LMFDB - Embedded newform 6036.2.a.i.1.5

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			1.5
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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.60444 q^{5} -1.53102 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.60444 q^{5} -1.53102 q^{7} +1.00000 q^{9} +4.96919 q^{11} +3.57535 q^{13} +2.60444 q^{15} -5.67149 q^{17} +7.50947 q^{19} +1.53102 q^{21} -5.35597 q^{23} +1.78310 q^{25} -1.00000 q^{27} +3.66790 q^{29} +4.23096 q^{31} -4.96919 q^{33} +3.98745 q^{35} -6.07248 q^{37} -3.57535 q^{39} -1.00478 q^{41} +1.75353 q^{43} -2.60444 q^{45} -2.20924 q^{47} -4.65597 q^{49} +5.67149 q^{51} +5.12050 q^{53} -12.9419 q^{55} -7.50947 q^{57} +1.75140 q^{59} +2.51859 q^{61} -1.53102 q^{63} -9.31177 q^{65} +12.8570 q^{67} +5.35597 q^{69} -7.52624 q^{71} -0.588232 q^{73} -1.78310 q^{75} -7.60793 q^{77} +3.18145 q^{79} +1.00000 q^{81} -1.21414 q^{83} +14.7710 q^{85} -3.66790 q^{87} -13.4526 q^{89} -5.47394 q^{91} -4.23096 q^{93} -19.5580 q^{95} -16.6053 q^{97} +4.96919 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.60444	−1.16474	−0.582370	−	0.812924i	$-0.697875\pi$
$5$	−2.60444	−1.16474	−0.582370	+	0.812924i	$0.697875\pi$
$6$	0	0
$7$	−1.53102	−0.578672	−0.289336	−	0.957228i	$-0.593434\pi$
$7$	−1.53102	−0.578672	−0.289336	+	0.957228i	$0.593434\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.96919	1.49827	0.749133	−	0.662420i	$-0.230470\pi$
$11$	4.96919	1.49827	0.749133	+	0.662420i	$0.230470\pi$
$12$	0	0
$13$	3.57535	0.991623	0.495811	−	0.868430i	$-0.334871\pi$
$13$	3.57535	0.991623	0.495811	+	0.868430i	$0.334871\pi$
$14$	0	0
$15$	2.60444	0.672463
$16$	0	0
$17$	−5.67149	−1.37554	−0.687769	−	0.725930i	$-0.741410\pi$
$17$	−5.67149	−1.37554	−0.687769	+	0.725930i	$0.741410\pi$
$18$	0	0
$19$	7.50947	1.72279	0.861396	−	0.507935i	$-0.169591\pi$
$19$	7.50947	1.72279	0.861396	+	0.507935i	$0.169591\pi$
$20$	0	0
$21$	1.53102	0.334096
$22$	0	0
$23$	−5.35597	−1.11680	−0.558399	−	0.829573i	$-0.688584\pi$
$23$	−5.35597	−1.11680	−0.558399	+	0.829573i	$0.688584\pi$
$24$	0	0
$25$	1.78310	0.356620
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.66790	0.681112	0.340556	−	0.940224i	$-0.389385\pi$
$29$	3.66790	0.681112	0.340556	+	0.940224i	$0.389385\pi$
$30$	0	0
$31$	4.23096	0.759903	0.379951	−	0.925006i	$-0.375941\pi$
$31$	4.23096	0.759903	0.379951	+	0.925006i	$0.375941\pi$
$32$	0	0
$33$	−4.96919	−0.865024
$34$	0	0
$35$	3.98745	0.674003
$36$	0	0
$37$	−6.07248	−0.998310	−0.499155	−	0.866513i	$-0.666356\pi$
$37$	−6.07248	−0.998310	−0.499155	+	0.866513i	$0.666356\pi$
$38$	0	0
$39$	−3.57535	−0.572514
$40$	0	0
$41$	−1.00478	−0.156921	−0.0784604	−	0.996917i	$-0.525000\pi$
$41$	−1.00478	−0.156921	−0.0784604	+	0.996917i	$0.525000\pi$
$42$	0	0
$43$	1.75353	0.267411	0.133706	−	0.991021i	$-0.457312\pi$
$43$	1.75353	0.267411	0.133706	+	0.991021i	$0.457312\pi$
$44$	0	0
$45$	−2.60444	−0.388247
$46$	0	0
$47$	−2.20924	−0.322251	−0.161126	−	0.986934i	$-0.551512\pi$
$47$	−2.20924	−0.322251	−0.161126	+	0.986934i	$0.551512\pi$
$48$	0	0
$49$	−4.65597	−0.665139
$50$	0	0
$51$	5.67149	0.794167
$52$	0	0
$53$	5.12050	0.703355	0.351678	−	0.936121i	$-0.385611\pi$
$53$	5.12050	0.703355	0.351678	+	0.936121i	$0.385611\pi$
$54$	0	0
$55$	−12.9419	−1.74509
$56$	0	0
$57$	−7.50947	−0.994654
$58$	0	0
$59$	1.75140	0.228013	0.114006	−	0.993480i	$-0.463632\pi$
$59$	1.75140	0.228013	0.114006	+	0.993480i	$0.463632\pi$
$60$	0	0
$61$	2.51859	0.322472	0.161236	−	0.986916i	$-0.448452\pi$
$61$	2.51859	0.322472	0.161236	+	0.986916i	$0.448452\pi$
$62$	0	0
$63$	−1.53102	−0.192891
$64$	0	0
$65$	−9.31177	−1.15498
$66$	0	0
$67$	12.8570	1.57074	0.785368	−	0.619030i	$-0.212474\pi$
$67$	12.8570	1.57074	0.785368	+	0.619030i	$0.212474\pi$
$68$	0	0
$69$	5.35597	0.644783
$70$	0	0
$71$	−7.52624	−0.893200	−0.446600	−	0.894734i	$-0.647365\pi$
$71$	−7.52624	−0.893200	−0.446600	+	0.894734i	$0.647365\pi$
$72$	0	0
$73$	−0.588232	−0.0688474	−0.0344237	−	0.999407i	$-0.510960\pi$
$73$	−0.588232	−0.0688474	−0.0344237	+	0.999407i	$0.510960\pi$
$74$	0	0
$75$	−1.78310	−0.205894
$76$	0	0
$77$	−7.60793	−0.867005
$78$	0	0
$79$	3.18145	0.357941	0.178970	−	0.983854i	$-0.442723\pi$
$79$	3.18145	0.357941	0.178970	+	0.983854i	$0.442723\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.21414	−0.133269	−0.0666346	−	0.997777i	$-0.521226\pi$
$83$	−1.21414	−0.133269	−0.0666346	+	0.997777i	$0.521226\pi$
