LMFDB - Embedded newform 4016.2.a.m.1.6

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:	$23$
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.51727 q^{3} -0.472191 q^{5} -4.16870 q^{7} +3.33662 q^{9} +O(q^{10})$	$q-2.51727 q^{3} -0.472191 q^{5} -4.16870 q^{7} +3.33662 q^{9} +0.304525 q^{11} +3.93359 q^{13} +1.18863 q^{15} +5.23074 q^{17} +3.05840 q^{19} +10.4937 q^{21} -7.82281 q^{23} -4.77704 q^{25} -0.847370 q^{27} -7.79678 q^{29} +9.12452 q^{31} -0.766569 q^{33} +1.96842 q^{35} -7.45897 q^{37} -9.90189 q^{39} -5.32993 q^{41} -10.2669 q^{43} -1.57552 q^{45} -8.20514 q^{47} +10.3781 q^{49} -13.1672 q^{51} -8.29482 q^{53} -0.143794 q^{55} -7.69881 q^{57} -3.03547 q^{59} -0.845264 q^{61} -13.9094 q^{63} -1.85741 q^{65} +5.94412 q^{67} +19.6921 q^{69} -6.20529 q^{71} +10.1038 q^{73} +12.0251 q^{75} -1.26947 q^{77} +9.74582 q^{79} -7.87682 q^{81} +4.98258 q^{83} -2.46991 q^{85} +19.6266 q^{87} +12.3965 q^{89} -16.3980 q^{91} -22.9688 q^{93} -1.44415 q^{95} +0.396603 q^{97} +1.01608 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * 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
$3$	−2.51727	−1.45334	−0.726672	−	0.686985i	$-0.758934\pi$
$3$	−2.51727	−1.45334	−0.726672	+	0.686985i	$0.758934\pi$
$4$	0	0
$5$	−0.472191	−0.211170	−0.105585	−	0.994410i	$-0.533672\pi$
$5$	−0.472191	−0.211170	−0.105585	+	0.994410i	$0.533672\pi$
$6$	0	0
$7$	−4.16870	−1.57562	−0.787811	−	0.615918i	$-0.788785\pi$
$7$	−4.16870	−1.57562	−0.787811	+	0.615918i	$0.788785\pi$
$8$	0	0
$9$	3.33662	1.11221
$10$	0	0
$11$	0.304525	0.0918176	0.0459088	−	0.998946i	$-0.485382\pi$
$11$	0.304525	0.0918176	0.0459088	+	0.998946i	$0.485382\pi$
$12$	0	0
$13$	3.93359	1.09098	0.545491	−	0.838117i	$-0.316343\pi$
$13$	3.93359	1.09098	0.545491	+	0.838117i	$0.316343\pi$
$14$	0	0
$15$	1.18863	0.306903
$16$	0	0
$17$	5.23074	1.26864	0.634320	−	0.773071i	$-0.281280\pi$
$17$	5.23074	1.26864	0.634320	+	0.773071i	$0.281280\pi$
$18$	0	0
$19$	3.05840	0.701646	0.350823	−	0.936442i	$-0.385902\pi$
$19$	3.05840	0.701646	0.350823	+	0.936442i	$0.385902\pi$
$20$	0	0
$21$	10.4937	2.28992
$22$	0	0
$23$	−7.82281	−1.63117	−0.815585	−	0.578638i	$-0.803584\pi$
$23$	−7.82281	−1.63117	−0.815585	+	0.578638i	$0.803584\pi$
$24$	0	0
$25$	−4.77704	−0.955407
$26$	0	0
$27$	−0.847370	−0.163076
$28$	0	0
$29$	−7.79678	−1.44782	−0.723912	−	0.689892i	$-0.757658\pi$
$29$	−7.79678	−1.44782	−0.723912	+	0.689892i	$0.757658\pi$
$30$	0	0
$31$	9.12452	1.63881	0.819406	−	0.573213i	$-0.194303\pi$
$31$	9.12452	1.63881	0.819406	+	0.573213i	$0.194303\pi$
$32$	0	0
$33$	−0.766569	−0.133443
$34$	0	0
$35$	1.96842	0.332724
$36$	0	0
$37$	−7.45897	−1.22625	−0.613123	−	0.789987i	$-0.710087\pi$
$37$	−7.45897	−1.22625	−0.613123	+	0.789987i	$0.710087\pi$
$38$	0	0
$39$	−9.90189	−1.58557
$40$	0	0
$41$	−5.32993	−0.832395	−0.416198	−	0.909274i	$-0.636638\pi$
$41$	−5.32993	−0.832395	−0.416198	+	0.909274i	$0.636638\pi$
$42$	0	0
$43$	−10.2669	−1.56569	−0.782845	−	0.622216i	$-0.786232\pi$
$43$	−10.2669	−1.56569	−0.782845	+	0.622216i	$0.786232\pi$
$44$	0	0
$45$	−1.57552	−0.234865
$46$	0	0
$47$	−8.20514	−1.19684	−0.598421	−	0.801182i	$-0.704205\pi$
$47$	−8.20514	−1.19684	−0.598421	+	0.801182i	$0.704205\pi$
$48$	0	0
$49$	10.3781	1.48258
$50$	0	0
$51$	−13.1672	−1.84377
$52$	0	0
$53$	−8.29482	−1.13938	−0.569691	−	0.821859i	$-0.692937\pi$
$53$	−8.29482	−1.13938	−0.569691	+	0.821859i	$0.692937\pi$
$54$	0	0
$55$	−0.143794	−0.0193891
$56$	0	0
$57$	−7.69881	−1.01973
$58$	0	0
$59$	−3.03547	−0.395184	−0.197592	−	0.980284i	$-0.563312\pi$
$59$	−3.03547	−0.395184	−0.197592	+	0.980284i	$0.563312\pi$
$60$	0	0
$61$	−0.845264	−0.108225	−0.0541125	−	0.998535i	$-0.517233\pi$
$61$	−0.845264	−0.108225	−0.0541125	+	0.998535i	$0.517233\pi$
$62$	0	0
$63$	−13.9094	−1.75242
$64$	0	0
$65$	−1.85741	−0.230383
$66$	0	0
$67$	5.94412	0.726190	0.363095	−	0.931752i	$-0.381720\pi$
$67$	5.94412	0.726190	0.363095	+	0.931752i	$0.381720\pi$
$68$	0	0
$69$	19.6921	2.37065
$70$	0	0
$71$	−6.20529	−0.736432	−0.368216	−	0.929740i	$-0.620031\pi$
$71$	−6.20529	−0.736432	−0.368216	+	0.929740i	$0.620031\pi$
$72$	0	0
$73$	10.1038	1.18256	0.591279	−	0.806467i	$-0.298623\pi$
$73$	10.1038	1.18256	0.591279	+	0.806467i	$0.298623\pi$
$74$	0	0
$75$	12.0251	1.38853
$76$	0	0
$77$	−1.26947	−0.144670
$78$	0	0
$79$	9.74582	1.09649	0.548245	−	0.836318i	$-0.315296\pi$
$79$	9.74582	1.09649	0.548245	+	0.836318i	$0.315296\pi$
$80$	0	0
$81$	−7.87682	−0.875202
