LMFDB - Embedded newform 4004.2.a.h.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14 $ x^9 - 3x^8 - 19x^7 + 51x^6 + 116x^5 - 247x^4 - 249x^3 + 288x^2 + 189x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.0675528$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0675528 q^{3} -1.58844 q^{5} +1.00000 q^{7} -2.99544 q^{9} +O(q^{10})$	$q+0.0675528 q^{3} -1.58844 q^{5} +1.00000 q^{7} -2.99544 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.107304 q^{15} -7.43757 q^{17} -0.425745 q^{19} +0.0675528 q^{21} +2.83856 q^{23} -2.47685 q^{25} -0.405009 q^{27} +2.09851 q^{29} +9.18316 q^{31} +0.0675528 q^{33} -1.58844 q^{35} +3.75566 q^{37} -0.0675528 q^{39} -3.36839 q^{41} -7.73439 q^{43} +4.75808 q^{45} +9.56426 q^{47} +1.00000 q^{49} -0.502429 q^{51} -3.70217 q^{53} -1.58844 q^{55} -0.0287603 q^{57} +14.9588 q^{59} +1.17020 q^{61} -2.99544 q^{63} +1.58844 q^{65} -0.996339 q^{67} +0.191753 q^{69} +6.57629 q^{71} -4.05427 q^{73} -0.167318 q^{75} +1.00000 q^{77} +8.23658 q^{79} +8.95895 q^{81} +17.0373 q^{83} +11.8141 q^{85} +0.141760 q^{87} -10.8466 q^{89} -1.00000 q^{91} +0.620348 q^{93} +0.676271 q^{95} +11.5981 q^{97} -2.99544 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9	$ 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + 5 * q^17 + 10 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^25 + 27 * q^27 + 14 * q^29 + 11 * q^31 + 3 * q^33 - 3 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 + 10 * q^47 + 9 * q^49 - 15 * q^51 + 21 * q^53 - 8 * q^57 + 23 * q^59 + 34 * q^61 + 20 * q^63 + 10 * q^67 - 16 * q^69 + 4 * q^71 + 9 * q^73 + 30 * q^75 + 9 * q^77 - 34 * q^79 + 69 * q^81 + 15 * q^83 + 5 * q^85 + 39 * q^87 - 9 * q^91 + 3 * q^93 - 64 * q^95 + 15 * q^97 + 20 * 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
$3$	0.0675528	0.0390016	0.0195008	−	0.999810i	$-0.493792\pi$
$3$	0.0675528	0.0390016	0.0195008	+	0.999810i	$0.493792\pi$
$4$	0	0
$5$	−1.58844	−0.710373	−0.355187	−	0.934795i	$-0.615583\pi$
$5$	−1.58844	−0.710373	−0.355187	+	0.934795i	$0.615583\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.99544	−0.998479
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.107304	−0.0277057
$16$	0	0
$17$	−7.43757	−1.80387	−0.901937	−	0.431867i	$-0.857855\pi$
$17$	−7.43757	−1.80387	−0.901937	+	0.431867i	$0.857855\pi$
$18$	0	0
$19$	−0.425745	−0.0976726	−0.0488363	−	0.998807i	$-0.515551\pi$
$19$	−0.425745	−0.0976726	−0.0488363	+	0.998807i	$0.515551\pi$
$20$	0	0
$21$	0.0675528	0.0147412
$22$	0	0
$23$	2.83856	0.591880	0.295940	−	0.955206i	$-0.404367\pi$
$23$	2.83856	0.591880	0.295940	+	0.955206i	$0.404367\pi$
$24$	0	0
$25$	−2.47685	−0.495370
$26$	0	0
$27$	−0.405009	−0.0779439
$28$	0	0
$29$	2.09851	0.389683	0.194841	−	0.980835i	$-0.437581\pi$
$29$	2.09851	0.389683	0.194841	+	0.980835i	$0.437581\pi$
$30$	0	0
$31$	9.18316	1.64934	0.824672	−	0.565611i	$-0.191360\pi$
$31$	9.18316	1.64934	0.824672	+	0.565611i	$0.191360\pi$
$32$	0	0
$33$	0.0675528	0.0117594
$34$	0	0
$35$	−1.58844	−0.268496
$36$	0	0
$37$	3.75566	0.617426	0.308713	−	0.951155i	$-0.400102\pi$
$37$	3.75566	0.617426	0.308713	+	0.951155i	$0.400102\pi$
$38$	0	0
$39$	−0.0675528	−0.0108171
$40$	0	0
$41$	−3.36839	−0.526055	−0.263027	−	0.964788i	$-0.584721\pi$
$41$	−3.36839	−0.526055	−0.263027	+	0.964788i	$0.584721\pi$
$42$	0	0
$43$	−7.73439	−1.17948	−0.589742	−	0.807592i	$-0.700771\pi$
$43$	−7.73439	−1.17948	−0.589742	+	0.807592i	$0.700771\pi$
$44$	0	0
$45$	4.75808	0.709293
$46$	0	0
$47$	9.56426	1.39509	0.697545	−	0.716541i	$-0.254276\pi$
$47$	9.56426	1.39509	0.697545	+	0.716541i	$0.254276\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.502429	−0.0703541
$52$	0	0
$53$	−3.70217	−0.508532	−0.254266	−	0.967134i	$-0.581834\pi$
$53$	−3.70217	−0.508532	−0.254266	+	0.967134i	$0.581834\pi$
$54$	0	0
$55$	−1.58844	−0.214186
$56$	0	0
$57$	−0.0287603	−0.00380939
$58$	0	0
$59$	14.9588	1.94747	0.973734	−	0.227689i	$-0.0731169\pi$
$59$	14.9588	1.94747	0.973734	+	0.227689i	$0.0731169\pi$
$60$	0	0
$61$	1.17020	0.149829	0.0749144	−	0.997190i	$-0.476132\pi$
$61$	1.17020	0.149829	0.0749144	+	0.997190i	$0.476132\pi$
$62$	0	0
$63$	−2.99544	−0.377390
$64$	0	0
$65$	1.58844	0.197022
$66$	0	0
$67$	−0.996339	−0.121722	−0.0608611	−	0.998146i	$-0.519385\pi$
$67$	−0.996339	−0.121722	−0.0608611	+	0.998146i	$0.519385\pi$
$68$	0	0
$69$	0.191753	0.0230843
$70$	0	0
$71$	6.57629	0.780462	0.390231	−	0.920717i	$-0.372395\pi$
$71$	6.57629	0.780462	0.390231	+	0.920717i	$0.372395\pi$
$72$	0	0
$73$	−4.05427	−0.474516	−0.237258	−	0.971447i	$-0.576249\pi$
$73$	−4.05427	−0.474516	−0.237258	+	0.971447i	$0.576249\pi$
