LMFDB - Embedded newform 8016.2.a.be.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.38119$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.38119 q^{5} +0.260099 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.38119 q^{5} +0.260099 q^{7} +1.00000 q^{9} +3.57901 q^{11} -1.29974 q^{13} -1.38119 q^{15} -4.24502 q^{17} -5.37321 q^{19} +0.260099 q^{21} -5.60388 q^{23} -3.09232 q^{25} +1.00000 q^{27} +2.23067 q^{29} +9.62228 q^{31} +3.57901 q^{33} -0.359246 q^{35} +0.0808905 q^{37} -1.29974 q^{39} +4.29012 q^{41} +10.5096 q^{43} -1.38119 q^{45} +6.20376 q^{47} -6.93235 q^{49} -4.24502 q^{51} +12.3984 q^{53} -4.94329 q^{55} -5.37321 q^{57} -3.18358 q^{59} -6.40971 q^{61} +0.260099 q^{63} +1.79518 q^{65} -2.58992 q^{67} -5.60388 q^{69} -10.9927 q^{71} +11.2300 q^{73} -3.09232 q^{75} +0.930895 q^{77} +11.9509 q^{79} +1.00000 q^{81} -4.33550 q^{83} +5.86317 q^{85} +2.23067 q^{87} +8.61186 q^{89} -0.338060 q^{91} +9.62228 q^{93} +7.42142 q^{95} +0.0718231 q^{97} +3.57901 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	$ 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) $ 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.38119	−0.617687	−0.308843	−	0.951113i	$-0.599942\pi$
$5$	−1.38119	−0.617687	−0.308843	+	0.951113i	$0.599942\pi$
$6$	0	0
$7$	0.260099	0.0983081	0.0491540	−	0.998791i	$-0.484347\pi$
$7$	0.260099	0.0983081	0.0491540	+	0.998791i	$0.484347\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.57901	1.07911	0.539556	−	0.841950i	$-0.318592\pi$
$11$	3.57901	1.07911	0.539556	+	0.841950i	$0.318592\pi$
$12$	0	0
$13$	−1.29974	−0.360482	−0.180241	−	0.983622i	$-0.557688\pi$
$13$	−1.29974	−0.360482	−0.180241	+	0.983622i	$0.557688\pi$
$14$	0	0
$15$	−1.38119	−0.356621
$16$	0	0
$17$	−4.24502	−1.02957	−0.514784	−	0.857320i	$-0.672128\pi$
$17$	−4.24502	−1.02957	−0.514784	+	0.857320i	$0.672128\pi$
$18$	0	0
$19$	−5.37321	−1.23270	−0.616350	−	0.787473i	$-0.711389\pi$
$19$	−5.37321	−1.23270	−0.616350	+	0.787473i	$0.711389\pi$
$20$	0	0
$21$	0.260099	0.0567582
$22$	0	0
$23$	−5.60388	−1.16849	−0.584245	−	0.811578i	$-0.698609\pi$
$23$	−5.60388	−1.16849	−0.584245	+	0.811578i	$0.698609\pi$
$24$	0	0
$25$	−3.09232	−0.618463
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.23067	0.414224	0.207112	−	0.978317i	$-0.433594\pi$
$29$	2.23067	0.414224	0.207112	+	0.978317i	$0.433594\pi$
$30$	0	0
$31$	9.62228	1.72821	0.864106	−	0.503309i	$-0.167884\pi$
$31$	9.62228	1.72821	0.864106	+	0.503309i	$0.167884\pi$
$32$	0	0
$33$	3.57901	0.623025
$34$	0	0
$35$	−0.359246	−0.0607236
$36$	0	0
$37$	0.0808905	0.0132983	0.00664916	−	0.999978i	$-0.497883\pi$
$37$	0.0808905	0.0132983	0.00664916	+	0.999978i	$0.497883\pi$
$38$	0	0
$39$	−1.29974	−0.208125
$40$	0	0
$41$	4.29012	0.670005	0.335002	−	0.942217i	$-0.391263\pi$
$41$	4.29012	0.670005	0.335002	+	0.942217i	$0.391263\pi$
$42$	0	0
$43$	10.5096	1.60270	0.801352	−	0.598193i	$-0.204114\pi$
$43$	10.5096	1.60270	0.801352	+	0.598193i	$0.204114\pi$
$44$	0	0
$45$	−1.38119	−0.205896
$46$	0	0
$47$	6.20376	0.904911	0.452455	−	0.891787i	$-0.350548\pi$
$47$	6.20376	0.904911	0.452455	+	0.891787i	$0.350548\pi$
$48$	0	0
$49$	−6.93235	−0.990336
$50$	0	0
$51$	−4.24502	−0.594421
$52$	0	0
$53$	12.3984	1.70305	0.851524	−	0.524316i	$-0.175679\pi$
$53$	12.3984	1.70305	0.851524	+	0.524316i	$0.175679\pi$
$54$	0	0
$55$	−4.94329	−0.666553
$56$	0	0
$57$	−5.37321	−0.711699
$58$	0	0
$59$	−3.18358	−0.414467	−0.207234	−	0.978292i	$-0.566446\pi$
$59$	−3.18358	−0.414467	−0.207234	+	0.978292i	$0.566446\pi$
$60$	0	0
$61$	−6.40971	−0.820679	−0.410339	−	0.911933i	$-0.634590\pi$
$61$	−6.40971	−0.820679	−0.410339	+	0.911933i	$0.634590\pi$
$62$	0	0
$63$	0.260099	0.0327694
$64$	0	0
$65$	1.79518	0.222665
$66$	0	0
$67$	−2.58992	−0.316409	−0.158204	−	0.987406i	$-0.550570\pi$
$67$	−2.58992	−0.316409	−0.158204	+	0.987406i	$0.550570\pi$
$68$	0	0
$69$	−5.60388	−0.674627
$70$	0	0
$71$	−10.9927	−1.30460	−0.652298	−	0.757963i	$-0.726195\pi$
$71$	−10.9927	−1.30460	−0.652298	+	0.757963i	$0.726195\pi$
$72$	0	0
$73$	11.2300	1.31437	0.657185	−	0.753729i	$-0.271747\pi$
$73$	11.2300	1.31437	0.657185	+	0.753729i	$0.271747\pi$
$74$	0	0
$75$	−3.09232	−0.357070
$76$	0	0
$77$	0.930895	0.106085
$78$	0	0
$79$	11.9509	1.34458	0.672290	−	0.740288i	$-0.265311\pi$
$79$	11.9509	1.34458	0.672290	+	0.740288i	$0.265311\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.33550	−0.475883	−0.237941	−	0.971280i	$-0.576473\pi$
$83$	−4.33550	−0.475883	−0.237941	+	0.971280i	$0.576473\pi$
$84$	0	0
$85$	5.86317	0.635950
$86$	0	0
$87$	2.23067	0.239152
$88$	0	0