$82$	0	0
$83$	4.98258	0.546909	0.273455	−	0.961885i	$-0.411834\pi$
$83$	4.98258	0.546909	0.273455	+	0.961885i	$0.411834\pi$
$84$	0	0
$85$	−2.46991	−0.267899
$86$	0	0
$87$	19.6266	2.10419
$88$	0	0
$89$	12.3965	1.31403	0.657013	−	0.753879i	$-0.271820\pi$
$89$	12.3965	1.31403	0.657013	+	0.753879i	$0.271820\pi$
$90$	0	0
$91$	−16.3980	−1.71897
$92$	0	0
$93$	−22.9688	−2.38176
$94$	0	0
$95$	−1.44415	−0.148167
$96$	0	0
$97$	0.396603	0.0402689	0.0201345	−	0.999797i	$-0.493591\pi$
$97$	0.396603	0.0402689	0.0201345	+	0.999797i	$0.493591\pi$
$98$	0	0
$99$	1.01608	0.102120
$100$	0	0
$101$	18.5637	1.84716	0.923579	−	0.383408i	$-0.125250\pi$
$101$	18.5637	1.84716	0.923579	+	0.383408i	$0.125250\pi$
$102$	0	0
$103$	0.248464	0.0244819	0.0122409	−	0.999925i	$-0.496103\pi$
$103$	0.248464	0.0244819	0.0122409	+	0.999925i	$0.496103\pi$
$104$	0	0
$105$	−4.95504	−0.483563
$106$	0	0
$107$	10.0489	0.971464	0.485732	−	0.874108i	$-0.338553\pi$
$107$	10.0489	0.971464	0.485732	+	0.874108i	$0.338553\pi$
$108$	0	0
$109$	0.607413	0.0581796	0.0290898	−	0.999577i	$-0.490739\pi$
$109$	0.607413	0.0581796	0.0290898	+	0.999577i	$0.490739\pi$
$110$	0	0
$111$	18.7762	1.78216
$112$	0	0
$113$	6.60924	0.621745	0.310872	−	0.950452i	$-0.399379\pi$
$113$	6.60924	0.621745	0.310872	+	0.950452i	$0.399379\pi$
$114$	0	0
$115$	3.69386	0.344454
$116$	0	0
$117$	13.1249	1.21340
$118$	0	0
$119$	−21.8054	−1.99890
$120$	0	0
$121$	−10.9073	−0.991570
$122$	0	0
$123$	13.4168	1.20976
$124$	0	0
$125$	4.61663	0.412924
$126$	0	0
$127$	−9.82850	−0.872139	−0.436069	−	0.899913i	$-0.643630\pi$
$127$	−9.82850	−0.872139	−0.436069	+	0.899913i	$0.643630\pi$
$128$	0	0
$129$	25.8446	2.27549
$130$	0	0
$131$	−13.6457	−1.19223	−0.596113	−	0.802900i	$-0.703289\pi$
$131$	−13.6457	−1.19223	−0.596113	+	0.802900i	$0.703289\pi$
$132$	0	0
$133$	−12.7496	−1.10553
$134$	0	0
$135$	0.400120	0.0344369
$136$	0	0
$137$	11.4512	0.978341	0.489171	−	0.872188i	$-0.337300\pi$
$137$	11.4512	0.978341	0.489171	+	0.872188i	$0.337300\pi$
$138$	0	0
$139$	9.46034	0.802416	0.401208	−	0.915987i	$-0.368590\pi$
$139$	9.46034	0.802416	0.401208	+	0.915987i	$0.368590\pi$
$140$	0	0
$141$	20.6545	1.73942
$142$	0	0
$143$	1.19788	0.100171
$144$	0	0
$145$	3.68157	0.305737
$146$	0	0
$147$	−26.1244	−2.15470
$148$	0	0
$149$	9.37862	0.768327	0.384163	−	0.923265i	$-0.374490\pi$
$149$	9.37862	0.768327	0.384163	+	0.923265i	$0.374490\pi$
$150$	0	0
$151$	11.0340	0.897938	0.448969	−	0.893547i	$-0.351791\pi$
$151$	11.0340	0.897938	0.448969	+	0.893547i	$0.351791\pi$
$152$	0	0
$153$	17.4530	1.41099
$154$	0	0
$155$	−4.30852	−0.346068
$156$	0	0
$157$	−20.0501	−1.60017	−0.800087	−	0.599884i	$-0.795213\pi$
$157$	−20.0501	−1.60017	−0.800087	+	0.599884i	$0.795213\pi$
$158$	0	0
$159$	20.8803	1.65591
$160$	0	0
$161$	32.6110	2.57010
$162$	0	0
$163$	11.5614	0.905561	0.452780	−	0.891622i	$-0.350432\pi$
$163$	11.5614	0.905561	0.452780	+	0.891622i	$0.350432\pi$
$164$	0	0
$165$	0.361967	0.0281791
$166$	0	0
$167$	16.8019	1.30017	0.650086	−	0.759861i	$-0.274733\pi$
$167$	16.8019	1.30017	0.650086	+	0.759861i	$0.274733\pi$
$168$	0	0
$169$	2.47315	0.190242
$170$	0	0
$171$	10.2047	0.780376
$172$	0	0
$173$	−24.7209	−1.87950	−0.939749	−	0.341864i	$-0.888942\pi$
$173$	−24.7209	−1.87950	−0.939749	+	0.341864i	$0.888942\pi$
$174$	0	0
$175$	19.9140	1.50536
$176$	0	0
$177$	7.64108	0.574339
$178$	0	0
$179$	8.59975	0.642776	0.321388	−	0.946948i	$-0.395851\pi$
$179$	8.59975	0.642776	0.321388	+	0.946948i	$0.395851\pi$
$180$	0	0
$181$	−22.4733	−1.67042	−0.835212	−	0.549928i	$-0.814655\pi$
$181$	−22.4733	−1.67042	−0.835212	+	0.549928i	$0.814655\pi$
$182$	0	0
$183$	2.12775	0.157288
$184$	0	0
$185$	3.52206	0.258947
$186$	0	0
$187$	1.59289	0.116483
$188$	0	0
$189$	3.53243	0.256947
$190$	0	0
$191$	9.86963	0.714141	0.357071	−	0.934077i	$-0.383776\pi$
$191$	9.86963	0.714141	0.357071	+	0.934077i	$0.383776\pi$
$192$	0	0
$193$	4.97722	0.358268	0.179134	−	0.983825i	$-0.442670\pi$
$193$	4.97722	0.358268	0.179134	+	0.983825i	$0.442670\pi$
$194$	0	0
$195$	4.67558	0.334826
$196$	0	0
$197$	−5.49936	−0.391813	−0.195907	−	0.980623i	$-0.562765\pi$
$197$	−5.49936	−0.391813	−0.195907	+	0.980623i	$0.562765\pi$
$198$	0	0
$199$	1.22600	0.0869085	0.0434543	−	0.999055i	$-0.486164\pi$
$199$	1.22600	0.0869085	0.0434543	+	0.999055i	$0.486164\pi$
$200$	0	0
$201$	−14.9629	−1.05540
$202$	0	0
$203$	32.5024	2.28122
$204$	0	0
$205$	2.51674	0.175777
$206$	0	0
$207$	−26.1018	−1.81420
$208$	0	0
