LMFDB - Embedded newform 4012.2.a.i.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 $ x^18 - 8x^17 - 4x^16 + 178x^15 - 265x^14 - 1405x^13 + 3503x^12 + 4295x^11 - 17374x^10 - 893x^9 + 38112x^8 - 18700x^7 - 32137x^6 + 26381x^5 + 3964x^4 - 5788x^3 - 108x^2 + 232*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$3.26206$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.26206 q^{3} +2.73134 q^{5} +1.98500 q^{7} +7.64104 q^{9} +O(q^{10})$	$q+3.26206 q^{3} +2.73134 q^{5} +1.98500 q^{7} +7.64104 q^{9} +5.04378 q^{11} -5.87655 q^{13} +8.90981 q^{15} -1.00000 q^{17} -4.90264 q^{19} +6.47518 q^{21} -2.08875 q^{23} +2.46024 q^{25} +15.1394 q^{27} -9.28121 q^{29} -1.56695 q^{31} +16.4531 q^{33} +5.42171 q^{35} +9.10324 q^{37} -19.1697 q^{39} +2.43179 q^{41} +1.35497 q^{43} +20.8703 q^{45} +7.13078 q^{47} -3.05979 q^{49} -3.26206 q^{51} +1.59624 q^{53} +13.7763 q^{55} -15.9927 q^{57} -1.00000 q^{59} +12.4420 q^{61} +15.1674 q^{63} -16.0509 q^{65} +2.11172 q^{67} -6.81362 q^{69} -7.89699 q^{71} +12.4783 q^{73} +8.02546 q^{75} +10.0119 q^{77} -14.4756 q^{79} +26.4624 q^{81} -6.11174 q^{83} -2.73134 q^{85} -30.2759 q^{87} +7.05525 q^{89} -11.6649 q^{91} -5.11150 q^{93} -13.3908 q^{95} -5.45991 q^{97} +38.5397 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * 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
$3$	3.26206	1.88335	0.941676	−	0.336521i	$-0.109250\pi$
$3$	3.26206	1.88335	0.941676	+	0.336521i	$0.109250\pi$
$4$	0	0
$5$	2.73134	1.22149	0.610747	−	0.791826i	$-0.290869\pi$
$5$	2.73134	1.22149	0.610747	+	0.791826i	$0.290869\pi$
$6$	0	0
$7$	1.98500	0.750258	0.375129	−	0.926973i	$-0.377598\pi$
$7$	1.98500	0.750258	0.375129	+	0.926973i	$0.377598\pi$
$8$	0	0
$9$	7.64104	2.54701
$10$	0	0
$11$	5.04378	1.52076	0.760378	−	0.649481i	$-0.225014\pi$
$11$	5.04378	1.52076	0.760378	+	0.649481i	$0.225014\pi$
$12$	0	0
$13$	−5.87655	−1.62986	−0.814931	−	0.579558i	$-0.803225\pi$
$13$	−5.87655	−1.62986	−0.814931	+	0.579558i	$0.803225\pi$
$14$	0	0
$15$	8.90981	2.30050
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.90264	−1.12474	−0.562372	−	0.826884i	$-0.690111\pi$
$19$	−4.90264	−1.12474	−0.562372	+	0.826884i	$0.690111\pi$
$20$	0	0
$21$	6.47518	1.41300
$22$	0	0
$23$	−2.08875	−0.435533	−0.217767	−	0.976001i	$-0.569877\pi$
$23$	−2.08875	−0.435533	−0.217767	+	0.976001i	$0.569877\pi$
$24$	0	0
$25$	2.46024	0.492048
$26$	0	0
$27$	15.1394	2.91357
$28$	0	0
$29$	−9.28121	−1.72348	−0.861739	−	0.507352i	$-0.830624\pi$
$29$	−9.28121	−1.72348	−0.861739	+	0.507352i	$0.830624\pi$
$30$	0	0
$31$	−1.56695	−0.281433	−0.140717	−	0.990050i	$-0.544941\pi$
$31$	−1.56695	−0.281433	−0.140717	+	0.990050i	$0.544941\pi$
$32$	0	0
$33$	16.4531	2.86412
$34$	0	0
$35$	5.42171	0.916436
$36$	0	0
$37$	9.10324	1.49656	0.748282	−	0.663381i	$-0.230879\pi$
$37$	9.10324	1.49656	0.748282	+	0.663381i	$0.230879\pi$
$38$	0	0
$39$	−19.1697	−3.06960
$40$	0	0
$41$	2.43179	0.379781	0.189891	−	0.981805i	$-0.439187\pi$
$41$	2.43179	0.379781	0.189891	+	0.981805i	$0.439187\pi$
$42$	0	0
$43$	1.35497	0.206631	0.103316	−	0.994649i	$-0.467055\pi$
$43$	1.35497	0.206631	0.103316	+	0.994649i	$0.467055\pi$
$44$	0	0
$45$	20.8703	3.11116
$46$	0	0
$47$	7.13078	1.04013	0.520065	−	0.854126i	$-0.325908\pi$
$47$	7.13078	1.04013	0.520065	+	0.854126i	$0.325908\pi$
$48$	0	0
$49$	−3.05979	−0.437113
$50$	0	0
$51$	−3.26206	−0.456780
$52$	0	0
$53$	1.59624	0.219260	0.109630	−	0.993972i	$-0.465033\pi$
$53$	1.59624	0.219260	0.109630	+	0.993972i	$0.465033\pi$
$54$	0	0
$55$	13.7763	1.85759
$56$	0	0
$57$	−15.9927	−2.11829
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	12.4420	1.59304	0.796520	−	0.604613i	$-0.206672\pi$
$61$	12.4420	1.59304	0.796520	+	0.604613i	$0.206672\pi$
$62$	0	0
$63$	15.1674	1.91092
$64$	0	0
$65$	−16.0509	−1.99087
$66$	0	0
$67$	2.11172	0.257988	0.128994	−	0.991645i	$-0.458825\pi$
$67$	2.11172	0.257988	0.128994	+	0.991645i	$0.458825\pi$
$68$	0	0
$69$	−6.81362	−0.820263
$70$	0	0
$71$	−7.89699	−0.937200	−0.468600	−	0.883410i	$-0.655241\pi$
$71$	−7.89699	−0.937200	−0.468600	+	0.883410i	$0.655241\pi$
$72$	0	0
$73$	12.4783	1.46047	0.730236	−	0.683194i	$-0.239410\pi$
$73$	12.4783	1.46047	0.730236	+	0.683194i	$0.239410\pi$
$74$	0	0
$75$	8.02546	0.926700
$76$	0	0
$77$	10.0119	1.14096
$78$	0	0
$79$	−14.4756	−1.62863	−0.814315	−	0.580423i	$-0.802887\pi$
$79$	−14.4756	−1.62863	−0.814315	+	0.580423i	$0.802887\pi$
$80$	0	0
$81$	26.4624	2.94027
