LMFDB - Embedded newform 4592.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4592,2,Mod(1,4592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4592, 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("4592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4592 = 2^{4} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4592.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:	$36.6673046082$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.633117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x^{2} + 6x - 3 $ x^5 - x^4 - 6x^3 + 4x^2 + 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.08727$ of defining polynomial
Character	$\chi$	$=$	4592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08727 q^{3} +0.209668 q^{5} -1.00000 q^{7} +1.35670 q^{9} +O(q^{10})$	$q-2.08727 q^{3} +0.209668 q^{5} -1.00000 q^{7} +1.35670 q^{9} -6.03819 q^{11} +3.67193 q^{13} -0.437633 q^{15} -5.37138 q^{17} +3.54285 q^{19} +2.08727 q^{21} +1.30362 q^{23} -4.95604 q^{25} +3.43002 q^{27} -8.00307 q^{29} +0.384208 q^{31} +12.6033 q^{33} -0.209668 q^{35} -3.68876 q^{37} -7.66432 q^{39} -1.00000 q^{41} -0.824527 q^{43} +0.284455 q^{45} -5.11625 q^{47} +1.00000 q^{49} +11.2115 q^{51} +1.53217 q^{53} -1.26601 q^{55} -7.39489 q^{57} -10.2669 q^{59} +9.36070 q^{61} -1.35670 q^{63} +0.769885 q^{65} -11.3638 q^{67} -2.72101 q^{69} +14.9494 q^{71} +7.77203 q^{73} +10.3446 q^{75} +6.03819 q^{77} +6.04703 q^{79} -11.2295 q^{81} -14.1871 q^{83} -1.12620 q^{85} +16.7046 q^{87} +0.520905 q^{89} -3.67193 q^{91} -0.801946 q^{93} +0.742821 q^{95} -3.65270 q^{97} -8.19200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9	$ 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9} - 2 q^{11} + 5 q^{13} + 5 q^{15} + 13 q^{17} + 4 q^{21} - 2 q^{23} + 22 q^{25} - 10 q^{27} - 5 q^{29} - 17 q^{31} + 3 q^{33} + 5 q^{35} - 7 q^{37} - 5 q^{39} - 5 q^{41} - q^{43} - 23 q^{45} - 9 q^{47} + 5 q^{49} - 5 q^{51} + 5 q^{53} - 33 q^{55} - 3 q^{57} - 7 q^{59} + 22 q^{61} - q^{63} - 31 q^{65} + 3 q^{67} - 22 q^{69} + 24 q^{71} + 40 q^{73} - 24 q^{75} + 2 q^{77} + 42 q^{79} + 9 q^{81} + 12 q^{83} - 23 q^{85} + 32 q^{87} + 8 q^{89} - 5 q^{91} - 11 q^{93} + 17 q^{95} + 16 q^{97} + 45 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9 - 2 * q^11 + 5 * q^13 + 5 * q^15 + 13 * q^17 + 4 * q^21 - 2 * q^23 + 22 * q^25 - 10 * q^27 - 5 * q^29 - 17 * q^31 + 3 * q^33 + 5 * q^35 - 7 * q^37 - 5 * q^39 - 5 * q^41 - q^43 - 23 * q^45 - 9 * q^47 + 5 * q^49 - 5 * q^51 + 5 * q^53 - 33 * q^55 - 3 * q^57 - 7 * q^59 + 22 * q^61 - q^63 - 31 * q^65 + 3 * q^67 - 22 * q^69 + 24 * q^71 + 40 * q^73 - 24 * q^75 + 2 * q^77 + 42 * q^79 + 9 * q^81 + 12 * q^83 - 23 * q^85 + 32 * q^87 + 8 * q^89 - 5 * q^91 - 11 * q^93 + 17 * q^95 + 16 * q^97 + 45 * 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
$3$	−2.08727	−1.20509	−0.602543	−	0.798086i	$-0.705846\pi$
$3$	−2.08727	−1.20509	−0.602543	+	0.798086i	$0.705846\pi$
$4$	0	0
$5$	0.209668	0.0937662	0.0468831	−	0.998900i	$-0.485071\pi$
$5$	0.209668	0.0937662	0.0468831	+	0.998900i	$0.485071\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.35670	0.452232
$10$	0	0
$11$	−6.03819	−1.82058	−0.910292	−	0.413967i	$-0.864143\pi$
$11$	−6.03819	−1.82058	−0.910292	+	0.413967i	$0.864143\pi$
$12$	0	0
$13$	3.67193	1.01841	0.509205	−	0.860645i	$-0.329939\pi$
$13$	3.67193	1.01841	0.509205	+	0.860645i	$0.329939\pi$
$14$	0	0
$15$	−0.437633	−0.112996
$16$	0	0
$17$	−5.37138	−1.30275	−0.651375	−	0.758756i	$-0.725808\pi$
$17$	−5.37138	−1.30275	−0.651375	+	0.758756i	$0.725808\pi$
$18$	0	0
$19$	3.54285	0.812786	0.406393	−	0.913698i	$-0.366786\pi$
$19$	3.54285	0.812786	0.406393	+	0.913698i	$0.366786\pi$
$20$	0	0
$21$	2.08727	0.455480
$22$	0	0
$23$	1.30362	0.271824	0.135912	−	0.990721i	$-0.456604\pi$
$23$	1.30362	0.271824	0.135912	+	0.990721i	$0.456604\pi$
$24$	0	0
$25$	−4.95604	−0.991208
$26$	0	0
$27$	3.43002	0.660107
$28$	0	0
$29$	−8.00307	−1.48613	−0.743066	−	0.669218i	$-0.766629\pi$
$29$	−8.00307	−1.48613	−0.743066	+	0.669218i	$0.766629\pi$
$30$	0	0
$31$	0.384208	0.0690058	0.0345029	−	0.999405i	$-0.489015\pi$
$31$	0.384208	0.0690058	0.0345029	+	0.999405i	$0.489015\pi$
$32$	0	0
$33$	12.6033	2.19396
$34$	0	0
$35$	−0.209668	−0.0354403
$36$	0	0
$37$	−3.68876	−0.606429	−0.303214	−	0.952922i	$-0.598060\pi$
$37$	−3.68876	−0.606429	−0.303214	+	0.952922i	$0.598060\pi$
$38$	0	0
$39$	−7.66432	−1.22727
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−0.824527	−0.125739	−0.0628696	−	0.998022i	$-0.520025\pi$
$43$	−0.824527	−0.125739	−0.0628696	+	0.998022i	$0.520025\pi$
$44$	0	0
$45$	0.284455	0.0424041
$46$	0	0
$47$	−5.11625	−0.746282	−0.373141	−	0.927775i	$-0.621719\pi$
$47$	−5.11625	−0.746282	−0.373141	+	0.927775i	$0.621719\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	11.2115	1.56993
$52$	0	0
