LMFDB - Embedded newform 6028.2.a.c.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0141418 q^{3} -1.21436 q^{5} -2.15622 q^{7} -2.99980 q^{9} +O(q^{10})$	$q+0.0141418 q^{3} -1.21436 q^{5} -2.15622 q^{7} -2.99980 q^{9} +1.00000 q^{11} -3.23007 q^{13} -0.0171733 q^{15} +6.94720 q^{17} +3.01957 q^{19} -0.0304929 q^{21} +5.18957 q^{23} -3.52532 q^{25} -0.0848482 q^{27} +5.13447 q^{29} -4.75259 q^{31} +0.0141418 q^{33} +2.61843 q^{35} +8.56757 q^{37} -0.0456791 q^{39} -5.14934 q^{41} -0.616718 q^{43} +3.64285 q^{45} -2.93661 q^{47} -2.35073 q^{49} +0.0982461 q^{51} +3.61560 q^{53} -1.21436 q^{55} +0.0427023 q^{57} -2.53054 q^{59} -9.02436 q^{61} +6.46822 q^{63} +3.92248 q^{65} -0.0146546 q^{67} +0.0733901 q^{69} +15.3901 q^{71} +11.3064 q^{73} -0.0498545 q^{75} -2.15622 q^{77} +2.11444 q^{79} +8.99820 q^{81} -3.55256 q^{83} -8.43642 q^{85} +0.0726109 q^{87} -3.82667 q^{89} +6.96473 q^{91} -0.0672103 q^{93} -3.66686 q^{95} -14.7336 q^{97} -2.99980 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * 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
$3$	0.0141418	0.00816479	0.00408240	−	0.999992i	$-0.498701\pi$
$3$	0.0141418	0.00816479	0.00408240	+	0.999992i	$0.498701\pi$
$4$	0	0
$5$	−1.21436	−0.543080	−0.271540	−	0.962427i	$-0.587533\pi$
$5$	−1.21436	−0.543080	−0.271540	+	0.962427i	$0.587533\pi$
$6$	0	0
$7$	−2.15622	−0.814973	−0.407487	−	0.913211i	$-0.633595\pi$
$7$	−2.15622	−0.814973	−0.407487	+	0.913211i	$0.633595\pi$
$8$	0	0
$9$	−2.99980	−0.999933
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.23007	−0.895860	−0.447930	−	0.894069i	$-0.647839\pi$
$13$	−3.23007	−0.895860	−0.447930	+	0.894069i	$0.647839\pi$
$14$	0	0
$15$	−0.0171733	−0.00443413
$16$	0	0
$17$	6.94720	1.68494	0.842471	−	0.538741i	$-0.181100\pi$
$17$	6.94720	1.68494	0.842471	+	0.538741i	$0.181100\pi$
$18$	0	0
$19$	3.01957	0.692737	0.346369	−	0.938098i	$-0.387415\pi$
$19$	3.01957	0.692737	0.346369	+	0.938098i	$0.387415\pi$
$20$	0	0
$21$	−0.0304929	−0.00665409
$22$	0	0
$23$	5.18957	1.08210	0.541050	−	0.840990i	$-0.318027\pi$
$23$	5.18957	1.08210	0.541050	+	0.840990i	$0.318027\pi$
$24$	0	0
$25$	−3.52532	−0.705064
$26$	0	0
$27$	−0.0848482	−0.0163290
$28$	0	0
$29$	5.13447	0.953448	0.476724	−	0.879053i	$-0.341824\pi$
$29$	5.13447	0.953448	0.476724	+	0.879053i	$0.341824\pi$
$30$	0	0
$31$	−4.75259	−0.853590	−0.426795	−	0.904348i	$-0.640357\pi$
$31$	−4.75259	−0.853590	−0.426795	+	0.904348i	$0.640357\pi$
$32$	0	0
$33$	0.0141418	0.00246178
$34$	0	0
$35$	2.61843	0.442596
$36$	0	0
$37$	8.56757	1.40850	0.704250	−	0.709952i	$-0.251284\pi$
$37$	8.56757	1.40850	0.704250	+	0.709952i	$0.251284\pi$
$38$	0	0
$39$	−0.0456791	−0.00731451
$40$	0	0
$41$	−5.14934	−0.804192	−0.402096	−	0.915597i	$-0.631718\pi$
$41$	−5.14934	−0.804192	−0.402096	+	0.915597i	$0.631718\pi$
$42$	0	0
$43$	−0.616718	−0.0940486	−0.0470243	−	0.998894i	$-0.514974\pi$
$43$	−0.616718	−0.0940486	−0.0470243	+	0.998894i	$0.514974\pi$
$44$	0	0
$45$	3.64285	0.543044
$46$	0	0
$47$	−2.93661	−0.428348	−0.214174	−	0.976795i	$-0.568706\pi$
$47$	−2.93661	−0.428348	−0.214174	+	0.976795i	$0.568706\pi$
$48$	0	0
$49$	−2.35073	−0.335818
$50$	0	0
$51$	0.0982461	0.0137572
$52$	0	0
$53$	3.61560	0.496640	0.248320	−	0.968678i	$-0.420122\pi$
$53$	3.61560	0.496640	0.248320	+	0.968678i	$0.420122\pi$
$54$	0	0
$55$	−1.21436	−0.163745
$56$	0	0
$57$	0.0427023	0.00565606
$58$	0	0
$59$	−2.53054	−0.329449	−0.164724	−	0.986340i	$-0.552673\pi$
$59$	−2.53054	−0.329449	−0.164724	+	0.986340i	$0.552673\pi$
$60$	0	0
$61$	−9.02436	−1.15545	−0.577726	−	0.816231i	$-0.696060\pi$
$61$	−9.02436	−1.15545	−0.577726	+	0.816231i	$0.696060\pi$
$62$	0	0
$63$	6.46822	0.814919
$64$	0	0
$65$	3.92248	0.486523
$66$	0	0
$67$	−0.0146546	−0.00179035	−0.000895173	−	1.00000i	$-0.500285\pi$
$67$	−0.0146546	−0.00179035	−0.000895173		1.00000i	$0.500285\pi$
$68$	0	0
$69$	0.0733901	0.00883513
$70$	0	0
$71$	15.3901	1.82647	0.913234	−	0.407436i	$-0.133577\pi$
$71$	15.3901	1.82647	0.913234	+	0.407436i	$0.133577\pi$
$72$	0	0
$73$	11.3064	1.32332	0.661659	−	0.749805i	$-0.269853\pi$
$73$	11.3064	1.32332	0.661659	+	0.749805i	$0.269853\pi$
$74$	0	0
$75$	−0.0498545	−0.00575670
$76$	0	0
$77$	−2.15622	−0.245724
$78$	0	0
$79$	2.11444	0.237893	0.118946	−	0.992901i	$-0.462048\pi$
$79$	2.11444	0.237893	0.118946	+	0.992901i	$0.462048\pi$
$80$	0	0
$81$	8.99820	0.999800
$82$	0	0
$83$	−3.55256	−0.389944	−0.194972	−	0.980809i	$-0.562462\pi$
$83$	−3.55256	−0.389944	−0.194972	+	0.980809i	$0.562462\pi$
$84$	0	0
