LMFDB - Embedded newform 4016.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.138917.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 2x^{2} + 3x + 1 $ x^5 - 6x^3 - 2x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.729679$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.88943 q^{3} +0.781072 q^{5} -2.79232 q^{7} +5.34878 q^{9} +O(q^{10})$	$q-2.88943 q^{3} +0.781072 q^{5} -2.79232 q^{7} +5.34878 q^{9} -2.64079 q^{11} -5.63036 q^{13} -2.25685 q^{15} -4.71925 q^{17} +1.48104 q^{19} +8.06821 q^{21} -7.99736 q^{23} -4.38993 q^{25} -6.78664 q^{27} -2.41618 q^{29} +6.65661 q^{31} +7.63036 q^{33} -2.18101 q^{35} -2.28504 q^{37} +16.2685 q^{39} -3.37850 q^{41} +4.82110 q^{43} +4.17779 q^{45} -12.8493 q^{47} +0.797070 q^{49} +13.6359 q^{51} +4.78928 q^{53} -2.06264 q^{55} -4.27935 q^{57} +4.11961 q^{59} +8.84107 q^{61} -14.9355 q^{63} -4.39771 q^{65} -12.9006 q^{67} +23.1078 q^{69} -0.115142 q^{71} -13.1142 q^{73} +12.6844 q^{75} +7.37393 q^{77} -16.0883 q^{79} +3.56314 q^{81} -1.52200 q^{83} -3.68607 q^{85} +6.98136 q^{87} -3.34854 q^{89} +15.7218 q^{91} -19.2338 q^{93} +1.15680 q^{95} -6.59243 q^{97} -14.1250 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9	$ 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9} - 7 q^{11} + 4 q^{13} - 3 q^{15} + 6 q^{17} + 10 q^{19} + 2 q^{21} - 13 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{29} + 13 q^{31} + 6 q^{33} - 13 q^{35} + 16 q^{37} + 16 q^{39} + 6 q^{41} + 8 q^{43} + 26 q^{45} - 29 q^{47} + 4 q^{49} + 7 q^{51} + 25 q^{53} - q^{55} + 19 q^{57} - 11 q^{59} + 11 q^{61} - 15 q^{63} + 7 q^{65} + 6 q^{67} - 12 q^{69} - 15 q^{71} - 9 q^{73} + 25 q^{75} + 12 q^{77} + 29 q^{79} - 7 q^{81} + 9 q^{83} - 7 q^{85} + 11 q^{87} - 9 q^{89} + 34 q^{91} - 8 q^{93} + 13 q^{95} - 6 q^{97} + q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9 - 7 * q^11 + 4 * q^13 - 3 * q^15 + 6 * q^17 + 10 * q^19 + 2 * q^21 - 13 * q^23 + 6 * q^25 - 8 * q^27 + 4 * q^29 + 13 * q^31 + 6 * q^33 - 13 * q^35 + 16 * q^37 + 16 * q^39 + 6 * q^41 + 8 * q^43 + 26 * q^45 - 29 * q^47 + 4 * q^49 + 7 * q^51 + 25 * q^53 - q^55 + 19 * q^57 - 11 * q^59 + 11 * q^61 - 15 * q^63 + 7 * q^65 + 6 * q^67 - 12 * q^69 - 15 * q^71 - 9 * q^73 + 25 * q^75 + 12 * q^77 + 29 * q^79 - 7 * q^81 + 9 * q^83 - 7 * q^85 + 11 * q^87 - 9 * q^89 + 34 * q^91 - 8 * q^93 + 13 * q^95 - 6 * q^97 + 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
$3$	−2.88943	−1.66821	−0.834106	−	0.551605i	$-0.814016\pi$
$3$	−2.88943	−1.66821	−0.834106	+	0.551605i	$0.814016\pi$
$4$	0	0
$5$	0.781072	0.349306	0.174653	−	0.984630i	$-0.444120\pi$
$5$	0.781072	0.349306	0.174653	+	0.984630i	$0.444120\pi$
$6$	0	0
$7$	−2.79232	−1.05540	−0.527700	−	0.849431i	$-0.676945\pi$
$7$	−2.79232	−1.05540	−0.527700	+	0.849431i	$0.676945\pi$
$8$	0	0
$9$	5.34878	1.78293
$10$	0	0
$11$	−2.64079	−0.796227	−0.398113	−	0.917336i	$-0.630335\pi$
$11$	−2.64079	−0.796227	−0.398113	+	0.917336i	$0.630335\pi$
$12$	0	0
$13$	−5.63036	−1.56158	−0.780790	−	0.624794i	$-0.785183\pi$
$13$	−5.63036	−1.56158	−0.780790	+	0.624794i	$0.785183\pi$
$14$	0	0
$15$	−2.25685	−0.582716
$16$	0	0
$17$	−4.71925	−1.14459	−0.572293	−	0.820049i	$-0.693946\pi$
$17$	−4.71925	−1.14459	−0.572293	+	0.820049i	$0.693946\pi$
$18$	0	0
$19$	1.48104	0.339774	0.169887	−	0.985464i	$-0.445660\pi$
$19$	1.48104	0.339774	0.169887	+	0.985464i	$0.445660\pi$
$20$	0	0
$21$	8.06821	1.76063
$22$	0	0
$23$	−7.99736	−1.66756	−0.833782	−	0.552093i	$-0.813829\pi$
$23$	−7.99736	−1.66756	−0.833782	+	0.552093i	$0.813829\pi$
$24$	0	0
$25$	−4.38993	−0.877985
$26$	0	0
$27$	−6.78664	−1.30609
$28$	0	0
$29$	−2.41618	−0.448673	−0.224336	−	0.974512i	$-0.572021\pi$
$29$	−2.41618	−0.448673	−0.224336	+	0.974512i	$0.572021\pi$
$30$	0	0
$31$	6.65661	1.19556	0.597781	−	0.801660i	$-0.296049\pi$
$31$	6.65661	1.19556	0.597781	+	0.801660i	$0.296049\pi$
$32$	0	0
$33$	7.63036	1.32827
$34$	0	0
$35$	−2.18101	−0.368657
$36$	0	0
$37$	−2.28504	−0.375658	−0.187829	−	0.982202i	$-0.560145\pi$
$37$	−2.28504	−0.375658	−0.187829	+	0.982202i	$0.560145\pi$
$38$	0	0
$39$	16.2685	2.60505
$40$	0	0
$41$	−3.37850	−0.527633	−0.263816	−	0.964573i	$-0.584981\pi$
$41$	−3.37850	−0.527633	−0.263816	+	0.964573i	$0.584981\pi$
$42$	0	0
$43$	4.82110	0.735211	0.367605	−	0.929982i	$-0.380178\pi$
$43$	4.82110	0.735211	0.367605	+	0.929982i	$0.380178\pi$
$44$	0	0
$45$	4.17779	0.622788
$46$	0	0
$47$	−12.8493	−1.87426	−0.937130	−	0.348979i	$-0.886528\pi$
$47$	−12.8493	−1.87426	−0.937130	+	0.348979i	$0.886528\pi$
$48$	0	0
$49$	0.797070	0.113867
$50$	0	0
$51$	13.6359	1.90941
$52$	0	0
$53$	4.78928	0.657858	0.328929	−	0.944355i	$-0.393312\pi$
$53$	4.78928	0.657858	0.328929	+	0.944355i	$0.393312\pi$
$54$	0	0
$55$	−2.06264	−0.278127
$56$	0	0
$57$	−4.27935	−0.566814
