LMFDB - Embedded newform 4012.2.a.j.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22137 q^{3} -1.27900 q^{5} +0.396716 q^{7} +1.93446 q^{9} +O(q^{10})$	$q+2.22137 q^{3} -1.27900 q^{5} +0.396716 q^{7} +1.93446 q^{9} -2.72422 q^{11} +6.91685 q^{13} -2.84113 q^{15} +1.00000 q^{17} +0.0994799 q^{19} +0.881251 q^{21} +6.04990 q^{23} -3.36416 q^{25} -2.36695 q^{27} -0.610013 q^{29} -6.40183 q^{31} -6.05148 q^{33} -0.507401 q^{35} +11.3427 q^{37} +15.3648 q^{39} +2.02962 q^{41} +9.63051 q^{43} -2.47418 q^{45} +13.2148 q^{47} -6.84262 q^{49} +2.22137 q^{51} -1.02927 q^{53} +3.48428 q^{55} +0.220981 q^{57} +1.00000 q^{59} +0.211649 q^{61} +0.767433 q^{63} -8.84666 q^{65} -8.77777 q^{67} +13.4390 q^{69} +5.34794 q^{71} -6.43008 q^{73} -7.47302 q^{75} -1.08074 q^{77} +3.68797 q^{79} -11.0612 q^{81} +12.1769 q^{83} -1.27900 q^{85} -1.35506 q^{87} +12.3468 q^{89} +2.74403 q^{91} -14.2208 q^{93} -0.127235 q^{95} +5.49270 q^{97} -5.26990 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.22137	1.28251	0.641253	−	0.767330i	$-0.278415\pi$
$3$	2.22137	1.28251	0.641253	+	0.767330i	$0.278415\pi$
$4$	0	0
$5$	−1.27900	−0.571987	−0.285993	−	0.958232i	$-0.592324\pi$
$5$	−1.27900	−0.571987	−0.285993	+	0.958232i	$0.592324\pi$
$6$	0	0
$7$	0.396716	0.149945	0.0749723	−	0.997186i	$-0.476113\pi$
$7$	0.396716	0.149945	0.0749723	+	0.997186i	$0.476113\pi$
$8$	0	0
$9$	1.93446	0.644821
$10$	0	0
$11$	−2.72422	−0.821383	−0.410691	−	0.911774i	$-0.634713\pi$
$11$	−2.72422	−0.821383	−0.410691	+	0.911774i	$0.634713\pi$
$12$	0	0
$13$	6.91685	1.91839	0.959195	−	0.282747i	$-0.0912456\pi$
$13$	6.91685	1.91839	0.959195	+	0.282747i	$0.0912456\pi$
$14$	0	0
$15$	−2.84113	−0.733576
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.0994799	0.0228222	0.0114111	−	0.999935i	$-0.496368\pi$
$19$	0.0994799	0.0228222	0.0114111	+	0.999935i	$0.496368\pi$
$20$	0	0
$21$	0.881251	0.192305
$22$	0	0
$23$	6.04990	1.26149	0.630745	−	0.775990i	$-0.282749\pi$
$23$	6.04990	1.26149	0.630745	+	0.775990i	$0.282749\pi$
$24$	0	0
$25$	−3.36416	−0.672831
$26$	0	0
$27$	−2.36695	−0.455519
$28$	0	0
$29$	−0.610013	−0.113277	−0.0566383	−	0.998395i	$-0.518038\pi$
$29$	−0.610013	−0.113277	−0.0566383	+	0.998395i	$0.518038\pi$
$30$	0	0
$31$	−6.40183	−1.14980	−0.574901	−	0.818223i	$-0.694959\pi$
$31$	−6.40183	−1.14980	−0.574901	+	0.818223i	$0.694959\pi$
$32$	0	0
$33$	−6.05148	−1.05343
$34$	0	0
$35$	−0.507401	−0.0857663
$36$	0	0
$37$	11.3427	1.86473	0.932367	−	0.361513i	$-0.117740\pi$
$37$	11.3427	1.86473	0.932367	+	0.361513i	$0.117740\pi$
$38$	0	0
$39$	15.3648	2.46034
$40$	0	0
$41$	2.02962	0.316973	0.158487	−	0.987361i	$-0.449339\pi$
$41$	2.02962	0.316973	0.158487	+	0.987361i	$0.449339\pi$
$42$	0	0
$43$	9.63051	1.46864	0.734319	−	0.678804i	$-0.237501\pi$
$43$	9.63051	1.46864	0.734319	+	0.678804i	$0.237501\pi$
$44$	0	0
$45$	−2.47418	−0.368829
$46$	0	0
$47$	13.2148	1.92758	0.963789	−	0.266664i	$-0.0859215\pi$
$47$	13.2148	1.92758	0.963789	+	0.266664i	$0.0859215\pi$
$48$	0	0
$49$	−6.84262	−0.977517
$50$	0	0
$51$	2.22137	0.311053
$52$	0	0
$53$	−1.02927	−0.141382	−0.0706909	−	0.997498i	$-0.522520\pi$
$53$	−1.02927	−0.141382	−0.0706909	+	0.997498i	$0.522520\pi$
$54$	0	0
$55$	3.48428	0.469820
$56$	0	0
$57$	0.220981	0.0292697
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.211649	0.0270989	0.0135494	−	0.999908i	$-0.495687\pi$
$61$	0.211649	0.0270989	0.0135494	+	0.999908i	$0.495687\pi$
$62$	0	0
$63$	0.767433	0.0966874
$64$	0	0
$65$	−8.84666	−1.09729
$66$	0	0
$67$	−8.77777	−1.07237	−0.536187	−	0.844099i	$-0.680136\pi$
$67$	−8.77777	−1.07237	−0.536187	+	0.844099i	$0.680136\pi$
$68$	0	0
$69$	13.4390	1.61787
$70$	0	0
$71$	5.34794	0.634683	0.317342	−	0.948311i	$-0.397210\pi$
$71$	5.34794	0.634683	0.317342	+	0.948311i	$0.397210\pi$
$72$	0	0
$73$	−6.43008	−0.752583	−0.376292	−	0.926501i	$-0.622801\pi$
$73$	−6.43008	−0.752583	−0.376292	+	0.926501i	$0.622801\pi$
$74$	0	0
$75$	−7.47302	−0.862910
$76$	0	0
$77$	−1.08074	−0.123162
$78$	0	0
$79$	3.68797	0.414929	0.207465	−	0.978243i	$-0.433479\pi$
$79$	3.68797	0.414929	0.207465	+	0.978243i	$0.433479\pi$
$80$	0	0
$81$	−11.0612	−1.22903
$82$	0	0
$83$	12.1769	1.33658	0.668292	−	0.743899i	$-0.267025\pi$
$83$	12.1769	1.33658	0.668292	+	0.743899i	$0.267025\pi$
$84$	0	0
$85$	−1.27900	−0.138727
$86$	0	0
$87$	−1.35506	−0.145278
$88$	0	0
$89$	12.3468	1.30876	0.654378	−	0.756168i	$-0.272931\pi$