$74$	0	0
$75$	−0.167318	−0.0193202
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	8.23658	0.926688	0.463344	−	0.886179i	$-0.346649\pi$
$79$	8.23658	0.926688	0.463344	+	0.886179i	$0.346649\pi$
$80$	0	0
$81$	8.95895	0.995439
$82$	0	0
$83$	17.0373	1.87009	0.935045	−	0.354530i	$-0.115359\pi$
$83$	17.0373	1.87009	0.935045	+	0.354530i	$0.115359\pi$
$84$	0	0
$85$	11.8141	1.28142
$86$	0	0
$87$	0.141760	0.0151983
$88$	0	0
$89$	−10.8466	−1.14974	−0.574870	−	0.818245i	$-0.694947\pi$
$89$	−10.8466	−1.14974	−0.574870	+	0.818245i	$0.694947\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0.620348	0.0643271
$94$	0	0
$95$	0.676271	0.0693840
$96$	0	0
$97$	11.5981	1.17761	0.588803	−	0.808277i	$-0.299600\pi$
$97$	11.5981	1.17761	0.588803	+	0.808277i	$0.299600\pi$
$98$	0	0
$99$	−2.99544	−0.301053
$100$	0	0
$101$	−18.0594	−1.79697	−0.898487	−	0.439000i	$-0.855333\pi$
$101$	−18.0594	−1.79697	−0.898487	+	0.439000i	$0.855333\pi$
$102$	0	0
$103$	4.82865	0.475781	0.237891	−	0.971292i	$-0.423544\pi$
$103$	4.82865	0.475781	0.237891	+	0.971292i	$0.423544\pi$
$104$	0	0
$105$	−0.107304	−0.0104718
$106$	0	0
$107$	−17.2340	−1.66608	−0.833038	−	0.553216i	$-0.813401\pi$
$107$	−17.2340	−1.66608	−0.833038	+	0.553216i	$0.813401\pi$
$108$	0	0
$109$	−1.28059	−0.122658	−0.0613291	−	0.998118i	$-0.519534\pi$
$109$	−1.28059	−0.122658	−0.0613291	+	0.998118i	$0.519534\pi$
$110$	0	0
$111$	0.253705	0.0240806
$112$	0	0
$113$	−5.18312	−0.487587	−0.243794	−	0.969827i	$-0.578392\pi$
$113$	−5.18312	−0.487587	−0.243794	+	0.969827i	$0.578392\pi$
$114$	0	0
$115$	−4.50889	−0.420456
$116$	0	0
$117$	2.99544	0.276928
$118$	0	0
$119$	−7.43757	−0.681801
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.227544	−0.0205170
$124$	0	0
$125$	11.8765	1.06227
$126$	0	0
$127$	7.43061	0.659360	0.329680	−	0.944093i	$-0.393059\pi$
$127$	7.43061	0.659360	0.329680	+	0.944093i	$0.393059\pi$
$128$	0	0
$129$	−0.522480	−0.0460018
$130$	0	0
$131$	6.70073	0.585446	0.292723	−	0.956197i	$-0.405439\pi$
$131$	6.70073	0.585446	0.292723	+	0.956197i	$0.405439\pi$
$132$	0	0
$133$	−0.425745	−0.0369168
$134$	0	0
$135$	0.643333	0.0553693
$136$	0	0
$137$	−12.8559	−1.09836	−0.549178	−	0.835705i	$-0.685059\pi$
$137$	−12.8559	−1.09836	−0.549178	+	0.835705i	$0.685059\pi$
$138$	0	0
$139$	23.4610	1.98994	0.994969	−	0.100184i	$-0.0319431\pi$
$139$	23.4610	1.98994	0.994969	+	0.100184i	$0.0319431\pi$
$140$	0	0
$141$	0.646092	0.0544108
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−3.33336	−0.276820
$146$	0	0
$147$	0.0675528	0.00557166
$148$	0	0
$149$	2.84807	0.233323	0.116662	−	0.993172i	$-0.462781\pi$
$149$	2.84807	0.233323	0.116662	+	0.993172i	$0.462781\pi$
$150$	0	0
$151$	11.1163	0.904635	0.452317	−	0.891857i	$-0.350597\pi$
$151$	11.1163	0.904635	0.452317	+	0.891857i	$0.350597\pi$
$152$	0	0
$153$	22.2788	1.80113
$154$	0	0
$155$	−14.5869	−1.17165
$156$	0	0
$157$	14.4903	1.15645	0.578226	−	0.815876i	$-0.303745\pi$
$157$	14.4903	1.15645	0.578226	+	0.815876i	$0.303745\pi$
$158$	0	0
$159$	−0.250092	−0.0198336
$160$	0	0
$161$	2.83856	0.223710
$162$	0	0
$163$	−10.9592	−0.858388	−0.429194	−	0.903212i	$-0.641202\pi$
$163$	−10.9592	−0.858388	−0.429194	+	0.903212i	$0.641202\pi$
$164$	0	0
$165$	−0.107304	−0.00835359
$166$	0	0
$167$	−2.06363	−0.159689	−0.0798443	−	0.996807i	$-0.525442\pi$
$167$	−2.06363	−0.159689	−0.0798443	+	0.996807i	$0.525442\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.27529	0.0975240
$172$	0	0
$173$	−2.83210	−0.215321	−0.107660	−	0.994188i	$-0.534336\pi$
$173$	−2.83210	−0.215321	−0.107660	+	0.994188i	$0.534336\pi$
$174$	0	0
$175$	−2.47685	−0.187232
$176$	0	0
$177$	1.01051	0.0759544
$178$	0	0
$179$	−21.7943	−1.62898	−0.814491	−	0.580177i	$-0.802984\pi$
$179$	−21.7943	−1.62898	−0.814491	+	0.580177i	$0.802984\pi$
$180$	0	0
$181$	21.9150	1.62893	0.814464	−	0.580214i	$-0.197031\pi$
$181$	21.9150	1.62893	0.814464	+	0.580214i	$0.197031\pi$
$182$	0	0
$183$	0.0790503	0.00584357
$184$	0	0
$185$	−5.96564	−0.438603
$186$	0	0
$187$	−7.43757	−0.543889
$188$	0	0
$189$	−0.405009	−0.0294600
$190$	0	0
$191$	20.4629	1.48065	0.740324	−	0.672251i	$-0.234672\pi$
$191$	20.4629	1.48065	0.740324	+	0.672251i	$0.234672\pi$
$192$	0	0
$193$	−0.540000	−0.0388701	−0.0194350	−	0.999811i	$-0.506187\pi$
$193$	−0.540000	−0.0388701	−0.0194350	+	0.999811i	$0.506187\pi$
$194$	0	0
$195$	0.107304	0.00768418
$196$	0	0
$197$	12.8438	0.915085	0.457542	−	0.889188i	$-0.348730\pi$
$197$	12.8438	0.915085	0.457542	+	0.889188i	$0.348730\pi$
$198$	0	0
$199$	12.7433	0.903351	0.451675	−	0.892182i	$-0.350826\pi$
$199$	12.7433	0.903351	0.451675	+	0.892182i	$0.350826\pi$