$82$	0	0
$83$	−6.11174	−0.670850	−0.335425	−	0.942067i	$-0.608880\pi$
$83$	−6.11174	−0.670850	−0.335425	+	0.942067i	$0.608880\pi$
$84$	0	0
$85$	−2.73134	−0.296256
$86$	0	0
$87$	−30.2759	−3.24591
$88$	0	0
$89$	7.05525	0.747855	0.373928	−	0.927458i	$-0.378011\pi$
$89$	7.05525	0.747855	0.373928	+	0.927458i	$0.378011\pi$
$90$	0	0
$91$	−11.6649	−1.22282
$92$	0	0
$93$	−5.11150	−0.530037
$94$	0	0
$95$	−13.3908	−1.37387
$96$	0	0
$97$	−5.45991	−0.554369	−0.277185	−	0.960817i	$-0.589401\pi$
$97$	−5.45991	−0.554369	−0.277185	+	0.960817i	$0.589401\pi$
$98$	0	0
$99$	38.5397	3.87339
$100$	0	0
$101$	6.93565	0.690123	0.345062	−	0.938580i	$-0.387858\pi$
$101$	6.93565	0.690123	0.345062	+	0.938580i	$0.387858\pi$
$102$	0	0
$103$	−16.8353	−1.65883	−0.829416	−	0.558631i	$-0.811327\pi$
$103$	−16.8353	−1.65883	−0.829416	+	0.558631i	$0.811327\pi$
$104$	0	0
$105$	17.6859	1.72597
$106$	0	0
$107$	−4.92403	−0.476024	−0.238012	−	0.971262i	$-0.576496\pi$
$107$	−4.92403	−0.476024	−0.238012	+	0.971262i	$0.576496\pi$
$108$	0	0
$109$	−17.6514	−1.69070	−0.845351	−	0.534212i	$-0.820608\pi$
$109$	−17.6514	−1.69070	−0.845351	+	0.534212i	$0.820608\pi$
$110$	0	0
$111$	29.6953	2.81856
$112$	0	0
$113$	−4.64988	−0.437424	−0.218712	−	0.975789i	$-0.570186\pi$
$113$	−4.64988	−0.437424	−0.218712	+	0.975789i	$0.570186\pi$
$114$	0	0
$115$	−5.70508	−0.532002
$116$	0	0
$117$	−44.9030	−4.15128
$118$	0	0
$119$	−1.98500	−0.181964
$120$	0	0
$121$	14.4397	1.31270
$122$	0	0
$123$	7.93263	0.715262
$124$	0	0
$125$	−6.93695	−0.620460
$126$	0	0
$127$	−3.37585	−0.299558	−0.149779	−	0.988719i	$-0.547856\pi$
$127$	−3.37585	−0.299558	−0.149779	+	0.988719i	$0.547856\pi$
$128$	0	0
$129$	4.42000	0.389159
$130$	0	0
$131$	−3.79039	−0.331168	−0.165584	−	0.986196i	$-0.552951\pi$
$131$	−3.79039	−0.331168	−0.165584	+	0.986196i	$0.552951\pi$
$132$	0	0
$133$	−9.73173	−0.843848
$134$	0	0
$135$	41.3508	3.55891
$136$	0	0
$137$	−16.0220	−1.36885	−0.684424	−	0.729084i	$-0.739946\pi$
$137$	−16.0220	−1.36885	−0.684424	+	0.729084i	$0.739946\pi$
$138$	0	0
$139$	14.9700	1.26974	0.634869	−	0.772620i	$-0.281054\pi$
$139$	14.9700	1.26974	0.634869	+	0.772620i	$0.281054\pi$
$140$	0	0
$141$	23.2610	1.95893
$142$	0	0
$143$	−29.6400	−2.47862
$144$	0	0
$145$	−25.3502	−2.10522
$146$	0	0
$147$	−9.98123	−0.823238
$148$	0	0
$149$	−15.6551	−1.28252	−0.641260	−	0.767324i	$-0.721588\pi$
$149$	−15.6551	−1.28252	−0.641260	+	0.767324i	$0.721588\pi$
$150$	0	0
$151$	7.56592	0.615706	0.307853	−	0.951434i	$-0.400390\pi$
$151$	7.56592	0.615706	0.307853	+	0.951434i	$0.400390\pi$
$152$	0	0
$153$	−7.64104	−0.617742
$154$	0	0
$155$	−4.27989	−0.343769
$156$	0	0
$157$	9.52764	0.760389	0.380194	−	0.924907i	$-0.375857\pi$
$157$	9.52764	0.760389	0.380194	+	0.924907i	$0.375857\pi$
$158$	0	0
$159$	5.20703	0.412945
$160$	0	0
$161$	−4.14615	−0.326762
$162$	0	0
$163$	7.77972	0.609355	0.304677	−	0.952456i	$-0.401451\pi$
$163$	7.77972	0.609355	0.304677	+	0.952456i	$0.401451\pi$
$164$	0	0
$165$	44.9391	3.49850
$166$	0	0
$167$	−9.26861	−0.717226	−0.358613	−	0.933486i	$-0.616750\pi$
$167$	−9.26861	−0.717226	−0.358613	+	0.933486i	$0.616750\pi$
$168$	0	0
$169$	21.5338	1.65645
$170$	0	0
$171$	−37.4613	−2.86474
$172$	0	0
$173$	−22.0516	−1.67655	−0.838274	−	0.545249i	$-0.816435\pi$
$173$	−22.0516	−1.67655	−0.838274	+	0.545249i	$0.816435\pi$
$174$	0	0
$175$	4.88357	0.369163
$176$	0	0
$177$	−3.26206	−0.245192
$178$	0	0
$179$	5.17419	0.386737	0.193368	−	0.981126i	$-0.438059\pi$
$179$	5.17419	0.386737	0.193368	+	0.981126i	$0.438059\pi$
$180$	0	0
$181$	23.2203	1.72595	0.862976	−	0.505245i	$-0.168598\pi$
$181$	23.2203	1.72595	0.862976	+	0.505245i	$0.168598\pi$
$182$	0	0
$183$	40.5867	3.00025
$184$	0	0
$185$	24.8641	1.82804
$186$	0	0
$187$	−5.04378	−0.368837
$188$	0	0
$189$	30.0516	2.18593
$190$	0	0
$191$	14.1371	1.02293	0.511463	−	0.859305i	$-0.329104\pi$
$191$	14.1371	1.02293	0.511463	+	0.859305i	$0.329104\pi$
$192$	0	0
$193$	6.19128	0.445658	0.222829	−	0.974857i	$-0.428471\pi$
$193$	6.19128	0.445658	0.222829	+	0.974857i	$0.428471\pi$
$194$	0	0
$195$	−52.3590	−3.74950
$196$	0	0
$197$	−19.4405	−1.38508	−0.692539	−	0.721381i	$-0.743508\pi$
$197$	−19.4405	−1.38508	−0.692539	+	0.721381i	$0.743508\pi$
$198$	0	0
$199$	−17.0261	−1.20695	−0.603474	−	0.797382i	$-0.706217\pi$
$199$	−17.0261	−1.20695	−0.603474	+	0.797382i	$0.706217\pi$
$200$	0	0
$201$	6.88858	0.485883
$202$	0	0
$203$	−18.4232	−1.29305
$204$	0	0
$205$	6.64204	0.463900
