LMFDB - Embedded newform 8020.2.a.c.1.14

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:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+0.0547853 q^{3} -1.00000 q^{5} +3.87235 q^{7} -2.99700 q^{9} +O(q^{10})$	$q+0.0547853 q^{3} -1.00000 q^{5} +3.87235 q^{7} -2.99700 q^{9} +3.25314 q^{11} -6.91309 q^{13} -0.0547853 q^{15} +0.379841 q^{17} +4.37288 q^{19} +0.212148 q^{21} +1.01656 q^{23} +1.00000 q^{25} -0.328547 q^{27} -3.37707 q^{29} -6.22792 q^{31} +0.178224 q^{33} -3.87235 q^{35} -0.791545 q^{37} -0.378736 q^{39} +2.36327 q^{41} +0.241812 q^{43} +2.99700 q^{45} -6.57722 q^{47} +7.99512 q^{49} +0.0208097 q^{51} +9.52720 q^{53} -3.25314 q^{55} +0.239569 q^{57} +1.22289 q^{59} -9.03747 q^{61} -11.6054 q^{63} +6.91309 q^{65} -12.4260 q^{67} +0.0556924 q^{69} -11.8150 q^{71} -1.03700 q^{73} +0.0547853 q^{75} +12.5973 q^{77} +1.42411 q^{79} +8.97300 q^{81} +10.0097 q^{83} -0.379841 q^{85} -0.185014 q^{87} -0.256993 q^{89} -26.7699 q^{91} -0.341199 q^{93} -4.37288 q^{95} +4.05385 q^{97} -9.74965 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	0.0547853	0.0316303	0.0158152	−	0.999875i	$-0.494966\pi$
$3$	0.0547853	0.0316303	0.0158152	+	0.999875i	$0.494966\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.87235	1.46361	0.731806	−	0.681513i	$-0.238678\pi$
$7$	3.87235	1.46361	0.731806	+	0.681513i	$0.238678\pi$
$8$	0	0
$9$	−2.99700	−0.999000
$10$	0	0
$11$	3.25314	0.980858	0.490429	−	0.871481i	$-0.336840\pi$
$11$	3.25314	0.980858	0.490429	+	0.871481i	$0.336840\pi$
$12$	0	0
$13$	−6.91309	−1.91735	−0.958674	−	0.284508i	$-0.908170\pi$
$13$	−6.91309	−1.91735	−0.958674	+	0.284508i	$0.908170\pi$
$14$	0	0
$15$	−0.0547853	−0.0141455
$16$	0	0
$17$	0.379841	0.0921251	0.0460625	−	0.998939i	$-0.485333\pi$
$17$	0.379841	0.0921251	0.0460625	+	0.998939i	$0.485333\pi$
$18$	0	0
$19$	4.37288	1.00321	0.501603	−	0.865098i	$-0.332744\pi$
$19$	4.37288	1.00321	0.501603	+	0.865098i	$0.332744\pi$
$20$	0	0
$21$	0.212148	0.0462945
$22$	0	0
$23$	1.01656	0.211967	0.105983	−	0.994368i	$-0.466201\pi$
$23$	1.01656	0.211967	0.105983	+	0.994368i	$0.466201\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.328547	−0.0632290
$28$	0	0
$29$	−3.37707	−0.627106	−0.313553	−	0.949571i	$-0.601519\pi$
$29$	−3.37707	−0.627106	−0.313553	+	0.949571i	$0.601519\pi$
$30$	0	0
$31$	−6.22792	−1.11857	−0.559284	−	0.828976i	$-0.688924\pi$
$31$	−6.22792	−1.11857	−0.559284	+	0.828976i	$0.688924\pi$
$32$	0	0
$33$	0.178224	0.0310248
$34$	0	0
$35$	−3.87235	−0.654547
$36$	0	0
$37$	−0.791545	−0.130129	−0.0650646	−	0.997881i	$-0.520725\pi$
$37$	−0.791545	−0.130129	−0.0650646	+	0.997881i	$0.520725\pi$
$38$	0	0
$39$	−0.378736	−0.0606463
$40$	0	0
$41$	2.36327	0.369080	0.184540	−	0.982825i	$-0.440920\pi$
$41$	2.36327	0.369080	0.184540	+	0.982825i	$0.440920\pi$
$42$	0	0
$43$	0.241812	0.0368760	0.0184380	−	0.999830i	$-0.494131\pi$
$43$	0.241812	0.0368760	0.0184380	+	0.999830i	$0.494131\pi$
$44$	0	0
$45$	2.99700	0.446766
$46$	0	0
$47$	−6.57722	−0.959386	−0.479693	−	0.877436i	$-0.659252\pi$
$47$	−6.57722	−0.959386	−0.479693	+	0.877436i	$0.659252\pi$
$48$	0	0
$49$	7.99512	1.14216
$50$	0	0
$51$	0.0208097	0.00291394
$52$	0	0
$53$	9.52720	1.30866	0.654331	−	0.756208i	$-0.272950\pi$
$53$	9.52720	1.30866	0.654331	+	0.756208i	$0.272950\pi$
$54$	0	0
$55$	−3.25314	−0.438653
$56$	0	0
$57$	0.239569	0.0317317
$58$	0	0
$59$	1.22289	0.159207	0.0796034	−	0.996827i	$-0.474635\pi$
$59$	1.22289	0.159207	0.0796034	+	0.996827i	$0.474635\pi$
$60$	0	0
$61$	−9.03747	−1.15713	−0.578565	−	0.815636i	$-0.696387\pi$
$61$	−9.03747	−1.15713	−0.578565	+	0.815636i	$0.696387\pi$
$62$	0	0
$63$	−11.6054	−1.46215
$64$	0	0
$65$	6.91309	0.857464
$66$	0	0
$67$	−12.4260	−1.51807	−0.759036	−	0.651048i	$-0.774330\pi$
$67$	−12.4260	−1.51807	−0.759036	+	0.651048i	$0.774330\pi$
$68$	0	0
$69$	0.0556924	0.00670458
$70$	0	0
$71$	−11.8150	−1.40219	−0.701093	−	0.713070i	$-0.747304\pi$
$71$	−11.8150	−1.40219	−0.701093	+	0.713070i	$0.747304\pi$
$72$	0	0
$73$	−1.03700	−0.121372	−0.0606861	−	0.998157i	$-0.519329\pi$
$73$	−1.03700	−0.121372	−0.0606861	+	0.998157i	$0.519329\pi$
$74$	0	0
$75$	0.0547853	0.00632606
$76$	0	0
$77$	12.5973	1.43560
$78$	0	0
$79$	1.42411	0.160225	0.0801124	−	0.996786i	$-0.474472\pi$
$79$	1.42411	0.160225	0.0801124	+	0.996786i	$0.474472\pi$
$80$	0	0
$81$	8.97300	0.997000
$82$	0	0
$83$	10.0097	1.09871	0.549356	−	0.835588i	$-0.314873\pi$
$83$	10.0097	1.09871	0.549356	+	0.835588i	$0.314873\pi$
$84$	0	0