$89$	12.3468	1.30876	0.654378	+	0.756168i	$0.272931\pi$
$90$	0	0
$91$	2.74403	0.287652
$92$	0	0
$93$	−14.2208	−1.47463
$94$	0	0
$95$	−0.127235	−0.0130540
$96$	0	0
$97$	5.49270	0.557699	0.278849	−	0.960335i	$-0.410047\pi$
$97$	5.49270	0.557699	0.278849	+	0.960335i	$0.410047\pi$
$98$	0	0
$99$	−5.26990	−0.529645
$100$	0	0
$101$	3.92668	0.390719	0.195360	−	0.980732i	$-0.437413\pi$
$101$	3.92668	0.390719	0.195360	+	0.980732i	$0.437413\pi$
$102$	0	0
$103$	−17.2153	−1.69628	−0.848139	−	0.529773i	$-0.822277\pi$
$103$	−17.2153	−1.69628	−0.848139	+	0.529773i	$0.822277\pi$
$104$	0	0
$105$	−1.12712	−0.109996
$106$	0	0
$107$	19.8408	1.91809	0.959043	−	0.283260i	$-0.0914159\pi$
$107$	19.8408	1.91809	0.959043	+	0.283260i	$0.0914159\pi$
$108$	0	0
$109$	−9.60167	−0.919673	−0.459836	−	0.888004i	$-0.652092\pi$
$109$	−9.60167	−0.919673	−0.459836	+	0.888004i	$0.652092\pi$
$110$	0	0
$111$	25.1964	2.39153
$112$	0	0
$113$	8.45903	0.795759	0.397879	−	0.917438i	$-0.369746\pi$
$113$	8.45903	0.795759	0.397879	+	0.917438i	$0.369746\pi$
$114$	0	0
$115$	−7.73783	−0.721556
$116$	0	0
$117$	13.3804	1.23702
$118$	0	0
$119$	0.396716	0.0363669
$120$	0	0
$121$	−3.57863	−0.325330
$122$	0	0
$123$	4.50853	0.406520
$124$	0	0
$125$	10.6978	0.956837
$126$	0	0
$127$	−3.74587	−0.332392	−0.166196	−	0.986093i	$-0.553148\pi$
$127$	−3.74587	−0.332392	−0.166196	+	0.986093i	$0.553148\pi$
$128$	0	0
$129$	21.3929	1.88354
$130$	0	0
$131$	1.87400	0.163732	0.0818659	−	0.996643i	$-0.473912\pi$
$131$	1.87400	0.163732	0.0818659	+	0.996643i	$0.473912\pi$
$132$	0	0
$133$	0.0394653	0.00342207
$134$	0	0
$135$	3.02733	0.260551
$136$	0	0
$137$	−10.5364	−0.900183	−0.450091	−	0.892982i	$-0.648609\pi$
$137$	−10.5364	−0.900183	−0.450091	+	0.892982i	$0.648609\pi$
$138$	0	0
$139$	20.2222	1.71523	0.857614	−	0.514294i	$-0.171946\pi$
$139$	20.2222	1.71523	0.857614	+	0.514294i	$0.171946\pi$
$140$	0	0
$141$	29.3549	2.47213
$142$	0	0
$143$	−18.8430	−1.57573
$144$	0	0
$145$	0.780208	0.0647927
$146$	0	0
$147$	−15.1999	−1.25367
$148$	0	0
$149$	15.4874	1.26877	0.634387	−	0.773015i	$-0.281252\pi$
$149$	15.4874	1.26877	0.634387	+	0.773015i	$0.281252\pi$
$150$	0	0
$151$	10.3085	0.838892	0.419446	−	0.907780i	$-0.362224\pi$
$151$	10.3085	0.838892	0.419446	+	0.907780i	$0.362224\pi$
$152$	0	0
$153$	1.93446	0.156392
$154$	0	0
$155$	8.18795	0.657672
$156$	0	0
$157$	−10.5321	−0.840553	−0.420276	−	0.907396i	$-0.638067\pi$
$157$	−10.5321	−0.840553	−0.420276	+	0.907396i	$0.638067\pi$
$158$	0	0
$159$	−2.28640	−0.181323
$160$	0	0
$161$	2.40009	0.189154
$162$	0	0
$163$	−15.0358	−1.17770	−0.588848	−	0.808244i	$-0.700418\pi$
$163$	−15.0358	−1.17770	−0.588848	+	0.808244i	$0.700418\pi$
$164$	0	0
$165$	7.73986	0.602547
$166$	0	0
$167$	8.55867	0.662290	0.331145	−	0.943580i	$-0.392565\pi$
$167$	8.55867	0.662290	0.331145	+	0.943580i	$0.392565\pi$
$168$	0	0
$169$	34.8428	2.68022
$170$	0	0
$171$	0.192440	0.0147163
$172$	0	0
$173$	−11.7227	−0.891258	−0.445629	−	0.895218i	$-0.647020\pi$
$173$	−11.7227	−0.891258	−0.445629	+	0.895218i	$0.647020\pi$
$174$	0	0
$175$	−1.33461	−0.100887
$176$	0	0
$177$	2.22137	0.166968
$178$	0	0
$179$	−2.43081	−0.181687	−0.0908437	−	0.995865i	$-0.528956\pi$
$179$	−2.43081	−0.181687	−0.0908437	+	0.995865i	$0.528956\pi$
$180$	0	0
$181$	3.29725	0.245082	0.122541	−	0.992463i	$-0.460896\pi$
$181$	3.29725	0.245082	0.122541	+	0.992463i	$0.460896\pi$
$182$	0	0
$183$	0.470150	0.0347545
$184$	0	0
$185$	−14.5074	−1.06660
$186$	0	0
$187$	−2.72422	−0.199215
$188$	0	0
$189$	−0.939006	−0.0683027
$190$	0	0
$191$	10.4008	0.752578	0.376289	−	0.926502i	$-0.377200\pi$
$191$	10.4008	0.752578	0.376289	+	0.926502i	$0.377200\pi$
$192$	0	0
$193$	0.264170	0.0190154	0.00950768	−	0.999955i	$-0.496974\pi$
$193$	0.264170	0.0190154	0.00950768	+	0.999955i	$0.496974\pi$
$194$	0	0
$195$	−19.6517	−1.40728
$196$	0	0
$197$	−21.5349	−1.53430	−0.767150	−	0.641468i	$-0.778326\pi$
$197$	−21.5349	−1.53430	−0.767150	+	0.641468i	$0.778326\pi$
$198$	0	0
$199$	−15.1327	−1.07273	−0.536365	−	0.843986i	$-0.680203\pi$
$199$	−15.1327	−1.07273	−0.536365	+	0.843986i	$0.680203\pi$
$200$	0	0
$201$	−19.4986	−1.37533
$202$	0	0
$203$	−0.242002	−0.0169852
$204$	0	0
$205$	−2.59589	−0.181305
$206$	0	0
$207$	11.7033	0.813435
$208$	0	0
$209$	−0.271005	−0.0187458
$210$	0	0
$211$	3.28566	0.226194	0.113097	−	0.993584i	$-0.463923\pi$
$211$	3.28566	0.226194	0.113097	+	0.993584i	$0.463923\pi$
$212$	0	0
$213$	11.8797	0.813985
$214$	0	0
