LMFDB - Embedded newform 8001.2.a.z.1.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.23
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18013 q^{2} -0.607291 q^{4} -4.43826 q^{5} -1.00000 q^{7} -3.07694 q^{8} +O(q^{10})$	\(q+1.18013 q^{2} -0.607291 q^{4} -4.43826 q^{5} -1.00000 q^{7} -3.07694 q^{8} -5.23773 q^{10} +5.41915 q^{11} -2.38471 q^{13} -1.18013 q^{14} -2.41662 q^{16} -2.36152 q^{17} -2.75497 q^{19} +2.69531 q^{20} +6.39530 q^{22} +5.73685 q^{23} +14.6981 q^{25} -2.81427 q^{26} +0.607291 q^{28} +3.90830 q^{29} +2.83738 q^{31} +3.30196 q^{32} -2.78691 q^{34} +4.43826 q^{35} -3.58729 q^{37} -3.25123 q^{38} +13.6563 q^{40} +0.707118 q^{41} +9.07783 q^{43} -3.29100 q^{44} +6.77024 q^{46} -4.92884 q^{47} +1.00000 q^{49} +17.3457 q^{50} +1.44821 q^{52} -10.8388 q^{53} -24.0516 q^{55} +3.07694 q^{56} +4.61231 q^{58} -8.70245 q^{59} +15.4770 q^{61} +3.34848 q^{62} +8.72998 q^{64} +10.5839 q^{65} +5.37268 q^{67} +1.43413 q^{68} +5.23773 q^{70} +8.79079 q^{71} -14.5845 q^{73} -4.23348 q^{74} +1.67307 q^{76} -5.41915 q^{77} -13.5647 q^{79} +10.7256 q^{80} +0.834492 q^{82} +5.85553 q^{83} +10.4810 q^{85} +10.7130 q^{86} -16.6744 q^{88} -5.77576 q^{89} +2.38471 q^{91} -3.48394 q^{92} -5.81668 q^{94} +12.2273 q^{95} +13.8577 q^{97} +1.18013 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

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 $

$2$	1.18013	0.834479	0.417239	−	0.908797i	$-0.362998\pi$
$2$	1.18013	0.834479	0.417239	+	0.908797i	$0.362998\pi$
$3$	0	0
$3$	0	0
$4$	−0.607291	−0.303645
$5$	−4.43826	−1.98485	−0.992425	−	0.122856i	$-0.960795\pi$
$5$	−4.43826	−1.98485	−0.992425	+	0.122856i	$0.960795\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.07694	−1.08786
$9$	0	0
$10$	−5.23773	−1.65631
$11$	5.41915	1.63393	0.816967	−	0.576685i	$-0.195654\pi$
$11$	5.41915	1.63393	0.816967	+	0.576685i	$0.195654\pi$
$12$	0	0
$13$	−2.38471	−0.661398	−0.330699	−	0.943736i	$-0.607285\pi$
$13$	−2.38471	−0.661398	−0.330699	+	0.943736i	$0.607285\pi$
$14$	−1.18013	−0.315403
$15$	0	0
$16$	−2.41662	−0.604154
$17$	−2.36152	−0.572753	−0.286377	−	0.958117i	$-0.592451\pi$
$17$	−2.36152	−0.572753	−0.286377	+	0.958117i	$0.592451\pi$
$18$	0	0
$19$	−2.75497	−0.632034	−0.316017	−	0.948753i	$-0.602346\pi$
$19$	−2.75497	−0.632034	−0.316017	+	0.948753i	$0.602346\pi$
$20$	2.69531	0.602690
$21$	0	0
$22$	6.39530	1.36348
$23$	5.73685	1.19622	0.598108	−	0.801416i	$-0.295919\pi$
$23$	5.73685	1.19622	0.598108	+	0.801416i	$0.295919\pi$
$24$	0	0
$25$	14.6981	2.93963
$26$	−2.81427	−0.551923
$27$	0	0
$28$	0.607291	0.114767
$29$	3.90830	0.725754	0.362877	−	0.931837i	$-0.381795\pi$
$29$	3.90830	0.725754	0.362877	+	0.931837i	$0.381795\pi$
$30$	0	0
$31$	2.83738	0.509609	0.254804	−	0.966993i	$-0.417989\pi$
$31$	2.83738	0.509609	0.254804	+	0.966993i	$0.417989\pi$
$32$	3.30196	0.583710
$33$	0	0
$34$	−2.78691	−0.477950
$35$	4.43826	0.750202
$36$	0	0
$37$	−3.58729	−0.589747	−0.294874	−	0.955536i	$-0.595278\pi$
$37$	−3.58729	−0.589747	−0.294874	+	0.955536i	$0.595278\pi$
$38$	−3.25123	−0.527419
$39$	0	0
$40$	13.6563	2.15925
$41$	0.707118	0.110433	0.0552166	−	0.998474i	$-0.482415\pi$
$41$	0.707118	0.110433	0.0552166	+	0.998474i	$0.482415\pi$
$42$	0	0
$43$	9.07783	1.38436	0.692178	−	0.721727i	$-0.256651\pi$
$43$	9.07783	1.38436	0.692178	+	0.721727i	$0.256651\pi$
$44$	−3.29100	−0.496136
$45$	0	0
$46$	6.77024	0.998217
$47$	−4.92884	−0.718945	−0.359473	−	0.933156i	$-0.617043\pi$
$47$	−4.92884	−0.718945	−0.359473	+	0.933156i	$0.617043\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	17.3457	2.45306
$51$	0	0
$52$	1.44821	0.200830
$53$	−10.8388	−1.48883	−0.744414	−	0.667719i	$-0.767271\pi$
$53$	−10.8388	−1.48883	−0.744414	+	0.667719i	$0.767271\pi$
$54$	0	0
$55$	−24.0516	−3.24311
$56$	3.07694	0.411174
$57$	0	0
$58$	4.61231	0.605626
$59$	−8.70245	−1.13296	−0.566481	−	0.824075i	$-0.691696\pi$
$59$	−8.70245	−1.13296	−0.566481	+	0.824075i	$0.691696\pi$
$60$	0	0
$61$	15.4770	1.98163	0.990816	−	0.135214i	$-0.0431723\pi$