$84$	0	0
$85$	14.7710	1.60214
$86$	0	0
$87$	−3.66790	−0.393240
$88$	0	0
$89$	−13.4526	−1.42597	−0.712984	−	0.701180i	$-0.752657\pi$
$89$	−13.4526	−1.42597	−0.712984	+	0.701180i	$0.752657\pi$
$90$	0	0
$91$	−5.47394	−0.573824
$92$	0	0
$93$	−4.23096	−0.438730
$94$	0	0
$95$	−19.5580	−2.00660
$96$	0	0
$97$	−16.6053	−1.68601	−0.843007	−	0.537902i	$-0.819217\pi$
$97$	−16.6053	−1.68601	−0.843007	+	0.537902i	$0.819217\pi$
$98$	0	0
$99$	4.96919	0.499422
$100$	0	0
$101$	14.3577	1.42865	0.714323	−	0.699816i	$-0.246735\pi$
$101$	14.3577	1.42865	0.714323	+	0.699816i	$0.246735\pi$
$102$	0	0
$103$	−0.930739	−0.0917084	−0.0458542	−	0.998948i	$-0.514601\pi$
$103$	−0.930739	−0.0917084	−0.0458542	+	0.998948i	$0.514601\pi$
$104$	0	0
$105$	−3.98745	−0.389136
$106$	0	0
$107$	−18.4954	−1.78802	−0.894008	−	0.448051i	$-0.852118\pi$
$107$	−18.4954	−1.78802	−0.894008	+	0.448051i	$0.852118\pi$
$108$	0	0
$109$	5.59426	0.535833	0.267917	−	0.963442i	$-0.413665\pi$
$109$	5.59426	0.535833	0.267917	+	0.963442i	$0.413665\pi$
$110$	0	0
$111$	6.07248	0.576375
$112$	0	0
$113$	−0.687519	−0.0646764	−0.0323382	−	0.999477i	$-0.510295\pi$
$113$	−0.687519	−0.0646764	−0.0323382	+	0.999477i	$0.510295\pi$
$114$	0	0
$115$	13.9493	1.30078
$116$	0	0
$117$	3.57535	0.330541
$118$	0	0
$119$	8.68318	0.795985
$120$	0	0
$121$	13.6928	1.24480
$122$	0	0
$123$	1.00478	0.0905982
$124$	0	0
$125$	8.37822	0.749371
$126$	0	0
$127$	−2.53928	−0.225324	−0.112662	−	0.993633i	$-0.535938\pi$
$127$	−2.53928	−0.225324	−0.112662	+	0.993633i	$0.535938\pi$
$128$	0	0
$129$	−1.75353	−0.154390
$130$	0	0
$131$	3.99950	0.349438	0.174719	−	0.984618i	$-0.444098\pi$
$131$	3.99950	0.349438	0.174719	+	0.984618i	$0.444098\pi$
$132$	0	0
$133$	−11.4972	−0.996931
$134$	0	0
$135$	2.60444	0.224154
$136$	0	0
$137$	−3.88366	−0.331803	−0.165902	−	0.986142i	$-0.553053\pi$
$137$	−3.88366	−0.331803	−0.165902	+	0.986142i	$0.553053\pi$
$138$	0	0
$139$	−11.2261	−0.952182	−0.476091	−	0.879396i	$-0.657947\pi$
$139$	−11.2261	−0.952182	−0.476091	+	0.879396i	$0.657947\pi$
$140$	0	0
$141$	2.20924	0.186052
$142$	0	0
$143$	17.7666	1.48571
$144$	0	0
$145$	−9.55281	−0.793318
$146$	0	0
$147$	4.65597	0.384018
$148$	0	0
$149$	−2.56685	−0.210284	−0.105142	−	0.994457i	$-0.533530\pi$
$149$	−2.56685	−0.210284	−0.105142	+	0.994457i	$0.533530\pi$
$150$	0	0
$151$	14.5577	1.18469	0.592346	−	0.805684i	$-0.298202\pi$
$151$	14.5577	1.18469	0.592346	+	0.805684i	$0.298202\pi$
$152$	0	0
$153$	−5.67149	−0.458513
$154$	0	0
$155$	−11.0193	−0.885089
$156$	0	0
$157$	17.7585	1.41728	0.708641	−	0.705569i	$-0.249309\pi$
$157$	17.7585	1.41728	0.708641	+	0.705569i	$0.249309\pi$
$158$	0	0
$159$	−5.12050	−0.406082
$160$	0	0
$161$	8.20011	0.646260
$162$	0	0
$163$	2.77417	0.217289	0.108645	−	0.994081i	$-0.465349\pi$
$163$	2.77417	0.217289	0.108645	+	0.994081i	$0.465349\pi$
$164$	0	0
$165$	12.9419	1.00753
$166$	0	0
$167$	−1.49278	−0.115515	−0.0577574	−	0.998331i	$-0.518395\pi$
$167$	−1.49278	−0.115515	−0.0577574	+	0.998331i	$0.518395\pi$
$168$	0	0
$169$	−0.216891	−0.0166839
$170$	0	0
$171$	7.50947	0.574264
$172$	0	0
$173$	12.3514	0.939059	0.469529	−	0.882917i	$-0.344424\pi$
$173$	12.3514	0.939059	0.469529	+	0.882917i	$0.344424\pi$
$174$	0	0
$175$	−2.72996	−0.206366
$176$	0	0
$177$	−1.75140	−0.131643
$178$	0	0
$179$	1.98419	0.148305	0.0741527	−	0.997247i	$-0.476375\pi$
$179$	1.98419	0.148305	0.0741527	+	0.997247i	$0.476375\pi$
$180$	0	0
$181$	4.44069	0.330074	0.165037	−	0.986287i	$-0.447226\pi$
$181$	4.44069	0.330074	0.165037	+	0.986287i	$0.447226\pi$
$182$	0	0
$183$	−2.51859	−0.186180
$184$	0	0
$185$	15.8154	1.16277
$186$	0	0
$187$	−28.1827	−2.06092
$188$	0	0
$189$	1.53102	0.111365
$190$	0	0
$191$	6.71869	0.486147	0.243074	−	0.970008i	$-0.421844\pi$
$191$	6.71869	0.486147	0.243074	+	0.970008i	$0.421844\pi$
$192$	0	0
$193$	−13.0665	−0.940547	−0.470273	−	0.882521i	$-0.655845\pi$
$193$	−13.0665	−0.940547	−0.470273	+	0.882521i	$0.655845\pi$
$194$	0	0
$195$	9.31177	0.666830
$196$	0	0
$197$	13.7962	0.982941	0.491470	−	0.870894i	$-0.336460\pi$
$197$	13.7962	0.982941	0.491470	+	0.870894i	$0.336460\pi$
$198$	0	0
$199$	1.87652	0.133023	0.0665113	−	0.997786i	$-0.478813\pi$
$199$	1.87652	0.133023	0.0665113	+	0.997786i	$0.478813\pi$
$200$	0	0
$201$	−12.8570	−0.906864
$202$	0	0
$203$	−5.61563	−0.394140
$204$	0	0
$205$	2.61690	0.182772
$206$	0	0
$207$	−5.35597	−0.372266
$208$	0	0
$209$	37.3160	2.58120
