LMFDB - Embedded newform 4012.2.a.g.1.3

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} + \cdots + 82 $ x^12 - 4x^11 - 13x^10 + 60x^9 + 48x^8 - 289x^7 - 89x^6 + 602x^5 + 161x^4 - 555x^3 - 197x^2 + 181*x + 82
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.36915$ 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)

$f(q)$	$=$	$q-2.36915 q^{3} +2.61284 q^{5} -2.25519 q^{7} +2.61289 q^{9} +O(q^{10})$	$q-2.36915 q^{3} +2.61284 q^{5} -2.25519 q^{7} +2.61289 q^{9} -6.06042 q^{11} +5.42929 q^{13} -6.19022 q^{15} -1.00000 q^{17} -4.71401 q^{19} +5.34289 q^{21} +9.23431 q^{23} +1.82693 q^{25} +0.917132 q^{27} +4.85561 q^{29} -0.261418 q^{31} +14.3581 q^{33} -5.89245 q^{35} +4.91282 q^{37} -12.8628 q^{39} -5.42987 q^{41} -5.25368 q^{43} +6.82705 q^{45} -1.34302 q^{47} -1.91412 q^{49} +2.36915 q^{51} +5.39130 q^{53} -15.8349 q^{55} +11.1682 q^{57} +1.00000 q^{59} -12.6581 q^{61} -5.89256 q^{63} +14.1859 q^{65} -3.44937 q^{67} -21.8775 q^{69} -2.07941 q^{71} -0.551991 q^{73} -4.32827 q^{75} +13.6674 q^{77} +15.1411 q^{79} -10.0115 q^{81} +7.01331 q^{83} -2.61284 q^{85} -11.5037 q^{87} -8.76169 q^{89} -12.2441 q^{91} +0.619339 q^{93} -12.3170 q^{95} -1.65497 q^{97} -15.8352 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.36915	−1.36783	−0.683916	−	0.729561i	$-0.739724\pi$
$3$	−2.36915	−1.36783	−0.683916	+	0.729561i	$0.739724\pi$
$4$	0	0
$5$	2.61284	1.16850	0.584249	−	0.811575i	$-0.301389\pi$
$5$	2.61284	1.16850	0.584249	+	0.811575i	$0.301389\pi$
$6$	0	0
$7$	−2.25519	−0.852382	−0.426191	−	0.904633i	$-0.640145\pi$
$7$	−2.25519	−0.852382	−0.426191	+	0.904633i	$0.640145\pi$
$8$	0	0
$9$	2.61289	0.870962
$10$	0	0
$11$	−6.06042	−1.82729	−0.913643	−	0.406517i	$-0.866743\pi$
$11$	−6.06042	−1.82729	−0.913643	+	0.406517i	$0.866743\pi$
$12$	0	0
$13$	5.42929	1.50581	0.752907	−	0.658127i	$-0.228651\pi$
$13$	5.42929	1.50581	0.752907	+	0.658127i	$0.228651\pi$
$14$	0	0
$15$	−6.19022	−1.59831
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.71401	−1.08147	−0.540734	−	0.841193i	$-0.681854\pi$
$19$	−4.71401	−1.08147	−0.540734	+	0.841193i	$0.681854\pi$
$20$	0	0
$21$	5.34289	1.16591
$22$	0	0
$23$	9.23431	1.92549	0.962744	−	0.270415i	$-0.0871609\pi$
$23$	9.23431	1.92549	0.962744	+	0.270415i	$0.0871609\pi$
$24$	0	0
$25$	1.82693	0.365385
$26$	0	0
$27$	0.917132	0.176502
$28$	0	0
$29$	4.85561	0.901664	0.450832	−	0.892609i	$-0.351127\pi$
$29$	4.85561	0.901664	0.450832	+	0.892609i	$0.351127\pi$
$30$	0	0
$31$	−0.261418	−0.0469521	−0.0234760	−	0.999724i	$-0.507473\pi$
$31$	−0.261418	−0.0469521	−0.0234760	+	0.999724i	$0.507473\pi$
$32$	0	0
$33$	14.3581	2.49942
$34$	0	0
$35$	−5.89245	−0.996006
$36$	0	0
$37$	4.91282	0.807663	0.403832	−	0.914833i	$-0.367678\pi$
$37$	4.91282	0.807663	0.403832	+	0.914833i	$0.367678\pi$
$38$	0	0
$39$	−12.8628	−2.05970
$40$	0	0
$41$	−5.42987	−0.848003	−0.424001	−	0.905662i	$-0.639375\pi$
$41$	−5.42987	−0.848003	−0.424001	+	0.905662i	$0.639375\pi$
$42$	0	0
$43$	−5.25368	−0.801179	−0.400589	−	0.916258i	$-0.631195\pi$
$43$	−5.25368	−0.801179	−0.400589	+	0.916258i	$0.631195\pi$
$44$	0	0
$45$	6.82705	1.01772
$46$	0	0
$47$	−1.34302	−0.195899	−0.0979495	−	0.995191i	$-0.531228\pi$
$47$	−1.34302	−0.195899	−0.0979495	+	0.995191i	$0.531228\pi$
$48$	0	0
$49$	−1.91412	−0.273445
$50$	0	0
$51$	2.36915	0.331748
$52$	0	0
$53$	5.39130	0.740553	0.370276	−	0.928922i	$-0.379263\pi$
$53$	5.39130	0.740553	0.370276	+	0.928922i	$0.379263\pi$
$54$	0	0
$55$	−15.8349	−2.13518
$56$	0	0
$57$	11.1682	1.47927
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−12.6581	−1.62070	−0.810350	−	0.585946i	$-0.800723\pi$
$61$	−12.6581	−1.62070	−0.810350	+	0.585946i	$0.800723\pi$
$62$	0	0
$63$	−5.89256	−0.742392
$64$	0	0
$65$	14.1859	1.75954
$66$	0	0
$67$	−3.44937	−0.421407	−0.210704	−	0.977550i	$-0.567576\pi$
$67$	−3.44937	−0.421407	−0.210704	+	0.977550i	$0.567576\pi$
$68$	0	0
$69$	−21.8775	−2.63374
$70$	0	0
$71$	−2.07941	−0.246780	−0.123390	−	0.992358i	$-0.539377\pi$
$71$	−2.07941	−0.246780	−0.123390	+	0.992358i	$0.539377\pi$
$72$	0	0
$73$	−0.551991	−0.0646057	−0.0323029	−	0.999478i	$-0.510284\pi$
$73$	−0.551991	−0.0646057	−0.0323029	+	0.999478i	$0.510284\pi$
$74$	0	0
$75$	−4.32827	−0.499786
$76$	0	0
$77$	13.6674	1.55755
$78$	0	0
$79$	15.1411	1.70350	0.851752	−	0.523945i	$-0.175540\pi$
$79$	15.1411	1.70350	0.851752	+	0.523945i	$0.175540\pi$
$80$	0	0
$81$	−10.0115	−1.11239
$82$	0	0
