LMFDB - Embedded newform 8020.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.35123 q^{3} +1.00000 q^{5} +0.606242 q^{7} -1.17417 q^{9} +O(q^{10})$	$q-1.35123 q^{3} +1.00000 q^{5} +0.606242 q^{7} -1.17417 q^{9} -0.513496 q^{11} -4.05546 q^{13} -1.35123 q^{15} +1.21613 q^{17} -3.41563 q^{19} -0.819174 q^{21} +5.56762 q^{23} +1.00000 q^{25} +5.64027 q^{27} -0.352723 q^{29} -3.19863 q^{31} +0.693852 q^{33} +0.606242 q^{35} +2.97893 q^{37} +5.47987 q^{39} +9.88938 q^{41} -5.95245 q^{43} -1.17417 q^{45} +6.73581 q^{47} -6.63247 q^{49} -1.64328 q^{51} +13.8930 q^{53} -0.513496 q^{55} +4.61530 q^{57} -9.57824 q^{59} +2.75403 q^{61} -0.711834 q^{63} -4.05546 q^{65} -0.461052 q^{67} -7.52314 q^{69} +16.2366 q^{71} -14.9847 q^{73} -1.35123 q^{75} -0.311303 q^{77} -7.62051 q^{79} -4.09879 q^{81} +13.6545 q^{83} +1.21613 q^{85} +0.476611 q^{87} -13.2210 q^{89} -2.45859 q^{91} +4.32209 q^{93} -3.41563 q^{95} -16.8774 q^{97} +0.602933 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.35123	−0.780134	−0.390067	−	0.920787i	$-0.627548\pi$
$3$	−1.35123	−0.780134	−0.390067	+	0.920787i	$0.627548\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.606242	0.229138	0.114569	−	0.993415i	$-0.463451\pi$
$7$	0.606242	0.229138	0.114569	+	0.993415i	$0.463451\pi$
$8$	0	0
$9$	−1.17417	−0.391391
$10$	0	0
$11$	−0.513496	−0.154825	−0.0774124	−	0.996999i	$-0.524666\pi$
$11$	−0.513496	−0.154825	−0.0774124	+	0.996999i	$0.524666\pi$
$12$	0	0
$13$	−4.05546	−1.12478	−0.562391	−	0.826871i	$-0.690118\pi$
$13$	−4.05546	−1.12478	−0.562391	+	0.826871i	$0.690118\pi$
$14$	0	0
$15$	−1.35123	−0.348886
$16$	0	0
$17$	1.21613	0.294956	0.147478	−	0.989065i	$-0.452884\pi$
$17$	1.21613	0.294956	0.147478	+	0.989065i	$0.452884\pi$
$18$	0	0
$19$	−3.41563	−0.783598	−0.391799	−	0.920051i	$-0.628147\pi$
$19$	−3.41563	−0.783598	−0.391799	+	0.920051i	$0.628147\pi$
$20$	0	0
$21$	−0.819174	−0.178758
$22$	0	0
$23$	5.56762	1.16093	0.580464	−	0.814286i	$-0.302871\pi$
$23$	5.56762	1.16093	0.580464	+	0.814286i	$0.302871\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.64027	1.08547
$28$	0	0
$29$	−0.352723	−0.0654991	−0.0327495	−	0.999464i	$-0.510426\pi$
$29$	−0.352723	−0.0654991	−0.0327495	+	0.999464i	$0.510426\pi$
$30$	0	0
$31$	−3.19863	−0.574491	−0.287245	−	0.957857i	$-0.592740\pi$
$31$	−3.19863	−0.574491	−0.287245	+	0.957857i	$0.592740\pi$
$32$	0	0
$33$	0.693852	0.120784
$34$	0	0
$35$	0.606242	0.102474
$36$	0	0
$37$	2.97893	0.489733	0.244866	−	0.969557i	$-0.421256\pi$
$37$	2.97893	0.489733	0.244866	+	0.969557i	$0.421256\pi$
$38$	0	0
$39$	5.47987	0.877481
$40$	0	0
$41$	9.88938	1.54446	0.772231	−	0.635342i	$-0.219141\pi$
$41$	9.88938	1.54446	0.772231	+	0.635342i	$0.219141\pi$
$42$	0	0
$43$	−5.95245	−0.907741	−0.453870	−	0.891068i	$-0.649957\pi$
$43$	−5.95245	−0.907741	−0.453870	+	0.891068i	$0.649957\pi$
$44$	0	0
$45$	−1.17417	−0.175036
$46$	0	0
$47$	6.73581	0.982519	0.491259	−	0.871013i	$-0.336537\pi$
$47$	6.73581	0.982519	0.491259	+	0.871013i	$0.336537\pi$
$48$	0	0
$49$	−6.63247	−0.947496
$50$	0	0
$51$	−1.64328	−0.230105
$52$	0	0
$53$	13.8930	1.90835	0.954177	−	0.299244i	$-0.0967344\pi$
$53$	13.8930	1.90835	0.954177	+	0.299244i	$0.0967344\pi$
$54$	0	0
$55$	−0.513496	−0.0692398
$56$	0	0
$57$	4.61530	0.611311
$58$	0	0
$59$	−9.57824	−1.24698	−0.623491	−	0.781831i	$-0.714286\pi$
$59$	−9.57824	−1.24698	−0.623491	+	0.781831i	$0.714286\pi$
$60$	0	0
$61$	2.75403	0.352617	0.176309	−	0.984335i	$-0.443584\pi$
$61$	2.75403	0.352617	0.176309	+	0.984335i	$0.443584\pi$
$62$	0	0
$63$	−0.711834	−0.0896827
$64$	0	0
$65$	−4.05546	−0.503018
$66$	0	0
$67$	−0.461052	−0.0563264	−0.0281632	−	0.999603i	$-0.508966\pi$
$67$	−0.461052	−0.0563264	−0.0281632	+	0.999603i	$0.508966\pi$
$68$	0	0
$69$	−7.52314	−0.905680
$70$	0	0
$71$	16.2366	1.92693	0.963463	−	0.267841i	$-0.0863100\pi$
$71$	16.2366	1.92693	0.963463	+	0.267841i	$0.0863100\pi$
$72$	0	0
$73$	−14.9847	−1.75383	−0.876915	−	0.480646i	$-0.840402\pi$
$73$	−14.9847	−1.75383	−0.876915	+	0.480646i	$0.840402\pi$
$74$	0	0
$75$	−1.35123	−0.156027
$76$	0	0
$77$	−0.311303	−0.0354763
$78$	0	0
$79$	−7.62051	−0.857374	−0.428687	−	0.903453i	$-0.641024\pi$
$79$	−7.62051	−0.857374	−0.428687	+	0.903453i	$0.641024\pi$
$80$	0	0
$81$	−4.09879	−0.455421
$82$	0	0
$83$	13.6545	1.49878	0.749389	−	0.662130i	$-0.230348\pi$
$83$	13.6545	1.49878	0.749389	+	0.662130i	$0.230348\pi$
$84$	0	0
$85$	1.21613	0.131908
$86$	0	0
$87$	0.476611	0.0510980