$85$	−0.379841	−0.0411996
$86$	0	0
$87$	−0.185014	−0.0198356
$88$	0	0
$89$	−0.256993	−0.0272412	−0.0136206	−	0.999907i	$-0.504336\pi$
$89$	−0.256993	−0.0272412	−0.0136206	+	0.999907i	$0.504336\pi$
$90$	0	0
$91$	−26.7699	−2.80625
$92$	0	0
$93$	−0.341199	−0.0353807
$94$	0	0
$95$	−4.37288	−0.448648
$96$	0	0
$97$	4.05385	0.411606	0.205803	−	0.978593i	$-0.434019\pi$
$97$	4.05385	0.411606	0.205803	+	0.978593i	$0.434019\pi$
$98$	0	0
$99$	−9.74965	−0.979876
$100$	0	0
$101$	−1.29284	−0.128642	−0.0643211	−	0.997929i	$-0.520488\pi$
$101$	−1.29284	−0.128642	−0.0643211	+	0.997929i	$0.520488\pi$
$102$	0	0
$103$	4.69720	0.462828	0.231414	−	0.972855i	$-0.425665\pi$
$103$	4.69720	0.462828	0.231414	+	0.972855i	$0.425665\pi$
$104$	0	0
$105$	−0.212148	−0.0207035
$106$	0	0
$107$	2.90373	0.280714	0.140357	−	0.990101i	$-0.455175\pi$
$107$	2.90373	0.280714	0.140357	+	0.990101i	$0.455175\pi$
$108$	0	0
$109$	−3.52516	−0.337649	−0.168824	−	0.985646i	$-0.553997\pi$
$109$	−3.52516	−0.337649	−0.168824	+	0.985646i	$0.553997\pi$
$110$	0	0
$111$	−0.0433650	−0.00411603
$112$	0	0
$113$	−4.53383	−0.426507	−0.213253	−	0.976997i	$-0.568406\pi$
$113$	−4.53383	−0.426507	−0.213253	+	0.976997i	$0.568406\pi$
$114$	0	0
$115$	−1.01656	−0.0947945
$116$	0	0
$117$	20.7185	1.91543
$118$	0	0
$119$	1.47088	0.134835
$120$	0	0
$121$	−0.417101	−0.0379183
$122$	0	0
$123$	0.129472	0.0116741
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.8505	−1.22903	−0.614514	−	0.788906i	$-0.710648\pi$
$127$	−13.8505	−1.22903	−0.614514	+	0.788906i	$0.710648\pi$
$128$	0	0
$129$	0.0132477	0.00116640
$130$	0	0
$131$	−11.4760	−1.00267	−0.501333	−	0.865255i	$-0.667157\pi$
$131$	−11.4760	−1.00267	−0.501333	+	0.865255i	$0.667157\pi$
$132$	0	0
$133$	16.9333	1.46831
$134$	0	0
$135$	0.328547	0.0282769
$136$	0	0
$137$	−13.2569	−1.13262	−0.566308	−	0.824194i	$-0.691629\pi$
$137$	−13.2569	−1.13262	−0.566308	+	0.824194i	$0.691629\pi$
$138$	0	0
$139$	−2.87929	−0.244218	−0.122109	−	0.992517i	$-0.538966\pi$
$139$	−2.87929	−0.244218	−0.122109	+	0.992517i	$0.538966\pi$
$140$	0	0
$141$	−0.360335	−0.0303457
$142$	0	0
$143$	−22.4892	−1.88064
$144$	0	0
$145$	3.37707	0.280450
$146$	0	0
$147$	0.438015	0.0361269
$148$	0	0
$149$	5.39273	0.441790	0.220895	−	0.975298i	$-0.429102\pi$
$149$	5.39273	0.441790	0.220895	+	0.975298i	$0.429102\pi$
$150$	0	0
$151$	−2.42119	−0.197033	−0.0985167	−	0.995135i	$-0.531410\pi$
$151$	−2.42119	−0.197033	−0.0985167	+	0.995135i	$0.531410\pi$
$152$	0	0
$153$	−1.13838	−0.0920329
$154$	0	0
$155$	6.22792	0.500239
$156$	0	0
$157$	0.904608	0.0721956	0.0360978	−	0.999348i	$-0.488507\pi$
$157$	0.904608	0.0721956	0.0360978	+	0.999348i	$0.488507\pi$
$158$	0	0
$159$	0.521951	0.0413934
$160$	0	0
$161$	3.93647	0.310237
$162$	0	0
$163$	6.73871	0.527817	0.263908	−	0.964548i	$-0.414988\pi$
$163$	6.73871	0.527817	0.263908	+	0.964548i	$0.414988\pi$
$164$	0	0
$165$	−0.178224	−0.0138747
$166$	0	0
$167$	−20.2600	−1.56777	−0.783885	−	0.620907i	$-0.786765\pi$
$167$	−20.2600	−1.56777	−0.783885	+	0.620907i	$0.786765\pi$
$168$	0	0
$169$	34.7909	2.67622
$170$	0	0
$171$	−13.1055	−1.00220
$172$	0	0
$173$	−5.63353	−0.428309	−0.214155	−	0.976800i	$-0.568700\pi$
$173$	−5.63353	−0.428309	−0.214155	+	0.976800i	$0.568700\pi$
$174$	0	0
$175$	3.87235	0.292722
$176$	0	0
$177$	0.0669965	0.00503576
$178$	0	0
$179$	14.2014	1.06147	0.530733	−	0.847539i	$-0.321917\pi$
$179$	14.2014	1.06147	0.530733	+	0.847539i	$0.321917\pi$
$180$	0	0
$181$	−20.5197	−1.52522	−0.762608	−	0.646860i	$-0.776082\pi$
$181$	−20.5197	−1.52522	−0.762608	+	0.646860i	$0.776082\pi$
$182$	0	0
$183$	−0.495121	−0.0366004
$184$	0	0
$185$	0.791545	0.0581955
$186$	0	0
$187$	1.23568	0.0903616
$188$	0	0
$189$	−1.27225	−0.0925427
$190$	0	0
$191$	−21.5800	−1.56147	−0.780736	−	0.624861i	$-0.785156\pi$
$191$	−21.5800	−1.56147	−0.780736	+	0.624861i	$0.785156\pi$
$192$	0	0
$193$	13.6158	0.980090	0.490045	−	0.871697i	$-0.336980\pi$
$193$	13.6158	0.980090	0.490045	+	0.871697i	$0.336980\pi$
$194$	0	0
$195$	0.378736	0.0271218
$196$	0	0
$197$	−17.2022	−1.22561	−0.612804	−	0.790235i	$-0.709958\pi$
$197$	−17.2022	−1.22561	−0.612804	+	0.790235i	$0.709958\pi$
$198$	0	0
$199$	19.4704	1.38022	0.690109	−	0.723705i	$-0.257562\pi$
$199$	19.4704	1.38022	0.690109	+	0.723705i	$0.257562\pi$
$200$	0	0
$201$	−0.680760	−0.0480171
$202$	0	0
$203$	−13.0772	−0.917840
$204$	0	0
$205$	−2.36327	−0.165058
$206$	0	0
$207$	−3.04662	−0.211755
$208$	0	0
$209$	14.2256	0.984003
$210$	0	0