$58$	0	0
$59$	4.11961	0.536327	0.268163	−	0.963373i	$-0.413583\pi$
$59$	4.11961	0.536327	0.268163	+	0.963373i	$0.413583\pi$
$60$	0	0
$61$	8.84107	1.13198	0.565992	−	0.824411i	$-0.308493\pi$
$61$	8.84107	1.13198	0.565992	+	0.824411i	$0.308493\pi$
$62$	0	0
$63$	−14.9355	−1.88170
$64$	0	0
$65$	−4.39771	−0.545469
$66$	0	0
$67$	−12.9006	−1.57605	−0.788027	−	0.615640i	$-0.788897\pi$
$67$	−12.9006	−1.57605	−0.788027	+	0.615640i	$0.788897\pi$
$68$	0	0
$69$	23.1078	2.78185
$70$	0	0
$71$	−0.115142	−0.0136649	−0.00683243	−	0.999977i	$-0.502175\pi$
$71$	−0.115142	−0.0136649	−0.00683243	+	0.999977i	$0.502175\pi$
$72$	0	0
$73$	−13.1142	−1.53490	−0.767449	−	0.641111i	$-0.778474\pi$
$73$	−13.1142	−1.53490	−0.767449	+	0.641111i	$0.778474\pi$
$74$	0	0
$75$	12.6844	1.46466
$76$	0	0
$77$	7.37393	0.840337
$78$	0	0
$79$	−16.0883	−1.81008	−0.905039	−	0.425329i	$-0.860158\pi$
$79$	−16.0883	−1.81008	−0.905039	+	0.425329i	$0.860158\pi$
$80$	0	0
$81$	3.56314	0.395905
$82$	0	0
$83$	−1.52200	−0.167061	−0.0835307	−	0.996505i	$-0.526620\pi$
$83$	−1.52200	−0.167061	−0.0835307	+	0.996505i	$0.526620\pi$
$84$	0	0
$85$	−3.68607	−0.399811
$86$	0	0
$87$	6.98136	0.748480
$88$	0	0
$89$	−3.34854	−0.354945	−0.177472	−	0.984126i	$-0.556792\pi$
$89$	−3.34854	−0.354945	−0.177472	+	0.984126i	$0.556792\pi$
$90$	0	0
$91$	15.7218	1.64809
$92$	0	0
$93$	−19.2338	−1.99445
$94$	0	0
$95$	1.15680	0.118685
$96$	0	0
$97$	−6.59243	−0.669360	−0.334680	−	0.942332i	$-0.608628\pi$
$97$	−6.59243	−0.669360	−0.334680	+	0.942332i	$0.608628\pi$
$98$	0	0
$99$	−14.1250	−1.41962
$100$	0	0
$101$	−7.43293	−0.739604	−0.369802	−	0.929111i	$-0.620575\pi$
$101$	−7.43293	−0.739604	−0.369802	+	0.929111i	$0.620575\pi$
$102$	0	0
$103$	16.4454	1.62041	0.810205	−	0.586147i	$-0.199356\pi$
$103$	16.4454	1.62041	0.810205	+	0.586147i	$0.199356\pi$
$104$	0	0
$105$	6.30186	0.614998
$106$	0	0
$107$	−14.0589	−1.35912	−0.679562	−	0.733618i	$-0.737830\pi$
$107$	−14.0589	−1.35912	−0.679562	+	0.733618i	$0.737830\pi$
$108$	0	0
$109$	10.4840	1.00418	0.502091	−	0.864815i	$-0.332564\pi$
$109$	10.4840	1.00418	0.502091	+	0.864815i	$0.332564\pi$
$110$	0	0
$111$	6.60244	0.626676
$112$	0	0
$113$	0.819265	0.0770699	0.0385350	−	0.999257i	$-0.487731\pi$
$113$	0.819265	0.0770699	0.0385350	+	0.999257i	$0.487731\pi$
$114$	0	0
$115$	−6.24651	−0.582490
$116$	0	0
$117$	−30.1156	−2.78419
$118$	0	0
$119$	13.1777	1.20800
$120$	0	0
$121$	−4.02625	−0.366023
$122$	0	0
$123$	9.76192	0.880203
$124$	0	0
$125$	−7.33421	−0.655992
$126$	0	0
$127$	18.7494	1.66374	0.831868	−	0.554973i	$-0.187271\pi$
$127$	18.7494	1.66374	0.831868	+	0.554973i	$0.187271\pi$
$128$	0	0
$129$	−13.9302	−1.22649
$130$	0	0
$131$	3.51914	0.307469	0.153734	−	0.988112i	$-0.450870\pi$
$131$	3.51914	0.307469	0.153734	+	0.988112i	$0.450870\pi$
$132$	0	0
$133$	−4.13554	−0.358597
$134$	0	0
$135$	−5.30086	−0.456225
$136$	0	0
$137$	7.93736	0.678134	0.339067	−	0.940762i	$-0.389889\pi$
$137$	7.93736	0.678134	0.339067	+	0.940762i	$0.389889\pi$
$138$	0	0
$139$	10.1597	0.861739	0.430869	−	0.902414i	$-0.358207\pi$
$139$	10.1597	0.861739	0.430869	+	0.902414i	$0.358207\pi$
$140$	0	0
$141$	37.1271	3.12666
$142$	0	0
$143$	14.8686	1.24337
$144$	0	0
$145$	−1.88721	−0.156724
$146$	0	0
$147$	−2.30307	−0.189954
$148$	0	0
$149$	−3.08821	−0.252996	−0.126498	−	0.991967i	$-0.540374\pi$
$149$	−3.08821	−0.252996	−0.126498	+	0.991967i	$0.540374\pi$
$150$	0	0
$151$	−8.77528	−0.714122	−0.357061	−	0.934081i	$-0.616221\pi$
$151$	−8.77528	−0.714122	−0.357061	+	0.934081i	$0.616221\pi$
$152$	0	0
$153$	−25.2423	−2.04072
$154$	0	0
$155$	5.19929	0.417617
$156$	0	0
$157$	14.4454	1.15287	0.576433	−	0.817145i	$-0.304444\pi$
$157$	14.4454	1.15287	0.576433	+	0.817145i	$0.304444\pi$
$158$	0	0
$159$	−13.8383	−1.09745
$160$	0	0
$161$	22.3312	1.75995
$162$	0	0
$163$	4.43260	0.347188	0.173594	−	0.984817i	$-0.444462\pi$
$163$	4.43260	0.347188	0.173594	+	0.984817i	$0.444462\pi$
$164$	0	0
$165$	5.95986	0.463974
$166$	0	0
$167$	5.62665	0.435403	0.217702	−	0.976015i	$-0.430144\pi$
$167$	5.62665	0.435403	0.217702	+	0.976015i	$0.430144\pi$
$168$	0	0
$169$	18.7009	1.43853
$170$	0	0
$171$	7.92176	0.605792
$172$	0	0
$173$	−14.3818	−1.09343	−0.546713	−	0.837320i	$-0.684121\pi$
$173$	−14.3818	−1.09343	−0.546713	+	0.837320i	$0.684121\pi$
$174$	0	0
$175$	12.2581	0.926625
$176$	0	0
$177$	−11.9033	−0.894707
$178$	0	0
$179$	−11.2670	−0.842134	−0.421067	−	0.907029i	$-0.638344\pi$
$179$	−11.2670	−0.842134	−0.421067	+	0.907029i	$0.638344\pi$
$180$	0	0
$181$	−19.2819	−1.43321	−0.716605	−	0.697480i	$-0.754305\pi$
$181$	−19.2819	−1.43321	−0.716605	+	0.697480i	$0.754305\pi$
$182$	0	0
$183$	−25.5456	−1.88839
$184$	0	0
$185$	−1.78478	−0.131219
$186$	0	0
$187$	12.4625	0.911350
$188$	0	0
$189$	18.9505	1.37845