$89$	8.61186	0.912855	0.456428	−	0.889760i	$-0.349129\pi$
$89$	8.61186	0.912855	0.456428	+	0.889760i	$0.349129\pi$
$90$	0	0
$91$	−0.338060	−0.0354383
$92$	0	0
$93$	9.62228	0.997784
$94$	0	0
$95$	7.42142	0.761422
$96$	0	0
$97$	0.0718231	0.00729253	0.00364627	−	0.999993i	$-0.498839\pi$
$97$	0.0718231	0.00729253	0.00364627	+	0.999993i	$0.498839\pi$
$98$	0	0
$99$	3.57901	0.359704
$100$	0	0
$101$	17.9592	1.78700	0.893501	−	0.449061i	$-0.148241\pi$
$101$	17.9592	1.78700	0.893501	+	0.449061i	$0.148241\pi$
$102$	0	0
$103$	−0.277767	−0.0273692	−0.0136846	−	0.999906i	$-0.504356\pi$
$103$	−0.277767	−0.0273692	−0.0136846	+	0.999906i	$0.504356\pi$
$104$	0	0
$105$	−0.359246	−0.0350588
$106$	0	0
$107$	5.01695	0.485007	0.242504	−	0.970150i	$-0.422031\pi$
$107$	5.01695	0.485007	0.242504	+	0.970150i	$0.422031\pi$
$108$	0	0
$109$	15.0343	1.44002	0.720012	−	0.693962i	$-0.244136\pi$
$109$	15.0343	1.44002	0.720012	+	0.693962i	$0.244136\pi$
$110$	0	0
$111$	0.0808905	0.00767779
$112$	0	0
$113$	10.2708	0.966198	0.483099	−	0.875566i	$-0.339511\pi$
$113$	10.2708	0.966198	0.483099	+	0.875566i	$0.339511\pi$
$114$	0	0
$115$	7.74001	0.721760
$116$	0	0
$117$	−1.29974	−0.120161
$118$	0	0
$119$	−1.10412	−0.101215
$120$	0	0
$121$	1.80930	0.164481
$122$	0	0
$123$	4.29012	0.386827
$124$	0	0
$125$	11.1770	0.999703
$126$	0	0
$127$	3.02975	0.268847	0.134423	−	0.990924i	$-0.457082\pi$
$127$	3.02975	0.268847	0.134423	+	0.990924i	$0.457082\pi$
$128$	0	0
$129$	10.5096	0.925322
$130$	0	0
$131$	17.2097	1.50361	0.751807	−	0.659383i	$-0.229182\pi$
$131$	17.2097	1.50361	0.751807	+	0.659383i	$0.229182\pi$
$132$	0	0
$133$	−1.39757	−0.121184
$134$	0	0
$135$	−1.38119	−0.118874
$136$	0	0
$137$	−7.36980	−0.629645	−0.314822	−	0.949151i	$-0.601945\pi$
$137$	−7.36980	−0.629645	−0.314822	+	0.949151i	$0.601945\pi$
$138$	0	0
$139$	−15.4085	−1.30693	−0.653465	−	0.756957i	$-0.726685\pi$
$139$	−15.4085	−1.30693	−0.653465	+	0.756957i	$0.726685\pi$
$140$	0	0
$141$	6.20376	0.522450
$142$	0	0
$143$	−4.65177	−0.389001
$144$	0	0
$145$	−3.08097	−0.255861
$146$	0	0
$147$	−6.93235	−0.571770
$148$	0	0
$149$	−19.7078	−1.61452	−0.807261	−	0.590194i	$-0.799051\pi$
$149$	−19.7078	−1.61452	−0.807261	+	0.590194i	$0.799051\pi$
$150$	0	0
$151$	6.39372	0.520314	0.260157	−	0.965566i	$-0.416226\pi$
$151$	6.39372	0.520314	0.260157	+	0.965566i	$0.416226\pi$
$152$	0	0
$153$	−4.24502	−0.343189
$154$	0	0
$155$	−13.2902	−1.06749
$156$	0	0
$157$	11.6913	0.933070	0.466535	−	0.884503i	$-0.345502\pi$
$157$	11.6913	0.933070	0.466535	+	0.884503i	$0.345502\pi$
$158$	0	0
$159$	12.3984	0.983255
$160$	0	0
$161$	−1.45756	−0.114872
$162$	0	0
$163$	−9.58622	−0.750851	−0.375425	−	0.926853i	$-0.622503\pi$
$163$	−9.58622	−0.750851	−0.375425	+	0.926853i	$0.622503\pi$
$164$	0	0
$165$	−4.94329	−0.384834
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−11.3107	−0.870052
$170$	0	0
$171$	−5.37321	−0.410900
$172$	0	0
$173$	25.1843	1.91472	0.957362	−	0.288890i	$-0.0932860\pi$
$173$	25.1843	1.91472	0.957362	+	0.288890i	$0.0932860\pi$
$174$	0	0
$175$	−0.804308	−0.0607999
$176$	0	0
$177$	−3.18358	−0.239293
$178$	0	0
$179$	0.125279	0.00936380	0.00468190	−	0.999989i	$-0.498510\pi$
$179$	0.125279	0.00936380	0.00468190	+	0.999989i	$0.498510\pi$
$180$	0	0
$181$	−15.0014	−1.11504	−0.557522	−	0.830162i	$-0.688248\pi$
$181$	−15.0014	−1.11504	−0.557522	+	0.830162i	$0.688248\pi$
$182$	0	0
$183$	−6.40971	−0.473819
$184$	0	0
$185$	−0.111725	−0.00821419
$186$	0	0
$187$	−15.1929	−1.11102
$188$	0	0
$189$	0.260099	0.0189194
$190$	0	0
$191$	4.07344	0.294744	0.147372	−	0.989081i	$-0.452919\pi$
$191$	4.07344	0.294744	0.147372	+	0.989081i	$0.452919\pi$
$192$	0	0
$193$	11.2459	0.809496	0.404748	−	0.914428i	$-0.367359\pi$
$193$	11.2459	0.809496	0.404748	+	0.914428i	$0.367359\pi$
$194$	0	0
$195$	1.79518	0.128556
$196$	0	0
$197$	−5.88075	−0.418986	−0.209493	−	0.977810i	$-0.567181\pi$
$197$	−5.88075	−0.418986	−0.209493	+	0.977810i	$0.567181\pi$
$198$	0	0
$199$	−19.8895	−1.40993	−0.704963	−	0.709244i	$-0.749037\pi$
$199$	−19.8895	−1.40993	−0.704963	+	0.709244i	$0.749037\pi$
$200$	0	0
$201$	−2.58992	−0.182679
$202$	0	0
$203$	0.580193	0.0407216
$204$	0	0
$205$	−5.92547	−0.413853
$206$	0	0
$207$	−5.60388	−0.389496
$208$	0	0
$209$	−19.2308	−1.33022
$210$	0	0
$211$	−14.3807	−0.990008	−0.495004	−	0.868891i	$-0.664833\pi$
$211$	−14.3807	−0.990008	−0.495004	+	0.868891i	$0.664833\pi$
$212$	0	0
$213$	−10.9927	−0.753209
$214$	0	0
$215$	−14.5158	−0.989969
$216$	0	0
$217$	2.50274	0.169897
$218$	0	0
$219$	11.2300	0.758852
$220$	0	0
$221$	5.51741	0.371141
$222$	0	0
$223$	28.9444	1.93826	0.969131	−	0.246546i	$-0.0792958\pi$