$85$	−8.43642	−0.915058
$86$	0	0
$87$	0.0726109	0.00778470
$88$	0	0
$89$	−3.82667	−0.405626	−0.202813	−	0.979217i	$-0.565008\pi$
$89$	−3.82667	−0.405626	−0.202813	+	0.979217i	$0.565008\pi$
$90$	0	0
$91$	6.96473	0.730102
$92$	0	0
$93$	−0.0672103	−0.00696938
$94$	0	0
$95$	−3.66686	−0.376212
$96$	0	0
$97$	−14.7336	−1.49597	−0.747987	−	0.663713i	$-0.768980\pi$
$97$	−14.7336	−1.49597	−0.747987	+	0.663713i	$0.768980\pi$
$98$	0	0
$99$	−2.99980	−0.301491
$100$	0	0
$101$	−13.3859	−1.33195	−0.665976	−	0.745973i	$-0.731985\pi$
$101$	−13.3859	−1.33195	−0.665976	+	0.745973i	$0.731985\pi$
$102$	0	0
$103$	−5.11866	−0.504357	−0.252178	−	0.967681i	$-0.581147\pi$
$103$	−5.11866	−0.504357	−0.252178	+	0.967681i	$0.581147\pi$
$104$	0	0
$105$	0.0370294	0.00361370
$106$	0	0
$107$	−0.848815	−0.0820581	−0.0410290	−	0.999158i	$-0.513064\pi$
$107$	−0.848815	−0.0820581	−0.0410290	+	0.999158i	$0.513064\pi$
$108$	0	0
$109$	−11.6170	−1.11271	−0.556354	−	0.830946i	$-0.687800\pi$
$109$	−11.6170	−1.11271	−0.556354	+	0.830946i	$0.687800\pi$
$110$	0	0
$111$	0.121161	0.0115001
$112$	0	0
$113$	−2.24589	−0.211275	−0.105638	−	0.994405i	$-0.533688\pi$
$113$	−2.24589	−0.211275	−0.105638	+	0.994405i	$0.533688\pi$
$114$	0	0
$115$	−6.30203	−0.587667
$116$	0	0
$117$	9.68956	0.895800
$118$	0	0
$119$	−14.9797	−1.37318
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.0728212	−0.00656606
$124$	0	0
$125$	10.3528	0.925986
$126$	0	0
$127$	−5.45327	−0.483899	−0.241950	−	0.970289i	$-0.577787\pi$
$127$	−5.45327	−0.483899	−0.241950	+	0.970289i	$0.577787\pi$
$128$	0	0
$129$	−0.00872152	−0.000767887 0
$130$	0	0
$131$	−6.79211	−0.593430	−0.296715	−	0.954966i	$-0.595891\pi$
$131$	−6.79211	−0.593430	−0.296715	+	0.954966i	$0.595891\pi$
$132$	0	0
$133$	−6.51085	−0.564563
$134$	0	0
$135$	0.103037	0.00886797
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	−22.4382	−1.90318	−0.951590	−	0.307370i	$-0.900551\pi$
$139$	−22.4382	−1.90318	−0.951590	+	0.307370i	$0.900551\pi$
$140$	0	0
$141$	−0.0415290	−0.00349737
$142$	0	0
$143$	−3.23007	−0.270112
$144$	0	0
$145$	−6.23512	−0.517798
$146$	0	0
$147$	−0.0332436	−0.00274189
$148$	0	0
$149$	−0.0134206	−0.00109946	−0.000549728	−	1.00000i	$-0.500175\pi$
$149$	−0.0134206	−0.00109946	−0.000549728		1.00000i	$0.500175\pi$
$150$	0	0
$151$	−8.94447	−0.727891	−0.363945	−	0.931420i	$-0.618571\pi$
$151$	−8.94447	−0.727891	−0.363945	+	0.931420i	$0.618571\pi$
$152$	0	0
$153$	−20.8402	−1.68483
$154$	0	0
$155$	5.77137	0.463568
$156$	0	0
$157$	−22.7345	−1.81441	−0.907206	−	0.420686i	$-0.861789\pi$
$157$	−22.7345	−1.81441	−0.907206	+	0.420686i	$0.861789\pi$
$158$	0	0
$159$	0.0511312	0.00405496
$160$	0	0
$161$	−11.1898	−0.881883
$162$	0	0
$163$	−2.31933	−0.181664	−0.0908319	−	0.995866i	$-0.528953\pi$
$163$	−2.31933	−0.181664	−0.0908319	+	0.995866i	$0.528953\pi$
$164$	0	0
$165$	−0.0171733	−0.00133694
$166$	0	0
$167$	20.8804	1.61578	0.807888	−	0.589335i	$-0.200610\pi$
$167$	20.8804	1.61578	0.807888	+	0.589335i	$0.200610\pi$
$168$	0	0
$169$	−2.56665	−0.197435
$170$	0	0
$171$	−9.05811	−0.692691
$172$	0	0
$173$	4.48267	0.340811	0.170405	−	0.985374i	$-0.445492\pi$
$173$	4.48267	0.340811	0.170405	+	0.985374i	$0.445492\pi$
$174$	0	0
$175$	7.60136	0.574609
$176$	0	0
$177$	−0.0357865	−0.00268988
$178$	0	0
$179$	12.5897	0.940998	0.470499	−	0.882400i	$-0.344074\pi$
$179$	12.5897	0.940998	0.470499	+	0.882400i	$0.344074\pi$
$180$	0	0
$181$	15.9697	1.18702	0.593508	−	0.804828i	$-0.297743\pi$
$181$	15.9697	1.18702	0.593508	+	0.804828i	$0.297743\pi$
$182$	0	0
$183$	−0.127621	−0.00943402
$184$	0	0
$185$	−10.4041	−0.764927
$186$	0	0
$187$	6.94720	0.508029
$188$	0	0
$189$	0.182951	0.0133077
$190$	0	0
$191$	3.85977	0.279283	0.139642	−	0.990202i	$-0.455405\pi$
$191$	3.85977	0.279283	0.139642	+	0.990202i	$0.455405\pi$
$192$	0	0
$193$	−10.5967	−0.762767	−0.381384	−	0.924417i	$-0.624552\pi$
$193$	−10.5967	−0.762767	−0.381384	+	0.924417i	$0.624552\pi$
$194$	0	0
$195$	0.0554710	0.00397236
$196$	0	0
$197$	−21.0021	−1.49634	−0.748170	−	0.663507i	$-0.769067\pi$
$197$	−21.0021	−1.49634	−0.748170	+	0.663507i	$0.769067\pi$
$198$	0	0
$199$	−2.44268	−0.173157	−0.0865786	−	0.996245i	$-0.527593\pi$
$199$	−2.44268	−0.173157	−0.0865786	+	0.996245i	$0.527593\pi$
$200$	0	0
$201$	−0.000207243 0	−1.46178e−5 0
$202$	0	0
$203$	−11.0710	−0.777035
$204$	0	0
$205$	6.25317	0.436741
$206$	0	0
$207$	−15.5677	−1.08203
$208$	0	0
$209$	3.01957	0.208868
$210$	0	0
$211$	−13.4516	−0.926044	−0.463022	−	0.886347i	$-0.653235\pi$
