−	0.941855i	$-0.609081\pi$
$409$	−13.5912	−0.672041	−0.336020	+	0.941855i	$0.609081\pi$
$410$	−19.4577	−0.960949
$411$	0	0
$412$	−19.5527	−0.963294
$413$	6.89501	0.339281
$414$	0	0
$415$	−13.4600	−0.660725
$416$	1.98316	0.0972325
$417$	0	0
$418$	0	0
$419$	1.38136	0.0674838	0.0337419	−	0.999431i	$-0.489258\pi$
$419$	1.38136	0.0674838	0.0337419	+	0.999431i	$0.489258\pi$
$420$	0	0
$421$	28.1477	1.37184	0.685919	−	0.727678i	$-0.259401\pi$
$421$	28.1477	1.37184	0.685919	+	0.727678i	$0.259401\pi$
$422$	−44.3563	−2.15923
$423$	0	0
$424$	−43.0845	−2.09237
$425$	−3.37786	−0.163851
$426$	0	0
$427$	8.50160	0.411421
$428$	−68.8768	−3.32928
$429$	0	0
$430$	36.8498	1.77705
$431$	10.7754	0.519034	0.259517	−	0.965738i	$-0.416437\pi$
$431$	10.7754	0.519034	0.259517	+	0.965738i	$0.416437\pi$
$432$	0	0
$433$	−14.7516	−0.708917	−0.354459	−	0.935072i	$-0.615335\pi$
$433$	−14.7516	−0.708917	−0.354459	+	0.935072i	$0.615335\pi$
$434$	−17.0064	−0.816331
$435$	0	0
$436$	49.1844	2.35550
$437$	16.4675	0.787745
$438$	0	0
$439$	−4.56545	−0.217897	−0.108949	−	0.994047i	$-0.534748\pi$
$439$	−4.56545	−0.217897	−0.108949	+	0.994047i	$0.534748\pi$
$440$	0	0
$441$	0	0
$442$	−6.39977	−0.304406
$443$	−19.1300	−0.908896	−0.454448	−	0.890773i	$-0.650163\pi$
$443$	−19.1300	−0.908896	−0.454448	+	0.890773i	$0.650163\pi$
$444$	0	0
$445$	−6.23749	−0.295685
$446$	−59.1109	−2.79898
$447$	0	0
$448$	−9.02320	−0.426306
$449$	27.4346	1.29472	0.647359	−	0.762185i	$-0.275873\pi$
$449$	27.4346	1.29472	0.647359	+	0.762185i	$0.275873\pi$
$450$	0	0
$451$	0	0
$452$	−12.5277	−0.589252
$453$	0	0
$454$	22.0743	1.03600
$455$	3.29272	0.154365
$456$	0	0
$457$	5.58244	0.261135	0.130568	−	0.991439i	$-0.458320\pi$
$457$	5.58244	0.261135	0.130568	+	0.991439i	$0.458320\pi$
$458$	2.99774	0.140075
$459$	0	0
$460$	16.8145	0.783981
$461$	−31.2918	−1.45740	−0.728701	−	0.684832i	$-0.759876\pi$
$461$	−31.2918	−1.45740	−0.728701	+	0.684832i	$0.759876\pi$
$462$	0	0
$463$	20.5780	0.956340	0.478170	−	0.878267i	$-0.341300\pi$
$463$	20.5780	0.956340	0.478170	+	0.878267i	$0.341300\pi$
$464$	−10.9717	−0.509350
$465$	0	0
$466$	−22.2184	−1.02925
$467$	40.8220	1.88902	0.944508	−	0.328489i	$-0.106539\pi$
$467$	40.8220	1.88902	0.944508	+	0.328489i	$0.106539\pi$
$468$	0	0
$469$	7.61405	0.351584
$470$	−45.0050	−2.07592
$471$	0	0
$472$	32.0022	1.47302
$473$	0	0
$474$	0	0
$475$	16.1750	0.742160
$476$	4.47252	0.204998
$477$	0	0
$478$	62.6554	2.86579
$479$	26.5969	1.21525	0.607623	−	0.794226i	$-0.292123\pi$
$479$	26.5969	1.21525	0.607623	+	0.794226i	$0.292123\pi$
$480$	0	0
$481$	18.7977	0.857100
$482$	−9.00937	−0.410366
$483$	0	0
$484$	0	0
$485$	9.38772	0.426274
$486$	0	0
$487$	23.6150	1.07010	0.535049	−	0.844821i	$-0.320293\pi$
$487$	23.6150	1.07010	0.535049	+	0.844821i	$0.320293\pi$
$488$	39.4590	1.78623