$88$	0	0
$89$	−13.2210	−1.40142	−0.700710	−	0.713446i	$-0.747133\pi$
$89$	−13.2210	−1.40142	−0.700710	+	0.713446i	$0.747133\pi$
$90$	0	0
$91$	−2.45859	−0.257730
$92$	0	0
$93$	4.32209	0.448180
$94$	0	0
$95$	−3.41563	−0.350436
$96$	0	0
$97$	−16.8774	−1.71365	−0.856823	−	0.515611i	$-0.827565\pi$
$97$	−16.8774	−1.71365	−0.856823	+	0.515611i	$0.827565\pi$
$98$	0	0
$99$	0.602933	0.0605971
$100$	0	0
$101$	1.35478	0.134806	0.0674028	−	0.997726i	$-0.478529\pi$
$101$	1.35478	0.134806	0.0674028	+	0.997726i	$0.478529\pi$
$102$	0	0
$103$	5.08356	0.500898	0.250449	−	0.968130i	$-0.419422\pi$
$103$	5.08356	0.500898	0.250449	+	0.968130i	$0.419422\pi$
$104$	0	0
$105$	−0.819174	−0.0799432
$106$	0	0
$107$	−17.2058	−1.66335	−0.831674	−	0.555265i	$-0.812617\pi$
$107$	−17.2058	−1.66335	−0.831674	+	0.555265i	$0.812617\pi$
$108$	0	0
$109$	8.45022	0.809385	0.404692	−	0.914453i	$-0.367379\pi$
$109$	8.45022	0.809385	0.404692	+	0.914453i	$0.367379\pi$
$110$	0	0
$111$	−4.02522	−0.382057
$112$	0	0
$113$	0.192700	0.0181277	0.00906387	−	0.999959i	$-0.497115\pi$
$113$	0.192700	0.0181277	0.00906387	+	0.999959i	$0.497115\pi$
$114$	0	0
$115$	5.56762	0.519183
$116$	0	0
$117$	4.76182	0.440230
$118$	0	0
$119$	0.737271	0.0675856
$120$	0	0
$121$	−10.7363	−0.976029
$122$	0	0
$123$	−13.3628	−1.20489
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.0814	−1.24952	−0.624759	−	0.780818i	$-0.714803\pi$
$127$	−14.0814	−1.24952	−0.624759	+	0.780818i	$0.714803\pi$
$128$	0	0
$129$	8.04314	0.708159
$130$	0	0
$131$	7.05974	0.616813	0.308406	−	0.951255i	$-0.400204\pi$
$131$	7.05974	0.616813	0.308406	+	0.951255i	$0.400204\pi$
$132$	0	0
$133$	−2.07070	−0.179552
$134$	0	0
$135$	5.64027	0.485438
$136$	0	0
$137$	−18.6083	−1.58981	−0.794907	−	0.606731i	$-0.792481\pi$
$137$	−18.6083	−1.58981	−0.794907	+	0.606731i	$0.792481\pi$
$138$	0	0
$139$	−0.381840	−0.0323872	−0.0161936	−	0.999869i	$-0.505155\pi$
$139$	−0.381840	−0.0323872	−0.0161936	+	0.999869i	$0.505155\pi$
$140$	0	0
$141$	−9.10164	−0.766496
$142$	0	0
$143$	2.08246	0.174144
$144$	0	0
$145$	−0.352723	−0.0292921
$146$	0	0
$147$	8.96200	0.739173
$148$	0	0
$149$	6.95647	0.569896	0.284948	−	0.958543i	$-0.408024\pi$
$149$	6.95647	0.569896	0.284948	+	0.958543i	$0.408024\pi$
$150$	0	0
$151$	1.74046	0.141636	0.0708181	−	0.997489i	$-0.477439\pi$
$151$	1.74046	0.141636	0.0708181	+	0.997489i	$0.477439\pi$
$152$	0	0
$153$	−1.42795	−0.115443
$154$	0	0
$155$	−3.19863	−0.256920
$156$	0	0
$157$	−10.6679	−0.851393	−0.425697	−	0.904866i	$-0.639971\pi$
$157$	−10.6679	−0.851393	−0.425697	+	0.904866i	$0.639971\pi$
$158$	0	0
$159$	−18.7727	−1.48877
$160$	0	0
$161$	3.37533	0.266013
$162$	0	0
$163$	−11.0911	−0.868722	−0.434361	−	0.900739i	$-0.643026\pi$
$163$	−11.0911	−0.868722	−0.434361	+	0.900739i	$0.643026\pi$
$164$	0	0
$165$	0.693852	0.0540163
$166$	0	0
$167$	3.08839	0.238987	0.119494	−	0.992835i	$-0.461873\pi$
$167$	3.08839	0.238987	0.119494	+	0.992835i	$0.461873\pi$
$168$	0	0
$169$	3.44676	0.265136
$170$	0	0
$171$	4.01054	0.306694
$172$	0	0
$173$	9.29434	0.706636	0.353318	−	0.935503i	$-0.385053\pi$
$173$	9.29434	0.706636	0.353318	+	0.935503i	$0.385053\pi$
$174$	0	0
$175$	0.606242	0.0458276
$176$	0	0
$177$	12.9424	0.972812
$178$	0	0
$179$	−3.66882	−0.274220	−0.137110	−	0.990556i	$-0.543781\pi$
$179$	−3.66882	−0.274220	−0.137110	+	0.990556i	$0.543781\pi$
$180$	0	0
$181$	7.87094	0.585042	0.292521	−	0.956259i	$-0.405506\pi$
$181$	7.87094	0.585042	0.292521	+	0.956259i	$0.405506\pi$
$182$	0	0
$183$	−3.72133	−0.275089
$184$	0	0
$185$	2.97893	0.219015
$186$	0	0
$187$	−0.624479	−0.0456665
$188$	0	0
$189$	3.41937	0.248723
$190$	0	0
$191$	−18.8532	−1.36417	−0.682086	−	0.731272i	$-0.738927\pi$
$191$	−18.8532	−1.36417	−0.682086	+	0.731272i	$0.738927\pi$
$192$	0	0
$193$	−0.300821	−0.0216536	−0.0108268	−	0.999941i	$-0.503446\pi$
$193$	−0.300821	−0.0216536	−0.0108268	+	0.999941i	$0.503446\pi$
$194$	0	0
$195$	5.47987	0.392421
$196$	0	0
$197$	−9.83775	−0.700911	−0.350455	−	0.936579i	$-0.613973\pi$
$197$	−9.83775	−0.700911	−0.350455	+	0.936579i	$0.613973\pi$
$198$	0	0
$199$	6.48116	0.459437	0.229719	−	0.973257i	$-0.426219\pi$
$199$	6.48116	0.459437	0.229719	+	0.973257i	$0.426219\pi$
$200$	0	0
$201$	0.622987	0.0439421
$202$	0	0
$203$	−0.213836	−0.0150083
$204$	0	0
$205$	9.88938	0.690704
$206$	0	0
$207$	−6.53735	−0.454378
$208$	0	0
$209$	1.75391	0.121320
$210$	0	0
$211$	−1.81870	−0.125205	−0.0626023	−	0.998039i	$-0.519940\pi$
