LMFDB - Embedded newform 4020.2.a.i.1.2

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 $ x^7 - 21x^5 - 12x^4 + 93x^3 + 18x^2 - 120*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.72664$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -2.80679 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -2.80679 q^{7} +1.00000 q^{9} -5.68373 q^{11} -2.36680 q^{13} +1.00000 q^{15} +3.76636 q^{17} +4.21558 q^{19} -2.80679 q^{21} +0.258489 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.06528 q^{29} +4.94313 q^{31} -5.68373 q^{33} -2.80679 q^{35} -1.71176 q^{37} -2.36680 q^{39} +7.05824 q^{41} +1.19164 q^{43} +1.00000 q^{45} +12.8176 q^{47} +0.878057 q^{49} +3.76636 q^{51} +6.06859 q^{53} -5.68373 q^{55} +4.21558 q^{57} -5.14948 q^{59} +0.0917133 q^{61} -2.80679 q^{63} -2.36680 q^{65} -1.00000 q^{67} +0.258489 q^{69} +7.04357 q^{71} +12.5871 q^{73} +1.00000 q^{75} +15.9530 q^{77} +4.55774 q^{79} +1.00000 q^{81} -11.9438 q^{83} +3.76636 q^{85} -1.06528 q^{87} +0.0460502 q^{89} +6.64310 q^{91} +4.94313 q^{93} +4.21558 q^{95} +8.86030 q^{97} -5.68373 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * 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
$7$	−2.80679	−1.06087	−0.530433	−	0.847727i	$-0.677971\pi$
$7$	−2.80679	−1.06087	−0.530433	+	0.847727i	$0.677971\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.68373	−1.71371	−0.856855	−	0.515558i	$-0.827585\pi$
$11$	−5.68373	−1.71371	−0.856855	+	0.515558i	$0.827585\pi$
$12$	0	0
$13$	−2.36680	−0.656432	−0.328216	−	0.944603i	$-0.606447\pi$
$13$	−2.36680	−0.656432	−0.328216	+	0.944603i	$0.606447\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	3.76636	0.913477	0.456738	−	0.889601i	$-0.349018\pi$
$17$	3.76636	0.913477	0.456738	+	0.889601i	$0.349018\pi$
$18$	0	0
$19$	4.21558	0.967121	0.483561	−	0.875311i	$-0.339343\pi$
$19$	4.21558	0.967121	0.483561	+	0.875311i	$0.339343\pi$
$20$	0	0
$21$	−2.80679	−0.612491
$22$	0	0
$23$	0.258489	0.0538987	0.0269494	−	0.999637i	$-0.491421\pi$
$23$	0.258489	0.0538987	0.0269494	+	0.999637i	$0.491421\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.06528	−0.197817	−0.0989085	−	0.995097i	$-0.531535\pi$
$29$	−1.06528	−0.197817	−0.0989085	+	0.995097i	$0.531535\pi$
$30$	0	0
$31$	4.94313	0.887812	0.443906	−	0.896073i	$-0.353592\pi$
$31$	4.94313	0.887812	0.443906	+	0.896073i	$0.353592\pi$
$32$	0	0
$33$	−5.68373	−0.989411
$34$	0	0
$35$	−2.80679	−0.474434
$36$	0	0
$37$	−1.71176	−0.281412	−0.140706	−	0.990051i	$-0.544937\pi$
$37$	−1.71176	−0.281412	−0.140706	+	0.990051i	$0.544937\pi$
$38$	0	0
$39$	−2.36680	−0.378991
$40$	0	0
$41$	7.05824	1.10231	0.551156	−	0.834402i	$-0.314187\pi$
$41$	7.05824	1.10231	0.551156	+	0.834402i	$0.314187\pi$
$42$	0	0
$43$	1.19164	0.181724	0.0908619	−	0.995864i	$-0.471038\pi$
$43$	1.19164	0.181724	0.0908619	+	0.995864i	$0.471038\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	12.8176	1.86964	0.934819	−	0.355124i	$-0.115561\pi$
$47$	12.8176	1.86964	0.934819	+	0.355124i	$0.115561\pi$
$48$	0	0
$49$	0.878057	0.125437
$50$	0	0
$51$	3.76636	0.527396
$52$	0	0
$53$	6.06859	0.833584	0.416792	−	0.909002i	$-0.363154\pi$
$53$	6.06859	0.833584	0.416792	+	0.909002i	$0.363154\pi$
$54$	0	0
$55$	−5.68373	−0.766394
$56$	0	0
$57$	4.21558	0.558368
$58$	0	0
$59$	−5.14948	−0.670405	−0.335203	−	0.942146i	$-0.608805\pi$
$59$	−5.14948	−0.670405	−0.335203	+	0.942146i	$0.608805\pi$
$60$	0	0
$61$	0.0917133	0.0117427	0.00587135	−	0.999983i	$-0.498131\pi$
$61$	0.0917133	0.0117427	0.00587135	+	0.999983i	$0.498131\pi$
$62$	0	0
$63$	−2.80679	−0.353622
$64$	0	0
$65$	−2.36680	−0.293565
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	0.258489	0.0311184
$70$	0	0
$71$	7.04357	0.835918	0.417959	−	0.908466i	$-0.362746\pi$
$71$	7.04357	0.835918	0.417959	+	0.908466i	$0.362746\pi$
$72$	0	0
$73$	12.5871	1.47321	0.736607	−	0.676321i	$-0.236427\pi$
$73$	12.5871	1.47321	0.736607	+	0.676321i	$0.236427\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	15.9530	1.81802
$78$	0	0
$79$	4.55774	0.512785	0.256393	−	0.966573i	$-0.417466\pi$
$79$	4.55774	0.512785	0.256393	+	0.966573i	$0.417466\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.9438	−1.31100	−0.655501	−	0.755194i	$-0.727542\pi$
$83$	−11.9438	−1.31100	−0.655501	+	0.755194i	$0.727542\pi$
$84$	0	0
$85$	3.76636	0.408519
$86$	0	0
$87$	−1.06528	−0.114210
$88$	0	0
$89$	0.0460502	0.00488131	0.00244066	−	0.999997i	$-0.499223\pi$