$209$	0.931359	0.0644234
$210$	0	0
$211$	−16.5606	−1.14008	−0.570041	−	0.821616i	$-0.693073\pi$
$211$	−16.5606	−1.14008	−0.570041	+	0.821616i	$0.693073\pi$
$212$	0	0
$213$	15.6204	1.07029
$214$	0	0
$215$	4.84795	0.330627
$216$	0	0
$217$	−38.0374	−2.58215
$218$	0	0
$219$	−25.4339	−1.71866
$220$	0	0
$221$	20.5756	1.38406
$222$	0	0
$223$	22.9580	1.53738	0.768692	−	0.639620i	$-0.220908\pi$
$223$	22.9580	1.53738	0.768692	+	0.639620i	$0.220908\pi$
$224$	0	0
$225$	−15.9392	−1.06261
$226$	0	0
$227$	−29.6756	−1.96964	−0.984821	−	0.173575i	$-0.944468\pi$
$227$	−29.6756	−1.96964	−0.984821	+	0.173575i	$0.944468\pi$
$228$	0	0
$229$	20.7268	1.36966	0.684832	−	0.728701i	$-0.259875\pi$
$229$	20.7268	1.36966	0.684832	+	0.728701i	$0.259875\pi$
$230$	0	0
$231$	3.19560	0.210255
$232$	0	0
$233$	0.322003	0.0210951	0.0105475	−	0.999944i	$-0.496643\pi$
$233$	0.322003	0.0210951	0.0105475	+	0.999944i	$0.496643\pi$
$234$	0	0
$235$	3.87439	0.252738
$236$	0	0
$237$	−24.5328	−1.59358
$238$	0	0
$239$	−7.73473	−0.500318	−0.250159	−	0.968205i	$-0.580483\pi$
$239$	−7.73473	−0.500318	−0.250159	+	0.968205i	$0.580483\pi$
$240$	0	0
$241$	24.8205	1.59883	0.799414	−	0.600780i	$-0.205143\pi$
$241$	24.8205	1.59883	0.799414	+	0.600780i	$0.205143\pi$
$242$	0	0
$243$	22.3701	1.43505
$244$	0	0
$245$	−4.90043	−0.313077
$246$	0	0
$247$	12.0305	0.765483
$248$	0	0
$249$	−12.5425	−0.794847
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−2.38224	−0.149770
$254$	0	0
$255$	6.21741	0.389349
$256$	0	0
$257$	29.0611	1.81278	0.906391	−	0.422440i	$-0.138826\pi$
$257$	29.0611	1.81278	0.906391	+	0.422440i	$0.138826\pi$
$258$	0	0
$259$	31.0942	1.93210
$260$	0	0
$261$	−26.0149	−1.61028
$262$	0	0
$263$	−4.26424	−0.262944	−0.131472	−	0.991320i	$-0.541970\pi$
$263$	−4.26424	−0.262944	−0.131472	+	0.991320i	$0.541970\pi$
$264$	0	0
$265$	3.91674	0.240603
$266$	0	0
$267$	−31.2053	−1.90973
$268$	0	0
$269$	−23.6535	−1.44218	−0.721089	−	0.692843i	$-0.756358\pi$
$269$	−23.6535	−1.44218	−0.721089	+	0.692843i	$0.756358\pi$
$270$	0	0
$271$	16.6254	1.00992	0.504961	−	0.863142i	$-0.331507\pi$
$271$	16.6254	1.00992	0.504961	+	0.863142i	$0.331507\pi$
$272$	0	0
$273$	41.2780	2.49826
$274$	0	0
$275$	−1.45472	−0.0877232
$276$	0	0
$277$	−8.78549	−0.527869	−0.263935	−	0.964541i	$-0.585020\pi$
$277$	−8.78549	−0.527869	−0.263935	+	0.964541i	$0.585020\pi$
$278$	0	0
$279$	30.4451	1.82270
$280$	0	0
$281$	11.4541	0.683293	0.341646	−	0.939829i	$-0.389015\pi$
$281$	11.4541	0.683293	0.341646	+	0.939829i	$0.389015\pi$
$282$	0	0
$283$	−25.8433	−1.53623	−0.768114	−	0.640314i	$-0.778804\pi$
$283$	−25.8433	−1.53623	−0.768114	+	0.640314i	$0.778804\pi$
$284$	0	0
$285$	3.63531	0.215337
$286$	0	0
$287$	22.2189	1.31154
$288$	0	0
$289$	10.3606	0.609448
$290$	0	0
$291$	−0.998355	−0.0585246
$292$	0	0
$293$	−0.0646721	−0.00377819	−0.00188909	−	0.999998i	$-0.500601\pi$
$293$	−0.0646721	−0.00377819	−0.00188909	+	0.999998i	$0.500601\pi$
$294$	0	0
$295$	1.43332	0.0834512
$296$	0	0
$297$	−0.258045	−0.0149733
$298$	0	0
$299$	−30.7718	−1.77958
$300$	0	0
$301$	42.7997	2.46694
$302$	0	0
$303$	−46.7298	−2.68456
$304$	0	0
$305$	0.399126	0.0228539
$306$	0	0
$307$	−7.89727	−0.450721	−0.225361	−	0.974275i	$-0.572356\pi$
$307$	−7.89727	−0.450721	−0.225361	+	0.974275i	$0.572356\pi$
$308$	0	0
$309$	−0.625450	−0.0355806
$310$	0	0
$311$	30.8494	1.74931	0.874656	−	0.484745i	$-0.161088\pi$
$311$	30.8494	1.74931	0.874656	+	0.484745i	$0.161088\pi$
$312$	0	0
$313$	−14.9817	−0.846814	−0.423407	−	0.905940i	$-0.639166\pi$
$313$	−14.9817	−0.846814	−0.423407	+	0.905940i	$0.639166\pi$
$314$	0	0
$315$	6.56789	0.370059
$316$	0	0
$317$	−6.55514	−0.368173	−0.184087	−	0.982910i	$-0.558933\pi$
$317$	−6.55514	−0.368173	−0.184087	+	0.982910i	$0.558933\pi$
$318$	0	0
$319$	−2.37431	−0.132936
$320$	0	0
$321$	−25.2958	−1.41187
$322$	0	0
$323$	15.9977	0.890136
$324$	0	0
$325$	−18.7909	−1.04233
$326$	0	0
$327$	−1.52902	−0.0845549
$328$	0	0
$329$	34.2048	1.88577
$330$	0	0
$331$	19.7693	1.08662	0.543310	−	0.839532i	$-0.317171\pi$
$331$	19.7693	1.08662	0.543310	+	0.839532i	$0.317171\pi$
$332$	0	0
$333$	−24.8878	−1.36384
$334$	0	0
$335$	−2.80676	−0.153350
$336$	0	0
$337$	−21.3765	−1.16445	−0.582226	−	0.813027i	$-0.697818\pi$
$337$	−21.3765	−1.16445	−0.582226	+	0.813027i	$0.697818\pi$
$338$	0	0
$339$	−16.6372	−0.903609
$340$	0	0
$341$	2.77864	0.150472
$342$	0	0
$343$	−14.0822	−0.760367
$344$	0	0
$345$	−9.29843	−0.500610
$346$	0	0