$211$	−1.81870	−0.125205	−0.0626023	+	0.998039i	$0.519940\pi$
$212$	0	0
$213$	−21.9394	−1.50326
$214$	0	0
$215$	−5.95245	−0.405954
$216$	0	0
$217$	−1.93914	−0.131638
$218$	0	0
$219$	20.2478	1.36822
$220$	0	0
$221$	−4.93198	−0.331761
$222$	0	0
$223$	−18.4401	−1.23484	−0.617421	−	0.786633i	$-0.711823\pi$
$223$	−18.4401	−1.23484	−0.617421	+	0.786633i	$0.711823\pi$
$224$	0	0
$225$	−1.17417	−0.0782783
$226$	0	0
$227$	5.40400	0.358676	0.179338	−	0.983788i	$-0.442604\pi$
$227$	5.40400	0.358676	0.179338	+	0.983788i	$0.442604\pi$
$228$	0	0
$229$	3.60956	0.238526	0.119263	−	0.992863i	$-0.461947\pi$
$229$	3.60956	0.238526	0.119263	+	0.992863i	$0.461947\pi$
$230$	0	0
$231$	0.420642	0.0276762
$232$	0	0
$233$	−6.57605	−0.430811	−0.215406	−	0.976525i	$-0.569107\pi$
$233$	−6.57605	−0.430811	−0.215406	+	0.976525i	$0.569107\pi$
$234$	0	0
$235$	6.73581	0.439396
$236$	0	0
$237$	10.2971	0.668867
$238$	0	0
$239$	−13.0971	−0.847181	−0.423591	−	0.905854i	$-0.639231\pi$
$239$	−13.0971	−0.847181	−0.423591	+	0.905854i	$0.639231\pi$
$240$	0	0
$241$	10.2302	0.658985	0.329493	−	0.944158i	$-0.393122\pi$
$241$	10.2302	0.658985	0.329493	+	0.944158i	$0.393122\pi$
$242$	0	0
$243$	−11.3824	−0.730182
$244$	0	0
$245$	−6.63247	−0.423733
$246$	0	0
$247$	13.8519	0.881378
$248$	0	0
$249$	−18.4504	−1.16925
$250$	0	0
$251$	21.6762	1.36819	0.684096	−	0.729392i	$-0.260197\pi$
$251$	21.6762	1.36819	0.684096	+	0.729392i	$0.260197\pi$
$252$	0	0
$253$	−2.85895	−0.179741
$254$	0	0
$255$	−1.64328	−0.102906
$256$	0	0
$257$	−27.8062	−1.73450	−0.867252	−	0.497870i	$-0.834116\pi$
$257$	−27.8062	−1.73450	−0.867252	+	0.497870i	$0.834116\pi$
$258$	0	0
$259$	1.80595	0.112216
$260$	0	0
$261$	0.414158	0.0256358
$262$	0	0
$263$	19.6195	1.20979	0.604896	−	0.796304i	$-0.293215\pi$
$263$	19.6195	1.20979	0.604896	+	0.796304i	$0.293215\pi$
$264$	0	0
$265$	13.8930	0.853442
$266$	0	0
$267$	17.8646	1.09330
$268$	0	0
$269$	1.05968	0.0646100	0.0323050	−	0.999478i	$-0.489715\pi$
$269$	1.05968	0.0646100	0.0323050	+	0.999478i	$0.489715\pi$
$270$	0	0
$271$	−1.87140	−0.113679	−0.0568397	−	0.998383i	$-0.518102\pi$
$271$	−1.87140	−0.113679	−0.0568397	+	0.998383i	$0.518102\pi$
$272$	0	0
$273$	3.32213	0.201064
$274$	0	0
$275$	−0.513496	−0.0309650
$276$	0	0
$277$	−3.40614	−0.204655	−0.102327	−	0.994751i	$-0.532629\pi$
$277$	−3.40614	−0.204655	−0.102327	+	0.994751i	$0.532629\pi$
$278$	0	0
$279$	3.75575	0.224851
$280$	0	0
$281$	−4.23600	−0.252699	−0.126349	−	0.991986i	$-0.540326\pi$
$281$	−4.23600	−0.252699	−0.126349	+	0.991986i	$0.540326\pi$
$282$	0	0
$283$	−3.68934	−0.219309	−0.109654	−	0.993970i	$-0.534974\pi$
$283$	−3.68934	−0.219309	−0.109654	+	0.993970i	$0.534974\pi$
$284$	0	0
$285$	4.61530	0.273387
$286$	0	0
$287$	5.99536	0.353895
$288$	0	0
$289$	−15.5210	−0.913001
$290$	0	0
$291$	22.8053	1.33687
$292$	0	0
$293$	−7.66371	−0.447719	−0.223859	−	0.974621i	$-0.571866\pi$
$293$	−7.66371	−0.447719	−0.223859	+	0.974621i	$0.571866\pi$
$294$	0	0
$295$	−9.57824	−0.557667
$296$	0	0
$297$	−2.89626	−0.168058
$298$	0	0
$299$	−22.5793	−1.30579
$300$	0	0
$301$	−3.60863	−0.207998
$302$	0	0
$303$	−1.83062	−0.105166
$304$	0	0
$305$	2.75403	0.157695
$306$	0	0
$307$	21.2147	1.21079	0.605395	−	0.795926i	$-0.293015\pi$
$307$	21.2147	1.21079	0.605395	+	0.795926i	$0.293015\pi$
$308$	0	0
$309$	−6.86906	−0.390767
$310$	0	0
$311$	13.8089	0.783028	0.391514	−	0.920172i	$-0.371951\pi$
$311$	13.8089	0.783028	0.391514	+	0.920172i	$0.371951\pi$
$312$	0	0
$313$	−9.79116	−0.553429	−0.276715	−	0.960952i	$-0.589246\pi$
$313$	−9.79116	−0.553429	−0.276715	+	0.960952i	$0.589246\pi$
$314$	0	0
$315$	−0.711834	−0.0401073
$316$	0	0
$317$	−8.81819	−0.495279	−0.247639	−	0.968852i	$-0.579655\pi$
$317$	−8.81819	−0.495279	−0.247639	+	0.968852i	$0.579655\pi$
$318$	0	0
$319$	0.181122	0.0101409
$320$	0	0
$321$	23.2490	1.29763
$322$	0	0
$323$	−4.15386	−0.231127
$324$	0	0
$325$	−4.05546	−0.224956
$326$	0	0
$327$	−11.4182	−0.631428
$328$	0	0
$329$	4.08353	0.225132
$330$	0	0
$331$	14.9908	0.823972	0.411986	−	0.911190i	$-0.364835\pi$
$331$	14.9908	0.823972	0.411986	+	0.911190i	$0.364835\pi$
$332$	0	0
$333$	−3.49778	−0.191677
$334$	0	0
$335$	−0.461052	−0.0251899
$336$	0	0
$337$	−18.6680	−1.01691	−0.508455	−	0.861089i	$-0.669783\pi$
$337$	−18.6680	−1.01691	−0.508455	+	0.861089i	$0.669783\pi$
$338$	0	0
$339$	−0.260383	−0.0141421
$340$	0	0
$341$	1.64248	0.0889454
$342$	0	0
$343$	−8.26458	−0.446245
$344$	0	0
$345$	−7.52314	−0.405032