$89$	0.0460502	0.00488131	0.00244066	+	0.999997i	$0.499223\pi$
$90$	0	0
$91$	6.64310	0.696386
$92$	0	0
$93$	4.94313	0.512579
$94$	0	0
$95$	4.21558	0.432510
$96$	0	0
$97$	8.86030	0.899627	0.449813	−	0.893123i	$-0.351490\pi$
$97$	8.86030	0.899627	0.449813	+	0.893123i	$0.351490\pi$
$98$	0	0
$99$	−5.68373	−0.571237
$100$	0	0
$101$	13.8542	1.37855	0.689273	−	0.724502i	$-0.257930\pi$
$101$	13.8542	1.37855	0.689273	+	0.724502i	$0.257930\pi$
$102$	0	0
$103$	−0.484618	−0.0477509	−0.0238754	−	0.999715i	$-0.507601\pi$
$103$	−0.484618	−0.0477509	−0.0238754	+	0.999715i	$0.507601\pi$
$104$	0	0
$105$	−2.80679	−0.273914
$106$	0	0
$107$	−3.49838	−0.338201	−0.169100	−	0.985599i	$-0.554086\pi$
$107$	−3.49838	−0.338201	−0.169100	+	0.985599i	$0.554086\pi$
$108$	0	0
$109$	10.1086	0.968232	0.484116	−	0.875004i	$-0.339141\pi$
$109$	10.1086	0.968232	0.484116	+	0.875004i	$0.339141\pi$
$110$	0	0
$111$	−1.71176	−0.162474
$112$	0	0
$113$	9.78509	0.920504	0.460252	−	0.887788i	$-0.347759\pi$
$113$	9.78509	0.920504	0.460252	+	0.887788i	$0.347759\pi$
$114$	0	0
$115$	0.258489	0.0241042
$116$	0	0
$117$	−2.36680	−0.218811
$118$	0	0
$119$	−10.5714	−0.969076
$120$	0	0
$121$	21.3048	1.93680
$122$	0	0
$123$	7.05824	0.636420
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.3323	1.09431	0.547157	−	0.837030i	$-0.315710\pi$
$127$	12.3323	1.09431	0.547157	+	0.837030i	$0.315710\pi$
$128$	0	0
$129$	1.19164	0.104918
$130$	0	0
$131$	−12.7775	−1.11638	−0.558188	−	0.829715i	$-0.688503\pi$
$131$	−12.7775	−1.11638	−0.558188	+	0.829715i	$0.688503\pi$
$132$	0	0
$133$	−11.8322	−1.02599
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	6.59394	0.563358	0.281679	−	0.959509i	$-0.409109\pi$
$137$	6.59394	0.563358	0.281679	+	0.959509i	$0.409109\pi$
$138$	0	0
$139$	14.7836	1.25393	0.626965	−	0.779047i	$-0.284297\pi$
$139$	14.7836	1.25393	0.626965	+	0.779047i	$0.284297\pi$
$140$	0	0
$141$	12.8176	1.07944
$142$	0	0
$143$	13.4522	1.12493
$144$	0	0
$145$	−1.06528	−0.0884664
$146$	0	0
$147$	0.878057	0.0724209
$148$	0	0
$149$	−4.77041	−0.390807	−0.195404	−	0.980723i	$-0.562602\pi$
$149$	−4.77041	−0.390807	−0.195404	+	0.980723i	$0.562602\pi$
$150$	0	0
$151$	−11.8082	−0.960935	−0.480467	−	0.877013i	$-0.659533\pi$
$151$	−11.8082	−0.960935	−0.480467	+	0.877013i	$0.659533\pi$
$152$	0	0
$153$	3.76636	0.304492
$154$	0	0
$155$	4.94313	0.397042
$156$	0	0
$157$	−16.7054	−1.33324	−0.666619	−	0.745399i	$-0.732259\pi$
$157$	−16.7054	−1.33324	−0.666619	+	0.745399i	$0.732259\pi$
$158$	0	0
$159$	6.06859	0.481270
$160$	0	0
$161$	−0.725524	−0.0571793
$162$	0	0
$163$	−5.92001	−0.463691	−0.231845	−	0.972753i	$-0.574476\pi$
$163$	−5.92001	−0.463691	−0.231845	+	0.972753i	$0.574476\pi$
$164$	0	0
$165$	−5.68373	−0.442478
$166$	0	0
$167$	−9.10527	−0.704587	−0.352293	−	0.935890i	$-0.614598\pi$
$167$	−9.10527	−0.704587	−0.352293	+	0.935890i	$0.614598\pi$
$168$	0	0
$169$	−7.39827	−0.569097
$170$	0	0
$171$	4.21558	0.322374
$172$	0	0
$173$	15.9650	1.21379	0.606897	−	0.794781i	$-0.292414\pi$
$173$	15.9650	1.21379	0.606897	+	0.794781i	$0.292414\pi$
$174$	0	0
$175$	−2.80679	−0.212173
$176$	0	0
$177$	−5.14948	−0.387059
$178$	0	0
$179$	−20.8683	−1.55977	−0.779886	−	0.625921i	$-0.784723\pi$
$179$	−20.8683	−1.55977	−0.779886	+	0.625921i	$0.784723\pi$
$180$	0	0
$181$	8.67885	0.645094	0.322547	−	0.946553i	$-0.395461\pi$
$181$	8.67885	0.645094	0.322547	+	0.946553i	$0.395461\pi$
$182$	0	0
$183$	0.0917133	0.00677965
$184$	0	0
$185$	−1.71176	−0.125851
$186$	0	0
$187$	−21.4070	−1.56543
$188$	0	0
$189$	−2.80679	−0.204164
$190$	0	0
$191$	0.474438	0.0343291	0.0171645	−	0.999853i	$-0.494536\pi$
$191$	0.474438	0.0343291	0.0171645	+	0.999853i	$0.494536\pi$
$192$	0	0
$193$	−15.3097	−1.10202	−0.551008	−	0.834500i	$-0.685756\pi$
$193$	−15.3097	−1.10202	−0.551008	+	0.834500i	$0.685756\pi$
$194$	0	0
$195$	−2.36680	−0.169490
$196$	0	0
$197$	3.22282	0.229616	0.114808	−	0.993388i	$-0.463375\pi$
$197$	3.22282	0.229616	0.114808	+	0.993388i	$0.463375\pi$
$198$	0	0
$199$	1.17536	0.0833191	0.0416595	−	0.999132i	$-0.486736\pi$
$199$	1.17536	0.0833191	0.0416595	+	0.999132i	$0.486736\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	2.99001	0.209857
$204$	0	0
$205$	7.05824	0.492969
$206$	0	0
$207$	0.258489	0.0179662
$208$	0	0
$209$	−23.9602	−1.65736
$210$	0	0
$211$	−14.6928	−1.01149	−0.505747	−	0.862682i	$-0.668783\pi$
$211$	−14.6928	−1.01149	−0.505747	+	0.862682i	$0.668783\pi$
$212$	0	0
$213$	7.04357	0.482618
