LMFDB - Embedded newform 8001.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.70334$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70334 q^{2} +5.30805 q^{4} +3.37987 q^{5} +1.00000 q^{7} -8.94278 q^{8} +O(q^{10})$	\(q-2.70334 q^{2} +5.30805 q^{4} +3.37987 q^{5} +1.00000 q^{7} -8.94278 q^{8} -9.13693 q^{10} -5.25067 q^{11} +4.49005 q^{13} -2.70334 q^{14} +13.5593 q^{16} +2.55842 q^{17} -6.86089 q^{19} +17.9405 q^{20} +14.1943 q^{22} -6.11893 q^{23} +6.42349 q^{25} -12.1381 q^{26} +5.30805 q^{28} +9.43242 q^{29} -0.617404 q^{31} -18.7698 q^{32} -6.91628 q^{34} +3.37987 q^{35} -10.3444 q^{37} +18.5473 q^{38} -30.2254 q^{40} -6.74914 q^{41} -8.27103 q^{43} -27.8708 q^{44} +16.5415 q^{46} -3.41047 q^{47} +1.00000 q^{49} -17.3649 q^{50} +23.8334 q^{52} +1.36566 q^{53} -17.7466 q^{55} -8.94278 q^{56} -25.4990 q^{58} -0.337596 q^{59} +5.71415 q^{61} +1.66905 q^{62} +23.6226 q^{64} +15.1758 q^{65} +6.68440 q^{67} +13.5802 q^{68} -9.13693 q^{70} +5.42767 q^{71} +1.39586 q^{73} +27.9644 q^{74} -36.4179 q^{76} -5.25067 q^{77} -6.00792 q^{79} +45.8286 q^{80} +18.2452 q^{82} +15.5854 q^{83} +8.64712 q^{85} +22.3594 q^{86} +46.9556 q^{88} +9.64990 q^{89} +4.49005 q^{91} -32.4796 q^{92} +9.21967 q^{94} -23.1889 q^{95} +3.82769 q^{97} -2.70334 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

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 $

$2$	−2.70334	−1.91155	−0.955775	−	0.294098i	$-0.904981\pi$
$2$	−2.70334	−1.91155	−0.955775	+	0.294098i	$0.904981\pi$
$3$	0	0
$3$	0	0
$4$	5.30805	2.65402
$5$	3.37987	1.51152	0.755761	−	0.654848i	$-0.227267\pi$
$5$	3.37987	1.51152	0.755761	+	0.654848i	$0.227267\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−8.94278	−3.16175
$9$	0	0
$10$	−9.13693	−2.88935
$11$	−5.25067	−1.58314	−0.791568	−	0.611081i	$-0.790735\pi$
$11$	−5.25067	−1.58314	−0.791568	+	0.611081i	$0.790735\pi$
$12$	0	0
$13$	4.49005	1.24532	0.622658	−	0.782494i	$-0.286053\pi$
$13$	4.49005	1.24532	0.622658	+	0.782494i	$0.286053\pi$
$14$	−2.70334	−0.722498
$15$	0	0
$16$	13.5593	3.38982
$17$	2.55842	0.620508	0.310254	−	0.950654i	$-0.399586\pi$
$17$	2.55842	0.620508	0.310254	+	0.950654i	$0.399586\pi$
$18$	0	0
$19$	−6.86089	−1.57400	−0.786998	−	0.616956i	$-0.788366\pi$
$19$	−6.86089	−1.57400	−0.786998	+	0.616956i	$0.788366\pi$
$20$	17.9405	4.01162
$21$	0	0
$22$	14.1943	3.02624
$23$	−6.11893	−1.27588	−0.637942	−	0.770084i	$-0.720214\pi$
$23$	−6.11893	−1.27588	−0.637942	+	0.770084i	$0.720214\pi$
$24$	0	0
$25$	6.42349	1.28470
$26$	−12.1381	−2.38048
$27$	0	0
$28$	5.30805	1.00313
$29$	9.43242	1.75156	0.875778	−	0.482713i	$-0.160349\pi$
$29$	9.43242	1.75156	0.875778	+	0.482713i	$0.160349\pi$
$30$	0	0
$31$	−0.617404	−0.110889	−0.0554445	−	0.998462i	$-0.517658\pi$
$31$	−0.617404	−0.110889	−0.0554445	+	0.998462i	$0.517658\pi$
$32$	−18.7698	−3.31807
$33$	0	0
$34$	−6.91628	−1.18613
$35$	3.37987	0.571302
$36$	0	0
$37$	−10.3444	−1.70061	−0.850304	−	0.526291i	$-0.823582\pi$
$37$	−10.3444	−1.70061	−0.850304	+	0.526291i	$0.823582\pi$
$38$	18.5473	3.00877
$39$	0	0
$40$	−30.2254	−4.77906
$41$	−6.74914	−1.05404	−0.527019	−	0.849853i	$-0.676690\pi$
$41$	−6.74914	−1.05404	−0.527019	+	0.849853i	$0.676690\pi$
$42$	0	0
$43$	−8.27103	−1.26132	−0.630660	−	0.776059i	$-0.717216\pi$
$43$	−8.27103	−1.26132	−0.630660	+	0.776059i	$0.717216\pi$
$44$	−27.8708	−4.20168
$45$	0	0
$46$	16.5415	2.43892
$47$	−3.41047	−0.497469	−0.248734	−	0.968572i	$-0.580015\pi$
$47$	−3.41047	−0.497469	−0.248734	+	0.968572i	$0.580015\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−17.3649	−2.45577
$51$	0	0
$52$	23.8334	3.30510
$53$	1.36566	0.187588	0.0937941	−	0.995592i	$-0.470100\pi$
$53$	1.36566	0.187588	0.0937941	+	0.995592i	$0.470100\pi$
$54$	0	0
$55$	−17.7466	−2.39294
$56$	−8.94278	−1.19503
$57$	0	0
$58$	−25.4990	−3.34819
$59$	−0.337596	−0.0439512	−0.0219756	−	0.999759i	$-0.506996\pi$
$59$	−0.337596	−0.0439512	−0.0219756	+	0.999759i	$0.506996\pi$
$60$	0	0
$61$	5.71415	0.731622	0.365811	−	0.930689i	$-0.380792\pi$
$61$	5.71415	0.731622	0.365811	+	0.930689i	$0.380792\pi$
$62$	1.66905	0.211970
$63$	0	0
$64$	23.6226	2.95283
$65$	15.1758	1.88232
$66$	0	0
$67$	6.68440	0.816629	0.408315	−	0.912841i	$-0.366117\pi$
$67$	6.68440	0.816629	0.408315	+	0.912841i	$0.366117\pi$
$68$	13.5802	1.64684
$69$	0	0
$70$	−9.13693	−1.09207
$71$	5.42767	0.644146	0.322073	−	0.946715i	$-0.395620\pi$
$71$	5.42767	0.644146	0.322073	+	0.946715i	$0.395620\pi$
$72$	0	0
$73$	1.39586	0.163373	0.0816867	−	0.996658i	$-0.473969\pi$
$73$	1.39586	0.163373	0.0816867	+	0.996658i	$0.473969\pi$
$74$	27.9644	3.25080
$75$	0	0
$76$	−36.4179	−4.17742
$77$	−5.25067	−0.598369
$78$	0	0
$79$	−6.00792	−0.675944	−0.337972	−	0.941156i	$-0.609741\pi$
$79$	−6.00792	−0.675944	−0.337972	+	0.941156i	$0.609741\pi$
$80$	45.8286	5.12379
