LMFDB - Embedded newform 4024.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52320 q^{3} -3.79071 q^{5} +1.89288 q^{7} +3.36655 q^{9} +O(q^{10})$	$q-2.52320 q^{3} -3.79071 q^{5} +1.89288 q^{7} +3.36655 q^{9} -2.21680 q^{11} -2.48038 q^{13} +9.56473 q^{15} -5.79626 q^{17} -7.56137 q^{19} -4.77611 q^{21} -1.14346 q^{23} +9.36947 q^{25} -0.924880 q^{27} +9.03614 q^{29} -3.50125 q^{31} +5.59344 q^{33} -7.17534 q^{35} -9.65722 q^{37} +6.25850 q^{39} -1.45354 q^{41} -10.2026 q^{43} -12.7616 q^{45} -3.76096 q^{47} -3.41702 q^{49} +14.6251 q^{51} -4.76389 q^{53} +8.40326 q^{55} +19.0789 q^{57} -11.6881 q^{59} -12.2538 q^{61} +6.37246 q^{63} +9.40240 q^{65} +0.920539 q^{67} +2.88518 q^{69} +6.29985 q^{71} -13.9966 q^{73} -23.6411 q^{75} -4.19613 q^{77} -5.70551 q^{79} -7.76599 q^{81} +11.5458 q^{83} +21.9719 q^{85} -22.8000 q^{87} +16.3195 q^{89} -4.69505 q^{91} +8.83435 q^{93} +28.6630 q^{95} -17.0014 q^{97} -7.46298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	−2.52320	−1.45677	−0.728386	−	0.685167i	$-0.759729\pi$
$3$	−2.52320	−1.45677	−0.728386	+	0.685167i	$0.759729\pi$
$4$	0	0
$5$	−3.79071	−1.69526	−0.847628	−	0.530591i	$-0.821970\pi$
$5$	−3.79071	−1.69526	−0.847628	+	0.530591i	$0.821970\pi$
$6$	0	0
$7$	1.89288	0.715440	0.357720	−	0.933829i	$-0.383554\pi$
$7$	1.89288	0.715440	0.357720	+	0.933829i	$0.383554\pi$
$8$	0	0
$9$	3.36655	1.12218
$10$	0	0
$11$	−2.21680	−0.668391	−0.334196	−	0.942504i	$-0.608465\pi$
$11$	−2.21680	−0.668391	−0.334196	+	0.942504i	$0.608465\pi$
$12$	0	0
$13$	−2.48038	−0.687934	−0.343967	−	0.938982i	$-0.611771\pi$
$13$	−2.48038	−0.687934	−0.343967	+	0.938982i	$0.611771\pi$
$14$	0	0
$15$	9.56473	2.46960
$16$	0	0
$17$	−5.79626	−1.40580	−0.702900	−	0.711289i	$-0.748112\pi$
$17$	−5.79626	−1.40580	−0.702900	+	0.711289i	$0.748112\pi$
$18$	0	0
$19$	−7.56137	−1.73470	−0.867349	−	0.497701i	$-0.834178\pi$
$19$	−7.56137	−1.73470	−0.867349	+	0.497701i	$0.834178\pi$
$20$	0	0
$21$	−4.77611	−1.04223
$22$	0	0
$23$	−1.14346	−0.238428	−0.119214	−	0.992869i	$-0.538037\pi$
$23$	−1.14346	−0.238428	−0.119214	+	0.992869i	$0.538037\pi$
$24$	0	0
$25$	9.36947	1.87389
$26$	0	0
$27$	−0.924880	−0.177993
$28$	0	0
$29$	9.03614	1.67797	0.838985	−	0.544155i	$-0.183150\pi$
$29$	9.03614	1.67797	0.838985	+	0.544155i	$0.183150\pi$
$30$	0	0
$31$	−3.50125	−0.628842	−0.314421	−	0.949284i	$-0.601810\pi$
$31$	−3.50125	−0.628842	−0.314421	+	0.949284i	$0.601810\pi$
$32$	0	0
$33$	5.59344	0.973694
$34$	0	0
$35$	−7.17534	−1.21285
$36$	0	0
$37$	−9.65722	−1.58764	−0.793818	−	0.608155i	$-0.791910\pi$
$37$	−9.65722	−1.58764	−0.793818	+	0.608155i	$0.791910\pi$
$38$	0	0
$39$	6.25850	1.00216
$40$	0	0
$41$	−1.45354	−0.227005	−0.113503	−	0.993538i	$-0.536207\pi$
$41$	−1.45354	−0.227005	−0.113503	+	0.993538i	$0.536207\pi$
$42$	0	0
$43$	−10.2026	−1.55589	−0.777943	−	0.628335i	$-0.783737\pi$
$43$	−10.2026	−1.55589	−0.777943	+	0.628335i	$0.783737\pi$
$44$	0	0
$45$	−12.7616	−1.90239
$46$	0	0
$47$	−3.76096	−0.548593	−0.274296	−	0.961645i	$-0.588445\pi$
$47$	−3.76096	−0.548593	−0.274296	+	0.961645i	$0.588445\pi$
$48$	0	0
$49$	−3.41702	−0.488146
$50$	0	0
$51$	14.6251	2.04793
$52$	0	0
$53$	−4.76389	−0.654371	−0.327186	−	0.944960i	$-0.606100\pi$
$53$	−4.76389	−0.654371	−0.327186	+	0.944960i	$0.606100\pi$
$54$	0	0
$55$	8.40326	1.13309
$56$	0	0
$57$	19.0789	2.52706
$58$	0	0
$59$	−11.6881	−1.52166	−0.760831	−	0.648951i	$-0.775208\pi$
$59$	−11.6881	−1.52166	−0.760831	+	0.648951i	$0.775208\pi$
$60$	0	0
$61$	−12.2538	−1.56894	−0.784470	−	0.620167i	$-0.787065\pi$
$61$	−12.2538	−1.56894	−0.784470	+	0.620167i	$0.787065\pi$
$62$	0	0
$63$	6.37246	0.802855
$64$	0	0
$65$	9.40240	1.16622
$66$	0	0
$67$	0.920539	0.112462	0.0562309	−	0.998418i	$-0.482092\pi$
$67$	0.920539	0.112462	0.0562309	+	0.998418i	$0.482092\pi$
$68$	0	0
$69$	2.88518	0.347334
$70$	0	0
$71$	6.29985	0.747655	0.373827	−	0.927498i	$-0.378045\pi$
$71$	6.29985	0.747655	0.373827	+	0.927498i	$0.378045\pi$
$72$	0	0
$73$	−13.9966	−1.63818	−0.819089	−	0.573666i	$-0.805521\pi$
$73$	−13.9966	−1.63818	−0.819089	+	0.573666i	$0.805521\pi$
$74$	0	0
$75$	−23.6411	−2.72984
$76$	0	0
$77$	−4.19613	−0.478194
$78$	0	0
$79$	−5.70551	−0.641920	−0.320960	−	0.947093i	$-0.604006\pi$
$79$	−5.70551	−0.641920	−0.320960	+	0.947093i	$0.604006\pi$
$80$	0	0
$81$	−7.76599	−0.862888
$82$	0	0
$83$	11.5458	1.26732	0.633660	−	0.773612i	$-0.281552\pi$
$83$	11.5458	1.26732	0.633660	+	0.773612i	$0.281552\pi$