$214$	0	0
$215$	1.19164	0.0812693
$216$	0	0
$217$	−13.8743	−0.941850
$218$	0	0
$219$	12.5871	0.850560
$220$	0	0
$221$	−8.91421	−0.599635
$222$	0	0
$223$	−1.67252	−0.112000	−0.0560000	−	0.998431i	$-0.517835\pi$
$223$	−1.67252	−0.112000	−0.0560000	+	0.998431i	$0.517835\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−15.7371	−1.04451	−0.522253	−	0.852790i	$-0.674908\pi$
$227$	−15.7371	−1.04451	−0.522253	+	0.852790i	$0.674908\pi$
$228$	0	0
$229$	−8.39165	−0.554536	−0.277268	−	0.960793i	$-0.589429\pi$
$229$	−8.39165	−0.554536	−0.277268	+	0.960793i	$0.589429\pi$
$230$	0	0
$231$	15.9530	1.04963
$232$	0	0
$233$	26.0120	1.70410	0.852050	−	0.523460i	$-0.175359\pi$
$233$	26.0120	1.70410	0.852050	+	0.523460i	$0.175359\pi$
$234$	0	0
$235$	12.8176	0.836128
$236$	0	0
$237$	4.55774	0.296057
$238$	0	0
$239$	26.8469	1.73658	0.868292	−	0.496054i	$-0.165218\pi$
$239$	26.8469	1.73658	0.868292	+	0.496054i	$0.165218\pi$
$240$	0	0
$241$	−5.58049	−0.359471	−0.179736	−	0.983715i	$-0.557524\pi$
$241$	−5.58049	−0.359471	−0.179736	+	0.983715i	$0.557524\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.878057	0.0560970
$246$	0	0
$247$	−9.97743	−0.634849
$248$	0	0
$249$	−11.9438	−0.756907
$250$	0	0
$251$	−8.83258	−0.557508	−0.278754	−	0.960363i	$-0.589921\pi$
$251$	−8.83258	−0.557508	−0.278754	+	0.960363i	$0.589921\pi$
$252$	0	0
$253$	−1.46918	−0.0923668
$254$	0	0
$255$	3.76636	0.235859
$256$	0	0
$257$	17.5885	1.09714	0.548569	−	0.836105i	$-0.315173\pi$
$257$	17.5885	1.09714	0.548569	+	0.836105i	$0.315173\pi$
$258$	0	0
$259$	4.80456	0.298541
$260$	0	0
$261$	−1.06528	−0.0659390
$262$	0	0
$263$	19.8419	1.22351	0.611753	−	0.791049i	$-0.290465\pi$
$263$	19.8419	1.22351	0.611753	+	0.791049i	$0.290465\pi$
$264$	0	0
$265$	6.06859	0.372790
$266$	0	0
$267$	0.0460502	0.00281823
$268$	0	0
$269$	−2.23221	−0.136100	−0.0680502	−	0.997682i	$-0.521678\pi$
$269$	−2.23221	−0.136100	−0.0680502	+	0.997682i	$0.521678\pi$
$270$	0	0
$271$	−23.7530	−1.44289	−0.721446	−	0.692470i	$-0.756522\pi$
$271$	−23.7530	−1.44289	−0.721446	+	0.692470i	$0.756522\pi$
$272$	0	0
$273$	6.64310	0.402059
$274$	0	0
$275$	−5.68373	−0.342742
$276$	0	0
$277$	4.63501	0.278491	0.139245	−	0.990258i	$-0.455532\pi$
$277$	4.63501	0.278491	0.139245	+	0.990258i	$0.455532\pi$
$278$	0	0
$279$	4.94313	0.295937
$280$	0	0
$281$	7.33875	0.437793	0.218896	−	0.975748i	$-0.429754\pi$
$281$	7.33875	0.437793	0.218896	+	0.975748i	$0.429754\pi$
$282$	0	0
$283$	13.0265	0.774348	0.387174	−	0.922007i	$-0.373451\pi$
$283$	13.0265	0.774348	0.387174	+	0.922007i	$0.373451\pi$
$284$	0	0
$285$	4.21558	0.249710
$286$	0	0
$287$	−19.8110	−1.16941
$288$	0	0
$289$	−2.81453	−0.165561
$290$	0	0
$291$	8.86030	0.519400
$292$	0	0
$293$	33.7211	1.97001	0.985003	−	0.172539i	$-0.0551971\pi$
$293$	33.7211	1.97001	0.985003	+	0.172539i	$0.0551971\pi$
$294$	0	0
$295$	−5.14948	−0.299814
$296$	0	0
$297$	−5.68373	−0.329804
$298$	0	0
$299$	−0.611792	−0.0353808
$300$	0	0
$301$	−3.34469	−0.192785
$302$	0	0
$303$	13.8542	0.795904
$304$	0	0
$305$	0.0917133	0.00525149
$306$	0	0
$307$	−14.0722	−0.803145	−0.401572	−	0.915827i	$-0.631536\pi$
$307$	−14.0722	−0.803145	−0.401572	+	0.915827i	$0.631536\pi$
$308$	0	0
$309$	−0.484618	−0.0275690
$310$	0	0
$311$	18.8597	1.06944	0.534718	−	0.845030i	$-0.320418\pi$
$311$	18.8597	1.06944	0.534718	+	0.845030i	$0.320418\pi$
$312$	0	0
$313$	−2.66158	−0.150441	−0.0752207	−	0.997167i	$-0.523966\pi$
$313$	−2.66158	−0.150441	−0.0752207	+	0.997167i	$0.523966\pi$
$314$	0	0
$315$	−2.80679	−0.158145
$316$	0	0
$317$	−14.7446	−0.828140	−0.414070	−	0.910245i	$-0.635893\pi$
$317$	−14.7446	−0.828140	−0.414070	+	0.910245i	$0.635893\pi$
$318$	0	0
$319$	6.05475	0.339001
$320$	0	0
$321$	−3.49838	−0.195260
$322$	0	0
$323$	15.8774	0.883442
$324$	0	0
$325$	−2.36680	−0.131286
$326$	0	0
$327$	10.1086	0.559009
$328$	0	0
$329$	−35.9763	−1.98344
$330$	0	0
$331$	−31.7341	−1.74427	−0.872133	−	0.489269i	$-0.837264\pi$
$331$	−31.7341	−1.74427	−0.872133	+	0.489269i	$0.837264\pi$
$332$	0	0
$333$	−1.71176	−0.0938041
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−6.81363	−0.371162	−0.185581	−	0.982629i	$-0.559417\pi$
$337$	−6.81363	−0.371162	−0.185581	+	0.982629i	$0.559417\pi$
$338$	0	0
$339$	9.78509	0.531453
$340$	0	0
$341$	−28.0954	−1.52145
$342$	0	0
$343$	17.1830	0.927794
$344$	0	0
$345$	0.258489	0.0139166
$346$	0	0
$347$	12.4384	0.667727	0.333864	−	0.942621i	$-0.391648\pi$