$83$	7.01331	0.769811	0.384906	−	0.922956i	$-0.374234\pi$
$83$	7.01331	0.769811	0.384906	+	0.922956i	$0.374234\pi$
$84$	0	0
$85$	−2.61284	−0.283402
$86$	0	0
$87$	−11.5037	−1.23332
$88$	0	0
$89$	−8.76169	−0.928737	−0.464369	−	0.885642i	$-0.653719\pi$
$89$	−8.76169	−0.928737	−0.464369	+	0.885642i	$0.653719\pi$
$90$	0	0
$91$	−12.2441	−1.28353
$92$	0	0
$93$	0.619339	0.0642225
$94$	0	0
$95$	−12.3170	−1.26369
$96$	0	0
$97$	−1.65497	−0.168037	−0.0840184	−	0.996464i	$-0.526775\pi$
$97$	−1.65497	−0.168037	−0.0840184	+	0.996464i	$0.526775\pi$
$98$	0	0
$99$	−15.8352	−1.59150
$100$	0	0
$101$	−10.3274	−1.02761	−0.513806	−	0.857906i	$-0.671765\pi$
$101$	−10.3274	−1.02761	−0.513806	+	0.857906i	$0.671765\pi$
$102$	0	0
$103$	6.42576	0.633149	0.316574	−	0.948568i	$-0.397467\pi$
$103$	6.42576	0.633149	0.316574	+	0.948568i	$0.397467\pi$
$104$	0	0
$105$	13.9601	1.36237
$106$	0	0
$107$	−10.5279	−1.01777	−0.508887	−	0.860833i	$-0.669943\pi$
$107$	−10.5279	−1.01777	−0.508887	+	0.860833i	$0.669943\pi$
$108$	0	0
$109$	−0.328812	−0.0314945	−0.0157473	−	0.999876i	$-0.505013\pi$
$109$	−0.328812	−0.0314945	−0.0157473	+	0.999876i	$0.505013\pi$
$110$	0	0
$111$	−11.6392	−1.10475
$112$	0	0
$113$	−2.58368	−0.243052	−0.121526	−	0.992588i	$-0.538779\pi$
$113$	−2.58368	−0.243052	−0.121526	+	0.992588i	$0.538779\pi$
$114$	0	0
$115$	24.1278	2.24993
$116$	0	0
$117$	14.1861	1.31151
$118$	0	0
$119$	2.25519	0.206733
$120$	0	0
$121$	25.7287	2.33897
$122$	0	0
$123$	12.8642	1.15992
$124$	0	0
$125$	−8.29073	−0.741545
$126$	0	0
$127$	2.01055	0.178407	0.0892037	−	0.996013i	$-0.471568\pi$
$127$	2.01055	0.178407	0.0892037	+	0.996013i	$0.471568\pi$
$128$	0	0
$129$	12.4468	1.09588
$130$	0	0
$131$	13.4608	1.17607	0.588036	−	0.808835i	$-0.299901\pi$
$131$	13.4608	1.17607	0.588036	+	0.808835i	$0.299901\pi$
$132$	0	0
$133$	10.6310	0.921824
$134$	0	0
$135$	2.39632	0.206242
$136$	0	0
$137$	−19.1535	−1.63639	−0.818197	−	0.574939i	$-0.805026\pi$
$137$	−19.1535	−1.63639	−0.818197	+	0.574939i	$0.805026\pi$
$138$	0	0
$139$	14.2452	1.20826	0.604129	−	0.796886i	$-0.293521\pi$
$139$	14.2452	1.20826	0.604129	+	0.796886i	$0.293521\pi$
$140$	0	0
$141$	3.18181	0.267957
$142$	0	0
$143$	−32.9038	−2.75155
$144$	0	0
$145$	12.6869	1.05359
$146$	0	0
$147$	4.53483	0.374027
$148$	0	0
$149$	6.55596	0.537085	0.268543	−	0.963268i	$-0.413458\pi$
$149$	6.55596	0.537085	0.268543	+	0.963268i	$0.413458\pi$
$150$	0	0
$151$	−22.2739	−1.81262	−0.906311	−	0.422612i	$-0.861113\pi$
$151$	−22.2739	−1.81262	−0.906311	+	0.422612i	$0.861113\pi$
$152$	0	0
$153$	−2.61289	−0.211239
$154$	0	0
$155$	−0.683043	−0.0548633
$156$	0	0
$157$	−9.47766	−0.756400	−0.378200	−	0.925724i	$-0.623457\pi$
$157$	−9.47766	−0.756400	−0.378200	+	0.925724i	$0.623457\pi$
$158$	0	0
$159$	−12.7728	−1.01295
$160$	0	0
$161$	−20.8251	−1.64125
$162$	0	0
$163$	4.15317	0.325301	0.162651	−	0.986684i	$-0.447996\pi$
$163$	4.15317	0.325301	0.162651	+	0.986684i	$0.447996\pi$
$164$	0	0
$165$	37.5153	2.92056
$166$	0	0
$167$	−4.32048	−0.334329	−0.167164	−	0.985929i	$-0.553461\pi$
$167$	−4.32048	−0.334329	−0.167164	+	0.985929i	$0.553461\pi$
$168$	0	0
$169$	16.4772	1.26748
$170$	0	0
$171$	−12.3172	−0.941918
$172$	0	0
$173$	6.97592	0.530369	0.265185	−	0.964198i	$-0.414567\pi$
$173$	6.97592	0.530369	0.265185	+	0.964198i	$0.414567\pi$
$174$	0	0
$175$	−4.12007	−0.311448
$176$	0	0
$177$	−2.36915	−0.178076
$178$	0	0
$179$	−18.5731	−1.38822	−0.694109	−	0.719870i	$-0.744201\pi$
$179$	−18.5731	−1.38822	−0.694109	+	0.719870i	$0.744201\pi$
$180$	0	0
$181$	−23.3636	−1.73660	−0.868300	−	0.496039i	$-0.834787\pi$
$181$	−23.3636	−1.73660	−0.868300	+	0.496039i	$0.834787\pi$
$182$	0	0
$183$	29.9889	2.21685
$184$	0	0
$185$	12.8364	0.943752
$186$	0	0
$187$	6.06042	0.443182
$188$	0	0
$189$	−2.06831	−0.150447
$190$	0	0
$191$	3.19794	0.231395	0.115697	−	0.993285i	$-0.463090\pi$
$191$	3.19794	0.231395	0.115697	+	0.993285i	$0.463090\pi$
$192$	0	0
$193$	−20.6927	−1.48950	−0.744748	−	0.667346i	$-0.767430\pi$
$193$	−20.6927	−1.48950	−0.744748	+	0.667346i	$0.767430\pi$
$194$	0	0
$195$	−33.6085	−2.40675
$196$	0	0
$197$	1.44238	0.102765	0.0513827	−	0.998679i	$-0.483637\pi$
$197$	1.44238	0.102765	0.0513827	+	0.998679i	$0.483637\pi$
$198$	0	0
$199$	−6.29939	−0.446552	−0.223276	−	0.974755i	$-0.571675\pi$
$199$	−6.29939	−0.446552	−0.223276	+	0.974755i	$0.571675\pi$
$200$	0	0
$201$	8.17208	0.576414
$202$	0	0
$203$	−10.9503	−0.768562
$204$	0	0
$205$	−14.1874	−0.990889
$206$	0	0