$61$	15.4770	1.98163	0.990816	+	0.135214i	$0.0431723\pi$
$62$	3.34848	0.425258
$63$	0	0
$64$	8.72998	1.09125
$65$	10.5839	1.31278
$66$	0	0
$67$	5.37268	0.656377	0.328189	−	0.944612i	$-0.393562\pi$
$67$	5.37268	0.656377	0.328189	+	0.944612i	$0.393562\pi$
$68$	1.43413	0.173914
$69$	0	0
$70$	5.23773	0.626028
$71$	8.79079	1.04327	0.521637	−	0.853167i	$-0.325321\pi$
$71$	8.79079	1.04327	0.521637	+	0.853167i	$0.325321\pi$
$72$	0	0
$73$	−14.5845	−1.70698	−0.853492	−	0.521107i	$-0.825519\pi$
$73$	−14.5845	−1.70698	−0.853492	+	0.521107i	$0.825519\pi$
$74$	−4.23348	−0.492132
$75$	0	0
$76$	1.67307	0.191914
$77$	−5.41915	−0.617569
$78$	0	0
$79$	−13.5647	−1.52615	−0.763077	−	0.646308i	$-0.776312\pi$
$79$	−13.5647	−1.52615	−0.763077	+	0.646308i	$0.776312\pi$
$80$	10.7256	1.19915
$81$	0	0
$82$	0.834492	0.0921542
$83$	5.85553	0.642728	0.321364	−	0.946956i	$-0.395859\pi$
$83$	5.85553	0.642728	0.321364	+	0.946956i	$0.395859\pi$
$84$	0	0
$85$	10.4810	1.13683
$86$	10.7130	1.15522
$87$	0	0
$88$	−16.6744	−1.77750
$89$	−5.77576	−0.612229	−0.306114	−	0.951995i	$-0.599029\pi$
$89$	−5.77576	−0.612229	−0.306114	+	0.951995i	$0.599029\pi$
$90$	0	0
$91$	2.38471	0.249985
$92$	−3.48394	−0.363225
$93$	0	0
$94$	−5.81668	−0.599944
$95$	12.2273	1.25449
$96$	0	0
$97$	13.8577	1.40703	0.703517	−	0.710678i	$-0.251612\pi$
$97$	13.8577	1.40703	0.703517	+	0.710678i	$0.251612\pi$
$98$	1.18013	0.119211
$99$	0	0
$100$	−8.92604	−0.892604
$101$	1.80702	0.179806	0.0899028	−	0.995951i	$-0.471344\pi$
$101$	1.80702	0.179806	0.0899028	+	0.995951i	$0.471344\pi$
$102$	0	0
$103$	−10.8246	−1.06658	−0.533288	−	0.845934i	$-0.679044\pi$
$103$	−10.8246	−1.06658	−0.533288	+	0.845934i	$0.679044\pi$
$104$	7.33761	0.719512
$105$	0	0
$106$	−12.7912	−1.24239
$107$	12.8202	1.23937	0.619686	−	0.784849i	$-0.287260\pi$
$107$	12.8202	1.23937	0.619686	+	0.784849i	$0.287260\pi$
$108$	0	0
$109$	−9.99649	−0.957490	−0.478745	−	0.877954i	$-0.658908\pi$
$109$	−9.99649	−0.957490	−0.478745	+	0.877954i	$0.658908\pi$
$110$	−28.3840	−2.70631
$111$	0	0
$112$	2.41662	0.228349
$113$	8.33535	0.784124	0.392062	−	0.919939i	$-0.371762\pi$
$113$	8.33535	0.784124	0.392062	+	0.919939i	$0.371762\pi$
$114$	0	0
$115$	−25.4616	−2.37431
$116$	−2.37348	−0.220372
$117$	0	0
$118$	−10.2700	−0.945433
$119$	2.36152	0.216480
$120$	0	0
$121$	18.3671	1.66974
$122$	18.2649	1.65363
$123$	0	0
$124$	−1.72311	−0.154740
$125$	−43.0428	−3.84986
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	3.69860	0.326913
$129$	0	0
$130$	12.4904	1.09548
$131$	9.14569	0.799062	0.399531	−	0.916720i	$-0.369173\pi$
$131$	9.14569	0.799062	0.399531	+	0.916720i	$0.369173\pi$
$132$	0	0
$133$	2.75497	0.238886
$134$	6.34047	0.547733
$135$	0	0
$136$	7.26627	0.623078
$137$	−13.9083	−1.18826	−0.594132	−	0.804368i	$-0.702504\pi$
$137$	−13.9083	−1.18826	−0.594132	+	0.804368i	$0.702504\pi$
$138$	0	0
$139$	−17.0334	−1.44476	−0.722379	−	0.691497i	$-0.756951\pi$
$139$	−17.0334	−1.44476	−0.722379	+	0.691497i	$0.756951\pi$
$140$	−2.69531	−0.227795
$141$	0	0
$142$	10.3743	0.870590
$143$	−12.9231	−1.08068
$144$	0	0
$145$	−17.3461	−1.44051
$146$	−17.2116	−1.42444
$147$	0	0
$148$	2.17853	0.179074
$149$	−12.3798	−1.01419	−0.507097	−	0.861889i	$-0.669281\pi$
$149$	−12.3798	−1.01419	−0.507097	+	0.861889i	$0.669281\pi$
$150$	0	0
$151$	−12.4328	−1.01176	−0.505882	−	0.862603i	$-0.668833\pi$
$151$	−12.4328	−1.01176	−0.505882	+	0.862603i	$0.668833\pi$
$152$	8.47690	0.687567
$153$	0	0
$154$	−6.39530	−0.515348
$155$	−12.5930	−1.01150
$156$	0	0
$157$	−19.3845	−1.54705	−0.773524	−	0.633767i	$-0.781508\pi$
$157$	−19.3845	−1.54705	−0.773524	+	0.633767i	$0.781508\pi$
$158$	−16.0082	−1.27354
$159$	0	0
$160$	−14.6550	−1.15858
$161$	−5.73685	−0.452127
$162$	0	0
$163$	−3.46010	−0.271016	−0.135508	−	0.990776i	$-0.543267\pi$
$163$	−3.46010	−0.271016	−0.135508	+	0.990776i	$0.543267\pi$
$164$	−0.429426	−0.0335325
$165$	0	0
$166$	6.91029	0.536343
$167$	19.5272	1.51106	0.755531	−	0.655113i	$-0.227379\pi$
$167$	19.5272	1.51106	0.755531	+	0.655113i	$0.227379\pi$
$168$	0	0
$169$	−7.31318	−0.562552
