LMFDB - Embedded newform 6036.2.a.i.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.04534 q^{5} +4.50218 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.04534 q^{5} +4.50218 q^{7} +1.00000 q^{9} +3.45850 q^{11} -2.07798 q^{13} -3.04534 q^{15} -5.67239 q^{17} +7.79821 q^{19} -4.50218 q^{21} -9.14325 q^{23} +4.27409 q^{25} -1.00000 q^{27} -8.61918 q^{29} +0.690524 q^{31} -3.45850 q^{33} +13.7107 q^{35} +4.51215 q^{37} +2.07798 q^{39} +6.24824 q^{41} +10.9511 q^{43} +3.04534 q^{45} +6.37328 q^{47} +13.2696 q^{49} +5.67239 q^{51} -1.10555 q^{53} +10.5323 q^{55} -7.79821 q^{57} -12.6748 q^{59} +8.83264 q^{61} +4.50218 q^{63} -6.32814 q^{65} +6.27458 q^{67} +9.14325 q^{69} +2.91573 q^{71} -4.14086 q^{73} -4.27409 q^{75} +15.5708 q^{77} -2.05544 q^{79} +1.00000 q^{81} +16.8389 q^{83} -17.2743 q^{85} +8.61918 q^{87} +15.4179 q^{89} -9.35542 q^{91} -0.690524 q^{93} +23.7482 q^{95} -2.33216 q^{97} +3.45850 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.04534	1.36192	0.680958	−	0.732322i	$-0.261563\pi$
$5$	3.04534	1.36192	0.680958	+	0.732322i	$0.261563\pi$
$6$	0	0
$7$	4.50218	1.70166	0.850832	−	0.525438i	$-0.176098\pi$
$7$	4.50218	1.70166	0.850832	+	0.525438i	$0.176098\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.45850	1.04278	0.521389	−	0.853319i	$-0.325414\pi$
$11$	3.45850	1.04278	0.521389	+	0.853319i	$0.325414\pi$
$12$	0	0
$13$	−2.07798	−0.576327	−0.288163	−	0.957581i	$-0.593045\pi$
$13$	−2.07798	−0.576327	−0.288163	+	0.957581i	$0.593045\pi$
$14$	0	0
$15$	−3.04534	−0.786303
$16$	0	0
$17$	−5.67239	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$17$	−5.67239	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$18$	0	0
$19$	7.79821	1.78903	0.894516	−	0.447037i	$-0.147521\pi$
$19$	7.79821	1.78903	0.894516	+	0.447037i	$0.147521\pi$
$20$	0	0
$21$	−4.50218	−0.982456
$22$	0	0
$23$	−9.14325	−1.90650	−0.953250	−	0.302183i	$-0.902285\pi$
$23$	−9.14325	−1.90650	−0.953250	+	0.302183i	$0.902285\pi$
$24$	0	0
$25$	4.27409	0.854817
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.61918	−1.60054	−0.800271	−	0.599638i	$-0.795311\pi$
$29$	−8.61918	−1.60054	−0.800271	+	0.599638i	$0.795311\pi$
$30$	0	0
$31$	0.690524	0.124022	0.0620109	−	0.998075i	$-0.480249\pi$
$31$	0.690524	0.124022	0.0620109	+	0.998075i	$0.480249\pi$
$32$	0	0
$33$	−3.45850	−0.602048
$34$	0	0
$35$	13.7107	2.31753
$36$	0	0
$37$	4.51215	0.741793	0.370896	−	0.928674i	$-0.379050\pi$
$37$	4.51215	0.741793	0.370896	+	0.928674i	$0.379050\pi$
$38$	0	0
$39$	2.07798	0.332742
$40$	0	0
$41$	6.24824	0.975812	0.487906	−	0.872896i	$-0.337761\pi$
$41$	6.24824	0.975812	0.487906	+	0.872896i	$0.337761\pi$
$42$	0	0
$43$	10.9511	1.67003	0.835016	−	0.550226i	$-0.185458\pi$
$43$	10.9511	1.67003	0.835016	+	0.550226i	$0.185458\pi$
$44$	0	0
$45$	3.04534	0.453972
$46$	0	0
$47$	6.37328	0.929638	0.464819	−	0.885406i	$-0.346119\pi$
$47$	6.37328	0.929638	0.464819	+	0.885406i	$0.346119\pi$
$48$	0	0
$49$	13.2696	1.89566
$50$	0	0
$51$	5.67239	0.794293
$52$	0	0
$53$	−1.10555	−0.151859	−0.0759293	−	0.997113i	$-0.524192\pi$
$53$	−1.10555	−0.151859	−0.0759293	+	0.997113i	$0.524192\pi$
$54$	0	0
$55$	10.5323	1.42018
$56$	0	0
$57$	−7.79821	−1.03290
$58$	0	0
$59$	−12.6748	−1.65012	−0.825059	−	0.565046i	$-0.808858\pi$
$59$	−12.6748	−1.65012	−0.825059	+	0.565046i	$0.808858\pi$
$60$	0	0
$61$	8.83264	1.13090	0.565452	−	0.824781i	$-0.308702\pi$
$61$	8.83264	1.13090	0.565452	+	0.824781i	$0.308702\pi$
$62$	0	0
$63$	4.50218	0.567221
$64$	0	0
$65$	−6.32814	−0.784909
$66$	0	0
$67$	6.27458	0.766562	0.383281	−	0.923632i	$-0.374794\pi$
$67$	6.27458	0.766562	0.383281	+	0.923632i	$0.374794\pi$
$68$	0	0
$69$	9.14325	1.10072
$70$	0	0
$71$	2.91573	0.346034	0.173017	−	0.984919i	$-0.444649\pi$
$71$	2.91573	0.346034	0.173017	+	0.984919i	$0.444649\pi$
$72$	0	0
$73$	−4.14086	−0.484651	−0.242325	−	0.970195i	$-0.577910\pi$
$73$	−4.14086	−0.484651	−0.242325	+	0.970195i	$0.577910\pi$
$74$	0	0
$75$	−4.27409	−0.493529
$76$	0	0
$77$	15.5708	1.77446
$78$	0	0
$79$	−2.05544	−0.231255	−0.115627	−	0.993293i	$-0.536888\pi$
$79$	−2.05544	−0.231255	−0.115627	+	0.993293i	$0.536888\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.8389	1.84831	0.924157	−	0.382013i	$-0.124769\pi$
$83$	16.8389	1.84831	0.924157	+	0.382013i	$0.124769\pi$
$84$	0	0
$85$	−17.2743	−1.87367
$86$	0	0
$87$	8.61918	0.924073
$88$	0	0
$89$	15.4179	1.63429	0.817147	−	0.576429i	$-0.195554\pi$
