LMFDB - Embedded newform 4016.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
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} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 8x^3 - 17x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.358013$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.358013 q^{3} +2.25358 q^{5} +4.66949 q^{7} -2.87183 q^{9} +O(q^{10})$	$q-0.358013 q^{3} +2.25358 q^{5} +4.66949 q^{7} -2.87183 q^{9} +0.681867 q^{11} +0.464791 q^{13} -0.806810 q^{15} -0.986388 q^{17} +2.32972 q^{19} -1.67174 q^{21} +5.37412 q^{23} +0.0786287 q^{25} +2.10219 q^{27} -4.75253 q^{29} +1.57250 q^{31} -0.244117 q^{33} +10.5231 q^{35} +3.55119 q^{37} -0.166401 q^{39} +0.278393 q^{41} +1.78076 q^{43} -6.47190 q^{45} -2.22141 q^{47} +14.8042 q^{49} +0.353139 q^{51} +2.88544 q^{53} +1.53664 q^{55} -0.834068 q^{57} +6.32720 q^{59} +0.0560602 q^{61} -13.4100 q^{63} +1.04744 q^{65} +5.83951 q^{67} -1.92400 q^{69} +2.74305 q^{71} +9.63680 q^{73} -0.0281501 q^{75} +3.18397 q^{77} -3.00611 q^{79} +7.86287 q^{81} +2.32667 q^{83} -2.22291 q^{85} +1.70147 q^{87} -10.3727 q^{89} +2.17034 q^{91} -0.562976 q^{93} +5.25021 q^{95} -4.01956 q^{97} -1.95820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7}+O(q^{10}) $ 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7	$ 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7} + 5 q^{11} - q^{13} + 6 q^{15} - 8 q^{17} + 15 q^{19} - 3 q^{21} + 5 q^{23} - 9 q^{25} + 9 q^{27} + 21 q^{31} + 7 q^{35} - q^{37} + 23 q^{39} - 10 q^{41} + 23 q^{43} - 4 q^{45} + 10 q^{47} - 13 q^{49} + 20 q^{51} - q^{53} + 23 q^{55} - 6 q^{57} + 4 q^{59} + 3 q^{61} + 4 q^{63} + 4 q^{65} + 28 q^{67} + 18 q^{69} + 18 q^{71} - 7 q^{73} - 11 q^{75} + 6 q^{77} + 30 q^{79} - 5 q^{81} - 13 q^{83} + q^{85} + 7 q^{87} + 18 q^{91} + 36 q^{93} - 2 q^{97} + 10 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7 + 5 * q^11 - q^13 + 6 * q^15 - 8 * q^17 + 15 * q^19 - 3 * q^21 + 5 * q^23 - 9 * q^25 + 9 * q^27 + 21 * q^31 + 7 * q^35 - q^37 + 23 * q^39 - 10 * q^41 + 23 * q^43 - 4 * q^45 + 10 * q^47 - 13 * q^49 + 20 * q^51 - q^53 + 23 * q^55 - 6 * q^57 + 4 * q^59 + 3 * q^61 + 4 * q^63 + 4 * q^65 + 28 * q^67 + 18 * q^69 + 18 * q^71 - 7 * q^73 - 11 * q^75 + 6 * q^77 + 30 * q^79 - 5 * q^81 - 13 * q^83 + q^85 + 7 * q^87 + 18 * q^91 + 36 * q^93 - 2 * q^97 + 10 * 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
$3$	−0.358013	−0.206699	−0.103349	−	0.994645i	$-0.532956\pi$
$3$	−0.358013	−0.206699	−0.103349	+	0.994645i	$0.532956\pi$
$4$	0	0
$5$	2.25358	1.00783	0.503916	−	0.863753i	$-0.331892\pi$
$5$	2.25358	1.00783	0.503916	+	0.863753i	$0.331892\pi$
$6$	0	0
$7$	4.66949	1.76490	0.882451	−	0.470404i	$-0.155892\pi$
$7$	4.66949	1.76490	0.882451	+	0.470404i	$0.155892\pi$
$8$	0	0
$9$	−2.87183	−0.957276
$10$	0	0
$11$	0.681867	0.205591	0.102795	−	0.994703i	$-0.467221\pi$
$11$	0.681867	0.205591	0.102795	+	0.994703i	$0.467221\pi$
$12$	0	0
$13$	0.464791	0.128910	0.0644550	−	0.997921i	$-0.479469\pi$
$13$	0.464791	0.128910	0.0644550	+	0.997921i	$0.479469\pi$
$14$	0	0
$15$	−0.806810	−0.208318
$16$	0	0
$17$	−0.986388	−0.239234	−0.119617	−	0.992820i	$-0.538167\pi$
$17$	−0.986388	−0.239234	−0.119617	+	0.992820i	$0.538167\pi$
$18$	0	0
$19$	2.32972	0.534474	0.267237	−	0.963631i	$-0.413889\pi$
$19$	2.32972	0.534474	0.267237	+	0.963631i	$0.413889\pi$
$20$	0	0
$21$	−1.67174	−0.364803
$22$	0	0
$23$	5.37412	1.12058	0.560290	−	0.828296i	$-0.310690\pi$
$23$	5.37412	1.12058	0.560290	+	0.828296i	$0.310690\pi$
$24$	0	0
$25$	0.0786287	0.0157257
$26$	0	0
$27$	2.10219	0.404566
$28$	0	0
$29$	−4.75253	−0.882523	−0.441261	−	0.897379i	$-0.645469\pi$
$29$	−4.75253	−0.882523	−0.441261	+	0.897379i	$0.645469\pi$
$30$	0	0
$31$	1.57250	0.282430	0.141215	−	0.989979i	$-0.454899\pi$
$31$	1.57250	0.282430	0.141215	+	0.989979i	$0.454899\pi$
$32$	0	0
$33$	−0.244117	−0.0424953
$34$	0	0
$35$	10.5231	1.77873
$36$	0	0
$37$	3.55119	0.583812	0.291906	−	0.956447i	$-0.405711\pi$
$37$	3.55119	0.583812	0.291906	+	0.956447i	$0.405711\pi$
$38$	0	0
$39$	−0.166401	−0.0266455
$40$	0	0
$41$	0.278393	0.0434777	0.0217389	−	0.999764i	$-0.493080\pi$
$41$	0.278393	0.0434777	0.0217389	+	0.999764i	$0.493080\pi$
$42$	0	0
$43$	1.78076	0.271563	0.135782	−	0.990739i	$-0.456645\pi$
$43$	1.78076	0.271563	0.135782	+	0.990739i	$0.456645\pi$
$44$	0	0
$45$	−6.47190	−0.964773
$46$	0	0
$47$	−2.22141	−0.324026	−0.162013	−	0.986789i	$-0.551799\pi$
$47$	−2.22141	−0.324026	−0.162013	+	0.986789i	$0.551799\pi$
$48$	0	0
$49$	14.8042	2.11488
$50$	0	0
$51$	0.353139	0.0494494
$52$	0	0
$53$	2.88544	0.396346	0.198173	−	0.980167i	$-0.436499\pi$
$53$	2.88544	0.396346	0.198173	+	0.980167i	$0.436499\pi$
$54$	0	0
$55$	1.53664	0.207201
$56$	0	0
$57$	−0.834068	−0.110475
$58$	0	0
$59$	6.32720	0.823731	0.411866	−	0.911245i	$-0.364877\pi$
