LMFDB - Embedded newform 4012.2.a.i.1.17

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			1.17
Root			$3.05303$ 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)

$f(q)$	$=$	$q+3.05303 q^{3} +2.19982 q^{5} -4.80385 q^{7} +6.32102 q^{9} +O(q^{10})$	$q+3.05303 q^{3} +2.19982 q^{5} -4.80385 q^{7} +6.32102 q^{9} +4.15004 q^{11} +6.02533 q^{13} +6.71612 q^{15} -1.00000 q^{17} +1.81168 q^{19} -14.6663 q^{21} -0.482167 q^{23} -0.160801 q^{25} +10.1392 q^{27} +1.65810 q^{29} -2.13354 q^{31} +12.6702 q^{33} -10.5676 q^{35} -3.95502 q^{37} +18.3955 q^{39} +0.592414 q^{41} -1.38453 q^{43} +13.9051 q^{45} +11.2033 q^{47} +16.0770 q^{49} -3.05303 q^{51} -11.0941 q^{53} +9.12933 q^{55} +5.53112 q^{57} -1.00000 q^{59} -8.98341 q^{61} -30.3652 q^{63} +13.2546 q^{65} -8.06527 q^{67} -1.47207 q^{69} +9.62281 q^{71} +4.21912 q^{73} -0.490932 q^{75} -19.9362 q^{77} +2.33941 q^{79} +11.9922 q^{81} +6.57638 q^{83} -2.19982 q^{85} +5.06223 q^{87} -6.86734 q^{89} -28.9448 q^{91} -6.51378 q^{93} +3.98537 q^{95} +12.6274 q^{97} +26.2325 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.05303	1.76267	0.881335	−	0.472492i	$-0.156645\pi$
$3$	3.05303	1.76267	0.881335	+	0.472492i	$0.156645\pi$
$4$	0	0
$5$	2.19982	0.983788	0.491894	−	0.870655i	$-0.336305\pi$
$5$	2.19982	0.983788	0.491894	+	0.870655i	$0.336305\pi$
$6$	0	0
$7$	−4.80385	−1.81568	−0.907842	−	0.419313i	$-0.862271\pi$
$7$	−4.80385	−1.81568	−0.907842	+	0.419313i	$0.862271\pi$
$8$	0	0
$9$	6.32102	2.10701
$10$	0	0
$11$	4.15004	1.25128	0.625642	−	0.780110i	$-0.284837\pi$
$11$	4.15004	1.25128	0.625642	+	0.780110i	$0.284837\pi$
$12$	0	0
$13$	6.02533	1.67113	0.835563	−	0.549395i	$-0.185142\pi$
$13$	6.02533	1.67113	0.835563	+	0.549395i	$0.185142\pi$
$14$	0	0
$15$	6.71612	1.73409
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.81168	0.415628	0.207814	−	0.978168i	$-0.433365\pi$
$19$	1.81168	0.415628	0.207814	+	0.978168i	$0.433365\pi$
$20$	0	0
$21$	−14.6663	−3.20045
$22$	0	0
$23$	−0.482167	−0.100539	−0.0502694	−	0.998736i	$-0.516008\pi$
$23$	−0.482167	−0.100539	−0.0502694	+	0.998736i	$0.516008\pi$
$24$	0	0
$25$	−0.160801	−0.0321603
$26$	0	0
$27$	10.1392	1.95129
$28$	0	0
$29$	1.65810	0.307901	0.153950	−	0.988079i	$-0.450800\pi$
$29$	1.65810	0.307901	0.153950	+	0.988079i	$0.450800\pi$
$30$	0	0
$31$	−2.13354	−0.383195	−0.191598	−	0.981474i	$-0.561367\pi$
$31$	−2.13354	−0.383195	−0.191598	+	0.981474i	$0.561367\pi$
$32$	0	0
$33$	12.6702	2.20560
$34$	0	0
$35$	−10.5676	−1.78625
$36$	0	0
$37$	−3.95502	−0.650202	−0.325101	−	0.945679i	$-0.605398\pi$
$37$	−3.95502	−0.650202	−0.325101	+	0.945679i	$0.605398\pi$
$38$	0	0
$39$	18.3955	2.94564
$40$	0	0
$41$	0.592414	0.0925195	0.0462598	−	0.998929i	$-0.485270\pi$
$41$	0.592414	0.0925195	0.0462598	+	0.998929i	$0.485270\pi$
$42$	0	0
$43$	−1.38453	−0.211138	−0.105569	−	0.994412i	$-0.533666\pi$
$43$	−1.38453	−0.211138	−0.105569	+	0.994412i	$0.533666\pi$
$44$	0	0
$45$	13.9051	2.07285
$46$	0	0
$47$	11.2033	1.63417	0.817085	−	0.576517i	$-0.195589\pi$
$47$	11.2033	1.63417	0.817085	+	0.576517i	$0.195589\pi$
$48$	0	0
$49$	16.0770	2.29671
$50$	0	0
$51$	−3.05303	−0.427510
$52$	0	0
$53$	−11.0941	−1.52390	−0.761948	−	0.647638i	$-0.775757\pi$
$53$	−11.0941	−1.52390	−0.761948	+	0.647638i	$0.775757\pi$
$54$	0	0
$55$	9.12933	1.23100
$56$	0	0
$57$	5.53112	0.732615
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−8.98341	−1.15021	−0.575104	−	0.818080i	$-0.695038\pi$
$61$	−8.98341	−1.15021	−0.575104	+	0.818080i	$0.695038\pi$
$62$	0	0
$63$	−30.3652	−3.82566
$64$	0	0
$65$	13.2546	1.64403
$66$	0	0
$67$	−8.06527	−0.985330	−0.492665	−	0.870219i	$-0.663977\pi$
$67$	−8.06527	−0.985330	−0.492665	+	0.870219i	$0.663977\pi$
$68$	0	0
$69$	−1.47207	−0.177217
$70$	0	0
$71$	9.62281	1.14202	0.571009	−	0.820944i	$-0.306552\pi$
$71$	9.62281	1.14202	0.571009	+	0.820944i	$0.306552\pi$
$72$	0	0
$73$	4.21912	0.493810	0.246905	−	0.969040i	$-0.420586\pi$
$73$	4.21912	0.493810	0.246905	+	0.969040i	$0.420586\pi$
$74$	0	0
$75$	−0.490932	−0.0566880
$76$	0	0
$77$	−19.9362	−2.27194
$78$	0	0
$79$	2.33941	0.263204	0.131602	−	0.991303i	$-0.457988\pi$
$79$	2.33941	0.263204	0.131602	+	0.991303i	$0.457988\pi$
$80$	0	0
$81$	11.9922	1.33247
$82$	0	0
$83$	6.57638	0.721851	0.360925	−	0.932595i	$-0.382461\pi$
$83$	6.57638	0.721851	0.360925	+	0.932595i	$0.382461\pi$
$84$	0	0
$85$	−2.19982	−0.238604
$86$	0	0
$87$	5.06223	0.542728
$88$	0	0