$200$	0	0
$201$	−0.0673055	−0.00474736
$202$	0	0
$203$	2.09851	0.147286
$204$	0	0
$205$	5.35050	0.373695
$206$	0	0
$207$	−8.50272	−0.590980
$208$	0	0
$209$	−0.425745	−0.0294494
$210$	0	0
$211$	17.7159	1.21961	0.609806	−	0.792551i	$-0.291247\pi$
$211$	17.7159	1.21961	0.609806	+	0.792551i	$0.291247\pi$
$212$	0	0
$213$	0.444247	0.0304393
$214$	0	0
$215$	12.2856	0.837874
$216$	0	0
$217$	9.18316	0.623394
$218$	0	0
$219$	−0.273877	−0.0185069
$220$	0	0
$221$	7.43757	0.500305
$222$	0	0
$223$	−20.2631	−1.35692	−0.678459	−	0.734638i	$-0.737352\pi$
$223$	−20.2631	−1.35692	−0.678459	+	0.734638i	$0.737352\pi$
$224$	0	0
$225$	7.41925	0.494617
$226$	0	0
$227$	17.4269	1.15666	0.578332	−	0.815801i	$-0.303704\pi$
$227$	17.4269	1.15666	0.578332	+	0.815801i	$0.303704\pi$
$228$	0	0
$229$	21.8651	1.44489	0.722443	−	0.691430i	$-0.243019\pi$
$229$	21.8651	1.44489	0.722443	+	0.691430i	$0.243019\pi$
$230$	0	0
$231$	0.0675528	0.00444465
$232$	0	0
$233$	5.21944	0.341937	0.170969	−	0.985276i	$-0.445310\pi$
$233$	5.21944	0.341937	0.170969	+	0.985276i	$0.445310\pi$
$234$	0	0
$235$	−15.1923	−0.991034
$236$	0	0
$237$	0.556404	0.0361423
$238$	0	0
$239$	14.3463	0.927986	0.463993	−	0.885839i	$-0.346416\pi$
$239$	14.3463	0.927986	0.463993	+	0.885839i	$0.346416\pi$
$240$	0	0
$241$	−11.2851	−0.726936	−0.363468	−	0.931607i	$-0.618407\pi$
$241$	−11.2851	−0.726936	−0.363468	+	0.931607i	$0.618407\pi$
$242$	0	0
$243$	1.82023	0.116768
$244$	0	0
$245$	−1.58844	−0.101482
$246$	0	0
$247$	0.425745	0.0270895
$248$	0	0
$249$	1.15092	0.0729365
$250$	0	0
$251$	−6.77924	−0.427902	−0.213951	−	0.976844i	$-0.568633\pi$
$251$	−6.77924	−0.427902	−0.213951	+	0.976844i	$0.568633\pi$
$252$	0	0
$253$	2.83856	0.178459
$254$	0	0
$255$	0.798079	0.0499776
$256$	0	0
$257$	21.2326	1.32445	0.662226	−	0.749305i	$-0.269612\pi$
$257$	21.2326	1.32445	0.662226	+	0.749305i	$0.269612\pi$
$258$	0	0
$259$	3.75566	0.233365
$260$	0	0
$261$	−6.28594	−0.389090
$262$	0	0
$263$	−2.18212	−0.134555	−0.0672777	−	0.997734i	$-0.521431\pi$
$263$	−2.18212	−0.134555	−0.0672777	+	0.997734i	$0.521431\pi$
$264$	0	0
$265$	5.88068	0.361248
$266$	0	0
$267$	−0.732720	−0.0448417
$268$	0	0
$269$	−15.1016	−0.920759	−0.460379	−	0.887722i	$-0.652287\pi$
$269$	−15.1016	−0.920759	−0.460379	+	0.887722i	$0.652287\pi$
$270$	0	0
$271$	−11.6635	−0.708510	−0.354255	−	0.935149i	$-0.615265\pi$
$271$	−11.6635	−0.708510	−0.354255	+	0.935149i	$0.615265\pi$
$272$	0	0
$273$	−0.0675528	−0.00408848
$274$	0	0
$275$	−2.47685	−0.149360
$276$	0	0
$277$	4.84360	0.291023	0.145512	−	0.989357i	$-0.453517\pi$
$277$	4.84360	0.291023	0.145512	+	0.989357i	$0.453517\pi$
$278$	0	0
$279$	−27.5076	−1.64684
$280$	0	0
$281$	9.95076	0.593613	0.296806	−	0.954938i	$-0.404078\pi$
$281$	9.95076	0.593613	0.296806	+	0.954938i	$0.404078\pi$
$282$	0	0
$283$	−13.6712	−0.812669	−0.406334	−	0.913724i	$-0.633193\pi$
$283$	−13.6712	−0.812669	−0.406334	+	0.913724i	$0.633193\pi$
$284$	0	0
$285$	0.0456840	0.00270609
$286$	0	0
$287$	−3.36839	−0.198830
$288$	0	0
$289$	38.3174	2.25396
$290$	0	0
$291$	0.783482	0.0459285
$292$	0	0
$293$	−15.1281	−0.883791	−0.441896	−	0.897067i	$-0.645694\pi$
$293$	−15.1281	−0.883791	−0.441896	+	0.897067i	$0.645694\pi$
$294$	0	0
$295$	−23.7612	−1.38343
$296$	0	0
$297$	−0.405009	−0.0235010
$298$	0	0
$299$	−2.83856	−0.164158
$300$	0	0
$301$	−7.73439	−0.445803
$302$	0	0
$303$	−1.21996	−0.0700849
$304$	0	0
$305$	−1.85880	−0.106434
$306$	0	0
$307$	4.08497	0.233141	0.116571	−	0.993182i	$-0.462810\pi$
$307$	4.08497	0.233141	0.116571	+	0.993182i	$0.462810\pi$
$308$	0	0
$309$	0.326189	0.0185563
$310$	0	0
$311$	−2.31574	−0.131314	−0.0656569	−	0.997842i	$-0.520914\pi$
$311$	−2.31574	−0.131314	−0.0656569	+	0.997842i	$0.520914\pi$
$312$	0	0
$313$	−30.5088	−1.72446	−0.862230	−	0.506516i	$-0.830933\pi$
$313$	−30.5088	−1.72446	−0.862230	+	0.506516i	$0.830933\pi$
$314$	0	0
$315$	4.75808	0.268087
$316$	0	0
$317$	−1.01605	−0.0570673	−0.0285336	−	0.999593i	$-0.509084\pi$
$317$	−1.01605	−0.0570673	−0.0285336	+	0.999593i	$0.509084\pi$
$318$	0	0
$319$	2.09851	0.117494
$320$	0	0
$321$	−1.16421	−0.0649797
$322$	0	0
$323$	3.16651	0.176189
$324$	0	0
$325$	2.47685	0.137391
$326$	0	0
$327$	−0.0865073	−0.00478387
$328$	0	0
$329$	9.56426	0.527294
$330$	0	0
$331$	20.2645	1.11384	0.556918	−	0.830567i	$-0.311984\pi$
$331$	20.2645	1.11384	0.556918	+	0.830567i	$0.311984\pi$
$332$	0	0
$333$	−11.2498	−0.616487
$334$	0	0
$335$	1.58263	0.0864682
$336$	0	0
$337$	2.10583	0.114712	0.0573560	−	0.998354i	$-0.481733\pi$
$337$	2.10583	0.114712	0.0573560	+	0.998354i	$0.481733\pi$
$338$	0	0
$339$	−0.350135	−0.0190167
$340$	0	0