$207$	24.1282	1.67703
$208$	0	0
$209$	28.5689	1.97615
$210$	0	0
$211$	2.51274	0.172984	0.0864920	−	0.996253i	$-0.472434\pi$
$211$	2.51274	0.172984	0.0864920	+	0.996253i	$0.472434\pi$
$212$	0	0
$213$	4.92644	0.337554
$214$	0	0
$215$	−13.7270	−0.936175
$216$	0	0
$217$	0.589547	0.0400211
$218$	0	0
$219$	1.30775	0.0883697
$220$	0	0
$221$	−5.42929	−0.365214
$222$	0	0
$223$	−1.19251	−0.0798561	−0.0399280	−	0.999203i	$-0.512713\pi$
$223$	−1.19251	−0.0798561	−0.0399280	+	0.999203i	$0.512713\pi$
$224$	0	0
$225$	4.77355	0.318237
$226$	0	0
$227$	−29.2551	−1.94173	−0.970865	−	0.239629i	$-0.922974\pi$
$227$	−29.2551	−1.94173	−0.970865	+	0.239629i	$0.922974\pi$
$228$	0	0
$229$	−16.7652	−1.10787	−0.553937	−	0.832559i	$-0.686875\pi$
$229$	−16.7652	−1.10787	−0.553937	+	0.832559i	$0.686875\pi$
$230$	0	0
$231$	−32.3802	−2.13046
$232$	0	0
$233$	11.8680	0.777498	0.388749	−	0.921344i	$-0.372907\pi$
$233$	11.8680	0.777498	0.388749	+	0.921344i	$0.372907\pi$
$234$	0	0
$235$	−3.50908	−0.228907
$236$	0	0
$237$	−35.8715	−2.33011
$238$	0	0
$239$	−19.0664	−1.23330	−0.616652	−	0.787236i	$-0.711511\pi$
$239$	−19.0664	−1.23330	−0.616652	+	0.787236i	$0.711511\pi$
$240$	0	0
$241$	10.4893	0.675677	0.337838	−	0.941204i	$-0.390304\pi$
$241$	10.4893	0.675677	0.337838	+	0.941204i	$0.390304\pi$
$242$	0	0
$243$	20.9673	1.34506
$244$	0	0
$245$	−5.00128	−0.319520
$246$	0	0
$247$	−25.5937	−1.62849
$248$	0	0
$249$	−16.6156	−1.05297
$250$	0	0
$251$	−1.91962	−0.121165	−0.0605825	−	0.998163i	$-0.519296\pi$
$251$	−1.91962	−0.121165	−0.0605825	+	0.998163i	$0.519296\pi$
$252$	0	0
$253$	−55.9638	−3.51842
$254$	0	0
$255$	6.19022	0.387646
$256$	0	0
$257$	−15.0099	−0.936293	−0.468146	−	0.883651i	$-0.655078\pi$
$257$	−15.0099	−0.936293	−0.468146	+	0.883651i	$0.655078\pi$
$258$	0	0
$259$	−11.0794	−0.688438
$260$	0	0
$261$	12.6872	0.785315
$262$	0	0
$263$	−7.83895	−0.483370	−0.241685	−	0.970355i	$-0.577700\pi$
$263$	−7.83895	−0.483370	−0.241685	+	0.970355i	$0.577700\pi$
$264$	0	0
$265$	14.0866	0.865334
$266$	0	0
$267$	20.7578	1.27036
$268$	0	0
$269$	−8.83638	−0.538764	−0.269382	−	0.963033i	$-0.586819\pi$
$269$	−8.83638	−0.538764	−0.269382	+	0.963033i	$0.586819\pi$
$270$	0	0
$271$	−27.0201	−1.64136	−0.820678	−	0.571391i	$-0.806404\pi$
$271$	−27.0201	−1.64136	−0.820678	+	0.571391i	$0.806404\pi$
$272$	0	0
$273$	29.0081	1.75565
$274$	0	0
$275$	−11.0720	−0.667664
$276$	0	0
$277$	−24.5367	−1.47427	−0.737133	−	0.675748i	$-0.763821\pi$
$277$	−24.5367	−1.47427	−0.737133	+	0.675748i	$0.763821\pi$
$278$	0	0
$279$	−0.683055	−0.0408935
$280$	0	0
$281$	12.8388	0.765896	0.382948	−	0.923770i	$-0.374909\pi$
$281$	12.8388	0.765896	0.382948	+	0.923770i	$0.374909\pi$
$282$	0	0
$283$	32.3095	1.92060	0.960302	−	0.278964i	$-0.0899911\pi$
$283$	32.3095	1.92060	0.960302	+	0.278964i	$0.0899911\pi$
$284$	0	0
$285$	29.1808	1.72852
$286$	0	0
$287$	12.2454	0.722822
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.92088	0.229846
$292$	0	0
$293$	7.83519	0.457737	0.228868	−	0.973457i	$-0.426497\pi$
$293$	7.83519	0.457737	0.228868	+	0.973457i	$0.426497\pi$
$294$	0	0
$295$	2.61284	0.152125
$296$	0	0
$297$	−5.55821	−0.322520
$298$	0	0
$299$	50.1358	2.89943
$300$	0	0
$301$	11.8481	0.682910
$302$	0	0
$303$	24.4671	1.40560
$304$	0	0
$305$	−33.0735	−1.89378
$306$	0	0
$307$	−23.2355	−1.32612	−0.663059	−	0.748567i	$-0.730742\pi$
$307$	−23.2355	−1.32612	−0.663059	+	0.748567i	$0.730742\pi$
$308$	0	0
$309$	−15.2236	−0.866041
$310$	0	0
$311$	24.8702	1.41026	0.705129	−	0.709079i	$-0.250889\pi$
$311$	24.8702	1.41026	0.705129	+	0.709079i	$0.250889\pi$
$312$	0	0
$313$	−19.7144	−1.11432	−0.557162	−	0.830404i	$-0.688110\pi$
$313$	−19.7144	−1.11432	−0.557162	+	0.830404i	$0.688110\pi$
$314$	0	0
$315$	−15.3963	−0.867483
$316$	0	0
$317$	1.20697	0.0677904	0.0338952	−	0.999425i	$-0.489209\pi$
$317$	1.20697	0.0677904	0.0338952	+	0.999425i	$0.489209\pi$
$318$	0	0
$319$	−29.4270	−1.64760
$320$	0	0
$321$	24.9423	1.39214
$322$	0	0
$323$	4.71401	0.262295
$324$	0	0
$325$	9.91892	0.550203
$326$	0	0
$327$	0.779007	0.0430792
$328$	0	0
$329$	3.02876	0.166981
$330$	0	0
$331$	−22.7735	−1.25175	−0.625873	−	0.779925i	$-0.715257\pi$
$331$	−22.7735	−1.25175	−0.625873	+	0.779925i	$0.715257\pi$
$332$	0	0
$333$	12.8366	0.703444
$334$	0	0
$335$	−9.01264	−0.492413
$336$	0	0
$337$	−31.6281	−1.72289	−0.861447	−	0.507848i	$-0.830441\pi$
$337$	−31.6281	−1.72289	−0.861447	+	0.507848i	$0.830441\pi$
$338$	0	0
$339$	6.12112	0.332454
