LMFDB - Embedded newform 4020.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 28x^{4} - 12x^{3} + 209x^{2} + 360x + 144 $ x^6 - x^5 - 28x^4 - 12x^3 + 209x^2 + 360x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.08027$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.08027 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +4.08027 q^{7} +1.00000 q^{9} -1.55469 q^{11} +6.10335 q^{13} -1.00000 q^{15} -3.65804 q^{17} -1.29240 q^{19} -4.08027 q^{21} -1.32893 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.88462 q^{29} +9.73831 q^{31} +1.55469 q^{33} +4.08027 q^{35} +3.32893 q^{37} -6.10335 q^{39} +0.488048 q^{41} -3.81798 q^{43} +1.00000 q^{45} +5.14691 q^{47} +9.64858 q^{49} +3.65804 q^{51} -7.45294 q^{53} -1.55469 q^{55} +1.29240 q^{57} -4.10335 q^{59} +3.29240 q^{61} +4.08027 q^{63} +6.10335 q^{65} +1.00000 q^{67} +1.32893 q^{69} -7.61530 q^{71} +8.17702 q^{73} -1.00000 q^{75} -6.34355 q^{77} +5.31305 q^{79} +1.00000 q^{81} -11.1376 q^{83} -3.65804 q^{85} -6.88462 q^{87} -1.05016 q^{89} +24.9033 q^{91} -9.73831 q^{93} -1.29240 q^{95} -5.60930 q^{97} -1.55469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.08027	1.54220	0.771098	−	0.636716i	$-0.219708\pi$
$7$	4.08027	1.54220	0.771098	+	0.636716i	$0.219708\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.55469	−0.468757	−0.234378	−	0.972145i	$-0.575305\pi$
$11$	−1.55469	−0.468757	−0.234378	+	0.972145i	$0.575305\pi$
$12$	0	0
$13$	6.10335	1.69276	0.846382	−	0.532576i	$-0.178776\pi$
$13$	6.10335	1.69276	0.846382	+	0.532576i	$0.178776\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−3.65804	−0.887205	−0.443602	−	0.896224i	$-0.646300\pi$
$17$	−3.65804	−0.887205	−0.443602	+	0.896224i	$0.646300\pi$
$18$	0	0
$19$	−1.29240	−0.296497	−0.148248	−	0.988950i	$-0.547364\pi$
$19$	−1.29240	−0.296497	−0.148248	+	0.988950i	$0.547364\pi$
$20$	0	0
$21$	−4.08027	−0.890387
$22$	0	0
$23$	−1.32893	−0.277102	−0.138551	−	0.990355i	$-0.544244\pi$
$23$	−1.32893	−0.277102	−0.138551	+	0.990355i	$0.544244\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.88462	1.27844	0.639221	−	0.769023i	$-0.279257\pi$
$29$	6.88462	1.27844	0.639221	+	0.769023i	$0.279257\pi$
$30$	0	0
$31$	9.73831	1.74905	0.874526	−	0.484979i	$-0.161173\pi$
$31$	9.73831	1.74905	0.874526	+	0.484979i	$0.161173\pi$
$32$	0	0
$33$	1.55469	0.270637
$34$	0	0
$35$	4.08027	0.689691
$36$	0	0
$37$	3.32893	0.547273	0.273637	−	0.961833i	$-0.411773\pi$
$37$	3.32893	0.547273	0.273637	+	0.961833i	$0.411773\pi$
$38$	0	0
$39$	−6.10335	−0.977318
$40$	0	0
$41$	0.488048	0.0762204	0.0381102	−	0.999274i	$-0.487866\pi$
$41$	0.488048	0.0762204	0.0381102	+	0.999274i	$0.487866\pi$
$42$	0	0
$43$	−3.81798	−0.582236	−0.291118	−	0.956687i	$-0.594027\pi$
$43$	−3.81798	−0.582236	−0.291118	+	0.956687i	$0.594027\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	5.14691	0.750754	0.375377	−	0.926872i	$-0.377513\pi$
$47$	5.14691	0.750754	0.375377	+	0.926872i	$0.377513\pi$
$48$	0	0
$49$	9.64858	1.37837
$50$	0	0
$51$	3.65804	0.512228
$52$	0	0
$53$	−7.45294	−1.02374	−0.511870	−	0.859063i	$-0.671047\pi$
$53$	−7.45294	−1.02374	−0.511870	+	0.859063i	$0.671047\pi$
$54$	0	0
$55$	−1.55469	−0.209634
$56$	0	0
$57$	1.29240	0.171183
$58$	0	0
$59$	−4.10335	−0.534210	−0.267105	−	0.963667i	$-0.586067\pi$
$59$	−4.10335	−0.534210	−0.267105	+	0.963667i	$0.586067\pi$
$60$	0	0
$61$	3.29240	0.421549	0.210774	−	0.977535i	$-0.432401\pi$
$61$	3.29240	0.421549	0.210774	+	0.977535i	$0.432401\pi$
$62$	0	0
$63$	4.08027	0.514065
$64$	0	0
$65$	6.10335	0.757027
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	1.32893	0.159985
$70$	0	0
$71$	−7.61530	−0.903770	−0.451885	−	0.892076i	$-0.649248\pi$
$71$	−7.61530	−0.903770	−0.451885	+	0.892076i	$0.649248\pi$
$72$	0	0
$73$	8.17702	0.957048	0.478524	−	0.878075i	$-0.341172\pi$
$73$	8.17702	0.957048	0.478524	+	0.878075i	$0.341172\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−6.34355	−0.722915
$78$	0	0
$79$	5.31305	0.597765	0.298883	−	0.954290i	$-0.403386\pi$
$79$	5.31305	0.597765	0.298883	+	0.954290i	$0.403386\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.1376	−1.22251	−0.611257	−	0.791432i	$-0.709336\pi$
$83$	−11.1376	−1.22251	−0.611257	+	0.791432i	$0.709336\pi$
$84$	0	0
$85$	−3.65804	−0.396770
$86$	0	0
$87$	−6.88462	−0.738109
$88$	0	0
$89$	−1.05016	−0.111317	−0.0556583	−	0.998450i	$-0.517726\pi$
$89$	−1.05016	−0.111317	−0.0556583	+	0.998450i	$0.517726\pi$
