LMFDB - Embedded newform 8001.2.a.q.1.10

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.15799$ 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)

$f(q)$	$=$	$q+1.15799 q^{2} -0.659052 q^{4} -3.10556 q^{5} -1.00000 q^{7} -3.07916 q^{8} +O(q^{10})$	\(q+1.15799 q^{2} -0.659052 q^{4} -3.10556 q^{5} -1.00000 q^{7} -3.07916 q^{8} -3.59621 q^{10} -2.14121 q^{11} +0.468697 q^{13} -1.15799 q^{14} -2.24754 q^{16} +3.75379 q^{17} +6.67029 q^{19} +2.04672 q^{20} -2.47950 q^{22} +2.72526 q^{23} +4.64448 q^{25} +0.542748 q^{26} +0.659052 q^{28} -1.21084 q^{29} +6.04597 q^{31} +3.55569 q^{32} +4.34686 q^{34} +3.10556 q^{35} -6.22635 q^{37} +7.72415 q^{38} +9.56252 q^{40} -7.66637 q^{41} -2.32678 q^{43} +1.41117 q^{44} +3.15583 q^{46} +6.31293 q^{47} +1.00000 q^{49} +5.37827 q^{50} -0.308896 q^{52} -7.01780 q^{53} +6.64963 q^{55} +3.07916 q^{56} -1.40215 q^{58} +5.33351 q^{59} +3.01407 q^{61} +7.00119 q^{62} +8.61255 q^{64} -1.45557 q^{65} -5.67105 q^{67} -2.47394 q^{68} +3.59621 q^{70} +3.57957 q^{71} -9.11747 q^{73} -7.21007 q^{74} -4.39607 q^{76} +2.14121 q^{77} +13.8152 q^{79} +6.97988 q^{80} -8.87760 q^{82} +15.2677 q^{83} -11.6576 q^{85} -2.69439 q^{86} +6.59312 q^{88} -13.0244 q^{89} -0.468697 q^{91} -1.79609 q^{92} +7.31033 q^{94} -20.7150 q^{95} -6.76517 q^{97} +1.15799 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

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 $

$2$	1.15799	0.818825	0.409412	−	0.912349i	$-0.365734\pi$
$2$	1.15799	0.818825	0.409412	+	0.912349i	$0.365734\pi$
$3$	0	0
$3$	0	0
$4$	−0.659052	−0.329526
$5$	−3.10556	−1.38885	−0.694423	−	0.719567i	$-0.744341\pi$
$5$	−3.10556	−1.38885	−0.694423	+	0.719567i	$0.744341\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.07916	−1.08865
$9$	0	0
$10$	−3.59621	−1.13722
$11$	−2.14121	−0.645598	−0.322799	−	0.946468i	$-0.604624\pi$
$11$	−2.14121	−0.645598	−0.322799	+	0.946468i	$0.604624\pi$
$12$	0	0
$13$	0.468697	0.129993	0.0649966	−	0.997885i	$-0.479296\pi$
$13$	0.468697	0.129993	0.0649966	+	0.997885i	$0.479296\pi$
$14$	−1.15799	−0.309487
$15$	0	0
$16$	−2.24754	−0.561886
$17$	3.75379	0.910427	0.455214	−	0.890382i	$-0.349563\pi$
$17$	3.75379	0.910427	0.455214	+	0.890382i	$0.349563\pi$
$18$	0	0
$19$	6.67029	1.53027	0.765135	−	0.643870i	$-0.222672\pi$
$19$	6.67029	1.53027	0.765135	+	0.643870i	$0.222672\pi$
$20$	2.04672	0.457661
$21$	0	0
$22$	−2.47950	−0.528631
$23$	2.72526	0.568255	0.284127	−	0.958787i	$-0.408296\pi$
$23$	2.72526	0.568255	0.284127	+	0.958787i	$0.408296\pi$
$24$	0	0
$25$	4.64448	0.928896
$26$	0.542748	0.106442
$27$	0	0
$28$	0.659052	0.124549
$29$	−1.21084	−0.224848	−0.112424	−	0.993660i	$-0.535861\pi$
$29$	−1.21084	−0.224848	−0.112424	+	0.993660i	$0.535861\pi$
$30$	0	0
$31$	6.04597	1.08589	0.542944	−	0.839769i	$-0.317310\pi$
$31$	6.04597	1.08589	0.542944	+	0.839769i	$0.317310\pi$
$32$	3.55569	0.628563
$33$	0	0
$34$	4.34686	0.745480
$35$	3.10556	0.524935
$36$	0	0
$37$	−6.22635	−1.02361	−0.511803	−	0.859103i	$-0.671022\pi$
$37$	−6.22635	−1.02361	−0.511803	+	0.859103i	$0.671022\pi$
$38$	7.72415	1.25302
$39$	0	0
$40$	9.56252	1.51197
$41$	−7.66637	−1.19729	−0.598643	−	0.801016i	$-0.704293\pi$
$41$	−7.66637	−1.19729	−0.598643	+	0.801016i	$0.704293\pi$
$42$	0	0
$43$	−2.32678	−0.354830	−0.177415	−	0.984136i	$-0.556773\pi$
$43$	−2.32678	−0.354830	−0.177415	+	0.984136i	$0.556773\pi$
$44$	1.41117	0.212741
$45$	0	0
$46$	3.15583	0.465301
$47$	6.31293	0.920836	0.460418	−	0.887702i	$-0.347700\pi$
$47$	6.31293	0.920836	0.460418	+	0.887702i	$0.347700\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.37827	0.760603
$51$	0	0
$52$	−0.308896	−0.0428362
$53$	−7.01780	−0.963969	−0.481984	−	0.876180i	$-0.660084\pi$
$53$	−7.01780	−0.963969	−0.481984	+	0.876180i	$0.660084\pi$
$54$	0	0
$55$	6.64963	0.896636
$56$	3.07916	0.411471
$57$	0	0
$58$	−1.40215	−0.184111
$59$	5.33351	0.694364	0.347182	−	0.937798i	$-0.387139\pi$
$59$	5.33351	0.694364	0.347182	+	0.937798i	$0.387139\pi$
$60$	0	0
$61$	3.01407	0.385913	0.192956	−	0.981207i	$-0.438192\pi$
$61$	3.01407	0.385913	0.192956	+	0.981207i	$0.438192\pi$
$62$	7.00119	0.889152
$63$	0	0
$64$	8.61255	1.07657
$65$	−1.45557	−0.180541
$66$	0	0
$67$	−5.67105	−0.692829	−0.346415	−	0.938082i	$-0.612601\pi$
$67$	−5.67105	−0.692829	−0.346415	+	0.938082i	$0.612601\pi$
$68$	−2.47394	−0.300010
$69$	0	0
$70$	3.59621	0.429830
$71$	3.57957	0.424817	0.212408	−	0.977181i	$-0.431869\pi$
$71$	3.57957	0.424817	0.212408	+	0.977181i	$0.431869\pi$
$72$	0	0
$73$	−9.11747	−1.06712	−0.533559	−	0.845763i	$-0.679146\pi$
$73$	−9.11747	−1.06712	−0.533559	+	0.845763i	$0.679146\pi$
$74$	−7.21007	−0.838153
$75$	0	0
$76$	−4.39607	−0.504264
$77$	2.14121	0.244013
$78$	0	0
$79$	13.8152	1.55433	0.777164	−	0.629298i	$-0.216658\pi$
$79$	13.8152	1.55433	0.777164	+	0.629298i	$0.216658\pi$