$89$	15.4179	1.63429	0.817147	+	0.576429i	$0.195554\pi$
$90$	0	0
$91$	−9.35542	−0.980715
$92$	0	0
$93$	−0.690524	−0.0716040
$94$	0	0
$95$	23.7482	2.43651
$96$	0	0
$97$	−2.33216	−0.236795	−0.118397	−	0.992966i	$-0.537776\pi$
$97$	−2.33216	−0.236795	−0.118397	+	0.992966i	$0.537776\pi$
$98$	0	0
$99$	3.45850	0.347593
$100$	0	0
$101$	6.95919	0.692465	0.346232	−	0.938149i	$-0.387461\pi$
$101$	6.95919	0.692465	0.346232	+	0.938149i	$0.387461\pi$
$102$	0	0
$103$	−7.91217	−0.779609	−0.389805	−	0.920898i	$-0.627457\pi$
$103$	−7.91217	−0.779609	−0.389805	+	0.920898i	$0.627457\pi$
$104$	0	0
$105$	−13.7107	−1.33802
$106$	0	0
$107$	−11.8843	−1.14889	−0.574447	−	0.818541i	$-0.694783\pi$
$107$	−11.8843	−1.14889	−0.574447	+	0.818541i	$0.694783\pi$
$108$	0	0
$109$	8.18818	0.784285	0.392143	−	0.919904i	$-0.371734\pi$
$109$	8.18818	0.784285	0.392143	+	0.919904i	$0.371734\pi$
$110$	0	0
$111$	−4.51215	−0.428274
$112$	0	0
$113$	12.5028	1.17617	0.588083	−	0.808801i	$-0.299883\pi$
$113$	12.5028	1.17617	0.588083	+	0.808801i	$0.299883\pi$
$114$	0	0
$115$	−27.8443	−2.59649
$116$	0	0
$117$	−2.07798	−0.192109
$118$	0	0
$119$	−25.5381	−2.34108
$120$	0	0
$121$	0.961244	0.0873858
$122$	0	0
$123$	−6.24824	−0.563385
$124$	0	0
$125$	−2.21065	−0.197727
$126$	0	0
$127$	4.92271	0.436820	0.218410	−	0.975857i	$-0.429913\pi$
$127$	4.92271	0.436820	0.218410	+	0.975857i	$0.429913\pi$
$128$	0	0
$129$	−10.9511	−0.964193
$130$	0	0
$131$	−6.64716	−0.580765	−0.290382	−	0.956911i	$-0.593782\pi$
$131$	−6.64716	−0.580765	−0.290382	+	0.956911i	$0.593782\pi$
$132$	0	0
$133$	35.1089	3.04433
$134$	0	0
$135$	−3.04534	−0.262101
$136$	0	0
$137$	−10.8868	−0.930118	−0.465059	−	0.885280i	$-0.653967\pi$
$137$	−10.8868	−0.930118	−0.465059	+	0.885280i	$0.653967\pi$
$138$	0	0
$139$	−16.6039	−1.40832	−0.704160	−	0.710041i	$-0.748676\pi$
$139$	−16.6039	−1.40832	−0.704160	+	0.710041i	$0.748676\pi$
$140$	0	0
$141$	−6.37328	−0.536727
$142$	0	0
$143$	−7.18669	−0.600981
$144$	0	0
$145$	−26.2483	−2.17981
$146$	0	0
$147$	−13.2696	−1.09446
$148$	0	0
$149$	20.0219	1.64026	0.820128	−	0.572181i	$-0.193902\pi$
$149$	20.0219	1.64026	0.820128	+	0.572181i	$0.193902\pi$
$150$	0	0
$151$	−12.3666	−1.00638	−0.503188	−	0.864177i	$-0.667840\pi$
$151$	−12.3666	−1.00638	−0.503188	+	0.864177i	$0.667840\pi$
$152$	0	0
$153$	−5.67239	−0.458585
$154$	0	0
$155$	2.10288	0.168907
$156$	0	0
$157$	1.17964	0.0941457	0.0470729	−	0.998891i	$-0.485011\pi$
$157$	1.17964	0.0941457	0.0470729	+	0.998891i	$0.485011\pi$
$158$	0	0
$159$	1.10555	0.0876756
$160$	0	0
$161$	−41.1646	−3.24422
$162$	0	0
$163$	10.4382	0.817579	0.408790	−	0.912629i	$-0.365951\pi$
$163$	10.4382	0.817579	0.408790	+	0.912629i	$0.365951\pi$
$164$	0	0
$165$	−10.5323	−0.819939
$166$	0	0
$167$	17.7240	1.37153	0.685763	−	0.727825i	$-0.259469\pi$
$167$	17.7240	1.37153	0.685763	+	0.727825i	$0.259469\pi$
$168$	0	0
$169$	−8.68202	−0.667847
$170$	0	0
$171$	7.79821	0.596344
$172$	0	0
$173$	−10.0897	−0.767103	−0.383551	−	0.923520i	$-0.625299\pi$
$173$	−10.0897	−0.767103	−0.383551	+	0.923520i	$0.625299\pi$
$174$	0	0
$175$	19.2427	1.45461
$176$	0	0
$177$	12.6748	0.952697
$178$	0	0
$179$	10.5502	0.788556	0.394278	−	0.918991i	$-0.370995\pi$
$179$	10.5502	0.788556	0.394278	+	0.918991i	$0.370995\pi$
$180$	0	0
$181$	8.89637	0.661262	0.330631	−	0.943760i	$-0.392738\pi$
$181$	8.89637	0.661262	0.330631	+	0.943760i	$0.392738\pi$
$182$	0	0
$183$	−8.83264	−0.652928
$184$	0	0
$185$	13.7410	1.01026
$186$	0	0
$187$	−19.6180	−1.43461
$188$	0	0
$189$	−4.50218	−0.327485
$190$	0	0
$191$	3.67401	0.265842	0.132921	−	0.991127i	$-0.457564\pi$
$191$	3.67401	0.265842	0.132921	+	0.991127i	$0.457564\pi$
$192$	0	0
$193$	16.5434	1.19082	0.595412	−	0.803421i	$-0.296989\pi$
$193$	16.5434	1.19082	0.595412	+	0.803421i	$0.296989\pi$
$194$	0	0
$195$	6.32814	0.453168
$196$	0	0
$197$	−18.2657	−1.30138	−0.650689	−	0.759345i	$-0.725520\pi$
$197$	−18.2657	−1.30138	−0.650689	+	0.759345i	$0.725520\pi$
$198$	0	0
$199$	11.3851	0.807072	0.403536	−	0.914964i	$-0.367781\pi$
$199$	11.3851	0.807072	0.403536	+	0.914964i	$0.367781\pi$
$200$	0	0
$201$	−6.27458	−0.442575
$202$	0	0
$203$	−38.8051	−2.72359
$204$	0	0
$205$	19.0280	1.32897
$206$	0	0
$207$	−9.14325	−0.635500
$208$	0	0
$209$	26.9701	1.86556
$210$	0	0
$211$	25.3564	1.74561	0.872803	−	0.488072i	$-0.162300\pi$
$211$	25.3564	1.74561	0.872803	+	0.488072i	$0.162300\pi$