$340$	0	0
$341$	1.58430	0.0857948
$342$	0	0
$343$	20.1030	1.08546
$344$	0	0
$345$	−57.1624	−3.07752
$346$	0	0
$347$	12.3370	0.662283	0.331141	−	0.943581i	$-0.392566\pi$
$347$	12.3370	0.662283	0.331141	+	0.943581i	$0.392566\pi$
$348$	0	0
$349$	17.5046	0.936999	0.468499	−	0.883464i	$-0.344795\pi$
$349$	17.5046	0.936999	0.468499	+	0.883464i	$0.344795\pi$
$350$	0	0
$351$	4.97938	0.265779
$352$	0	0
$353$	−12.6537	−0.673488	−0.336744	−	0.941596i	$-0.609326\pi$
$353$	−12.6537	−0.673488	−0.336744	+	0.941596i	$0.609326\pi$
$354$	0	0
$355$	−5.43316	−0.288362
$356$	0	0
$357$	−5.34289	−0.282776
$358$	0	0
$359$	26.6098	1.40441	0.702206	−	0.711974i	$-0.252199\pi$
$359$	26.6098	1.40441	0.702206	+	0.711974i	$0.252199\pi$
$360$	0	0
$361$	3.22193	0.169575
$362$	0	0
$363$	−60.9553	−3.19932
$364$	0	0
$365$	−1.44226	−0.0754916
$366$	0	0
$367$	16.8798	0.881119	0.440560	−	0.897723i	$-0.354780\pi$
$367$	16.8798	0.881119	0.440560	+	0.897723i	$0.354780\pi$
$368$	0	0
$369$	−14.1876	−0.738578
$370$	0	0
$371$	−12.1584	−0.631234
$372$	0	0
$373$	−28.4599	−1.47360	−0.736800	−	0.676111i	$-0.763664\pi$
$373$	−28.4599	−1.47360	−0.736800	+	0.676111i	$0.763664\pi$
$374$	0	0
$375$	19.6420	1.01431
$376$	0	0
$377$	26.3625	1.35774
$378$	0	0
$379$	29.1179	1.49569	0.747843	−	0.663876i	$-0.231090\pi$
$379$	29.1179	1.49569	0.747843	+	0.663876i	$0.231090\pi$
$380$	0	0
$381$	−4.76330	−0.244031
$382$	0	0
$383$	−32.0664	−1.63852	−0.819258	−	0.573425i	$-0.805614\pi$
$383$	−32.0664	−1.63852	−0.819258	+	0.573425i	$0.805614\pi$
$384$	0	0
$385$	35.7107	1.81999
$386$	0	0
$387$	−13.7273	−0.697796
$388$	0	0
$389$	−19.3753	−0.982369	−0.491184	−	0.871056i	$-0.663436\pi$
$389$	−19.3753	−0.982369	−0.491184	+	0.871056i	$0.663436\pi$
$390$	0	0
$391$	−9.23431	−0.466999
$392$	0	0
$393$	−31.8906	−1.60867
$394$	0	0
$395$	39.5612	1.99054
$396$	0	0
$397$	6.83768	0.343173	0.171587	−	0.985169i	$-0.445111\pi$
$397$	6.83768	0.343173	0.171587	+	0.985169i	$0.445111\pi$
$398$	0	0
$399$	−25.1865	−1.26090
$400$	0	0
$401$	−10.4316	−0.520930	−0.260465	−	0.965483i	$-0.583876\pi$
$401$	−10.4316	−0.520930	−0.260465	+	0.965483i	$0.583876\pi$
$402$	0	0
$403$	−1.41931	−0.0707011
$404$	0	0
$405$	−26.1584	−1.29982
$406$	0	0
$407$	−29.7738	−1.47583
$408$	0	0
$409$	−13.7407	−0.679436	−0.339718	−	0.940527i	$-0.610332\pi$
$409$	−13.7407	−0.679436	−0.339718	+	0.940527i	$0.610332\pi$
$410$	0	0
$411$	45.3776	2.23831
$412$	0	0
$413$	−2.25519	−0.110971
$414$	0	0
$415$	18.3247	0.899522
$416$	0	0
$417$	−33.7490	−1.65269
$418$	0	0
$419$	−9.01763	−0.440540	−0.220270	−	0.975439i	$-0.570694\pi$
$419$	−9.01763	−0.440540	−0.220270	+	0.975439i	$0.570694\pi$
$420$	0	0
$421$	40.4142	1.96967	0.984833	−	0.173507i	$-0.0555100\pi$
$421$	40.4142	1.96967	0.984833	+	0.173507i	$0.0555100\pi$
$422$	0	0
$423$	−3.50915	−0.170621
$424$	0	0
$425$	−1.82693	−0.0886190
$426$	0	0
$427$	28.5464	1.38146
$428$	0	0
$429$	77.9541	3.76366
$430$	0	0
$431$	−10.1119	−0.487074	−0.243537	−	0.969892i	$-0.578308\pi$
$431$	−10.1119	−0.487074	−0.243537	+	0.969892i	$0.578308\pi$
$432$	0	0
$433$	14.9409	0.718014	0.359007	−	0.933335i	$-0.383115\pi$
$433$	14.9409	0.718014	0.359007	+	0.933335i	$0.383115\pi$
$434$	0	0
$435$	−30.0573	−1.44114
$436$	0	0
$437$	−43.5307	−2.08236
$438$	0	0
$439$	−22.7125	−1.08401	−0.542004	−	0.840376i	$-0.682334\pi$
$439$	−22.7125	−1.08401	−0.542004	+	0.840376i	$0.682334\pi$
$440$	0	0
$441$	−5.00137	−0.238160
$442$	0	0
$443$	−1.53378	−0.0728720	−0.0364360	−	0.999336i	$-0.511601\pi$
$443$	−1.53378	−0.0728720	−0.0364360	+	0.999336i	$0.511601\pi$
$444$	0	0
$445$	−22.8929	−1.08523
$446$	0	0
$447$	−15.5321	−0.734642
$448$	0	0
$449$	−13.8416	−0.653228	−0.326614	−	0.945158i	$-0.605908\pi$
$449$	−13.8416	−0.653228	−0.326614	+	0.945158i	$0.605908\pi$
$450$	0	0
$451$	32.9073	1.54954
$452$	0	0
$453$	52.7702	2.47936
$454$	0	0
$455$	−31.9918	−1.49980
$456$	0	0
$457$	−30.4282	−1.42337	−0.711685	−	0.702499i	$-0.752068\pi$
$457$	−30.4282	−1.42337	−0.711685	+	0.702499i	$0.752068\pi$
$458$	0	0
$459$	−0.917132	−0.0428081
$460$	0	0
$461$	16.5661	0.771561	0.385780	−	0.922591i	$-0.373932\pi$
$461$	16.5661	0.771561	0.385780	+	0.922591i	$0.373932\pi$
$462$	0	0
$463$	14.7244	0.684300	0.342150	−	0.939645i	$-0.388845\pi$
$463$	14.7244	0.684300	0.342150	+	0.939645i	$0.388845\pi$
$464$	0	0
$465$	1.61823	0.0750438
$466$	0	0
$467$	−0.384116	−0.0177747	−0.00888737	−	0.999961i	$-0.502829\pi$
$467$	−0.384116	−0.0177747	−0.00888737	+	0.999961i	$0.502829\pi$