$89$	−6.86734	−0.727937	−0.363968	−	0.931411i	$-0.618578\pi$
$89$	−6.86734	−0.727937	−0.363968	+	0.931411i	$0.618578\pi$
$90$	0	0
$91$	−28.9448	−3.03424
$92$	0	0
$93$	−6.51378	−0.675447
$94$	0	0
$95$	3.98537	0.408890
$96$	0	0
$97$	12.6274	1.28212	0.641061	−	0.767490i	$-0.278495\pi$
$97$	12.6274	1.28212	0.641061	+	0.767490i	$0.278495\pi$
$98$	0	0
$99$	26.2325	2.63646
$100$	0	0
$101$	−14.0313	−1.39617	−0.698083	−	0.716016i	$-0.745964\pi$
$101$	−14.0313	−1.39617	−0.698083	+	0.716016i	$0.745964\pi$
$102$	0	0
$103$	10.8543	1.06951	0.534755	−	0.845007i	$-0.320404\pi$
$103$	10.8543	1.06951	0.534755	+	0.845007i	$0.320404\pi$
$104$	0	0
$105$	−32.2632	−3.14857
$106$	0	0
$107$	5.75983	0.556824	0.278412	−	0.960462i	$-0.410192\pi$
$107$	5.75983	0.556824	0.278412	+	0.960462i	$0.410192\pi$
$108$	0	0
$109$	−11.2481	−1.07738	−0.538689	−	0.842505i	$-0.681080\pi$
$109$	−11.2481	−1.07738	−0.538689	+	0.842505i	$0.681080\pi$
$110$	0	0
$111$	−12.0748	−1.14609
$112$	0	0
$113$	−9.99479	−0.940231	−0.470115	−	0.882605i	$-0.655788\pi$
$113$	−9.99479	−0.940231	−0.470115	+	0.882605i	$0.655788\pi$
$114$	0	0
$115$	−1.06068	−0.0989090
$116$	0	0
$117$	38.0862	3.52107
$118$	0	0
$119$	4.80385	0.440368
$120$	0	0
$121$	6.22282	0.565711
$122$	0	0
$123$	1.80866	0.163081
$124$	0	0
$125$	−11.3528	−1.01543
$126$	0	0
$127$	19.6199	1.74098	0.870491	−	0.492185i	$-0.163802\pi$
$127$	19.6199	1.74098	0.870491	+	0.492185i	$0.163802\pi$
$128$	0	0
$129$	−4.22701	−0.372167
$130$	0	0
$131$	9.13382	0.798025	0.399013	−	0.916945i	$-0.369353\pi$
$131$	9.13382	0.798025	0.399013	+	0.916945i	$0.369353\pi$
$132$	0	0
$133$	−8.70304	−0.754649
$134$	0	0
$135$	22.3044	1.91965
$136$	0	0
$137$	−1.50005	−0.128158	−0.0640788	−	0.997945i	$-0.520411\pi$
$137$	−1.50005	−0.128158	−0.0640788	+	0.997945i	$0.520411\pi$
$138$	0	0
$139$	18.2152	1.54500	0.772499	−	0.635016i	$-0.219007\pi$
$139$	18.2152	1.54500	0.772499	+	0.635016i	$0.219007\pi$
$140$	0	0
$141$	34.2041	2.88050
$142$	0	0
$143$	25.0054	2.09105
$144$	0	0
$145$	3.64751	0.302909
$146$	0	0
$147$	49.0835	4.04834
$148$	0	0
$149$	−14.9208	−1.22236	−0.611178	−	0.791493i	$-0.709304\pi$
$149$	−14.9208	−1.22236	−0.611178	+	0.791493i	$0.709304\pi$
$150$	0	0
$151$	−20.4881	−1.66730	−0.833649	−	0.552295i	$-0.813752\pi$
$151$	−20.4881	−1.66730	−0.833649	+	0.552295i	$0.813752\pi$
$152$	0	0
$153$	−6.32102	−0.511024
$154$	0	0
$155$	−4.69340	−0.376983
$156$	0	0
$157$	−1.04478	−0.0833824	−0.0416912	−	0.999131i	$-0.513275\pi$
$157$	−1.04478	−0.0833824	−0.0416912	+	0.999131i	$0.513275\pi$
$158$	0	0
$159$	−33.8708	−2.68613
$160$	0	0
$161$	2.31626	0.182547
$162$	0	0
$163$	19.8300	1.55321	0.776603	−	0.629991i	$-0.216941\pi$
$163$	19.8300	1.55321	0.776603	+	0.629991i	$0.216941\pi$
$164$	0	0
$165$	27.8722	2.16984
$166$	0	0
$167$	−6.24385	−0.483163	−0.241582	−	0.970380i	$-0.577666\pi$
$167$	−6.24385	−0.483163	−0.241582	+	0.970380i	$0.577666\pi$
$168$	0	0
$169$	23.3046	1.79266
$170$	0	0
$171$	11.4517	0.875731
$172$	0	0
$173$	−11.6868	−0.888532	−0.444266	−	0.895895i	$-0.646536\pi$
$173$	−11.6868	−0.888532	−0.444266	+	0.895895i	$0.646536\pi$
$174$	0	0
$175$	0.772465	0.0583929
$176$	0	0
$177$	−3.05303	−0.229480
$178$	0	0
$179$	22.9408	1.71468	0.857338	−	0.514754i	$-0.172117\pi$
$179$	22.9408	1.71468	0.857338	+	0.514754i	$0.172117\pi$
$180$	0	0
$181$	4.62015	0.343413	0.171707	−	0.985148i	$-0.445072\pi$
$181$	4.62015	0.343413	0.171707	+	0.985148i	$0.445072\pi$
$182$	0	0
$183$	−27.4267	−2.02744
$184$	0	0
$185$	−8.70033	−0.639661
$186$	0	0
$187$	−4.15004	−0.303481
$188$	0	0
$189$	−48.7071	−3.54292
$190$	0	0
$191$	9.32994	0.675090	0.337545	−	0.941309i	$-0.390403\pi$
$191$	9.32994	0.675090	0.337545	+	0.941309i	$0.390403\pi$
$192$	0	0
$193$	−14.6171	−1.05216	−0.526081	−	0.850435i	$-0.676339\pi$
$193$	−14.6171	−1.05216	−0.526081	+	0.850435i	$0.676339\pi$
$194$	0	0
$195$	40.4668	2.89789
$196$	0	0
$197$	9.97502	0.710691	0.355345	−	0.934735i	$-0.384363\pi$
$197$	9.97502	0.710691	0.355345	+	0.934735i	$0.384363\pi$
$198$	0	0
$199$	15.4259	1.09352	0.546758	−	0.837291i	$-0.315862\pi$
$199$	15.4259	1.09352	0.546758	+	0.837291i	$0.315862\pi$
$200$	0	0
$201$	−24.6236	−1.73681
$202$	0	0
$203$	−7.96525	−0.559051
$204$	0	0
$205$	1.30320	0.0910196
$206$	0	0
$207$	−3.04779	−0.211836
$208$	0	0
$209$	7.51854	0.520069
$210$	0	0
$211$	−2.19316	−0.150984	−0.0754918	−	0.997146i	$-0.524053\pi$
$211$	−2.19316	−0.150984	−0.0754918	+	0.997146i	$0.524053\pi$
$212$	0	0