$81$	0	0
$82$	18.2452	2.01485
$83$	15.5854	1.71072	0.855358	−	0.518037i	$-0.173337\pi$
$83$	15.5854	1.71072	0.855358	+	0.518037i	$0.173337\pi$
$84$	0	0
$85$	8.64712	0.937911
$86$	22.3594	2.41108
$87$	0	0
$88$	46.9556	5.00548
$89$	9.64990	1.02289	0.511444	−	0.859317i	$-0.329111\pi$
$89$	9.64990	1.02289	0.511444	+	0.859317i	$0.329111\pi$
$90$	0	0
$91$	4.49005	0.470685
$92$	−32.4796	−3.38623
$93$	0	0
$94$	9.21967	0.950937
$95$	−23.1889	−2.37913
$96$	0	0
$97$	3.82769	0.388643	0.194322	−	0.980938i	$-0.437749\pi$
$97$	3.82769	0.388643	0.194322	+	0.980938i	$0.437749\pi$
$98$	−2.70334	−0.273079
$99$	0	0
$100$	34.0962	3.40962
$101$	−6.95323	−0.691872	−0.345936	−	0.938258i	$-0.612439\pi$
$101$	−6.95323	−0.691872	−0.345936	+	0.938258i	$0.612439\pi$
$102$	0	0
$103$	−8.95576	−0.882437	−0.441218	−	0.897400i	$-0.645454\pi$
$103$	−8.95576	−0.882437	−0.441218	+	0.897400i	$0.645454\pi$
$104$	−40.1535	−3.93738
$105$	0	0
$106$	−3.69185	−0.358584
$107$	−5.94244	−0.574478	−0.287239	−	0.957859i	$-0.592737\pi$
$107$	−5.94244	−0.574478	−0.287239	+	0.957859i	$0.592737\pi$
$108$	0	0
$109$	−9.18703	−0.879958	−0.439979	−	0.898008i	$-0.645014\pi$
$109$	−9.18703	−0.879958	−0.439979	+	0.898008i	$0.645014\pi$
$110$	47.9750	4.57423
$111$	0	0
$112$	13.5593	1.28123
$113$	−19.6841	−1.85173	−0.925863	−	0.377859i	$-0.876660\pi$
$113$	−19.6841	−1.85173	−0.925863	+	0.377859i	$0.876660\pi$
$114$	0	0
$115$	−20.6812	−1.92853
$116$	50.0678	4.64868
$117$	0	0
$118$	0.912636	0.0840149
$119$	2.55842	0.234530
$120$	0	0
$121$	16.5695	1.50632
$122$	−15.4473	−1.39853
$123$	0	0
$124$	−3.27721	−0.294302
$125$	4.81121	0.430327
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−26.3203	−2.32641
$129$	0	0
$130$	−41.0252	−3.59815
$131$	−3.12882	−0.273367	−0.136683	−	0.990615i	$-0.543644\pi$
$131$	−3.12882	−0.273367	−0.136683	+	0.990615i	$0.543644\pi$
$132$	0	0
$133$	−6.86089	−0.594914
$134$	−18.0702	−1.56103
$135$	0	0
$136$	−22.8794	−1.96189
$137$	1.05209	0.0898865	0.0449433	−	0.998990i	$-0.485689\pi$
$137$	1.05209	0.0898865	0.0449433	+	0.998990i	$0.485689\pi$
$138$	0	0
$139$	2.33299	0.197882	0.0989410	−	0.995093i	$-0.468454\pi$
$139$	2.33299	0.197882	0.0989410	+	0.995093i	$0.468454\pi$
$140$	17.9405	1.51625
$141$	0	0
$142$	−14.6728	−1.23132
$143$	−23.5758	−1.97150
$144$	0	0
$145$	31.8803	2.64752
$146$	−3.77349	−0.312297
$147$	0	0
$148$	−54.9086	−4.51346
$149$	−15.2050	−1.24564	−0.622821	−	0.782365i	$-0.714013\pi$
$149$	−15.2050	−1.24564	−0.622821	+	0.782365i	$0.714013\pi$
$150$	0	0
$151$	4.99981	0.406879	0.203439	−	0.979088i	$-0.434788\pi$
$151$	4.99981	0.406879	0.203439	+	0.979088i	$0.434788\pi$
$152$	61.3554	4.97658
$153$	0	0
$154$	14.1943	1.14381
$155$	−2.08674	−0.167611
$156$	0	0
$157$	18.3318	1.46304	0.731520	−	0.681820i	$-0.238811\pi$
$157$	18.3318	1.46304	0.731520	+	0.681820i	$0.238811\pi$
$158$	16.2415	1.29210
$159$	0	0
$160$	−63.4394	−5.01533
$161$	−6.11893	−0.482239
$162$	0	0
$163$	−3.16547	−0.247939	−0.123970	−	0.992286i	$-0.539563\pi$
$163$	−3.16547	−0.247939	−0.123970	+	0.992286i	$0.539563\pi$
$164$	−35.8248	−2.79744
$165$	0	0
$166$	−42.1326	−3.27012
$167$	−7.87265	−0.609204	−0.304602	−	0.952480i	$-0.598523\pi$
$167$	−7.87265	−0.609204	−0.304602	+	0.952480i	$0.598523\pi$
$168$	0	0
$169$	7.16053	0.550810
$170$	−23.3761	−1.79286
$171$	0	0
$172$	−43.9030	−3.34757
$173$	−18.3086	−1.39198	−0.695990	−	0.718052i	$-0.745034\pi$
$173$	−18.3086	−1.39198	−0.695990	+	0.718052i	$0.745034\pi$
$174$	0	0
$175$	6.42349	0.485570
$176$	−71.1954	−5.36655
$177$	0	0
$178$	−26.0870	−1.95530
$179$	15.1624	1.13329	0.566645	−	0.823962i	$-0.308241\pi$
$179$	15.1624	1.13329	0.566645	+	0.823962i	$0.308241\pi$
$180$	0	0
$181$	−12.7500	−0.947697	−0.473849	−	0.880606i	$-0.657136\pi$
$181$	−12.7500	−0.947697	−0.473849	+	0.880606i	$0.657136\pi$
$182$	−12.1381	−0.899738
$183$	0	0
$184$	54.7203	4.03403
$185$	−34.9627	−2.57051
$186$	0	0
$187$	−13.4334	−0.982349
$188$	−18.1030	−1.32029
$189$	0	0
$190$	62.6874	4.54782
$191$	−17.0688	−1.23506	−0.617528	−	0.786549i	$-0.711866\pi$
$191$	−17.0688	−1.23506	−0.617528	+	0.786549i	$0.711866\pi$
$192$	0	0
$193$	9.72396	0.699946	0.349973	−	0.936760i	$-0.386191\pi$
$193$	9.72396	0.699946	0.349973	+	0.936760i	$0.386191\pi$
$194$	−10.3476	−0.742911
$195$	0	0
$196$	5.30805	0.379146
$197$	−14.9898	−1.06798	−0.533989	−	0.845491i	$-0.679308\pi$
$197$	−14.9898	−1.06798	−0.533989	+	0.845491i	$0.679308\pi$
$198$	0	0
$199$	−10.3640	−0.734684	−0.367342	−	0.930086i	$-0.619732\pi$
$199$	−10.3640	−0.734684	−0.367342	+	0.930086i	$0.619732\pi$
$200$	−57.4439	−4.06190
$201$	0	0
$202$	18.7969	1.32255
$203$	9.43242	0.662026
$204$	0	0
$205$	−22.8112	−1.59320
$206$	24.2105	1.68682
$207$	0	0
$208$	60.8819	4.22140
$209$	36.0243	2.49185
$210$	0	0
$211$	−10.4342	−0.718323	−0.359162	−	0.933275i	$-0.616937\pi$