$90$	0	0
$91$	24.9033	2.61057
$92$	0	0
$93$	−9.73831	−1.00982
$94$	0	0
$95$	−1.29240	−0.132597
$96$	0	0
$97$	−5.60930	−0.569538	−0.284769	−	0.958596i	$-0.591917\pi$
$97$	−5.60930	−0.569538	−0.284769	+	0.958596i	$0.591917\pi$
$98$	0	0
$99$	−1.55469	−0.156252
$100$	0	0
$101$	−12.0743	−1.20143	−0.600717	−	0.799462i	$-0.705118\pi$
$101$	−12.0743	−1.20143	−0.600717	+	0.799462i	$0.705118\pi$
$102$	0	0
$103$	−4.47584	−0.441018	−0.220509	−	0.975385i	$-0.570772\pi$
$103$	−4.47584	−0.441018	−0.220509	+	0.975385i	$0.570772\pi$
$104$	0	0
$105$	−4.08027	−0.398193
$106$	0	0
$107$	1.06664	0.103116	0.0515581	−	0.998670i	$-0.483581\pi$
$107$	1.06664	0.103116	0.0515581	+	0.998670i	$0.483581\pi$
$108$	0	0
$109$	−5.69518	−0.545499	−0.272750	−	0.962085i	$-0.587933\pi$
$109$	−5.69518	−0.545499	−0.272750	+	0.962085i	$0.587933\pi$
$110$	0	0
$111$	−3.32893	−0.315968
$112$	0	0
$113$	17.7624	1.67094	0.835472	−	0.549533i	$-0.185194\pi$
$113$	17.7624	1.67094	0.835472	+	0.549533i	$0.185194\pi$
$114$	0	0
$115$	−1.32893	−0.123924
$116$	0	0
$117$	6.10335	0.564255
$118$	0	0
$119$	−14.9258	−1.36824
$120$	0	0
$121$	−8.58294	−0.780267
$122$	0	0
$123$	−0.488048	−0.0440058
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.2143	−1.35005	−0.675026	−	0.737794i	$-0.735868\pi$
$127$	−15.2143	−1.35005	−0.675026	+	0.737794i	$0.735868\pi$
$128$	0	0
$129$	3.81798	0.336154
$130$	0	0
$131$	1.65804	0.144863	0.0724317	−	0.997373i	$-0.476924\pi$
$131$	1.65804	0.144863	0.0724317	+	0.997373i	$0.476924\pi$
$132$	0	0
$133$	−5.27334	−0.457256
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	13.5418	1.15696	0.578479	−	0.815698i	$-0.303647\pi$
$137$	13.5418	1.15696	0.578479	+	0.815698i	$0.303647\pi$
$138$	0	0
$139$	4.64541	0.394018	0.197009	−	0.980402i	$-0.436877\pi$
$139$	4.64541	0.394018	0.197009	+	0.980402i	$0.436877\pi$
$140$	0	0
$141$	−5.14691	−0.433448
$142$	0	0
$143$	−9.48882	−0.793495
$144$	0	0
$145$	6.88462	0.571737
$146$	0	0
$147$	−9.64858	−0.795802
$148$	0	0
$149$	21.5263	1.76351	0.881754	−	0.471710i	$-0.156363\pi$
$149$	21.5263	1.76351	0.881754	+	0.471710i	$0.156363\pi$
$150$	0	0
$151$	−11.4174	−0.929135	−0.464567	−	0.885538i	$-0.653790\pi$
$151$	−11.4174	−0.929135	−0.464567	+	0.885538i	$0.653790\pi$
$152$	0	0
$153$	−3.65804	−0.295735
$154$	0	0
$155$	9.73831	0.782200
$156$	0	0
$157$	−8.50651	−0.678894	−0.339447	−	0.940625i	$-0.610240\pi$
$157$	−8.50651	−0.678894	−0.339447	+	0.940625i	$0.610240\pi$
$158$	0	0
$159$	7.45294	0.591056
$160$	0	0
$161$	−5.42240	−0.427345
$162$	0	0
$163$	6.26229	0.490500	0.245250	−	0.969460i	$-0.421130\pi$
$163$	6.26229	0.490500	0.245250	+	0.969460i	$0.421130\pi$
$164$	0	0
$165$	1.55469	0.121033
$166$	0	0
$167$	19.9725	1.54552	0.772760	−	0.634699i	$-0.218876\pi$
$167$	19.9725	1.54552	0.772760	+	0.634699i	$0.218876\pi$
$168$	0	0
$169$	24.2509	1.86545
$170$	0	0
$171$	−1.29240	−0.0988323
$172$	0	0
$173$	−4.49650	−0.341862	−0.170931	−	0.985283i	$-0.554678\pi$
$173$	−4.49650	−0.341862	−0.170931	+	0.985283i	$0.554678\pi$
$174$	0	0
$175$	4.08027	0.308439
$176$	0	0
$177$	4.10335	0.308427
$178$	0	0
$179$	3.78144	0.282638	0.141319	−	0.989964i	$-0.454866\pi$
$179$	3.78144	0.282638	0.141319	+	0.989964i	$0.454866\pi$
$180$	0	0
$181$	11.8025	0.877272	0.438636	−	0.898665i	$-0.355462\pi$
$181$	11.8025	0.877272	0.438636	+	0.898665i	$0.355462\pi$
$182$	0	0
$183$	−3.29240	−0.243381
$184$	0	0
$185$	3.32893	0.244748
$186$	0	0
$187$	5.68712	0.415883
$188$	0	0
$189$	−4.08027	−0.296796
$190$	0	0
$191$	5.20092	0.376326	0.188163	−	0.982138i	$-0.439747\pi$
$191$	5.20092	0.376326	0.188163	+	0.982138i	$0.439747\pi$
$192$	0	0
$193$	18.3863	1.32347	0.661737	−	0.749736i	$-0.269819\pi$
$193$	18.3863	1.32347	0.661737	+	0.749736i	$0.269819\pi$
$194$	0	0
$195$	−6.10335	−0.437070
$196$	0	0
$197$	9.05736	0.645310	0.322655	−	0.946517i	$-0.395425\pi$
$197$	9.05736	0.645310	0.322655	+	0.946517i	$0.395425\pi$
$198$	0	0
$199$	−2.92308	−0.207212	−0.103606	−	0.994618i	$-0.533038\pi$
$199$	−2.92308	−0.207212	−0.103606	+	0.994618i	$0.533038\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	28.0911	1.97161
$204$	0	0
$205$	0.488048	0.0340868
$206$	0	0
$207$	−1.32893	−0.0923672
$208$	0	0
$209$	2.00928	0.138985
$210$	0	0
$211$	5.56189	0.382896	0.191448	−	0.981503i	$-0.438682\pi$
$211$	5.56189	0.382896	0.191448	+	0.981503i	$0.438682\pi$
$212$	0	0
$213$	7.61530	0.521792
$214$	0	0
$215$	−3.81798	−0.260384