$84$	0	0
$85$	21.9719	2.38319
$86$	0	0
$87$	−22.8000	−2.44442
$88$	0	0
$89$	16.3195	1.72986	0.864931	−	0.501890i	$-0.167362\pi$
$89$	16.3195	1.72986	0.864931	+	0.501890i	$0.167362\pi$
$90$	0	0
$91$	−4.69505	−0.492175
$92$	0	0
$93$	8.83435	0.916079
$94$	0	0
$95$	28.6630	2.94076
$96$	0	0
$97$	−17.0014	−1.72624	−0.863118	−	0.505003i	$-0.831491\pi$
$97$	−17.0014	−1.72624	−0.863118	+	0.505003i	$0.831491\pi$
$98$	0	0
$99$	−7.46298	−0.750058
$100$	0	0
$101$	−18.1884	−1.80981	−0.904905	−	0.425614i	$-0.860058\pi$
$101$	−18.1884	−1.80981	−0.904905	+	0.425614i	$0.860058\pi$
$102$	0	0
$103$	10.2340	1.00839	0.504194	−	0.863590i	$-0.331790\pi$
$103$	10.2340	1.00839	0.504194	+	0.863590i	$0.331790\pi$
$104$	0	0
$105$	18.1048	1.76685
$106$	0	0
$107$	−0.384493	−0.0371703	−0.0185852	−	0.999827i	$-0.505916\pi$
$107$	−0.384493	−0.0371703	−0.0185852	+	0.999827i	$0.505916\pi$
$108$	0	0
$109$	−1.52554	−0.146120	−0.0730602	−	0.997328i	$-0.523277\pi$
$109$	−1.52554	−0.146120	−0.0730602	+	0.997328i	$0.523277\pi$
$110$	0	0
$111$	24.3671	2.31282
$112$	0	0
$113$	−13.3347	−1.25442	−0.627212	−	0.778849i	$-0.715804\pi$
$113$	−13.3347	−1.25442	−0.627212	+	0.778849i	$0.715804\pi$
$114$	0	0
$115$	4.33452	0.404196
$116$	0	0
$117$	−8.35033	−0.771988
$118$	0	0
$119$	−10.9716	−1.00576
$120$	0	0
$121$	−6.08578	−0.553253
$122$	0	0
$123$	3.66758	0.330695
$124$	0	0
$125$	−16.5634	−1.48148
$126$	0	0
$127$	3.46583	0.307543	0.153771	−	0.988106i	$-0.450858\pi$
$127$	3.46583	0.307543	0.153771	+	0.988106i	$0.450858\pi$
$128$	0	0
$129$	25.7433	2.26657
$130$	0	0
$131$	14.9941	1.31004	0.655019	−	0.755613i	$-0.272661\pi$
$131$	14.9941	1.31004	0.655019	+	0.755613i	$0.272661\pi$
$132$	0	0
$133$	−14.3127	−1.24107
$134$	0	0
$135$	3.50595	0.301744
$136$	0	0
$137$	−10.7085	−0.914892	−0.457446	−	0.889237i	$-0.651236\pi$
$137$	−10.7085	−0.914892	−0.457446	+	0.889237i	$0.651236\pi$
$138$	0	0
$139$	−4.76068	−0.403795	−0.201898	−	0.979407i	$-0.564711\pi$
$139$	−4.76068	−0.403795	−0.201898	+	0.979407i	$0.564711\pi$
$140$	0	0
$141$	9.48967	0.799174
$142$	0	0
$143$	5.49852	0.459809
$144$	0	0
$145$	−34.2534	−2.84459
$146$	0	0
$147$	8.62184	0.711117
$148$	0	0
$149$	−2.68037	−0.219585	−0.109792	−	0.993955i	$-0.535019\pi$
$149$	−2.68037	−0.219585	−0.109792	+	0.993955i	$0.535019\pi$
$150$	0	0
$151$	12.6956	1.03315	0.516576	−	0.856241i	$-0.327206\pi$
$151$	12.6956	1.03315	0.516576	+	0.856241i	$0.327206\pi$
$152$	0	0
$153$	−19.5134	−1.57756
$154$	0	0
$155$	13.2722	1.06605
$156$	0	0
$157$	−3.67319	−0.293153	−0.146576	−	0.989199i	$-0.546825\pi$
$157$	−3.67319	−0.293153	−0.146576	+	0.989199i	$0.546825\pi$
$158$	0	0
$159$	12.0203	0.953269
$160$	0	0
$161$	−2.16442	−0.170581
$162$	0	0
$163$	0.276639	0.0216681	0.0108340	−	0.999941i	$-0.496551\pi$
$163$	0.276639	0.0216681	0.0108340	+	0.999941i	$0.496551\pi$
$164$	0	0
$165$	−21.2031	−1.65066
$166$	0	0
$167$	9.48589	0.734040	0.367020	−	0.930213i	$-0.380378\pi$
$167$	9.48589	0.734040	0.367020	+	0.930213i	$0.380378\pi$
$168$	0	0
$169$	−6.84771	−0.526747
$170$	0	0
$171$	−25.4557	−1.94665
$172$	0	0
$173$	14.3584	1.09165	0.545826	−	0.837898i	$-0.316216\pi$
$173$	14.3584	1.09165	0.545826	+	0.837898i	$0.316216\pi$
$174$	0	0
$175$	17.7353	1.34066
$176$	0	0
$177$	29.4914	2.21671
$178$	0	0
$179$	21.7901	1.62867	0.814334	−	0.580397i	$-0.197103\pi$
$179$	21.7901	1.62867	0.814334	+	0.580397i	$0.197103\pi$
$180$	0	0
$181$	9.75072	0.724765	0.362383	−	0.932029i	$-0.381963\pi$
$181$	9.75072	0.724765	0.362383	+	0.932029i	$0.381963\pi$
$182$	0	0
$183$	30.9188	2.28559
$184$	0	0
$185$	36.6077	2.69145
$186$	0	0
$187$	12.8492	0.939624
$188$	0	0
$189$	−1.75068	−0.127344
$190$	0	0
$191$	9.45315	0.684006	0.342003	−	0.939699i	$-0.388895\pi$
$191$	9.45315	0.684006	0.342003	+	0.939699i	$0.388895\pi$
$192$	0	0
$193$	12.8853	0.927506	0.463753	−	0.885964i	$-0.346502\pi$
$193$	12.8853	0.927506	0.463753	+	0.885964i	$0.346502\pi$
$194$	0	0
$195$	−23.7242	−1.69892
$196$	0	0
$197$	20.4085	1.45405	0.727024	−	0.686612i	$-0.240903\pi$
$197$	20.4085	1.45405	0.727024	+	0.686612i	$0.240903\pi$
$198$	0	0
$199$	−19.5518	−1.38599	−0.692997	−	0.720941i	$-0.743710\pi$
$199$	−19.5518	−1.38599	−0.692997	+	0.720941i	$0.743710\pi$
$200$	0	0
$201$	−2.32271	−0.163831
$202$	0	0
$203$	17.1043	1.20049
$204$	0	0
$205$	5.50996	0.384832
$206$	0	0
$207$	−3.84951	−0.267559
$208$	0	0
$209$	16.7621	1.15946
$210$	0	0