$212$	0	0
$213$	−2.91573	−0.199783
$214$	0	0
$215$	33.3499	2.27444
$216$	0	0
$217$	3.10887	0.211044
$218$	0	0
$219$	4.14086	0.279813
$220$	0	0
$221$	11.7871	0.792885
$222$	0	0
$223$	−24.5159	−1.64171	−0.820855	−	0.571137i	$-0.806502\pi$
$223$	−24.5159	−1.64171	−0.820855	+	0.571137i	$0.806502\pi$
$224$	0	0
$225$	4.27409	0.284939
$226$	0	0
$227$	−17.6165	−1.16925	−0.584624	−	0.811304i	$-0.698758\pi$
$227$	−17.6165	−1.16925	−0.584624	+	0.811304i	$0.698758\pi$
$228$	0	0
$229$	20.7425	1.37070	0.685351	−	0.728213i	$-0.259649\pi$
$229$	20.7425	1.37070	0.685351	+	0.728213i	$0.259649\pi$
$230$	0	0
$231$	−15.5708	−1.02448
$232$	0	0
$233$	6.29247	0.412233	0.206117	−	0.978527i	$-0.433917\pi$
$233$	6.29247	0.412233	0.206117	+	0.978527i	$0.433917\pi$
$234$	0	0
$235$	19.4088	1.26609
$236$	0	0
$237$	2.05544	0.133515
$238$	0	0
$239$	15.5951	1.00876	0.504382	−	0.863481i	$-0.331720\pi$
$239$	15.5951	1.00876	0.504382	+	0.863481i	$0.331720\pi$
$240$	0	0
$241$	−8.74016	−0.563003	−0.281502	−	0.959561i	$-0.590832\pi$
$241$	−8.74016	−0.563003	−0.281502	+	0.959561i	$0.590832\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	40.4105	2.58173
$246$	0	0
$247$	−16.2045	−1.03107
$248$	0	0
$249$	−16.8389	−1.06712
$250$	0	0
$251$	−17.4168	−1.09934	−0.549670	−	0.835382i	$-0.685247\pi$
$251$	−17.4168	−1.09934	−0.549670	+	0.835382i	$0.685247\pi$
$252$	0	0
$253$	−31.6220	−1.98806
$254$	0	0
$255$	17.2743	1.08176
$256$	0	0
$257$	−17.8974	−1.11641	−0.558203	−	0.829704i	$-0.688509\pi$
$257$	−17.8974	−1.11641	−0.558203	+	0.829704i	$0.688509\pi$
$258$	0	0
$259$	20.3145	1.26228
$260$	0	0
$261$	−8.61918	−0.533514
$262$	0	0
$263$	−27.0763	−1.66959	−0.834797	−	0.550558i	$-0.814415\pi$
$263$	−27.0763	−1.66959	−0.834797	+	0.550558i	$0.814415\pi$
$264$	0	0
$265$	−3.36677	−0.206819
$266$	0	0
$267$	−15.4179	−0.943560
$268$	0	0
$269$	30.6675	1.86983	0.934914	−	0.354874i	$-0.115476\pi$
$269$	30.6675	1.86983	0.934914	+	0.354874i	$0.115476\pi$
$270$	0	0
$271$	−9.49811	−0.576969	−0.288485	−	0.957485i	$-0.593151\pi$
$271$	−9.49811	−0.576969	−0.288485	+	0.957485i	$0.593151\pi$
$272$	0	0
$273$	9.35542	0.566216
$274$	0	0
$275$	14.7819	0.891385
$276$	0	0
$277$	7.56035	0.454257	0.227129	−	0.973865i	$-0.427066\pi$
$277$	7.56035	0.454257	0.227129	+	0.973865i	$0.427066\pi$
$278$	0	0
$279$	0.690524	0.0413406
$280$	0	0
$281$	−30.8149	−1.83826	−0.919131	−	0.393951i	$-0.871108\pi$
$281$	−30.8149	−1.83826	−0.919131	+	0.393951i	$0.871108\pi$
$282$	0	0
$283$	−12.2127	−0.725970	−0.362985	−	0.931795i	$-0.618242\pi$
$283$	−12.2127	−0.725970	−0.362985	+	0.931795i	$0.618242\pi$
$284$	0	0
$285$	−23.7482	−1.40672
$286$	0	0
$287$	28.1307	1.66050
$288$	0	0
$289$	15.1760	0.892706
$290$	0	0
$291$	2.33216	0.136714
$292$	0	0
$293$	18.2017	1.06336	0.531678	−	0.846947i	$-0.321562\pi$
$293$	18.2017	1.06336	0.531678	+	0.846947i	$0.321562\pi$
$294$	0	0
$295$	−38.5991	−2.24732
$296$	0	0
$297$	−3.45850	−0.200683
$298$	0	0
$299$	18.9995	1.09877
$300$	0	0
$301$	49.3040	2.84183
$302$	0	0
$303$	−6.95919	−0.399795
$304$	0	0
$305$	26.8984	1.54020
$306$	0	0
$307$	−21.4170	−1.22233	−0.611167	−	0.791502i	$-0.709300\pi$
$307$	−21.4170	−1.22233	−0.611167	+	0.791502i	$0.709300\pi$
$308$	0	0
$309$	7.91217	0.450108
$310$	0	0
$311$	−12.9860	−0.736366	−0.368183	−	0.929753i	$-0.620020\pi$
$311$	−12.9860	−0.736366	−0.368183	+	0.929753i	$0.620020\pi$
$312$	0	0
$313$	−34.5858	−1.95490	−0.977452	−	0.211156i	$-0.932277\pi$
$313$	−34.5858	−1.95490	−0.977452	+	0.211156i	$0.932277\pi$
$314$	0	0
$315$	13.7107	0.772508
$316$	0	0
$317$	−2.39779	−0.134673	−0.0673367	−	0.997730i	$-0.521450\pi$
$317$	−2.39779	−0.134673	−0.0673367	+	0.997730i	$0.521450\pi$
$318$	0	0
$319$	−29.8095	−1.66901
$320$	0	0
$321$	11.8843	0.663315
$322$	0	0
$323$	−44.2345	−2.46127
$324$	0	0
$325$	−8.88145	−0.492654
$326$	0	0
$327$	−8.18818	−0.452807
$328$	0	0
$329$	28.6936	1.58193
$330$	0	0
$331$	−3.36842	−0.185145	−0.0925726	−	0.995706i	$-0.529509\pi$
$331$	−3.36842	−0.185145	−0.0925726	+	0.995706i	$0.529509\pi$
$332$	0	0
$333$	4.51215	0.247264
$334$	0	0
$335$	19.1082	1.04399
$336$	0	0
$337$	10.4589	0.569732	0.284866	−	0.958567i	$-0.408051\pi$
$337$	10.4589	0.569732	0.284866	+	0.958567i	$0.408051\pi$
$338$	0	0
$339$	−12.5028	−0.679059
$340$	0	0
$341$	2.38818	0.129327
$342$	0	0
$343$	28.2270	1.52412
$344$	0	0
$345$	27.8443	1.49909
$346$	0	0
$347$	12.9347	0.694374	0.347187	−	0.937796i	$-0.387137\pi$