$190$	0	0
$191$	13.4851	0.975744	0.487872	−	0.872915i	$-0.337773\pi$
$191$	13.4851	0.975744	0.487872	+	0.872915i	$0.337773\pi$
$192$	0	0
$193$	−9.12358	−0.656730	−0.328365	−	0.944551i	$-0.606498\pi$
$193$	−9.12358	−0.656730	−0.328365	+	0.944551i	$0.606498\pi$
$194$	0	0
$195$	12.7069	0.909958
$196$	0	0
$197$	22.7473	1.62068	0.810341	−	0.585959i	$-0.199282\pi$
$197$	22.7473	1.62068	0.810341	+	0.585959i	$0.199282\pi$
$198$	0	0
$199$	−0.391275	−0.0277368	−0.0138684	−	0.999904i	$-0.504415\pi$
$199$	−0.391275	−0.0277368	−0.0138684	+	0.999904i	$0.504415\pi$
$200$	0	0
$201$	37.2752	2.62919
$202$	0	0
$203$	6.74674	0.473529
$204$	0	0
$205$	−2.63885	−0.184305
$206$	0	0
$207$	−42.7762	−2.97315
$208$	0	0
$209$	−3.91111	−0.270537
$210$	0	0
$211$	4.47764	0.308254	0.154127	−	0.988051i	$-0.450744\pi$
$211$	4.47764	0.308254	0.154127	+	0.988051i	$0.450744\pi$
$212$	0	0
$213$	0.332695	0.0227959
$214$	0	0
$215$	3.76563	0.256814
$216$	0	0
$217$	−18.5874	−1.26179
$218$	0	0
$219$	37.8924	2.56053
$220$	0	0
$221$	26.5711	1.78736
$222$	0	0
$223$	−29.2029	−1.95557	−0.977787	−	0.209603i	$-0.932783\pi$
$223$	−29.2029	−1.95557	−0.977787	+	0.209603i	$0.932783\pi$
$224$	0	0
$225$	−23.4808	−1.56538
$226$	0	0
$227$	5.62427	0.373296	0.186648	−	0.982427i	$-0.440238\pi$
$227$	5.62427	0.373296	0.186648	+	0.982427i	$0.440238\pi$
$228$	0	0
$229$	13.2923	0.878381	0.439190	−	0.898394i	$-0.355265\pi$
$229$	13.2923	0.878381	0.439190	+	0.898394i	$0.355265\pi$
$230$	0	0
$231$	−21.3064	−1.40186
$232$	0	0
$233$	5.23085	0.342684	0.171342	−	0.985212i	$-0.445190\pi$
$233$	5.23085	0.342684	0.171342	+	0.985212i	$0.445190\pi$
$234$	0	0
$235$	−10.0362	−0.654691
$236$	0	0
$237$	46.4860	3.01959
$238$	0	0
$239$	21.9442	1.41945	0.709727	−	0.704477i	$-0.248818\pi$
$239$	21.9442	1.41945	0.709727	+	0.704477i	$0.248818\pi$
$240$	0	0
$241$	−0.526158	−0.0338928	−0.0169464	−	0.999856i	$-0.505394\pi$
$241$	−0.526158	−0.0338928	−0.0169464	+	0.999856i	$0.505394\pi$
$242$	0	0
$243$	10.0645	0.645637
$244$	0	0
$245$	0.622569	0.0397745
$246$	0	0
$247$	−8.33878	−0.530584
$248$	0	0
$249$	4.39771	0.278694
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	21.1193	1.32776
$254$	0	0
$255$	10.6506	0.666969
$256$	0	0
$257$	9.39077	0.585780	0.292890	−	0.956146i	$-0.405383\pi$
$257$	9.39077	0.585780	0.292890	+	0.956146i	$0.405383\pi$
$258$	0	0
$259$	6.38056	0.396469
$260$	0	0
$261$	−12.9236	−0.799951
$262$	0	0
$263$	−12.5145	−0.771679	−0.385839	−	0.922566i	$-0.626088\pi$
$263$	−12.5145	−0.771679	−0.385839	+	0.922566i	$0.626088\pi$
$264$	0	0
$265$	3.74077	0.229794
$266$	0	0
$267$	9.67536	0.592123
$268$	0	0
$269$	−16.4029	−1.00010	−0.500052	−	0.865995i	$-0.666686\pi$
$269$	−16.4029	−1.00010	−0.500052	+	0.865995i	$0.666686\pi$
$270$	0	0
$271$	−22.5884	−1.37215	−0.686074	−	0.727532i	$-0.740667\pi$
$271$	−22.5884	−1.37215	−0.686074	+	0.727532i	$0.740667\pi$
$272$	0	0
$273$	−45.4269	−2.74936
$274$	0	0
$275$	11.5929	0.699076
$276$	0	0
$277$	17.4346	1.04754	0.523772	−	0.851859i	$-0.324524\pi$
$277$	17.4346	1.04754	0.523772	+	0.851859i	$0.324524\pi$
$278$	0	0
$279$	35.6048	2.13160
$280$	0	0
$281$	32.1477	1.91777	0.958886	−	0.283791i	$-0.0915922\pi$
$281$	32.1477	1.91777	0.958886	+	0.283791i	$0.0915922\pi$
$282$	0	0
$283$	14.8919	0.885232	0.442616	−	0.896711i	$-0.354050\pi$
$283$	14.8919	0.885232	0.442616	+	0.896711i	$0.354050\pi$
$284$	0	0
$285$	−3.34248	−0.197992
$286$	0	0
$287$	9.43386	0.556863
$288$	0	0
$289$	5.27132	0.310078
$290$	0	0
$291$	19.0484	1.11663
$292$	0	0
$293$	29.2706	1.71001	0.855003	−	0.518623i	$-0.173555\pi$
$293$	29.2706	1.71001	0.855003	+	0.518623i	$0.173555\pi$
$294$	0	0
$295$	3.21771	0.187342
$296$	0	0
$297$	17.9221	1.03994
$298$	0	0
$299$	45.0280	2.60404
$300$	0	0
$301$	−13.4621	−0.775941
$302$	0	0
$303$	21.4769	1.23382
$304$	0	0
$305$	6.90552	0.395409
$306$	0	0
$307$	−31.8806	−1.81952	−0.909762	−	0.415130i	$-0.863736\pi$
$307$	−31.8806	−1.81952	−0.909762	+	0.415130i	$0.863736\pi$
$308$	0	0
$309$	−47.5177	−2.70319
$310$	0	0
$311$	−26.6253	−1.50978	−0.754892	−	0.655850i	$-0.772311\pi$
$311$	−26.6253	−1.50978	−0.754892	+	0.655850i	$0.772311\pi$
$312$	0	0
$313$	4.86438	0.274951	0.137476	−	0.990505i	$-0.456101\pi$
$313$	4.86438	0.274951	0.137476	+	0.990505i	$0.456101\pi$
$314$	0	0
$315$	−11.6657	−0.657289
$316$	0	0
$317$	0.753643	0.0423288	0.0211644	−	0.999776i	$-0.493263\pi$
$317$	0.753643	0.0423288	0.0211644	+	0.999776i	$0.493263\pi$
$318$	0	0
$319$	6.38060	0.357245
$320$	0	0
$321$	40.6221	2.26731
$322$	0	0
$323$	−6.98939	−0.388900
$324$	0	0
$325$	24.7169	1.37104
$326$	0	0
$327$	−30.2926	−1.67519
$328$	0	0
$329$	35.8794	1.97809
$330$	0	0
$331$	18.4761	1.01554	0.507768	−	0.861494i	$-0.330471\pi$
$331$	18.4761	1.01554	0.507768	+	0.861494i	$0.330471\pi$
$332$	0	0