$347$	17.0719	0.916467	0.458234	−	0.888832i	$-0.348482\pi$
$347$	17.0719	0.916467	0.458234	+	0.888832i	$0.348482\pi$
$348$	0	0
$349$	21.5795	1.15512	0.577562	−	0.816347i	$-0.304004\pi$
$349$	21.5795	1.15512	0.577562	+	0.816347i	$0.304004\pi$
$350$	0	0
$351$	−3.33321	−0.177913
$352$	0	0
$353$	−7.78068	−0.414124	−0.207062	−	0.978328i	$-0.566390\pi$
$353$	−7.78068	−0.414124	−0.207062	+	0.978328i	$0.566390\pi$
$354$	0	0
$355$	2.93008	0.155513
$356$	0	0
$357$	54.8899	2.90508
$358$	0	0
$359$	13.3156	0.702773	0.351387	−	0.936230i	$-0.385710\pi$
$359$	13.3156	0.702773	0.351387	+	0.936230i	$0.385710\pi$
$360$	0	0
$361$	−9.64617	−0.507693
$362$	0	0
$363$	27.4565	1.44109
$364$	0	0
$365$	−4.77091	−0.249721
$366$	0	0
$367$	13.1444	0.686132	0.343066	−	0.939311i	$-0.388535\pi$
$367$	13.1444	0.686132	0.343066	+	0.939311i	$0.388535\pi$
$368$	0	0
$369$	−17.7840	−0.925796
$370$	0	0
$371$	34.5786	1.79523
$372$	0	0
$373$	37.3600	1.93443	0.967215	−	0.253960i	$-0.0817333\pi$
$373$	37.3600	1.93443	0.967215	+	0.253960i	$0.0817333\pi$
$374$	0	0
$375$	−11.6213	−0.600120
$376$	0	0
$377$	−30.6693	−1.57955
$378$	0	0
$379$	2.82223	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$379$	2.82223	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$380$	0	0
$381$	24.7410	1.26752
$382$	0	0
$383$	11.2386	0.574268	0.287134	−	0.957890i	$-0.407298\pi$
$383$	11.2386	0.574268	0.287134	+	0.957890i	$0.407298\pi$
$384$	0	0
$385$	0.599433	0.0305499
$386$	0	0
$387$	−34.2568	−1.74137
$388$	0	0
$389$	−23.0334	−1.16784	−0.583919	−	0.811812i	$-0.698482\pi$
$389$	−23.0334	−1.16784	−0.583919	+	0.811812i	$0.698482\pi$
$390$	0	0
$391$	−40.9191	−2.06937
$392$	0	0
$393$	34.3497	1.73271
$394$	0	0
$395$	−4.60189	−0.231546
$396$	0	0
$397$	39.2231	1.96855	0.984275	−	0.176642i	$-0.0565234\pi$
$397$	39.2231	1.96855	0.984275	+	0.176642i	$0.0565234\pi$
$398$	0	0
$399$	32.0940	1.60671
$400$	0	0
$401$	−14.5312	−0.725654	−0.362827	−	0.931857i	$-0.618188\pi$
$401$	−14.5312	−0.725654	−0.362827	+	0.931857i	$0.618188\pi$
$402$	0	0
$403$	35.8921	1.78792
$404$	0	0
$405$	3.71936	0.184817
$406$	0	0
$407$	−2.27144	−0.112591
$408$	0	0
$409$	28.0104	1.38503	0.692513	−	0.721405i	$-0.256503\pi$
$409$	28.0104	1.38503	0.692513	+	0.721405i	$0.256503\pi$
$410$	0	0
$411$	−28.8257	−1.42187
$412$	0	0
$413$	12.6540	0.622661
$414$	0	0
$415$	−2.35273	−0.115491
$416$	0	0
$417$	−23.8142	−1.16619
$418$	0	0
$419$	4.01120	0.195960	0.0979799	−	0.995188i	$-0.468762\pi$
$419$	4.01120	0.195960	0.0979799	+	0.995188i	$0.468762\pi$
$420$	0	0
$421$	−18.1701	−0.885555	−0.442777	−	0.896632i	$-0.646007\pi$
$421$	−18.1701	−0.885555	−0.442777	+	0.896632i	$0.646007\pi$
$422$	0	0
$423$	−27.3775	−1.33114
$424$	0	0
$425$	−24.9874	−1.21207
$426$	0	0
$427$	3.52365	0.170522
$428$	0	0
$429$	−3.01537	−0.145583
$430$	0	0
$431$	−30.3177	−1.46035	−0.730175	−	0.683260i	$-0.760562\pi$
$431$	−30.3177	−1.46035	−0.730175	+	0.683260i	$0.760562\pi$
$432$	0	0
$433$	35.4115	1.70177	0.850883	−	0.525355i	$-0.176067\pi$
$433$	35.4115	1.70177	0.850883	+	0.525355i	$0.176067\pi$
$434$	0	0
$435$	−9.26748	−0.444342
$436$	0	0
$437$	−23.9253	−1.14450
$438$	0	0
$439$	25.5194	1.21798	0.608988	−	0.793180i	$-0.291576\pi$
$439$	25.5194	1.21798	0.608988	+	0.793180i	$0.291576\pi$
$440$	0	0
$441$	34.6277	1.64894
$442$	0	0
$443$	−32.1210	−1.52611	−0.763057	−	0.646331i	$-0.776302\pi$
$443$	−32.1210	−1.52611	−0.763057	+	0.646331i	$0.776302\pi$
$444$	0	0
$445$	−5.85351	−0.277483
$446$	0	0
$447$	−23.6085	−1.11664
$448$	0	0
$449$	5.85110	0.276130	0.138065	−	0.990423i	$-0.455912\pi$
$449$	5.85110	0.276130	0.138065	+	0.990423i	$0.455912\pi$
$450$	0	0
$451$	−1.62309	−0.0764285
$452$	0	0
$453$	−27.7756	−1.30501
$454$	0	0
$455$	7.74298	0.362996
$456$	0	0
$457$	−18.4553	−0.863301	−0.431650	−	0.902041i	$-0.642069\pi$
$457$	−18.4553	−0.863301	−0.431650	+	0.902041i	$0.642069\pi$
$458$	0	0
$459$	−4.43237	−0.206885
$460$	0	0
$461$	3.42093	0.159329	0.0796644	−	0.996822i	$-0.474615\pi$
$461$	3.42093	0.159329	0.0796644	+	0.996822i	$0.474615\pi$
$462$	0	0
$463$	−0.00333581	−0.000155028 0	−7.75142e−5	−	1.00000i	$-0.500025\pi$
$463$	−0.00333581	−0.000155028 0	−7.75142e−5		1.00000i	$0.500025\pi$
$464$	0	0
$465$	10.8457	0.502956
$466$	0	0
$467$	15.7806	0.730238	0.365119	−	0.930961i	$-0.381028\pi$
$467$	15.7806	0.730238	0.365119	+	0.930961i	$0.381028\pi$
$468$	0	0
$469$	−24.7793	−1.14420
$470$	0	0
$471$	50.4715	2.32560
$472$	0	0
$473$	−3.12653	−0.143758
$474$	0	0