$211$	−13.4516	−0.926044	−0.463022	+	0.886347i	$0.653235\pi$
$212$	0	0
$213$	0.217644	0.0149127
$214$	0	0
$215$	0.748919	0.0510759
$216$	0	0
$217$	10.2476	0.695653
$218$	0	0
$219$	0.159894	0.0108046
$220$	0	0
$221$	−22.4399	−1.50947
$222$	0	0
$223$	−2.35212	−0.157510	−0.0787548	−	0.996894i	$-0.525094\pi$
$223$	−2.35212	−0.157510	−0.0787548	+	0.996894i	$0.525094\pi$
$224$	0	0
$225$	10.5753	0.705017
$226$	0	0
$227$	−4.14634	−0.275202	−0.137601	−	0.990488i	$-0.543939\pi$
$227$	−4.14634	−0.275202	−0.137601	+	0.990488i	$0.543939\pi$
$228$	0	0
$229$	−20.5187	−1.35591	−0.677956	−	0.735102i	$-0.737134\pi$
$229$	−20.5187	−1.35591	−0.677956	+	0.735102i	$0.737134\pi$
$230$	0	0
$231$	−0.0304929	−0.00200628
$232$	0	0
$233$	10.1443	0.664573	0.332287	−	0.943178i	$-0.392180\pi$
$233$	10.1443	0.664573	0.332287	+	0.943178i	$0.392180\pi$
$234$	0	0
$235$	3.56611	0.232627
$236$	0	0
$237$	0.0299020	0.00194235
$238$	0	0
$239$	−0.553546	−0.0358059	−0.0179030	−	0.999840i	$-0.505699\pi$
$239$	−0.553546	−0.0358059	−0.0179030	+	0.999840i	$0.505699\pi$
$240$	0	0
$241$	−9.87578	−0.636155	−0.318077	−	0.948065i	$-0.603037\pi$
$241$	−9.87578	−0.636155	−0.318077	+	0.948065i	$0.603037\pi$
$242$	0	0
$243$	0.381796	0.0244922
$244$	0	0
$245$	2.85464	0.182376
$246$	0	0
$247$	−9.75343	−0.620596
$248$	0	0
$249$	−0.0502397	−0.00318381
$250$	0	0
$251$	−29.6346	−1.87052	−0.935258	−	0.353966i	$-0.884833\pi$
$251$	−29.6346	−1.87052	−0.935258	+	0.353966i	$0.884833\pi$
$252$	0	0
$253$	5.18957	0.326266
$254$	0	0
$255$	−0.119306	−0.00747126
$256$	0	0
$257$	−10.6427	−0.663873	−0.331937	−	0.943302i	$-0.607702\pi$
$257$	−10.6427	−0.663873	−0.331937	+	0.943302i	$0.607702\pi$
$258$	0	0
$259$	−18.4735	−1.14789
$260$	0	0
$261$	−15.4024	−0.953384
$262$	0	0
$263$	−12.4520	−0.767823	−0.383912	−	0.923370i	$-0.625423\pi$
$263$	−12.4520	−0.767823	−0.383912	+	0.923370i	$0.625423\pi$
$264$	0	0
$265$	−4.39065	−0.269715
$266$	0	0
$267$	−0.0541161	−0.00331185
$268$	0	0
$269$	31.0359	1.89229	0.946146	−	0.323741i	$-0.104941\pi$
$269$	31.0359	1.89229	0.946146	+	0.323741i	$0.104941\pi$
$270$	0	0
$271$	−25.3891	−1.54228	−0.771138	−	0.636668i	$-0.780312\pi$
$271$	−25.3891	−1.54228	−0.771138	+	0.636668i	$0.780312\pi$
$272$	0	0
$273$	0.0984941	0.00596113
$274$	0	0
$275$	−3.52532	−0.212585
$276$	0	0
$277$	7.29137	0.438096	0.219048	−	0.975714i	$-0.429705\pi$
$277$	7.29137	0.438096	0.219048	+	0.975714i	$0.429705\pi$
$278$	0	0
$279$	14.2568	0.853533
$280$	0	0
$281$	11.2832	0.673101	0.336550	−	0.941665i	$-0.390740\pi$
$281$	11.2832	0.673101	0.336550	+	0.941665i	$0.390740\pi$
$282$	0	0
$283$	5.53326	0.328918	0.164459	−	0.986384i	$-0.447412\pi$
$283$	5.53326	0.328918	0.164459	+	0.986384i	$0.447412\pi$
$284$	0	0
$285$	−0.0518561	−0.00307169
$286$	0	0
$287$	11.1031	0.655395
$288$	0	0
$289$	31.2635	1.83903
$290$	0	0
$291$	−0.208361	−0.0122143
$292$	0	0
$293$	21.4797	1.25486	0.627429	−	0.778674i	$-0.284107\pi$
$293$	21.4797	1.25486	0.627429	+	0.778674i	$0.284107\pi$
$294$	0	0
$295$	3.07300	0.178917
$296$	0	0
$297$	−0.0848482	−0.00492339
$298$	0	0
$299$	−16.7627	−0.969411
$300$	0	0
$301$	1.32978	0.0766471
$302$	0	0
$303$	−0.189302	−0.0108751
$304$	0	0
$305$	10.9589	0.627502
$306$	0	0
$307$	6.37949	0.364097	0.182048	−	0.983290i	$-0.441727\pi$
$307$	6.37949	0.364097	0.182048	+	0.983290i	$0.441727\pi$
$308$	0	0
$309$	−0.0723872	−0.00411797
$310$	0	0
$311$	11.6231	0.659086	0.329543	−	0.944141i	$-0.393105\pi$
$311$	11.6231	0.659086	0.329543	+	0.944141i	$0.393105\pi$
$312$	0	0
$313$	5.12695	0.289792	0.144896	−	0.989447i	$-0.453715\pi$
$313$	5.12695	0.289792	0.144896	+	0.989447i	$0.453715\pi$
$314$	0	0
$315$	−7.85477	−0.442566
$316$	0	0
$317$	22.6389	1.27153	0.635763	−	0.771884i	$-0.280686\pi$
$317$	22.6389	1.27153	0.635763	+	0.771884i	$0.280686\pi$
$318$	0	0
$319$	5.13447	0.287475
$320$	0	0
$321$	−0.0120038	−0.000669987 0
$322$	0	0
$323$	20.9776	1.16722
$324$	0	0
$325$	11.3870	0.631639
$326$	0	0
$327$	−0.164286	−0.00908502
$328$	0	0
$329$	6.33196	0.349092
$330$	0	0
$331$	−7.76340	−0.426715	−0.213358	−	0.976974i	$-0.568440\pi$
$331$	−7.76340	−0.426715	−0.213358	+	0.976974i	$0.568440\pi$
$332$	0	0
$333$	−25.7010	−1.40841
$334$	0	0
$335$	0.0177960	0.000972301 0
$336$	0	0
$337$	16.0510	0.874355	0.437178	−	0.899375i	$-0.355978\pi$
$337$	16.0510	0.874355	0.437178	+	0.899375i	$0.355978\pi$
$338$	0	0
$339$	−0.0317610	−0.00172502
$340$	0	0
$341$	−4.75259	−0.257367
$342$	0	0
$343$	20.1622	1.08866
$344$	0	0
$345$	−0.0891222	−0.00479818
$346$	0	0