$211$	−10.4342	−0.718323	−0.359162	+	0.933275i	$0.616937\pi$
$212$	7.24900	0.497864
$213$	0	0
$214$	16.0644	1.09814
$215$	−27.9550	−1.90651
$216$	0	0
$217$	−0.617404	−0.0419121
$218$	24.8357	1.68208
$219$	0	0
$220$	−94.1996	−6.35094
$221$	11.4874	0.772728
$222$	0	0
$223$	7.78334	0.521211	0.260605	−	0.965445i	$-0.416078\pi$
$223$	7.78334	0.521211	0.260605	+	0.965445i	$0.416078\pi$
$224$	−18.7698	−1.25411
$225$	0	0
$226$	53.2129	3.53967
$227$	−15.3057	−1.01588	−0.507938	−	0.861394i	$-0.669592\pi$
$227$	−15.3057	−1.01588	−0.507938	+	0.861394i	$0.669592\pi$
$228$	0	0
$229$	13.3312	0.880950	0.440475	−	0.897765i	$-0.354810\pi$
$229$	13.3312	0.880950	0.440475	+	0.897765i	$0.354810\pi$
$230$	55.9082	3.68648
$231$	0	0
$232$	−84.3521	−5.53799
$233$	0.980505	0.0642351	0.0321175	−	0.999484i	$-0.489775\pi$
$233$	0.980505	0.0642351	0.0321175	+	0.999484i	$0.489775\pi$
$234$	0	0
$235$	−11.5269	−0.751935
$236$	−1.79197	−0.116648
$237$	0	0
$238$	−6.91628	−0.448316
$239$	−15.3472	−0.992726	−0.496363	−	0.868115i	$-0.665332\pi$
$239$	−15.3472	−0.992726	−0.496363	+	0.868115i	$0.665332\pi$
$240$	0	0
$241$	−14.8878	−0.959008	−0.479504	−	0.877540i	$-0.659183\pi$
$241$	−14.8878	−0.959008	−0.479504	+	0.877540i	$0.659183\pi$
$242$	−44.7931	−2.87941
$243$	0	0
$244$	30.3310	1.94174
$245$	3.37987	0.215932
$246$	0	0
$247$	−30.8057	−1.96012
$248$	5.52131	0.350603
$249$	0	0
$250$	−13.0063	−0.822592
$251$	7.46391	0.471118	0.235559	−	0.971860i	$-0.424308\pi$
$251$	7.46391	0.471118	0.235559	+	0.971860i	$0.424308\pi$
$252$	0	0
$253$	32.1285	2.01990
$254$	2.70334	0.169623
$255$	0	0
$256$	23.9076	1.49422
$257$	14.5952	0.910424	0.455212	−	0.890383i	$-0.349564\pi$
$257$	14.5952	0.910424	0.455212	+	0.890383i	$0.349564\pi$
$258$	0	0
$259$	−10.3444	−0.642770
$260$	80.5537	4.99573
$261$	0	0
$262$	8.45827	0.522554
$263$	−3.51267	−0.216600	−0.108300	−	0.994118i	$-0.534541\pi$
$263$	−3.51267	−0.216600	−0.108300	+	0.994118i	$0.534541\pi$
$264$	0	0
$265$	4.61575	0.283544
$266$	18.5473	1.13721
$267$	0	0
$268$	35.4811	2.16735
$269$	−28.4308	−1.73346	−0.866730	−	0.498778i	$-0.833782\pi$
$269$	−28.4308	−1.73346	−0.866730	+	0.498778i	$0.833782\pi$
$270$	0	0
$271$	−14.5540	−0.884091	−0.442046	−	0.896993i	$-0.645747\pi$
$271$	−14.5540	−0.884091	−0.442046	+	0.896993i	$0.645747\pi$
$272$	34.6904	2.10341
$273$	0	0
$274$	−2.84417	−0.171823
$275$	−33.7276	−2.03385
$276$	0	0
$277$	−4.94513	−0.297124	−0.148562	−	0.988903i	$-0.547464\pi$
$277$	−4.94513	−0.297124	−0.148562	+	0.988903i	$0.547464\pi$
$278$	−6.30688	−0.378261
$279$	0	0
$280$	−30.2254	−1.80631
$281$	6.32037	0.377041	0.188521	−	0.982069i	$-0.439631\pi$
$281$	6.32037	0.377041	0.188521	+	0.982069i	$0.439631\pi$
$282$	0	0
$283$	−0.0334489	−0.00198833	−0.000994166	−	1.00000i	$-0.500316\pi$
$283$	−0.0334489	−0.00198833	−0.000994166		1.00000i	$0.500316\pi$
$284$	28.8104	1.70958
$285$	0	0
$286$	63.7333	3.76863
$287$	−6.74914	−0.398389
$288$	0	0
$289$	−10.4545	−0.614970
$290$	−86.1834	−5.06086
$291$	0	0
$292$	7.40931	0.433597
$293$	−2.54882	−0.148904	−0.0744518	−	0.997225i	$-0.523721\pi$
$293$	−2.54882	−0.148904	−0.0744518	+	0.997225i	$0.523721\pi$
$294$	0	0
$295$	−1.14103	−0.0664332
$296$	92.5077	5.37690
$297$	0	0
$298$	41.1043	2.38111
$299$	−27.4743	−1.58888
$300$	0	0
$301$	−8.27103	−0.476734
$302$	−13.5162	−0.777769
$303$	0	0
$304$	−93.0288	−5.33557
$305$	19.3131	1.10586
$306$	0	0
$307$	11.5638	0.659978	0.329989	−	0.943985i	$-0.392955\pi$
$307$	11.5638	0.659978	0.329989	+	0.943985i	$0.392955\pi$
$308$	−27.8708	−1.58809
$309$	0	0
$310$	5.64117	0.320397
$311$	4.48029	0.254054	0.127027	−	0.991899i	$-0.459457\pi$
$311$	4.48029	0.254054	0.127027	+	0.991899i	$0.459457\pi$
$312$	0	0
$313$	4.70683	0.266046	0.133023	−	0.991113i	$-0.457532\pi$
$313$	4.70683	0.266046	0.133023	+	0.991113i	$0.457532\pi$
$314$	−49.5572	−2.79668
$315$	0	0
$316$	−31.8903	−1.79397
$317$	17.3945	0.976973	0.488487	−	0.872571i	$-0.337549\pi$
$317$	17.3945	0.976973	0.488487	+	0.872571i	$0.337549\pi$
$318$	0	0
$319$	−49.5265	−2.77295
$320$	79.8412	4.46326
$321$	0	0
$322$	16.5415	0.921824
$323$	−17.5530	−0.976677
$324$	0	0
$325$	28.8418	1.59985
$326$	8.55735	0.473948
$327$	0	0
$328$	60.3561	3.33261
$329$	−3.41047	−0.188026
$330$	0	0
$331$	−30.3051	−1.66572	−0.832858	−	0.553486i	$-0.813297\pi$
$331$	−30.3051	−1.66572	−0.832858	+	0.553486i	$0.813297\pi$
$332$	82.7279	4.54028
$333$	0	0
$334$	21.2824	1.16452
$335$	22.5924	1.23435
$336$	0	0
$337$	−16.7859	−0.914387	−0.457193	−	0.889367i	$-0.651145\pi$
$337$	−16.7859	−0.914387	−0.457193	+	0.889367i	$0.651145\pi$
$338$	−19.3574	−1.05290
$339$	0	0
$340$	45.8993	2.48924
$341$	3.24178	0.175552
$342$	0	0
$343$	1.00000	0.0539949
$344$	73.9660	3.98798
$345$	0	0
$346$	49.4945	2.66084
$347$	−10.5664	−0.567237	−0.283618	−	0.958937i	$-0.591535\pi$