$211$	13.7532	0.946808	0.473404	−	0.880845i	$-0.343025\pi$
$211$	13.7532	0.946808	0.473404	+	0.880845i	$0.343025\pi$
$212$	0	0
$213$	−15.8958	−1.08916
$214$	0	0
$215$	38.6752	2.63762
$216$	0	0
$217$	−6.62742	−0.449899
$218$	0	0
$219$	35.3163	2.38645
$220$	0	0
$221$	14.3769	0.967097
$222$	0	0
$223$	9.06144	0.606799	0.303399	−	0.952864i	$-0.401878\pi$
$223$	9.06144	0.606799	0.303399	+	0.952864i	$0.401878\pi$
$224$	0	0
$225$	31.5428	2.10285
$226$	0	0
$227$	−21.2861	−1.41281	−0.706403	−	0.707809i	$-0.749684\pi$
$227$	−21.2861	−1.41281	−0.706403	+	0.707809i	$0.749684\pi$
$228$	0	0
$229$	14.7935	0.977582	0.488791	−	0.872401i	$-0.337438\pi$
$229$	14.7935	0.977582	0.488791	+	0.872401i	$0.337438\pi$
$230$	0	0
$231$	10.5877	0.696619
$232$	0	0
$233$	5.60775	0.367376	0.183688	−	0.982985i	$-0.441196\pi$
$233$	5.60775	0.367376	0.183688	+	0.982985i	$0.441196\pi$
$234$	0	0
$235$	14.2567	0.930005
$236$	0	0
$237$	14.3962	0.935131
$238$	0	0
$239$	11.2766	0.729426	0.364713	−	0.931120i	$-0.381167\pi$
$239$	11.2766	0.729426	0.364713	+	0.931120i	$0.381167\pi$
$240$	0	0
$241$	−3.95633	−0.254850	−0.127425	−	0.991848i	$-0.540671\pi$
$241$	−3.95633	−0.254850	−0.127425	+	0.991848i	$0.540671\pi$
$242$	0	0
$243$	22.3698	1.43502
$244$	0	0
$245$	12.9529	0.827533
$246$	0	0
$247$	18.7551	1.19336
$248$	0	0
$249$	−29.1325	−1.84620
$250$	0	0
$251$	−20.5277	−1.29570	−0.647850	−	0.761768i	$-0.724331\pi$
$251$	−20.5277	−1.29570	−0.647850	+	0.761768i	$0.724331\pi$
$252$	0	0
$253$	2.53482	0.159363
$254$	0	0
$255$	−55.4396	−3.47176
$256$	0	0
$257$	10.6943	0.667092	0.333546	−	0.942734i	$-0.391755\pi$
$257$	10.6943	0.667092	0.333546	+	0.942734i	$0.391755\pi$
$258$	0	0
$259$	−18.2799	−1.13586
$260$	0	0
$261$	30.4206	1.88299
$262$	0	0
$263$	−23.1451	−1.42719	−0.713594	−	0.700560i	$-0.752934\pi$
$263$	−23.1451	−1.42719	−0.713594	+	0.700560i	$0.752934\pi$
$264$	0	0
$265$	18.0585	1.10933
$266$	0	0
$267$	−41.1774	−2.52001
$268$	0	0
$269$	−1.16039	−0.0707500	−0.0353750	−	0.999374i	$-0.511263\pi$
$269$	−1.16039	−0.0707500	−0.0353750	+	0.999374i	$0.511263\pi$
$270$	0	0
$271$	3.53798	0.214917	0.107458	−	0.994210i	$-0.465729\pi$
$271$	3.53798	0.214917	0.107458	+	0.994210i	$0.465729\pi$
$272$	0	0
$273$	11.8466	0.716987
$274$	0	0
$275$	−20.7703	−1.25250
$276$	0	0
$277$	−25.1057	−1.50845	−0.754226	−	0.656615i	$-0.771988\pi$
$277$	−25.1057	−1.50845	−0.754226	+	0.656615i	$0.771988\pi$
$278$	0	0
$279$	−11.7871	−0.705676
$280$	0	0
$281$	−16.8395	−1.00456	−0.502280	−	0.864705i	$-0.667505\pi$
$281$	−16.8395	−1.00456	−0.502280	+	0.864705i	$0.667505\pi$
$282$	0	0
$283$	−19.0704	−1.13362	−0.566808	−	0.823850i	$-0.691822\pi$
$283$	−19.0704	−1.13362	−0.566808	+	0.823850i	$0.691822\pi$
$284$	0	0
$285$	−72.3224	−4.28401
$286$	0	0
$287$	−2.75138	−0.162409
$288$	0	0
$289$	16.5966	0.976271
$290$	0	0
$291$	42.8981	2.51473
$292$	0	0
$293$	7.86752	0.459625	0.229813	−	0.973235i	$-0.426189\pi$
$293$	7.86752	0.459625	0.229813	+	0.973235i	$0.426189\pi$
$294$	0	0
$295$	44.3062	2.57961
$296$	0	0
$297$	2.05028	0.118969
$298$	0	0
$299$	2.83621	0.164022
$300$	0	0
$301$	−19.3123	−1.11314
$302$	0	0
$303$	45.8929	2.63648
$304$	0	0
$305$	46.4506	2.65976
$306$	0	0
$307$	15.4675	0.882778	0.441389	−	0.897316i	$-0.354486\pi$
$307$	15.4675	0.882778	0.441389	+	0.897316i	$0.354486\pi$
$308$	0	0
$309$	−25.8225	−1.46899
$310$	0	0
$311$	−14.6722	−0.831983	−0.415991	−	0.909369i	$-0.636565\pi$
$311$	−14.6722	−0.831983	−0.415991	+	0.909369i	$0.636565\pi$
$312$	0	0
$313$	25.4651	1.43937	0.719685	−	0.694301i	$-0.244286\pi$
$313$	25.4651	1.43937	0.719685	+	0.694301i	$0.244286\pi$
$314$	0	0
$315$	−24.1561	−1.36104
$316$	0	0
$317$	−4.97289	−0.279305	−0.139653	−	0.990201i	$-0.544599\pi$
$317$	−4.97289	−0.279305	−0.139653	+	0.990201i	$0.544599\pi$
$318$	0	0
$319$	−20.0314	−1.12154
$320$	0	0
$321$	0.970153	0.0541487
$322$	0	0
$323$	43.8277	2.43864
$324$	0	0
$325$	−23.2399	−1.28912
$326$	0	0
$327$	3.84925	0.212864
$328$	0	0
$329$	−7.11903	−0.392485
$330$	0	0
$331$	12.6857	0.697271	0.348635	−	0.937258i	$-0.386645\pi$
$331$	12.6857	0.697271	0.348635	+	0.937258i	$0.386645\pi$
$332$	0	0
$333$	−32.5115	−1.78162
$334$	0	0
$335$	−3.48950	−0.190652
$336$	0	0
$337$	−20.1424	−1.09722	−0.548612	−	0.836077i	$-0.684844\pi$
$337$	−20.1424	−1.09722	−0.548612	+	0.836077i	$0.684844\pi$
$338$	0	0
$339$	33.6462	1.82741
$340$	0	0
$341$	7.76157	0.420313
$342$	0	0
$343$	−19.7181	−1.06468
$344$	0	0