$333$	−12.2222	−0.669770
$334$	0	0
$335$	−10.0763	−0.550525
$336$	0	0
$337$	16.0252	0.872946	0.436473	−	0.899717i	$-0.356227\pi$
$337$	16.0252	0.872946	0.436473	+	0.899717i	$0.356227\pi$
$338$	0	0
$339$	−2.36720	−0.128569
$340$	0	0
$341$	−17.5787	−0.951938
$342$	0	0
$343$	17.3206	0.935224
$344$	0	0
$345$	18.0488	0.971717
$346$	0	0
$347$	−5.46504	−0.293379	−0.146689	−	0.989183i	$-0.546862\pi$
$347$	−5.46504	−0.293379	−0.146689	+	0.989183i	$0.546862\pi$
$348$	0	0
$349$	−16.7402	−0.896084	−0.448042	−	0.894013i	$-0.647878\pi$
$349$	−16.7402	−0.896084	−0.448042	+	0.894013i	$0.647878\pi$
$350$	0	0
$351$	38.2112	2.03956
$352$	0	0
$353$	4.89126	0.260336	0.130168	−	0.991492i	$-0.458448\pi$
$353$	4.89126	0.260336	0.130168	+	0.991492i	$0.458448\pi$
$354$	0	0
$355$	−0.0899343	−0.00477322
$356$	0	0
$357$	−38.0759	−2.01519
$358$	0	0
$359$	15.6679	0.826918	0.413459	−	0.910523i	$-0.364321\pi$
$359$	15.6679	0.826918	0.413459	+	0.910523i	$0.364321\pi$
$360$	0	0
$361$	−16.8065	−0.884554
$362$	0	0
$363$	11.6335	0.610603
$364$	0	0
$365$	−10.2431	−0.536149
$366$	0	0
$367$	14.3330	0.748177	0.374088	−	0.927393i	$-0.377956\pi$
$367$	14.3330	0.748177	0.374088	+	0.927393i	$0.377956\pi$
$368$	0	0
$369$	−18.0709	−0.940731
$370$	0	0
$371$	−13.3732	−0.694303
$372$	0	0
$373$	18.6080	0.963486	0.481743	−	0.876312i	$-0.340004\pi$
$373$	18.6080	0.963486	0.481743	+	0.876312i	$0.340004\pi$
$374$	0	0
$375$	21.1917	1.09433
$376$	0	0
$377$	13.6039	0.700638
$378$	0	0
$379$	21.4796	1.10333	0.551667	−	0.834064i	$-0.313992\pi$
$379$	21.4796	1.10333	0.551667	+	0.834064i	$0.313992\pi$
$380$	0	0
$381$	−54.1749	−2.77546
$382$	0	0
$383$	−14.9595	−0.764393	−0.382197	−	0.924081i	$-0.624832\pi$
$383$	−14.9595	−0.764393	−0.382197	+	0.924081i	$0.624832\pi$
$384$	0	0
$385$	5.75957	0.293535
$386$	0	0
$387$	25.7870	1.31083
$388$	0	0
$389$	15.6435	0.793159	0.396579	−	0.918000i	$-0.370197\pi$
$389$	15.6435	0.793159	0.396579	+	0.918000i	$0.370197\pi$
$390$	0	0
$391$	37.7415	1.90867
$392$	0	0
$393$	−10.1683	−0.512923
$394$	0	0
$395$	−12.5661	−0.632271
$396$	0	0
$397$	4.43839	0.222757	0.111378	−	0.993778i	$-0.464473\pi$
$397$	4.43839	0.222757	0.111378	+	0.993778i	$0.464473\pi$
$398$	0	0
$399$	11.9493	0.598215
$400$	0	0
$401$	2.80409	0.140030	0.0700148	−	0.997546i	$-0.477695\pi$
$401$	2.80409	0.140030	0.0700148	+	0.997546i	$0.477695\pi$
$402$	0	0
$403$	−37.4791	−1.86697
$404$	0	0
$405$	2.78307	0.138292
$406$	0	0
$407$	6.03429	0.299109
$408$	0	0
$409$	−33.9136	−1.67692	−0.838459	−	0.544965i	$-0.816543\pi$
$409$	−33.9136	−1.67692	−0.838459	+	0.544965i	$0.816543\pi$
$410$	0	0
$411$	−22.9344	−1.13127
$412$	0	0
$413$	−11.5033	−0.566039
$414$	0	0
$415$	−1.18879	−0.0583556
$416$	0	0
$417$	−29.3558	−1.43756
$418$	0	0
$419$	5.64806	0.275926	0.137963	−	0.990437i	$-0.455945\pi$
$419$	5.64806	0.275926	0.137963	+	0.990437i	$0.455945\pi$
$420$	0	0
$421$	4.46310	0.217518	0.108759	−	0.994068i	$-0.465312\pi$
$421$	4.46310	0.217518	0.108759	+	0.994068i	$0.465312\pi$
$422$	0	0
$423$	−68.7281	−3.34167
$424$	0	0
$425$	20.7172	1.00493
$426$	0	0
$427$	−24.6871	−1.19469
$428$	0	0
$429$	−42.9616	−2.07421
$430$	0	0
$431$	−28.3618	−1.36614	−0.683070	−	0.730353i	$-0.739356\pi$
$431$	−28.3618	−1.36614	−0.683070	+	0.730353i	$0.739356\pi$
$432$	0	0
$433$	3.44256	0.165439	0.0827195	−	0.996573i	$-0.473639\pi$
$433$	3.44256	0.165439	0.0827195	+	0.996573i	$0.473639\pi$
$434$	0	0
$435$	5.45295	0.261449
$436$	0	0
$437$	−11.8444	−0.566594
$438$	0	0
$439$	2.67693	0.127763	0.0638814	−	0.997957i	$-0.479652\pi$
$439$	2.67693	0.127763	0.0638814	+	0.997957i	$0.479652\pi$
$440$	0	0
$441$	4.26336	0.203017
$442$	0	0
$443$	36.2149	1.72062	0.860312	−	0.509768i	$-0.170269\pi$
$443$	36.2149	1.72062	0.860312	+	0.509768i	$0.170269\pi$
$444$	0	0
$445$	−2.61545	−0.123984
$446$	0	0
$447$	8.92316	0.422051
$448$	0	0
$449$	−30.8533	−1.45606	−0.728029	−	0.685546i	$-0.759564\pi$
$449$	−30.8533	−1.45606	−0.728029	+	0.685546i	$0.759564\pi$
$450$	0	0
$451$	8.92189	0.420115
$452$	0	0
$453$	25.3555	1.19131
$454$	0	0
$455$	12.2798	0.575688
$456$	0	0
$457$	0.671004	0.0313882	0.0156941	−	0.999877i	$-0.495004\pi$
$457$	0.671004	0.0313882	0.0156941	+	0.999877i	$0.495004\pi$
$458$	0	0
$459$	32.0279	1.49493
$460$	0	0
$461$	−19.5454	−0.910319	−0.455160	−	0.890410i	$-0.650418\pi$
$461$	−19.5454	−0.910319	−0.455160	+	0.890410i	$0.650418\pi$
$462$	0	0
$463$	−25.9998	−1.20831	−0.604156	−	0.796866i	$-0.706490\pi$
$463$	−25.9998	−1.20831	−0.604156	+	0.796866i	$0.706490\pi$
$464$	0	0
$465$	−15.0230	−0.696673
$466$	0	0
$467$	−31.4809	−1.45676	−0.728382	−	0.685171i	$-0.759727\pi$
$467$	−31.4809	−1.45676	−0.728382	+	0.685171i	$0.759727\pi$
$468$	0	0
$469$	36.0225	1.66337
$470$	0	0
$471$	−41.7388	−1.92322
$472$	0	0
$473$	−12.7315	−0.585395