$347$	12.9347	0.694374	0.347187	+	0.937796i	$0.387137\pi$
$348$	0	0
$349$	30.4545	1.63019	0.815095	−	0.579327i	$-0.196685\pi$
$349$	30.4545	1.63019	0.815095	+	0.579327i	$0.196685\pi$
$350$	0	0
$351$	2.07798	0.110914
$352$	0	0
$353$	−30.5253	−1.62470	−0.812348	−	0.583173i	$-0.801811\pi$
$353$	−30.5253	−1.62470	−0.812348	+	0.583173i	$0.801811\pi$
$354$	0	0
$355$	8.87939	0.471269
$356$	0	0
$357$	25.5381	1.35162
$358$	0	0
$359$	4.92692	0.260033	0.130016	−	0.991512i	$-0.458497\pi$
$359$	4.92692	0.260033	0.130016	+	0.991512i	$0.458497\pi$
$360$	0	0
$361$	41.8120	2.20063
$362$	0	0
$363$	−0.961244	−0.0504522
$364$	0	0
$365$	−12.6103	−0.660054
$366$	0	0
$367$	−8.68035	−0.453110	−0.226555	−	0.973998i	$-0.572746\pi$
$367$	−8.68035	−0.453110	−0.226555	+	0.973998i	$0.572746\pi$
$368$	0	0
$369$	6.24824	0.325271
$370$	0	0
$371$	−4.97737	−0.258412
$372$	0	0
$373$	−12.2098	−0.632201	−0.316100	−	0.948726i	$-0.602374\pi$
$373$	−12.2098	−0.632201	−0.316100	+	0.948726i	$0.602374\pi$
$374$	0	0
$375$	2.21065	0.114157
$376$	0	0
$377$	17.9105	0.922435
$378$	0	0
$379$	−13.7593	−0.706769	−0.353384	−	0.935478i	$-0.614969\pi$
$379$	−13.7593	−0.706769	−0.353384	+	0.935478i	$0.614969\pi$
$380$	0	0
$381$	−4.92271	−0.252198
$382$	0	0
$383$	11.5810	0.591760	0.295880	−	0.955225i	$-0.404387\pi$
$383$	11.5810	0.591760	0.295880	+	0.955225i	$0.404387\pi$
$384$	0	0
$385$	47.4184	2.41666
$386$	0	0
$387$	10.9511	0.556677
$388$	0	0
$389$	−9.05551	−0.459133	−0.229566	−	0.973293i	$-0.573731\pi$
$389$	−9.05551	−0.459133	−0.229566	+	0.973293i	$0.573731\pi$
$390$	0	0
$391$	51.8641	2.62288
$392$	0	0
$393$	6.64716	0.335305
$394$	0	0
$395$	−6.25950	−0.314950
$396$	0	0
$397$	−13.1976	−0.662368	−0.331184	−	0.943566i	$-0.607448\pi$
$397$	−13.1976	−0.662368	−0.331184	+	0.943566i	$0.607448\pi$
$398$	0	0
$399$	−35.1089	−1.75764
$400$	0	0
$401$	−3.33139	−0.166362	−0.0831809	−	0.996534i	$-0.526508\pi$
$401$	−3.33139	−0.166362	−0.0831809	+	0.996534i	$0.526508\pi$
$402$	0	0
$403$	−1.43489	−0.0714771
$404$	0	0
$405$	3.04534	0.151324
$406$	0	0
$407$	15.6053	0.773525
$408$	0	0
$409$	−25.3221	−1.25210	−0.626049	−	0.779784i	$-0.715329\pi$
$409$	−25.3221	−1.25210	−0.626049	+	0.779784i	$0.715329\pi$
$410$	0	0
$411$	10.8868	0.537004
$412$	0	0
$413$	−57.0643	−2.80795
$414$	0	0
$415$	51.2803	2.51725
$416$	0	0
$417$	16.6039	0.813094
$418$	0	0
$419$	8.46014	0.413305	0.206653	−	0.978414i	$-0.433743\pi$
$419$	8.46014	0.413305	0.206653	+	0.978414i	$0.433743\pi$
$420$	0	0
$421$	24.5613	1.19705	0.598523	−	0.801106i	$-0.295755\pi$
$421$	24.5613	1.19705	0.598523	+	0.801106i	$0.295755\pi$
$422$	0	0
$423$	6.37328	0.309879
$424$	0	0
$425$	−24.2443	−1.17602
$426$	0	0
$427$	39.7661	1.92442
$428$	0	0
$429$	7.18669	0.346977
$430$	0	0
$431$	−20.9314	−1.00823	−0.504115	−	0.863636i	$-0.668181\pi$
$431$	−20.9314	−1.00823	−0.504115	+	0.863636i	$0.668181\pi$
$432$	0	0
$433$	0.757334	0.0363951	0.0181976	−	0.999834i	$-0.494207\pi$
$433$	0.757334	0.0363951	0.0181976	+	0.999834i	$0.494207\pi$
$434$	0	0
$435$	26.2483	1.25851
$436$	0	0
$437$	−71.3010	−3.41079
$438$	0	0
$439$	−4.86386	−0.232139	−0.116070	−	0.993241i	$-0.537030\pi$
$439$	−4.86386	−0.232139	−0.116070	+	0.993241i	$0.537030\pi$
$440$	0	0
$441$	13.2696	0.631887
$442$	0	0
$443$	−8.29499	−0.394107	−0.197053	−	0.980393i	$-0.563137\pi$
$443$	−8.29499	−0.394107	−0.197053	+	0.980393i	$0.563137\pi$
$444$	0	0
$445$	46.9527	2.22577
$446$	0	0
$447$	−20.0219	−0.947002
$448$	0	0
$449$	−4.34862	−0.205224	−0.102612	−	0.994721i	$-0.532720\pi$
$449$	−4.34862	−0.205224	−0.102612	+	0.994721i	$0.532720\pi$
$450$	0	0
$451$	21.6096	1.01756
$452$	0	0
$453$	12.3666	0.581031
$454$	0	0
$455$	−28.4904	−1.33565
$456$	0	0
$457$	42.5590	1.99082	0.995412	−	0.0956804i	$-0.0305027\pi$
$457$	42.5590	1.99082	0.995412	+	0.0956804i	$0.0305027\pi$
$458$	0	0
$459$	5.67239	0.264764
$460$	0	0
$461$	15.1505	0.705628	0.352814	−	0.935694i	$-0.385225\pi$
$461$	15.1505	0.705628	0.352814	+	0.935694i	$0.385225\pi$
$462$	0	0
$463$	27.9602	1.29942	0.649710	−	0.760182i	$-0.274890\pi$
$463$	27.9602	1.29942	0.649710	+	0.760182i	$0.274890\pi$
$464$	0	0
$465$	−2.10288	−0.0975187
$466$	0	0
$467$	30.2636	1.40043	0.700217	−	0.713930i	$-0.253086\pi$
$467$	30.2636	1.40043	0.700217	+	0.713930i	$0.253086\pi$
$468$	0	0
$469$	28.2493	1.30443
$470$	0	0
$471$	−1.17964	−0.0543551
$472$	0	0
$473$	37.8745	1.74147
$474$	0	0