$210$	0	0
$211$	−7.48033	−0.514967	−0.257483	−	0.966283i	$-0.582893\pi$
$211$	−7.48033	−0.514967	−0.257483	+	0.966283i	$0.582893\pi$
$212$	0	0
$213$	7.52624	0.515689
$214$	0	0
$215$	−4.56697	−0.311465
$216$	0	0
$217$	−6.47769	−0.439734
$218$	0	0
$219$	0.588232	0.0397490
$220$	0	0
$221$	−20.2775	−1.36401
$222$	0	0
$223$	24.3790	1.63254	0.816269	−	0.577672i	$-0.196039\pi$
$223$	24.3790	1.63254	0.816269	+	0.577672i	$0.196039\pi$
$224$	0	0
$225$	1.78310	0.118873
$226$	0	0
$227$	−6.35944	−0.422091	−0.211045	−	0.977476i	$-0.567687\pi$
$227$	−6.35944	−0.422091	−0.211045	+	0.977476i	$0.567687\pi$
$228$	0	0
$229$	19.0492	1.25880	0.629402	−	0.777080i	$-0.283300\pi$
$229$	19.0492	1.25880	0.629402	+	0.777080i	$0.283300\pi$
$230$	0	0
$231$	7.60793	0.500565
$232$	0	0
$233$	15.3385	1.00486	0.502430	−	0.864618i	$-0.332440\pi$
$233$	15.3385	1.00486	0.502430	+	0.864618i	$0.332440\pi$
$234$	0	0
$235$	5.75384	0.375339
$236$	0	0
$237$	−3.18145	−0.206657
$238$	0	0
$239$	20.1708	1.30474	0.652369	−	0.757901i	$-0.273775\pi$
$239$	20.1708	1.30474	0.652369	+	0.757901i	$0.273775\pi$
$240$	0	0
$241$	22.8061	1.46907	0.734535	−	0.678571i	$-0.237400\pi$
$241$	22.8061	1.46907	0.734535	+	0.678571i	$0.237400\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	12.1262	0.774714
$246$	0	0
$247$	26.8490	1.70836
$248$	0	0
$249$	1.21414	0.0769430
$250$	0	0
$251$	27.2851	1.72222	0.861111	−	0.508417i	$-0.169769\pi$
$251$	27.2851	1.72222	0.861111	+	0.508417i	$0.169769\pi$
$252$	0	0
$253$	−26.6148	−1.67326
$254$	0	0
$255$	−14.7710	−0.924998
$256$	0	0
$257$	29.0322	1.81098	0.905488	−	0.424372i	$-0.139505\pi$
$257$	29.0322	1.81098	0.905488	+	0.424372i	$0.139505\pi$
$258$	0	0
$259$	9.29711	0.577694
$260$	0	0
$261$	3.66790	0.227037
$262$	0	0
$263$	−3.16211	−0.194984	−0.0974919	−	0.995236i	$-0.531082\pi$
$263$	−3.16211	−0.194984	−0.0974919	+	0.995236i	$0.531082\pi$
$264$	0	0
$265$	−13.3360	−0.819226
$266$	0	0
$267$	13.4526	0.823284
$268$	0	0
$269$	−4.24758	−0.258979	−0.129490	−	0.991581i	$-0.541334\pi$
$269$	−4.24758	−0.258979	−0.129490	+	0.991581i	$0.541334\pi$
$270$	0	0
$271$	12.7020	0.771594	0.385797	−	0.922584i	$-0.373926\pi$
$271$	12.7020	0.771594	0.385797	+	0.922584i	$0.373926\pi$
$272$	0	0
$273$	5.47394	0.331298
$274$	0	0
$275$	8.86054	0.534311
$276$	0	0
$277$	11.3458	0.681702	0.340851	−	0.940117i	$-0.389285\pi$
$277$	11.3458	0.681702	0.340851	+	0.940117i	$0.389285\pi$
$278$	0	0
$279$	4.23096	0.253301
$280$	0	0
$281$	−8.24035	−0.491578	−0.245789	−	0.969323i	$-0.579047\pi$
$281$	−8.24035	−0.491578	−0.245789	+	0.969323i	$0.579047\pi$
$282$	0	0
$283$	16.5964	0.986555	0.493277	−	0.869872i	$-0.335799\pi$
$283$	16.5964	0.986555	0.493277	+	0.869872i	$0.335799\pi$
$284$	0	0
$285$	19.5580	1.15851
$286$	0	0
$287$	1.53835	0.0908056
$288$	0	0
$289$	15.1658	0.892105
$290$	0	0
$291$	16.6053	0.973421
$292$	0	0
$293$	−3.57802	−0.209030	−0.104515	−	0.994523i	$-0.533329\pi$
$293$	−3.57802	−0.209030	−0.104515	+	0.994523i	$0.533329\pi$
$294$	0	0
$295$	−4.56141	−0.265575
$296$	0	0
$297$	−4.96919	−0.288341
$298$	0	0
$299$	−19.1495	−1.10744
$300$	0	0
$301$	−2.68470	−0.154744
$302$	0	0
$303$	−14.3577	−0.824829
$304$	0	0
$305$	−6.55951	−0.375597
$306$	0	0
$307$	19.1089	1.09060	0.545301	−	0.838240i	$-0.316415\pi$
$307$	19.1089	1.09060	0.545301	+	0.838240i	$0.316415\pi$
$308$	0	0
$309$	0.930739	0.0529479
$310$	0	0
$311$	0.227763	0.0129153	0.00645764	−	0.999979i	$-0.497944\pi$
$311$	0.227763	0.0129153	0.00645764	+	0.999979i	$0.497944\pi$
$312$	0	0
$313$	−11.6885	−0.660671	−0.330335	−	0.943864i	$-0.607162\pi$
$313$	−11.6885	−0.660671	−0.330335	+	0.943864i	$0.607162\pi$
$314$	0	0
$315$	3.98745	0.224668
$316$	0	0
$317$	0.785578	0.0441225	0.0220612	−	0.999757i	$-0.492977\pi$
$317$	0.785578	0.0441225	0.0220612	+	0.999757i	$0.492977\pi$
$318$	0	0
$319$	18.2265	1.02049
$320$	0	0
$321$	18.4954	1.03231
$322$	0	0
$323$	−42.5899	−2.36976
$324$	0	0
$325$	6.37519	0.353632
$326$	0	0
$327$	−5.59426	−0.309363
$328$	0	0
$329$	3.38240	0.186478
$330$	0	0
$331$	−14.5158	−0.797861	−0.398931	−	0.916981i	$-0.630619\pi$
$331$	−14.5158	−0.797861	−0.398931	+	0.916981i	$0.630619\pi$
$332$	0	0
$333$	−6.07248	−0.332770
$334$	0	0
$335$	−33.4853	−1.82950
$336$	0	0
$337$	2.87622	0.156678	0.0783389	−	0.996927i	$-0.475038\pi$
$337$	2.87622	0.156678	0.0783389	+	0.996927i	$0.475038\pi$
$338$	0	0
$339$	0.687519	0.0373409
$340$	0	0
$341$	21.0244	1.13854