$474$	0	0
$475$	−6.50165	−0.298316
$476$	0	0
$477$	25.6168	1.17291
$478$	0	0
$479$	−34.8031	−1.59020	−0.795098	−	0.606481i	$-0.792581\pi$
$479$	−34.8031	−1.59020	−0.795098	+	0.606481i	$0.792581\pi$
$480$	0	0
$481$	12.8656	0.586619
$482$	0	0
$483$	−64.5244	−2.93596
$484$	0	0
$485$	−5.14917	−0.233812
$486$	0	0
$487$	−0.231294	−0.0104809	−0.00524045	−	0.999986i	$-0.501668\pi$
$487$	−0.231294	−0.0104809	−0.00524045	+	0.999986i	$0.501668\pi$
$488$	0	0
$489$	−12.8077	−0.579182
$490$	0	0
$491$	−23.7554	−1.07207	−0.536033	−	0.844197i	$-0.680078\pi$
$491$	−23.7554	−1.07207	−0.536033	+	0.844197i	$0.680078\pi$
$492$	0	0
$493$	11.4025	0.513544
$494$	0	0
$495$	−11.0326	−0.495880
$496$	0	0
$497$	0.321514	0.0144219
$498$	0	0
$499$	−39.9082	−1.78654	−0.893268	−	0.449524i	$-0.851594\pi$
$499$	−39.9082	−1.78654	−0.893268	+	0.449524i	$0.851594\pi$
$500$	0	0
$501$	−16.2578	−0.726344
$502$	0	0
$503$	26.8139	1.19557	0.597786	−	0.801656i	$-0.296047\pi$
$503$	26.8139	1.19557	0.597786	+	0.801656i	$0.296047\pi$
$504$	0	0
$505$	−5.80565	−0.258348
$506$	0	0
$507$	−54.0349	−2.39978
$508$	0	0
$509$	−1.61309	−0.0714991	−0.0357496	−	0.999361i	$-0.511382\pi$
$509$	−1.61309	−0.0714991	−0.0357496	+	0.999361i	$0.511382\pi$
$510$	0	0
$511$	36.6190	1.61993
$512$	0	0
$513$	−10.0513	−0.443775
$514$	0	0
$515$	12.8450	0.566019
$516$	0	0
$517$	33.9322	1.49234
$518$	0	0
$519$	41.5551	1.82407
$520$	0	0
$521$	−10.6989	−0.468729	−0.234364	−	0.972149i	$-0.575301\pi$
$521$	−10.6989	−0.468729	−0.234364	+	0.972149i	$0.575301\pi$
$522$	0	0
$523$	18.7901	0.821636	0.410818	−	0.911717i	$-0.365243\pi$
$523$	18.7901	0.821636	0.410818	+	0.911717i	$0.365243\pi$
$524$	0	0
$525$	−35.4189	−1.54581
$526$	0	0
$527$	−31.4142	−1.36842
$528$	0	0
$529$	40.9577	1.78077
$530$	0	0
$531$	22.0349	0.956233
$532$	0	0
$533$	19.0221	0.823941
$534$	0	0
$535$	−10.9810	−0.474750
$536$	0	0
$537$	32.5551	1.40486
$538$	0	0
$539$	−2.10489	−0.0906641
$540$	0	0
$541$	27.6325	1.18801	0.594007	−	0.804460i	$-0.297545\pi$
$541$	27.6325	1.18801	0.594007	+	0.804460i	$0.297545\pi$
$542$	0	0
$543$	55.7135	2.39090
$544$	0	0
$545$	8.18873	0.350767
$546$	0	0
$547$	25.6756	1.09781	0.548905	−	0.835885i	$-0.315045\pi$
$547$	25.6756	1.09781	0.548905	+	0.835885i	$0.315045\pi$
$548$	0	0
$549$	47.2890	2.01825
$550$	0	0
$551$	−3.57845	−0.152447
$552$	0	0
$553$	44.9238	1.91035
$554$	0	0
$555$	5.15698	0.218902
$556$	0	0
$557$	18.1322	0.768284	0.384142	−	0.923274i	$-0.374497\pi$
$557$	18.1322	0.768284	0.384142	+	0.923274i	$0.374497\pi$
$558$	0	0
$559$	−27.1445	−1.14809
$560$	0	0
$561$	−36.0096	−1.52032
$562$	0	0
$563$	−30.9320	−1.30363	−0.651813	−	0.758379i	$-0.725991\pi$
$563$	−30.9320	−1.30363	−0.651813	+	0.758379i	$0.725991\pi$
$564$	0	0
$565$	0.639905	0.0269210
$566$	0	0
$567$	−9.94945	−0.417838
$568$	0	0
$569$	−38.2962	−1.60546	−0.802730	−	0.596343i	$-0.796620\pi$
$569$	−38.2962	−1.60546	−0.802730	+	0.596343i	$0.796620\pi$
$570$	0	0
$571$	−17.2551	−0.722105	−0.361053	−	0.932545i	$-0.617583\pi$
$571$	−17.2551	−0.722105	−0.361053	+	0.932545i	$0.617583\pi$
$572$	0	0
$573$	−38.9641	−1.62775
$574$	0	0
$575$	35.1078	1.46410
$576$	0	0
$577$	12.9425	0.538802	0.269401	−	0.963028i	$-0.413174\pi$
$577$	12.9425	0.538802	0.269401	+	0.963028i	$0.413174\pi$
$578$	0	0
$579$	26.3619	1.09556
$580$	0	0
$581$	4.24992	0.176317
$582$	0	0
$583$	−12.6475	−0.523805
$584$	0	0
$585$	−23.5224	−0.972533
$586$	0	0
$587$	5.80358	0.239540	0.119770	−	0.992802i	$-0.461784\pi$
$587$	5.80358	0.239540	0.119770	+	0.992802i	$0.461784\pi$
$588$	0	0
$589$	9.85869	0.406220
$590$	0	0
$591$	−65.7268	−2.70364
$592$	0	0
$593$	−20.4230	−0.838672	−0.419336	−	0.907831i	$-0.637737\pi$
$593$	−20.4230	−0.838672	−0.419336	+	0.907831i	$0.637737\pi$
$594$	0	0
$595$	10.2927	0.421960
$596$	0	0
$597$	1.13056	0.0462708
$598$	0	0
$599$	0.0499318	0.00204016	0.00102008	−	0.999999i	$-0.499675\pi$
$599$	0.0499318	0.00204016	0.00102008	+	0.999999i	$0.499675\pi$
$600$	0	0
$601$	−3.80073	−0.155035	−0.0775175	−	0.996991i	$-0.524699\pi$
$601$	−3.80073	−0.155035	−0.0775175	+	0.996991i	$0.524699\pi$
$602$	0	0
$603$	−69.0023	−2.80999
$604$	0	0
$605$	−3.14479	−0.127854
$606$	0	0
$607$	14.0871	0.571776	0.285888	−	0.958263i	$-0.407711\pi$
$607$	14.0871	0.571776	0.285888	+	0.958263i	$0.407711\pi$
$608$	0	0
$609$	−19.4942	−0.789946
$610$	0	0
$611$	72.3461	2.92681
$612$	0	0
$613$	2.97608	0.120203	0.0601015	−	0.998192i	$-0.480858\pi$
$613$	2.97608	0.120203	0.0601015	+	0.998192i	$0.480858\pi$
$614$	0	0
$615$	7.62476	0.307460
$616$	0	0
$617$	36.0901	1.45293	0.726466	−	0.687203i	$-0.241162\pi$
$617$	36.0901	1.45293	0.726466	+	0.687203i	$0.241162\pi$
$618$	0	0
$619$	−18.8252	−0.756650	−0.378325	−	0.925673i	$-0.623500\pi$
$619$	−18.8252	−0.756650	−0.378325	+	0.925673i	$0.623500\pi$