$215$	−12.3174	−0.840042
$216$	0	0
$217$	−2.53971	−0.172407
$218$	0	0
$219$	−14.2835	−0.965193
$220$	0	0
$221$	6.91685	0.465278
$222$	0	0
$223$	−16.0572	−1.07527	−0.537636	−	0.843177i	$-0.680683\pi$
$223$	−16.0572	−1.07527	−0.537636	+	0.843177i	$0.680683\pi$
$224$	0	0
$225$	−6.50783	−0.433856
$226$	0	0
$227$	22.1971	1.47328	0.736638	−	0.676287i	$-0.236412\pi$
$227$	22.1971	1.47328	0.736638	+	0.676287i	$0.236412\pi$
$228$	0	0
$229$	4.96428	0.328049	0.164024	−	0.986456i	$-0.447552\pi$
$229$	4.96428	0.328049	0.164024	+	0.986456i	$0.447552\pi$
$230$	0	0
$231$	−2.40072	−0.157956
$232$	0	0
$233$	18.7205	1.22642	0.613211	−	0.789919i	$-0.289877\pi$
$233$	18.7205	1.22642	0.613211	+	0.789919i	$0.289877\pi$
$234$	0	0
$235$	−16.9018	−1.10255
$236$	0	0
$237$	8.19233	0.532149
$238$	0	0
$239$	−17.8155	−1.15239	−0.576194	−	0.817313i	$-0.695463\pi$
$239$	−17.8155	−1.15239	−0.576194	+	0.817313i	$0.695463\pi$
$240$	0	0
$241$	21.8145	1.40520	0.702598	−	0.711587i	$-0.252023\pi$
$241$	21.8145	1.40520	0.702598	+	0.711587i	$0.252023\pi$
$242$	0	0
$243$	−17.4702	−1.12071
$244$	0	0
$245$	8.75172	0.559127
$246$	0	0
$247$	0.688087	0.0437819
$248$	0	0
$249$	27.0493	1.71418
$250$	0	0
$251$	−10.3241	−0.651651	−0.325825	−	0.945430i	$-0.605642\pi$
$251$	−10.3241	−0.651651	−0.325825	+	0.945430i	$0.605642\pi$
$252$	0	0
$253$	−16.4812	−1.03617
$254$	0	0
$255$	−2.84113	−0.177918
$256$	0	0
$257$	27.2547	1.70010	0.850050	−	0.526702i	$-0.176572\pi$
$257$	27.2547	1.70010	0.850050	+	0.526702i	$0.176572\pi$
$258$	0	0
$259$	4.49985	0.279607
$260$	0	0
$261$	−1.18005	−0.0730431
$262$	0	0
$263$	−7.06647	−0.435737	−0.217869	−	0.975978i	$-0.569910\pi$
$263$	−7.06647	−0.435737	−0.217869	+	0.975978i	$0.569910\pi$
$264$	0	0
$265$	1.31644	0.0808685
$266$	0	0
$267$	27.4267	1.67849
$268$	0	0
$269$	20.7138	1.26294	0.631471	−	0.775399i	$-0.282451\pi$
$269$	20.7138	1.26294	0.631471	+	0.775399i	$0.282451\pi$
$270$	0	0
$271$	−17.3390	−1.05327	−0.526636	−	0.850091i	$-0.676547\pi$
$271$	−17.3390	−1.05327	−0.526636	+	0.850091i	$0.676547\pi$
$272$	0	0
$273$	6.09548	0.368916
$274$	0	0
$275$	9.16469	0.552652
$276$	0	0
$277$	1.95353	0.117376	0.0586881	−	0.998276i	$-0.481308\pi$
$277$	1.95353	0.117376	0.0586881	+	0.998276i	$0.481308\pi$
$278$	0	0
$279$	−12.3841	−0.741416
$280$	0	0
$281$	−20.1283	−1.20075	−0.600375	−	0.799718i	$-0.704982\pi$
$281$	−20.1283	−1.20075	−0.600375	+	0.799718i	$0.704982\pi$
$282$	0	0
$283$	−12.0396	−0.715681	−0.357841	−	0.933783i	$-0.616487\pi$
$283$	−12.0396	−0.715681	−0.357841	+	0.933783i	$0.616487\pi$
$284$	0	0
$285$	−0.282635	−0.0167419
$286$	0	0
$287$	0.805183	0.0475284
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.2013	0.715252
$292$	0	0
$293$	−14.5655	−0.850926	−0.425463	−	0.904976i	$-0.639889\pi$
$293$	−14.5655	−0.850926	−0.425463	+	0.904976i	$0.639889\pi$
$294$	0	0
$295$	−1.27900	−0.0744663
$296$	0	0
$297$	6.44808	0.374156
$298$	0	0
$299$	41.8462	2.42003
$300$	0	0
$301$	3.82058	0.220214
$302$	0	0
$303$	8.72258	0.501099
$304$	0	0
$305$	−0.270699	−0.0155002
$306$	0	0
$307$	7.47762	0.426770	0.213385	−	0.976968i	$-0.431551\pi$
$307$	7.47762	0.426770	0.213385	+	0.976968i	$0.431551\pi$
$308$	0	0
$309$	−38.2416	−2.17549
$310$	0	0
$311$	8.00138	0.453717	0.226858	−	0.973928i	$-0.427155\pi$
$311$	8.00138	0.453717	0.226858	+	0.973928i	$0.427155\pi$
$312$	0	0
$313$	−21.7642	−1.23019	−0.615093	−	0.788455i	$-0.710881\pi$
$313$	−21.7642	−1.23019	−0.615093	+	0.788455i	$0.710881\pi$
$314$	0	0
$315$	−0.981547	−0.0553039
$316$	0	0
$317$	−14.0400	−0.788563	−0.394282	−	0.918990i	$-0.629007\pi$
$317$	−14.0400	−0.788563	−0.394282	+	0.918990i	$0.629007\pi$
$318$	0	0
$319$	1.66181	0.0930435
$320$	0	0
$321$	44.0737	2.45996
$322$	0	0
$323$	0.0994799	0.00553521
$324$	0	0
$325$	−23.2694	−1.29075
$326$	0	0
$327$	−21.3288	−1.17949
$328$	0	0
$329$	5.24253	0.289030
$330$	0	0
$331$	14.1586	0.778226	0.389113	−	0.921190i	$-0.372781\pi$
$331$	14.1586	0.778226	0.389113	+	0.921190i	$0.372781\pi$
$332$	0	0
$333$	21.9421	1.20242
$334$	0	0
$335$	11.2268	0.613384
$336$	0	0
$337$	−14.1952	−0.773260	−0.386630	−	0.922235i	$-0.626361\pi$
$337$	−14.1952	−0.773260	−0.386630	+	0.922235i	$0.626361\pi$
$338$	0	0
$339$	18.7906	1.02057
$340$	0	0
$341$	17.4400	0.944428
$342$	0	0
$343$	−5.49159	−0.296518
$344$	0	0
$345$	−17.1885	−0.925400
$346$	0	0
$347$	15.5947	0.837169	0.418584	−	0.908178i	$-0.362526\pi$
$347$	15.5947	0.837169	0.418584	+	0.908178i	$0.362526\pi$