$347$	−14.1425	−0.759210	−0.379605	−	0.925149i	$-0.623940\pi$
$347$	−14.1425	−0.759210	−0.379605	+	0.925149i	$0.623940\pi$
$348$	0	0
$349$	9.87676	0.528691	0.264345	−	0.964428i	$-0.414844\pi$
$349$	9.87676	0.528691	0.264345	+	0.964428i	$0.414844\pi$
$350$	0	0
$351$	0.274065	0.0146285
$352$	0	0
$353$	−21.7541	−1.15786	−0.578928	−	0.815379i	$-0.696529\pi$
$353$	−21.7541	−1.15786	−0.578928	+	0.815379i	$0.696529\pi$
$354$	0	0
$355$	−18.6892	−0.991918
$356$	0	0
$357$	−0.211840	−0.0112118
$358$	0	0
$359$	−10.7171	−0.565628	−0.282814	−	0.959175i	$-0.591268\pi$
$359$	−10.7171	−0.565628	−0.282814	+	0.959175i	$0.591268\pi$
$360$	0	0
$361$	−9.88218	−0.520115
$362$	0	0
$363$	0.0141418	0.000742254 0
$364$	0	0
$365$	−13.7301	−0.718667
$366$	0	0
$367$	−18.7480	−0.978636	−0.489318	−	0.872105i	$-0.662754\pi$
$367$	−18.7480	−0.978636	−0.489318	+	0.872105i	$0.662754\pi$
$368$	0	0
$369$	15.4470	0.804139
$370$	0	0
$371$	−7.79601	−0.404749
$372$	0	0
$373$	7.57204	0.392065	0.196033	−	0.980597i	$-0.437194\pi$
$373$	7.57204	0.392065	0.196033	+	0.980597i	$0.437194\pi$
$374$	0	0
$375$	0.146408	0.00756048
$376$	0	0
$377$	−16.5847	−0.854156
$378$	0	0
$379$	14.6962	0.754895	0.377448	−	0.926031i	$-0.376802\pi$
$379$	14.6962	0.754895	0.377448	+	0.926031i	$0.376802\pi$
$380$	0	0
$381$	−0.0771192	−0.00395094
$382$	0	0
$383$	−34.0998	−1.74242	−0.871208	−	0.490914i	$-0.836663\pi$
$383$	−34.0998	−1.74242	−0.871208	+	0.490914i	$0.836663\pi$
$384$	0	0
$385$	2.61843	0.133448
$386$	0	0
$387$	1.85003	0.0940423
$388$	0	0
$389$	1.44335	0.0731808	0.0365904	−	0.999330i	$-0.488350\pi$
$389$	1.44335	0.0731808	0.0365904	+	0.999330i	$0.488350\pi$
$390$	0	0
$391$	36.0530	1.82328
$392$	0	0
$393$	−0.0960529	−0.00484523
$394$	0	0
$395$	−2.56770	−0.129195
$396$	0	0
$397$	0.420015	0.0210799	0.0105400	−	0.999944i	$-0.496645\pi$
$397$	0.420015	0.0210799	0.0105400	+	0.999944i	$0.496645\pi$
$398$	0	0
$399$	−0.0920754	−0.00460954
$400$	0	0
$401$	3.48578	0.174071	0.0870357	−	0.996205i	$-0.472261\pi$
$401$	3.48578	0.174071	0.0870357	+	0.996205i	$0.472261\pi$
$402$	0	0
$403$	15.3512	0.764697
$404$	0	0
$405$	−10.9271	−0.542971
$406$	0	0
$407$	8.56757	0.424679
$408$	0	0
$409$	−22.6442	−1.11968	−0.559841	−	0.828600i	$-0.689138\pi$
$409$	−22.6442	−1.11968	−0.559841	+	0.828600i	$0.689138\pi$
$410$	0	0
$411$	−0.0141418	−0.000697565 0
$412$	0	0
$413$	5.45640	0.268492
$414$	0	0
$415$	4.31410	0.211771
$416$	0	0
$417$	−0.317317	−0.0155391
$418$	0	0
$419$	−32.2778	−1.57688	−0.788438	−	0.615114i	$-0.789110\pi$
$419$	−32.2778	−1.57688	−0.788438	+	0.615114i	$0.789110\pi$
$420$	0	0
$421$	30.7459	1.49846	0.749231	−	0.662309i	$-0.230423\pi$
$421$	30.7459	1.49846	0.749231	+	0.662309i	$0.230423\pi$
$422$	0	0
$423$	8.80924	0.428320
$424$	0	0
$425$	−24.4911	−1.18799
$426$	0	0
$427$	19.4585	0.941662
$428$	0	0
$429$	−0.0456791	−0.00220541
$430$	0	0
$431$	1.89977	0.0915086	0.0457543	−	0.998953i	$-0.485431\pi$
$431$	1.89977	0.0915086	0.0457543	+	0.998953i	$0.485431\pi$
$432$	0	0
$433$	1.76613	0.0848748	0.0424374	−	0.999099i	$-0.486488\pi$
$433$	1.76613	0.0848748	0.0424374	+	0.999099i	$0.486488\pi$
$434$	0	0
$435$	−0.0881760	−0.00422771
$436$	0	0
$437$	15.6703	0.749611
$438$	0	0
$439$	−2.02673	−0.0967304	−0.0483652	−	0.998830i	$-0.515401\pi$
$439$	−2.02673	−0.0967304	−0.0483652	+	0.998830i	$0.515401\pi$
$440$	0	0
$441$	7.05171	0.335796
$442$	0	0
$443$	0.610916	0.0290255	0.0145127	−	0.999895i	$-0.495380\pi$
$443$	0.610916	0.0290255	0.0145127	+	0.999895i	$0.495380\pi$
$444$	0	0
$445$	4.64697	0.220287
$446$	0	0
$447$	−0.000189792 0	−8.97684e−6 0
$448$	0	0
$449$	7.10453	0.335284	0.167642	−	0.985848i	$-0.446385\pi$
$449$	7.10453	0.335284	0.167642	+	0.985848i	$0.446385\pi$
$450$	0	0
$451$	−5.14934	−0.242473
$452$	0	0
$453$	−0.126491	−0.00594308
$454$	0	0
$455$	−8.45771	−0.396504
$456$	0	0
$457$	−20.9512	−0.980058	−0.490029	−	0.871706i	$-0.663014\pi$
$457$	−20.9512	−0.980058	−0.490029	+	0.871706i	$0.663014\pi$
$458$	0	0
$459$	−0.589457	−0.0275135
$460$	0	0
$461$	−33.2278	−1.54757	−0.773786	−	0.633447i	$-0.781639\pi$
$461$	−33.2278	−1.54757	−0.773786	+	0.633447i	$0.781639\pi$
$462$	0	0
$463$	−37.0326	−1.72105	−0.860525	−	0.509407i	$-0.829865\pi$
$463$	−37.0326	−1.72105	−0.860525	+	0.509407i	$0.829865\pi$
$464$	0	0
$465$	0.0816177	0.00378493
$466$	0	0
$467$	−19.3580	−0.895779	−0.447890	−	0.894089i	$-0.647824\pi$
$467$	−19.3580	−0.895779	−0.447890	+	0.894089i	$0.647824\pi$
$468$	0	0
$469$	0.0315985	0.00145908
$470$	0	0
$471$	−0.321508	−0.0148143
$472$	0	0