$489$	0	0
$490$	−3.47841	−0.157138
$491$	40.8045	1.84148	0.920741	−	0.390174i	$-0.127585\pi$
$491$	40.8045	1.84148	0.920741	+	0.390174i	$0.127585\pi$
$492$	0	0
$493$	−3.62357	−0.163197
$494$	30.6455	1.37880
$495$	0	0
$496$	−24.2345	−1.08816
$497$	1.92362	0.0862860
$498$	0	0
$499$	9.57197	0.428500	0.214250	−	0.976779i	$-0.431269\pi$
$499$	9.57197	0.428500	0.214250	+	0.976779i	$0.431269\pi$
$500$	44.4852	1.98944
$501$	0	0
$502$	43.1131	1.92423
$503$	20.1131	0.896797	0.448399	−	0.893834i	$-0.351995\pi$
$503$	20.1131	0.896797	0.448399	+	0.893834i	$0.351995\pi$
$504$	0	0
$505$	−14.4863	−0.644634
$506$	0	0
$507$	0	0
$508$	−18.6679	−0.828255
$509$	−6.27889	−0.278307	−0.139154	−	0.990271i	$-0.544438\pi$
$509$	−6.27889	−0.278307	−0.139154	+	0.990271i	$0.544438\pi$
$510$	0	0
$511$	−4.83704	−0.213978
$512$	−35.1417	−1.55306
$513$	0	0
$514$	−20.5029	−0.904344
$515$	7.15680	0.315366
$516$	0	0
$517$	0	0
$518$	−19.8577	−0.872497
$519$	0	0
$520$	15.2827	0.670192
$521$	23.5173	1.03031	0.515156	−	0.857097i	$-0.327734\pi$
$521$	23.5173	1.03031	0.515156	+	0.857097i	$0.327734\pi$
$522$	0	0
$523$	7.38211	0.322797	0.161399	−	0.986889i	$-0.448400\pi$
$523$	7.38211	0.322797	0.161399	+	0.986889i	$0.448400\pi$
$524$	−64.7825	−2.83004
$525$	0	0
$526$	−7.51701	−0.327757
$527$	−8.00377	−0.348650
$528$	0	0
$529$	−13.9647	−0.607159
$530$	32.2891	1.40255
$531$	0	0
$532$	−21.4168	−0.928536
$533$	−12.8723	−0.557561
$534$	0	0
$535$	25.2107	1.08995
$536$	35.3396	1.52644
$537$	0	0
$538$	30.9864	1.33592
$539$	0	0
$540$	0	0
$541$	−10.5161	−0.452123	−0.226061	−	0.974113i	$-0.572585\pi$
$541$	−10.5161	−0.452123	−0.226061	+	0.974113i	$0.572585\pi$
$542$	−55.1805	−2.37020
$543$	0	0
$544$	−0.985976	−0.0422734
$545$	−18.0027	−0.771152
$546$	0	0
$547$	−9.42803	−0.403114	−0.201557	−	0.979477i	$-0.564600\pi$
$547$	−9.42803	−0.403114	−0.201557	+	0.979477i	$0.564600\pi$
$548$	−37.7141	−1.61107
$549$	0	0
$550$	0	0
$551$	17.3516	0.739202
$552$	0	0
$553$	−13.5348	−0.575558
$554$	−15.6472	−0.664788
$555$	0	0
$556$	81.9578	3.47578
$557$	2.83580	0.120157	0.0600784	−	0.998194i	$-0.480865\pi$
$557$	2.83580	0.120157	0.0600784	+	0.998194i	$0.480865\pi$
$558$	0	0
$559$	24.3780	1.03108
$560$	−4.95681	−0.209463
$561$	0	0
$562$	−4.14359	−0.174787
$563$	−13.8127	−0.582138	−0.291069	−	0.956702i	$-0.594011\pi$
$563$	−13.8127	−0.582138	−0.291069	+	0.956702i	$0.594011\pi$
$564$	0	0
$565$	4.58545	0.192911
$566$	0.0209417	0.000880246 0
$567$	0	0
$568$	8.92820	0.374619
$569$	−20.9491	−0.878230	−0.439115	−	0.898431i	$-0.644708\pi$
$569$	−20.9491	−0.878230	−0.439115	+	0.898431i	$0.644708\pi$
$570$	0	0
$571$	−30.2109	−1.26429	−0.632144	−	0.774851i	$-0.717825\pi$
$571$	−30.2109	−1.26429	−0.632144	+	0.774851i	$0.717825\pi$
$572$	0	0
$573$	0	0
$574$	13.5982	0.567577