$348$	0	0
$349$	−35.2859	−1.88881	−0.944404	−	0.328786i	$-0.893360\pi$
$349$	−35.2859	−1.88881	−0.944404	+	0.328786i	$0.893360\pi$
$350$	0	0
$351$	−16.3718	−0.873863
$352$	0	0
$353$	−6.27353	−0.333906	−0.166953	−	0.985965i	$-0.553393\pi$
$353$	−6.27353	−0.333906	−0.166953	+	0.985965i	$0.553393\pi$
$354$	0	0
$355$	−6.84002	−0.363031
$356$	0	0
$357$	0.881251	0.0466408
$358$	0	0
$359$	−34.2386	−1.80705	−0.903523	−	0.428539i	$-0.859028\pi$
$359$	−34.2386	−1.80705	−0.903523	+	0.428539i	$0.859028\pi$
$360$	0	0
$361$	−18.9901	−0.999479
$362$	0	0
$363$	−7.94945	−0.417238
$364$	0	0
$365$	8.22408	0.430468
$366$	0	0
$367$	−26.4773	−1.38210	−0.691052	−	0.722805i	$-0.742852\pi$
$367$	−26.4773	−1.38210	−0.691052	+	0.722805i	$0.742852\pi$
$368$	0	0
$369$	3.92622	0.204391
$370$	0	0
$371$	−0.408330	−0.0211994
$372$	0	0
$373$	10.2414	0.530281	0.265140	−	0.964210i	$-0.414582\pi$
$373$	10.2414	0.530281	0.265140	+	0.964210i	$0.414582\pi$
$374$	0	0
$375$	23.7636	1.22715
$376$	0	0
$377$	−4.21937	−0.217309
$378$	0	0
$379$	−21.3706	−1.09774	−0.548868	−	0.835909i	$-0.684941\pi$
$379$	−21.3706	−1.09774	−0.548868	+	0.835909i	$0.684941\pi$
$380$	0	0
$381$	−8.32094	−0.426295
$382$	0	0
$383$	−15.5666	−0.795418	−0.397709	−	0.917512i	$-0.630195\pi$
$383$	−15.5666	−0.795418	−0.397709	+	0.917512i	$0.630195\pi$
$384$	0	0
$385$	1.38227	0.0704470
$386$	0	0
$387$	18.6299	0.947009
$388$	0	0
$389$	−30.8120	−1.56223	−0.781115	−	0.624387i	$-0.785349\pi$
$389$	−30.8120	−1.56223	−0.781115	+	0.624387i	$0.785349\pi$
$390$	0	0
$391$	6.04990	0.305956
$392$	0	0
$393$	4.16283	0.209987
$394$	0	0
$395$	−4.71692	−0.237334
$396$	0	0
$397$	−14.5447	−0.729976	−0.364988	−	0.931012i	$-0.618927\pi$
$397$	−14.5447	−0.729976	−0.364988	+	0.931012i	$0.618927\pi$
$398$	0	0
$399$	0.0876668	0.00438883
$400$	0	0
$401$	−6.07170	−0.303206	−0.151603	−	0.988441i	$-0.548444\pi$
$401$	−6.07170	−0.303206	−0.151603	+	0.988441i	$0.548444\pi$
$402$	0	0
$403$	−44.2805	−2.20577
$404$	0	0
$405$	14.1473	0.702987
$406$	0	0
$407$	−30.9001	−1.53166
$408$	0	0
$409$	10.8845	0.538206	0.269103	−	0.963111i	$-0.413273\pi$
$409$	10.8845	0.538206	0.269103	+	0.963111i	$0.413273\pi$
$410$	0	0
$411$	−23.4051	−1.15449
$412$	0	0
$413$	0.396716	0.0195211
$414$	0	0
$415$	−15.5742	−0.764509
$416$	0	0
$417$	44.9210	2.19979
$418$	0	0
$419$	13.5499	0.661958	0.330979	−	0.943638i	$-0.392621\pi$
$419$	13.5499	0.661958	0.330979	+	0.943638i	$0.392621\pi$
$420$	0	0
$421$	−5.98192	−0.291541	−0.145771	−	0.989318i	$-0.546566\pi$
$421$	−5.98192	−0.291541	−0.145771	+	0.989318i	$0.546566\pi$
$422$	0	0
$423$	25.5636	1.24294
$424$	0	0
$425$	−3.36416	−0.163186
$426$	0	0
$427$	0.0839646	0.00406333
$428$	0	0
$429$	−41.8572	−2.02089
$430$	0	0
$431$	26.8451	1.29308	0.646542	−	0.762879i	$-0.276215\pi$
$431$	26.8451	1.29308	0.646542	+	0.762879i	$0.276215\pi$
$432$	0	0
$433$	8.87008	0.426269	0.213135	−	0.977023i	$-0.431633\pi$
$433$	8.87008	0.426269	0.213135	+	0.977023i	$0.431633\pi$
$434$	0	0
$435$	1.73313	0.0830970
$436$	0	0
$437$	0.601843	0.0287900
$438$	0	0
$439$	−14.1443	−0.675069	−0.337535	−	0.941313i	$-0.609593\pi$
$439$	−14.1443	−0.675069	−0.337535	+	0.941313i	$0.609593\pi$
$440$	0	0
$441$	−13.2368	−0.630323
$442$	0	0
$443$	−22.7074	−1.07886	−0.539430	−	0.842031i	$-0.681360\pi$
$443$	−22.7074	−1.07886	−0.539430	+	0.842031i	$0.681360\pi$
$444$	0	0
$445$	−15.7915	−0.748591
$446$	0	0
$447$	34.4031	1.62721
$448$	0	0
$449$	−22.0320	−1.03976	−0.519878	−	0.854241i	$-0.674023\pi$
$449$	−22.0320	−1.03976	−0.519878	+	0.854241i	$0.674023\pi$
$450$	0	0
$451$	−5.52913	−0.260356
$452$	0	0
$453$	22.8989	1.07588
$454$	0	0
$455$	−3.50961	−0.164533
$456$	0	0
$457$	18.9216	0.885117	0.442558	−	0.896740i	$-0.354071\pi$
$457$	18.9216	0.885117	0.442558	+	0.896740i	$0.354071\pi$
$458$	0	0
$459$	−2.36695	−0.110480
$460$	0	0
$461$	−12.3814	−0.576661	−0.288331	−	0.957531i	$-0.593100\pi$
$461$	−12.3814	−0.576661	−0.288331	+	0.957531i	$0.593100\pi$
$462$	0	0
$463$	7.60847	0.353596	0.176798	−	0.984247i	$-0.443426\pi$
$463$	7.60847	0.353596	0.176798	+	0.984247i	$0.443426\pi$
$464$	0	0
$465$	18.1884	0.843468
$466$	0	0
$467$	−9.37688	−0.433910	−0.216955	−	0.976182i	$-0.569613\pi$
$467$	−9.37688	−0.433910	−0.216955	+	0.976182i	$0.569613\pi$
$468$	0	0
$469$	−3.48228	−0.160797
$470$	0	0
$471$	−23.3956	−1.07801
$472$	0	0
$473$	−26.2356	−1.20631
$474$	0	0
$475$	−0.334666	−0.0153555
$476$	0	0
$477$	−1.99109	−0.0911659
