LMFDB - Embedded newform 8037.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70026 q^{2} +5.29141 q^{4} -2.20508 q^{5} +0.381908 q^{7} -8.88768 q^{8} +O(q^{10})$	\(q-2.70026 q^{2} +5.29141 q^{4} -2.20508 q^{5} +0.381908 q^{7} -8.88768 q^{8} +5.95430 q^{10} +0.541461 q^{11} -4.39015 q^{13} -1.03125 q^{14} +13.4162 q^{16} +7.56447 q^{17} +1.00000 q^{19} -11.6680 q^{20} -1.46209 q^{22} -4.83305 q^{23} -0.137605 q^{25} +11.8545 q^{26} +2.02083 q^{28} +2.92584 q^{29} +10.0216 q^{31} -18.4520 q^{32} -20.4261 q^{34} -0.842139 q^{35} -6.82713 q^{37} -2.70026 q^{38} +19.5981 q^{40} +1.66489 q^{41} -8.47623 q^{43} +2.86509 q^{44} +13.0505 q^{46} -1.00000 q^{47} -6.85415 q^{49} +0.371569 q^{50} -23.2301 q^{52} -5.12841 q^{53} -1.19397 q^{55} -3.39428 q^{56} -7.90055 q^{58} +6.38720 q^{59} +9.65918 q^{61} -27.0609 q^{62} +22.9928 q^{64} +9.68064 q^{65} +6.65611 q^{67} +40.0268 q^{68} +2.27400 q^{70} -8.57056 q^{71} -8.65829 q^{73} +18.4350 q^{74} +5.29141 q^{76} +0.206788 q^{77} -2.49612 q^{79} -29.5839 q^{80} -4.49563 q^{82} -5.42636 q^{83} -16.6803 q^{85} +22.8880 q^{86} -4.81233 q^{88} +12.6622 q^{89} -1.67663 q^{91} -25.5737 q^{92} +2.70026 q^{94} -2.20508 q^{95} -8.67832 q^{97} +18.5080 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) $ 23 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 2 * q^7 - 12 * q^8	$ 23 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 2 q^{7} - 12 q^{8} + q^{10} - 13 q^{11} + 5 q^{13} - 11 q^{14} + 33 q^{16} - 22 q^{17} + 23 q^{19} - 23 q^{20} + q^{22} - 23 q^{23} + 31 q^{25} - 21 q^{26} - 22 q^{28} - 24 q^{29} + 20 q^{31} - 19 q^{32} - 14 q^{34} - 15 q^{35} + 8 q^{37} - 5 q^{38} - 15 q^{40} - 28 q^{41} - 6 q^{43} - q^{44} + 29 q^{46} - 23 q^{47} + 39 q^{49} - 27 q^{50} + 12 q^{52} - 22 q^{53} - 28 q^{55} - 30 q^{56} - 5 q^{58} - 53 q^{59} + 4 q^{61} - 38 q^{62} + 30 q^{64} - 27 q^{65} + 6 q^{68} + 30 q^{70} - 54 q^{71} + 15 q^{73} + 12 q^{74} + 25 q^{76} - 33 q^{77} - 8 q^{79} - 45 q^{80} - 69 q^{82} - 20 q^{83} - 25 q^{85} - 50 q^{86} + 48 q^{88} - 26 q^{89} + 11 q^{91} - 81 q^{92} + 5 q^{94} - 12 q^{95} - 4 q^{97} + 55 q^{98}+O(q^{100}) $ 23 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 2 * q^7 - 12 * q^8 + q^10 - 13 * q^11 + 5 * q^13 - 11 * q^14 + 33 * q^16 - 22 * q^17 + 23 * q^19 - 23 * q^20 + q^22 - 23 * q^23 + 31 * q^25 - 21 * q^26 - 22 * q^28 - 24 * q^29 + 20 * q^31 - 19 * q^32 - 14 * q^34 - 15 * q^35 + 8 * q^37 - 5 * q^38 - 15 * q^40 - 28 * q^41 - 6 * q^43 - q^44 + 29 * q^46 - 23 * q^47 + 39 * q^49 - 27 * q^50 + 12 * q^52 - 22 * q^53 - 28 * q^55 - 30 * q^56 - 5 * q^58 - 53 * q^59 + 4 * q^61 - 38 * q^62 + 30 * q^64 - 27 * q^65 + 6 * q^68 + 30 * q^70 - 54 * q^71 + 15 * q^73 + 12 * q^74 + 25 * q^76 - 33 * q^77 - 8 * q^79 - 45 * q^80 - 69 * q^82 - 20 * q^83 - 25 * q^85 - 50 * q^86 + 48 * q^88 - 26 * q^89 + 11 * q^91 - 81 * q^92 + 5 * q^94 - 12 * q^95 - 4 * q^97 + 55 * 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 $

$2$	−2.70026	−1.90937	−0.954687	−	0.297612i	$-0.903810\pi$
$2$	−2.70026	−1.90937	−0.954687	+	0.297612i	$0.903810\pi$
$3$	0	0
$3$	0	0
$4$	5.29141	2.64571
$5$	−2.20508	−0.986144	−0.493072	−	0.869989i	$-0.664126\pi$
$5$	−2.20508	−0.986144	−0.493072	+	0.869989i	$0.664126\pi$
$6$	0	0
$7$	0.381908	0.144348	0.0721738	−	0.997392i	$-0.477006\pi$
$7$	0.381908	0.144348	0.0721738	+	0.997392i	$0.477006\pi$
$8$	−8.88768	−3.14227
$9$	0	0
$10$	5.95430	1.88292
$11$	0.541461	0.163256	0.0816282	−	0.996663i	$-0.473988\pi$
$11$	0.541461	0.163256	0.0816282	+	0.996663i	$0.473988\pi$
$12$	0	0
$13$	−4.39015	−1.21761	−0.608804	−	0.793321i	$-0.708350\pi$
$13$	−4.39015	−1.21761	−0.608804	+	0.793321i	$0.708350\pi$
$14$	−1.03125	−0.275614
$15$	0	0
$16$	13.4162	3.35406
$17$	7.56447	1.83465	0.917327	−	0.398135i	$-0.130342\pi$
$17$	7.56447	1.83465	0.917327	+	0.398135i	$0.130342\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−11.6680	−2.60905
$21$	0	0
$22$	−1.46209	−0.311718
$23$	−4.83305	−1.00776	−0.503880	−	0.863774i	$-0.668095\pi$
$23$	−4.83305	−1.00776	−0.503880	+	0.863774i	$0.668095\pi$
$24$	0	0
$25$	−0.137605	−0.0275210
$26$	11.8545	2.32487
$27$	0	0
$28$	2.02083	0.381902
$29$	2.92584	0.543316	0.271658	−	0.962394i	$-0.412428\pi$
$29$	2.92584	0.543316	0.271658	+	0.962394i	$0.412428\pi$
$30$	0	0
$31$	10.0216	1.79993	0.899963	−	0.435965i	$-0.143593\pi$
$31$	10.0216	1.79993	0.899963	+	0.435965i	$0.143593\pi$
$32$	−18.4520	−3.26188
$33$	0	0
$34$	−20.4261	−3.50304
$35$	−0.842139	−0.142348
$36$	0	0
$37$	−6.82713	−1.12237	−0.561186	−	0.827690i	$-0.689655\pi$
$37$	−6.82713	−1.12237	−0.561186	+	0.827690i	$0.689655\pi$
$38$	−2.70026	−0.438040
$39$	0	0
$40$	19.5981	3.09873
$41$	1.66489	0.260012	0.130006	−	0.991513i	$-0.458500\pi$
$41$	1.66489	0.260012	0.130006	+	0.991513i	$0.458500\pi$
$42$	0	0
$43$	−8.47623	−1.29261	−0.646306	−	0.763078i	$-0.723687\pi$
$43$	−8.47623	−1.29261	−0.646306	+	0.763078i	$0.723687\pi$
$44$	2.86509	0.431929
$45$	0	0
$46$	13.0505	1.92419
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.85415	−0.979164
$50$	0.371569	0.0525478
$51$	0	0
$52$	−23.2301	−3.22143
$53$	−5.12841	−0.704441	−0.352221	−	0.935917i	$-0.614573\pi$
$53$	−5.12841	−0.704441	−0.352221	+	0.935917i	$0.614573\pi$
