LMFDB - Embedded newform 4014.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 12x^{4} + 50x^{3} - 36x^{2} - 38x + 18 $ x^7 - x^6 - 14x^5 + 12x^4 + 50x^3 - 36x^2 - 38*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.08762$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.580657 q^{5} -1.39541 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -0.580657 q^{5} -1.39541 q^{7} -1.00000 q^{8} +0.580657 q^{10} +3.94828 q^{11} -5.15722 q^{13} +1.39541 q^{14} +1.00000 q^{16} +1.15262 q^{17} -6.17523 q^{19} -0.580657 q^{20} -3.94828 q^{22} -8.81546 q^{23} -4.66284 q^{25} +5.15722 q^{26} -1.39541 q^{28} +3.09956 q^{29} +4.25998 q^{31} -1.00000 q^{32} -1.15262 q^{34} +0.810256 q^{35} +11.3755 q^{37} +6.17523 q^{38} +0.580657 q^{40} +2.30330 q^{41} +6.45934 q^{43} +3.94828 q^{44} +8.81546 q^{46} +4.58791 q^{47} -5.05282 q^{49} +4.66284 q^{50} -5.15722 q^{52} -7.69266 q^{53} -2.29260 q^{55} +1.39541 q^{56} -3.09956 q^{58} +8.70194 q^{59} +14.9904 q^{61} -4.25998 q^{62} +1.00000 q^{64} +2.99457 q^{65} +7.23327 q^{67} +1.15262 q^{68} -0.810256 q^{70} -8.42034 q^{71} -4.40613 q^{73} -11.3755 q^{74} -6.17523 q^{76} -5.50948 q^{77} +2.59714 q^{79} -0.580657 q^{80} -2.30330 q^{82} -6.07040 q^{83} -0.669276 q^{85} -6.45934 q^{86} -3.94828 q^{88} -2.89241 q^{89} +7.19645 q^{91} -8.81546 q^{92} -4.58791 q^{94} +3.58569 q^{95} -3.84047 q^{97} +5.05282 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8} + 2 q^{10} - 9 q^{11} - 2 q^{13} - 6 q^{14} + 7 q^{16} + 7 q^{17} - 2 q^{19} - 2 q^{20} + 9 q^{22} - 15 q^{23} + 13 q^{25} + 2 q^{26} + 6 q^{28} - 9 q^{29} - 2 q^{31} - 7 q^{32} - 7 q^{34} + 4 q^{35} + 5 q^{37} + 2 q^{38} + 2 q^{40} + 33 q^{41} + 20 q^{43} - 9 q^{44} + 15 q^{46} + 2 q^{47} + 3 q^{49} - 13 q^{50} - 2 q^{52} + 13 q^{53} - 18 q^{55} - 6 q^{56} + 9 q^{58} - 9 q^{59} + 8 q^{61} + 2 q^{62} + 7 q^{64} + 44 q^{65} + 29 q^{67} + 7 q^{68} - 4 q^{70} - 37 q^{73} - 5 q^{74} - 2 q^{76} + 18 q^{77} + 32 q^{79} - 2 q^{80} - 33 q^{82} + 6 q^{83} - 4 q^{85} - 20 q^{86} + 9 q^{88} + 17 q^{89} - 4 q^{91} - 15 q^{92} - 2 q^{94} + 12 q^{95} + 12 q^{97} - 3 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8 + 2 * q^10 - 9 * q^11 - 2 * q^13 - 6 * q^14 + 7 * q^16 + 7 * q^17 - 2 * q^19 - 2 * q^20 + 9 * q^22 - 15 * q^23 + 13 * q^25 + 2 * q^26 + 6 * q^28 - 9 * q^29 - 2 * q^31 - 7 * q^32 - 7 * q^34 + 4 * q^35 + 5 * q^37 + 2 * q^38 + 2 * q^40 + 33 * q^41 + 20 * q^43 - 9 * q^44 + 15 * q^46 + 2 * q^47 + 3 * q^49 - 13 * q^50 - 2 * q^52 + 13 * q^53 - 18 * q^55 - 6 * q^56 + 9 * q^58 - 9 * q^59 + 8 * q^61 + 2 * q^62 + 7 * q^64 + 44 * q^65 + 29 * q^67 + 7 * q^68 - 4 * q^70 - 37 * q^73 - 5 * q^74 - 2 * q^76 + 18 * q^77 + 32 * q^79 - 2 * q^80 - 33 * q^82 + 6 * q^83 - 4 * q^85 - 20 * q^86 + 9 * q^88 + 17 * q^89 - 4 * q^91 - 15 * q^92 - 2 * q^94 + 12 * q^95 + 12 * q^97 - 3 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.580657	−0.259678	−0.129839	−	0.991535i	$-0.541446\pi$
$5$	−0.580657	−0.259678	−0.129839	+	0.991535i	$0.541446\pi$
$6$	0	0
$7$	−1.39541	−0.527416	−0.263708	−	0.964603i	$-0.584946\pi$
$7$	−1.39541	−0.527416	−0.263708	+	0.964603i	$0.584946\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.580657	0.183620
$11$	3.94828	1.19045	0.595226	−	0.803558i	$-0.297062\pi$
$11$	3.94828	1.19045	0.595226	+	0.803558i	$0.297062\pi$
$12$	0	0
$13$	−5.15722	−1.43036	−0.715178	−	0.698943i	$-0.753654\pi$
$13$	−5.15722	−1.43036	−0.715178	+	0.698943i	$0.753654\pi$
$14$	1.39541	0.372940
$15$	0	0
$16$	1.00000	0.250000
$17$	1.15262	0.279551	0.139776	−	0.990183i	$-0.455362\pi$
$17$	1.15262	0.279551	0.139776	+	0.990183i	$0.455362\pi$
$18$	0	0
$19$	−6.17523	−1.41670	−0.708348	−	0.705864i	$-0.750559\pi$
$19$	−6.17523	−1.41670	−0.708348	+	0.705864i	$0.750559\pi$
$20$	−0.580657	−0.129839
$21$	0	0
$22$	−3.94828	−0.841777
$23$	−8.81546	−1.83815	−0.919075	−	0.394083i	$-0.871062\pi$
$23$	−8.81546	−1.83815	−0.919075	+	0.394083i	$0.871062\pi$
$24$	0	0
$25$	−4.66284	−0.932568
$26$	5.15722	1.01141
$27$	0	0
$28$	−1.39541	−0.263708
$29$	3.09956	0.575574	0.287787	−	0.957694i	$-0.407080\pi$
$29$	3.09956	0.575574	0.287787	+	0.957694i	$0.407080\pi$
$30$	0	0
$31$	4.25998	0.765115	0.382557	−	0.923932i	$-0.375043\pi$
$31$	4.25998	0.765115	0.382557	+	0.923932i	$0.375043\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.15262	−0.197673
$35$	0.810256	0.136958
$36$	0	0
$37$	11.3755	1.87012	0.935059	−	0.354493i	$-0.115347\pi$
$37$	11.3755	1.87012	0.935059	+	0.354493i	$0.115347\pi$
$38$	6.17523	1.00175
$39$	0	0
$40$	0.580657	0.0918099
$41$	2.30330	0.359714	0.179857	−	0.983693i	$-0.442436\pi$
$41$	2.30330	0.359714	0.179857	+	0.983693i	$0.442436\pi$
$42$	0	0
$43$	6.45934	0.985040	0.492520	−	0.870301i	$-0.336076\pi$
$43$	6.45934	0.985040	0.492520	+	0.870301i	$0.336076\pi$
$44$	3.94828	0.595226
$45$	0	0
$46$	8.81546	1.29977
$47$	4.58791	0.669215	0.334608	−	0.942358i	$-0.391396\pi$
$47$	4.58791	0.669215	0.334608	+	0.942358i	$0.391396\pi$
$48$	0	0
$49$	−5.05282	−0.721832
$50$	4.66284	0.659425
$51$	0	0
$52$	−5.15722	−0.715178
$53$	−7.69266	−1.05667	−0.528334	−	0.849036i	$-0.677183\pi$
$53$	−7.69266	−1.05667	−0.528334	+	0.849036i	$0.677183\pi$
$54$	0	0