$478$	0	0
$479$	37.9244	1.73281	0.866406	−	0.499340i	$-0.166424\pi$
$479$	37.9244	1.73281	0.866406	+	0.499340i	$0.166424\pi$
$480$	0	0
$481$	78.4560	3.57729
$482$	0	0
$483$	5.33148	0.242591
$484$	0	0
$485$	−7.02517	−0.318996
$486$	0	0
$487$	32.3367	1.46531	0.732657	−	0.680598i	$-0.238280\pi$
$487$	32.3367	1.46531	0.732657	+	0.680598i	$0.238280\pi$
$488$	0	0
$489$	−33.4000	−1.51040
$490$	0	0
$491$	4.86262	0.219447	0.109723	−	0.993962i	$-0.465003\pi$
$491$	4.86262	0.219447	0.109723	+	0.993962i	$0.465003\pi$
$492$	0	0
$493$	−0.610013	−0.0274736
$494$	0	0
$495$	6.74021	0.302950
$496$	0	0
$497$	2.12161	0.0951674
$498$	0	0
$499$	15.7513	0.705127	0.352564	−	0.935788i	$-0.385310\pi$
$499$	15.7513	0.705127	0.352564	+	0.935788i	$0.385310\pi$
$500$	0	0
$501$	19.0119	0.849390
$502$	0	0
$503$	13.5110	0.602426	0.301213	−	0.953557i	$-0.402608\pi$
$503$	13.5110	0.602426	0.301213	+	0.953557i	$0.402608\pi$
$504$	0	0
$505$	−5.02223	−0.223486
$506$	0	0
$507$	77.3986	3.43739
$508$	0	0
$509$	24.5881	1.08985	0.544923	−	0.838486i	$-0.316559\pi$
$509$	24.5881	1.08985	0.544923	+	0.838486i	$0.316559\pi$
$510$	0	0
$511$	−2.55091	−0.112846
$512$	0	0
$513$	−0.235464	−0.0103960
$514$	0	0
$515$	22.0185	0.970249
$516$	0	0
$517$	−36.0000	−1.58328
$518$	0	0
$519$	−26.0403	−1.14304
$520$	0	0
$521$	−22.1171	−0.968970	−0.484485	−	0.874800i	$-0.660993\pi$
$521$	−22.1171	−0.968970	−0.484485	+	0.874800i	$0.660993\pi$
$522$	0	0
$523$	−37.0750	−1.62118	−0.810588	−	0.585617i	$-0.800852\pi$
$523$	−37.0750	−1.62118	−0.810588	+	0.585617i	$0.800852\pi$
$524$	0	0
$525$	−2.96467	−0.129389
$526$	0	0
$527$	−6.40183	−0.278868
$528$	0	0
$529$	13.6012	0.591358
$530$	0	0
$531$	1.93446	0.0839485
$532$	0	0
$533$	14.0386	0.608078
$534$	0	0
$535$	−25.3765	−1.09712
$536$	0	0
$537$	−5.39972	−0.233015
$538$	0	0
$539$	18.6408	0.802915
$540$	0	0
$541$	28.6764	1.23290	0.616448	−	0.787396i	$-0.288571\pi$
$541$	28.6764	1.23290	0.616448	+	0.787396i	$0.288571\pi$
$542$	0	0
$543$	7.32439	0.314319
$544$	0	0
$545$	12.2805	0.526041
$546$	0	0
$547$	34.5247	1.47617	0.738084	−	0.674708i	$-0.235731\pi$
$547$	34.5247	1.47617	0.738084	+	0.674708i	$0.235731\pi$
$548$	0	0
$549$	0.409427	0.0174739
$550$	0	0
$551$	−0.0606840	−0.00258523
$552$	0	0
$553$	1.46308	0.0622164
$554$	0	0
$555$	−32.2262	−1.36793
$556$	0	0
$557$	−15.9631	−0.676378	−0.338189	−	0.941078i	$-0.609814\pi$
$557$	−15.9631	−0.676378	−0.338189	+	0.941078i	$0.609814\pi$
$558$	0	0
$559$	66.6128	2.81742
$560$	0	0
$561$	−6.05148	−0.255494
$562$	0	0
$563$	−42.8342	−1.80525	−0.902623	−	0.430431i	$-0.858362\pi$
$563$	−42.8342	−1.80525	−0.902623	+	0.430431i	$0.858362\pi$
$564$	0	0
$565$	−10.8191	−0.455164
$566$	0	0
$567$	−4.38817	−0.184286
$568$	0	0
$569$	−26.0189	−1.09077	−0.545385	−	0.838186i	$-0.683616\pi$
$569$	−26.0189	−1.09077	−0.545385	+	0.838186i	$0.683616\pi$
$570$	0	0
$571$	0.609323	0.0254994	0.0127497	−	0.999919i	$-0.495942\pi$
$571$	0.609323	0.0254994	0.0127497	+	0.999919i	$0.495942\pi$
$572$	0	0
$573$	23.1040	0.965185
$574$	0	0
$575$	−20.3528	−0.848770
$576$	0	0
$577$	−42.6570	−1.77583	−0.887917	−	0.460003i	$-0.847848\pi$
$577$	−42.6570	−1.77583	−0.887917	+	0.460003i	$0.847848\pi$
$578$	0	0
$579$	0.586817	0.0243873
$580$	0	0
$581$	4.83076	0.200414
$582$	0	0
$583$	2.80397	0.116129
$584$	0	0
$585$	−17.1135	−0.707558
$586$	0	0
$587$	−34.1910	−1.41121	−0.705607	−	0.708604i	$-0.749325\pi$
$587$	−34.1910	−1.41121	−0.705607	+	0.708604i	$0.749325\pi$
$588$	0	0
$589$	−0.636853	−0.0262411
$590$	0	0
$591$	−47.8369	−1.96775
$592$	0	0
$593$	29.1643	1.19763	0.598817	−	0.800886i	$-0.295638\pi$
$593$	29.1643	1.19763	0.598817	+	0.800886i	$0.295638\pi$
$594$	0	0
$595$	−0.507401	−0.0208014
$596$	0	0
$597$	−33.6153	−1.37578
$598$	0	0
$599$	−7.27404	−0.297209	−0.148605	−	0.988897i	$-0.547478\pi$
$599$	−7.27404	−0.297209	−0.148605	+	0.988897i	$0.547478\pi$
$600$	0	0
$601$	−15.4093	−0.628560	−0.314280	−	0.949330i	$-0.601763\pi$
$601$	−15.4093	−0.628560	−0.314280	+	0.949330i	$0.601763\pi$
$602$	0	0
$603$	−16.9803	−0.691490
$604$	0	0
$605$	4.57708	0.186085
$606$	0	0
$607$	44.3398	1.79969	0.899847	−	0.436205i	$-0.143678\pi$
$607$	44.3398	1.79969	0.899847	+	0.436205i	$0.143678\pi$
$608$	0	0
$609$	−0.537575	−0.0217836
$610$	0	0
$611$	91.4049	3.69785
$612$	0	0
$613$	16.5539	0.668605	0.334303	−	0.942466i	$-0.391499\pi$
$613$	16.5539	0.668605	0.334303	+	0.942466i	$0.391499\pi$
$614$	0	0
$615$	−5.76641	−0.232524