$346$	0	0
$347$	15.1547	0.813547	0.406773	−	0.913529i	$-0.366654\pi$
$347$	15.1547	0.813547	0.406773	+	0.913529i	$0.366654\pi$
$348$	0	0
$349$	0.805271	0.0431052	0.0215526	−	0.999768i	$-0.493139\pi$
$349$	0.805271	0.0431052	0.0215526	+	0.999768i	$0.493139\pi$
$350$	0	0
$351$	−22.8739	−1.22092
$352$	0	0
$353$	11.9920	0.638268	0.319134	−	0.947710i	$-0.396608\pi$
$353$	11.9920	0.638268	0.319134	+	0.947710i	$0.396608\pi$
$354$	0	0
$355$	16.2366	0.861748
$356$	0	0
$357$	−0.996224	−0.0527258
$358$	0	0
$359$	2.30187	0.121488	0.0607441	−	0.998153i	$-0.480653\pi$
$359$	2.30187	0.121488	0.0607441	+	0.998153i	$0.480653\pi$
$360$	0	0
$361$	−7.33350	−0.385974
$362$	0	0
$363$	14.5073	0.761433
$364$	0	0
$365$	−14.9847	−0.784336
$366$	0	0
$367$	19.8192	1.03455	0.517277	−	0.855818i	$-0.326946\pi$
$367$	19.8192	1.03455	0.517277	+	0.855818i	$0.326946\pi$
$368$	0	0
$369$	−11.6118	−0.604489
$370$	0	0
$371$	8.42254	0.437276
$372$	0	0
$373$	−16.1130	−0.834297	−0.417149	−	0.908838i	$-0.636971\pi$
$373$	−16.1130	−0.834297	−0.417149	+	0.908838i	$0.636971\pi$
$374$	0	0
$375$	−1.35123	−0.0697773
$376$	0	0
$377$	1.43046	0.0736722
$378$	0	0
$379$	−0.258072	−0.0132563	−0.00662815	−	0.999978i	$-0.502110\pi$
$379$	−0.258072	−0.0132563	−0.00662815	+	0.999978i	$0.502110\pi$
$380$	0	0
$381$	19.0272	0.974791
$382$	0	0
$383$	−26.5346	−1.35586	−0.677928	−	0.735128i	$-0.737122\pi$
$383$	−26.5346	−1.35586	−0.677928	+	0.735128i	$0.737122\pi$
$384$	0	0
$385$	−0.311303	−0.0158655
$386$	0	0
$387$	6.98922	0.355282
$388$	0	0
$389$	−11.0038	−0.557915	−0.278957	−	0.960303i	$-0.589989\pi$
$389$	−11.0038	−0.557915	−0.278957	+	0.960303i	$0.589989\pi$
$390$	0	0
$391$	6.77097	0.342423
$392$	0	0
$393$	−9.53935	−0.481196
$394$	0	0
$395$	−7.62051	−0.383429
$396$	0	0
$397$	1.33962	0.0672335	0.0336168	−	0.999435i	$-0.489297\pi$
$397$	1.33962	0.0672335	0.0336168	+	0.999435i	$0.489297\pi$
$398$	0	0
$399$	2.79799	0.140075
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	12.9719	0.646177
$404$	0	0
$405$	−4.09879	−0.203671
$406$	0	0
$407$	−1.52967	−0.0758228
$408$	0	0
$409$	1.10382	0.0545805	0.0272902	−	0.999628i	$-0.491312\pi$
$409$	1.10382	0.0545805	0.0272902	+	0.999628i	$0.491312\pi$
$410$	0	0
$411$	25.1441	1.24027
$412$	0	0
$413$	−5.80674	−0.285731
$414$	0	0
$415$	13.6545	0.670274
$416$	0	0
$417$	0.515954	0.0252664
$418$	0	0
$419$	−5.64055	−0.275559	−0.137780	−	0.990463i	$-0.543997\pi$
$419$	−5.64055	−0.275559	−0.137780	+	0.990463i	$0.543997\pi$
$420$	0	0
$421$	1.44108	0.0702340	0.0351170	−	0.999383i	$-0.488820\pi$
$421$	1.44108	0.0702340	0.0351170	+	0.999383i	$0.488820\pi$
$422$	0	0
$423$	−7.90901	−0.384549
$424$	0	0
$425$	1.21613	0.0589911
$426$	0	0
$427$	1.66961	0.0807981
$428$	0	0
$429$	−2.81389	−0.135856
$430$	0	0
$431$	−16.3145	−0.785840	−0.392920	−	0.919573i	$-0.628535\pi$
$431$	−16.3145	−0.785840	−0.392920	+	0.919573i	$0.628535\pi$
$432$	0	0
$433$	−33.9287	−1.63051	−0.815255	−	0.579102i	$-0.803403\pi$
$433$	−33.9287	−1.63051	−0.815255	+	0.579102i	$0.803403\pi$
$434$	0	0
$435$	0.476611	0.0228517
$436$	0	0
$437$	−19.0169	−0.909702
$438$	0	0
$439$	2.02962	0.0968684	0.0484342	−	0.998826i	$-0.484577\pi$
$439$	2.02962	0.0968684	0.0484342	+	0.998826i	$0.484577\pi$
$440$	0	0
$441$	7.78767	0.370842
$442$	0	0
$443$	−20.0894	−0.954476	−0.477238	−	0.878774i	$-0.658362\pi$
$443$	−20.0894	−0.954476	−0.477238	+	0.878774i	$0.658362\pi$
$444$	0	0
$445$	−13.2210	−0.626734
$446$	0	0
$447$	−9.39980	−0.444595
$448$	0	0
$449$	−6.82741	−0.322205	−0.161103	−	0.986938i	$-0.551505\pi$
$449$	−6.82741	−0.322205	−0.161103	+	0.986938i	$0.551505\pi$
$450$	0	0
$451$	−5.07815	−0.239121
$452$	0	0
$453$	−2.35176	−0.110495
$454$	0	0
$455$	−2.45859	−0.115261
$456$	0	0
$457$	−28.7368	−1.34425	−0.672125	−	0.740437i	$-0.734618\pi$
$457$	−28.7368	−1.34425	−0.672125	+	0.740437i	$0.734618\pi$
$458$	0	0
$459$	6.85933	0.320166
$460$	0	0
$461$	−7.94545	−0.370057	−0.185028	−	0.982733i	$-0.559238\pi$
$461$	−7.94545	−0.370057	−0.185028	+	0.982733i	$0.559238\pi$
$462$	0	0
$463$	36.4474	1.69385	0.846927	−	0.531709i	$-0.178450\pi$
$463$	36.4474	1.69385	0.846927	+	0.531709i	$0.178450\pi$
$464$	0	0
$465$	4.32209	0.200432
$466$	0	0
$467$	−6.50055	−0.300810	−0.150405	−	0.988625i	$-0.548058\pi$
$467$	−6.50055	−0.300810	−0.150405	+	0.988625i	$0.548058\pi$
$468$	0	0
$469$	−0.279509	−0.0129065
$470$	0	0
$471$	14.4148	0.664201
$472$	0	0
$473$	3.05656	0.140541
$474$	0	0
$475$	−3.41563	−0.156720