$80$	6.97988	0.780374
$81$	0	0
$82$	−8.87760	−0.980367
$83$	15.2677	1.67584	0.837922	−	0.545790i	$-0.183770\pi$
$83$	15.2677	1.67584	0.837922	+	0.545790i	$0.183770\pi$
$84$	0	0
$85$	−11.6576	−1.26444
$86$	−2.69439	−0.290544
$87$	0	0
$88$	6.59312	0.702829
$89$	−13.0244	−1.38058	−0.690292	−	0.723530i	$-0.742518\pi$
$89$	−13.0244	−1.38058	−0.690292	+	0.723530i	$0.742518\pi$
$90$	0	0
$91$	−0.468697	−0.0491328
$92$	−1.79609	−0.187255
$93$	0	0
$94$	7.31033	0.754003
$95$	−20.7150	−2.12531
$96$	0	0
$97$	−6.76517	−0.686899	−0.343449	−	0.939171i	$-0.611595\pi$
$97$	−6.76517	−0.686899	−0.343449	+	0.939171i	$0.611595\pi$
$98$	1.15799	0.116975
$99$	0	0
$100$	−3.06096	−0.306096
$101$	−4.46150	−0.443936	−0.221968	−	0.975054i	$-0.571248\pi$
$101$	−4.46150	−0.443936	−0.221968	+	0.975054i	$0.571248\pi$
$102$	0	0
$103$	8.61554	0.848914	0.424457	−	0.905448i	$-0.360465\pi$
$103$	8.61554	0.848914	0.424457	+	0.905448i	$0.360465\pi$
$104$	−1.44320	−0.141517
$105$	0	0
$106$	−8.12656	−0.789321
$107$	−11.0174	−1.06509	−0.532546	−	0.846401i	$-0.678765\pi$
$107$	−11.0174	−1.06509	−0.532546	+	0.846401i	$0.678765\pi$
$108$	0	0
$109$	11.5318	1.10454	0.552271	−	0.833665i	$-0.313761\pi$
$109$	11.5318	1.10454	0.552271	+	0.833665i	$0.313761\pi$
$110$	7.70023	0.734188
$111$	0	0
$112$	2.24754	0.212373
$113$	−4.42711	−0.416468	−0.208234	−	0.978079i	$-0.566772\pi$
$113$	−4.42711	−0.416468	−0.208234	+	0.978079i	$0.566772\pi$
$114$	0	0
$115$	−8.46343	−0.789219
$116$	0.798009	0.0740932
$117$	0	0
$118$	6.17616	0.568562
$119$	−3.75379	−0.344109
$120$	0	0
$121$	−6.41524	−0.583204
$122$	3.49028	0.315995
$123$	0	0
$124$	−3.98461	−0.357829
$125$	1.10409	0.0987528
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	2.86190	0.252958
$129$	0	0
$130$	−1.68554	−0.147831
$131$	4.44996	0.388795	0.194397	−	0.980923i	$-0.437725\pi$
$131$	4.44996	0.388795	0.194397	+	0.980923i	$0.437725\pi$
$132$	0	0
$133$	−6.67029	−0.578388
$134$	−6.56704	−0.567306
$135$	0	0
$136$	−11.5585	−0.991136
$137$	14.6563	1.25217	0.626087	−	0.779753i	$-0.284655\pi$
$137$	14.6563	1.25217	0.626087	+	0.779753i	$0.284655\pi$
$138$	0	0
$139$	−8.67628	−0.735913	−0.367956	−	0.929843i	$-0.619942\pi$
$139$	−8.67628	−0.735913	−0.367956	+	0.929843i	$0.619942\pi$
$140$	−2.04672	−0.172980
$141$	0	0
$142$	4.14511	0.347850
$143$	−1.00358	−0.0839233
$144$	0	0
$145$	3.76034	0.312279
$146$	−10.5580	−0.873783
$147$	0	0
$148$	4.10349	0.337305
$149$	−7.06758	−0.578999	−0.289499	−	0.957178i	$-0.593489\pi$
$149$	−7.06758	−0.578999	−0.289499	+	0.957178i	$0.593489\pi$
$150$	0	0
$151$	−8.11585	−0.660459	−0.330229	−	0.943901i	$-0.607126\pi$
$151$	−8.11585	−0.660459	−0.330229	+	0.943901i	$0.607126\pi$
$152$	−20.5389	−1.66593
$153$	0	0
$154$	2.47950	0.199804
$155$	−18.7761	−1.50813
$156$	0	0
$157$	−11.5119	−0.918747	−0.459374	−	0.888243i	$-0.651926\pi$
$157$	−11.5119	−0.918747	−0.459374	+	0.888243i	$0.651926\pi$
$158$	15.9979	1.27272
$159$	0	0
$160$	−11.0424	−0.872977
$161$	−2.72526	−0.214780
$162$	0	0
$163$	12.5121	0.980020	0.490010	−	0.871717i	$-0.336993\pi$
$163$	12.5121	0.980020	0.490010	+	0.871717i	$0.336993\pi$
$164$	5.05254	0.394537
$165$	0	0
$166$	17.6798	1.37222
$167$	−15.6744	−1.21292	−0.606460	−	0.795114i	$-0.707411\pi$
$167$	−15.6744	−1.21292	−0.606460	+	0.795114i	$0.707411\pi$
$168$	0	0
$169$	−12.7803	−0.983102
$170$	−13.4994	−1.03536
$171$	0	0
$172$	1.53347	0.116926
$173$	−9.74832	−0.741151	−0.370576	−	0.928802i	$-0.620840\pi$
$173$	−9.74832	−0.741151	−0.370576	+	0.928802i	$0.620840\pi$
$174$	0	0
$175$	−4.64448	−0.351090
$176$	4.81245	0.362752
$177$	0	0
$178$	−15.0822	−1.13046
$179$	−2.34103	−0.174977	−0.0874883	−	0.996166i	$-0.527884\pi$
$179$	−2.34103	−0.174977	−0.0874883	+	0.996166i	$0.527884\pi$
$180$	0	0
$181$	−25.8405	−1.92071	−0.960353	−	0.278787i	$-0.910068\pi$
$181$	−25.8405	−1.92071	−0.960353	+	0.278787i	$0.910068\pi$
$182$	−0.542748	−0.0402312
$183$	0	0
$184$	−8.39151	−0.618630
$185$	19.3363	1.42163
$186$	0	0
$187$	−8.03763	−0.587770
$188$	−4.16055	−0.303439
$189$	0	0
$190$	−23.9878	−1.74026
$191$	−9.99169	−0.722973	−0.361487	−	0.932377i	$-0.617731\pi$
$191$	−9.99169	−0.722973	−0.361487	+	0.932377i	$0.617731\pi$
$192$	0	0
$193$	−22.1036	−1.59105	−0.795527	−	0.605918i	$-0.792806\pi$
$193$	−22.1036	−1.59105	−0.795527	+	0.605918i	$0.792806\pi$
$194$	−7.83401	−0.562449
$195$	0	0
$196$	−0.659052	−0.0470752
$197$	5.69619	0.405837	0.202918	−	0.979196i	$-0.434957\pi$
$197$	5.69619	0.405837	0.202918	+	0.979196i	$0.434957\pi$
$198$	0	0
$199$	−6.41938	−0.455058	−0.227529	−	0.973771i	$-0.573065\pi$
$199$	−6.41938	−0.455058	−0.227529	+	0.973771i	$0.573065\pi$
$200$	−14.3011	−1.01124
$201$	0	0
$202$	−5.16638	−0.363505
$203$	1.21084	0.0849845
$204$	0	0
$205$	23.8083	1.66285
$206$	9.97673	0.695112
$207$	0	0
$208$	−1.05342	−0.0730414
$209$	−14.2825	−0.987939