$216$	0	0
$217$	39.7349	2.69738
$218$	0	0
$219$	−8.17702	−0.552552
$220$	0	0
$221$	−22.3263	−1.50183
$222$	0	0
$223$	−20.6245	−1.38112	−0.690560	−	0.723275i	$-0.742636\pi$
$223$	−20.6245	−1.38112	−0.690560	+	0.723275i	$0.742636\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.0345	0.666010	0.333005	−	0.942925i	$-0.391937\pi$
$227$	10.0345	0.666010	0.333005	+	0.942925i	$0.391937\pi$
$228$	0	0
$229$	21.6169	1.42848	0.714242	−	0.699899i	$-0.246772\pi$
$229$	21.6169	1.42848	0.714242	+	0.699899i	$0.246772\pi$
$230$	0	0
$231$	6.34355	0.417375
$232$	0	0
$233$	−0.522003	−0.0341976	−0.0170988	−	0.999854i	$-0.505443\pi$
$233$	−0.522003	−0.0341976	−0.0170988	+	0.999854i	$0.505443\pi$
$234$	0	0
$235$	5.14691	0.335747
$236$	0	0
$237$	−5.31305	−0.345120
$238$	0	0
$239$	25.4871	1.64862	0.824312	−	0.566136i	$-0.191562\pi$
$239$	25.4871	1.64862	0.824312	+	0.566136i	$0.191562\pi$
$240$	0	0
$241$	−26.6707	−1.71801	−0.859005	−	0.511968i	$-0.828917\pi$
$241$	−26.6707	−1.71801	−0.859005	+	0.511968i	$0.828917\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	9.64858	0.616425
$246$	0	0
$247$	−7.88797	−0.501899
$248$	0	0
$249$	11.1376	0.705818
$250$	0	0
$251$	−17.6394	−1.11339	−0.556694	−	0.830718i	$-0.687930\pi$
$251$	−17.6394	−1.11339	−0.556694	+	0.830718i	$0.687930\pi$
$252$	0	0
$253$	2.06608	0.129893
$254$	0	0
$255$	3.65804	0.229075
$256$	0	0
$257$	−5.39674	−0.336640	−0.168320	−	0.985732i	$-0.553834\pi$
$257$	−5.39674	−0.336640	−0.168320	+	0.985732i	$0.553834\pi$
$258$	0	0
$259$	13.5829	0.844003
$260$	0	0
$261$	6.88462	0.426147
$262$	0	0
$263$	14.0369	0.865551	0.432775	−	0.901502i	$-0.357534\pi$
$263$	14.0369	0.865551	0.432775	+	0.901502i	$0.357534\pi$
$264$	0	0
$265$	−7.45294	−0.457830
$266$	0	0
$267$	1.05016	0.0642686
$268$	0	0
$269$	3.49789	0.213270	0.106635	−	0.994298i	$-0.465992\pi$
$269$	3.49789	0.213270	0.106635	+	0.994298i	$0.465992\pi$
$270$	0	0
$271$	−9.48387	−0.576104	−0.288052	−	0.957615i	$-0.593008\pi$
$271$	−9.48387	−0.576104	−0.288052	+	0.957615i	$0.593008\pi$
$272$	0	0
$273$	−24.9033	−1.50722
$274$	0	0
$275$	−1.55469	−0.0937514
$276$	0	0
$277$	−12.4100	−0.745643	−0.372821	−	0.927903i	$-0.621610\pi$
$277$	−12.4100	−0.745643	−0.372821	+	0.927903i	$0.621610\pi$
$278$	0	0
$279$	9.73831	0.583017
$280$	0	0
$281$	−15.6699	−0.934788	−0.467394	−	0.884049i	$-0.654807\pi$
$281$	−15.6699	−0.934788	−0.467394	+	0.884049i	$0.654807\pi$
$282$	0	0
$283$	28.5433	1.69672	0.848360	−	0.529420i	$-0.177590\pi$
$283$	28.5433	1.69672	0.848360	+	0.529420i	$0.177590\pi$
$284$	0	0
$285$	1.29240	0.0765552
$286$	0	0
$287$	1.99137	0.117547
$288$	0	0
$289$	−3.61875	−0.212868
$290$	0	0
$291$	5.60930	0.328823
$292$	0	0
$293$	10.6430	0.621770	0.310885	−	0.950448i	$-0.399375\pi$
$293$	10.6430	0.621770	0.310885	+	0.950448i	$0.399375\pi$
$294$	0	0
$295$	−4.10335	−0.238906
$296$	0	0
$297$	1.55469	0.0902123
$298$	0	0
$299$	−8.11094	−0.469068
$300$	0	0
$301$	−15.5784	−0.897922
$302$	0	0
$303$	12.0743	0.693649
$304$	0	0
$305$	3.29240	0.188522
$306$	0	0
$307$	−32.6978	−1.86616	−0.933081	−	0.359667i	$-0.882890\pi$
$307$	−32.6978	−1.86616	−0.933081	+	0.359667i	$0.882890\pi$
$308$	0	0
$309$	4.47584	0.254622
$310$	0	0
$311$	−24.9475	−1.41464	−0.707320	−	0.706893i	$-0.750096\pi$
$311$	−24.9475	−1.41464	−0.707320	+	0.706893i	$0.750096\pi$
$312$	0	0
$313$	29.5938	1.67274	0.836371	−	0.548164i	$-0.184673\pi$
$313$	29.5938	1.67274	0.836371	+	0.548164i	$0.184673\pi$
$314$	0	0
$315$	4.08027	0.229897
$316$	0	0
$317$	−9.95721	−0.559252	−0.279626	−	0.960109i	$-0.590211\pi$
$317$	−9.95721	−0.559252	−0.279626	+	0.960109i	$0.590211\pi$
$318$	0	0
$319$	−10.7035	−0.599278
$320$	0	0
$321$	−1.06664	−0.0595342
$322$	0	0
$323$	4.72765	0.263053
$324$	0	0
$325$	6.10335	0.338553
$326$	0	0
$327$	5.69518	0.314944
$328$	0	0
$329$	21.0008	1.15781
$330$	0	0
$331$	13.3687	0.734810	0.367405	−	0.930061i	$-0.380246\pi$
$331$	13.3687	0.734810	0.367405	+	0.930061i	$0.380246\pi$
$332$	0	0
$333$	3.32893	0.182424
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	17.9996	0.980501	0.490250	−	0.871582i	$-0.336905\pi$
$337$	17.9996	0.980501	0.490250	+	0.871582i	$0.336905\pi$
$338$	0	0
$339$	−17.7624	−0.964720
$340$	0	0
$341$	−15.1401	−0.819880
$342$	0	0
$343$	10.8069	0.583519
$344$	0	0
$345$	1.32893	0.0715473
$346$	0	0
$347$	14.1938	0.761965	0.380983	−	0.924582i	$-0.375586\pi$
$347$	14.1938	0.761965	0.380983	+	0.924582i	$0.375586\pi$
$348$	0	0