$341$	9.18316	0.497296
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.304588	−0.0163985
$346$	0	0
$347$	15.5619	0.835408	0.417704	−	0.908583i	$-0.362835\pi$
$347$	15.5619	0.835408	0.417704	+	0.908583i	$0.362835\pi$
$348$	0	0
$349$	−9.86285	−0.527946	−0.263973	−	0.964530i	$-0.585033\pi$
$349$	−9.86285	−0.527946	−0.263973	+	0.964530i	$0.585033\pi$
$350$	0	0
$351$	0.405009	0.0216178
$352$	0	0
$353$	13.9926	0.744749	0.372375	−	0.928082i	$-0.378544\pi$
$353$	13.9926	0.744749	0.372375	+	0.928082i	$0.378544\pi$
$354$	0	0
$355$	−10.4461	−0.554420
$356$	0	0
$357$	−0.502429	−0.0265913
$358$	0	0
$359$	−7.46116	−0.393785	−0.196892	−	0.980425i	$-0.563085\pi$
$359$	−7.46116	−0.393785	−0.196892	+	0.980425i	$0.563085\pi$
$360$	0	0
$361$	−18.8187	−0.990460
$362$	0	0
$363$	0.0675528	0.00354560
$364$	0	0
$365$	6.43997	0.337083
$366$	0	0
$367$	34.0202	1.77584	0.887920	−	0.459998i	$-0.152150\pi$
$367$	34.0202	1.77584	0.887920	+	0.459998i	$0.152150\pi$
$368$	0	0
$369$	10.0898	0.525254
$370$	0	0
$371$	−3.70217	−0.192207
$372$	0	0
$373$	1.12780	0.0583954	0.0291977	−	0.999574i	$-0.490705\pi$
$373$	1.12780	0.0583954	0.0291977	+	0.999574i	$0.490705\pi$
$374$	0	0
$375$	0.802294	0.0414303
$376$	0	0
$377$	−2.09851	−0.108079
$378$	0	0
$379$	21.9091	1.12539	0.562696	−	0.826664i	$-0.309764\pi$
$379$	21.9091	1.12539	0.562696	+	0.826664i	$0.309764\pi$
$380$	0	0
$381$	0.501959	0.0257161
$382$	0	0
$383$	−18.5866	−0.949730	−0.474865	−	0.880059i	$-0.657503\pi$
$383$	−18.5866	−0.949730	−0.474865	+	0.880059i	$0.657503\pi$
$384$	0	0
$385$	−1.58844	−0.0809545
$386$	0	0
$387$	23.1679	1.17769
$388$	0	0
$389$	−31.9345	−1.61915	−0.809573	−	0.587020i	$-0.800301\pi$
$389$	−31.9345	−1.61915	−0.809573	+	0.587020i	$0.800301\pi$
$390$	0	0
$391$	−21.1120	−1.06768
$392$	0	0
$393$	0.452653	0.0228333
$394$	0	0
$395$	−13.0833	−0.658294
$396$	0	0
$397$	5.55088	0.278591	0.139295	−	0.990251i	$-0.455516\pi$
$397$	5.55088	0.278591	0.139295	+	0.990251i	$0.455516\pi$
$398$	0	0
$399$	−0.0287603	−0.00143981
$400$	0	0
$401$	2.48083	0.123887	0.0619433	−	0.998080i	$-0.480270\pi$
$401$	2.48083	0.123887	0.0619433	+	0.998080i	$0.480270\pi$
$402$	0	0
$403$	−9.18316	−0.457446
$404$	0	0
$405$	−14.2308	−0.707133
$406$	0	0
$407$	3.75566	0.186161
$408$	0	0
$409$	25.7733	1.27441	0.637203	−	0.770696i	$-0.280091\pi$
$409$	25.7733	1.27441	0.637203	+	0.770696i	$0.280091\pi$
$410$	0	0
$411$	−0.868454	−0.0428377
$412$	0	0
$413$	14.9588	0.736074
$414$	0	0
$415$	−27.0628	−1.32846
$416$	0	0
$417$	1.58486	0.0776108
$418$	0	0
$419$	1.11114	0.0542826	0.0271413	−	0.999632i	$-0.491360\pi$
$419$	1.11114	0.0542826	0.0271413	+	0.999632i	$0.491360\pi$
$420$	0	0
$421$	−4.66704	−0.227458	−0.113729	−	0.993512i	$-0.536279\pi$
$421$	−4.66704	−0.227458	−0.113729	+	0.993512i	$0.536279\pi$
$422$	0	0
$423$	−28.6491	−1.39297
$424$	0	0
$425$	18.4217	0.893586
$426$	0	0
$427$	1.17020	0.0566299
$428$	0	0
$429$	−0.0675528	−0.00326148
$430$	0	0
$431$	−17.4631	−0.841170	−0.420585	−	0.907253i	$-0.638175\pi$
$431$	−17.4631	−0.841170	−0.420585	+	0.907253i	$0.638175\pi$
$432$	0	0
$433$	30.5066	1.46605	0.733026	−	0.680201i	$-0.238107\pi$
$433$	30.5066	1.46605	0.733026	+	0.680201i	$0.238107\pi$
$434$	0	0
$435$	−0.225178	−0.0107964
$436$	0	0
$437$	−1.20850	−0.0578105
$438$	0	0
$439$	−8.02657	−0.383087	−0.191544	−	0.981484i	$-0.561349\pi$
$439$	−8.02657	−0.383087	−0.191544	+	0.981484i	$0.561349\pi$
$440$	0	0
$441$	−2.99544	−0.142640
$442$	0	0
$443$	−30.2096	−1.43530	−0.717650	−	0.696404i	$-0.754782\pi$
$443$	−30.2096	−1.43530	−0.717650	+	0.696404i	$0.754782\pi$
$444$	0	0
$445$	17.2292	0.816744
$446$	0	0
$447$	0.192395	0.00909999
$448$	0	0
$449$	−8.74051	−0.412490	−0.206245	−	0.978500i	$-0.566124\pi$
$449$	−8.74051	−0.412490	−0.206245	+	0.978500i	$0.566124\pi$
$450$	0	0
$451$	−3.36839	−0.158611
$452$	0	0
$453$	0.750940	0.0352822
$454$	0	0
$455$	1.58844	0.0744673
$456$	0	0
$457$	−23.1668	−1.08370	−0.541848	−	0.840476i	$-0.682275\pi$
$457$	−23.1668	−1.08370	−0.541848	+	0.840476i	$0.682275\pi$
$458$	0	0
$459$	3.01228	0.140601
$460$	0	0
$461$	19.3541	0.901411	0.450705	−	0.892673i	$-0.351173\pi$
$461$	19.3541	0.901411	0.450705	+	0.892673i	$0.351173\pi$
$462$	0	0
$463$	30.8365	1.43309	0.716547	−	0.697539i	$-0.245721\pi$
$463$	30.8365	1.43309	0.716547	+	0.697539i	$0.245721\pi$
$464$	0	0
$465$	−0.985388	−0.0456963
$466$	0	0
$467$	39.7469	1.83927	0.919633	−	0.392780i	$-0.128486\pi$
$467$	39.7469	1.83927	0.919633	+	0.392780i	$0.128486\pi$
$468$	0	0
$469$	−0.996339	−0.0460067
$470$	0	0
$471$	0.978861	0.0451035
$472$	0	0
$473$	−7.73439	−0.355628
$474$	0	0
$475$	1.05451	0.0483841