$206$	0	0
$207$	−15.9602	−1.10931
$208$	0	0
$209$	−24.7278	−1.71046
$210$	0	0
$211$	−3.21759	−0.221508	−0.110754	−	0.993848i	$-0.535327\pi$
$211$	−3.21759	−0.221508	−0.110754	+	0.993848i	$0.535327\pi$
$212$	0	0
$213$	−25.7605	−1.76508
$214$	0	0
$215$	3.70089	0.252399
$216$	0	0
$217$	−3.11040	−0.211147
$218$	0	0
$219$	40.7049	2.75058
$220$	0	0
$221$	5.87655	0.395299
$222$	0	0
$223$	−1.32981	−0.0890508	−0.0445254	−	0.999008i	$-0.514178\pi$
$223$	−1.32981	−0.0890508	−0.0445254	+	0.999008i	$0.514178\pi$
$224$	0	0
$225$	18.7988	1.25325
$226$	0	0
$227$	18.9827	1.25992	0.629961	−	0.776626i	$-0.283071\pi$
$227$	18.9827	1.25992	0.629961	+	0.776626i	$0.283071\pi$
$228$	0	0
$229$	2.74218	0.181208	0.0906041	−	0.995887i	$-0.471120\pi$
$229$	2.74218	0.181208	0.0906041	+	0.995887i	$0.471120\pi$
$230$	0	0
$231$	32.6593	2.14883
$232$	0	0
$233$	24.0968	1.57863	0.789317	−	0.613985i	$-0.210435\pi$
$233$	24.0968	1.57863	0.789317	+	0.613985i	$0.210435\pi$
$234$	0	0
$235$	19.4766	1.27051
$236$	0	0
$237$	−47.2202	−3.06728
$238$	0	0
$239$	14.4821	0.936770	0.468385	−	0.883525i	$-0.344836\pi$
$239$	14.4821	0.936770	0.468385	+	0.883525i	$0.344836\pi$
$240$	0	0
$241$	−1.44747	−0.0932394	−0.0466197	−	0.998913i	$-0.514845\pi$
$241$	−1.44747	−0.0932394	−0.0466197	+	0.998913i	$0.514845\pi$
$242$	0	0
$243$	40.9040	2.62399
$244$	0	0
$245$	−8.35734	−0.533931
$246$	0	0
$247$	28.8106	1.83318
$248$	0	0
$249$	−19.9369	−1.26345
$250$	0	0
$251$	−13.0184	−0.821711	−0.410856	−	0.911700i	$-0.634770\pi$
$251$	−13.0184	−0.821711	−0.410856	+	0.911700i	$0.634770\pi$
$252$	0	0
$253$	−10.5352	−0.662340
$254$	0	0
$255$	−8.90981	−0.557954
$256$	0	0
$257$	−21.3745	−1.33331	−0.666653	−	0.745368i	$-0.732274\pi$
$257$	−21.3745	−1.33331	−0.666653	+	0.745368i	$0.732274\pi$
$258$	0	0
$259$	18.0699	1.12281
$260$	0	0
$261$	−70.9181	−4.38972
$262$	0	0
$263$	26.3530	1.62500	0.812498	−	0.582964i	$-0.198107\pi$
$263$	26.3530	1.62500	0.812498	+	0.582964i	$0.198107\pi$
$264$	0	0
$265$	4.35988	0.267825
$266$	0	0
$267$	23.0147	1.40847
$268$	0	0
$269$	8.84396	0.539226	0.269613	−	0.962969i	$-0.413104\pi$
$269$	8.84396	0.539226	0.269613	+	0.962969i	$0.413104\pi$
$270$	0	0
$271$	−18.1107	−1.10015	−0.550075	−	0.835116i	$-0.685401\pi$
$271$	−18.1107	−1.10015	−0.550075	+	0.835116i	$0.685401\pi$
$272$	0	0
$273$	−38.0517	−2.30299
$274$	0	0
$275$	12.4089	0.748285
$276$	0	0
$277$	20.0416	1.20418	0.602091	−	0.798427i	$-0.294334\pi$
$277$	20.0416	1.20418	0.602091	+	0.798427i	$0.294334\pi$
$278$	0	0
$279$	−11.9732	−0.716814
$280$	0	0
$281$	12.5447	0.748354	0.374177	−	0.927357i	$-0.377925\pi$
$281$	12.5447	0.748354	0.374177	+	0.927357i	$0.377925\pi$
$282$	0	0
$283$	3.64813	0.216859	0.108429	−	0.994104i	$-0.465418\pi$
$283$	3.64813	0.216859	0.108429	+	0.994104i	$0.465418\pi$
$284$	0	0
$285$	−43.6816	−2.58748
$286$	0	0
$287$	4.82708	0.284934
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−17.8105	−1.04407
$292$	0	0
$293$	−13.5621	−0.792305	−0.396153	−	0.918185i	$-0.629655\pi$
$293$	−13.5621	−0.792305	−0.396153	+	0.918185i	$0.629655\pi$
$294$	0	0
$295$	−2.73134	−0.159025
$296$	0	0
$297$	76.3596	4.43083
$298$	0	0
$299$	12.2746	0.709859
$300$	0	0
$301$	2.68961	0.155027
$302$	0	0
$303$	22.6245	1.29975
$304$	0	0
$305$	33.9835	1.94589
$306$	0	0
$307$	−23.0509	−1.31558	−0.657792	−	0.753199i	$-0.728510\pi$
$307$	−23.0509	−1.31558	−0.657792	+	0.753199i	$0.728510\pi$
$308$	0	0
$309$	−54.9178	−3.12417
$310$	0	0
$311$	7.73547	0.438638	0.219319	−	0.975653i	$-0.429616\pi$
$311$	7.73547	0.438638	0.219319	+	0.975653i	$0.429616\pi$
$312$	0	0
$313$	29.9320	1.69186	0.845928	−	0.533296i	$-0.179047\pi$
$313$	29.9320	1.69186	0.845928	+	0.533296i	$0.179047\pi$
$314$	0	0
$315$	41.4275	2.33418
$316$	0	0
$317$	−20.9691	−1.17774	−0.588871	−	0.808227i	$-0.700428\pi$
$317$	−20.9691	−1.17774	−0.588871	+	0.808227i	$0.700428\pi$
$318$	0	0
$319$	−46.8123	−2.62099
$320$	0	0
$321$	−16.0625	−0.896521
$322$	0	0
$323$	4.90264	0.272790
$324$	0	0
$325$	−14.4577	−0.801971
$326$	0	0
$327$	−57.5801	−3.18419
$328$	0	0
$329$	14.1546	0.780366
$330$	0	0
$331$	−5.98654	−0.329050	−0.164525	−	0.986373i	$-0.552609\pi$
$331$	−5.98654	−0.329050	−0.164525	+	0.986373i	$0.552609\pi$
$332$	0	0
$333$	69.5583	3.81177
$334$	0	0
$335$	5.76785	0.315131
$336$	0	0
$337$	2.36318	0.128730	0.0643652	−	0.997926i	$-0.479498\pi$
$337$	2.36318	0.128730	0.0643652	+	0.997926i	$0.479498\pi$
$338$	0	0
$339$	−15.1682	−0.823824