$620$	0	0
$621$	54.2752	2.17799
$622$	0	0
$623$	9.35021	0.374608
$624$	0	0
$625$	16.2211	0.648843
$626$	0	0
$627$	11.3009	0.451313
$628$	0	0
$629$	10.7837	0.429972
$630$	0	0
$631$	17.7946	0.708390	0.354195	−	0.935172i	$-0.384755\pi$
$631$	17.7946	0.708390	0.354195	+	0.935172i	$0.384755\pi$
$632$	0	0
$633$	−12.9378	−0.514232
$634$	0	0
$635$	14.6446	0.581153
$636$	0	0
$637$	−4.48779	−0.177813
$638$	0	0
$639$	−0.615870	−0.0243635
$640$	0	0
$641$	−49.0261	−1.93642	−0.968208	−	0.250148i	$-0.919521\pi$
$641$	−49.0261	−1.93642	−0.968208	+	0.250148i	$0.919521\pi$
$642$	0	0
$643$	−8.45615	−0.333478	−0.166739	−	0.986001i	$-0.553324\pi$
$643$	−8.45615	−0.333478	−0.166739	+	0.986001i	$0.553324\pi$
$644$	0	0
$645$	−10.8805	−0.428419
$646$	0	0
$647$	44.6161	1.75404	0.877021	−	0.480453i	$-0.159528\pi$
$647$	44.6161	1.75404	0.877021	+	0.480453i	$0.159528\pi$
$648$	0	0
$649$	−10.8790	−0.427038
$650$	0	0
$651$	53.7069	2.10494
$652$	0	0
$653$	−31.0409	−1.21473	−0.607363	−	0.794425i	$-0.707773\pi$
$653$	−31.0409	−1.21473	−0.607363	+	0.794425i	$0.707773\pi$
$654$	0	0
$655$	2.74870	0.107401
$656$	0	0
$657$	−70.1449	−2.73661
$658$	0	0
$659$	31.2861	1.21873	0.609367	−	0.792888i	$-0.291424\pi$
$659$	31.2861	1.21873	0.609367	+	0.792888i	$0.291424\pi$
$660$	0	0
$661$	19.4005	0.754591	0.377295	−	0.926093i	$-0.376854\pi$
$661$	19.4005	0.754591	0.377295	+	0.926093i	$0.376854\pi$
$662$	0	0
$663$	−76.7751	−2.98170
$664$	0	0
$665$	−3.23015	−0.125260
$666$	0	0
$667$	19.3230	0.748190
$668$	0	0
$669$	84.3797	3.26231
$670$	0	0
$671$	−23.3474	−0.901316
$672$	0	0
$673$	−44.9157	−1.73137	−0.865687	−	0.500586i	$-0.833118\pi$
$673$	−44.9157	−1.73137	−0.865687	+	0.500586i	$0.833118\pi$
$674$	0	0
$675$	29.7929	1.14673
$676$	0	0
$677$	−24.5289	−0.942723	−0.471361	−	0.881940i	$-0.656237\pi$
$677$	−24.5289	−0.942723	−0.471361	+	0.881940i	$0.656237\pi$
$678$	0	0
$679$	18.4082	0.706442
$680$	0	0
$681$	−16.2509	−0.622737
$682$	0	0
$683$	−2.56472	−0.0981363	−0.0490681	−	0.998795i	$-0.515625\pi$
$683$	−2.56472	−0.0981363	−0.0490681	+	0.998795i	$0.515625\pi$
$684$	0	0
$685$	6.19965	0.236876
$686$	0	0
$687$	−38.4071	−1.46532
$688$	0	0
$689$	−26.9654	−1.02730
$690$	0	0
$691$	−12.6135	−0.479841	−0.239921	−	0.970793i	$-0.577121\pi$
$691$	−12.6135	−0.479841	−0.239921	+	0.970793i	$0.577121\pi$
$692$	0	0
$693$	39.4416	1.49826
$694$	0	0
$695$	7.93549	0.301010
$696$	0	0
$697$	15.9440	0.603921
$698$	0	0
$699$	−15.1141	−0.571669
$700$	0	0
$701$	−27.9256	−1.05474	−0.527368	−	0.849637i	$-0.676821\pi$
$701$	−27.9256	−1.05474	−0.527368	+	0.849637i	$0.676821\pi$
$702$	0	0
$703$	−3.38423	−0.127639
$704$	0	0
$705$	28.9989	1.09216
$706$	0	0
$707$	20.7551	0.780578
$708$	0	0
$709$	27.5821	1.03587	0.517933	−	0.855421i	$-0.326702\pi$
$709$	27.5821	1.03587	0.517933	+	0.855421i	$0.326702\pi$
$710$	0	0
$711$	−86.0530	−3.22724
$712$	0	0
$713$	−53.2353	−1.99368
$714$	0	0
$715$	11.6134	0.434317
$716$	0	0
$717$	−63.4062	−2.36795
$718$	0	0
$719$	−32.8405	−1.22474	−0.612372	−	0.790570i	$-0.709784\pi$
$719$	−32.8405	−1.22474	−0.612372	+	0.790570i	$0.709784\pi$
$720$	0	0
$721$	−45.9208	−1.71018
$722$	0	0
$723$	1.52030	0.0565404
$724$	0	0
$725$	10.6068	0.393928
$726$	0	0
$727$	−48.1836	−1.78703	−0.893516	−	0.449032i	$-0.851769\pi$
$727$	−48.1836	−1.78703	−0.893516	+	0.449032i	$0.851769\pi$
$728$	0	0
$729$	−39.7700	−1.47296
$730$	0	0
$731$	−22.7520	−0.841512
$732$	0	0
$733$	−20.2090	−0.746438	−0.373219	−	0.927743i	$-0.621746\pi$
$733$	−20.2090	−0.746438	−0.373219	+	0.927743i	$0.621746\pi$
$734$	0	0
$735$	−1.79887	−0.0663522
$736$	0	0
$737$	34.0676	1.25490
$738$	0	0
$739$	20.7673	0.763938	0.381969	−	0.924175i	$-0.375246\pi$
$739$	20.7673	0.763938	0.381969	+	0.924175i	$0.375246\pi$
$740$	0	0
$741$	24.0943	0.885126
$742$	0	0
$743$	−2.28224	−0.0837272	−0.0418636	−	0.999123i	$-0.513329\pi$
$743$	−2.28224	−0.0837272	−0.0418636	+	0.999123i	$0.513329\pi$
$744$	0	0
$745$	−2.41211	−0.0883730
$746$	0	0
$747$	−8.14086	−0.297859
$748$	0	0
$749$	39.2570	1.43442
$750$	0	0
$751$	52.9750	1.93309	0.966543	−	0.256505i	$-0.0825710\pi$
$751$	52.9750	1.93309	0.966543	+	0.256505i	$0.0825710\pi$
$752$	0	0
$753$	2.88943	0.105297
$754$	0	0
$755$	−6.85412	−0.249447
$756$	0	0
$757$	8.08122	0.293717	0.146859	−	0.989158i	$-0.453084\pi$
$757$	8.08122	0.293717	0.146859	+	0.989158i	$0.453084\pi$
$758$	0	0
$759$	−61.0227	−2.21498
$760$	0	0
$761$	4.81830	0.174663	0.0873316	−	0.996179i	$-0.472166\pi$
$761$	4.81830	0.174663	0.0873316	+	0.996179i	$0.472166\pi$
$762$	0	0
$763$	−29.2746	−1.05981
$764$	0	0
$765$	−19.7160	−0.712834
$766$	0	0
$767$	−23.1949	−0.837517
$768$	0	0
$769$	9.95020	0.358813	0.179407	−	0.983775i	$-0.442582\pi$
$769$	9.95020	0.358813	0.179407	+	0.983775i	$0.442582\pi$
$770$	0	0
$771$	−27.1339	−0.977205