$210$	0	0
$211$	7.45995	0.513564	0.256782	−	0.966469i	$-0.417338\pi$
$211$	7.45995	0.513564	0.256782	+	0.966469i	$0.417338\pi$
$212$	4.62510	0.317653
$213$	0	0
$214$	−12.7581	−0.872124
$215$	7.22593	0.492805
$216$	0	0
$217$	−6.04597	−0.410427
$218$	13.3537	0.904426
$219$	0	0
$220$	−4.38246	−0.295465
$221$	1.75939	0.118349
$222$	0	0
$223$	−15.0024	−1.00463	−0.502316	−	0.864684i	$-0.667519\pi$
$223$	−15.0024	−1.00463	−0.502316	+	0.864684i	$0.667519\pi$
$224$	−3.55569	−0.237574
$225$	0	0
$226$	−5.12656	−0.341014
$227$	7.06734	0.469076	0.234538	−	0.972107i	$-0.424642\pi$
$227$	7.06734	0.469076	0.234538	+	0.972107i	$0.424642\pi$
$228$	0	0
$229$	−29.6548	−1.95964	−0.979821	−	0.199877i	$-0.935946\pi$
$229$	−29.6548	−1.95964	−0.979821	+	0.199877i	$0.935946\pi$
$230$	−9.80060	−0.646232
$231$	0	0
$232$	3.72838	0.244780
$233$	2.61087	0.171043	0.0855217	−	0.996336i	$-0.472744\pi$
$233$	2.61087	0.171043	0.0855217	+	0.996336i	$0.472744\pi$
$234$	0	0
$235$	−19.6052	−1.27890
$236$	−3.51506	−0.228811
$237$	0	0
$238$	−4.34686	−0.281765
$239$	16.5873	1.07294	0.536471	−	0.843918i	$-0.319757\pi$
$239$	16.5873	1.07294	0.536471	+	0.843918i	$0.319757\pi$
$240$	0	0
$241$	−15.4114	−0.992735	−0.496368	−	0.868112i	$-0.665333\pi$
$241$	−15.4114	−0.992735	−0.496368	+	0.868112i	$0.665333\pi$
$242$	−7.42880	−0.477542
$243$	0	0
$244$	−1.98643	−0.127168
$245$	−3.10556	−0.198407
$246$	0	0
$247$	3.12635	0.198925
$248$	−18.6165	−1.18215
$249$	0	0
$250$	1.27853	0.0808613
$251$	22.4325	1.41593	0.707964	−	0.706249i	$-0.249614\pi$
$251$	22.4325	1.41593	0.707964	+	0.706249i	$0.249614\pi$
$252$	0	0
$253$	−5.83533	−0.366864
$254$	1.15799	0.0726589
$255$	0	0
$256$	−13.9110	−0.869440
$257$	−9.00800	−0.561904	−0.280952	−	0.959722i	$-0.590650\pi$
$257$	−9.00800	−0.561904	−0.280952	+	0.959722i	$0.590650\pi$
$258$	0	0
$259$	6.22635	0.386886
$260$	0.959294	0.0594929
$261$	0	0
$262$	5.15302	0.318355
$263$	18.9433	1.16809	0.584046	−	0.811720i	$-0.301469\pi$
$263$	18.9433	1.16809	0.584046	+	0.811720i	$0.301469\pi$
$264$	0	0
$265$	21.7942	1.33880
$266$	−7.72415	−0.473598
$267$	0	0
$268$	3.73752	0.228305
$269$	12.8419	0.782986	0.391493	−	0.920181i	$-0.371959\pi$
$269$	12.8419	0.782986	0.391493	+	0.920181i	$0.371959\pi$
$270$	0	0
$271$	−26.0748	−1.58393	−0.791967	−	0.610564i	$-0.790943\pi$
$271$	−26.0748	−1.58393	−0.791967	+	0.610564i	$0.790943\pi$
$272$	−8.43681	−0.511557
$273$	0	0
$274$	16.9719	1.02531
$275$	−9.94478	−0.599693
$276$	0	0
$277$	−21.6706	−1.30206	−0.651030	−	0.759052i	$-0.725663\pi$
$277$	−21.6706	−1.30206	−0.651030	+	0.759052i	$0.725663\pi$
$278$	−10.0471	−0.602583
$279$	0	0
$280$	−9.56252	−0.571470
$281$	−18.1086	−1.08027	−0.540133	−	0.841579i	$-0.681626\pi$
$281$	−18.1086	−1.08027	−0.540133	+	0.841579i	$0.681626\pi$
$282$	0	0
$283$	31.2786	1.85932	0.929659	−	0.368420i	$-0.120101\pi$
$283$	31.2786	1.85932	0.929659	+	0.368420i	$0.120101\pi$
$284$	−2.35912	−0.139988
$285$	0	0
$286$	−1.16214	−0.0687185
$287$	7.66637	0.452531
$288$	0	0
$289$	−2.90907	−0.171122
$290$	4.35445	0.255702
$291$	0	0
$292$	6.00889	0.351644
$293$	1.52898	0.0893241	0.0446620	−	0.999002i	$-0.485779\pi$
$293$	1.52898	0.0893241	0.0446620	+	0.999002i	$0.485779\pi$
$294$	0	0
$295$	−16.5635	−0.964365
$296$	19.1719	1.11435
$297$	0	0
$298$	−8.18420	−0.474098
$299$	1.27732	0.0738693
$300$	0	0
$301$	2.32678	0.134113
$302$	−9.39810	−0.540800
$303$	0	0
$304$	−14.9918	−0.859838
$305$	−9.36038	−0.535974
$306$	0	0
$307$	15.3296	0.874905	0.437453	−	0.899242i	$-0.355881\pi$
$307$	15.3296	0.874905	0.437453	+	0.899242i	$0.355881\pi$
$308$	−1.41117	−0.0804087
$309$	0	0
$310$	−21.7426	−1.23490
$311$	−16.0187	−0.908337	−0.454169	−	0.890916i	$-0.650064\pi$
$311$	−16.0187	−0.908337	−0.454169	+	0.890916i	$0.650064\pi$
$312$	0	0
$313$	−22.5129	−1.27251	−0.636253	−	0.771481i	$-0.719516\pi$
$313$	−22.5129	−1.27251	−0.636253	+	0.771481i	$0.719516\pi$
$314$	−13.3307	−0.752293
$315$	0	0
$316$	−9.10492	−0.512192
$317$	5.91303	0.332109	0.166054	−	0.986117i	$-0.446897\pi$
$317$	5.91303	0.332109	0.166054	+	0.986117i	$0.446897\pi$
$318$	0	0
$319$	2.59266	0.145161
$320$	−26.7468	−1.49519
$321$	0	0
$322$	−3.15583	−0.175867
$323$	25.0389	1.39320
$324$	0	0
$325$	2.17686	0.120750
$326$	14.4889	0.802464
$327$	0	0
$328$	23.6060	1.30342
$329$	−6.31293	−0.348043
$330$	0	0
$331$	24.8030	1.36330	0.681648	−	0.731680i	$-0.261263\pi$
$331$	24.8030	1.36330	0.681648	+	0.731680i	$0.261263\pi$
$332$	−10.0622	−0.552235
$333$	0	0
$334$	−18.1508	−0.993168
$335$	17.6118	0.962234
$336$	0	0
$337$	−16.4773	−0.897578	−0.448789	−	0.893638i	$-0.648145\pi$
$337$	−16.4773	−0.897578	−0.448789	+	0.893638i	$0.648145\pi$
$338$	−14.7995	−0.804988
$339$	0	0
$340$	7.68297	0.416668
$341$	−12.9457	−0.701047
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.16453	0.386285
$345$	0	0
$346$	−11.2885	−0.606873