$342$	0	0
$343$	17.8456	0.963569
$344$	0	0
$345$	−13.9493	−0.751005
$346$	0	0
$347$	−19.4818	−1.04584	−0.522918	−	0.852383i	$-0.675157\pi$
$347$	−19.4818	−1.04584	−0.522918	+	0.852383i	$0.675157\pi$
$348$	0	0
$349$	35.7705	1.91475	0.957376	−	0.288845i	$-0.0932713\pi$
$349$	35.7705	1.91475	0.957376	+	0.288845i	$0.0932713\pi$
$350$	0	0
$351$	−3.57535	−0.190838
$352$	0	0
$353$	28.4320	1.51328	0.756640	−	0.653831i	$-0.226839\pi$
$353$	28.4320	1.51328	0.756640	+	0.653831i	$0.226839\pi$
$354$	0	0
$355$	19.6016	1.04035
$356$	0	0
$357$	−8.68318	−0.459562
$358$	0	0
$359$	23.2239	1.22571	0.612855	−	0.790195i	$-0.290021\pi$
$359$	23.2239	1.22571	0.612855	+	0.790195i	$0.290021\pi$
$360$	0	0
$361$	37.3922	1.96801
$362$	0	0
$363$	−13.6928	−0.718686
$364$	0	0
$365$	1.53201	0.0801893
$366$	0	0
$367$	3.39674	0.177308	0.0886542	−	0.996062i	$-0.471743\pi$
$367$	3.39674	0.177308	0.0886542	+	0.996062i	$0.471743\pi$
$368$	0	0
$369$	−1.00478	−0.0523069
$370$	0	0
$371$	−7.83960	−0.407012
$372$	0	0
$373$	24.1298	1.24939	0.624697	−	0.780867i	$-0.285223\pi$
$373$	24.1298	1.24939	0.624697	+	0.780867i	$0.285223\pi$
$374$	0	0
$375$	−8.37822	−0.432650
$376$	0	0
$377$	13.1140	0.675406
$378$	0	0
$379$	−24.5331	−1.26018	−0.630090	−	0.776522i	$-0.716982\pi$
$379$	−24.5331	−1.26018	−0.630090	+	0.776522i	$0.716982\pi$
$380$	0	0
$381$	2.53928	0.130091
$382$	0	0
$383$	−2.14678	−0.109695	−0.0548477	−	0.998495i	$-0.517467\pi$
$383$	−2.14678	−0.109695	−0.0548477	+	0.998495i	$0.517467\pi$
$384$	0	0
$385$	19.8144	1.00983
$386$	0	0
$387$	1.75353	0.0891371
$388$	0	0
$389$	−17.1885	−0.871493	−0.435746	−	0.900069i	$-0.643516\pi$
$389$	−17.1885	−0.871493	−0.435746	+	0.900069i	$0.643516\pi$
$390$	0	0
$391$	30.3763	1.53620
$392$	0	0
$393$	−3.99950	−0.201748
$394$	0	0
$395$	−8.28588	−0.416908
$396$	0	0
$397$	−19.8842	−0.997961	−0.498981	−	0.866613i	$-0.666292\pi$
$397$	−19.8842	−0.997961	−0.498981	+	0.866613i	$0.666292\pi$
$398$	0	0
$399$	11.4972	0.575578
$400$	0	0
$401$	13.6547	0.681883	0.340941	−	0.940085i	$-0.389254\pi$
$401$	13.6547	0.681883	0.340941	+	0.940085i	$0.389254\pi$
$402$	0	0
$403$	15.1271	0.753537
$404$	0	0
$405$	−2.60444	−0.129416
$406$	0	0
$407$	−30.1753	−1.49573
$408$	0	0
$409$	10.3286	0.510715	0.255358	−	0.966847i	$-0.417807\pi$
$409$	10.3286	0.510715	0.255358	+	0.966847i	$0.417807\pi$
$410$	0	0
$411$	3.88366	0.191567
$412$	0	0
$413$	−2.68143	−0.131945
$414$	0	0
$415$	3.16215	0.155224
$416$	0	0
$417$	11.2261	0.549742
$418$	0	0
$419$	2.17066	0.106044	0.0530219	−	0.998593i	$-0.483115\pi$
$419$	2.17066	0.106044	0.0530219	+	0.998593i	$0.483115\pi$
$420$	0	0
$421$	−3.17225	−0.154606	−0.0773031	−	0.997008i	$-0.524631\pi$
$421$	−3.17225	−0.154606	−0.0773031	+	0.997008i	$0.524631\pi$
$422$	0	0
$423$	−2.20924	−0.107417
$424$	0	0
$425$	−10.1128	−0.490544
$426$	0	0
$427$	−3.85602	−0.186606
$428$	0	0
$429$	−17.7666	−0.857778
$430$	0	0
$431$	−9.12452	−0.439512	−0.219756	−	0.975555i	$-0.570526\pi$
$431$	−9.12452	−0.439512	−0.219756	+	0.975555i	$0.570526\pi$
$432$	0	0
$433$	7.17758	0.344933	0.172466	−	0.985015i	$-0.444826\pi$
$433$	7.17758	0.344933	0.172466	+	0.985015i	$0.444826\pi$
$434$	0	0
$435$	9.55281	0.458022
$436$	0	0
$437$	−40.2205	−1.92401
$438$	0	0
$439$	−31.6570	−1.51091	−0.755453	−	0.655202i	$-0.772583\pi$
$439$	−31.6570	−1.51091	−0.755453	+	0.655202i	$0.772583\pi$
$440$	0	0
$441$	−4.65597	−0.221713
$442$	0	0
$443$	−36.1949	−1.71967	−0.859837	−	0.510569i	$-0.829435\pi$
$443$	−36.1949	−1.71967	−0.859837	+	0.510569i	$0.829435\pi$
$444$	0	0
$445$	35.0364	1.66088
$446$	0	0
$447$	2.56685	0.121408
$448$	0	0
$449$	−20.4497	−0.965082	−0.482541	−	0.875873i	$-0.660286\pi$
$449$	−20.4497	−0.965082	−0.482541	+	0.875873i	$0.660286\pi$
$450$	0	0
$451$	−4.99295	−0.235109
$452$	0	0
$453$	−14.5577	−0.683982
$454$	0	0
$455$	14.2565	0.668356
$456$	0	0
$457$	−4.52653	−0.211742	−0.105871	−	0.994380i	$-0.533763\pi$
$457$	−4.52653	−0.211742	−0.105871	+	0.994380i	$0.533763\pi$
$458$	0	0
$459$	5.67149	0.264722
$460$	0	0
$461$	−5.65304	−0.263288	−0.131644	−	0.991297i	$-0.542026\pi$
$461$	−5.65304	−0.263288	−0.131644	+	0.991297i	$0.542026\pi$
$462$	0	0
$463$	12.5351	0.582555	0.291277	−	0.956639i	$-0.405920\pi$
$463$	12.5351	0.582555	0.291277	+	0.956639i	$0.405920\pi$
$464$	0	0
$465$	11.0193	0.511006
$466$	0	0
$467$	0.431350	0.0199605	0.00998024	−	0.999950i	$-0.496823\pi$
$467$	0.431350	0.0199605	0.00998024	+	0.999950i	$0.496823\pi$