$476$	0	0
$477$	−16.3128	−0.746913
$478$	0	0
$479$	−21.7454	−0.993571	−0.496786	−	0.867873i	$-0.665486\pi$
$479$	−21.7454	−0.993571	−0.496786	+	0.867873i	$0.665486\pi$
$480$	0	0
$481$	−12.0809	−0.550843
$482$	0	0
$483$	−4.56085	−0.207526
$484$	0	0
$485$	−16.8774	−0.766365
$486$	0	0
$487$	−30.0984	−1.36389	−0.681944	−	0.731404i	$-0.738865\pi$
$487$	−30.0984	−1.36389	−0.681944	+	0.731404i	$0.738865\pi$
$488$	0	0
$489$	14.9866	0.677719
$490$	0	0
$491$	19.5954	0.884326	0.442163	−	0.896935i	$-0.354211\pi$
$491$	19.5954	0.884326	0.442163	+	0.896935i	$0.354211\pi$
$492$	0	0
$493$	−0.428958	−0.0193193
$494$	0	0
$495$	0.602933	0.0270998
$496$	0	0
$497$	9.84330	0.441532
$498$	0	0
$499$	11.5906	0.518865	0.259432	−	0.965761i	$-0.416465\pi$
$499$	11.5906	0.518865	0.259432	+	0.965761i	$0.416465\pi$
$500$	0	0
$501$	−4.17313	−0.186442
$502$	0	0
$503$	10.3163	0.459983	0.229992	−	0.973193i	$-0.426130\pi$
$503$	10.3163	0.459983	0.229992	+	0.973193i	$0.426130\pi$
$504$	0	0
$505$	1.35478	0.0602869
$506$	0	0
$507$	−4.65737	−0.206841
$508$	0	0
$509$	−0.0107739	−0.000477543 0	−0.000238771	−	1.00000i	$-0.500076\pi$
$509$	−0.0107739	−0.000477543 0	−0.000238771		1.00000i	$0.500076\pi$
$510$	0	0
$511$	−9.08438	−0.401869
$512$	0	0
$513$	−19.2651	−0.850574
$514$	0	0
$515$	5.08356	0.224008
$516$	0	0
$517$	−3.45881	−0.152118
$518$	0	0
$519$	−12.5588	−0.551270
$520$	0	0
$521$	−32.7674	−1.43557	−0.717783	−	0.696266i	$-0.754843\pi$
$521$	−32.7674	−1.43557	−0.717783	+	0.696266i	$0.754843\pi$
$522$	0	0
$523$	0.424493	0.0185618	0.00928089	−	0.999957i	$-0.497046\pi$
$523$	0.424493	0.0185618	0.00928089	+	0.999957i	$0.497046\pi$
$524$	0	0
$525$	−0.819174	−0.0357517
$526$	0	0
$527$	−3.88996	−0.169449
$528$	0	0
$529$	7.99839	0.347756
$530$	0	0
$531$	11.2465	0.488058
$532$	0	0
$533$	−40.1060	−1.73718
$534$	0	0
$535$	−17.2058	−0.743872
$536$	0	0
$537$	4.95742	0.213929
$538$	0	0
$539$	3.40575	0.146696
$540$	0	0
$541$	20.3644	0.875535	0.437768	−	0.899088i	$-0.355769\pi$
$541$	20.3644	0.875535	0.437768	+	0.899088i	$0.355769\pi$
$542$	0	0
$543$	−10.6355	−0.456411
$544$	0	0
$545$	8.45022	0.361968
$546$	0	0
$547$	−39.5191	−1.68971	−0.844857	−	0.534993i	$-0.820314\pi$
$547$	−39.5191	−1.68971	−0.844857	+	0.534993i	$0.820314\pi$
$548$	0	0
$549$	−3.23371	−0.138011
$550$	0	0
$551$	1.20477	0.0513250
$552$	0	0
$553$	−4.61988	−0.196457
$554$	0	0
$555$	−4.02522	−0.170861
$556$	0	0
$557$	−38.8261	−1.64512	−0.822558	−	0.568682i	$-0.807454\pi$
$557$	−38.8261	−1.64512	−0.822558	+	0.568682i	$0.807454\pi$
$558$	0	0
$559$	24.1399	1.02101
$560$	0	0
$561$	0.843816	0.0356259
$562$	0	0
$563$	8.85778	0.373311	0.186656	−	0.982425i	$-0.440235\pi$
$563$	8.85778	0.373311	0.186656	+	0.982425i	$0.440235\pi$
$564$	0	0
$565$	0.192700	0.00810697
$566$	0	0
$567$	−2.48486	−0.104354
$568$	0	0
$569$	1.34442	0.0563611	0.0281806	−	0.999603i	$-0.491029\pi$
$569$	1.34442	0.0563611	0.0281806	+	0.999603i	$0.491029\pi$
$570$	0	0
$571$	−28.4542	−1.19077	−0.595385	−	0.803440i	$-0.703001\pi$
$571$	−28.4542	−1.19077	−0.595385	+	0.803440i	$0.703001\pi$
$572$	0	0
$573$	25.4751	1.06424
$574$	0	0
$575$	5.56762	0.232186
$576$	0	0
$577$	−2.69718	−0.112285	−0.0561426	−	0.998423i	$-0.517880\pi$
$577$	−2.69718	−0.112285	−0.0561426	+	0.998423i	$0.517880\pi$
$578$	0	0
$579$	0.406478	0.0168927
$580$	0	0
$581$	8.27794	0.343427
$582$	0	0
$583$	−7.13401	−0.295460
$584$	0	0
$585$	4.76182	0.196877
$586$	0	0
$587$	−32.0125	−1.32130	−0.660649	−	0.750695i	$-0.729719\pi$
$587$	−32.0125	−1.32130	−0.660649	+	0.750695i	$0.729719\pi$
$588$	0	0
$589$	10.9253	0.450170
$590$	0	0
$591$	13.2931	0.546804
$592$	0	0
$593$	14.2380	0.584684	0.292342	−	0.956314i	$-0.405565\pi$
$593$	14.2380	0.584684	0.292342	+	0.956314i	$0.405565\pi$
$594$	0	0
$595$	0.737271	0.0302252
$596$	0	0
$597$	−8.75755	−0.358422
$598$	0	0
$599$	27.3223	1.11636	0.558179	−	0.829720i	$-0.311500\pi$
$599$	27.3223	1.11636	0.558179	+	0.829720i	$0.311500\pi$
$600$	0	0
$601$	−9.94346	−0.405602	−0.202801	−	0.979220i	$-0.565004\pi$
$601$	−9.94346	−0.405602	−0.202801	+	0.979220i	$0.565004\pi$
$602$	0	0
$603$	0.541355	0.0220457
$604$	0	0
$605$	−10.7363	−0.436494
$606$	0	0
$607$	21.1269	0.857515	0.428758	−	0.903420i	$-0.358951\pi$
$607$	21.1269	0.857515	0.428758	+	0.903420i	$0.358951\pi$
$608$	0	0
$609$	0.288942	0.0117085
$610$	0	0
$611$	−27.3168	−1.10512
$612$	0	0
$613$	−7.21101	−0.291250	−0.145625	−	0.989340i	$-0.546519\pi$
$613$	−7.21101	−0.291250	−0.145625	+	0.989340i	$0.546519\pi$