$170$	12.3690	0.948659
$171$	0	0
$172$	−5.51288	−0.420353
$173$	19.1759	1.45792	0.728959	−	0.684558i	$-0.240005\pi$
$173$	19.1759	1.45792	0.728959	+	0.684558i	$0.240005\pi$
$174$	0	0
$175$	−14.6981	−1.11107
$176$	−13.0960	−0.987148
$177$	0	0
$178$	−6.81615	−0.510892
$179$	7.02739	0.525252	0.262626	−	0.964898i	$-0.415412\pi$
$179$	7.02739	0.525252	0.262626	+	0.964898i	$0.415412\pi$
$180$	0	0
$181$	−15.3645	−1.14203	−0.571017	−	0.820938i	$-0.693451\pi$
$181$	−15.3645	−1.14203	−0.571017	+	0.820938i	$0.693451\pi$
$182$	2.81427	0.208607
$183$	0	0
$184$	−17.6520	−1.30132
$185$	15.9213	1.17056
$186$	0	0
$187$	−12.7974	−0.935841
$188$	2.99324	0.218304
$189$	0	0
$190$	14.4298	1.04685
$191$	7.76445	0.561816	0.280908	−	0.959735i	$-0.409364\pi$
$191$	7.76445	0.561816	0.280908	+	0.959735i	$0.409364\pi$
$192$	0	0
$193$	−24.7433	−1.78106	−0.890529	−	0.454926i	$-0.849666\pi$
$193$	−24.7433	−1.78106	−0.890529	+	0.454926i	$0.849666\pi$
$194$	16.3539	1.17414
$195$	0	0
$196$	−0.607291	−0.0433779
$197$	10.5794	0.753754	0.376877	−	0.926263i	$-0.376998\pi$
$197$	10.5794	0.753754	0.376877	+	0.926263i	$0.376998\pi$
$198$	0	0
$199$	0.813046	0.0576353	0.0288177	−	0.999585i	$-0.490826\pi$
$199$	0.813046	0.0576353	0.0288177	+	0.999585i	$0.490826\pi$
$200$	−45.2253	−3.19791
$201$	0	0
$202$	2.13253	0.150044
$203$	−3.90830	−0.274309
$204$	0	0
$205$	−3.13837	−0.219193
$206$	−12.7744	−0.890035
$207$	0	0
$208$	5.76292	0.399587
$209$	−14.9296	−1.03270
$210$	0	0
$211$	−2.74396	−0.188902	−0.0944509	−	0.995530i	$-0.530110\pi$
$211$	−2.74396	−0.188902	−0.0944509	+	0.995530i	$0.530110\pi$
$212$	6.58232	0.452075
$213$	0	0
$214$	15.1295	1.03423
$215$	−40.2897	−2.74774
$216$	0	0
$217$	−2.83738	−0.192614
$218$	−11.7972	−0.799005
$219$	0	0
$220$	14.6063	0.984756
$221$	5.63153	0.378818
$222$	0	0
$223$	19.4867	1.30493	0.652464	−	0.757820i	$-0.273735\pi$
$223$	19.4867	1.30493	0.652464	+	0.757820i	$0.273735\pi$
$224$	−3.30196	−0.220622
$225$	0	0
$226$	9.83680	0.654334
$227$	−12.5024	−0.829813	−0.414906	−	0.909864i	$-0.636186\pi$
$227$	−12.5024	−0.829813	−0.414906	+	0.909864i	$0.636186\pi$
$228$	0	0
$229$	−25.3292	−1.67380	−0.836900	−	0.547356i	$-0.815634\pi$
$229$	−25.3292	−1.67380	−0.836900	+	0.547356i	$0.815634\pi$
$230$	−30.0480	−1.98131
$231$	0	0
$232$	−12.0256	−0.789521
$233$	9.91538	0.649578	0.324789	−	0.945786i	$-0.394707\pi$
$233$	9.91538	0.649578	0.324789	+	0.945786i	$0.394707\pi$
$234$	0	0
$235$	21.8755	1.42700
$236$	5.28491	0.344019
$237$	0	0
$238$	2.78691	0.180648
$239$	−18.4248	−1.19180	−0.595900	−	0.803058i	$-0.703205\pi$
$239$	−18.4248	−1.19180	−0.595900	+	0.803058i	$0.703205\pi$
$240$	0	0
$241$	8.87131	0.571451	0.285725	−	0.958312i	$-0.407765\pi$
$241$	8.87131	0.571451	0.285725	+	0.958312i	$0.407765\pi$
$242$	21.6756	1.39336
$243$	0	0
$244$	−9.39906	−0.601713
$245$	−4.43826	−0.283550
$246$	0	0
$247$	6.56980	0.418026
$248$	−8.73046	−0.554385
$249$	0	0
$250$	−50.7961	−3.21263
$251$	−0.561631	−0.0354498	−0.0177249	−	0.999843i	$-0.505642\pi$
$251$	−0.561631	−0.0354498	−0.0177249	+	0.999843i	$0.505642\pi$
$252$	0	0
$253$	31.0888	1.95454
$254$	−1.18013	−0.0740480
$255$	0	0
$256$	−13.0951	−0.818446
$257$	3.53641	0.220595	0.110298	−	0.993899i	$-0.464820\pi$
$257$	3.53641	0.220595	0.110298	+	0.993899i	$0.464820\pi$
$258$	0	0
$259$	3.58729	0.222904
$260$	−6.42753	−0.398618
$261$	0	0
$262$	10.7931	0.666801
$263$	10.8675	0.670118	0.335059	−	0.942197i	$-0.391244\pi$
$263$	10.8675	0.670118	0.335059	+	0.942197i	$0.391244\pi$
$264$	0	0
$265$	48.1055	2.95510
$266$	3.25123	0.199346
$267$	0	0
$268$	−3.26278	−0.199306
$269$	15.7001	0.957254	0.478627	−	0.878018i	$-0.341135\pi$
$269$	15.7001	0.957254	0.478627	+	0.878018i	$0.341135\pi$
$270$	0	0
$271$	−27.0181	−1.64123	−0.820617	−	0.571478i	$-0.806370\pi$
$271$	−27.0181	−1.64123	−0.820617	+	0.571478i	$0.806370\pi$
$272$	5.70689	0.346031
$273$	0	0
$274$	−16.4136	−0.991581
$275$	79.6513	4.80315
$276$	0	0
$277$	−20.7983	−1.24965	−0.624823	−	0.780766i	$-0.714829\pi$
$277$	−20.7983	−1.24965	−0.624823	+	0.780766i	$0.714829\pi$
