LMFDB - Embedded newform 8048.2.a.v.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11208 q^{3} +2.26631 q^{5} +4.06526 q^{7} +1.46088 q^{9} +O(q^{10})$	$q-2.11208 q^{3} +2.26631 q^{5} +4.06526 q^{7} +1.46088 q^{9} +1.00233 q^{11} -5.09690 q^{13} -4.78662 q^{15} +1.45394 q^{17} -6.53873 q^{19} -8.58614 q^{21} +4.54695 q^{23} +0.136156 q^{25} +3.25075 q^{27} -6.23661 q^{29} +10.1104 q^{31} -2.11699 q^{33} +9.21313 q^{35} +3.44313 q^{37} +10.7651 q^{39} -9.13103 q^{41} -2.78427 q^{43} +3.31079 q^{45} -6.12193 q^{47} +9.52632 q^{49} -3.07084 q^{51} -3.12167 q^{53} +2.27158 q^{55} +13.8103 q^{57} -2.23107 q^{59} -13.6990 q^{61} +5.93883 q^{63} -11.5512 q^{65} -14.1564 q^{67} -9.60351 q^{69} -13.8371 q^{71} +0.606303 q^{73} -0.287572 q^{75} +4.07472 q^{77} +14.1692 q^{79} -11.2485 q^{81} +6.43050 q^{83} +3.29508 q^{85} +13.1722 q^{87} +16.6860 q^{89} -20.7202 q^{91} -21.3540 q^{93} -14.8188 q^{95} -17.0100 q^{97} +1.46427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	−2.11208	−1.21941	−0.609705	−	0.792629i	$-0.708712\pi$
$3$	−2.11208	−1.21941	−0.609705	+	0.792629i	$0.708712\pi$
$4$	0	0
$5$	2.26631	1.01352	0.506762	−	0.862086i	$-0.330842\pi$
$5$	2.26631	1.01352	0.506762	+	0.862086i	$0.330842\pi$
$6$	0	0
$7$	4.06526	1.53652	0.768261	−	0.640136i	$-0.221122\pi$
$7$	4.06526	1.53652	0.768261	+	0.640136i	$0.221122\pi$
$8$	0	0
$9$	1.46088	0.486958
$10$	0	0
$11$	1.00233	0.302213	0.151106	−	0.988517i	$-0.451716\pi$
$11$	1.00233	0.302213	0.151106	+	0.988517i	$0.451716\pi$
$12$	0	0
$13$	−5.09690	−1.41363	−0.706813	−	0.707400i	$-0.749868\pi$
$13$	−5.09690	−1.41363	−0.706813	+	0.707400i	$0.749868\pi$
$14$	0	0
$15$	−4.78662	−1.23590
$16$	0	0
$17$	1.45394	0.352633	0.176316	−	0.984334i	$-0.443582\pi$
$17$	1.45394	0.352633	0.176316	+	0.984334i	$0.443582\pi$
$18$	0	0
$19$	−6.53873	−1.50009	−0.750044	−	0.661388i	$-0.769968\pi$
$19$	−6.53873	−1.50009	−0.750044	+	0.661388i	$0.769968\pi$
$20$	0	0
$21$	−8.58614	−1.87365
$22$	0	0
$23$	4.54695	0.948104	0.474052	−	0.880497i	$-0.342791\pi$
$23$	4.54695	0.948104	0.474052	+	0.880497i	$0.342791\pi$
$24$	0	0
$25$	0.136156	0.0272311
$26$	0	0
$27$	3.25075	0.625608
$28$	0	0
$29$	−6.23661	−1.15811	−0.579055	−	0.815288i	$-0.696578\pi$
$29$	−6.23661	−1.15811	−0.579055	+	0.815288i	$0.696578\pi$
$30$	0	0
$31$	10.1104	1.81588	0.907942	−	0.419095i	$-0.137653\pi$
$31$	10.1104	1.81588	0.907942	+	0.419095i	$0.137653\pi$
$32$	0	0
$33$	−2.11699	−0.368521
$34$	0	0
$35$	9.21313	1.55730
$36$	0	0
$37$	3.44313	0.566047	0.283024	−	0.959113i	$-0.408663\pi$
$37$	3.44313	0.566047	0.283024	+	0.959113i	$0.408663\pi$
$38$	0	0
$39$	10.7651	1.72379
$40$	0	0
$41$	−9.13103	−1.42603	−0.713013	−	0.701150i	$-0.752670\pi$
$41$	−9.13103	−1.42603	−0.713013	+	0.701150i	$0.752670\pi$
$42$	0	0
$43$	−2.78427	−0.424597	−0.212299	−	0.977205i	$-0.568095\pi$
$43$	−2.78427	−0.424597	−0.212299	+	0.977205i	$0.568095\pi$
$44$	0	0
$45$	3.31079	0.493544
$46$	0	0
$47$	−6.12193	−0.892976	−0.446488	−	0.894790i	$-0.647325\pi$
$47$	−6.12193	−0.892976	−0.446488	+	0.894790i	$0.647325\pi$
$48$	0	0
$49$	9.52632	1.36090
$50$	0	0
$51$	−3.07084	−0.430004
$52$	0	0
$53$	−3.12167	−0.428794	−0.214397	−	0.976747i	$-0.568779\pi$
$53$	−3.12167	−0.428794	−0.214397	+	0.976747i	$0.568779\pi$
$54$	0	0
$55$	2.27158	0.306300
$56$	0	0
$57$	13.8103	1.82922
$58$	0	0
$59$	−2.23107	−0.290461	−0.145230	−	0.989398i	$-0.546392\pi$
$59$	−2.23107	−0.290461	−0.145230	+	0.989398i	$0.546392\pi$
$60$	0	0
$61$	−13.6990	−1.75398	−0.876992	−	0.480506i	$-0.840453\pi$
$61$	−13.6990	−1.75398	−0.876992	+	0.480506i	$0.840453\pi$
$62$	0	0
$63$	5.93883	0.748223
$64$	0	0
$65$	−11.5512	−1.43274
$66$	0	0
$67$	−14.1564	−1.72948	−0.864742	−	0.502216i	$-0.832518\pi$
$67$	−14.1564	−1.72948	−0.864742	+	0.502216i	$0.832518\pi$
$68$	0	0
$69$	−9.60351	−1.15613
$70$	0	0
$71$	−13.8371	−1.64216	−0.821078	−	0.570816i	$-0.806627\pi$
$71$	−13.8371	−1.64216	−0.821078	+	0.570816i	$0.806627\pi$
$72$	0	0
$73$	0.606303	0.0709624	0.0354812	−	0.999370i	$-0.488704\pi$
$73$	0.606303	0.0709624	0.0354812	+	0.999370i	$0.488704\pi$
$74$	0	0
$75$	−0.287572	−0.0332059
$76$	0	0
$77$	4.07472	0.464357
$78$	0	0
$79$	14.1692	1.59416	0.797082	−	0.603871i	$-0.206376\pi$
$79$	14.1692	1.59416	0.797082	+	0.603871i	$0.206376\pi$
$80$	0	0
$81$	−11.2485	−1.24983
$82$	0	0
$83$	6.43050	0.705839	0.352920	−	0.935654i	$-0.385189\pi$
$83$	6.43050	0.705839	0.352920	+	0.935654i	$0.385189\pi$