$614$	0	0
$615$	−13.3628	−0.538842
$616$	0	0
$617$	−24.1082	−0.970558	−0.485279	−	0.874359i	$-0.661282\pi$
$617$	−24.1082	−0.970558	−0.485279	+	0.874359i	$0.661282\pi$
$618$	0	0
$619$	−33.2831	−1.33776	−0.668881	−	0.743370i	$-0.733226\pi$
$619$	−33.2831	−1.33776	−0.668881	+	0.743370i	$0.733226\pi$
$620$	0	0
$621$	31.4029	1.26016
$622$	0	0
$623$	−8.01512	−0.321119
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.36994	−0.0946462
$628$	0	0
$629$	3.62277	0.144449
$630$	0	0
$631$	−26.2142	−1.04357	−0.521786	−	0.853076i	$-0.674734\pi$
$631$	−26.2142	−1.04357	−0.521786	+	0.853076i	$0.674734\pi$
$632$	0	0
$633$	2.45749	0.0976764
$634$	0	0
$635$	−14.0814	−0.558802
$636$	0	0
$637$	26.8977	1.06573
$638$	0	0
$639$	−19.0646	−0.754182
$640$	0	0
$641$	1.99830	0.0789281	0.0394641	−	0.999221i	$-0.487435\pi$
$641$	1.99830	0.0789281	0.0394641	+	0.999221i	$0.487435\pi$
$642$	0	0
$643$	−32.0283	−1.26307	−0.631536	−	0.775347i	$-0.717575\pi$
$643$	−32.0283	−1.26307	−0.631536	+	0.775347i	$0.717575\pi$
$644$	0	0
$645$	8.04314	0.316698
$646$	0	0
$647$	20.9388	0.823188	0.411594	−	0.911367i	$-0.364972\pi$
$647$	20.9388	0.823188	0.411594	+	0.911367i	$0.364972\pi$
$648$	0	0
$649$	4.91839	0.193064
$650$	0	0
$651$	2.62023	0.102695
$652$	0	0
$653$	−22.7532	−0.890402	−0.445201	−	0.895431i	$-0.646868\pi$
$653$	−22.7532	−0.890402	−0.445201	+	0.895431i	$0.646868\pi$
$654$	0	0
$655$	7.05974	0.275847
$656$	0	0
$657$	17.5947	0.686434
$658$	0	0
$659$	5.57035	0.216990	0.108495	−	0.994097i	$-0.465397\pi$
$659$	5.57035	0.216990	0.108495	+	0.994097i	$0.465397\pi$
$660$	0	0
$661$	−27.4326	−1.06701	−0.533503	−	0.845798i	$-0.679125\pi$
$661$	−27.4326	−1.06701	−0.533503	+	0.845798i	$0.679125\pi$
$662$	0	0
$663$	6.66425	0.258818
$664$	0	0
$665$	−2.07070	−0.0802982
$666$	0	0
$667$	−1.96383	−0.0760398
$668$	0	0
$669$	24.9169	0.963342
$670$	0	0
$671$	−1.41418	−0.0545939
$672$	0	0
$673$	−19.2708	−0.742834	−0.371417	−	0.928466i	$-0.621128\pi$
$673$	−19.2708	−0.742834	−0.371417	+	0.928466i	$0.621128\pi$
$674$	0	0
$675$	5.64027	0.217094
$676$	0	0
$677$	30.8792	1.18678	0.593392	−	0.804914i	$-0.297789\pi$
$677$	30.8792	1.18678	0.593392	+	0.804914i	$0.297789\pi$
$678$	0	0
$679$	−10.2318	−0.392661
$680$	0	0
$681$	−7.30206	−0.279815
$682$	0	0
$683$	47.2044	1.80623	0.903113	−	0.429402i	$-0.141276\pi$
$683$	47.2044	1.80623	0.903113	+	0.429402i	$0.141276\pi$
$684$	0	0
$685$	−18.6083	−0.710987
$686$	0	0
$687$	−4.87735	−0.186083
$688$	0	0
$689$	−56.3426	−2.14648
$690$	0	0
$691$	−15.5424	−0.591261	−0.295631	−	0.955302i	$-0.595530\pi$
$691$	−15.5424	−0.591261	−0.295631	+	0.955302i	$0.595530\pi$
$692$	0	0
$693$	0.365524	0.0138851
$694$	0	0
$695$	−0.381840	−0.0144840
$696$	0	0
$697$	12.0268	0.455548
$698$	0	0
$699$	8.88576	0.336090
$700$	0	0
$701$	29.7092	1.12210	0.561050	−	0.827782i	$-0.310398\pi$
$701$	29.7092	1.12210	0.561050	+	0.827782i	$0.310398\pi$
$702$	0	0
$703$	−10.1749	−0.383754
$704$	0	0
$705$	−9.10164	−0.342787
$706$	0	0
$707$	0.821325	0.0308891
$708$	0	0
$709$	8.00686	0.300704	0.150352	−	0.988633i	$-0.451959\pi$
$709$	8.00686	0.300704	0.150352	+	0.988633i	$0.451959\pi$
$710$	0	0
$711$	8.94780	0.335569
$712$	0	0
$713$	−17.8087	−0.666943
$714$	0	0
$715$	2.08246	0.0778797
$716$	0	0
$717$	17.6972	0.660915
$718$	0	0
$719$	1.28015	0.0477417	0.0238708	−	0.999715i	$-0.492401\pi$
$719$	1.28015	0.0477417	0.0238708	+	0.999715i	$0.492401\pi$
$720$	0	0
$721$	3.08187	0.114775
$722$	0	0
$723$	−13.8234	−0.514096
$724$	0	0
$725$	−0.352723	−0.0130998
$726$	0	0
$727$	5.19885	0.192815	0.0964073	−	0.995342i	$-0.469265\pi$
$727$	5.19885	0.192815	0.0964073	+	0.995342i	$0.469265\pi$
$728$	0	0
$729$	27.6766	1.02506
$730$	0	0
$731$	−7.23898	−0.267743
$732$	0	0
$733$	4.04235	0.149308	0.0746538	−	0.997210i	$-0.476215\pi$
$733$	4.04235	0.149308	0.0746538	+	0.997210i	$0.476215\pi$
$734$	0	0
$735$	8.96200	0.330568
$736$	0	0
$737$	0.236748	0.00872073
$738$	0	0
$739$	−10.7039	−0.393748	−0.196874	−	0.980429i	$-0.563079\pi$
$739$	−10.7039	−0.393748	−0.196874	+	0.980429i	$0.563079\pi$
$740$	0	0
$741$	−18.7172	−0.687592
$742$	0	0
$743$	−19.8808	−0.729358	−0.364679	−	0.931133i	$-0.618821\pi$
$743$	−19.8808	−0.729358	−0.364679	+	0.931133i	$0.618821\pi$
$744$	0	0
$745$	6.95647	0.254865
$746$	0	0
$747$	−16.0328	−0.586608
$748$	0	0
$749$	−10.4309	−0.381136
$750$	0	0
$751$	20.1013	0.733505	0.366753	−	0.930318i	$-0.380470\pi$
$751$	20.1013	0.733505	0.366753	+	0.930318i	$0.380470\pi$