$347$	12.4384	0.667727	0.333864	+	0.942621i	$0.391648\pi$
$348$	0	0
$349$	−18.5479	−0.992847	−0.496424	−	0.868080i	$-0.665354\pi$
$349$	−18.5479	−0.992847	−0.496424	+	0.868080i	$0.665354\pi$
$350$	0	0
$351$	−2.36680	−0.126330
$352$	0	0
$353$	−15.6265	−0.831716	−0.415858	−	0.909430i	$-0.636519\pi$
$353$	−15.6265	−0.831716	−0.415858	+	0.909430i	$0.636519\pi$
$354$	0	0
$355$	7.04357	0.373834
$356$	0	0
$357$	−10.5714	−0.559496
$358$	0	0
$359$	28.0289	1.47931	0.739654	−	0.672987i	$-0.234989\pi$
$359$	28.0289	1.47931	0.739654	+	0.672987i	$0.234989\pi$
$360$	0	0
$361$	−1.22886	−0.0646769
$362$	0	0
$363$	21.3048	1.11821
$364$	0	0
$365$	12.5871	0.658841
$366$	0	0
$367$	−17.0075	−0.887784	−0.443892	−	0.896080i	$-0.646403\pi$
$367$	−17.0075	−0.887784	−0.443892	+	0.896080i	$0.646403\pi$
$368$	0	0
$369$	7.05824	0.367437
$370$	0	0
$371$	−17.0332	−0.884321
$372$	0	0
$373$	38.3864	1.98757	0.993786	−	0.111307i	$-0.0355036\pi$
$373$	38.3864	1.98757	0.993786	+	0.111307i	$0.0355036\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	2.52130	0.129853
$378$	0	0
$379$	25.8888	1.32982	0.664910	−	0.746924i	$-0.268470\pi$
$379$	25.8888	1.32982	0.664910	+	0.746924i	$0.268470\pi$
$380$	0	0
$381$	12.3323	0.631803
$382$	0	0
$383$	−15.7505	−0.804814	−0.402407	−	0.915461i	$-0.631826\pi$
$383$	−15.7505	−0.804814	−0.402407	+	0.915461i	$0.631826\pi$
$384$	0	0
$385$	15.9530	0.813042
$386$	0	0
$387$	1.19164	0.0605746
$388$	0	0
$389$	−19.1239	−0.969622	−0.484811	−	0.874619i	$-0.661112\pi$
$389$	−19.1239	−0.969622	−0.484811	+	0.874619i	$0.661112\pi$
$390$	0	0
$391$	0.973564	0.0492352
$392$	0	0
$393$	−12.7775	−0.644540
$394$	0	0
$395$	4.55774	0.229325
$396$	0	0
$397$	18.7030	0.938675	0.469337	−	0.883019i	$-0.344493\pi$
$397$	18.7030	0.938675	0.469337	+	0.883019i	$0.344493\pi$
$398$	0	0
$399$	−11.8322	−0.592353
$400$	0	0
$401$	34.3001	1.71286	0.856432	−	0.516260i	$-0.172676\pi$
$401$	34.3001	1.71286	0.856432	+	0.516260i	$0.172676\pi$
$402$	0	0
$403$	−11.6994	−0.582788
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	9.72921	0.482259
$408$	0	0
$409$	−3.31237	−0.163786	−0.0818930	−	0.996641i	$-0.526097\pi$
$409$	−3.31237	−0.163786	−0.0818930	+	0.996641i	$0.526097\pi$
$410$	0	0
$411$	6.59394	0.325255
$412$	0	0
$413$	14.4535	0.711210
$414$	0	0
$415$	−11.9438	−0.586298
$416$	0	0
$417$	14.7836	0.723957
$418$	0	0
$419$	−7.90380	−0.386126	−0.193063	−	0.981186i	$-0.561842\pi$
$419$	−7.90380	−0.386126	−0.193063	+	0.981186i	$0.561842\pi$
$420$	0	0
$421$	3.69039	0.179859	0.0899294	−	0.995948i	$-0.471336\pi$
$421$	3.69039	0.179859	0.0899294	+	0.995948i	$0.471336\pi$
$422$	0	0
$423$	12.8176	0.623213
$424$	0	0
$425$	3.76636	0.182695
$426$	0	0
$427$	−0.257420	−0.0124574
$428$	0	0
$429$	13.4522	0.649481
$430$	0	0
$431$	22.8918	1.10266	0.551331	−	0.834287i	$-0.314120\pi$
$431$	22.8918	1.10266	0.551331	+	0.834287i	$0.314120\pi$
$432$	0	0
$433$	−11.2296	−0.539658	−0.269829	−	0.962908i	$-0.586967\pi$
$433$	−11.2296	−0.539658	−0.269829	+	0.962908i	$0.586967\pi$
$434$	0	0
$435$	−1.06528	−0.0510761
$436$	0	0
$437$	1.08968	0.0521266
$438$	0	0
$439$	−3.00281	−0.143316	−0.0716582	−	0.997429i	$-0.522829\pi$
$439$	−3.00281	−0.143316	−0.0716582	+	0.997429i	$0.522829\pi$
$440$	0	0
$441$	0.878057	0.0418123
$442$	0	0
$443$	5.12075	0.243294	0.121647	−	0.992573i	$-0.461182\pi$
$443$	5.12075	0.243294	0.121647	+	0.992573i	$0.461182\pi$
$444$	0	0
$445$	0.0460502	0.00218299
$446$	0	0
$447$	−4.77041	−0.225633
$448$	0	0
$449$	21.0054	0.991307	0.495654	−	0.868520i	$-0.334929\pi$
$449$	21.0054	0.991307	0.495654	+	0.868520i	$0.334929\pi$
$450$	0	0
$451$	−40.1171	−1.88904
$452$	0	0
$453$	−11.8082	−0.554796
$454$	0	0
$455$	6.64310	0.311433
$456$	0	0
$457$	−22.7747	−1.06536	−0.532679	−	0.846317i	$-0.678815\pi$
$457$	−22.7747	−1.06536	−0.532679	+	0.846317i	$0.678815\pi$
$458$	0	0
$459$	3.76636	0.175799
$460$	0	0
$461$	−28.7075	−1.33704	−0.668521	−	0.743694i	$-0.733072\pi$
$461$	−28.7075	−1.33704	−0.668521	+	0.743694i	$0.733072\pi$
$462$	0	0
$463$	−22.9828	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$463$	−22.9828	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$464$	0	0
$465$	4.94313	0.229232
$466$	0	0
$467$	18.6892	0.864832	0.432416	−	0.901674i	$-0.357661\pi$
$467$	18.6892	0.864832	0.432416	+	0.901674i	$0.357661\pi$
$468$	0	0
$469$	2.80679	0.129605
$470$	0	0
$471$	−16.7054	−0.769745
$472$	0	0
$473$	−6.77298	−0.311422
$474$	0	0
$475$	4.21558	0.193424