$84$	0	0
$85$	3.29508	0.357402
$86$	0	0
$87$	13.1722	1.41221
$88$	0	0
$89$	16.6860	1.76872	0.884358	−	0.466808i	$-0.154596\pi$
$89$	16.6860	1.76872	0.884358	+	0.466808i	$0.154596\pi$
$90$	0	0
$91$	−20.7202	−2.17207
$92$	0	0
$93$	−21.3540	−2.21431
$94$	0	0
$95$	−14.8188	−1.52038
$96$	0	0
$97$	−17.0100	−1.72710	−0.863552	−	0.504260i	$-0.831765\pi$
$97$	−17.0100	−1.72710	−0.863552	+	0.504260i	$0.831765\pi$
$98$	0	0
$99$	1.46427	0.147165
$100$	0	0
$101$	−7.32097	−0.728463	−0.364232	−	0.931308i	$-0.618668\pi$
$101$	−7.32097	−0.728463	−0.364232	+	0.931308i	$0.618668\pi$
$102$	0	0
$103$	−3.44714	−0.339657	−0.169828	−	0.985474i	$-0.554321\pi$
$103$	−3.44714	−0.339657	−0.169828	+	0.985474i	$0.554321\pi$
$104$	0	0
$105$	−19.4589	−1.89899
$106$	0	0
$107$	5.62609	0.543895	0.271948	−	0.962312i	$-0.412332\pi$
$107$	5.62609	0.543895	0.271948	+	0.962312i	$0.412332\pi$
$108$	0	0
$109$	−1.02306	−0.0979914	−0.0489957	−	0.998799i	$-0.515602\pi$
$109$	−1.02306	−0.0979914	−0.0489957	+	0.998799i	$0.515602\pi$
$110$	0	0
$111$	−7.27216	−0.690243
$112$	0	0
$113$	−2.18647	−0.205686	−0.102843	−	0.994698i	$-0.532794\pi$
$113$	−2.18647	−0.205686	−0.102843	+	0.994698i	$0.532794\pi$
$114$	0	0
$115$	10.3048	0.960926
$116$	0	0
$117$	−7.44594	−0.688377
$118$	0	0
$119$	5.91065	0.541828
$120$	0	0
$121$	−9.99534	−0.908667
$122$	0	0
$123$	19.2854	1.73891
$124$	0	0
$125$	−11.0230	−0.985925
$126$	0	0
$127$	12.8538	1.14059	0.570296	−	0.821439i	$-0.306829\pi$
$127$	12.8538	1.14059	0.570296	+	0.821439i	$0.306829\pi$
$128$	0	0
$129$	5.88060	0.517758
$130$	0	0
$131$	4.41645	0.385867	0.192933	−	0.981212i	$-0.438200\pi$
$131$	4.41645	0.385867	0.192933	+	0.981212i	$0.438200\pi$
$132$	0	0
$133$	−26.5816	−2.30492
$134$	0	0
$135$	7.36721	0.634068
$136$	0	0
$137$	−9.10428	−0.777831	−0.388915	−	0.921273i	$-0.627150\pi$
$137$	−9.10428	−0.777831	−0.388915	+	0.921273i	$0.627150\pi$
$138$	0	0
$139$	−13.5113	−1.14602	−0.573008	−	0.819549i	$-0.694224\pi$
$139$	−13.5113	−1.14602	−0.573008	+	0.819549i	$0.694224\pi$
$140$	0	0
$141$	12.9300	1.08890
$142$	0	0
$143$	−5.10876	−0.427216
$144$	0	0
$145$	−14.1341	−1.17377
$146$	0	0
$147$	−20.1203	−1.65950
$148$	0	0
$149$	7.19494	0.589433	0.294716	−	0.955585i	$-0.404775\pi$
$149$	7.19494	0.589433	0.294716	+	0.955585i	$0.404775\pi$
$150$	0	0
$151$	−10.9314	−0.889582	−0.444791	−	0.895634i	$-0.646722\pi$
$151$	−10.9314	−0.889582	−0.444791	+	0.895634i	$0.646722\pi$
$152$	0	0
$153$	2.12403	0.171717
$154$	0	0
$155$	22.9133	1.84044
$156$	0	0
$157$	9.72181	0.775885	0.387942	−	0.921684i	$-0.373186\pi$
$157$	9.72181	0.775885	0.387942	+	0.921684i	$0.373186\pi$
$158$	0	0
$159$	6.59321	0.522876
$160$	0	0
$161$	18.4845	1.45678
$162$	0	0
$163$	8.78257	0.687904	0.343952	−	0.938987i	$-0.388234\pi$
$163$	8.78257	0.687904	0.343952	+	0.938987i	$0.388234\pi$
$164$	0	0
$165$	−4.79776	−0.373505
$166$	0	0
$167$	−22.3161	−1.72687	−0.863436	−	0.504458i	$-0.831692\pi$
$167$	−22.3161	−1.72687	−0.863436	+	0.504458i	$0.831692\pi$
$168$	0	0
$169$	12.9784	0.998339
$170$	0	0
$171$	−9.55228	−0.730481
$172$	0	0
$173$	17.9889	1.36767	0.683836	−	0.729635i	$-0.260310\pi$
$173$	17.9889	1.36767	0.683836	+	0.729635i	$0.260310\pi$
$174$	0	0
$175$	0.553508	0.0418413
$176$	0	0
$177$	4.71219	0.354190
$178$	0	0
$179$	−15.9614	−1.19301	−0.596505	−	0.802609i	$-0.703445\pi$
$179$	−15.9614	−1.19301	−0.596505	+	0.802609i	$0.703445\pi$
$180$	0	0
$181$	8.14218	0.605204	0.302602	−	0.953117i	$-0.402145\pi$
$181$	8.14218	0.605204	0.302602	+	0.953117i	$0.402145\pi$
$182$	0	0
$183$	28.9335	2.13882
$184$	0	0
$185$	7.80320	0.573702
$186$	0	0
$187$	1.45733	0.106570
$188$	0	0
$189$	13.2151	0.961260
$190$	0	0
$191$	−7.81409	−0.565407	−0.282704	−	0.959207i	$-0.591231\pi$
$191$	−7.81409	−0.565407	−0.282704	+	0.959207i	$0.591231\pi$
$192$	0	0
$193$	−23.6828	−1.70473	−0.852363	−	0.522951i	$-0.824831\pi$
$193$	−23.6828	−1.70473	−0.852363	+	0.522951i	$0.824831\pi$
$194$	0	0
$195$	24.3969	1.74710
$196$	0	0
$197$	−3.36345	−0.239636	−0.119818	−	0.992796i	$-0.538231\pi$
$197$	−3.36345	−0.239636	−0.119818	+	0.992796i	$0.538231\pi$
$198$	0	0
$199$	18.0958	1.28278	0.641389	−	0.767216i	$-0.278358\pi$
$199$	18.0958	1.28278	0.641389	+	0.767216i	$0.278358\pi$
$200$	0	0
$201$	29.8995	2.10895
$202$	0	0
$203$	−25.3534	−1.77946
$204$	0	0
$205$	−20.6937	−1.44531
$206$	0	0
$207$	6.64252	0.461687
$208$	0	0
$209$	−6.55395	−0.453346
$210$	0	0