$468$	0	0
$469$	−19.6844	−0.908941
$470$	0	0
$471$	−17.7585	−0.818268
$472$	0	0
$473$	8.71363	0.400653
$474$	0	0
$475$	13.3901	0.614381
$476$	0	0
$477$	5.12050	0.234452
$478$	0	0
$479$	−23.5609	−1.07652	−0.538262	−	0.842778i	$-0.680919\pi$
$479$	−23.5609	−1.07652	−0.538262	+	0.842778i	$0.680919\pi$
$480$	0	0
$481$	−21.7112	−0.989947
$482$	0	0
$483$	−8.20011	−0.373118
$484$	0	0
$485$	43.2475	1.96377
$486$	0	0
$487$	−19.9979	−0.906191	−0.453096	−	0.891462i	$-0.649680\pi$
$487$	−19.9979	−0.906191	−0.453096	+	0.891462i	$0.649680\pi$
$488$	0	0
$489$	−2.77417	−0.125452
$490$	0	0
$491$	−24.9543	−1.12617	−0.563086	−	0.826398i	$-0.690386\pi$
$491$	−24.9543	−1.12617	−0.563086	+	0.826398i	$0.690386\pi$
$492$	0	0
$493$	−20.8024	−0.936895
$494$	0	0
$495$	−12.9419	−0.581697
$496$	0	0
$497$	11.5228	0.516870
$498$	0	0
$499$	28.7102	1.28525	0.642623	−	0.766182i	$-0.277846\pi$
$499$	28.7102	1.28525	0.642623	+	0.766182i	$0.277846\pi$
$500$	0	0
$501$	1.49278	0.0666925
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−37.3938	−1.66400
$506$	0	0
$507$	0.216891	0.00963247
$508$	0	0
$509$	40.9096	1.81329	0.906643	−	0.421899i	$-0.138636\pi$
$509$	40.9096	1.81329	0.906643	+	0.421899i	$0.138636\pi$
$510$	0	0
$511$	0.900596	0.0398400
$512$	0	0
$513$	−7.50947	−0.331551
$514$	0	0
$515$	2.42405	0.106816
$516$	0	0
$517$	−10.9781	−0.482818
$518$	0	0
$519$	−12.3514	−0.542166
$520$	0	0
$521$	−36.0090	−1.57758	−0.788792	−	0.614660i	$-0.789293\pi$
$521$	−36.0090	−1.57758	−0.788792	+	0.614660i	$0.789293\pi$
$522$	0	0
$523$	19.0800	0.834310	0.417155	−	0.908835i	$-0.363027\pi$
$523$	19.0800	0.834310	0.417155	+	0.908835i	$0.363027\pi$
$524$	0	0
$525$	2.72996	0.119145
$526$	0	0
$527$	−23.9958	−1.04527
$528$	0	0
$529$	5.68644	0.247237
$530$	0	0
$531$	1.75140	0.0760042
$532$	0	0
$533$	−3.59245	−0.155606
$534$	0	0
$535$	48.1701	2.08257
$536$	0	0
$537$	−1.98419	−0.0856242
$538$	0	0
$539$	−23.1364	−0.996554
$540$	0	0
$541$	16.2772	0.699810	0.349905	−	0.936785i	$-0.386214\pi$
$541$	16.2772	0.699810	0.349905	+	0.936785i	$0.386214\pi$
$542$	0	0
$543$	−4.44069	−0.190568
$544$	0	0
$545$	−14.5699	−0.624106
$546$	0	0
$547$	5.83056	0.249297	0.124648	−	0.992201i	$-0.460220\pi$
$547$	5.83056	0.249297	0.124648	+	0.992201i	$0.460220\pi$
$548$	0	0
$549$	2.51859	0.107491
$550$	0	0
$551$	27.5440	1.17341
$552$	0	0
$553$	−4.87087	−0.207130
$554$	0	0
$555$	−15.8154	−0.671327
$556$	0	0
$557$	30.8077	1.30536	0.652681	−	0.757633i	$-0.273644\pi$
$557$	30.8077	1.30536	0.652681	+	0.757633i	$0.273644\pi$
$558$	0	0
$559$	6.26949	0.265171
$560$	0	0
$561$	28.1827	1.18987
$562$	0	0
$563$	−18.7838	−0.791641	−0.395820	−	0.918328i	$-0.629540\pi$
$563$	−18.7838	−0.791641	−0.395820	+	0.918328i	$0.629540\pi$
$564$	0	0
$565$	1.79060	0.0753312
$566$	0	0
$567$	−1.53102	−0.0642969
$568$	0	0
$569$	16.9298	0.709732	0.354866	−	0.934917i	$-0.384526\pi$
$569$	16.9298	0.709732	0.354866	+	0.934917i	$0.384526\pi$
$570$	0	0
$571$	10.7282	0.448962	0.224481	−	0.974478i	$-0.427931\pi$
$571$	10.7282	0.448962	0.224481	+	0.974478i	$0.427931\pi$
$572$	0	0
$573$	−6.71869	−0.280677
$574$	0	0
$575$	−9.55022	−0.398272
$576$	0	0
$577$	−13.8899	−0.578244	−0.289122	−	0.957292i	$-0.593363\pi$
$577$	−13.8899	−0.578244	−0.289122	+	0.957292i	$0.593363\pi$
$578$	0	0
$579$	13.0665	0.543025
$580$	0	0
$581$	1.85888	0.0771191
$582$	0	0
$583$	25.4447	1.05381
$584$	0	0
$585$	−9.31177	−0.384994
$586$	0	0
$587$	−42.5350	−1.75561	−0.877804	−	0.479019i	$-0.840992\pi$
$587$	−42.5350	−1.75561	−0.877804	+	0.479019i	$0.840992\pi$
$588$	0	0
$589$	31.7723	1.30915
$590$	0	0
$591$	−13.7962	−0.567501
$592$	0	0
$593$	16.5124	0.678082	0.339041	−	0.940772i	$-0.389897\pi$
$593$	16.5124	0.678082	0.339041	+	0.940772i	$0.389897\pi$
$594$	0	0
$595$	−22.6148	−0.927116
$596$	0	0
$597$	−1.87652	−0.0768007
$598$	0	0
$599$	−33.9060	−1.38536	−0.692682	−	0.721243i	$-0.743571\pi$
$599$	−33.9060	−1.38536	−0.692682	+	0.721243i	$0.743571\pi$
$600$	0	0
$601$	27.0745	1.10439	0.552196	−	0.833714i	$-0.313790\pi$
$601$	27.0745	1.10439	0.552196	+	0.833714i	$0.313790\pi$
$602$	0	0
$603$	12.8570	0.523578
$604$	0	0
$605$	−35.6621	−1.44987
$606$	0	0
$607$	−36.1212	−1.46611	−0.733057	−	0.680168i	$-0.761907\pi$
$607$	−36.1212	−1.46611	−0.733057	+	0.680168i	$0.761907\pi$
$608$	0	0
$609$	5.61563	0.227557
$610$	0	0
$611$	−7.89881	−0.319552
$612$	0	0
$613$	−12.4341	−0.502209	−0.251105	−	0.967960i	$-0.580794\pi$