$54$	0	0
$55$	−1.19397	−0.160994
$56$	−3.39428	−0.453579
$57$	0	0
$58$	−7.90055	−1.03739
$59$	6.38720	0.831543	0.415771	−	0.909469i	$-0.363512\pi$
$59$	6.38720	0.831543	0.415771	+	0.909469i	$0.363512\pi$
$60$	0	0
$61$	9.65918	1.23673	0.618366	−	0.785890i	$-0.287795\pi$
$61$	9.65918	1.23673	0.618366	+	0.785890i	$0.287795\pi$
$62$	−27.0609	−3.43673
$63$	0	0
$64$	22.9928	2.87410
$65$	9.68064	1.20074
$66$	0	0
$67$	6.65611	0.813174	0.406587	−	0.913612i	$-0.366719\pi$
$67$	6.65611	0.813174	0.406587	+	0.913612i	$0.366719\pi$
$68$	40.0268	4.85396
$69$	0	0
$70$	2.27400	0.271795
$71$	−8.57056	−1.01714	−0.508569	−	0.861021i	$-0.669825\pi$
$71$	−8.57056	−1.01714	−0.508569	+	0.861021i	$0.669825\pi$
$72$	0	0
$73$	−8.65829	−1.01338	−0.506688	−	0.862130i	$-0.669130\pi$
$73$	−8.65829	−1.01338	−0.506688	+	0.862130i	$0.669130\pi$
$74$	18.4350	2.14303
$75$	0	0
$76$	5.29141	0.606967
$77$	0.206788	0.0235657
$78$	0	0
$79$	−2.49612	−0.280836	−0.140418	−	0.990092i	$-0.544845\pi$
$79$	−2.49612	−0.280836	−0.140418	+	0.990092i	$0.544845\pi$
$80$	−29.5839	−3.30759
$81$	0	0
$82$	−4.49563	−0.496459
$83$	−5.42636	−0.595621	−0.297810	−	0.954625i	$-0.596256\pi$
$83$	−5.42636	−0.595621	−0.297810	+	0.954625i	$0.596256\pi$
$84$	0	0
$85$	−16.6803	−1.80923
$86$	22.8880	2.46808
$87$	0	0
$88$	−4.81233	−0.512996
$89$	12.6622	1.34219	0.671094	−	0.741372i	$-0.265825\pi$
$89$	12.6622	1.34219	0.671094	+	0.741372i	$0.265825\pi$
$90$	0	0
$91$	−1.67663	−0.175759
$92$	−25.5737	−2.66624
$93$	0	0
$94$	2.70026	0.278511
$95$	−2.20508	−0.226237
$96$	0	0
$97$	−8.67832	−0.881150	−0.440575	−	0.897716i	$-0.645225\pi$
$97$	−8.67832	−0.881150	−0.440575	+	0.897716i	$0.645225\pi$
$98$	18.5080	1.86959
$99$	0	0
$100$	−0.728124	−0.0728124
$101$	−9.77259	−0.972409	−0.486204	−	0.873845i	$-0.661619\pi$
$101$	−9.77259	−0.972409	−0.486204	+	0.873845i	$0.661619\pi$
$102$	0	0
$103$	11.0004	1.08391	0.541953	−	0.840409i	$-0.317685\pi$
$103$	11.0004	1.08391	0.541953	+	0.840409i	$0.317685\pi$
$104$	39.0182	3.82605
$105$	0	0
$106$	13.8480	1.34504
$107$	7.79770	0.753832	0.376916	−	0.926247i	$-0.376985\pi$
$107$	7.79770	0.753832	0.376916	+	0.926247i	$0.376985\pi$
$108$	0	0
$109$	16.7420	1.60359	0.801796	−	0.597597i	$-0.203878\pi$
$109$	16.7420	1.60359	0.801796	+	0.597597i	$0.203878\pi$
$110$	3.22402	0.307398
$111$	0	0
$112$	5.12377	0.484151
$113$	14.0364	1.32043	0.660216	−	0.751076i	$-0.270465\pi$
$113$	14.0364	1.32043	0.660216	+	0.751076i	$0.270465\pi$
$114$	0	0
$115$	10.6573	0.993796
$116$	15.4819	1.43745
$117$	0	0
$118$	−17.2471	−1.58773
$119$	2.88893	0.264828
$120$	0	0
$121$	−10.7068	−0.973347
$122$	−26.0823	−2.36138
$123$	0	0
$124$	53.0283	4.76208
$125$	11.3288	1.01328
$126$	0	0
$127$	−11.7591	−1.04345	−0.521727	−	0.853112i	$-0.674712\pi$
$127$	−11.7591	−1.04345	−0.521727	+	0.853112i	$0.674712\pi$
$128$	−25.1825	−2.22584
$129$	0	0
$130$	−26.1403	−2.29265
$131$	6.71779	0.586936	0.293468	−	0.955969i	$-0.405191\pi$
$131$	6.71779	0.586936	0.293468	+	0.955969i	$0.405191\pi$
$132$	0	0
$133$	0.381908	0.0331156
$134$	−17.9733	−1.55265
$135$	0	0
$136$	−67.2306	−5.76498
$137$	−12.5363	−1.07104	−0.535522	−	0.844521i	$-0.679885\pi$
$137$	−12.5363	−1.07104	−0.535522	+	0.844521i	$0.679885\pi$
$138$	0	0
$139$	−8.93314	−0.757699	−0.378849	−	0.925458i	$-0.623680\pi$
$139$	−8.93314	−0.757699	−0.378849	+	0.925458i	$0.623680\pi$
$140$	−4.45611	−0.376610
$141$	0	0
$142$	23.1428	1.94210
$143$	−2.37709	−0.198782
$144$	0	0
$145$	−6.45173	−0.535787
$146$	23.3796	1.93491
$147$	0	0
$148$	−36.1252	−2.96947
$149$	−4.20717	−0.344664	−0.172332	−	0.985039i	$-0.555130\pi$
$149$	−4.20717	−0.344664	−0.172332	+	0.985039i	$0.555130\pi$
$150$	0	0
$151$	18.0236	1.46674	0.733369	−	0.679831i	$-0.237947\pi$
$151$	18.0236	1.46674	0.733369	+	0.679831i	$0.237947\pi$
$152$	−8.88768	−0.720886
$153$	0	0
$154$	−0.558382	−0.0449957
$155$	−22.0984	−1.77499
$156$	0	0
$157$	17.2290	1.37503	0.687514	−	0.726171i	$-0.258702\pi$
$157$	17.2290	1.37503	0.687514	+	0.726171i	$0.258702\pi$
$158$	6.74019	0.536221
$159$	0	0
$160$	40.6882	3.21669
$161$	−1.84578	−0.145468
$162$	0	0
$163$	6.95091	0.544437	0.272219	−	0.962235i	$-0.412243\pi$
$163$	6.95091	0.544437	0.272219	+	0.962235i	$0.412243\pi$
$164$	8.80960	0.687914
$165$	0	0
$166$	14.6526	1.13726
$167$	9.99053	0.773090	0.386545	−	0.922270i	$-0.373668\pi$
$167$	9.99053	0.773090	0.386545	+	0.922270i	$0.373668\pi$
$168$	0	0
$169$	6.27337	0.482567
$170$	45.0412	3.45450
$171$	0	0
$172$	−44.8512	−3.41987
$173$	8.92035	0.678202	0.339101	−	0.940750i	$-0.389877\pi$
$173$	8.92035	0.678202	0.339101	+	0.940750i	$0.389877\pi$
$174$	0	0
$175$	−0.0525524	−0.00397259
$176$	7.26437	0.547572
$177$	0	0
$178$	−34.1912	−2.56274
$179$	−19.7358	−1.47512	−0.737561	−	0.675281i	$-0.764022\pi$
$179$	−19.7358	−1.47512	−0.737561	+	0.675281i	$0.764022\pi$
$180$	0	0
$181$	−2.51847	−0.187196	−0.0935982	−	0.995610i	$-0.529837\pi$
$181$	−2.51847	−0.187196	−0.0935982	+	0.995610i	$0.529837\pi$
$182$	4.52735	0.335589
$183$	0	0
$184$	42.9546	3.16666
$185$	15.0544	1.10682
$186$	0	0
$187$	4.09586	0.299519
$188$	−5.29141	−0.385916
$189$	0	0
$190$	5.95430	0.431971
$191$	−18.3243	−1.32590	−0.662952	−	0.748662i	$-0.730697\pi$