$223$	28.9444	1.93826	0.969131	+	0.246546i	$0.0792958\pi$
$224$	0	0
$225$	−3.09232	−0.206154
$226$	0	0
$227$	−20.1056	−1.33445	−0.667227	−	0.744855i	$-0.732519\pi$
$227$	−20.1056	−1.33445	−0.667227	+	0.744855i	$0.732519\pi$
$228$	0	0
$229$	26.3261	1.73968	0.869838	−	0.493337i	$-0.164223\pi$
$229$	26.3261	1.73968	0.869838	+	0.493337i	$0.164223\pi$
$230$	0	0
$231$	0.930895	0.0612484
$232$	0	0
$233$	8.10123	0.530729	0.265365	−	0.964148i	$-0.414508\pi$
$233$	8.10123	0.530729	0.265365	+	0.964148i	$0.414508\pi$
$234$	0	0
$235$	−8.56856	−0.558951
$236$	0	0
$237$	11.9509	0.776294
$238$	0	0
$239$	13.5900	0.879062	0.439531	−	0.898227i	$-0.355145\pi$
$239$	13.5900	0.879062	0.439531	+	0.898227i	$0.355145\pi$
$240$	0	0
$241$	25.7782	1.66052	0.830261	−	0.557375i	$-0.188191\pi$
$241$	25.7782	1.66052	0.830261	+	0.557375i	$0.188191\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	9.57488	0.611717
$246$	0	0
$247$	6.98377	0.444366
$248$	0	0
$249$	−4.33550	−0.274751
$250$	0	0
$251$	−11.4113	−0.720274	−0.360137	−	0.932899i	$-0.617270\pi$
$251$	−11.4113	−0.720274	−0.360137	+	0.932899i	$0.617270\pi$
$252$	0	0
$253$	−20.0563	−1.26093
$254$	0	0
$255$	5.86317	0.367166
$256$	0	0
$257$	7.63267	0.476113	0.238056	−	0.971251i	$-0.423490\pi$
$257$	7.63267	0.476113	0.238056	+	0.971251i	$0.423490\pi$
$258$	0	0
$259$	0.0210395	0.00130733
$260$	0	0
$261$	2.23067	0.138075
$262$	0	0
$263$	−12.6577	−0.780510	−0.390255	−	0.920707i	$-0.627613\pi$
$263$	−12.6577	−0.780510	−0.390255	+	0.920707i	$0.627613\pi$
$264$	0	0
$265$	−17.1245	−1.05195
$266$	0	0
$267$	8.61186	0.527037
$268$	0	0
$269$	−0.272939	−0.0166414	−0.00832070	−	0.999965i	$-0.502649\pi$
$269$	−0.272939	−0.0166414	−0.00832070	+	0.999965i	$0.502649\pi$
$270$	0	0
$271$	−3.48099	−0.211455	−0.105727	−	0.994395i	$-0.533717\pi$
$271$	−3.48099	−0.211455	−0.105727	+	0.994395i	$0.533717\pi$
$272$	0	0
$273$	−0.338060	−0.0204603
$274$	0	0
$275$	−11.0674	−0.667391
$276$	0	0
$277$	9.17422	0.551225	0.275613	−	0.961269i	$-0.411119\pi$
$277$	9.17422	0.551225	0.275613	+	0.961269i	$0.411119\pi$
$278$	0	0
$279$	9.62228	0.576071
$280$	0	0
$281$	27.2804	1.62741	0.813706	−	0.581277i	$-0.197447\pi$
$281$	27.2804	1.62741	0.813706	+	0.581277i	$0.197447\pi$
$282$	0	0
$283$	−15.4054	−0.915759	−0.457879	−	0.889014i	$-0.651391\pi$
$283$	−15.4054	−0.915759	−0.457879	+	0.889014i	$0.651391\pi$
$284$	0	0
$285$	7.42142	0.439607
$286$	0	0
$287$	1.11586	0.0658669
$288$	0	0
$289$	1.02017	0.0600100
$290$	0	0
$291$	0.0718231	0.00421034
$292$	0	0
$293$	15.4315	0.901519	0.450759	−	0.892645i	$-0.351153\pi$
$293$	15.4315	0.901519	0.450759	+	0.892645i	$0.351153\pi$
$294$	0	0
$295$	4.39713	0.256011
$296$	0	0
$297$	3.57901	0.207675
$298$	0	0
$299$	7.28357	0.421220
$300$	0	0
$301$	2.73354	0.157559
$302$	0	0
$303$	17.9592	1.03173
$304$	0	0
$305$	8.85302	0.506922
$306$	0	0
$307$	−8.51510	−0.485982	−0.242991	−	0.970029i	$-0.578129\pi$
$307$	−8.51510	−0.485982	−0.242991	+	0.970029i	$0.578129\pi$
$308$	0	0
$309$	−0.277767	−0.0158016
$310$	0	0
$311$	−23.1383	−1.31205	−0.656026	−	0.754739i	$-0.727764\pi$
$311$	−23.1383	−1.31205	−0.656026	+	0.754739i	$0.727764\pi$
$312$	0	0
$313$	16.5786	0.937079	0.468539	−	0.883443i	$-0.344780\pi$
$313$	16.5786	0.937079	0.468539	+	0.883443i	$0.344780\pi$
$314$	0	0
$315$	−0.359246	−0.0202412
$316$	0	0
$317$	15.1097	0.848645	0.424322	−	0.905511i	$-0.360512\pi$
$317$	15.1097	0.848645	0.424322	+	0.905511i	$0.360512\pi$
$318$	0	0
$319$	7.98357	0.446994
$320$	0	0
$321$	5.01695	0.280019
$322$	0	0
$323$	22.8094	1.26915
$324$	0	0
$325$	4.01920	0.222945
$326$	0	0
$327$	15.0343	0.831398
$328$	0	0
$329$	1.61359	0.0889600
$330$	0	0
$331$	−21.7163	−1.19364	−0.596818	−	0.802377i	$-0.703568\pi$
$331$	−21.7163	−1.19364	−0.596818	+	0.802377i	$0.703568\pi$
$332$	0	0
$333$	0.0808905	0.00443277
$334$	0	0
$335$	3.57716	0.195441
$336$	0	0
$337$	−31.3088	−1.70550	−0.852750	−	0.522319i	$-0.825067\pi$
$337$	−31.3088	−1.70550	−0.852750	+	0.522319i	$0.825067\pi$
$338$	0	0
$339$	10.2708	0.557835
$340$	0	0
$341$	34.4382	1.86493
$342$	0	0
$343$	−3.62379	−0.195666
$344$	0	0
$345$	7.74001	0.416708
$346$	0	0
$347$	1.35131	0.0725419	0.0362710	−	0.999342i	$-0.488452\pi$
$347$	1.35131	0.0725419	0.0362710	+	0.999342i	$0.488452\pi$
$348$	0	0
$349$	−0.0510230	−0.00273120	−0.00136560	−	0.999999i	$-0.500435\pi$
$349$	−0.0510230	−0.00273120	−0.00136560	+	0.999999i	$0.500435\pi$
$350$	0	0
$351$	−1.29974	−0.0693749
$352$	0	0
$353$	19.0639	1.01467	0.507334	−	0.861749i	$-0.330631\pi$
$353$	19.0639	1.01467	0.507334	+	0.861749i	$0.330631\pi$
$354$	0	0
$355$	15.1830	0.805831