$211$	−12.7402	−0.877072	−0.438536	−	0.898714i	$-0.644503\pi$
$211$	−12.7402	−0.877072	−0.438536	+	0.898714i	$0.644503\pi$
$212$	0	0
$213$	−0.647290	−0.0443516
$214$	0	0
$215$	−0.241812	−0.0164914
$216$	0	0
$217$	−24.1167	−1.63715
$218$	0	0
$219$	−0.0568126	−0.00383904
$220$	0	0
$221$	−2.62588	−0.176636
$222$	0	0
$223$	−2.58615	−0.173181	−0.0865906	−	0.996244i	$-0.527597\pi$
$223$	−2.58615	−0.173181	−0.0865906	+	0.996244i	$0.527597\pi$
$224$	0	0
$225$	−2.99700	−0.199800
$226$	0	0
$227$	15.8804	1.05402	0.527009	−	0.849860i	$-0.323313\pi$
$227$	15.8804	1.05402	0.527009	+	0.849860i	$0.323313\pi$
$228$	0	0
$229$	−4.44561	−0.293774	−0.146887	−	0.989153i	$-0.546925\pi$
$229$	−4.44561	−0.293774	−0.146887	+	0.989153i	$0.546925\pi$
$230$	0	0
$231$	0.690147	0.0454083
$232$	0	0
$233$	−14.1850	−0.929292	−0.464646	−	0.885497i	$-0.653818\pi$
$233$	−14.1850	−0.929292	−0.464646	+	0.885497i	$0.653818\pi$
$234$	0	0
$235$	6.57722	0.429051
$236$	0	0
$237$	0.0780202	0.00506796
$238$	0	0
$239$	−7.56239	−0.489170	−0.244585	−	0.969628i	$-0.578652\pi$
$239$	−7.56239	−0.489170	−0.244585	+	0.969628i	$0.578652\pi$
$240$	0	0
$241$	9.87453	0.636074	0.318037	−	0.948078i	$-0.396976\pi$
$241$	9.87453	0.636074	0.318037	+	0.948078i	$0.396976\pi$
$242$	0	0
$243$	1.47723	0.0947644
$244$	0	0
$245$	−7.99512	−0.510789
$246$	0	0
$247$	−30.2301	−1.92350
$248$	0	0
$249$	0.548387	0.0347526
$250$	0	0
$251$	−22.0552	−1.39211	−0.696056	−	0.717988i	$-0.745063\pi$
$251$	−22.0552	−1.39211	−0.696056	+	0.717988i	$0.745063\pi$
$252$	0	0
$253$	3.30700	0.207909
$254$	0	0
$255$	−0.0208097	−0.00130316
$256$	0	0
$257$	17.3588	1.08281	0.541405	−	0.840762i	$-0.317892\pi$
$257$	17.3588	1.08281	0.541405	+	0.840762i	$0.317892\pi$
$258$	0	0
$259$	−3.06514	−0.190459
$260$	0	0
$261$	10.1211	0.626478
$262$	0	0
$263$	−18.2110	−1.12294	−0.561468	−	0.827498i	$-0.689763\pi$
$263$	−18.2110	−1.12294	−0.561468	+	0.827498i	$0.689763\pi$
$264$	0	0
$265$	−9.52720	−0.585251
$266$	0	0
$267$	−0.0140795	−0.000861648 0
$268$	0	0
$269$	4.83629	0.294874	0.147437	−	0.989071i	$-0.452898\pi$
$269$	4.83629	0.294874	0.147437	+	0.989071i	$0.452898\pi$
$270$	0	0
$271$	14.0413	0.852949	0.426474	−	0.904500i	$-0.359755\pi$
$271$	14.0413	0.852949	0.426474	+	0.904500i	$0.359755\pi$
$272$	0	0
$273$	−1.46660	−0.0887626
$274$	0	0
$275$	3.25314	0.196172
$276$	0	0
$277$	−16.8843	−1.01448	−0.507238	−	0.861806i	$-0.669334\pi$
$277$	−16.8843	−1.01448	−0.507238	+	0.861806i	$0.669334\pi$
$278$	0	0
$279$	18.6651	1.11745
$280$	0	0
$281$	−17.8389	−1.06418	−0.532091	−	0.846687i	$-0.678593\pi$
$281$	−17.8389	−1.06418	−0.532091	+	0.846687i	$0.678593\pi$
$282$	0	0
$283$	29.4099	1.74824	0.874119	−	0.485711i	$-0.161439\pi$
$283$	29.4099	1.74824	0.874119	+	0.485711i	$0.161439\pi$
$284$	0	0
$285$	−0.239569	−0.0141909
$286$	0	0
$287$	9.15141	0.540191
$288$	0	0
$289$	−16.8557	−0.991513
$290$	0	0
$291$	0.222091	0.0130192
$292$	0	0
$293$	−11.2205	−0.655507	−0.327754	−	0.944763i	$-0.606292\pi$
$293$	−11.2205	−0.655507	−0.327754	+	0.944763i	$0.606292\pi$
$294$	0	0
$295$	−1.22289	−0.0711995
$296$	0	0
$297$	−1.06881	−0.0620186
$298$	0	0
$299$	−7.02756	−0.406414
$300$	0	0
$301$	0.936382	0.0539722
$302$	0	0
$303$	−0.0708285	−0.00406899
$304$	0	0
$305$	9.03747	0.517484
$306$	0	0
$307$	16.0714	0.917244	0.458622	−	0.888631i	$-0.348343\pi$
$307$	16.0714	0.917244	0.458622	+	0.888631i	$0.348343\pi$
$308$	0	0
$309$	0.257337	0.0146394
$310$	0	0
$311$	−23.5345	−1.33452	−0.667259	−	0.744825i	$-0.732533\pi$
$311$	−23.5345	−1.33452	−0.667259	+	0.744825i	$0.732533\pi$
$312$	0	0
$313$	−2.46325	−0.139231	−0.0696155	−	0.997574i	$-0.522177\pi$
$313$	−2.46325	−0.139231	−0.0696155	+	0.997574i	$0.522177\pi$
$314$	0	0
$315$	11.6054	0.653892
$316$	0	0
$317$	2.74184	0.153997	0.0769984	−	0.997031i	$-0.475466\pi$
$317$	2.74184	0.153997	0.0769984	+	0.997031i	$0.475466\pi$
$318$	0	0
$319$	−10.9861	−0.615102
$320$	0	0
$321$	0.159082	0.00887908
$322$	0	0
$323$	1.66100	0.0924205
$324$	0	0
$325$	−6.91309	−0.383469
$326$	0	0
$327$	−0.193127	−0.0106799
$328$	0	0
$329$	−25.4693	−1.40417
$330$	0	0
$331$	20.1919	1.10985	0.554924	−	0.831901i	$-0.312747\pi$
$331$	20.1919	1.10985	0.554924	+	0.831901i	$0.312747\pi$
$332$	0	0
$333$	2.37226	0.129999
$334$	0	0
$335$	12.4260	0.678903
$336$	0	0
$337$	26.5295	1.44516	0.722578	−	0.691290i	$-0.242957\pi$
$337$	26.5295	1.44516	0.722578	+	0.691290i	$0.242957\pi$
$338$	0	0
$339$	−0.248387	−0.0134905
$340$	0	0
$341$	−20.2603	−1.09716
$342$	0	0
$343$	3.85346	0.208067