$278$	−20.1017	−1.20562
$279$	0	0
$280$	−13.6563	−0.816118
$281$	−11.3243	−0.675554	−0.337777	−	0.941226i	$-0.609675\pi$
$281$	−11.3243	−0.675554	−0.337777	+	0.941226i	$0.609675\pi$
$282$	0	0
$283$	7.09116	0.421526	0.210763	−	0.977537i	$-0.432405\pi$
$283$	7.09116	0.421526	0.210763	+	0.977537i	$0.432405\pi$
$284$	−5.33856	−0.316785
$285$	0	0
$286$	−15.2509	−0.901805
$287$	−0.707118	−0.0417398
$288$	0	0
$289$	−11.4232	−0.671954
$290$	−20.4706	−1.20208
$291$	0	0
$292$	8.85701	0.518317
$293$	−1.07254	−0.0626582	−0.0313291	−	0.999509i	$-0.509974\pi$
$293$	−1.07254	−0.0626582	−0.0313291	+	0.999509i	$0.509974\pi$
$294$	0	0
$295$	38.6237	2.24876
$296$	11.0379	0.641565
$297$	0	0
$298$	−14.6098	−0.846324
$299$	−13.6807	−0.791175
$300$	0	0
$301$	−9.07783	−0.523237
$302$	−14.6723	−0.844296
$303$	0	0
$304$	6.65772	0.381846
$305$	−68.6911	−3.93324
$306$	0	0
$307$	18.1259	1.03450	0.517249	−	0.855835i	$-0.326956\pi$
$307$	18.1259	1.03450	0.517249	+	0.855835i	$0.326956\pi$
$308$	3.29100	0.187522
$309$	0	0
$310$	−14.8614	−0.844072
$311$	−6.03589	−0.342264	−0.171132	−	0.985248i	$-0.554742\pi$
$311$	−6.03589	−0.342264	−0.171132	+	0.985248i	$0.554742\pi$
$312$	0	0
$313$	16.9041	0.955475	0.477737	−	0.878503i	$-0.341457\pi$
$313$	16.9041	0.955475	0.477737	+	0.878503i	$0.341457\pi$
$314$	−22.8762	−1.29098
$315$	0	0
$316$	8.23774	0.463409
$317$	18.1372	1.01868	0.509342	−	0.860564i	$-0.329889\pi$
$317$	18.1372	1.01868	0.509342	+	0.860564i	$0.329889\pi$
$318$	0	0
$319$	21.1797	1.18583
$320$	−38.7459	−2.16596
$321$	0	0
$322$	−6.77024	−0.377290
$323$	6.50593	0.362000
$324$	0	0
$325$	−35.0507	−1.94426
$326$	−4.08337	−0.226157
$327$	0	0
$328$	−2.17576	−0.120136
$329$	4.92884	0.271736
$330$	0	0
$331$	−8.15730	−0.448366	−0.224183	−	0.974547i	$-0.571971\pi$
$331$	−8.15730	−0.448366	−0.224183	+	0.974547i	$0.571971\pi$
$332$	−3.55601	−0.195161
$333$	0	0
$334$	23.0447	1.26095
$335$	−23.8453	−1.30281
$336$	0	0
$337$	−9.35375	−0.509531	−0.254765	−	0.967003i	$-0.581998\pi$
$337$	−9.35375	−0.509531	−0.254765	+	0.967003i	$0.581998\pi$
$338$	−8.63051	−0.469438
$339$	0	0
$340$	−6.36504	−0.345193
$341$	15.3762	0.832667
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−27.9320	−1.50599
$345$	0	0
$346$	22.6301	1.21660
$347$	2.98500	0.160243	0.0801216	−	0.996785i	$-0.474469\pi$
$347$	2.98500	0.160243	0.0801216	+	0.996785i	$0.474469\pi$
$348$	0	0
$349$	30.6058	1.63829	0.819146	−	0.573585i	$-0.194448\pi$
$349$	30.6058	1.63829	0.819146	+	0.573585i	$0.194448\pi$
$350$	−17.3457	−0.927168
$351$	0	0
$352$	17.8938	0.953744
$353$	−6.64509	−0.353682	−0.176841	−	0.984239i	$-0.556588\pi$
$353$	−6.64509	−0.353682	−0.176841	+	0.984239i	$0.556588\pi$
$354$	0	0
$355$	−39.0158	−2.07074
$356$	3.50756	0.185900
$357$	0	0
$358$	8.29324	0.438311
$359$	32.2341	1.70125	0.850626	−	0.525772i	$-0.176223\pi$
$359$	32.2341	1.70125	0.850626	+	0.525772i	$0.176223\pi$
$360$	0	0
$361$	−11.4101	−0.600533
$362$	−18.1321	−0.953003
$363$	0	0
$364$	−1.44821	−0.0759068
$365$	64.7296	3.38810
$366$	0	0
$367$	12.1394	0.633670	0.316835	−	0.948481i	$-0.397380\pi$
$367$	12.1394	0.633670	0.316835	+	0.948481i	$0.397380\pi$
$368$	−13.8638	−0.722699
$369$	0	0
$370$	18.7893	0.976807
$371$	10.8388	0.562724
$372$	0	0
$373$	−32.5972	−1.68782	−0.843910	−	0.536485i	$-0.819752\pi$
$373$	−32.5972	−1.68782	−0.843910	+	0.536485i	$0.819752\pi$
$374$	−15.1026	−0.780939
$375$	0	0
$376$	15.1658	0.782115
$377$	−9.32015	−0.480012
$378$	0	0
$379$	21.7712	1.11831	0.559155	−	0.829063i	$-0.311126\pi$
$379$	21.7712	1.11831	0.559155	+	0.829063i	$0.311126\pi$
$380$	−7.42551	−0.380921
$381$	0	0
$382$	9.16307	0.468824
$383$	−3.86635	−0.197561	−0.0987806	−	0.995109i	$-0.531494\pi$
$383$	−3.86635	−0.197561	−0.0987806	+	0.995109i	$0.531494\pi$
$384$	0	0
$385$	24.0516	1.22578
$386$	−29.2003	−1.48626
$387$	0	0
$388$	−8.41564	−0.427239
$389$	−17.3609	−0.880230	−0.440115	−	0.897941i	$-0.645062\pi$
$389$	−17.3609	−0.880230	−0.440115	+	0.897941i	$0.645062\pi$
$390$	0	0
$391$	−13.5477	−0.685137