$53$	1.53217	0.210460	0.105230	−	0.994448i	$-0.466442\pi$
$53$	1.53217	0.210460	0.105230	+	0.994448i	$0.466442\pi$
$54$	0	0
$55$	−1.26601	−0.170709
$56$	0	0
$57$	−7.39489	−0.979477
$58$	0	0
$59$	−10.2669	−1.33664	−0.668320	−	0.743874i	$-0.732986\pi$
$59$	−10.2669	−1.33664	−0.668320	+	0.743874i	$0.732986\pi$
$60$	0	0
$61$	9.36070	1.19851	0.599257	−	0.800557i	$-0.295463\pi$
$61$	9.36070	1.19851	0.599257	+	0.800557i	$0.295463\pi$
$62$	0	0
$63$	−1.35670	−0.170928
$64$	0	0
$65$	0.769885	0.0954925
$66$	0	0
$67$	−11.3638	−1.38830	−0.694152	−	0.719828i	$-0.744221\pi$
$67$	−11.3638	−1.38830	−0.694152	+	0.719828i	$0.744221\pi$
$68$	0	0
$69$	−2.72101	−0.327571
$70$	0	0
$71$	14.9494	1.77416	0.887081	−	0.461614i	$-0.152729\pi$
$71$	14.9494	1.77416	0.887081	+	0.461614i	$0.152729\pi$
$72$	0	0
$73$	7.77203	0.909648	0.454824	−	0.890581i	$-0.349702\pi$
$73$	7.77203	0.909648	0.454824	+	0.890581i	$0.349702\pi$
$74$	0	0
$75$	10.3446	1.19449
$76$	0	0
$77$	6.03819	0.688116
$78$	0	0
$79$	6.04703	0.680344	0.340172	−	0.940363i	$-0.389515\pi$
$79$	6.04703	0.680344	0.340172	+	0.940363i	$0.389515\pi$
$80$	0	0
$81$	−11.2295	−1.24772
$82$	0	0
$83$	−14.1871	−1.55723	−0.778617	−	0.627500i	$-0.784078\pi$
$83$	−14.1871	−1.55723	−0.778617	+	0.627500i	$0.784078\pi$
$84$	0	0
$85$	−1.12620	−0.122154
$86$	0	0
$87$	16.7046	1.79092
$88$	0	0
$89$	0.520905	0.0552159	0.0276079	−	0.999619i	$-0.491211\pi$
$89$	0.520905	0.0552159	0.0276079	+	0.999619i	$0.491211\pi$
$90$	0	0
$91$	−3.67193	−0.384923
$92$	0	0
$93$	−0.801946	−0.0831580
$94$	0	0
$95$	0.742821	0.0762118
$96$	0	0
$97$	−3.65270	−0.370876	−0.185438	−	0.982656i	$-0.559370\pi$
$97$	−3.65270	−0.370876	−0.185438	+	0.982656i	$0.559370\pi$
$98$	0	0
$99$	−8.19200	−0.823327
$100$	0	0
$101$	−2.45465	−0.244247	−0.122123	−	0.992515i	$-0.538970\pi$
$101$	−2.45465	−0.244247	−0.122123	+	0.992515i	$0.538970\pi$
$102$	0	0
$103$	10.2479	1.00975	0.504876	−	0.863192i	$-0.331538\pi$
$103$	10.2479	1.00975	0.504876	+	0.863192i	$0.331538\pi$
$104$	0	0
$105$	0.437633	0.0427086
$106$	0	0
$107$	5.33318	0.515578	0.257789	−	0.966201i	$-0.417006\pi$
$107$	5.33318	0.515578	0.257789	+	0.966201i	$0.417006\pi$
$108$	0	0
$109$	8.82638	0.845413	0.422707	−	0.906267i	$-0.361080\pi$
$109$	8.82638	0.845413	0.422707	+	0.906267i	$0.361080\pi$
$110$	0	0
$111$	7.69944	0.730799
$112$	0	0
$113$	−18.2045	−1.71253	−0.856267	−	0.516534i	$-0.827222\pi$
$113$	−18.2045	−1.71253	−0.856267	+	0.516534i	$0.827222\pi$
$114$	0	0
$115$	0.273327	0.0254879
$116$	0	0
$117$	4.98170	0.460559
$118$	0	0
$119$	5.37138	0.492393
$120$	0	0
$121$	25.4598	2.31452
$122$	0	0
$123$	2.08727	0.188203
$124$	0	0
$125$	−2.08746	−0.186708
$126$	0	0
$127$	13.1751	1.16910	0.584551	−	0.811357i	$-0.301271\pi$
$127$	13.1751	1.16910	0.584551	+	0.811357i	$0.301271\pi$
$128$	0	0
$129$	1.72101	0.151527
$130$	0	0
$131$	−13.0506	−1.14023	−0.570117	−	0.821563i	$-0.693102\pi$
$131$	−13.0506	−1.14023	−0.570117	+	0.821563i	$0.693102\pi$
$132$	0	0
$133$	−3.54285	−0.307204
$134$	0	0
$135$	0.719163	0.0618957
$136$	0	0
$137$	12.7699	1.09100	0.545502	−	0.838109i	$-0.316339\pi$
$137$	12.7699	1.09100	0.545502	+	0.838109i	$0.316339\pi$
$138$	0	0
$139$	−13.9759	−1.18542	−0.592712	−	0.805415i	$-0.701943\pi$
$139$	−13.9759	−1.18542	−0.592712	+	0.805415i	$0.701943\pi$
$140$	0	0
$141$	10.6790	0.899334
$142$	0	0
$143$	−22.1718	−1.85410
$144$	0	0
$145$	−1.67798	−0.139349
$146$	0	0
$147$	−2.08727	−0.172155
$148$	0	0
$149$	−2.02136	−0.165597	−0.0827983	−	0.996566i	$-0.526386\pi$
$149$	−2.02136	−0.165597	−0.0827983	+	0.996566i	$0.526386\pi$
$150$	0	0
$151$	−11.7648	−0.957406	−0.478703	−	0.877977i	$-0.658893\pi$
$151$	−11.7648	−0.957406	−0.478703	+	0.877977i	$0.658893\pi$
$152$	0	0
$153$	−7.28733	−0.589146
$154$	0	0
$155$	0.0805560	0.00647041
$156$	0	0
$157$	18.8859	1.50726	0.753631	−	0.657297i	$-0.228300\pi$
$157$	18.8859	1.50726	0.753631	+	0.657297i	$0.228300\pi$
$158$	0	0
$159$	−3.19805	−0.253622
$160$	0	0
$161$	−1.30362	−0.102740
$162$	0	0
$163$	−16.3263	−1.27877	−0.639387	−	0.768885i	$-0.720812\pi$
$163$	−16.3263	−1.27877	−0.639387	+	0.768885i	$0.720812\pi$
$164$	0	0
$165$	2.64251	0.205719
$166$	0	0
$167$	22.5846	1.74765	0.873824	−	0.486242i	$-0.161633\pi$
$167$	22.5846	1.74765	0.873824	+	0.486242i	$0.161633\pi$
$168$	0	0
$169$	0.483092	0.0371609
$170$	0	0
$171$	4.80658	0.367568
$172$	0	0
$173$	−22.2230	−1.68958	−0.844790	−	0.535097i	$-0.820275\pi$
$173$	−22.2230	−1.68958	−0.844790	+	0.535097i	$0.820275\pi$
$174$	0	0
$175$	4.95604	0.374641
$176$	0	0
$177$	21.4299	1.61077
$178$	0	0
$179$	10.0095	0.748142	0.374071	−	0.927400i	$-0.377962\pi$
$179$	10.0095	0.748142	0.374071	+	0.927400i	$0.377962\pi$
$180$	0	0
$181$	10.8101	0.803511	0.401755	−	0.915747i	$-0.368400\pi$
$181$	10.8101	0.803511	0.401755	+	0.915747i	$0.368400\pi$
$182$	0	0
$183$	−19.5383	−1.44431
$184$	0	0
$185$	−0.773414	−0.0568625
$186$	0	0