$340$	0	0
$341$	−7.90336	−0.427991
$342$	0	0
$343$	−19.9686	−1.07821
$344$	0	0
$345$	−18.6103	−1.00195
$346$	0	0
$347$	−9.52056	−0.511091	−0.255545	−	0.966797i	$-0.582255\pi$
$347$	−9.52056	−0.511091	−0.255545	+	0.966797i	$0.582255\pi$
$348$	0	0
$349$	−20.4361	−1.09392	−0.546959	−	0.837159i	$-0.684215\pi$
$349$	−20.4361	−1.09392	−0.546959	+	0.837159i	$0.684215\pi$
$350$	0	0
$351$	−88.9673	−4.74872
$352$	0	0
$353$	−32.8878	−1.75044	−0.875220	−	0.483726i	$-0.839283\pi$
$353$	−32.8878	−1.75044	−0.875220	+	0.483726i	$0.839283\pi$
$354$	0	0
$355$	−21.5694	−1.14478
$356$	0	0
$357$	−6.47518	−0.342703
$358$	0	0
$359$	−14.2949	−0.754457	−0.377228	−	0.926120i	$-0.623123\pi$
$359$	−14.2949	−0.754457	−0.377228	+	0.926120i	$0.623123\pi$
$360$	0	0
$361$	5.03592	0.265048
$362$	0	0
$363$	47.1031	2.47227
$364$	0	0
$365$	34.0825	1.78396
$366$	0	0
$367$	7.08122	0.369637	0.184818	−	0.982773i	$-0.440830\pi$
$367$	7.08122	0.369637	0.184818	+	0.982773i	$0.440830\pi$
$368$	0	0
$369$	18.5814	0.967308
$370$	0	0
$371$	3.16853	0.164502
$372$	0	0
$373$	−9.16911	−0.474758	−0.237379	−	0.971417i	$-0.576288\pi$
$373$	−9.16911	−0.474758	−0.237379	+	0.971417i	$0.576288\pi$
$374$	0	0
$375$	−22.6288	−1.16854
$376$	0	0
$377$	54.5415	2.80903
$378$	0	0
$379$	−6.00488	−0.308450	−0.154225	−	0.988036i	$-0.549288\pi$
$379$	−6.00488	−0.308450	−0.154225	+	0.988036i	$0.549288\pi$
$380$	0	0
$381$	−11.0122	−0.564174
$382$	0	0
$383$	33.5413	1.71388	0.856939	−	0.515418i	$-0.172363\pi$
$383$	33.5413	1.71388	0.856939	+	0.515418i	$0.172363\pi$
$384$	0	0
$385$	27.3459	1.39367
$386$	0	0
$387$	10.3534	0.526293
$388$	0	0
$389$	24.1064	1.22224	0.611122	−	0.791536i	$-0.290718\pi$
$389$	24.1064	1.22224	0.611122	+	0.791536i	$0.290718\pi$
$390$	0	0
$391$	2.08875	0.105632
$392$	0	0
$393$	−12.3645	−0.623705
$394$	0	0
$395$	−39.5378	−1.98936
$396$	0	0
$397$	−27.8197	−1.39623	−0.698115	−	0.715986i	$-0.745978\pi$
$397$	−27.8197	−1.39623	−0.698115	+	0.715986i	$0.745978\pi$
$398$	0	0
$399$	−31.7455	−1.58926
$400$	0	0
$401$	3.21664	0.160632	0.0803158	−	0.996769i	$-0.474407\pi$
$401$	3.21664	0.160632	0.0803158	+	0.996769i	$0.474407\pi$
$402$	0	0
$403$	9.20828	0.458697
$404$	0	0
$405$	72.2780	3.59152
$406$	0	0
$407$	45.9147	2.27591
$408$	0	0
$409$	13.1370	0.649582	0.324791	−	0.945786i	$-0.394706\pi$
$409$	13.1370	0.649582	0.324791	+	0.945786i	$0.394706\pi$
$410$	0	0
$411$	−52.2646	−2.57802
$412$	0	0
$413$	−1.98500	−0.0976753
$414$	0	0
$415$	−16.6933	−0.819440
$416$	0	0
$417$	48.8330	2.39136
$418$	0	0
$419$	11.9956	0.586022	0.293011	−	0.956109i	$-0.405343\pi$
$419$	11.9956	0.586022	0.293011	+	0.956109i	$0.405343\pi$
$420$	0	0
$421$	26.7971	1.30601	0.653005	−	0.757354i	$-0.273508\pi$
$421$	26.7971	1.30601	0.653005	+	0.757354i	$0.273508\pi$
$422$	0	0
$423$	54.4866	2.64923
$424$	0	0
$425$	−2.46024	−0.119339
$426$	0	0
$427$	24.6974	1.19519
$428$	0	0
$429$	−96.6875	−4.66812
$430$	0	0
$431$	5.68900	0.274029	0.137015	−	0.990569i	$-0.456249\pi$
$431$	5.68900	0.274029	0.137015	+	0.990569i	$0.456249\pi$
$432$	0	0
$433$	−14.6479	−0.703934	−0.351967	−	0.936012i	$-0.614487\pi$
$433$	−14.6479	−0.703934	−0.351967	+	0.936012i	$0.614487\pi$
$434$	0	0
$435$	−82.6938	−3.96487
$436$	0	0
$437$	10.2404	0.489864
$438$	0	0
$439$	27.8909	1.33116	0.665580	−	0.746327i	$-0.268184\pi$
$439$	27.8909	1.33116	0.665580	+	0.746327i	$0.268184\pi$
$440$	0	0
$441$	−23.3800	−1.11333
$442$	0	0
$443$	−35.3239	−1.67829	−0.839143	−	0.543910i	$-0.816943\pi$
$443$	−35.3239	−1.67829	−0.839143	+	0.543910i	$0.816943\pi$
$444$	0	0
$445$	19.2703	0.913501
$446$	0	0
$447$	−51.0680	−2.41544
$448$	0	0
$449$	35.6565	1.68273	0.841367	−	0.540464i	$-0.181751\pi$
$449$	35.6565	1.68273	0.841367	+	0.540464i	$0.181751\pi$
$450$	0	0
$451$	12.2654	0.577554
$452$	0	0
$453$	24.6805	1.15959
$454$	0	0
$455$	−31.8609	−1.49366
$456$	0	0
$457$	−23.0294	−1.07727	−0.538636	−	0.842539i	$-0.681060\pi$
$457$	−23.0294	−1.07727	−0.538636	+	0.842539i	$0.681060\pi$
$458$	0	0
$459$	−15.1394	−0.706645
$460$	0	0
$461$	−1.34054	−0.0624350	−0.0312175	−	0.999513i	$-0.509938\pi$
$461$	−1.34054	−0.0624350	−0.0312175	+	0.999513i	$0.509938\pi$
$462$	0	0
$463$	−11.4926	−0.534105	−0.267053	−	0.963682i	$-0.586050\pi$
$463$	−11.4926	−0.534105	−0.267053	+	0.963682i	$0.586050\pi$
$464$	0	0
$465$	−13.9613	−0.647438
$466$	0	0
$467$	6.11357	0.282902	0.141451	−	0.989945i	$-0.454823\pi$
$467$	6.11357	0.282902	0.141451	+	0.989945i	$0.454823\pi$