$211$	−4.83062	−0.332554	−0.166277	−	0.986079i	$-0.553175\pi$
$211$	−4.83062	−0.332554	−0.166277	+	0.986079i	$0.553175\pi$
$212$	0	0
$213$	29.2249	2.00246
$214$	0	0
$215$	−6.31002	−0.430340
$216$	0	0
$217$	41.1014	2.79015
$218$	0	0
$219$	−1.28056	−0.0865322
$220$	0	0
$221$	−7.41060	−0.498491
$222$	0	0
$223$	24.8400	1.66341	0.831705	−	0.555217i	$-0.187365\pi$
$223$	24.8400	1.66341	0.831705	+	0.555217i	$0.187365\pi$
$224$	0	0
$225$	0.198907	0.0132604
$226$	0	0
$227$	−3.83084	−0.254262	−0.127131	−	0.991886i	$-0.540577\pi$
$227$	−3.83084	−0.254262	−0.127131	+	0.991886i	$0.540577\pi$
$228$	0	0
$229$	22.2951	1.47330	0.736651	−	0.676273i	$-0.236406\pi$
$229$	22.2951	1.47330	0.736651	+	0.676273i	$0.236406\pi$
$230$	0	0
$231$	−8.60612	−0.566241
$232$	0	0
$233$	20.5463	1.34603	0.673017	−	0.739627i	$-0.264998\pi$
$233$	20.5463	1.34603	0.673017	+	0.739627i	$0.264998\pi$
$234$	0	0
$235$	−13.8742	−0.905052
$236$	0	0
$237$	−29.9266	−1.94394
$238$	0	0
$239$	22.2656	1.44024	0.720120	−	0.693849i	$-0.244087\pi$
$239$	22.2656	1.44024	0.720120	+	0.693849i	$0.244087\pi$
$240$	0	0
$241$	3.08389	0.198651	0.0993253	−	0.995055i	$-0.468332\pi$
$241$	3.08389	0.198651	0.0993253	+	0.995055i	$0.468332\pi$
$242$	0	0
$243$	14.0054	0.898446
$244$	0	0
$245$	21.5896	1.37931
$246$	0	0
$247$	33.3273	2.12056
$248$	0	0
$249$	−13.5817	−0.860707
$250$	0	0
$251$	−9.34145	−0.589627	−0.294814	−	0.955555i	$-0.595258\pi$
$251$	−9.34145	−0.589627	−0.294814	+	0.955555i	$0.595258\pi$
$252$	0	0
$253$	4.55753	0.286529
$254$	0	0
$255$	−6.95947	−0.435819
$256$	0	0
$257$	−4.88888	−0.304960	−0.152480	−	0.988307i	$-0.548726\pi$
$257$	−4.88888	−0.304960	−0.152480	+	0.988307i	$0.548726\pi$
$258$	0	0
$259$	13.9972	0.869744
$260$	0	0
$261$	−9.11092	−0.563952
$262$	0	0
$263$	5.72421	0.352970	0.176485	−	0.984303i	$-0.443527\pi$
$263$	5.72421	0.352970	0.176485	+	0.984303i	$0.443527\pi$
$264$	0	0
$265$	−7.07467	−0.434594
$266$	0	0
$267$	−35.2422	−2.15679
$268$	0	0
$269$	−16.5404	−1.00848	−0.504242	−	0.863562i	$-0.668228\pi$
$269$	−16.5404	−1.00848	−0.504242	+	0.863562i	$0.668228\pi$
$270$	0	0
$271$	−3.11644	−0.189310	−0.0946550	−	0.995510i	$-0.530175\pi$
$271$	−3.11644	−0.189310	−0.0946550	+	0.995510i	$0.530175\pi$
$272$	0	0
$273$	43.7627	2.64864
$274$	0	0
$275$	0.136473	0.00822960
$276$	0	0
$277$	−6.30954	−0.379103	−0.189552	−	0.981871i	$-0.560703\pi$
$277$	−6.30954	−0.379103	−0.189552	+	0.981871i	$0.560703\pi$
$278$	0	0
$279$	14.7701	0.884260
$280$	0	0
$281$	−21.7959	−1.30023	−0.650117	−	0.759834i	$-0.725280\pi$
$281$	−21.7959	−1.30023	−0.650117	+	0.759834i	$0.725280\pi$
$282$	0	0
$283$	−0.424595	−0.0252396	−0.0126198	−	0.999920i	$-0.504017\pi$
$283$	−0.424595	−0.0252396	−0.0126198	+	0.999920i	$0.504017\pi$
$284$	0	0
$285$	31.2984	1.85396
$286$	0	0
$287$	−37.1200	−2.19112
$288$	0	0
$289$	−14.8861	−0.875650
$290$	0	0
$291$	35.9265	2.10605
$292$	0	0
$293$	9.78177	0.571457	0.285728	−	0.958311i	$-0.407764\pi$
$293$	9.78177	0.571457	0.285728	+	0.958311i	$0.407764\pi$
$294$	0	0
$295$	−5.05629	−0.294389
$296$	0	0
$297$	3.25832	0.189067
$298$	0	0
$299$	−23.1753	−1.34026
$300$	0	0
$301$	−11.3188	−0.652404
$302$	0	0
$303$	15.4625	0.888295
$304$	0	0
$305$	−31.0463	−1.77770
$306$	0	0
$307$	−1.13123	−0.0645625	−0.0322813	−	0.999479i	$-0.510277\pi$
$307$	−1.13123	−0.0645625	−0.0322813	+	0.999479i	$0.510277\pi$
$308$	0	0
$309$	7.28063	0.414181
$310$	0	0
$311$	−10.9793	−0.622578	−0.311289	−	0.950315i	$-0.600761\pi$
$311$	−10.9793	−0.622578	−0.311289	+	0.950315i	$0.600761\pi$
$312$	0	0
$313$	19.4180	1.09757	0.548784	−	0.835964i	$-0.315091\pi$
$313$	19.4180	1.09757	0.548784	+	0.835964i	$0.315091\pi$
$314$	0	0
$315$	13.4592	0.758342
$316$	0	0
$317$	30.5541	1.71609	0.858043	−	0.513577i	$-0.171680\pi$
$317$	30.5541	1.71609	0.858043	+	0.513577i	$0.171680\pi$
$318$	0	0
$319$	−6.25113	−0.349996
$320$	0	0
$321$	−11.8828	−0.663230
$322$	0	0
$323$	−9.50694	−0.528980
$324$	0	0
$325$	−0.693972	−0.0384947
$326$	0	0
$327$	2.16078	0.119492
$328$	0	0
$329$	−24.8872	−1.37208
$330$	0	0
$331$	8.49611	0.466989	0.233494	−	0.972358i	$-0.424984\pi$
$331$	8.49611	0.466989	0.233494	+	0.972358i	$0.424984\pi$
$332$	0	0
$333$	5.02998	0.275641
$334$	0	0
$335$	−32.0829	−1.75287
$336$	0	0
$337$	−7.03611	−0.383282	−0.191641	−	0.981465i	$-0.561381\pi$
$337$	−7.03611	−0.383282	−0.191641	+	0.981465i	$0.561381\pi$
$338$	0	0
$339$	4.61799	0.250815
$340$	0	0
$341$	10.1339	0.548784
$342$	0	0
$343$	10.2701	0.554535