$213$	29.3788	2.01300
$214$	0	0
$215$	−3.04571	−0.207715
$216$	0	0
$217$	10.2492	0.695762
$218$	0	0
$219$	12.8811	0.870424
$220$	0	0
$221$	−6.02533	−0.405308
$222$	0	0
$223$	−2.82266	−0.189019	−0.0945095	−	0.995524i	$-0.530128\pi$
$223$	−2.82266	−0.189019	−0.0945095	+	0.995524i	$0.530128\pi$
$224$	0	0
$225$	−1.01643	−0.0677619
$226$	0	0
$227$	−22.9017	−1.52004	−0.760021	−	0.649899i	$-0.774811\pi$
$227$	−22.9017	−1.52004	−0.760021	+	0.649899i	$0.774811\pi$
$228$	0	0
$229$	5.52089	0.364831	0.182415	−	0.983222i	$-0.441608\pi$
$229$	5.52089	0.364831	0.182415	+	0.983222i	$0.441608\pi$
$230$	0	0
$231$	−60.8658	−4.00467
$232$	0	0
$233$	18.0735	1.18403	0.592017	−	0.805926i	$-0.298332\pi$
$233$	18.0735	1.18403	0.592017	+	0.805926i	$0.298332\pi$
$234$	0	0
$235$	24.6452	1.60768
$236$	0	0
$237$	7.14230	0.463942
$238$	0	0
$239$	−8.34362	−0.539704	−0.269852	−	0.962902i	$-0.586975\pi$
$239$	−8.34362	−0.539704	−0.269852	+	0.962902i	$0.586975\pi$
$240$	0	0
$241$	−17.3221	−1.11581	−0.557907	−	0.829904i	$-0.688395\pi$
$241$	−17.3221	−1.11581	−0.557907	+	0.829904i	$0.688395\pi$
$242$	0	0
$243$	6.19511	0.397417
$244$	0	0
$245$	35.3664	2.25947
$246$	0	0
$247$	10.9160	0.694567
$248$	0	0
$249$	20.0779	1.27239
$250$	0	0
$251$	−22.5330	−1.42227	−0.711136	−	0.703054i	$-0.751819\pi$
$251$	−22.5330	−1.42227	−0.711136	+	0.703054i	$0.751819\pi$
$252$	0	0
$253$	−2.00101	−0.125803
$254$	0	0
$255$	−6.71612	−0.420580
$256$	0	0
$257$	10.8254	0.675270	0.337635	−	0.941277i	$-0.390373\pi$
$257$	10.8254	0.675270	0.337635	+	0.941277i	$0.390373\pi$
$258$	0	0
$259$	18.9993	1.18056
$260$	0	0
$261$	10.4809	0.648749
$262$	0	0
$263$	−27.8885	−1.71968	−0.859840	−	0.510563i	$-0.829437\pi$
$263$	−27.8885	−1.71968	−0.859840	+	0.510563i	$0.829437\pi$
$264$	0	0
$265$	−24.4051	−1.49919
$266$	0	0
$267$	−20.9662	−1.28311
$268$	0	0
$269$	−27.3193	−1.66569	−0.832844	−	0.553507i	$-0.813289\pi$
$269$	−27.3193	−1.66569	−0.832844	+	0.553507i	$0.813289\pi$
$270$	0	0
$271$	18.0727	1.09784	0.548918	−	0.835876i	$-0.315040\pi$
$271$	18.0727	1.09784	0.548918	+	0.835876i	$0.315040\pi$
$272$	0	0
$273$	−88.3694	−5.34836
$274$	0	0
$275$	−0.667332	−0.0402416
$276$	0	0
$277$	−1.45434	−0.0873826	−0.0436913	−	0.999045i	$-0.513912\pi$
$277$	−1.45434	−0.0873826	−0.0436913	+	0.999045i	$0.513912\pi$
$278$	0	0
$279$	−13.4862	−0.807395
$280$	0	0
$281$	−1.12069	−0.0668549	−0.0334275	−	0.999441i	$-0.510642\pi$
$281$	−1.12069	−0.0668549	−0.0334275	+	0.999441i	$0.510642\pi$
$282$	0	0
$283$	−30.8870	−1.83604	−0.918020	−	0.396534i	$-0.870213\pi$
$283$	−30.8870	−1.83604	−0.918020	+	0.396534i	$0.870213\pi$
$284$	0	0
$285$	12.1675	0.720738
$286$	0	0
$287$	−2.84587	−0.167986
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	38.5520	2.25996
$292$	0	0
$293$	−33.9038	−1.98068	−0.990340	−	0.138660i	$-0.955721\pi$
$293$	−33.9038	−1.98068	−0.990340	+	0.138660i	$0.955721\pi$
$294$	0	0
$295$	−2.19982	−0.128078
$296$	0	0
$297$	42.0780	2.44161
$298$	0	0
$299$	−2.90522	−0.168013
$300$	0	0
$301$	6.65105	0.383360
$302$	0	0
$303$	−42.8381	−2.46098
$304$	0	0
$305$	−19.7619	−1.13156
$306$	0	0
$307$	6.62816	0.378289	0.189145	−	0.981949i	$-0.439429\pi$
$307$	6.62816	0.378289	0.189145	+	0.981949i	$0.439429\pi$
$308$	0	0
$309$	33.1386	1.88519
$310$	0	0
$311$	−10.9758	−0.622383	−0.311192	−	0.950347i	$-0.600728\pi$
$311$	−10.9758	−0.622383	−0.311192	+	0.950347i	$0.600728\pi$
$312$	0	0
$313$	−7.34680	−0.415265	−0.207633	−	0.978207i	$-0.566576\pi$
$313$	−7.34680	−0.415265	−0.207633	+	0.978207i	$0.566576\pi$
$314$	0	0
$315$	−66.7979	−3.76364
$316$	0	0
$317$	−17.1629	−0.963964	−0.481982	−	0.876181i	$-0.660083\pi$
$317$	−17.1629	−0.963964	−0.481982	+	0.876181i	$0.660083\pi$
$318$	0	0
$319$	6.88117	0.385271
$320$	0	0
$321$	17.5850	0.981496
$322$	0	0
$323$	−1.81168	−0.100805
$324$	0	0
$325$	−0.968881	−0.0537439
$326$	0	0
$327$	−34.3410	−1.89906
$328$	0	0
$329$	−53.8190	−2.96714
$330$	0	0
$331$	2.38876	0.131298	0.0656492	−	0.997843i	$-0.479088\pi$
$331$	2.38876	0.131298	0.0656492	+	0.997843i	$0.479088\pi$
$332$	0	0
$333$	−24.9998	−1.36998
$334$	0	0
$335$	−17.7421	−0.969356
$336$	0	0
$337$	18.9556	1.03258	0.516288	−	0.856415i	$-0.327314\pi$
$337$	18.9556	1.03258	0.516288	+	0.856415i	$0.327314\pi$
$338$	0	0
$339$	−30.5144	−1.65732
$340$	0	0
$341$	−8.85428	−0.479486
$342$	0	0
$343$	−43.6043	−2.35441
$344$	0	0
$345$	−3.23829	−0.174344
$346$	0	0
$347$	−6.47254	−0.347464	−0.173732	−	0.984793i	$-0.555583\pi$