$344$	0	0
$345$	−0.0556924	−0.00299838
$346$	0	0
$347$	−1.16729	−0.0626634	−0.0313317	−	0.999509i	$-0.509975\pi$
$347$	−1.16729	−0.0626634	−0.0313317	+	0.999509i	$0.509975\pi$
$348$	0	0
$349$	13.6879	0.732698	0.366349	−	0.930477i	$-0.380608\pi$
$349$	13.6879	0.732698	0.366349	+	0.930477i	$0.380608\pi$
$350$	0	0
$351$	2.27128	0.121232
$352$	0	0
$353$	−0.960426	−0.0511183	−0.0255591	−	0.999673i	$-0.508137\pi$
$353$	−0.960426	−0.0511183	−0.0255591	+	0.999673i	$0.508137\pi$
$354$	0	0
$355$	11.8150	0.627077
$356$	0	0
$357$	0.0805826	0.00426488
$358$	0	0
$359$	20.7589	1.09561	0.547807	−	0.836605i	$-0.315463\pi$
$359$	20.7589	1.09561	0.547807	+	0.836605i	$0.315463\pi$
$360$	0	0
$361$	0.122051	0.00642376
$362$	0	0
$363$	−0.0228510	−0.00119937
$364$	0	0
$365$	1.03700	0.0542793
$366$	0	0
$367$	−23.1120	−1.20643	−0.603217	−	0.797577i	$-0.706115\pi$
$367$	−23.1120	−1.20643	−0.603217	+	0.797577i	$0.706115\pi$
$368$	0	0
$369$	−7.08271	−0.368711
$370$	0	0
$371$	36.8927	1.91537
$372$	0	0
$373$	−1.77344	−0.0918255	−0.0459127	−	0.998945i	$-0.514620\pi$
$373$	−1.77344	−0.0918255	−0.0459127	+	0.998945i	$0.514620\pi$
$374$	0	0
$375$	−0.0547853	−0.00282910
$376$	0	0
$377$	23.3460	1.20238
$378$	0	0
$379$	−12.4511	−0.639572	−0.319786	−	0.947490i	$-0.603611\pi$
$379$	−12.4511	−0.639572	−0.319786	+	0.947490i	$0.603611\pi$
$380$	0	0
$381$	−0.758801	−0.0388746
$382$	0	0
$383$	−11.6751	−0.596569	−0.298284	−	0.954477i	$-0.596414\pi$
$383$	−11.6751	−0.596569	−0.298284	+	0.954477i	$0.596414\pi$
$384$	0	0
$385$	−12.5973	−0.642018
$386$	0	0
$387$	−0.724711	−0.0368391
$388$	0	0
$389$	22.6275	1.14726	0.573629	−	0.819115i	$-0.305535\pi$
$389$	22.6275	1.14726	0.573629	+	0.819115i	$0.305535\pi$
$390$	0	0
$391$	0.386131	0.0195275
$392$	0	0
$393$	−0.628718	−0.0317146
$394$	0	0
$395$	−1.42411	−0.0716547
$396$	0	0
$397$	20.7266	1.04024	0.520118	−	0.854094i	$-0.325888\pi$
$397$	20.7266	1.04024	0.520118	+	0.854094i	$0.325888\pi$
$398$	0	0
$399$	0.927697	0.0464430
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	43.0542	2.14468
$404$	0	0
$405$	−8.97300	−0.445872
$406$	0	0
$407$	−2.57500	−0.127638
$408$	0	0
$409$	−14.5502	−0.719461	−0.359730	−	0.933056i	$-0.617131\pi$
$409$	−14.5502	−0.719461	−0.359730	+	0.933056i	$0.617131\pi$
$410$	0	0
$411$	−0.726285	−0.0358250
$412$	0	0
$413$	4.73547	0.233017
$414$	0	0
$415$	−10.0097	−0.491359
$416$	0	0
$417$	−0.157743	−0.00772469
$418$	0	0
$419$	0.495617	0.0242125	0.0121062	−	0.999927i	$-0.496146\pi$
$419$	0.495617	0.0242125	0.0121062	+	0.999927i	$0.496146\pi$
$420$	0	0
$421$	−36.0564	−1.75728	−0.878640	−	0.477486i	$-0.841548\pi$
$421$	−36.0564	−1.75728	−0.878640	+	0.477486i	$0.841548\pi$
$422$	0	0
$423$	19.7119	0.958427
$424$	0	0
$425$	0.379841	0.0184250
$426$	0	0
$427$	−34.9963	−1.69359
$428$	0	0
$429$	−1.23208	−0.0594854
$430$	0	0
$431$	25.2202	1.21481	0.607406	−	0.794391i	$-0.292210\pi$
$431$	25.2202	1.21481	0.607406	+	0.794391i	$0.292210\pi$
$432$	0	0
$433$	11.8649	0.570193	0.285096	−	0.958499i	$-0.407974\pi$
$433$	11.8649	0.570193	0.285096	+	0.958499i	$0.407974\pi$
$434$	0	0
$435$	0.185014	0.00887073
$436$	0	0
$437$	4.44528	0.212647
$438$	0	0
$439$	16.7449	0.799191	0.399595	−	0.916692i	$-0.369151\pi$
$439$	16.7449	0.799191	0.399595	+	0.916692i	$0.369151\pi$
$440$	0	0
$441$	−23.9614	−1.14102
$442$	0	0
$443$	17.4045	0.826912	0.413456	−	0.910524i	$-0.364322\pi$
$443$	17.4045	0.826912	0.413456	+	0.910524i	$0.364322\pi$
$444$	0	0
$445$	0.256993	0.0121826
$446$	0	0
$447$	0.295443	0.0139740
$448$	0	0
$449$	−4.06051	−0.191627	−0.0958136	−	0.995399i	$-0.530545\pi$
$449$	−4.06051	−0.191627	−0.0958136	+	0.995399i	$0.530545\pi$
$450$	0	0
$451$	7.68803	0.362015
$452$	0	0
$453$	−0.132645	−0.00623223
$454$	0	0
$455$	26.7699	1.25499
$456$	0	0
$457$	−23.7234	−1.10973	−0.554867	−	0.831939i	$-0.687231\pi$
$457$	−23.7234	−1.10973	−0.554867	+	0.831939i	$0.687231\pi$
$458$	0	0
$459$	−0.124796	−0.00582497
$460$	0	0
$461$	2.02394	0.0942642	0.0471321	−	0.998889i	$-0.484992\pi$
$461$	2.02394	0.0942642	0.0471321	+	0.998889i	$0.484992\pi$
$462$	0	0
$463$	−12.4483	−0.578520	−0.289260	−	0.957251i	$-0.593409\pi$
$463$	−12.4483	−0.578520	−0.289260	+	0.957251i	$0.593409\pi$
$464$	0	0
$465$	0.341199	0.0158227
$466$	0	0
$467$	−21.4461	−0.992406	−0.496203	−	0.868206i	$-0.665273\pi$
$467$	−21.4461	−0.992406	−0.496203	+	0.868206i	$0.665273\pi$
$468$	0	0
$469$	−48.1177	−2.22187
$470$	0	0
$471$	0.0495592	0.00228357
$472$	0	0
$473$	0.786648	0.0361701
$474$	0	0