$356$	0	0
$357$	−1.10412	−0.0584364
$358$	0	0
$359$	20.9791	1.10724	0.553618	−	0.832771i	$-0.313247\pi$
$359$	20.9791	1.10724	0.553618	+	0.832771i	$0.313247\pi$
$360$	0	0
$361$	9.87140	0.519547
$362$	0	0
$363$	1.80930	0.0949634
$364$	0	0
$365$	−15.5107	−0.811869
$366$	0	0
$367$	21.3923	1.11667	0.558335	−	0.829616i	$-0.311440\pi$
$367$	21.3923	1.11667	0.558335	+	0.829616i	$0.311440\pi$
$368$	0	0
$369$	4.29012	0.223335
$370$	0	0
$371$	3.22480	0.167423
$372$	0	0
$373$	35.6765	1.84726	0.923628	−	0.383289i	$-0.125209\pi$
$373$	35.6765	1.84726	0.923628	+	0.383289i	$0.125209\pi$
$374$	0	0
$375$	11.1770	0.577179
$376$	0	0
$377$	−2.89928	−0.149321
$378$	0	0
$379$	−2.77234	−0.142405	−0.0712027	−	0.997462i	$-0.522684\pi$
$379$	−2.77234	−0.142405	−0.0712027	+	0.997462i	$0.522684\pi$
$380$	0	0
$381$	3.02975	0.155219
$382$	0	0
$383$	−22.0347	−1.12592	−0.562960	−	0.826484i	$-0.690337\pi$
$383$	−22.0347	−1.12592	−0.562960	+	0.826484i	$0.690337\pi$
$384$	0	0
$385$	−1.28574	−0.0655275
$386$	0	0
$387$	10.5096	0.534235
$388$	0	0
$389$	17.7239	0.898635	0.449318	−	0.893372i	$-0.351667\pi$
$389$	17.7239	0.898635	0.449318	+	0.893372i	$0.351667\pi$
$390$	0	0
$391$	23.7886	1.20304
$392$	0	0
$393$	17.2097	0.868112
$394$	0	0
$395$	−16.5064	−0.830529
$396$	0	0
$397$	−24.1866	−1.21389	−0.606944	−	0.794744i	$-0.707605\pi$
$397$	−24.1866	−1.21389	−0.606944	+	0.794744i	$0.707605\pi$
$398$	0	0
$399$	−1.39757	−0.0699658
$400$	0	0
$401$	−14.3128	−0.714748	−0.357374	−	0.933961i	$-0.616328\pi$
$401$	−14.3128	−0.714748	−0.357374	+	0.933961i	$0.616328\pi$
$402$	0	0
$403$	−12.5064	−0.622990
$404$	0	0
$405$	−1.38119	−0.0686318
$406$	0	0
$407$	0.289508	0.0143504
$408$	0	0
$409$	−21.1631	−1.04645	−0.523223	−	0.852196i	$-0.675271\pi$
$409$	−21.1631	−1.04645	−0.523223	+	0.852196i	$0.675271\pi$
$410$	0	0
$411$	−7.36980	−0.363526
$412$	0	0
$413$	−0.828046	−0.0407455
$414$	0	0
$415$	5.98814	0.293946
$416$	0	0
$417$	−15.4085	−0.754556
$418$	0	0
$419$	19.8760	0.971004	0.485502	−	0.874235i	$-0.338637\pi$
$419$	19.8760	0.971004	0.485502	+	0.874235i	$0.338637\pi$
$420$	0	0
$421$	−31.1319	−1.51728	−0.758639	−	0.651511i	$-0.774135\pi$
$421$	−31.1319	−1.51728	−0.758639	+	0.651511i	$0.774135\pi$
$422$	0	0
$423$	6.20376	0.301637
$424$	0	0
$425$	13.1269	0.636750
$426$	0	0
$427$	−1.66716	−0.0806794
$428$	0	0
$429$	−4.65177	−0.224590
$430$	0	0
$431$	21.1346	1.01802	0.509008	−	0.860762i	$-0.330012\pi$
$431$	21.1346	1.01802	0.509008	+	0.860762i	$0.330012\pi$
$432$	0	0
$433$	−28.1322	−1.35195	−0.675974	−	0.736926i	$-0.736277\pi$
$433$	−28.1322	−1.35195	−0.675974	+	0.736926i	$0.736277\pi$
$434$	0	0
$435$	−3.08097	−0.147721
$436$	0	0
$437$	30.1108	1.44040
$438$	0	0
$439$	29.5191	1.40887	0.704435	−	0.709769i	$-0.251201\pi$
$439$	29.5191	1.40887	0.704435	+	0.709769i	$0.251201\pi$
$440$	0	0
$441$	−6.93235	−0.330112
$442$	0	0
$443$	29.2445	1.38945	0.694724	−	0.719276i	$-0.255526\pi$
$443$	29.2445	1.38945	0.694724	+	0.719276i	$0.255526\pi$
$444$	0	0
$445$	−11.8946	−0.563859
$446$	0	0
$447$	−19.7078	−0.932145
$448$	0	0
$449$	31.0316	1.46447	0.732236	−	0.681051i	$-0.238477\pi$
$449$	31.0316	1.46447	0.732236	+	0.681051i	$0.238477\pi$
$450$	0	0
$451$	15.3544	0.723010
$452$	0	0
$453$	6.39372	0.300403
$454$	0	0
$455$	0.466925	0.0218898
$456$	0	0
$457$	−1.48784	−0.0695981	−0.0347991	−	0.999394i	$-0.511079\pi$
$457$	−1.48784	−0.0695981	−0.0347991	+	0.999394i	$0.511079\pi$
$458$	0	0
$459$	−4.24502	−0.198140
$460$	0	0
$461$	−5.65328	−0.263299	−0.131650	−	0.991296i	$-0.542027\pi$
$461$	−5.65328	−0.263299	−0.131650	+	0.991296i	$0.542027\pi$
$462$	0	0
$463$	−10.1653	−0.472421	−0.236210	−	0.971702i	$-0.575905\pi$
$463$	−10.1653	−0.472421	−0.236210	+	0.971702i	$0.575905\pi$
$464$	0	0
$465$	−13.2902	−0.616318
$466$	0	0
$467$	17.3327	0.802062	0.401031	−	0.916064i	$-0.368652\pi$
$467$	17.3327	0.802062	0.401031	+	0.916064i	$0.368652\pi$
$468$	0	0
$469$	−0.673634	−0.0311055
$470$	0	0
$471$	11.6913	0.538708
$472$	0	0
$473$	37.6141	1.72950
$474$	0	0
$475$	16.6157	0.762379
$476$	0	0
$477$	12.3984	0.567683
$478$	0	0
$479$	32.5368	1.48664	0.743322	−	0.668934i	$-0.233249\pi$
$479$	32.5368	1.48664	0.743322	+	0.668934i	$0.233249\pi$
$480$	0	0
$481$	−0.105136	−0.00479381
$482$	0	0
$483$	−1.45756	−0.0663213
$484$	0	0
$485$	−0.0992013	−0.00450450
$486$	0	0
$487$	−20.4665	−0.927426	−0.463713	−	0.885985i	$-0.653483\pi$
$487$	−20.4665	−0.927426	−0.463713	+	0.885985i	$0.653483\pi$
$488$	0	0
$489$	−9.58622	−0.433504
$490$	0	0
$491$	34.2008	1.54346	0.771730	−	0.635951i	$-0.219392\pi$
$491$	34.2008	1.54346	0.771730	+	0.635951i	$0.219392\pi$
$492$	0	0