$575$	8.87488	0.370108
$576$	0	0
$577$	44.1976	1.83997	0.919985	−	0.391955i	$-0.128201\pi$
$577$	44.1976	1.83997	0.919985	+	0.391955i	$0.128201\pi$
$578$	−38.1436	−1.58657
$579$	0	0
$580$	17.7173	0.735669
$581$	9.40661	0.390252
$582$	0	0
$583$	0	0
$584$	−22.4505	−0.929007
$585$	0	0
$586$	7.27158	0.300386
$587$	7.43693	0.306955	0.153477	−	0.988152i	$-0.450953\pi$
$587$	7.43693	0.306955	0.153477	+	0.988152i	$0.450953\pi$
$588$	0	0
$589$	38.3263	1.57921
$590$	−23.9836	−0.987391
$591$	0	0
$592$	−28.2977	−1.16303
$593$	−30.7149	−1.26131	−0.630656	−	0.776063i	$-0.717214\pi$
$593$	−30.7149	−1.26131	−0.630656	+	0.776063i	$0.717214\pi$
$594$	0	0
$595$	−1.63706	−0.0671128
$596$	48.4063	1.98280
$597$	0	0
$598$	16.8145	0.687597
$599$	−11.3859	−0.465217	−0.232609	−	0.972570i	$-0.574726\pi$
$599$	−11.3859	−0.465217	−0.232609	+	0.972570i	$0.574726\pi$
$600$	0	0
$601$	−4.93519	−0.201311	−0.100655	−	0.994921i	$-0.532094\pi$
$601$	−4.93519	−0.201311	−0.100655	+	0.994921i	$0.532094\pi$
$602$	−25.7527	−1.04960
$603$	0	0
$604$	76.9948	3.13287
$605$	0	0
$606$	0	0
$607$	2.24652	0.0911836	0.0455918	−	0.998960i	$-0.485483\pi$
$607$	2.24652	0.0911836	0.0455918	+	0.998960i	$0.485483\pi$
$608$	4.72138	0.191477
$609$	0	0
$610$	−29.5720	−1.19734
$611$	−29.7731	−1.20449
$612$	0	0
$613$	−26.3261	−1.06330	−0.531651	−	0.846964i	$-0.678428\pi$
$613$	−26.3261	−1.06330	−0.531651	+	0.846964i	$0.678428\pi$
$614$	−40.3811	−1.62965
$615$	0	0
$616$	0	0
$617$	−6.08323	−0.244901	−0.122451	−	0.992475i	$-0.539075\pi$
$617$	−6.08323	−0.244901	−0.122451	+	0.992475i	$0.539075\pi$
$618$	0	0
$619$	−17.6732	−0.710345	−0.355172	−	0.934801i	$-0.615578\pi$
$619$	−17.6732	−0.710345	−0.355172	+	0.934801i	$0.615578\pi$
$620$	39.1341	1.57166
$621$	0	0
$622$	−43.8244	−1.75720
$623$	4.35911	0.174644
$624$	0	0
$625$	−1.52021	−0.0608085
$626$	78.1173	3.12219
$627$	0	0
$628$	−6.02038	−0.240239
$629$	−9.34571	−0.372638
$630$	0	0
$631$	−15.2270	−0.606178	−0.303089	−	0.952962i	$-0.598018\pi$
$631$	−15.2270	−0.606178	−0.303089	+	0.952962i	$0.598018\pi$
$632$	−62.8199	−2.49884
$633$	0	0
$634$	63.0310	2.50328
$635$	6.83293	0.271157
$636$	0	0
$637$	−2.30114	−0.0911746
$638$	0	0
$639$	0	0
$640$	28.9200	1.14316
$641$	−40.3827	−1.59502	−0.797510	−	0.603305i	$-0.793850\pi$
$641$	−40.3827	−1.59502	−0.797510	+	0.603305i	$0.793850\pi$
$642$	0	0
$643$	−0.813217	−0.0320701	−0.0160351	−	0.999871i	$-0.505104\pi$
$643$	−0.813217	−0.0320701	−0.0160351	+	0.999871i	$0.505104\pi$
$644$	−11.7509	−0.463052
$645$	0	0
$646$	−15.2361	−0.599457
$647$	−30.9476	−1.21667	−0.608337	−	0.793679i	$-0.708163\pi$
$647$	−30.9476	−1.21667	−0.608337	+	0.793679i	$0.708163\pi$
$648$	0	0
$649$	0	0
$650$	16.5159	0.647807
$651$	0	0
$652$	96.6941	3.78683
$653$	−7.68593	−0.300774	−0.150387	−	0.988627i	$-0.548052\pi$