$476$	0	0
$477$	6.06859	0.277861
$478$	0	0
$479$	−18.6270	−0.851088	−0.425544	−	0.904938i	$-0.639917\pi$
$479$	−18.6270	−0.851088	−0.425544	+	0.904938i	$0.639917\pi$
$480$	0	0
$481$	4.05140	0.184728
$482$	0	0
$483$	−0.725524	−0.0330125
$484$	0	0
$485$	8.86030	0.402325
$486$	0	0
$487$	−37.0797	−1.68024	−0.840122	−	0.542398i	$-0.817516\pi$
$487$	−37.0797	−1.68024	−0.840122	+	0.542398i	$0.817516\pi$
$488$	0	0
$489$	−5.92001	−0.267712
$490$	0	0
$491$	25.3512	1.14409	0.572043	−	0.820224i	$-0.306151\pi$
$491$	25.3512	1.14409	0.572043	+	0.820224i	$0.306151\pi$
$492$	0	0
$493$	−4.01222	−0.180701
$494$	0	0
$495$	−5.68373	−0.255465
$496$	0	0
$497$	−19.7698	−0.886797
$498$	0	0
$499$	5.87867	0.263166	0.131583	−	0.991305i	$-0.457994\pi$
$499$	5.87867	0.263166	0.131583	+	0.991305i	$0.457994\pi$
$500$	0	0
$501$	−9.10527	−0.406793
$502$	0	0
$503$	2.70035	0.120403	0.0602013	−	0.998186i	$-0.480826\pi$
$503$	2.70035	0.120403	0.0602013	+	0.998186i	$0.480826\pi$
$504$	0	0
$505$	13.8542	0.616504
$506$	0	0
$507$	−7.39827	−0.328569
$508$	0	0
$509$	−6.77916	−0.300481	−0.150241	−	0.988649i	$-0.548005\pi$
$509$	−6.77916	−0.300481	−0.150241	+	0.988649i	$0.548005\pi$
$510$	0	0
$511$	−35.3294	−1.56288
$512$	0	0
$513$	4.21558	0.186123
$514$	0	0
$515$	−0.484618	−0.0213548
$516$	0	0
$517$	−72.8518	−3.20402
$518$	0	0
$519$	15.9650	0.700784
$520$	0	0
$521$	−2.63008	−0.115226	−0.0576129	−	0.998339i	$-0.518349\pi$
$521$	−2.63008	−0.115226	−0.0576129	+	0.998339i	$0.518349\pi$
$522$	0	0
$523$	26.2742	1.14889	0.574445	−	0.818543i	$-0.305218\pi$
$523$	26.2742	1.14889	0.574445	+	0.818543i	$0.305218\pi$
$524$	0	0
$525$	−2.80679	−0.122498
$526$	0	0
$527$	18.6176	0.810996
$528$	0	0
$529$	−22.9332	−0.997095
$530$	0	0
$531$	−5.14948	−0.223468
$532$	0	0
$533$	−16.7054	−0.723592
$534$	0	0
$535$	−3.49838	−0.151248
$536$	0	0
$537$	−20.8683	−0.900535
$538$	0	0
$539$	−4.99064	−0.214962
$540$	0	0
$541$	19.8451	0.853206	0.426603	−	0.904439i	$-0.359710\pi$
$541$	19.8451	0.853206	0.426603	+	0.904439i	$0.359710\pi$
$542$	0	0
$543$	8.67885	0.372445
$544$	0	0
$545$	10.1086	0.433007
$546$	0	0
$547$	11.5773	0.495009	0.247505	−	0.968887i	$-0.420389\pi$
$547$	11.5773	0.495009	0.247505	+	0.968887i	$0.420389\pi$
$548$	0	0
$549$	0.0917133	0.00391423
$550$	0	0
$551$	−4.49076	−0.191313
$552$	0	0
$553$	−12.7926	−0.543997
$554$	0	0
$555$	−1.71176	−0.0726604
$556$	0	0
$557$	3.35857	0.142307	0.0711537	−	0.997465i	$-0.477332\pi$
$557$	3.35857	0.142307	0.0711537	+	0.997465i	$0.477332\pi$
$558$	0	0
$559$	−2.82038	−0.119289
$560$	0	0
$561$	−21.4070	−0.903803
$562$	0	0
$563$	21.0774	0.888306	0.444153	−	0.895951i	$-0.353505\pi$
$563$	21.0774	0.888306	0.444153	+	0.895951i	$0.353505\pi$
$564$	0	0
$565$	9.78509	0.411662
$566$	0	0
$567$	−2.80679	−0.117874
$568$	0	0
$569$	35.4893	1.48779	0.743893	−	0.668298i	$-0.232977\pi$
$569$	35.4893	1.48779	0.743893	+	0.668298i	$0.232977\pi$
$570$	0	0
$571$	−25.4024	−1.06306	−0.531529	−	0.847040i	$-0.678382\pi$
$571$	−25.4024	−1.06306	−0.531529	+	0.847040i	$0.678382\pi$
$572$	0	0
$573$	0.474438	0.0198199
$574$	0	0
$575$	0.258489	0.0107797
$576$	0	0
$577$	18.6833	0.777795	0.388897	−	0.921281i	$-0.372856\pi$
$577$	18.6833	0.777795	0.388897	+	0.921281i	$0.372856\pi$
$578$	0	0
$579$	−15.3097	−0.636249
$580$	0	0
$581$	33.5237	1.39080
$582$	0	0
$583$	−34.4922	−1.42852
$584$	0	0
$585$	−2.36680	−0.0978551
$586$	0	0
$587$	5.53876	0.228609	0.114305	−	0.993446i	$-0.463536\pi$
$587$	5.53876	0.228609	0.114305	+	0.993446i	$0.463536\pi$
$588$	0	0
$589$	20.8382	0.858622
$590$	0	0
$591$	3.22282	0.132569
$592$	0	0
$593$	−7.09447	−0.291335	−0.145667	−	0.989334i	$-0.546533\pi$
$593$	−7.09447	−0.291335	−0.145667	+	0.989334i	$0.546533\pi$
$594$	0	0
$595$	−10.5714	−0.433384
$596$	0	0
$597$	1.17536	0.0481043
$598$	0	0
$599$	−2.21245	−0.0903983	−0.0451992	−	0.998978i	$-0.514392\pi$
$599$	−2.21245	−0.0903983	−0.0451992	+	0.998978i	$0.514392\pi$
$600$	0	0
$601$	−3.38263	−0.137980	−0.0689902	−	0.997617i	$-0.521978\pi$
$601$	−3.38263	−0.137980	−0.0689902	+	0.997617i	$0.521978\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	21.3048	0.866163
$606$	0	0
$607$	8.64234	0.350782	0.175391	−	0.984499i	$-0.443881\pi$
$607$	8.64234	0.350782	0.175391	+	0.984499i	$0.443881\pi$
$608$	0	0
$609$	2.99001	0.121161
$610$	0	0
$611$	−30.3367	−1.22729
$612$	0	0
$613$	8.53692	0.344803	0.172402	−	0.985027i	$-0.444847\pi$
$613$	8.53692	0.344803	0.172402	+	0.985027i	$0.444847\pi$