$468$	0	0
$469$	7.77898	0.359200
$470$	0	0
$471$	22.4540	1.03463
$472$	0	0
$473$	31.8395	1.46398
$474$	0	0
$475$	−8.61216	−0.395153
$476$	0	0
$477$	14.0869	0.644993
$478$	0	0
$479$	29.5628	1.35076	0.675380	−	0.737469i	$-0.263979\pi$
$479$	29.5628	1.35076	0.675380	+	0.737469i	$0.263979\pi$
$480$	0	0
$481$	26.6731	1.21619
$482$	0	0
$483$	49.3379	2.24495
$484$	0	0
$485$	−4.32417	−0.196351
$486$	0	0
$487$	8.39436	0.380385	0.190192	−	0.981747i	$-0.439089\pi$
$487$	8.39436	0.380385	0.190192	+	0.981747i	$0.439089\pi$
$488$	0	0
$489$	−9.83949	−0.444957
$490$	0	0
$491$	−42.2433	−1.90642	−0.953208	−	0.302316i	$-0.902240\pi$
$491$	−42.2433	−1.90642	−0.953208	+	0.302316i	$0.902240\pi$
$492$	0	0
$493$	−4.85561	−0.218686
$494$	0	0
$495$	−41.3748	−1.85966
$496$	0	0
$497$	4.68946	0.210351
$498$	0	0
$499$	−10.8615	−0.486227	−0.243114	−	0.969998i	$-0.578169\pi$
$499$	−10.8615	−0.486227	−0.243114	+	0.969998i	$0.578169\pi$
$500$	0	0
$501$	10.2359	0.457305
$502$	0	0
$503$	−36.9612	−1.64802	−0.824008	−	0.566578i	$-0.808267\pi$
$503$	−36.9612	−1.64802	−0.824008	+	0.566578i	$0.808267\pi$
$504$	0	0
$505$	−26.9838	−1.20076
$506$	0	0
$507$	−39.0370	−1.73369
$508$	0	0
$509$	9.67091	0.428655	0.214328	−	0.976762i	$-0.431244\pi$
$509$	9.67091	0.428655	0.214328	+	0.976762i	$0.431244\pi$
$510$	0	0
$511$	1.24485	0.0550687
$512$	0	0
$513$	−4.32337	−0.190882
$514$	0	0
$515$	16.7895	0.739833
$516$	0	0
$517$	8.13924	0.357963
$518$	0	0
$519$	−16.5270	−0.725456
$520$	0	0
$521$	6.39688	0.280253	0.140126	−	0.990134i	$-0.455249\pi$
$521$	6.39688	0.280253	0.140126	+	0.990134i	$0.455249\pi$
$522$	0	0
$523$	−23.1391	−1.01180	−0.505902	−	0.862591i	$-0.668840\pi$
$523$	−23.1391	−1.01180	−0.505902	+	0.862591i	$0.668840\pi$
$524$	0	0
$525$	9.76107	0.426008
$526$	0	0
$527$	0.261418	0.0113875
$528$	0	0
$529$	62.2726	2.70750
$530$	0	0
$531$	2.61289	0.113390
$532$	0	0
$533$	−29.4803	−1.27693
$534$	0	0
$535$	−27.5078	−1.18927
$536$	0	0
$537$	44.0025	1.89885
$538$	0	0
$539$	11.6004	0.499663
$540$	0	0
$541$	−26.6644	−1.14639	−0.573196	−	0.819418i	$-0.694297\pi$
$541$	−26.6644	−1.14639	−0.573196	+	0.819418i	$0.694297\pi$
$542$	0	0
$543$	55.3519	2.37538
$544$	0	0
$545$	−0.859134	−0.0368012
$546$	0	0
$547$	21.5239	0.920294	0.460147	−	0.887843i	$-0.347797\pi$
$547$	21.5239	0.920294	0.460147	+	0.887843i	$0.347797\pi$
$548$	0	0
$549$	−33.0741	−1.41157
$550$	0	0
$551$	−22.8894	−0.975121
$552$	0	0
$553$	−34.1460	−1.45204
$554$	0	0
$555$	−30.4114	−1.29089
$556$	0	0
$557$	28.3005	1.19913	0.599565	−	0.800326i	$-0.295340\pi$
$557$	28.3005	1.19913	0.599565	+	0.800326i	$0.295340\pi$
$558$	0	0
$559$	−28.5238	−1.20643
$560$	0	0
$561$	−14.3581	−0.606198
$562$	0	0
$563$	−0.369301	−0.0155642	−0.00778209	−	0.999970i	$-0.502477\pi$
$563$	−0.369301	−0.0155642	−0.00778209	+	0.999970i	$0.502477\pi$
$564$	0	0
$565$	−6.75073	−0.284005
$566$	0	0
$567$	22.5778	0.948179
$568$	0	0
$569$	14.2674	0.598120	0.299060	−	0.954234i	$-0.403327\pi$
$569$	14.2674	0.598120	0.299060	+	0.954234i	$0.403327\pi$
$570$	0	0
$571$	29.6239	1.23972	0.619860	−	0.784713i	$-0.287189\pi$
$571$	29.6239	1.23972	0.619860	+	0.784713i	$0.287189\pi$
$572$	0	0
$573$	−7.57640	−0.316509
$574$	0	0
$575$	16.8704	0.703545
$576$	0	0
$577$	−10.7653	−0.448167	−0.224084	−	0.974570i	$-0.571939\pi$
$577$	−10.7653	−0.448167	−0.224084	+	0.974570i	$0.571939\pi$
$578$	0	0
$579$	49.0243	2.03738
$580$	0	0
$581$	−15.8164	−0.656173
$582$	0	0
$583$	−32.6736	−1.35320
$584$	0	0
$585$	37.0660	1.53249
$586$	0	0
$587$	−9.82168	−0.405384	−0.202692	−	0.979243i	$-0.564969\pi$
$587$	−9.82168	−0.405384	−0.202692	+	0.979243i	$0.564969\pi$
$588$	0	0
$589$	1.23233	0.0507772
$590$	0	0
$591$	−3.41722	−0.140566
$592$	0	0
$593$	33.6900	1.38348	0.691741	−	0.722146i	$-0.256844\pi$
$593$	33.6900	1.38348	0.691741	+	0.722146i	$0.256844\pi$
$594$	0	0
$595$	5.89245	0.241567
$596$	0	0
$597$	14.9242	0.610808
$598$	0	0
$599$	−9.78884	−0.399961	−0.199981	−	0.979800i	$-0.564088\pi$
$599$	−9.78884	−0.399961	−0.199981	+	0.979800i	$0.564088\pi$
$600$	0	0
$601$	2.64087	0.107723	0.0538616	−	0.998548i	$-0.482847\pi$
$601$	2.64087	0.107723	0.0538616	+	0.998548i	$0.482847\pi$
$602$	0	0
$603$	−9.01281	−0.367030
$604$	0	0
$605$	67.2250	2.73308
$606$	0	0
$607$	15.4789	0.628268	0.314134	−	0.949379i	$-0.398286\pi$
$607$	15.4789	0.628268	0.314134	+	0.949379i	$0.398286\pi$
$608$	0	0
$609$	25.9430	1.05126
$610$	0	0
$611$	−7.29162	−0.294987