$653$	−7.68593	−0.300774	−0.150387	+	0.988627i	$0.548052\pi$
$654$	0	0
$655$	23.7121	0.926507
$656$	19.3777	0.756573
$657$	0	0
$658$	31.4520	1.22613
$659$	8.17999	0.318647	0.159324	−	0.987226i	$-0.449069\pi$
$659$	8.17999	0.318647	0.159324	+	0.987226i	$0.449069\pi$
$660$	0	0
$661$	30.5381	1.18780	0.593898	−	0.804540i	$-0.297588\pi$
$661$	30.5381	1.18780	0.593898	+	0.804540i	$0.297588\pi$
$662$	−25.8886	−1.00619
$663$	0	0
$664$	43.6595	1.69432
$665$	7.83909	0.303987
$666$	0	0
$667$	9.52043	0.368633
$668$	−47.7016	−1.84563
$669$	0	0
$670$	−26.4848	−1.02320
$671$	0	0
$672$	0	0
$673$	−34.4546	−1.32813	−0.664063	−	0.747676i	$-0.731169\pi$
$673$	−34.4546	−1.32813	−0.664063	+	0.747676i	$0.731169\pi$
$674$	−35.2840	−1.35909
$675$	0	0
$676$	−30.1202	−1.15847
$677$	−40.6807	−1.56349	−0.781744	−	0.623600i	$-0.785669\pi$
$677$	−40.6807	−1.56349	−0.781744	+	0.623600i	$0.785669\pi$
$678$	0	0
$679$	−6.56067	−0.251776
$680$	−7.59817	−0.291377
$681$	0	0
$682$	0	0
$683$	38.4349	1.47067	0.735336	−	0.677703i	$-0.237024\pi$
$683$	38.4349	1.47067	0.735336	+	0.677703i	$0.237024\pi$
$684$	0	0
$685$	13.8043	0.527436
$686$	2.43091	0.0928125
$687$	0	0
$688$	−36.6982	−1.39911
$689$	21.3609	0.813785
$690$	0	0
$691$	−13.8850	−0.528211	−0.264105	−	0.964494i	$-0.585077\pi$
$691$	−13.8850	−0.528211	−0.264105	+	0.964494i	$0.585077\pi$
$692$	−48.2747	−1.83513
$693$	0	0
$694$	35.7498	1.35704
$695$	−29.9986	−1.13791
$696$	0	0
$697$	6.39977	0.242408
$698$	4.60775	0.174406
$699$	0	0
$700$	−11.5423	−0.436256
$701$	44.2982	1.67312	0.836560	−	0.547876i	$-0.184563\pi$
$701$	44.2982	1.67312	0.836560	+	0.547876i	$0.184563\pi$
$702$	0	0
$703$	44.7522	1.68786
$704$	0	0
$705$	0	0
$706$	−56.1573	−2.11351
$707$	10.1239	0.380748
$708$	0	0
$709$	40.8278	1.53332	0.766660	−	0.642053i	$-0.221917\pi$
$709$	40.8278	1.53332	0.766660	+	0.642053i	$0.221917\pi$
$710$	−6.69112	−0.251113
$711$	0	0
$712$	20.2322	0.758234
$713$	21.0288	0.787536
$714$	0	0
$715$	0	0
$716$	−69.2091	−2.58646
$717$	0	0
$718$	−10.2066	−0.380908
$719$	30.4882	1.13702	0.568510	−	0.822676i	$-0.307520\pi$
$719$	30.4882	1.13702	0.568510	+	0.822676i	$0.307520\pi$
$720$	0	0
$721$	−5.00158	−0.186268
$722$	26.7714	0.996328
$723$	0	0
$724$	−22.0750	−0.820411
$725$	9.35136	0.347301
$726$	0	0
$727$	26.4661	0.981572	0.490786	−	0.871280i	$-0.336710\pi$
$727$	26.4661	0.981572	0.490786	+	0.871280i	$0.336710\pi$
$728$	−10.6804	−0.395843
$729$	0	0
$730$	16.8252	0.622728
$731$	−12.1201	−0.448278
$732$	0	0
$733$	−1.68895	−0.0623826	−0.0311913	−	0.999513i	$-0.509930\pi$
$733$	−1.68895	−0.0623826	−0.0311913	+	0.999513i	$0.509930\pi$
$734$	−7.17295	−0.264759
$735$	0	0
$736$	2.59052	0.0954878
$737$	0	0
$738$	0	0
$739$	−43.5644	−1.60254	−0.801270	−	0.598302i	$-0.795842\pi$
$739$	−43.5644	−1.60254	−0.801270	+	0.598302i	$0.795842\pi$