$392$	−3.07694	−0.155409
$393$	0	0
$394$	12.4851	0.628991
$395$	60.2038	3.02918
$396$	0	0
$397$	−36.5315	−1.83346	−0.916732	−	0.399502i	$-0.869183\pi$
$397$	−36.5315	−1.83346	−0.916732	+	0.399502i	$0.869183\pi$
$398$	0.959501	0.0480955
$399$	0	0
$400$	−35.5197	−1.77599
$401$	−15.3087	−0.764482	−0.382241	−	0.924063i	$-0.624848\pi$
$401$	−15.3087	−0.764482	−0.382241	+	0.924063i	$0.624848\pi$
$402$	0	0
$403$	−6.76632	−0.337054
$404$	−1.09739	−0.0545971
$405$	0	0
$406$	−4.61231	−0.228905
$407$	−19.4401	−0.963608
$408$	0	0
$409$	27.7504	1.37217	0.686083	−	0.727523i	$-0.259328\pi$
$409$	27.7504	1.37217	0.686083	+	0.727523i	$0.259328\pi$
$410$	−3.70369	−0.182912
$411$	0	0
$412$	6.57366	0.323861
$413$	8.70245	0.428219
$414$	0	0
$415$	−25.9883	−1.27572
$416$	−7.87421	−0.386065
$417$	0	0
$418$	−17.6189	−0.861768
$419$	7.51179	0.366975	0.183488	−	0.983022i	$-0.441261\pi$
$419$	7.51179	0.366975	0.183488	+	0.983022i	$0.441261\pi$
$420$	0	0
$421$	6.05425	0.295066	0.147533	−	0.989057i	$-0.452867\pi$
$421$	6.05425	0.295066	0.147533	+	0.989057i	$0.452867\pi$
$422$	−3.23823	−0.157634
$423$	0	0
$424$	33.3505	1.61964
$425$	−34.7100	−1.68368
$426$	0	0
$427$	−15.4770	−0.748987
$428$	−7.78557	−0.376330
$429$	0	0
$430$	−47.5472	−2.29293
$431$	−33.5629	−1.61667	−0.808334	−	0.588724i	$-0.799631\pi$
$431$	−33.5629	−1.61667	−0.808334	+	0.588724i	$0.799631\pi$
$432$	0	0
$433$	14.8488	0.713586	0.356793	−	0.934183i	$-0.383870\pi$
$433$	14.8488	0.713586	0.356793	+	0.934183i	$0.383870\pi$
$434$	−3.34848	−0.160732
$435$	0	0
$436$	6.07078	0.290737
$437$	−15.8049	−0.756049
$438$	0	0
$439$	18.5869	0.887103	0.443552	−	0.896249i	$-0.353718\pi$
$439$	18.5869	0.887103	0.443552	+	0.896249i	$0.353718\pi$
$440$	74.0053	3.52807
$441$	0	0
$442$	6.64595	0.316116
$443$	−2.21728	−0.105346	−0.0526730	−	0.998612i	$-0.516774\pi$
$443$	−2.21728	−0.105346	−0.0526730	+	0.998612i	$0.516774\pi$
$444$	0	0
$445$	25.6343	1.21518
$446$	22.9969	1.08893
$447$	0	0
$448$	−8.72998	−0.412453
$449$	−12.9587	−0.611558	−0.305779	−	0.952102i	$-0.598917\pi$
$449$	−12.9587	−0.611558	−0.305779	+	0.952102i	$0.598917\pi$
$450$	0	0
$451$	3.83197	0.180441
$452$	−5.06198	−0.238095
$453$	0	0
$454$	−14.7545	−0.692461
$455$	−10.5839	−0.496183
$456$	0	0
$457$	−36.1883	−1.69282	−0.846410	−	0.532532i	$-0.821240\pi$
$457$	−36.1883	−1.69282	−0.846410	+	0.532532i	$0.821240\pi$
$458$	−29.8918	−1.39675
$459$	0	0
$460$	15.4626	0.720948
$461$	−41.2133	−1.91949	−0.959747	−	0.280866i	$-0.909378\pi$
$461$	−41.2133	−1.91949	−0.959747	+	0.280866i	$0.909378\pi$
$462$	0	0
$463$	−39.1721	−1.82048	−0.910240	−	0.414080i	$-0.864103\pi$
$463$	−39.1721	−1.82048	−0.910240	+	0.414080i	$0.864103\pi$
$464$	−9.44487	−0.438467
$465$	0	0
$466$	11.7014	0.542059
$467$	27.5007	1.27258	0.636291	−	0.771449i	$-0.280468\pi$
$467$	27.5007	1.27258	0.636291	+	0.771449i	$0.280468\pi$
$468$	0	0
$469$	−5.37268	−0.248087
$470$	25.8159	1.19080
$471$	0	0
$472$	26.7769	1.23251
$473$	49.1941	2.26195
$474$	0	0
$475$	−40.4930	−1.85794
$476$	−1.43413	−0.0657332
$477$	0	0
$478$	−21.7437	−0.994532
$479$	2.25308	0.102946	0.0514728	−	0.998674i	$-0.483608\pi$
$479$	2.25308	0.102946	0.0514728	+	0.998674i	$0.483608\pi$
$480$	0	0
$481$	8.55464	0.390058
$482$	10.4693	0.476864
$483$	0	0
$484$	−11.1542	−0.507009
$485$	−61.5039	−2.79275
$486$	0	0
$487$	4.33373	0.196380	0.0981901	−	0.995168i	$-0.468695\pi$
$487$	4.33373	0.196380	0.0981901	+	0.995168i	$0.468695\pi$
$488$	−47.6220	−2.15575
$489$	0	0
$490$	−5.23773	−0.236616
$491$	3.53012	0.159312	0.0796560	−	0.996822i	$-0.474618\pi$
$491$	3.53012	0.159312	0.0796560	+	0.996822i	$0.474618\pi$
$492$	0	0
$493$	−9.22954	−0.415678
$494$	7.75323	0.348834
$495$	0	0
$496$	−6.85686	−0.307882
$497$	−8.79079	−0.394321
$498$	0	0
$499$	−19.5211	−0.873884	−0.436942	−	0.899490i	$-0.643939\pi$
$499$	−19.5211	−0.873884	−0.436942	+	0.899490i	$0.643939\pi$
$500$	26.1395	1.16899
$501$	0	0
$502$	−0.662798	−0.0295821
$503$	25.9921	1.15893	0.579465	−	0.814997i	$-0.303261\pi$
$503$	25.9921	1.15893	0.579465	+	0.814997i	$0.303261\pi$