$191$	−18.3243	−1.32590	−0.662952	+	0.748662i	$0.730697\pi$
$192$	0	0
$193$	24.2325	1.74429	0.872146	−	0.489246i	$-0.162728\pi$
$193$	24.2325	1.74429	0.872146	+	0.489246i	$0.162728\pi$
$194$	23.4337	1.68245
$195$	0	0
$196$	−36.2681	−2.59058
$197$	−15.3067	−1.09055	−0.545277	−	0.838256i	$-0.683576\pi$
$197$	−15.3067	−1.09055	−0.545277	+	0.838256i	$0.683576\pi$
$198$	0	0
$199$	15.1884	1.07668	0.538339	−	0.842728i	$-0.319052\pi$
$199$	15.1884	1.07668	0.538339	+	0.842728i	$0.319052\pi$
$200$	1.22299	0.0864783
$201$	0	0
$202$	26.3885	1.85669
$203$	1.11740	0.0784264
$204$	0	0
$205$	−3.67121	−0.256409
$206$	−29.7041	−2.06958
$207$	0	0
$208$	−58.8993	−4.08393
$209$	0.541461	0.0374536
$210$	0	0
$211$	−10.0965	−0.695070	−0.347535	−	0.937667i	$-0.612981\pi$
$211$	−10.0965	−0.695070	−0.347535	+	0.937667i	$0.612981\pi$
$212$	−27.1365	−1.86375
$213$	0	0
$214$	−21.0558	−1.43935
$215$	18.6908	1.27470
$216$	0	0
$217$	3.82732	0.259815
$218$	−45.2078	−3.06186
$219$	0	0
$220$	−6.31777	−0.425944
$221$	−33.2091	−2.23389
$222$	0	0
$223$	19.4848	1.30480	0.652400	−	0.757875i	$-0.273762\pi$
$223$	19.4848	1.30480	0.652400	+	0.757875i	$0.273762\pi$
$224$	−7.04697	−0.470845
$225$	0	0
$226$	−37.9019	−2.52120
$227$	−14.1735	−0.940727	−0.470363	−	0.882473i	$-0.655877\pi$
$227$	−14.1735	−0.940727	−0.470363	+	0.882473i	$0.655877\pi$
$228$	0	0
$229$	−15.0866	−0.996951	−0.498475	−	0.866904i	$-0.666107\pi$
$229$	−15.0866	−0.996951	−0.498475	+	0.866904i	$0.666107\pi$
$230$	−28.7774	−1.89753
$231$	0	0
$232$	−26.0040	−1.70724
$233$	−29.5646	−1.93684	−0.968421	−	0.249321i	$-0.919792\pi$
$233$	−29.5646	−1.93684	−0.968421	+	0.249321i	$0.919792\pi$
$234$	0	0
$235$	2.20508	0.143844
$236$	33.7973	2.20002
$237$	0	0
$238$	−7.80087	−0.505656
$239$	−2.73697	−0.177040	−0.0885199	−	0.996074i	$-0.528214\pi$
$239$	−2.73697	−0.177040	−0.0885199	+	0.996074i	$0.528214\pi$
$240$	0	0
$241$	−13.2806	−0.855477	−0.427738	−	0.903903i	$-0.640690\pi$
$241$	−13.2806	−0.855477	−0.427738	+	0.903903i	$0.640690\pi$
$242$	28.9112	1.85848
$243$	0	0
$244$	51.1107	3.27203
$245$	15.1140	0.965596
$246$	0	0
$247$	−4.39015	−0.279338
$248$	−89.0685	−5.65586
$249$	0	0
$250$	−30.5909	−1.93474
$251$	13.6733	0.863049	0.431525	−	0.902101i	$-0.357976\pi$
$251$	13.6733	0.863049	0.431525	+	0.902101i	$0.357976\pi$
$252$	0	0
$253$	−2.61690	−0.164523
$254$	31.7527	1.99234
$255$	0	0
$256$	22.0138	1.37586
$257$	11.9308	0.744222	0.372111	−	0.928188i	$-0.378634\pi$
$257$	11.9308	0.744222	0.372111	+	0.928188i	$0.378634\pi$
$258$	0	0
$259$	−2.60733	−0.162012
$260$	51.2243	3.17679
$261$	0	0
$262$	−18.1398	−1.12068
$263$	20.6566	1.27374	0.636871	−	0.770970i	$-0.280228\pi$
$263$	20.6566	1.27374	0.636871	+	0.770970i	$0.280228\pi$
$264$	0	0
$265$	11.3086	0.694680
$266$	−1.03125	−0.0632301
$267$	0	0
$268$	35.2203	2.15142
$269$	28.6700	1.74804	0.874020	−	0.485889i	$-0.161504\pi$
$269$	28.6700	1.74804	0.874020	+	0.485889i	$0.161504\pi$
$270$	0	0
$271$	9.20417	0.559114	0.279557	−	0.960129i	$-0.409812\pi$
$271$	9.20417	0.559114	0.279557	+	0.960129i	$0.409812\pi$
$272$	101.487	6.15354
$273$	0	0
$274$	33.8512	2.04502
$275$	−0.0745076	−0.00449298
$276$	0	0
$277$	−19.8425	−1.19222	−0.596111	−	0.802902i	$-0.703288\pi$
$277$	−19.8425	−1.19222	−0.596111	+	0.802902i	$0.703288\pi$
$278$	24.1218	1.44673
$279$	0	0
$280$	7.48467	0.447294
$281$	−32.7137	−1.95153	−0.975767	−	0.218811i	$-0.929782\pi$
$281$	−32.7137	−1.95153	−0.975767	+	0.218811i	$0.929782\pi$
$282$	0	0
$283$	−26.7479	−1.59000	−0.794999	−	0.606611i	$-0.792528\pi$
$283$	−26.7479	−1.59000	−0.794999	+	0.606611i	$0.792528\pi$
$284$	−45.3504	−2.69105
$285$	0	0
$286$	6.41877	0.379550
$287$	0.635833	0.0375321
$288$	0	0
$289$	40.2212	2.36595
$290$	17.4214	1.02302
$291$	0	0
$292$	−45.8146	−2.68110
$293$	5.15932	0.301411	0.150705	−	0.988579i	$-0.451846\pi$
$293$	5.15932	0.301411	0.150705	+	0.988579i	$0.451846\pi$
$294$	0	0
$295$	−14.0843	−0.820021
$296$	60.6773	3.52680
$297$	0	0
$298$	11.3604	0.658093
$299$	21.2178	1.22706
$300$	0	0
$301$	−3.23714	−0.186586
$302$	−48.6684	−2.80055
$303$	0	0
$304$	13.4162	0.769474
$305$	−21.2993	−1.21959
$306$	0	0
$307$	5.44081	0.310523	0.155262	−	0.987873i	$-0.450378\pi$
$307$	5.44081	0.310523	0.155262	+	0.987873i	$0.450378\pi$
$308$	1.09420	0.0623479
$309$	0	0
$310$	59.6715	3.38911
$311$	−21.0326	−1.19265	−0.596323	−	0.802744i	$-0.703372\pi$
$311$	−21.0326	−1.19265	−0.596323	+	0.802744i	$0.703372\pi$
$312$	0	0
$313$	−15.1762	−0.857808	−0.428904	−	0.903350i	$-0.641100\pi$
$313$	−15.1762	−0.857808	−0.428904	+	0.903350i	$0.641100\pi$
$314$	−46.5229	−2.62544
$315$	0	0
$316$	−13.2080	−0.743010
$317$	4.90155	0.275298	0.137649	−	0.990481i	$-0.456045\pi$
$317$	4.90155	0.275298	0.137649	+	0.990481i	$0.456045\pi$
$318$	0	0
$319$	1.58423	0.0886998
$320$	−50.7010	−2.83427
$321$	0	0
$322$	4.98409	0.277752
$323$	7.56447	0.420898
$324$	0	0
$325$	0.604105	0.0335097
$326$	−18.7693	−1.03953
$327$	0	0
$328$	−14.7970	−0.817027
$329$	−0.381908	−0.0210553
$330$	0	0
$331$	−36.1520	−1.98709	−0.993545	−	0.113434i	$-0.963815\pi$
$331$	−36.1520	−1.98709	−0.993545	+	0.113434i	$0.963815\pi$
$332$	−28.7131	−1.57584
$333$	0	0
$334$	−26.9770	−1.47612
$335$	−14.6773	−0.801906
$336$	0	0