$475$	−14.6101	−0.670357
$476$	0	0
$477$	−27.6767	−1.26723
$478$	0	0
$479$	−23.4049	−1.06940	−0.534700	−	0.845042i	$-0.679575\pi$
$479$	−23.4049	−1.06940	−0.534700	+	0.845042i	$0.679575\pi$
$480$	0	0
$481$	−29.3405	−1.33781
$482$	0	0
$483$	−82.0905	−3.73525
$484$	0	0
$485$	−0.187272	−0.00850360
$486$	0	0
$487$	7.72620	0.350108	0.175054	−	0.984559i	$-0.443990\pi$
$487$	7.72620	0.350108	0.175054	+	0.984559i	$0.443990\pi$
$488$	0	0
$489$	−29.1032	−1.31609
$490$	0	0
$491$	1.96908	0.0888635	0.0444318	−	0.999012i	$-0.485852\pi$
$491$	1.96908	0.0888635	0.0444318	+	0.999012i	$0.485852\pi$
$492$	0	0
$493$	−40.7829	−1.83677
$494$	0	0
$495$	−0.479786	−0.0215648
$496$	0	0
$497$	25.8680	1.16034
$498$	0	0
$499$	23.8066	1.06573	0.532865	−	0.846200i	$-0.321115\pi$
$499$	23.8066	1.06573	0.532865	+	0.846200i	$0.321115\pi$
$500$	0	0
$501$	−42.2949	−1.88960
$502$	0	0
$503$	35.6782	1.59081	0.795406	−	0.606076i	$-0.207257\pi$
$503$	35.6782	1.59081	0.795406	+	0.606076i	$0.207257\pi$
$504$	0	0
$505$	−8.76562	−0.390065
$506$	0	0
$507$	−6.22557	−0.276487
$508$	0	0
$509$	37.6993	1.67099	0.835496	−	0.549496i	$-0.185180\pi$
$509$	37.6993	1.67099	0.835496	+	0.549496i	$0.185180\pi$
$510$	0	0
$511$	−42.1197	−1.86326
$512$	0	0
$513$	−2.59160	−0.114422
$514$	0	0
$515$	−0.117322	−0.00516984
$516$	0	0
$517$	−2.49867	−0.109891
$518$	0	0
$519$	62.2292	2.73156
$520$	0	0
$521$	−2.68569	−0.117662	−0.0588310	−	0.998268i	$-0.518737\pi$
$521$	−2.68569	−0.117662	−0.0588310	+	0.998268i	$0.518737\pi$
$522$	0	0
$523$	−16.2403	−0.710138	−0.355069	−	0.934840i	$-0.615543\pi$
$523$	−16.2403	−0.710138	−0.355069	+	0.934840i	$0.615543\pi$
$524$	0	0
$525$	−50.1289	−2.18781
$526$	0	0
$527$	47.7280	2.07906
$528$	0	0
$529$	38.1964	1.66071
$530$	0	0
$531$	−10.1282	−0.439527
$532$	0	0
$533$	−20.9658	−0.908128
$534$	0	0
$535$	−4.74500	−0.205144
$536$	0	0
$537$	−21.6479	−0.934174
$538$	0	0
$539$	3.16038	0.136127
$540$	0	0
$541$	32.7476	1.40793	0.703964	−	0.710236i	$-0.251412\pi$
$541$	32.7476	1.40793	0.703964	+	0.710236i	$0.251412\pi$
$542$	0	0
$543$	56.5711	2.42770
$544$	0	0
$545$	−0.286815	−0.0122858
$546$	0	0
$547$	19.2228	0.821908	0.410954	−	0.911656i	$-0.365196\pi$
$547$	19.2228	0.821908	0.410954	+	0.911656i	$0.365196\pi$
$548$	0	0
$549$	−2.82033	−0.120369
$550$	0	0
$551$	−23.8457	−1.01586
$552$	0	0
$553$	−40.6274	−1.72765
$554$	0	0
$555$	−8.86595	−0.376339
$556$	0	0
$557$	18.9698	0.803778	0.401889	−	0.915689i	$-0.368354\pi$
$557$	18.9698	0.803778	0.401889	+	0.915689i	$0.368354\pi$
$558$	0	0
$559$	−40.3859	−1.70814
$560$	0	0
$561$	−4.00972	−0.169291
$562$	0	0
$563$	2.77977	0.117153	0.0585767	−	0.998283i	$-0.481344\pi$
$563$	2.77977	0.117153	0.0585767	+	0.998283i	$0.481344\pi$
$564$	0	0
$565$	−3.12082	−0.131294
$566$	0	0
$567$	32.8361	1.37899
$568$	0	0
$569$	24.2475	1.01651	0.508254	−	0.861207i	$-0.330291\pi$
$569$	24.2475	1.01651	0.508254	+	0.861207i	$0.330291\pi$
$570$	0	0
$571$	24.1351	1.01002	0.505012	−	0.863112i	$-0.331488\pi$
$571$	24.1351	1.01002	0.505012	+	0.863112i	$0.331488\pi$
$572$	0	0
$573$	−24.8445	−1.03789
$574$	0	0
$575$	37.3699	1.55843
$576$	0	0
$577$	−8.81763	−0.367083	−0.183541	−	0.983012i	$-0.558756\pi$
$577$	−8.81763	−0.367083	−0.183541	+	0.983012i	$0.558756\pi$
$578$	0	0
$579$	−12.5290	−0.520687
$580$	0	0
$581$	−20.7709	−0.861722
$582$	0	0
$583$	−2.52598	−0.104615
$584$	0	0
$585$	−6.19747	−0.256234
$586$	0	0
$587$	15.0645	0.621780	0.310890	−	0.950446i	$-0.399373\pi$
$587$	15.0645	0.621780	0.310890	+	0.950446i	$0.399373\pi$
$588$	0	0
$589$	27.9065	1.14987
$590$	0	0
$591$	13.8434	0.569440
$592$	0	0
$593$	28.7573	1.18092	0.590461	−	0.807066i	$-0.298946\pi$
$593$	28.7573	1.18092	0.590461	+	0.807066i	$0.298946\pi$
$594$	0	0
$595$	10.2963	0.422107
$596$	0	0
$597$	−3.08616	−0.126308
$598$	0	0
$599$	25.8141	1.05474	0.527368	−	0.849637i	$-0.323179\pi$
$599$	25.8141	1.05474	0.527368	+	0.849637i	$0.323179\pi$
$600$	0	0
$601$	2.64163	0.107754	0.0538771	−	0.998548i	$-0.482842\pi$
$601$	2.64163	0.107754	0.0538771	+	0.998548i	$0.482842\pi$
$602$	0	0
$603$	19.8333	0.807674
$604$	0	0
$605$	5.15031	0.209390
$606$	0	0
$607$	26.5488	1.07758	0.538792	−	0.842439i	$-0.318881\pi$
$607$	26.5488	1.07758	0.538792	+	0.842439i	$0.318881\pi$
$608$	0	0
$609$	−81.8172	−3.31540
$610$	0	0
$611$	−32.2757	−1.30573
$612$	0	0
$613$	−36.7305	−1.48353	−0.741765	−	0.670659i	$-0.766011\pi$
$613$	−36.7305	−1.48353	−0.741765	+	0.670659i	$0.766011\pi$
$614$	0	0
$615$	−6.33531	−0.255464