$55$	−2.29260	−0.309134
$56$	1.39541	0.186470
$57$	0	0
$58$	−3.09956	−0.406992
$59$	8.70194	1.13290	0.566448	−	0.824098i	$-0.308317\pi$
$59$	8.70194	1.13290	0.566448	+	0.824098i	$0.308317\pi$
$60$	0	0
$61$	14.9904	1.91932	0.959662	−	0.281158i	$-0.0907184\pi$
$61$	14.9904	1.91932	0.959662	+	0.281158i	$0.0907184\pi$
$62$	−4.25998	−0.541018
$63$	0	0
$64$	1.00000	0.125000
$65$	2.99457	0.371431
$66$	0	0
$67$	7.23327	0.883685	0.441842	−	0.897093i	$-0.354325\pi$
$67$	7.23327	0.883685	0.441842	+	0.897093i	$0.354325\pi$
$68$	1.15262	0.139776
$69$	0	0
$70$	−0.810256	−0.0968441
$71$	−8.42034	−0.999310	−0.499655	−	0.866224i	$-0.666540\pi$
$71$	−8.42034	−0.999310	−0.499655	+	0.866224i	$0.666540\pi$
$72$	0	0
$73$	−4.40613	−0.515698	−0.257849	−	0.966185i	$-0.583014\pi$
$73$	−4.40613	−0.515698	−0.257849	+	0.966185i	$0.583014\pi$
$74$	−11.3755	−1.32237
$75$	0	0
$76$	−6.17523	−0.708348
$77$	−5.50948	−0.627864
$78$	0	0
$79$	2.59714	0.292201	0.146101	−	0.989270i	$-0.453328\pi$
$79$	2.59714	0.292201	0.146101	+	0.989270i	$0.453328\pi$
$80$	−0.580657	−0.0649194
$81$	0	0
$82$	−2.30330	−0.254357
$83$	−6.07040	−0.666313	−0.333156	−	0.942872i	$-0.608114\pi$
$83$	−6.07040	−0.666313	−0.333156	+	0.942872i	$0.608114\pi$
$84$	0	0
$85$	−0.669276	−0.0725932
$86$	−6.45934	−0.696528
$87$	0	0
$88$	−3.94828	−0.420888
$89$	−2.89241	−0.306594	−0.153297	−	0.988180i	$-0.548989\pi$
$89$	−2.89241	−0.306594	−0.153297	+	0.988180i	$0.548989\pi$
$90$	0	0
$91$	7.19645	0.754393
$92$	−8.81546	−0.919075
$93$	0	0
$94$	−4.58791	−0.473207
$95$	3.58569	0.367884
$96$	0	0
$97$	−3.84047	−0.389941	−0.194970	−	0.980809i	$-0.562461\pi$
$97$	−3.84047	−0.389941	−0.194970	+	0.980809i	$0.562461\pi$
$98$	5.05282	0.510412
$99$	0	0
$100$	−4.66284	−0.466284
$101$	13.8790	1.38101	0.690506	−	0.723327i	$-0.257388\pi$
$101$	13.8790	1.38101	0.690506	+	0.723327i	$0.257388\pi$
$102$	0	0
$103$	−14.9175	−1.46987	−0.734935	−	0.678138i	$-0.762787\pi$
$103$	−14.9175	−1.46987	−0.734935	+	0.678138i	$0.762787\pi$
$104$	5.15722	0.505707
$105$	0	0
$106$	7.69266	0.747178
$107$	2.20691	0.213350	0.106675	−	0.994294i	$-0.465980\pi$
$107$	2.20691	0.213350	0.106675	+	0.994294i	$0.465980\pi$
$108$	0	0
$109$	−5.10930	−0.489382	−0.244691	−	0.969601i	$-0.578687\pi$
$109$	−5.10930	−0.489382	−0.244691	+	0.969601i	$0.578687\pi$
$110$	2.29260	0.218591
$111$	0	0
$112$	−1.39541	−0.131854
$113$	4.65162	0.437587	0.218794	−	0.975771i	$-0.429788\pi$
$113$	4.65162	0.437587	0.218794	+	0.975771i	$0.429788\pi$
$114$	0	0
$115$	5.11875	0.477326
$116$	3.09956	0.287787
$117$	0	0
$118$	−8.70194	−0.801078
$119$	−1.60838	−0.147440
$120$	0	0
$121$	4.58894	0.417177
$122$	−14.9904	−1.35717
$123$	0	0
$124$	4.25998	0.382557
$125$	5.61079	0.501844
$126$	0	0
$127$	12.1807	1.08086	0.540430	−	0.841389i	$-0.318262\pi$
$127$	12.1807	1.08086	0.540430	+	0.841389i	$0.318262\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.99457	−0.262641
$131$	1.14392	0.0999452	0.0499726	−	0.998751i	$-0.484087\pi$
$131$	1.14392	0.0999452	0.0499726	+	0.998751i	$0.484087\pi$
$132$	0	0
$133$	8.61700	0.747188
$134$	−7.23327	−0.624860
$135$	0	0
$136$	−1.15262	−0.0988363
$137$	18.9183	1.61630	0.808149	−	0.588978i	$-0.200470\pi$
$137$	18.9183	1.61630	0.808149	+	0.588978i	$0.200470\pi$
$138$	0	0
$139$	3.35904	0.284910	0.142455	−	0.989801i	$-0.454500\pi$
$139$	3.35904	0.284910	0.142455	+	0.989801i	$0.454500\pi$
$140$	0.810256	0.0684791
$141$	0	0
$142$	8.42034	0.706619
$143$	−20.3622	−1.70277
$144$	0	0
$145$	−1.79978	−0.149464
$146$	4.40613	0.364654
$147$	0	0
$148$	11.3755	0.935059
$149$	4.75135	0.389246	0.194623	−	0.980878i	$-0.437652\pi$
$149$	4.75135	0.389246	0.194623	+	0.980878i	$0.437652\pi$
$150$	0	0
$151$	−11.4044	−0.928078	−0.464039	−	0.885815i	$-0.653600\pi$
$151$	−11.4044	−0.928078	−0.464039	+	0.885815i	$0.653600\pi$
$152$	6.17523	0.500877
$153$	0	0
$154$	5.50948	0.443967
$155$	−2.47359	−0.198683
$156$	0	0
$157$	−8.12086	−0.648116	−0.324058	−	0.946037i	$-0.605047\pi$
$157$	−8.12086	−0.648116	−0.324058	+	0.946037i	$0.605047\pi$
$158$	−2.59714	−0.206617
$159$	0	0
$160$	0.580657	0.0459049
$161$	12.3012	0.969470
$162$	0	0
$163$	15.1661	1.18790	0.593950	−	0.804502i	$-0.297568\pi$
$163$	15.1661	1.18790	0.593950	+	0.804502i	$0.297568\pi$
$164$	2.30330	0.179857
$165$	0	0
$166$	6.07040	0.471154
$167$	15.9248	1.23229	0.616147	−	0.787631i	$-0.288693\pi$
$167$	15.9248	1.23229	0.616147	+	0.787631i	$0.288693\pi$
$168$	0	0
$169$	13.5969	1.04592
$170$	0.669276	0.0513311
$171$	0	0
$172$	6.45934	0.492520
$173$	−8.19787	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$173$	−8.19787	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$174$	0	0
$175$	6.50658	0.491851
$176$	3.94828	0.297613
$177$	0	0
$178$	2.89241	0.216795
$179$	10.4109	0.778146	0.389073	−	0.921207i	$-0.372795\pi$
$179$	10.4109	0.778146	0.389073	+	0.921207i	$0.372795\pi$
$180$	0	0
$181$	11.3037	0.840199	0.420100	−	0.907478i	$-0.361995\pi$
$181$	11.3037	0.840199	0.420100	+	0.907478i	$0.361995\pi$
$182$	−7.19645	−0.533436
$183$	0	0
$184$	8.81546	0.649884
$185$	−6.60525	−0.485627
$186$	0	0
$187$	4.55087	0.332792
$188$	4.58791	0.334608
$189$	0	0
$190$	−3.58569	−0.260133