$752$	0	0
$753$	−29.2896	−1.06737
$754$	0	0
$755$	1.74046	0.0633417
$756$	0	0
$757$	−43.3432	−1.57534	−0.787669	−	0.616099i	$-0.788712\pi$
$757$	−43.3432	−1.57534	−0.787669	+	0.616099i	$0.788712\pi$
$758$	0	0
$759$	3.86310	0.140222
$760$	0	0
$761$	27.5161	0.997459	0.498730	−	0.866758i	$-0.333800\pi$
$761$	27.5161	0.997459	0.498730	+	0.866758i	$0.333800\pi$
$762$	0	0
$763$	5.12288	0.185461
$764$	0	0
$765$	−1.42795	−0.0516277
$766$	0	0
$767$	38.8442	1.40258
$768$	0	0
$769$	0.781969	0.0281985	0.0140993	−	0.999901i	$-0.495512\pi$
$769$	0.781969	0.0281985	0.0140993	+	0.999901i	$0.495512\pi$
$770$	0	0
$771$	37.5726	1.35314
$772$	0	0
$773$	−18.3231	−0.659035	−0.329518	−	0.944149i	$-0.606886\pi$
$773$	−18.3231	−0.659035	−0.329518	+	0.944149i	$0.606886\pi$
$774$	0	0
$775$	−3.19863	−0.114898
$776$	0	0
$777$	−2.44026	−0.0875438
$778$	0	0
$779$	−33.7784	−1.21024
$780$	0	0
$781$	−8.33741	−0.298336
$782$	0	0
$783$	−1.98946	−0.0710974
$784$	0	0
$785$	−10.6679	−0.380755
$786$	0	0
$787$	29.4284	1.04901	0.524504	−	0.851408i	$-0.324251\pi$
$787$	29.4284	1.04901	0.524504	+	0.851408i	$0.324251\pi$
$788$	0	0
$789$	−26.5105	−0.943800
$790$	0	0
$791$	0.116823	0.00415375
$792$	0	0
$793$	−11.1689	−0.396618
$794$	0	0
$795$	−18.7727	−0.665799
$796$	0	0
$797$	31.5547	1.11772	0.558862	−	0.829261i	$-0.311238\pi$
$797$	31.5547	1.11772	0.558862	+	0.829261i	$0.311238\pi$
$798$	0	0
$799$	8.19164	0.289799
$800$	0	0
$801$	15.5237	0.548504
$802$	0	0
$803$	7.69459	0.271536
$804$	0	0
$805$	3.37533	0.118965
$806$	0	0
$807$	−1.43188	−0.0504045
$808$	0	0
$809$	−40.6008	−1.42745	−0.713725	−	0.700426i	$-0.752993\pi$
$809$	−40.6008	−1.42745	−0.713725	+	0.700426i	$0.752993\pi$
$810$	0	0
$811$	30.5995	1.07449	0.537247	−	0.843425i	$-0.319464\pi$
$811$	30.5995	1.07449	0.537247	+	0.843425i	$0.319464\pi$
$812$	0	0
$813$	2.52869	0.0886851
$814$	0	0
$815$	−11.0911	−0.388504
$816$	0	0
$817$	20.3314	0.711304
$818$	0	0
$819$	2.88681	0.100873
$820$	0	0
$821$	−44.7574	−1.56204	−0.781022	−	0.624503i	$-0.785302\pi$
$821$	−44.7574	−1.56204	−0.781022	+	0.624503i	$0.785302\pi$
$822$	0	0
$823$	35.8903	1.25106	0.625528	−	0.780202i	$-0.284884\pi$
$823$	35.8903	1.25106	0.625528	+	0.780202i	$0.284884\pi$
$824$	0	0
$825$	0.693852	0.0241568
$826$	0	0
$827$	8.49205	0.295297	0.147649	−	0.989040i	$-0.452829\pi$
$827$	8.49205	0.295297	0.147649	+	0.989040i	$0.452829\pi$
$828$	0	0
$829$	5.67364	0.197054	0.0985268	−	0.995134i	$-0.468587\pi$
$829$	5.67364	0.197054	0.0985268	+	0.995134i	$0.468587\pi$
$830$	0	0
$831$	4.60248	0.159658
$832$	0	0
$833$	−8.06597	−0.279469
$834$	0	0
$835$	3.08839	0.106878
$836$	0	0
$837$	−18.0411	−0.623593
$838$	0	0
$839$	15.5868	0.538115	0.269057	−	0.963124i	$-0.413288\pi$
$839$	15.5868	0.538115	0.269057	+	0.963124i	$0.413288\pi$
$840$	0	0
$841$	−28.8756	−0.995710
$842$	0	0
$843$	5.72382	0.197139
$844$	0	0
$845$	3.44676	0.118572
$846$	0	0
$847$	−6.50881	−0.223645
$848$	0	0
$849$	4.98516	0.171090
$850$	0	0
$851$	16.5855	0.568545
$852$	0	0
$853$	−6.86186	−0.234946	−0.117473	−	0.993076i	$-0.537479\pi$
$853$	−6.86186	−0.234946	−0.117473	+	0.993076i	$0.537479\pi$
$854$	0	0
$855$	4.01054	0.137158
$856$	0	0
$857$	−24.6735	−0.842830	−0.421415	−	0.906868i	$-0.638466\pi$
$857$	−24.6735	−0.842830	−0.421415	+	0.906868i	$0.638466\pi$
$858$	0	0
$859$	19.0722	0.650735	0.325368	−	0.945588i	$-0.394512\pi$
$859$	19.0722	0.650735	0.325368	+	0.945588i	$0.394512\pi$
$860$	0	0
$861$	−8.10112	−0.276085
$862$	0	0
$863$	35.8497	1.22034	0.610169	−	0.792271i	$-0.291102\pi$
$863$	35.8497	1.22034	0.610169	+	0.792271i	$0.291102\pi$
$864$	0	0
$865$	9.29434	0.316017
$866$	0	0
$867$	20.9725	0.712263
$868$	0	0
$869$	3.91310	0.132743
$870$	0	0
$871$	1.86978	0.0633550
$872$	0	0
$873$	19.8171	0.670706
$874$	0	0
$875$	0.606242	0.0204947
$876$	0	0
$877$	−21.3777	−0.721874	−0.360937	−	0.932590i	$-0.617543\pi$
$877$	−21.3777	−0.721874	−0.360937	+	0.932590i	$0.617543\pi$
$878$	0	0
$879$	10.3554	0.349280
$880$	0	0
$881$	40.9093	1.37827	0.689134	−	0.724634i	$-0.257991\pi$
$881$	40.9093	1.37827	0.689134	+	0.724634i	$0.257991\pi$
$882$	0	0
$883$	−13.5093	−0.454624	−0.227312	−	0.973822i	$-0.572994\pi$
$883$	−13.5093	−0.454624	−0.227312	+	0.973822i	$0.572994\pi$
$884$	0	0
$885$	12.9424	0.435055
$886$	0	0
$887$	−7.14262	−0.239826	−0.119913	−	0.992784i	$-0.538262\pi$
$887$	−7.14262	−0.239826	−0.119913	+	0.992784i	$0.538262\pi$
$888$	0	0
$889$	−8.53671	−0.286312
$890$	0	0