$59$	6.32720	0.823731	0.411866	+	0.911245i	$0.364877\pi$
$60$	0	0
$61$	0.0560602	0.00717778	0.00358889	−	0.999994i	$-0.498858\pi$
$61$	0.0560602	0.00717778	0.00358889	+	0.999994i	$0.498858\pi$
$62$	0	0
$63$	−13.4100	−1.68950
$64$	0	0
$65$	1.04744	0.129920
$66$	0	0
$67$	5.83951	0.713410	0.356705	−	0.934217i	$-0.383900\pi$
$67$	5.83951	0.713410	0.356705	+	0.934217i	$0.383900\pi$
$68$	0	0
$69$	−1.92400	−0.231623
$70$	0	0
$71$	2.74305	0.325540	0.162770	−	0.986664i	$-0.447957\pi$
$71$	2.74305	0.325540	0.162770	+	0.986664i	$0.447957\pi$
$72$	0	0
$73$	9.63680	1.12790	0.563951	−	0.825808i	$-0.309281\pi$
$73$	9.63680	1.12790	0.563951	+	0.825808i	$0.309281\pi$
$74$	0	0
$75$	−0.0281501	−0.00325049
$76$	0	0
$77$	3.18397	0.362847
$78$	0	0
$79$	−3.00611	−0.338214	−0.169107	−	0.985598i	$-0.554088\pi$
$79$	−3.00611	−0.338214	−0.169107	+	0.985598i	$0.554088\pi$
$80$	0	0
$81$	7.86287	0.873652
$82$	0	0
$83$	2.32667	0.255385	0.127693	−	0.991814i	$-0.459243\pi$
$83$	2.32667	0.255385	0.127693	+	0.991814i	$0.459243\pi$
$84$	0	0
$85$	−2.22291	−0.241108
$86$	0	0
$87$	1.70147	0.182416
$88$	0	0
$89$	−10.3727	−1.09950	−0.549750	−	0.835329i	$-0.685277\pi$
$89$	−10.3727	−1.09950	−0.549750	+	0.835329i	$0.685277\pi$
$90$	0	0
$91$	2.17034	0.227513
$92$	0	0
$93$	−0.562976	−0.0583778
$94$	0	0
$95$	5.25021	0.538660
$96$	0	0
$97$	−4.01956	−0.408124	−0.204062	−	0.978958i	$-0.565414\pi$
$97$	−4.01956	−0.408124	−0.204062	+	0.978958i	$0.565414\pi$
$98$	0	0
$99$	−1.95820	−0.196807
$100$	0	0
$101$	−11.3006	−1.12445	−0.562226	−	0.826983i	$-0.690055\pi$
$101$	−11.3006	−1.12445	−0.562226	+	0.826983i	$0.690055\pi$
$102$	0	0
$103$	−1.11246	−0.109614	−0.0548071	−	0.998497i	$-0.517454\pi$
$103$	−1.11246	−0.109614	−0.0548071	+	0.998497i	$0.517454\pi$
$104$	0	0
$105$	−3.76740	−0.367660
$106$	0	0
$107$	9.37893	0.906696	0.453348	−	0.891334i	$-0.350230\pi$
$107$	9.37893	0.906696	0.453348	+	0.891334i	$0.350230\pi$
$108$	0	0
$109$	−3.44018	−0.329510	−0.164755	−	0.986335i	$-0.552683\pi$
$109$	−3.44018	−0.329510	−0.164755	+	0.986335i	$0.552683\pi$
$110$	0	0
$111$	−1.27137	−0.120673
$112$	0	0
$113$	14.3670	1.35153	0.675767	−	0.737115i	$-0.263813\pi$
$113$	14.3670	1.35153	0.675767	+	0.737115i	$0.263813\pi$
$114$	0	0
$115$	12.1110	1.12936
$116$	0	0
$117$	−1.33480	−0.123402
$118$	0	0
$119$	−4.60593	−0.422225
$120$	0	0
$121$	−10.5351	−0.957732
$122$	0	0
$123$	−0.0996683	−0.00898679
$124$	0	0
$125$	−11.0907	−0.991983
$126$	0	0
$127$	−0.360627	−0.0320005	−0.0160002	−	0.999872i	$-0.505093\pi$
$127$	−0.360627	−0.0320005	−0.0160002	+	0.999872i	$0.505093\pi$
$128$	0	0
$129$	−0.637534	−0.0561318
$130$	0	0
$131$	−17.6353	−1.54080	−0.770402	−	0.637558i	$-0.779945\pi$
$131$	−17.6353	−1.54080	−0.770402	+	0.637558i	$0.779945\pi$
$132$	0	0
$133$	10.8786	0.943295
$134$	0	0
$135$	4.73745	0.407735
$136$	0	0
$137$	−0.948673	−0.0810506	−0.0405253	−	0.999179i	$-0.512903\pi$
$137$	−0.948673	−0.0810506	−0.0405253	+	0.999179i	$0.512903\pi$
$138$	0	0
$139$	13.1604	1.11625	0.558126	−	0.829756i	$-0.311520\pi$
$139$	13.1604	1.11625	0.558126	+	0.829756i	$0.311520\pi$
$140$	0	0
$141$	0.795293	0.0669758
$142$	0	0
$143$	0.316926	0.0265027
$144$	0	0
$145$	−10.7102	−0.889435
$146$	0	0
$147$	−5.30008	−0.437143
$148$	0	0
$149$	9.21002	0.754515	0.377257	−	0.926108i	$-0.376867\pi$
$149$	9.21002	0.754515	0.377257	+	0.926108i	$0.376867\pi$
$150$	0	0
$151$	−15.1103	−1.22966	−0.614828	−	0.788661i	$-0.710774\pi$
$151$	−15.1103	−1.22966	−0.614828	+	0.788661i	$0.710774\pi$
$152$	0	0
$153$	2.83274	0.229013
$154$	0	0
$155$	3.54376	0.284642
$156$	0	0
$157$	3.02040	0.241054	0.120527	−	0.992710i	$-0.461542\pi$
$157$	3.02040	0.241054	0.120527	+	0.992710i	$0.461542\pi$
$158$	0	0
$159$	−1.03302	−0.0819241
$160$	0	0
$161$	25.0944	1.97772
$162$	0	0
$163$	−6.10519	−0.478195	−0.239098	−	0.970996i	$-0.576852\pi$
$163$	−6.10519	−0.478195	−0.239098	+	0.970996i	$0.576852\pi$
$164$	0	0
$165$	−0.550137	−0.0428281
$166$	0	0
$167$	−0.604730	−0.0467954	−0.0233977	−	0.999726i	$-0.507448\pi$
$167$	−0.604730	−0.0467954	−0.0233977	+	0.999726i	$0.507448\pi$
$168$	0	0
$169$	−12.7840	−0.983382
$170$	0	0
$171$	−6.69055	−0.511639
$172$	0	0
$173$	2.88294	0.219186	0.109593	−	0.993977i	$-0.465045\pi$
$173$	2.88294	0.219186	0.109593	+	0.993977i	$0.465045\pi$
$174$	0	0
$175$	0.367156	0.0277544
$176$	0	0
$177$	−2.26522	−0.170264
$178$	0	0
$179$	−9.12375	−0.681941	−0.340971	−	0.940074i	$-0.610756\pi$
$179$	−9.12375	−0.681941	−0.340971	+	0.940074i	$0.610756\pi$
$180$	0	0
$181$	12.1054	0.899788	0.449894	−	0.893082i	$-0.351462\pi$
$181$	12.1054	0.899788	0.449894	+	0.893082i	$0.351462\pi$
$182$	0	0
$183$	−0.0200703	−0.00148364
$184$	0	0
$185$	8.00289	0.588385
$186$	0	0
$187$	−0.672585	−0.0491843
$188$	0	0
$189$	9.81615	0.714020
$190$	0	0