$613$	−12.4341	−0.502209	−0.251105	+	0.967960i	$0.580794\pi$
$614$	0	0
$615$	−2.61690	−0.105523
$616$	0	0
$617$	−24.9748	−1.00545	−0.502723	−	0.864447i	$-0.667669\pi$
$617$	−24.9748	−1.00545	−0.502723	+	0.864447i	$0.667669\pi$
$618$	0	0
$619$	2.49189	0.100157	0.0500787	−	0.998745i	$-0.484053\pi$
$619$	2.49189	0.100157	0.0500787	+	0.998745i	$0.484053\pi$
$620$	0	0
$621$	5.35597	0.214928
$622$	0	0
$623$	20.5962	0.825168
$624$	0	0
$625$	−30.7361	−1.22944
$626$	0	0
$627$	−37.3160	−1.49026
$628$	0	0
$629$	34.4400	1.37321
$630$	0	0
$631$	34.8386	1.38690	0.693451	−	0.720504i	$-0.256089\pi$
$631$	34.8386	1.38690	0.693451	+	0.720504i	$0.256089\pi$
$632$	0	0
$633$	7.48033	0.297316
$634$	0	0
$635$	6.61339	0.262444
$636$	0	0
$637$	−16.6467	−0.659567
$638$	0	0
$639$	−7.52624	−0.297733
$640$	0	0
$641$	−27.1526	−1.07247	−0.536233	−	0.844070i	$-0.680153\pi$
$641$	−27.1526	−1.07247	−0.536233	+	0.844070i	$0.680153\pi$
$642$	0	0
$643$	12.7032	0.500965	0.250483	−	0.968121i	$-0.419411\pi$
$643$	12.7032	0.500965	0.250483	+	0.968121i	$0.419411\pi$
$644$	0	0
$645$	4.56697	0.179824
$646$	0	0
$647$	20.4500	0.803974	0.401987	−	0.915645i	$-0.368320\pi$
$647$	20.4500	0.803974	0.401987	+	0.915645i	$0.368320\pi$
$648$	0	0
$649$	8.70302	0.341624
$650$	0	0
$651$	6.47769	0.253881
$652$	0	0
$653$	−10.0442	−0.393061	−0.196530	−	0.980498i	$-0.562967\pi$
$653$	−10.0442	−0.393061	−0.196530	+	0.980498i	$0.562967\pi$
$654$	0	0
$655$	−10.4165	−0.407005
$656$	0	0
$657$	−0.588232	−0.0229491
$658$	0	0
$659$	−41.2908	−1.60846	−0.804231	−	0.594317i	$-0.797422\pi$
$659$	−41.2908	−1.60846	−0.804231	+	0.594317i	$0.797422\pi$
$660$	0	0
$661$	2.52344	0.0981504	0.0490752	−	0.998795i	$-0.484373\pi$
$661$	2.52344	0.0981504	0.0490752	+	0.998795i	$0.484373\pi$
$662$	0	0
$663$	20.2775	0.787514
$664$	0	0
$665$	29.9437	1.16117
$666$	0	0
$667$	−19.6452	−0.760664
$668$	0	0
$669$	−24.3790	−0.942547
$670$	0	0
$671$	12.5153	0.483149
$672$	0	0
$673$	−16.8799	−0.650670	−0.325335	−	0.945599i	$-0.605477\pi$
$673$	−16.8799	−0.650670	−0.325335	+	0.945599i	$0.605477\pi$
$674$	0	0
$675$	−1.78310	−0.0686315
$676$	0	0
$677$	18.7503	0.720632	0.360316	−	0.932830i	$-0.382669\pi$
$677$	18.7503	0.720632	0.360316	+	0.932830i	$0.382669\pi$
$678$	0	0
$679$	25.4231	0.975649
$680$	0	0
$681$	6.35944	0.243694
$682$	0	0
$683$	47.8176	1.82969	0.914844	−	0.403807i	$-0.132313\pi$
$683$	47.8176	1.82969	0.914844	+	0.403807i	$0.132313\pi$
$684$	0	0
$685$	10.1147	0.386465
$686$	0	0
$687$	−19.0492	−0.726771
$688$	0	0
$689$	18.3076	0.697463
$690$	0	0
$691$	20.8495	0.793151	0.396575	−	0.918002i	$-0.370199\pi$
$691$	20.8495	0.793151	0.396575	+	0.918002i	$0.370199\pi$
$692$	0	0
$693$	−7.60793	−0.289002
$694$	0	0
$695$	29.2376	1.10904
$696$	0	0
$697$	5.69861	0.215850
$698$	0	0
$699$	−15.3385	−0.580156
$700$	0	0
$701$	29.7457	1.12348	0.561741	−	0.827313i	$-0.310132\pi$
$701$	29.7457	1.12348	0.561741	+	0.827313i	$0.310132\pi$
$702$	0	0
$703$	−45.6012	−1.71988
$704$	0	0
$705$	−5.75384	−0.216702
$706$	0	0
$707$	−21.9820	−0.826717
$708$	0	0
$709$	33.9443	1.27480	0.637402	−	0.770532i	$-0.280009\pi$
$709$	33.9443	1.27480	0.637402	+	0.770532i	$0.280009\pi$
$710$	0	0
$711$	3.18145	0.119314
$712$	0	0
$713$	−22.6609	−0.848657
$714$	0	0
$715$	−46.2719	−1.73047
$716$	0	0
$717$	−20.1708	−0.753291
$718$	0	0
$719$	−7.94929	−0.296459	−0.148229	−	0.988953i	$-0.547357\pi$
$719$	−7.94929	−0.296459	−0.148229	+	0.988953i	$0.547357\pi$
$720$	0	0
$721$	1.42498	0.0530691
$722$	0	0
$723$	−22.8061	−0.848168
$724$	0	0
$725$	6.54022	0.242898
$726$	0	0
$727$	−39.6257	−1.46964	−0.734818	−	0.678264i	$-0.762733\pi$
$727$	−39.6257	−1.46964	−0.734818	+	0.678264i	$0.762733\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.94515	−0.367835
$732$	0	0
$733$	35.3545	1.30585	0.652923	−	0.757424i	$-0.273542\pi$
$733$	35.3545	1.30585	0.652923	+	0.757424i	$0.273542\pi$
$734$	0	0
$735$	−12.1262	−0.447281
$736$	0	0
$737$	63.8889	2.35338
$738$	0	0
$739$	31.1775	1.14688	0.573441	−	0.819247i	$-0.305608\pi$
$739$	31.1775	1.14688	0.573441	+	0.819247i	$0.305608\pi$
$740$	0	0
$741$	−26.8490	−0.986322
$742$	0	0
$743$	−6.53170	−0.239625	−0.119812	−	0.992797i	$-0.538229\pi$
$743$	−6.53170	−0.239625	−0.119812	+	0.992797i	$0.538229\pi$
$744$	0	0
$745$	6.68520	0.244927
$746$	0	0
$747$	−1.21414	−0.0444230
$748$	0	0
$749$	28.3168	1.03468
$750$	0	0
$751$	47.7077	1.74088	0.870439	−	0.492277i	$-0.163835\pi$