$475$	4.37288	0.200641
$476$	0	0
$477$	−28.5530	−1.30735
$478$	0	0
$479$	−31.3565	−1.43271	−0.716357	−	0.697734i	$-0.754192\pi$
$479$	−31.3565	−1.43271	−0.716357	+	0.697734i	$0.754192\pi$
$480$	0	0
$481$	5.47202	0.249503
$482$	0	0
$483$	0.215661	0.00981290
$484$	0	0
$485$	−4.05385	−0.184076
$486$	0	0
$487$	−34.6622	−1.57069	−0.785347	−	0.619056i	$-0.787515\pi$
$487$	−34.6622	−1.57069	−0.785347	+	0.619056i	$0.787515\pi$
$488$	0	0
$489$	0.369182	0.0166950
$490$	0	0
$491$	−26.3685	−1.18999	−0.594997	−	0.803728i	$-0.702847\pi$
$491$	−26.3685	−1.18999	−0.594997	+	0.803728i	$0.702847\pi$
$492$	0	0
$493$	−1.28275	−0.0577722
$494$	0	0
$495$	9.74965	0.438214
$496$	0	0
$497$	−45.7520	−2.05226
$498$	0	0
$499$	−31.7512	−1.42138	−0.710689	−	0.703506i	$-0.751617\pi$
$499$	−31.7512	−1.42138	−0.710689	+	0.703506i	$0.751617\pi$
$500$	0	0
$501$	−1.10995	−0.0495890
$502$	0	0
$503$	−23.5992	−1.05224	−0.526118	−	0.850411i	$-0.676353\pi$
$503$	−23.5992	−1.05224	−0.526118	+	0.850411i	$0.676353\pi$
$504$	0	0
$505$	1.29284	0.0575305
$506$	0	0
$507$	1.90603	0.0846497
$508$	0	0
$509$	33.4949	1.48463	0.742317	−	0.670048i	$-0.233727\pi$
$509$	33.4949	1.48463	0.742317	+	0.670048i	$0.233727\pi$
$510$	0	0
$511$	−4.01565	−0.177642
$512$	0	0
$513$	−1.43670	−0.0634317
$514$	0	0
$515$	−4.69720	−0.206983
$516$	0	0
$517$	−21.3966	−0.941022
$518$	0	0
$519$	−0.308635	−0.0135476
$520$	0	0
$521$	−40.2923	−1.76524	−0.882620	−	0.470087i	$-0.844222\pi$
$521$	−40.2923	−1.76524	−0.882620	+	0.470087i	$0.844222\pi$
$522$	0	0
$523$	−41.1087	−1.79756	−0.898778	−	0.438404i	$-0.855544\pi$
$523$	−41.1087	−1.79756	−0.898778	+	0.438404i	$0.855544\pi$
$524$	0	0
$525$	0.212148	0.00925890
$526$	0	0
$527$	−2.36562	−0.103048
$528$	0	0
$529$	−21.9666	−0.955070
$530$	0	0
$531$	−3.66500	−0.159048
$532$	0	0
$533$	−16.3375	−0.707655
$534$	0	0
$535$	−2.90373	−0.125539
$536$	0	0
$537$	0.778030	0.0335745
$538$	0	0
$539$	26.0092	1.12030
$540$	0	0
$541$	−15.0005	−0.644923	−0.322461	−	0.946583i	$-0.604510\pi$
$541$	−15.0005	−0.644923	−0.322461	+	0.946583i	$0.604510\pi$
$542$	0	0
$543$	−1.12418	−0.0482431
$544$	0	0
$545$	3.52516	0.151001
$546$	0	0
$547$	8.48196	0.362662	0.181331	−	0.983422i	$-0.441959\pi$
$547$	8.48196	0.362662	0.181331	+	0.983422i	$0.441959\pi$
$548$	0	0
$549$	27.0853	1.15597
$550$	0	0
$551$	−14.7675	−0.629117
$552$	0	0
$553$	5.51465	0.234507
$554$	0	0
$555$	0.0433650	0.00184074
$556$	0	0
$557$	−12.8347	−0.543823	−0.271911	−	0.962322i	$-0.587656\pi$
$557$	−12.8347	−0.543823	−0.271911	+	0.962322i	$0.587656\pi$
$558$	0	0
$559$	−1.67167	−0.0707041
$560$	0	0
$561$	0.0676969	0.00285816
$562$	0	0
$563$	−37.1619	−1.56619	−0.783094	−	0.621903i	$-0.786360\pi$
$563$	−37.1619	−1.56619	−0.783094	+	0.621903i	$0.786360\pi$
$564$	0	0
$565$	4.53383	0.190740
$566$	0	0
$567$	34.7466	1.45922
$568$	0	0
$569$	−10.6682	−0.447234	−0.223617	−	0.974677i	$-0.571786\pi$
$569$	−10.6682	−0.447234	−0.223617	+	0.974677i	$0.571786\pi$
$570$	0	0
$571$	2.83451	0.118620	0.0593102	−	0.998240i	$-0.481110\pi$
$571$	2.83451	0.118620	0.0593102	+	0.998240i	$0.481110\pi$
$572$	0	0
$573$	−1.18227	−0.0493898
$574$	0	0
$575$	1.01656	0.0423934
$576$	0	0
$577$	−8.54980	−0.355933	−0.177966	−	0.984037i	$-0.556952\pi$
$577$	−8.54980	−0.355933	−0.177966	+	0.984037i	$0.556952\pi$
$578$	0	0
$579$	0.745948	0.0310005
$580$	0	0
$581$	38.7613	1.60809
$582$	0	0
$583$	30.9933	1.28361
$584$	0	0
$585$	−20.7185	−0.856606
$586$	0	0
$587$	−28.2948	−1.16785	−0.583926	−	0.811807i	$-0.698484\pi$
$587$	−28.2948	−1.16785	−0.583926	+	0.811807i	$0.698484\pi$
$588$	0	0
$589$	−27.2339	−1.12216
$590$	0	0
$591$	−0.942429	−0.0387663
$592$	0	0
$593$	−2.04491	−0.0839743	−0.0419872	−	0.999118i	$-0.513369\pi$
$593$	−2.04491	−0.0839743	−0.0419872	+	0.999118i	$0.513369\pi$
$594$	0	0
$595$	−1.47088	−0.0603002
$596$	0	0
$597$	1.06669	0.0436567
$598$	0	0
$599$	−0.317784	−0.0129843	−0.00649215	−	0.999979i	$-0.502067\pi$
$599$	−0.317784	−0.0129843	−0.00649215	+	0.999979i	$0.502067\pi$
$600$	0	0
$601$	7.97661	0.325373	0.162686	−	0.986678i	$-0.447984\pi$
$601$	7.97661	0.325373	0.162686	+	0.986678i	$0.447984\pi$
$602$	0	0
$603$	37.2406	1.51655
$604$	0	0
$605$	0.417101	0.0169576
$606$	0	0
$607$	38.9256	1.57994	0.789971	−	0.613144i	$-0.210096\pi$
$607$	38.9256	1.57994	0.789971	+	0.613144i	$0.210096\pi$
$608$	0	0
$609$	−0.716438	−0.0290315
$610$	0	0
$611$	45.4690	1.83948
$612$	0	0
$613$	−36.6279	−1.47939	−0.739694	−	0.672943i	$-0.765030\pi$