$191$	−21.7332	−1.57256	−0.786279	−	0.617872i	$-0.787995\pi$
$191$	−21.7332	−1.57256	−0.786279	+	0.617872i	$0.787995\pi$
$192$	0	0
$193$	13.9952	1.00739	0.503697	−	0.863880i	$-0.331973\pi$
$193$	13.9952	1.00739	0.503697	+	0.863880i	$0.331973\pi$
$194$	0	0
$195$	−0.374998	−0.0268542
$196$	0	0
$197$	−8.06139	−0.574350	−0.287175	−	0.957878i	$-0.592716\pi$
$197$	−8.06139	−0.574350	−0.287175	+	0.957878i	$0.592716\pi$
$198$	0	0
$199$	19.9548	1.41456	0.707281	−	0.706933i	$-0.249922\pi$
$199$	19.9548	1.41456	0.707281	+	0.706933i	$0.249922\pi$
$200$	0	0
$201$	−2.09062	−0.147461
$202$	0	0
$203$	−22.1919	−1.55757
$204$	0	0
$205$	0.627382	0.0438183
$206$	0	0
$207$	−15.4335	−1.07270
$208$	0	0
$209$	1.58856	0.109883
$210$	0	0
$211$	21.5982	1.48688	0.743439	−	0.668803i	$-0.233193\pi$
$211$	21.5982	1.48688	0.743439	+	0.668803i	$0.233193\pi$
$212$	0	0
$213$	−0.982045	−0.0672886
$214$	0	0
$215$	4.01309	0.273690
$216$	0	0
$217$	7.34279	0.498461
$218$	0	0
$219$	−3.45009	−0.233136
$220$	0	0
$221$	−0.458465	−0.0308397
$222$	0	0
$223$	−23.1367	−1.54935	−0.774674	−	0.632361i	$-0.782086\pi$
$223$	−23.1367	−1.54935	−0.774674	+	0.632361i	$0.782086\pi$
$224$	0	0
$225$	−0.225808	−0.0150539
$226$	0	0
$227$	5.98118	0.396985	0.198492	−	0.980102i	$-0.436396\pi$
$227$	5.98118	0.396985	0.198492	+	0.980102i	$0.436396\pi$
$228$	0	0
$229$	24.9844	1.65101	0.825506	−	0.564393i	$-0.190890\pi$
$229$	24.9844	1.65101	0.825506	+	0.564393i	$0.190890\pi$
$230$	0	0
$231$	−1.13990	−0.0750001
$232$	0	0
$233$	8.09995	0.530646	0.265323	−	0.964160i	$-0.414521\pi$
$233$	8.09995	0.530646	0.265323	+	0.964160i	$0.414521\pi$
$234$	0	0
$235$	−5.00613	−0.326564
$236$	0	0
$237$	1.07623	0.0699084
$238$	0	0
$239$	14.5792	0.943052	0.471526	−	0.881852i	$-0.343703\pi$
$239$	14.5792	0.943052	0.471526	+	0.881852i	$0.343703\pi$
$240$	0	0
$241$	−30.4446	−1.96111	−0.980553	−	0.196254i	$-0.937122\pi$
$241$	−30.4446	−1.96111	−0.980553	+	0.196254i	$0.937122\pi$
$242$	0	0
$243$	−9.12157	−0.585149
$244$	0	0
$245$	33.3624	2.13145
$246$	0	0
$247$	1.08283	0.0688990
$248$	0	0
$249$	−0.832978	−0.0527878
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	3.66443	0.230381
$254$	0	0
$255$	0.795828	0.0498367
$256$	0	0
$257$	7.48685	0.467017	0.233508	−	0.972355i	$-0.424979\pi$
$257$	7.48685	0.467017	0.233508	+	0.972355i	$0.424979\pi$
$258$	0	0
$259$	16.5823	1.03037
$260$	0	0
$261$	13.6484	0.844818
$262$	0	0
$263$	−1.11608	−0.0688202	−0.0344101	−	0.999408i	$-0.510955\pi$
$263$	−1.11608	−0.0688202	−0.0344101	+	0.999408i	$0.510955\pi$
$264$	0	0
$265$	6.50257	0.399450
$266$	0	0
$267$	3.71354	0.227265
$268$	0	0
$269$	−8.28310	−0.505029	−0.252515	−	0.967593i	$-0.581258\pi$
$269$	−8.28310	−0.505029	−0.252515	+	0.967593i	$0.581258\pi$
$270$	0	0
$271$	22.2358	1.35073	0.675364	−	0.737484i	$-0.263986\pi$
$271$	22.2358	1.35073	0.675364	+	0.737484i	$0.263986\pi$
$272$	0	0
$273$	−0.777009	−0.0470267
$274$	0	0
$275$	0.0536143	0.00323306
$276$	0	0
$277$	−10.6054	−0.637219	−0.318609	−	0.947886i	$-0.603216\pi$
$277$	−10.6054	−0.637219	−0.318609	+	0.947886i	$0.603216\pi$
$278$	0	0
$279$	−4.51596	−0.270363
$280$	0	0
$281$	8.18734	0.488416	0.244208	−	0.969723i	$-0.421472\pi$
$281$	8.18734	0.488416	0.244208	+	0.969723i	$0.421472\pi$
$282$	0	0
$283$	27.4202	1.62996	0.814980	−	0.579490i	$-0.196748\pi$
$283$	27.4202	1.62996	0.814980	+	0.579490i	$0.196748\pi$
$284$	0	0
$285$	−1.87964	−0.111340
$286$	0	0
$287$	1.29996	0.0767340
$288$	0	0
$289$	−16.0270	−0.942767
$290$	0	0
$291$	1.43905	0.0843587
$292$	0	0
$293$	−3.15462	−0.184295	−0.0921475	−	0.995745i	$-0.529373\pi$
$293$	−3.15462	−0.184295	−0.0921475	+	0.995745i	$0.529373\pi$
$294$	0	0
$295$	14.2589	0.830183
$296$	0	0
$297$	1.43341	0.0831750
$298$	0	0
$299$	2.49784	0.144454
$300$	0	0
$301$	8.31525	0.479283
$302$	0	0
$303$	4.04576	0.232423
$304$	0	0
$305$	0.126336	0.00723400
$306$	0	0
$307$	18.4553	1.05330	0.526650	−	0.850082i	$-0.323448\pi$
$307$	18.4553	1.05330	0.526650	+	0.850082i	$0.323448\pi$
$308$	0	0
$309$	0.398275	0.0226571
$310$	0	0
$311$	−25.0171	−1.41859	−0.709296	−	0.704911i	$-0.750987\pi$
$311$	−25.0171	−1.41859	−0.709296	+	0.704911i	$0.750987\pi$
$312$	0	0
$313$	10.0867	0.570132	0.285066	−	0.958508i	$-0.407984\pi$
$313$	10.0867	0.570132	0.285066	+	0.958508i	$0.407984\pi$
$314$	0	0
$315$	−30.2205	−1.70273
$316$	0	0
$317$	22.3677	1.25629	0.628147	−	0.778095i	$-0.283814\pi$
$317$	22.3677	1.25629	0.628147	+	0.778095i	$0.283814\pi$
$318$	0	0
$319$	−3.24059	−0.181438
$320$	0	0
$321$	−3.35777	−0.187413
$322$	0	0
$323$	−2.29801	−0.127864
$324$	0	0
$325$	0.0365459	0.00202720
$326$	0	0
$327$	1.23163	0.0681092
$328$	0	0
$329$	−10.3729	−0.571875
$330$	0	0
$331$	19.0247	1.04569	0.522846	−	0.852427i	$-0.324870\pi$
$331$	19.0247	1.04569	0.522846	+	0.852427i	$0.324870\pi$
$332$	0	0
$333$	−10.1984	−0.558869