$347$	−6.47254	−0.347464	−0.173732	+	0.984793i	$0.555583\pi$
$348$	0	0
$349$	−28.9546	−1.54990	−0.774952	−	0.632020i	$-0.782226\pi$
$349$	−28.9546	−1.54990	−0.774952	+	0.632020i	$0.782226\pi$
$350$	0	0
$351$	61.0919	3.26085
$352$	0	0
$353$	29.6321	1.57716	0.788578	−	0.614935i	$-0.210818\pi$
$353$	29.6321	1.57716	0.788578	+	0.614935i	$0.210818\pi$
$354$	0	0
$355$	21.1684	1.12350
$356$	0	0
$357$	14.6663	0.776224
$358$	0	0
$359$	10.1375	0.535038	0.267519	−	0.963553i	$-0.413796\pi$
$359$	10.1375	0.535038	0.267519	+	0.963553i	$0.413796\pi$
$360$	0	0
$361$	−15.7178	−0.827253
$362$	0	0
$363$	18.9985	0.997162
$364$	0	0
$365$	9.28129	0.485805
$366$	0	0
$367$	−15.3360	−0.800531	−0.400265	−	0.916399i	$-0.631082\pi$
$367$	−15.3360	−0.800531	−0.400265	+	0.916399i	$0.631082\pi$
$368$	0	0
$369$	3.74466	0.194939
$370$	0	0
$371$	53.2945	2.76691
$372$	0	0
$373$	−19.1410	−0.991084	−0.495542	−	0.868584i	$-0.665031\pi$
$373$	−19.1410	−0.991084	−0.495542	+	0.868584i	$0.665031\pi$
$374$	0	0
$375$	−34.6606	−1.78986
$376$	0	0
$377$	9.99059	0.514541
$378$	0	0
$379$	−34.0652	−1.74981	−0.874907	−	0.484292i	$-0.839077\pi$
$379$	−34.0652	−1.74981	−0.874907	+	0.484292i	$0.839077\pi$
$380$	0	0
$381$	59.9001	3.06878
$382$	0	0
$383$	17.3513	0.886612	0.443306	−	0.896370i	$-0.353806\pi$
$383$	17.3513	0.886612	0.443306	+	0.896370i	$0.353806\pi$
$384$	0	0
$385$	−43.8559	−2.23510
$386$	0	0
$387$	−8.75162	−0.444870
$388$	0	0
$389$	5.58759	0.283302	0.141651	−	0.989917i	$-0.454759\pi$
$389$	5.58759	0.283302	0.141651	+	0.989917i	$0.454759\pi$
$390$	0	0
$391$	0.482167	0.0243843
$392$	0	0
$393$	27.8859	1.40666
$394$	0	0
$395$	5.14628	0.258937
$396$	0	0
$397$	17.2695	0.866730	0.433365	−	0.901219i	$-0.357326\pi$
$397$	17.2695	0.866730	0.433365	+	0.901219i	$0.357326\pi$
$398$	0	0
$399$	−26.5707	−1.33020
$400$	0	0
$401$	−2.71382	−0.135522	−0.0677609	−	0.997702i	$-0.521585\pi$
$401$	−2.71382	−0.135522	−0.0677609	+	0.997702i	$0.521585\pi$
$402$	0	0
$403$	−12.8553	−0.640368
$404$	0	0
$405$	26.3807	1.31087
$406$	0	0
$407$	−16.4135	−0.813587
$408$	0	0
$409$	14.5814	0.721004	0.360502	−	0.932758i	$-0.382605\pi$
$409$	14.5814	0.721004	0.360502	+	0.932758i	$0.382605\pi$
$410$	0	0
$411$	−4.57969	−0.225900
$412$	0	0
$413$	4.80385	0.236382
$414$	0	0
$415$	14.4668	0.710149
$416$	0	0
$417$	55.6118	2.72332
$418$	0	0
$419$	−14.7397	−0.720081	−0.360040	−	0.932937i	$-0.617237\pi$
$419$	−14.7397	−0.720081	−0.360040	+	0.932937i	$0.617237\pi$
$420$	0	0
$421$	−6.37465	−0.310681	−0.155341	−	0.987861i	$-0.549648\pi$
$421$	−6.37465	−0.310681	−0.155341	+	0.987861i	$0.549648\pi$
$422$	0	0
$423$	70.8163	3.44321
$424$	0	0
$425$	0.160801	0.00780001
$426$	0	0
$427$	43.1549	2.08841
$428$	0	0
$429$	76.3422	3.68584
$430$	0	0
$431$	−8.17297	−0.393678	−0.196839	−	0.980436i	$-0.563068\pi$
$431$	−8.17297	−0.393678	−0.196839	+	0.980436i	$0.563068\pi$
$432$	0	0
$433$	36.7763	1.76736	0.883679	−	0.468094i	$-0.155059\pi$
$433$	36.7763	1.76736	0.883679	+	0.468094i	$0.155059\pi$
$434$	0	0
$435$	11.1360	0.533929
$436$	0	0
$437$	−0.873533	−0.0417868
$438$	0	0
$439$	−29.5470	−1.41020	−0.705100	−	0.709108i	$-0.749098\pi$
$439$	−29.5470	−1.41020	−0.705100	+	0.709108i	$0.749098\pi$
$440$	0	0
$441$	101.623	4.83918
$442$	0	0
$443$	18.3433	0.871517	0.435759	−	0.900064i	$-0.356480\pi$
$443$	18.3433	0.871517	0.435759	+	0.900064i	$0.356480\pi$
$444$	0	0
$445$	−15.1069	−0.716136
$446$	0	0
$447$	−45.5536	−2.15461
$448$	0	0
$449$	4.15755	0.196207	0.0981035	−	0.995176i	$-0.468722\pi$
$449$	4.15755	0.196207	0.0981035	+	0.995176i	$0.468722\pi$
$450$	0	0
$451$	2.45854	0.115768
$452$	0	0
$453$	−62.5508	−2.93889
$454$	0	0
$455$	−63.6732	−2.98505
$456$	0	0
$457$	12.4475	0.582271	0.291135	−	0.956682i	$-0.405967\pi$
$457$	12.4475	0.582271	0.291135	+	0.956682i	$0.405967\pi$
$458$	0	0
$459$	−10.1392	−0.473257
$460$	0	0
$461$	5.27717	0.245782	0.122891	−	0.992420i	$-0.460783\pi$
$461$	5.27717	0.245782	0.122891	+	0.992420i	$0.460783\pi$
$462$	0	0
$463$	26.1904	1.21717	0.608585	−	0.793488i	$-0.291737\pi$
$463$	26.1904	1.21717	0.608585	+	0.793488i	$0.291737\pi$
$464$	0	0
$465$	−14.3291	−0.664497
$466$	0	0
$467$	11.3205	0.523848	0.261924	−	0.965088i	$-0.415643\pi$
$467$	11.3205	0.523848	0.261924	+	0.965088i	$0.415643\pi$
$468$	0	0
$469$	38.7443	1.78905
$470$	0	0
$471$	−3.18975	−0.146976
$472$	0	0
$473$	−5.74584	−0.264194
$474$	0	0
$475$	−0.291321	−0.0133667
$476$	0	0
$477$	−70.1262	−3.21086