$187$	32.4334	2.37177
$188$	0	0
$189$	−3.43002	−0.249497
$190$	0	0
$191$	19.7693	1.43046	0.715229	−	0.698890i	$-0.246322\pi$
$191$	19.7693	1.43046	0.715229	+	0.698890i	$0.246322\pi$
$192$	0	0
$193$	18.0955	1.30254	0.651270	−	0.758846i	$-0.274237\pi$
$193$	18.0955	1.30254	0.651270	+	0.758846i	$0.274237\pi$
$194$	0	0
$195$	−1.60696	−0.115077
$196$	0	0
$197$	4.47538	0.318858	0.159429	−	0.987209i	$-0.449035\pi$
$197$	4.47538	0.318858	0.159429	+	0.987209i	$0.449035\pi$
$198$	0	0
$199$	−10.1956	−0.722744	−0.361372	−	0.932422i	$-0.617692\pi$
$199$	−10.1956	−0.722744	−0.361372	+	0.932422i	$0.617692\pi$
$200$	0	0
$201$	23.7192	1.67303
$202$	0	0
$203$	8.00307	0.561705
$204$	0	0
$205$	−0.209668	−0.0146438
$206$	0	0
$207$	1.76862	0.122928
$208$	0	0
$209$	−21.3924	−1.47974
$210$	0	0
$211$	14.0477	0.967081	0.483540	−	0.875322i	$-0.339351\pi$
$211$	14.0477	0.967081	0.483540	+	0.875322i	$0.339351\pi$
$212$	0	0
$213$	−31.2033	−2.13802
$214$	0	0
$215$	−0.172877	−0.0117901
$216$	0	0
$217$	−0.384208	−0.0260818
$218$	0	0
$219$	−16.2223	−1.09620
$220$	0	0
$221$	−19.7233	−1.32674
$222$	0	0
$223$	0.701496	0.0469756	0.0234878	−	0.999724i	$-0.492523\pi$
$223$	0.701496	0.0469756	0.0234878	+	0.999724i	$0.492523\pi$
$224$	0	0
$225$	−6.72385	−0.448256
$226$	0	0
$227$	−9.36869	−0.621822	−0.310911	−	0.950439i	$-0.600634\pi$
$227$	−9.36869	−0.621822	−0.310911	+	0.950439i	$0.600634\pi$
$228$	0	0
$229$	12.4012	0.819496	0.409748	−	0.912199i	$-0.365617\pi$
$229$	12.4012	0.819496	0.409748	+	0.912199i	$0.365617\pi$
$230$	0	0
$231$	−12.6033	−0.829239
$232$	0	0
$233$	9.61739	0.630056	0.315028	−	0.949082i	$-0.397986\pi$
$233$	9.61739	0.630056	0.315028	+	0.949082i	$0.397986\pi$
$234$	0	0
$235$	−1.07271	−0.0699760
$236$	0	0
$237$	−12.6218	−0.819873
$238$	0	0
$239$	16.5603	1.07120	0.535600	−	0.844472i	$-0.320086\pi$
$239$	16.5603	1.07120	0.535600	+	0.844472i	$0.320086\pi$
$240$	0	0
$241$	9.29684	0.598862	0.299431	−	0.954118i	$-0.403203\pi$
$241$	9.29684	0.598862	0.299431	+	0.954118i	$0.403203\pi$
$242$	0	0
$243$	13.1489	0.843501
$244$	0	0
$245$	0.209668	0.0133952
$246$	0	0
$247$	13.0091	0.827750
$248$	0	0
$249$	29.6123	1.87660
$250$	0	0
$251$	−2.89503	−0.182733	−0.0913664	−	0.995817i	$-0.529123\pi$
$251$	−2.89503	−0.182733	−0.0913664	+	0.995817i	$0.529123\pi$
$252$	0	0
$253$	−7.87152	−0.494878
$254$	0	0
$255$	2.35069	0.147206
$256$	0	0
$257$	5.41470	0.337760	0.168880	−	0.985637i	$-0.445985\pi$
$257$	5.41470	0.337760	0.168880	+	0.985637i	$0.445985\pi$
$258$	0	0
$259$	3.68876	0.229209
$260$	0	0
$261$	−10.8577	−0.672077
$262$	0	0
$263$	26.7339	1.64848	0.824242	−	0.566238i	$-0.191602\pi$
$263$	26.7339	1.64848	0.824242	+	0.566238i	$0.191602\pi$
$264$	0	0
$265$	0.321246	0.0197340
$266$	0	0
$267$	−1.08727	−0.0665399
$268$	0	0
$269$	−14.6488	−0.893151	−0.446576	−	0.894746i	$-0.647357\pi$
$269$	−14.6488	−0.893151	−0.446576	+	0.894746i	$0.647357\pi$
$270$	0	0
$271$	−22.8241	−1.38646	−0.693232	−	0.720714i	$-0.743814\pi$
$271$	−22.8241	−1.38646	−0.693232	+	0.720714i	$0.743814\pi$
$272$	0	0
$273$	7.66432	0.463866
$274$	0	0
$275$	29.9255	1.80458
$276$	0	0
$277$	−8.31190	−0.499414	−0.249707	−	0.968321i	$-0.580334\pi$
$277$	−8.31190	−0.499414	−0.249707	+	0.968321i	$0.580334\pi$
$278$	0	0
$279$	0.521254	0.0312067
$280$	0	0
$281$	17.3137	1.03285	0.516423	−	0.856333i	$-0.327263\pi$
$281$	17.3137	1.03285	0.516423	+	0.856333i	$0.327263\pi$
$282$	0	0
$283$	14.7818	0.878686	0.439343	−	0.898319i	$-0.355211\pi$
$283$	14.7818	0.878686	0.439343	+	0.898319i	$0.355211\pi$
$284$	0	0
$285$	−1.55047	−0.0918418
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	11.8517	0.697158
$290$	0	0
$291$	7.62418	0.446937
$292$	0	0
$293$	4.99638	0.291892	0.145946	−	0.989293i	$-0.453377\pi$
$293$	4.99638	0.291892	0.145946	+	0.989293i	$0.453377\pi$
$294$	0	0
$295$	−2.15264	−0.125332
$296$	0	0
$297$	−20.7111	−1.20178
$298$	0	0
$299$	4.78681	0.276828
$300$	0	0
$301$	0.824527	0.0475250
$302$	0	0
$303$	5.12352	0.294338
$304$	0	0
$305$	1.96263	0.112380
$306$	0	0
$307$	7.85295	0.448192	0.224096	−	0.974567i	$-0.428057\pi$
$307$	7.85295	0.448192	0.224096	+	0.974567i	$0.428057\pi$
$308$	0	0
$309$	−21.3901	−1.21684
$310$	0	0
$311$	27.0379	1.53318	0.766589	−	0.642138i	$-0.221952\pi$
$311$	27.0379	1.53318	0.766589	+	0.642138i	$0.221952\pi$
$312$	0	0
$313$	−0.168393	−0.00951815	−0.00475907	−	0.999989i	$-0.501515\pi$
$313$	−0.168393	−0.00951815	−0.00475907	+	0.999989i	$0.501515\pi$
$314$	0	0
$315$	−0.284455	−0.0160272
$316$	0	0
$317$	5.33690	0.299750	0.149875	−	0.988705i	$-0.452113\pi$
$317$	5.33690	0.299750	0.149875	+	0.988705i	$0.452113\pi$
$318$	0	0
$319$	48.3240	2.70563
$320$	0	0
$321$	−11.1318	−0.621316
$322$	0	0
$323$	−19.0300	−1.05886
$324$	0	0
$325$	−18.1982	−1.00946
$326$	0	0
$327$	−18.4230	−1.01880
$328$	0	0
$329$	5.11625	0.282068
$330$	0	0
$331$	8.09405	0.444889	0.222445	−	0.974945i	$-0.428596\pi$