$334$	0	0
$335$	13.1598	0.718998
$336$	0	0
$337$	16.3647	0.891445	0.445722	−	0.895171i	$-0.352947\pi$
$337$	16.3647	0.891445	0.445722	+	0.895171i	$0.352947\pi$
$338$	0	0
$339$	−5.14357	−0.279360
$340$	0	0
$341$	1.07224	0.0580649
$342$	0	0
$343$	36.4415	1.96766
$344$	0	0
$345$	−4.33589	−0.233437
$346$	0	0
$347$	−0.559753	−0.0300491	−0.0150246	−	0.999887i	$-0.504783\pi$
$347$	−0.559753	−0.0300491	−0.0150246	+	0.999887i	$0.504783\pi$
$348$	0	0
$349$	−11.4850	−0.614777	−0.307389	−	0.951584i	$-0.599455\pi$
$349$	−11.4850	−0.614777	−0.307389	+	0.951584i	$0.599455\pi$
$350$	0	0
$351$	0.977079	0.0521526
$352$	0	0
$353$	−0.0283464	−0.00150872	−0.000754362	−	1.00000i	$-0.500240\pi$
$353$	−0.0283464	−0.00150872	−0.000754362		1.00000i	$0.500240\pi$
$354$	0	0
$355$	6.18168	0.328089
$356$	0	0
$357$	1.64898	0.0872734
$358$	0	0
$359$	6.12611	0.323324	0.161662	−	0.986846i	$-0.448315\pi$
$359$	6.12611	0.323324	0.161662	+	0.986846i	$0.448315\pi$
$360$	0	0
$361$	−13.5724	−0.714338
$362$	0	0
$363$	3.77168	0.197962
$364$	0	0
$365$	21.7173	1.13674
$366$	0	0
$367$	5.29169	0.276224	0.138112	−	0.990417i	$-0.455897\pi$
$367$	5.29169	0.276224	0.138112	+	0.990417i	$0.455897\pi$
$368$	0	0
$369$	−0.799498	−0.0416202
$370$	0	0
$371$	13.4735	0.699511
$372$	0	0
$373$	27.7468	1.43668	0.718338	−	0.695694i	$-0.244903\pi$
$373$	27.7468	1.43668	0.718338	+	0.695694i	$0.244903\pi$
$374$	0	0
$375$	3.97061	0.205042
$376$	0	0
$377$	−2.20893	−0.113766
$378$	0	0
$379$	−4.91895	−0.252669	−0.126335	−	0.991988i	$-0.540321\pi$
$379$	−4.91895	−0.252669	−0.126335	+	0.991988i	$0.540321\pi$
$380$	0	0
$381$	0.129109	0.00661446
$382$	0	0
$383$	5.62298	0.287321	0.143661	−	0.989627i	$-0.454113\pi$
$383$	5.62298	0.287321	0.143661	+	0.989627i	$0.454113\pi$
$384$	0	0
$385$	7.17534	0.365689
$386$	0	0
$387$	−5.11403	−0.259961
$388$	0	0
$389$	32.4279	1.64416	0.822080	−	0.569372i	$-0.192813\pi$
$389$	32.4279	1.64416	0.822080	+	0.569372i	$0.192813\pi$
$390$	0	0
$391$	−5.30096	−0.268081
$392$	0	0
$393$	6.31366	0.318482
$394$	0	0
$395$	−6.77452	−0.340863
$396$	0	0
$397$	−4.65060	−0.233407	−0.116704	−	0.993167i	$-0.537233\pi$
$397$	−4.65060	−0.233407	−0.116704	+	0.993167i	$0.537233\pi$
$398$	0	0
$399$	−3.89468	−0.194978
$400$	0	0
$401$	−19.4879	−0.973180	−0.486590	−	0.873630i	$-0.661759\pi$
$401$	−19.4879	−0.973180	−0.486590	+	0.873630i	$0.661759\pi$
$402$	0	0
$403$	0.730885	0.0364080
$404$	0	0
$405$	17.7196	0.880495
$406$	0	0
$407$	2.42144	0.120026
$408$	0	0
$409$	−20.7056	−1.02383	−0.511913	−	0.859037i	$-0.671063\pi$
$409$	−20.7056	−1.02383	−0.511913	+	0.859037i	$0.671063\pi$
$410$	0	0
$411$	0.339637	0.0167531
$412$	0	0
$413$	29.5448	1.45381
$414$	0	0
$415$	5.24334	0.257386
$416$	0	0
$417$	−4.71160	−0.230728
$418$	0	0
$419$	−30.0719	−1.46911	−0.734555	−	0.678549i	$-0.762609\pi$
$419$	−30.0719	−1.46911	−0.734555	+	0.678549i	$0.762609\pi$
$420$	0	0
$421$	−24.2237	−1.18059	−0.590295	−	0.807188i	$-0.700988\pi$
$421$	−24.2237	−1.18059	−0.590295	+	0.807188i	$0.700988\pi$
$422$	0	0
$423$	6.37951	0.310182
$424$	0	0
$425$	−0.0775584	−0.00376214
$426$	0	0
$427$	0.261773	0.0126681
$428$	0	0
$429$	−0.113463	−0.00547807
$430$	0	0
$431$	−12.0779	−0.581774	−0.290887	−	0.956757i	$-0.593950\pi$
$431$	−12.0779	−0.581774	−0.290887	+	0.956757i	$0.593950\pi$
$432$	0	0
$433$	19.5412	0.939089	0.469544	−	0.882909i	$-0.344418\pi$
$433$	19.5412	0.939089	0.469544	+	0.882909i	$0.344418\pi$
$434$	0	0
$435$	3.83439	0.183845
$436$	0	0
$437$	12.5202	0.598921
$438$	0	0
$439$	12.2606	0.585168	0.292584	−	0.956240i	$-0.405485\pi$
$439$	12.2606	0.585168	0.292584	+	0.956240i	$0.405485\pi$
$440$	0	0
$441$	−42.5150	−2.02452
$442$	0	0
$443$	−24.8669	−1.18146	−0.590731	−	0.806869i	$-0.701160\pi$
$443$	−24.8669	−1.18146	−0.590731	+	0.806869i	$0.701160\pi$
$444$	0	0
$445$	−23.3756	−1.10811
$446$	0	0
$447$	−3.29730	−0.155957
$448$	0	0
$449$	−20.3163	−0.958785	−0.479393	−	0.877601i	$-0.659143\pi$
$449$	−20.3163	−0.958785	−0.479393	+	0.877601i	$0.659143\pi$
$450$	0	0
$451$	0.189827	0.00893862
$452$	0	0
$453$	5.40966	0.254168
$454$	0	0
$455$	4.89104	0.229295
$456$	0	0
$457$	−26.0262	−1.21746	−0.608728	−	0.793379i	$-0.708320\pi$
$457$	−26.0262	−1.21746	−0.608728	+	0.793379i	$0.708320\pi$
$458$	0	0
$459$	−2.07357	−0.0967861
$460$	0	0
$461$	−10.3380	−0.481490	−0.240745	−	0.970588i	$-0.577392\pi$
$461$	−10.3380	−0.481490	−0.240745	+	0.970588i	$0.577392\pi$
$462$	0	0
$463$	−1.96910	−0.0915117	−0.0457559	−	0.998953i	$-0.514570\pi$
$463$	−1.96910	−0.0915117	−0.0457559	+	0.998953i	$0.514570\pi$
$464$	0	0
$465$	−1.26871	−0.0588351
$466$	0	0
$467$	34.7843	1.60963	0.804814	−	0.593528i	$-0.202265\pi$
$467$	34.7843	1.60963	0.804814	+	0.593528i	$0.202265\pi$
$468$	0	0
$469$	27.2676	1.25910
$470$	0	0
$471$	−1.08134	−0.0498256
$472$	0	0
$473$	1.21424	0.0558309