$478$	0	0
$479$	−42.8295	−1.95693	−0.978464	−	0.206417i	$-0.933820\pi$
$479$	−42.8295	−1.95693	−0.978464	+	0.206417i	$0.933820\pi$
$480$	0	0
$481$	−23.8303	−1.08657
$482$	0	0
$483$	7.07162	0.321770
$484$	0	0
$485$	27.7780	1.26134
$486$	0	0
$487$	−25.8166	−1.16986	−0.584931	−	0.811083i	$-0.698879\pi$
$487$	−25.8166	−1.16986	−0.584931	+	0.811083i	$0.698879\pi$
$488$	0	0
$489$	60.5417	2.73779
$490$	0	0
$491$	−23.2625	−1.04982	−0.524911	−	0.851157i	$-0.675901\pi$
$491$	−23.2625	−1.04982	−0.524911	+	0.851157i	$0.675901\pi$
$492$	0	0
$493$	−1.65810	−0.0746770
$494$	0	0
$495$	57.7067	2.59372
$496$	0	0
$497$	−46.2265	−2.07354
$498$	0	0
$499$	3.42171	0.153177	0.0765884	−	0.997063i	$-0.475597\pi$
$499$	3.42171	0.153177	0.0765884	+	0.997063i	$0.475597\pi$
$500$	0	0
$501$	−19.0627	−0.851658
$502$	0	0
$503$	27.2849	1.21657	0.608287	−	0.793717i	$-0.291857\pi$
$503$	27.2849	1.21657	0.608287	+	0.793717i	$0.291857\pi$
$504$	0	0
$505$	−30.8663	−1.37353
$506$	0	0
$507$	71.1498	3.15987
$508$	0	0
$509$	−1.55883	−0.0690938	−0.0345469	−	0.999403i	$-0.510999\pi$
$509$	−1.55883	−0.0690938	−0.0345469	+	0.999403i	$0.510999\pi$
$510$	0	0
$511$	−20.2680	−0.896603
$512$	0	0
$513$	18.3690	0.811010
$514$	0	0
$515$	23.8776	1.05217
$516$	0	0
$517$	46.4941	2.04481
$518$	0	0
$519$	−35.6803	−1.56619
$520$	0	0
$521$	8.35338	0.365968	0.182984	−	0.983116i	$-0.441424\pi$
$521$	8.35338	0.365968	0.182984	+	0.983116i	$0.441424\pi$
$522$	0	0
$523$	−18.1366	−0.793057	−0.396529	−	0.918022i	$-0.629785\pi$
$523$	−18.1366	−0.793057	−0.396529	+	0.918022i	$0.629785\pi$
$524$	0	0
$525$	2.35836	0.102927
$526$	0	0
$527$	2.13354	0.0929385
$528$	0	0
$529$	−22.7675	−0.989892
$530$	0	0
$531$	−6.32102	−0.274309
$532$	0	0
$533$	3.56949	0.154612
$534$	0	0
$535$	12.6706	0.547797
$536$	0	0
$537$	70.0391	3.02241
$538$	0	0
$539$	66.7200	2.87383
$540$	0	0
$541$	−17.8203	−0.766154	−0.383077	−	0.923716i	$-0.625136\pi$
$541$	−17.8203	−0.766154	−0.383077	+	0.923716i	$0.625136\pi$
$542$	0	0
$543$	14.1055	0.605324
$544$	0	0
$545$	−24.7439	−1.05991
$546$	0	0
$547$	−11.6143	−0.496593	−0.248297	−	0.968684i	$-0.579871\pi$
$547$	−11.6143	−0.496593	−0.248297	+	0.968684i	$0.579871\pi$
$548$	0	0
$549$	−56.7843	−2.42349
$550$	0	0
$551$	3.00394	0.127972
$552$	0	0
$553$	−11.2382	−0.477896
$554$	0	0
$555$	−26.5624	−1.12751
$556$	0	0
$557$	3.81299	0.161562	0.0807808	−	0.996732i	$-0.474259\pi$
$557$	3.81299	0.161562	0.0807808	+	0.996732i	$0.474259\pi$
$558$	0	0
$559$	−8.34223	−0.352839
$560$	0	0
$561$	−12.6702	−0.534937
$562$	0	0
$563$	−41.1713	−1.73516	−0.867582	−	0.497295i	$-0.834327\pi$
$563$	−41.1713	−1.73516	−0.867582	+	0.497295i	$0.834327\pi$
$564$	0	0
$565$	−21.9867	−0.924988
$566$	0	0
$567$	−57.6088	−2.41934
$568$	0	0
$569$	−10.2212	−0.428494	−0.214247	−	0.976779i	$-0.568730\pi$
$569$	−10.2212	−0.428494	−0.214247	+	0.976779i	$0.568730\pi$
$570$	0	0
$571$	30.3082	1.26836	0.634178	−	0.773187i	$-0.281338\pi$
$571$	30.3082	1.26836	0.634178	+	0.773187i	$0.281338\pi$
$572$	0	0
$573$	28.4846	1.18996
$574$	0	0
$575$	0.0775332	0.00323336
$576$	0	0
$577$	45.1878	1.88119	0.940596	−	0.339527i	$-0.110267\pi$
$577$	45.1878	1.88119	0.940596	+	0.339527i	$0.110267\pi$
$578$	0	0
$579$	−44.6265	−1.85461
$580$	0	0
$581$	−31.5919	−1.31065
$582$	0	0
$583$	−46.0411	−1.90683
$584$	0	0
$585$	83.7828	3.46399
$586$	0	0
$587$	−26.5358	−1.09525	−0.547625	−	0.836724i	$-0.684468\pi$
$587$	−26.5358	−1.09525	−0.547625	+	0.836724i	$0.684468\pi$
$588$	0	0
$589$	−3.86530	−0.159267
$590$	0	0
$591$	30.4541	1.25271
$592$	0	0
$593$	−15.3659	−0.631004	−0.315502	−	0.948925i	$-0.602173\pi$
$593$	−15.3659	−0.631004	−0.315502	+	0.948925i	$0.602173\pi$
$594$	0	0
$595$	10.5676	0.433229
$596$	0	0
$597$	47.0959	1.92751
$598$	0	0
$599$	8.35312	0.341299	0.170650	−	0.985332i	$-0.445413\pi$
$599$	8.35312	0.341299	0.170650	+	0.985332i	$0.445413\pi$
$600$	0	0
$601$	−9.67099	−0.394488	−0.197244	−	0.980354i	$-0.563199\pi$
$601$	−9.67099	−0.394488	−0.197244	+	0.980354i	$0.563199\pi$
$602$	0	0
$603$	−50.9807	−2.07610
$604$	0	0
$605$	13.6891	0.556540
$606$	0	0
$607$	−35.5261	−1.44196	−0.720979	−	0.692957i	$-0.756308\pi$
$607$	−35.5261	−1.44196	−0.720979	+	0.692957i	$0.756308\pi$
$608$	0	0
$609$	−24.3182	−0.985422
$610$	0	0
$611$	67.5036	2.73090
$612$	0	0
$613$	9.37254	0.378553	0.189277	−	0.981924i	$-0.439386\pi$
$613$	9.37254	0.378553	0.189277	+	0.981924i	$0.439386\pi$
$614$	0	0
$615$	3.97872	0.160438