$751$	47.7077	1.74088	0.870439	+	0.492277i	$0.163835\pi$
$752$	0	0
$753$	−27.2851	−0.994325
$754$	0	0
$755$	−37.9147	−1.37986
$756$	0	0
$757$	−47.8335	−1.73854	−0.869269	−	0.494339i	$-0.835410\pi$
$757$	−47.8335	−1.73854	−0.869269	+	0.494339i	$0.835410\pi$
$758$	0	0
$759$	26.6148	0.966057
$760$	0	0
$761$	−0.784035	−0.0284212	−0.0142106	−	0.999899i	$-0.504524\pi$
$761$	−0.784035	−0.0284212	−0.0142106	+	0.999899i	$0.504524\pi$
$762$	0	0
$763$	−8.56494	−0.310072
$764$	0	0
$765$	14.7710	0.534048
$766$	0	0
$767$	6.26186	0.226103
$768$	0	0
$769$	42.1434	1.51973	0.759864	−	0.650082i	$-0.225265\pi$
$769$	42.1434	1.51973	0.759864	+	0.650082i	$0.225265\pi$
$770$	0	0
$771$	−29.0322	−1.04557
$772$	0	0
$773$	−13.2457	−0.476413	−0.238207	−	0.971214i	$-0.576560\pi$
$773$	−13.2457	−0.476413	−0.238207	+	0.971214i	$0.576560\pi$
$774$	0	0
$775$	7.54421	0.270996
$776$	0	0
$777$	−9.29711	−0.333532
$778$	0	0
$779$	−7.54539	−0.270342
$780$	0	0
$781$	−37.3993	−1.33825
$782$	0	0
$783$	−3.66790	−0.131080
$784$	0	0
$785$	−46.2509	−1.65077
$786$	0	0
$787$	31.5151	1.12339	0.561696	−	0.827343i	$-0.310149\pi$
$787$	31.5151	1.12339	0.561696	+	0.827343i	$0.310149\pi$
$788$	0	0
$789$	3.16211	0.112574
$790$	0	0
$791$	1.05261	0.0374264
$792$	0	0
$793$	9.00483	0.319771
$794$	0	0
$795$	13.3360	0.472980
$796$	0	0
$797$	6.25486	0.221559	0.110779	−	0.993845i	$-0.464665\pi$
$797$	6.25486	0.221559	0.110779	+	0.993845i	$0.464665\pi$
$798$	0	0
$799$	12.5297	0.443269
$800$	0	0
$801$	−13.4526	−0.475323
$802$	0	0
$803$	−2.92303	−0.103152
$804$	0	0
$805$	−21.3567	−0.752724
$806$	0	0
$807$	4.24758	0.149522
$808$	0	0
$809$	2.05770	0.0723448	0.0361724	−	0.999346i	$-0.488483\pi$
$809$	2.05770	0.0723448	0.0361724	+	0.999346i	$0.488483\pi$
$810$	0	0
$811$	4.69233	0.164770	0.0823851	−	0.996601i	$-0.473746\pi$
$811$	4.69233	0.164770	0.0823851	+	0.996601i	$0.473746\pi$
$812$	0	0
$813$	−12.7020	−0.445480
$814$	0	0
$815$	−7.22514	−0.253086
$816$	0	0
$817$	13.1681	0.460694
$818$	0	0
$819$	−5.47394	−0.191275
$820$	0	0
$821$	−11.4084	−0.398157	−0.199078	−	0.979984i	$-0.563795\pi$
$821$	−11.4084	−0.398157	−0.199078	+	0.979984i	$0.563795\pi$
$822$	0	0
$823$	37.7270	1.31508	0.657540	−	0.753420i	$-0.271597\pi$
$823$	37.7270	1.31508	0.657540	+	0.753420i	$0.271597\pi$
$824$	0	0
$825$	−8.86054	−0.308485
$826$	0	0
$827$	32.1800	1.11901	0.559504	−	0.828827i	$-0.310991\pi$
$827$	32.1800	1.11901	0.559504	+	0.828827i	$0.310991\pi$
$828$	0	0
$829$	−52.4790	−1.82267	−0.911336	−	0.411664i	$-0.864948\pi$
$829$	−52.4790	−1.82267	−0.911336	+	0.411664i	$0.864948\pi$
$830$	0	0
$831$	−11.3458	−0.393581
$832$	0	0
$833$	26.4063	0.914923
$834$	0	0
$835$	3.88785	0.134545
$836$	0	0
$837$	−4.23096	−0.146243
$838$	0	0
$839$	−11.8577	−0.409372	−0.204686	−	0.978828i	$-0.565617\pi$
$839$	−11.8577	−0.409372	−0.204686	+	0.978828i	$0.565617\pi$
$840$	0	0
$841$	−15.5465	−0.536087
$842$	0	0
$843$	8.24035	0.283813
$844$	0	0
$845$	0.564880	0.0194324
$846$	0	0
$847$	−20.9640	−0.720331
$848$	0	0
$849$	−16.5964	−0.569588
$850$	0	0
$851$	32.5241	1.11491
$852$	0	0
$853$	24.6461	0.843868	0.421934	−	0.906627i	$-0.361351\pi$
$853$	24.6461	0.843868	0.421934	+	0.906627i	$0.361351\pi$
$854$	0	0
$855$	−19.5580	−0.668868
$856$	0	0
$857$	24.2410	0.828057	0.414028	−	0.910264i	$-0.364121\pi$
$857$	24.2410	0.828057	0.414028	+	0.910264i	$0.364121\pi$
$858$	0	0
$859$	−17.8246	−0.608166	−0.304083	−	0.952646i	$-0.598350\pi$
$859$	−17.8246	−0.608166	−0.304083	+	0.952646i	$0.598350\pi$
$860$	0	0
$861$	−1.53835	−0.0524267
$862$	0	0
$863$	−29.9302	−1.01883	−0.509417	−	0.860520i	$-0.670139\pi$
$863$	−29.9302	−1.01883	−0.509417	+	0.860520i	$0.670139\pi$
$864$	0	0
$865$	−32.1684	−1.09376
$866$	0	0
$867$	−15.1658	−0.515057
$868$	0	0
$869$	15.8092	0.536290
$870$	0	0
$871$	45.9683	1.55758
$872$	0	0
$873$	−16.6053	−0.562005
$874$	0	0
$875$	−12.8272	−0.433640
$876$	0	0
$877$	38.9181	1.31417	0.657085	−	0.753816i	$-0.271789\pi$
$877$	38.9181	1.31417	0.657085	+	0.753816i	$0.271789\pi$
$878$	0	0
$879$	3.57802	0.120684
$880$	0	0
$881$	10.2233	0.344431	0.172216	−	0.985059i	$-0.444907\pi$
$881$	10.2233	0.344431	0.172216	+	0.985059i	$0.444907\pi$
$882$	0	0
$883$	−0.0661155	−0.00222497	−0.00111248	−	0.999999i	$-0.500354\pi$
$883$	−0.0661155	−0.00222497	−0.00111248	+	0.999999i	$0.500354\pi$
$884$	0	0
$885$	4.56141	0.153330
$886$	0	0
$887$	−13.5336	−0.454412	−0.227206	−	0.973847i	$-0.572959\pi$