$616$	0	0
$617$	6.19526	0.249412	0.124706	−	0.992194i	$-0.460201\pi$
$617$	6.19526	0.249412	0.124706	+	0.992194i	$0.460201\pi$
$618$	0	0
$619$	0.382739	0.0153836	0.00769179	−	0.999970i	$-0.497552\pi$
$619$	0.382739	0.0153836	0.00769179	+	0.999970i	$0.497552\pi$
$620$	0	0
$621$	6.62881	0.266005
$622$	0	0
$623$	−51.6773	−2.07041
$624$	0	0
$625$	21.7052	0.868210
$626$	0	0
$627$	−2.34448	−0.0936294
$628$	0	0
$629$	−39.0159	−1.55567
$630$	0	0
$631$	27.6617	1.10120	0.550598	−	0.834771i	$-0.314400\pi$
$631$	27.6617	1.10120	0.550598	+	0.834771i	$0.314400\pi$
$632$	0	0
$633$	41.6875	1.65693
$634$	0	0
$635$	4.64093	0.184170
$636$	0	0
$637$	40.8231	1.61747
$638$	0	0
$639$	−20.7047	−0.819066
$640$	0	0
$641$	32.6196	1.28840	0.644199	−	0.764858i	$-0.277191\pi$
$641$	32.6196	1.28840	0.644199	+	0.764858i	$0.277191\pi$
$642$	0	0
$643$	31.3759	1.23735	0.618673	−	0.785648i	$-0.287670\pi$
$643$	31.3759	1.23735	0.618673	+	0.785648i	$0.287670\pi$
$644$	0	0
$645$	−12.2036	−0.480515
$646$	0	0
$647$	−26.6413	−1.04738	−0.523688	−	0.851910i	$-0.675444\pi$
$647$	−26.6413	−1.04738	−0.523688	+	0.851910i	$0.675444\pi$
$648$	0	0
$649$	−0.924375	−0.0362849
$650$	0	0
$651$	95.7502	3.75275
$652$	0	0
$653$	0.501126	0.0196106	0.00980529	−	0.999952i	$-0.496879\pi$
$653$	0.501126	0.0196106	0.00980529	+	0.999952i	$0.496879\pi$
$654$	0	0
$655$	6.44336	0.251763
$656$	0	0
$657$	33.7125	1.31525
$658$	0	0
$659$	−5.06507	−0.197307	−0.0986536	−	0.995122i	$-0.531454\pi$
$659$	−5.06507	−0.197307	−0.0986536	+	0.995122i	$0.531454\pi$
$660$	0	0
$661$	−12.8951	−0.501561	−0.250780	−	0.968044i	$-0.580687\pi$
$661$	−12.8951	−0.501561	−0.250780	+	0.968044i	$0.580687\pi$
$662$	0	0
$663$	−51.7942	−2.01152
$664$	0	0
$665$	6.02023	0.233455
$666$	0	0
$667$	60.9927	2.36165
$668$	0	0
$669$	−57.7914	−2.23435
$670$	0	0
$671$	−0.257404	−0.00993696
$672$	0	0
$673$	−15.8230	−0.609933	−0.304966	−	0.952363i	$-0.598645\pi$
$673$	−15.8230	−0.609933	−0.304966	+	0.952363i	$0.598645\pi$
$674$	0	0
$675$	4.04792	0.155804
$676$	0	0
$677$	−20.2826	−0.779523	−0.389761	−	0.920916i	$-0.627443\pi$
$677$	−20.2826	−0.779523	−0.389761	+	0.920916i	$0.627443\pi$
$678$	0	0
$679$	−1.65332	−0.0634486
$680$	0	0
$681$	74.7014	2.86257
$682$	0	0
$683$	9.57134	0.366237	0.183119	−	0.983091i	$-0.441381\pi$
$683$	9.57134	0.366237	0.183119	+	0.983091i	$0.441381\pi$
$684$	0	0
$685$	−5.40715	−0.206597
$686$	0	0
$687$	−52.1748	−1.99059
$688$	0	0
$689$	−32.6284	−1.24304
$690$	0	0
$691$	−10.1320	−0.385441	−0.192721	−	0.981254i	$-0.561731\pi$
$691$	−10.1320	−0.385441	−0.192721	+	0.981254i	$0.561731\pi$
$692$	0	0
$693$	−4.23575	−0.160903
$694$	0	0
$695$	−4.46709	−0.169446
$696$	0	0
$697$	−27.8795	−1.05601
$698$	0	0
$699$	−0.810566	−0.0306584
$700$	0	0
$701$	−5.86605	−0.221558	−0.110779	−	0.993845i	$-0.535335\pi$
$701$	−5.86605	−0.221558	−0.110779	+	0.993845i	$0.535335\pi$
$702$	0	0
$703$	−22.8125	−0.860391
$704$	0	0
$705$	−9.75287	−0.367314
$706$	0	0
$707$	−77.3866	−2.91042
$708$	0	0
$709$	−16.3570	−0.614301	−0.307151	−	0.951661i	$-0.599376\pi$
$709$	−16.3570	−0.614301	−0.307151	+	0.951661i	$0.599376\pi$
$710$	0	0
$711$	32.5181	1.21953
$712$	0	0
$713$	−71.3794	−2.67318
$714$	0	0
$715$	−0.565626	−0.0211532
$716$	0	0
$717$	19.4704	0.727134
$718$	0	0
$719$	−8.70874	−0.324781	−0.162391	−	0.986727i	$-0.551920\pi$
$719$	−8.70874	−0.324781	−0.162391	+	0.986727i	$0.551920\pi$
$720$	0	0
$721$	−1.03577	−0.0385742
$722$	0	0
$723$	−62.4797	−2.32365
$724$	0	0
$725$	37.2455	1.38326
$726$	0	0
$727$	−30.9035	−1.14615	−0.573074	−	0.819504i	$-0.694249\pi$
$727$	−30.9035	−1.14615	−0.573074	+	0.819504i	$0.694249\pi$
$728$	0	0
$729$	−32.6811	−1.21041
$730$	0	0
$731$	−53.7036	−1.98630
$732$	0	0
$733$	21.9242	0.809790	0.404895	−	0.914363i	$-0.367308\pi$
$733$	21.9242	0.809790	0.404895	+	0.914363i	$0.367308\pi$
$734$	0	0
$735$	12.3357	0.455009
$736$	0	0
$737$	1.81013	0.0666770
$738$	0	0
$739$	−24.7830	−0.911656	−0.455828	−	0.890068i	$-0.650657\pi$
$739$	−24.7830	−0.911656	−0.455828	+	0.890068i	$0.650657\pi$
$740$	0	0
$741$	−30.2840	−1.11251
$742$	0	0
$743$	−11.7429	−0.430806	−0.215403	−	0.976525i	$-0.569107\pi$
$743$	−11.7429	−0.430806	−0.215403	+	0.976525i	$0.569107\pi$
$744$	0	0
$745$	−4.42850	−0.162248
$746$	0	0
$747$	16.6250	0.608277
$748$	0	0
$749$	−41.8909	−1.53066
$750$	0	0
$751$	18.5022	0.675153	0.337577	−	0.941298i	$-0.390393\pi$
$751$	18.5022	0.675153	0.337577	+	0.941298i	$0.390393\pi$
$752$	0	0
$753$	−2.51727	−0.0917342