$347$	−10.5664	−0.567237	−0.283618	+	0.958937i	$0.591535\pi$
$348$	0	0
$349$	−23.9202	−1.28042	−0.640210	−	0.768200i	$-0.721153\pi$
$349$	−23.9202	−1.28042	−0.640210	+	0.768200i	$0.721153\pi$
$350$	−17.3649	−0.928192
$351$	0	0
$352$	98.5541	5.25295
$353$	−16.7448	−0.891234	−0.445617	−	0.895224i	$-0.647016\pi$
$353$	−16.7448	−0.891234	−0.445617	+	0.895224i	$0.647016\pi$
$354$	0	0
$355$	18.3448	0.973641
$356$	51.2222	2.71477
$357$	0	0
$358$	−40.9891	−2.16634
$359$	−20.7481	−1.09504	−0.547521	−	0.836792i	$-0.684428\pi$
$359$	−20.7481	−1.09504	−0.547521	+	0.836792i	$0.684428\pi$
$360$	0	0
$361$	28.0718	1.47746
$362$	34.4675	1.81157
$363$	0	0
$364$	23.8334	1.24921
$365$	4.71783	0.246942
$366$	0	0
$367$	16.6774	0.870552	0.435276	−	0.900297i	$-0.356651\pi$
$367$	16.6774	0.870552	0.435276	+	0.900297i	$0.356651\pi$
$368$	−82.9683	−4.32502
$369$	0	0
$370$	94.5160	4.91365
$371$	1.36566	0.0709017
$372$	0	0
$373$	−17.9735	−0.930630	−0.465315	−	0.885145i	$-0.654059\pi$
$373$	−17.9735	−0.930630	−0.465315	+	0.885145i	$0.654059\pi$
$374$	36.3151	1.87781
$375$	0	0
$376$	30.4991	1.57287
$377$	42.3520	2.18124
$378$	0	0
$379$	24.7475	1.27119	0.635596	−	0.772022i	$-0.280754\pi$
$379$	24.7475	1.27119	0.635596	+	0.772022i	$0.280754\pi$
$380$	−123.088	−6.31427
$381$	0	0
$382$	46.1428	2.36087
$383$	−12.0933	−0.617937	−0.308969	−	0.951072i	$-0.599984\pi$
$383$	−12.0933	−0.617937	−0.308969	+	0.951072i	$0.599984\pi$
$384$	0	0
$385$	−17.7466	−0.904448
$386$	−26.2872	−1.33798
$387$	0	0
$388$	20.3176	1.03147
$389$	17.9749	0.911366	0.455683	−	0.890142i	$-0.349395\pi$
$389$	17.9749	0.911366	0.455683	+	0.890142i	$0.349395\pi$
$390$	0	0
$391$	−15.6548	−0.791697
$392$	−8.94278	−0.451679
$393$	0	0
$394$	40.5225	2.04149
$395$	−20.3060	−1.02170
$396$	0	0
$397$	−20.0933	−1.00845	−0.504227	−	0.863571i	$-0.668223\pi$
$397$	−20.0933	−1.00845	−0.504227	+	0.863571i	$0.668223\pi$
$398$	28.0174	1.40438
$399$	0	0
$400$	87.0980	4.35490
$401$	12.8155	0.639977	0.319988	−	0.947421i	$-0.396321\pi$
$401$	12.8155	0.639977	0.319988	+	0.947421i	$0.396321\pi$
$402$	0	0
$403$	−2.77217	−0.138092
$404$	−36.9081	−1.83625
$405$	0	0
$406$	−25.4990	−1.26550
$407$	54.3150	2.69230
$408$	0	0
$409$	−36.7896	−1.81913	−0.909565	−	0.415561i	$-0.863585\pi$
$409$	−36.7896	−1.81913	−0.909565	+	0.415561i	$0.863585\pi$
$410$	61.6664	3.04548
$411$	0	0
$412$	−47.5376	−2.34201
$413$	−0.337596	−0.0166120
$414$	0	0
$415$	52.6764	2.58578
$416$	−84.2774	−4.13204
$417$	0	0
$418$	−97.3858	−4.76330
$419$	−6.79609	−0.332011	−0.166005	−	0.986125i	$-0.553087\pi$
$419$	−6.79609	−0.332011	−0.166005	+	0.986125i	$0.553087\pi$
$420$	0	0
$421$	−3.22713	−0.157281	−0.0786403	−	0.996903i	$-0.525058\pi$
$421$	−3.22713	−0.157281	−0.0786403	+	0.996903i	$0.525058\pi$
$422$	28.2073	1.37311
$423$	0	0
$424$	−12.2128	−0.593107
$425$	16.4340	0.797165
$426$	0	0
$427$	5.71415	0.276527
$428$	−31.5428	−1.52468
$429$	0	0
$430$	75.5718	3.64439
$431$	−0.898205	−0.0432650	−0.0216325	−	0.999766i	$-0.506886\pi$
$431$	−0.898205	−0.0432650	−0.0216325	+	0.999766i	$0.506886\pi$
$432$	0	0
$433$	−9.28781	−0.446344	−0.223172	−	0.974779i	$-0.571641\pi$
$433$	−9.28781	−0.446344	−0.223172	+	0.974779i	$0.571641\pi$
$434$	1.66905	0.0801171
$435$	0	0
$436$	−48.7652	−2.33543
$437$	41.9813	2.00824
$438$	0	0
$439$	31.8478	1.52001	0.760007	−	0.649915i	$-0.225195\pi$
$439$	31.8478	1.52001	0.760007	+	0.649915i	$0.225195\pi$
$440$	158.704	7.56590
$441$	0	0
$442$	−31.0544	−1.47711
$443$	30.0862	1.42944	0.714719	−	0.699412i	$-0.246555\pi$
$443$	30.0862	1.42944	0.714719	+	0.699412i	$0.246555\pi$
$444$	0	0
$445$	32.6154	1.54612
$446$	−21.0410	−0.996321
$447$	0	0
$448$	23.6226	1.11606
$449$	4.23952	0.200075	0.100038	−	0.994984i	$-0.468104\pi$
$449$	4.23952	0.200075	0.100038	+	0.994984i	$0.468104\pi$
$450$	0	0
$451$	35.4375	1.66869
$452$	−104.484	−4.91453
$453$	0	0
$454$	41.3765	1.94190
$455$	15.1758	0.711451
$456$	0	0
$457$	33.7568	1.57907	0.789537	−	0.613702i	$-0.210321\pi$
$457$	33.7568	1.57907	0.789537	+	0.613702i	$0.210321\pi$
$458$	−36.0388	−1.68398
$459$	0	0
$460$	−109.777	−5.11836
$461$	−26.6862	−1.24290	−0.621450	−	0.783454i	$-0.713456\pi$
$461$	−26.6862	−1.24290	−0.621450	+	0.783454i	$0.713456\pi$
$462$	0	0
$463$	7.67632	0.356749	0.178374	−	0.983963i	$-0.442916\pi$
$463$	7.67632	0.356749	0.178374	+	0.983963i	$0.442916\pi$
$464$	127.897	5.93747
$465$	0	0
$466$	−2.65064	−0.122789
$467$	19.7939	0.915951	0.457976	−	0.888965i	$-0.348575\pi$
$467$	19.7939	0.915951	0.457976	+	0.888965i	$0.348575\pi$
$468$	0	0
$469$	6.68440	0.308657
$470$	31.1613	1.43736
$471$	0	0
$472$	3.01904	0.138963
$473$	43.4284	1.99684
$474$	0	0
$475$	−44.0708	−2.02211
$476$	13.5802	0.622448
$477$	0	0
$478$	41.4886	1.89765
$479$	−17.6105	−0.804645	−0.402323	−	0.915498i	$-0.631797\pi$