$614$	0	0
$615$	7.05824	0.284616
$616$	0	0
$617$	−4.49918	−0.181130	−0.0905651	−	0.995891i	$-0.528867\pi$
$617$	−4.49918	−0.181130	−0.0905651	+	0.995891i	$0.528867\pi$
$618$	0	0
$619$	−10.6422	−0.427748	−0.213874	−	0.976861i	$-0.568608\pi$
$619$	−10.6422	−0.427748	−0.213874	+	0.976861i	$0.568608\pi$
$620$	0	0
$621$	0.258489	0.0103728
$622$	0	0
$623$	−0.129253	−0.00517842
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−23.9602	−0.956880
$628$	0	0
$629$	−6.44712	−0.257064
$630$	0	0
$631$	43.6283	1.73681	0.868407	−	0.495851i	$-0.165144\pi$
$631$	43.6283	1.73681	0.868407	+	0.495851i	$0.165144\pi$
$632$	0	0
$633$	−14.6928	−0.583987
$634$	0	0
$635$	12.3323	0.489392
$636$	0	0
$637$	−2.07818	−0.0823407
$638$	0	0
$639$	7.04357	0.278639
$640$	0	0
$641$	2.44057	0.0963968	0.0481984	−	0.998838i	$-0.484652\pi$
$641$	2.44057	0.0963968	0.0481984	+	0.998838i	$0.484652\pi$
$642$	0	0
$643$	−21.2088	−0.836393	−0.418196	−	0.908357i	$-0.637338\pi$
$643$	−21.2088	−0.836393	−0.418196	+	0.908357i	$0.637338\pi$
$644$	0	0
$645$	1.19164	0.0469209
$646$	0	0
$647$	20.7101	0.814198	0.407099	−	0.913384i	$-0.366540\pi$
$647$	20.7101	0.814198	0.407099	+	0.913384i	$0.366540\pi$
$648$	0	0
$649$	29.2683	1.14888
$650$	0	0
$651$	−13.8743	−0.543777
$652$	0	0
$653$	41.4162	1.62074	0.810372	−	0.585916i	$-0.199265\pi$
$653$	41.4162	1.62074	0.810372	+	0.585916i	$0.199265\pi$
$654$	0	0
$655$	−12.7775	−0.499259
$656$	0	0
$657$	12.5871	0.491071
$658$	0	0
$659$	23.8329	0.928396	0.464198	−	0.885731i	$-0.346343\pi$
$659$	23.8329	0.928396	0.464198	+	0.885731i	$0.346343\pi$
$660$	0	0
$661$	−13.5011	−0.525133	−0.262566	−	0.964914i	$-0.584569\pi$
$661$	−13.5011	−0.525133	−0.262566	+	0.964914i	$0.584569\pi$
$662$	0	0
$663$	−8.91421	−0.346199
$664$	0	0
$665$	−11.8322	−0.458835
$666$	0	0
$667$	−0.275363	−0.0106621
$668$	0	0
$669$	−1.67252	−0.0646632
$670$	0	0
$671$	−0.521274	−0.0201236
$672$	0	0
$673$	−37.5601	−1.44784	−0.723918	−	0.689886i	$-0.757661\pi$
$673$	−37.5601	−1.44784	−0.723918	+	0.689886i	$0.757661\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	12.2477	0.470717	0.235358	−	0.971909i	$-0.424374\pi$
$677$	12.2477	0.470717	0.235358	+	0.971909i	$0.424374\pi$
$678$	0	0
$679$	−24.8690	−0.954384
$680$	0	0
$681$	−15.7371	−0.603046
$682$	0	0
$683$	20.1534	0.771147	0.385574	−	0.922677i	$-0.374004\pi$
$683$	20.1534	0.771147	0.385574	+	0.922677i	$0.374004\pi$
$684$	0	0
$685$	6.59394	0.251942
$686$	0	0
$687$	−8.39165	−0.320161
$688$	0	0
$689$	−14.3631	−0.547191
$690$	0	0
$691$	−37.3701	−1.42163	−0.710814	−	0.703380i	$-0.751673\pi$
$691$	−37.3701	−1.42163	−0.710814	+	0.703380i	$0.751673\pi$
$692$	0	0
$693$	15.9530	0.606005
$694$	0	0
$695$	14.7836	0.560775
$696$	0	0
$697$	26.5839	1.00694
$698$	0	0
$699$	26.0120	0.983863
$700$	0	0
$701$	−35.3093	−1.33361	−0.666806	−	0.745231i	$-0.732339\pi$
$701$	−35.3093	−1.33361	−0.666806	+	0.745231i	$0.732339\pi$
$702$	0	0
$703$	−7.21609	−0.272160
$704$	0	0
$705$	12.8176	0.482739
$706$	0	0
$707$	−38.8858	−1.46245
$708$	0	0
$709$	24.9373	0.936540	0.468270	−	0.883585i	$-0.344877\pi$
$709$	24.9373	0.936540	0.468270	+	0.883585i	$0.344877\pi$
$710$	0	0
$711$	4.55774	0.170928
$712$	0	0
$713$	1.27775	0.0478520
$714$	0	0
$715$	13.4522	0.503085
$716$	0	0
$717$	26.8469	1.00262
$718$	0	0
$719$	11.7518	0.438267	0.219133	−	0.975695i	$-0.429677\pi$
$719$	11.7518	0.438267	0.219133	+	0.975695i	$0.429677\pi$
$720$	0	0
$721$	1.36022	0.0506573
$722$	0	0
$723$	−5.58049	−0.207541
$724$	0	0
$725$	−1.06528	−0.0395634
$726$	0	0
$727$	−24.5515	−0.910563	−0.455282	−	0.890348i	$-0.650461\pi$
$727$	−24.5515	−0.910563	−0.455282	+	0.890348i	$0.650461\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.48816	0.166000
$732$	0	0
$733$	31.2990	1.15605	0.578027	−	0.816018i	$-0.303823\pi$
$733$	31.2990	1.15605	0.578027	+	0.816018i	$0.303823\pi$
$734$	0	0
$735$	0.878057	0.0323876
$736$	0	0
$737$	5.68373	0.209363
$738$	0	0
$739$	−24.4603	−0.899786	−0.449893	−	0.893082i	$-0.648538\pi$
$739$	−24.4603	−0.899786	−0.449893	+	0.893082i	$0.648538\pi$
$740$	0	0
$741$	−9.97743	−0.366530
$742$	0	0
$743$	22.3799	0.821038	0.410519	−	0.911852i	$-0.365348\pi$
$743$	22.3799	0.821038	0.410519	+	0.911852i	$0.365348\pi$
$744$	0	0
$745$	−4.77041	−0.174774
$746$	0	0
$747$	−11.9438	−0.437001
$748$	0	0
$749$	9.81920	0.358786
$750$	0	0
$751$	−9.76149	−0.356202	−0.178101	−	0.984012i	$-0.556995\pi$
$751$	−9.76149	−0.356202	−0.178101	+	0.984012i	$0.556995\pi$