$344$	0	0
$345$	−21.7645	−1.17176
$346$	0	0
$347$	−1.64781	−0.0884591	−0.0442295	−	0.999021i	$-0.514083\pi$
$347$	−1.64781	−0.0884591	−0.0442295	+	0.999021i	$0.514083\pi$
$348$	0	0
$349$	−20.2197	−1.08234	−0.541168	−	0.840915i	$-0.682018\pi$
$349$	−20.2197	−1.08234	−0.541168	+	0.840915i	$0.682018\pi$
$350$	0	0
$351$	−16.5688	−0.884375
$352$	0	0
$353$	−15.7630	−0.838982	−0.419491	−	0.907760i	$-0.637791\pi$
$353$	−15.7630	−0.838982	−0.419491	+	0.907760i	$0.637791\pi$
$354$	0	0
$355$	−31.3590	−1.66437
$356$	0	0
$357$	−12.4838	−0.660710
$358$	0	0
$359$	−19.6558	−1.03739	−0.518697	−	0.854958i	$-0.673583\pi$
$359$	−19.6558	−1.03739	−0.518697	+	0.854958i	$0.673583\pi$
$360$	0	0
$361$	23.7550	1.25027
$362$	0	0
$363$	21.1109	1.10804
$364$	0	0
$365$	1.37407	0.0719221
$366$	0	0
$367$	27.9198	1.45740	0.728701	−	0.684832i	$-0.240125\pi$
$367$	27.9198	1.45740	0.728701	+	0.684832i	$0.240125\pi$
$368$	0	0
$369$	−13.3393	−0.694416
$370$	0	0
$371$	−12.6904	−0.658852
$372$	0	0
$373$	29.8859	1.54743	0.773717	−	0.633531i	$-0.218395\pi$
$373$	29.8859	1.54743	0.773717	+	0.633531i	$0.218395\pi$
$374$	0	0
$375$	23.2814	1.20225
$376$	0	0
$377$	31.7874	1.63713
$378$	0	0
$379$	−24.0226	−1.23396	−0.616980	−	0.786979i	$-0.711644\pi$
$379$	−24.0226	−1.23396	−0.616980	+	0.786979i	$0.711644\pi$
$380$	0	0
$381$	−27.1483	−1.39085
$382$	0	0
$383$	−10.7433	−0.548958	−0.274479	−	0.961593i	$-0.588505\pi$
$383$	−10.7433	−0.548958	−0.274479	+	0.961593i	$0.588505\pi$
$384$	0	0
$385$	9.23457	0.470637
$386$	0	0
$387$	−4.06747	−0.206761
$388$	0	0
$389$	−17.5814	−0.891412	−0.445706	−	0.895179i	$-0.647047\pi$
$389$	−17.5814	−0.891412	−0.445706	+	0.895179i	$0.647047\pi$
$390$	0	0
$391$	6.61100	0.334332
$392$	0	0
$393$	−9.32788	−0.470529
$394$	0	0
$395$	32.1119	1.61572
$396$	0	0
$397$	3.03562	0.152353	0.0761766	−	0.997094i	$-0.475729\pi$
$397$	3.03562	0.152353	0.0761766	+	0.997094i	$0.475729\pi$
$398$	0	0
$399$	56.1425	2.81064
$400$	0	0
$401$	−24.3368	−1.21532	−0.607661	−	0.794197i	$-0.707892\pi$
$401$	−24.3368	−1.21532	−0.607661	+	0.794197i	$0.707892\pi$
$402$	0	0
$403$	−51.5318	−2.56698
$404$	0	0
$405$	−25.4925	−1.26673
$406$	0	0
$407$	3.45114	0.171067
$408$	0	0
$409$	−32.4090	−1.60252	−0.801262	−	0.598314i	$-0.795838\pi$
$409$	−32.4090	−1.60252	−0.801262	+	0.598314i	$0.795838\pi$
$410$	0	0
$411$	19.2289	0.948494
$412$	0	0
$413$	−9.06987	−0.446299
$414$	0	0
$415$	14.5735	0.715385
$416$	0	0
$417$	28.5370	1.39746
$418$	0	0
$419$	22.9156	1.11950	0.559749	−	0.828662i	$-0.310897\pi$
$419$	22.9156	1.11950	0.559749	+	0.828662i	$0.310897\pi$
$420$	0	0
$421$	−14.6625	−0.714606	−0.357303	−	0.933989i	$-0.616304\pi$
$421$	−14.6625	−0.714606	−0.357303	+	0.933989i	$0.616304\pi$
$422$	0	0
$423$	−8.94338	−0.434842
$424$	0	0
$425$	0.197963	0.00960259
$426$	0	0
$427$	−55.6902	−2.69504
$428$	0	0
$429$	10.7901	0.520951
$430$	0	0
$431$	12.0617	0.580989	0.290495	−	0.956877i	$-0.406180\pi$
$431$	12.0617	0.580989	0.290495	+	0.956877i	$0.406180\pi$
$432$	0	0
$433$	−32.4757	−1.56068	−0.780341	−	0.625354i	$-0.784955\pi$
$433$	−32.4757	−1.56068	−0.780341	+	0.625354i	$0.784955\pi$
$434$	0	0
$435$	29.8523	1.43131
$436$	0	0
$437$	−29.7313	−1.42224
$438$	0	0
$439$	−15.8820	−0.758008	−0.379004	−	0.925395i	$-0.623733\pi$
$439$	−15.8820	−0.758008	−0.379004	+	0.925395i	$0.623733\pi$
$440$	0	0
$441$	13.9168	0.662703
$442$	0	0
$443$	−19.8338	−0.942334	−0.471167	−	0.882044i	$-0.656167\pi$
$443$	−19.8338	−0.942334	−0.471167	+	0.882044i	$0.656167\pi$
$444$	0	0
$445$	37.8157	1.79264
$446$	0	0
$447$	−15.1963	−0.718759
$448$	0	0
$449$	−20.1451	−0.950705	−0.475352	−	0.879796i	$-0.657679\pi$
$449$	−20.1451	−0.950705	−0.475352	+	0.879796i	$0.657679\pi$
$450$	0	0
$451$	−9.15227	−0.430964
$452$	0	0
$453$	23.0879	1.08476
$454$	0	0
$455$	−46.9584	−2.20144
$456$	0	0
$457$	29.8675	1.39714	0.698572	−	0.715540i	$-0.253819\pi$
$457$	29.8675	1.39714	0.698572	+	0.715540i	$0.253819\pi$
$458$	0	0
$459$	4.72640	0.220610
$460$	0	0
$461$	26.1591	1.21835	0.609176	−	0.793035i	$-0.291500\pi$
$461$	26.1591	1.21835	0.609176	+	0.793035i	$0.291500\pi$
$462$	0	0
$463$	−19.3246	−0.898090	−0.449045	−	0.893509i	$-0.648236\pi$
$463$	−19.3246	−0.898090	−0.449045	+	0.893509i	$0.648236\pi$
$464$	0	0
$465$	−48.3947	−2.24425
$466$	0	0
$467$	5.38944	0.249393	0.124697	−	0.992195i	$-0.460204\pi$
$467$	5.38944	0.249393	0.124697	+	0.992195i	$0.460204\pi$
$468$	0	0
$469$	−57.5496	−2.65739