$479$	−17.6105	−0.804645	−0.402323	+	0.915498i	$0.631797\pi$
$480$	0	0
$481$	−46.4468	−2.11779
$482$	40.2468	1.83319
$483$	0	0
$484$	87.9519	3.99781
$485$	12.9371	0.587442
$486$	0	0
$487$	23.7475	1.07610	0.538051	−	0.842912i	$-0.319161\pi$
$487$	23.7475	1.07610	0.538051	+	0.842912i	$0.319161\pi$
$488$	−51.1004	−2.31321
$489$	0	0
$490$	−9.13693	−0.412764
$491$	19.9562	0.900610	0.450305	−	0.892875i	$-0.351315\pi$
$491$	19.9562	0.900610	0.450305	+	0.892875i	$0.351315\pi$
$492$	0	0
$493$	24.1321	1.08686
$494$	83.2783	3.74687
$495$	0	0
$496$	−8.37156	−0.375894
$497$	5.42767	0.243464
$498$	0	0
$499$	8.66691	0.387984	0.193992	−	0.981003i	$-0.437856\pi$
$499$	8.66691	0.387984	0.193992	+	0.981003i	$0.437856\pi$
$500$	25.5381	1.14210
$501$	0	0
$502$	−20.1775	−0.900566
$503$	4.93173	0.219895	0.109948	−	0.993937i	$-0.464932\pi$
$503$	4.93173	0.219895	0.109948	+	0.993937i	$0.464932\pi$
$504$	0	0
$505$	−23.5010	−1.04578
$506$	−86.8542	−3.86114
$507$	0	0
$508$	−5.30805	−0.235507
$509$	−31.0693	−1.37712	−0.688561	−	0.725178i	$-0.741757\pi$
$509$	−31.0693	−1.37712	−0.688561	+	0.725178i	$0.741757\pi$
$510$	0	0
$511$	1.39586	0.0617494
$512$	−11.9897	−0.529875
$513$	0	0
$514$	−39.4558	−1.74032
$515$	−30.2693	−1.33382
$516$	0	0
$517$	17.9073	0.787561
$518$	27.9644	1.22869
$519$	0	0
$520$	−135.714	−5.95143
$521$	19.9687	0.874845	0.437423	−	0.899256i	$-0.355891\pi$
$521$	19.9687	0.874845	0.437423	+	0.899256i	$0.355891\pi$
$522$	0	0
$523$	−35.4320	−1.54933	−0.774666	−	0.632370i	$-0.782082\pi$
$523$	−35.4320	−1.54933	−0.774666	+	0.632370i	$0.782082\pi$
$524$	−16.6079	−0.725522
$525$	0	0
$526$	9.49594	0.414043
$527$	−1.57958	−0.0688075
$528$	0	0
$529$	14.4413	0.627882
$530$	−12.4780	−0.542008
$531$	0	0
$532$	−36.4179	−1.57892
$533$	−30.3040	−1.31261
$534$	0	0
$535$	−20.0847	−0.868335
$536$	−59.7771	−2.58198
$537$	0	0
$538$	76.8583	3.31359
$539$	−5.25067	−0.226162
$540$	0	0
$541$	31.7052	1.36311	0.681557	−	0.731765i	$-0.261303\pi$
$541$	31.7052	1.36311	0.681557	+	0.731765i	$0.261303\pi$
$542$	39.3444	1.68999
$543$	0	0
$544$	−48.0211	−2.05889
$545$	−31.0509	−1.33008
$546$	0	0
$547$	4.49178	0.192055	0.0960274	−	0.995379i	$-0.469386\pi$
$547$	4.49178	0.192055	0.0960274	+	0.995379i	$0.469386\pi$
$548$	5.58457	0.238561
$549$	0	0
$550$	91.1772	3.88781
$551$	−64.7148	−2.75694
$552$	0	0
$553$	−6.00792	−0.255483
$554$	13.3684	0.567968
$555$	0	0
$556$	12.3837	0.525184
$557$	40.0124	1.69538	0.847691	−	0.530491i	$-0.177992\pi$
$557$	40.0124	1.69538	0.847691	+	0.530491i	$0.177992\pi$
$558$	0	0
$559$	−37.1373	−1.57074
$560$	45.8286	1.93661
$561$	0	0
$562$	−17.0861	−0.720734
$563$	3.72850	0.157138	0.0785689	−	0.996909i	$-0.474965\pi$
$563$	3.72850	0.157138	0.0785689	+	0.996909i	$0.474965\pi$
$564$	0	0
$565$	−66.5297	−2.79892
$566$	0.0904239	0.00380080
$567$	0	0
$568$	−48.5385	−2.03663
$569$	16.1147	0.675565	0.337782	−	0.941224i	$-0.390323\pi$
$569$	16.1147	0.675565	0.337782	+	0.941224i	$0.390323\pi$
$570$	0	0
$571$	16.1134	0.674323	0.337162	−	0.941447i	$-0.390533\pi$
$571$	16.1134	0.674323	0.337162	+	0.941447i	$0.390533\pi$
$572$	−125.141	−5.23242
$573$	0	0
$574$	18.2452	0.761541
$575$	−39.3049	−1.63913
$576$	0	0
$577$	−15.0898	−0.628195	−0.314097	−	0.949391i	$-0.601702\pi$
$577$	−15.0898	−0.628195	−0.314097	+	0.949391i	$0.601702\pi$
$578$	28.2620	1.17555
$579$	0	0
$580$	169.222	7.02657
$581$	15.5854	0.646590
$582$	0	0
$583$	−7.17064	−0.296978
$584$	−12.4829	−0.516546
$585$	0	0
$586$	6.89033	0.284637
$587$	−33.6856	−1.39035	−0.695176	−	0.718839i	$-0.744674\pi$
$587$	−33.6856	−1.39035	−0.695176	+	0.718839i	$0.744674\pi$
$588$	0	0
$589$	4.23594	0.174539
$590$	3.08459	0.126990
$591$	0	0
$592$	−140.263	−5.76476
$593$	32.7387	1.34442	0.672208	−	0.740363i	$-0.265346\pi$
$593$	32.7387	1.34442	0.672208	+	0.740363i	$0.265346\pi$
$594$	0	0
$595$	8.64712	0.354497
$596$	−80.7088	−3.30596
$597$	0	0
$598$	74.2724	3.03722
$599$	9.12458	0.372820	0.186410	−	0.982472i	$-0.440315\pi$
$599$	9.12458	0.372820	0.186410	+	0.982472i	$0.440315\pi$
$600$	0	0
$601$	42.9337	1.75130	0.875651	−	0.482944i	$-0.160433\pi$
$601$	42.9337	1.75130	0.875651	+	0.482944i	$0.160433\pi$
$602$	22.3594	0.911301
$603$	0	0
$604$	26.5392	1.07987
$605$	56.0028	2.27684
$606$	0	0
$607$	35.8019	1.45315	0.726576	−	0.687086i	$-0.241110\pi$
$607$	35.8019	1.45315	0.726576	+	0.687086i	$0.241110\pi$
$608$	128.778	5.22262
$609$	0	0
$610$	−52.2097	−2.11391
$611$	−15.3132	−0.619506
$612$	0	0
$613$	14.5183	0.586390	0.293195	−	0.956053i	$-0.405281\pi$
$613$	14.5183	0.586390	0.293195	+	0.956053i	$0.405281\pi$
$614$	−31.2608	−1.26158
$615$	0	0
$616$	46.9556	1.89190
$617$	−20.4296	−0.822465	−0.411233	−	0.911530i	$-0.634902\pi$
$617$	−20.4296	−0.822465	−0.411233	+	0.911530i	$0.634902\pi$
$618$	0	0
$619$	−21.9525	−0.882344	−0.441172	−	0.897423i	$-0.645437\pi$