$474$	0	0
$475$	0.183183	0.00840500
$476$	0	0
$477$	−8.28648	−0.379412
$478$	0	0
$479$	−10.7833	−0.492701	−0.246351	−	0.969181i	$-0.579232\pi$
$479$	−10.7833	−0.492701	−0.246351	+	0.969181i	$0.579232\pi$
$480$	0	0
$481$	1.65056	0.0752592
$482$	0	0
$483$	−8.98411	−0.408791
$484$	0	0
$485$	−9.05840	−0.411321
$486$	0	0
$487$	−21.7901	−0.987402	−0.493701	−	0.869632i	$-0.664356\pi$
$487$	−21.7901	−0.987402	−0.493701	+	0.869632i	$0.664356\pi$
$488$	0	0
$489$	2.18573	0.0988423
$490$	0	0
$491$	−23.6974	−1.06945	−0.534725	−	0.845026i	$-0.679585\pi$
$491$	−23.6974	−1.06945	−0.534725	+	0.845026i	$0.679585\pi$
$492$	0	0
$493$	4.68784	0.211130
$494$	0	0
$495$	−4.41297	−0.198348
$496$	0	0
$497$	12.8086	0.574546
$498$	0	0
$499$	5.33650	0.238895	0.119447	−	0.992841i	$-0.461888\pi$
$499$	5.33650	0.238895	0.119447	+	0.992841i	$0.461888\pi$
$500$	0	0
$501$	0.216501	0.00967255
$502$	0	0
$503$	21.7718	0.970755	0.485378	−	0.874305i	$-0.338682\pi$
$503$	21.7718	0.970755	0.485378	+	0.874305i	$0.338682\pi$
$504$	0	0
$505$	−25.4668	−1.13326
$506$	0	0
$507$	4.57682	0.203264
$508$	0	0
$509$	4.15914	0.184351	0.0921753	−	0.995743i	$-0.470618\pi$
$509$	4.15914	0.184351	0.0921753	+	0.995743i	$0.470618\pi$
$510$	0	0
$511$	44.9990	1.99064
$512$	0	0
$513$	4.89750	0.216230
$514$	0	0
$515$	−2.50702	−0.110473
$516$	0	0
$517$	−1.51471	−0.0666167
$518$	0	0
$519$	−1.03213	−0.0453055
$520$	0	0
$521$	−37.9596	−1.66304	−0.831520	−	0.555494i	$-0.812529\pi$
$521$	−37.9596	−1.66304	−0.831520	+	0.555494i	$0.812529\pi$
$522$	0	0
$523$	27.9866	1.22377	0.611884	−	0.790947i	$-0.290412\pi$
$523$	27.9866	1.22377	0.611884	+	0.790947i	$0.290412\pi$
$524$	0	0
$525$	−0.131447	−0.00573680
$526$	0	0
$527$	−1.55110	−0.0675669
$528$	0	0
$529$	5.88113	0.255701
$530$	0	0
$531$	−18.1706	−0.788538
$532$	0	0
$533$	0.129395	0.00560471
$534$	0	0
$535$	21.1362	0.913797
$536$	0	0
$537$	3.26642	0.140956
$538$	0	0
$539$	10.0945	0.434800
$540$	0	0
$541$	−25.0936	−1.07886	−0.539429	−	0.842031i	$-0.681360\pi$
$541$	−25.0936	−1.07886	−0.539429	+	0.842031i	$0.681360\pi$
$542$	0	0
$543$	−4.33389	−0.185985
$544$	0	0
$545$	−7.75273	−0.332091
$546$	0	0
$547$	12.9856	0.555225	0.277613	−	0.960693i	$-0.410457\pi$
$547$	12.9856	0.555225	0.277613	+	0.960693i	$0.410457\pi$
$548$	0	0
$549$	−0.160995	−0.00687111
$550$	0	0
$551$	−11.0721	−0.471685
$552$	0	0
$553$	−14.0370	−0.596915
$554$	0	0
$555$	−2.86514	−0.121618
$556$	0	0
$557$	−43.4396	−1.84060	−0.920298	−	0.391219i	$-0.872054\pi$
$557$	−43.4396	−1.84060	−0.920298	+	0.391219i	$0.872054\pi$
$558$	0	0
$559$	0.827682	0.0350072
$560$	0	0
$561$	0.240794	0.0101663
$562$	0	0
$563$	31.9340	1.34586	0.672929	−	0.739707i	$-0.265036\pi$
$563$	31.9340	1.34586	0.672929	+	0.739707i	$0.265036\pi$
$564$	0	0
$565$	32.3772	1.36212
$566$	0	0
$567$	36.7156	1.54191
$568$	0	0
$569$	−22.4340	−0.940480	−0.470240	−	0.882539i	$-0.655833\pi$
$569$	−22.4340	−0.940480	−0.470240	+	0.882539i	$0.655833\pi$
$570$	0	0
$571$	−15.4435	−0.646291	−0.323146	−	0.946349i	$-0.604740\pi$
$571$	−15.4435	−0.646291	−0.323146	+	0.946349i	$0.604740\pi$
$572$	0	0
$573$	7.78075	0.325045
$574$	0	0
$575$	0.422560	0.0176220
$576$	0	0
$577$	−34.0505	−1.41754	−0.708771	−	0.705439i	$-0.750750\pi$
$577$	−34.0505	−1.41754	−0.708771	+	0.705439i	$0.750750\pi$
$578$	0	0
$579$	−5.01044	−0.208227
$580$	0	0
$581$	10.8644	0.450730
$582$	0	0
$583$	1.96749	0.0814849
$584$	0	0
$585$	−3.00808	−0.124369
$586$	0	0
$587$	−21.8100	−0.900195	−0.450098	−	0.892979i	$-0.648611\pi$
$587$	−21.8100	−0.900195	−0.450098	+	0.892979i	$0.648611\pi$
$588$	0	0
$589$	3.66349	0.150951
$590$	0	0
$591$	2.88608	0.118717
$592$	0	0
$593$	−25.9000	−1.06359	−0.531794	−	0.846874i	$-0.678482\pi$
$593$	−25.9000	−1.06359	−0.531794	+	0.846874i	$0.678482\pi$
$594$	0	0
$595$	−10.3798	−0.425532
$596$	0	0
$597$	−7.14409	−0.292388
$598$	0	0
$599$	48.0636	1.96383	0.981913	−	0.189331i	$-0.0606319\pi$
$599$	48.0636	1.96383	0.981913	+	0.189331i	$0.0606319\pi$
$600$	0	0
$601$	3.37140	0.137522	0.0687611	−	0.997633i	$-0.478095\pi$
$601$	3.37140	0.137522	0.0687611	+	0.997633i	$0.478095\pi$
$602$	0	0
$603$	−16.7701	−0.682930
$604$	0	0
$605$	−23.7416	−0.965234
$606$	0	0
$607$	−13.6598	−0.554433	−0.277216	−	0.960808i	$-0.589412\pi$
$607$	−13.6598	−0.554433	−0.277216	+	0.960808i	$0.589412\pi$
$608$	0	0
$609$	7.94498	0.321947
$610$	0	0
$611$	−1.03249	−0.0417702
$612$	0	0
$613$	−10.0736	−0.406870	−0.203435	−	0.979088i	$-0.565211\pi$
$613$	−10.0736	−0.406870	−0.203435	+	0.979088i	$0.565211\pi$
$614$	0	0
$615$	−0.224611	−0.00905717
$616$	0	0
$617$	22.2686	0.896500	0.448250	−	0.893908i	$-0.352048\pi$
$617$	22.2686	0.896500	0.448250	+	0.893908i	$0.352048\pi$
$618$	0	0
$619$	−2.92761	−0.117671	−0.0588354	−	0.998268i	$-0.518739\pi$
$619$	−2.92761	−0.117671	−0.0588354	+	0.998268i	$0.518739\pi$