$470$	0	0
$471$	−20.5332	−0.946121
$472$	0	0
$473$	−2.79075	−0.128319
$474$	0	0
$475$	−0.890286	−0.0408491
$476$	0	0
$477$	−4.56037	−0.208805
$478$	0	0
$479$	3.08980	0.141177	0.0705884	−	0.997506i	$-0.477512\pi$
$479$	3.08980	0.141177	0.0705884	+	0.997506i	$0.477512\pi$
$480$	0	0
$481$	−17.5493	−0.800179
$482$	0	0
$483$	−39.0407	−1.77641
$484$	0	0
$485$	−38.5499	−1.75046
$486$	0	0
$487$	−11.5620	−0.523923	−0.261961	−	0.965078i	$-0.584369\pi$
$487$	−11.5620	−0.523923	−0.261961	+	0.965078i	$0.584369\pi$
$488$	0	0
$489$	−18.5495	−0.838836
$490$	0	0
$491$	−22.2460	−1.00395	−0.501975	−	0.864882i	$-0.667393\pi$
$491$	−22.2460	−1.00395	−0.501975	+	0.864882i	$0.667393\pi$
$492$	0	0
$493$	−9.06768	−0.408388
$494$	0	0
$495$	3.31850	0.149155
$496$	0	0
$497$	−56.2512	−2.52321
$498$	0	0
$499$	−41.1617	−1.84265	−0.921326	−	0.388792i	$-0.872892\pi$
$499$	−41.1617	−1.84265	−0.921326	+	0.388792i	$0.872892\pi$
$500$	0	0
$501$	47.1334	2.10576
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−16.5916	−0.738315
$506$	0	0
$507$	−27.4114	−1.21738
$508$	0	0
$509$	32.0313	1.41976	0.709881	−	0.704322i	$-0.248749\pi$
$509$	32.0313	1.41976	0.709881	+	0.704322i	$0.248749\pi$
$510$	0	0
$511$	2.46478	0.109035
$512$	0	0
$513$	−21.2558	−0.938467
$514$	0	0
$515$	−7.81229	−0.344250
$516$	0	0
$517$	−6.13618	−0.269869
$518$	0	0
$519$	−37.9940	−1.66775
$520$	0	0
$521$	−21.5660	−0.944823	−0.472412	−	0.881378i	$-0.656616\pi$
$521$	−21.5660	−0.944823	−0.472412	+	0.881378i	$0.656616\pi$
$522$	0	0
$523$	27.0448	1.18259	0.591294	−	0.806456i	$-0.298617\pi$
$523$	27.0448	1.18259	0.591294	+	0.806456i	$0.298617\pi$
$524$	0	0
$525$	−1.16905	−0.0510216
$526$	0	0
$527$	14.7000	0.640340
$528$	0	0
$529$	−2.32529	−0.101099
$530$	0	0
$531$	−3.25931	−0.141442
$532$	0	0
$533$	46.5399	2.01587
$534$	0	0
$535$	12.7505	0.551251
$536$	0	0
$537$	33.7117	1.45477
$538$	0	0
$539$	9.54849	0.411282
$540$	0	0
$541$	28.7357	1.23545	0.617723	−	0.786396i	$-0.288055\pi$
$541$	28.7357	1.23545	0.617723	+	0.786396i	$0.288055\pi$
$542$	0	0
$543$	−17.1969	−0.737991
$544$	0	0
$545$	−2.31857	−0.0993167
$546$	0	0
$547$	16.5139	0.706083	0.353041	−	0.935608i	$-0.385148\pi$
$547$	16.5139	0.706083	0.353041	+	0.935608i	$0.385148\pi$
$548$	0	0
$549$	−20.0126	−0.854117
$550$	0	0
$551$	40.7796	1.73727
$552$	0	0
$553$	57.6016	2.44947
$554$	0	0
$555$	−16.4810	−0.699578
$556$	0	0
$557$	−30.5389	−1.29397	−0.646986	−	0.762502i	$-0.723971\pi$
$557$	−30.5389	−1.29397	−0.646986	+	0.762502i	$0.723971\pi$
$558$	0	0
$559$	14.1912	0.600222
$560$	0	0
$561$	−3.07798	−0.129953
$562$	0	0
$563$	−27.7696	−1.17035	−0.585174	−	0.810908i	$-0.698974\pi$
$563$	−27.7696	−1.17035	−0.585174	+	0.810908i	$0.698974\pi$
$564$	0	0
$565$	−4.95521	−0.208467
$566$	0	0
$567$	−45.7279	−1.92039
$568$	0	0
$569$	6.01578	0.252194	0.126097	−	0.992018i	$-0.459755\pi$
$569$	6.01578	0.252194	0.126097	+	0.992018i	$0.459755\pi$
$570$	0	0
$571$	13.6672	0.571956	0.285978	−	0.958236i	$-0.407682\pi$
$571$	13.6672	0.571956	0.285978	+	0.958236i	$0.407682\pi$
$572$	0	0
$573$	16.5040	0.689463
$574$	0	0
$575$	0.619093	0.0258179
$576$	0	0
$577$	−23.8180	−0.991555	−0.495778	−	0.868449i	$-0.665117\pi$
$577$	−23.8180	−0.991555	−0.495778	+	0.868449i	$0.665117\pi$
$578$	0	0
$579$	50.0199	2.07876
$580$	0	0
$581$	26.1416	1.08454
$582$	0	0
$583$	−3.12893	−0.129587
$584$	0	0
$585$	−16.8748	−0.697687
$586$	0	0
$587$	−40.5651	−1.67430	−0.837151	−	0.546972i	$-0.815780\pi$
$587$	−40.5651	−1.67430	−0.837151	+	0.546972i	$0.815780\pi$
$588$	0	0
$589$	−66.1093	−2.72399
$590$	0	0
$591$	7.10388	0.292215
$592$	0	0
$593$	10.2283	0.420028	0.210014	−	0.977698i	$-0.432649\pi$
$593$	10.2283	0.420028	0.210014	+	0.977698i	$0.432649\pi$
$594$	0	0
$595$	13.3954	0.549156
$596$	0	0
$597$	−38.2198	−1.56423
$598$	0	0
$599$	42.5115	1.73697	0.868486	−	0.495714i	$-0.165094\pi$
$599$	42.5115	1.73697	0.868486	+	0.495714i	$0.165094\pi$
$600$	0	0
$601$	15.7198	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$601$	15.7198	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$602$	0	0
$603$	−20.6808	−0.842187
$604$	0	0
$605$	−22.6525	−0.920956
$606$	0	0
$607$	−15.9609	−0.647832	−0.323916	−	0.946086i	$-0.605000\pi$
$607$	−15.9609	−0.647832	−0.323916	+	0.946086i	$0.605000\pi$
$608$	0	0
$609$	53.5485	2.16989
$610$	0	0
$611$	31.2029	1.26233
$612$	0	0
$613$	−2.46409	−0.0995238	−0.0497619	−	0.998761i	$-0.515846\pi$