$345$	−10.9369	−0.588821
$346$	0	0
$347$	−22.8003	−1.22398	−0.611992	−	0.790864i	$-0.709632\pi$
$347$	−22.8003	−1.22398	−0.611992	+	0.790864i	$0.709632\pi$
$348$	0	0
$349$	−31.7096	−1.69738	−0.848688	−	0.528894i	$-0.822607\pi$
$349$	−31.7096	−1.69738	−0.848688	+	0.528894i	$0.822607\pi$
$350$	0	0
$351$	2.29406	0.122448
$352$	0	0
$353$	15.1902	0.808494	0.404247	−	0.914650i	$-0.367534\pi$
$353$	15.1902	0.808494	0.404247	+	0.914650i	$0.367534\pi$
$354$	0	0
$355$	−23.8809	−1.26747
$356$	0	0
$357$	27.6836	1.46517
$358$	0	0
$359$	3.32110	0.175281	0.0876405	−	0.996152i	$-0.472067\pi$
$359$	3.32110	0.175281	0.0876405	+	0.996152i	$0.472067\pi$
$360$	0	0
$361$	38.1743	2.00918
$362$	0	0
$363$	15.3557	0.805963
$364$	0	0
$365$	53.0570	2.77713
$366$	0	0
$367$	−0.799433	−0.0417300	−0.0208650	−	0.999782i	$-0.506642\pi$
$367$	−0.799433	−0.0417300	−0.0208650	+	0.999782i	$0.506642\pi$
$368$	0	0
$369$	−4.89343	−0.254742
$370$	0	0
$371$	−9.01746	−0.468163
$372$	0	0
$373$	−34.6685	−1.79507	−0.897533	−	0.440947i	$-0.854643\pi$
$373$	−34.6685	−1.79507	−0.897533	+	0.440947i	$0.854643\pi$
$374$	0	0
$375$	41.7928	2.15817
$376$	0	0
$377$	−22.4131	−1.15433
$378$	0	0
$379$	13.9354	0.715813	0.357906	−	0.933758i	$-0.383491\pi$
$379$	13.9354	0.715813	0.357906	+	0.933758i	$0.383491\pi$
$380$	0	0
$381$	−8.74499	−0.448020
$382$	0	0
$383$	−36.1308	−1.84620	−0.923100	−	0.384560i	$-0.874353\pi$
$383$	−36.1308	−1.84620	−0.923100	+	0.384560i	$0.874353\pi$
$384$	0	0
$385$	15.9063	0.810661
$386$	0	0
$387$	−34.3476	−1.74599
$388$	0	0
$389$	20.0184	1.01498	0.507488	−	0.861659i	$-0.330574\pi$
$389$	20.0184	1.01498	0.507488	+	0.861659i	$0.330574\pi$
$390$	0	0
$391$	6.62778	0.335181
$392$	0	0
$393$	−37.8331	−1.90842
$394$	0	0
$395$	21.6279	1.08822
$396$	0	0
$397$	−27.5138	−1.38088	−0.690439	−	0.723390i	$-0.742583\pi$
$397$	−27.5138	−1.38088	−0.690439	+	0.723390i	$0.742583\pi$
$398$	0	0
$399$	36.1139	1.80796
$400$	0	0
$401$	1.70779	0.0852831	0.0426415	−	0.999090i	$-0.486423\pi$
$401$	1.70779	0.0852831	0.0426415	+	0.999090i	$0.486423\pi$
$402$	0	0
$403$	8.68442	0.432602
$404$	0	0
$405$	29.4386	1.46282
$406$	0	0
$407$	21.4082	1.06116
$408$	0	0
$409$	26.5348	1.31206	0.656030	−	0.754735i	$-0.272235\pi$
$409$	26.5348	1.31206	0.656030	+	0.754735i	$0.272235\pi$
$410$	0	0
$411$	27.0198	1.33279
$412$	0	0
$413$	−22.1241	−1.08866
$414$	0	0
$415$	−43.7669	−2.14843
$416$	0	0
$417$	12.0121	0.588237
$418$	0	0
$419$	1.75677	0.0858237	0.0429118	−	0.999079i	$-0.486337\pi$
$419$	1.75677	0.0858237	0.0429118	+	0.999079i	$0.486337\pi$
$420$	0	0
$421$	−33.4203	−1.62881	−0.814404	−	0.580299i	$-0.802936\pi$
$421$	−33.4203	−1.62881	−0.814404	+	0.580299i	$0.802936\pi$
$422$	0	0
$423$	−12.6615	−0.615622
$424$	0	0
$425$	−54.3079	−2.63432
$426$	0	0
$427$	−23.1949	−1.12248
$428$	0	0
$429$	−13.8739	−0.669837
$430$	0	0
$431$	5.13608	0.247396	0.123698	−	0.992320i	$-0.460525\pi$
$431$	5.13608	0.247396	0.123698	+	0.992320i	$0.460525\pi$
$432$	0	0
$433$	12.8816	0.619048	0.309524	−	0.950892i	$-0.399830\pi$
$433$	12.8816	0.619048	0.309524	+	0.950892i	$0.399830\pi$
$434$	0	0
$435$	86.4282	4.14392
$436$	0	0
$437$	8.64611	0.413600
$438$	0	0
$439$	21.2257	1.01305	0.506523	−	0.862227i	$-0.330931\pi$
$439$	21.2257	1.01305	0.506523	+	0.862227i	$0.330931\pi$
$440$	0	0
$441$	−11.5036	−0.547789
$442$	0	0
$443$	31.2706	1.48571	0.742855	−	0.669452i	$-0.233471\pi$
$443$	31.2706	1.48571	0.742855	+	0.669452i	$0.233471\pi$
$444$	0	0
$445$	−61.8624	−2.93256
$446$	0	0
$447$	6.76312	0.319884
$448$	0	0
$449$	−16.0298	−0.756494	−0.378247	−	0.925705i	$-0.623473\pi$
$449$	−16.0298	−0.756494	−0.378247	+	0.925705i	$0.623473\pi$
$450$	0	0
$451$	3.22222	0.151728
$452$	0	0
$453$	−32.0335	−1.50507
$454$	0	0
$455$	17.7976	0.834363
$456$	0	0
$457$	−36.1396	−1.69054	−0.845270	−	0.534339i	$-0.820561\pi$
$457$	−36.1396	−1.69054	−0.845270	+	0.534339i	$0.820561\pi$
$458$	0	0
$459$	5.36085	0.250223
$460$	0	0
$461$	20.4137	0.950761	0.475380	−	0.879780i	$-0.342310\pi$
$461$	20.4137	0.950761	0.475380	+	0.879780i	$0.342310\pi$
$462$	0	0
$463$	−5.06221	−0.235261	−0.117630	−	0.993057i	$-0.537530\pi$
$463$	−5.06221	−0.235261	−0.117630	+	0.993057i	$0.537530\pi$
$464$	0	0
$465$	−33.4885	−1.55299
$466$	0	0
$467$	8.95934	0.414589	0.207294	−	0.978279i	$-0.433534\pi$
$467$	8.95934	0.414589	0.207294	+	0.978279i	$0.433534\pi$
$468$	0	0
$469$	1.74247	0.0804596
$470$	0	0
$471$	9.26821	0.427056
$472$	0	0