$337$	−19.5250	−1.06359	−0.531796	−	0.846873i	$-0.678483\pi$
$337$	−19.5250	−1.06359	−0.531796	+	0.846873i	$0.678483\pi$
$338$	−16.9398	−0.921401
$339$	0	0
$340$	−88.2623	−4.78670
$341$	5.42628	0.293850
$342$	0	0
$343$	−5.29101	−0.285688
$344$	75.3340	4.06174
$345$	0	0
$346$	−24.0873	−1.29494
$347$	−1.91679	−0.102899	−0.0514493	−	0.998676i	$-0.516384\pi$
$347$	−1.91679	−0.102899	−0.0514493	+	0.998676i	$0.516384\pi$
$348$	0	0
$349$	17.4874	0.936079	0.468040	−	0.883708i	$-0.344960\pi$
$349$	17.4874	0.936079	0.468040	+	0.883708i	$0.344960\pi$
$350$	0.141905	0.00758515
$351$	0	0
$352$	−9.99103	−0.532524
$353$	13.1334	0.699021	0.349510	−	0.936933i	$-0.386348\pi$
$353$	13.1334	0.699021	0.349510	+	0.936933i	$0.386348\pi$
$354$	0	0
$355$	18.8988	1.00304
$356$	67.0008	3.55104
$357$	0	0
$358$	53.2918	2.81656
$359$	16.6782	0.880241	0.440121	−	0.897939i	$-0.354936\pi$
$359$	16.6782	0.880241	0.440121	+	0.897939i	$0.354936\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.80053	0.357428
$363$	0	0
$364$	−8.87175	−0.465006
$365$	19.0923	0.999334
$366$	0	0
$367$	−1.77145	−0.0924688	−0.0462344	−	0.998931i	$-0.514722\pi$
$367$	−1.77145	−0.0924688	−0.0462344	+	0.998931i	$0.514722\pi$
$368$	−64.8413	−3.38009
$369$	0	0
$370$	−40.6508	−2.11333
$371$	−1.95858	−0.101684
$372$	0	0
$373$	−2.63760	−0.136570	−0.0682848	−	0.997666i	$-0.521753\pi$
$373$	−2.63760	−0.136570	−0.0682848	+	0.997666i	$0.521753\pi$
$374$	−11.0599	−0.571894
$375$	0	0
$376$	8.88768	0.458347
$377$	−12.8449	−0.661545
$378$	0	0
$379$	7.01437	0.360304	0.180152	−	0.983639i	$-0.442341\pi$
$379$	7.01437	0.360304	0.180152	+	0.983639i	$0.442341\pi$
$380$	−11.6680	−0.598556
$381$	0	0
$382$	49.4805	2.53164
$383$	19.4874	0.995761	0.497880	−	0.867246i	$-0.334112\pi$
$383$	19.4874	0.995761	0.497880	+	0.867246i	$0.334112\pi$
$384$	0	0
$385$	−0.455985	−0.0232392
$386$	−65.4340	−3.33050
$387$	0	0
$388$	−45.9206	−2.33127
$389$	−5.96803	−0.302591	−0.151296	−	0.988489i	$-0.548345\pi$
$389$	−5.96803	−0.302591	−0.151296	+	0.988489i	$0.548345\pi$
$390$	0	0
$391$	−36.5594	−1.84889
$392$	60.9175	3.07680
$393$	0	0
$394$	41.3320	2.08228
$395$	5.50416	0.276944
$396$	0	0
$397$	6.34949	0.318672	0.159336	−	0.987224i	$-0.449065\pi$
$397$	6.34949	0.318672	0.159336	+	0.987224i	$0.449065\pi$
$398$	−41.0127	−2.05578
$399$	0	0
$400$	−1.84614	−0.0923070
$401$	−20.4053	−1.01899	−0.509496	−	0.860473i	$-0.670168\pi$
$401$	−20.4053	−1.01899	−0.509496	+	0.860473i	$0.670168\pi$
$402$	0	0
$403$	−43.9961	−2.19160
$404$	−51.7108	−2.57271
$405$	0	0
$406$	−3.01728	−0.149745
$407$	−3.69662	−0.183235
$408$	0	0
$409$	12.3652	0.611421	0.305710	−	0.952125i	$-0.401106\pi$
$409$	12.3652	0.611421	0.305710	+	0.952125i	$0.401106\pi$
$410$	9.91324	0.489580
$411$	0	0
$412$	58.2079	2.86770
$413$	2.43932	0.120031
$414$	0	0
$415$	11.9656	0.587368
$416$	81.0070	3.97169
$417$	0	0
$418$	−1.46209	−0.0715129
$419$	13.4073	0.654992	0.327496	−	0.944853i	$-0.393795\pi$
$419$	13.4073	0.654992	0.327496	+	0.944853i	$0.393795\pi$
$420$	0	0
$421$	9.55704	0.465782	0.232891	−	0.972503i	$-0.425182\pi$
$421$	9.55704	0.465782	0.232891	+	0.972503i	$0.425182\pi$
$422$	27.2631	1.32715
$423$	0	0
$424$	45.5797	2.21354
$425$	−1.04091	−0.0504914
$426$	0	0
$427$	3.68892	0.178519
$428$	41.2609	1.99442
$429$	0	0
$430$	−50.4700	−2.43388
$431$	12.6183	0.607804	0.303902	−	0.952703i	$-0.401711\pi$
$431$	12.6183	0.607804	0.303902	+	0.952703i	$0.401711\pi$
$432$	0	0
$433$	16.0281	0.770262	0.385131	−	0.922862i	$-0.374156\pi$
$433$	16.0281	0.770262	0.385131	+	0.922862i	$0.374156\pi$
$434$	−10.3348	−0.496084
$435$	0	0
$436$	88.5889	4.24264
$437$	−4.83305	−0.231196
$438$	0	0
$439$	3.06186	0.146135	0.0730674	−	0.997327i	$-0.476721\pi$
$439$	3.06186	0.146135	0.0730674	+	0.997327i	$0.476721\pi$
$440$	10.6116	0.505888
$441$	0	0
$442$	89.6733	4.26533
$443$	−9.52124	−0.452368	−0.226184	−	0.974085i	$-0.572625\pi$
$443$	−9.52124	−0.452368	−0.226184	+	0.974085i	$0.572625\pi$
$444$	0	0
$445$	−27.9212	−1.32359
$446$	−52.6141	−2.49135
$447$	0	0
$448$	8.78112	0.414869
$449$	−31.2958	−1.47694	−0.738469	−	0.674287i	$-0.764451\pi$
$449$	−31.2958	−1.47694	−0.738469	+	0.674287i	$0.764451\pi$
$450$	0	0
$451$	0.901470	0.0424486
$452$	74.2724	3.49348
$453$	0	0
$454$	38.2721	1.79620
$455$	3.69711	0.173323
$456$	0	0
$457$	3.14927	0.147317	0.0736583	−	0.997284i	$-0.476533\pi$
$457$	3.14927	0.147317	0.0736583	+	0.997284i	$0.476533\pi$
$458$	40.7378	1.90355
$459$	0	0
$460$	56.3921	2.62929
$461$	−36.5366	−1.70168	−0.850840	−	0.525424i	$-0.823907\pi$
$461$	−36.5366	−1.70168	−0.850840	+	0.525424i	$0.823907\pi$
$462$	0	0
$463$	2.42169	0.112546	0.0562728	−	0.998415i	$-0.482078\pi$
$463$	2.42169	0.112546	0.0562728	+	0.998415i	$0.482078\pi$
$464$	39.2538	1.82231
$465$	0	0
$466$	79.8322	3.69815
$467$	−9.54643	−0.441756	−0.220878	−	0.975301i	$-0.570892\pi$
$467$	−9.54643	−0.441756	−0.220878	+	0.975301i	$0.570892\pi$
$468$	0	0
$469$	2.54202	0.117380
$470$	−5.95430	−0.274652
$471$	0	0
$472$	−56.7674	−2.61293
$473$	−4.58954	−0.211027
$474$	0	0
$475$	−0.137605	−0.00631374
$476$	15.2865	0.700657
$477$	0	0
$478$	7.39053	0.338035
$479$	13.5642	0.619763	0.309881	−	0.950775i	$-0.399711\pi$
$479$	13.5642	0.619763	0.309881	+	0.950775i	$0.399711\pi$