$468$	0	0
$469$	4.19176	0.193558
$470$	0	0
$471$	31.0798	1.43208
$472$	0	0
$473$	6.83417	0.314236
$474$	0	0
$475$	−12.0617	−0.553428
$476$	0	0
$477$	12.1969	0.558460
$478$	0	0
$479$	−32.7397	−1.49591	−0.747957	−	0.663747i	$-0.768965\pi$
$479$	−32.7397	−1.49591	−0.747957	+	0.663747i	$0.768965\pi$
$480$	0	0
$481$	−53.4957	−2.43919
$482$	0	0
$483$	−13.5250	−0.615409
$484$	0	0
$485$	−14.9129	−0.677159
$486$	0	0
$487$	1.58108	0.0716454	0.0358227	−	0.999358i	$-0.488595\pi$
$487$	1.58108	0.0716454	0.0358227	+	0.999358i	$0.488595\pi$
$488$	0	0
$489$	25.3779	1.14763
$490$	0	0
$491$	6.94132	0.313258	0.156629	−	0.987658i	$-0.449937\pi$
$491$	6.94132	0.313258	0.156629	+	0.987658i	$0.449937\pi$
$492$	0	0
$493$	9.28121	0.418005
$494$	0	0
$495$	105.265	4.73132
$496$	0	0
$497$	−15.6755	−0.703142
$498$	0	0
$499$	−7.05477	−0.315815	−0.157907	−	0.987454i	$-0.550475\pi$
$499$	−7.05477	−0.315815	−0.157907	+	0.987454i	$0.550475\pi$
$500$	0	0
$501$	−30.2348	−1.35079
$502$	0	0
$503$	−35.6081	−1.58769	−0.793843	−	0.608123i	$-0.791923\pi$
$503$	−35.6081	−1.58769	−0.793843	+	0.608123i	$0.791923\pi$
$504$	0	0
$505$	18.9437	0.842982
$506$	0	0
$507$	70.2447	3.11968
$508$	0	0
$509$	25.0512	1.11037	0.555187	−	0.831725i	$-0.312647\pi$
$509$	25.0512	1.11037	0.555187	+	0.831725i	$0.312647\pi$
$510$	0	0
$511$	24.7693	1.09573
$512$	0	0
$513$	−74.2230	−3.27702
$514$	0	0
$515$	−45.9830	−2.02625
$516$	0	0
$517$	35.9660	1.58178
$518$	0	0
$519$	−71.9335	−3.15753
$520$	0	0
$521$	−7.69560	−0.337151	−0.168575	−	0.985689i	$-0.553917\pi$
$521$	−7.69560	−0.337151	−0.168575	+	0.985689i	$0.553917\pi$
$522$	0	0
$523$	3.75116	0.164027	0.0820134	−	0.996631i	$-0.473865\pi$
$523$	3.75116	0.164027	0.0820134	+	0.996631i	$0.473865\pi$
$524$	0	0
$525$	15.9305	0.695264
$526$	0	0
$527$	1.56695	0.0682575
$528$	0	0
$529$	−18.6371	−0.810311
$530$	0	0
$531$	−7.64104	−0.331593
$532$	0	0
$533$	−14.2905	−0.618991
$534$	0	0
$535$	−13.4492	−0.581461
$536$	0	0
$537$	16.8785	0.728362
$538$	0	0
$539$	−15.4329	−0.664742
$540$	0	0
$541$	−29.9452	−1.28745	−0.643723	−	0.765258i	$-0.722611\pi$
$541$	−29.9452	−1.28745	−0.643723	+	0.765258i	$0.722611\pi$
$542$	0	0
$543$	75.7460	3.25057
$544$	0	0
$545$	−48.2122	−2.06518
$546$	0	0
$547$	20.6565	0.883207	0.441603	−	0.897210i	$-0.354410\pi$
$547$	20.6565	0.883207	0.441603	+	0.897210i	$0.354410\pi$
$548$	0	0
$549$	95.0701	4.05749
$550$	0	0
$551$	45.5025	1.93847
$552$	0	0
$553$	−28.7340	−1.22189
$554$	0	0
$555$	81.1082	3.44285
$556$	0	0
$557$	25.7648	1.09169	0.545846	−	0.837886i	$-0.316209\pi$
$557$	25.7648	1.09169	0.545846	+	0.837886i	$0.316209\pi$
$558$	0	0
$559$	−7.96256	−0.336780
$560$	0	0
$561$	−16.4531	−0.694651
$562$	0	0
$563$	6.36255	0.268149	0.134075	−	0.990971i	$-0.457194\pi$
$563$	6.36255	0.268149	0.134075	+	0.990971i	$0.457194\pi$
$564$	0	0
$565$	−12.7004	−0.534311
$566$	0	0
$567$	52.5278	2.20596
$568$	0	0
$569$	−9.79063	−0.410445	−0.205222	−	0.978715i	$-0.565792\pi$
$569$	−9.79063	−0.410445	−0.205222	+	0.978715i	$0.565792\pi$
$570$	0	0
$571$	1.94488	0.0813905	0.0406952	−	0.999172i	$-0.487043\pi$
$571$	1.94488	0.0813905	0.0406952	+	0.999172i	$0.487043\pi$
$572$	0	0
$573$	46.1162	1.92653
$574$	0	0
$575$	−5.13882	−0.214304
$576$	0	0
$577$	−21.1215	−0.879300	−0.439650	−	0.898169i	$-0.644898\pi$
$577$	−21.1215	−0.879300	−0.439650	+	0.898169i	$0.644898\pi$
$578$	0	0
$579$	20.1964	0.839332
$580$	0	0
$581$	−12.1318	−0.503311
$582$	0	0
$583$	8.05108	0.333442
$584$	0	0
$585$	−122.645	−5.07077
$586$	0	0
$587$	41.7072	1.72144	0.860719	−	0.509080i	$-0.170014\pi$
$587$	41.7072	1.72144	0.860719	+	0.509080i	$0.170014\pi$
$588$	0	0
$589$	7.68221	0.316540
$590$	0	0
$591$	−63.4161	−2.60859
$592$	0	0
$593$	28.9090	1.18715	0.593575	−	0.804778i	$-0.297716\pi$
$593$	28.9090	1.18715	0.593575	+	0.804778i	$0.297716\pi$
$594$	0	0
$595$	−5.42171	−0.222268
$596$	0	0
$597$	−55.5402	−2.27311
$598$	0	0
$599$	36.6244	1.49643	0.748216	−	0.663455i	$-0.230911\pi$
$599$	36.6244	1.49643	0.748216	+	0.663455i	$0.230911\pi$
$600$	0	0
$601$	21.1248	0.861699	0.430850	−	0.902424i	$-0.358214\pi$
$601$	21.1248	0.861699	0.430850	+	0.902424i	$0.358214\pi$
$602$	0	0
$603$	16.1358	0.657100
$604$	0	0
$605$	39.4397	1.60345
$606$	0	0
$607$	13.1930	0.535487	0.267743	−	0.963490i	$-0.413722\pi$
$607$	13.1930	0.535487	0.267743	+	0.963490i	$0.413722\pi$
$608$	0	0
$609$	−60.0975	−2.43527
$610$	0	0
$611$	−41.9044	−1.69527
$612$	0	0