$616$	0	0
$617$	−29.0175	−1.16820	−0.584100	−	0.811682i	$-0.698552\pi$
$617$	−29.0175	−1.16820	−0.584100	+	0.811682i	$0.698552\pi$
$618$	0	0
$619$	20.8516	0.838097	0.419048	−	0.907964i	$-0.362364\pi$
$619$	20.8516	0.838097	0.419048	+	0.907964i	$0.362364\pi$
$620$	0	0
$621$	−14.3198	−0.574633
$622$	0	0
$623$	4.89817	0.196241
$624$	0	0
$625$	3.13832	0.125533
$626$	0	0
$627$	−0.602001	−0.0240416
$628$	0	0
$629$	11.3427	0.452264
$630$	0	0
$631$	4.53794	0.180652	0.0903262	−	0.995912i	$-0.471209\pi$
$631$	4.53794	0.180652	0.0903262	+	0.995912i	$0.471209\pi$
$632$	0	0
$633$	7.29865	0.290095
$634$	0	0
$635$	4.79097	0.190124
$636$	0	0
$637$	−47.3294	−1.87526
$638$	0	0
$639$	10.3454	0.409257
$640$	0	0
$641$	3.73898	0.147681	0.0738405	−	0.997270i	$-0.476474\pi$
$641$	3.73898	0.147681	0.0738405	+	0.997270i	$0.476474\pi$
$642$	0	0
$643$	−10.1748	−0.401256	−0.200628	−	0.979667i	$-0.564298\pi$
$643$	−10.1748	−0.401256	−0.200628	+	0.979667i	$0.564298\pi$
$644$	0	0
$645$	−27.3615	−1.07736
$646$	0	0
$647$	10.2698	0.403747	0.201874	−	0.979412i	$-0.435297\pi$
$647$	10.2698	0.403747	0.201874	+	0.979412i	$0.435297\pi$
$648$	0	0
$649$	−2.72422	−0.106935
$650$	0	0
$651$	−5.64162	−0.221113
$652$	0	0
$653$	−18.5592	−0.726279	−0.363139	−	0.931735i	$-0.618295\pi$
$653$	−18.5592	−0.726279	−0.363139	+	0.931735i	$0.618295\pi$
$654$	0	0
$655$	−2.39684	−0.0936525
$656$	0	0
$657$	−12.4387	−0.485282
$658$	0	0
$659$	16.0530	0.625334	0.312667	−	0.949863i	$-0.398778\pi$
$659$	16.0530	0.625334	0.312667	+	0.949863i	$0.398778\pi$
$660$	0	0
$661$	−45.4546	−1.76798	−0.883990	−	0.467505i	$-0.845153\pi$
$661$	−45.4546	−1.76798	−0.883990	+	0.467505i	$0.845153\pi$
$662$	0	0
$663$	15.3648	0.596721
$664$	0	0
$665$	−0.0504761	−0.00195738
$666$	0	0
$667$	−3.69052	−0.142897
$668$	0	0
$669$	−35.6690	−1.37904
$670$	0	0
$671$	−0.576578	−0.0222585
$672$	0	0
$673$	−21.0557	−0.811637	−0.405819	−	0.913954i	$-0.633014\pi$
$673$	−21.0557	−0.811637	−0.405819	+	0.913954i	$0.633014\pi$
$674$	0	0
$675$	7.96278	0.306487
$676$	0	0
$677$	−11.5944	−0.445609	−0.222804	−	0.974863i	$-0.571521\pi$
$677$	−11.5944	−0.445609	−0.222804	+	0.974863i	$0.571521\pi$
$678$	0	0
$679$	2.17904	0.0836239
$680$	0	0
$681$	49.3080	1.88948
$682$	0	0
$683$	−10.6092	−0.405950	−0.202975	−	0.979184i	$-0.565061\pi$
$683$	−10.6092	−0.405950	−0.202975	+	0.979184i	$0.565061\pi$
$684$	0	0
$685$	13.4760	0.514893
$686$	0	0
$687$	11.0275	0.420724
$688$	0	0
$689$	−7.11934	−0.271225
$690$	0	0
$691$	23.9652	0.911680	0.455840	−	0.890062i	$-0.349339\pi$
$691$	23.9652	0.911680	0.455840	+	0.890062i	$0.349339\pi$
$692$	0	0
$693$	−2.09065	−0.0794174
$694$	0	0
$695$	−25.8643	−0.981088
$696$	0	0
$697$	2.02962	0.0768773
$698$	0	0
$699$	41.5851	1.57289
$700$	0	0
$701$	11.5614	0.436666	0.218333	−	0.975874i	$-0.429938\pi$
$701$	11.5614	0.436666	0.218333	+	0.975874i	$0.429938\pi$
$702$	0	0
$703$	1.12837	0.0425574
$704$	0	0
$705$	−37.5450	−1.41403
$706$	0	0
$707$	1.55778	0.0585862
$708$	0	0
$709$	−27.2618	−1.02384	−0.511920	−	0.859033i	$-0.671066\pi$
$709$	−27.2618	−1.02384	−0.511920	+	0.859033i	$0.671066\pi$
$710$	0	0
$711$	7.13424	0.267555
$712$	0	0
$713$	−38.7304	−1.45046
$714$	0	0
$715$	24.1002	0.901298
$716$	0	0
$717$	−39.5747	−1.47794
$718$	0	0
$719$	1.68485	0.0628343	0.0314172	−	0.999506i	$-0.489998\pi$
$719$	1.68485	0.0628343	0.0314172	+	0.999506i	$0.489998\pi$
$720$	0	0
$721$	−6.82961	−0.254348
$722$	0	0
$723$	48.4580	1.80217
$724$	0	0
$725$	2.05218	0.0762160
$726$	0	0
$727$	45.1113	1.67309	0.836544	−	0.547900i	$-0.184573\pi$
$727$	45.1113	1.67309	0.836544	+	0.547900i	$0.184573\pi$
$728$	0	0
$729$	−5.62400	−0.208296
$730$	0	0
$731$	9.63051	0.356197
$732$	0	0
$733$	−14.9390	−0.551783	−0.275892	−	0.961189i	$-0.588973\pi$
$733$	−14.9390	−0.551783	−0.275892	+	0.961189i	$0.588973\pi$
$734$	0	0
$735$	19.4408	0.717083
$736$	0	0
$737$	23.9126	0.880830
$738$	0	0
$739$	−32.2234	−1.18536	−0.592679	−	0.805439i	$-0.701930\pi$
$739$	−32.2234	−1.18536	−0.592679	+	0.805439i	$0.701930\pi$
$740$	0	0
$741$	1.52849	0.0561506
$742$	0	0
$743$	−14.7123	−0.539743	−0.269871	−	0.962896i	$-0.586981\pi$
$743$	−14.7123	−0.539743	−0.269871	+	0.962896i	$0.586981\pi$
$744$	0	0
$745$	−19.8084	−0.725723
$746$	0	0
$747$	23.5557	0.861858
$748$	0	0
$749$	7.87118	0.287607
$750$	0	0
$751$	2.41554	0.0881444	0.0440722	−	0.999028i	$-0.485967\pi$
$751$	2.41554	0.0881444	0.0440722	+	0.999028i	$0.485967\pi$
$752$	0	0