$740$	45.6954	1.67980
$741$	0	0
$742$	−22.5655	−0.828404
$743$	−9.89726	−0.363095	−0.181548	−	0.983382i	$-0.558111\pi$
$743$	−9.89726	−0.363095	−0.181548	+	0.983382i	$0.558111\pi$
$744$	0	0
$745$	−17.7179	−0.649135
$746$	52.6695	1.92837
$747$	0	0
$748$	0	0
$749$	−17.6186	−0.643771
$750$	0	0
$751$	−26.8450	−0.979589	−0.489795	−	0.871838i	$-0.662928\pi$
$751$	−26.8450	−0.979589	−0.489795	+	0.871838i	$0.662928\pi$
$752$	44.8199	1.63441
$753$	0	0
$754$	17.7173	0.645225
$755$	−28.1820	−1.02565
$756$	0	0
$757$	21.5277	0.782437	0.391218	−	0.920298i	$-0.372054\pi$
$757$	21.5277	0.782437	0.391218	+	0.920298i	$0.372054\pi$
$758$	71.2658	2.58849
$759$	0	0
$760$	36.3841	1.31979
$761$	−29.3264	−1.06308	−0.531540	−	0.847033i	$-0.678387\pi$
$761$	−29.3264	−1.06308	−0.531540	+	0.847033i	$0.678387\pi$
$762$	0	0
$763$	12.5813	0.455475
$764$	26.8268	0.970561
$765$	0	0
$766$	−90.5089	−3.27022
$767$	−15.8664	−0.572903
$768$	0	0
$769$	30.6982	1.10701	0.553503	−	0.832847i	$-0.313291\pi$
$769$	30.6982	1.10701	0.553503	+	0.832847i	$0.313291\pi$
$770$	0	0
$771$	0	0
$772$	−3.14249	−0.113101
$773$	23.7252	0.853336	0.426668	−	0.904408i	$-0.359687\pi$
$773$	23.7252	0.853336	0.426668	+	0.904408i	$0.359687\pi$
$774$	0	0
$775$	20.6554	0.741963
$776$	−30.4505	−1.09311
$777$	0	0
$778$	71.4368	2.56113
$779$	−30.6455	−1.09799
$780$	0	0
$781$	0	0
$782$	−8.35974	−0.298944
$783$	0	0
$784$	3.46410	0.123718
$785$	2.20361	0.0786503
$786$	0	0
$787$	−1.87454	−0.0668202	−0.0334101	−	0.999442i	$-0.510637\pi$
$787$	−1.87454	−0.0668202	−0.0334101	+	0.999442i	$0.510637\pi$
$788$	−64.3012	−2.29064
$789$	0	0
$790$	47.0795	1.67501
$791$	−3.20457	−0.113941
$792$	0	0
$793$	−19.5634	−0.694717
$794$	−45.1023	−1.60062
$795$	0	0
$796$	−14.2751	−0.505966
$797$	9.58066	0.339365	0.169682	−	0.985499i	$-0.445726\pi$
$797$	9.58066	0.339365	0.169682	+	0.985499i	$0.445726\pi$
$798$	0	0
$799$	14.8024	0.523672
$800$	2.54451	0.0899621
$801$	0	0
$802$	−7.53364	−0.266022
$803$	0	0
$804$	0	0
$805$	4.30114	0.151595
$806$	39.1341	1.37844
$807$	0	0
$808$	46.9886	1.65305
$809$	−38.1237	−1.34036	−0.670179	−	0.742199i	$-0.733783\pi$
$809$	−38.1237	−1.34036	−0.670179	+	0.742199i	$0.733783\pi$
$810$	0	0
$811$	−39.4724	−1.38606	−0.693032	−	0.720907i	$-0.743726\pi$
$811$	−39.4724	−1.38606	−0.693032	+	0.720907i	$0.743726\pi$
$812$	−12.3818	−0.434517
$813$	0	0
$814$	0	0
$815$	−35.3925	−1.23975
$816$	0	0
$817$	58.0375	2.03047
$818$	−33.0389	−1.15518
$819$	0	0
$820$	−31.2913	−1.09274
$821$	28.2469	0.985822	0.492911	−	0.870080i	$-0.335933\pi$
$821$	28.2469	0.985822	0.492911	+	0.870080i	$0.335933\pi$
$822$	0	0
$823$	37.4954	1.30701	0.653503	−	0.756924i	$-0.273299\pi$
$823$	37.4954	1.30701	0.653503	+	0.756924i	$0.273299\pi$
$824$	−23.2141	−0.808703
$825$	0	0
$826$	16.7611	0.583194