$613$	45.6621	1.84427	0.922137	−	0.386863i	$-0.126441\pi$
$613$	45.6621	1.84427	0.922137	+	0.386863i	$0.126441\pi$
$614$	0	0
$615$	21.6668	0.873688
$616$	0	0
$617$	10.2173	0.411334	0.205667	−	0.978622i	$-0.434064\pi$
$617$	10.2173	0.411334	0.205667	+	0.978622i	$0.434064\pi$
$618$	0	0
$619$	4.36232	0.175336	0.0876681	−	0.996150i	$-0.472058\pi$
$619$	4.36232	0.175336	0.0876681	+	0.996150i	$0.472058\pi$
$620$	0	0
$621$	−31.6223	−1.26896
$622$	0	0
$623$	14.0046	0.561084
$624$	0	0
$625$	−31.2484	−1.24994
$626$	0	0
$627$	−80.6637	−3.22140
$628$	0	0
$629$	−9.10324	−0.362970
$630$	0	0
$631$	18.9876	0.755886	0.377943	−	0.925829i	$-0.376632\pi$
$631$	18.9876	0.755886	0.377943	+	0.925829i	$0.376632\pi$
$632$	0	0
$633$	−10.4960	−0.417178
$634$	0	0
$635$	−9.22061	−0.365909
$636$	0	0
$637$	17.9810	0.712434
$638$	0	0
$639$	−60.3412	−2.38706
$640$	0	0
$641$	26.8272	1.05961	0.529806	−	0.848119i	$-0.322265\pi$
$641$	26.8272	1.05961	0.529806	+	0.848119i	$0.322265\pi$
$642$	0	0
$643$	10.4962	0.413929	0.206965	−	0.978348i	$-0.433642\pi$
$643$	10.4962	0.413929	0.206965	+	0.978348i	$0.433642\pi$
$644$	0	0
$645$	12.0725	0.475356
$646$	0	0
$647$	29.2803	1.15113	0.575564	−	0.817756i	$-0.304782\pi$
$647$	29.2803	1.15113	0.575564	+	0.817756i	$0.304782\pi$
$648$	0	0
$649$	−5.04378	−0.197986
$650$	0	0
$651$	−10.1463	−0.397665
$652$	0	0
$653$	0.138871	0.00543443	0.00271722	−	0.999996i	$-0.499135\pi$
$653$	0.138871	0.00543443	0.00271722	+	0.999996i	$0.499135\pi$
$654$	0	0
$655$	−10.3529	−0.404519
$656$	0	0
$657$	95.3471	3.71985
$658$	0	0
$659$	−32.9659	−1.28417	−0.642084	−	0.766634i	$-0.721930\pi$
$659$	−32.9659	−1.28417	−0.642084	+	0.766634i	$0.721930\pi$
$660$	0	0
$661$	−43.8751	−1.70654	−0.853272	−	0.521466i	$-0.825385\pi$
$661$	−43.8751	−1.70654	−0.853272	+	0.521466i	$0.825385\pi$
$662$	0	0
$663$	19.1697	0.744488
$664$	0	0
$665$	−26.5807	−1.03076
$666$	0	0
$667$	19.3861	0.750632
$668$	0	0
$669$	−4.33793	−0.167714
$670$	0	0
$671$	62.7548	2.42262
$672$	0	0
$673$	−15.8775	−0.612034	−0.306017	−	0.952026i	$-0.598996\pi$
$673$	−15.8775	−0.612034	−0.306017	+	0.952026i	$0.598996\pi$
$674$	0	0
$675$	37.2465	1.43362
$676$	0	0
$677$	34.5007	1.32597	0.662985	−	0.748632i	$-0.269289\pi$
$677$	34.5007	1.32597	0.662985	+	0.748632i	$0.269289\pi$
$678$	0	0
$679$	−10.8379	−0.415920
$680$	0	0
$681$	61.9226	2.37288
$682$	0	0
$683$	37.5796	1.43794	0.718972	−	0.695039i	$-0.244613\pi$
$683$	37.5796	1.43794	0.718972	+	0.695039i	$0.244613\pi$
$684$	0	0
$685$	−43.7615	−1.67204
$686$	0	0
$687$	8.94515	0.341279
$688$	0	0
$689$	−9.38039	−0.357364
$690$	0	0
$691$	0.812044	0.0308916	0.0154458	−	0.999881i	$-0.495083\pi$
$691$	0.812044	0.0308916	0.0154458	+	0.999881i	$0.495083\pi$
$692$	0	0
$693$	76.5012	2.90604
$694$	0	0
$695$	40.8882	1.55098
$696$	0	0
$697$	−2.43179	−0.0921104
$698$	0	0
$699$	78.6053	2.97313
$700$	0	0
$701$	5.45158	0.205903	0.102952	−	0.994686i	$-0.467171\pi$
$701$	5.45158	0.205903	0.102952	+	0.994686i	$0.467171\pi$
$702$	0	0
$703$	−44.6300	−1.68325
$704$	0	0
$705$	63.5339	2.39282
$706$	0	0
$707$	13.7672	0.517771
$708$	0	0
$709$	37.8612	1.42191	0.710953	−	0.703239i	$-0.248264\pi$
$709$	37.8612	1.42191	0.710953	+	0.703239i	$0.248264\pi$
$710$	0	0
$711$	−110.609	−4.14814
$712$	0	0
$713$	3.27297	0.122574
$714$	0	0
$715$	−80.9570	−3.02762
$716$	0	0
$717$	47.2415	1.76427
$718$	0	0
$719$	−7.87209	−0.293579	−0.146790	−	0.989168i	$-0.546894\pi$
$719$	−7.87209	−0.293579	−0.146790	+	0.989168i	$0.546894\pi$
$720$	0	0
$721$	−33.4180	−1.24455
$722$	0	0
$723$	−4.72172	−0.175603
$724$	0	0
$725$	−22.8340	−0.848034
$726$	0	0
$727$	38.2968	1.42035	0.710174	−	0.704026i	$-0.248616\pi$
$727$	38.2968	1.42035	0.710174	+	0.704026i	$0.248616\pi$
$728$	0	0
$729$	54.0439	2.00163
$730$	0	0
$731$	−1.35497	−0.0501154
$732$	0	0
$733$	−45.6001	−1.68428	−0.842138	−	0.539261i	$-0.818703\pi$
$733$	−45.6001	−1.68428	−0.842138	+	0.539261i	$0.818703\pi$
$734$	0	0
$735$	−27.2622	−1.00558
$736$	0	0
$737$	10.6511	0.392337
$738$	0	0
$739$	39.1775	1.44117	0.720584	−	0.693367i	$-0.243874\pi$
$739$	39.1775	1.44117	0.720584	+	0.693367i	$0.243874\pi$
$740$	0	0
$741$	93.9820	3.45252
$742$	0	0
$743$	−31.2294	−1.14569	−0.572847	−	0.819662i	$-0.694161\pi$
$743$	−31.2294	−1.14569	−0.572847	+	0.819662i	$0.694161\pi$
$744$	0	0
$745$	−42.7596	−1.56659
$746$	0	0
$747$	−46.7001	−1.70867
$748$	0	0
$749$	−9.77418	−0.357141
$750$	0	0
$751$	18.7468	0.684079	0.342039	−	0.939686i	$-0.388882\pi$