$613$	−36.6279	−1.47939	−0.739694	+	0.672943i	$0.765030\pi$
$614$	0	0
$615$	−0.129472	−0.00522083
$616$	0	0
$617$	−23.0675	−0.928662	−0.464331	−	0.885662i	$-0.653705\pi$
$617$	−23.0675	−0.928662	−0.464331	+	0.885662i	$0.653705\pi$
$618$	0	0
$619$	13.1149	0.527131	0.263565	−	0.964642i	$-0.415102\pi$
$619$	13.1149	0.527131	0.263565	+	0.964642i	$0.415102\pi$
$620$	0	0
$621$	−0.333987	−0.0134024
$622$	0	0
$623$	−0.995169	−0.0398706
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.779352	0.0311243
$628$	0	0
$629$	−0.300662	−0.0119882
$630$	0	0
$631$	22.8199	0.908444	0.454222	−	0.890888i	$-0.349917\pi$
$631$	22.8199	0.908444	0.454222	+	0.890888i	$0.349917\pi$
$632$	0	0
$633$	−0.697976	−0.0277420
$634$	0	0
$635$	13.8505	0.549638
$636$	0	0
$637$	−55.2710	−2.18992
$638$	0	0
$639$	35.4096	1.40078
$640$	0	0
$641$	−2.76533	−0.109224	−0.0546121	−	0.998508i	$-0.517392\pi$
$641$	−2.76533	−0.109224	−0.0546121	+	0.998508i	$0.517392\pi$
$642$	0	0
$643$	48.3052	1.90497	0.952485	−	0.304585i	$-0.0985179\pi$
$643$	48.3052	1.90497	0.952485	+	0.304585i	$0.0985179\pi$
$644$	0	0
$645$	−0.0132477	−0.000521630 0
$646$	0	0
$647$	−23.1138	−0.908699	−0.454349	−	0.890824i	$-0.650128\pi$
$647$	−23.1138	−0.908699	−0.454349	+	0.890824i	$0.650128\pi$
$648$	0	0
$649$	3.97823	0.156159
$650$	0	0
$651$	−1.32124	−0.0517835
$652$	0	0
$653$	−39.2971	−1.53781	−0.768907	−	0.639360i	$-0.779199\pi$
$653$	−39.2971	−1.53781	−0.768907	+	0.639360i	$0.779199\pi$
$654$	0	0
$655$	11.4760	0.448405
$656$	0	0
$657$	3.10790	0.121251
$658$	0	0
$659$	−20.7465	−0.808171	−0.404085	−	0.914721i	$-0.632410\pi$
$659$	−20.7465	−0.808171	−0.404085	+	0.914721i	$0.632410\pi$
$660$	0	0
$661$	27.1256	1.05506	0.527531	−	0.849536i	$-0.323118\pi$
$661$	27.1256	1.05506	0.527531	+	0.849536i	$0.323118\pi$
$662$	0	0
$663$	−0.143860	−0.00558704
$664$	0	0
$665$	−16.9333	−0.656646
$666$	0	0
$667$	−3.43298	−0.132926
$668$	0	0
$669$	−0.141683	−0.00547778
$670$	0	0
$671$	−29.4001	−1.13498
$672$	0	0
$673$	4.95240	0.190901	0.0954505	−	0.995434i	$-0.469571\pi$
$673$	4.95240	0.190901	0.0954505	+	0.995434i	$0.469571\pi$
$674$	0	0
$675$	−0.328547	−0.0126458
$676$	0	0
$677$	35.1674	1.35159	0.675796	−	0.737089i	$-0.263800\pi$
$677$	35.1674	1.35159	0.675796	+	0.737089i	$0.263800\pi$
$678$	0	0
$679$	15.6979	0.602431
$680$	0	0
$681$	0.870011	0.0333389
$682$	0	0
$683$	−33.0119	−1.26317	−0.631584	−	0.775308i	$-0.717595\pi$
$683$	−33.0119	−1.26317	−0.631584	+	0.775308i	$0.717595\pi$
$684$	0	0
$685$	13.2569	0.506521
$686$	0	0
$687$	−0.243554	−0.00929216
$688$	0	0
$689$	−65.8624	−2.50916
$690$	0	0
$691$	−0.394264	−0.0149985	−0.00749926	−	0.999972i	$-0.502387\pi$
$691$	−0.394264	−0.0149985	−0.00749926	+	0.999972i	$0.502387\pi$
$692$	0	0
$693$	−37.7541	−1.43416
$694$	0	0
$695$	2.87929	0.109218
$696$	0	0
$697$	0.897667	0.0340016
$698$	0	0
$699$	−0.777131	−0.0293938
$700$	0	0
$701$	30.2131	1.14113	0.570566	−	0.821252i	$-0.306724\pi$
$701$	30.2131	1.14113	0.570566	+	0.821252i	$0.306724\pi$
$702$	0	0
$703$	−3.46133	−0.130546
$704$	0	0
$705$	0.360335	0.0135710
$706$	0	0
$707$	−5.00633	−0.188282
$708$	0	0
$709$	−13.2139	−0.496260	−0.248130	−	0.968727i	$-0.579816\pi$
$709$	−13.2139	−0.496260	−0.248130	+	0.968727i	$0.579816\pi$
$710$	0	0
$711$	−4.26805	−0.160064
$712$	0	0
$713$	−6.33104	−0.237099
$714$	0	0
$715$	22.4892	0.841050
$716$	0	0
$717$	−0.414308	−0.0154726
$718$	0	0
$719$	9.77985	0.364727	0.182363	−	0.983231i	$-0.441625\pi$
$719$	9.77985	0.364727	0.182363	+	0.983231i	$0.441625\pi$
$720$	0	0
$721$	18.1892	0.677401
$722$	0	0
$723$	0.540979	0.0201192
$724$	0	0
$725$	−3.37707	−0.125421
$726$	0	0
$727$	13.0727	0.484839	0.242420	−	0.970172i	$-0.422059\pi$
$727$	13.0727	0.484839	0.242420	+	0.970172i	$0.422059\pi$
$728$	0	0
$729$	−26.8381	−0.994002
$730$	0	0
$731$	0.0918503	0.00339720
$732$	0	0
$733$	−1.89623	−0.0700389	−0.0350194	−	0.999387i	$-0.511149\pi$
$733$	−1.89623	−0.0700389	−0.0350194	+	0.999387i	$0.511149\pi$
$734$	0	0
$735$	−0.438015	−0.0161564
$736$	0	0
$737$	−40.4233	−1.48901
$738$	0	0
$739$	−7.42230	−0.273034	−0.136517	−	0.990638i	$-0.543591\pi$
$739$	−7.42230	−0.273034	−0.136517	+	0.990638i	$0.543591\pi$
$740$	0	0
$741$	−1.65617	−0.0608408
$742$	0	0
$743$	−26.4765	−0.971328	−0.485664	−	0.874146i	$-0.661422\pi$
$743$	−26.4765	−0.971328	−0.485664	+	0.874146i	$0.661422\pi$
$744$	0	0
$745$	−5.39273	−0.197574
$746$	0	0
$747$	−29.9992	−1.09761
$748$	0	0
$749$	11.2443	0.410857
$750$	0	0
$751$	36.3315	1.32576	0.662879	−	0.748727i	$-0.269335\pi$