$473$	22.6172	1.03994
$474$	0	0
$475$	−70.8461	−3.25064
$476$	0	0
$477$	−16.0379	−0.734324
$478$	0	0
$479$	−19.4682	−0.889525	−0.444762	−	0.895649i	$-0.646712\pi$
$479$	−19.4682	−0.889525	−0.444762	+	0.895649i	$0.646712\pi$
$480$	0	0
$481$	23.9536	1.09219
$482$	0	0
$483$	5.46128	0.248497
$484$	0	0
$485$	64.4475	2.92641
$486$	0	0
$487$	11.1434	0.504957	0.252479	−	0.967602i	$-0.418754\pi$
$487$	11.1434	0.504957	0.252479	+	0.967602i	$0.418754\pi$
$488$	0	0
$489$	−0.698017	−0.0315654
$490$	0	0
$491$	−25.5595	−1.15348	−0.576742	−	0.816927i	$-0.695676\pi$
$491$	−25.5595	−1.15348	−0.576742	+	0.816927i	$0.695676\pi$
$492$	0	0
$493$	−52.3758	−2.35889
$494$	0	0
$495$	28.2900	1.27154
$496$	0	0
$497$	11.9248	0.534902
$498$	0	0
$499$	32.8963	1.47264	0.736320	−	0.676633i	$-0.236562\pi$
$499$	32.8963	1.47264	0.736320	+	0.676633i	$0.236562\pi$
$500$	0	0
$501$	−23.9348	−1.06933
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	68.9468	3.06809
$506$	0	0
$507$	17.2782	0.767350
$508$	0	0
$509$	−34.6387	−1.53533	−0.767666	−	0.640850i	$-0.778582\pi$
$509$	−34.6387	−1.53533	−0.767666	+	0.640850i	$0.778582\pi$
$510$	0	0
$511$	−26.4938	−1.17202
$512$	0	0
$513$	6.99336	0.308765
$514$	0	0
$515$	−38.7942	−1.70948
$516$	0	0
$517$	8.33731	0.366675
$518$	0	0
$519$	−36.2293	−1.59029
$520$	0	0
$521$	28.2777	1.23887	0.619435	−	0.785048i	$-0.287362\pi$
$521$	28.2777	1.23887	0.619435	+	0.785048i	$0.287362\pi$
$522$	0	0
$523$	−4.05596	−0.177355	−0.0886774	−	0.996060i	$-0.528264\pi$
$523$	−4.05596	−0.177355	−0.0886774	+	0.996060i	$0.528264\pi$
$524$	0	0
$525$	−44.7496	−1.95303
$526$	0	0
$527$	20.2941	0.884026
$528$	0	0
$529$	−21.6925	−0.943152
$530$	0	0
$531$	−39.3486	−1.70758
$532$	0	0
$533$	3.60534	0.156165
$534$	0	0
$535$	1.45750	0.0630132
$536$	0	0
$537$	−54.9808	−2.37260
$538$	0	0
$539$	7.57487	0.326273
$540$	0	0
$541$	28.4259	1.22213	0.611063	−	0.791582i	$-0.290742\pi$
$541$	28.4259	1.22213	0.611063	+	0.791582i	$0.290742\pi$
$542$	0	0
$543$	−24.6030	−1.05582
$544$	0	0
$545$	5.78288	0.247711
$546$	0	0
$547$	32.9987	1.41092	0.705462	−	0.708748i	$-0.250740\pi$
$547$	32.9987	1.41092	0.705462	+	0.708748i	$0.250740\pi$
$548$	0	0
$549$	−41.2531	−1.76064
$550$	0	0
$551$	−68.3256	−2.91077
$552$	0	0
$553$	−10.7998	−0.459255
$554$	0	0
$555$	−92.3686	−3.92083
$556$	0	0
$557$	−1.66112	−0.0703837	−0.0351919	−	0.999381i	$-0.511204\pi$
$557$	−1.66112	−0.0703837	−0.0351919	+	0.999381i	$0.511204\pi$
$558$	0	0
$559$	25.3064	1.07035
$560$	0	0
$561$	−32.4210	−1.36882
$562$	0	0
$563$	11.2155	0.472675	0.236337	−	0.971671i	$-0.424053\pi$
$563$	11.2155	0.472675	0.236337	+	0.971671i	$0.424053\pi$
$564$	0	0
$565$	50.5480	2.12657
$566$	0	0
$567$	−14.7001	−0.617344
$568$	0	0
$569$	−34.4245	−1.44315	−0.721575	−	0.692337i	$-0.756581\pi$
$569$	−34.4245	−1.44315	−0.721575	+	0.692337i	$0.756581\pi$
$570$	0	0
$571$	−5.19841	−0.217547	−0.108773	−	0.994067i	$-0.534692\pi$
$571$	−5.19841	−0.217547	−0.108773	+	0.994067i	$0.534692\pi$
$572$	0	0
$573$	−23.8522	−0.996441
$574$	0	0
$575$	−10.7136	−0.446788
$576$	0	0
$577$	−2.83324	−0.117949	−0.0589746	−	0.998259i	$-0.518783\pi$
$577$	−2.83324	−0.117949	−0.0589746	+	0.998259i	$0.518783\pi$
$578$	0	0
$579$	−32.5123	−1.35116
$580$	0	0
$581$	21.8548	0.906691
$582$	0	0
$583$	10.5606	0.437376
$584$	0	0
$585$	31.6537	1.30872
$586$	0	0
$587$	−23.5244	−0.970954	−0.485477	−	0.874249i	$-0.661354\pi$
$587$	−23.5244	−0.970954	−0.485477	+	0.874249i	$0.661354\pi$
$588$	0	0
$589$	26.4742	1.09085
$590$	0	0
$591$	−51.4949	−2.11822
$592$	0	0
$593$	−39.1688	−1.60847	−0.804234	−	0.594313i	$-0.797424\pi$
$593$	−39.1688	−1.60847	−0.804234	+	0.594313i	$0.797424\pi$
$594$	0	0
$595$	41.5901	1.70503
$596$	0	0
$597$	49.3333	2.01908
$598$	0	0
$599$	7.07652	0.289139	0.144569	−	0.989495i	$-0.453820\pi$
$599$	7.07652	0.289139	0.144569	+	0.989495i	$0.453820\pi$
$600$	0	0
$601$	2.14568	0.0875242	0.0437621	−	0.999042i	$-0.486066\pi$
$601$	2.14568	0.0875242	0.0437621	+	0.999042i	$0.486066\pi$
$602$	0	0
$603$	3.09904	0.126203
$604$	0	0
$605$	23.0694	0.937906
$606$	0	0
$607$	−7.18334	−0.291563	−0.145781	−	0.989317i	$-0.546570\pi$
$607$	−7.18334	−0.291563	−0.145781	+	0.989317i	$0.546570\pi$
$608$	0	0
$609$	−43.1576	−1.74883
$610$	0	0
$611$	9.32862	0.377396
$612$	0	0
$613$	18.9920	0.767080	0.383540	−	0.923524i	$-0.374705\pi$
$613$	18.9920	0.767080	0.383540	+	0.923524i	$0.374705\pi$
$614$	0	0
$615$	−13.9027	−0.560613
$616$	0	0