$473$	−0.616718	−0.0283567
$474$	0	0
$475$	−10.6450	−0.488424
$476$	0	0
$477$	−10.8461	−0.496607
$478$	0	0
$479$	35.2738	1.61170	0.805850	−	0.592120i	$-0.201709\pi$
$479$	35.2738	1.61170	0.805850	+	0.592120i	$0.201709\pi$
$480$	0	0
$481$	−27.6738	−1.26182
$482$	0	0
$483$	−0.158245	−0.00720039
$484$	0	0
$485$	17.8920	0.812434
$486$	0	0
$487$	−5.63581	−0.255383	−0.127691	−	0.991814i	$-0.540757\pi$
$487$	−5.63581	−0.255383	−0.127691	+	0.991814i	$0.540757\pi$
$488$	0	0
$489$	−0.0327995	−0.00148325
$490$	0	0
$491$	−12.0332	−0.543052	−0.271526	−	0.962431i	$-0.587528\pi$
$491$	−12.0332	−0.543052	−0.271526	+	0.962431i	$0.587528\pi$
$492$	0	0
$493$	35.6702	1.60650
$494$	0	0
$495$	3.64285	0.163734
$496$	0	0
$497$	−33.1844	−1.48852
$498$	0	0
$499$	−0.684214	−0.0306296	−0.0153148	−	0.999883i	$-0.504875\pi$
$499$	−0.684214	−0.0306296	−0.0153148	+	0.999883i	$0.504875\pi$
$500$	0	0
$501$	0.295288	0.0131925
$502$	0	0
$503$	6.56818	0.292861	0.146430	−	0.989221i	$-0.453222\pi$
$503$	6.56818	0.292861	0.146430	+	0.989221i	$0.453222\pi$
$504$	0	0
$505$	16.2554	0.723356
$506$	0	0
$507$	−0.0362972	−0.00161201
$508$	0	0
$509$	−0.908047	−0.0402485	−0.0201242	−	0.999797i	$-0.506406\pi$
$509$	−0.908047	−0.0402485	−0.0201242	+	0.999797i	$0.506406\pi$
$510$	0	0
$511$	−24.3791	−1.07847
$512$	0	0
$513$	−0.256205	−0.0113117
$514$	0	0
$515$	6.21591	0.273906
$516$	0	0
$517$	−2.93661	−0.129152
$518$	0	0
$519$	0.0633931	0.00278265
$520$	0	0
$521$	5.63732	0.246975	0.123488	−	0.992346i	$-0.460592\pi$
$521$	5.63732	0.246975	0.123488	+	0.992346i	$0.460592\pi$
$522$	0	0
$523$	−8.42242	−0.368287	−0.184143	−	0.982899i	$-0.558951\pi$
$523$	−8.42242	−0.368287	−0.184143	+	0.982899i	$0.558951\pi$
$524$	0	0
$525$	0.107497	0.00469156
$526$	0	0
$527$	−33.0172	−1.43825
$528$	0	0
$529$	3.93166	0.170942
$530$	0	0
$531$	7.59112	0.329427
$532$	0	0
$533$	16.6327	0.720444
$534$	0	0
$535$	1.03077	0.0445641
$536$	0	0
$537$	0.178041	0.00768305
$538$	0	0
$539$	−2.35073	−0.101253
$540$	0	0
$541$	35.1567	1.51151	0.755753	−	0.654857i	$-0.227271\pi$
$541$	35.1567	1.51151	0.755753	+	0.654857i	$0.227271\pi$
$542$	0	0
$543$	0.225840	0.00969173
$544$	0	0
$545$	14.1073	0.604289
$546$	0	0
$547$	0.210383	0.00899531	0.00449766	−	0.999990i	$-0.498568\pi$
$547$	0.210383	0.00899531	0.00449766	+	0.999990i	$0.498568\pi$
$548$	0	0
$549$	27.0713	1.15537
$550$	0	0
$551$	15.5039	0.660489
$552$	0	0
$553$	−4.55919	−0.193876
$554$	0	0
$555$	−0.147134	−0.00624547
$556$	0	0
$557$	16.7103	0.708038	0.354019	−	0.935238i	$-0.384815\pi$
$557$	16.7103	0.708038	0.354019	+	0.935238i	$0.384815\pi$
$558$	0	0
$559$	1.99204	0.0842544
$560$	0	0
$561$	0.0982461	0.00414795
$562$	0	0
$563$	24.2127	1.02044	0.510222	−	0.860042i	$-0.329563\pi$
$563$	24.2127	1.02044	0.510222	+	0.860042i	$0.329563\pi$
$564$	0	0
$565$	2.72732	0.114739
$566$	0	0
$567$	−19.4021	−0.814810
$568$	0	0
$569$	−27.2004	−1.14030	−0.570149	−	0.821541i	$-0.693115\pi$
$569$	−27.2004	−1.14030	−0.570149	+	0.821541i	$0.693115\pi$
$570$	0	0
$571$	−28.6650	−1.19959	−0.599796	−	0.800153i	$-0.704752\pi$
$571$	−28.6650	−1.19959	−0.599796	+	0.800153i	$0.704752\pi$
$572$	0	0
$573$	0.0545843	0.00228029
$574$	0	0
$575$	−18.2949	−0.762951
$576$	0	0
$577$	−43.1143	−1.79487	−0.897436	−	0.441146i	$-0.854572\pi$
$577$	−43.1143	−1.79487	−0.897436	+	0.441146i	$0.854572\pi$
$578$	0	0
$579$	−0.149857	−0.00622784
$580$	0	0
$581$	7.66009	0.317794
$582$	0	0
$583$	3.61560	0.149743
$584$	0	0
$585$	−11.7666	−0.486491
$586$	0	0
$587$	−29.6335	−1.22311	−0.611553	−	0.791203i	$-0.709455\pi$
$587$	−29.6335	−1.22311	−0.611553	+	0.791203i	$0.709455\pi$
$588$	0	0
$589$	−14.3508	−0.591314
$590$	0	0
$591$	−0.297009	−0.0122173
$592$	0	0
$593$	5.06839	0.208134	0.104067	−	0.994570i	$-0.466814\pi$
$593$	5.06839	0.208134	0.104067	+	0.994570i	$0.466814\pi$
$594$	0	0
$595$	18.1908	0.745748
$596$	0	0
$597$	−0.0345440	−0.00141379
$598$	0	0
$599$	−21.8568	−0.893046	−0.446523	−	0.894772i	$-0.647338\pi$
$599$	−21.8568	−0.893046	−0.446523	+	0.894772i	$0.647338\pi$
$600$	0	0
$601$	36.5349	1.49029	0.745145	−	0.666902i	$-0.232380\pi$
$601$	36.5349	1.49029	0.745145	+	0.666902i	$0.232380\pi$
$602$	0	0
$603$	0.0439609	0.00179023
$604$	0	0
$605$	−1.21436	−0.0493709
$606$	0	0
$607$	29.0488	1.17905	0.589527	−	0.807749i	$-0.299314\pi$
$607$	29.0488	1.17905	0.589527	+	0.807749i	$0.299314\pi$
$608$	0	0
$609$	−0.156565	−0.00634432
$610$	0	0
$611$	9.48545	0.383740
$612$	0	0
$613$	1.15348	0.0465886	0.0232943	−	0.999729i	$-0.492585\pi$