$616$	0	0
$617$	33.1331	1.33389	0.666944	−	0.745107i	$-0.267602\pi$
$617$	33.1331	1.33389	0.666944	+	0.745107i	$0.267602\pi$
$618$	0	0
$619$	−35.1025	−1.41089	−0.705443	−	0.708766i	$-0.749252\pi$
$619$	−35.1025	−1.41089	−0.705443	+	0.708766i	$0.749252\pi$
$620$	0	0
$621$	−4.88878	−0.196180
$622$	0	0
$623$	32.9897	1.32170
$624$	0	0
$625$	−24.1701	−0.966805
$626$	0	0
$627$	22.9544	0.916709
$628$	0	0
$629$	3.95502	0.157697
$630$	0	0
$631$	−16.4516	−0.654927	−0.327463	−	0.944864i	$-0.606194\pi$
$631$	−16.4516	−0.654927	−0.327463	+	0.944864i	$0.606194\pi$
$632$	0	0
$633$	−6.69581	−0.266134
$634$	0	0
$635$	43.1601	1.71276
$636$	0	0
$637$	96.8690	3.83809
$638$	0	0
$639$	60.8259	2.40624
$640$	0	0
$641$	−39.5236	−1.56109	−0.780543	−	0.625101i	$-0.785058\pi$
$641$	−39.5236	−1.56109	−0.780543	+	0.625101i	$0.785058\pi$
$642$	0	0
$643$	−11.3027	−0.445736	−0.222868	−	0.974849i	$-0.571542\pi$
$643$	−11.3027	−0.445736	−0.222868	+	0.974849i	$0.571542\pi$
$644$	0	0
$645$	−9.29865	−0.366134
$646$	0	0
$647$	12.9424	0.508817	0.254409	−	0.967097i	$-0.418119\pi$
$647$	12.9424	0.508817	0.254409	+	0.967097i	$0.418119\pi$
$648$	0	0
$649$	−4.15004	−0.162903
$650$	0	0
$651$	31.2912	1.22640
$652$	0	0
$653$	−32.7556	−1.28182	−0.640912	−	0.767614i	$-0.721444\pi$
$653$	−32.7556	−1.28182	−0.640912	+	0.767614i	$0.721444\pi$
$654$	0	0
$655$	20.0927	0.785088
$656$	0	0
$657$	26.6691	1.04046
$658$	0	0
$659$	8.78131	0.342071	0.171036	−	0.985265i	$-0.445289\pi$
$659$	8.78131	0.342071	0.171036	+	0.985265i	$0.445289\pi$
$660$	0	0
$661$	−38.3024	−1.48979	−0.744896	−	0.667181i	$-0.767501\pi$
$661$	−38.3024	−1.48979	−0.744896	+	0.667181i	$0.767501\pi$
$662$	0	0
$663$	−18.3955	−0.714424
$664$	0	0
$665$	−19.1451	−0.742415
$666$	0	0
$667$	−0.799481	−0.0309560
$668$	0	0
$669$	−8.61767	−0.333178
$670$	0	0
$671$	−37.2815	−1.43924
$672$	0	0
$673$	40.1454	1.54749	0.773745	−	0.633497i	$-0.218381\pi$
$673$	40.1454	1.54749	0.773745	+	0.633497i	$0.218381\pi$
$674$	0	0
$675$	−1.63039	−0.0627539
$676$	0	0
$677$	−27.0991	−1.04150	−0.520752	−	0.853708i	$-0.674348\pi$
$677$	−27.0991	−1.04150	−0.520752	+	0.853708i	$0.674348\pi$
$678$	0	0
$679$	−60.6603	−2.32793
$680$	0	0
$681$	−69.9198	−2.67933
$682$	0	0
$683$	24.6558	0.943429	0.471715	−	0.881751i	$-0.343635\pi$
$683$	24.6558	0.943429	0.471715	+	0.881751i	$0.343635\pi$
$684$	0	0
$685$	−3.29983	−0.126080
$686$	0	0
$687$	16.8555	0.643076
$688$	0	0
$689$	−66.8458	−2.54662
$690$	0	0
$691$	18.5300	0.704913	0.352456	−	0.935828i	$-0.385346\pi$
$691$	18.5300	0.704913	0.352456	+	0.935828i	$0.385346\pi$
$692$	0	0
$693$	−126.017	−4.78698
$694$	0	0
$695$	40.0702	1.51995
$696$	0	0
$697$	−0.592414	−0.0224393
$698$	0	0
$699$	55.1790	2.08706
$700$	0	0
$701$	−2.46721	−0.0931851	−0.0465925	−	0.998914i	$-0.514836\pi$
$701$	−2.46721	−0.0931851	−0.0465925	+	0.998914i	$0.514836\pi$
$702$	0	0
$703$	−7.16524	−0.270242
$704$	0	0
$705$	75.2427	2.83381
$706$	0	0
$707$	67.4042	2.53500
$708$	0	0
$709$	51.5322	1.93533	0.967666	−	0.252234i	$-0.0811653\pi$
$709$	51.5322	1.93533	0.967666	+	0.252234i	$0.0811653\pi$
$710$	0	0
$711$	14.7875	0.554573
$712$	0	0
$713$	1.02872	0.0385260
$714$	0	0
$715$	55.0072	2.05715
$716$	0	0
$717$	−25.4733	−0.951320
$718$	0	0
$719$	−13.6337	−0.508449	−0.254225	−	0.967145i	$-0.581820\pi$
$719$	−13.6337	−0.508449	−0.254225	+	0.967145i	$0.581820\pi$
$720$	0	0
$721$	−52.1426	−1.94189
$722$	0	0
$723$	−52.8849	−1.96681
$724$	0	0
$725$	−0.266624	−0.00990218
$726$	0	0
$727$	29.1559	1.08133	0.540666	−	0.841237i	$-0.318172\pi$
$727$	29.1559	1.08133	0.540666	+	0.841237i	$0.318172\pi$
$728$	0	0
$729$	−17.0628	−0.631955
$730$	0	0
$731$	1.38453	0.0512086
$732$	0	0
$733$	27.5368	1.01709	0.508547	−	0.861034i	$-0.330183\pi$
$733$	27.5368	1.01709	0.508547	+	0.861034i	$0.330183\pi$
$734$	0	0
$735$	107.975	3.98271
$736$	0	0
$737$	−33.4712	−1.23293
$738$	0	0
$739$	−11.2460	−0.413691	−0.206846	−	0.978374i	$-0.566320\pi$
$739$	−11.2460	−0.413691	−0.206846	+	0.978374i	$0.566320\pi$
$740$	0	0
$741$	33.3268	1.22429
$742$	0	0
$743$	32.5920	1.19568	0.597842	−	0.801614i	$-0.296025\pi$
$743$	32.5920	1.19568	0.597842	+	0.801614i	$0.296025\pi$
$744$	0	0
$745$	−32.8230	−1.20254
$746$	0	0
$747$	41.5694	1.52094
$748$	0	0
$749$	−27.6693	−1.01102
$750$	0	0
$751$	40.3132	1.47105	0.735525	−	0.677498i	$-0.236936\pi$
$751$	40.3132	1.47105	0.735525	+	0.677498i	$0.236936\pi$
$752$	0	0
$753$	−68.7941	−2.50700
$754$	0	0
$755$	−45.0701	−1.64027