$827$	19.9210	0.692722	0.346361	−	0.938101i	$-0.387417\pi$
$827$	19.9210	0.692722	0.346361	+	0.938101i	$0.387417\pi$
$828$	0	0
$829$	49.7189	1.72681	0.863404	−	0.504513i	$-0.168328\pi$
$829$	49.7189	1.72681	0.863404	+	0.504513i	$0.168328\pi$
$830$	−32.7200	−1.13573
$831$	0	0
$832$	20.7637	0.719851
$833$	1.14407	0.0396396
$834$	0	0
$835$	17.4600	0.604228
$836$	0	0
$837$	0	0
$838$	3.35796	0.115999
$839$	−2.57431	−0.0888749	−0.0444375	−	0.999012i	$-0.514150\pi$
$839$	−2.57431	−0.0888749	−0.0444375	+	0.999012i	$0.514150\pi$
$840$	0	0
$841$	−18.9684	−0.654084
$842$	68.4246	2.35807
$843$	0	0
$844$	−71.3325	−2.45536
$845$	11.0248	0.379264
$846$	0	0
$847$	0	0
$848$	−32.1563	−1.10425
$849$	0	0
$850$	−8.21128	−0.281645
$851$	24.5546	0.841721
$852$	0	0
$853$	−40.3968	−1.38316	−0.691580	−	0.722300i	$-0.743085\pi$
$853$	−40.3968	−1.38316	−0.691580	+	0.722300i	$0.743085\pi$
$854$	20.6666	0.707197
$855$	0	0
$856$	−81.7745	−2.79500
$857$	−3.24467	−0.110836	−0.0554179	−	0.998463i	$-0.517649\pi$
$857$	−3.24467	−0.110836	−0.0554179	+	0.998463i	$0.517649\pi$
$858$	0	0
$859$	6.75730	0.230556	0.115278	−	0.993333i	$-0.463224\pi$
$859$	6.75730	0.230556	0.115278	+	0.993333i	$0.463224\pi$
$860$	59.2607	2.02077
$861$	0	0
$862$	26.1941	0.892174
$863$	−13.7259	−0.467234	−0.233617	−	0.972329i	$-0.575056\pi$
$863$	−13.7259	−0.467234	−0.233617	+	0.972329i	$0.575056\pi$
$864$	0	0
$865$	17.6698	0.600790
$866$	−35.8598	−1.21857
$867$	0	0
$868$	−27.3491	−0.928289
$869$	0	0
$870$	0	0
$871$	−17.5210	−0.593677
$872$	58.3946	1.97749
$873$	0	0
$874$	40.0309	1.35406
$875$	11.3793	0.384691
$876$	0	0
$877$	9.63610	0.325388	0.162694	−	0.986677i	$-0.447982\pi$
$877$	9.63610	0.325388	0.162694	+	0.986677i	$0.447982\pi$
$878$	−11.0982	−0.374546
$879$	0	0
$880$	0	0
$881$	−24.2777	−0.817937	−0.408968	−	0.912549i	$-0.634111\pi$
$881$	−24.2777	−0.817937	−0.408968	+	0.912549i	$0.634111\pi$
$882$	0	0
$883$	−12.8564	−0.432653	−0.216326	−	0.976321i	$-0.569407\pi$
$883$	−12.8564	−0.432653	−0.216326	+	0.976321i	$0.569407\pi$
$884$	−10.2919	−0.346154
$885$	0	0
$886$	−46.5034	−1.56231
$887$	−47.1594	−1.58346	−0.791729	−	0.610873i	$-0.790819\pi$
$887$	−47.1594	−1.58346	−0.791729	+	0.610873i	$0.790819\pi$
$888$	0	0
$889$	−4.77524	−0.160156
$890$	−15.1628	−0.508257
$891$	0	0
$892$	−95.0604	−3.18286
$893$	−70.8818	−2.37197
$894$	0	0
$895$	25.3323	0.846765
$896$	−20.2109	−0.675200
$897$	0	0
$898$	66.6910	2.22551
$899$	22.1578	0.739005
$900$	0	0
$901$	−10.6201	−0.353806
$902$	0	0
$903$	0	0
$904$	−14.8736	−0.494688
$905$	8.08001	0.268589
$906$	0	0
$907$	−4.99890	−0.165986	−0.0829928	−	0.996550i	$-0.526448\pi$
$907$	−4.99890	−0.165986	−0.0829928	+	0.996550i	$0.526448\pi$
$908$	35.4993	1.17808
$909$	0	0
$910$	8.00431	0.265340
$911$	−24.2373	−0.803017	−0.401509	−	0.915855i	$-0.631514\pi$