$613$	−2.46409	−0.0995238	−0.0497619	+	0.998761i	$0.515846\pi$
$614$	0	0
$615$	43.7068	1.76243
$616$	0	0
$617$	5.72545	0.230498	0.115249	−	0.993337i	$-0.463233\pi$
$617$	5.72545	0.230498	0.115249	+	0.993337i	$0.463233\pi$
$618$	0	0
$619$	5.54382	0.222825	0.111412	−	0.993774i	$-0.464463\pi$
$619$	5.54382	0.222825	0.111412	+	0.993774i	$0.464463\pi$
$620$	0	0
$621$	14.7810	0.593141
$622$	0	0
$623$	67.8331	2.71767
$624$	0	0
$625$	−25.6622	−1.02649
$626$	0	0
$627$	13.8425	0.552814
$628$	0	0
$629$	5.00611	0.199607
$630$	0	0
$631$	14.2290	0.566448	0.283224	−	0.959054i	$-0.408596\pi$
$631$	14.2290	0.566448	0.283224	+	0.959054i	$0.408596\pi$
$632$	0	0
$633$	10.2027	0.405519
$634$	0	0
$635$	29.1307	1.15602
$636$	0	0
$637$	−48.5547	−1.92381
$638$	0	0
$639$	−20.2142	−0.799662
$640$	0	0
$641$	−22.9795	−0.907636	−0.453818	−	0.891094i	$-0.649938\pi$
$641$	−22.9795	−0.907636	−0.453818	+	0.891094i	$0.649938\pi$
$642$	0	0
$643$	−42.2301	−1.66539	−0.832696	−	0.553730i	$-0.813204\pi$
$643$	−42.2301	−1.66539	−0.832696	+	0.553730i	$0.813204\pi$
$644$	0	0
$645$	13.3273	0.524760
$646$	0	0
$647$	44.3781	1.74468	0.872341	−	0.488898i	$-0.162601\pi$
$647$	44.3781	1.74468	0.872341	+	0.488898i	$0.162601\pi$
$648$	0	0
$649$	−2.23626	−0.0877809
$650$	0	0
$651$	−86.8095	−3.40233
$652$	0	0
$653$	15.9984	0.626065	0.313032	−	0.949742i	$-0.398655\pi$
$653$	15.9984	0.626065	0.313032	+	0.949742i	$0.398655\pi$
$654$	0	0
$655$	10.0090	0.391085
$656$	0	0
$657$	0.885733	0.0345558
$658$	0	0
$659$	4.85181	0.189000	0.0944998	−	0.995525i	$-0.469875\pi$
$659$	4.85181	0.189000	0.0944998	+	0.995525i	$0.469875\pi$
$660$	0	0
$661$	51.1792	1.99064	0.995321	−	0.0966234i	$-0.0308042\pi$
$661$	51.1792	1.99064	0.995321	+	0.0966234i	$0.0308042\pi$
$662$	0	0
$663$	15.6518	0.607864
$664$	0	0
$665$	−60.2422	−2.33609
$666$	0	0
$667$	−28.3575	−1.09801
$668$	0	0
$669$	−52.4641	−2.02838
$670$	0	0
$671$	−13.7309	−0.530076
$672$	0	0
$673$	−24.8027	−0.956075	−0.478038	−	0.878339i	$-0.658652\pi$
$673$	−24.8027	−0.956075	−0.478038	+	0.878339i	$0.658652\pi$
$674$	0	0
$675$	0.442608	0.0170360
$676$	0	0
$677$	−13.0820	−0.502781	−0.251391	−	0.967886i	$-0.580888\pi$
$677$	−13.0820	−0.502781	−0.251391	+	0.967886i	$0.580888\pi$
$678$	0	0
$679$	−69.1500	−2.65374
$680$	0	0
$681$	8.09104	0.310049
$682$	0	0
$683$	−13.8577	−0.530249	−0.265125	−	0.964214i	$-0.585413\pi$
$683$	−13.8577	−0.530249	−0.265125	+	0.964214i	$0.585413\pi$
$684$	0	0
$685$	−20.6331	−0.788350
$686$	0	0
$687$	−47.0890	−1.79656
$688$	0	0
$689$	15.9108	0.606155
$690$	0	0
$691$	4.85400	0.184655	0.0923275	−	0.995729i	$-0.470569\pi$
$691$	4.85400	0.184655	0.0923275	+	0.995729i	$0.470569\pi$
$692$	0	0
$693$	5.95265	0.226123
$694$	0	0
$695$	−30.6209	−1.16152
$696$	0	0
$697$	−13.2760	−0.502864
$698$	0	0
$699$	−43.3954	−1.64137
$700$	0	0
$701$	−26.4296	−0.998233	−0.499116	−	0.866535i	$-0.666342\pi$
$701$	−26.4296	−0.998233	−0.499116	+	0.866535i	$0.666342\pi$
$702$	0	0
$703$	−22.5137	−0.849121
$704$	0	0
$705$	29.3034	1.10363
$706$	0	0
$707$	−29.7616	−1.11930
$708$	0	0
$709$	−3.89825	−0.146402	−0.0732010	−	0.997317i	$-0.523321\pi$
$709$	−3.89825	−0.146402	−0.0732010	+	0.997317i	$0.523321\pi$
$710$	0	0
$711$	20.6995	0.776292
$712$	0	0
$713$	45.9715	1.72165
$714$	0	0
$715$	−11.5780	−0.432994
$716$	0	0
$717$	−47.0266	−1.75624
$718$	0	0
$719$	2.53133	0.0944026	0.0472013	−	0.998885i	$-0.484970\pi$
$719$	2.53133	0.0944026	0.0472013	+	0.998885i	$0.484970\pi$
$720$	0	0
$721$	−14.0135	−0.521891
$722$	0	0
$723$	−6.51341	−0.242236
$724$	0	0
$725$	−0.849151	−0.0315367
$726$	0	0
$727$	−5.56956	−0.206564	−0.103282	−	0.994652i	$-0.532934\pi$
$727$	−5.56956	−0.206564	−0.103282	+	0.994652i	$0.532934\pi$
$728$	0	0
$729$	4.16492	0.154256
$730$	0	0
$731$	−4.04817	−0.149727
$732$	0	0
$733$	15.1483	0.559516	0.279758	−	0.960071i	$-0.409746\pi$
$733$	15.1483	0.559516	0.279758	+	0.960071i	$0.409746\pi$
$734$	0	0
$735$	−45.5989	−1.68194
$736$	0	0
$737$	−14.1894	−0.522673
$738$	0	0
$739$	43.6156	1.60443	0.802213	−	0.597038i	$-0.203656\pi$
$739$	43.6156	1.60443	0.802213	+	0.597038i	$0.203656\pi$
$740$	0	0
$741$	−70.3898	−2.58584
$742$	0	0
$743$	47.4032	1.73905	0.869527	−	0.493885i	$-0.164424\pi$
$743$	47.4032	1.73905	0.869527	+	0.493885i	$0.164424\pi$
$744$	0	0
$745$	16.3060	0.597404
$746$	0	0
$747$	9.39416	0.343714
$748$	0	0
$749$	22.8715	0.835707
$750$	0	0
$751$	−14.1780	−0.517364	−0.258682	−	0.965963i	$-0.583288\pi$