$476$	0	0
$477$	11.0896	0.507759
$478$	0	0
$479$	−13.9110	−0.635608	−0.317804	−	0.948156i	$-0.602945\pi$
$479$	−13.9110	−0.635608	−0.317804	+	0.948156i	$0.602945\pi$
$480$	0	0
$481$	−3.75566	−0.171243
$482$	0	0
$483$	0.191753	0.00872504
$484$	0	0
$485$	−18.4229	−0.836539
$486$	0	0
$487$	−13.3680	−0.605763	−0.302881	−	0.953028i	$-0.597949\pi$
$487$	−13.3680	−0.605763	−0.302881	+	0.953028i	$0.597949\pi$
$488$	0	0
$489$	−0.740322	−0.0334785
$490$	0	0
$491$	−43.1700	−1.94824	−0.974119	−	0.226038i	$-0.927423\pi$
$491$	−43.1700	−1.94824	−0.974119	+	0.226038i	$0.927423\pi$
$492$	0	0
$493$	−15.6078	−0.702939
$494$	0	0
$495$	4.75808	0.213860
$496$	0	0
$497$	6.57629	0.294987
$498$	0	0
$499$	6.33655	0.283663	0.141831	−	0.989891i	$-0.454701\pi$
$499$	6.33655	0.283663	0.141831	+	0.989891i	$0.454701\pi$
$500$	0	0
$501$	−0.139404	−0.00622812
$502$	0	0
$503$	18.9594	0.845356	0.422678	−	0.906280i	$-0.361090\pi$
$503$	18.9594	0.845356	0.422678	+	0.906280i	$0.361090\pi$
$504$	0	0
$505$	28.6863	1.27652
$506$	0	0
$507$	0.0675528	0.00300013
$508$	0	0
$509$	0.411693	0.0182480	0.00912398	−	0.999958i	$-0.497096\pi$
$509$	0.411693	0.0182480	0.00912398	+	0.999958i	$0.497096\pi$
$510$	0	0
$511$	−4.05427	−0.179350
$512$	0	0
$513$	0.172430	0.00761299
$514$	0	0
$515$	−7.67004	−0.337982
$516$	0	0
$517$	9.56426	0.420635
$518$	0	0
$519$	−0.191316	−0.00839785
$520$	0	0
$521$	37.7432	1.65356	0.826780	−	0.562525i	$-0.190170\pi$
$521$	37.7432	1.65356	0.826780	+	0.562525i	$0.190170\pi$
$522$	0	0
$523$	−3.56070	−0.155698	−0.0778492	−	0.996965i	$-0.524805\pi$
$523$	−3.56070	−0.155698	−0.0778492	+	0.996965i	$0.524805\pi$
$524$	0	0
$525$	−0.167318	−0.00730236
$526$	0	0
$527$	−68.3004	−2.97521
$528$	0	0
$529$	−14.9426	−0.649678
$530$	0	0
$531$	−44.8081	−1.94451
$532$	0	0
$533$	3.36839	0.145901
$534$	0	0
$535$	27.3753	1.18354
$536$	0	0
$537$	−1.47227	−0.0635329
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	3.33131	0.143224	0.0716120	−	0.997433i	$-0.477186\pi$
$541$	3.33131	0.143224	0.0716120	+	0.997433i	$0.477186\pi$
$542$	0	0
$543$	1.48042	0.0635309
$544$	0	0
$545$	2.03414	0.0871330
$546$	0	0
$547$	37.5290	1.60463	0.802313	−	0.596904i	$-0.203603\pi$
$547$	37.5290	1.60463	0.802313	+	0.596904i	$0.203603\pi$
$548$	0	0
$549$	−3.50526	−0.149601
$550$	0	0
$551$	−0.893428	−0.0380613
$552$	0	0
$553$	8.23658	0.350255
$554$	0	0
$555$	−0.402996	−0.0171062
$556$	0	0
$557$	−40.8148	−1.72938	−0.864689	−	0.502308i	$-0.832484\pi$
$557$	−40.8148	−1.72938	−0.864689	+	0.502308i	$0.832484\pi$
$558$	0	0
$559$	7.73439	0.327130
$560$	0	0
$561$	−0.502429	−0.0212125
$562$	0	0
$563$	−17.3523	−0.731313	−0.365657	−	0.930750i	$-0.619156\pi$
$563$	−17.3523	−0.731313	−0.365657	+	0.930750i	$0.619156\pi$
$564$	0	0
$565$	8.23309	0.346369
$566$	0	0
$567$	8.95895	0.376241
$568$	0	0
$569$	19.5388	0.819109	0.409554	−	0.912286i	$-0.365684\pi$
$569$	19.5388	0.819109	0.409554	+	0.912286i	$0.365684\pi$
$570$	0	0
$571$	−0.759394	−0.0317797	−0.0158898	−	0.999874i	$-0.505058\pi$
$571$	−0.759394	−0.0317797	−0.0158898	+	0.999874i	$0.505058\pi$
$572$	0	0
$573$	1.38233	0.0577477
$574$	0	0
$575$	−7.03068	−0.293200
$576$	0	0
$577$	−12.8918	−0.536693	−0.268347	−	0.963322i	$-0.586477\pi$
$577$	−12.8918	−0.536693	−0.268347	+	0.963322i	$0.586477\pi$
$578$	0	0
$579$	−0.0364785	−0.00151600
$580$	0	0
$581$	17.0373	0.706827
$582$	0	0
$583$	−3.70217	−0.153328
$584$	0	0
$585$	−4.75808	−0.196722
$586$	0	0
$587$	30.0798	1.24153	0.620763	−	0.783998i	$-0.286823\pi$
$587$	30.0798	1.24153	0.620763	+	0.783998i	$0.286823\pi$
$588$	0	0
$589$	−3.90969	−0.161096
$590$	0	0
$591$	0.867637	0.0356898
$592$	0	0
$593$	−42.4846	−1.74463	−0.872316	−	0.488942i	$-0.837383\pi$
$593$	−42.4846	−1.74463	−0.872316	+	0.488942i	$0.837383\pi$
$594$	0	0
$595$	11.8141	0.484333
$596$	0	0
$597$	0.860848	0.0352322
$598$	0	0
$599$	−37.0547	−1.51401	−0.757006	−	0.653408i	$-0.773339\pi$
$599$	−37.0547	−1.51401	−0.757006	+	0.653408i	$0.773339\pi$
$600$	0	0
$601$	−20.8365	−0.849940	−0.424970	−	0.905207i	$-0.639715\pi$
$601$	−20.8365	−0.849940	−0.424970	+	0.905207i	$0.639715\pi$
$602$	0	0
$603$	2.98447	0.121537
$604$	0	0
$605$	−1.58844	−0.0645794
$606$	0	0
$607$	−17.2473	−0.700045	−0.350022	−	0.936741i	$-0.613826\pi$
$607$	−17.2473	−0.700045	−0.350022	+	0.936741i	$0.613826\pi$
$608$	0	0
$609$	0.141760	0.00574440
$610$	0	0
$611$	−9.56426	−0.386928
$612$	0	0
$613$	35.8001	1.44595	0.722975	−	0.690874i	$-0.242774\pi$
$613$	35.8001	1.44595	0.722975	+	0.690874i	$0.242774\pi$
$614$	0	0
$615$	0.361441	0.0145747
$616$	0	0
$617$	38.3363	1.54336	0.771680	−	0.636011i	$-0.219417\pi$