$751$	18.7468	0.684079	0.342039	+	0.939686i	$0.388882\pi$
$752$	0	0
$753$	−42.4667	−1.54757
$754$	0	0
$755$	20.6651	0.752081
$756$	0	0
$757$	−47.1556	−1.71390	−0.856949	−	0.515401i	$-0.827643\pi$
$757$	−47.1556	−1.71390	−0.856949	+	0.515401i	$0.827643\pi$
$758$	0	0
$759$	−34.3663	−1.24742
$760$	0	0
$761$	−34.5479	−1.25236	−0.626181	−	0.779678i	$-0.715383\pi$
$761$	−34.5479	−1.25236	−0.626181	+	0.779678i	$0.715383\pi$
$762$	0	0
$763$	−35.0380	−1.26846
$764$	0	0
$765$	−20.8703	−0.754568
$766$	0	0
$767$	5.87655	0.212190
$768$	0	0
$769$	−13.7171	−0.494653	−0.247326	−	0.968932i	$-0.579552\pi$
$769$	−13.7171	−0.494653	−0.247326	+	0.968932i	$0.579552\pi$
$770$	0	0
$771$	−69.7250	−2.51109
$772$	0	0
$773$	−18.6330	−0.670183	−0.335092	−	0.942186i	$-0.608767\pi$
$773$	−18.6330	−0.670183	−0.335092	+	0.942186i	$0.608767\pi$
$774$	0	0
$775$	−3.85508	−0.138479
$776$	0	0
$777$	58.9451	2.11464
$778$	0	0
$779$	−11.9222	−0.427156
$780$	0	0
$781$	−39.8306	−1.42525
$782$	0	0
$783$	−140.512	−5.02148
$784$	0	0
$785$	26.0233	0.928810
$786$	0	0
$787$	−3.61111	−0.128722	−0.0643611	−	0.997927i	$-0.520501\pi$
$787$	−3.61111	−0.128722	−0.0643611	+	0.997927i	$0.520501\pi$
$788$	0	0
$789$	85.9651	3.06044
$790$	0	0
$791$	−9.23000	−0.328181
$792$	0	0
$793$	−73.1162	−2.59643
$794$	0	0
$795$	14.2222	0.504410
$796$	0	0
$797$	−26.0644	−0.923248	−0.461624	−	0.887076i	$-0.652733\pi$
$797$	−26.0644	−0.923248	−0.461624	+	0.887076i	$0.652733\pi$
$798$	0	0
$799$	−7.13078	−0.252269
$800$	0	0
$801$	53.9095	1.90480
$802$	0	0
$803$	62.9377	2.22102
$804$	0	0
$805$	−11.3246	−0.399138
$806$	0	0
$807$	28.8496	1.01555
$808$	0	0
$809$	37.5765	1.32112	0.660559	−	0.750774i	$-0.270319\pi$
$809$	37.5765	1.32112	0.660559	+	0.750774i	$0.270319\pi$
$810$	0	0
$811$	35.8582	1.25915	0.629575	−	0.776940i	$-0.283229\pi$
$811$	35.8582	1.25915	0.629575	+	0.776940i	$0.283229\pi$
$812$	0	0
$813$	−59.0784	−2.07197
$814$	0	0
$815$	21.2491	0.744323
$816$	0	0
$817$	−6.64294	−0.232407
$818$	0	0
$819$	−89.1322	−3.11453
$820$	0	0
$821$	18.8727	0.658660	0.329330	−	0.944215i	$-0.393177\pi$
$821$	18.8727	0.658660	0.329330	+	0.944215i	$0.393177\pi$
$822$	0	0
$823$	41.0825	1.43205	0.716024	−	0.698076i	$-0.245960\pi$
$823$	41.0825	1.43205	0.716024	+	0.698076i	$0.245960\pi$
$824$	0	0
$825$	40.4786	1.40928
$826$	0	0
$827$	49.5289	1.72229	0.861144	−	0.508361i	$-0.169748\pi$
$827$	49.5289	1.72229	0.861144	+	0.508361i	$0.169748\pi$
$828$	0	0
$829$	−16.2025	−0.562736	−0.281368	−	0.959600i	$-0.590788\pi$
$829$	−16.2025	−0.562736	−0.281368	+	0.959600i	$0.590788\pi$
$830$	0	0
$831$	65.3769	2.26790
$832$	0	0
$833$	3.05979	0.106015
$834$	0	0
$835$	−25.3158	−0.876088
$836$	0	0
$837$	−23.7227	−0.819976
$838$	0	0
$839$	−5.62866	−0.194323	−0.0971615	−	0.995269i	$-0.530976\pi$
$839$	−5.62866	−0.194323	−0.0971615	+	0.995269i	$0.530976\pi$
$840$	0	0
$841$	57.1409	1.97037
$842$	0	0
$843$	40.9216	1.40941
$844$	0	0
$845$	58.8163	2.02334
$846$	0	0
$847$	28.6627	0.984862
$848$	0	0
$849$	11.9004	0.408422
$850$	0	0
$851$	−19.0144	−0.651804
$852$	0	0
$853$	5.82933	0.199592	0.0997961	−	0.995008i	$-0.468181\pi$
$853$	5.82933	0.199592	0.0997961	+	0.995008i	$0.468181\pi$
$854$	0	0
$855$	−102.320	−3.49926
$856$	0	0
$857$	−12.5286	−0.427970	−0.213985	−	0.976837i	$-0.568644\pi$
$857$	−12.5286	−0.427970	−0.213985	+	0.976837i	$0.568644\pi$
$858$	0	0
$859$	−8.82229	−0.301013	−0.150506	−	0.988609i	$-0.548090\pi$
$859$	−8.82229	−0.301013	−0.150506	+	0.988609i	$0.548090\pi$
$860$	0	0
$861$	15.7462	0.536631
$862$	0	0
$863$	−25.7430	−0.876303	−0.438152	−	0.898901i	$-0.644367\pi$
$863$	−25.7430	−0.876303	−0.438152	+	0.898901i	$0.644367\pi$
$864$	0	0
$865$	−60.2304	−2.04789
$866$	0	0
$867$	3.26206	0.110785
$868$	0	0
$869$	−73.0116	−2.47675
$870$	0	0
$871$	−12.4097	−0.420485
$872$	0	0
$873$	−41.7194	−1.41199
$874$	0	0
$875$	−13.7698	−0.465505
$876$	0	0
$877$	−1.81581	−0.0613157	−0.0306578	−	0.999530i	$-0.509760\pi$
$877$	−1.81581	−0.0613157	−0.0306578	+	0.999530i	$0.509760\pi$
$878$	0	0
$879$	−44.2404	−1.49219
$880$	0	0
$881$	8.30096	0.279666	0.139833	−	0.990175i	$-0.455343\pi$
$881$	8.30096	0.279666	0.139833	+	0.990175i	$0.455343\pi$
$882$	0	0
$883$	−28.6914	−0.965544	−0.482772	−	0.875746i	$-0.660370\pi$
$883$	−28.6914	−0.965544	−0.482772	+	0.875746i	$0.660370\pi$
$884$	0	0
$885$	−8.90981	−0.299500
$886$	0	0
$887$	7.55970	0.253830	0.126915	−	0.991914i	$-0.459492\pi$
$887$	7.55970	0.253830	0.126915	+	0.991914i	$0.459492\pi$