$504$	0	0
$505$	−8.02004	−0.356887
$506$	36.6889	1.63102
$507$	0	0
$508$	0.607291	0.0269442
$509$	−39.4799	−1.74992	−0.874959	−	0.484198i	$-0.839112\pi$
$509$	−39.4799	−1.74992	−0.874959	+	0.484198i	$0.839112\pi$
$510$	0	0
$511$	14.5845	0.645179
$512$	−22.8512	−1.00989
$513$	0	0
$514$	4.17343	0.184082
$515$	48.0422	2.11699
$516$	0	0
$517$	−26.7101	−1.17471
$518$	4.23348	0.186008
$519$	0	0
$520$	−32.5662	−1.42812
$521$	10.6512	0.466639	0.233320	−	0.972400i	$-0.425041\pi$
$521$	10.6512	0.466639	0.233320	+	0.972400i	$0.425041\pi$
$522$	0	0
$523$	25.7574	1.12629	0.563147	−	0.826357i	$-0.309591\pi$
$523$	25.7574	1.12629	0.563147	+	0.826357i	$0.309591\pi$
$524$	−5.55409	−0.242632
$525$	0	0
$526$	12.8251	0.559199
$527$	−6.70054	−0.291880
$528$	0	0
$529$	9.91145	0.430933
$530$	56.7708	2.46597
$531$	0	0
$532$	−1.67307	−0.0725368
$533$	−1.68627	−0.0730404
$534$	0	0
$535$	−56.8992	−2.45997
$536$	−16.5314	−0.714049
$537$	0	0
$538$	18.5282	0.798808
$539$	5.41915	0.233419
$540$	0	0
$541$	−27.6386	−1.18828	−0.594138	−	0.804363i	$-0.702507\pi$
$541$	−27.6386	−1.18828	−0.594138	+	0.804363i	$0.702507\pi$
$542$	−31.8849	−1.36958
$543$	0	0
$544$	−7.79766	−0.334322
$545$	44.3670	1.90047
$546$	0	0
$547$	19.2255	0.822024	0.411012	−	0.911630i	$-0.365175\pi$
$547$	19.2255	0.822024	0.411012	+	0.911630i	$0.365175\pi$
$548$	8.44636	0.360811
$549$	0	0
$550$	93.9990	4.00813
$551$	−10.7673	−0.458701
$552$	0	0
$553$	13.5647	0.576832
$554$	−24.5447	−1.04280
$555$	0	0
$556$	10.3442	0.438694
$557$	29.8537	1.26494	0.632471	−	0.774584i	$-0.282041\pi$
$557$	29.8537	1.26494	0.632471	+	0.774584i	$0.282041\pi$
$558$	0	0
$559$	−21.6480	−0.915611
$560$	−10.7256	−0.453238
$561$	0	0
$562$	−13.3642	−0.563735
$563$	32.2423	1.35885	0.679425	−	0.733745i	$-0.262229\pi$
$563$	32.2423	1.35885	0.679425	+	0.733745i	$0.262229\pi$
$564$	0	0
$565$	−36.9944	−1.55637
$566$	8.36850	0.351754
$567$	0	0
$568$	−27.0488	−1.13494
$569$	−27.2451	−1.14217	−0.571086	−	0.820890i	$-0.693478\pi$
$569$	−27.2451	−1.14217	−0.571086	+	0.820890i	$0.693478\pi$
$570$	0	0
$571$	−23.3646	−0.977779	−0.488890	−	0.872346i	$-0.662598\pi$
$571$	−23.3646	−0.977779	−0.488890	+	0.872346i	$0.662598\pi$
$572$	7.84806	0.328144
$573$	0	0
$574$	−0.834492	−0.0348310
$575$	84.3210	3.51643
$576$	0	0
$577$	−14.4610	−0.602018	−0.301009	−	0.953621i	$-0.597323\pi$
$577$	−14.4610	−0.602018	−0.301009	+	0.953621i	$0.597323\pi$
$578$	−13.4809	−0.560731
$579$	0	0
$580$	10.5341	0.437404
$581$	−5.85553	−0.242928
$582$	0	0
$583$	−58.7372	−2.43265
$584$	44.8756	1.85697
$585$	0	0
$586$	−1.26573	−0.0522870
$587$	−15.5314	−0.641049	−0.320524	−	0.947240i	$-0.603859\pi$
$587$	−15.5314	−0.641049	−0.320524	+	0.947240i	$0.603859\pi$
$588$	0	0
$589$	−7.81691	−0.322090
$590$	45.5810	1.87654
$591$	0	0
$592$	8.66911	0.356298
$593$	−15.6281	−0.641769	−0.320884	−	0.947118i	$-0.603980\pi$
$593$	−15.6281	−0.641769	−0.320884	+	0.947118i	$0.603980\pi$
$594$	0	0
$595$	−10.4810	−0.429681
$596$	7.51815	0.307955
$597$	0	0
$598$	−16.1450	−0.660219
$599$	−8.04131	−0.328559	−0.164280	−	0.986414i	$-0.552530\pi$
$599$	−8.04131	−0.328559	−0.164280	+	0.986414i	$0.552530\pi$
$600$	0	0
$601$	−46.7643	−1.90755	−0.953777	−	0.300514i	$-0.902842\pi$
$601$	−46.7643	−1.90755	−0.953777	+	0.300514i	$0.902842\pi$
$602$	−10.7130	−0.436630
$603$	0	0
$604$	7.55030	0.307217
$605$	−81.5181	−3.31418
$606$	0	0
$607$	4.05313	0.164511	0.0822557	−	0.996611i	$-0.473788\pi$
$607$	4.05313	0.164511	0.0822557	+	0.996611i	$0.473788\pi$
$608$	−9.09682	−0.368925
$609$	0	0
$610$	−81.0645	−3.28221
$611$	11.7538	0.475509
$612$	0	0
$613$	7.60595	0.307201	0.153601	−	0.988133i	$-0.450913\pi$
$613$	7.60595	0.307201	0.153601	+	0.988133i	$0.450913\pi$
$614$	21.3909	0.863267
$615$	0	0
$616$	16.6744	0.671831
$617$	−23.6786	−0.953265	−0.476632	−	0.879103i	$-0.658143\pi$
$617$	−23.6786	−0.953265	−0.476632	+	0.879103i	$0.658143\pi$
$618$	0	0
$619$	30.7464	1.23580	0.617901	−	0.786256i	$-0.287983\pi$
$619$	30.7464	1.23580	0.617901	+	0.786256i	$0.287983\pi$
$620$	7.64763	0.307136