$613$	1.15348	0.0465886	0.0232943	+	0.999729i	$0.492585\pi$
$614$	0	0
$615$	0.0884313	0.00356590
$616$	0	0
$617$	−34.0989	−1.37277	−0.686386	−	0.727238i	$-0.740804\pi$
$617$	−34.0989	−1.37277	−0.686386	+	0.727238i	$0.740804\pi$
$618$	0	0
$619$	−28.3746	−1.14047	−0.570235	−	0.821481i	$-0.693148\pi$
$619$	−28.3746	−1.14047	−0.570235	+	0.821481i	$0.693148\pi$
$620$	0	0
$621$	−0.440326	−0.0176697
$622$	0	0
$623$	8.25113	0.330575
$624$	0	0
$625$	5.05450	0.202180
$626$	0	0
$627$	0.0427023	0.00170536
$628$	0	0
$629$	59.5206	2.37324
$630$	0	0
$631$	33.9571	1.35181	0.675905	−	0.736988i	$-0.263753\pi$
$631$	33.9571	1.35181	0.675905	+	0.736988i	$0.263753\pi$
$632$	0	0
$633$	−0.190230	−0.00756096
$634$	0	0
$635$	6.62225	0.262796
$636$	0	0
$637$	7.59302	0.300846
$638$	0	0
$639$	−46.1672	−1.82635
$640$	0	0
$641$	4.48819	0.177273	0.0886364	−	0.996064i	$-0.471749\pi$
$641$	4.48819	0.177273	0.0886364	+	0.996064i	$0.471749\pi$
$642$	0	0
$643$	−0.0261263	−0.00103032	−0.000515161	−	1.00000i	$-0.500164\pi$
$643$	−0.0261263	−0.00103032	−0.000515161		1.00000i	$0.500164\pi$
$644$	0	0
$645$	0.0105911	0.000417024 0
$646$	0	0
$647$	2.86305	0.112558	0.0562791	−	0.998415i	$-0.482076\pi$
$647$	2.86305	0.112558	0.0562791	+	0.998415i	$0.482076\pi$
$648$	0	0
$649$	−2.53054	−0.0993325
$650$	0	0
$651$	0.144920	0.00567986
$652$	0	0
$653$	33.2013	1.29927	0.649635	−	0.760247i	$-0.274922\pi$
$653$	33.2013	1.29927	0.649635	+	0.760247i	$0.274922\pi$
$654$	0	0
$655$	8.24809	0.322280
$656$	0	0
$657$	−33.9170	−1.32323
$658$	0	0
$659$	−36.7521	−1.43166	−0.715830	−	0.698275i	$-0.753951\pi$
$659$	−36.7521	−1.43166	−0.715830	+	0.698275i	$0.753951\pi$
$660$	0	0
$661$	25.2691	0.982853	0.491427	−	0.870919i	$-0.336476\pi$
$661$	25.2691	0.982853	0.491427	+	0.870919i	$0.336476\pi$
$662$	0	0
$663$	−0.317342	−0.0123245
$664$	0	0
$665$	7.90654	0.306602
$666$	0	0
$667$	26.6457	1.03173
$668$	0	0
$669$	−0.0332633	−0.00128603
$670$	0	0
$671$	−9.02436	−0.348382
$672$	0	0
$673$	20.9966	0.809360	0.404680	−	0.914458i	$-0.367383\pi$
$673$	20.9966	0.809360	0.404680	+	0.914458i	$0.367383\pi$
$674$	0	0
$675$	0.299117	0.0115130
$676$	0	0
$677$	−3.20747	−0.123273	−0.0616366	−	0.998099i	$-0.519632\pi$
$677$	−3.20747	−0.123273	−0.0616366	+	0.998099i	$0.519632\pi$
$678$	0	0
$679$	31.7689	1.21918
$680$	0	0
$681$	−0.0586368	−0.00224697
$682$	0	0
$683$	−35.5306	−1.35954	−0.679771	−	0.733425i	$-0.737921\pi$
$683$	−35.5306	−1.35954	−0.679771	+	0.733425i	$0.737921\pi$
$684$	0	0
$685$	1.21436	0.0463984
$686$	0	0
$687$	−0.290172	−0.0110707
$688$	0	0
$689$	−11.6786	−0.444920
$690$	0	0
$691$	48.8173	1.85710	0.928548	−	0.371212i	$-0.121058\pi$
$691$	48.8173	1.85710	0.928548	+	0.371212i	$0.121058\pi$
$692$	0	0
$693$	6.46822	0.245707
$694$	0	0
$695$	27.2481	1.03358
$696$	0	0
$697$	−35.7735	−1.35502
$698$	0	0
$699$	0.143459	0.00542610
$700$	0	0
$701$	32.0592	1.21086	0.605430	−	0.795898i	$-0.293001\pi$
$701$	32.0592	1.21086	0.605430	+	0.795898i	$0.293001\pi$
$702$	0	0
$703$	25.8704	0.975720
$704$	0	0
$705$	0.0504313	0.00189935
$706$	0	0
$707$	28.8630	1.08551
$708$	0	0
$709$	0.820299	0.0308070	0.0154035	−	0.999881i	$-0.495097\pi$
$709$	0.820299	0.0308070	0.0154035	+	0.999881i	$0.495097\pi$
$710$	0	0
$711$	−6.34289	−0.237877
$712$	0	0
$713$	−24.6639	−0.923670
$714$	0	0
$715$	3.92248	0.146692
$716$	0	0
$717$	−0.00782816	−0.000292348 0
$718$	0	0
$719$	−52.1408	−1.94452	−0.972262	−	0.233896i	$-0.924852\pi$
$719$	−52.1408	−1.94452	−0.972262	+	0.233896i	$0.924852\pi$
$720$	0	0
$721$	11.0369	0.411037
$722$	0	0
$723$	−0.139662	−0.00519407
$724$	0	0
$725$	−18.1007	−0.672242
$726$	0	0
$727$	11.0453	0.409649	0.204825	−	0.978799i	$-0.434338\pi$
$727$	11.0453	0.409649	0.204825	+	0.978799i	$0.434338\pi$
$728$	0	0
$729$	−26.9892	−0.999600
$730$	0	0
$731$	−4.28446	−0.158466
$732$	0	0
$733$	−7.60923	−0.281053	−0.140527	−	0.990077i	$-0.544880\pi$
$733$	−7.60923	−0.281053	−0.140527	+	0.990077i	$0.544880\pi$
$734$	0	0
$735$	0.0403698	0.00148906
$736$	0	0
$737$	−0.0146546	−0.000539810 0
$738$	0	0
$739$	40.8072	1.50112	0.750558	−	0.660804i	$-0.229785\pi$
$739$	40.8072	1.50112	0.750558	+	0.660804i	$0.229785\pi$
$740$	0	0
$741$	−0.137931	−0.00506703
$742$	0	0
$743$	−13.5015	−0.495323	−0.247661	−	0.968847i	$-0.579662\pi$
$743$	−13.5015	−0.495323	−0.247661	+	0.968847i	$0.579662\pi$
$744$	0	0
$745$	0.0162975	0.000597093 0
$746$	0	0
$747$	10.6570	0.389918
$748$	0	0
$749$	1.83023	0.0668752
$750$	0	0
$751$	−2.82551	−0.103104	−0.0515521	−	0.998670i	$-0.516417\pi$