$620$	0	0
$621$	11.2974	0.453349
$622$	0	0
$623$	−48.4351	−1.94051
$624$	0	0
$625$	−25.3870	−1.01548
$626$	0	0
$627$	−0.568724	−0.0227126
$628$	0	0
$629$	−3.50285	−0.139668
$630$	0	0
$631$	1.95941	0.0780027	0.0390014	−	0.999239i	$-0.487582\pi$
$631$	1.95941	0.0780027	0.0390014	+	0.999239i	$0.487582\pi$
$632$	0	0
$633$	−7.73241	−0.307336
$634$	0	0
$635$	−0.812703	−0.0322511
$636$	0	0
$637$	6.88085	0.272629
$638$	0	0
$639$	−7.87756	−0.311631
$640$	0	0
$641$	−43.3664	−1.71287	−0.856435	−	0.516255i	$-0.827326\pi$
$641$	−43.3664	−1.71287	−0.856435	+	0.516255i	$0.827326\pi$
$642$	0	0
$643$	−20.0953	−0.792480	−0.396240	−	0.918147i	$-0.629685\pi$
$643$	−20.0953	−0.792480	−0.396240	+	0.918147i	$0.629685\pi$
$644$	0	0
$645$	−1.43674	−0.0565714
$646$	0	0
$647$	9.18944	0.361274	0.180637	−	0.983550i	$-0.442184\pi$
$647$	9.18944	0.361274	0.180637	+	0.983550i	$0.442184\pi$
$648$	0	0
$649$	4.31431	0.169351
$650$	0	0
$651$	−2.62881	−0.103031
$652$	0	0
$653$	7.73955	0.302872	0.151436	−	0.988467i	$-0.451610\pi$
$653$	7.73955	0.302872	0.151436	+	0.988467i	$0.451610\pi$
$654$	0	0
$655$	−39.7426	−1.55287
$656$	0	0
$657$	−27.6752	−1.07971
$658$	0	0
$659$	10.9678	0.427243	0.213622	−	0.976916i	$-0.431474\pi$
$659$	10.9678	0.427243	0.213622	+	0.976916i	$0.431474\pi$
$660$	0	0
$661$	−22.9753	−0.893636	−0.446818	−	0.894625i	$-0.647443\pi$
$661$	−22.9753	−0.893636	−0.446818	+	0.894625i	$0.647443\pi$
$662$	0	0
$663$	0.164136	0.00637452
$664$	0	0
$665$	24.5158	0.950683
$666$	0	0
$667$	−25.5407	−0.988938
$668$	0	0
$669$	8.28323	0.320248
$670$	0	0
$671$	0.0382256	0.00147568
$672$	0	0
$673$	−40.6928	−1.56859	−0.784296	−	0.620387i	$-0.786976\pi$
$673$	−40.6928	−1.56859	−0.784296	+	0.620387i	$0.786976\pi$
$674$	0	0
$675$	0.165292	0.00636210
$676$	0	0
$677$	−3.47467	−0.133543	−0.0667713	−	0.997768i	$-0.521270\pi$
$677$	−3.47467	−0.133543	−0.0667713	+	0.997768i	$0.521270\pi$
$678$	0	0
$679$	−18.7693	−0.720299
$680$	0	0
$681$	−2.14134	−0.0820562
$682$	0	0
$683$	−13.6429	−0.522033	−0.261016	−	0.965334i	$-0.584058\pi$
$683$	−13.6429	−0.522033	−0.261016	+	0.965334i	$0.584058\pi$
$684$	0	0
$685$	−2.13791	−0.0816854
$686$	0	0
$687$	−8.94471	−0.341262
$688$	0	0
$689$	1.34113	0.0510929
$690$	0	0
$691$	−51.5022	−1.95924	−0.979619	−	0.200867i	$-0.935624\pi$
$691$	−51.5022	−1.95924	−0.979619	+	0.200867i	$0.935624\pi$
$692$	0	0
$693$	−9.14382	−0.347345
$694$	0	0
$695$	29.6581	1.12500
$696$	0	0
$697$	−0.274604	−0.0104014
$698$	0	0
$699$	−2.89988	−0.109684
$700$	0	0
$701$	−29.0166	−1.09594	−0.547971	−	0.836497i	$-0.684600\pi$
$701$	−29.0166	−1.09594	−0.547971	+	0.836497i	$0.684600\pi$
$702$	0	0
$703$	8.27327	0.312032
$704$	0	0
$705$	1.79226	0.0675003
$706$	0	0
$707$	−52.7681	−1.98455
$708$	0	0
$709$	−18.5227	−0.695634	−0.347817	−	0.937563i	$-0.613077\pi$
$709$	−18.5227	−0.695634	−0.347817	+	0.937563i	$0.613077\pi$
$710$	0	0
$711$	8.63303	0.323764
$712$	0	0
$713$	8.45081	0.316485
$714$	0	0
$715$	0.714218	0.0267102
$716$	0	0
$717$	−5.21955	−0.194927
$718$	0	0
$719$	16.7133	0.623300	0.311650	−	0.950197i	$-0.399118\pi$
$719$	16.7133	0.623300	0.311650	+	0.950197i	$0.399118\pi$
$720$	0	0
$721$	−5.19463	−0.193458
$722$	0	0
$723$	10.8995	0.405358
$724$	0	0
$725$	−0.373685	−0.0138783
$726$	0	0
$727$	−5.88214	−0.218156	−0.109078	−	0.994033i	$-0.534790\pi$
$727$	−5.88214	−0.218156	−0.109078	+	0.994033i	$0.534790\pi$
$728$	0	0
$729$	−20.3230	−0.752703
$730$	0	0
$731$	−1.75652	−0.0649672
$732$	0	0
$733$	0.631585	0.0233281	0.0116641	−	0.999932i	$-0.496287\pi$
$733$	0.631585	0.0233281	0.0116641	+	0.999932i	$0.496287\pi$
$734$	0	0
$735$	−11.9442	−0.440567
$736$	0	0
$737$	3.98177	0.146670
$738$	0	0
$739$	−24.4246	−0.898473	−0.449236	−	0.893413i	$-0.648304\pi$
$739$	−24.4246	−0.898473	−0.449236	+	0.893413i	$0.648304\pi$
$740$	0	0
$741$	−0.387668	−0.0142413
$742$	0	0
$743$	−42.5291	−1.56024	−0.780120	−	0.625629i	$-0.784842\pi$
$743$	−42.5291	−1.56024	−0.780120	+	0.625629i	$0.784842\pi$
$744$	0	0
$745$	20.7555	0.760424
$746$	0	0
$747$	−6.68180	−0.244474
$748$	0	0
$749$	43.7949	1.60023
$750$	0	0
$751$	−28.3865	−1.03584	−0.517919	−	0.855429i	$-0.673293\pi$
$751$	−28.3865	−1.03584	−0.517919	+	0.855429i	$0.673293\pi$
$752$	0	0
$753$	0.358013	0.0130467
$754$	0	0
$755$	−34.0522	−1.23929
$756$	0	0
$757$	23.1842	0.842645	0.421323	−	0.906911i	$-0.361566\pi$
$757$	23.1842	0.842645	0.421323	+	0.906911i	$0.361566\pi$
$758$	0	0
$759$	−1.31191	−0.0476194
$760$	0	0
$761$	−29.7476	−1.07835	−0.539174	−	0.842194i	$-0.681264\pi$
$761$	−29.7476	−1.07835	−0.539174	+	0.842194i	$0.681264\pi$
$762$	0	0
$763$	−16.0639	−0.581553
$764$	0	0
$765$	6.38380	0.230807
$766$	0	0
$767$	2.94083	0.106187
$768$	0	0
$769$	30.6145	1.10399	0.551993	−	0.833849i	$-0.313867\pi$
$769$	30.6145	1.10399	0.551993	+	0.833849i	$0.313867\pi$