$480$	0	0
$481$	29.9721	1.36661
$482$	35.8610	1.63342
$483$	0	0
$484$	−56.6542	−2.57519
$485$	19.1364	0.868941
$486$	0	0
$487$	−27.7764	−1.25867	−0.629335	−	0.777134i	$-0.716672\pi$
$487$	−27.7764	−1.25867	−0.629335	+	0.777134i	$0.716672\pi$
$488$	−85.8478	−3.88615
$489$	0	0
$490$	−40.8117	−1.84368
$491$	13.7552	0.620762	0.310381	−	0.950612i	$-0.399543\pi$
$491$	13.7552	0.620762	0.310381	+	0.950612i	$0.399543\pi$
$492$	0	0
$493$	22.1325	0.996796
$494$	11.8545	0.533361
$495$	0	0
$496$	134.452	6.03706
$497$	−3.27317	−0.146822
$498$	0	0
$499$	12.9853	0.581304	0.290652	−	0.956829i	$-0.406128\pi$
$499$	12.9853	0.581304	0.290652	+	0.956829i	$0.406128\pi$
$500$	59.9456	2.68085
$501$	0	0
$502$	−36.9214	−1.64788
$503$	−0.878196	−0.0391568	−0.0195784	−	0.999808i	$-0.506232\pi$
$503$	−0.878196	−0.0391568	−0.0195784	+	0.999808i	$0.506232\pi$
$504$	0	0
$505$	21.5494	0.958935
$506$	7.06633	0.314137
$507$	0	0
$508$	−62.2225	−2.76067
$509$	−21.1285	−0.936502	−0.468251	−	0.883595i	$-0.655116\pi$
$509$	−21.1285	−0.936502	−0.468251	+	0.883595i	$0.655116\pi$
$510$	0	0
$511$	−3.30667	−0.146278
$512$	−9.07795	−0.401193
$513$	0	0
$514$	−32.2162	−1.42100
$515$	−24.2569	−1.06889
$516$	0	0
$517$	−0.541461	−0.0238134
$518$	7.04049	0.309341
$519$	0	0
$520$	−86.0384	−3.77304
$521$	21.4364	0.939146	0.469573	−	0.882894i	$-0.344408\pi$
$521$	21.4364	0.939146	0.469573	+	0.882894i	$0.344408\pi$
$522$	0	0
$523$	8.62069	0.376957	0.188478	−	0.982077i	$-0.439644\pi$
$523$	8.62069	0.376957	0.188478	+	0.982077i	$0.439644\pi$
$524$	35.5466	1.55286
$525$	0	0
$526$	−55.7783	−2.43205
$527$	75.8079	3.30224
$528$	0	0
$529$	0.358354	0.0155806
$530$	−30.5361	−1.32640
$531$	0	0
$532$	2.02083	0.0876143
$533$	−7.30909	−0.316592
$534$	0	0
$535$	−17.1946	−0.743387
$536$	−59.1574	−2.55521
$537$	0	0
$538$	−77.4165	−3.33766
$539$	−3.71125	−0.159855
$540$	0	0
$541$	36.7599	1.58043	0.790216	−	0.612828i	$-0.209968\pi$
$541$	36.7599	1.58043	0.790216	+	0.612828i	$0.209968\pi$
$542$	−24.8537	−1.06756
$543$	0	0
$544$	−139.580	−5.98443
$545$	−36.9175	−1.58137
$546$	0	0
$547$	−20.6459	−0.882756	−0.441378	−	0.897321i	$-0.645510\pi$
$547$	−20.6459	−0.882756	−0.441378	+	0.897321i	$0.645510\pi$
$548$	−66.3345	−2.83367
$549$	0	0
$550$	0.201190	0.00857877
$551$	2.92584	0.124645
$552$	0	0
$553$	−0.953290	−0.0405380
$554$	53.5800	2.27640
$555$	0	0
$556$	−47.2689	−2.00465
$557$	23.6903	1.00379	0.501896	−	0.864928i	$-0.332636\pi$
$557$	23.6903	1.00379	0.501896	+	0.864928i	$0.332636\pi$
$558$	0	0
$559$	37.2119	1.57389
$560$	−11.2983	−0.477442
$561$	0	0
$562$	88.3355	3.72621
$563$	−9.84108	−0.414752	−0.207376	−	0.978261i	$-0.566492\pi$
$563$	−9.84108	−0.414752	−0.207376	+	0.978261i	$0.566492\pi$
$564$	0	0
$565$	−30.9514	−1.30214
$566$	72.2263	3.03590
$567$	0	0
$568$	76.1724	3.19612
$569$	−1.63516	−0.0685495	−0.0342748	−	0.999412i	$-0.510912\pi$
$569$	−1.63516	−0.0685495	−0.0342748	+	0.999412i	$0.510912\pi$
$570$	0	0
$571$	−3.22029	−0.134765	−0.0673824	−	0.997727i	$-0.521465\pi$
$571$	−3.22029	−0.134765	−0.0673824	+	0.997727i	$0.521465\pi$
$572$	−12.5782	−0.525920
$573$	0	0
$574$	−1.71692	−0.0716627
$575$	0.665051	0.0277345
$576$	0	0
$577$	16.4921	0.686577	0.343288	−	0.939230i	$-0.388459\pi$
$577$	16.4921	0.686577	0.343288	+	0.939230i	$0.388459\pi$
$578$	−108.608	−4.51749
$579$	0	0
$580$	−34.1388	−1.41754
$581$	−2.07237	−0.0859765
$582$	0	0
$583$	−2.77683	−0.115005
$584$	76.9521	3.18430
$585$	0	0
$586$	−13.9315	−0.575506
$587$	−18.4883	−0.763095	−0.381548	−	0.924349i	$-0.624609\pi$
$587$	−18.4883	−0.763095	−0.381548	+	0.924349i	$0.624609\pi$
$588$	0	0
$589$	10.0216	0.412932
$590$	38.0313	1.56573
$591$	0	0
$592$	−91.5944	−3.76451
$593$	43.4614	1.78475	0.892373	−	0.451299i	$-0.149039\pi$
$593$	43.4614	1.78475	0.892373	+	0.451299i	$0.149039\pi$
$594$	0	0
$595$	−6.37034	−0.261158
$596$	−22.2619	−0.911881
$597$	0	0
$598$	−57.2936	−2.34291
$599$	−23.3190	−0.952788	−0.476394	−	0.879232i	$-0.658056\pi$
$599$	−23.3190	−0.952788	−0.476394	+	0.879232i	$0.658056\pi$
$600$	0	0
$601$	−34.0943	−1.39074	−0.695368	−	0.718654i	$-0.744759\pi$
$601$	−34.0943	−1.39074	−0.695368	+	0.718654i	$0.744759\pi$
$602$	8.74112	0.356262
$603$	0	0
$604$	95.3703	3.88056
$605$	23.6094	0.959860
$606$	0	0
$607$	−11.0254	−0.447507	−0.223754	−	0.974646i	$-0.571831\pi$
$607$	−11.0254	−0.447507	−0.223754	+	0.974646i	$0.571831\pi$
$608$	−18.4520	−0.748328
$609$	0	0
$610$	57.5137	2.32866
$611$	4.39015	0.177606
$612$	0	0
$613$	−4.46689	−0.180416	−0.0902081	−	0.995923i	$-0.528753\pi$
$613$	−4.46689	−0.180416	−0.0902081	+	0.995923i	$0.528753\pi$
$614$	−14.6916	−0.592905
$615$	0	0
$616$	−1.83787	−0.0740498
$617$	−6.29671	−0.253496	−0.126748	−	0.991935i	$-0.540454\pi$
$617$	−6.29671	−0.253496	−0.126748	+	0.991935i	$0.540454\pi$
$618$	0	0
$619$	−36.2640	−1.45757	−0.728787	−	0.684741i	$-0.759915\pi$
$619$	−36.2640	−1.45757	−0.728787	+	0.684741i	$0.759915\pi$
$620$	−116.932	−4.69609
$621$	0	0
$622$	56.7934	2.27721
$623$	4.83579	0.193742
$624$	0	0
$625$	−24.2930	−0.971722
$626$	40.9796	1.63788
$627$	0	0
$628$	91.1660	3.63792
$629$	−51.6436	−2.05916
$630$	0	0
$631$	−10.7987	−0.429891	−0.214945	−	0.976626i	$-0.568957\pi$