$617$	38.3363	1.54336	0.771680	+	0.636011i	$0.219417\pi$
$618$	0	0
$619$	29.1047	1.16982	0.584908	−	0.811099i	$-0.301130\pi$
$619$	29.1047	1.16982	0.584908	+	0.811099i	$0.301130\pi$
$620$	0	0
$621$	−1.14964	−0.0461335
$622$	0	0
$623$	−10.8466	−0.434561
$624$	0	0
$625$	−6.48096	−0.259239
$626$	0	0
$627$	−0.0287603	−0.00114857
$628$	0	0
$629$	−27.9329	−1.11376
$630$	0	0
$631$	−35.0433	−1.39505	−0.697527	−	0.716559i	$-0.745716\pi$
$631$	−35.0433	−1.39505	−0.697527	+	0.716559i	$0.745716\pi$
$632$	0	0
$633$	1.19676	0.0475669
$634$	0	0
$635$	−11.8031	−0.468392
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−19.6989	−0.779275
$640$	0	0
$641$	−14.2553	−0.563050	−0.281525	−	0.959554i	$-0.590840\pi$
$641$	−14.2553	−0.563050	−0.281525	+	0.959554i	$0.590840\pi$
$642$	0	0
$643$	7.03655	0.277494	0.138747	−	0.990328i	$-0.455692\pi$
$643$	7.03655	0.277494	0.138747	+	0.990328i	$0.455692\pi$
$644$	0	0
$645$	0.829929	0.0326784
$646$	0	0
$647$	11.8246	0.464874	0.232437	−	0.972611i	$-0.425330\pi$
$647$	11.8246	0.464874	0.232437	+	0.972611i	$0.425330\pi$
$648$	0	0
$649$	14.9588	0.587184
$650$	0	0
$651$	0.620348	0.0243134
$652$	0	0
$653$	−30.0220	−1.17485	−0.587426	−	0.809278i	$-0.699858\pi$
$653$	−30.0220	−1.17485	−0.587426	+	0.809278i	$0.699858\pi$
$654$	0	0
$655$	−10.6437	−0.415885
$656$	0	0
$657$	12.1443	0.473794
$658$	0	0
$659$	−1.43416	−0.0558670	−0.0279335	−	0.999610i	$-0.508893\pi$
$659$	−1.43416	−0.0558670	−0.0279335	+	0.999610i	$0.508893\pi$
$660$	0	0
$661$	32.9782	1.28270	0.641352	−	0.767247i	$-0.278374\pi$
$661$	32.9782	1.28270	0.641352	+	0.767247i	$0.278374\pi$
$662$	0	0
$663$	0.502429	0.0195127
$664$	0	0
$665$	0.676271	0.0262247
$666$	0	0
$667$	5.95673	0.230646
$668$	0	0
$669$	−1.36883	−0.0529220
$670$	0	0
$671$	1.17020	0.0451751
$672$	0	0
$673$	−23.7876	−0.916946	−0.458473	−	0.888708i	$-0.651603\pi$
$673$	−23.7876	−0.916946	−0.458473	+	0.888708i	$0.651603\pi$
$674$	0	0
$675$	1.00315	0.0386111
$676$	0	0
$677$	8.90969	0.342427	0.171214	−	0.985234i	$-0.445231\pi$
$677$	8.90969	0.342427	0.171214	+	0.985234i	$0.445231\pi$
$678$	0	0
$679$	11.5981	0.445093
$680$	0	0
$681$	1.17724	0.0451118
$682$	0	0
$683$	−20.7251	−0.793025	−0.396512	−	0.918029i	$-0.629780\pi$
$683$	−20.7251	−0.793025	−0.396512	+	0.918029i	$0.629780\pi$
$684$	0	0
$685$	20.4209	0.780242
$686$	0	0
$687$	1.47705	0.0563529
$688$	0	0
$689$	3.70217	0.141041
$690$	0	0
$691$	−31.5931	−1.20186	−0.600929	−	0.799302i	$-0.705203\pi$
$691$	−31.5931	−1.20186	−0.600929	+	0.799302i	$0.705203\pi$
$692$	0	0
$693$	−2.99544	−0.113787
$694$	0	0
$695$	−37.2665	−1.41360
$696$	0	0
$697$	25.0526	0.948937
$698$	0	0
$699$	0.352588	0.0133361
$700$	0	0
$701$	13.4436	0.507759	0.253880	−	0.967236i	$-0.418293\pi$
$701$	13.4436	0.507759	0.253880	+	0.967236i	$0.418293\pi$
$702$	0	0
$703$	−1.59895	−0.0603056
$704$	0	0
$705$	−1.02628	−0.0386520
$706$	0	0
$707$	−18.0594	−0.679192
$708$	0	0
$709$	4.67309	0.175502	0.0877508	−	0.996142i	$-0.472032\pi$
$709$	4.67309	0.175502	0.0877508	+	0.996142i	$0.472032\pi$
$710$	0	0
$711$	−24.6722	−0.925278
$712$	0	0
$713$	26.0669	0.976215
$714$	0	0
$715$	1.58844	0.0594044
$716$	0	0
$717$	0.969134	0.0361930
$718$	0	0
$719$	−18.9044	−0.705017	−0.352508	−	0.935809i	$-0.614671\pi$
$719$	−18.9044	−0.705017	−0.352508	+	0.935809i	$0.614671\pi$
$720$	0	0
$721$	4.82865	0.179828
$722$	0	0
$723$	−0.762339	−0.0283517
$724$	0	0
$725$	−5.19768	−0.193037
$726$	0	0
$727$	−36.2313	−1.34375	−0.671873	−	0.740667i	$-0.734510\pi$
$727$	−36.2313	−1.34375	−0.671873	+	0.740667i	$0.734510\pi$
$728$	0	0
$729$	−26.7539	−0.990885
$730$	0	0
$731$	57.5251	2.12764
$732$	0	0
$733$	11.3557	0.419432	0.209716	−	0.977762i	$-0.432746\pi$
$733$	11.3557	0.419432	0.209716	+	0.977762i	$0.432746\pi$
$734$	0	0
$735$	−0.107304	−0.00395796
$736$	0	0
$737$	−0.996339	−0.0367006
$738$	0	0
$739$	−1.19225	−0.0438575	−0.0219288	−	0.999760i	$-0.506981\pi$
$739$	−1.19225	−0.0438575	−0.0219288	+	0.999760i	$0.506981\pi$
$740$	0	0
$741$	0.0287603	0.00105653
$742$	0	0
$743$	30.4947	1.11874	0.559372	−	0.828917i	$-0.311042\pi$
$743$	30.4947	1.11874	0.559372	+	0.828917i	$0.311042\pi$
$744$	0	0
$745$	−4.52400	−0.165747
$746$	0	0
$747$	−51.0342	−1.86724
$748$	0	0
$749$	−17.2340	−0.629718
$750$	0	0
$751$	−3.79238	−0.138386	−0.0691930	−	0.997603i	$-0.522042\pi$
$751$	−3.79238	−0.138386	−0.0691930	+	0.997603i	$0.522042\pi$
$752$	0	0
$753$	−0.457957	−0.0166889
$754$	0	0
$755$	−17.6577	−0.642628
$756$	0	0
$757$	−20.5063	−0.745312	−0.372656	−	0.927969i	$-0.621553\pi$
$757$	−20.5063	−0.745312	−0.372656	+	0.927969i	$0.621553\pi$
$758$	0	0
$759$	0.191753	0.00696018
$760$	0	0