$331$	8.09405	0.444889	0.222445	+	0.974945i	$0.428596\pi$
$332$	0	0
$333$	−5.00453	−0.274247
$334$	0	0
$335$	−2.38261	−0.130176
$336$	0	0
$337$	20.1540	1.09786	0.548929	−	0.835869i	$-0.315036\pi$
$337$	20.1540	1.09786	0.548929	+	0.835869i	$0.315036\pi$
$338$	0	0
$339$	37.9977	2.06375
$340$	0	0
$341$	−2.31992	−0.125631
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.570508	−0.0307151
$346$	0	0
$347$	−18.2807	−0.981359	−0.490679	−	0.871340i	$-0.663251\pi$
$347$	−18.2807	−0.981359	−0.490679	+	0.871340i	$0.663251\pi$
$348$	0	0
$349$	9.74567	0.521674	0.260837	−	0.965383i	$-0.416001\pi$
$349$	9.74567	0.521674	0.260837	+	0.965383i	$0.416001\pi$
$350$	0	0
$351$	12.5948	0.672260
$352$	0	0
$353$	26.2143	1.39525	0.697624	−	0.716464i	$-0.254241\pi$
$353$	26.2143	1.39525	0.697624	+	0.716464i	$0.254241\pi$
$354$	0	0
$355$	3.13439	0.166356
$356$	0	0
$357$	−11.2115	−0.593376
$358$	0	0
$359$	−0.473108	−0.0249697	−0.0124849	−	0.999922i	$-0.503974\pi$
$359$	−0.473108	−0.0249697	−0.0124849	+	0.999922i	$0.503974\pi$
$360$	0	0
$361$	−6.44820	−0.339379
$362$	0	0
$363$	−53.1414	−2.78920
$364$	0	0
$365$	1.62954	0.0852942
$366$	0	0
$367$	−10.2670	−0.535932	−0.267966	−	0.963428i	$-0.586352\pi$
$367$	−10.2670	−0.535932	−0.267966	+	0.963428i	$0.586352\pi$
$368$	0	0
$369$	−1.35670	−0.0706268
$370$	0	0
$371$	−1.53217	−0.0795463
$372$	0	0
$373$	20.1653	1.04412	0.522061	−	0.852908i	$-0.325163\pi$
$373$	20.1653	1.04412	0.522061	+	0.852908i	$0.325163\pi$
$374$	0	0
$375$	4.35709	0.224999
$376$	0	0
$377$	−29.3867	−1.51349
$378$	0	0
$379$	26.8293	1.37813	0.689063	−	0.724701i	$-0.258022\pi$
$379$	26.8293	1.37813	0.689063	+	0.724701i	$0.258022\pi$
$380$	0	0
$381$	−27.5000	−1.40887
$382$	0	0
$383$	24.1679	1.23492	0.617461	−	0.786602i	$-0.288161\pi$
$383$	24.1679	1.23492	0.617461	+	0.786602i	$0.288161\pi$
$384$	0	0
$385$	1.26601	0.0645220
$386$	0	0
$387$	−1.11863	−0.0568634
$388$	0	0
$389$	15.9595	0.809180	0.404590	−	0.914498i	$-0.367414\pi$
$389$	15.9595	0.809180	0.404590	+	0.914498i	$0.367414\pi$
$390$	0	0
$391$	−7.00224	−0.354119
$392$	0	0
$393$	27.2401	1.37408
$394$	0	0
$395$	1.26786	0.0637932
$396$	0	0
$397$	−8.73399	−0.438346	−0.219173	−	0.975686i	$-0.570336\pi$
$397$	−8.73399	−0.438346	−0.219173	+	0.975686i	$0.570336\pi$
$398$	0	0
$399$	7.39489	0.370208
$400$	0	0
$401$	1.77238	0.0885086	0.0442543	−	0.999020i	$-0.485909\pi$
$401$	1.77238	0.0885086	0.0442543	+	0.999020i	$0.485909\pi$
$402$	0	0
$403$	1.41079	0.0702763
$404$	0	0
$405$	−2.35445	−0.116994
$406$	0	0
$407$	22.2735	1.10405
$408$	0	0
$409$	−27.0895	−1.33949	−0.669746	−	0.742590i	$-0.733597\pi$
$409$	−27.0895	−1.33949	−0.669746	+	0.742590i	$0.733597\pi$
$410$	0	0
$411$	−26.6542	−1.31475
$412$	0	0
$413$	10.2669	0.505203
$414$	0	0
$415$	−2.97457	−0.146016
$416$	0	0
$417$	29.1716	1.42854
$418$	0	0
$419$	18.3386	0.895898	0.447949	−	0.894059i	$-0.352155\pi$
$419$	18.3386	0.895898	0.447949	+	0.894059i	$0.352155\pi$
$420$	0	0
$421$	16.7202	0.814894	0.407447	−	0.913229i	$-0.366419\pi$
$421$	16.7202	0.814894	0.407447	+	0.913229i	$0.366419\pi$
$422$	0	0
$423$	−6.94120	−0.337493
$424$	0	0
$425$	26.6208	1.29130
$426$	0	0
$427$	−9.36070	−0.452996
$428$	0	0
$429$	46.2786	2.23435
$430$	0	0
$431$	−0.204738	−0.00986186	−0.00493093	−	0.999988i	$-0.501570\pi$
$431$	−0.204738	−0.00986186	−0.00493093	+	0.999988i	$0.501570\pi$
$432$	0	0
$433$	31.9997	1.53781	0.768903	−	0.639366i	$-0.220803\pi$
$433$	31.9997	1.53781	0.768903	+	0.639366i	$0.220803\pi$
$434$	0	0
$435$	3.50240	0.167927
$436$	0	0
$437$	4.61854	0.220935
$438$	0	0
$439$	−24.9052	−1.18866	−0.594330	−	0.804221i	$-0.702583\pi$
$439$	−24.9052	−1.18866	−0.594330	+	0.804221i	$0.702583\pi$
$440$	0	0
$441$	1.35670	0.0646046
$442$	0	0
$443$	−19.0300	−0.904142	−0.452071	−	0.891982i	$-0.649315\pi$
$443$	−19.0300	−0.904142	−0.452071	+	0.891982i	$0.649315\pi$
$444$	0	0
$445$	0.109217	0.00517738
$446$	0	0
$447$	4.21913	0.199558
$448$	0	0
$449$	−5.95010	−0.280802	−0.140401	−	0.990095i	$-0.544839\pi$
$449$	−5.95010	−0.280802	−0.140401	+	0.990095i	$0.544839\pi$
$450$	0	0
$451$	6.03819	0.284327
$452$	0	0
$453$	24.5563	1.15376
$454$	0	0
$455$	−0.769885	−0.0360928
$456$	0	0
$457$	−31.8410	−1.48946	−0.744729	−	0.667367i	$-0.767421\pi$
$457$	−31.8410	−1.48946	−0.744729	+	0.667367i	$0.767421\pi$
$458$	0	0
$459$	−18.4239	−0.859955
$460$	0	0
$461$	4.99638	0.232705	0.116352	−	0.993208i	$-0.462880\pi$
$461$	4.99638	0.232705	0.116352	+	0.993208i	$0.462880\pi$
$462$	0	0
$463$	−17.6038	−0.818119	−0.409060	−	0.912508i	$-0.634143\pi$
$463$	−17.6038	−0.818119	−0.409060	+	0.912508i	$0.634143\pi$
$464$	0	0
$465$	−0.168142	−0.00779740
$466$	0	0
$467$	42.9444	1.98723	0.993614	−	0.112833i	$-0.0359925\pi$
$467$	42.9444	1.98723	0.993614	+	0.112833i	$0.0359925\pi$
$468$	0	0
$469$	11.3638	0.524730
$470$	0	0
$471$	−39.4201	−1.81638
$472$	0	0