$756$	0	0
$757$	−0.132218	−0.00480553	−0.00240276	−	0.999997i	$-0.500765\pi$
$757$	−0.132218	−0.00480553	−0.00240276	+	0.999997i	$0.500765\pi$
$758$	0	0
$759$	−6.10916	−0.221749
$760$	0	0
$761$	14.1592	0.513270	0.256635	−	0.966508i	$-0.417386\pi$
$761$	14.1592	0.513270	0.256635	+	0.966508i	$0.417386\pi$
$762$	0	0
$763$	54.0344	1.95618
$764$	0	0
$765$	−13.9051	−0.502740
$766$	0	0
$767$	−6.02533	−0.217562
$768$	0	0
$769$	−15.0744	−0.543598	−0.271799	−	0.962354i	$-0.587619\pi$
$769$	−15.0744	−0.543598	−0.271799	+	0.962354i	$0.587619\pi$
$770$	0	0
$771$	33.0503	1.19028
$772$	0	0
$773$	18.3496	0.659990	0.329995	−	0.943983i	$-0.392953\pi$
$773$	18.3496	0.659990	0.329995	+	0.943983i	$0.392953\pi$
$774$	0	0
$775$	0.343076	0.0123237
$776$	0	0
$777$	58.0056	2.08094
$778$	0	0
$779$	1.07326	0.0384537
$780$	0	0
$781$	39.9350	1.42899
$782$	0	0
$783$	16.8118	0.600803
$784$	0	0
$785$	−2.29832	−0.0820307
$786$	0	0
$787$	26.8246	0.956193	0.478097	−	0.878307i	$-0.341327\pi$
$787$	26.8246	0.956193	0.478097	+	0.878307i	$0.341327\pi$
$788$	0	0
$789$	−85.1446	−3.03123
$790$	0	0
$791$	48.0135	1.70716
$792$	0	0
$793$	−54.1280	−1.92214
$794$	0	0
$795$	−74.5095	−2.64258
$796$	0	0
$797$	42.9521	1.52144	0.760720	−	0.649080i	$-0.224846\pi$
$797$	42.9521	1.52144	0.760720	+	0.649080i	$0.224846\pi$
$798$	0	0
$799$	−11.2033	−0.396344
$800$	0	0
$801$	−43.4086	−1.53377
$802$	0	0
$803$	17.5095	0.617897
$804$	0	0
$805$	5.09535	0.179587
$806$	0	0
$807$	−83.4068	−2.93606
$808$	0	0
$809$	39.2235	1.37903	0.689513	−	0.724274i	$-0.257825\pi$
$809$	39.2235	1.37903	0.689513	+	0.724274i	$0.257825\pi$
$810$	0	0
$811$	−54.5193	−1.91443	−0.957216	−	0.289374i	$-0.906553\pi$
$811$	−54.5193	−1.91443	−0.957216	+	0.289374i	$0.906553\pi$
$812$	0	0
$813$	55.1765	1.93512
$814$	0	0
$815$	43.6224	1.52803
$816$	0	0
$817$	−2.50832	−0.0877550
$818$	0	0
$819$	−182.960	−6.39316
$820$	0	0
$821$	−2.95765	−0.103223	−0.0516113	−	0.998667i	$-0.516436\pi$
$821$	−2.95765	−0.103223	−0.0516113	+	0.998667i	$0.516436\pi$
$822$	0	0
$823$	8.31781	0.289941	0.144970	−	0.989436i	$-0.453691\pi$
$823$	8.31781	0.289941	0.144970	+	0.989436i	$0.453691\pi$
$824$	0	0
$825$	−2.03739	−0.0709327
$826$	0	0
$827$	−36.0010	−1.25188	−0.625939	−	0.779872i	$-0.715284\pi$
$827$	−36.0010	−1.25188	−0.625939	+	0.779872i	$0.715284\pi$
$828$	0	0
$829$	−26.8614	−0.932934	−0.466467	−	0.884539i	$-0.654473\pi$
$829$	−26.8614	−0.932934	−0.466467	+	0.884539i	$0.654473\pi$
$830$	0	0
$831$	−4.44014	−0.154027
$832$	0	0
$833$	−16.0770	−0.557033
$834$	0	0
$835$	−13.7353	−0.475331
$836$	0	0
$837$	−21.6324	−0.747724
$838$	0	0
$839$	8.19906	0.283063	0.141531	−	0.989934i	$-0.454797\pi$
$839$	8.19906	0.283063	0.141531	+	0.989934i	$0.454797\pi$
$840$	0	0
$841$	−26.2507	−0.905197
$842$	0	0
$843$	−3.42151	−0.117843
$844$	0	0
$845$	51.2659	1.76360
$846$	0	0
$847$	−29.8935	−1.02715
$848$	0	0
$849$	−94.2990	−3.23633
$850$	0	0
$851$	1.90698	0.0653705
$852$	0	0
$853$	−3.10163	−0.106198	−0.0530988	−	0.998589i	$-0.516910\pi$
$853$	−3.10163	−0.106198	−0.0530988	+	0.998589i	$0.516910\pi$
$854$	0	0
$855$	25.1916	0.861534
$856$	0	0
$857$	−19.8850	−0.679258	−0.339629	−	0.940560i	$-0.610301\pi$
$857$	−19.8850	−0.679258	−0.339629	+	0.940560i	$0.610301\pi$
$858$	0	0
$859$	24.7532	0.844569	0.422285	−	0.906463i	$-0.361228\pi$
$859$	24.7532	0.844569	0.422285	+	0.906463i	$0.361228\pi$
$860$	0	0
$861$	−8.68853	−0.296104
$862$	0	0
$863$	−24.5576	−0.835950	−0.417975	−	0.908458i	$-0.637260\pi$
$863$	−24.5576	−0.835950	−0.417975	+	0.908458i	$0.637260\pi$
$864$	0	0
$865$	−25.7089	−0.874128
$866$	0	0
$867$	3.05303	0.103686
$868$	0	0
$869$	9.70864	0.329343
$870$	0	0
$871$	−48.5959	−1.64661
$872$	0	0
$873$	79.8182	2.70144
$874$	0	0
$875$	54.5372	1.84369
$876$	0	0
$877$	−43.3716	−1.46456	−0.732278	−	0.681006i	$-0.761543\pi$
$877$	−43.3716	−1.46456	−0.732278	+	0.681006i	$0.761543\pi$
$878$	0	0
$879$	−103.509	−3.49129
$880$	0	0
$881$	−15.9924	−0.538798	−0.269399	−	0.963029i	$-0.586825\pi$
$881$	−15.9924	−0.538798	−0.269399	+	0.963029i	$0.586825\pi$
$882$	0	0
$883$	−4.22359	−0.142135	−0.0710676	−	0.997472i	$-0.522641\pi$
$883$	−4.22359	−0.142135	−0.0710676	+	0.997472i	$0.522641\pi$
$884$	0	0
$885$	−6.71612	−0.225760
$886$	0	0
$887$	8.88946	0.298479	0.149239	−	0.988801i	$-0.452317\pi$
$887$	8.88946	0.298479	0.149239	+	0.988801i	$0.452317\pi$
$888$	0	0
$889$	−94.2508	−3.16107
$890$	0	0
$891$	49.7682	1.66730
$892$	0	0