$752$	0	0
$753$	−8.83258	−0.321877
$754$	0	0
$755$	−11.8082	−0.429743
$756$	0	0
$757$	51.5225	1.87262	0.936308	−	0.351180i	$-0.114219\pi$
$757$	51.5225	1.87262	0.936308	+	0.351180i	$0.114219\pi$
$758$	0	0
$759$	−1.46918	−0.0533280
$760$	0	0
$761$	48.9423	1.77416	0.887078	−	0.461619i	$-0.152731\pi$
$761$	48.9423	1.77416	0.887078	+	0.461619i	$0.152731\pi$
$762$	0	0
$763$	−28.3728	−1.02716
$764$	0	0
$765$	3.76636	0.136173
$766$	0	0
$767$	12.1878	0.440075
$768$	0	0
$769$	15.3340	0.552959	0.276480	−	0.961020i	$-0.410832\pi$
$769$	15.3340	0.552959	0.276480	+	0.961020i	$0.410832\pi$
$770$	0	0
$771$	17.5885	0.633433
$772$	0	0
$773$	19.7652	0.710903	0.355452	−	0.934695i	$-0.384327\pi$
$773$	19.7652	0.710903	0.355452	+	0.934695i	$0.384327\pi$
$774$	0	0
$775$	4.94313	0.177562
$776$	0	0
$777$	4.80456	0.172363
$778$	0	0
$779$	29.7546	1.06607
$780$	0	0
$781$	−40.0338	−1.43252
$782$	0	0
$783$	−1.06528	−0.0380699
$784$	0	0
$785$	−16.7054	−0.596242
$786$	0	0
$787$	14.9086	0.531435	0.265717	−	0.964051i	$-0.414391\pi$
$787$	14.9086	0.531435	0.265717	+	0.964051i	$0.414391\pi$
$788$	0	0
$789$	19.8419	0.706392
$790$	0	0
$791$	−27.4647	−0.976531
$792$	0	0
$793$	−0.217067	−0.00770827
$794$	0	0
$795$	6.06859	0.215231
$796$	0	0
$797$	−11.1894	−0.396350	−0.198175	−	0.980167i	$-0.563501\pi$
$797$	−11.1894	−0.396350	−0.198175	+	0.980167i	$0.563501\pi$
$798$	0	0
$799$	48.2757	1.70787
$800$	0	0
$801$	0.0460502	0.00162710
$802$	0	0
$803$	−71.5419	−2.52466
$804$	0	0
$805$	−0.725524	−0.0255714
$806$	0	0
$807$	−2.23221	−0.0785776
$808$	0	0
$809$	−45.2716	−1.59166	−0.795832	−	0.605518i	$-0.792966\pi$
$809$	−45.2716	−1.59166	−0.795832	+	0.605518i	$0.792966\pi$
$810$	0	0
$811$	−20.7847	−0.729851	−0.364926	−	0.931037i	$-0.618906\pi$
$811$	−20.7847	−0.729851	−0.364926	+	0.931037i	$0.618906\pi$
$812$	0	0
$813$	−23.7530	−0.833054
$814$	0	0
$815$	−5.92001	−0.207369
$816$	0	0
$817$	5.02347	0.175749
$818$	0	0
$819$	6.64310	0.232129
$820$	0	0
$821$	−48.9768	−1.70930	−0.854652	−	0.519202i	$-0.826229\pi$
$821$	−48.9768	−1.70930	−0.854652	+	0.519202i	$0.826229\pi$
$822$	0	0
$823$	6.57688	0.229255	0.114628	−	0.993409i	$-0.463432\pi$
$823$	6.57688	0.229255	0.114628	+	0.993409i	$0.463432\pi$
$824$	0	0
$825$	−5.68373	−0.197882
$826$	0	0
$827$	−23.8120	−0.828026	−0.414013	−	0.910271i	$-0.635873\pi$
$827$	−23.8120	−0.828026	−0.414013	+	0.910271i	$0.635873\pi$
$828$	0	0
$829$	−38.0189	−1.32045	−0.660225	−	0.751068i	$-0.729539\pi$
$829$	−38.0189	−1.32045	−0.660225	+	0.751068i	$0.729539\pi$
$830$	0	0
$831$	4.63501	0.160787
$832$	0	0
$833$	3.30708	0.114584
$834$	0	0
$835$	−9.10527	−0.315101
$836$	0	0
$837$	4.94313	0.170860
$838$	0	0
$839$	−46.9848	−1.62209	−0.811047	−	0.584981i	$-0.801102\pi$
$839$	−46.9848	−1.62209	−0.811047	+	0.584981i	$0.801102\pi$
$840$	0	0
$841$	−27.8652	−0.960868
$842$	0	0
$843$	7.33875	0.252760
$844$	0	0
$845$	−7.39827	−0.254508
$846$	0	0
$847$	−59.7981	−2.05469
$848$	0	0
$849$	13.0265	0.447070
$850$	0	0
$851$	−0.442473	−0.0151678
$852$	0	0
$853$	24.1191	0.825821	0.412911	−	0.910771i	$-0.364512\pi$
$853$	24.1191	0.825821	0.412911	+	0.910771i	$0.364512\pi$
$854$	0	0
$855$	4.21558	0.144170
$856$	0	0
$857$	−31.5180	−1.07663	−0.538317	−	0.842742i	$-0.680940\pi$
$857$	−31.5180	−1.07663	−0.538317	+	0.842742i	$0.680940\pi$
$858$	0	0
$859$	−24.6422	−0.840780	−0.420390	−	0.907344i	$-0.638107\pi$
$859$	−24.6422	−0.840780	−0.420390	+	0.907344i	$0.638107\pi$
$860$	0	0
$861$	−19.8110	−0.675156
$862$	0	0
$863$	−3.07851	−0.104794	−0.0523968	−	0.998626i	$-0.516686\pi$
$863$	−3.07851	−0.104794	−0.0523968	+	0.998626i	$0.516686\pi$
$864$	0	0
$865$	15.9650	0.542825
$866$	0	0
$867$	−2.81453	−0.0955864
$868$	0	0
$869$	−25.9049	−0.878765
$870$	0	0
$871$	2.36680	0.0801959
$872$	0	0
$873$	8.86030	0.299876
$874$	0	0
$875$	−2.80679	−0.0948867
$876$	0	0
$877$	−37.6000	−1.26966	−0.634830	−	0.772652i	$-0.718930\pi$
$877$	−37.6000	−1.26966	−0.634830	+	0.772652i	$0.718930\pi$
$878$	0	0
$879$	33.7211	1.13738
$880$	0	0
$881$	−23.9307	−0.806246	−0.403123	−	0.915146i	$-0.632075\pi$
$881$	−23.9307	−0.806246	−0.403123	+	0.915146i	$0.632075\pi$
$882$	0	0
$883$	17.9178	0.602982	0.301491	−	0.953469i	$-0.402516\pi$
$883$	17.9178	0.602982	0.301491	+	0.953469i	$0.402516\pi$
$884$	0	0
$885$	−5.14948	−0.173098
$886$	0	0
$887$	−48.8908	−1.64159	−0.820797	−	0.571221i	$-0.806470\pi$
$887$	−48.8908	−1.64159	−0.820797	+	0.571221i	$0.806470\pi$