$347$	4.03505	0.216613	0.108306	−	0.994118i	$-0.465457\pi$
$347$	4.03505	0.216613	0.108306	+	0.994118i	$0.465457\pi$
$348$	0	0
$349$	30.5362	1.63457	0.817284	−	0.576236i	$-0.195479\pi$
$349$	30.5362	1.63457	0.817284	+	0.576236i	$0.195479\pi$
$350$	−5.37827	−0.287481
$351$	0	0
$352$	−7.61345	−0.405798
$353$	17.4369	0.928074	0.464037	−	0.885816i	$-0.346401\pi$
$353$	17.4369	0.928074	0.464037	+	0.885816i	$0.346401\pi$
$354$	0	0
$355$	−11.1165	−0.590005
$356$	8.58377	0.454939
$357$	0	0
$358$	−2.71089	−0.143275
$359$	−27.2023	−1.43568	−0.717841	−	0.696208i	$-0.754869\pi$
$359$	−27.2023	−1.43568	−0.717841	+	0.696208i	$0.754869\pi$
$360$	0	0
$361$	25.4928	1.34173
$362$	−29.9231	−1.57272
$363$	0	0
$364$	0.308896	0.0161906
$365$	28.3148	1.48206
$366$	0	0
$367$	2.15562	0.112523	0.0562613	−	0.998416i	$-0.482082\pi$
$367$	2.15562	0.112523	0.0562613	+	0.998416i	$0.482082\pi$
$368$	−6.12513	−0.319295
$369$	0	0
$370$	22.3913	1.16407
$371$	7.01780	0.364346
$372$	0	0
$373$	32.0958	1.66186	0.830930	−	0.556378i	$-0.187809\pi$
$373$	32.0958	1.66186	0.830930	+	0.556378i	$0.187809\pi$
$374$	−9.30752	−0.481280
$375$	0	0
$376$	−19.4385	−1.00247
$377$	−0.567519	−0.0292287
$378$	0	0
$379$	−9.18622	−0.471865	−0.235932	−	0.971770i	$-0.575814\pi$
$379$	−9.18622	−0.471865	−0.235932	+	0.971770i	$0.575814\pi$
$380$	13.6523	0.700346
$381$	0	0
$382$	−11.5703	−0.591988
$383$	−14.9946	−0.766188	−0.383094	−	0.923709i	$-0.625141\pi$
$383$	−14.9946	−0.766188	−0.383094	+	0.923709i	$0.625141\pi$
$384$	0	0
$385$	−6.64963	−0.338897
$386$	−25.5959	−1.30279
$387$	0	0
$388$	4.45860	0.226351
$389$	10.5166	0.533212	0.266606	−	0.963806i	$-0.414098\pi$
$389$	10.5166	0.533212	0.266606	+	0.963806i	$0.414098\pi$
$390$	0	0
$391$	10.2300	0.517355
$392$	−3.07916	−0.155521
$393$	0	0
$394$	6.59615	0.332309
$395$	−42.9038	−2.15872
$396$	0	0
$397$	8.51648	0.427430	0.213715	−	0.976896i	$-0.431444\pi$
$397$	8.51648	0.427430	0.213715	+	0.976896i	$0.431444\pi$
$398$	−7.43360	−0.372613
$399$	0	0
$400$	−10.4387	−0.521934
$401$	1.15487	0.0576717	0.0288358	−	0.999584i	$-0.490820\pi$
$401$	1.15487	0.0576717	0.0288358	+	0.999584i	$0.490820\pi$
$402$	0	0
$403$	2.83373	0.141158
$404$	2.94036	0.146288
$405$	0	0
$406$	1.40215	0.0695874
$407$	13.3319	0.660837
$408$	0	0
$409$	−0.293377	−0.0145065	−0.00725327	−	0.999974i	$-0.502309\pi$
$409$	−0.293377	−0.0145065	−0.00725327	+	0.999974i	$0.502309\pi$
$410$	27.5699	1.36158
$411$	0	0
$412$	−5.67809	−0.279739
$413$	−5.33351	−0.262445
$414$	0	0
$415$	−47.4146	−2.32749
$416$	1.66654	0.0817089
$417$	0	0
$418$	−16.5390	−0.808949
$419$	−23.1361	−1.13027	−0.565135	−	0.824998i	$-0.691176\pi$
$419$	−23.1361	−1.13027	−0.565135	+	0.824998i	$0.691176\pi$
$420$	0	0
$421$	28.3750	1.38291	0.691457	−	0.722417i	$-0.256969\pi$
$421$	28.3750	1.38291	0.691457	+	0.722417i	$0.256969\pi$
$422$	8.63857	0.420519
$423$	0	0
$424$	21.6089	1.04942
$425$	17.4344	0.845692
$426$	0	0
$427$	−3.01407	−0.145861
$428$	7.26105	0.350976
$429$	0	0
$430$	8.36758	0.403521
$431$	−23.5515	−1.13444	−0.567218	−	0.823568i	$-0.691980\pi$
$431$	−23.5515	−1.13444	−0.567218	+	0.823568i	$0.691980\pi$
$432$	0	0
$433$	19.9832	0.960329	0.480165	−	0.877178i	$-0.340577\pi$
$433$	19.9832	0.960329	0.480165	+	0.877178i	$0.340577\pi$
$434$	−7.00119	−0.336068
$435$	0	0
$436$	−7.60003	−0.363976
$437$	18.1783	0.869584
$438$	0	0
$439$	−4.57517	−0.218361	−0.109180	−	0.994022i	$-0.534823\pi$
$439$	−4.57517	−0.218361	−0.109180	+	0.994022i	$0.534823\pi$
$440$	−20.4753	−0.976122
$441$	0	0
$442$	2.03736	0.0969074
$443$	−3.22831	−0.153382	−0.0766909	−	0.997055i	$-0.524435\pi$
$443$	−3.22831	−0.153382	−0.0766909	+	0.997055i	$0.524435\pi$
$444$	0	0
$445$	40.4480	1.91742
$446$	−17.3726	−0.822618
$447$	0	0
$448$	−8.61255	−0.406905
$449$	14.2423	0.672135	0.336067	−	0.941838i	$-0.390903\pi$
$449$	14.2423	0.672135	0.336067	+	0.941838i	$0.390903\pi$
$450$	0	0
$451$	16.4153	0.772965
$452$	2.91770	0.137237
$453$	0	0
$454$	8.18393	0.384091
$455$	1.45557	0.0682380
$456$	0	0
$457$	41.9295	1.96138	0.980688	−	0.195576i	$-0.0626577\pi$
$457$	41.9295	1.96138	0.980688	+	0.195576i	$0.0626577\pi$
$458$	−34.3400	−1.60460
$459$	0	0
$460$	5.57785	0.260068
$461$	39.7100	1.84948	0.924740	−	0.380600i	$-0.124282\pi$
$461$	39.7100	1.84948	0.924740	+	0.380600i	$0.124282\pi$
$462$	0	0
$463$	−0.133956	−0.00622545	−0.00311272	−	0.999995i	$-0.500991\pi$
$463$	−0.133956	−0.00622545	−0.00311272	+	0.999995i	$0.500991\pi$
$464$	2.72142	0.126339
$465$	0	0
$466$	3.02336	0.140055
$467$	−22.0889	−1.02215	−0.511076	−	0.859536i	$-0.670753\pi$
$467$	−22.0889	−1.02215	−0.511076	+	0.859536i	$0.670753\pi$
$468$	0	0
$469$	5.67105	0.261865
$470$	−22.7026	−1.04719
$471$	0	0
$472$	−16.4227	−0.755918
$473$	4.98211	0.229077
$474$	0	0
$475$	30.9800	1.42146
$476$	2.47394	0.113393
$477$	0	0