$191$	13.7113	0.992118	0.496059	−	0.868289i	$-0.334780\pi$
$191$	13.7113	0.992118	0.496059	+	0.868289i	$0.334780\pi$
$192$	0	0
$193$	−7.51938	−0.541256	−0.270628	−	0.962684i	$-0.587231\pi$
$193$	−7.51938	−0.541256	−0.270628	+	0.962684i	$0.587231\pi$
$194$	3.84047	0.275730
$195$	0	0
$196$	−5.05282	−0.360916
$197$	9.28097	0.661242	0.330621	−	0.943764i	$-0.392742\pi$
$197$	9.28097	0.661242	0.330621	+	0.943764i	$0.392742\pi$
$198$	0	0
$199$	6.68850	0.474136	0.237068	−	0.971493i	$-0.423814\pi$
$199$	6.68850	0.474136	0.237068	+	0.971493i	$0.423814\pi$
$200$	4.66284	0.329712
$201$	0	0
$202$	−13.8790	−0.976523
$203$	−4.32517	−0.303567
$204$	0	0
$205$	−1.33742	−0.0934098
$206$	14.9175	1.03935
$207$	0	0
$208$	−5.15722	−0.357589
$209$	−24.3816	−1.68651
$210$	0	0
$211$	25.0574	1.72502	0.862511	−	0.506038i	$-0.168890\pi$
$211$	25.0574	1.72502	0.862511	+	0.506038i	$0.168890\pi$
$212$	−7.69266	−0.528334
$213$	0	0
$214$	−2.20691	−0.150861
$215$	−3.75066	−0.255793
$216$	0	0
$217$	−5.94443	−0.403534
$218$	5.10930	0.346046
$219$	0	0
$220$	−2.29260	−0.154567
$221$	−5.94431	−0.399857
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	1.39541	0.0932349
$225$	0	0
$226$	−4.65162	−0.309421
$227$	−20.6803	−1.37260	−0.686299	−	0.727320i	$-0.740766\pi$
$227$	−20.6803	−1.37260	−0.686299	+	0.727320i	$0.740766\pi$
$228$	0	0
$229$	16.7199	1.10488	0.552442	−	0.833552i	$-0.313696\pi$
$229$	16.7199	1.10488	0.552442	+	0.833552i	$0.313696\pi$
$230$	−5.11875	−0.337521
$231$	0	0
$232$	−3.09956	−0.203496
$233$	−16.7194	−1.09532	−0.547661	−	0.836700i	$-0.684482\pi$
$233$	−16.7194	−1.09532	−0.547661	+	0.836700i	$0.684482\pi$
$234$	0	0
$235$	−2.66400	−0.173780
$236$	8.70194	0.566448
$237$	0	0
$238$	1.60838	0.104256
$239$	0.452261	0.0292543	0.0146272	−	0.999893i	$-0.495344\pi$
$239$	0.452261	0.0292543	0.0146272	+	0.999893i	$0.495344\pi$
$240$	0	0
$241$	−5.63657	−0.363083	−0.181542	−	0.983383i	$-0.558109\pi$
$241$	−5.63657	−0.363083	−0.181542	+	0.983383i	$0.558109\pi$
$242$	−4.58894	−0.294988
$243$	0	0
$244$	14.9904	0.959662
$245$	2.93396	0.187444
$246$	0	0
$247$	31.8470	2.02638
$248$	−4.25998	−0.270509
$249$	0	0
$250$	−5.61079	−0.354858
$251$	−22.8236	−1.44061	−0.720307	−	0.693655i	$-0.755999\pi$
$251$	−22.8236	−1.44061	−0.720307	+	0.693655i	$0.755999\pi$
$252$	0	0
$253$	−34.8059	−2.18823
$254$	−12.1807	−0.764283
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.1676	1.32040	0.660198	−	0.751092i	$-0.270472\pi$
$257$	21.1676	1.32040	0.660198	+	0.751092i	$0.270472\pi$
$258$	0	0
$259$	−15.8735	−0.986330
$260$	2.99457	0.185716
$261$	0	0
$262$	−1.14392	−0.0706719
$263$	−14.2781	−0.880423	−0.440211	−	0.897894i	$-0.645096\pi$
$263$	−14.2781	−0.880423	−0.440211	+	0.897894i	$0.645096\pi$
$264$	0	0
$265$	4.46680	0.274393
$266$	−8.61700	−0.528342
$267$	0	0
$268$	7.23327	0.441842
$269$	−8.70356	−0.530666	−0.265333	−	0.964157i	$-0.585482\pi$
$269$	−8.70356	−0.530666	−0.265333	+	0.964157i	$0.585482\pi$
$270$	0	0
$271$	17.3264	1.05250	0.526252	−	0.850328i	$-0.323597\pi$
$271$	17.3264	1.05250	0.526252	+	0.850328i	$0.323597\pi$
$272$	1.15262	0.0698878
$273$	0	0
$274$	−18.9183	−1.14290
$275$	−18.4102	−1.11018
$276$	0	0
$277$	−19.2916	−1.15912	−0.579560	−	0.814930i	$-0.696775\pi$
$277$	−19.2916	−1.15912	−0.579560	+	0.814930i	$0.696775\pi$
$278$	−3.35904	−0.201462
$279$	0	0
$280$	−0.810256	−0.0484220
$281$	27.3598	1.63215	0.816073	−	0.577949i	$-0.196147\pi$
$281$	27.3598	1.63215	0.816073	+	0.577949i	$0.196147\pi$
$282$	0	0
$283$	30.0939	1.78890	0.894448	−	0.447171i	$-0.147569\pi$
$283$	30.0939	1.78890	0.894448	+	0.447171i	$0.147569\pi$
$284$	−8.42034	−0.499655
$285$	0	0
$286$	20.3622	1.20404
$287$	−3.21405	−0.189719
$288$	0	0
$289$	−15.6715	−0.921851
$290$	1.79978	0.105687
$291$	0	0
$292$	−4.40613	−0.257849
$293$	23.1851	1.35449	0.677243	−	0.735760i	$-0.263175\pi$
$293$	23.1851	1.35449	0.677243	+	0.735760i	$0.263175\pi$
$294$	0	0
$295$	−5.05284	−0.294187
$296$	−11.3755	−0.661186
$297$	0	0
$298$	−4.75135	−0.275238
$299$	45.4632	2.62921
$300$	0	0
$301$	−9.01344	−0.519526
$302$	11.4044	0.656250
$303$	0	0
$304$	−6.17523	−0.354174
$305$	−8.70427	−0.498405
$306$	0	0
$307$	1.57953	0.0901487	0.0450744	−	0.998984i	$-0.485648\pi$
$307$	1.57953	0.0901487	0.0450744	+	0.998984i	$0.485648\pi$
$308$	−5.50948	−0.313932
$309$	0	0
$310$	2.47359	0.140490
$311$	18.7052	1.06067	0.530337	−	0.847787i	$-0.322065\pi$
$311$	18.7052	1.06067	0.530337	+	0.847787i	$0.322065\pi$
$312$	0	0
$313$	31.9489	1.80586	0.902930	−	0.429788i	$-0.141412\pi$
$313$	31.9489	1.80586	0.902930	+	0.429788i	$0.141412\pi$
$314$	8.12086	0.458287
$315$	0	0
$316$	2.59714	0.146101
$317$	−2.07785	−0.116703	−0.0583517	−	0.998296i	$-0.518584\pi$
$317$	−2.07785	−0.116703	−0.0583517	+	0.998296i	$0.518584\pi$
$318$	0	0
$319$	12.2380	0.685194
$320$	−0.580657	−0.0324597
$321$	0	0
$322$	−12.3012	−0.685519
$323$	−7.11769	−0.396039
$324$	0	0
$325$	24.0473	1.33390
$326$	−15.1661	−0.839973
$327$	0	0
$328$	−2.30330	−0.127178
$329$	−6.40202	−0.352955
$330$	0	0
$331$	−4.10039	−0.225378	−0.112689	−	0.993630i	$-0.535946\pi$
$331$	−4.10039	−0.225378	−0.112689	+	0.993630i	$0.535946\pi$
$332$	−6.07040	−0.333156
$333$	0	0
$334$	−15.9248	−0.871364