$619$	−21.9525	−0.882344	−0.441172	+	0.897423i	$0.645437\pi$
$620$	−11.0765	−0.444844
$621$	0	0
$622$	−12.1117	−0.485637
$623$	9.64990	0.386615
$624$	0	0
$625$	−15.8562	−0.634249
$626$	−12.7242	−0.508560
$627$	0	0
$628$	97.3064	3.88295
$629$	−26.4653	−1.05524
$630$	0	0
$631$	−24.2276	−0.964484	−0.482242	−	0.876038i	$-0.660177\pi$
$631$	−24.2276	−0.964484	−0.482242	+	0.876038i	$0.660177\pi$
$632$	53.7275	2.13717
$633$	0	0
$634$	−47.0233	−1.86753
$635$	−3.37987	−0.134126
$636$	0	0
$637$	4.49005	0.177902
$638$	133.887	5.30064
$639$	0	0
$640$	−88.9592	−3.51642
$641$	−12.2334	−0.483190	−0.241595	−	0.970377i	$-0.577670\pi$
$641$	−12.2334	−0.483190	−0.241595	+	0.970377i	$0.577670\pi$
$642$	0	0
$643$	−12.0429	−0.474927	−0.237463	−	0.971396i	$-0.576316\pi$
$643$	−12.0429	−0.474927	−0.237463	+	0.971396i	$0.576316\pi$
$644$	−32.4796	−1.27987
$645$	0	0
$646$	47.4518	1.86697
$647$	1.39209	0.0547288	0.0273644	−	0.999626i	$-0.491289\pi$
$647$	1.39209	0.0547288	0.0273644	+	0.999626i	$0.491289\pi$
$648$	0	0
$649$	1.77260	0.0695807
$650$	−77.9692	−3.05820
$651$	0	0
$652$	−16.8025	−0.658036
$653$	−2.45928	−0.0962392	−0.0481196	−	0.998842i	$-0.515323\pi$
$653$	−2.45928	−0.0962392	−0.0481196	+	0.998842i	$0.515323\pi$
$654$	0	0
$655$	−10.5750	−0.413200
$656$	−91.5135	−3.57300
$657$	0	0
$658$	9.21967	0.359420
$659$	−20.6958	−0.806193	−0.403097	−	0.915157i	$-0.632066\pi$
$659$	−20.6958	−0.806193	−0.403097	+	0.915157i	$0.632066\pi$
$660$	0	0
$661$	18.6274	0.724520	0.362260	−	0.932077i	$-0.382005\pi$
$661$	18.6274	0.724520	0.362260	+	0.932077i	$0.382005\pi$
$662$	81.9249	3.18410
$663$	0	0
$664$	−139.377	−5.40886
$665$	−23.1889	−0.899226
$666$	0	0
$667$	−57.7163	−2.23479
$668$	−41.7884	−1.61684
$669$	0	0
$670$	−61.0749	−2.35953
$671$	−30.0031	−1.15826
$672$	0	0
$673$	−7.42948	−0.286385	−0.143193	−	0.989695i	$-0.545737\pi$
$673$	−7.42948	−0.286385	−0.143193	+	0.989695i	$0.545737\pi$
$674$	45.3780	1.74790
$675$	0	0
$676$	38.0085	1.46186
$677$	−26.4077	−1.01493	−0.507465	−	0.861672i	$-0.669417\pi$
$677$	−26.4077	−1.01493	−0.507465	+	0.861672i	$0.669417\pi$
$678$	0	0
$679$	3.82769	0.146893
$680$	−77.3293	−2.96544
$681$	0	0
$682$	−8.76364	−0.335577
$683$	29.6157	1.13321	0.566607	−	0.823988i	$-0.308256\pi$
$683$	29.6157	1.13321	0.566607	+	0.823988i	$0.308256\pi$
$684$	0	0
$685$	3.55594	0.135865
$686$	−2.70334	−0.103214
$687$	0	0
$688$	−112.149	−4.27565
$689$	6.13189	0.233606
$690$	0	0
$691$	−38.9659	−1.48233	−0.741166	−	0.671322i	$-0.765727\pi$
$691$	−38.9659	−1.48233	−0.741166	+	0.671322i	$0.765727\pi$
$692$	−97.1831	−3.69435
$693$	0	0
$694$	28.5647	1.08430
$695$	7.88521	0.299103
$696$	0	0
$697$	−17.2671	−0.654039
$698$	64.6645	2.44759
$699$	0	0
$700$	34.0962	1.28872
$701$	−0.751645	−0.0283893	−0.0141946	−	0.999899i	$-0.504518\pi$
$701$	−0.751645	−0.0283893	−0.0141946	+	0.999899i	$0.504518\pi$
$702$	0	0
$703$	70.9718	2.67675
$704$	−124.034	−4.67473
$705$	0	0
$706$	45.2668	1.70364
$707$	−6.95323	−0.261503
$708$	0	0
$709$	−8.20397	−0.308106	−0.154053	−	0.988063i	$-0.549233\pi$
$709$	−8.20397	−0.308106	−0.154053	+	0.988063i	$0.549233\pi$
$710$	−49.5922	−1.86116
$711$	0	0
$712$	−86.2970	−3.23412
$713$	3.77785	0.141482
$714$	0	0
$715$	−79.6829	−2.97997
$716$	80.4826	3.00778
$717$	0	0
$718$	56.0891	2.09323
$719$	−47.3025	−1.76409	−0.882043	−	0.471169i	$-0.843832\pi$
$719$	−47.3025	−1.76409	−0.882043	+	0.471169i	$0.843832\pi$
$720$	0	0
$721$	−8.95576	−0.333530
$722$	−75.8876	−2.82424
$723$	0	0
$724$	−67.6774	−2.51521
$725$	60.5891	2.25022
$726$	0	0
$727$	−16.4466	−0.609972	−0.304986	−	0.952357i	$-0.598652\pi$
$727$	−16.4466	−0.609972	−0.304986	+	0.952357i	$0.598652\pi$
$728$	−40.1535	−1.48819
$729$	0	0
$730$	−12.7539	−0.472043
$731$	−21.1608	−0.782659
$732$	0	0
$733$	2.21251	0.0817209	0.0408604	−	0.999165i	$-0.486990\pi$
$733$	2.21251	0.0817209	0.0408604	+	0.999165i	$0.486990\pi$
$734$	−45.0846	−1.66410
$735$	0	0
$736$	114.851	4.23347
$737$	−35.0976	−1.29284
$738$	0	0
$739$	−37.2766	−1.37124	−0.685622	−	0.727958i	$-0.740470\pi$
$739$	−37.2766	−1.37124	−0.685622	+	0.727958i	$0.740470\pi$
$740$	−185.584	−6.82219
$741$	0	0
$742$	−3.69185	−0.135532
$743$	−19.2950	−0.707865	−0.353933	−	0.935271i	$-0.615156\pi$
$743$	−19.2950	−0.707865	−0.353933	+	0.935271i	$0.615156\pi$
$744$	0	0
$745$	−51.3908	−1.88281
$746$	48.5884	1.77895
$747$	0	0
$748$	−71.3052	−2.60718
$749$	−5.94244	−0.217132
$750$	0	0
$751$	−0.882173	−0.0321909	−0.0160955	−	0.999870i	$-0.505124\pi$
$751$	−0.882173	−0.0321909	−0.0160955	+	0.999870i	$0.505124\pi$
$752$	−46.2436	−1.68633
$753$	0	0
$754$	−114.492	−4.16955
$755$	16.8987	0.615006
$756$	0	0
$757$	0.911581	0.0331320	0.0165660	−	0.999863i	$-0.494727\pi$
$757$	0.911581	0.0331320	0.0165660	+	0.999863i	$0.494727\pi$
$758$	−66.9009	−2.42995