$349$	0.000650308 0	3.48102e−5 0	1.74051e−5	−	1.00000i	$-0.499994\pi$
$349$	0.000650308 0	3.48102e−5 0	1.74051e−5		1.00000i	$0.499994\pi$
$350$	0	0
$351$	−6.10335	−0.325773
$352$	0	0
$353$	19.4207	1.03366	0.516829	−	0.856088i	$-0.327112\pi$
$353$	19.4207	1.03366	0.516829	+	0.856088i	$0.327112\pi$
$354$	0	0
$355$	−7.61530	−0.404178
$356$	0	0
$357$	14.9258	0.789956
$358$	0	0
$359$	−21.6044	−1.14024	−0.570119	−	0.821562i	$-0.693103\pi$
$359$	−21.6044	−1.14024	−0.570119	+	0.821562i	$0.693103\pi$
$360$	0	0
$361$	−17.3297	−0.912090
$362$	0	0
$363$	8.58294	0.450487
$364$	0	0
$365$	8.17702	0.428005
$366$	0	0
$367$	14.7191	0.768330	0.384165	−	0.923265i	$-0.374489\pi$
$367$	14.7191	0.768330	0.384165	+	0.923265i	$0.374489\pi$
$368$	0	0
$369$	0.488048	0.0254068
$370$	0	0
$371$	−30.4100	−1.57881
$372$	0	0
$373$	22.3422	1.15683	0.578417	−	0.815742i	$-0.303671\pi$
$373$	22.3422	1.15683	0.578417	+	0.815742i	$0.303671\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	42.0192	2.16410
$378$	0	0
$379$	−10.2569	−0.526861	−0.263431	−	0.964678i	$-0.584854\pi$
$379$	−10.2569	−0.526861	−0.263431	+	0.964678i	$0.584854\pi$
$380$	0	0
$381$	15.2143	0.779453
$382$	0	0
$383$	−12.6506	−0.646414	−0.323207	−	0.946328i	$-0.604761\pi$
$383$	−12.6506	−0.646414	−0.323207	+	0.946328i	$0.604761\pi$
$384$	0	0
$385$	−6.34355	−0.323297
$386$	0	0
$387$	−3.81798	−0.194079
$388$	0	0
$389$	−14.4606	−0.733180	−0.366590	−	0.930383i	$-0.619475\pi$
$389$	−14.4606	−0.733180	−0.366590	+	0.930383i	$0.619475\pi$
$390$	0	0
$391$	4.86129	0.245846
$392$	0	0
$393$	−1.65804	−0.0836370
$394$	0	0
$395$	5.31305	0.267329
$396$	0	0
$397$	26.6829	1.33918	0.669588	−	0.742733i	$-0.266471\pi$
$397$	26.6829	1.33918	0.669588	+	0.742733i	$0.266471\pi$
$398$	0	0
$399$	5.27334	0.263997
$400$	0	0
$401$	25.5884	1.27782	0.638912	−	0.769280i	$-0.279385\pi$
$401$	25.5884	1.27782	0.638912	+	0.769280i	$0.279385\pi$
$402$	0	0
$403$	59.4363	2.96073
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−5.17546	−0.256538
$408$	0	0
$409$	22.7786	1.12633	0.563166	−	0.826344i	$-0.309583\pi$
$409$	22.7786	1.12633	0.563166	+	0.826344i	$0.309583\pi$
$410$	0	0
$411$	−13.5418	−0.667969
$412$	0	0
$413$	−16.7428	−0.823857
$414$	0	0
$415$	−11.1376	−0.546725
$416$	0	0
$417$	−4.64541	−0.227487
$418$	0	0
$419$	−29.1461	−1.42388	−0.711939	−	0.702241i	$-0.752183\pi$
$419$	−29.1461	−1.42388	−0.711939	+	0.702241i	$0.752183\pi$
$420$	0	0
$421$	−9.79713	−0.477483	−0.238741	−	0.971083i	$-0.576735\pi$
$421$	−9.79713	−0.477483	−0.238741	+	0.971083i	$0.576735\pi$
$422$	0	0
$423$	5.14691	0.250251
$424$	0	0
$425$	−3.65804	−0.177441
$426$	0	0
$427$	13.4339	0.650111
$428$	0	0
$429$	9.48882	0.458124
$430$	0	0
$431$	34.1182	1.64341	0.821707	−	0.569910i	$-0.193022\pi$
$431$	34.1182	1.64341	0.821707	+	0.569910i	$0.193022\pi$
$432$	0	0
$433$	−24.6949	−1.18676	−0.593381	−	0.804922i	$-0.702207\pi$
$433$	−24.6949	−1.18676	−0.593381	+	0.804922i	$0.702207\pi$
$434$	0	0
$435$	−6.88462	−0.330092
$436$	0	0
$437$	1.71751	0.0821598
$438$	0	0
$439$	15.8347	0.755747	0.377874	−	0.925857i	$-0.376655\pi$
$439$	15.8347	0.755747	0.377874	+	0.925857i	$0.376655\pi$
$440$	0	0
$441$	9.64858	0.459456
$442$	0	0
$443$	−1.51652	−0.0720521	−0.0360260	−	0.999351i	$-0.511470\pi$
$443$	−1.51652	−0.0720521	−0.0360260	+	0.999351i	$0.511470\pi$
$444$	0	0
$445$	−1.05016	−0.0497823
$446$	0	0
$447$	−21.5263	−1.01816
$448$	0	0
$449$	−9.57837	−0.452031	−0.226016	−	0.974124i	$-0.572570\pi$
$449$	−9.57837	−0.452031	−0.226016	+	0.974124i	$0.572570\pi$
$450$	0	0
$451$	−0.758764	−0.0357288
$452$	0	0
$453$	11.4174	0.536436
$454$	0	0
$455$	24.9033	1.16748
$456$	0	0
$457$	22.4690	1.05105	0.525527	−	0.850777i	$-0.323868\pi$
$457$	22.4690	1.05105	0.525527	+	0.850777i	$0.323868\pi$
$458$	0	0
$459$	3.65804	0.170743
$460$	0	0
$461$	−22.8997	−1.06654	−0.533272	−	0.845944i	$-0.679038\pi$
$461$	−22.8997	−1.06654	−0.533272	+	0.845944i	$0.679038\pi$
$462$	0	0
$463$	1.81579	0.0843868	0.0421934	−	0.999109i	$-0.486565\pi$
$463$	1.81579	0.0843868	0.0421934	+	0.999109i	$0.486565\pi$
$464$	0	0
$465$	−9.73831	−0.451603
$466$	0	0
$467$	−4.91153	−0.227279	−0.113639	−	0.993522i	$-0.536251\pi$
$467$	−4.91153	−0.227279	−0.113639	+	0.993522i	$0.536251\pi$
$468$	0	0
$469$	4.08027	0.188409
$470$	0	0
$471$	8.50651	0.391960
$472$	0	0
$473$	5.93577	0.272927
$474$	0	0
$475$	−1.29240	−0.0592994
$476$	0	0
$477$	−7.45294	−0.341246
$478$	0	0