$612$	0	0
$613$	16.0780	0.649382	0.324691	−	0.945820i	$-0.394740\pi$
$613$	16.0780	0.649382	0.324691	+	0.945820i	$0.394740\pi$
$614$	0	0
$615$	33.6120	1.35537
$616$	0	0
$617$	−42.1336	−1.69623	−0.848116	−	0.529810i	$-0.822263\pi$
$617$	−42.1336	−1.69623	−0.848116	+	0.529810i	$0.822263\pi$
$618$	0	0
$619$	−17.6412	−0.709061	−0.354530	−	0.935045i	$-0.615359\pi$
$619$	−17.6412	−0.709061	−0.354530	+	0.935045i	$0.615359\pi$
$620$	0	0
$621$	8.46909	0.339853
$622$	0	0
$623$	19.7593	0.791639
$624$	0	0
$625$	−30.7970	−1.23188
$626$	0	0
$627$	−67.6841	−2.70304
$628$	0	0
$629$	−4.91282	−0.195887
$630$	0	0
$631$	40.2793	1.60350	0.801748	−	0.597663i	$-0.203904\pi$
$631$	40.2793	1.60350	0.801748	+	0.597663i	$0.203904\pi$
$632$	0	0
$633$	−5.95306	−0.236613
$634$	0	0
$635$	5.25324	0.208468
$636$	0	0
$637$	−10.3923	−0.411758
$638$	0	0
$639$	−5.43326	−0.214936
$640$	0	0
$641$	−6.22392	−0.245830	−0.122915	−	0.992417i	$-0.539224\pi$
$641$	−6.22392	−0.245830	−0.122915	+	0.992417i	$0.539224\pi$
$642$	0	0
$643$	40.6123	1.60159	0.800796	−	0.598937i	$-0.204410\pi$
$643$	40.6123	1.60159	0.800796	+	0.598937i	$0.204410\pi$
$644$	0	0
$645$	32.5214	1.28053
$646$	0	0
$647$	−42.4761	−1.66991	−0.834953	−	0.550321i	$-0.814505\pi$
$647$	−42.4761	−1.66991	−0.834953	+	0.550321i	$0.814505\pi$
$648$	0	0
$649$	−6.06042	−0.237892
$650$	0	0
$651$	−1.39673	−0.0547421
$652$	0	0
$653$	−1.45550	−0.0569582	−0.0284791	−	0.999594i	$-0.509066\pi$
$653$	−1.45550	−0.0569582	−0.0284791	+	0.999594i	$0.509066\pi$
$654$	0	0
$655$	35.1708	1.37424
$656$	0	0
$657$	−1.44229	−0.0562691
$658$	0	0
$659$	−12.1468	−0.473172	−0.236586	−	0.971611i	$-0.576028\pi$
$659$	−12.1468	−0.473172	−0.236586	+	0.971611i	$0.576028\pi$
$660$	0	0
$661$	−24.1812	−0.940541	−0.470271	−	0.882522i	$-0.655844\pi$
$661$	−24.1812	−0.940541	−0.470271	+	0.882522i	$0.655844\pi$
$662$	0	0
$663$	12.8628	0.499550
$664$	0	0
$665$	27.7771	1.07715
$666$	0	0
$667$	44.8382	1.73614
$668$	0	0
$669$	2.82523	0.109230
$670$	0	0
$671$	76.7133	2.96148
$672$	0	0
$673$	28.6407	1.10402	0.552009	−	0.833838i	$-0.313861\pi$
$673$	28.6407	1.10402	0.552009	+	0.833838i	$0.313861\pi$
$674$	0	0
$675$	1.67553	0.0644913
$676$	0	0
$677$	30.3915	1.16804	0.584020	−	0.811739i	$-0.301479\pi$
$677$	30.3915	1.16804	0.584020	+	0.811739i	$0.301479\pi$
$678$	0	0
$679$	3.73228	0.143232
$680$	0	0
$681$	69.3098	2.65596
$682$	0	0
$683$	−42.1625	−1.61330	−0.806651	−	0.591028i	$-0.798722\pi$
$683$	−42.1625	−1.61330	−0.806651	+	0.591028i	$0.798722\pi$
$684$	0	0
$685$	−50.0450	−1.91212
$686$	0	0
$687$	39.7192	1.51538
$688$	0	0
$689$	29.2710	1.11513
$690$	0	0
$691$	−6.37017	−0.242333	−0.121166	−	0.992632i	$-0.538663\pi$
$691$	−6.37017	−0.242333	−0.121166	+	0.992632i	$0.538663\pi$
$692$	0	0
$693$	35.7114	1.35656
$694$	0	0
$695$	37.2203	1.41185
$696$	0	0
$697$	5.42987	0.205671
$698$	0	0
$699$	−28.1171	−1.06349
$700$	0	0
$701$	−27.0562	−1.02190	−0.510948	−	0.859611i	$-0.670706\pi$
$701$	−27.0562	−1.02190	−0.510948	+	0.859611i	$0.670706\pi$
$702$	0	0
$703$	−23.1591	−0.873463
$704$	0	0
$705$	8.31355	0.313107
$706$	0	0
$707$	23.2902	0.875918
$708$	0	0
$709$	26.8377	1.00791	0.503955	−	0.863730i	$-0.331878\pi$
$709$	26.8377	1.00791	0.503955	+	0.863730i	$0.331878\pi$
$710$	0	0
$711$	39.5619	1.48369
$712$	0	0
$713$	−2.41402	−0.0904056
$714$	0	0
$715$	−85.9723	−3.21518
$716$	0	0
$717$	45.1712	1.68695
$718$	0	0
$719$	−14.3047	−0.533476	−0.266738	−	0.963769i	$-0.585946\pi$
$719$	−14.3047	−0.533476	−0.266738	+	0.963769i	$0.585946\pi$
$720$	0	0
$721$	−14.4913	−0.539685
$722$	0	0
$723$	−24.8508	−0.924211
$724$	0	0
$725$	8.87084	0.329455
$726$	0	0
$727$	−23.1343	−0.858005	−0.429003	−	0.903303i	$-0.641135\pi$
$727$	−23.1343	−0.858005	−0.429003	+	0.903303i	$0.641135\pi$
$728$	0	0
$729$	−19.6404	−0.727422
$730$	0	0
$731$	5.25368	0.194314
$732$	0	0
$733$	45.4479	1.67866	0.839329	−	0.543624i	$-0.182948\pi$
$733$	45.4479	1.67866	0.839329	+	0.543624i	$0.182948\pi$
$734$	0	0
$735$	11.8488	0.437049
$736$	0	0
$737$	20.9046	0.770032
$738$	0	0
$739$	−39.6465	−1.45842	−0.729210	−	0.684290i	$-0.760112\pi$
$739$	−39.6465	−1.45842	−0.729210	+	0.684290i	$0.760112\pi$
$740$	0	0
$741$	60.6355	2.22750
$742$	0	0
$743$	−42.5081	−1.55947	−0.779736	−	0.626108i	$-0.784647\pi$
$743$	−42.5081	−1.55947	−0.779736	+	0.626108i	$0.784647\pi$
$744$	0	0
$745$	17.1297	0.627582
$746$	0	0
$747$	18.3250	0.670476
$748$	0	0
$749$	23.7425	0.867533
$750$	0	0