$335$	−4.20005	−0.229473
$336$	0	0
$337$	18.5440	1.01016	0.505079	−	0.863073i	$-0.331463\pi$
$337$	18.5440	1.01016	0.505079	+	0.863073i	$0.331463\pi$
$338$	−13.5969	−0.739574
$339$	0	0
$340$	−0.669276	−0.0362966
$341$	16.8196	0.910833
$342$	0	0
$343$	16.8187	0.908122
$344$	−6.45934	−0.348264
$345$	0	0
$346$	8.19787	0.440720
$347$	−25.6707	−1.37808	−0.689038	−	0.724726i	$-0.741967\pi$
$347$	−25.6707	−1.37808	−0.689038	+	0.724726i	$0.741967\pi$
$348$	0	0
$349$	−6.59919	−0.353247	−0.176623	−	0.984279i	$-0.556517\pi$
$349$	−6.59919	−0.353247	−0.176623	+	0.984279i	$0.556517\pi$
$350$	−6.50658	−0.347791
$351$	0	0
$352$	−3.94828	−0.210444
$353$	17.8120	0.948037	0.474019	−	0.880515i	$-0.342803\pi$
$353$	17.8120	0.948037	0.474019	+	0.880515i	$0.342803\pi$
$354$	0	0
$355$	4.88932	0.259498
$356$	−2.89241	−0.153297
$357$	0	0
$358$	−10.4109	−0.550232
$359$	11.0263	0.581947	0.290974	−	0.956731i	$-0.406021\pi$
$359$	11.0263	0.581947	0.290974	+	0.956731i	$0.406021\pi$
$360$	0	0
$361$	19.1335	1.00703
$362$	−11.3037	−0.594111
$363$	0	0
$364$	7.19645	0.377196
$365$	2.55845	0.133915
$366$	0	0
$367$	−28.4975	−1.48756	−0.743779	−	0.668426i	$-0.766969\pi$
$367$	−28.4975	−1.48756	−0.743779	+	0.668426i	$0.766969\pi$
$368$	−8.81546	−0.459537
$369$	0	0
$370$	6.60525	0.343390
$371$	10.7344	0.557304
$372$	0	0
$373$	4.83959	0.250584	0.125292	−	0.992120i	$-0.460013\pi$
$373$	4.83959	0.250584	0.125292	+	0.992120i	$0.460013\pi$
$374$	−4.55087	−0.235320
$375$	0	0
$376$	−4.58791	−0.236603
$377$	−15.9851	−0.823276
$378$	0	0
$379$	21.8684	1.12331	0.561653	−	0.827373i	$-0.310166\pi$
$379$	21.8684	1.12331	0.561653	+	0.827373i	$0.310166\pi$
$380$	3.58569	0.183942
$381$	0	0
$382$	−13.7113	−0.701533
$383$	−24.1983	−1.23647	−0.618237	−	0.785991i	$-0.712153\pi$
$383$	−24.1983	−1.23647	−0.618237	+	0.785991i	$0.712153\pi$
$384$	0	0
$385$	3.19912	0.163042
$386$	7.51938	0.382726
$387$	0	0
$388$	−3.84047	−0.194970
$389$	−9.35574	−0.474355	−0.237177	−	0.971466i	$-0.576222\pi$
$389$	−9.35574	−0.474355	−0.237177	+	0.971466i	$0.576222\pi$
$390$	0	0
$391$	−10.1609	−0.513857
$392$	5.05282	0.255206
$393$	0	0
$394$	−9.28097	−0.467569
$395$	−1.50805	−0.0758781
$396$	0	0
$397$	−6.09722	−0.306011	−0.153005	−	0.988225i	$-0.548895\pi$
$397$	−6.09722	−0.306011	−0.153005	+	0.988225i	$0.548895\pi$
$398$	−6.68850	−0.335264
$399$	0	0
$400$	−4.66284	−0.233142
$401$	34.5898	1.72733	0.863665	−	0.504066i	$-0.168163\pi$
$401$	34.5898	1.72733	0.863665	+	0.504066i	$0.168163\pi$
$402$	0	0
$403$	−21.9696	−1.09439
$404$	13.8790	0.690506
$405$	0	0
$406$	4.32517	0.214654
$407$	44.9136	2.22629
$408$	0	0
$409$	24.1859	1.19592	0.597959	−	0.801527i	$-0.295979\pi$
$409$	24.1859	1.19592	0.597959	+	0.801527i	$0.295979\pi$
$410$	1.33742	0.0660507
$411$	0	0
$412$	−14.9175	−0.734935
$413$	−12.1428	−0.597508
$414$	0	0
$415$	3.52482	0.173026
$416$	5.15722	0.252853
$417$	0	0
$418$	24.3816	1.19254
$419$	26.2162	1.28075	0.640373	−	0.768064i	$-0.278780\pi$
$419$	26.2162	1.28075	0.640373	+	0.768064i	$0.278780\pi$
$420$	0	0
$421$	−14.6003	−0.711575	−0.355788	−	0.934567i	$-0.615787\pi$
$421$	−14.6003	−0.711575	−0.355788	+	0.934567i	$0.615787\pi$
$422$	−25.0574	−1.21978
$423$	0	0
$424$	7.69266	0.373589
$425$	−5.37448	−0.260700
$426$	0	0
$427$	−20.9178	−1.01228
$428$	2.20691	0.106675
$429$	0	0
$430$	3.75066	0.180873
$431$	−11.2807	−0.543373	−0.271686	−	0.962386i	$-0.587581\pi$
$431$	−11.2807	−0.543373	−0.271686	+	0.962386i	$0.587581\pi$
$432$	0	0
$433$	−27.2986	−1.31189	−0.655944	−	0.754810i	$-0.727729\pi$
$433$	−27.2986	−1.31189	−0.655944	+	0.754810i	$0.727729\pi$
$434$	5.94443	0.285342
$435$	0	0
$436$	−5.10930	−0.244691
$437$	54.4375	2.60410
$438$	0	0
$439$	−4.58851	−0.218998	−0.109499	−	0.993987i	$-0.534925\pi$
$439$	−4.58851	−0.218998	−0.109499	+	0.993987i	$0.534925\pi$
$440$	2.29260	0.109295
$441$	0	0
$442$	5.94431	0.282742
$443$	23.3875	1.11118	0.555588	−	0.831458i	$-0.312493\pi$
$443$	23.3875	1.11118	0.555588	+	0.831458i	$0.312493\pi$
$444$	0	0
$445$	1.67950	0.0796157
$446$	1.00000	0.0473514
$447$	0	0
$448$	−1.39541	−0.0659270
$449$	23.8370	1.12494	0.562468	−	0.826819i	$-0.309852\pi$
$449$	23.8370	1.12494	0.562468	+	0.826819i	$0.309852\pi$
$450$	0	0
$451$	9.09407	0.428223
$452$	4.65162	0.218794
$453$	0	0
$454$	20.6803	0.970573
$455$	−4.17867	−0.195899
$456$	0	0
$457$	26.9981	1.26292	0.631460	−	0.775408i	$-0.282456\pi$
$457$	26.9981	1.26292	0.631460	+	0.775408i	$0.282456\pi$
$458$	−16.7199	−0.781270
$459$	0	0
$460$	5.11875	0.238663
$461$	15.8293	0.737242	0.368621	−	0.929580i	$-0.379830\pi$
$461$	15.8293	0.737242	0.368621	+	0.929580i	$0.379830\pi$
$462$	0	0
$463$	−30.1239	−1.39998	−0.699988	−	0.714155i	$-0.746811\pi$
$463$	−30.1239	−1.39998	−0.699988	+	0.714155i	$0.746811\pi$
$464$	3.09956	0.143894
$465$	0	0
$466$	16.7194	0.774510
$467$	22.1329	1.02419	0.512095	−	0.858929i	$-0.328870\pi$
$467$	22.1329	1.02419	0.512095	+	0.858929i	$0.328870\pi$
$468$	0	0
$469$	−10.0934	−0.466070
$470$	2.66400	0.122881
$471$	0	0
$472$	−8.70194	−0.400539
$473$	25.5033	1.17264
$474$	0	0
$475$	28.7941	1.32116
$476$	−1.60838	−0.0737199
$477$	0	0
$478$	−0.452261	−0.0206859
$479$	−24.6860	−1.12793	−0.563967	−	0.825797i	$-0.690726\pi$