$475$	33.3302	1.52930
$476$	0	0
$477$	−1.10555	−0.0506195
$478$	0	0
$479$	−3.35167	−0.153142	−0.0765709	−	0.997064i	$-0.524397\pi$
$479$	−3.35167	−0.153142	−0.0765709	+	0.997064i	$0.524397\pi$
$480$	0	0
$481$	−9.37614	−0.427515
$482$	0	0
$483$	41.1646	1.87305
$484$	0	0
$485$	−7.10221	−0.322495
$486$	0	0
$487$	−37.1978	−1.68559	−0.842797	−	0.538232i	$-0.819092\pi$
$487$	−37.1978	−1.68559	−0.842797	+	0.538232i	$0.819092\pi$
$488$	0	0
$489$	−10.4382	−0.472030
$490$	0	0
$491$	−16.2825	−0.734817	−0.367409	−	0.930060i	$-0.619755\pi$
$491$	−16.2825	−0.734817	−0.367409	+	0.930060i	$0.619755\pi$
$492$	0	0
$493$	48.8914	2.20196
$494$	0	0
$495$	10.5323	0.473392
$496$	0	0
$497$	13.1271	0.588833
$498$	0	0
$499$	−14.4546	−0.647077	−0.323538	−	0.946215i	$-0.604873\pi$
$499$	−14.4546	−0.647077	−0.323538	+	0.946215i	$0.604873\pi$
$500$	0	0
$501$	−17.7240	−0.791851
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	21.1931	0.943080
$506$	0	0
$507$	8.68202	0.385582
$508$	0	0
$509$	−20.0804	−0.890047	−0.445023	−	0.895519i	$-0.646805\pi$
$509$	−20.0804	−0.890047	−0.445023	+	0.895519i	$0.646805\pi$
$510$	0	0
$511$	−18.6429	−0.824713
$512$	0	0
$513$	−7.79821	−0.344299
$514$	0	0
$515$	−24.0952	−1.06176
$516$	0	0
$517$	22.0420	0.969406
$518$	0	0
$519$	10.0897	0.442887
$520$	0	0
$521$	−33.6048	−1.47225	−0.736127	−	0.676843i	$-0.763348\pi$
$521$	−33.6048	−1.47225	−0.736127	+	0.676843i	$0.763348\pi$
$522$	0	0
$523$	−29.6591	−1.29690	−0.648451	−	0.761256i	$-0.724583\pi$
$523$	−29.6591	−1.29690	−0.648451	+	0.761256i	$0.724583\pi$
$524$	0	0
$525$	−19.2427	−0.839821
$526$	0	0
$527$	−3.91692	−0.170624
$528$	0	0
$529$	60.5991	2.63474
$530$	0	0
$531$	−12.6748	−0.550040
$532$	0	0
$533$	−12.9837	−0.562387
$534$	0	0
$535$	−36.1916	−1.56470
$536$	0	0
$537$	−10.5502	−0.455273
$538$	0	0
$539$	45.8931	1.97675
$540$	0	0
$541$	−28.2899	−1.21628	−0.608139	−	0.793830i	$-0.708084\pi$
$541$	−28.2899	−1.21628	−0.608139	+	0.793830i	$0.708084\pi$
$542$	0	0
$543$	−8.89637	−0.381780
$544$	0	0
$545$	24.9358	1.06813
$546$	0	0
$547$	−6.08221	−0.260057	−0.130028	−	0.991510i	$-0.541507\pi$
$547$	−6.08221	−0.260057	−0.130028	+	0.991510i	$0.541507\pi$
$548$	0	0
$549$	8.83264	0.376968
$550$	0	0
$551$	−67.2142	−2.86342
$552$	0	0
$553$	−9.25395	−0.393518
$554$	0	0
$555$	−13.7410	−0.583274
$556$	0	0
$557$	3.44809	0.146100	0.0730500	−	0.997328i	$-0.476727\pi$
$557$	3.44809	0.146100	0.0730500	+	0.997328i	$0.476727\pi$
$558$	0	0
$559$	−22.7562	−0.962484
$560$	0	0
$561$	19.6180	0.828272
$562$	0	0
$563$	12.3883	0.522103	0.261052	−	0.965325i	$-0.415931\pi$
$563$	12.3883	0.522103	0.261052	+	0.965325i	$0.415931\pi$
$564$	0	0
$565$	38.0753	1.60184
$566$	0	0
$567$	4.50218	0.189074
$568$	0	0
$569$	38.8053	1.62680	0.813401	−	0.581703i	$-0.197614\pi$
$569$	38.8053	1.62680	0.813401	+	0.581703i	$0.197614\pi$
$570$	0	0
$571$	−39.2095	−1.64087	−0.820434	−	0.571741i	$-0.806268\pi$
$571$	−39.2095	−1.64087	−0.820434	+	0.571741i	$0.806268\pi$
$572$	0	0
$573$	−3.67401	−0.153484
$574$	0	0
$575$	−39.0791	−1.62971
$576$	0	0
$577$	−2.63260	−0.109596	−0.0547982	−	0.998497i	$-0.517452\pi$
$577$	−2.63260	−0.109596	−0.0547982	+	0.998497i	$0.517452\pi$
$578$	0	0
$579$	−16.5434	−0.687522
$580$	0	0
$581$	75.8120	3.14521
$582$	0	0
$583$	−3.82354	−0.158355
$584$	0	0
$585$	−6.32814	−0.261636
$586$	0	0
$587$	−8.45733	−0.349071	−0.174536	−	0.984651i	$-0.555842\pi$
$587$	−8.45733	−0.349071	−0.174536	+	0.984651i	$0.555842\pi$
$588$	0	0
$589$	5.38485	0.221879
$590$	0	0
$591$	18.2657	0.751350
$592$	0	0
$593$	12.2092	0.501371	0.250685	−	0.968069i	$-0.419344\pi$
$593$	12.2092	0.501371	0.250685	+	0.968069i	$0.419344\pi$
$594$	0	0
$595$	−77.7722	−3.18835
$596$	0	0
$597$	−11.3851	−0.465963
$598$	0	0
$599$	14.4106	0.588801	0.294400	−	0.955682i	$-0.404880\pi$
$599$	14.4106	0.588801	0.294400	+	0.955682i	$0.404880\pi$
$600$	0	0
$601$	13.4577	0.548953	0.274476	−	0.961594i	$-0.411495\pi$
$601$	13.4577	0.548953	0.274476	+	0.961594i	$0.411495\pi$
$602$	0	0
$603$	6.27458	0.255521
$604$	0	0
$605$	2.92731	0.119012
$606$	0	0
$607$	18.1319	0.735952	0.367976	−	0.929835i	$-0.380051\pi$
$607$	18.1319	0.735952	0.367976	+	0.929835i	$0.380051\pi$
$608$	0	0
$609$	38.8051	1.57246
$610$	0	0
$611$	−13.2435	−0.535775
$612$	0	0
$613$	15.2502	0.615949	0.307975	−	0.951395i	$-0.400349\pi$
$613$	15.2502	0.615949	0.307975	+	0.951395i	$0.400349\pi$
$614$	0	0
$615$	−19.0280	−0.767284