$751$	−2.82551	−0.103104	−0.0515521	+	0.998670i	$0.516417\pi$
$752$	0	0
$753$	−0.419087	−0.0152724
$754$	0	0
$755$	10.8618	0.395303
$756$	0	0
$757$	1.33537	0.0485347	0.0242674	−	0.999706i	$-0.492275\pi$
$757$	1.33537	0.0485347	0.0242674	+	0.999706i	$0.492275\pi$
$758$	0	0
$759$	0.0733901	0.00266389
$760$	0	0
$761$	−3.30540	−0.119821	−0.0599103	−	0.998204i	$-0.519081\pi$
$761$	−3.30540	−0.119821	−0.0599103	+	0.998204i	$0.519081\pi$
$762$	0	0
$763$	25.0488	0.906827
$764$	0	0
$765$	25.3076	0.914997
$766$	0	0
$767$	8.17383	0.295140
$768$	0	0
$769$	37.5171	1.35290	0.676451	−	0.736488i	$-0.263517\pi$
$769$	37.5171	1.35290	0.676451	+	0.736488i	$0.263517\pi$
$770$	0	0
$771$	−0.150507	−0.00542039
$772$	0	0
$773$	−46.2033	−1.66182	−0.830909	−	0.556409i	$-0.812179\pi$
$773$	−46.2033	−1.66182	−0.830909	+	0.556409i	$0.812179\pi$
$774$	0	0
$775$	16.7544	0.601836
$776$	0	0
$777$	−0.261250	−0.00937228
$778$	0	0
$779$	−15.5488	−0.557094
$780$	0	0
$781$	15.3901	0.550701
$782$	0	0
$783$	−0.435651	−0.0155689
$784$	0	0
$785$	27.6080	0.985371
$786$	0	0
$787$	27.6125	0.984281	0.492140	−	0.870516i	$-0.336215\pi$
$787$	27.6125	0.984281	0.492140	+	0.870516i	$0.336215\pi$
$788$	0	0
$789$	−0.176094	−0.00626912
$790$	0	0
$791$	4.84262	0.172184
$792$	0	0
$793$	29.1493	1.03512
$794$	0	0
$795$	−0.0620918	−0.00220217
$796$	0	0
$797$	−20.9524	−0.742171	−0.371086	−	0.928599i	$-0.621014\pi$
$797$	−20.9524	−0.742171	−0.371086	+	0.928599i	$0.621014\pi$
$798$	0	0
$799$	−20.4012	−0.721742
$800$	0	0
$801$	11.4792	0.405599
$802$	0	0
$803$	11.3064	0.398995
$804$	0	0
$805$	13.5885	0.478933
$806$	0	0
$807$	0.438904	0.0154502
$808$	0	0
$809$	−8.40968	−0.295668	−0.147834	−	0.989012i	$-0.547230\pi$
$809$	−8.40968	−0.295668	−0.147834	+	0.989012i	$0.547230\pi$
$810$	0	0
$811$	−27.9523	−0.981539	−0.490769	−	0.871289i	$-0.663284\pi$
$811$	−27.9523	−0.981539	−0.490769	+	0.871289i	$0.663284\pi$
$812$	0	0
$813$	−0.359048	−0.0125924
$814$	0	0
$815$	2.81651	0.0986579
$816$	0	0
$817$	−1.86222	−0.0651510
$818$	0	0
$819$	−20.8928	−0.730053
$820$	0	0
$821$	−40.0449	−1.39758	−0.698788	−	0.715329i	$-0.746277\pi$
$821$	−40.0449	−1.39758	−0.698788	+	0.715329i	$0.746277\pi$
$822$	0	0
$823$	−51.7441	−1.80369	−0.901844	−	0.432062i	$-0.857786\pi$
$823$	−51.7441	−1.80369	−0.901844	+	0.432062i	$0.857786\pi$
$824$	0	0
$825$	−0.0498545	−0.00173571
$826$	0	0
$827$	−34.1248	−1.18664	−0.593319	−	0.804968i	$-0.702183\pi$
$827$	−34.1248	−1.18664	−0.593319	+	0.804968i	$0.702183\pi$
$828$	0	0
$829$	43.2970	1.50377	0.751884	−	0.659295i	$-0.229145\pi$
$829$	43.2970	1.50377	0.751884	+	0.659295i	$0.229145\pi$
$830$	0	0
$831$	0.103113	0.00357696
$832$	0	0
$833$	−16.3310	−0.565835
$834$	0	0
$835$	−25.3564	−0.877496
$836$	0	0
$837$	0.403248	0.0139383
$838$	0	0
$839$	−12.2246	−0.422041	−0.211021	−	0.977482i	$-0.567679\pi$
$839$	−12.2246	−0.422041	−0.211021	+	0.977482i	$0.567679\pi$
$840$	0	0
$841$	−2.63719	−0.0909375
$842$	0	0
$843$	0.159566	0.00549573
$844$	0	0
$845$	3.11685	0.107223
$846$	0	0
$847$	−2.15622	−0.0740885
$848$	0	0
$849$	0.0782504	0.00268555
$850$	0	0
$851$	44.4620	1.52414
$852$	0	0
$853$	12.8668	0.440550	0.220275	−	0.975438i	$-0.429305\pi$
$853$	12.8668	0.440550	0.220275	+	0.975438i	$0.429305\pi$
$854$	0	0
$855$	10.9998	0.376187
$856$	0	0
$857$	−20.3070	−0.693673	−0.346836	−	0.937926i	$-0.612744\pi$
$857$	−20.3070	−0.693673	−0.346836	+	0.937926i	$0.612744\pi$
$858$	0	0
$859$	−7.73959	−0.264071	−0.132036	−	0.991245i	$-0.542151\pi$
$859$	−7.73959	−0.264071	−0.132036	+	0.991245i	$0.542151\pi$
$860$	0	0
$861$	0.157018	0.00535117
$862$	0	0
$863$	12.4119	0.422507	0.211254	−	0.977431i	$-0.432245\pi$
$863$	12.4119	0.422507	0.211254	+	0.977431i	$0.432245\pi$
$864$	0	0
$865$	−5.44359	−0.185087
$866$	0	0
$867$	0.442124	0.0150153
$868$	0	0
$869$	2.11444	0.0717274
$870$	0	0
$871$	0.0473354	0.00160390
$872$	0	0
$873$	44.1980	1.49588
$874$	0	0
$875$	−22.3230	−0.754654
$876$	0	0
$877$	−6.73618	−0.227465	−0.113732	−	0.993511i	$-0.536281\pi$
$877$	−6.73618	−0.227465	−0.113732	+	0.993511i	$0.536281\pi$
$878$	0	0
$879$	0.303762	0.0102457
$880$	0	0
$881$	48.5930	1.63714	0.818571	−	0.574406i	$-0.194767\pi$
$881$	48.5930	1.63714	0.818571	+	0.574406i	$0.194767\pi$
$882$	0	0
$883$	−41.5064	−1.39680	−0.698400	−	0.715707i	$-0.746104\pi$
$883$	−41.5064	−1.39680	−0.698400	+	0.715707i	$0.746104\pi$
$884$	0	0
$885$	0.0434578	0.00146082
$886$	0	0
$887$	6.97338	0.234143	0.117072	−	0.993123i	$-0.462649\pi$
$887$	6.97338	0.234143	0.117072	+	0.993123i	$0.462649\pi$
$888$	0	0