$617$	−6.51159	−0.262147	−0.131073	−	0.991373i	$-0.541842\pi$
$617$	−6.51159	−0.262147	−0.131073	+	0.991373i	$0.541842\pi$
$618$	0	0
$619$	−47.9457	−1.92710	−0.963550	−	0.267528i	$-0.913793\pi$
$619$	−47.9457	−1.92710	−0.963550	+	0.267528i	$0.913793\pi$
$620$	0	0
$621$	1.05756	0.0424385
$622$	0	0
$623$	30.8908	1.23761
$624$	0	0
$625$	15.9397	0.637587
$626$	0	0
$627$	−42.2941	−1.68906
$628$	0	0
$629$	55.9757	2.23190
$630$	0	0
$631$	−23.5666	−0.938172	−0.469086	−	0.883152i	$-0.655417\pi$
$631$	−23.5666	−0.938172	−0.469086	+	0.883152i	$0.655417\pi$
$632$	0	0
$633$	−34.7021	−1.37928
$634$	0	0
$635$	−13.1380	−0.521364
$636$	0	0
$637$	8.47551	0.335812
$638$	0	0
$639$	21.2088	0.839006
$640$	0	0
$641$	−35.4911	−1.40182	−0.700908	−	0.713252i	$-0.747222\pi$
$641$	−35.4911	−1.40182	−0.700908	+	0.713252i	$0.747222\pi$
$642$	0	0
$643$	−12.2751	−0.484084	−0.242042	−	0.970266i	$-0.577817\pi$
$643$	−12.2751	−0.484084	−0.242042	+	0.970266i	$0.577817\pi$
$644$	0	0
$645$	−97.5853	−3.84242
$646$	0	0
$647$	28.2240	1.10960	0.554800	−	0.831984i	$-0.312795\pi$
$647$	28.2240	1.10960	0.554800	+	0.831984i	$0.312795\pi$
$648$	0	0
$649$	25.9102	1.01707
$650$	0	0
$651$	16.7223	0.655400
$652$	0	0
$653$	−14.6699	−0.574077	−0.287039	−	0.957919i	$-0.592671\pi$
$653$	−14.6699	−0.574077	−0.287039	+	0.957919i	$0.592671\pi$
$654$	0	0
$655$	−56.8381	−2.22085
$656$	0	0
$657$	−47.1203	−1.83834
$658$	0	0
$659$	19.8579	0.773552	0.386776	−	0.922174i	$-0.373589\pi$
$659$	19.8579	0.773552	0.386776	+	0.922174i	$0.373589\pi$
$660$	0	0
$661$	50.1792	1.95174	0.975872	−	0.218343i	$-0.0700650\pi$
$661$	50.1792	1.95174	0.975872	+	0.218343i	$0.0700650\pi$
$662$	0	0
$663$	−36.2759	−1.40884
$664$	0	0
$665$	54.2554	2.10393
$666$	0	0
$667$	−10.3325	−0.400074
$668$	0	0
$669$	−22.8638	−0.883967
$670$	0	0
$671$	27.1643	1.04867
$672$	0	0
$673$	−15.9650	−0.615406	−0.307703	−	0.951482i	$-0.599560\pi$
$673$	−15.9650	−0.615406	−0.307703	+	0.951482i	$0.599560\pi$
$674$	0	0
$675$	−8.66564	−0.333541
$676$	0	0
$677$	−22.4304	−0.862072	−0.431036	−	0.902335i	$-0.641852\pi$
$677$	−22.4304	−0.862072	−0.431036	+	0.902335i	$0.641852\pi$
$678$	0	0
$679$	−32.1816	−1.23502
$680$	0	0
$681$	53.7091	2.05814
$682$	0	0
$683$	44.5677	1.70534	0.852669	−	0.522452i	$-0.174983\pi$
$683$	44.5677	1.70534	0.852669	+	0.522452i	$0.174983\pi$
$684$	0	0
$685$	40.5929	1.55098
$686$	0	0
$687$	−37.3270	−1.42411
$688$	0	0
$689$	11.8163	0.450164
$690$	0	0
$691$	19.7555	0.751536	0.375768	−	0.926714i	$-0.377379\pi$
$691$	19.7555	0.751536	0.375768	+	0.926714i	$0.377379\pi$
$692$	0	0
$693$	−14.1265	−0.536621
$694$	0	0
$695$	18.0463	0.684537
$696$	0	0
$697$	8.42511	0.319124
$698$	0	0
$699$	−14.1495	−0.535183
$700$	0	0
$701$	12.8445	0.485131	0.242566	−	0.970135i	$-0.422011\pi$
$701$	12.8445	0.485131	0.242566	+	0.970135i	$0.422011\pi$
$702$	0	0
$703$	73.0218	2.75407
$704$	0	0
$705$	−35.9726	−1.35481
$706$	0	0
$707$	−34.4283	−1.29481
$708$	0	0
$709$	7.26507	0.272845	0.136423	−	0.990651i	$-0.456439\pi$
$709$	7.26507	0.272845	0.136423	+	0.990651i	$0.456439\pi$
$710$	0	0
$711$	−19.2079	−0.720352
$712$	0	0
$713$	4.00353	0.149933
$714$	0	0
$715$	−20.8433	−0.779494
$716$	0	0
$717$	−28.4533	−1.06261
$718$	0	0
$719$	−24.1311	−0.899937	−0.449969	−	0.893044i	$-0.648565\pi$
$719$	−24.1311	−0.899937	−0.449969	+	0.893044i	$0.648565\pi$
$720$	0	0
$721$	19.3717	0.721441
$722$	0	0
$723$	9.98262	0.371258
$724$	0	0
$725$	84.6639	3.14434
$726$	0	0
$727$	5.02237	0.186269	0.0931347	−	0.995654i	$-0.470311\pi$
$727$	5.02237	0.186269	0.0931347	+	0.995654i	$0.470311\pi$
$728$	0	0
$729$	−33.1456	−1.22761
$730$	0	0
$731$	59.1370	2.18726
$732$	0	0
$733$	24.2138	0.894359	0.447179	−	0.894444i	$-0.352429\pi$
$733$	24.2138	0.894359	0.447179	+	0.894444i	$0.352429\pi$
$734$	0	0
$735$	−32.6829	−1.20553
$736$	0	0
$737$	−2.04065	−0.0751685
$738$	0	0
$739$	−23.4716	−0.863417	−0.431708	−	0.902013i	$-0.642089\pi$
$739$	−23.4716	−0.863417	−0.431708	+	0.902013i	$0.642089\pi$
$740$	0	0
$741$	−47.3229	−1.73845
$742$	0	0
$743$	−50.0260	−1.83527	−0.917637	−	0.397419i	$-0.869906\pi$
$743$	−50.0260	−1.83527	−0.917637	+	0.397419i	$0.869906\pi$
$744$	0	0
$745$	10.1605	0.372252
$746$	0	0
$747$	38.8696	1.42216
$748$	0	0
$749$	−0.727797	−0.0265931
$750$	0	0
$751$	11.8739	0.433285	0.216643	−	0.976251i	$-0.430489\pi$
$751$	11.8739	0.433285	0.216643	+	0.976251i	$0.430489\pi$
$752$	0	0
$753$	51.7956	1.88754
$754$	0	0