$616$	0	0
$617$	33.3746	1.34361	0.671805	−	0.740728i	$-0.265519\pi$
$617$	33.3746	1.34361	0.671805	+	0.740728i	$0.265519\pi$
$618$	0	0
$619$	−24.3563	−0.978962	−0.489481	−	0.872014i	$-0.662814\pi$
$619$	−24.3563	−0.978962	−0.489481	+	0.872014i	$0.662814\pi$
$620$	0	0
$621$	9.14325	0.366906
$622$	0	0
$623$	69.4142	2.78102
$624$	0	0
$625$	−28.1026	−1.12410
$626$	0	0
$627$	−26.9701	−1.07708
$628$	0	0
$629$	−25.5947	−1.02053
$630$	0	0
$631$	29.2056	1.16266	0.581328	−	0.813669i	$-0.302533\pi$
$631$	29.2056	1.16266	0.581328	+	0.813669i	$0.302533\pi$
$632$	0	0
$633$	−25.3564	−1.00783
$634$	0	0
$635$	14.9913	0.594913
$636$	0	0
$637$	−27.5740	−1.09252
$638$	0	0
$639$	2.91573	0.115345
$640$	0	0
$641$	−29.6360	−1.17055	−0.585275	−	0.810835i	$-0.699013\pi$
$641$	−29.6360	−1.17055	−0.585275	+	0.810835i	$0.699013\pi$
$642$	0	0
$643$	29.6329	1.16861	0.584303	−	0.811536i	$-0.301368\pi$
$643$	29.6329	1.16861	0.584303	+	0.811536i	$0.301368\pi$
$644$	0	0
$645$	−33.3499	−1.31315
$646$	0	0
$647$	−5.79403	−0.227787	−0.113893	−	0.993493i	$-0.536332\pi$
$647$	−5.79403	−0.227787	−0.113893	+	0.993493i	$0.536332\pi$
$648$	0	0
$649$	−43.8358	−1.72071
$650$	0	0
$651$	−3.10887	−0.121846
$652$	0	0
$653$	0.150068	0.00587261	0.00293630	−	0.999996i	$-0.499065\pi$
$653$	0.150068	0.00587261	0.00293630	+	0.999996i	$0.499065\pi$
$654$	0	0
$655$	−20.2428	−0.790953
$656$	0	0
$657$	−4.14086	−0.161550
$658$	0	0
$659$	26.2902	1.02412	0.512060	−	0.858950i	$-0.328883\pi$
$659$	26.2902	1.02412	0.512060	+	0.858950i	$0.328883\pi$
$660$	0	0
$661$	−8.94901	−0.348076	−0.174038	−	0.984739i	$-0.555682\pi$
$661$	−8.94901	−0.348076	−0.174038	+	0.984739i	$0.555682\pi$
$662$	0	0
$663$	−11.7871	−0.457773
$664$	0	0
$665$	106.919	4.14612
$666$	0	0
$667$	78.8074	3.05143
$668$	0	0
$669$	24.5159	0.947841
$670$	0	0
$671$	30.5477	1.17928
$672$	0	0
$673$	19.7062	0.759618	0.379809	−	0.925065i	$-0.375990\pi$
$673$	19.7062	0.759618	0.379809	+	0.925065i	$0.375990\pi$
$674$	0	0
$675$	−4.27409	−0.164510
$676$	0	0
$677$	−35.1754	−1.35190	−0.675951	−	0.736946i	$-0.736267\pi$
$677$	−35.1754	−1.35190	−0.675951	+	0.736946i	$0.736267\pi$
$678$	0	0
$679$	−10.4998	−0.402945
$680$	0	0
$681$	17.6165	0.675066
$682$	0	0
$683$	−43.5694	−1.66714	−0.833568	−	0.552416i	$-0.813706\pi$
$683$	−43.5694	−1.66714	−0.833568	+	0.552416i	$0.813706\pi$
$684$	0	0
$685$	−33.1539	−1.26674
$686$	0	0
$687$	−20.7425	−0.791375
$688$	0	0
$689$	2.29730	0.0875202
$690$	0	0
$691$	−9.55345	−0.363430	−0.181715	−	0.983351i	$-0.558165\pi$
$691$	−9.55345	−0.363430	−0.181715	+	0.983351i	$0.558165\pi$
$692$	0	0
$693$	15.5708	0.591486
$694$	0	0
$695$	−50.5644	−1.91802
$696$	0	0
$697$	−35.4425	−1.34248
$698$	0	0
$699$	−6.29247	−0.238003
$700$	0	0
$701$	9.40944	0.355390	0.177695	−	0.984086i	$-0.443136\pi$
$701$	9.40944	0.355390	0.177695	+	0.984086i	$0.443136\pi$
$702$	0	0
$703$	35.1867	1.32709
$704$	0	0
$705$	−19.4088	−0.730977
$706$	0	0
$707$	31.3315	1.17834
$708$	0	0
$709$	−31.2934	−1.17525	−0.587624	−	0.809134i	$-0.699937\pi$
$709$	−31.2934	−1.17525	−0.587624	+	0.809134i	$0.699937\pi$
$710$	0	0
$711$	−2.05544	−0.0770849
$712$	0	0
$713$	−6.31364	−0.236448
$714$	0	0
$715$	−21.8859	−0.818486
$716$	0	0
$717$	−15.5951	−0.582410
$718$	0	0
$719$	−2.27385	−0.0848003	−0.0424002	−	0.999101i	$-0.513500\pi$
$719$	−2.27385	−0.0848003	−0.0424002	+	0.999101i	$0.513500\pi$
$720$	0	0
$721$	−35.6220	−1.32663
$722$	0	0
$723$	8.74016	0.325050
$724$	0	0
$725$	−36.8391	−1.36817
$726$	0	0
$727$	38.8592	1.44121	0.720604	−	0.693347i	$-0.243865\pi$
$727$	38.8592	1.44121	0.720604	+	0.693347i	$0.243865\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−62.1191	−2.29756
$732$	0	0
$733$	−49.2057	−1.81746	−0.908728	−	0.417389i	$-0.862945\pi$
$733$	−49.2057	−1.81746	−0.908728	+	0.417389i	$0.862945\pi$
$734$	0	0
$735$	−40.4105	−1.49056
$736$	0	0
$737$	21.7007	0.799354
$738$	0	0
$739$	−17.8278	−0.655806	−0.327903	−	0.944711i	$-0.606342\pi$
$739$	−17.8278	−0.655806	−0.327903	+	0.944711i	$0.606342\pi$
$740$	0	0
$741$	16.2045	0.595287
$742$	0	0
$743$	−5.31009	−0.194808	−0.0974042	−	0.995245i	$-0.531054\pi$
$743$	−5.31009	−0.194808	−0.0974042	+	0.995245i	$0.531054\pi$
$744$	0	0
$745$	60.9734	2.23389
$746$	0	0
$747$	16.8389	0.616105
$748$	0	0
$749$	−53.5051	−1.95503
$750$	0	0
$751$	−36.8775	−1.34568	−0.672839	−	0.739789i	$-0.734925\pi$
$751$	−36.8775	−1.34568	−0.672839	+	0.739789i	$0.734925\pi$