$751$	−14.1780	−0.517364	−0.258682	+	0.965963i	$0.583288\pi$
$752$	0	0
$753$	19.7299	0.718997
$754$	0	0
$755$	−24.7739	−0.901613
$756$	0	0
$757$	−41.6574	−1.51406	−0.757032	−	0.653377i	$-0.773352\pi$
$757$	−41.6574	−1.51406	−0.757032	+	0.653377i	$0.773352\pi$
$758$	0	0
$759$	−9.62585	−0.349396
$760$	0	0
$761$	−30.2244	−1.09563	−0.547817	−	0.836598i	$-0.684541\pi$
$761$	−30.2244	−1.09563	−0.547817	+	0.836598i	$0.684541\pi$
$762$	0	0
$763$	−4.15901	−0.150566
$764$	0	0
$765$	4.81370	0.174040
$766$	0	0
$767$	11.3715	0.410603
$768$	0	0
$769$	−7.00069	−0.252451	−0.126226	−	0.992002i	$-0.540286\pi$
$769$	−7.00069	−0.252451	−0.126226	+	0.992002i	$0.540286\pi$
$770$	0	0
$771$	10.3257	0.371871
$772$	0	0
$773$	−38.7751	−1.39464	−0.697322	−	0.716758i	$-0.745625\pi$
$773$	−38.7751	−1.39464	−0.697322	+	0.716758i	$0.745625\pi$
$774$	0	0
$775$	1.37659	0.0494486
$776$	0	0
$777$	−29.5632	−1.06057
$778$	0	0
$779$	59.7054	2.13917
$780$	0	0
$781$	−13.8693	−0.496281
$782$	0	0
$783$	−20.2737	−0.724523
$784$	0	0
$785$	22.0326	0.786378
$786$	0	0
$787$	−37.1123	−1.32291	−0.661455	−	0.749984i	$-0.730061\pi$
$787$	−37.1123	−1.32291	−0.661455	+	0.749984i	$0.730061\pi$
$788$	0	0
$789$	−12.0900	−0.430415
$790$	0	0
$791$	−8.88855	−0.316041
$792$	0	0
$793$	69.8227	2.47948
$794$	0	0
$795$	14.9423	0.529947
$796$	0	0
$797$	−45.0577	−1.59603	−0.798013	−	0.602641i	$-0.794115\pi$
$797$	−45.0577	−1.59603	−0.798013	+	0.602641i	$0.794115\pi$
$798$	0	0
$799$	−8.90093	−0.314892
$800$	0	0
$801$	24.3762	0.861292
$802$	0	0
$803$	0.607714	0.0214458
$804$	0	0
$805$	41.8916	1.47648
$806$	0	0
$807$	34.9345	1.22975
$808$	0	0
$809$	52.2972	1.83867	0.919335	−	0.393475i	$-0.128727\pi$
$809$	52.2972	1.83867	0.919335	+	0.393475i	$0.128727\pi$
$810$	0	0
$811$	51.0213	1.79160	0.895799	−	0.444459i	$-0.146604\pi$
$811$	51.0213	1.79160	0.895799	+	0.444459i	$0.146604\pi$
$812$	0	0
$813$	6.58216	0.230846
$814$	0	0
$815$	19.9040	0.697207
$816$	0	0
$817$	18.2056	0.636934
$818$	0	0
$819$	−30.2697	−1.05771
$820$	0	0
$821$	−22.6715	−0.791242	−0.395621	−	0.918414i	$-0.629471\pi$
$821$	−22.6715	−0.791242	−0.395621	+	0.918414i	$0.629471\pi$
$822$	0	0
$823$	−36.8627	−1.28495	−0.642477	−	0.766305i	$-0.722093\pi$
$823$	−36.8627	−1.28495	−0.642477	+	0.766305i	$0.722093\pi$
$824$	0	0
$825$	−0.288241	−0.0100353
$826$	0	0
$827$	13.5507	0.471202	0.235601	−	0.971850i	$-0.424294\pi$
$827$	13.5507	0.471202	0.235601	+	0.971850i	$0.424294\pi$
$828$	0	0
$829$	12.5840	0.437060	0.218530	−	0.975830i	$-0.429874\pi$
$829$	12.5840	0.437060	0.218530	+	0.975830i	$0.429874\pi$
$830$	0	0
$831$	13.3262	0.462282
$832$	0	0
$833$	13.8507	0.479899
$834$	0	0
$835$	−50.5752	−1.75023
$836$	0	0
$837$	32.8665	1.13603
$838$	0	0
$839$	40.1777	1.38709	0.693545	−	0.720414i	$-0.256048\pi$
$839$	40.1777	1.38709	0.693545	+	0.720414i	$0.256048\pi$
$840$	0	0
$841$	9.89537	0.341220
$842$	0	0
$843$	46.0346	1.58552
$844$	0	0
$845$	29.4131	1.01184
$846$	0	0
$847$	−40.6336	−1.39619
$848$	0	0
$849$	0.896779	0.0307774
$850$	0	0
$851$	15.6557	0.536671
$852$	0	0
$853$	−8.72638	−0.298786	−0.149393	−	0.988778i	$-0.547732\pi$
$853$	−8.72638	−0.298786	−0.149393	+	0.988778i	$0.547732\pi$
$854$	0	0
$855$	−21.6484	−0.740360
$856$	0	0
$857$	34.4294	1.17609	0.588044	−	0.808829i	$-0.299898\pi$
$857$	34.4294	1.17609	0.588044	+	0.808829i	$0.299898\pi$
$858$	0	0
$859$	−20.9612	−0.715188	−0.357594	−	0.933877i	$-0.616403\pi$
$859$	−20.9612	−0.715188	−0.357594	+	0.933877i	$0.616403\pi$
$860$	0	0
$861$	78.4003	2.67188
$862$	0	0
$863$	−57.0318	−1.94138	−0.970692	−	0.240327i	$-0.922745\pi$
$863$	−57.0318	−1.94138	−0.970692	+	0.240327i	$0.922745\pi$
$864$	0	0
$865$	40.7685	1.38617
$866$	0	0
$867$	31.4405	1.06778
$868$	0	0
$869$	14.2022	0.481777
$870$	0	0
$871$	72.1540	2.44484
$872$	0	0
$873$	−24.8495	−0.841028
$874$	0	0
$875$	−44.8112	−1.51490
$876$	0	0
$877$	−38.0437	−1.28465	−0.642323	−	0.766434i	$-0.722029\pi$
$877$	−38.0437	−1.28465	−0.642323	+	0.766434i	$0.722029\pi$
$878$	0	0
$879$	−20.6599	−0.696840
$880$	0	0
$881$	35.7757	1.20531	0.602657	−	0.798000i	$-0.294109\pi$
$881$	35.7757	1.20531	0.602657	+	0.798000i	$0.294109\pi$
$882$	0	0
$883$	26.1190	0.878976	0.439488	−	0.898248i	$-0.355160\pi$
$883$	26.1190	0.878976	0.439488	+	0.898248i	$0.355160\pi$
$884$	0	0
$885$	10.6793	0.358980
$886$	0	0
$887$	6.17839	0.207450	0.103725	−	0.994606i	$-0.466924\pi$
$887$	6.17839	0.207450	0.103725	+	0.994606i	$0.466924\pi$
$888$	0	0