$761$	−18.1855	−0.659224	−0.329612	−	0.944117i	$-0.606918\pi$
$761$	−18.1855	−0.659224	−0.329612	+	0.944117i	$0.606918\pi$
$762$	0	0
$763$	−1.28059	−0.0463604
$764$	0	0
$765$	−35.3885	−1.27948
$766$	0	0
$767$	−14.9588	−0.540130
$768$	0	0
$769$	−26.7490	−0.964593	−0.482296	−	0.876008i	$-0.660197\pi$
$769$	−26.7490	−0.964593	−0.482296	+	0.876008i	$0.660197\pi$
$770$	0	0
$771$	1.43432	0.0516558
$772$	0	0
$773$	−30.3256	−1.09074	−0.545368	−	0.838197i	$-0.683610\pi$
$773$	−30.3256	−1.09074	−0.545368	+	0.838197i	$0.683610\pi$
$774$	0	0
$775$	−22.7453	−0.817036
$776$	0	0
$777$	0.253705	0.00910162
$778$	0	0
$779$	1.43408	0.0513811
$780$	0	0
$781$	6.57629	0.235318
$782$	0	0
$783$	−0.849913	−0.0303734
$784$	0	0
$785$	−23.0170	−0.821513
$786$	0	0
$787$	−19.3139	−0.688468	−0.344234	−	0.938884i	$-0.611861\pi$
$787$	−19.3139	−0.688468	−0.344234	+	0.938884i	$0.611861\pi$
$788$	0	0
$789$	−0.147409	−0.00524788
$790$	0	0
$791$	−5.18312	−0.184291
$792$	0	0
$793$	−1.17020	−0.0415550
$794$	0	0
$795$	0.397257	0.0140892
$796$	0	0
$797$	7.11285	0.251950	0.125975	−	0.992033i	$-0.459794\pi$
$797$	7.11285	0.251950	0.125975	+	0.992033i	$0.459794\pi$
$798$	0	0
$799$	−71.1348	−2.51657
$800$	0	0
$801$	32.4904	1.14799
$802$	0	0
$803$	−4.05427	−0.143072
$804$	0	0
$805$	−4.50889	−0.158917
$806$	0	0
$807$	−1.02015	−0.0359111
$808$	0	0
$809$	25.4125	0.893455	0.446727	−	0.894670i	$-0.352589\pi$
$809$	25.4125	0.893455	0.446727	+	0.894670i	$0.352589\pi$
$810$	0	0
$811$	41.8983	1.47125	0.735624	−	0.677390i	$-0.236889\pi$
$811$	41.8983	1.47125	0.735624	+	0.677390i	$0.236889\pi$
$812$	0	0
$813$	−0.787905	−0.0276330
$814$	0	0
$815$	17.4080	0.609776
$816$	0	0
$817$	3.29288	0.115203
$818$	0	0
$819$	2.99544	0.104669
$820$	0	0
$821$	49.9666	1.74385	0.871923	−	0.489643i	$-0.162873\pi$
$821$	49.9666	1.74385	0.871923	+	0.489643i	$0.162873\pi$
$822$	0	0
$823$	5.98652	0.208677	0.104338	−	0.994542i	$-0.466727\pi$
$823$	5.98652	0.208677	0.104338	+	0.994542i	$0.466727\pi$
$824$	0	0
$825$	−0.167318	−0.00582527
$826$	0	0
$827$	35.0200	1.21776	0.608881	−	0.793261i	$-0.291618\pi$
$827$	35.0200	1.21776	0.608881	+	0.793261i	$0.291618\pi$
$828$	0	0
$829$	−36.1595	−1.25587	−0.627936	−	0.778265i	$-0.716100\pi$
$829$	−36.1595	−1.25587	−0.627936	+	0.778265i	$0.716100\pi$
$830$	0	0
$831$	0.327198	0.0113504
$832$	0	0
$833$	−7.43757	−0.257696
$834$	0	0
$835$	3.27796	0.113439
$836$	0	0
$837$	−3.71926	−0.128556
$838$	0	0
$839$	37.7191	1.30221	0.651104	−	0.758989i	$-0.274306\pi$
$839$	37.7191	1.30221	0.651104	+	0.758989i	$0.274306\pi$
$840$	0	0
$841$	−24.5963	−0.848147
$842$	0	0
$843$	0.672202	0.0231519
$844$	0	0
$845$	−1.58844	−0.0546441
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−0.923528	−0.0316954
$850$	0	0
$851$	10.6606	0.365442
$852$	0	0
$853$	23.1562	0.792855	0.396427	−	0.918066i	$-0.370250\pi$
$853$	23.1562	0.792855	0.396427	+	0.918066i	$0.370250\pi$
$854$	0	0
$855$	−2.02573	−0.0692784
$856$	0	0
$857$	1.79943	0.0614675	0.0307337	−	0.999528i	$-0.490216\pi$
$857$	1.79943	0.0614675	0.0307337	+	0.999528i	$0.490216\pi$
$858$	0	0
$859$	−20.5436	−0.700937	−0.350469	−	0.936574i	$-0.613978\pi$
$859$	−20.5436	−0.700937	−0.350469	+	0.936574i	$0.613978\pi$
$860$	0	0
$861$	−0.227544	−0.00775469
$862$	0	0
$863$	−31.9178	−1.08649	−0.543247	−	0.839573i	$-0.682805\pi$
$863$	−31.9178	−1.08649	−0.543247	+	0.839573i	$0.682805\pi$
$864$	0	0
$865$	4.49863	0.152958
$866$	0	0
$867$	2.58845	0.0879083
$868$	0	0
$869$	8.23658	0.279407
$870$	0	0
$871$	0.996339	0.0337597
$872$	0	0
$873$	−34.7413	−1.17581
$874$	0	0
$875$	11.8765	0.401501
$876$	0	0
$877$	24.1846	0.816655	0.408328	−	0.912835i	$-0.366112\pi$
$877$	24.1846	0.816655	0.408328	+	0.912835i	$0.366112\pi$
$878$	0	0
$879$	−1.02194	−0.0344693
$880$	0	0
$881$	4.17866	0.140782	0.0703912	−	0.997519i	$-0.477575\pi$
$881$	4.17866	0.140782	0.0703912	+	0.997519i	$0.477575\pi$
$882$	0	0
$883$	27.2532	0.917143	0.458572	−	0.888657i	$-0.348361\pi$
$883$	27.2532	0.917143	0.458572	+	0.888657i	$0.348361\pi$
$884$	0	0
$885$	−1.60513	−0.0539560
$886$	0	0
$887$	21.8146	0.732464	0.366232	−	0.930524i	$-0.380648\pi$
$887$	21.8146	0.732464	0.366232	+	0.930524i	$0.380648\pi$
$888$	0	0
$889$	7.43061	0.249215
$890$	0	0
$891$	8.95895	0.300136
$892$	0	0
$893$	−4.07193	−0.136262
$894$	0	0
$895$	34.6190	1.15718
$896$	0	0
$897$	−0.191753	−0.00640243
$898$	0	0
$899$	19.2709	0.642721
$900$	0	0
$901$	27.5351	0.917328
$902$	0	0
$903$	−0.522480	−0.0173870
$904$	0	0
$905$	−34.8107	−1.15715
$906$	0	0
$907$	46.3762	1.53990	0.769948	−	0.638107i	$-0.220282\pi$
$907$	46.3762	1.53990	0.769948	+	0.638107i	$0.220282\pi$
$908$	0	0