$478$	19.2080	0.878552
$479$	−30.2419	−1.38179	−0.690895	−	0.722955i	$-0.742783\pi$
$479$	−30.2419	−1.38179	−0.690895	+	0.722955i	$0.742783\pi$
$480$	0	0
$481$	−2.91827	−0.133062
$482$	−17.8463	−0.812876
$483$	0	0
$484$	4.22798	0.192181
$485$	21.0096	0.953997
$486$	0	0
$487$	−4.39894	−0.199335	−0.0996675	−	0.995021i	$-0.531778\pi$
$487$	−4.39894	−0.199335	−0.0996675	+	0.995021i	$0.531778\pi$
$488$	−9.28083	−0.420123
$489$	0	0
$490$	−3.59621	−0.162460
$491$	−12.2940	−0.554820	−0.277410	−	0.960752i	$-0.589476\pi$
$491$	−12.2940	−0.554820	−0.277410	+	0.960752i	$0.589476\pi$
$492$	0	0
$493$	−4.54525	−0.204708
$494$	3.62029	0.162885
$495$	0	0
$496$	−13.5886	−0.610145
$497$	−3.57957	−0.160566
$498$	0	0
$499$	−24.3172	−1.08859	−0.544294	−	0.838894i	$-0.683203\pi$
$499$	−24.3172	−1.08859	−0.544294	+	0.838894i	$0.683203\pi$
$500$	−0.727653	−0.0325417
$501$	0	0
$502$	25.9767	1.15940
$503$	14.3993	0.642034	0.321017	−	0.947074i	$-0.395975\pi$
$503$	14.3993	0.642034	0.321017	+	0.947074i	$0.395975\pi$
$504$	0	0
$505$	13.8554	0.616559
$506$	−6.75727	−0.300397
$507$	0	0
$508$	−0.659052	−0.0292407
$509$	−42.5399	−1.88555	−0.942775	−	0.333431i	$-0.891794\pi$
$509$	−42.5399	−1.88555	−0.942775	+	0.333431i	$0.891794\pi$
$510$	0	0
$511$	9.11747	0.403333
$512$	−21.8327	−0.964877
$513$	0	0
$514$	−10.4312	−0.460101
$515$	−26.7560	−1.17901
$516$	0	0
$517$	−13.5173	−0.594489
$518$	7.21007	0.316792
$519$	0	0
$520$	4.48193	0.196545
$521$	22.0824	0.967447	0.483723	−	0.875221i	$-0.339284\pi$
$521$	22.0824	0.967447	0.483723	+	0.875221i	$0.339284\pi$
$522$	0	0
$523$	−30.2691	−1.32358	−0.661788	−	0.749691i	$-0.730202\pi$
$523$	−30.2691	−1.32358	−0.661788	+	0.749691i	$0.730202\pi$
$524$	−2.93276	−0.128118
$525$	0	0
$526$	21.9362	0.956463
$527$	22.6953	0.988622
$528$	0	0
$529$	−15.5730	−0.677086
$530$	25.2375	1.09625
$531$	0	0
$532$	4.39607	0.190594
$533$	−3.59321	−0.155639
$534$	0	0
$535$	34.2152	1.47925
$536$	17.4621	0.754248
$537$	0	0
$538$	14.8709	0.641128
$539$	−2.14121	−0.0922282
$540$	0	0
$541$	8.72865	0.375274	0.187637	−	0.982238i	$-0.439917\pi$
$541$	8.72865	0.375274	0.187637	+	0.982238i	$0.439917\pi$
$542$	−30.1945	−1.29696
$543$	0	0
$544$	13.3473	0.572261
$545$	−35.8125	−1.53404
$546$	0	0
$547$	33.1576	1.41772	0.708859	−	0.705350i	$-0.249210\pi$
$547$	33.1576	1.41772	0.708859	+	0.705350i	$0.249210\pi$
$548$	−9.65929	−0.412624
$549$	0	0
$550$	−11.5160	−0.491043
$551$	−8.07668	−0.344078
$552$	0	0
$553$	−13.8152	−0.587481
$554$	−25.0944	−1.06616
$555$	0	0
$556$	5.71813	0.242503
$557$	9.85412	0.417532	0.208766	−	0.977966i	$-0.433055\pi$
$557$	9.85412	0.417532	0.208766	+	0.977966i	$0.433055\pi$
$558$	0	0
$559$	−1.09055	−0.0461255
$560$	−6.97988	−0.294954
$561$	0	0
$562$	−20.9696	−0.884549
$563$	−17.0134	−0.717031	−0.358515	−	0.933524i	$-0.616717\pi$
$563$	−17.0134	−0.717031	−0.358515	+	0.933524i	$0.616717\pi$
$564$	0	0
$565$	13.7486	0.578410
$566$	36.2204	1.52246
$567$	0	0
$568$	−11.0221	−0.462476
$569$	17.7855	0.745607	0.372803	−	0.927910i	$-0.378397\pi$
$569$	17.7855	0.745607	0.372803	+	0.927910i	$0.378397\pi$
$570$	0	0
$571$	8.27061	0.346114	0.173057	−	0.984912i	$-0.444635\pi$
$571$	8.27061	0.346114	0.173057	+	0.984912i	$0.444635\pi$
$572$	0.661410	0.0276549
$573$	0	0
$574$	8.87760	0.370544
$575$	12.6574	0.527850
$576$	0	0
$577$	−3.85103	−0.160321	−0.0801603	−	0.996782i	$-0.525543\pi$
$577$	−3.85103	−0.160321	−0.0801603	+	0.996782i	$0.525543\pi$
$578$	−3.36869	−0.140119
$579$	0	0
$580$	−2.47826	−0.102904
$581$	−15.2677	−0.633409
$582$	0	0
$583$	15.0265	0.622336
$584$	28.0742	1.16172
$585$	0	0
$586$	1.77055	0.0731407
$587$	23.6394	0.975704	0.487852	−	0.872926i	$-0.337781\pi$
$587$	23.6394	0.975704	0.487852	+	0.872926i	$0.337781\pi$
$588$	0	0
$589$	40.3284	1.66170
$590$	−19.1804	−0.789646
$591$	0	0
$592$	13.9940	0.575150
$593$	−34.3345	−1.40995	−0.704974	−	0.709233i	$-0.749041\pi$
$593$	−34.3345	−1.40995	−0.704974	+	0.709233i	$0.749041\pi$
$594$	0	0
$595$	11.6576	0.477915
$596$	4.65790	0.190795
$597$	0	0
$598$	1.47913	0.0604860
$599$	−17.4272	−0.712055	−0.356028	−	0.934475i	$-0.615869\pi$
$599$	−17.4272	−0.712055	−0.356028	+	0.934475i	$0.615869\pi$
$600$	0	0
$601$	−33.2905	−1.35795	−0.678975	−	0.734162i	$-0.737575\pi$
$601$	−33.2905	−1.35795	−0.678975	+	0.734162i	$0.737575\pi$
$602$	2.69439	0.109815
$603$	0	0
$604$	5.34877	0.217638
$605$	19.9229	0.809981
$606$	0	0
$607$	−1.79020	−0.0726620	−0.0363310	−	0.999340i	$-0.511567\pi$
$607$	−1.79020	−0.0726620	−0.0363310	+	0.999340i	$0.511567\pi$
$608$	23.7175	0.961871
$609$	0	0
$610$	−10.8393	−0.438868
$611$	2.95885	0.119702
$612$	0	0
$613$	41.0907	1.65964	0.829819	−	0.558033i	$-0.188444\pi$
$613$	41.0907	1.65964	0.829819	+	0.558033i	$0.188444\pi$
$614$	17.7515	0.716394
$615$	0	0
$616$	−6.59312	−0.265644
$617$	11.6854	0.470437	0.235219	−	0.971942i	$-0.424419\pi$