$889$	52.2541	1.75254
$890$	0	0
$891$	−11.2746	−0.377715
$892$	0	0
$893$	40.0297	1.33954
$894$	0	0
$895$	−36.1735	−1.20915
$896$	0	0
$897$	48.9481	1.63433
$898$	0	0
$899$	−63.0548	−2.10299
$900$	0	0
$901$	−4.53873	−0.151207
$902$	0	0
$903$	23.9062	0.795547
$904$	0	0
$905$	18.4527	0.613388
$906$	0	0
$907$	−43.7016	−1.45109	−0.725544	−	0.688176i	$-0.758412\pi$
$907$	−43.7016	−1.45109	−0.725544	+	0.688176i	$0.758412\pi$
$908$	0	0
$909$	−10.6950	−0.354731
$910$	0	0
$911$	−16.3022	−0.540116	−0.270058	−	0.962844i	$-0.587043\pi$
$911$	−16.3022	−0.540116	−0.270058	+	0.962844i	$0.587043\pi$
$912$	0	0
$913$	6.44546	0.213314
$914$	0	0
$915$	65.5722	2.16775
$916$	0	0
$917$	17.9540	0.592893
$918$	0	0
$919$	50.9542	1.68083	0.840413	−	0.541947i	$-0.182313\pi$
$919$	50.9542	1.68083	0.840413	+	0.541947i	$0.182313\pi$
$920$	0	0
$921$	2.38924	0.0787281
$922$	0	0
$923$	70.5261	2.32140
$924$	0	0
$925$	0.468802	0.0154141
$926$	0	0
$927$	−5.03584	−0.165399
$928$	0	0
$929$	−15.0110	−0.492494	−0.246247	−	0.969207i	$-0.579197\pi$
$929$	−15.0110	−0.492494	−0.246247	+	0.969207i	$0.579197\pi$
$930$	0	0
$931$	−62.2901	−2.04147
$932$	0	0
$933$	23.1891	0.759177
$934$	0	0
$935$	3.30275	0.108011
$936$	0	0
$937$	−26.8600	−0.877477	−0.438738	−	0.898615i	$-0.644575\pi$
$937$	−26.8600	−0.877477	−0.438738	+	0.898615i	$0.644575\pi$
$938$	0	0
$939$	−41.0123	−1.33838
$940$	0	0
$941$	−38.5130	−1.25549	−0.627744	−	0.778420i	$-0.716021\pi$
$941$	−38.5130	−1.25549	−0.627744	+	0.778420i	$0.716021\pi$
$942$	0	0
$943$	−41.5183	−1.35202
$944$	0	0
$945$	29.9496	0.974260
$946$	0	0
$947$	46.6430	1.51569	0.757846	−	0.652433i	$-0.226252\pi$
$947$	46.6430	1.51569	0.757846	+	0.652433i	$0.226252\pi$
$948$	0	0
$949$	−3.09027	−0.100314
$950$	0	0
$951$	−64.5326	−2.09261
$952$	0	0
$953$	45.0755	1.46014	0.730070	−	0.683373i	$-0.239488\pi$
$953$	45.0755	1.46014	0.730070	+	0.683373i	$0.239488\pi$
$954$	0	0
$955$	−17.7091	−0.573054
$956$	0	0
$957$	13.2029	0.426788
$958$	0	0
$959$	−37.0112	−1.19515
$960$	0	0
$961$	71.2205	2.29744
$962$	0	0
$963$	8.21902	0.264854
$964$	0	0
$965$	−53.6725	−1.72778
$966$	0	0
$967$	−17.5880	−0.565591	−0.282795	−	0.959180i	$-0.591262\pi$
$967$	−17.5880	−0.565591	−0.282795	+	0.959180i	$0.591262\pi$
$968$	0	0
$969$	20.0794	0.645043
$970$	0	0
$971$	−28.5401	−0.915895	−0.457948	−	0.888979i	$-0.651415\pi$
$971$	−28.5401	−0.915895	−0.457948	+	0.888979i	$0.651415\pi$
$972$	0	0
$973$	−54.9271	−1.76088
$974$	0	0
$975$	1.46572	0.0469407
$976$	0	0
$977$	20.4418	0.653992	0.326996	−	0.945026i	$-0.393964\pi$
$977$	20.4418	0.653992	0.326996	+	0.945026i	$0.393964\pi$
$978$	0	0
$979$	16.7249	0.534529
$980$	0	0
$981$	−1.49456	−0.0477178
$982$	0	0
$983$	12.8951	0.411291	0.205645	−	0.978627i	$-0.434071\pi$
$983$	12.8951	0.411291	0.205645	+	0.978627i	$0.434071\pi$
$984$	0	0
$985$	−7.62263	−0.242877
$986$	0	0
$987$	52.5638	1.67312
$988$	0	0
$989$	−12.6599	−0.402562
$990$	0	0
$991$	−40.3531	−1.28186	−0.640929	−	0.767600i	$-0.721451\pi$
$991$	−40.3531	−1.28186	−0.640929	+	0.767600i	$0.721451\pi$
$992$	0	0
$993$	−17.9445	−0.569450
$994$	0	0
$995$	41.0107	1.30013
$996$	0	0
$997$	26.4183	0.836677	0.418339	−	0.908291i	$-0.362613\pi$
$997$	26.4183	0.836677	0.418339	+	0.908291i	$0.362613\pi$
$998$	0	0
$999$	11.1928	0.354123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.5		28
4.3	odd	2		4024.2.a.d.1.24	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.24	✓	28	4.3	odd	2
8048.2.a.v.1.5		28	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11208 q^{3} +2.26631 q^{5} +4.06526 q^{7} +1.46088 q^{9} +O(q^{10})\)	\(q-2.11208 q^{3} +2.26631 q^{5} +4.06526 q^{7} +1.46088 q^{9} +1.00233 q^{11} -5.09690 q^{13} -4.78662 q^{15} +1.45394 q^{17} -6.53873 q^{19} -8.58614 q^{21} +4.54695 q^{23} +0.136156 q^{25} +3.25075 q^{27} -6.23661 q^{29} +10.1104 q^{31} -2.11699 q^{33} +9.21313 q^{35} +3.44313 q^{37} +10.7651 q^{39} -9.13103 q^{41} -2.78427 q^{43} +3.31079 q^{45} -6.12193 q^{47} +9.52632 q^{49} -3.07084 q^{51} -3.12167 q^{53} +2.27158 q^{55} +13.8103 q^{57} -2.23107 q^{59} -13.6990 q^{61} +5.93883 q^{63} -11.5512 q^{65} -14.1564 q^{67} -9.60351 q^{69} -13.8371 q^{71} +0.606303 q^{73} -0.287572 q^{75} +4.07472 q^{77} +14.1692 q^{79} -11.2485 q^{81} +6.43050 q^{83} +3.29508 q^{85} +13.1722 q^{87} +16.6860 q^{89} -20.7202 q^{91} -21.3540 q^{93} -14.8188 q^{95} -17.0100 q^{97} +1.46427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more