$911$	−24.2373	−0.803017	−0.401509	+	0.915855i	$0.631514\pi$
$912$	0	0
$913$	0	0
$914$	13.5704	0.448869
$915$	0	0
$916$	4.82088	0.159286
$917$	−16.5713	−0.547234
$918$	0	0
$919$	−37.5240	−1.23780	−0.618901	−	0.785469i	$-0.712422\pi$
$919$	−37.5240	−1.23780	−0.618901	+	0.785469i	$0.712422\pi$
$920$	19.9632	0.658166
$921$	0	0
$922$	−76.0674	−2.50515
$923$	−4.42652	−0.145701
$924$	0	0
$925$	24.1185	0.793012
$926$	50.0232	1.64386
$927$	0	0
$928$	2.72960	0.0896035
$929$	20.8638	0.684519	0.342259	−	0.939606i	$-0.388808\pi$
$929$	20.8638	0.684519	0.342259	+	0.939606i	$0.388808\pi$
$930$	0	0
$931$	−5.47841	−0.179547
$932$	−35.7309	−1.17041
$933$	0	0
$934$	99.2345	3.24705
$935$	0	0
$936$	0	0
$937$	0.463822	0.0151524	0.00757620	−	0.999971i	$-0.497588\pi$
$937$	0.463822	0.0151524	0.00757620	+	0.999971i	$0.497588\pi$
$938$	18.5091	0.604342
$939$	0	0
$940$	−72.3757	−2.36063
$941$	23.6522	0.771039	0.385519	−	0.922700i	$-0.374022\pi$
$941$	23.6522	0.771039	0.385519	+	0.922700i	$0.374022\pi$
$942$	0	0
$943$	−16.8145	−0.547556
$944$	23.8850	0.777391
$945$	0	0
$946$	0	0
$947$	28.3269	0.920499	0.460250	−	0.887789i	$-0.347760\pi$
$947$	28.3269	0.920499	0.460250	+	0.887789i	$0.347760\pi$
$948$	0	0
$949$	11.1307	0.361319
$950$	39.3199	1.27571
$951$	0	0
$952$	5.31004	0.172099
$953$	−48.2418	−1.56270	−0.781352	−	0.624090i	$-0.785470\pi$
$953$	−48.2418	−1.56270	−0.781352	+	0.624090i	$0.785470\pi$
$954$	0	0
$955$	−9.81931	−0.317745
$956$	100.760	3.25883
$957$	0	0
$958$	64.6547	2.08890
$959$	−9.64725	−0.311526
$960$	0	0
$961$	17.9424	0.578789
$962$	45.6954	1.47328
$963$	0	0
$964$	−14.4886	−0.466646
$965$	1.15023	0.0370273
$966$	0	0
$967$	−35.1632	−1.13077	−0.565387	−	0.824826i	$-0.691273\pi$
$967$	−35.1632	−1.13077	−0.565387	+	0.824826i	$0.691273\pi$
$968$	0	0
$969$	0	0
$970$	22.8207	0.732728
$971$	54.0629	1.73496	0.867480	−	0.497471i	$-0.165738\pi$
$971$	54.0629	1.73496	0.867480	+	0.497471i	$0.165738\pi$
$972$	0	0
$973$	20.9647	0.672099
$974$	57.4059	1.83940
$975$	0	0
$976$	29.4504	0.942685
$977$	42.2318	1.35111	0.675557	−	0.737307i	$-0.263903\pi$
$977$	42.2318	1.35111	0.675557	+	0.737307i	$0.263903\pi$
$978$	0	0
$979$	0	0
$980$	−5.59387	−0.178690
$981$	0	0
$982$	99.1920	3.16534
$983$	0.876060	0.0279420	0.0139710	−	0.999902i	$-0.495553\pi$
$983$	0.876060	0.0279420	0.0139710	+	0.999902i	$0.495553\pi$
$984$	0	0
$985$	23.5359	0.749916
$986$	−8.80856	−0.280522
$987$	0	0
$988$	49.2831	1.56790
$989$	31.8439	1.01258
$990$	0	0
$991$	−41.7171	−1.32519	−0.662594	−	0.748979i	$-0.730544\pi$
$991$	−41.7171	−1.32519	−0.662594	+	0.748979i	$0.730544\pi$
$992$	6.02917	0.191426
$993$	0	0
$994$	4.67613	0.148318
$995$	5.22504	0.165645
$996$	0	0
$997$	−7.75731	−0.245676	−0.122838	−	0.992427i	$-0.539200\pi$
$997$	−7.75731	−0.245676	−0.122838	+	0.992427i	$0.539200\pi$