$755$	−48.1253	−1.75146
$756$	0	0
$757$	−10.1501	−0.368912	−0.184456	−	0.982841i	$-0.559052\pi$
$757$	−10.1501	−0.368912	−0.184456	+	0.982841i	$0.559052\pi$
$758$	0	0
$759$	−6.39587	−0.232155
$760$	0	0
$761$	28.0765	1.01777	0.508886	−	0.860834i	$-0.330057\pi$
$761$	28.0765	1.01777	0.508886	+	0.860834i	$0.330057\pi$
$762$	0	0
$763$	−2.88766	−0.104540
$764$	0	0
$765$	73.9696	2.67438
$766$	0	0
$767$	28.9909	1.04680
$768$	0	0
$769$	23.7823	0.857613	0.428807	−	0.903396i	$-0.358934\pi$
$769$	23.7823	0.857613	0.428807	+	0.903396i	$0.358934\pi$
$770$	0	0
$771$	−26.9839	−0.971800
$772$	0	0
$773$	−32.9227	−1.18415	−0.592074	−	0.805883i	$-0.701691\pi$
$773$	−32.9227	−1.18415	−0.592074	+	0.805883i	$0.701691\pi$
$774$	0	0
$775$	−32.8048	−1.17838
$776$	0	0
$777$	46.1239	1.65469
$778$	0	0
$779$	10.9908	0.393786
$780$	0	0
$781$	−13.9655	−0.499726
$782$	0	0
$783$	−8.35735	−0.298667
$784$	0	0
$785$	13.9240	0.496969
$786$	0	0
$787$	17.0756	0.608679	0.304339	−	0.952564i	$-0.401564\pi$
$787$	17.0756	0.608679	0.304339	+	0.952564i	$0.401564\pi$
$788$	0	0
$789$	58.3998	2.07909
$790$	0	0
$791$	−25.2409	−0.897465
$792$	0	0
$793$	30.3941	1.07933
$794$	0	0
$795$	−45.5653	−1.61604
$796$	0	0
$797$	−42.1644	−1.49354	−0.746771	−	0.665082i	$-0.768397\pi$
$797$	−42.1644	−1.49354	−0.746771	+	0.665082i	$0.768397\pi$
$798$	0	0
$799$	21.7995	0.771211
$800$	0	0
$801$	54.9404	1.94122
$802$	0	0
$803$	31.0277	1.09494
$804$	0	0
$805$	8.20470	0.289178
$806$	0	0
$807$	2.92789	0.103067
$808$	0	0
$809$	−34.1708	−1.20138	−0.600691	−	0.799481i	$-0.705108\pi$
$809$	−34.1708	−1.20138	−0.600691	+	0.799481i	$0.705108\pi$
$810$	0	0
$811$	20.8557	0.732341	0.366171	−	0.930548i	$-0.380669\pi$
$811$	20.8557	0.732341	0.366171	+	0.930548i	$0.380669\pi$
$812$	0	0
$813$	−8.92703	−0.313085
$814$	0	0
$815$	−1.04866	−0.0367329
$816$	0	0
$817$	77.1458	2.69899
$818$	0	0
$819$	−15.8061	−0.552311
$820$	0	0
$821$	−5.62797	−0.196417	−0.0982087	−	0.995166i	$-0.531311\pi$
$821$	−5.62797	−0.196417	−0.0982087	+	0.995166i	$0.531311\pi$
$822$	0	0
$823$	13.3149	0.464129	0.232064	−	0.972700i	$-0.425452\pi$
$823$	13.3149	0.464129	0.232064	+	0.972700i	$0.425452\pi$
$824$	0	0
$825$	52.4076	1.82460
$826$	0	0
$827$	−2.77382	−0.0964553	−0.0482277	−	0.998836i	$-0.515357\pi$
$827$	−2.77382	−0.0964553	−0.0482277	+	0.998836i	$0.515357\pi$
$828$	0	0
$829$	−35.4798	−1.23226	−0.616132	−	0.787643i	$-0.711301\pi$
$829$	−35.4798	−1.23226	−0.616132	+	0.787643i	$0.711301\pi$
$830$	0	0
$831$	63.3466	2.19747
$832$	0	0
$833$	19.8059	0.686235
$834$	0	0
$835$	−35.9583	−1.24439
$836$	0	0
$837$	3.23823	0.111930
$838$	0	0
$839$	−4.51677	−0.155936	−0.0779681	−	0.996956i	$-0.524843\pi$
$839$	−4.51677	−0.155936	−0.0779681	+	0.996956i	$0.524843\pi$
$840$	0	0
$841$	52.6519	1.81558
$842$	0	0
$843$	42.4895	1.46342
$844$	0	0
$845$	25.9577	0.892971
$846$	0	0
$847$	−11.5196	−0.395819
$848$	0	0
$849$	48.1184	1.65142
$850$	0	0
$851$	11.0426	0.378536
$852$	0	0
$853$	−15.2890	−0.523487	−0.261743	−	0.965137i	$-0.584297\pi$
$853$	−15.2890	−0.523487	−0.261743	+	0.965137i	$0.584297\pi$
$854$	0	0
$855$	96.4953	3.30007
$856$	0	0
$857$	44.6333	1.52465	0.762323	−	0.647197i	$-0.224059\pi$
$857$	44.6333	1.52465	0.762323	+	0.647197i	$0.224059\pi$
$858$	0	0
$859$	−40.3821	−1.37782	−0.688910	−	0.724847i	$-0.741910\pi$
$859$	−40.3821	−1.37782	−0.688910	+	0.724847i	$0.741910\pi$
$860$	0	0
$861$	6.94228	0.236592
$862$	0	0
$863$	4.89789	0.166726	0.0833631	−	0.996519i	$-0.473434\pi$
$863$	4.89789	0.166726	0.0833631	+	0.996519i	$0.473434\pi$
$864$	0	0
$865$	−54.4287	−1.85063
$866$	0	0
$867$	−41.8766	−1.42220
$868$	0	0
$869$	12.6480	0.429054
$870$	0	0
$871$	−2.28329	−0.0773663
$872$	0	0
$873$	−57.2362	−1.93715
$874$	0	0
$875$	−31.3525	−1.05991
$876$	0	0
$877$	−31.5657	−1.06590	−0.532948	−	0.846148i	$-0.678916\pi$
$877$	−31.5657	−1.06590	−0.532948	+	0.846148i	$0.678916\pi$
$878$	0	0
$879$	−19.8513	−0.669569
$880$	0	0
$881$	46.2882	1.55949	0.779745	−	0.626098i	$-0.215349\pi$
$881$	46.2882	1.55949	0.779745	+	0.626098i	$0.215349\pi$
$882$	0	0
$883$	32.3464	1.08854	0.544271	−	0.838909i	$-0.316806\pi$
$883$	32.3464	1.08854	0.544271	+	0.838909i	$0.316806\pi$
$884$	0	0
$885$	−111.793	−3.75790
$886$	0	0
$887$	21.9502	0.737015	0.368507	−	0.929625i	$-0.379869\pi$
$887$	21.9502	0.737015	0.368507	+	0.929625i	$0.379869\pi$
$888$	0	0
$889$	6.56039	0.220028
$890$	0	0
$891$	17.2157	0.576747
$892$	0	0
$893$	28.4380	0.951642