$493$	−9.46921	−0.426472
$494$	0	0
$495$	−4.94329	−0.222184
$496$	0	0
$497$	−2.85919	−0.128252
$498$	0	0
$499$	−19.6637	−0.880269	−0.440134	−	0.897932i	$-0.645069\pi$
$499$	−19.6637	−0.880269	−0.440134	+	0.897932i	$0.645069\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	38.2959	1.70753	0.853764	−	0.520660i	$-0.174314\pi$
$503$	38.2959	1.70753	0.853764	+	0.520660i	$0.174314\pi$
$504$	0	0
$505$	−24.8050	−1.10381
$506$	0	0
$507$	−11.3107	−0.502325
$508$	0	0
$509$	27.9600	1.23931	0.619653	−	0.784876i	$-0.287273\pi$
$509$	27.9600	1.23931	0.619653	+	0.784876i	$0.287273\pi$
$510$	0	0
$511$	2.92091	0.129213
$512$	0	0
$513$	−5.37321	−0.237233
$514$	0	0
$515$	0.383648	0.0169056
$516$	0	0
$517$	22.2033	0.976499
$518$	0	0
$519$	25.1843	1.10547
$520$	0	0
$521$	−4.32725	−0.189580	−0.0947900	−	0.995497i	$-0.530218\pi$
$521$	−4.32725	−0.189580	−0.0947900	+	0.995497i	$0.530218\pi$
$522$	0	0
$523$	−4.71291	−0.206081	−0.103041	−	0.994677i	$-0.532857\pi$
$523$	−4.71291	−0.206081	−0.103041	+	0.994677i	$0.532857\pi$
$524$	0	0
$525$	−0.804308	−0.0351029
$526$	0	0
$527$	−40.8467	−1.77931
$528$	0	0
$529$	8.40344	0.365367
$530$	0	0
$531$	−3.18358	−0.138156
$532$	0	0
$533$	−5.57604	−0.241525
$534$	0	0
$535$	−6.92936	−0.299582
$536$	0	0
$537$	0.125279	0.00540619
$538$	0	0
$539$	−24.8109	−1.06868
$540$	0	0
$541$	−24.6119	−1.05815	−0.529075	−	0.848575i	$-0.677461\pi$
$541$	−24.6119	−1.05815	−0.529075	+	0.848575i	$0.677461\pi$
$542$	0	0
$543$	−15.0014	−0.643771
$544$	0	0
$545$	−20.7652	−0.889483
$546$	0	0
$547$	−29.3102	−1.25321	−0.626607	−	0.779335i	$-0.715557\pi$
$547$	−29.3102	−1.25321	−0.626607	+	0.779335i	$0.715557\pi$
$548$	0	0
$549$	−6.40971	−0.273560
$550$	0	0
$551$	−11.9858	−0.510614
$552$	0	0
$553$	3.10841	0.132183
$554$	0	0
$555$	−0.111725	−0.00474247
$556$	0	0
$557$	−32.1370	−1.36169	−0.680844	−	0.732428i	$-0.738387\pi$
$557$	−32.1370	−1.36169	−0.680844	+	0.732428i	$0.738387\pi$
$558$	0	0
$559$	−13.6598	−0.577747
$560$	0	0
$561$	−15.1929	−0.641447
$562$	0	0
$563$	−36.8692	−1.55385	−0.776926	−	0.629591i	$-0.783222\pi$
$563$	−36.8692	−1.55385	−0.776926	+	0.629591i	$0.783222\pi$
$564$	0	0
$565$	−14.1860	−0.596808
$566$	0	0
$567$	0.260099	0.0109231
$568$	0	0
$569$	39.4146	1.65234	0.826172	−	0.563418i	$-0.190514\pi$
$569$	39.4146	1.65234	0.826172	+	0.563418i	$0.190514\pi$
$570$	0	0
$571$	−21.3581	−0.893809	−0.446905	−	0.894582i	$-0.647474\pi$
$571$	−21.3581	−0.893809	−0.446905	+	0.894582i	$0.647474\pi$
$572$	0	0
$573$	4.07344	0.170170
$574$	0	0
$575$	17.3290	0.722668
$576$	0	0
$577$	36.0651	1.50141	0.750704	−	0.660639i	$-0.229714\pi$
$577$	36.0651	1.50141	0.750704	+	0.660639i	$0.229714\pi$
$578$	0	0
$579$	11.2459	0.467363
$580$	0	0
$581$	−1.12766	−0.0467831
$582$	0	0
$583$	44.3739	1.83778
$584$	0	0
$585$	1.79518	0.0742217
$586$	0	0
$587$	−6.65047	−0.274494	−0.137247	−	0.990537i	$-0.543825\pi$
$587$	−6.65047	−0.274494	−0.137247	+	0.990537i	$0.543825\pi$
$588$	0	0
$589$	−51.7025	−2.13037
$590$	0	0
$591$	−5.88075	−0.241902
$592$	0	0
$593$	0.0351096	0.00144178	0.000720889	−	1.00000i	$-0.499771\pi$
$593$	0.0351096	0.00144178	0.000720889		1.00000i	$0.499771\pi$
$594$	0	0
$595$	1.52500	0.0625190
$596$	0	0
$597$	−19.8895	−0.814022
$598$	0	0
$599$	29.1516	1.19110	0.595550	−	0.803318i	$-0.296934\pi$
$599$	29.1516	1.19110	0.595550	+	0.803318i	$0.296934\pi$
$600$	0	0
$601$	41.1130	1.67703	0.838517	−	0.544876i	$-0.183423\pi$
$601$	41.1130	1.67703	0.838517	+	0.544876i	$0.183423\pi$
$602$	0	0
$603$	−2.58992	−0.105470
$604$	0	0
$605$	−2.49898	−0.101598
$606$	0	0
$607$	−5.72077	−0.232199	−0.116099	−	0.993238i	$-0.537039\pi$
$607$	−5.72077	−0.232199	−0.116099	+	0.993238i	$0.537039\pi$
$608$	0	0
$609$	0.580193	0.0235106
$610$	0	0
$611$	−8.06326	−0.326204
$612$	0	0
$613$	5.16466	0.208599	0.104299	−	0.994546i	$-0.466740\pi$
$613$	5.16466	0.208599	0.104299	+	0.994546i	$0.466740\pi$
$614$	0	0
$615$	−5.92547	−0.238938
$616$	0	0
$617$	−35.1409	−1.41472	−0.707360	−	0.706854i	$-0.750114\pi$
$617$	−35.1409	−1.41472	−0.707360	+	0.706854i	$0.750114\pi$
$618$	0	0
$619$	22.6982	0.912318	0.456159	−	0.889898i	$-0.349225\pi$
$619$	22.6982	0.912318	0.456159	+	0.889898i	$0.349225\pi$
$620$	0	0
$621$	−5.60388	−0.224876
$622$	0	0
$623$	2.23993	0.0897411
$624$	0	0
$625$	0.0240062	0.000960247 0
$626$	0	0
$627$	−19.2308	−0.768003
$628$	0	0
$629$	−0.343382	−0.0136915
$630$	0	0
$631$	35.0964	1.39717	0.698583	−	0.715529i	$-0.253814\pi$
$631$	35.0964	1.39717	0.698583	+	0.715529i	$0.253814\pi$
$632$	0	0
$633$	−14.3807	−0.571581
$634$	0	0
$635$	−4.18465	−0.166063
$636$	0	0
$637$	9.01024	0.356999