$772$	0	0
$773$	−43.4582	−1.56308	−0.781542	−	0.623853i	$-0.785566\pi$
$773$	−43.4582	−1.56308	−0.781542	+	0.623853i	$0.785566\pi$
$774$	0	0
$775$	−29.2220	−1.04969
$776$	0	0
$777$	−18.4362	−0.661393
$778$	0	0
$779$	−5.00369	−0.179276
$780$	0	0
$781$	0.304066	0.0108803
$782$	0	0
$783$	16.3977	0.586007
$784$	0	0
$785$	11.2829	0.402703
$786$	0	0
$787$	−39.1994	−1.39731	−0.698655	−	0.715459i	$-0.746218\pi$
$787$	−39.1994	−1.39731	−0.698655	+	0.715459i	$0.746218\pi$
$788$	0	0
$789$	36.1598	1.28732
$790$	0	0
$791$	−2.28765	−0.0813395
$792$	0	0
$793$	−49.7784	−1.76768
$794$	0	0
$795$	−10.8087	−0.383345
$796$	0	0
$797$	9.68352	0.343008	0.171504	−	0.985183i	$-0.445137\pi$
$797$	9.68352	0.343008	0.171504	+	0.985183i	$0.445137\pi$
$798$	0	0
$799$	60.6390	2.14525
$800$	0	0
$801$	−17.9106	−0.632841
$802$	0	0
$803$	34.6317	1.22213
$804$	0	0
$805$	17.4423	0.614760
$806$	0	0
$807$	47.3951	1.66838
$808$	0	0
$809$	6.88557	0.242084	0.121042	−	0.992647i	$-0.461376\pi$
$809$	6.88557	0.242084	0.121042	+	0.992647i	$0.461376\pi$
$810$	0	0
$811$	−55.7304	−1.95696	−0.978480	−	0.206343i	$-0.933844\pi$
$811$	−55.7304	−1.95696	−0.978480	+	0.206343i	$0.933844\pi$
$812$	0	0
$813$	65.2675	2.28903
$814$	0	0
$815$	3.46218	0.121275
$816$	0	0
$817$	7.14024	0.249805
$818$	0	0
$819$	84.0924	2.93843
$820$	0	0
$821$	−8.92213	−0.311384	−0.155692	−	0.987806i	$-0.549761\pi$
$821$	−8.92213	−0.311384	−0.155692	+	0.987806i	$0.549761\pi$
$822$	0	0
$823$	−51.2598	−1.78680	−0.893402	−	0.449258i	$-0.851688\pi$
$823$	−51.2598	−1.78680	−0.893402	+	0.449258i	$0.851688\pi$
$824$	0	0
$825$	−33.4967	−1.16621
$826$	0	0
$827$	−54.7928	−1.90533	−0.952666	−	0.304019i	$-0.901671\pi$
$827$	−54.7928	−1.90533	−0.952666	+	0.304019i	$0.901671\pi$
$828$	0	0
$829$	−4.24929	−0.147584	−0.0737919	−	0.997274i	$-0.523510\pi$
$829$	−4.24929	−0.147584	−0.0737919	+	0.997274i	$0.523510\pi$
$830$	0	0
$831$	−50.3760	−1.74752
$832$	0	0
$833$	−3.76157	−0.130331
$834$	0	0
$835$	4.39482	0.152089
$836$	0	0
$837$	−45.1760	−1.56151
$838$	0	0
$839$	19.5248	0.674070	0.337035	−	0.941492i	$-0.390576\pi$
$839$	19.5248	0.674070	0.337035	+	0.941492i	$0.390576\pi$
$840$	0	0
$841$	−23.1621	−0.798693
$842$	0	0
$843$	−92.8885	−3.19925
$844$	0	0
$845$	14.6068	0.502488
$846$	0	0
$847$	11.2426	0.386300
$848$	0	0
$849$	−43.0291	−1.47675
$850$	0	0
$851$	18.2742	0.626433
$852$	0	0
$853$	−38.3582	−1.31336	−0.656680	−	0.754169i	$-0.728040\pi$
$853$	−38.3582	−1.31336	−0.656680	+	0.754169i	$0.728040\pi$
$854$	0	0
$855$	6.18746	0.211607
$856$	0	0
$857$	−27.3987	−0.935923	−0.467962	−	0.883749i	$-0.655011\pi$
$857$	−27.3987	−0.935923	−0.467962	+	0.883749i	$0.655011\pi$
$858$	0	0
$859$	−9.46444	−0.322922	−0.161461	−	0.986879i	$-0.551621\pi$
$859$	−9.46444	−0.322922	−0.161461	+	0.986879i	$0.551621\pi$
$860$	0	0
$861$	−27.2584	−0.928965
$862$	0	0
$863$	−8.09245	−0.275470	−0.137735	−	0.990469i	$-0.543982\pi$
$863$	−8.09245	−0.275470	−0.137735	+	0.990469i	$0.543982\pi$
$864$	0	0
$865$	−11.2332	−0.381941
$866$	0	0
$867$	−15.2311	−0.517275
$868$	0	0
$869$	42.4858	1.44123
$870$	0	0
$871$	72.6348	2.46114
$872$	0	0
$873$	−35.2615	−1.19342
$874$	0	0
$875$	20.4795	0.692333
$876$	0	0
$877$	39.1573	1.32225	0.661125	−	0.750276i	$-0.270079\pi$
$877$	39.1573	1.32225	0.661125	+	0.750276i	$0.270079\pi$
$878$	0	0
$879$	−84.5752	−2.85265
$880$	0	0
$881$	−0.430756	−0.0145125	−0.00725626	−	0.999974i	$-0.502310\pi$
$881$	−0.430756	−0.0145125	−0.00725626	+	0.999974i	$0.502310\pi$
$882$	0	0
$883$	−27.2856	−0.918235	−0.459117	−	0.888376i	$-0.651834\pi$
$883$	−27.2856	−0.918235	−0.459117	+	0.888376i	$0.651834\pi$
$884$	0	0
$885$	−9.29733	−0.312526
$886$	0	0
$887$	−46.9276	−1.57567	−0.787837	−	0.615884i	$-0.788799\pi$
$887$	−46.9276	−1.57567	−0.787837	+	0.615884i	$0.788799\pi$
$888$	0	0
$889$	−52.3543	−1.75591
$890$	0	0
$891$	−9.40950	−0.315230
$892$	0	0
$893$	−19.0303	−0.636824
$894$	0	0
$895$	−8.80033	−0.294163
$896$	0	0
$897$	−130.105	−4.34408
$898$	0	0
$899$	−16.0835	−0.536416
$900$	0	0
$901$	−22.6018	−0.752976
$902$	0	0
$903$	38.8977	1.29443
$904$	0	0
$905$	−15.0605	−0.500629
$906$	0	0
$907$	11.4138	0.378988	0.189494	−	0.981882i	$-0.439315\pi$
$907$	11.4138	0.378988	0.189494	+	0.981882i	$0.439315\pi$
$908$	0	0
$909$	−39.7572	−1.31866
$910$	0	0
$911$	10.5420	0.349272	0.174636	−	0.984633i	$-0.444125\pi$
$911$	10.5420	0.349272	0.174636	+	0.984633i	$0.444125\pi$
$912$	0	0
$913$	4.01928	0.133019
$914$	0	0
$915$	−19.9530	−0.659625
$916$	0	0
$917$	−9.82657	−0.324502
$918$	0	0
$919$	28.0648	0.925774	0.462887	−	0.886417i	$-0.346814\pi$
$919$	28.0648	0.925774	0.462887	+	0.886417i	$0.346814\pi$
$920$	0	0
$921$	92.1168	3.03535
$922$	0	0
$923$	0.648291	0.0213388
$924$	0	0
$925$	10.0311	0.329822
$926$	0	0
$927$	87.9627	2.88907