$888$	0	0
$889$	−34.6141	−1.16092
$890$	0	0
$891$	−5.68373	−0.190412
$892$	0	0
$893$	54.0336	1.80817
$894$	0	0
$895$	−20.8683	−0.697551
$896$	0	0
$897$	−0.611792	−0.0204271
$898$	0	0
$899$	−5.26580	−0.175624
$900$	0	0
$901$	22.8565	0.761460
$902$	0	0
$903$	−3.34469	−0.111304
$904$	0	0
$905$	8.67885	0.288495
$906$	0	0
$907$	14.3493	0.476459	0.238230	−	0.971209i	$-0.423433\pi$
$907$	14.3493	0.476459	0.238230	+	0.971209i	$0.423433\pi$
$908$	0	0
$909$	13.8542	0.459515
$910$	0	0
$911$	−17.2210	−0.570558	−0.285279	−	0.958445i	$-0.592086\pi$
$911$	−17.2210	−0.570558	−0.285279	+	0.958445i	$0.592086\pi$
$912$	0	0
$913$	67.8853	2.24668
$914$	0	0
$915$	0.0917133	0.00303195
$916$	0	0
$917$	35.8638	1.18433
$918$	0	0
$919$	−35.2390	−1.16243	−0.581213	−	0.813752i	$-0.697422\pi$
$919$	−35.2390	−1.16243	−0.581213	+	0.813752i	$0.697422\pi$
$920$	0	0
$921$	−14.0722	−0.463696
$922$	0	0
$923$	−16.6707	−0.548723
$924$	0	0
$925$	−1.71176	−0.0562825
$926$	0	0
$927$	−0.484618	−0.0159170
$928$	0	0
$929$	−41.8671	−1.37362	−0.686808	−	0.726839i	$-0.740989\pi$
$929$	−41.8671	−1.37362	−0.686808	+	0.726839i	$0.740989\pi$
$930$	0	0
$931$	3.70152	0.121313
$932$	0	0
$933$	18.8597	0.617439
$934$	0	0
$935$	−21.4070	−0.700083
$936$	0	0
$937$	32.3481	1.05677	0.528384	−	0.849006i	$-0.322798\pi$
$937$	32.3481	1.05677	0.528384	+	0.849006i	$0.322798\pi$
$938$	0	0
$939$	−2.66158	−0.0868574
$940$	0	0
$941$	32.3211	1.05364	0.526819	−	0.849977i	$-0.323385\pi$
$941$	32.3211	1.05364	0.526819	+	0.849977i	$0.323385\pi$
$942$	0	0
$943$	1.82448	0.0594132
$944$	0	0
$945$	−2.80679	−0.0913048
$946$	0	0
$947$	−21.4021	−0.695476	−0.347738	−	0.937592i	$-0.613050\pi$
$947$	−21.4021	−0.695476	−0.347738	+	0.937592i	$0.613050\pi$
$948$	0	0
$949$	−29.7912	−0.967064
$950$	0	0
$951$	−14.7446	−0.478127
$952$	0	0
$953$	−32.3282	−1.04721	−0.523607	−	0.851960i	$-0.675414\pi$
$953$	−32.3282	−1.04721	−0.523607	+	0.851960i	$0.675414\pi$
$954$	0	0
$955$	0.474438	0.0153524
$956$	0	0
$957$	6.05475	0.195722
$958$	0	0
$959$	−18.5078	−0.597648
$960$	0	0
$961$	−6.56547	−0.211789
$962$	0	0
$963$	−3.49838	−0.112734
$964$	0	0
$965$	−15.3097	−0.492836
$966$	0	0
$967$	−18.1284	−0.582969	−0.291485	−	0.956576i	$-0.594149\pi$
$967$	−18.1284	−0.582969	−0.291485	+	0.956576i	$0.594149\pi$
$968$	0	0
$969$	15.8774	0.510056
$970$	0	0
$971$	38.2909	1.22881	0.614407	−	0.788990i	$-0.289395\pi$
$971$	38.2909	1.22881	0.614407	+	0.788990i	$0.289395\pi$
$972$	0	0
$973$	−41.4945	−1.33025
$974$	0	0
$975$	−2.36680	−0.0757982
$976$	0	0
$977$	25.8487	0.826973	0.413486	−	0.910510i	$-0.364311\pi$
$977$	25.8487	0.826973	0.413486	+	0.910510i	$0.364311\pi$
$978$	0	0
$979$	−0.261737	−0.00836515
$980$	0	0
$981$	10.1086	0.322744
$982$	0	0
$983$	21.7164	0.692646	0.346323	−	0.938115i	$-0.387430\pi$
$983$	21.7164	0.692646	0.346323	+	0.938115i	$0.387430\pi$
$984$	0	0
$985$	3.22282	0.102688
$986$	0	0
$987$	−35.9763	−1.14514
$988$	0	0
$989$	0.308027	0.00979468
$990$	0	0
$991$	−56.5150	−1.79526	−0.897629	−	0.440752i	$-0.854712\pi$
$991$	−56.5150	−1.79526	−0.897629	+	0.440752i	$0.854712\pi$
$992$	0	0
$993$	−31.7341	−1.00705
$994$	0	0
$995$	1.17536	0.0372614
$996$	0	0
$997$	0.649453	0.0205684	0.0102842	−	0.999947i	$-0.496726\pi$
$997$	0.649453	0.0205684	0.0102842	+	0.999947i	$0.496726\pi$
$998$	0	0
$999$	−1.71176	−0.0541578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.i.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.i.1.2	✓	7	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.80679 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.80679 q^{7} +1.00000 q^{9} -5.68373 q^{11} -2.36680 q^{13} +1.00000 q^{15} +3.76636 q^{17} +4.21558 q^{19} -2.80679 q^{21} +0.258489 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.06528 q^{29} +4.94313 q^{31} -5.68373 q^{33} -2.80679 q^{35} -1.71176 q^{37} -2.36680 q^{39} +7.05824 q^{41} +1.19164 q^{43} +1.00000 q^{45} +12.8176 q^{47} +0.878057 q^{49} +3.76636 q^{51} +6.06859 q^{53} -5.68373 q^{55} +4.21558 q^{57} -5.14948 q^{59} +0.0917133 q^{61} -2.80679 q^{63} -2.36680 q^{65} -1.00000 q^{67} +0.258489 q^{69} +7.04357 q^{71} +12.5871 q^{73} +1.00000 q^{75} +15.9530 q^{77} +4.55774 q^{79} +1.00000 q^{81} -11.9438 q^{83} +3.76636 q^{85} -1.06528 q^{87} +0.0460502 q^{89} +6.64310 q^{91} +4.94313 q^{93} +4.21558 q^{95} +8.86030 q^{97} -5.68373 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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