$638$	0	0
$639$	−10.9927	−0.434865
$640$	0	0
$641$	5.49562	0.217064	0.108532	−	0.994093i	$-0.465385\pi$
$641$	5.49562	0.217064	0.108532	+	0.994093i	$0.465385\pi$
$642$	0	0
$643$	30.9837	1.22188	0.610940	−	0.791677i	$-0.290792\pi$
$643$	30.9837	1.22188	0.610940	+	0.791677i	$0.290792\pi$
$644$	0	0
$645$	−14.5158	−0.571559
$646$	0	0
$647$	−15.3651	−0.604065	−0.302033	−	0.953298i	$-0.597665\pi$
$647$	−15.3651	−0.604065	−0.302033	+	0.953298i	$0.597665\pi$
$648$	0	0
$649$	−11.3941	−0.447256
$650$	0	0
$651$	2.50274	0.0980902
$652$	0	0
$653$	−49.3594	−1.93158	−0.965791	−	0.259321i	$-0.916501\pi$
$653$	−49.3594	−1.93158	−0.965791	+	0.259321i	$0.916501\pi$
$654$	0	0
$655$	−23.7698	−0.928762
$656$	0	0
$657$	11.2300	0.438124
$658$	0	0
$659$	18.8524	0.734385	0.367192	−	0.930145i	$-0.380319\pi$
$659$	18.8524	0.734385	0.367192	+	0.930145i	$0.380319\pi$
$660$	0	0
$661$	−36.9080	−1.43556	−0.717778	−	0.696272i	$-0.754841\pi$
$661$	−36.9080	−1.43556	−0.717778	+	0.696272i	$0.754841\pi$
$662$	0	0
$663$	5.51741	0.214278
$664$	0	0
$665$	1.93030	0.0748539
$666$	0	0
$667$	−12.5004	−0.484017
$668$	0	0
$669$	28.9444	1.11906
$670$	0	0
$671$	−22.9404	−0.885604
$672$	0	0
$673$	−4.15105	−0.160011	−0.0800057	−	0.996794i	$-0.525494\pi$
$673$	−4.15105	−0.160011	−0.0800057	+	0.996794i	$0.525494\pi$
$674$	0	0
$675$	−3.09232	−0.119023
$676$	0	0
$677$	−45.6659	−1.75508	−0.877541	−	0.479502i	$-0.840817\pi$
$677$	−45.6659	−1.75508	−0.877541	+	0.479502i	$0.840817\pi$
$678$	0	0
$679$	0.0186811	0.000716915 0
$680$	0	0
$681$	−20.1056	−0.770447
$682$	0	0
$683$	−34.4197	−1.31704	−0.658518	−	0.752565i	$-0.728816\pi$
$683$	−34.4197	−1.31704	−0.658518	+	0.752565i	$0.728816\pi$
$684$	0	0
$685$	10.1791	0.388923
$686$	0	0
$687$	26.3261	1.00440
$688$	0	0
$689$	−16.1146	−0.613919
$690$	0	0
$691$	28.3874	1.07991	0.539954	−	0.841694i	$-0.318441\pi$
$691$	28.3874	1.07991	0.539954	+	0.841694i	$0.318441\pi$
$692$	0	0
$693$	0.930895	0.0353618
$694$	0	0
$695$	21.2820	0.807273
$696$	0	0
$697$	−18.2117	−0.689815
$698$	0	0
$699$	8.10123	0.306417
$700$	0	0
$701$	20.7096	0.782189	0.391095	−	0.920350i	$-0.372096\pi$
$701$	20.7096	0.782189	0.391095	+	0.920350i	$0.372096\pi$
$702$	0	0
$703$	−0.434642	−0.0163928
$704$	0	0
$705$	−8.56856	−0.322711
$706$	0	0
$707$	4.67115	0.175677
$708$	0	0
$709$	12.3505	0.463833	0.231917	−	0.972736i	$-0.425500\pi$
$709$	12.3505	0.463833	0.231917	+	0.972736i	$0.425500\pi$
$710$	0	0
$711$	11.9509	0.448193
$712$	0	0
$713$	−53.9221	−2.01940
$714$	0	0
$715$	6.42498	0.240281
$716$	0	0
$717$	13.5900	0.507527
$718$	0	0
$719$	40.7997	1.52157	0.760786	−	0.649003i	$-0.224814\pi$
$719$	40.7997	1.52157	0.760786	+	0.649003i	$0.224814\pi$
$720$	0	0
$721$	−0.0722468	−0.00269061
$722$	0	0
$723$	25.7782	0.958703
$724$	0	0
$725$	−6.89793	−0.256183
$726$	0	0
$727$	−25.3456	−0.940015	−0.470007	−	0.882662i	$-0.655749\pi$
$727$	−25.3456	−0.940015	−0.470007	+	0.882662i	$0.655749\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−44.6136	−1.65009
$732$	0	0
$733$	−2.51219	−0.0927899	−0.0463949	−	0.998923i	$-0.514773\pi$
$733$	−2.51219	−0.0927899	−0.0463949	+	0.998923i	$0.514773\pi$
$734$	0	0
$735$	9.57488	0.353175
$736$	0	0
$737$	−9.26933	−0.341440
$738$	0	0
$739$	−35.0409	−1.28900	−0.644499	−	0.764605i	$-0.722934\pi$
$739$	−35.0409	−1.28900	−0.644499	+	0.764605i	$0.722934\pi$
$740$	0	0
$741$	6.98377	0.256555
$742$	0	0
$743$	−11.5421	−0.423440	−0.211720	−	0.977330i	$-0.567906\pi$
$743$	−11.5421	−0.423440	−0.211720	+	0.977330i	$0.567906\pi$
$744$	0	0
$745$	27.2201	0.997269
$746$	0	0
$747$	−4.33550	−0.158628
$748$	0	0
$749$	1.30490	0.0476801
$750$	0	0
$751$	37.9598	1.38517	0.692586	−	0.721336i	$-0.256471\pi$
$751$	37.9598	1.38517	0.692586	+	0.721336i	$0.256471\pi$
$752$	0	0
$753$	−11.4113	−0.415851
$754$	0	0
$755$	−8.83094	−0.321391
$756$	0	0
$757$	8.58630	0.312074	0.156037	−	0.987751i	$-0.450128\pi$
$757$	8.58630	0.312074	0.156037	+	0.987751i	$0.450128\pi$
$758$	0	0
$759$	−20.0563	−0.727998
$760$	0	0
$761$	−1.05601	−0.0382802	−0.0191401	−	0.999817i	$-0.506093\pi$
$761$	−1.05601	−0.0382802	−0.0191401	+	0.999817i	$0.506093\pi$
$762$	0	0
$763$	3.91040	0.141566
$764$	0	0
$765$	5.86317	0.211983
$766$	0	0
$767$	4.13782	0.149408
$768$	0	0
$769$	42.6410	1.53767	0.768836	−	0.639446i	$-0.220836\pi$
$769$	42.6410	1.53767	0.768836	+	0.639446i	$0.220836\pi$
$770$	0	0
$771$	7.63267	0.274884
$772$	0	0
$773$	32.7347	1.17738	0.588692	−	0.808357i	$-0.299643\pi$
$773$	32.7347	1.17738	0.588692	+	0.808357i	$0.299643\pi$
$774$	0	0
$775$	−29.7551	−1.06884
$776$	0	0
$777$	0.0210395	0.000754789 0
$778$	0	0
$779$	−23.0517	−0.825914