$752$	0	0
$753$	17.4168	0.634705
$754$	0	0
$755$	−37.6603	−1.37060
$756$	0	0
$757$	26.6854	0.969896	0.484948	−	0.874543i	$-0.338839\pi$
$757$	26.6854	0.969896	0.484948	+	0.874543i	$0.338839\pi$
$758$	0	0
$759$	31.6220	1.14780
$760$	0	0
$761$	−1.72623	−0.0625758	−0.0312879	−	0.999510i	$-0.509961\pi$
$761$	−1.72623	−0.0625758	−0.0312879	+	0.999510i	$0.509961\pi$
$762$	0	0
$763$	36.8647	1.33459
$764$	0	0
$765$	−17.2743	−0.624555
$766$	0	0
$767$	26.3379	0.951008
$768$	0	0
$769$	22.8733	0.824833	0.412416	−	0.910995i	$-0.364685\pi$
$769$	22.8733	0.824833	0.412416	+	0.910995i	$0.364685\pi$
$770$	0	0
$771$	17.8974	0.644558
$772$	0	0
$773$	−32.4712	−1.16791	−0.583954	−	0.811787i	$-0.698495\pi$
$773$	−32.4712	−1.16791	−0.583954	+	0.811787i	$0.698495\pi$
$774$	0	0
$775$	2.95136	0.106016
$776$	0	0
$777$	−20.3145	−0.728779
$778$	0	0
$779$	48.7251	1.74576
$780$	0	0
$781$	10.0841	0.360836
$782$	0	0
$783$	8.61918	0.308024
$784$	0	0
$785$	3.59241	0.128219
$786$	0	0
$787$	−49.3151	−1.75789	−0.878947	−	0.476919i	$-0.841754\pi$
$787$	−49.3151	−1.75789	−0.878947	+	0.476919i	$0.841754\pi$
$788$	0	0
$789$	27.0763	0.963941
$790$	0	0
$791$	56.2899	2.00144
$792$	0	0
$793$	−18.3540	−0.651770
$794$	0	0
$795$	3.36677	0.119407
$796$	0	0
$797$	−21.5754	−0.764241	−0.382120	−	0.924113i	$-0.624806\pi$
$797$	−21.5754	−0.764241	−0.382120	+	0.924113i	$0.624806\pi$
$798$	0	0
$799$	−36.1517	−1.27896
$800$	0	0
$801$	15.4179	0.544765
$802$	0	0
$803$	−14.3212	−0.505383
$804$	0	0
$805$	−125.360	−4.41836
$806$	0	0
$807$	−30.6675	−1.07955
$808$	0	0
$809$	24.1700	0.849773	0.424886	−	0.905247i	$-0.360314\pi$
$809$	24.1700	0.849773	0.424886	+	0.905247i	$0.360314\pi$
$810$	0	0
$811$	7.45814	0.261891	0.130945	−	0.991390i	$-0.458199\pi$
$811$	7.45814	0.261891	0.130945	+	0.991390i	$0.458199\pi$
$812$	0	0
$813$	9.49811	0.333113
$814$	0	0
$815$	31.7877	1.11348
$816$	0	0
$817$	85.3992	2.98774
$818$	0	0
$819$	−9.35542	−0.326905
$820$	0	0
$821$	14.4722	0.505083	0.252541	−	0.967586i	$-0.418734\pi$
$821$	14.4722	0.505083	0.252541	+	0.967586i	$0.418734\pi$
$822$	0	0
$823$	−40.3551	−1.40669	−0.703346	−	0.710848i	$-0.748311\pi$
$823$	−40.3551	−1.40669	−0.703346	+	0.710848i	$0.748311\pi$
$824$	0	0
$825$	−14.7819	−0.514641
$826$	0	0
$827$	18.0750	0.628528	0.314264	−	0.949336i	$-0.398242\pi$
$827$	18.0750	0.628528	0.314264	+	0.949336i	$0.398242\pi$
$828$	0	0
$829$	−5.09200	−0.176852	−0.0884262	−	0.996083i	$-0.528184\pi$
$829$	−5.09200	−0.176852	−0.0884262	+	0.996083i	$0.528184\pi$
$830$	0	0
$831$	−7.56035	−0.262266
$832$	0	0
$833$	−75.2705	−2.60797
$834$	0	0
$835$	53.9756	1.86790
$836$	0	0
$837$	−0.690524	−0.0238680
$838$	0	0
$839$	31.2231	1.07794	0.538971	−	0.842325i	$-0.318813\pi$
$839$	31.2231	1.07794	0.538971	+	0.842325i	$0.318813\pi$
$840$	0	0
$841$	45.2903	1.56173
$842$	0	0
$843$	30.8149	1.06132
$844$	0	0
$845$	−26.4397	−0.909553
$846$	0	0
$847$	4.32769	0.148701
$848$	0	0
$849$	12.2127	0.419139
$850$	0	0
$851$	−41.2557	−1.41423
$852$	0	0
$853$	−23.1392	−0.792271	−0.396135	−	0.918192i	$-0.629649\pi$
$853$	−23.1392	−0.792271	−0.396135	+	0.918192i	$0.629649\pi$
$854$	0	0
$855$	23.7482	0.812171
$856$	0	0
$857$	31.8095	1.08659	0.543296	−	0.839541i	$-0.317176\pi$
$857$	31.8095	1.08659	0.543296	+	0.839541i	$0.317176\pi$
$858$	0	0
$859$	21.7956	0.743657	0.371828	−	0.928302i	$-0.378731\pi$
$859$	21.7956	0.743657	0.371828	+	0.928302i	$0.378731\pi$
$860$	0	0
$861$	−28.1307	−0.958693
$862$	0	0
$863$	−41.9026	−1.42638	−0.713190	−	0.700971i	$-0.752750\pi$
$863$	−41.9026	−1.42638	−0.713190	+	0.700971i	$0.752750\pi$
$864$	0	0
$865$	−30.7264	−1.04473
$866$	0	0
$867$	−15.1760	−0.515404
$868$	0	0
$869$	−7.10874	−0.241147
$870$	0	0
$871$	−13.0384	−0.441791
$872$	0	0
$873$	−2.33216	−0.0789316
$874$	0	0
$875$	−9.95274	−0.336464
$876$	0	0
$877$	−26.3603	−0.890123	−0.445061	−	0.895500i	$-0.646818\pi$
$877$	−26.3603	−0.890123	−0.445061	+	0.895500i	$0.646818\pi$
$878$	0	0
$879$	−18.2017	−0.613928
$880$	0	0
$881$	1.34774	0.0454065	0.0227032	−	0.999742i	$-0.492773\pi$
$881$	1.34774	0.0454065	0.0227032	+	0.999742i	$0.492773\pi$
$882$	0	0
$883$	−46.0807	−1.55074	−0.775369	−	0.631508i	$-0.782436\pi$
$883$	−46.0807	−1.55074	−0.775369	+	0.631508i	$0.782436\pi$
$884$	0	0
$885$	38.5991	1.29749
$886$	0	0
$887$	48.7892	1.63818	0.819091	−	0.573664i	$-0.194478\pi$
$887$	48.7892	1.63818	0.819091	+	0.573664i	$0.194478\pi$
$888$	0	0