$770$	0	0
$771$	−2.68038	−0.0965317
$772$	0	0
$773$	24.8766	0.894748	0.447374	−	0.894347i	$-0.352359\pi$
$773$	24.8766	0.894748	0.447374	+	0.894347i	$0.352359\pi$
$774$	0	0
$775$	0.123644	0.00444142
$776$	0	0
$777$	−5.93666	−0.212976
$778$	0	0
$779$	0.648578	0.0232377
$780$	0	0
$781$	1.87039	0.0669279
$782$	0	0
$783$	−9.99071	−0.357039
$784$	0	0
$785$	6.80672	0.242942
$786$	0	0
$787$	−32.1374	−1.14557	−0.572787	−	0.819704i	$-0.694138\pi$
$787$	−32.1374	−1.14557	−0.572787	+	0.819704i	$0.694138\pi$
$788$	0	0
$789$	0.399569	0.0142250
$790$	0	0
$791$	67.0866	2.38533
$792$	0	0
$793$	0.0260563	0.000925287 0
$794$	0	0
$795$	−2.32800	−0.0825657
$796$	0	0
$797$	1.16154	0.0411440	0.0205720	−	0.999788i	$-0.493451\pi$
$797$	1.16154	0.0411440	0.0205720	+	0.999788i	$0.493451\pi$
$798$	0	0
$799$	2.19117	0.0775182
$800$	0	0
$801$	29.7885	1.05252
$802$	0	0
$803$	6.57101	0.231886
$804$	0	0
$805$	56.5523	1.99321
$806$	0	0
$807$	2.96545	0.104389
$808$	0	0
$809$	27.8304	0.978465	0.489232	−	0.872153i	$-0.337277\pi$
$809$	27.8304	0.978465	0.489232	+	0.872153i	$0.337277\pi$
$810$	0	0
$811$	28.3598	0.995847	0.497924	−	0.867221i	$-0.334096\pi$
$811$	28.3598	0.995847	0.497924	+	0.867221i	$0.334096\pi$
$812$	0	0
$813$	−7.96069	−0.279194
$814$	0	0
$815$	−13.7585	−0.481941
$816$	0	0
$817$	4.14867	0.145144
$818$	0	0
$819$	−6.23284	−0.217793
$820$	0	0
$821$	7.07414	0.246889	0.123445	−	0.992351i	$-0.460606\pi$
$821$	7.07414	0.246889	0.123445	+	0.992351i	$0.460606\pi$
$822$	0	0
$823$	32.8771	1.14602	0.573011	−	0.819548i	$-0.305775\pi$
$823$	32.8771	1.14602	0.573011	+	0.819548i	$0.305775\pi$
$824$	0	0
$825$	−0.0191946	−0.000668270 0
$826$	0	0
$827$	35.8898	1.24801	0.624006	−	0.781420i	$-0.285504\pi$
$827$	35.8898	1.24801	0.624006	+	0.781420i	$0.285504\pi$
$828$	0	0
$829$	32.7648	1.13797	0.568984	−	0.822349i	$-0.307337\pi$
$829$	32.7648	1.13797	0.568984	+	0.822349i	$0.307337\pi$
$830$	0	0
$831$	3.79688	0.131712
$832$	0	0
$833$	−14.6027	−0.505952
$834$	0	0
$835$	−1.36281	−0.0471619
$836$	0	0
$837$	3.30570	0.114262
$838$	0	0
$839$	8.55189	0.295244	0.147622	−	0.989044i	$-0.452838\pi$
$839$	8.55189	0.295244	0.147622	+	0.989044i	$0.452838\pi$
$840$	0	0
$841$	−6.41345	−0.221154
$842$	0	0
$843$	−2.93117	−0.100955
$844$	0	0
$845$	−28.8097	−0.991084
$846$	0	0
$847$	−49.1934	−1.69030
$848$	0	0
$849$	−9.81676	−0.336910
$850$	0	0
$851$	19.0845	0.654208
$852$	0	0
$853$	−40.7800	−1.39628	−0.698140	−	0.715961i	$-0.745989\pi$
$853$	−40.7800	−1.39628	−0.698140	+	0.715961i	$0.745989\pi$
$854$	0	0
$855$	−15.0777	−0.515646
$856$	0	0
$857$	−22.2800	−0.761069	−0.380535	−	0.924767i	$-0.624260\pi$
$857$	−22.2800	−0.761069	−0.380535	+	0.924767i	$0.624260\pi$
$858$	0	0
$859$	44.1604	1.50673	0.753367	−	0.657600i	$-0.228428\pi$
$859$	44.1604	1.50673	0.753367	+	0.657600i	$0.228428\pi$
$860$	0	0
$861$	−0.465401	−0.0158608
$862$	0	0
$863$	−54.9486	−1.87047	−0.935236	−	0.354024i	$-0.884813\pi$
$863$	−54.9486	−1.87047	−0.935236	+	0.354024i	$0.884813\pi$
$864$	0	0
$865$	6.49695	0.220903
$866$	0	0
$867$	5.73788	0.194869
$868$	0	0
$869$	−2.04977	−0.0695336
$870$	0	0
$871$	2.71415	0.0919656
$872$	0	0
$873$	11.5435	0.390687
$874$	0	0
$875$	−51.7880	−1.75075
$876$	0	0
$877$	52.8730	1.78540	0.892698	−	0.450656i	$-0.148810\pi$
$877$	52.8730	1.78540	0.892698	+	0.450656i	$0.148810\pi$
$878$	0	0
$879$	1.12939	0.0380935
$880$	0	0
$881$	−36.9873	−1.24613	−0.623067	−	0.782168i	$-0.714114\pi$
$881$	−36.9873	−1.24613	−0.623067	+	0.782168i	$0.714114\pi$
$882$	0	0
$883$	55.4842	1.86719	0.933597	−	0.358326i	$-0.116652\pi$
$883$	55.4842	1.86719	0.933597	+	0.358326i	$0.116652\pi$
$884$	0	0
$885$	−5.10485	−0.171598
$886$	0	0
$887$	−27.5877	−0.926306	−0.463153	−	0.886278i	$-0.653282\pi$
$887$	−27.5877	−0.926306	−0.463153	+	0.886278i	$0.653282\pi$
$888$	0	0
$889$	−1.68395	−0.0564777
$890$	0	0
$891$	5.36143	0.179615
$892$	0	0
$893$	−5.17526	−0.173184
$894$	0	0
$895$	−20.5611	−0.687282
$896$	0	0
$897$	−0.894259	−0.0298584
$898$	0	0
$899$	−7.47337	−0.249251
$900$	0	0
$901$	−2.84616	−0.0948194
$902$	0	0
$903$	−2.97696	−0.0990671
$904$	0	0
$905$	27.2805	0.906836
$906$	0	0
$907$	31.2818	1.03870	0.519348	−	0.854563i	$-0.326175\pi$
$907$	31.2818	1.03870	0.519348	+	0.854563i	$0.326175\pi$
$908$	0	0
$909$	32.4534	1.07641
$910$	0	0
$911$	−41.7288	−1.38254	−0.691268	−	0.722598i	$-0.742948\pi$
$911$	−41.7288	−1.38254	−0.691268	+	0.722598i	$0.742948\pi$
$912$	0	0
$913$	1.58648	0.0525048
$914$	0	0
$915$	−0.0452300	−0.00149526
$916$	0	0
$917$	−82.3480	−2.71937
$918$	0	0
$919$	34.8630	1.15002	0.575012	−	0.818145i	$-0.304997\pi$
$919$	34.8630	1.15002	0.575012	+	0.818145i	$0.304997\pi$
$920$	0	0
$921$	−6.60723	−0.217716
$922$	0	0
$923$	1.27494	0.0419653
$924$	0	0
$925$	0.279225	0.00918088
$926$	0	0
$927$	3.19480	0.104931
$928$	0	0