$780$	0	0
$781$	−39.3430	−1.40780
$782$	0	0
$783$	2.23067	0.0797175
$784$	0	0
$785$	−16.1479	−0.576345
$786$	0	0
$787$	−38.4271	−1.36978	−0.684890	−	0.728647i	$-0.740150\pi$
$787$	−38.4271	−1.36978	−0.684890	+	0.728647i	$0.740150\pi$
$788$	0	0
$789$	−12.6577	−0.450627
$790$	0	0
$791$	2.67143	0.0949851
$792$	0	0
$793$	8.33094	0.295840
$794$	0	0
$795$	−17.1245	−0.607344
$796$	0	0
$797$	46.5485	1.64883	0.824416	−	0.565985i	$-0.191504\pi$
$797$	46.5485	1.64883	0.824416	+	0.565985i	$0.191504\pi$
$798$	0	0
$799$	−26.3350	−0.931667
$800$	0	0
$801$	8.61186	0.304285
$802$	0	0
$803$	40.1922	1.41835
$804$	0	0
$805$	2.01317	0.0709548
$806$	0	0
$807$	−0.272939	−0.00960791
$808$	0	0
$809$	38.3586	1.34862	0.674308	−	0.738450i	$-0.264442\pi$
$809$	38.3586	1.34862	0.674308	+	0.738450i	$0.264442\pi$
$810$	0	0
$811$	35.3307	1.24063	0.620314	−	0.784354i	$-0.287005\pi$
$811$	35.3307	1.24063	0.620314	+	0.784354i	$0.287005\pi$
$812$	0	0
$813$	−3.48099	−0.122084
$814$	0	0
$815$	13.2404	0.463790
$816$	0	0
$817$	−56.4705	−1.97565
$818$	0	0
$819$	−0.338060	−0.0118128
$820$	0	0
$821$	−6.32816	−0.220854	−0.110427	−	0.993884i	$-0.535222\pi$
$821$	−6.32816	−0.220854	−0.110427	+	0.993884i	$0.535222\pi$
$822$	0	0
$823$	−13.5763	−0.473239	−0.236620	−	0.971602i	$-0.576040\pi$
$823$	−13.5763	−0.473239	−0.236620	+	0.971602i	$0.576040\pi$
$824$	0	0
$825$	−11.0674	−0.385318
$826$	0	0
$827$	9.00252	0.313048	0.156524	−	0.987674i	$-0.449971\pi$
$827$	9.00252	0.313048	0.156524	+	0.987674i	$0.449971\pi$
$828$	0	0
$829$	26.7081	0.927610	0.463805	−	0.885937i	$-0.346484\pi$
$829$	26.7081	0.927610	0.463805	+	0.885937i	$0.346484\pi$
$830$	0	0
$831$	9.17422	0.318250
$832$	0	0
$833$	29.4279	1.01962
$834$	0	0
$835$	1.38119	0.0477980
$836$	0	0
$837$	9.62228	0.332595
$838$	0	0
$839$	−38.2297	−1.31983	−0.659917	−	0.751338i	$-0.729409\pi$
$839$	−38.2297	−1.31983	−0.659917	+	0.751338i	$0.729409\pi$
$840$	0	0
$841$	−24.0241	−0.828418
$842$	0	0
$843$	27.2804	0.939587
$844$	0	0
$845$	15.6222	0.537420
$846$	0	0
$847$	0.470595	0.0161699
$848$	0	0
$849$	−15.4054	−0.528713
$850$	0	0
$851$	−0.453301	−0.0155389
$852$	0	0
$853$	−27.6328	−0.946129	−0.473065	−	0.881028i	$-0.656852\pi$
$853$	−27.6328	−0.946129	−0.473065	+	0.881028i	$0.656852\pi$
$854$	0	0
$855$	7.42142	0.253807
$856$	0	0
$857$	−47.1251	−1.60976	−0.804882	−	0.593436i	$-0.797771\pi$
$857$	−47.1251	−1.60976	−0.804882	+	0.593436i	$0.797771\pi$
$858$	0	0
$859$	49.2154	1.67921	0.839604	−	0.543199i	$-0.182787\pi$
$859$	49.2154	1.67921	0.839604	+	0.543199i	$0.182787\pi$
$860$	0	0
$861$	1.11586	0.0380283
$862$	0	0
$863$	−23.9655	−0.815795	−0.407897	−	0.913028i	$-0.633738\pi$
$863$	−23.9655	−0.815795	−0.407897	+	0.913028i	$0.633738\pi$
$864$	0	0
$865$	−34.7842	−1.18270
$866$	0	0
$867$	1.02017	0.0346468
$868$	0	0
$869$	42.7723	1.45095
$870$	0	0
$871$	3.36621	0.114060
$872$	0	0
$873$	0.0718231	0.00243084
$874$	0	0
$875$	2.90713	0.0982789
$876$	0	0
$877$	6.72694	0.227152	0.113576	−	0.993529i	$-0.463769\pi$
$877$	6.72694	0.227152	0.113576	+	0.993529i	$0.463769\pi$
$878$	0	0
$879$	15.4315	0.520492
$880$	0	0
$881$	−2.21204	−0.0745254	−0.0372627	−	0.999306i	$-0.511864\pi$
$881$	−2.21204	−0.0745254	−0.0372627	+	0.999306i	$0.511864\pi$
$882$	0	0
$883$	−40.9253	−1.37725	−0.688623	−	0.725119i	$-0.741785\pi$
$883$	−40.9253	−1.37725	−0.688623	+	0.725119i	$0.741785\pi$
$884$	0	0
$885$	4.39713	0.147808
$886$	0	0
$887$	2.67010	0.0896532	0.0448266	−	0.998995i	$-0.485726\pi$
$887$	2.67010	0.0896532	0.0448266	+	0.998995i	$0.485726\pi$
$888$	0	0
$889$	0.788033	0.0264298
$890$	0	0
$891$	3.57901	0.119901
$892$	0	0
$893$	−33.3341	−1.11548
$894$	0	0
$895$	−0.173034	−0.00578389
$896$	0	0
$897$	7.28357	0.243191
$898$	0	0
$899$	21.4641	0.715868
$900$	0	0
$901$	−52.6313	−1.75340
$902$	0	0
$903$	2.73354	0.0909666
$904$	0	0
$905$	20.7198	0.688748
$906$	0	0
$907$	31.5598	1.04793	0.523963	−	0.851741i	$-0.324453\pi$
$907$	31.5598	1.04793	0.523963	+	0.851741i	$0.324453\pi$
$908$	0	0
$909$	17.9592	0.595667
$910$	0	0
$911$	−1.18983	−0.0394210	−0.0197105	−	0.999806i	$-0.506274\pi$
$911$	−1.18983	−0.0394210	−0.0197105	+	0.999806i	$0.506274\pi$
$912$	0	0
$913$	−15.5168	−0.513530
$914$	0	0
$915$	8.85302	0.292672
$916$	0	0
$917$	4.47621	0.147817
$918$	0	0
$919$	18.7543	0.618648	0.309324	−	0.950957i	$-0.399897\pi$
$919$	18.7543	0.618648	0.309324	+	0.950957i	$0.399897\pi$
$920$	0	0
$921$	−8.51510	−0.280582
$922$	0	0
$923$	14.2877	0.470284
$924$	0	0
$925$	−0.250139	−0.00822452
$926$	0	0
$927$	−0.277767	−0.00912306
$928$	0	0
$929$	17.6507	0.579102	0.289551	−	0.957163i	$-0.406494\pi$