$754$	0	0
$755$	−5.21018	−0.189618
$756$	0	0
$757$	35.1308	1.27685	0.638425	−	0.769684i	$-0.279586\pi$
$757$	35.1308	1.27685	0.638425	+	0.769684i	$0.279586\pi$
$758$	0	0
$759$	5.99673	0.217667
$760$	0	0
$761$	−9.80837	−0.355553	−0.177777	−	0.984071i	$-0.556890\pi$
$761$	−9.80837	−0.355553	−0.177777	+	0.984071i	$0.556890\pi$
$762$	0	0
$763$	−2.53212	−0.0916690
$764$	0	0
$765$	−8.24115	−0.297959
$766$	0	0
$767$	−11.9403	−0.431139
$768$	0	0
$769$	14.7856	0.533181	0.266590	−	0.963810i	$-0.414103\pi$
$769$	14.7856	0.533181	0.266590	+	0.963810i	$0.414103\pi$
$770$	0	0
$771$	−73.1545	−2.63459
$772$	0	0
$773$	3.90318	0.140388	0.0701939	−	0.997533i	$-0.477638\pi$
$773$	3.90318	0.140388	0.0701939	+	0.997533i	$0.477638\pi$
$774$	0	0
$775$	−43.5882	−1.56573
$776$	0	0
$777$	−78.2724	−2.80801
$778$	0	0
$779$	−16.3011	−0.584046
$780$	0	0
$781$	−1.88966	−0.0676175
$782$	0	0
$783$	6.60675	0.236106
$784$	0	0
$785$	9.46749	0.337909
$786$	0	0
$787$	47.6348	1.69800	0.848999	−	0.528394i	$-0.177206\pi$
$787$	47.6348	1.69800	0.848999	+	0.528394i	$0.177206\pi$
$788$	0	0
$789$	10.7342	0.382148
$790$	0	0
$791$	−27.5519	−0.979634
$792$	0	0
$793$	−3.32492	−0.118071
$794$	0	0
$795$	−9.85947	−0.349679
$796$	0	0
$797$	−10.2414	−0.362769	−0.181384	−	0.983412i	$-0.558058\pi$
$797$	−10.2414	−0.362769	−0.181384	+	0.983412i	$0.558058\pi$
$798$	0	0
$799$	−42.9189	−1.51836
$800$	0	0
$801$	41.3624	1.46147
$802$	0	0
$803$	3.07685	0.108580
$804$	0	0
$805$	−15.3986	−0.542730
$806$	0	0
$807$	59.5420	2.09598
$808$	0	0
$809$	4.87595	0.171429	0.0857146	−	0.996320i	$-0.472683\pi$
$809$	4.87595	0.171429	0.0857146	+	0.996320i	$0.472683\pi$
$810$	0	0
$811$	23.0368	0.808932	0.404466	−	0.914553i	$-0.367457\pi$
$811$	23.0368	0.808932	0.404466	+	0.914553i	$0.367457\pi$
$812$	0	0
$813$	−41.8506	−1.46776
$814$	0	0
$815$	−5.45920	−0.191227
$816$	0	0
$817$	−31.4004	−1.09856
$818$	0	0
$819$	−54.7139	−1.91186
$820$	0	0
$821$	−27.1590	−0.947857	−0.473928	−	0.880563i	$-0.657164\pi$
$821$	−27.1590	−0.947857	−0.473928	+	0.880563i	$0.657164\pi$
$822$	0	0
$823$	−17.5575	−0.612017	−0.306009	−	0.952029i	$-0.598994\pi$
$823$	−17.5575	−0.612017	−0.306009	+	0.952029i	$0.598994\pi$
$824$	0	0
$825$	3.66193	0.127492
$826$	0	0
$827$	−4.55165	−0.158276	−0.0791381	−	0.996864i	$-0.525217\pi$
$827$	−4.55165	−0.158276	−0.0791381	+	0.996864i	$0.525217\pi$
$828$	0	0
$829$	8.01539	0.278386	0.139193	−	0.990265i	$-0.455549\pi$
$829$	8.01539	0.278386	0.139193	+	0.990265i	$0.455549\pi$
$830$	0	0
$831$	22.1154	0.767175
$832$	0	0
$833$	54.2850	1.88086
$834$	0	0
$835$	−7.93372	−0.274558
$836$	0	0
$837$	−7.73184	−0.267252
$838$	0	0
$839$	−25.0653	−0.865351	−0.432675	−	0.901550i	$-0.642430\pi$
$839$	−25.0653	−0.865351	−0.432675	+	0.901550i	$0.642430\pi$
$840$	0	0
$841$	31.7897	1.09620
$842$	0	0
$843$	−28.8329	−0.993059
$844$	0	0
$845$	−1.16780	−0.0401735
$846$	0	0
$847$	45.4691	1.56234
$848$	0	0
$849$	65.0545	2.23267
$850$	0	0
$851$	58.3501	2.00022
$852$	0	0
$853$	1.45894	0.0499531	0.0249765	−	0.999688i	$-0.492049\pi$
$853$	1.45894	0.0499531	0.0249765	+	0.999688i	$0.492049\pi$
$854$	0	0
$855$	−4.81858	−0.164792
$856$	0	0
$857$	−31.1230	−1.06314	−0.531571	−	0.847014i	$-0.678398\pi$
$857$	−31.1230	−1.06314	−0.531571	+	0.847014i	$0.678398\pi$
$858$	0	0
$859$	7.21675	0.246232	0.123116	−	0.992392i	$-0.460711\pi$
$859$	7.21675	0.246232	0.123116	+	0.992392i	$0.460711\pi$
$860$	0	0
$861$	−55.9308	−1.90612
$862$	0	0
$863$	−43.9265	−1.49527	−0.747637	−	0.664107i	$-0.768812\pi$
$863$	−43.9265	−1.49527	−0.747637	+	0.664107i	$0.768812\pi$
$864$	0	0
$865$	11.6730	0.396894
$866$	0	0
$867$	−26.0804	−0.885737
$868$	0	0
$869$	2.96784	0.100677
$870$	0	0
$871$	23.3817	0.792260
$872$	0	0
$873$	1.32332	0.0447874
$874$	0	0
$875$	−19.2453	−0.650611
$876$	0	0
$877$	−28.0980	−0.948802	−0.474401	−	0.880309i	$-0.657335\pi$
$877$	−28.0980	−0.948802	−0.474401	+	0.880309i	$0.657335\pi$
$878$	0	0
$879$	0.162797	0.00549100
$880$	0	0
$881$	−35.2907	−1.18897	−0.594486	−	0.804106i	$-0.702645\pi$
$881$	−35.2907	−1.18897	−0.594486	+	0.804106i	$0.702645\pi$
$882$	0	0
$883$	−12.2754	−0.413101	−0.206550	−	0.978436i	$-0.566224\pi$
$883$	−12.2754	−0.413101	−0.206550	+	0.978436i	$0.566224\pi$
$884$	0	0
$885$	−3.60805	−0.121283
$886$	0	0
$887$	−50.1769	−1.68478	−0.842388	−	0.538871i	$-0.818851\pi$
$887$	−50.1769	−1.68478	−0.842388	+	0.538871i	$0.818851\pi$
$888$	0	0
$889$	40.9721	1.37416
$890$	0	0
$891$	−2.39868	−0.0803589