$631$	−10.7987	−0.429891	−0.214945	+	0.976626i	$0.568957\pi$
$632$	22.1848	0.882462
$633$	0	0
$634$	−13.2355	−0.525648
$635$	25.9299	1.02900
$636$	0	0
$637$	30.0907	1.19224
$638$	−4.27783	−0.169361
$639$	0	0
$640$	55.5295	2.19499
$641$	−27.5198	−1.08697	−0.543483	−	0.839420i	$-0.682895\pi$
$641$	−27.5198	−1.08697	−0.543483	+	0.839420i	$0.682895\pi$
$642$	0	0
$643$	9.75948	0.384876	0.192438	−	0.981309i	$-0.438361\pi$
$643$	9.75948	0.384876	0.192438	+	0.981309i	$0.438361\pi$
$644$	−9.76679	−0.384865
$645$	0	0
$646$	−20.4261	−0.803652
$647$	−24.2245	−0.952363	−0.476181	−	0.879347i	$-0.657979\pi$
$647$	−24.2245	−0.952363	−0.476181	+	0.879347i	$0.657979\pi$
$648$	0	0
$649$	3.45842	0.135755
$650$	−1.63124	−0.0639826
$651$	0	0
$652$	36.7801	1.44042
$653$	−12.6373	−0.494535	−0.247268	−	0.968947i	$-0.579533\pi$
$653$	−12.6373	−0.494535	−0.247268	+	0.968947i	$0.579533\pi$
$654$	0	0
$655$	−14.8133	−0.578803
$656$	22.3365	0.872094
$657$	0	0
$658$	1.03125	0.0402024
$659$	−29.8421	−1.16248	−0.581241	−	0.813731i	$-0.697433\pi$
$659$	−29.8421	−1.16248	−0.581241	+	0.813731i	$0.697433\pi$
$660$	0	0
$661$	−8.01684	−0.311819	−0.155909	−	0.987771i	$-0.549831\pi$
$661$	−8.01684	−0.311819	−0.155909	+	0.987771i	$0.549831\pi$
$662$	97.6197	3.79410
$663$	0	0
$664$	48.2278	1.87160
$665$	−0.842139	−0.0326568
$666$	0	0
$667$	−14.1407	−0.547532
$668$	52.8640	2.04537
$669$	0	0
$670$	39.6325	1.53114
$671$	5.23007	0.201904
$672$	0	0
$673$	−38.4718	−1.48298	−0.741489	−	0.670965i	$-0.765880\pi$
$673$	−38.4718	−1.48298	−0.741489	+	0.670965i	$0.765880\pi$
$674$	52.7225	2.03079
$675$	0	0
$676$	33.1950	1.27673
$677$	10.1601	0.390486	0.195243	−	0.980755i	$-0.437450\pi$
$677$	10.1601	0.390486	0.195243	+	0.980755i	$0.437450\pi$
$678$	0	0
$679$	−3.31432	−0.127192
$680$	148.249	5.68510
$681$	0	0
$682$	−14.6524	−0.561069
$683$	26.6293	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$683$	26.6293	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$684$	0	0
$685$	27.6435	1.05620
$686$	14.2871	0.545485
$687$	0	0
$688$	−113.719	−4.33550
$689$	22.5145	0.857733
$690$	0	0
$691$	−22.8073	−0.867631	−0.433816	−	0.901002i	$-0.642833\pi$
$691$	−22.8073	−0.867631	−0.433816	+	0.901002i	$0.642833\pi$
$692$	47.2013	1.79432
$693$	0	0
$694$	5.17583	0.196472
$695$	19.6983	0.747200
$696$	0	0
$697$	12.5940	0.477031
$698$	−47.2206	−1.78732
$699$	0	0
$700$	−0.278077	−0.0105103
$701$	37.4962	1.41621	0.708105	−	0.706107i	$-0.249550\pi$
$701$	37.4962	1.41621	0.708105	+	0.706107i	$0.249550\pi$
$702$	0	0
$703$	−6.82713	−0.257490
$704$	12.4497	0.469215
$705$	0	0
$706$	−35.4636	−1.33469
$707$	−3.73223	−0.140365
$708$	0	0
$709$	−19.6326	−0.737318	−0.368659	−	0.929565i	$-0.620183\pi$
$709$	−19.6326	−0.737318	−0.368659	+	0.929565i	$0.620183\pi$
$710$	−51.0317	−1.91519
$711$	0	0
$712$	−112.537	−4.21752
$713$	−48.4347	−1.81389
$714$	0	0
$715$	5.24168	0.196028
$716$	−104.430	−3.90274
$717$	0	0
$718$	−45.0355	−1.68071
$719$	3.20241	0.119430	0.0597150	−	0.998215i	$-0.480981\pi$
$719$	3.20241	0.119430	0.0597150	+	0.998215i	$0.480981\pi$
$720$	0	0
$721$	4.20115	0.156459
$722$	−2.70026	−0.100493
$723$	0	0
$724$	−13.3263	−0.495267
$725$	−0.402610	−0.0149526
$726$	0	0
$727$	20.1186	0.746157	0.373078	−	0.927800i	$-0.378302\pi$
$727$	20.1186	0.746157	0.373078	+	0.927800i	$0.378302\pi$
$728$	14.9014	0.552282
$729$	0	0
$730$	−51.5541	−1.90810
$731$	−64.1182	−2.37150
$732$	0	0
$733$	−42.8614	−1.58312	−0.791560	−	0.611091i	$-0.790731\pi$
$733$	−42.8614	−1.58312	−0.791560	+	0.611091i	$0.790731\pi$
$734$	4.78337	0.176557
$735$	0	0
$736$	89.1794	3.28720
$737$	3.60402	0.132756
$738$	0	0
$739$	43.3153	1.59338	0.796689	−	0.604389i	$-0.206583\pi$
$739$	43.3153	1.59338	0.796689	+	0.604389i	$0.206583\pi$
$740$	79.6590	2.92832
$741$	0	0
$742$	5.28868	0.194154
$743$	15.7700	0.578544	0.289272	−	0.957247i	$-0.406587\pi$
$743$	15.7700	0.578544	0.289272	+	0.957247i	$0.406587\pi$
$744$	0	0
$745$	9.27715	0.339889
$746$	7.12220	0.260762
$747$	0	0
$748$	21.6729	0.792440
$749$	2.97800	0.108814
$750$	0	0
$751$	25.1146	0.916445	0.458223	−	0.888837i	$-0.348486\pi$
$751$	25.1146	0.916445	0.458223	+	0.888837i	$0.348486\pi$
$752$	−13.4162	−0.489240
$753$	0	0
$754$	34.6845	1.26314
$755$	−39.7435	−1.44641
$756$	0	0
$757$	−48.6599	−1.76857	−0.884287	−	0.466944i	$-0.845355\pi$
$757$	−48.6599	−1.76857	−0.884287	+	0.466944i	$0.845355\pi$
$758$	−18.9407	−0.687955
$759$	0	0
$760$	19.5981	0.710897
$761$	−15.5719	−0.564481	−0.282241	−	0.959344i	$-0.591078\pi$
$761$	−15.5719	−0.564481	−0.282241	+	0.959344i	$0.591078\pi$
$762$	0	0
$763$	6.39390	0.231475
$764$	−96.9617	−3.50795
$765$	0	0
$766$	−52.6212	−1.90128
$767$	−28.0407	−1.01249
$768$	0	0
$769$	−24.5292	−0.884545	−0.442273	−	0.896881i	$-0.645828\pi$
$769$	−24.5292	−0.884545	−0.442273	+	0.896881i	$0.645828\pi$
$770$	1.23128	0.0443722
$771$	0	0
$772$	128.224	4.61488
$773$	−24.2794	−0.873271	−0.436635	−	0.899639i	$-0.643830\pi$
$773$	−24.2794	−0.873271	−0.436635	+	0.899639i	$0.643830\pi$
$774$	0	0
$775$	−1.37902	−0.0495357
$776$	77.1302	2.76881
$777$	0	0
$778$	16.1152	0.577760
$779$	1.66489	0.0596507
$780$	0	0
$781$	−4.64062	−0.166054
$782$	98.7201	3.53022
$783$	0	0
$784$	−91.9569	−3.28417
$785$	−37.9915	−1.35597
$786$	0	0