$998$	23.2686	0.736554
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cn.1.4		4
3.2	odd	2		2541.2.a.bl.1.1	✓	4
11.10	odd	2		7623.2.a.cg.1.1		4
33.32	even	2		2541.2.a.bp.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bl.1.1	✓	4	3.2	odd	2
2541.2.a.bp.1.4	yes	4	33.32	even	2
7623.2.a.cg.1.1		4	11.10	odd	2
7623.2.a.cn.1.4		4	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+2.43091 q^{2} +3.90931 q^{4} -1.43091 q^{5} +1.00000 q^{7} +4.64136 q^{8} +O(q^{10})\)	\(q+2.43091 q^{2} +3.90931 q^{4} -1.43091 q^{5} +1.00000 q^{7} +4.64136 q^{8} -3.47841 q^{10} -2.30114 q^{13} +2.43091 q^{14} +3.46410 q^{16} +1.14407 q^{17} -5.47841 q^{19} -5.59387 q^{20} -3.00588 q^{23} -2.95250 q^{25} -5.59387 q^{26} +3.90931 q^{28} -3.16727 q^{29} -6.99589 q^{31} -0.861816 q^{32} +2.78112 q^{34} -1.43091 q^{35} -8.16884 q^{37} -13.3175 q^{38} -6.64136 q^{40} +5.59387 q^{41} -10.5939 q^{43} -7.30703 q^{46} +12.9384 q^{47} +1.00000 q^{49} -7.17726 q^{50} -8.99589 q^{52} -9.28273 q^{53} +4.64136 q^{56} -7.69933 q^{58} +6.89501 q^{59} +8.50160 q^{61} -17.0064 q^{62} -9.02320 q^{64} +3.29272 q^{65} +7.61405 q^{67} +4.47252 q^{68} -3.47841 q^{70} +1.92362 q^{71} -4.83704 q^{73} -19.8577 q^{74} -21.4168 q^{76} -13.5348 q^{79} -4.95681 q^{80} +13.5982 q^{82} +9.40661 q^{83} -1.63706 q^{85} -25.7527 q^{86} +4.35911 q^{89} -2.30114 q^{91} -11.7509 q^{92} +31.4520 q^{94} +7.83909 q^{95} -6.56067 q^{97} +2.43091 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7	\( 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} - 10 q^{10} - 10 q^{13} + 2 q^{14} + 6 q^{17} - 18 q^{19} + 2 q^{23} - 8 q^{25} + 4 q^{28} + 6 q^{29} + 12 q^{32} - 2 q^{34} + 2 q^{35} - 4 q^{37} - 8 q^{40} - 20 q^{43} - 16 q^{46} + 6 q^{47} + 4 q^{49} - 24 q^{50} - 8 q^{52} + 24 q^{58} + 6 q^{59} + 10 q^{61} - 16 q^{64} - 10 q^{65} + 4 q^{67} + 28 q^{68} - 10 q^{70} + 6 q^{71} - 34 q^{73} - 36 q^{74} - 36 q^{76} - 24 q^{79} - 12 q^{80} + 28 q^{82} + 6 q^{83} + 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} + 6 q^{94} - 18 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 - 10 * q^10 - 10 * q^13 + 2 * q^14 + 6 * q^17 - 18 * q^19 + 2 * q^23 - 8 * q^25 + 4 * q^28 + 6 * q^29 + 12 * q^32 - 2 * q^34 + 2 * q^35 - 4 * q^37 - 8 * q^40 - 20 * q^43 - 16 * q^46 + 6 * q^47 + 4 * q^49 - 24 * q^50 - 8 * q^52 + 24 * q^58 + 6 * q^59 + 10 * q^61 - 16 * q^64 - 10 * q^65 + 4 * q^67 + 28 * q^68 - 10 * q^70 + 6 * q^71 - 34 * q^73 - 36 * q^74 - 36 * q^76 - 24 * q^79 - 12 * q^80 + 28 * q^82 + 6 * q^83 + 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 + 6 * q^94 - 18 * q^95 - 10 * q^97 + 2 * q^98

Introduction

L-functions

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Database

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