$479$	−28.6886	−1.31081	−0.655407	−	0.755276i	$-0.727503\pi$
$479$	−28.6886	−1.31081	−0.655407	+	0.755276i	$0.727503\pi$
$480$	0	0
$481$	20.3176	0.926404
$482$	0	0
$483$	5.42240	0.246728
$484$	0	0
$485$	−5.60930	−0.254705
$486$	0	0
$487$	−34.7005	−1.57243	−0.786215	−	0.617953i	$-0.787962\pi$
$487$	−34.7005	−1.57243	−0.786215	+	0.617953i	$0.787962\pi$
$488$	0	0
$489$	−6.26229	−0.283191
$490$	0	0
$491$	−38.3582	−1.73108	−0.865541	−	0.500838i	$-0.833025\pi$
$491$	−38.3582	−1.73108	−0.865541	+	0.500838i	$0.833025\pi$
$492$	0	0
$493$	−25.1842	−1.13424
$494$	0	0
$495$	−1.55469	−0.0698781
$496$	0	0
$497$	−31.0725	−1.39379
$498$	0	0
$499$	−6.43221	−0.287945	−0.143973	−	0.989582i	$-0.545988\pi$
$499$	−6.43221	−0.287945	−0.143973	+	0.989582i	$0.545988\pi$
$500$	0	0
$501$	−19.9725	−0.892306
$502$	0	0
$503$	35.7299	1.59312	0.796559	−	0.604561i	$-0.206652\pi$
$503$	35.7299	1.59312	0.796559	+	0.604561i	$0.206652\pi$
$504$	0	0
$505$	−12.0743	−0.537298
$506$	0	0
$507$	−24.2509	−1.07702
$508$	0	0
$509$	−3.14134	−0.139238	−0.0696188	−	0.997574i	$-0.522178\pi$
$509$	−3.14134	−0.139238	−0.0696188	+	0.997574i	$0.522178\pi$
$510$	0	0
$511$	33.3644	1.47596
$512$	0	0
$513$	1.29240	0.0570609
$514$	0	0
$515$	−4.47584	−0.197229
$516$	0	0
$517$	−8.00185	−0.351921
$518$	0	0
$519$	4.49650	0.197374
$520$	0	0
$521$	20.4993	0.898090	0.449045	−	0.893509i	$-0.351764\pi$
$521$	20.4993	0.898090	0.449045	+	0.893509i	$0.351764\pi$
$522$	0	0
$523$	1.68256	0.0735731	0.0367866	−	0.999323i	$-0.488288\pi$
$523$	1.68256	0.0735731	0.0367866	+	0.999323i	$0.488288\pi$
$524$	0	0
$525$	−4.08027	−0.178077
$526$	0	0
$527$	−35.6231	−1.55177
$528$	0	0
$529$	−21.2339	−0.923215
$530$	0	0
$531$	−4.10335	−0.178070
$532$	0	0
$533$	2.97873	0.129023
$534$	0	0
$535$	1.06664	0.0461150
$536$	0	0
$537$	−3.78144	−0.163181
$538$	0	0
$539$	−15.0006	−0.646120
$540$	0	0
$541$	6.06934	0.260941	0.130471	−	0.991452i	$-0.458351\pi$
$541$	6.06934	0.260941	0.130471	+	0.991452i	$0.458351\pi$
$542$	0	0
$543$	−11.8025	−0.506493
$544$	0	0
$545$	−5.69518	−0.243955
$546$	0	0
$547$	−28.2421	−1.20755	−0.603773	−	0.797156i	$-0.706337\pi$
$547$	−28.2421	−1.20755	−0.603773	+	0.797156i	$0.706337\pi$
$548$	0	0
$549$	3.29240	0.140516
$550$	0	0
$551$	−8.89768	−0.379054
$552$	0	0
$553$	21.6787	0.921871
$554$	0	0
$555$	−3.32893	−0.141305
$556$	0	0
$557$	−18.0859	−0.766326	−0.383163	−	0.923681i	$-0.625165\pi$
$557$	−18.0859	−0.766326	−0.383163	+	0.923681i	$0.625165\pi$
$558$	0	0
$559$	−23.3024	−0.985588
$560$	0	0
$561$	−5.68712	−0.240110
$562$	0	0
$563$	−40.2081	−1.69457	−0.847285	−	0.531139i	$-0.821764\pi$
$563$	−40.2081	−1.69457	−0.847285	+	0.531139i	$0.821764\pi$
$564$	0	0
$565$	17.7624	0.747269
$566$	0	0
$567$	4.08027	0.171355
$568$	0	0
$569$	10.1367	0.424953	0.212476	−	0.977166i	$-0.431847\pi$
$569$	10.1367	0.424953	0.212476	+	0.977166i	$0.431847\pi$
$570$	0	0
$571$	−41.6424	−1.74268	−0.871341	−	0.490679i	$-0.836749\pi$
$571$	−41.6424	−1.74268	−0.871341	+	0.490679i	$0.836749\pi$
$572$	0	0
$573$	−5.20092	−0.217272
$574$	0	0
$575$	−1.32893	−0.0554203
$576$	0	0
$577$	45.8986	1.91078	0.955392	−	0.295341i	$-0.0954332\pi$
$577$	45.8986	1.91078	0.955392	+	0.295341i	$0.0954332\pi$
$578$	0	0
$579$	−18.3863	−0.764108
$580$	0	0
$581$	−45.4445	−1.88536
$582$	0	0
$583$	11.5870	0.479885
$584$	0	0
$585$	6.10335	0.252342
$586$	0	0
$587$	−7.91874	−0.326841	−0.163421	−	0.986556i	$-0.552253\pi$
$587$	−7.91874	−0.326841	−0.163421	+	0.986556i	$0.552253\pi$
$588$	0	0
$589$	−12.5858	−0.518588
$590$	0	0
$591$	−9.05736	−0.372570
$592$	0	0
$593$	15.7590	0.647143	0.323571	−	0.946204i	$-0.395116\pi$
$593$	15.7590	0.647143	0.323571	+	0.946204i	$0.395116\pi$
$594$	0	0
$595$	−14.9258	−0.611897
$596$	0	0
$597$	2.92308	0.119634
$598$	0	0
$599$	11.0815	0.452779	0.226390	−	0.974037i	$-0.427308\pi$
$599$	11.0815	0.452779	0.226390	+	0.974037i	$0.427308\pi$
$600$	0	0
$601$	−7.43090	−0.303113	−0.151556	−	0.988449i	$-0.548428\pi$
$601$	−7.43090	−0.303113	−0.151556	+	0.988449i	$0.548428\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−8.58294	−0.348946
$606$	0	0
$607$	2.88725	0.117190	0.0585950	−	0.998282i	$-0.481338\pi$
$607$	2.88725	0.117190	0.0585950	+	0.998282i	$0.481338\pi$
$608$	0	0
$609$	−28.0911	−1.13831
$610$	0	0
$611$	31.4134	1.27085
$612$	0	0
$613$	1.34542	0.0543409	0.0271704	−	0.999631i	$-0.491350\pi$
$613$	1.34542	0.0543409	0.0271704	+	0.999631i	$0.491350\pi$
$614$	0	0