$787$	−26.0922	−0.930086	−0.465043	−	0.885288i	$-0.653961\pi$
$787$	−26.0922	−0.930086	−0.465043	+	0.885288i	$0.653961\pi$
$788$	−80.9939	−2.88529
$789$	0	0
$790$	−14.8627	−0.528791
$791$	5.36061	0.190601
$792$	0	0
$793$	−42.4052	−1.50585
$794$	−17.1453	−0.608463
$795$	0	0
$796$	80.3682	2.84858
$797$	−12.5351	−0.444017	−0.222009	−	0.975045i	$-0.571261\pi$
$797$	−12.5351	−0.444017	−0.222009	+	0.975045i	$0.571261\pi$
$798$	0	0
$799$	−7.56447	−0.267612
$800$	2.53909	0.0897702
$801$	0	0
$802$	55.0997	1.94564
$803$	−4.68812	−0.165440
$804$	0	0
$805$	4.07010	0.143452
$806$	118.801	4.18459
$807$	0	0
$808$	86.8556	3.05557
$809$	−8.65520	−0.304301	−0.152150	−	0.988357i	$-0.548620\pi$
$809$	−8.65520	−0.304301	−0.152150	+	0.988357i	$0.548620\pi$
$810$	0	0
$811$	32.2900	1.13385	0.566927	−	0.823768i	$-0.308132\pi$
$811$	32.2900	1.13385	0.566927	+	0.823768i	$0.308132\pi$
$812$	5.91265	0.207493
$813$	0	0
$814$	9.98184	0.349863
$815$	−15.3273	−0.536893
$816$	0	0
$817$	−8.47623	−0.296546
$818$	−33.3893	−1.16743
$819$	0	0
$820$	−19.4259	−0.678382
$821$	−47.9178	−1.67234	−0.836172	−	0.548468i	$-0.815211\pi$
$821$	−47.9178	−1.67234	−0.836172	+	0.548468i	$0.815211\pi$
$822$	0	0
$823$	47.1327	1.64294	0.821471	−	0.570250i	$-0.193154\pi$
$823$	47.1327	1.64294	0.821471	+	0.570250i	$0.193154\pi$
$824$	−97.7684	−3.40592
$825$	0	0
$826$	−6.58681	−0.229185
$827$	−46.0328	−1.60072	−0.800358	−	0.599522i	$-0.795357\pi$
$827$	−46.0328	−1.60072	−0.800358	+	0.599522i	$0.795357\pi$
$828$	0	0
$829$	−18.0705	−0.627616	−0.313808	−	0.949486i	$-0.601605\pi$
$829$	−18.0705	−0.627616	−0.313808	+	0.949486i	$0.601605\pi$
$830$	−32.3102	−1.12150
$831$	0	0
$832$	−100.942	−3.49952
$833$	−51.8480	−1.79643
$834$	0	0
$835$	−22.0300	−0.762378
$836$	2.86509	0.0990913
$837$	0	0
$838$	−36.2034	−1.25062
$839$	8.19390	0.282885	0.141442	−	0.989946i	$-0.454826\pi$
$839$	8.19390	0.282885	0.141442	+	0.989946i	$0.454826\pi$
$840$	0	0
$841$	−20.4394	−0.704808
$842$	−25.8065	−0.889351
$843$	0	0
$844$	−53.4246	−1.83895
$845$	−13.8333	−0.475881
$846$	0	0
$847$	−4.08902	−0.140500
$848$	−68.8040	−2.36274
$849$	0	0
$850$	2.81072	0.0964070
$851$	32.9958	1.13108
$852$	0	0
$853$	23.0917	0.790646	0.395323	−	0.918542i	$-0.370633\pi$
$853$	23.0917	0.790646	0.395323	+	0.918542i	$0.370633\pi$
$854$	−9.96105	−0.340860
$855$	0	0
$856$	−69.3035	−2.36874
$857$	−29.1910	−0.997146	−0.498573	−	0.866848i	$-0.666142\pi$
$857$	−29.1910	−0.997146	−0.498573	+	0.866848i	$0.666142\pi$
$858$	0	0
$859$	−45.4744	−1.55157	−0.775783	−	0.630999i	$-0.782645\pi$
$859$	−45.4744	−1.55157	−0.775783	+	0.630999i	$0.782645\pi$
$860$	98.9007	3.37249
$861$	0	0
$862$	−34.0728	−1.16052
$863$	−31.0730	−1.05774	−0.528869	−	0.848703i	$-0.677384\pi$
$863$	−31.0730	−1.05774	−0.528869	+	0.848703i	$0.677384\pi$
$864$	0	0
$865$	−19.6701	−0.668804
$866$	−43.2801	−1.47072
$867$	0	0
$868$	20.2519	0.687395
$869$	−1.35155	−0.0458483
$870$	0	0
$871$	−29.2213	−0.990126
$872$	−148.798	−5.03892
$873$	0	0
$874$	13.0505	0.441440
$875$	4.32658	0.146265
$876$	0	0
$877$	−4.04121	−0.136462	−0.0682311	−	0.997670i	$-0.521736\pi$
$877$	−4.04121	−0.136462	−0.0682311	+	0.997670i	$0.521736\pi$
$878$	−8.26783	−0.279026
$879$	0	0
$880$	−16.0185	−0.539985
$881$	24.3680	0.820980	0.410490	−	0.911865i	$-0.365358\pi$
$881$	24.3680	0.820980	0.410490	+	0.911865i	$0.365358\pi$
$882$	0	0
$883$	−45.8443	−1.54278	−0.771391	−	0.636362i	$-0.780439\pi$
$883$	−45.8443	−1.54278	−0.771391	+	0.636362i	$0.780439\pi$
$884$	−175.723	−5.91021
$885$	0	0
$886$	25.7098	0.863739
$887$	−25.2072	−0.846375	−0.423187	−	0.906042i	$-0.639089\pi$
$887$	−25.2072	−0.846375	−0.423187	+	0.906042i	$0.639089\pi$
$888$	0	0
$889$	−4.49091	−0.150620
$890$	75.3944	2.52723
$891$	0	0
$892$	103.102	3.45212
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	43.5191	1.45468
$896$	−9.61739	−0.321294
$897$	0	0
$898$	84.5068	2.82003
$899$	29.3216	0.977928
$900$	0	0
$901$	−38.7937	−1.29241
$902$	−2.43421	−0.0810502
$903$	0	0
$904$	−124.751	−4.14916
$905$	5.55344	0.184603
$906$	0	0
$907$	−28.6832	−0.952409	−0.476205	−	0.879335i	$-0.657988\pi$
$907$	−28.6832	−0.952409	−0.476205	+	0.879335i	$0.657988\pi$
$908$	−74.9978	−2.48889
$909$	0	0
$910$	−9.98318	−0.330939
$911$	−44.0242	−1.45859	−0.729293	−	0.684201i	$-0.760151\pi$
$911$	−44.0242	−1.45859	−0.729293	+	0.684201i	$0.760151\pi$
$912$	0	0
$913$	−2.93816	−0.0972390
$914$	−8.50385	−0.281282
$915$	0	0
$916$	−79.8295	−2.63764
$917$	2.56558	0.0847229
$918$	0	0
$919$	−23.2387	−0.766573	−0.383287	−	0.923629i	$-0.625208\pi$
$919$	−23.2387	−0.766573	−0.383287	+	0.923629i	$0.625208\pi$
$920$	−94.7185	−3.12278
$921$	0	0
$922$	98.6585	3.24914
$923$	37.6260	1.23848
$924$	0	0
$925$	0.939446	0.0308888
$926$	−6.53920	−0.214892
$927$	0	0
$928$	−53.9877	−1.77223
$929$	41.4909	1.36127	0.680636	−	0.732622i	$-0.261704\pi$
$929$	41.4909	1.36127	0.680636	+	0.732622i	$0.261704\pi$
$930$	0	0
$931$	−6.85415	−0.224636
$932$	−156.439	−5.12432
$933$	0	0
$934$	25.7779	0.843477
$935$	−9.03172	−0.295369
$936$	0	0
$937$	−20.6869	−0.675812	−0.337906	−	0.941180i	$-0.609719\pi$
$937$	−20.6869	−0.675812	−0.337906	+	0.941180i	$0.609719\pi$
$938$	−6.86413	−0.224122
$939$	0	0
$940$	11.6680	0.380569