$929$	38.2158	1.25382	0.626910	−	0.779092i	$-0.284320\pi$
$929$	38.2158	1.25382	0.626910	+	0.779092i	$0.284320\pi$
$930$	0	0
$931$	34.4895	1.13035
$932$	0	0
$933$	8.95645	0.293221
$934$	0	0
$935$	−1.51573	−0.0495695
$936$	0	0
$937$	−23.1614	−0.756651	−0.378326	−	0.925673i	$-0.623500\pi$
$937$	−23.1614	−0.756651	−0.378326	+	0.925673i	$0.623500\pi$
$938$	0	0
$939$	−3.61115	−0.117846
$940$	0	0
$941$	−53.5932	−1.74709	−0.873544	−	0.486744i	$-0.838184\pi$
$941$	−53.5932	−1.74709	−0.873544	+	0.486744i	$0.838184\pi$
$942$	0	0
$943$	1.49612	0.0487203
$944$	0	0
$945$	22.1215	0.719612
$946$	0	0
$947$	52.7975	1.71569	0.857844	−	0.513911i	$-0.171804\pi$
$947$	52.7975	1.71569	0.857844	+	0.513911i	$0.171804\pi$
$948$	0	0
$949$	4.47910	0.145398
$950$	0	0
$951$	−8.00791	−0.259674
$952$	0	0
$953$	−5.49662	−0.178053	−0.0890264	−	0.996029i	$-0.528376\pi$
$953$	−5.49662	−0.178053	−0.0890264	+	0.996029i	$0.528376\pi$
$954$	0	0
$955$	−48.9775	−1.58487
$956$	0	0
$957$	1.16017	0.0375031
$958$	0	0
$959$	−4.42982	−0.143046
$960$	0	0
$961$	−28.5272	−0.920233
$962$	0	0
$963$	−26.9347	−0.867958
$964$	0	0
$965$	31.5392	1.01528
$966$	0	0
$967$	−18.1431	−0.583442	−0.291721	−	0.956503i	$-0.594228\pi$
$967$	−18.1431	−0.583442	−0.291721	+	0.956503i	$0.594228\pi$
$968$	0	0
$969$	0.822715	0.0264294
$970$	0	0
$971$	15.9803	0.512832	0.256416	−	0.966567i	$-0.417458\pi$
$971$	15.9803	0.512832	0.256416	+	0.966567i	$0.417458\pi$
$972$	0	0
$973$	61.4525	1.97008
$974$	0	0
$975$	−0.0130839	−0.000419020 0
$976$	0	0
$977$	34.0089	1.08804	0.544020	−	0.839072i	$-0.316901\pi$
$977$	34.0089	1.08804	0.544020	+	0.839072i	$0.316901\pi$
$978$	0	0
$979$	−7.07278	−0.226047
$980$	0	0
$981$	9.87961	0.315432
$982$	0	0
$983$	−2.66699	−0.0850637	−0.0425318	−	0.999095i	$-0.513542\pi$
$983$	−2.66699	−0.0850637	−0.0425318	+	0.999095i	$0.513542\pi$
$984$	0	0
$985$	−18.1670	−0.578849
$986$	0	0
$987$	3.71362	0.118206
$988$	0	0
$989$	9.57001	0.304309
$990$	0	0
$991$	23.2217	0.737660	0.368830	−	0.929497i	$-0.379758\pi$
$991$	23.2217	0.737660	0.368830	+	0.929497i	$0.379758\pi$
$992$	0	0
$993$	−6.81107	−0.216143
$994$	0	0
$995$	44.9699	1.42564
$996$	0	0
$997$	25.3804	0.803805	0.401903	−	0.915682i	$-0.368349\pi$
$997$	25.3804	0.803805	0.401903	+	0.915682i	$0.368349\pi$
$998$	0	0
$999$	7.46527	0.236191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.g.1.3		7
4.3	odd	2		1004.2.a.a.1.5	✓	7
12.11	even	2		9036.2.a.i.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.a.1.5	✓	7	4.3	odd	2
4016.2.a.g.1.3		7	1.1	even	1	trivial
9036.2.a.i.1.2		7	12.11	even	2

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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-0.358013 q^{3} +2.25358 q^{5} +4.66949 q^{7} -2.87183 q^{9} +O(q^{10})\)	\(q-0.358013 q^{3} +2.25358 q^{5} +4.66949 q^{7} -2.87183 q^{9} +0.681867 q^{11} +0.464791 q^{13} -0.806810 q^{15} -0.986388 q^{17} +2.32972 q^{19} -1.67174 q^{21} +5.37412 q^{23} +0.0786287 q^{25} +2.10219 q^{27} -4.75253 q^{29} +1.57250 q^{31} -0.244117 q^{33} +10.5231 q^{35} +3.55119 q^{37} -0.166401 q^{39} +0.278393 q^{41} +1.78076 q^{43} -6.47190 q^{45} -2.22141 q^{47} +14.8042 q^{49} +0.353139 q^{51} +2.88544 q^{53} +1.53664 q^{55} -0.834068 q^{57} +6.32720 q^{59} +0.0560602 q^{61} -13.4100 q^{63} +1.04744 q^{65} +5.83951 q^{67} -1.92400 q^{69} +2.74305 q^{71} +9.63680 q^{73} -0.0281501 q^{75} +3.18397 q^{77} -3.00611 q^{79} +7.86287 q^{81} +2.32667 q^{83} -2.22291 q^{85} +1.70147 q^{87} -10.3727 q^{89} +2.17034 q^{91} -0.562976 q^{93} +5.25021 q^{95} -4.01956 q^{97} -1.95820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7}+O(q^{10}) \) 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7	\( 7 q + 3 q^{3} - 2 q^{5} + 6 q^{7} + 5 q^{11} - q^{13} + 6 q^{15} - 8 q^{17} + 15 q^{19} - 3 q^{21} + 5 q^{23} - 9 q^{25} + 9 q^{27} + 21 q^{31} + 7 q^{35} - q^{37} + 23 q^{39} - 10 q^{41} + 23 q^{43} - 4 q^{45} + 10 q^{47} - 13 q^{49} + 20 q^{51} - q^{53} + 23 q^{55} - 6 q^{57} + 4 q^{59} + 3 q^{61} + 4 q^{63} + 4 q^{65} + 28 q^{67} + 18 q^{69} + 18 q^{71} - 7 q^{73} - 11 q^{75} + 6 q^{77} + 30 q^{79} - 5 q^{81} - 13 q^{83} + q^{85} + 7 q^{87} + 18 q^{91} + 36 q^{93} - 2 q^{97} + 10 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 2 * q^5 + 6 * q^7 + 5 * q^11 - q^13 + 6 * q^15 - 8 * q^17 + 15 * q^19 - 3 * q^21 + 5 * q^23 - 9 * q^25 + 9 * q^27 + 21 * q^31 + 7 * q^35 - q^37 + 23 * q^39 - 10 * q^41 + 23 * q^43 - 4 * q^45 + 10 * q^47 - 13 * q^49 + 20 * q^51 - q^53 + 23 * q^55 - 6 * q^57 + 4 * q^59 + 3 * q^61 + 4 * q^63 + 4 * q^65 + 28 * q^67 + 18 * q^69 + 18 * q^71 - 7 * q^73 - 11 * q^75 + 6 * q^77 + 30 * q^79 - 5 * q^81 - 13 * q^83 + q^85 + 7 * q^87 + 18 * q^91 + 36 * q^93 - 2 * q^97 + 10 * q^99

Introduction

L-functions

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