$909$	54.0957	1.79424
$910$	0	0
$911$	26.2699	0.870359	0.435180	−	0.900344i	$-0.356685\pi$
$911$	26.2699	0.870359	0.435180	+	0.900344i	$0.356685\pi$
$912$	0	0
$913$	17.0373	0.563853
$914$	0	0
$915$	−0.125567	−0.00415111
$916$	0	0
$917$	6.70073	0.221278
$918$	0	0
$919$	38.5699	1.27230	0.636151	−	0.771565i	$-0.280525\pi$
$919$	38.5699	1.27230	0.636151	+	0.771565i	$0.280525\pi$
$920$	0	0
$921$	0.275951	0.00909290
$922$	0	0
$923$	−6.57629	−0.216461
$924$	0	0
$925$	−9.30220	−0.305854
$926$	0	0
$927$	−14.4639	−0.475058
$928$	0	0
$929$	−27.1241	−0.889914	−0.444957	−	0.895552i	$-0.646781\pi$
$929$	−27.1241	−0.889914	−0.444957	+	0.895552i	$0.646781\pi$
$930$	0	0
$931$	−0.425745	−0.0139532
$932$	0	0
$933$	−0.156435	−0.00512145
$934$	0	0
$935$	11.8141	0.386364
$936$	0	0
$937$	−30.1625	−0.985367	−0.492684	−	0.870208i	$-0.663984\pi$
$937$	−30.1625	−0.985367	−0.492684	+	0.870208i	$0.663984\pi$
$938$	0	0
$939$	−2.06096	−0.0672568
$940$	0	0
$941$	11.2137	0.365556	0.182778	−	0.983154i	$-0.441491\pi$
$941$	11.2137	0.365556	0.182778	+	0.983154i	$0.441491\pi$
$942$	0	0
$943$	−9.56138	−0.311361
$944$	0	0
$945$	0.643333	0.0209276
$946$	0	0
$947$	−20.9812	−0.681798	−0.340899	−	0.940100i	$-0.610731\pi$
$947$	−20.9812	−0.681798	−0.340899	+	0.940100i	$0.610731\pi$
$948$	0	0
$949$	4.05427	0.131607
$950$	0	0
$951$	−0.0686373	−0.00222572
$952$	0	0
$953$	−43.4463	−1.40736	−0.703682	−	0.710515i	$-0.748462\pi$
$953$	−43.4463	−1.40736	−0.703682	+	0.710515i	$0.748462\pi$
$954$	0	0
$955$	−32.5042	−1.05181
$956$	0	0
$957$	0.141760	0.00458245
$958$	0	0
$959$	−12.8559	−0.415139
$960$	0	0
$961$	53.3305	1.72034
$962$	0	0
$963$	51.6234	1.66354
$964$	0	0
$965$	0.857759	0.0276123
$966$	0	0
$967$	−49.2041	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$967$	−49.2041	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$968$	0	0
$969$	0.213906	0.00687166
$970$	0	0
$971$	32.6157	1.04669	0.523344	−	0.852122i	$-0.324684\pi$
$971$	32.6157	1.04669	0.523344	+	0.852122i	$0.324684\pi$
$972$	0	0
$973$	23.4610	0.752126
$974$	0	0
$975$	0.167318	0.00535847
$976$	0	0
$977$	−57.6091	−1.84308	−0.921539	−	0.388287i	$-0.873067\pi$
$977$	−57.6091	−1.84308	−0.921539	+	0.388287i	$0.873067\pi$
$978$	0	0
$979$	−10.8466	−0.346659
$980$	0	0
$981$	3.83592	0.122472
$982$	0	0
$983$	16.9209	0.539693	0.269847	−	0.962903i	$-0.413027\pi$
$983$	16.9209	0.539693	0.269847	+	0.962903i	$0.413027\pi$
$984$	0	0
$985$	−20.4017	−0.650052
$986$	0	0
$987$	0.646092	0.0205653
$988$	0	0
$989$	−21.9545	−0.698113
$990$	0	0
$991$	−47.1798	−1.49872	−0.749358	−	0.662165i	$-0.769638\pi$
$991$	−47.1798	−1.49872	−0.749358	+	0.662165i	$0.769638\pi$
$992$	0	0
$993$	1.36892	0.0434415
$994$	0	0
$995$	−20.2421	−0.641716
$996$	0	0
$997$	58.7980	1.86215	0.931076	−	0.364826i	$-0.118872\pi$
$997$	58.7980	1.86215	0.931076	+	0.364826i	$0.118872\pi$
$998$	0	0
$999$	−1.52107	−0.0481246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.h.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.h.1.5	✓	9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0675528 q^{3} -1.58844 q^{5} +1.00000 q^{7} -2.99544 q^{9} +O(q^{10})\)	\(q+0.0675528 q^{3} -1.58844 q^{5} +1.00000 q^{7} -2.99544 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.107304 q^{15} -7.43757 q^{17} -0.425745 q^{19} +0.0675528 q^{21} +2.83856 q^{23} -2.47685 q^{25} -0.405009 q^{27} +2.09851 q^{29} +9.18316 q^{31} +0.0675528 q^{33} -1.58844 q^{35} +3.75566 q^{37} -0.0675528 q^{39} -3.36839 q^{41} -7.73439 q^{43} +4.75808 q^{45} +9.56426 q^{47} +1.00000 q^{49} -0.502429 q^{51} -3.70217 q^{53} -1.58844 q^{55} -0.0287603 q^{57} +14.9588 q^{59} +1.17020 q^{61} -2.99544 q^{63} +1.58844 q^{65} -0.996339 q^{67} +0.191753 q^{69} +6.57629 q^{71} -4.05427 q^{73} -0.167318 q^{75} +1.00000 q^{77} +8.23658 q^{79} +8.95895 q^{81} +17.0373 q^{83} +11.8141 q^{85} +0.141760 q^{87} -10.8466 q^{89} -1.00000 q^{91} +0.620348 q^{93} +0.676271 q^{95} +11.5981 q^{97} -2.99544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9	\( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + 5 * q^17 + 10 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^25 + 27 * q^27 + 14 * q^29 + 11 * q^31 + 3 * q^33 - 3 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 + 10 * q^47 + 9 * q^49 - 15 * q^51 + 21 * q^53 - 8 * q^57 + 23 * q^59 + 34 * q^61 + 20 * q^63 + 10 * q^67 - 16 * q^69 + 4 * q^71 + 9 * q^73 + 30 * q^75 + 9 * q^77 - 34 * q^79 + 69 * q^81 + 15 * q^83 + 5 * q^85 + 39 * q^87 - 9 * q^91 + 3 * q^93 - 64 * q^95 + 15 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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