$941$	−42.9252	−1.39932	−0.699661	−	0.714475i	$-0.746666\pi$
$941$	−42.9252	−1.39932	−0.699661	+	0.714475i	$0.746666\pi$
$942$	0	0
$943$	−8.04648	−0.262029
$944$	85.6923	2.78905
$945$	0	0
$946$	12.3930	0.402930
$947$	−54.2239	−1.76204	−0.881020	−	0.473079i	$-0.843143\pi$
$947$	−54.2239	−1.76204	−0.881020	+	0.473079i	$0.843143\pi$
$948$	0	0
$949$	38.0111	1.23389
$950$	0.371569	0.0120553
$951$	0	0
$952$	−25.6759	−0.832161
$953$	−29.3572	−0.950972	−0.475486	−	0.879723i	$-0.657728\pi$
$953$	−29.3572	−0.950972	−0.475486	+	0.879723i	$0.657728\pi$
$954$	0	0
$955$	40.4067	1.30753
$956$	−14.4824	−0.468395
$957$	0	0
$958$	−36.6268	−1.18336
$959$	−4.78770	−0.154603
$960$	0	0
$961$	69.4318	2.23974
$962$	−80.9325	−2.60937
$963$	0	0
$964$	−70.2730	−2.26334
$965$	−53.4346	−1.72012
$966$	0	0
$967$	−15.2193	−0.489420	−0.244710	−	0.969596i	$-0.578693\pi$
$967$	−15.2193	−0.489420	−0.244710	+	0.969596i	$0.578693\pi$
$968$	95.1588	3.05852
$969$	0	0
$970$	−51.6734	−1.65913
$971$	−0.850186	−0.0272838	−0.0136419	−	0.999907i	$-0.504342\pi$
$971$	−0.850186	−0.0272838	−0.0136419	+	0.999907i	$0.504342\pi$
$972$	0	0
$973$	−3.41164	−0.109372
$974$	75.0036	2.40327
$975$	0	0
$976$	129.590	4.14807
$977$	−1.29206	−0.0413366	−0.0206683	−	0.999786i	$-0.506579\pi$
$977$	−1.29206	−0.0413366	−0.0206683	+	0.999786i	$0.506579\pi$
$978$	0	0
$979$	6.85607	0.219121
$980$	79.9743	2.55468
$981$	0	0
$982$	−37.1426	−1.18527
$983$	−24.2367	−0.773032	−0.386516	−	0.922283i	$-0.626322\pi$
$983$	−24.2367	−0.773032	−0.386516	+	0.922283i	$0.626322\pi$
$984$	0	0
$985$	33.7525	1.07544
$986$	−59.7635	−1.90326
$987$	0	0
$988$	−23.2301	−0.739047
$989$	40.9660	1.30264
$990$	0	0
$991$	41.8952	1.33084	0.665422	−	0.746467i	$-0.268251\pi$
$991$	41.8952	1.33084	0.665422	+	0.746467i	$0.268251\pi$
$992$	−184.918	−5.87115
$993$	0	0
$994$	8.83841	0.280337
$995$	−33.4917	−1.06176
$996$	0	0
$997$	−13.4242	−0.425149	−0.212574	−	0.977145i	$-0.568185\pi$
$997$	−13.4242	−0.425149	−0.212574	+	0.977145i	$0.568185\pi$
$998$	−35.0638	−1.10993
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.p.1.1		23
3.2	odd	2		2679.2.a.n.1.23	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.n.1.23	✓	23	3.2	odd	2
8037.2.a.p.1.1		23	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70026 q^{2} +5.29141 q^{4} -2.20508 q^{5} +0.381908 q^{7} -8.88768 q^{8} +O(q^{10})\)	\(q-2.70026 q^{2} +5.29141 q^{4} -2.20508 q^{5} +0.381908 q^{7} -8.88768 q^{8} +5.95430 q^{10} +0.541461 q^{11} -4.39015 q^{13} -1.03125 q^{14} +13.4162 q^{16} +7.56447 q^{17} +1.00000 q^{19} -11.6680 q^{20} -1.46209 q^{22} -4.83305 q^{23} -0.137605 q^{25} +11.8545 q^{26} +2.02083 q^{28} +2.92584 q^{29} +10.0216 q^{31} -18.4520 q^{32} -20.4261 q^{34} -0.842139 q^{35} -6.82713 q^{37} -2.70026 q^{38} +19.5981 q^{40} +1.66489 q^{41} -8.47623 q^{43} +2.86509 q^{44} +13.0505 q^{46} -1.00000 q^{47} -6.85415 q^{49} +0.371569 q^{50} -23.2301 q^{52} -5.12841 q^{53} -1.19397 q^{55} -3.39428 q^{56} -7.90055 q^{58} +6.38720 q^{59} +9.65918 q^{61} -27.0609 q^{62} +22.9928 q^{64} +9.68064 q^{65} +6.65611 q^{67} +40.0268 q^{68} +2.27400 q^{70} -8.57056 q^{71} -8.65829 q^{73} +18.4350 q^{74} +5.29141 q^{76} +0.206788 q^{77} -2.49612 q^{79} -29.5839 q^{80} -4.49563 q^{82} -5.42636 q^{83} -16.6803 q^{85} +22.8880 q^{86} -4.81233 q^{88} +12.6622 q^{89} -1.67663 q^{91} -25.5737 q^{92} +2.70026 q^{94} -2.20508 q^{95} -8.67832 q^{97} +18.5080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) 23 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 2 * q^7 - 12 * q^8	\( 23 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 2 q^{7} - 12 q^{8} + q^{10} - 13 q^{11} + 5 q^{13} - 11 q^{14} + 33 q^{16} - 22 q^{17} + 23 q^{19} - 23 q^{20} + q^{22} - 23 q^{23} + 31 q^{25} - 21 q^{26} - 22 q^{28} - 24 q^{29} + 20 q^{31} - 19 q^{32} - 14 q^{34} - 15 q^{35} + 8 q^{37} - 5 q^{38} - 15 q^{40} - 28 q^{41} - 6 q^{43} - q^{44} + 29 q^{46} - 23 q^{47} + 39 q^{49} - 27 q^{50} + 12 q^{52} - 22 q^{53} - 28 q^{55} - 30 q^{56} - 5 q^{58} - 53 q^{59} + 4 q^{61} - 38 q^{62} + 30 q^{64} - 27 q^{65} + 6 q^{68} + 30 q^{70} - 54 q^{71} + 15 q^{73} + 12 q^{74} + 25 q^{76} - 33 q^{77} - 8 q^{79} - 45 q^{80} - 69 q^{82} - 20 q^{83} - 25 q^{85} - 50 q^{86} + 48 q^{88} - 26 q^{89} + 11 q^{91} - 81 q^{92} + 5 q^{94} - 12 q^{95} - 4 q^{97} + 55 q^{98}+O(q^{100}) \) 23 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 2 * q^7 - 12 * q^8 + q^10 - 13 * q^11 + 5 * q^13 - 11 * q^14 + 33 * q^16 - 22 * q^17 + 23 * q^19 - 23 * q^20 + q^22 - 23 * q^23 + 31 * q^25 - 21 * q^26 - 22 * q^28 - 24 * q^29 + 20 * q^31 - 19 * q^32 - 14 * q^34 - 15 * q^35 + 8 * q^37 - 5 * q^38 - 15 * q^40 - 28 * q^41 - 6 * q^43 - q^44 + 29 * q^46 - 23 * q^47 + 39 * q^49 - 27 * q^50 + 12 * q^52 - 22 * q^53 - 28 * q^55 - 30 * q^56 - 5 * q^58 - 53 * q^59 + 4 * q^61 - 38 * q^62 + 30 * q^64 - 27 * q^65 + 6 * q^68 + 30 * q^70 - 54 * q^71 + 15 * q^73 + 12 * q^74 + 25 * q^76 - 33 * q^77 - 8 * q^79 - 45 * q^80 - 69 * q^82 - 20 * q^83 - 25 * q^85 - 50 * q^86 + 48 * q^88 - 26 * q^89 + 11 * q^91 - 81 * q^92 + 5 * q^94 - 12 * q^95 - 4 * q^97 + 55 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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