$929$	17.6507	0.579102	0.289551	+	0.957163i	$0.406494\pi$
$930$	0	0
$931$	37.2490	1.22079
$932$	0	0
$933$	−23.1383	−0.757513
$934$	0	0
$935$	20.9843	0.686261
$936$	0	0
$937$	−48.8423	−1.59561	−0.797805	−	0.602916i	$-0.794005\pi$
$937$	−48.8423	−1.59561	−0.797805	+	0.602916i	$0.794005\pi$
$938$	0	0
$939$	16.5786	0.541023
$940$	0	0
$941$	−3.14208	−0.102429	−0.0512145	−	0.998688i	$-0.516309\pi$
$941$	−3.14208	−0.102429	−0.0512145	+	0.998688i	$0.516309\pi$
$942$	0	0
$943$	−24.0413	−0.782893
$944$	0	0
$945$	−0.359246	−0.0116863
$946$	0	0
$947$	−10.0506	−0.326602	−0.163301	−	0.986576i	$-0.552214\pi$
$947$	−10.0506	−0.326602	−0.163301	+	0.986576i	$0.552214\pi$
$948$	0	0
$949$	−14.5960	−0.473808
$950$	0	0
$951$	15.1097	0.489965
$952$	0	0
$953$	−7.69926	−0.249403	−0.124702	−	0.992194i	$-0.539797\pi$
$953$	−7.69926	−0.249403	−0.124702	+	0.992194i	$0.539797\pi$
$954$	0	0
$955$	−5.62619	−0.182059
$956$	0	0
$957$	7.98357	0.258072
$958$	0	0
$959$	−1.91688	−0.0618992
$960$	0	0
$961$	61.5883	1.98672
$962$	0	0
$963$	5.01695	0.161669
$964$	0	0
$965$	−15.5327	−0.500015
$966$	0	0
$967$	−26.5188	−0.852789	−0.426394	−	0.904537i	$-0.640216\pi$
$967$	−26.5188	−0.852789	−0.426394	+	0.904537i	$0.640216\pi$
$968$	0	0
$969$	22.8094	0.732743
$970$	0	0
$971$	−47.1360	−1.51267	−0.756333	−	0.654187i	$-0.773011\pi$
$971$	−47.1360	−1.51267	−0.756333	+	0.654187i	$0.773011\pi$
$972$	0	0
$973$	−4.00773	−0.128482
$974$	0	0
$975$	4.01920	0.128717
$976$	0	0
$977$	1.34004	0.0428718	0.0214359	−	0.999770i	$-0.493176\pi$
$977$	1.34004	0.0428718	0.0214359	+	0.999770i	$0.493176\pi$
$978$	0	0
$979$	30.8219	0.985073
$980$	0	0
$981$	15.0343	0.480008
$982$	0	0
$983$	18.9209	0.603485	0.301742	−	0.953390i	$-0.402432\pi$
$983$	18.9209	0.603485	0.301742	+	0.953390i	$0.402432\pi$
$984$	0	0
$985$	8.12242	0.258802
$986$	0	0
$987$	1.61359	0.0513611
$988$	0	0
$989$	−58.8947	−1.87274
$990$	0	0
$991$	−53.1851	−1.68948	−0.844740	−	0.535178i	$-0.820245\pi$
$991$	−53.1851	−1.68948	−0.844740	+	0.535178i	$0.820245\pi$
$992$	0	0
$993$	−21.7163	−0.689146
$994$	0	0
$995$	27.4711	0.870893
$996$	0	0
$997$	5.58170	0.176774	0.0883870	−	0.996086i	$-0.471829\pi$
$997$	5.58170	0.176774	0.0883870	+	0.996086i	$0.471829\pi$
$998$	0	0
$999$	0.0808905	0.00255926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.be.1.4		11
4.3	odd	2		4008.2.a.k.1.4	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.4	✓	11	4.3	odd	2
8016.2.a.be.1.4		11	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.38119 q^{5} +0.260099 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.38119 q^{5} +0.260099 q^{7} +1.00000 q^{9} +3.57901 q^{11} -1.29974 q^{13} -1.38119 q^{15} -4.24502 q^{17} -5.37321 q^{19} +0.260099 q^{21} -5.60388 q^{23} -3.09232 q^{25} +1.00000 q^{27} +2.23067 q^{29} +9.62228 q^{31} +3.57901 q^{33} -0.359246 q^{35} +0.0808905 q^{37} -1.29974 q^{39} +4.29012 q^{41} +10.5096 q^{43} -1.38119 q^{45} +6.20376 q^{47} -6.93235 q^{49} -4.24502 q^{51} +12.3984 q^{53} -4.94329 q^{55} -5.37321 q^{57} -3.18358 q^{59} -6.40971 q^{61} +0.260099 q^{63} +1.79518 q^{65} -2.58992 q^{67} -5.60388 q^{69} -10.9927 q^{71} +11.2300 q^{73} -3.09232 q^{75} +0.930895 q^{77} +11.9509 q^{79} +1.00000 q^{81} -4.33550 q^{83} +5.86317 q^{85} +2.23067 q^{87} +8.61186 q^{89} -0.338060 q^{91} +9.62228 q^{93} +7.42142 q^{95} +0.0718231 q^{97} +3.57901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9	\( 11 q + 11 q^{3} + 10 q^{5} + q^{7} + 11 q^{9} + q^{11} + 10 q^{13} + 10 q^{15} + 17 q^{17} - 2 q^{19} + q^{21} + 3 q^{23} + 21 q^{25} + 11 q^{27} + 17 q^{29} + 15 q^{31} + q^{33} - 11 q^{35} + 4 q^{37} + 10 q^{39} + 16 q^{41} - 10 q^{43} + 10 q^{45} + 16 q^{47} + 22 q^{49} + 17 q^{51} + 42 q^{53} + 5 q^{55} - 2 q^{57} + 2 q^{59} + 12 q^{61} + q^{63} + 10 q^{65} + q^{67} + 3 q^{69} + 9 q^{71} + 24 q^{73} + 21 q^{75} + 22 q^{77} + 30 q^{79} + 11 q^{81} - 16 q^{83} + 25 q^{85} + 17 q^{87} + 37 q^{89} - q^{91} + 15 q^{93} - 5 q^{95} + 4 q^{97} + q^{99}+O(q^{100}) \) 11 * q + 11 * q^3 + 10 * q^5 + q^7 + 11 * q^9 + q^11 + 10 * q^13 + 10 * q^15 + 17 * q^17 - 2 * q^19 + q^21 + 3 * q^23 + 21 * q^25 + 11 * q^27 + 17 * q^29 + 15 * q^31 + q^33 - 11 * q^35 + 4 * q^37 + 10 * q^39 + 16 * q^41 - 10 * q^43 + 10 * q^45 + 16 * q^47 + 22 * q^49 + 17 * q^51 + 42 * q^53 + 5 * q^55 - 2 * q^57 + 2 * q^59 + 12 * q^61 + q^63 + 10 * q^65 + q^67 + 3 * q^69 + 9 * q^71 + 24 * q^73 + 21 * q^75 + 22 * q^77 + 30 * q^79 + 11 * q^81 - 16 * q^83 + 25 * q^85 + 17 * q^87 + 37 * q^89 - q^91 + 15 * q^93 - 5 * q^95 + 4 * q^97 + q^99

Introduction

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