$893$	20.2968	0.679207
$894$	0	0
$895$	50.4656	1.68688
$896$	0	0
$897$	−8.86973	−0.296152
$898$	0	0
$899$	−3.53762	−0.117986
$900$	0	0
$901$	11.0941	0.369599
$902$	0	0
$903$	20.3059	0.675738
$904$	0	0
$905$	10.1635	0.337846
$906$	0	0
$907$	−27.8536	−0.924864	−0.462432	−	0.886655i	$-0.653023\pi$
$907$	−27.8536	−0.924864	−0.462432	+	0.886655i	$0.653023\pi$
$908$	0	0
$909$	−88.6921	−2.94173
$910$	0	0
$911$	37.1987	1.23245	0.616223	−	0.787571i	$-0.288662\pi$
$911$	37.1987	1.23245	0.616223	+	0.787571i	$0.288662\pi$
$912$	0	0
$913$	27.2922	0.903240
$914$	0	0
$915$	−60.3337	−1.99457
$916$	0	0
$917$	−43.8775	−1.44896
$918$	0	0
$919$	11.6608	0.384653	0.192326	−	0.981331i	$-0.438397\pi$
$919$	11.6608	0.384653	0.192326	+	0.981331i	$0.438397\pi$
$920$	0	0
$921$	20.2360	0.666799
$922$	0	0
$923$	57.9806	1.90845
$924$	0	0
$925$	0.635973	0.0209107
$926$	0	0
$927$	68.6104	2.25346
$928$	0	0
$929$	29.5564	0.969715	0.484857	−	0.874593i	$-0.338872\pi$
$929$	29.5564	0.969715	0.484857	+	0.874593i	$0.338872\pi$
$930$	0	0
$931$	29.1263	0.954576
$932$	0	0
$933$	−33.5096	−1.09706
$934$	0	0
$935$	−9.12933	−0.298561
$936$	0	0
$937$	35.8428	1.17093	0.585466	−	0.810697i	$-0.300911\pi$
$937$	35.8428	1.17093	0.585466	+	0.810697i	$0.300911\pi$
$938$	0	0
$939$	−22.4300	−0.731976
$940$	0	0
$941$	18.8299	0.613836	0.306918	−	0.951736i	$-0.400702\pi$
$941$	18.8299	0.613836	0.306918	+	0.951736i	$0.400702\pi$
$942$	0	0
$943$	−0.285643	−0.00930181
$944$	0	0
$945$	−107.147	−3.48548
$946$	0	0
$947$	14.0622	0.456960	0.228480	−	0.973549i	$-0.426625\pi$
$947$	14.0622	0.456960	0.228480	+	0.973549i	$0.426625\pi$
$948$	0	0
$949$	25.4216	0.825219
$950$	0	0
$951$	−52.3989	−1.69915
$952$	0	0
$953$	−5.33834	−0.172926	−0.0864629	−	0.996255i	$-0.527556\pi$
$953$	−5.33834	−0.172926	−0.0864629	+	0.996255i	$0.527556\pi$
$954$	0	0
$955$	20.5242	0.664146
$956$	0	0
$957$	21.0084	0.679107
$958$	0	0
$959$	7.20600	0.232694
$960$	0	0
$961$	−26.4480	−0.853161
$962$	0	0
$963$	36.4080	1.17323
$964$	0	0
$965$	−32.1549	−1.03510
$966$	0	0
$967$	38.8859	1.25049	0.625243	−	0.780430i	$-0.285000\pi$
$967$	38.8859	1.25049	0.625243	+	0.780430i	$0.285000\pi$
$968$	0	0
$969$	−5.53112	−0.177685
$970$	0	0
$971$	−4.31619	−0.138513	−0.0692566	−	0.997599i	$-0.522063\pi$
$971$	−4.31619	−0.138513	−0.0692566	+	0.997599i	$0.522063\pi$
$972$	0	0
$973$	−87.5033	−2.80523
$974$	0	0
$975$	−2.95803	−0.0947327
$976$	0	0
$977$	57.7753	1.84839	0.924197	−	0.381915i	$-0.124735\pi$
$977$	57.7753	1.84839	0.924197	+	0.381915i	$0.124735\pi$
$978$	0	0
$979$	−28.4997	−0.910855
$980$	0	0
$981$	−71.0997	−2.27004
$982$	0	0
$983$	−2.20202	−0.0702334	−0.0351167	−	0.999383i	$-0.511180\pi$
$983$	−2.20202	−0.0702334	−0.0351167	+	0.999383i	$0.511180\pi$
$984$	0	0
$985$	21.9432	0.699169
$986$	0	0
$987$	−164.311	−5.23008
$988$	0	0
$989$	0.667574	0.0212276
$990$	0	0
$991$	4.93997	0.156923	0.0784617	−	0.996917i	$-0.474999\pi$
$991$	4.93997	0.156923	0.0784617	+	0.996917i	$0.474999\pi$
$992$	0	0
$993$	7.29298	0.231436
$994$	0	0
$995$	33.9342	1.07579
$996$	0	0
$997$	−28.5755	−0.904997	−0.452498	−	0.891765i	$-0.649467\pi$
$997$	−28.5755	−0.904997	−0.452498	+	0.891765i	$0.649467\pi$
$998$	0	0
$999$	−40.1007	−1.26873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.17	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.17	✓	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)

\(f(q)\)	\(=\)	\(q+3.05303 q^{3} +2.19982 q^{5} -4.80385 q^{7} +6.32102 q^{9} +O(q^{10})\)	\(q+3.05303 q^{3} +2.19982 q^{5} -4.80385 q^{7} +6.32102 q^{9} +4.15004 q^{11} +6.02533 q^{13} +6.71612 q^{15} -1.00000 q^{17} +1.81168 q^{19} -14.6663 q^{21} -0.482167 q^{23} -0.160801 q^{25} +10.1392 q^{27} +1.65810 q^{29} -2.13354 q^{31} +12.6702 q^{33} -10.5676 q^{35} -3.95502 q^{37} +18.3955 q^{39} +0.592414 q^{41} -1.38453 q^{43} +13.9051 q^{45} +11.2033 q^{47} +16.0770 q^{49} -3.05303 q^{51} -11.0941 q^{53} +9.12933 q^{55} +5.53112 q^{57} -1.00000 q^{59} -8.98341 q^{61} -30.3652 q^{63} +13.2546 q^{65} -8.06527 q^{67} -1.47207 q^{69} +9.62281 q^{71} +4.21912 q^{73} -0.490932 q^{75} -19.9362 q^{77} +2.33941 q^{79} +11.9922 q^{81} +6.57638 q^{83} -2.19982 q^{85} +5.06223 q^{87} -6.86734 q^{89} -28.9448 q^{91} -6.51378 q^{93} +3.98537 q^{95} +12.6274 q^{97} +26.2325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 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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