$759$	0	0
$760$	207.373	7.52221
$761$	−17.7994	−0.645228	−0.322614	−	0.946531i	$-0.604561\pi$
$761$	−17.7994	−0.645228	−0.322614	+	0.946531i	$0.604561\pi$
$762$	0	0
$763$	−9.18703	−0.332593
$764$	−90.6022	−3.27787
$765$	0	0
$766$	32.6922	1.18122
$767$	−1.51582	−0.0547331
$768$	0	0
$769$	−15.9737	−0.576028	−0.288014	−	0.957626i	$-0.592995\pi$
$769$	−15.9737	−0.576028	−0.288014	+	0.957626i	$0.592995\pi$
$770$	47.9750	1.72890
$771$	0	0
$772$	51.6153	1.85767
$773$	42.1447	1.51584	0.757919	−	0.652349i	$-0.226216\pi$
$773$	42.1447	1.51584	0.757919	+	0.652349i	$0.226216\pi$
$774$	0	0
$775$	−3.96589	−0.142459
$776$	−34.2302	−1.22879
$777$	0	0
$778$	−48.5924	−1.74212
$779$	46.3051	1.65905
$780$	0	0
$781$	−28.4989	−1.01977
$782$	42.3202	1.51337
$783$	0	0
$784$	13.5593	0.484260
$785$	61.9592	2.21142
$786$	0	0
$787$	−26.5620	−0.946833	−0.473417	−	0.880839i	$-0.656979\pi$
$787$	−26.5620	−0.946833	−0.473417	+	0.880839i	$0.656979\pi$
$788$	−79.5665	−2.83444
$789$	0	0
$790$	54.8939	1.95304
$791$	−19.6841	−0.699887
$792$	0	0
$793$	25.6568	0.911100
$794$	54.3191	1.92771
$795$	0	0
$796$	−55.0126	−1.94987
$797$	−26.1504	−0.926295	−0.463148	−	0.886281i	$-0.653280\pi$
$797$	−26.1504	−0.926295	−0.463148	+	0.886281i	$0.653280\pi$
$798$	0	0
$799$	−8.72543	−0.308683
$800$	−120.568	−4.26271
$801$	0	0
$802$	−34.6447	−1.22335
$803$	−7.32922	−0.258642
$804$	0	0
$805$	−20.6812	−0.728915
$806$	7.49413	0.263969
$807$	0	0
$808$	62.1812	2.18753
$809$	−14.9716	−0.526372	−0.263186	−	0.964745i	$-0.584773\pi$
$809$	−14.9716	−0.526372	−0.263186	+	0.964745i	$0.584773\pi$
$810$	0	0
$811$	−17.8044	−0.625198	−0.312599	−	0.949885i	$-0.601200\pi$
$811$	−17.8044	−0.625198	−0.312599	+	0.949885i	$0.601200\pi$
$812$	50.0678	1.75703
$813$	0	0
$814$	−146.832	−5.14646
$815$	−10.6989	−0.374765
$816$	0	0
$817$	56.7466	1.98531
$818$	99.4549	3.47736
$819$	0	0
$820$	−121.083	−4.22840
$821$	21.6246	0.754704	0.377352	−	0.926070i	$-0.376835\pi$
$821$	21.6246	0.754704	0.377352	+	0.926070i	$0.376835\pi$
$822$	0	0
$823$	33.7893	1.17782	0.588911	−	0.808198i	$-0.299557\pi$
$823$	33.7893	1.17782	0.588911	+	0.808198i	$0.299557\pi$
$824$	80.0894	2.79005
$825$	0	0
$826$	0.912636	0.0317547
$827$	20.3700	0.708335	0.354167	−	0.935182i	$-0.384764\pi$
$827$	20.3700	0.708335	0.354167	+	0.935182i	$0.384764\pi$
$828$	0	0
$829$	19.1516	0.665163	0.332581	−	0.943075i	$-0.392080\pi$
$829$	19.1516	0.665163	0.332581	+	0.943075i	$0.392080\pi$
$830$	−142.402	−4.94286
$831$	0	0
$832$	106.067	3.67720
$833$	2.55842	0.0886440
$834$	0	0
$835$	−26.6085	−0.920825
$836$	191.219	6.61343
$837$	0	0
$838$	18.3722	0.634656
$839$	19.0227	0.656737	0.328368	−	0.944550i	$-0.393501\pi$
$839$	19.0227	0.656737	0.328368	+	0.944550i	$0.393501\pi$
$840$	0	0
$841$	59.9706	2.06795
$842$	8.72402	0.300650
$843$	0	0
$844$	−55.3855	−1.90645
$845$	24.2016	0.832562
$846$	0	0
$847$	16.5695	0.569336
$848$	18.5174	0.635891
$849$	0	0
$850$	−44.4267	−1.52382
$851$	63.2966	2.16978
$852$	0	0
$853$	−28.4911	−0.975517	−0.487758	−	0.872979i	$-0.662185\pi$
$853$	−28.4911	−0.975517	−0.487758	+	0.872979i	$0.662185\pi$
$854$	−15.4473	−0.528595
$855$	0	0
$856$	53.1420	1.81636
$857$	−11.1578	−0.381143	−0.190571	−	0.981673i	$-0.561034\pi$
$857$	−11.1578	−0.381143	−0.190571	+	0.981673i	$0.561034\pi$
$858$	0	0
$859$	−17.6130	−0.600948	−0.300474	−	0.953790i	$-0.597145\pi$
$859$	−17.6130	−0.600948	−0.300474	+	0.953790i	$0.597145\pi$
$860$	−148.386	−5.05993
$861$	0	0
$862$	2.42815	0.0827033
$863$	−28.3944	−0.966555	−0.483278	−	0.875467i	$-0.660554\pi$
$863$	−28.3944	−0.966555	−0.483278	+	0.875467i	$0.660554\pi$
$864$	0	0
$865$	−61.8807	−2.10401
$866$	25.1081	0.853208
$867$	0	0
$868$	−3.27721	−0.111236
$869$	31.5456	1.07011
$870$	0	0
$871$	30.0133	1.01696
$872$	82.1576	2.78221
$873$	0	0
$874$	−113.490	−3.83885
$875$	4.81121	0.162648
$876$	0	0
$877$	−15.9532	−0.538703	−0.269351	−	0.963042i	$-0.586809\pi$
$877$	−15.9532	−0.538703	−0.269351	+	0.963042i	$0.586809\pi$
$878$	−86.0955	−2.90558
$879$	0	0
$880$	−240.631	−8.11166
$881$	10.2652	0.345842	0.172921	−	0.984936i	$-0.444680\pi$
$881$	10.2652	0.345842	0.172921	+	0.984936i	$0.444680\pi$
$882$	0	0
$883$	28.8873	0.972134	0.486067	−	0.873921i	$-0.338431\pi$
$883$	28.8873	0.972134	0.486067	+	0.873921i	$0.338431\pi$
$884$	60.9758	2.05084
$885$	0	0
$886$	−81.3332	−2.73244
$887$	18.2617	0.613168	0.306584	−	0.951844i	$-0.400814\pi$
$887$	18.2617	0.613168	0.306584	+	0.951844i	$0.400814\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−88.1705	−2.95548
$891$	0	0
$892$	41.3143	1.38331
$893$	23.3989	0.783014
$894$	0	0
$895$	51.2468	1.71299
$896$	−26.3203	−0.879300
$897$	0	0
$898$	−11.4609	−0.382454
$899$	−5.82361	−0.194228
$900$	0	0
$901$	3.49394	0.116400
$902$	−95.7996	−3.18978
$903$	0	0
$904$	176.031	5.85470