$751$	15.6674	0.571712	0.285856	−	0.958273i	$-0.407722\pi$
$751$	15.6674	0.571712	0.285856	+	0.958273i	$0.407722\pi$
$752$	0	0
$753$	4.54786	0.165733
$754$	0	0
$755$	−58.1980	−2.11804
$756$	0	0
$757$	42.5551	1.54669	0.773345	−	0.633985i	$-0.218582\pi$
$757$	42.5551	1.54669	0.773345	+	0.633985i	$0.218582\pi$
$758$	0	0
$759$	132.587	4.81260
$760$	0	0
$761$	36.1326	1.30981	0.654903	−	0.755713i	$-0.272709\pi$
$761$	36.1326	1.30981	0.654903	+	0.755713i	$0.272709\pi$
$762$	0	0
$763$	0.741535	0.0268454
$764$	0	0
$765$	−6.82705	−0.246833
$766$	0	0
$767$	5.42929	0.196040
$768$	0	0
$769$	23.5262	0.848378	0.424189	−	0.905574i	$-0.360559\pi$
$769$	23.5262	0.848378	0.424189	+	0.905574i	$0.360559\pi$
$770$	0	0
$771$	35.5608	1.28069
$772$	0	0
$773$	−8.44290	−0.303670	−0.151835	−	0.988406i	$-0.548518\pi$
$773$	−8.44290	−0.303670	−0.151835	+	0.988406i	$0.548518\pi$
$774$	0	0
$775$	−0.477592	−0.0171556
$776$	0	0
$777$	26.2487	0.941666
$778$	0	0
$779$	25.5965	0.917089
$780$	0	0
$781$	12.6021	0.450938
$782$	0	0
$783$	4.45323	0.159146
$784$	0	0
$785$	−24.7636	−0.883851
$786$	0	0
$787$	52.5998	1.87498	0.937491	−	0.348009i	$-0.113142\pi$
$787$	52.5998	1.87498	0.937491	+	0.348009i	$0.113142\pi$
$788$	0	0
$789$	18.5717	0.661169
$790$	0	0
$791$	5.82668	0.207173
$792$	0	0
$793$	−68.7244	−2.44047
$794$	0	0
$795$	−33.3733	−1.18363
$796$	0	0
$797$	−5.46355	−0.193529	−0.0967645	−	0.995307i	$-0.530849\pi$
$797$	−5.46355	−0.193529	−0.0967645	+	0.995307i	$0.530849\pi$
$798$	0	0
$799$	1.34302	0.0475125
$800$	0	0
$801$	−22.8933	−0.808895
$802$	0	0
$803$	3.34530	0.118053
$804$	0	0
$805$	−54.4127	−1.91780
$806$	0	0
$807$	20.9347	0.736938
$808$	0	0
$809$	7.54210	0.265166	0.132583	−	0.991172i	$-0.457673\pi$
$809$	7.54210	0.265166	0.132583	+	0.991172i	$0.457673\pi$
$810$	0	0
$811$	−37.2599	−1.30837	−0.654187	−	0.756333i	$-0.726989\pi$
$811$	−37.2599	−1.30837	−0.654187	+	0.756333i	$0.726989\pi$
$812$	0	0
$813$	64.0148	2.24510
$814$	0	0
$815$	10.8516	0.380114
$816$	0	0
$817$	24.7659	0.866450
$818$	0	0
$819$	−31.9924	−1.11790
$820$	0	0
$821$	42.2056	1.47299	0.736494	−	0.676445i	$-0.236480\pi$
$821$	42.2056	1.47299	0.736494	+	0.676445i	$0.236480\pi$
$822$	0	0
$823$	−8.40358	−0.292930	−0.146465	−	0.989216i	$-0.546790\pi$
$823$	−8.40358	−0.292930	−0.146465	+	0.989216i	$0.546790\pi$
$824$	0	0
$825$	26.2311	0.913251
$826$	0	0
$827$	40.2908	1.40105	0.700525	−	0.713628i	$-0.252949\pi$
$827$	40.2908	1.40105	0.700525	+	0.713628i	$0.252949\pi$
$828$	0	0
$829$	32.4352	1.12652	0.563261	−	0.826279i	$-0.309547\pi$
$829$	32.4352	1.12652	0.563261	+	0.826279i	$0.309547\pi$
$830$	0	0
$831$	58.1311	2.01655
$832$	0	0
$833$	1.91412	0.0663202
$834$	0	0
$835$	−11.2887	−0.390662
$836$	0	0
$837$	−0.239755	−0.00828714
$838$	0	0
$839$	−38.7784	−1.33878	−0.669390	−	0.742911i	$-0.733445\pi$
$839$	−38.7784	−1.33878	−0.669390	+	0.742911i	$0.733445\pi$
$840$	0	0
$841$	−5.42307	−0.187002
$842$	0	0
$843$	−30.4170	−1.04762
$844$	0	0
$845$	43.0522	1.48104
$846$	0	0
$847$	−58.0232	−1.99370
$848$	0	0
$849$	−76.5463	−2.62706
$850$	0	0
$851$	45.3666	1.55515
$852$	0	0
$853$	25.0484	0.857640	0.428820	−	0.903390i	$-0.358929\pi$
$853$	25.0484	0.857640	0.428820	+	0.903390i	$0.358929\pi$
$854$	0	0
$855$	−32.1828	−1.10063
$856$	0	0
$857$	36.6693	1.25260	0.626300	−	0.779582i	$-0.284569\pi$
$857$	36.6693	1.25260	0.626300	+	0.779582i	$0.284569\pi$
$858$	0	0
$859$	−12.8294	−0.437734	−0.218867	−	0.975755i	$-0.570236\pi$
$859$	−12.8294	−0.437734	−0.218867	+	0.975755i	$0.570236\pi$
$860$	0	0
$861$	−29.0112	−0.988699
$862$	0	0
$863$	−35.0474	−1.19303	−0.596513	−	0.802603i	$-0.703448\pi$
$863$	−35.0474	−1.19303	−0.596513	+	0.802603i	$0.703448\pi$
$864$	0	0
$865$	18.2270	0.619735
$866$	0	0
$867$	−2.36915	−0.0804607
$868$	0	0
$869$	−91.7613	−3.11279
$870$	0	0
$871$	−18.7276	−0.634561
$872$	0	0
$873$	−4.32425	−0.146354
$874$	0	0
$875$	18.6972	0.632080
$876$	0	0
$877$	15.6076	0.527032	0.263516	−	0.964655i	$-0.415118\pi$
$877$	15.6076	0.527032	0.263516	+	0.964655i	$0.415118\pi$
$878$	0	0
$879$	−18.5628	−0.626106
$880$	0	0
$881$	20.6422	0.695453	0.347726	−	0.937596i	$-0.386954\pi$
$881$	20.6422	0.695453	0.347726	+	0.937596i	$0.386954\pi$
$882$	0	0
$883$	19.8510	0.668040	0.334020	−	0.942566i	$-0.391595\pi$
$883$	19.8510	0.668040	0.334020	+	0.942566i	$0.391595\pi$
$884$	0	0
$885$	−6.19022	−0.208082
$886$	0	0
$887$	−0.0138328	−0.000464460 0	−0.000232230	−	1.00000i	$-0.500074\pi$