$753$	−22.9336	−0.835746
$754$	0	0
$755$	−13.1846	−0.479835
$756$	0	0
$757$	31.7957	1.15563	0.577816	−	0.816167i	$-0.303905\pi$
$757$	31.7957	1.15563	0.577816	+	0.816167i	$0.303905\pi$
$758$	0	0
$759$	−36.6108	−1.32889
$760$	0	0
$761$	−4.87766	−0.176815	−0.0884075	−	0.996084i	$-0.528178\pi$
$761$	−4.87766	−0.176815	−0.0884075	+	0.996084i	$0.528178\pi$
$762$	0	0
$763$	−3.80914	−0.137900
$764$	0	0
$765$	−2.47418	−0.0894542
$766$	0	0
$767$	6.91685	0.249753
$768$	0	0
$769$	49.2749	1.77690	0.888449	−	0.458975i	$-0.151783\pi$
$769$	49.2749	1.77690	0.888449	+	0.458975i	$0.151783\pi$
$770$	0	0
$771$	60.5426	2.18039
$772$	0	0
$773$	−39.5744	−1.42339	−0.711696	−	0.702487i	$-0.752073\pi$
$773$	−39.5744	−1.42339	−0.711696	+	0.702487i	$0.752073\pi$
$774$	0	0
$775$	21.5367	0.773623
$776$	0	0
$777$	9.99580	0.358597
$778$	0	0
$779$	0.201906	0.00723404
$780$	0	0
$781$	−14.5690	−0.521318
$782$	0	0
$783$	1.44387	0.0515997
$784$	0	0
$785$	13.4706	0.480785
$786$	0	0
$787$	18.0961	0.645057	0.322528	−	0.946560i	$-0.395467\pi$
$787$	18.0961	0.645057	0.322528	+	0.946560i	$0.395467\pi$
$788$	0	0
$789$	−15.6972	−0.558835
$790$	0	0
$791$	3.35583	0.119320
$792$	0	0
$793$	1.46394	0.0519862
$794$	0	0
$795$	2.92430	0.103714
$796$	0	0
$797$	16.6346	0.589227	0.294614	−	0.955616i	$-0.404809\pi$
$797$	16.6346	0.589227	0.294614	+	0.955616i	$0.404809\pi$
$798$	0	0
$799$	13.2148	0.467507
$800$	0	0
$801$	23.8844	0.843913
$802$	0	0
$803$	17.5169	0.618159
$804$	0	0
$805$	−3.06972	−0.108193
$806$	0	0
$807$	46.0129	1.61973
$808$	0	0
$809$	−16.6645	−0.585894	−0.292947	−	0.956129i	$-0.594636\pi$
$809$	−16.6645	−0.585894	−0.292947	+	0.956129i	$0.594636\pi$
$810$	0	0
$811$	−14.2726	−0.501180	−0.250590	−	0.968093i	$-0.580625\pi$
$811$	−14.2726	−0.501180	−0.250590	+	0.968093i	$0.580625\pi$
$812$	0	0
$813$	−38.5163	−1.35083
$814$	0	0
$815$	19.2308	0.673626
$816$	0	0
$817$	0.958041	0.0335176
$818$	0	0
$819$	5.30822	0.185484
$820$	0	0
$821$	22.9263	0.800133	0.400066	−	0.916486i	$-0.368987\pi$
$821$	22.9263	0.800133	0.400066	+	0.916486i	$0.368987\pi$
$822$	0	0
$823$	46.5450	1.62246	0.811229	−	0.584729i	$-0.198799\pi$
$823$	46.5450	1.62246	0.811229	+	0.584729i	$0.198799\pi$
$824$	0	0
$825$	20.3581	0.708779
$826$	0	0
$827$	−3.80090	−0.132170	−0.0660851	−	0.997814i	$-0.521051\pi$
$827$	−3.80090	−0.132170	−0.0660851	+	0.997814i	$0.521051\pi$
$828$	0	0
$829$	14.7169	0.511138	0.255569	−	0.966791i	$-0.417737\pi$
$829$	14.7169	0.511138	0.255569	+	0.966791i	$0.417737\pi$
$830$	0	0
$831$	4.33950	0.150536
$832$	0	0
$833$	−6.84262	−0.237083
$834$	0	0
$835$	−10.9465	−0.378821
$836$	0	0
$837$	15.1528	0.523757
$838$	0	0
$839$	17.7640	0.613283	0.306641	−	0.951825i	$-0.400795\pi$
$839$	17.7640	0.613283	0.306641	+	0.951825i	$0.400795\pi$
$840$	0	0
$841$	−28.6279	−0.987168
$842$	0	0
$843$	−44.7122	−1.53997
$844$	0	0
$845$	−44.5640	−1.53305
$846$	0	0
$847$	−1.41970	−0.0487815
$848$	0	0
$849$	−26.7444	−0.917865
$850$	0	0
$851$	68.6224	2.35234
$852$	0	0
$853$	6.35715	0.217665	0.108832	−	0.994060i	$-0.465289\pi$
$853$	6.35715	0.217665	0.108832	+	0.994060i	$0.465289\pi$
$854$	0	0
$855$	−0.246131	−0.00841751
$856$	0	0
$857$	−19.1071	−0.652685	−0.326343	−	0.945252i	$-0.605816\pi$
$857$	−19.1071	−0.652685	−0.326343	+	0.945252i	$0.605816\pi$
$858$	0	0
$859$	−48.9245	−1.66928	−0.834641	−	0.550795i	$-0.814325\pi$
$859$	−48.9245	−1.66928	−0.834641	+	0.550795i	$0.814325\pi$
$860$	0	0
$861$	1.78861	0.0609555
$862$	0	0
$863$	−22.3956	−0.762355	−0.381177	−	0.924502i	$-0.624481\pi$
$863$	−22.3956	−0.762355	−0.381177	+	0.924502i	$0.624481\pi$
$864$	0	0
$865$	14.9933	0.509788
$866$	0	0
$867$	2.22137	0.0754415
$868$	0	0
$869$	−10.0468	−0.340816
$870$	0	0
$871$	−60.7145	−2.05723
$872$	0	0
$873$	10.6254	0.359616
$874$	0	0
$875$	4.24398	0.143473
$876$	0	0
$877$	−14.4743	−0.488761	−0.244381	−	0.969679i	$-0.578585\pi$
$877$	−14.4743	−0.488761	−0.244381	+	0.969679i	$0.578585\pi$
$878$	0	0
$879$	−32.3553	−1.09132
$880$	0	0
$881$	−39.8222	−1.34164	−0.670822	−	0.741619i	$-0.734058\pi$
$881$	−39.8222	−1.34164	−0.670822	+	0.741619i	$0.734058\pi$
$882$	0	0
$883$	7.57968	0.255076	0.127538	−	0.991834i	$-0.459292\pi$
$883$	7.57968	0.255076	0.127538	+	0.991834i	$0.459292\pi$
$884$	0	0
$885$	−2.84113	−0.0955035
$886$	0	0
$887$	−27.9178	−0.937388	−0.468694	−	0.883360i	$-0.655275\pi$
$887$	−27.9178	−0.937388	−0.468694	+	0.883360i	$0.655275\pi$
$888$	0	0
$889$	−1.48605	−0.0498404
$890$	0	0