$887$	−13.5336	−0.454412	−0.227206	+	0.973847i	$0.572959\pi$
$888$	0	0
$889$	3.88769	0.130389
$890$	0	0
$891$	4.96919	0.166474
$892$	0	0
$893$	−16.5902	−0.555171
$894$	0	0
$895$	−5.16770	−0.172737
$896$	0	0
$897$	19.1495	0.639382
$898$	0	0
$899$	15.5187	0.517578
$900$	0	0
$901$	−29.0409	−0.967492
$902$	0	0
$903$	2.68470	0.0893412
$904$	0	0
$905$	−11.5655	−0.384450
$906$	0	0
$907$	23.1097	0.767345	0.383672	−	0.923469i	$-0.374659\pi$
$907$	23.1097	0.767345	0.383672	+	0.923469i	$0.374659\pi$
$908$	0	0
$909$	14.3577	0.476215
$910$	0	0
$911$	33.0818	1.09605	0.548024	−	0.836463i	$-0.315380\pi$
$911$	33.0818	1.09605	0.548024	+	0.836463i	$0.315380\pi$
$912$	0	0
$913$	−6.03329	−0.199673
$914$	0	0
$915$	6.55951	0.216851
$916$	0	0
$917$	−6.12333	−0.202210
$918$	0	0
$919$	−40.3561	−1.33122	−0.665612	−	0.746298i	$-0.731830\pi$
$919$	−40.3561	−1.33122	−0.665612	+	0.746298i	$0.731830\pi$
$920$	0	0
$921$	−19.1089	−0.629660
$922$	0	0
$923$	−26.9089	−0.885718
$924$	0	0
$925$	−10.8278	−0.356017
$926$	0	0
$927$	−0.930739	−0.0305695
$928$	0	0
$929$	12.2446	0.401732	0.200866	−	0.979619i	$-0.435624\pi$
$929$	12.2446	0.401732	0.200866	+	0.979619i	$0.435624\pi$
$930$	0	0
$931$	−34.9639	−1.14590
$932$	0	0
$933$	−0.227763	−0.00745664
$934$	0	0
$935$	73.4000	2.40044
$936$	0	0
$937$	22.6841	0.741058	0.370529	−	0.928821i	$-0.379176\pi$
$937$	22.6841	0.741058	0.370529	+	0.928821i	$0.379176\pi$
$938$	0	0
$939$	11.6885	0.381438
$940$	0	0
$941$	29.4377	0.959640	0.479820	−	0.877367i	$-0.340702\pi$
$941$	29.4377	0.959640	0.479820	+	0.877367i	$0.340702\pi$
$942$	0	0
$943$	5.38159	0.175249
$944$	0	0
$945$	−3.98745	−0.129712
$946$	0	0
$947$	15.6199	0.507580	0.253790	−	0.967259i	$-0.418323\pi$
$947$	15.6199	0.507580	0.253790	+	0.967259i	$0.418323\pi$
$948$	0	0
$949$	−2.10313	−0.0682706
$950$	0	0
$951$	−0.785578	−0.0254741
$952$	0	0
$953$	−23.0916	−0.748011	−0.374005	−	0.927427i	$-0.622016\pi$
$953$	−23.0916	−0.748011	−0.374005	+	0.927427i	$0.622016\pi$
$954$	0	0
$955$	−17.4984	−0.566235
$956$	0	0
$957$	−18.2265	−0.589178
$958$	0	0
$959$	5.94597	0.192005
$960$	0	0
$961$	−13.0990	−0.422548
$962$	0	0
$963$	−18.4954	−0.596005
$964$	0	0
$965$	34.0309	1.09549
$966$	0	0
$967$	−34.2961	−1.10289	−0.551444	−	0.834212i	$-0.685923\pi$
$967$	−34.2961	−1.10289	−0.551444	+	0.834212i	$0.685923\pi$
$968$	0	0
$969$	42.5899	1.36818
$970$	0	0
$971$	34.7796	1.11613	0.558065	−	0.829797i	$-0.311544\pi$
$971$	34.7796	1.11613	0.558065	+	0.829797i	$0.311544\pi$
$972$	0	0
$973$	17.1873	0.551001
$974$	0	0
$975$	−6.37519	−0.204170
$976$	0	0
$977$	40.6246	1.29970	0.649848	−	0.760065i	$-0.274833\pi$
$977$	40.6246	1.29970	0.649848	+	0.760065i	$0.274833\pi$
$978$	0	0
$979$	−66.8483	−2.13648
$980$	0	0
$981$	5.59426	0.178611
$982$	0	0
$983$	60.5150	1.93013	0.965064	−	0.262013i	$-0.0843864\pi$
$983$	60.5150	1.93013	0.965064	+	0.262013i	$0.0843864\pi$
$984$	0	0
$985$	−35.9314	−1.14487
$986$	0	0
$987$	−3.38240	−0.107663
$988$	0	0
$989$	−9.39188	−0.298644
$990$	0	0
$991$	−27.6457	−0.878193	−0.439097	−	0.898440i	$-0.644701\pi$
$991$	−27.6457	−0.878193	−0.439097	+	0.898440i	$0.644701\pi$
$992$	0	0
$993$	14.5158	0.460645
$994$	0	0
$995$	−4.88727	−0.154937
$996$	0	0
$997$	52.5392	1.66393	0.831966	−	0.554826i	$-0.187215\pi$
$997$	52.5392	1.66393	0.831966	+	0.554826i	$0.187215\pi$
$998$	0	0
$999$	6.07248	0.192125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.5	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.60444 q^{5} -1.53102 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.60444 q^{5} -1.53102 q^{7} +1.00000 q^{9} +4.96919 q^{11} +3.57535 q^{13} +2.60444 q^{15} -5.67149 q^{17} +7.50947 q^{19} +1.53102 q^{21} -5.35597 q^{23} +1.78310 q^{25} -1.00000 q^{27} +3.66790 q^{29} +4.23096 q^{31} -4.96919 q^{33} +3.98745 q^{35} -6.07248 q^{37} -3.57535 q^{39} -1.00478 q^{41} +1.75353 q^{43} -2.60444 q^{45} -2.20924 q^{47} -4.65597 q^{49} +5.67149 q^{51} +5.12050 q^{53} -12.9419 q^{55} -7.50947 q^{57} +1.75140 q^{59} +2.51859 q^{61} -1.53102 q^{63} -9.31177 q^{65} +12.8570 q^{67} +5.35597 q^{69} -7.52624 q^{71} -0.588232 q^{73} -1.78310 q^{75} -7.60793 q^{77} +3.18145 q^{79} +1.00000 q^{81} -1.21414 q^{83} +14.7710 q^{85} -3.66790 q^{87} -13.4526 q^{89} -5.47394 q^{91} -4.23096 q^{93} -19.5580 q^{95} -16.6053 q^{97} +4.96919 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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