$751$	36.3315	1.32576	0.662879	+	0.748727i	$0.269335\pi$
$752$	0	0
$753$	−1.20830	−0.0440329
$754$	0	0
$755$	2.42119	0.0881160
$756$	0	0
$757$	40.6604	1.47783	0.738913	−	0.673801i	$-0.235340\pi$
$757$	40.6604	1.47783	0.738913	+	0.673801i	$0.235340\pi$
$758$	0	0
$759$	0.181175	0.00657623
$760$	0	0
$761$	−27.3976	−0.993162	−0.496581	−	0.867990i	$-0.665411\pi$
$761$	−27.3976	−0.993162	−0.496581	+	0.867990i	$0.665411\pi$
$762$	0	0
$763$	−13.6507	−0.494187
$764$	0	0
$765$	1.13838	0.0411584
$766$	0	0
$767$	−8.45396	−0.305255
$768$	0	0
$769$	−26.4027	−0.952104	−0.476052	−	0.879417i	$-0.657933\pi$
$769$	−26.4027	−0.952104	−0.476052	+	0.879417i	$0.657933\pi$
$770$	0	0
$771$	0.951006	0.0342496
$772$	0	0
$773$	45.6528	1.64202	0.821008	−	0.570917i	$-0.193412\pi$
$773$	45.6528	1.64202	0.821008	+	0.570917i	$0.193412\pi$
$774$	0	0
$775$	−6.22792	−0.223714
$776$	0	0
$777$	−0.167925	−0.00602426
$778$	0	0
$779$	10.3343	0.370264
$780$	0	0
$781$	−38.4359	−1.37534
$782$	0	0
$783$	1.10953	0.0396513
$784$	0	0
$785$	−0.904608	−0.0322869
$786$	0	0
$787$	4.23031	0.150794	0.0753971	−	0.997154i	$-0.475978\pi$
$787$	4.23031	0.150794	0.0753971	+	0.997154i	$0.475978\pi$
$788$	0	0
$789$	−0.997694	−0.0355188
$790$	0	0
$791$	−17.5566	−0.624241
$792$	0	0
$793$	62.4769	2.21862
$794$	0	0
$795$	−0.521951	−0.0185117
$796$	0	0
$797$	22.6939	0.803860	0.401930	−	0.915670i	$-0.368340\pi$
$797$	22.6939	0.803860	0.401930	+	0.915670i	$0.368340\pi$
$798$	0	0
$799$	−2.49830	−0.0883836
$800$	0	0
$801$	0.770208	0.0272140
$802$	0	0
$803$	−3.37352	−0.119049
$804$	0	0
$805$	−3.93647	−0.138742
$806$	0	0
$807$	0.264957	0.00932694
$808$	0	0
$809$	−44.9292	−1.57963	−0.789814	−	0.613347i	$-0.789823\pi$
$809$	−44.9292	−1.57963	−0.789814	+	0.613347i	$0.789823\pi$
$810$	0	0
$811$	−38.6415	−1.35689	−0.678443	−	0.734653i	$-0.737345\pi$
$811$	−38.6415	−1.35689	−0.678443	+	0.734653i	$0.737345\pi$
$812$	0	0
$813$	0.769257	0.0269790
$814$	0	0
$815$	−6.73871	−0.236047
$816$	0	0
$817$	1.05741	0.0369943
$818$	0	0
$819$	80.2295	2.80344
$820$	0	0
$821$	11.7155	0.408873	0.204437	−	0.978880i	$-0.434464\pi$
$821$	11.7155	0.408873	0.204437	+	0.978880i	$0.434464\pi$
$822$	0	0
$823$	31.5615	1.10017	0.550083	−	0.835110i	$-0.314596\pi$
$823$	31.5615	1.10017	0.550083	+	0.835110i	$0.314596\pi$
$824$	0	0
$825$	0.178224	0.00620497
$826$	0	0
$827$	35.7491	1.24312	0.621559	−	0.783368i	$-0.286500\pi$
$827$	35.7491	1.24312	0.621559	+	0.783368i	$0.286500\pi$
$828$	0	0
$829$	29.0953	1.01052	0.505261	−	0.862967i	$-0.331396\pi$
$829$	29.0953	1.01052	0.505261	+	0.862967i	$0.331396\pi$
$830$	0	0
$831$	−0.925009	−0.0320882
$832$	0	0
$833$	3.03688	0.105222
$834$	0	0
$835$	20.2600	0.701128
$836$	0	0
$837$	2.04617	0.0707259
$838$	0	0
$839$	−30.2324	−1.04374	−0.521870	−	0.853025i	$-0.674765\pi$
$839$	−30.2324	−1.04374	−0.521870	+	0.853025i	$0.674765\pi$
$840$	0	0
$841$	−17.5954	−0.606738
$842$	0	0
$843$	−0.977311	−0.0336604
$844$	0	0
$845$	−34.7909	−1.19684
$846$	0	0
$847$	−1.61516	−0.0554976
$848$	0	0
$849$	1.61123	0.0552973
$850$	0	0
$851$	−0.804651	−0.0275831
$852$	0	0
$853$	34.3299	1.17543	0.587716	−	0.809067i	$-0.300027\pi$
$853$	34.3299	1.17543	0.587716	+	0.809067i	$0.300027\pi$
$854$	0	0
$855$	13.1055	0.448199
$856$	0	0
$857$	52.8787	1.80630	0.903152	−	0.429322i	$-0.141247\pi$
$857$	52.8787	1.80630	0.903152	+	0.429322i	$0.141247\pi$
$858$	0	0
$859$	4.06326	0.138637	0.0693183	−	0.997595i	$-0.477918\pi$
$859$	4.06326	0.138637	0.0693183	+	0.997595i	$0.477918\pi$
$860$	0	0
$861$	0.501363	0.0170864
$862$	0	0
$863$	41.0841	1.39852	0.699260	−	0.714867i	$-0.253513\pi$
$863$	41.0841	1.39852	0.699260	+	0.714867i	$0.253513\pi$
$864$	0	0
$865$	5.63353	0.191546
$866$	0	0
$867$	−0.923446	−0.0313619
$868$	0	0
$869$	4.63282	0.157158
$870$	0	0
$871$	85.9018	2.91067
$872$	0	0
$873$	−12.1494	−0.411194
$874$	0	0
$875$	−3.87235	−0.130909
$876$	0	0
$877$	42.3791	1.43104	0.715521	−	0.698592i	$-0.246190\pi$
$877$	42.3791	1.43104	0.715521	+	0.698592i	$0.246190\pi$
$878$	0	0
$879$	−0.614717	−0.0207339
$880$	0	0
$881$	8.82835	0.297435	0.148717	−	0.988880i	$-0.452486\pi$
$881$	8.82835	0.297435	0.148717	+	0.988880i	$0.452486\pi$
$882$	0	0
$883$	9.06826	0.305171	0.152586	−	0.988290i	$-0.451240\pi$
$883$	9.06826	0.305171	0.152586	+	0.988290i	$0.451240\pi$
$884$	0	0
$885$	−0.0669965	−0.00225206
$886$	0	0
$887$	5.34816	0.179574	0.0897869	−	0.995961i	$-0.471381\pi$
$887$	5.34816	0.179574	0.0897869	+	0.995961i	$0.471381\pi$
$888$	0	0
$889$	−53.6338	−1.79882