$617$	11.6854	0.470437	0.235219	+	0.971942i	$0.424419\pi$
$618$	0	0
$619$	2.83061	0.113772	0.0568858	−	0.998381i	$-0.481883\pi$
$619$	2.83061	0.113772	0.0568858	+	0.998381i	$0.481883\pi$
$620$	12.3744	0.496969
$621$	0	0
$622$	−18.5495	−0.743769
$623$	13.0244	0.521812
$624$	0	0
$625$	−26.6512	−1.06605
$626$	−26.0698	−1.04196
$627$	0	0
$628$	7.58693	0.302751
$629$	−23.3724	−0.931918
$630$	0	0
$631$	−13.2297	−0.526666	−0.263333	−	0.964705i	$-0.584822\pi$
$631$	−13.2297	−0.526666	−0.263333	+	0.964705i	$0.584822\pi$
$632$	−42.5392	−1.69212
$633$	0	0
$634$	6.84725	0.271939
$635$	−3.10556	−0.123240
$636$	0	0
$637$	0.468697	0.0185705
$638$	3.00228	0.118862
$639$	0	0
$640$	−8.88779	−0.351321
$641$	28.6816	1.13286	0.566428	−	0.824111i	$-0.308325\pi$
$641$	28.6816	1.13286	0.566428	+	0.824111i	$0.308325\pi$
$642$	0	0
$643$	0.0362656	0.00143018	0.000715089	−	1.00000i	$-0.499772\pi$
$643$	0.0362656	0.00143018	0.000715089		1.00000i	$0.499772\pi$
$644$	1.79609	0.0707757
$645$	0	0
$646$	28.9948	1.14079
$647$	−28.1840	−1.10803	−0.554013	−	0.832508i	$-0.686904\pi$
$647$	−28.1840	−1.10803	−0.554013	+	0.832508i	$0.686904\pi$
$648$	0	0
$649$	−11.4201	−0.448279
$650$	2.52078	0.0988732
$651$	0	0
$652$	−8.24610	−0.322942
$653$	18.1448	0.710061	0.355031	−	0.934855i	$-0.384470\pi$
$653$	18.1448	0.710061	0.355031	+	0.934855i	$0.384470\pi$
$654$	0	0
$655$	−13.8196	−0.539976
$656$	17.2305	0.672738
$657$	0	0
$658$	−7.31033	−0.284986
$659$	28.8004	1.12190	0.560952	−	0.827848i	$-0.310435\pi$
$659$	28.8004	1.12190	0.560952	+	0.827848i	$0.310435\pi$
$660$	0	0
$661$	−23.1591	−0.900786	−0.450393	−	0.892830i	$-0.648716\pi$
$661$	−23.1591	−0.900786	−0.450393	+	0.892830i	$0.648716\pi$
$662$	28.7217	1.11630
$663$	0	0
$664$	−47.0116	−1.82441
$665$	20.7150	0.803292
$666$	0	0
$667$	−3.29985	−0.127771
$668$	10.3302	0.399689
$669$	0	0
$670$	20.3943	0.787901
$671$	−6.45375	−0.249144
$672$	0	0
$673$	−50.5163	−1.94726	−0.973630	−	0.228132i	$-0.926738\pi$
$673$	−50.5163	−1.94726	−0.973630	+	0.228132i	$0.926738\pi$
$674$	−19.0806	−0.734959
$675$	0	0
$676$	8.42290	0.323958
$677$	−41.7371	−1.60409	−0.802044	−	0.597266i	$-0.796254\pi$
$677$	−41.7371	−1.60409	−0.802044	+	0.597266i	$0.796254\pi$
$678$	0	0
$679$	6.76517	0.259623
$680$	35.8957	1.37654
$681$	0	0
$682$	−14.9910	−0.574034
$683$	−44.6286	−1.70767	−0.853833	−	0.520547i	$-0.825728\pi$
$683$	−44.6286	−1.70767	−0.853833	+	0.520547i	$0.825728\pi$
$684$	0	0
$685$	−45.5161	−1.73908
$686$	−1.15799	−0.0442124
$687$	0	0
$688$	5.22953	0.199374
$689$	−3.28922	−0.125309
$690$	0	0
$691$	6.87400	0.261499	0.130750	−	0.991415i	$-0.458262\pi$
$691$	6.87400	0.261499	0.130750	+	0.991415i	$0.458262\pi$
$692$	6.42466	0.244229
$693$	0	0
$694$	4.67256	0.177368
$695$	26.9447	1.02207
$696$	0	0
$697$	−28.7779	−1.09004
$698$	35.3607	1.33842
$699$	0	0
$700$	3.06096	0.115693
$701$	−26.7360	−1.00980	−0.504902	−	0.863176i	$-0.668472\pi$
$701$	−26.7360	−1.00980	−0.504902	+	0.863176i	$0.668472\pi$
$702$	0	0
$703$	−41.5316	−1.56639
$704$	−18.4412	−0.695030
$705$	0	0
$706$	20.1918	0.759930
$707$	4.46150	0.167792
$708$	0	0
$709$	−29.7059	−1.11563	−0.557814	−	0.829966i	$-0.688360\pi$
$709$	−29.7059	−1.11563	−0.557814	+	0.829966i	$0.688360\pi$
$710$	−12.8729	−0.483111
$711$	0	0
$712$	40.1043	1.50297
$713$	16.4768	0.617061
$714$	0	0
$715$	3.11667	0.116557
$716$	1.54286	0.0576594
$717$	0	0
$718$	−31.5000	−1.17557
$719$	−6.44781	−0.240463	−0.120231	−	0.992746i	$-0.538364\pi$
$719$	−6.44781	−0.240463	−0.120231	+	0.992746i	$0.538364\pi$
$720$	0	0
$721$	−8.61554	−0.320859
$722$	29.5205	1.09864
$723$	0	0
$724$	17.0302	0.632923
$725$	−5.62373	−0.208860
$726$	0	0
$727$	22.9507	0.851196	0.425598	−	0.904912i	$-0.360064\pi$
$727$	22.9507	0.851196	0.425598	+	0.904912i	$0.360064\pi$
$728$	1.44320	0.0534884
$729$	0	0
$730$	32.7883	1.21355
$731$	−8.73423	−0.323047
$732$	0	0
$733$	−41.8804	−1.54689	−0.773445	−	0.633864i	$-0.781468\pi$
$733$	−41.8804	−1.54689	−0.773445	+	0.633864i	$0.781468\pi$
$734$	2.49619	0.0921362
$735$	0	0
$736$	9.69015	0.357184
$737$	12.1429	0.447289
$738$	0	0
$739$	−3.40288	−0.125177	−0.0625884	−	0.998039i	$-0.519936\pi$
$739$	−3.40288	−0.125177	−0.0625884	+	0.998039i	$0.519936\pi$
$740$	−12.7436	−0.468465
$741$	0	0
$742$	8.12656	0.298335
$743$	−27.4070	−1.00546	−0.502732	−	0.864442i	$-0.667672\pi$
$743$	−27.4070	−1.00546	−0.502732	+	0.864442i	$0.667672\pi$
$744$	0	0
$745$	21.9488	0.804140
$746$	37.1667	1.36077
$747$	0	0
$748$	5.29722	0.193686
$749$	11.0174	0.402567
$750$	0	0
$751$	−1.69965	−0.0620212	−0.0310106	−	0.999519i	$-0.509873\pi$
$751$	−1.69965	−0.0620212	−0.0310106	+	0.999519i	$0.509873\pi$
$752$	−14.1886	−0.517405
$753$	0	0
$754$	−0.657183	−0.0239332
$755$	25.2042	0.917276
$756$	0	0
$757$	−26.3465	−0.957581	−0.478791	−	0.877929i	$-0.658925\pi$