$889$	22.1629	0.743321
$890$	0	0
$891$	3.45850	0.115864
$892$	0	0
$893$	49.7001	1.66315
$894$	0	0
$895$	32.1288	1.07395
$896$	0	0
$897$	−18.9995	−0.634373
$898$	0	0
$899$	−5.95175	−0.198502
$900$	0	0
$901$	6.27109	0.208920
$902$	0	0
$903$	−49.3040	−1.64073
$904$	0	0
$905$	27.0925	0.900584
$906$	0	0
$907$	−13.9725	−0.463950	−0.231975	−	0.972722i	$-0.574519\pi$
$907$	−13.9725	−0.463950	−0.231975	+	0.972722i	$0.574519\pi$
$908$	0	0
$909$	6.95919	0.230822
$910$	0	0
$911$	−16.4529	−0.545110	−0.272555	−	0.962140i	$-0.587869\pi$
$911$	−16.4529	−0.545110	−0.272555	+	0.962140i	$0.587869\pi$
$912$	0	0
$913$	58.2375	1.92738
$914$	0	0
$915$	−26.8984	−0.889233
$916$	0	0
$917$	−29.9267	−0.988267
$918$	0	0
$919$	15.6972	0.517803	0.258901	−	0.965904i	$-0.416640\pi$
$919$	15.6972	0.517803	0.258901	+	0.965904i	$0.416640\pi$
$920$	0	0
$921$	21.4170	0.705715
$922$	0	0
$923$	−6.05882	−0.199428
$924$	0	0
$925$	19.2853	0.634098
$926$	0	0
$927$	−7.91217	−0.259870
$928$	0	0
$929$	−32.6747	−1.07202	−0.536011	−	0.844211i	$-0.680069\pi$
$929$	−32.6747	−1.07202	−0.536011	+	0.844211i	$0.680069\pi$
$930$	0	0
$931$	103.479	3.39140
$932$	0	0
$933$	12.9860	0.425141
$934$	0	0
$935$	−59.7434	−1.95382
$936$	0	0
$937$	11.8851	0.388270	0.194135	−	0.980975i	$-0.437810\pi$
$937$	11.8851	0.388270	0.194135	+	0.980975i	$0.437810\pi$
$938$	0	0
$939$	34.5858	1.12866
$940$	0	0
$941$	−19.6082	−0.639208	−0.319604	−	0.947551i	$-0.603550\pi$
$941$	−19.6082	−0.639208	−0.319604	+	0.947551i	$0.603550\pi$
$942$	0	0
$943$	−57.1293	−1.86039
$944$	0	0
$945$	−13.7107	−0.446008
$946$	0	0
$947$	−16.1645	−0.525277	−0.262638	−	0.964894i	$-0.584593\pi$
$947$	−16.1645	−0.525277	−0.262638	+	0.964894i	$0.584593\pi$
$948$	0	0
$949$	8.60461	0.279317
$950$	0	0
$951$	2.39779	0.0777537
$952$	0	0
$953$	−30.9853	−1.00371	−0.501857	−	0.864951i	$-0.667350\pi$
$953$	−30.9853	−1.00371	−0.501857	+	0.864951i	$0.667350\pi$
$954$	0	0
$955$	11.1886	0.362055
$956$	0	0
$957$	29.8095	0.963603
$958$	0	0
$959$	−49.0141	−1.58275
$960$	0	0
$961$	−30.5232	−0.984619
$962$	0	0
$963$	−11.8843	−0.382965
$964$	0	0
$965$	50.3804	1.62180
$966$	0	0
$967$	−5.56418	−0.178932	−0.0894660	−	0.995990i	$-0.528516\pi$
$967$	−5.56418	−0.178932	−0.0894660	+	0.995990i	$0.528516\pi$
$968$	0	0
$969$	44.2345	1.42102
$970$	0	0
$971$	−14.5792	−0.467868	−0.233934	−	0.972252i	$-0.575160\pi$
$971$	−14.5792	−0.467868	−0.233934	+	0.972252i	$0.575160\pi$
$972$	0	0
$973$	−74.7536	−2.39649
$974$	0	0
$975$	8.88145	0.284434
$976$	0	0
$977$	14.3997	0.460688	0.230344	−	0.973109i	$-0.426015\pi$
$977$	14.3997	0.460688	0.230344	+	0.973109i	$0.426015\pi$
$978$	0	0
$979$	53.3229	1.70421
$980$	0	0
$981$	8.18818	0.261428
$982$	0	0
$983$	−52.3052	−1.66828	−0.834138	−	0.551556i	$-0.814034\pi$
$983$	−52.3052	−1.66828	−0.834138	+	0.551556i	$0.814034\pi$
$984$	0	0
$985$	−55.6252	−1.77237
$986$	0	0
$987$	−28.6936	−0.913329
$988$	0	0
$989$	−100.129	−3.18391
$990$	0	0
$991$	25.5941	0.813022	0.406511	−	0.913646i	$-0.366745\pi$
$991$	25.5941	0.813022	0.406511	+	0.913646i	$0.366745\pi$
$992$	0	0
$993$	3.36842	0.106894
$994$	0	0
$995$	34.6716	1.09916
$996$	0	0
$997$	14.9021	0.471955	0.235977	−	0.971759i	$-0.424171\pi$
$997$	14.9021	0.471955	0.235977	+	0.971759i	$0.424171\pi$
$998$	0	0
$999$	−4.51215	−0.142758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.21	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.21	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.04534 q^{5} +4.50218 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.04534 q^{5} +4.50218 q^{7} +1.00000 q^{9} +3.45850 q^{11} -2.07798 q^{13} -3.04534 q^{15} -5.67239 q^{17} +7.79821 q^{19} -4.50218 q^{21} -9.14325 q^{23} +4.27409 q^{25} -1.00000 q^{27} -8.61918 q^{29} +0.690524 q^{31} -3.45850 q^{33} +13.7107 q^{35} +4.51215 q^{37} +2.07798 q^{39} +6.24824 q^{41} +10.9511 q^{43} +3.04534 q^{45} +6.37328 q^{47} +13.2696 q^{49} +5.67239 q^{51} -1.10555 q^{53} +10.5323 q^{55} -7.79821 q^{57} -12.6748 q^{59} +8.83264 q^{61} +4.50218 q^{63} -6.32814 q^{65} +6.27458 q^{67} +9.14325 q^{69} +2.91573 q^{71} -4.14086 q^{73} -4.27409 q^{75} +15.5708 q^{77} -2.05544 q^{79} +1.00000 q^{81} +16.8389 q^{83} -17.2743 q^{85} +8.61918 q^{87} +15.4179 q^{89} -9.35542 q^{91} -0.690524 q^{93} +23.7482 q^{95} -2.33216 q^{97} +3.45850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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