$479$	−24.6860	−1.12793	−0.563967	+	0.825797i	$0.690726\pi$
$480$	0	0
$481$	−58.6658	−2.67493
$482$	5.63657	0.256739
$483$	0	0
$484$	4.58894	0.208588
$485$	2.23000	0.101259
$486$	0	0
$487$	−0.395103	−0.0179038	−0.00895192	−	0.999960i	$-0.502850\pi$
$487$	−0.395103	−0.0179038	−0.00895192	+	0.999960i	$0.502850\pi$
$488$	−14.9904	−0.678583
$489$	0	0
$490$	−2.93396	−0.132543
$491$	−19.7602	−0.891764	−0.445882	−	0.895092i	$-0.647110\pi$
$491$	−19.7602	−0.891764	−0.445882	+	0.895092i	$0.647110\pi$
$492$	0	0
$493$	3.57261	0.160902
$494$	−31.8470	−1.43287
$495$	0	0
$496$	4.25998	0.191279
$497$	11.7498	0.527053
$498$	0	0
$499$	30.0382	1.34470	0.672348	−	0.740235i	$-0.265286\pi$
$499$	30.0382	1.34470	0.672348	+	0.740235i	$0.265286\pi$
$500$	5.61079	0.250922
$501$	0	0
$502$	22.8236	1.01867
$503$	−19.0677	−0.850189	−0.425094	−	0.905149i	$-0.639759\pi$
$503$	−19.0677	−0.850189	−0.425094	+	0.905149i	$0.639759\pi$
$504$	0	0
$505$	−8.05893	−0.358618
$506$	34.8059	1.54731
$507$	0	0
$508$	12.1807	0.540430
$509$	−23.5572	−1.04415	−0.522077	−	0.852898i	$-0.674843\pi$
$509$	−23.5572	−1.04415	−0.522077	+	0.852898i	$0.674843\pi$
$510$	0	0
$511$	6.14836	0.271988
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.1676	−0.933661
$515$	8.66197	0.381692
$516$	0	0
$517$	18.1144	0.796669
$518$	15.8735	0.697441
$519$	0	0
$520$	−2.99457	−0.131321
$521$	−6.62648	−0.290311	−0.145156	−	0.989409i	$-0.546368\pi$
$521$	−6.62648	−0.290311	−0.145156	+	0.989409i	$0.546368\pi$
$522$	0	0
$523$	−32.6107	−1.42596	−0.712982	−	0.701182i	$-0.752656\pi$
$523$	−32.6107	−1.42596	−0.712982	+	0.701182i	$0.752656\pi$
$524$	1.14392	0.0499726
$525$	0	0
$526$	14.2781	0.622553
$527$	4.91013	0.213889
$528$	0	0
$529$	54.7123	2.37879
$530$	−4.46680	−0.194025
$531$	0	0
$532$	8.61700	0.373594
$533$	−11.8786	−0.514519
$534$	0	0
$535$	−1.28146	−0.0554022
$536$	−7.23327	−0.312430
$537$	0	0
$538$	8.70356	0.375237
$539$	−19.9500	−0.859306
$540$	0	0
$541$	−29.8277	−1.28239	−0.641197	−	0.767377i	$-0.721562\pi$
$541$	−29.8277	−1.28239	−0.641197	+	0.767377i	$0.721562\pi$
$542$	−17.3264	−0.744233
$543$	0	0
$544$	−1.15262	−0.0494181
$545$	2.96675	0.127082
$546$	0	0
$547$	−29.7106	−1.27033	−0.635166	−	0.772376i	$-0.719068\pi$
$547$	−29.7106	−1.27033	−0.635166	+	0.772376i	$0.719068\pi$
$548$	18.9183	0.808149
$549$	0	0
$550$	18.4102	0.785014
$551$	−19.1405	−0.815413
$552$	0	0
$553$	−3.62408	−0.154112
$554$	19.2916	0.819621
$555$	0	0
$556$	3.35904	0.142455
$557$	−45.0401	−1.90841	−0.954204	−	0.299156i	$-0.903295\pi$
$557$	−45.0401	−1.90841	−0.954204	+	0.299156i	$0.903295\pi$
$558$	0	0
$559$	−33.3122	−1.40896
$560$	0.810256	0.0342395
$561$	0	0
$562$	−27.3598	−1.15410
$563$	−20.7847	−0.875970	−0.437985	−	0.898982i	$-0.644308\pi$
$563$	−20.7847	−0.875970	−0.437985	+	0.898982i	$0.644308\pi$
$564$	0	0
$565$	−2.70099	−0.113632
$566$	−30.0939	−1.26494
$567$	0	0
$568$	8.42034	0.353309
$569$	26.7016	1.11939	0.559694	−	0.828700i	$-0.310919\pi$
$569$	26.7016	1.11939	0.559694	+	0.828700i	$0.310919\pi$
$570$	0	0
$571$	20.4027	0.853826	0.426913	−	0.904293i	$-0.359601\pi$
$571$	20.4027	0.853826	0.426913	+	0.904293i	$0.359601\pi$
$572$	−20.3622	−0.851385
$573$	0	0
$574$	3.21405	0.134152
$575$	41.1050	1.71420
$576$	0	0
$577$	−26.0871	−1.08602	−0.543010	−	0.839726i	$-0.682715\pi$
$577$	−26.0871	−1.08602	−0.543010	+	0.839726i	$0.682715\pi$
$578$	15.6715	0.651847
$579$	0	0
$580$	−1.79978	−0.0747319
$581$	8.47071	0.351424
$582$	0	0
$583$	−30.3728	−1.25791
$584$	4.40613	0.182327
$585$	0	0
$586$	−23.1851	−0.957766
$587$	9.79406	0.404244	0.202122	−	0.979360i	$-0.435216\pi$
$587$	9.79406	0.404244	0.202122	+	0.979360i	$0.435216\pi$
$588$	0	0
$589$	−26.3064	−1.08393
$590$	5.05284	0.208022
$591$	0	0
$592$	11.3755	0.467529
$593$	10.6769	0.438450	0.219225	−	0.975674i	$-0.429647\pi$
$593$	10.6769	0.438450	0.219225	+	0.975674i	$0.429647\pi$
$594$	0	0
$595$	0.933916	0.0382868
$596$	4.75135	0.194623
$597$	0	0
$598$	−45.4632	−1.85913
$599$	−13.0950	−0.535048	−0.267524	−	0.963551i	$-0.586205\pi$
$599$	−13.0950	−0.535048	−0.267524	+	0.963551i	$0.586205\pi$
$600$	0	0
$601$	48.2839	1.96954	0.984770	−	0.173862i	$-0.0556247\pi$
$601$	48.2839	1.96954	0.984770	+	0.173862i	$0.0556247\pi$
$602$	9.01344	0.367360
$603$	0	0
$604$	−11.4044	−0.464039
$605$	−2.66460	−0.108331
$606$	0	0
$607$	2.90944	0.118090	0.0590452	−	0.998255i	$-0.481194\pi$
$607$	2.90944	0.118090	0.0590452	+	0.998255i	$0.481194\pi$
$608$	6.17523	0.250439
$609$	0	0
$610$	8.70427	0.352426
$611$	−23.6608	−0.957215
$612$	0	0
$613$	−16.8708	−0.681404	−0.340702	−	0.940171i	$-0.610665\pi$
$613$	−16.8708	−0.681404	−0.340702	+	0.940171i	$0.610665\pi$
$614$	−1.57953	−0.0637448
$615$	0	0
$616$	5.50948	0.221983
$617$	23.1657	0.932615	0.466308	−	0.884623i	$-0.345584\pi$
$617$	23.1657	0.932615	0.466308	+	0.884623i	$0.345584\pi$
$618$	0	0
$619$	−37.6947	−1.51508	−0.757540	−	0.652789i	$-0.773599\pi$
$619$	−37.6947	−1.51508	−0.757540	+	0.652789i	$0.773599\pi$
$620$	−2.47359	−0.0993416
$621$	0	0
$622$	−18.7052	−0.750009
$623$	4.03610	0.161703
$624$	0	0
$625$	20.0562	0.802250
$626$	−31.9489	−1.27694
$627$	0	0
$628$	−8.12086	−0.324058
$629$	13.1116	0.522793