$615$	−0.488048	−0.0196800
$616$	0	0
$617$	31.7240	1.27716	0.638580	−	0.769555i	$-0.279522\pi$
$617$	31.7240	1.27716	0.638580	+	0.769555i	$0.279522\pi$
$618$	0	0
$619$	28.5844	1.14891	0.574453	−	0.818538i	$-0.305215\pi$
$619$	28.5844	1.14891	0.574453	+	0.818538i	$0.305215\pi$
$620$	0	0
$621$	1.32893	0.0533282
$622$	0	0
$623$	−4.28493	−0.171672
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.00928	−0.0802430
$628$	0	0
$629$	−12.1774	−0.485543
$630$	0	0
$631$	−15.5728	−0.619942	−0.309971	−	0.950746i	$-0.600319\pi$
$631$	−15.5728	−0.619942	−0.309971	+	0.950746i	$0.600319\pi$
$632$	0	0
$633$	−5.56189	−0.221065
$634$	0	0
$635$	−15.2143	−0.603762
$636$	0	0
$637$	58.8887	2.33325
$638$	0	0
$639$	−7.61530	−0.301257
$640$	0	0
$641$	40.0659	1.58251	0.791254	−	0.611488i	$-0.209429\pi$
$641$	40.0659	1.58251	0.791254	+	0.611488i	$0.209429\pi$
$642$	0	0
$643$	11.3700	0.448390	0.224195	−	0.974544i	$-0.428025\pi$
$643$	11.3700	0.448390	0.224195	+	0.974544i	$0.428025\pi$
$644$	0	0
$645$	3.81798	0.150333
$646$	0	0
$647$	−39.1584	−1.53947	−0.769737	−	0.638361i	$-0.779613\pi$
$647$	−39.1584	−1.53947	−0.769737	+	0.638361i	$0.779613\pi$
$648$	0	0
$649$	6.37944	0.250415
$650$	0	0
$651$	−39.7349	−1.55733
$652$	0	0
$653$	5.68568	0.222498	0.111249	−	0.993793i	$-0.464515\pi$
$653$	5.68568	0.222498	0.111249	+	0.993793i	$0.464515\pi$
$654$	0	0
$655$	1.65804	0.0647849
$656$	0	0
$657$	8.17702	0.319016
$658$	0	0
$659$	−41.3214	−1.60965	−0.804827	−	0.593509i	$-0.797742\pi$
$659$	−41.3214	−1.60965	−0.804827	+	0.593509i	$0.797742\pi$
$660$	0	0
$661$	−13.5351	−0.526456	−0.263228	−	0.964734i	$-0.584787\pi$
$661$	−13.5351	−0.526456	−0.263228	+	0.964734i	$0.584787\pi$
$662$	0	0
$663$	22.3263	0.867081
$664$	0	0
$665$	−5.27334	−0.204491
$666$	0	0
$667$	−9.14920	−0.354258
$668$	0	0
$669$	20.6245	0.797390
$670$	0	0
$671$	−5.11866	−0.197604
$672$	0	0
$673$	24.6153	0.948850	0.474425	−	0.880296i	$-0.342656\pi$
$673$	24.6153	0.948850	0.474425	+	0.880296i	$0.342656\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	12.5413	0.482003	0.241001	−	0.970525i	$-0.422524\pi$
$677$	12.5413	0.482003	0.241001	+	0.970525i	$0.422524\pi$
$678$	0	0
$679$	−22.8874	−0.878340
$680$	0	0
$681$	−10.0345	−0.384521
$682$	0	0
$683$	4.45957	0.170641	0.0853204	−	0.996354i	$-0.472809\pi$
$683$	4.45957	0.170641	0.0853204	+	0.996354i	$0.472809\pi$
$684$	0	0
$685$	13.5418	0.517407
$686$	0	0
$687$	−21.6169	−0.824736
$688$	0	0
$689$	−45.4879	−1.73295
$690$	0	0
$691$	−9.73527	−0.370347	−0.185174	−	0.982706i	$-0.559285\pi$
$691$	−9.73527	−0.370347	−0.185174	+	0.982706i	$0.559285\pi$
$692$	0	0
$693$	−6.34355	−0.240972
$694$	0	0
$695$	4.64541	0.176210
$696$	0	0
$697$	−1.78530	−0.0676231
$698$	0	0
$699$	0.522003	0.0197440
$700$	0	0
$701$	−41.7913	−1.57843	−0.789217	−	0.614114i	$-0.789514\pi$
$701$	−41.7913	−1.57843	−0.789217	+	0.614114i	$0.789514\pi$
$702$	0	0
$703$	−4.30231	−0.162265
$704$	0	0
$705$	−5.14691	−0.193844
$706$	0	0
$707$	−49.2662	−1.85285
$708$	0	0
$709$	−38.1078	−1.43117	−0.715584	−	0.698527i	$-0.753839\pi$
$709$	−38.1078	−1.43117	−0.715584	+	0.698527i	$0.753839\pi$
$710$	0	0
$711$	5.31305	0.199255
$712$	0	0
$713$	−12.9416	−0.484665
$714$	0	0
$715$	−9.48882	−0.354862
$716$	0	0
$717$	−25.4871	−0.951833
$718$	0	0
$719$	−15.6430	−0.583387	−0.291694	−	0.956512i	$-0.594219\pi$
$719$	−15.6430	−0.583387	−0.291694	+	0.956512i	$0.594219\pi$
$720$	0	0
$721$	−18.2626	−0.680136
$722$	0	0
$723$	26.6707	0.991893
$724$	0	0
$725$	6.88462	0.255688
$726$	0	0
$727$	−41.7851	−1.54972	−0.774862	−	0.632130i	$-0.782181\pi$
$727$	−41.7851	−1.54972	−0.774862	+	0.632130i	$0.782181\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.9663	0.516563
$732$	0	0
$733$	24.3013	0.897588	0.448794	−	0.893635i	$-0.351854\pi$
$733$	24.3013	0.897588	0.448794	+	0.893635i	$0.351854\pi$
$734$	0	0
$735$	−9.64858	−0.355893
$736$	0	0
$737$	−1.55469	−0.0572678
$738$	0	0
$739$	28.4701	1.04729	0.523646	−	0.851936i	$-0.324572\pi$
$739$	28.4701	1.04729	0.523646	+	0.851936i	$0.324572\pi$
$740$	0	0
$741$	7.88797	0.289772
$742$	0	0
$743$	−42.5677	−1.56166	−0.780829	−	0.624745i	$-0.785203\pi$
$743$	−42.5677	−1.56166	−0.780829	+	0.624745i	$0.785203\pi$
$744$	0	0
$745$	21.5263	0.788664
$746$	0	0
$747$	−11.1376	−0.407504
$748$	0	0
$749$	4.35219	0.159025
$750$	0	0
$751$	6.69440	0.244282	0.122141	−	0.992513i	$-0.461024\pi$
$751$	6.69440	0.244282	0.122141	+	0.992513i	$0.461024\pi$
$752$	0	0