$891$	2.10471	0.0705105
$892$	0	0
$893$	−23.0070	−0.769900
$894$	0	0
$895$	−3.66882	−0.122635
$896$	0	0
$897$	30.5098	1.01869
$898$	0	0
$899$	1.12823	0.0376286
$900$	0	0
$901$	16.8958	0.562880
$902$	0	0
$903$	4.87609	0.162266
$904$	0	0
$905$	7.87094	0.261639
$906$	0	0
$907$	−0.156907	−0.00521001	−0.00260501	−	0.999997i	$-0.500829\pi$
$907$	−0.156907	−0.00521001	−0.00260501	+	0.999997i	$0.500829\pi$
$908$	0	0
$909$	−1.59075	−0.0527618
$910$	0	0
$911$	41.2640	1.36714	0.683568	−	0.729886i	$-0.260427\pi$
$911$	41.2640	1.36714	0.683568	+	0.729886i	$0.260427\pi$
$912$	0	0
$913$	−7.01153	−0.232048
$914$	0	0
$915$	−3.72133	−0.123023
$916$	0	0
$917$	4.27992	0.141335
$918$	0	0
$919$	37.3237	1.23120	0.615598	−	0.788060i	$-0.288915\pi$
$919$	37.3237	1.23120	0.615598	+	0.788060i	$0.288915\pi$
$920$	0	0
$921$	−28.6660	−0.944577
$922$	0	0
$923$	−65.8468	−2.16737
$924$	0	0
$925$	2.97893	0.0979465
$926$	0	0
$927$	−5.96898	−0.196047
$928$	0	0
$929$	−45.3747	−1.48870	−0.744348	−	0.667792i	$-0.767240\pi$
$929$	−45.3747	−1.48870	−0.744348	+	0.667792i	$0.767240\pi$
$930$	0	0
$931$	22.6540	0.742456
$932$	0	0
$933$	−18.6590	−0.610867
$934$	0	0
$935$	−0.624479	−0.0204227
$936$	0	0
$937$	−10.2456	−0.334708	−0.167354	−	0.985897i	$-0.553522\pi$
$937$	−10.2456	−0.334708	−0.167354	+	0.985897i	$0.553522\pi$
$938$	0	0
$939$	13.2301	0.431749
$940$	0	0
$941$	−10.0036	−0.326107	−0.163054	−	0.986617i	$-0.552134\pi$
$941$	−10.0036	−0.326107	−0.163054	+	0.986617i	$0.552134\pi$
$942$	0	0
$943$	55.0603	1.79301
$944$	0	0
$945$	3.41937	0.111232
$946$	0	0
$947$	11.9561	0.388521	0.194261	−	0.980950i	$-0.437769\pi$
$947$	11.9561	0.388521	0.194261	+	0.980950i	$0.437769\pi$
$948$	0	0
$949$	60.7700	1.97268
$950$	0	0
$951$	11.9154	0.386384
$952$	0	0
$953$	−52.4471	−1.69893	−0.849464	−	0.527646i	$-0.823075\pi$
$953$	−52.4471	−1.69893	−0.849464	+	0.527646i	$0.823075\pi$
$954$	0	0
$955$	−18.8532	−0.610077
$956$	0	0
$957$	−0.244738	−0.00791124
$958$	0	0
$959$	−11.2811	−0.364287
$960$	0	0
$961$	−20.7688	−0.669960
$962$	0	0
$963$	20.2026	0.651020
$964$	0	0
$965$	−0.300821	−0.00968376
$966$	0	0
$967$	−13.0265	−0.418903	−0.209451	−	0.977819i	$-0.567168\pi$
$967$	−13.0265	−0.418903	−0.209451	+	0.977819i	$0.567168\pi$
$968$	0	0
$969$	5.61282	0.180310
$970$	0	0
$971$	−10.2746	−0.329728	−0.164864	−	0.986316i	$-0.552719\pi$
$971$	−10.2746	−0.329728	−0.164864	+	0.986316i	$0.552719\pi$
$972$	0	0
$973$	−0.231487	−0.00742114
$974$	0	0
$975$	5.47987	0.175496
$976$	0	0
$977$	−13.1490	−0.420675	−0.210337	−	0.977629i	$-0.567456\pi$
$977$	−13.1490	−0.420675	−0.210337	+	0.977629i	$0.567456\pi$
$978$	0	0
$979$	6.78892	0.216975
$980$	0	0
$981$	−9.92203	−0.316786
$982$	0	0
$983$	−21.2903	−0.679054	−0.339527	−	0.940596i	$-0.610267\pi$
$983$	−21.2903	−0.679054	−0.339527	+	0.940596i	$0.610267\pi$
$984$	0	0
$985$	−9.83775	−0.313457
$986$	0	0
$987$	−5.51780	−0.175633
$988$	0	0
$989$	−33.1410	−1.05382
$990$	0	0
$991$	6.78392	0.215498	0.107749	−	0.994178i	$-0.465636\pi$
$991$	6.78392	0.215498	0.107749	+	0.994178i	$0.465636\pi$
$992$	0	0
$993$	−20.2561	−0.642808
$994$	0	0
$995$	6.48116	0.205467
$996$	0	0
$997$	−15.4226	−0.488439	−0.244220	−	0.969720i	$-0.578532\pi$
$997$	−15.4226	−0.488439	−0.244220	+	0.969720i	$0.578532\pi$
$998$	0	0
$999$	16.8020	0.531591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.9	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.9	✓	29	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.35123 q^{3} +1.00000 q^{5} +0.606242 q^{7} -1.17417 q^{9} +O(q^{10})\)	\(q-1.35123 q^{3} +1.00000 q^{5} +0.606242 q^{7} -1.17417 q^{9} -0.513496 q^{11} -4.05546 q^{13} -1.35123 q^{15} +1.21613 q^{17} -3.41563 q^{19} -0.819174 q^{21} +5.56762 q^{23} +1.00000 q^{25} +5.64027 q^{27} -0.352723 q^{29} -3.19863 q^{31} +0.693852 q^{33} +0.606242 q^{35} +2.97893 q^{37} +5.47987 q^{39} +9.88938 q^{41} -5.95245 q^{43} -1.17417 q^{45} +6.73581 q^{47} -6.63247 q^{49} -1.64328 q^{51} +13.8930 q^{53} -0.513496 q^{55} +4.61530 q^{57} -9.57824 q^{59} +2.75403 q^{61} -0.711834 q^{63} -4.05546 q^{65} -0.461052 q^{67} -7.52314 q^{69} +16.2366 q^{71} -14.9847 q^{73} -1.35123 q^{75} -0.311303 q^{77} -7.62051 q^{79} -4.09879 q^{81} +13.6545 q^{83} +1.21613 q^{85} +0.476611 q^{87} -13.2210 q^{89} -2.45859 q^{91} +4.32209 q^{93} -3.41563 q^{95} -16.8774 q^{97} +0.602933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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