$630$	0	0
$631$	16.6225	0.661731	0.330866	−	0.943678i	$-0.392659\pi$
$631$	16.6225	0.661731	0.330866	+	0.943678i	$0.392659\pi$
$632$	−2.59714	−0.103309
$633$	0	0
$634$	2.07785	0.0825218
$635$	−7.07279	−0.280675
$636$	0	0
$637$	26.0585	1.03248
$638$	−12.2380	−0.484505
$639$	0	0
$640$	0.580657	0.0229525
$641$	40.7340	1.60889	0.804447	−	0.594024i	$-0.202462\pi$
$641$	40.7340	1.60889	0.804447	+	0.594024i	$0.202462\pi$
$642$	0	0
$643$	25.2616	0.996219	0.498110	−	0.867114i	$-0.334028\pi$
$643$	25.2616	0.996219	0.498110	+	0.867114i	$0.334028\pi$
$644$	12.3012	0.484735
$645$	0	0
$646$	7.11769	0.280042
$647$	−10.1592	−0.399399	−0.199700	−	0.979857i	$-0.563997\pi$
$647$	−10.1592	−0.399399	−0.199700	+	0.979857i	$0.563997\pi$
$648$	0	0
$649$	34.3577	1.34866
$650$	−24.0473	−0.943212
$651$	0	0
$652$	15.1661	0.593950
$653$	39.7438	1.55529	0.777647	−	0.628701i	$-0.216413\pi$
$653$	39.7438	1.55529	0.777647	+	0.628701i	$0.216413\pi$
$654$	0	0
$655$	−0.664228	−0.0259535
$656$	2.30330	0.0899286
$657$	0	0
$658$	6.40202	0.249577
$659$	−10.6983	−0.416745	−0.208373	−	0.978050i	$-0.566817\pi$
$659$	−10.6983	−0.416745	−0.208373	+	0.978050i	$0.566817\pi$
$660$	0	0
$661$	5.33286	0.207424	0.103712	−	0.994607i	$-0.466928\pi$
$661$	5.33286	0.207424	0.103712	+	0.994607i	$0.466928\pi$
$662$	4.10039	0.159366
$663$	0	0
$664$	6.07040	0.235577
$665$	−5.00352	−0.194028
$666$	0	0
$667$	−27.3241	−1.05799
$668$	15.9248	0.616147
$669$	0	0
$670$	4.20005	0.162262
$671$	59.1863	2.28486
$672$	0	0
$673$	41.5944	1.60335	0.801673	−	0.597763i	$-0.203944\pi$
$673$	41.5944	1.60335	0.801673	+	0.597763i	$0.203944\pi$
$674$	−18.5440	−0.714290
$675$	0	0
$676$	13.5969	0.522958
$677$	9.52252	0.365980	0.182990	−	0.983115i	$-0.441422\pi$
$677$	9.52252	0.365980	0.182990	+	0.983115i	$0.441422\pi$
$678$	0	0
$679$	5.35904	0.205661
$680$	0.669276	0.0256656
$681$	0	0
$682$	−16.8196	−0.644056
$683$	12.4356	0.475834	0.237917	−	0.971286i	$-0.423535\pi$
$683$	12.4356	0.475834	0.237917	+	0.971286i	$0.423535\pi$
$684$	0	0
$685$	−10.9850	−0.419716
$686$	−16.8187	−0.642139
$687$	0	0
$688$	6.45934	0.246260
$689$	39.6728	1.51141
$690$	0	0
$691$	−13.5075	−0.513849	−0.256925	−	0.966431i	$-0.582709\pi$
$691$	−13.5075	−0.513849	−0.256925	+	0.966431i	$0.582709\pi$
$692$	−8.19787	−0.311636
$693$	0	0
$694$	25.6707	0.974446
$695$	−1.95045	−0.0739846
$696$	0	0
$697$	2.65482	0.100559
$698$	6.59919	0.249783
$699$	0	0
$700$	6.50658	0.245926
$701$	−35.9391	−1.35740	−0.678701	−	0.734415i	$-0.737457\pi$
$701$	−35.9391	−1.35740	−0.678701	+	0.734415i	$0.737457\pi$
$702$	0	0
$703$	−70.2462	−2.64939
$704$	3.94828	0.148807
$705$	0	0
$706$	−17.8120	−0.670364
$707$	−19.3669	−0.728368
$708$	0	0
$709$	45.1933	1.69727	0.848636	−	0.528978i	$-0.177424\pi$
$709$	45.1933	1.69727	0.848636	+	0.528978i	$0.177424\pi$
$710$	−4.88932	−0.183493
$711$	0	0
$712$	2.89241	0.108398
$713$	−37.5537	−1.40640
$714$	0	0
$715$	11.8234	0.442171
$716$	10.4109	0.389073
$717$	0	0
$718$	−11.0263	−0.411499
$719$	−25.9238	−0.966793	−0.483396	−	0.875402i	$-0.660597\pi$
$719$	−25.9238	−0.966793	−0.483396	+	0.875402i	$0.660597\pi$
$720$	0	0
$721$	20.8161	0.775233
$722$	−19.1335	−0.712075
$723$	0	0
$724$	11.3037	0.420100
$725$	−14.4528	−0.536762
$726$	0	0
$727$	−29.2027	−1.08307	−0.541533	−	0.840679i	$-0.682156\pi$
$727$	−29.2027	−1.08307	−0.541533	+	0.840679i	$0.682156\pi$
$728$	−7.19645	−0.266718
$729$	0	0
$730$	−2.55845	−0.0946923
$731$	7.44515	0.275369
$732$	0	0
$733$	−2.52876	−0.0934019	−0.0467009	−	0.998909i	$-0.514871\pi$
$733$	−2.52876	−0.0934019	−0.0467009	+	0.998909i	$0.514871\pi$
$734$	28.4975	1.05186
$735$	0	0
$736$	8.81546	0.324942
$737$	28.5590	1.05198
$738$	0	0
$739$	−2.00454	−0.0737381	−0.0368690	−	0.999320i	$-0.511738\pi$
$739$	−2.00454	−0.0737381	−0.0368690	+	0.999320i	$0.511738\pi$
$740$	−6.60525	−0.242814
$741$	0	0
$742$	−10.7344	−0.394074
$743$	−17.5945	−0.645481	−0.322740	−	0.946488i	$-0.604604\pi$
$743$	−17.5945	−0.645481	−0.322740	+	0.946488i	$0.604604\pi$
$744$	0	0
$745$	−2.75890	−0.101078
$746$	−4.83959	−0.177190
$747$	0	0
$748$	4.55087	0.166396
$749$	−3.07955	−0.112524
$750$	0	0
$751$	−4.80540	−0.175352	−0.0876758	−	0.996149i	$-0.527944\pi$
$751$	−4.80540	−0.175352	−0.0876758	+	0.996149i	$0.527944\pi$
$752$	4.58791	0.167304
$753$	0	0
$754$	15.9851	0.582144
$755$	6.62205	0.241001
$756$	0	0
$757$	−7.91502	−0.287676	−0.143838	−	0.989601i	$-0.545944\pi$
$757$	−7.91502	−0.287676	−0.143838	+	0.989601i	$0.545944\pi$
$758$	−21.8684	−0.794297
$759$	0	0
$760$	−3.58569	−0.130067
$761$	−37.8785	−1.37310	−0.686548	−	0.727085i	$-0.740875\pi$
$761$	−37.8785	−1.37310	−0.686548	+	0.727085i	$0.740875\pi$
$762$	0	0
$763$	7.12958	0.258108
$764$	13.7113	0.496059
$765$	0	0
$766$	24.1983	0.874320
$767$	−44.8778	−1.62044
$768$	0	0
$769$	−32.1940	−1.16095	−0.580473	−	0.814280i	$-0.697132\pi$
$769$	−32.1940	−1.16095	−0.580473	+	0.814280i	$0.697132\pi$
$770$	−3.19912	−0.115288
$771$	0	0
$772$	−7.51938	−0.270628
$773$	−2.32801	−0.0837326	−0.0418663	−	0.999123i	$-0.513330\pi$
$773$	−2.32801	−0.0837326	−0.0418663	+	0.999123i	$0.513330\pi$
$774$	0	0
$775$	−19.8636	−0.713521
$776$	3.84047	0.137865
$777$	0	0
$778$	9.35574	0.335419