$757$	−26.3465	−0.957581	−0.478791	+	0.877929i	$0.658925\pi$
$758$	−10.6376	−0.386374
$759$	0	0
$760$	63.7848	2.31372
$761$	−25.5784	−0.927218	−0.463609	−	0.886040i	$-0.653446\pi$
$761$	−25.5784	−0.927218	−0.463609	+	0.886040i	$0.653446\pi$
$762$	0	0
$763$	−11.5318	−0.417478
$764$	6.58505	0.238239
$765$	0	0
$766$	−17.3636	−0.627373
$767$	2.49980	0.0902626
$768$	0	0
$769$	50.6863	1.82779	0.913897	−	0.405947i	$-0.133058\pi$
$769$	50.6863	1.82779	0.913897	+	0.405947i	$0.133058\pi$
$770$	−7.70023	−0.277497
$771$	0	0
$772$	14.5675	0.524294
$773$	0.874887	0.0314675	0.0157337	−	0.999876i	$-0.494992\pi$
$773$	0.874887	0.0314675	0.0157337	+	0.999876i	$0.494992\pi$
$774$	0	0
$775$	28.0804	1.00868
$776$	20.8311	0.747791
$777$	0	0
$778$	12.1781	0.436607
$779$	−51.1369	−1.83217
$780$	0	0
$781$	−7.66459	−0.274261
$782$	11.8463	0.423623
$783$	0	0
$784$	−2.24754	−0.0802695
$785$	35.7508	1.27600
$786$	0	0
$787$	−19.2006	−0.684426	−0.342213	−	0.939622i	$-0.611176\pi$
$787$	−19.2006	−0.684426	−0.342213	+	0.939622i	$0.611176\pi$
$788$	−3.75409	−0.133734
$789$	0	0
$790$	−49.6823	−1.76762
$791$	4.42711	0.157410
$792$	0	0
$793$	1.41269	0.0501661
$794$	9.86202	0.349990
$795$	0	0
$796$	4.23071	0.149954
$797$	−35.6125	−1.26146	−0.630731	−	0.776002i	$-0.717245\pi$
$797$	−35.6125	−1.26146	−0.630731	+	0.776002i	$0.717245\pi$
$798$	0	0
$799$	23.6974	0.838354
$800$	16.5143	0.583869
$801$	0	0
$802$	1.33734	0.0472230
$803$	19.5224	0.688929
$804$	0	0
$805$	8.46343	0.298297
$806$	3.28144	0.115584
$807$	0	0
$808$	13.7377	0.483290
$809$	30.2876	1.06485	0.532427	−	0.846476i	$-0.321280\pi$
$809$	30.2876	1.06485	0.532427	+	0.846476i	$0.321280\pi$
$810$	0	0
$811$	35.6416	1.25155	0.625773	−	0.780005i	$-0.284784\pi$
$811$	35.6416	1.25155	0.625773	+	0.780005i	$0.284784\pi$
$812$	−0.798009	−0.0280046
$813$	0	0
$814$	15.4382	0.541110
$815$	−38.8569	−1.36110
$816$	0	0
$817$	−15.5203	−0.542986
$818$	−0.339728	−0.0118783
$819$	0	0
$820$	−15.6909	−0.547952
$821$	−40.1457	−1.40109	−0.700547	−	0.713607i	$-0.747060\pi$
$821$	−40.1457	−1.40109	−0.700547	+	0.713607i	$0.747060\pi$
$822$	0	0
$823$	42.6161	1.48550	0.742751	−	0.669567i	$-0.233521\pi$
$823$	42.6161	1.48550	0.742751	+	0.669567i	$0.233521\pi$
$824$	−26.5287	−0.924169
$825$	0	0
$826$	−6.17616	−0.214896
$827$	−26.4841	−0.920942	−0.460471	−	0.887675i	$-0.652319\pi$
$827$	−26.4841	−0.920942	−0.460471	+	0.887675i	$0.652319\pi$
$828$	0	0
$829$	37.4412	1.30039	0.650194	−	0.759768i	$-0.274688\pi$
$829$	37.4412	1.30039	0.650194	+	0.759768i	$0.274688\pi$
$830$	−54.9058	−1.90581
$831$	0	0
$832$	4.03668	0.139947
$833$	3.75379	0.130061
$834$	0	0
$835$	48.6776	1.68456
$836$	9.41290	0.325552
$837$	0	0
$838$	−26.7914	−0.925493
$839$	−10.0505	−0.346980	−0.173490	−	0.984836i	$-0.555504\pi$
$839$	−10.0505	−0.346980	−0.173490	+	0.984836i	$0.555504\pi$
$840$	0	0
$841$	−27.5339	−0.949443
$842$	32.8581	1.13236
$843$	0	0
$844$	−4.91650	−0.169233
$845$	39.6900	1.36538
$846$	0	0
$847$	6.41524	0.220430
$848$	15.7728	0.541641
$849$	0	0
$850$	20.1889	0.692474
$851$	−16.9684	−0.581669
$852$	0	0
$853$	24.4567	0.837381	0.418691	−	0.908129i	$-0.362489\pi$
$853$	24.4567	0.837381	0.418691	+	0.908129i	$0.362489\pi$
$854$	−3.49028	−0.119435
$855$	0	0
$856$	33.9244	1.15951
$857$	25.1552	0.859284	0.429642	−	0.902999i	$-0.358640\pi$
$857$	25.1552	0.859284	0.429642	+	0.902999i	$0.358640\pi$
$858$	0	0
$859$	31.7574	1.08355	0.541774	−	0.840524i	$-0.317753\pi$
$859$	31.7574	1.08355	0.541774	+	0.840524i	$0.317753\pi$
$860$	−4.76227	−0.162392
$861$	0	0
$862$	−27.2725	−0.928904
$863$	−28.9441	−0.985268	−0.492634	−	0.870237i	$-0.663966\pi$
$863$	−28.9441	−0.985268	−0.492634	+	0.870237i	$0.663966\pi$
$864$	0	0
$865$	30.2740	1.02935
$866$	23.1404	0.786341
$867$	0	0
$868$	3.98461	0.135246
$869$	−29.5811	−1.00347
$870$	0	0
$871$	−2.65801	−0.0900631
$872$	−35.5082	−1.20246
$873$	0	0
$874$	21.0503	0.712037
$875$	−1.10409	−0.0373251
$876$	0	0
$877$	50.0204	1.68907	0.844534	−	0.535501i	$-0.179877\pi$
$877$	50.0204	1.68907	0.844534	+	0.535501i	$0.179877\pi$
$878$	−5.29801	−0.178799
$879$	0	0
$880$	−14.9453	−0.503808
$881$	−34.2282	−1.15318	−0.576589	−	0.817034i	$-0.695617\pi$
$881$	−34.2282	−1.15318	−0.576589	+	0.817034i	$0.695617\pi$
$882$	0	0
$883$	−21.9663	−0.739224	−0.369612	−	0.929186i	$-0.620509\pi$
$883$	−21.9663	−0.739224	−0.369612	+	0.929186i	$0.620509\pi$
$884$	−1.15953	−0.0389992
$885$	0	0
$886$	−3.73836	−0.125593
$887$	−48.9497	−1.64357	−0.821784	−	0.569799i	$-0.807021\pi$
$887$	−48.9497	−1.64357	−0.821784	+	0.569799i	$0.807021\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	46.8385	1.57003
$891$	0	0
$892$	9.88734	0.331053
$893$	42.1091	1.40913
$894$	0	0
$895$	7.27019	0.243016
$896$	−2.86190	−0.0956093
$897$	0	0
$898$	16.4925	0.550360
$899$	−7.32071	−0.244159
$900$	0	0
$901$	−26.3433	−0.877623