$473$	4.97865	0.228919
$474$	0	0
$475$	−17.5585	−0.805640
$476$	0	0
$477$	2.07869	0.0951767
$478$	0	0
$479$	−2.85249	−0.130334	−0.0651669	−	0.997874i	$-0.520758\pi$
$479$	−2.85249	−0.130334	−0.0651669	+	0.997874i	$0.520758\pi$
$480$	0	0
$481$	−13.5449	−0.617594
$482$	0	0
$483$	2.72101	0.123810
$484$	0	0
$485$	−0.765853	−0.0347756
$486$	0	0
$487$	6.23225	0.282410	0.141205	−	0.989980i	$-0.454902\pi$
$487$	6.23225	0.282410	0.141205	+	0.989980i	$0.454902\pi$
$488$	0	0
$489$	34.0774	1.54103
$490$	0	0
$491$	2.13846	0.0965071	0.0482536	−	0.998835i	$-0.484634\pi$
$491$	2.13846	0.0965071	0.0482536	+	0.998835i	$0.484634\pi$
$492$	0	0
$493$	42.9875	1.93606
$494$	0	0
$495$	−1.71760	−0.0772002
$496$	0	0
$497$	−14.9494	−0.670570
$498$	0	0
$499$	−25.1110	−1.12412	−0.562060	−	0.827096i	$-0.689991\pi$
$499$	−25.1110	−1.12412	−0.562060	+	0.827096i	$0.689991\pi$
$500$	0	0
$501$	−47.1402	−2.10607
$502$	0	0
$503$	−37.1641	−1.65707	−0.828534	−	0.559939i	$-0.810824\pi$
$503$	−37.1641	−1.65707	−0.828534	+	0.559939i	$0.810824\pi$
$504$	0	0
$505$	−0.514660	−0.0229021
$506$	0	0
$507$	−1.00834	−0.0447821
$508$	0	0
$509$	−2.36319	−0.104747	−0.0523734	−	0.998628i	$-0.516679\pi$
$509$	−2.36319	−0.104747	−0.0523734	+	0.998628i	$0.516679\pi$
$510$	0	0
$511$	−7.77203	−0.343815
$512$	0	0
$513$	12.1520	0.536526
$514$	0	0
$515$	2.14864	0.0946805
$516$	0	0
$517$	30.8929	1.35867
$518$	0	0
$519$	46.3853	2.03609
$520$	0	0
$521$	−2.27368	−0.0996116	−0.0498058	−	0.998759i	$-0.515860\pi$
$521$	−2.27368	−0.0996116	−0.0498058	+	0.998759i	$0.515860\pi$
$522$	0	0
$523$	35.7432	1.56294	0.781471	−	0.623941i	$-0.214469\pi$
$523$	35.7432	1.56294	0.781471	+	0.623941i	$0.214469\pi$
$524$	0	0
$525$	−10.3446	−0.451475
$526$	0	0
$527$	−2.06373	−0.0898974
$528$	0	0
$529$	−21.3006	−0.926112
$530$	0	0
$531$	−13.9291	−0.604472
$532$	0	0
$533$	−3.67193	−0.159049
$534$	0	0
$535$	1.11820	0.0483438
$536$	0	0
$537$	−20.8925	−0.901576
$538$	0	0
$539$	−6.03819	−0.260083
$540$	0	0
$541$	−22.5224	−0.968314	−0.484157	−	0.874981i	$-0.660874\pi$
$541$	−22.5224	−0.968314	−0.484157	+	0.874981i	$0.660874\pi$
$542$	0	0
$543$	−22.5637	−0.968299
$544$	0	0
$545$	1.85060	0.0792712
$546$	0	0
$547$	−3.63915	−0.155599	−0.0777994	−	0.996969i	$-0.524789\pi$
$547$	−3.63915	−0.155599	−0.0777994	+	0.996969i	$0.524789\pi$
$548$	0	0
$549$	12.6996	0.542007
$550$	0	0
$551$	−28.3537	−1.20791
$552$	0	0
$553$	−6.04703	−0.257146
$554$	0	0
$555$	1.61432	0.0685242
$556$	0	0
$557$	4.80008	0.203386	0.101693	−	0.994816i	$-0.467574\pi$
$557$	4.80008	0.203386	0.101693	+	0.994816i	$0.467574\pi$
$558$	0	0
$559$	−3.02761	−0.128054
$560$	0	0
$561$	−67.6973	−2.85818
$562$	0	0
$563$	−31.7115	−1.33648	−0.668241	−	0.743945i	$-0.732952\pi$
$563$	−31.7115	−1.33648	−0.668241	+	0.743945i	$0.732952\pi$
$564$	0	0
$565$	−3.81689	−0.160578
$566$	0	0
$567$	11.2295	0.471593
$568$	0	0
$569$	−35.4970	−1.48811	−0.744055	−	0.668118i	$-0.767100\pi$
$569$	−35.4970	−1.48811	−0.744055	+	0.668118i	$0.767100\pi$
$570$	0	0
$571$	38.7912	1.62336	0.811681	−	0.584100i	$-0.198553\pi$
$571$	38.7912	1.62336	0.811681	+	0.584100i	$0.198553\pi$
$572$	0	0
$573$	−41.2639	−1.72383
$574$	0	0
$575$	−6.46080	−0.269434
$576$	0	0
$577$	−0.335828	−0.0139807	−0.00699035	−	0.999976i	$-0.502225\pi$
$577$	−0.335828	−0.0139807	−0.00699035	+	0.999976i	$0.502225\pi$
$578$	0	0
$579$	−37.7701	−1.56967
$580$	0	0
$581$	14.1871	0.588579
$582$	0	0
$583$	−9.25154	−0.383160
$584$	0	0
$585$	1.04450	0.0431848
$586$	0	0
$587$	−8.19386	−0.338197	−0.169098	−	0.985599i	$-0.554086\pi$
$587$	−8.19386	−0.338197	−0.169098	+	0.985599i	$0.554086\pi$
$588$	0	0
$589$	1.36119	0.0560870
$590$	0	0
$591$	−9.34132	−0.384251
$592$	0	0
$593$	30.1222	1.23697	0.618486	−	0.785796i	$-0.287746\pi$
$593$	30.1222	1.23697	0.618486	+	0.785796i	$0.287746\pi$
$594$	0	0
$595$	1.12620	0.0461698
$596$	0	0
$597$	21.2809	0.870968
$598$	0	0
$599$	39.6072	1.61831	0.809153	−	0.587599i	$-0.199927\pi$
$599$	39.6072	1.61831	0.809153	+	0.587599i	$0.199927\pi$
$600$	0	0
$601$	−35.0329	−1.42902	−0.714511	−	0.699624i	$-0.753351\pi$
$601$	−35.0329	−1.42902	−0.714511	+	0.699624i	$0.753351\pi$
$602$	0	0
$603$	−15.4172	−0.627836
$604$	0	0
$605$	5.33809	0.217024
$606$	0	0
$607$	−13.4657	−0.546555	−0.273278	−	0.961935i	$-0.588108\pi$
$607$	−13.4657	−0.546555	−0.273278	+	0.961935i	$0.588108\pi$
$608$	0	0
$609$	−16.7046	−0.676903
$610$	0	0
$611$	−18.7865	−0.760022
$612$	0	0
$613$	−6.49820	−0.262460	−0.131230	−	0.991352i	$-0.541893\pi$
$613$	−6.49820	−0.262460	−0.131230	+	0.991352i	$0.541893\pi$
$614$	0	0
$615$	0.437633	0.0176471
$616$	0	0
$617$	6.63045	0.266932	0.133466	−	0.991053i	$-0.457389\pi$
$617$	6.63045	0.266932	0.133466	+	0.991053i	$0.457389\pi$
$618$	0	0
$619$	4.50756	0.181174	0.0905871	−	0.995889i	$-0.471126\pi$
$619$	4.50756	0.181174	0.0905871	+	0.995889i	$0.471126\pi$
$620$	0	0
$621$	4.47144	0.179433