$621$	0	0
$622$	−7.12314	−0.285612
$623$	5.77576	0.231401
$624$	0	0
$625$	117.544	4.70177
$626$	19.9490	0.797323
$627$	0	0
$628$	11.7720	0.469754
$629$	8.47147	0.337780
$630$	0	0
$631$	−39.5487	−1.57441	−0.787205	−	0.616691i	$-0.788473\pi$
$631$	−39.5487	−1.57441	−0.787205	+	0.616691i	$0.788473\pi$
$632$	41.7380	1.66025
$633$	0	0
$634$	21.4042	0.850071
$635$	4.43826	0.176127
$636$	0	0
$637$	−2.38471	−0.0944855
$638$	24.9948	0.989553
$639$	0	0
$640$	−16.4153	−0.648873
$641$	21.3454	0.843091	0.421546	−	0.906807i	$-0.361488\pi$
$641$	21.3454	0.843091	0.421546	+	0.906807i	$0.361488\pi$
$642$	0	0
$643$	9.79675	0.386346	0.193173	−	0.981165i	$-0.438122\pi$
$643$	9.79675	0.386346	0.193173	+	0.981165i	$0.438122\pi$
$644$	3.48394	0.137286
$645$	0	0
$646$	7.67785	0.302081
$647$	−9.33818	−0.367122	−0.183561	−	0.983008i	$-0.558762\pi$
$647$	−9.33818	−0.367122	−0.183561	+	0.983008i	$0.558762\pi$
$648$	0	0
$649$	−47.1598	−1.85118
$650$	−41.3644	−1.62245
$651$	0	0
$652$	2.10129	0.0822927
$653$	−1.02438	−0.0400872	−0.0200436	−	0.999799i	$-0.506380\pi$
$653$	−1.02438	−0.0400872	−0.0200436	+	0.999799i	$0.506380\pi$
$654$	0	0
$655$	−40.5909	−1.58602
$656$	−1.70883	−0.0667187
$657$	0	0
$658$	5.81668	0.226758
$659$	18.9099	0.736623	0.368312	−	0.929702i	$-0.379936\pi$
$659$	18.9099	0.736623	0.368312	+	0.929702i	$0.379936\pi$
$660$	0	0
$661$	−27.3932	−1.06547	−0.532736	−	0.846282i	$-0.678836\pi$
$661$	−27.3932	−1.06547	−0.532736	+	0.846282i	$0.678836\pi$
$662$	−9.62668	−0.374152
$663$	0	0
$664$	−18.0171	−0.699200
$665$	−12.2273	−0.474154
$666$	0	0
$667$	22.4213	0.868158
$668$	−11.8587	−0.458827
$669$	0	0
$670$	−28.1406	−1.08717
$671$	83.8724	3.23786
$672$	0	0
$673$	11.6979	0.450920	0.225460	−	0.974252i	$-0.427611\pi$
$673$	11.6979	0.450920	0.225460	+	0.974252i	$0.427611\pi$
$674$	−11.0386	−0.425193
$675$	0	0
$676$	4.44122	0.170816
$677$	−12.2230	−0.469768	−0.234884	−	0.972023i	$-0.575471\pi$
$677$	−12.2230	−0.469768	−0.234884	+	0.972023i	$0.575471\pi$
$678$	0	0
$679$	−13.8577	−0.531809
$680$	−32.2496	−1.23672
$681$	0	0
$682$	18.1459	0.694843
$683$	45.0284	1.72296	0.861482	−	0.507788i	$-0.169537\pi$
$683$	45.0284	1.72296	0.861482	+	0.507788i	$0.169537\pi$
$684$	0	0
$685$	61.7285	2.35852
$686$	−1.18013	−0.0450576
$687$	0	0
$688$	−21.9376	−0.836365
$689$	25.8474	0.984708
$690$	0	0
$691$	18.8479	0.717009	0.358504	−	0.933528i	$-0.383287\pi$
$691$	18.8479	0.717009	0.358504	+	0.933528i	$0.383287\pi$
$692$	−11.6453	−0.442690
$693$	0	0
$694$	3.52269	0.133719
$695$	75.5988	2.86763
$696$	0	0
$697$	−1.66987	−0.0632510
$698$	36.1189	1.36712
$699$	0	0
$700$	8.92604	0.337372
$701$	6.02478	0.227553	0.113776	−	0.993506i	$-0.463705\pi$
$701$	6.02478	0.227553	0.113776	+	0.993506i	$0.463705\pi$
$702$	0	0
$703$	9.88290	0.372740
$704$	47.3091	1.78303
$705$	0	0
$706$	−7.84208	−0.295140
$707$	−1.80702	−0.0679601
$708$	0	0
$709$	2.99684	0.112549	0.0562744	−	0.998415i	$-0.482078\pi$
$709$	2.99684	0.112549	0.0562744	+	0.998415i	$0.482078\pi$
$710$	−46.0437	−1.72799
$711$	0	0
$712$	17.7717	0.666022
$713$	16.2776	0.609602
$714$	0	0
$715$	57.3559	2.14499
$716$	−4.26767	−0.159490
$717$	0	0
$718$	38.0405	1.41966
$719$	36.4922	1.36093	0.680465	−	0.732781i	$-0.261778\pi$
$719$	36.4922	1.36093	0.680465	+	0.732781i	$0.261778\pi$
$720$	0	0
$721$	10.8246	0.403128
$722$	−13.4654	−0.501132
$723$	0	0
$724$	9.33071	0.346773
$725$	57.4447	2.13344
$726$	0	0
$727$	8.36457	0.310225	0.155112	−	0.987897i	$-0.450426\pi$
$727$	8.36457	0.310225	0.155112	+	0.987897i	$0.450426\pi$
$728$	−7.33761	−0.271950
$729$	0	0
$730$	76.3895	2.82730
$731$	−21.4375	−0.792894
$732$	0	0
$733$	28.0240	1.03509	0.517545	−	0.855656i	$-0.326846\pi$
$733$	28.0240	1.03509	0.517545	+	0.855656i	$0.326846\pi$
$734$	14.3260	0.528784
$735$	0	0
$736$	18.9429	0.698244
$737$	29.1153	1.07248
$738$	0	0
$739$	−44.3664	−1.63204	−0.816022	−	0.578021i	$-0.803825\pi$
$739$	−44.3664	−1.63204	−0.816022	+	0.578021i	$0.803825\pi$
$740$	−9.66887	−0.355435
$741$	0	0
$742$	12.7912	0.469581