$891$	30.1332	1.00950
$892$	0	0
$893$	1.31461	0.0439917
$894$	0	0
$895$	3.10901	0.103923
$896$	0	0
$897$	92.9557	3.10370
$898$	0	0
$899$	3.90520	0.130246
$900$	0	0
$901$	−1.02927	−0.0342901
$902$	0	0
$903$	8.48690	0.282426
$904$	0	0
$905$	−4.21718	−0.140184
$906$	0	0
$907$	55.6241	1.84697	0.923484	−	0.383636i	$-0.125328\pi$
$907$	55.6241	1.84697	0.923484	+	0.383636i	$0.125328\pi$
$908$	0	0
$909$	7.59601	0.251944
$910$	0	0
$911$	−43.6289	−1.44549	−0.722745	−	0.691115i	$-0.757120\pi$
$911$	−43.6289	−1.44549	−0.722745	+	0.691115i	$0.757120\pi$
$912$	0	0
$913$	−33.1724	−1.09785
$914$	0	0
$915$	−0.601322	−0.0198791
$916$	0	0
$917$	0.743445	0.0245507
$918$	0	0
$919$	10.9904	0.362539	0.181270	−	0.983433i	$-0.441979\pi$
$919$	10.9904	0.362539	0.181270	+	0.983433i	$0.441979\pi$
$920$	0	0
$921$	16.6105	0.547335
$922$	0	0
$923$	36.9909	1.21757
$924$	0	0
$925$	−38.1587	−1.25465
$926$	0	0
$927$	−33.3024	−1.09380
$928$	0	0
$929$	−52.7940	−1.73211	−0.866057	−	0.499945i	$-0.833354\pi$
$929$	−52.7940	−1.73211	−0.866057	+	0.499945i	$0.833354\pi$
$930$	0	0
$931$	−0.680702	−0.0223091
$932$	0	0
$933$	17.7740	0.581894
$934$	0	0
$935$	3.48428	0.113948
$936$	0	0
$937$	−3.43513	−0.112221	−0.0561104	−	0.998425i	$-0.517870\pi$
$937$	−3.43513	−0.112221	−0.0561104	+	0.998425i	$0.517870\pi$
$938$	0	0
$939$	−48.3462	−1.57772
$940$	0	0
$941$	−43.4484	−1.41638	−0.708189	−	0.706023i	$-0.750488\pi$
$941$	−43.4484	−1.41638	−0.708189	+	0.706023i	$0.750488\pi$
$942$	0	0
$943$	12.2790	0.399859
$944$	0	0
$945$	1.20099	0.0390682
$946$	0	0
$947$	4.28586	0.139272	0.0696359	−	0.997572i	$-0.477816\pi$
$947$	4.28586	0.139272	0.0696359	+	0.997572i	$0.477816\pi$
$948$	0	0
$949$	−44.4759	−1.44375
$950$	0	0
$951$	−31.1879	−1.01134
$952$	0	0
$953$	−0.844400	−0.0273528	−0.0136764	−	0.999906i	$-0.504353\pi$
$953$	−0.844400	−0.0273528	−0.0136764	+	0.999906i	$0.504353\pi$
$954$	0	0
$955$	−13.3027	−0.430465
$956$	0	0
$957$	3.69149	0.119329
$958$	0	0
$959$	−4.17995	−0.134978
$960$	0	0
$961$	9.98339	0.322045
$962$	0	0
$963$	38.3814	1.23682
$964$	0	0
$965$	−0.337873	−0.0108765
$966$	0	0
$967$	15.8215	0.508786	0.254393	−	0.967101i	$-0.418124\pi$
$967$	15.8215	0.508786	0.254393	+	0.967101i	$0.418124\pi$
$968$	0	0
$969$	0.220981	0.00709893
$970$	0	0
$971$	−39.5299	−1.26857	−0.634287	−	0.773098i	$-0.718706\pi$
$971$	−39.5299	−1.26857	−0.634287	+	0.773098i	$0.718706\pi$
$972$	0	0
$973$	8.02249	0.257189
$974$	0	0
$975$	−51.6897	−1.65540
$976$	0	0
$977$	−7.71814	−0.246925	−0.123463	−	0.992349i	$-0.539400\pi$
$977$	−7.71814	−0.246925	−0.123463	+	0.992349i	$0.539400\pi$
$978$	0	0
$979$	−33.6353	−1.07499
$980$	0	0
$981$	−18.5741	−0.593024
$982$	0	0
$983$	25.0438	0.798772	0.399386	−	0.916783i	$-0.369223\pi$
$983$	25.0438	0.798772	0.399386	+	0.916783i	$0.369223\pi$
$984$	0	0
$985$	27.5432	0.877599
$986$	0	0
$987$	11.6456	0.370683
$988$	0	0
$989$	58.2636	1.85267
$990$	0	0
$991$	−28.2669	−0.897928	−0.448964	−	0.893550i	$-0.648207\pi$
$991$	−28.2669	−0.897928	−0.448964	+	0.893550i	$0.648207\pi$
$992$	0	0
$993$	31.4514	0.998080
$994$	0	0
$995$	19.3548	0.613588
$996$	0	0
$997$	−0.0420885	−0.00133296	−0.000666478	−	1.00000i	$-0.500212\pi$
$997$	−0.0420885	−0.00133296	−0.000666478		1.00000i	$0.500212\pi$
$998$	0	0
$999$	−26.8477	−0.849422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.16	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.16	✓	21	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22137 q^{3} -1.27900 q^{5} +0.396716 q^{7} +1.93446 q^{9} +O(q^{10})\)	\(q+2.22137 q^{3} -1.27900 q^{5} +0.396716 q^{7} +1.93446 q^{9} -2.72422 q^{11} +6.91685 q^{13} -2.84113 q^{15} +1.00000 q^{17} +0.0994799 q^{19} +0.881251 q^{21} +6.04990 q^{23} -3.36416 q^{25} -2.36695 q^{27} -0.610013 q^{29} -6.40183 q^{31} -6.05148 q^{33} -0.507401 q^{35} +11.3427 q^{37} +15.3648 q^{39} +2.02962 q^{41} +9.63051 q^{43} -2.47418 q^{45} +13.2148 q^{47} -6.84262 q^{49} +2.22137 q^{51} -1.02927 q^{53} +3.48428 q^{55} +0.220981 q^{57} +1.00000 q^{59} +0.211649 q^{61} +0.767433 q^{63} -8.84666 q^{65} -8.77777 q^{67} +13.4390 q^{69} +5.34794 q^{71} -6.43008 q^{73} -7.47302 q^{75} -1.08074 q^{77} +3.68797 q^{79} -11.0612 q^{81} +12.1769 q^{83} -1.27900 q^{85} -1.35506 q^{87} +12.3468 q^{89} +2.74403 q^{91} -14.2208 q^{93} -0.127235 q^{95} +5.49270 q^{97} -5.26990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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