$753$	17.6394	0.642815
$754$	0	0
$755$	−11.4174	−0.415522
$756$	0	0
$757$	−40.3265	−1.46569	−0.732846	−	0.680394i	$-0.761808\pi$
$757$	−40.3265	−1.46569	−0.732846	+	0.680394i	$0.761808\pi$
$758$	0	0
$759$	−2.06608	−0.0749939
$760$	0	0
$761$	−1.55484	−0.0563630	−0.0281815	−	0.999603i	$-0.508972\pi$
$761$	−1.55484	−0.0563630	−0.0281815	+	0.999603i	$0.508972\pi$
$762$	0	0
$763$	−23.2378	−0.841267
$764$	0	0
$765$	−3.65804	−0.132257
$766$	0	0
$767$	−25.0442	−0.904292
$768$	0	0
$769$	2.21926	0.0800286	0.0400143	−	0.999199i	$-0.487260\pi$
$769$	2.21926	0.0800286	0.0400143	+	0.999199i	$0.487260\pi$
$770$	0	0
$771$	5.39674	0.194359
$772$	0	0
$773$	17.3459	0.623889	0.311945	−	0.950100i	$-0.399020\pi$
$773$	17.3459	0.623889	0.311945	+	0.950100i	$0.399020\pi$
$774$	0	0
$775$	9.73831	0.349810
$776$	0	0
$777$	−13.5829	−0.487285
$778$	0	0
$779$	−0.630754	−0.0225991
$780$	0	0
$781$	11.8394	0.423648
$782$	0	0
$783$	−6.88462	−0.246036
$784$	0	0
$785$	−8.50651	−0.303611
$786$	0	0
$787$	−8.02626	−0.286105	−0.143053	−	0.989715i	$-0.545692\pi$
$787$	−8.02626	−0.286105	−0.143053	+	0.989715i	$0.545692\pi$
$788$	0	0
$789$	−14.0369	−0.499726
$790$	0	0
$791$	72.4753	2.57692
$792$	0	0
$793$	20.0947	0.713582
$794$	0	0
$795$	7.45294	0.264328
$796$	0	0
$797$	28.1659	0.997688	0.498844	−	0.866692i	$-0.333758\pi$
$797$	28.1659	0.997688	0.498844	+	0.866692i	$0.333758\pi$
$798$	0	0
$799$	−18.8276	−0.666072
$800$	0	0
$801$	−1.05016	−0.0371055
$802$	0	0
$803$	−12.7127	−0.448623
$804$	0	0
$805$	−5.42240	−0.191115
$806$	0	0
$807$	−3.49789	−0.123132
$808$	0	0
$809$	0.298810	0.0105056	0.00525280	−	0.999986i	$-0.498328\pi$
$809$	0.298810	0.0105056	0.00525280	+	0.999986i	$0.498328\pi$
$810$	0	0
$811$	−19.3546	−0.679634	−0.339817	−	0.940492i	$-0.610365\pi$
$811$	−19.3546	−0.679634	−0.339817	+	0.940492i	$0.610365\pi$
$812$	0	0
$813$	9.48387	0.332614
$814$	0	0
$815$	6.26229	0.219358
$816$	0	0
$817$	4.93435	0.172631
$818$	0	0
$819$	24.9033	0.870191
$820$	0	0
$821$	−17.3430	−0.605274	−0.302637	−	0.953106i	$-0.597867\pi$
$821$	−17.3430	−0.605274	−0.302637	+	0.953106i	$0.597867\pi$
$822$	0	0
$823$	56.6563	1.97491	0.987456	−	0.157892i	$-0.0504697\pi$
$823$	56.6563	1.97491	0.987456	+	0.157892i	$0.0504697\pi$
$824$	0	0
$825$	1.55469	0.0541274
$826$	0	0
$827$	−5.68350	−0.197635	−0.0988173	−	0.995106i	$-0.531506\pi$
$827$	−5.68350	−0.197635	−0.0988173	+	0.995106i	$0.531506\pi$
$828$	0	0
$829$	−16.5517	−0.574864	−0.287432	−	0.957801i	$-0.592802\pi$
$829$	−16.5517	−0.574864	−0.287432	+	0.957801i	$0.592802\pi$
$830$	0	0
$831$	12.4100	0.430497
$832$	0	0
$833$	−35.2949	−1.22290
$834$	0	0
$835$	19.9725	0.691177
$836$	0	0
$837$	−9.73831	−0.336605
$838$	0	0
$839$	53.2300	1.83770	0.918852	−	0.394602i	$-0.129118\pi$
$839$	53.2300	1.83770	0.918852	+	0.394602i	$0.129118\pi$
$840$	0	0
$841$	18.3980	0.634413
$842$	0	0
$843$	15.6699	0.539700
$844$	0	0
$845$	24.2509	0.834255
$846$	0	0
$847$	−35.0207	−1.20332
$848$	0	0
$849$	−28.5433	−0.979602
$850$	0	0
$851$	−4.42393	−0.151650
$852$	0	0
$853$	29.1651	0.998593	0.499297	−	0.866431i	$-0.333592\pi$
$853$	29.1651	0.998593	0.499297	+	0.866431i	$0.333592\pi$
$854$	0	0
$855$	−1.29240	−0.0441991
$856$	0	0
$857$	−24.5121	−0.837318	−0.418659	−	0.908143i	$-0.637500\pi$
$857$	−24.5121	−0.837318	−0.418659	+	0.908143i	$0.637500\pi$
$858$	0	0
$859$	−38.4218	−1.31093	−0.655467	−	0.755223i	$-0.727528\pi$
$859$	−38.4218	−1.31093	−0.655467	+	0.755223i	$0.727528\pi$
$860$	0	0
$861$	−1.99137	−0.0678656
$862$	0	0
$863$	−23.3046	−0.793297	−0.396648	−	0.917971i	$-0.629827\pi$
$863$	−23.3046	−0.793297	−0.396648	+	0.917971i	$0.629827\pi$
$864$	0	0
$865$	−4.49650	−0.152885
$866$	0	0
$867$	3.61875	0.122899
$868$	0	0
$869$	−8.26016	−0.280207
$870$	0	0
$871$	6.10335	0.206804
$872$	0	0
$873$	−5.60930	−0.189846
$874$	0	0
$875$	4.08027	0.137938
$876$	0	0
$877$	−30.1034	−1.01652	−0.508259	−	0.861204i	$-0.669711\pi$
$877$	−30.1034	−1.01652	−0.508259	+	0.861204i	$0.669711\pi$
$878$	0	0
$879$	−10.6430	−0.358979
$880$	0	0
$881$	32.7159	1.10223	0.551114	−	0.834430i	$-0.314203\pi$
$881$	32.7159	1.10223	0.551114	+	0.834430i	$0.314203\pi$
$882$	0	0
$883$	23.8206	0.801627	0.400814	−	0.916160i	$-0.368727\pi$
$883$	23.8206	0.801627	0.400814	+	0.916160i	$0.368727\pi$
$884$	0	0
$885$	4.10335	0.137933
$886$	0	0
$887$	41.7535	1.40195	0.700973	−	0.713188i	$-0.252750\pi$
$887$	41.7535	1.40195	0.700973	+	0.713188i	$0.252750\pi$
$888$	0	0