$622$	0	0
$623$	−0.520905	−0.0208696
$624$	0	0
$625$	24.3425	0.973701
$626$	0	0
$627$	44.6518	1.78322
$628$	0	0
$629$	19.8137	0.790025
$630$	0	0
$631$	24.2677	0.966081	0.483040	−	0.875598i	$-0.339532\pi$
$631$	24.2677	0.966081	0.483040	+	0.875598i	$0.339532\pi$
$632$	0	0
$633$	−29.3213	−1.16542
$634$	0	0
$635$	2.76239	0.109622
$636$	0	0
$637$	3.67193	0.145487
$638$	0	0
$639$	20.2818	0.802334
$640$	0	0
$641$	22.2964	0.880653	0.440327	−	0.897838i	$-0.354863\pi$
$641$	22.2964	0.880653	0.440327	+	0.897838i	$0.354863\pi$
$642$	0	0
$643$	16.2405	0.640464	0.320232	−	0.947339i	$-0.396239\pi$
$643$	16.2405	0.640464	0.320232	+	0.947339i	$0.396239\pi$
$644$	0	0
$645$	0.360840	0.0142081
$646$	0	0
$647$	−13.5814	−0.533939	−0.266969	−	0.963705i	$-0.586022\pi$
$647$	−13.5814	−0.533939	−0.266969	+	0.963705i	$0.586022\pi$
$648$	0	0
$649$	61.9937	2.43347
$650$	0	0
$651$	0.801946	0.0314308
$652$	0	0
$653$	−47.1222	−1.84403	−0.922016	−	0.387151i	$-0.873459\pi$
$653$	−47.1222	−1.84403	−0.922016	+	0.387151i	$0.873459\pi$
$654$	0	0
$655$	−2.73628	−0.106915
$656$	0	0
$657$	10.5443	0.411372
$658$	0	0
$659$	−5.27483	−0.205478	−0.102739	−	0.994708i	$-0.532761\pi$
$659$	−5.27483	−0.205478	−0.102739	+	0.994708i	$0.532761\pi$
$660$	0	0
$661$	−47.8044	−1.85938	−0.929689	−	0.368346i	$-0.879924\pi$
$661$	−47.8044	−1.85938	−0.929689	+	0.368346i	$0.879924\pi$
$662$	0	0
$663$	41.1679	1.59883
$664$	0	0
$665$	−0.742821	−0.0288054
$666$	0	0
$667$	−10.4330	−0.403966
$668$	0	0
$669$	−1.46421	−0.0566097
$670$	0	0
$671$	−56.5217	−2.18200
$672$	0	0
$673$	−5.11960	−0.197346	−0.0986730	−	0.995120i	$-0.531460\pi$
$673$	−5.11960	−0.197346	−0.0986730	+	0.995120i	$0.531460\pi$
$674$	0	0
$675$	−16.9993	−0.654303
$676$	0	0
$677$	30.7985	1.18368	0.591842	−	0.806054i	$-0.298401\pi$
$677$	30.7985	1.18368	0.591842	+	0.806054i	$0.298401\pi$
$678$	0	0
$679$	3.65270	0.140178
$680$	0	0
$681$	19.5550	0.749349
$682$	0	0
$683$	18.8534	0.721407	0.360703	−	0.932681i	$-0.382537\pi$
$683$	18.8534	0.721407	0.360703	+	0.932681i	$0.382537\pi$
$684$	0	0
$685$	2.67743	0.102299
$686$	0	0
$687$	−25.8847	−0.987563
$688$	0	0
$689$	5.62603	0.214335
$690$	0	0
$691$	6.22638	0.236863	0.118431	−	0.992962i	$-0.462213\pi$
$691$	6.22638	0.236863	0.118431	+	0.992962i	$0.462213\pi$
$692$	0	0
$693$	8.19200	0.311188
$694$	0	0
$695$	−2.93030	−0.111153
$696$	0	0
$697$	5.37138	0.203455
$698$	0	0
$699$	−20.0741	−0.759272
$700$	0	0
$701$	40.6743	1.53625	0.768124	−	0.640301i	$-0.221190\pi$
$701$	40.6743	1.53625	0.768124	+	0.640301i	$0.221190\pi$
$702$	0	0
$703$	−13.0687	−0.492897
$704$	0	0
$705$	2.23904	0.0843271
$706$	0	0
$707$	2.45465	0.0923166
$708$	0	0
$709$	−47.5660	−1.78638	−0.893189	−	0.449681i	$-0.851538\pi$
$709$	−47.5660	−1.78638	−0.893189	+	0.449681i	$0.851538\pi$
$710$	0	0
$711$	8.20398	0.307673
$712$	0	0
$713$	0.500862	0.0187574
$714$	0	0
$715$	−4.64871	−0.173852
$716$	0	0
$717$	−34.5659	−1.29089
$718$	0	0
$719$	0.700457	0.0261226	0.0130613	−	0.999915i	$-0.495842\pi$
$719$	0.700457	0.0261226	0.0130613	+	0.999915i	$0.495842\pi$
$720$	0	0
$721$	−10.2479	−0.381650
$722$	0	0
$723$	−19.4050	−0.721680
$724$	0	0
$725$	39.6635	1.47307
$726$	0	0
$727$	−38.6447	−1.43325	−0.716625	−	0.697458i	$-0.754314\pi$
$727$	−38.6447	−1.43325	−0.716625	+	0.697458i	$0.754314\pi$
$728$	0	0
$729$	6.24313	0.231227
$730$	0	0
$731$	4.42885	0.163807
$732$	0	0
$733$	−31.3572	−1.15820	−0.579101	−	0.815255i	$-0.696596\pi$
$733$	−31.3572	−1.15820	−0.579101	+	0.815255i	$0.696596\pi$
$734$	0	0
$735$	−0.437633	−0.0161423
$736$	0	0
$737$	68.6166	2.52752
$738$	0	0
$739$	−51.5448	−1.89611	−0.948054	−	0.318110i	$-0.896952\pi$
$739$	−51.5448	−1.89611	−0.948054	+	0.318110i	$0.896952\pi$
$740$	0	0
$741$	−27.1535	−0.997510
$742$	0	0
$743$	15.7347	0.577251	0.288626	−	0.957442i	$-0.406802\pi$
$743$	15.7347	0.577251	0.288626	+	0.957442i	$0.406802\pi$
$744$	0	0
$745$	−0.423814	−0.0155274
$746$	0	0
$747$	−19.2476	−0.704231
$748$	0	0
$749$	−5.33318	−0.194870
$750$	0	0
$751$	12.1375	0.442904	0.221452	−	0.975171i	$-0.428920\pi$
$751$	12.1375	0.442904	0.221452	+	0.975171i	$0.428920\pi$
$752$	0	0
$753$	6.04271	0.220209
$754$	0	0
$755$	−2.46670	−0.0897723
$756$	0	0
$757$	−17.9846	−0.653660	−0.326830	−	0.945083i	$-0.605981\pi$
$757$	−17.9846	−0.653660	−0.326830	+	0.945083i	$0.605981\pi$
$758$	0	0
$759$	16.4300	0.596371
$760$	0	0
$761$	23.2298	0.842079	0.421039	−	0.907042i	$-0.361665\pi$
$761$	23.2298	0.842079	0.421039	+	0.907042i	$0.361665\pi$
$762$	0	0
$763$	−8.82638	−0.319536
$764$	0	0
$765$	−1.52792	−0.0552420
$766$	0	0
$767$	−37.6995	−1.36125
$768$	0	0
$769$	−34.9703	−1.26106	−0.630530	−	0.776165i	$-0.717162\pi$
$769$	−34.9703	−1.26106	−0.630530	+	0.776165i	$0.717162\pi$
$770$	0	0
$771$	−11.3019	−0.407030
$772$	0	0
$773$	−26.7805	−0.963228	−0.481614	−	0.876384i	$-0.659949\pi$
$773$	−26.7805	−0.963228	−0.481614	+	0.876384i	$0.659949\pi$