$889$	11.7584	0.394365
$890$	0	0
$891$	8.99820	0.301451
$892$	0	0
$893$	−8.86730	−0.296733
$894$	0	0
$895$	−15.2885	−0.511037
$896$	0	0
$897$	−0.237055	−0.00791504
$898$	0	0
$899$	−24.4020	−0.813854
$900$	0	0
$901$	25.1183	0.836810
$902$	0	0
$903$	0.0188055	0.000625808 0
$904$	0	0
$905$	−19.3930	−0.644644
$906$	0	0
$907$	42.0005	1.39460	0.697302	−	0.716777i	$-0.254384\pi$
$907$	42.0005	1.39460	0.697302	+	0.716777i	$0.254384\pi$
$908$	0	0
$909$	40.1552	1.33186
$910$	0	0
$911$	−31.7236	−1.05105	−0.525525	−	0.850778i	$-0.676131\pi$
$911$	−31.7236	−1.05105	−0.525525	+	0.850778i	$0.676131\pi$
$912$	0	0
$913$	−3.55256	−0.117573
$914$	0	0
$915$	0.154978	0.00512343
$916$	0	0
$917$	14.6453	0.483629
$918$	0	0
$919$	−51.5325	−1.69990	−0.849950	−	0.526863i	$-0.823368\pi$
$919$	−51.5325	−1.69990	−0.849950	+	0.526863i	$0.823368\pi$
$920$	0	0
$921$	0.0902177	0.00297278
$922$	0	0
$923$	−49.7111	−1.63626
$924$	0	0
$925$	−30.2034	−0.993083
$926$	0	0
$927$	15.3550	0.504323
$928$	0	0
$929$	−11.6735	−0.382995	−0.191497	−	0.981493i	$-0.561334\pi$
$929$	−11.6735	−0.382995	−0.191497	+	0.981493i	$0.561334\pi$
$930$	0	0
$931$	−7.09819	−0.232634
$932$	0	0
$933$	0.164372	0.00538130
$934$	0	0
$935$	−8.43642	−0.275900
$936$	0	0
$937$	−4.48496	−0.146517	−0.0732587	−	0.997313i	$-0.523340\pi$
$937$	−4.48496	−0.146517	−0.0732587	+	0.997313i	$0.523340\pi$
$938$	0	0
$939$	0.0725045	0.00236609
$940$	0	0
$941$	4.94340	0.161150	0.0805752	−	0.996749i	$-0.474324\pi$
$941$	4.94340	0.161150	0.0805752	+	0.996749i	$0.474324\pi$
$942$	0	0
$943$	−26.7229	−0.870217
$944$	0	0
$945$	−0.222169	−0.00722716
$946$	0	0
$947$	2.45163	0.0796673	0.0398336	−	0.999206i	$-0.487317\pi$
$947$	2.45163	0.0796673	0.0398336	+	0.999206i	$0.487317\pi$
$948$	0	0
$949$	−36.5206	−1.18551
$950$	0	0
$951$	0.320155	0.0103817
$952$	0	0
$953$	−7.50246	−0.243028	−0.121514	−	0.992590i	$-0.538775\pi$
$953$	−7.50246	−0.243028	−0.121514	+	0.992590i	$0.538775\pi$
$954$	0	0
$955$	−4.68717	−0.151673
$956$	0	0
$957$	0.0726109	0.00234718
$958$	0	0
$959$	2.15622	0.0696279
$960$	0	0
$961$	−8.41290	−0.271384
$962$	0	0
$963$	2.54628	0.0820526
$964$	0	0
$965$	12.8682	0.414243
$966$	0	0
$967$	30.5218	0.981515	0.490758	−	0.871296i	$-0.336720\pi$
$967$	30.5218	0.981515	0.490758	+	0.871296i	$0.336720\pi$
$968$	0	0
$969$	0.296661	0.00953013
$970$	0	0
$971$	4.96878	0.159456	0.0797279	−	0.996817i	$-0.474595\pi$
$971$	4.96878	0.159456	0.0797279	+	0.996817i	$0.474595\pi$
$972$	0	0
$973$	48.3815	1.55104
$974$	0	0
$975$	0.161034	0.00515720
$976$	0	0
$977$	40.4514	1.29415	0.647077	−	0.762425i	$-0.275991\pi$
$977$	40.4514	1.29415	0.647077	+	0.762425i	$0.275991\pi$
$978$	0	0
$979$	−3.82667	−0.122301
$980$	0	0
$981$	34.8487	1.11263
$982$	0	0
$983$	−36.8886	−1.17656	−0.588282	−	0.808656i	$-0.700196\pi$
$983$	−36.8886	−1.17656	−0.588282	+	0.808656i	$0.700196\pi$
$984$	0	0
$985$	25.5042	0.812632
$986$	0	0
$987$	0.0895456	0.00285027
$988$	0	0
$989$	−3.20050	−0.101770
$990$	0	0
$991$	26.8880	0.854125	0.427062	−	0.904222i	$-0.359548\pi$
$991$	26.8880	0.854125	0.427062	+	0.904222i	$0.359548\pi$
$992$	0	0
$993$	−0.109789	−0.00348404
$994$	0	0
$995$	2.96630	0.0940382
$996$	0	0
$997$	−5.22699	−0.165541	−0.0827703	−	0.996569i	$-0.526377\pi$
$997$	−5.22699	−0.165541	−0.0827703	+	0.996569i	$0.526377\pi$
$998$	0	0
$999$	−0.726942	−0.0229994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.c.1.15	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.c.1.15	✓	25	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0141418 q^{3} -1.21436 q^{5} -2.15622 q^{7} -2.99980 q^{9} +O(q^{10})\)	\(q+0.0141418 q^{3} -1.21436 q^{5} -2.15622 q^{7} -2.99980 q^{9} +1.00000 q^{11} -3.23007 q^{13} -0.0171733 q^{15} +6.94720 q^{17} +3.01957 q^{19} -0.0304929 q^{21} +5.18957 q^{23} -3.52532 q^{25} -0.0848482 q^{27} +5.13447 q^{29} -4.75259 q^{31} +0.0141418 q^{33} +2.61843 q^{35} +8.56757 q^{37} -0.0456791 q^{39} -5.14934 q^{41} -0.616718 q^{43} +3.64285 q^{45} -2.93661 q^{47} -2.35073 q^{49} +0.0982461 q^{51} +3.61560 q^{53} -1.21436 q^{55} +0.0427023 q^{57} -2.53054 q^{59} -9.02436 q^{61} +6.46822 q^{63} +3.92248 q^{65} -0.0146546 q^{67} +0.0733901 q^{69} +15.3901 q^{71} +11.3064 q^{73} -0.0498545 q^{75} -2.15622 q^{77} +2.11444 q^{79} +8.99820 q^{81} -3.55256 q^{83} -8.43642 q^{85} +0.0726109 q^{87} -3.82667 q^{89} +6.96473 q^{91} -0.0672103 q^{93} -3.66686 q^{95} -14.7336 q^{97} -2.99980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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