$887$	−0.0138328	−0.000464460 0	−0.000232230		1.00000i	$0.500074\pi$
$888$	0	0
$889$	−4.53417	−0.152071
$890$	0	0
$891$	60.6738	2.03265
$892$	0	0
$893$	6.33099	0.211859
$894$	0	0
$895$	−48.5285	−1.62213
$896$	0	0
$897$	−118.779	−3.96593
$898$	0	0
$899$	−1.26934	−0.0423350
$900$	0	0
$901$	−5.39130	−0.179610
$902$	0	0
$903$	−28.0698	−0.934106
$904$	0	0
$905$	−61.0452	−2.02921
$906$	0	0
$907$	5.63211	0.187011	0.0935056	−	0.995619i	$-0.470193\pi$
$907$	5.63211	0.187011	0.0935056	+	0.995619i	$0.470193\pi$
$908$	0	0
$909$	−26.9842	−0.895011
$910$	0	0
$911$	13.6162	0.451124	0.225562	−	0.974229i	$-0.427578\pi$
$911$	13.6162	0.451124	0.225562	+	0.974229i	$0.427578\pi$
$912$	0	0
$913$	−42.5036	−1.40667
$914$	0	0
$915$	78.3562	2.59038
$916$	0	0
$917$	−30.3566	−1.00246
$918$	0	0
$919$	14.6397	0.482918	0.241459	−	0.970411i	$-0.422374\pi$
$919$	14.6397	0.482918	0.241459	+	0.970411i	$0.422374\pi$
$920$	0	0
$921$	55.0483	1.81390
$922$	0	0
$923$	−11.2897	−0.371605
$924$	0	0
$925$	8.97537	0.295108
$926$	0	0
$927$	16.7898	0.551449
$928$	0	0
$929$	−54.5903	−1.79105	−0.895525	−	0.445012i	$-0.853200\pi$
$929$	−54.5903	−1.79105	−0.895525	+	0.445012i	$0.853200\pi$
$930$	0	0
$931$	9.02317	0.295722
$932$	0	0
$933$	−58.9213	−1.92900
$934$	0	0
$935$	15.8349	0.517857
$936$	0	0
$937$	23.6633	0.773047	0.386524	−	0.922280i	$-0.373676\pi$
$937$	23.6633	0.773047	0.386524	+	0.922280i	$0.373676\pi$
$938$	0	0
$939$	46.7064	1.52421
$940$	0	0
$941$	8.03358	0.261887	0.130944	−	0.991390i	$-0.458199\pi$
$941$	8.03358	0.261887	0.130944	+	0.991390i	$0.458199\pi$
$942$	0	0
$943$	−50.1411	−1.63282
$944$	0	0
$945$	−5.40416	−0.175797
$946$	0	0
$947$	57.9876	1.88435	0.942173	−	0.335128i	$-0.108780\pi$
$947$	57.9876	1.88435	0.942173	+	0.335128i	$0.108780\pi$
$948$	0	0
$949$	−2.99692	−0.0972842
$950$	0	0
$951$	−2.85950	−0.0927258
$952$	0	0
$953$	30.7105	0.994809	0.497405	−	0.867519i	$-0.334286\pi$
$953$	30.7105	0.994809	0.497405	+	0.867519i	$0.334286\pi$
$954$	0	0
$955$	8.35569	0.270384
$956$	0	0
$957$	69.7171	2.25364
$958$	0	0
$959$	43.1948	1.39483
$960$	0	0
$961$	−30.9317	−0.997796
$962$	0	0
$963$	−27.5083	−0.886443
$964$	0	0
$965$	−54.0668	−1.74047
$966$	0	0
$967$	−27.3529	−0.879610	−0.439805	−	0.898093i	$-0.644953\pi$
$967$	−27.3529	−0.879610	−0.439805	+	0.898093i	$0.644953\pi$
$968$	0	0
$969$	−11.1682	−0.358775
$970$	0	0
$971$	13.1142	0.420855	0.210428	−	0.977609i	$-0.432514\pi$
$971$	13.1142	0.420855	0.210428	+	0.977609i	$0.432514\pi$
$972$	0	0
$973$	−32.1256	−1.02990
$974$	0	0
$975$	−23.4994	−0.752584
$976$	0	0
$977$	−19.5068	−0.624078	−0.312039	−	0.950069i	$-0.601012\pi$
$977$	−19.5068	−0.624078	−0.312039	+	0.950069i	$0.601012\pi$
$978$	0	0
$979$	53.0996	1.69707
$980$	0	0
$981$	−0.859149	−0.0274305
$982$	0	0
$983$	17.2597	0.550500	0.275250	−	0.961373i	$-0.411239\pi$
$983$	17.2597	0.550500	0.275250	+	0.961373i	$0.411239\pi$
$984$	0	0
$985$	3.76871	0.120081
$986$	0	0
$987$	−7.17559	−0.228401
$988$	0	0
$989$	−48.5141	−1.54266
$990$	0	0
$991$	−41.2804	−1.31132	−0.655658	−	0.755058i	$-0.727609\pi$
$991$	−41.2804	−1.31132	−0.655658	+	0.755058i	$0.727609\pi$
$992$	0	0
$993$	53.9540	1.71218
$994$	0	0
$995$	−16.4593	−0.521795
$996$	0	0
$997$	29.9074	0.947178	0.473589	−	0.880746i	$-0.342958\pi$
$997$	29.9074	0.947178	0.473589	+	0.880746i	$0.342958\pi$
$998$	0	0
$999$	4.50571	0.142554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.3	✓	12	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)

\(f(q)\)	\(=\)	\(q-2.36915 q^{3} +2.61284 q^{5} -2.25519 q^{7} +2.61289 q^{9} +O(q^{10})\)	\(q-2.36915 q^{3} +2.61284 q^{5} -2.25519 q^{7} +2.61289 q^{9} -6.06042 q^{11} +5.42929 q^{13} -6.19022 q^{15} -1.00000 q^{17} -4.71401 q^{19} +5.34289 q^{21} +9.23431 q^{23} +1.82693 q^{25} +0.917132 q^{27} +4.85561 q^{29} -0.261418 q^{31} +14.3581 q^{33} -5.89245 q^{35} +4.91282 q^{37} -12.8628 q^{39} -5.42987 q^{41} -5.25368 q^{43} +6.82705 q^{45} -1.34302 q^{47} -1.91412 q^{49} +2.36915 q^{51} +5.39130 q^{53} -15.8349 q^{55} +11.1682 q^{57} +1.00000 q^{59} -12.6581 q^{61} -5.89256 q^{63} +14.1859 q^{65} -3.44937 q^{67} -21.8775 q^{69} -2.07941 q^{71} -0.551991 q^{73} -4.32827 q^{75} +13.6674 q^{77} +15.1411 q^{79} -10.0115 q^{81} +7.01331 q^{83} -2.61284 q^{85} -11.5037 q^{87} -8.76169 q^{89} -12.2441 q^{91} +0.619339 q^{93} -12.3170 q^{95} -1.65497 q^{97} -15.8352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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