$774$	0	0
$775$	−1.90415	−0.0683991
$776$	0	0
$777$	−7.69944	−0.276216
$778$	0	0
$779$	−3.54285	−0.126936
$780$	0	0
$781$	−90.2671	−3.23001
$782$	0	0
$783$	−27.4506	−0.981006
$784$	0	0
$785$	3.95977	0.141330
$786$	0	0
$787$	−0.177103	−0.00631303	−0.00315652	−	0.999995i	$-0.501005\pi$
$787$	−0.177103	−0.00631303	−0.00315652	+	0.999995i	$0.501005\pi$
$788$	0	0
$789$	−55.8009	−1.98657
$790$	0	0
$791$	18.2045	0.647277
$792$	0	0
$793$	34.3718	1.22058
$794$	0	0
$795$	−0.670528	−0.0237812
$796$	0	0
$797$	−15.4863	−0.548555	−0.274277	−	0.961651i	$-0.588439\pi$
$797$	−15.4863	−0.548555	−0.274277	+	0.961651i	$0.588439\pi$
$798$	0	0
$799$	27.4813	0.972219
$800$	0	0
$801$	0.706711	0.0249704
$802$	0	0
$803$	−46.9290	−1.65609
$804$	0	0
$805$	−0.273327	−0.00963352
$806$	0	0
$807$	30.5759	1.07632
$808$	0	0
$809$	−12.8695	−0.452467	−0.226234	−	0.974073i	$-0.572641\pi$
$809$	−12.8695	−0.452467	−0.226234	+	0.974073i	$0.572641\pi$
$810$	0	0
$811$	−24.8951	−0.874185	−0.437092	−	0.899417i	$-0.643992\pi$
$811$	−24.8951	−0.874185	−0.437092	+	0.899417i	$0.643992\pi$
$812$	0	0
$813$	47.6400	1.67081
$814$	0	0
$815$	−3.42309	−0.119906
$816$	0	0
$817$	−2.92118	−0.102199
$818$	0	0
$819$	−4.98170	−0.174075
$820$	0	0
$821$	−21.8002	−0.760834	−0.380417	−	0.924815i	$-0.624220\pi$
$821$	−21.8002	−0.760834	−0.380417	+	0.924815i	$0.624220\pi$
$822$	0	0
$823$	1.76071	0.0613744	0.0306872	−	0.999529i	$-0.490230\pi$
$823$	1.76071	0.0613744	0.0306872	+	0.999529i	$0.490230\pi$
$824$	0	0
$825$	−62.4627	−2.17467
$826$	0	0
$827$	−9.93134	−0.345347	−0.172673	−	0.984979i	$-0.555240\pi$
$827$	−9.93134	−0.345347	−0.172673	+	0.984979i	$0.555240\pi$
$828$	0	0
$829$	4.85768	0.168714	0.0843571	−	0.996436i	$-0.473116\pi$
$829$	4.85768	0.168714	0.0843571	+	0.996436i	$0.473116\pi$
$830$	0	0
$831$	17.3492	0.601837
$832$	0	0
$833$	−5.37138	−0.186107
$834$	0	0
$835$	4.73526	0.163870
$836$	0	0
$837$	1.31784	0.0455512
$838$	0	0
$839$	−15.9065	−0.549152	−0.274576	−	0.961565i	$-0.588537\pi$
$839$	−15.9065	−0.549152	−0.274576	+	0.961565i	$0.588537\pi$
$840$	0	0
$841$	35.0491	1.20859
$842$	0	0
$843$	−36.1383	−1.24467
$844$	0	0
$845$	0.101289	0.00348444
$846$	0	0
$847$	−25.4598	−0.874808
$848$	0	0
$849$	−30.8536	−1.05889
$850$	0	0
$851$	−4.80875	−0.164842
$852$	0	0
$853$	1.08814	0.0372572	0.0186286	−	0.999826i	$-0.494070\pi$
$853$	1.08814	0.0372572	0.0186286	+	0.999826i	$0.494070\pi$
$854$	0	0
$855$	1.00778	0.0344655
$856$	0	0
$857$	−1.53174	−0.0523232	−0.0261616	−	0.999658i	$-0.508328\pi$
$857$	−1.53174	−0.0523232	−0.0261616	+	0.999658i	$0.508328\pi$
$858$	0	0
$859$	−10.9042	−0.372047	−0.186024	−	0.982545i	$-0.559560\pi$
$859$	−10.9042	−0.372047	−0.186024	+	0.982545i	$0.559560\pi$
$860$	0	0
$861$	−2.08727	−0.0711340
$862$	0	0
$863$	54.7487	1.86367	0.931834	−	0.362884i	$-0.118208\pi$
$863$	54.7487	1.86367	0.931834	+	0.362884i	$0.118208\pi$
$864$	0	0
$865$	−4.65943	−0.158426
$866$	0	0
$867$	−24.7377	−0.840136
$868$	0	0
$869$	−36.5131	−1.23862
$870$	0	0
$871$	−41.7270	−1.41386
$872$	0	0
$873$	−4.95561	−0.167722
$874$	0	0
$875$	2.08746	0.0705690
$876$	0	0
$877$	−0.0143469	−0.000484459 0	−0.000242229	−	1.00000i	$-0.500077\pi$
$877$	−0.0143469	−0.000484459 0	−0.000242229		1.00000i	$0.500077\pi$
$878$	0	0
$879$	−10.4288	−0.351755
$880$	0	0
$881$	−56.3960	−1.90003	−0.950014	−	0.312206i	$-0.898932\pi$
$881$	−56.3960	−1.90003	−0.950014	+	0.312206i	$0.898932\pi$
$882$	0	0
$883$	26.5606	0.893835	0.446917	−	0.894575i	$-0.352522\pi$
$883$	26.5606	0.893835	0.446917	+	0.894575i	$0.352522\pi$
$884$	0	0
$885$	4.49314	0.151035
$886$	0	0
$887$	29.3990	0.987121	0.493560	−	0.869712i	$-0.335695\pi$
$887$	29.3990	0.987121	0.493560	+	0.869712i	$0.335695\pi$
$888$	0	0
$889$	−13.1751	−0.441879
$890$	0	0
$891$	67.8057	2.27158
$892$	0	0
$893$	−18.1261	−0.606567
$894$	0	0
$895$	2.09866	0.0701504
$896$	0	0
$897$	−9.99137	−0.333602
$898$	0	0
$899$	−3.07484	−0.102552
$900$	0	0
$901$	−8.22986	−0.274177
$902$	0	0
$903$	−1.72101	−0.0572717
$904$	0	0
$905$	2.26653	0.0753421
$906$	0	0
$907$	12.9258	0.429196	0.214598	−	0.976702i	$-0.431156\pi$
$907$	12.9258	0.429196	0.214598	+	0.976702i	$0.431156\pi$
$908$	0	0
$909$	−3.33022	−0.110456
$910$	0	0
$911$	−33.8152	−1.12035	−0.560174	−	0.828375i	$-0.689266\pi$
$911$	−33.8152	−1.12035	−0.560174	+	0.828375i	$0.689266\pi$
$912$	0	0
$913$	85.6643	2.83507
$914$	0	0
$915$	−4.09655	−0.135428
$916$	0	0
$917$	13.0506	0.430968
$918$	0	0
$919$	2.96555	0.0978246	0.0489123	−	0.998803i	$-0.484425\pi$
$919$	2.96555	0.0978246	0.0489123	+	0.998803i	$0.484425\pi$
$920$	0	0
$921$	−16.3912	−0.540110
$922$	0	0
$923$	54.8930	1.80683
$924$	0	0
$925$	18.2817	0.601097
$926$	0	0
$927$	13.9032	0.456642
$928$	0	0
$929$	−24.4362	−0.801726	−0.400863	−	0.916138i	$-0.631290\pi$
$929$	−24.4362	−0.801726	−0.400863	+	0.916138i	$0.631290\pi$
$930$	0	0