$894$	0	0
$895$	−82.5999	−2.76101
$896$	0	0
$897$	−7.15634	−0.238943
$898$	0	0
$899$	−31.6378	−1.05518
$900$	0	0
$901$	27.6128	0.919914
$902$	0	0
$903$	48.7288	1.62159
$904$	0	0
$905$	−36.9621	−1.22866
$906$	0	0
$907$	−55.6647	−1.84832	−0.924158	−	0.382009i	$-0.875232\pi$
$907$	−55.6647	−1.84832	−0.924158	+	0.382009i	$0.875232\pi$
$908$	0	0
$909$	−61.2320	−2.03094
$910$	0	0
$911$	−43.5874	−1.44412	−0.722058	−	0.691833i	$-0.756804\pi$
$911$	−43.5874	−1.44412	−0.722058	+	0.691833i	$0.756804\pi$
$912$	0	0
$913$	−25.5948	−0.847066
$914$	0	0
$915$	−117.204	−3.87466
$916$	0	0
$917$	28.3819	0.937253
$918$	0	0
$919$	−30.6797	−1.01203	−0.506015	−	0.862525i	$-0.668882\pi$
$919$	−30.6797	−1.01203	−0.506015	+	0.862525i	$0.668882\pi$
$920$	0	0
$921$	−39.0277	−1.28601
$922$	0	0
$923$	−15.6260	−0.514337
$924$	0	0
$925$	−90.4831	−2.97506
$926$	0	0
$927$	34.4534	1.13160
$928$	0	0
$929$	4.19116	0.137508	0.0687538	−	0.997634i	$-0.478098\pi$
$929$	4.19116	0.137508	0.0687538	+	0.997634i	$0.478098\pi$
$930$	0	0
$931$	25.8374	0.846786
$932$	0	0
$933$	37.0209	1.21201
$934$	0	0
$935$	−48.7074	−1.59290
$936$	0	0
$937$	−24.6629	−0.805702	−0.402851	−	0.915266i	$-0.631981\pi$
$937$	−24.6629	−0.805702	−0.402851	+	0.915266i	$0.631981\pi$
$938$	0	0
$939$	−64.2535	−2.09683
$940$	0	0
$941$	42.7879	1.39485	0.697424	−	0.716659i	$-0.254330\pi$
$941$	42.7879	1.39485	0.697424	+	0.716659i	$0.254330\pi$
$942$	0	0
$943$	1.66207	0.0541243
$944$	0	0
$945$	6.63633	0.215880
$946$	0	0
$947$	−42.6326	−1.38537	−0.692686	−	0.721239i	$-0.743573\pi$
$947$	−42.6326	−1.38537	−0.692686	+	0.721239i	$0.743573\pi$
$948$	0	0
$949$	34.7169	1.12696
$950$	0	0
$951$	12.5476	0.406884
$952$	0	0
$953$	−11.2997	−0.366034	−0.183017	−	0.983110i	$-0.558586\pi$
$953$	−11.2997	−0.366034	−0.183017	+	0.983110i	$0.558586\pi$
$954$	0	0
$955$	−35.8341	−1.15957
$956$	0	0
$957$	50.5432	1.63383
$958$	0	0
$959$	−20.2699	−0.654550
$960$	0	0
$961$	−18.7413	−0.604557
$962$	0	0
$963$	−1.29441	−0.0417119
$964$	0	0
$965$	−48.8445	−1.57236
$966$	0	0
$967$	−11.6288	−0.373957	−0.186979	−	0.982364i	$-0.559869\pi$
$967$	−11.6288	−0.373957	−0.186979	+	0.982364i	$0.559869\pi$
$968$	0	0
$969$	−110.586	−3.55254
$970$	0	0
$971$	−5.57678	−0.178967	−0.0894837	−	0.995988i	$-0.528522\pi$
$971$	−5.57678	−0.178967	−0.0894837	+	0.995988i	$0.528522\pi$
$972$	0	0
$973$	−9.01137	−0.288891
$974$	0	0
$975$	58.6389	1.87795
$976$	0	0
$977$	−32.0530	−1.02547	−0.512733	−	0.858548i	$-0.671367\pi$
$977$	−32.0530	−1.02547	−0.512733	+	0.858548i	$0.671367\pi$
$978$	0	0
$979$	−36.1771	−1.15623
$980$	0	0
$981$	−5.13581	−0.163974
$982$	0	0
$983$	20.2969	0.647371	0.323685	−	0.946165i	$-0.395078\pi$
$983$	20.2969	0.647371	0.323685	+	0.946165i	$0.395078\pi$
$984$	0	0
$985$	−77.3628	−2.46498
$986$	0	0
$987$	17.9628	0.571761
$988$	0	0
$989$	11.6663	0.370966
$990$	0	0
$991$	41.8111	1.32817	0.664087	−	0.747655i	$-0.268820\pi$
$991$	41.8111	1.32817	0.664087	+	0.747655i	$0.268820\pi$
$992$	0	0
$993$	−32.0087	−1.01576
$994$	0	0
$995$	74.1154	2.34962
$996$	0	0
$997$	−18.8226	−0.596118	−0.298059	−	0.954548i	$-0.596339\pi$
$997$	−18.8226	−0.596118	−0.298059	+	0.954548i	$0.596339\pi$
$998$	0	0
$999$	8.93177	0.282589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.4	✓	33
4.3	odd	2		8048.2.a.x.1.30		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.4	✓	33	1.1	even	1	trivial
8048.2.a.x.1.30		33	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52320 q^{3} -3.79071 q^{5} +1.89288 q^{7} +3.36655 q^{9} +O(q^{10})\)	\(q-2.52320 q^{3} -3.79071 q^{5} +1.89288 q^{7} +3.36655 q^{9} -2.21680 q^{11} -2.48038 q^{13} +9.56473 q^{15} -5.79626 q^{17} -7.56137 q^{19} -4.77611 q^{21} -1.14346 q^{23} +9.36947 q^{25} -0.924880 q^{27} +9.03614 q^{29} -3.50125 q^{31} +5.59344 q^{33} -7.17534 q^{35} -9.65722 q^{37} +6.25850 q^{39} -1.45354 q^{41} -10.2026 q^{43} -12.7616 q^{45} -3.76096 q^{47} -3.41702 q^{49} +14.6251 q^{51} -4.76389 q^{53} +8.40326 q^{55} +19.0789 q^{57} -11.6881 q^{59} -12.2538 q^{61} +6.37246 q^{63} +9.40240 q^{65} +0.920539 q^{67} +2.88518 q^{69} +6.29985 q^{71} -13.9966 q^{73} -23.6411 q^{75} -4.19613 q^{77} -5.70551 q^{79} -7.76599 q^{81} +11.5458 q^{83} +21.9719 q^{85} -22.8000 q^{87} +16.3195 q^{89} -4.69505 q^{91} +8.83435 q^{93} +28.6630 q^{95} -17.0014 q^{97} -7.46298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

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