$889$	−62.0785	−2.08205
$890$	0	0
$891$	−1.55469	−0.0520841
$892$	0	0
$893$	−6.65187	−0.222596
$894$	0	0
$895$	3.78144	0.126400
$896$	0	0
$897$	8.11094	0.270816
$898$	0	0
$899$	67.0445	2.23606
$900$	0	0
$901$	27.2631	0.908266
$902$	0	0
$903$	15.5784	0.518416
$904$	0	0
$905$	11.8025	0.392328
$906$	0	0
$907$	−28.7866	−0.955845	−0.477922	−	0.878402i	$-0.658610\pi$
$907$	−28.7866	−0.955845	−0.477922	+	0.878402i	$0.658610\pi$
$908$	0	0
$909$	−12.0743	−0.400478
$910$	0	0
$911$	13.5833	0.450036	0.225018	−	0.974355i	$-0.427756\pi$
$911$	13.5833	0.450036	0.225018	+	0.974355i	$0.427756\pi$
$912$	0	0
$913$	17.3156	0.573062
$914$	0	0
$915$	−3.29240	−0.108843
$916$	0	0
$917$	6.76524	0.223408
$918$	0	0
$919$	−2.04149	−0.0673426	−0.0336713	−	0.999433i	$-0.510720\pi$
$919$	−2.04149	−0.0673426	−0.0336713	+	0.999433i	$0.510720\pi$
$920$	0	0
$921$	32.6978	1.07743
$922$	0	0
$923$	−46.4788	−1.52987
$924$	0	0
$925$	3.32893	0.109455
$926$	0	0
$927$	−4.47584	−0.147006
$928$	0	0
$929$	6.96768	0.228602	0.114301	−	0.993446i	$-0.463537\pi$
$929$	6.96768	0.228602	0.114301	+	0.993446i	$0.463537\pi$
$930$	0	0
$931$	−12.4698	−0.408682
$932$	0	0
$933$	24.9475	0.816743
$934$	0	0
$935$	5.68712	0.185989
$936$	0	0
$937$	27.7469	0.906450	0.453225	−	0.891396i	$-0.350273\pi$
$937$	27.7469	0.906450	0.453225	+	0.891396i	$0.350273\pi$
$938$	0	0
$939$	−29.5938	−0.965758
$940$	0	0
$941$	−30.0606	−0.979946	−0.489973	−	0.871737i	$-0.662993\pi$
$941$	−30.0606	−0.979946	−0.489973	+	0.871737i	$0.662993\pi$
$942$	0	0
$943$	−0.648584	−0.0211208
$944$	0	0
$945$	−4.08027	−0.132731
$946$	0	0
$947$	−13.2577	−0.430818	−0.215409	−	0.976524i	$-0.569108\pi$
$947$	−13.2577	−0.430818	−0.215409	+	0.976524i	$0.569108\pi$
$948$	0	0
$949$	49.9072	1.62006
$950$	0	0
$951$	9.95721	0.322885
$952$	0	0
$953$	−28.1941	−0.913296	−0.456648	−	0.889647i	$-0.650950\pi$
$953$	−28.1941	−0.913296	−0.456648	+	0.889647i	$0.650950\pi$
$954$	0	0
$955$	5.20092	0.168298
$956$	0	0
$957$	10.7035	0.345994
$958$	0	0
$959$	55.2543	1.78425
$960$	0	0
$961$	63.8346	2.05918
$962$	0	0
$963$	1.06664	0.0343721
$964$	0	0
$965$	18.3863	0.591876
$966$	0	0
$967$	23.1587	0.744735	0.372367	−	0.928085i	$-0.378546\pi$
$967$	23.1587	0.744735	0.372367	+	0.928085i	$0.378546\pi$
$968$	0	0
$969$	−4.72765	−0.151874
$970$	0	0
$971$	−19.9250	−0.639422	−0.319711	−	0.947515i	$-0.603586\pi$
$971$	−19.9250	−0.639422	−0.319711	+	0.947515i	$0.603586\pi$
$972$	0	0
$973$	18.9545	0.607654
$974$	0	0
$975$	−6.10335	−0.195464
$976$	0	0
$977$	−46.1374	−1.47607	−0.738033	−	0.674764i	$-0.764245\pi$
$977$	−46.1374	−1.47607	−0.738033	+	0.674764i	$0.764245\pi$
$978$	0	0
$979$	1.63267	0.0521804
$980$	0	0
$981$	−5.69518	−0.181833
$982$	0	0
$983$	−44.7930	−1.42867	−0.714337	−	0.699802i	$-0.753272\pi$
$983$	−44.7930	−1.42867	−0.714337	+	0.699802i	$0.753272\pi$
$984$	0	0
$985$	9.05736	0.288591
$986$	0	0
$987$	−21.0008	−0.668462
$988$	0	0
$989$	5.07384	0.161339
$990$	0	0
$991$	−9.10755	−0.289311	−0.144655	−	0.989482i	$-0.546207\pi$
$991$	−9.10755	−0.289311	−0.144655	+	0.989482i	$0.546207\pi$
$992$	0	0
$993$	−13.3687	−0.424243
$994$	0	0
$995$	−2.92308	−0.0926679
$996$	0	0
$997$	−46.7464	−1.48047	−0.740237	−	0.672346i	$-0.765287\pi$
$997$	−46.7464	−1.48047	−0.740237	+	0.672346i	$0.765287\pi$
$998$	0	0
$999$	−3.32893	−0.105323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.f.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.f.1.5	✓	6	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.08027 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.08027 q^{7} +1.00000 q^{9} -1.55469 q^{11} +6.10335 q^{13} -1.00000 q^{15} -3.65804 q^{17} -1.29240 q^{19} -4.08027 q^{21} -1.32893 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.88462 q^{29} +9.73831 q^{31} +1.55469 q^{33} +4.08027 q^{35} +3.32893 q^{37} -6.10335 q^{39} +0.488048 q^{41} -3.81798 q^{43} +1.00000 q^{45} +5.14691 q^{47} +9.64858 q^{49} +3.65804 q^{51} -7.45294 q^{53} -1.55469 q^{55} +1.29240 q^{57} -4.10335 q^{59} +3.29240 q^{61} +4.08027 q^{63} +6.10335 q^{65} +1.00000 q^{67} +1.32893 q^{69} -7.61530 q^{71} +8.17702 q^{73} -1.00000 q^{75} -6.34355 q^{77} +5.31305 q^{79} +1.00000 q^{81} -11.1376 q^{83} -3.65804 q^{85} -6.88462 q^{87} -1.05016 q^{89} +24.9033 q^{91} -9.73831 q^{93} -1.29240 q^{95} -5.60930 q^{97} -1.55469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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