$902$	19.0088	0.632923
$903$	0	0
$904$	13.6318	0.453387
$905$	80.2490	2.66757
$906$	0	0
$907$	−9.54324	−0.316878	−0.158439	−	0.987369i	$-0.550646\pi$
$907$	−9.54324	−0.316878	−0.158439	+	0.987369i	$0.550646\pi$
$908$	−4.65775	−0.154573
$909$	0	0
$910$	1.68554	0.0558749
$911$	−23.3701	−0.774286	−0.387143	−	0.922020i	$-0.626538\pi$
$911$	−23.3701	−0.774286	−0.387143	+	0.922020i	$0.626538\pi$
$912$	0	0
$913$	−32.6912	−1.08192
$914$	48.5540	1.60602
$915$	0	0
$916$	19.5440	0.645753
$917$	−4.44996	−0.146951
$918$	0	0
$919$	2.23386	0.0736883	0.0368442	−	0.999321i	$-0.488269\pi$
$919$	2.23386	0.0736883	0.0368442	+	0.999321i	$0.488269\pi$
$920$	26.0603	0.859183
$921$	0	0
$922$	45.9839	1.51440
$923$	1.67773	0.0552233
$924$	0	0
$925$	−28.9181	−0.950823
$926$	−0.155120	−0.00509755
$927$	0	0
$928$	−4.30538	−0.141331
$929$	55.8632	1.83281	0.916406	−	0.400250i	$-0.131077\pi$
$929$	55.8632	1.83281	0.916406	+	0.400250i	$0.131077\pi$
$930$	0	0
$931$	6.67029	0.218610
$932$	−1.72070	−0.0563633
$933$	0	0
$934$	−25.5788	−0.836963
$935$	24.9613	0.816322
$936$	0	0
$937$	−35.2165	−1.15047	−0.575236	−	0.817988i	$-0.695090\pi$
$937$	−35.2165	−1.15047	−0.575236	+	0.817988i	$0.695090\pi$
$938$	6.56704	0.214421
$939$	0	0
$940$	12.9208	0.421431
$941$	−53.4314	−1.74181	−0.870906	−	0.491449i	$-0.836467\pi$
$941$	−53.4314	−1.74181	−0.870906	+	0.491449i	$0.836467\pi$
$942$	0	0
$943$	−20.8928	−0.680364
$944$	−11.9873	−0.390153
$945$	0	0
$946$	5.76924	0.187574
$947$	5.46398	0.177555	0.0887777	−	0.996051i	$-0.471704\pi$
$947$	5.46398	0.177555	0.0887777	+	0.996051i	$0.471704\pi$
$948$	0	0
$949$	−4.27333	−0.138718
$950$	35.8747	1.16393
$951$	0	0
$952$	11.5585	0.374614
$953$	47.0954	1.52557	0.762785	−	0.646652i	$-0.223832\pi$
$953$	47.0954	1.52557	0.762785	+	0.646652i	$0.223832\pi$
$954$	0	0
$955$	31.0298	1.00410
$956$	−10.9319	−0.353563
$957$	0	0
$958$	−35.0199	−1.13144
$959$	−14.6563	−0.473278
$960$	0	0
$961$	5.55372	0.179152
$962$	−3.37934	−0.108954
$963$	0	0
$964$	10.1569	0.327132
$965$	68.6441	2.20973
$966$	0	0
$967$	−25.4308	−0.817799	−0.408900	−	0.912579i	$-0.634087\pi$
$967$	−25.4308	−0.817799	−0.408900	+	0.912579i	$0.634087\pi$
$968$	19.7536	0.634904
$969$	0	0
$970$	24.3290	0.781156
$971$	33.4029	1.07195	0.535974	−	0.844234i	$-0.319944\pi$
$971$	33.4029	1.07195	0.535974	+	0.844234i	$0.319944\pi$
$972$	0	0
$973$	8.67628	0.278149
$974$	−5.09394	−0.163220
$975$	0	0
$976$	−6.77427	−0.216839
$977$	−48.0263	−1.53650	−0.768248	−	0.640152i	$-0.778871\pi$
$977$	−48.0263	−1.53650	−0.768248	+	0.640152i	$0.778871\pi$
$978$	0	0
$979$	27.8879	0.891302
$980$	2.04672	0.0653802
$981$	0	0
$982$	−14.2364	−0.454300
$983$	30.8042	0.982501	0.491250	−	0.871018i	$-0.336540\pi$
$983$	30.8042	0.982501	0.491250	+	0.871018i	$0.336540\pi$
$984$	0	0
$985$	−17.6898	−0.563645
$986$	−5.26336	−0.167620
$987$	0	0
$988$	−2.06043	−0.0655510
$989$	−6.34106	−0.201634
$990$	0	0
$991$	37.3771	1.18732	0.593661	−	0.804715i	$-0.297682\pi$
$991$	37.3771	1.18732	0.593661	+	0.804715i	$0.297682\pi$
$992$	21.4976	0.682548
$993$	0	0
$994$	−4.14511	−0.131475
$995$	19.9358	0.632006
$996$	0	0
$997$	48.9581	1.55052	0.775259	−	0.631643i	$-0.217619\pi$
$997$	48.9581	1.55052	0.775259	+	0.631643i	$0.217619\pi$
$998$	−28.1592	−0.891363
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.10		15
3.2	odd	2		889.2.a.b.1.6	✓	15
21.20	even	2		6223.2.a.j.1.6		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.6	✓	15	3.2	odd	2
6223.2.a.j.1.6		15	21.20	even	2
8001.2.a.q.1.10		15	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)

\(f(q)\)	\(=\)	\(q+1.15799 q^{2} -0.659052 q^{4} -3.10556 q^{5} -1.00000 q^{7} -3.07916 q^{8} +O(q^{10})\)	\(q+1.15799 q^{2} -0.659052 q^{4} -3.10556 q^{5} -1.00000 q^{7} -3.07916 q^{8} -3.59621 q^{10} -2.14121 q^{11} +0.468697 q^{13} -1.15799 q^{14} -2.24754 q^{16} +3.75379 q^{17} +6.67029 q^{19} +2.04672 q^{20} -2.47950 q^{22} +2.72526 q^{23} +4.64448 q^{25} +0.542748 q^{26} +0.659052 q^{28} -1.21084 q^{29} +6.04597 q^{31} +3.55569 q^{32} +4.34686 q^{34} +3.10556 q^{35} -6.22635 q^{37} +7.72415 q^{38} +9.56252 q^{40} -7.66637 q^{41} -2.32678 q^{43} +1.41117 q^{44} +3.15583 q^{46} +6.31293 q^{47} +1.00000 q^{49} +5.37827 q^{50} -0.308896 q^{52} -7.01780 q^{53} +6.64963 q^{55} +3.07916 q^{56} -1.40215 q^{58} +5.33351 q^{59} +3.01407 q^{61} +7.00119 q^{62} +8.61255 q^{64} -1.45557 q^{65} -5.67105 q^{67} -2.47394 q^{68} +3.59621 q^{70} +3.57957 q^{71} -9.11747 q^{73} -7.21007 q^{74} -4.39607 q^{76} +2.14121 q^{77} +13.8152 q^{79} +6.97988 q^{80} -8.87760 q^{82} +15.2677 q^{83} -11.6576 q^{85} -2.69439 q^{86} +6.59312 q^{88} -13.0244 q^{89} -0.468697 q^{91} -1.79609 q^{92} +7.31033 q^{94} -20.7150 q^{95} -6.76517 q^{97} +1.15799 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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