$905$	−43.0932	−1.43247
$906$	0	0
$907$	44.4961	1.47747	0.738735	−	0.673996i	$-0.235424\pi$
$907$	44.4961	1.47747	0.738735	+	0.673996i	$0.235424\pi$
$908$	−81.2434	−2.69616
$909$	0	0
$910$	−41.0252	−1.35997
$911$	10.0562	0.333176	0.166588	−	0.986027i	$-0.446725\pi$
$911$	10.0562	0.333176	0.166588	+	0.986027i	$0.446725\pi$
$912$	0	0
$913$	−81.8336	−2.70830
$914$	−91.2560	−3.01848
$915$	0	0
$916$	70.7627	2.33806
$917$	−3.12882	−0.103323
$918$	0	0
$919$	3.74615	0.123574	0.0617870	−	0.998089i	$-0.480320\pi$
$919$	3.74615	0.123574	0.0617870	+	0.998089i	$0.480320\pi$
$920$	184.947	6.09753
$921$	0	0
$922$	72.1419	2.37587
$923$	24.3705	0.802165
$924$	0	0
$925$	−66.4471	−2.18477
$926$	−20.7517	−0.681944
$927$	0	0
$928$	−177.045	−5.81178
$929$	0.579663	0.0190181	0.00950906	−	0.999955i	$-0.496973\pi$
$929$	0.579663	0.0190181	0.00950906	+	0.999955i	$0.496973\pi$
$930$	0	0
$931$	−6.86089	−0.224857
$932$	5.20457	0.170481
$933$	0	0
$934$	−53.5096	−1.75089
$935$	−45.4031	−1.48484
$936$	0	0
$937$	32.3440	1.05663	0.528315	−	0.849048i	$-0.322824\pi$
$937$	32.3440	1.05663	0.528315	+	0.849048i	$0.322824\pi$
$938$	−18.0702	−0.590013
$939$	0	0
$940$	−61.1856	−1.99565
$941$	−41.9432	−1.36731	−0.683654	−	0.729806i	$-0.739610\pi$
$941$	−41.9432	−1.36731	−0.683654	+	0.729806i	$0.739610\pi$
$942$	0	0
$943$	41.2975	1.34483
$944$	−4.57756	−0.148987
$945$	0	0
$946$	−117.402	−3.81706
$947$	8.39075	0.272663	0.136331	−	0.990663i	$-0.456469\pi$
$947$	8.39075	0.272663	0.136331	+	0.990663i	$0.456469\pi$
$948$	0	0
$949$	6.26749	0.203451
$950$	119.139	3.86536
$951$	0	0
$952$	−22.8794	−0.741526
$953$	54.9918	1.78136	0.890680	−	0.454631i	$-0.150229\pi$
$953$	54.9918	1.78136	0.890680	+	0.454631i	$0.150229\pi$
$954$	0	0
$955$	−57.6903	−1.86682
$956$	−81.4636	−2.63472
$957$	0	0
$958$	47.6072	1.53812
$959$	1.05209	0.0339739
$960$	0	0
$961$	−30.6188	−0.987704
$962$	125.562	4.04827
$963$	0	0
$964$	−79.0252	−2.54523
$965$	32.8657	1.05798
$966$	0	0
$967$	−45.4952	−1.46303	−0.731514	−	0.681826i	$-0.761186\pi$
$967$	−45.4952	−1.46303	−0.731514	+	0.681826i	$0.761186\pi$
$968$	−148.178	−4.76261
$969$	0	0
$970$	−34.9733	−1.12293
$971$	6.90693	0.221654	0.110827	−	0.993840i	$-0.464650\pi$
$971$	6.90693	0.221654	0.110827	+	0.993840i	$0.464650\pi$
$972$	0	0
$973$	2.33299	0.0747924
$974$	−64.1976	−2.05702
$975$	0	0
$976$	77.4798	2.48007
$977$	−0.719851	−0.0230301	−0.0115150	−	0.999934i	$-0.503665\pi$
$977$	−0.719851	−0.0230301	−0.0115150	+	0.999934i	$0.503665\pi$
$978$	0	0
$979$	−50.6684	−1.61937
$980$	17.9405	0.573088
$981$	0	0
$982$	−53.9484	−1.72156
$983$	27.5040	0.877243	0.438621	−	0.898672i	$-0.355467\pi$
$983$	27.5040	0.877243	0.438621	+	0.898672i	$0.355467\pi$
$984$	0	0
$985$	−50.6635	−1.61427
$986$	−65.2373	−2.07758
$987$	0	0
$988$	−163.518	−5.20221
$989$	50.6098	1.60930
$990$	0	0
$991$	7.91961	0.251575	0.125787	−	0.992057i	$-0.459854\pi$
$991$	7.91961	0.251575	0.125787	+	0.992057i	$0.459854\pi$
$992$	11.5886	0.367937
$993$	0	0
$994$	−14.6728	−0.465394
$995$	−35.0289	−1.11049
$996$	0	0
$997$	−24.9260	−0.789414	−0.394707	−	0.918807i	$-0.629154\pi$
$997$	−24.9260	−0.789414	−0.394707	+	0.918807i	$0.629154\pi$
$998$	−23.4296	−0.741652
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.2		20
3.2	odd	2		889.2.a.d.1.19	✓	20
21.20	even	2		6223.2.a.l.1.19		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.19	✓	20	3.2	odd	2
6223.2.a.l.1.19		20	21.20	even	2
8001.2.a.w.1.2		20	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70334 q^{2} +5.30805 q^{4} +3.37987 q^{5} +1.00000 q^{7} -8.94278 q^{8} +O(q^{10})\)	\(q-2.70334 q^{2} +5.30805 q^{4} +3.37987 q^{5} +1.00000 q^{7} -8.94278 q^{8} -9.13693 q^{10} -5.25067 q^{11} +4.49005 q^{13} -2.70334 q^{14} +13.5593 q^{16} +2.55842 q^{17} -6.86089 q^{19} +17.9405 q^{20} +14.1943 q^{22} -6.11893 q^{23} +6.42349 q^{25} -12.1381 q^{26} +5.30805 q^{28} +9.43242 q^{29} -0.617404 q^{31} -18.7698 q^{32} -6.91628 q^{34} +3.37987 q^{35} -10.3444 q^{37} +18.5473 q^{38} -30.2254 q^{40} -6.74914 q^{41} -8.27103 q^{43} -27.8708 q^{44} +16.5415 q^{46} -3.41047 q^{47} +1.00000 q^{49} -17.3649 q^{50} +23.8334 q^{52} +1.36566 q^{53} -17.7466 q^{55} -8.94278 q^{56} -25.4990 q^{58} -0.337596 q^{59} +5.71415 q^{61} +1.66905 q^{62} +23.6226 q^{64} +15.1758 q^{65} +6.68440 q^{67} +13.5802 q^{68} -9.13693 q^{70} +5.42767 q^{71} +1.39586 q^{73} +27.9644 q^{74} -36.4179 q^{76} -5.25067 q^{77} -6.00792 q^{79} +45.8286 q^{80} +18.2452 q^{82} +15.5854 q^{83} +8.64712 q^{85} +22.3594 q^{86} +46.9556 q^{88} +9.64990 q^{89} +4.49005 q^{91} -32.4796 q^{92} +9.21967 q^{94} -23.1889 q^{95} +3.82769 q^{97} -2.70334 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

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Database

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