$888$	0	0
$889$	−6.70105	−0.224746
$890$	0	0
$891$	133.471	4.47143
$892$	0	0
$893$	−34.9597	−1.16988
$894$	0	0
$895$	14.1325	0.472397
$896$	0	0
$897$	40.0405	1.33691
$898$	0	0
$899$	14.5432	0.485044
$900$	0	0
$901$	−1.59624	−0.0531785
$902$	0	0
$903$	8.77368	0.291970
$904$	0	0
$905$	63.4226	2.10824
$906$	0	0
$907$	32.3825	1.07524	0.537622	−	0.843186i	$-0.319323\pi$
$907$	32.3825	1.07524	0.537622	+	0.843186i	$0.319323\pi$
$908$	0	0
$909$	52.9956	1.75775
$910$	0	0
$911$	−48.7786	−1.61611	−0.808053	−	0.589109i	$-0.799479\pi$
$911$	−48.7786	−1.61611	−0.808053	+	0.589109i	$0.799479\pi$
$912$	0	0
$913$	−30.8262	−1.02020
$914$	0	0
$915$	110.856	3.66479
$916$	0	0
$917$	−7.52390	−0.248461
$918$	0	0
$919$	−45.4502	−1.49927	−0.749633	−	0.661854i	$-0.769770\pi$
$919$	−45.4502	−1.49927	−0.749633	+	0.661854i	$0.769770\pi$
$920$	0	0
$921$	−75.1935	−2.47771
$922$	0	0
$923$	46.4070	1.52751
$924$	0	0
$925$	22.3962	0.736382
$926$	0	0
$927$	−128.639	−4.22507
$928$	0	0
$929$	−22.7427	−0.746162	−0.373081	−	0.927799i	$-0.621699\pi$
$929$	−22.7427	−0.746162	−0.373081	+	0.927799i	$0.621699\pi$
$930$	0	0
$931$	15.0011	0.491640
$932$	0	0
$933$	25.2336	0.826110
$934$	0	0
$935$	−13.7763	−0.450533
$936$	0	0
$937$	10.7593	0.351491	0.175745	−	0.984436i	$-0.443766\pi$
$937$	10.7593	0.351491	0.175745	+	0.984436i	$0.443766\pi$
$938$	0	0
$939$	97.6400	3.18636
$940$	0	0
$941$	31.5621	1.02890	0.514448	−	0.857522i	$-0.327997\pi$
$941$	31.5621	1.02890	0.514448	+	0.857522i	$0.327997\pi$
$942$	0	0
$943$	−5.07938	−0.165407
$944$	0	0
$945$	82.0813	2.67010
$946$	0	0
$947$	−27.9806	−0.909246	−0.454623	−	0.890684i	$-0.650226\pi$
$947$	−27.9806	−0.909246	−0.454623	+	0.890684i	$0.650226\pi$
$948$	0	0
$949$	−73.3293	−2.38037
$950$	0	0
$951$	−68.4025	−2.21810
$952$	0	0
$953$	−50.0268	−1.62053	−0.810264	−	0.586066i	$-0.800676\pi$
$953$	−50.0268	−1.62053	−0.810264	+	0.586066i	$0.800676\pi$
$954$	0	0
$955$	38.6134	1.24950
$956$	0	0
$957$	−152.705	−4.93624
$958$	0	0
$959$	−31.8035	−1.02699
$960$	0	0
$961$	−28.5447	−0.920795
$962$	0	0
$963$	−37.6247	−1.21244
$964$	0	0
$965$	16.9105	0.544369
$966$	0	0
$967$	−1.18071	−0.0379691	−0.0189845	−	0.999820i	$-0.506043\pi$
$967$	−1.18071	−0.0379691	−0.0189845	+	0.999820i	$0.506043\pi$
$968$	0	0
$969$	15.9927	0.513760
$970$	0	0
$971$	34.9116	1.12037	0.560184	−	0.828368i	$-0.310730\pi$
$971$	34.9116	1.12037	0.560184	+	0.828368i	$0.310730\pi$
$972$	0	0
$973$	29.7154	0.952630
$974$	0	0
$975$	−47.1620	−1.51039
$976$	0	0
$977$	−1.41127	−0.0451506	−0.0225753	−	0.999745i	$-0.507187\pi$
$977$	−1.41127	−0.0451506	−0.0225753	+	0.999745i	$0.507187\pi$
$978$	0	0
$979$	35.5851	1.13730
$980$	0	0
$981$	−134.875	−4.30624
$982$	0	0
$983$	41.8551	1.33497	0.667485	−	0.744624i	$-0.267371\pi$
$983$	41.8551	1.33497	0.667485	+	0.744624i	$0.267371\pi$
$984$	0	0
$985$	−53.0987	−1.69186
$986$	0	0
$987$	46.1731	1.46970
$988$	0	0
$989$	−2.83019	−0.0899948
$990$	0	0
$991$	18.4166	0.585022	0.292511	−	0.956262i	$-0.405509\pi$
$991$	18.4166	0.585022	0.292511	+	0.956262i	$0.405509\pi$
$992$	0	0
$993$	−19.5285	−0.619717
$994$	0	0
$995$	−46.5042	−1.47428
$996$	0	0
$997$	−46.3765	−1.46876	−0.734379	−	0.678739i	$-0.762527\pi$
$997$	−46.3765	−1.46876	−0.734379	+	0.678739i	$0.762527\pi$
$998$	0	0
$999$	137.817	4.36035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.18	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.18	✓	18	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+3.26206 q^{3} +2.73134 q^{5} +1.98500 q^{7} +7.64104 q^{9} +O(q^{10})\)	\(q+3.26206 q^{3} +2.73134 q^{5} +1.98500 q^{7} +7.64104 q^{9} +5.04378 q^{11} -5.87655 q^{13} +8.90981 q^{15} -1.00000 q^{17} -4.90264 q^{19} +6.47518 q^{21} -2.08875 q^{23} +2.46024 q^{25} +15.1394 q^{27} -9.28121 q^{29} -1.56695 q^{31} +16.4531 q^{33} +5.42171 q^{35} +9.10324 q^{37} -19.1697 q^{39} +2.43179 q^{41} +1.35497 q^{43} +20.8703 q^{45} +7.13078 q^{47} -3.05979 q^{49} -3.26206 q^{51} +1.59624 q^{53} +13.7763 q^{55} -15.9927 q^{57} -1.00000 q^{59} +12.4420 q^{61} +15.1674 q^{63} -16.0509 q^{65} +2.11172 q^{67} -6.81362 q^{69} -7.89699 q^{71} +12.4783 q^{73} +8.02546 q^{75} +10.0119 q^{77} -14.4756 q^{79} +26.4624 q^{81} -6.11174 q^{83} -2.73134 q^{85} -30.2759 q^{87} +7.05525 q^{89} -11.6649 q^{91} -5.11150 q^{93} -13.3908 q^{95} -5.45991 q^{97} +38.5397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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