$779$	−14.2234	−0.509606
$780$	0	0
$781$	−33.2459	−1.18963
$782$	10.1609	0.363352
$783$	0	0
$784$	−5.05282	−0.180458
$785$	4.71543	0.168301
$786$	0	0
$787$	−19.8909	−0.709036	−0.354518	−	0.935049i	$-0.615355\pi$
$787$	−19.8909	−0.709036	−0.354518	+	0.935049i	$0.615355\pi$
$788$	9.28097	0.330621
$789$	0	0
$790$	1.50805	0.0536539
$791$	−6.49093	−0.230791
$792$	0	0
$793$	−77.3087	−2.74531
$794$	6.09722	0.216382
$795$	0	0
$796$	6.68850	0.237068
$797$	1.86978	0.0662309	0.0331154	−	0.999452i	$-0.489457\pi$
$797$	1.86978	0.0662309	0.0331154	+	0.999452i	$0.489457\pi$
$798$	0	0
$799$	5.28811	0.187080
$800$	4.66284	0.164856
$801$	0	0
$802$	−34.5898	−1.22141
$803$	−17.3966	−0.613914
$804$	0	0
$805$	−7.14277	−0.251750
$806$	21.9696	0.773848
$807$	0	0
$808$	−13.8790	−0.488261
$809$	25.0801	0.881768	0.440884	−	0.897564i	$-0.354665\pi$
$809$	25.0801	0.881768	0.440884	+	0.897564i	$0.354665\pi$
$810$	0	0
$811$	−7.10810	−0.249599	−0.124800	−	0.992182i	$-0.539829\pi$
$811$	−7.10810	−0.249599	−0.124800	+	0.992182i	$0.539829\pi$
$812$	−4.32517	−0.151784
$813$	0	0
$814$	−44.9136	−1.57422
$815$	−8.80630	−0.308471
$816$	0	0
$817$	−39.8879	−1.39550
$818$	−24.1859	−0.845641
$819$	0	0
$820$	−1.33742	−0.0467049
$821$	−29.6567	−1.03502	−0.517512	−	0.855676i	$-0.673142\pi$
$821$	−29.6567	−1.03502	−0.517512	+	0.855676i	$0.673142\pi$
$822$	0	0
$823$	23.0438	0.803256	0.401628	−	0.915803i	$-0.368445\pi$
$823$	23.0438	0.803256	0.401628	+	0.915803i	$0.368445\pi$
$824$	14.9175	0.519677
$825$	0	0
$826$	12.1428	0.422502
$827$	41.8367	1.45481	0.727403	−	0.686210i	$-0.240727\pi$
$827$	41.8367	1.45481	0.727403	+	0.686210i	$0.240727\pi$
$828$	0	0
$829$	13.5065	0.469099	0.234549	−	0.972104i	$-0.424639\pi$
$829$	13.5065	0.469099	0.234549	+	0.972104i	$0.424639\pi$
$830$	−3.52482	−0.122348
$831$	0	0
$832$	−5.15722	−0.178794
$833$	−5.82398	−0.201789
$834$	0	0
$835$	−9.24682	−0.319999
$836$	−24.3816	−0.843254
$837$	0	0
$838$	−26.2162	−0.905624
$839$	23.3938	0.807645	0.403823	−	0.914837i	$-0.367681\pi$
$839$	23.3938	0.807645	0.403823	+	0.914837i	$0.367681\pi$
$840$	0	0
$841$	−19.3927	−0.668714
$842$	14.6003	0.503160
$843$	0	0
$844$	25.0574	0.862511
$845$	−7.89513	−0.271601
$846$	0	0
$847$	−6.40347	−0.220026
$848$	−7.69266	−0.264167
$849$	0	0
$850$	5.37448	0.184343
$851$	−100.280	−3.43756
$852$	0	0
$853$	29.0497	0.994643	0.497322	−	0.867566i	$-0.334317\pi$
$853$	29.0497	0.994643	0.497322	+	0.867566i	$0.334317\pi$
$854$	20.9178	0.715792
$855$	0	0
$856$	−2.20691	−0.0754306
$857$	3.83133	0.130876	0.0654378	−	0.997857i	$-0.479156\pi$
$857$	3.83133	0.130876	0.0654378	+	0.997857i	$0.479156\pi$
$858$	0	0
$859$	−24.7635	−0.844918	−0.422459	−	0.906382i	$-0.638833\pi$
$859$	−24.7635	−0.844918	−0.422459	+	0.906382i	$0.638833\pi$
$860$	−3.75066	−0.127896
$861$	0	0
$862$	11.2807	0.384223
$863$	−5.60683	−0.190859	−0.0954293	−	0.995436i	$-0.530422\pi$
$863$	−5.60683	−0.190859	−0.0954293	+	0.995436i	$0.530422\pi$
$864$	0	0
$865$	4.76015	0.161850
$866$	27.2986	0.927644
$867$	0	0
$868$	−5.94443	−0.201767
$869$	10.2543	0.347852
$870$	0	0
$871$	−37.3036	−1.26398
$872$	5.10930	0.173023
$873$	0	0
$874$	−54.4375	−1.84138
$875$	−7.82937	−0.264681
$876$	0	0
$877$	28.8054	0.972691	0.486345	−	0.873767i	$-0.338330\pi$
$877$	28.8054	0.972691	0.486345	+	0.873767i	$0.338330\pi$
$878$	4.58851	0.154855
$879$	0	0
$880$	−2.29260	−0.0772834
$881$	22.0034	0.741315	0.370657	−	0.928770i	$-0.379132\pi$
$881$	22.0034	0.741315	0.370657	+	0.928770i	$0.379132\pi$
$882$	0	0
$883$	6.61604	0.222647	0.111324	−	0.993784i	$-0.464491\pi$
$883$	6.61604	0.222647	0.111324	+	0.993784i	$0.464491\pi$
$884$	−5.94431	−0.199929
$885$	0	0
$886$	−23.3875	−0.785720
$887$	−12.5887	−0.422688	−0.211344	−	0.977412i	$-0.567784\pi$
$887$	−12.5887	−0.422688	−0.211344	+	0.977412i	$0.567784\pi$
$888$	0	0
$889$	−16.9971	−0.570063
$890$	−1.67950	−0.0562968
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−28.3314	−0.948074
$894$	0	0
$895$	−6.04515	−0.202067
$896$	1.39541	0.0466175
$897$	0	0
$898$	−23.8370	−0.795450
$899$	13.2041	0.440380
$900$	0	0
$901$	−8.86671	−0.295393
$902$	−9.09407	−0.302799
$903$	0	0
$904$	−4.65162	−0.154711
$905$	−6.56358	−0.218181
$906$	0	0
$907$	5.54345	0.184067	0.0920337	−	0.995756i	$-0.470663\pi$
$907$	5.54345	0.184067	0.0920337	+	0.995756i	$0.470663\pi$
$908$	−20.6803	−0.686299
$909$	0	0
$910$	4.17867	0.138521
$911$	−47.8508	−1.58537	−0.792684	−	0.609632i	$-0.791317\pi$
$911$	−47.8508	−1.58537	−0.792684	+	0.609632i	$0.791317\pi$
$912$	0	0
$913$	−23.9676	−0.793213
$914$	−26.9981	−0.893019
$915$	0	0
$916$	16.7199	0.552442
$917$	−1.59625	−0.0527127
$918$	0	0
$919$	55.9958	1.84713	0.923566	−	0.383439i	$-0.125260\pi$
$919$	55.9958	1.84713	0.923566	+	0.383439i	$0.125260\pi$
$920$	−5.11875	−0.168760
$921$	0	0
$922$	−15.8293	−0.521309
$923$	43.4255	1.42937
$924$	0	0
$925$	−53.0420	−1.74401
$926$	30.1239	0.989932
$927$	0	0
$928$	−3.09956	−0.101748
$929$	48.2593	1.58334	0.791668	−	0.610951i	$-0.209213\pi$
$929$	48.2593	1.58334	0.791668	+	0.610951i	$0.209213\pi$
$930$	0	0
$931$	31.2024	1.02262
$932$	−16.7194	−0.547661
$933$	0	0
$934$	−22.1329	−0.724211
$935$	−2.64249	−0.0864187
$936$	0	0