$890$	0	0
$891$	29.1904	0.977915
$892$	0	0
$893$	−28.7614	−0.962463
$894$	0	0
$895$	−14.2014	−0.474702
$896$	0	0
$897$	−0.385007	−0.0128550
$898$	0	0
$899$	21.0321	0.701461
$900$	0	0
$901$	3.61883	0.120561
$902$	0	0
$903$	0.0513000	0.00170716
$904$	0	0
$905$	20.5197	0.682098
$906$	0	0
$907$	−12.5206	−0.415740	−0.207870	−	0.978156i	$-0.566653\pi$
$907$	−12.5206	−0.415740	−0.207870	+	0.978156i	$0.566653\pi$
$908$	0	0
$909$	3.87463	0.128514
$910$	0	0
$911$	−19.9114	−0.659695	−0.329847	−	0.944034i	$-0.606997\pi$
$911$	−19.9114	−0.659695	−0.329847	+	0.944034i	$0.606997\pi$
$912$	0	0
$913$	32.5631	1.07768
$914$	0	0
$915$	0.495121	0.0163682
$916$	0	0
$917$	−44.4392	−1.46751
$918$	0	0
$919$	−7.90519	−0.260768	−0.130384	−	0.991464i	$-0.541621\pi$
$919$	−7.90519	−0.260768	−0.130384	+	0.991464i	$0.541621\pi$
$920$	0	0
$921$	0.880477	0.0290127
$922$	0	0
$923$	81.6784	2.68848
$924$	0	0
$925$	−0.791545	−0.0260258
$926$	0	0
$927$	−14.0775	−0.462365
$928$	0	0
$929$	22.7571	0.746637	0.373319	−	0.927703i	$-0.378220\pi$
$929$	22.7571	0.746637	0.373319	+	0.927703i	$0.378220\pi$
$930$	0	0
$931$	34.9617	1.14582
$932$	0	0
$933$	−1.28934	−0.0422112
$934$	0	0
$935$	−1.23568	−0.0404109
$936$	0	0
$937$	−6.31048	−0.206154	−0.103077	−	0.994673i	$-0.532869\pi$
$937$	−6.31048	−0.206154	−0.103077	+	0.994673i	$0.532869\pi$
$938$	0	0
$939$	−0.134950	−0.00440392
$940$	0	0
$941$	9.34624	0.304679	0.152339	−	0.988328i	$-0.451319\pi$
$941$	9.34624	0.304679	0.152339	+	0.988328i	$0.451319\pi$
$942$	0	0
$943$	2.40240	0.0782328
$944$	0	0
$945$	1.27225	0.0413863
$946$	0	0
$947$	3.05680	0.0993326	0.0496663	−	0.998766i	$-0.484184\pi$
$947$	3.05680	0.0993326	0.0496663	+	0.998766i	$0.484184\pi$
$948$	0	0
$949$	7.16890	0.232712
$950$	0	0
$951$	0.150212	0.00487097
$952$	0	0
$953$	−14.6885	−0.475808	−0.237904	−	0.971289i	$-0.576460\pi$
$953$	−14.6885	−0.475808	−0.237904	+	0.971289i	$0.576460\pi$
$954$	0	0
$955$	21.5800	0.698312
$956$	0	0
$957$	−0.601875	−0.0194559
$958$	0	0
$959$	−51.3355	−1.65771
$960$	0	0
$961$	7.78703	0.251195
$962$	0	0
$963$	−8.70248	−0.280433
$964$	0	0
$965$	−13.6158	−0.438309
$966$	0	0
$967$	−3.07041	−0.0987377	−0.0493689	−	0.998781i	$-0.515721\pi$
$967$	−3.07041	−0.0987377	−0.0493689	+	0.998781i	$0.515721\pi$
$968$	0	0
$969$	0.0909984	0.00292329
$970$	0	0
$971$	26.1214	0.838275	0.419137	−	0.907923i	$-0.362333\pi$
$971$	26.1214	0.838275	0.419137	+	0.907923i	$0.362333\pi$
$972$	0	0
$973$	−11.1496	−0.357440
$974$	0	0
$975$	−0.378736	−0.0121293
$976$	0	0
$977$	22.7437	0.727634	0.363817	−	0.931470i	$-0.381473\pi$
$977$	22.7437	0.727634	0.363817	+	0.931470i	$0.381473\pi$
$978$	0	0
$979$	−0.836034	−0.0267198
$980$	0	0
$981$	10.5649	0.337311
$982$	0	0
$983$	11.1823	0.356660	0.178330	−	0.983971i	$-0.442931\pi$
$983$	11.1823	0.356660	0.178330	+	0.983971i	$0.442931\pi$
$984$	0	0
$985$	17.2022	0.548108
$986$	0	0
$987$	−1.39534	−0.0444143
$988$	0	0
$989$	0.245816	0.00781649
$990$	0	0
$991$	−39.2303	−1.24619	−0.623096	−	0.782145i	$-0.714125\pi$
$991$	−39.2303	−1.24619	−0.623096	+	0.782145i	$0.714125\pi$
$992$	0	0
$993$	1.10622	0.0351048
$994$	0	0
$995$	−19.4704	−0.617253
$996$	0	0
$997$	44.3720	1.40527	0.702637	−	0.711549i	$-0.252006\pi$
$997$	44.3720	1.40527	0.702637	+	0.711549i	$0.252006\pi$
$998$	0	0
$999$	0.260060	0.00822793

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.14	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.14	✓	28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0547853 q^{3} -1.00000 q^{5} +3.87235 q^{7} -2.99700 q^{9} +O(q^{10})\)	\(q+0.0547853 q^{3} -1.00000 q^{5} +3.87235 q^{7} -2.99700 q^{9} +3.25314 q^{11} -6.91309 q^{13} -0.0547853 q^{15} +0.379841 q^{17} +4.37288 q^{19} +0.212148 q^{21} +1.01656 q^{23} +1.00000 q^{25} -0.328547 q^{27} -3.37707 q^{29} -6.22792 q^{31} +0.178224 q^{33} -3.87235 q^{35} -0.791545 q^{37} -0.378736 q^{39} +2.36327 q^{41} +0.241812 q^{43} +2.99700 q^{45} -6.57722 q^{47} +7.99512 q^{49} +0.0208097 q^{51} +9.52720 q^{53} -3.25314 q^{55} +0.239569 q^{57} +1.22289 q^{59} -9.03747 q^{61} -11.6054 q^{63} +6.91309 q^{65} -12.4260 q^{67} +0.0556924 q^{69} -11.8150 q^{71} -1.03700 q^{73} +0.0547853 q^{75} +12.5973 q^{77} +1.42411 q^{79} +8.97300 q^{81} +10.0097 q^{83} -0.379841 q^{85} -0.185014 q^{87} -0.256993 q^{89} -26.7699 q^{91} -0.341199 q^{93} -4.37288 q^{95} +4.05385 q^{97} -9.74965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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