$931$	3.54285	0.116112
$932$	0	0
$933$	−56.4354	−1.84761
$934$	0	0
$935$	6.80023	0.222391
$936$	0	0
$937$	35.9240	1.17359	0.586793	−	0.809737i	$-0.300390\pi$
$937$	35.9240	1.17359	0.586793	+	0.809737i	$0.300390\pi$
$938$	0	0
$939$	0.351482	0.0114702
$940$	0	0
$941$	−42.9108	−1.39885	−0.699426	−	0.714705i	$-0.746561\pi$
$941$	−42.9108	−1.39885	−0.699426	+	0.714705i	$0.746561\pi$
$942$	0	0
$943$	−1.30362	−0.0424518
$944$	0	0
$945$	−0.719163	−0.0233944
$946$	0	0
$947$	−44.6039	−1.44943	−0.724716	−	0.689048i	$-0.758029\pi$
$947$	−44.6039	−1.44943	−0.724716	+	0.689048i	$0.758029\pi$
$948$	0	0
$949$	28.5384	0.926395
$950$	0	0
$951$	−11.1395	−0.361225
$952$	0	0
$953$	−12.8800	−0.417224	−0.208612	−	0.977999i	$-0.566895\pi$
$953$	−12.8800	−0.417224	−0.208612	+	0.977999i	$0.566895\pi$
$954$	0	0
$955$	4.14499	0.134129
$956$	0	0
$957$	−100.865	−3.26051
$958$	0	0
$959$	−12.7699	−0.412361
$960$	0	0
$961$	−30.8524	−0.995238
$962$	0	0
$963$	7.23552	0.233161
$964$	0	0
$965$	3.79403	0.122134
$966$	0	0
$967$	−49.0909	−1.57866	−0.789329	−	0.613970i	$-0.789571\pi$
$967$	−49.0909	−1.57866	−0.789329	+	0.613970i	$0.789571\pi$
$968$	0	0
$969$	39.7207	1.27601
$970$	0	0
$971$	5.24060	0.168179	0.0840895	−	0.996458i	$-0.473202\pi$
$971$	5.24060	0.168179	0.0840895	+	0.996458i	$0.473202\pi$
$972$	0	0
$973$	13.9759	0.448048
$974$	0	0
$975$	37.9847	1.21648
$976$	0	0
$977$	44.0772	1.41015	0.705077	−	0.709130i	$-0.250912\pi$
$977$	44.0772	1.41015	0.705077	+	0.709130i	$0.250912\pi$
$978$	0	0
$979$	−3.14533	−0.100525
$980$	0	0
$981$	11.9747	0.382323
$982$	0	0
$983$	54.1938	1.72851	0.864256	−	0.503052i	$-0.167789\pi$
$983$	54.1938	1.72851	0.864256	+	0.503052i	$0.167789\pi$
$984$	0	0
$985$	0.938341	0.0298980
$986$	0	0
$987$	−10.6790	−0.339916
$988$	0	0
$989$	−1.07487	−0.0341789
$990$	0	0
$991$	0.908400	0.0288563	0.0144281	−	0.999896i	$-0.495407\pi$
$991$	0.908400	0.0288563	0.0144281	+	0.999896i	$0.495407\pi$
$992$	0	0
$993$	−16.8945	−0.536130
$994$	0	0
$995$	−2.13768	−0.0677689
$996$	0	0
$997$	36.8530	1.16715	0.583573	−	0.812061i	$-0.301654\pi$
$997$	36.8530	1.16715	0.583573	+	0.812061i	$0.301654\pi$
$998$	0	0
$999$	−12.6525	−0.400308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4592.2.a.bb.1.2		5
4.3	odd	2		287.2.a.e.1.4	✓	5
12.11	even	2		2583.2.a.r.1.2		5
20.19	odd	2		7175.2.a.n.1.2		5
28.27	even	2		2009.2.a.n.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.e.1.4	✓	5	4.3	odd	2
2009.2.a.n.1.4		5	28.27	even	2
2583.2.a.r.1.2		5	12.11	even	2
4592.2.a.bb.1.2		5	1.1	even	1	trivial
7175.2.a.n.1.2		5	20.19	odd	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:	\( N \)	\(=\)	\( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4592.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08727 q^{3} +0.209668 q^{5} -1.00000 q^{7} +1.35670 q^{9} +O(q^{10})\)	\(q-2.08727 q^{3} +0.209668 q^{5} -1.00000 q^{7} +1.35670 q^{9} -6.03819 q^{11} +3.67193 q^{13} -0.437633 q^{15} -5.37138 q^{17} +3.54285 q^{19} +2.08727 q^{21} +1.30362 q^{23} -4.95604 q^{25} +3.43002 q^{27} -8.00307 q^{29} +0.384208 q^{31} +12.6033 q^{33} -0.209668 q^{35} -3.68876 q^{37} -7.66432 q^{39} -1.00000 q^{41} -0.824527 q^{43} +0.284455 q^{45} -5.11625 q^{47} +1.00000 q^{49} +11.2115 q^{51} +1.53217 q^{53} -1.26601 q^{55} -7.39489 q^{57} -10.2669 q^{59} +9.36070 q^{61} -1.35670 q^{63} +0.769885 q^{65} -11.3638 q^{67} -2.72101 q^{69} +14.9494 q^{71} +7.77203 q^{73} +10.3446 q^{75} +6.03819 q^{77} +6.04703 q^{79} -11.2295 q^{81} -14.1871 q^{83} -1.12620 q^{85} +16.7046 q^{87} +0.520905 q^{89} -3.67193 q^{91} -0.801946 q^{93} +0.742821 q^{95} -3.65270 q^{97} -8.19200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9	\( 5 q - 4 q^{3} - 5 q^{5} - 5 q^{7} + q^{9} - 2 q^{11} + 5 q^{13} + 5 q^{15} + 13 q^{17} + 4 q^{21} - 2 q^{23} + 22 q^{25} - 10 q^{27} - 5 q^{29} - 17 q^{31} + 3 q^{33} + 5 q^{35} - 7 q^{37} - 5 q^{39} - 5 q^{41} - q^{43} - 23 q^{45} - 9 q^{47} + 5 q^{49} - 5 q^{51} + 5 q^{53} - 33 q^{55} - 3 q^{57} - 7 q^{59} + 22 q^{61} - q^{63} - 31 q^{65} + 3 q^{67} - 22 q^{69} + 24 q^{71} + 40 q^{73} - 24 q^{75} + 2 q^{77} + 42 q^{79} + 9 q^{81} + 12 q^{83} - 23 q^{85} + 32 q^{87} + 8 q^{89} - 5 q^{91} - 11 q^{93} + 17 q^{95} + 16 q^{97} + 45 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 5 * q^5 - 5 * q^7 + q^9 - 2 * q^11 + 5 * q^13 + 5 * q^15 + 13 * q^17 + 4 * q^21 - 2 * q^23 + 22 * q^25 - 10 * q^27 - 5 * q^29 - 17 * q^31 + 3 * q^33 + 5 * q^35 - 7 * q^37 - 5 * q^39 - 5 * q^41 - q^43 - 23 * q^45 - 9 * q^47 + 5 * q^49 - 5 * q^51 + 5 * q^53 - 33 * q^55 - 3 * q^57 - 7 * q^59 + 22 * q^61 - q^63 - 31 * q^65 + 3 * q^67 - 22 * q^69 + 24 * q^71 + 40 * q^73 - 24 * q^75 + 2 * q^77 + 42 * q^79 + 9 * q^81 + 12 * q^83 - 23 * q^85 + 32 * q^87 + 8 * q^89 - 5 * q^91 - 11 * q^93 + 17 * q^95 + 16 * q^97 + 45 * q^99

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