$937$	−57.7293	−1.88593	−0.942967	−	0.332887i	$-0.891977\pi$
$937$	−57.7293	−1.88593	−0.942967	+	0.332887i	$0.891977\pi$
$938$	10.0934	0.329561
$939$	0	0
$940$	−2.66400	−0.0868901
$941$	−27.1250	−0.884251	−0.442125	−	0.896953i	$-0.645775\pi$
$941$	−27.1250	−0.884251	−0.442125	+	0.896953i	$0.645775\pi$
$942$	0	0
$943$	−20.3046	−0.661209
$944$	8.70194	0.283224
$945$	0	0
$946$	−25.5033	−0.829184
$947$	−4.14898	−0.134824	−0.0674119	−	0.997725i	$-0.521474\pi$
$947$	−4.14898	−0.134824	−0.0674119	+	0.997725i	$0.521474\pi$
$948$	0	0
$949$	22.7234	0.737631
$950$	−28.7941	−0.934204
$951$	0	0
$952$	1.60838	0.0521279
$953$	27.0795	0.877192	0.438596	−	0.898684i	$-0.355476\pi$
$953$	27.0795	0.877192	0.438596	+	0.898684i	$0.355476\pi$
$954$	0	0
$955$	−7.96158	−0.257631
$956$	0.452261	0.0146272
$957$	0	0
$958$	24.6860	0.797570
$959$	−26.3988	−0.852462
$960$	0	0
$961$	−12.8526	−0.414599
$962$	58.6658	1.89146
$963$	0	0
$964$	−5.63657	−0.181542
$965$	4.36617	0.140552
$966$	0	0
$967$	−45.4887	−1.46282	−0.731409	−	0.681939i	$-0.761137\pi$
$967$	−45.4887	−1.46282	−0.731409	+	0.681939i	$0.761137\pi$
$968$	−4.58894	−0.147494
$969$	0	0
$970$	−2.23000	−0.0716009
$971$	−46.1520	−1.48109	−0.740544	−	0.672007i	$-0.765432\pi$
$971$	−46.1520	−1.48109	−0.740544	+	0.672007i	$0.765432\pi$
$972$	0	0
$973$	−4.68724	−0.150266
$974$	0.395103	0.0126599
$975$	0	0
$976$	14.9904	0.479831
$977$	14.3241	0.458268	0.229134	−	0.973395i	$-0.426411\pi$
$977$	14.3241	0.458268	0.229134	+	0.973395i	$0.426411\pi$
$978$	0	0
$979$	−11.4200	−0.364986
$980$	2.93396	0.0937218
$981$	0	0
$982$	19.7602	0.630572
$983$	−11.0937	−0.353834	−0.176917	−	0.984226i	$-0.556612\pi$
$983$	−11.0937	−0.353834	−0.176917	+	0.984226i	$0.556612\pi$
$984$	0	0
$985$	−5.38906	−0.171710
$986$	−3.57261	−0.113775
$987$	0	0
$988$	31.8470	1.01319
$989$	−56.9420	−1.81065
$990$	0	0
$991$	−42.0086	−1.33445	−0.667223	−	0.744858i	$-0.732517\pi$
$991$	−42.0086	−1.33445	−0.667223	+	0.744858i	$0.732517\pi$
$992$	−4.25998	−0.135254
$993$	0	0
$994$	−11.7498	−0.372682
$995$	−3.88372	−0.123122
$996$	0	0
$997$	−43.9668	−1.39244	−0.696221	−	0.717828i	$-0.745137\pi$
$997$	−43.9668	−1.39244	−0.696221	+	0.717828i	$0.745137\pi$
$998$	−30.0382	−0.950843
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.w.1.4		7
3.2	odd	2		446.2.a.e.1.7	✓	7
12.11	even	2		3568.2.a.l.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.e.1.7	✓	7	3.2	odd	2
3568.2.a.l.1.1		7	12.11	even	2
4014.2.a.w.1.4		7	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.580657 q^{5} -1.39541 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.580657 q^{5} -1.39541 q^{7} -1.00000 q^{8} +0.580657 q^{10} +3.94828 q^{11} -5.15722 q^{13} +1.39541 q^{14} +1.00000 q^{16} +1.15262 q^{17} -6.17523 q^{19} -0.580657 q^{20} -3.94828 q^{22} -8.81546 q^{23} -4.66284 q^{25} +5.15722 q^{26} -1.39541 q^{28} +3.09956 q^{29} +4.25998 q^{31} -1.00000 q^{32} -1.15262 q^{34} +0.810256 q^{35} +11.3755 q^{37} +6.17523 q^{38} +0.580657 q^{40} +2.30330 q^{41} +6.45934 q^{43} +3.94828 q^{44} +8.81546 q^{46} +4.58791 q^{47} -5.05282 q^{49} +4.66284 q^{50} -5.15722 q^{52} -7.69266 q^{53} -2.29260 q^{55} +1.39541 q^{56} -3.09956 q^{58} +8.70194 q^{59} +14.9904 q^{61} -4.25998 q^{62} +1.00000 q^{64} +2.99457 q^{65} +7.23327 q^{67} +1.15262 q^{68} -0.810256 q^{70} -8.42034 q^{71} -4.40613 q^{73} -11.3755 q^{74} -6.17523 q^{76} -5.50948 q^{77} +2.59714 q^{79} -0.580657 q^{80} -2.30330 q^{82} -6.07040 q^{83} -0.669276 q^{85} -6.45934 q^{86} -3.94828 q^{88} -2.89241 q^{89} +7.19645 q^{91} -8.81546 q^{92} -4.58791 q^{94} +3.58569 q^{95} -3.84047 q^{97} +5.05282 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8} + 2 q^{10} - 9 q^{11} - 2 q^{13} - 6 q^{14} + 7 q^{16} + 7 q^{17} - 2 q^{19} - 2 q^{20} + 9 q^{22} - 15 q^{23} + 13 q^{25} + 2 q^{26} + 6 q^{28} - 9 q^{29} - 2 q^{31} - 7 q^{32} - 7 q^{34} + 4 q^{35} + 5 q^{37} + 2 q^{38} + 2 q^{40} + 33 q^{41} + 20 q^{43} - 9 q^{44} + 15 q^{46} + 2 q^{47} + 3 q^{49} - 13 q^{50} - 2 q^{52} + 13 q^{53} - 18 q^{55} - 6 q^{56} + 9 q^{58} - 9 q^{59} + 8 q^{61} + 2 q^{62} + 7 q^{64} + 44 q^{65} + 29 q^{67} + 7 q^{68} - 4 q^{70} - 37 q^{73} - 5 q^{74} - 2 q^{76} + 18 q^{77} + 32 q^{79} - 2 q^{80} - 33 q^{82} + 6 q^{83} - 4 q^{85} - 20 q^{86} + 9 q^{88} + 17 q^{89} - 4 q^{91} - 15 q^{92} - 2 q^{94} + 12 q^{95} + 12 q^{97} - 3 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8 + 2 * q^10 - 9 * q^11 - 2 * q^13 - 6 * q^14 + 7 * q^16 + 7 * q^17 - 2 * q^19 - 2 * q^20 + 9 * q^22 - 15 * q^23 + 13 * q^25 + 2 * q^26 + 6 * q^28 - 9 * q^29 - 2 * q^31 - 7 * q^32 - 7 * q^34 + 4 * q^35 + 5 * q^37 + 2 * q^38 + 2 * q^40 + 33 * q^41 + 20 * q^43 - 9 * q^44 + 15 * q^46 + 2 * q^47 + 3 * q^49 - 13 * q^50 - 2 * q^52 + 13 * q^53 - 18 * q^55 - 6 * q^56 + 9 * q^58 - 9 * q^59 + 8 * q^61 + 2 * q^62 + 7 * q^64 + 44 * q^65 + 29 * q^67 + 7 * q^68 - 4 * q^70 - 37 * q^73 - 5 * q^74 - 2 * q^76 + 18 * q^77 + 32 * q^79 - 2 * q^80 - 33 * q^82 + 6 * q^83 - 4 * q^85 - 20 * q^86 + 9 * q^88 + 17 * q^89 - 4 * q^91 - 15 * q^92 - 2 * q^94 + 12 * q^95 + 12 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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