$928$	0	0
$929$	−17.4591	−0.572815	−0.286408	−	0.958108i	$-0.592461\pi$
$929$	−17.4591	−0.572815	−0.286408	+	0.958108i	$0.592461\pi$
$930$	0	0
$931$	1.18049	0.0386890
$932$	0	0
$933$	76.9319	2.51864
$934$	0	0
$935$	9.73413	0.318340
$936$	0	0
$937$	55.8318	1.82395	0.911973	−	0.410250i	$-0.134558\pi$
$937$	55.8318	1.82395	0.911973	+	0.410250i	$0.134558\pi$
$938$	0	0
$939$	−14.0553	−0.458677
$940$	0	0
$941$	−25.7239	−0.838576	−0.419288	−	0.907853i	$-0.637720\pi$
$941$	−25.7239	−0.838576	−0.419288	+	0.907853i	$0.637720\pi$
$942$	0	0
$943$	27.0191	0.879861
$944$	0	0
$945$	14.8017	0.481499
$946$	0	0
$947$	−9.01405	−0.292917	−0.146459	−	0.989217i	$-0.546788\pi$
$947$	−9.01405	−0.292917	−0.146459	+	0.989217i	$0.546788\pi$
$948$	0	0
$949$	73.8374	2.39686
$950$	0	0
$951$	−2.17760	−0.0706134
$952$	0	0
$953$	27.4373	0.888781	0.444390	−	0.895833i	$-0.353420\pi$
$953$	27.4373	0.888781	0.444390	+	0.895833i	$0.353420\pi$
$954$	0	0
$955$	10.5328	0.340833
$956$	0	0
$957$	−18.4363	−0.595960
$958$	0	0
$959$	−22.1637	−0.715702
$960$	0	0
$961$	13.3104	0.429368
$962$	0	0
$963$	−75.1980	−2.42322
$964$	0	0
$965$	−7.12618	−0.229400
$966$	0	0
$967$	23.3450	0.750723	0.375362	−	0.926878i	$-0.377518\pi$
$967$	23.3450	0.750723	0.375362	+	0.926878i	$0.377518\pi$
$968$	0	0
$969$	20.1953	0.648768
$970$	0	0
$971$	−38.6772	−1.24121	−0.620605	−	0.784123i	$-0.713113\pi$
$971$	−38.6772	−1.24121	−0.620605	+	0.784123i	$0.713113\pi$
$972$	0	0
$973$	−28.3693	−0.909478
$974$	0	0
$975$	−71.4175	−2.28719
$976$	0	0
$977$	14.4914	0.463620	0.231810	−	0.972761i	$-0.425535\pi$
$977$	14.4914	0.463620	0.231810	+	0.972761i	$0.425535\pi$
$978$	0	0
$979$	8.84278	0.282616
$980$	0	0
$981$	56.0765	1.79038
$982$	0	0
$983$	44.7427	1.42707	0.713535	−	0.700619i	$-0.247093\pi$
$983$	44.7427	1.42707	0.713535	+	0.700619i	$0.247093\pi$
$984$	0	0
$985$	17.7673	0.566114
$986$	0	0
$987$	−103.671	−3.29988
$988$	0	0
$989$	−38.5561	−1.22601
$990$	0	0
$991$	48.3006	1.53432	0.767159	−	0.641457i	$-0.221670\pi$
$991$	48.3006	1.53432	0.767159	+	0.641457i	$0.221670\pi$
$992$	0	0
$993$	−53.3852	−1.69413
$994$	0	0
$995$	−0.305614	−0.00968863
$996$	0	0
$997$	42.3665	1.34176	0.670881	−	0.741565i	$-0.265916\pi$
$997$	42.3665	1.34176	0.670881	+	0.741565i	$0.265916\pi$
$998$	0	0
$999$	15.5077	0.490642

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.d.1.1		5
4.3	odd	2		502.2.a.d.1.5	✓	5
12.11	even	2		4518.2.a.t.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.d.1.5	✓	5	4.3	odd	2
4016.2.a.d.1.1		5	1.1	even	1	trivial
4518.2.a.t.1.4		5	12.11	even	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.88943 q^{3} +0.781072 q^{5} -2.79232 q^{7} +5.34878 q^{9} +O(q^{10})\)	\(q-2.88943 q^{3} +0.781072 q^{5} -2.79232 q^{7} +5.34878 q^{9} -2.64079 q^{11} -5.63036 q^{13} -2.25685 q^{15} -4.71925 q^{17} +1.48104 q^{19} +8.06821 q^{21} -7.99736 q^{23} -4.38993 q^{25} -6.78664 q^{27} -2.41618 q^{29} +6.65661 q^{31} +7.63036 q^{33} -2.18101 q^{35} -2.28504 q^{37} +16.2685 q^{39} -3.37850 q^{41} +4.82110 q^{43} +4.17779 q^{45} -12.8493 q^{47} +0.797070 q^{49} +13.6359 q^{51} +4.78928 q^{53} -2.06264 q^{55} -4.27935 q^{57} +4.11961 q^{59} +8.84107 q^{61} -14.9355 q^{63} -4.39771 q^{65} -12.9006 q^{67} +23.1078 q^{69} -0.115142 q^{71} -13.1142 q^{73} +12.6844 q^{75} +7.37393 q^{77} -16.0883 q^{79} +3.56314 q^{81} -1.52200 q^{83} -3.68607 q^{85} +6.98136 q^{87} -3.34854 q^{89} +15.7218 q^{91} -19.2338 q^{93} +1.15680 q^{95} -6.59243 q^{97} -14.1250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9	\( 5 q - 2 q^{3} + 7 q^{5} - q^{7} + 7 q^{9} - 7 q^{11} + 4 q^{13} - 3 q^{15} + 6 q^{17} + 10 q^{19} + 2 q^{21} - 13 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{29} + 13 q^{31} + 6 q^{33} - 13 q^{35} + 16 q^{37} + 16 q^{39} + 6 q^{41} + 8 q^{43} + 26 q^{45} - 29 q^{47} + 4 q^{49} + 7 q^{51} + 25 q^{53} - q^{55} + 19 q^{57} - 11 q^{59} + 11 q^{61} - 15 q^{63} + 7 q^{65} + 6 q^{67} - 12 q^{69} - 15 q^{71} - 9 q^{73} + 25 q^{75} + 12 q^{77} + 29 q^{79} - 7 q^{81} + 9 q^{83} - 7 q^{85} + 11 q^{87} - 9 q^{89} + 34 q^{91} - 8 q^{93} + 13 q^{95} - 6 q^{97} + q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 7 * q^5 - q^7 + 7 * q^9 - 7 * q^11 + 4 * q^13 - 3 * q^15 + 6 * q^17 + 10 * q^19 + 2 * q^21 - 13 * q^23 + 6 * q^25 - 8 * q^27 + 4 * q^29 + 13 * q^31 + 6 * q^33 - 13 * q^35 + 16 * q^37 + 16 * q^39 + 6 * q^41 + 8 * q^43 + 26 * q^45 - 29 * q^47 + 4 * q^49 + 7 * q^51 + 25 * q^53 - q^55 + 19 * q^57 - 11 * q^59 + 11 * q^61 - 15 * q^63 + 7 * q^65 + 6 * q^67 - 12 * q^69 - 15 * q^71 - 9 * q^73 + 25 * q^75 + 12 * q^77 + 29 * q^79 - 7 * q^81 + 9 * q^83 - 7 * q^85 + 11 * q^87 - 9 * q^89 + 34 * q^91 - 8 * q^93 + 13 * q^95 - 6 * q^97 + q^99

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