$892$	0	0
$893$	−25.0946	−0.839760
$894$	0	0
$895$	−4.06073	−0.135735
$896$	0	0
$897$	77.4607	2.58634
$898$	0	0
$899$	−71.1418	−2.37271
$900$	0	0
$901$	−43.3880	−1.44546
$902$	0	0
$903$	−107.738	−3.58531
$904$	0	0
$905$	10.6117	0.352744
$906$	0	0
$907$	2.35584	0.0782244	0.0391122	−	0.999235i	$-0.487547\pi$
$907$	2.35584	0.0782244	0.0391122	+	0.999235i	$0.487547\pi$
$908$	0	0
$909$	61.9401	2.05442
$910$	0	0
$911$	−2.89328	−0.0958588	−0.0479294	−	0.998851i	$-0.515262\pi$
$911$	−2.89328	−0.0958588	−0.0479294	+	0.998851i	$0.515262\pi$
$912$	0	0
$913$	1.51732	0.0502159
$914$	0	0
$915$	−1.00471	−0.0332145
$916$	0	0
$917$	56.8847	1.87850
$918$	0	0
$919$	−58.8755	−1.94212	−0.971062	−	0.238828i	$-0.923237\pi$
$919$	−58.8755	−1.94212	−0.971062	+	0.238828i	$0.923237\pi$
$920$	0	0
$921$	19.8795	0.655053
$922$	0	0
$923$	−24.4091	−0.803435
$924$	0	0
$925$	35.6318	1.17157
$926$	0	0
$927$	0.829030	0.0272289
$928$	0	0
$929$	46.4055	1.52252	0.761258	−	0.648449i	$-0.224582\pi$
$929$	46.4055	1.52252	0.761258	+	0.648449i	$0.224582\pi$
$930$	0	0
$931$	31.7403	1.04025
$932$	0	0
$933$	−77.6562	−2.54235
$934$	0	0
$935$	−0.752147	−0.0245978
$936$	0	0
$937$	18.6150	0.608126	0.304063	−	0.952652i	$-0.401657\pi$
$937$	18.6150	0.608126	0.304063	+	0.952652i	$0.401657\pi$
$938$	0	0
$939$	37.7128	1.23071
$940$	0	0
$941$	−14.0524	−0.458094	−0.229047	−	0.973415i	$-0.573561\pi$
$941$	−14.0524	−0.458094	−0.229047	+	0.973415i	$0.573561\pi$
$942$	0	0
$943$	41.6950	1.35778
$944$	0	0
$945$	−1.66798	−0.0542595
$946$	0	0
$947$	14.6936	0.477479	0.238740	−	0.971084i	$-0.423266\pi$
$947$	14.6936	0.477479	0.238740	+	0.971084i	$0.423266\pi$
$948$	0	0
$949$	39.7442	1.29015
$950$	0	0
$951$	16.5010	0.535082
$952$	0	0
$953$	−41.2058	−1.33479	−0.667394	−	0.744705i	$-0.732590\pi$
$953$	−41.2058	−1.33479	−0.667394	+	0.744705i	$0.732590\pi$
$954$	0	0
$955$	−4.66035	−0.150805
$956$	0	0
$957$	5.97677	0.193201
$958$	0	0
$959$	−47.7366	−1.54150
$960$	0	0
$961$	52.2569	1.68571
$962$	0	0
$963$	33.5294	1.08047
$964$	0	0
$965$	−2.35020	−0.0756556
$966$	0	0
$967$	13.6081	0.437607	0.218804	−	0.975769i	$-0.429785\pi$
$967$	13.6081	0.437607	0.218804	+	0.975769i	$0.429785\pi$
$968$	0	0
$969$	−40.2705	−1.29367
$970$	0	0
$971$	23.3940	0.750748	0.375374	−	0.926873i	$-0.377514\pi$
$971$	23.3940	0.750748	0.375374	+	0.926873i	$0.377514\pi$
$972$	0	0
$973$	−39.4374	−1.26430
$974$	0	0
$975$	47.3017	1.51487
$976$	0	0
$977$	−7.24136	−0.231672	−0.115836	−	0.993268i	$-0.536955\pi$
$977$	−7.24136	−0.231672	−0.115836	+	0.993268i	$0.536955\pi$
$978$	0	0
$979$	3.77504	0.120651
$980$	0	0
$981$	2.02671	0.0647078
$982$	0	0
$983$	4.90576	0.156469	0.0782347	−	0.996935i	$-0.475072\pi$
$983$	4.90576	0.156469	0.0782347	+	0.996935i	$0.475072\pi$
$984$	0	0
$985$	2.59675	0.0827393
$986$	0	0
$987$	−86.1025	−2.74067
$988$	0	0
$989$	80.3162	2.55391
$990$	0	0
$991$	−35.1041	−1.11512	−0.557559	−	0.830137i	$-0.688262\pi$
$991$	−35.1041	−1.11512	−0.557559	+	0.830137i	$0.688262\pi$
$992$	0	0
$993$	−49.7646	−1.57923
$994$	0	0
$995$	−0.578904	−0.0183525
$996$	0	0
$997$	−44.9594	−1.42388	−0.711939	−	0.702241i	$-0.752183\pi$
$997$	−44.9594	−1.42388	−0.711939	+	0.702241i	$0.752183\pi$
$998$	0	0
$999$	6.32051	0.199972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.m.1.6		23
4.3	odd	2		2008.2.a.d.1.18	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.18	✓	23	4.3	odd	2
4016.2.a.m.1.6		23	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51727 q^{3} -0.472191 q^{5} -4.16870 q^{7} +3.33662 q^{9} +O(q^{10})\)	\(q-2.51727 q^{3} -0.472191 q^{5} -4.16870 q^{7} +3.33662 q^{9} +0.304525 q^{11} +3.93359 q^{13} +1.18863 q^{15} +5.23074 q^{17} +3.05840 q^{19} +10.4937 q^{21} -7.82281 q^{23} -4.77704 q^{25} -0.847370 q^{27} -7.79678 q^{29} +9.12452 q^{31} -0.766569 q^{33} +1.96842 q^{35} -7.45897 q^{37} -9.90189 q^{39} -5.32993 q^{41} -10.2669 q^{43} -1.57552 q^{45} -8.20514 q^{47} +10.3781 q^{49} -13.1672 q^{51} -8.29482 q^{53} -0.143794 q^{55} -7.69881 q^{57} -3.03547 q^{59} -0.845264 q^{61} -13.9094 q^{63} -1.85741 q^{65} +5.94412 q^{67} +19.6921 q^{69} -6.20529 q^{71} +10.1038 q^{73} +12.0251 q^{75} -1.26947 q^{77} +9.74582 q^{79} -7.87682 q^{81} +4.98258 q^{83} -2.46991 q^{85} +19.6266 q^{87} +12.3965 q^{89} -16.3980 q^{91} -22.9688 q^{93} -1.44415 q^{95} +0.396603 q^{97} +1.01608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

Introduction

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