$743$	27.2520	0.999778	0.499889	−	0.866089i	$-0.333374\pi$
$743$	27.2520	0.999778	0.499889	+	0.866089i	$0.333374\pi$
$744$	0	0
$745$	54.9448	2.01302
$746$	−38.4690	−1.40845
$747$	0	0
$748$	7.77176	0.284164
$749$	−12.8202	−0.468439
$750$	0	0
$751$	20.4756	0.747167	0.373583	−	0.927597i	$-0.378129\pi$
$751$	20.4756	0.747167	0.373583	+	0.927597i	$0.378129\pi$
$752$	11.9111	0.434354
$753$	0	0
$754$	−10.9990	−0.400560
$755$	55.1798	2.00820
$756$	0	0
$757$	3.88978	0.141376	0.0706882	−	0.997498i	$-0.477480\pi$
$757$	3.88978	0.141376	0.0706882	+	0.997498i	$0.477480\pi$
$758$	25.6928	0.933206
$759$	0	0
$760$	−37.6227	−1.36472
$761$	−2.39284	−0.0867405	−0.0433702	−	0.999059i	$-0.513810\pi$
$761$	−2.39284	−0.0867405	−0.0433702	+	0.999059i	$0.513810\pi$
$762$	0	0
$763$	9.99649	0.361897
$764$	−4.71528	−0.170593
$765$	0	0
$766$	−4.56280	−0.164861
$767$	20.7528	0.749339
$768$	0	0
$769$	17.0826	0.616014	0.308007	−	0.951384i	$-0.400338\pi$
$769$	17.0826	0.616014	0.308007	+	0.951384i	$0.400338\pi$
$770$	28.3840	1.02289
$771$	0	0
$772$	15.0263	0.540810
$773$	−38.7362	−1.39324	−0.696621	−	0.717439i	$-0.745314\pi$
$773$	−38.7362	−1.39324	−0.696621	+	0.717439i	$0.745314\pi$
$774$	0	0
$775$	41.7042	1.49806
$776$	−42.6393	−1.53066
$777$	0	0
$778$	−20.4881	−0.734533
$779$	−1.94809	−0.0697976
$780$	0	0
$781$	47.6385	1.70464
$782$	−15.9881	−0.571732
$783$	0	0
$784$	−2.41662	−0.0863077
$785$	86.0332	3.07066
$786$	0	0
$787$	13.3375	0.475430	0.237715	−	0.971335i	$-0.423602\pi$
$787$	13.3375	0.475430	0.237715	+	0.971335i	$0.423602\pi$
$788$	−6.42479	−0.228874
$789$	0	0
$790$	71.0484	2.52779
$791$	−8.33535	−0.296371
$792$	0	0
$793$	−36.9082	−1.31065
$794$	−43.1120	−1.52999
$795$	0	0
$796$	−0.493755	−0.0175007
$797$	5.73847	0.203267	0.101633	−	0.994822i	$-0.467593\pi$
$797$	5.73847	0.203267	0.101633	+	0.994822i	$0.467593\pi$
$798$	0	0
$799$	11.6396	0.411778
$800$	48.5327	1.71589
$801$	0	0
$802$	−18.0663	−0.637944
$803$	−79.0354	−2.78910
$804$	0	0
$805$	25.4616	0.897404
$806$	−7.98514	−0.281265
$807$	0	0
$808$	−5.56011	−0.195604
$809$	−32.9630	−1.15892	−0.579458	−	0.815002i	$-0.696736\pi$
$809$	−32.9630	−1.15892	−0.579458	+	0.815002i	$0.696736\pi$
$810$	0	0
$811$	35.8058	1.25731	0.628656	−	0.777684i	$-0.283605\pi$
$811$	35.8058	1.25731	0.628656	+	0.777684i	$0.283605\pi$
$812$	2.37348	0.0832927
$813$	0	0
$814$	−22.9418	−0.804110
$815$	15.3568	0.537926
$816$	0	0
$817$	−25.0092	−0.874960
$818$	32.7491	1.14504
$819$	0	0
$820$	1.90590	0.0665570
$821$	45.5266	1.58889	0.794445	−	0.607336i	$-0.207762\pi$
$821$	45.5266	1.58889	0.794445	+	0.607336i	$0.207762\pi$
$822$	0	0
$823$	4.84441	0.168866	0.0844328	−	0.996429i	$-0.473092\pi$
$823$	4.84441	0.168866	0.0844328	+	0.996429i	$0.473092\pi$
$824$	33.3066	1.16029
$825$	0	0
$826$	10.2700	0.357340
$827$	−29.6775	−1.03199	−0.515993	−	0.856593i	$-0.672577\pi$
$827$	−29.6775	−1.03199	−0.515993	+	0.856593i	$0.672577\pi$
$828$	0	0
$829$	−38.9206	−1.35177	−0.675884	−	0.737008i	$-0.736238\pi$
$829$	−38.9206	−1.35177	−0.675884	+	0.737008i	$0.736238\pi$
$830$	−30.6696	−1.06456
$831$	0	0
$832$	−20.8184	−0.721750
$833$	−2.36152	−0.0818219
$834$	0	0
$835$	−86.6668	−2.99923
$836$	9.06661	0.313575
$837$	0	0
$838$	8.86490	0.306233
$839$	−11.2390	−0.388015	−0.194007	−	0.981000i	$-0.562149\pi$
$839$	−11.2390	−0.388015	−0.194007	+	0.981000i	$0.562149\pi$
$840$	0	0
$841$	−13.7252	−0.473282
$842$	7.14480	0.246226
$843$	0	0
$844$	1.66638	0.0573591
$845$	32.4578	1.11658
$846$	0	0
$847$	−18.3671	−0.631102
$848$	26.1933	0.899481
$849$	0	0
$850$	−40.9623	−1.40500
$851$	−20.5798	−0.705465
$852$	0	0
$853$	−3.28040	−0.112319	−0.0561594	−	0.998422i	$-0.517885\pi$
$853$	−3.28040	−0.112319	−0.0561594	+	0.998422i	$0.517885\pi$
$854$	−18.2649	−0.625014
$855$	0	0
$856$	−39.4470	−1.34827
$857$	18.3698	0.627499	0.313750	−	0.949506i	$-0.398415\pi$
$857$	18.3698	0.627499	0.313750	+	0.949506i	$0.398415\pi$
$858$	0	0
$859$	−39.9179	−1.36198	−0.680991	−	0.732292i	$-0.738451\pi$
$859$	−39.9179	−1.36198	−0.680991	+	0.732292i	$0.738451\pi$
$860$	24.4676	0.834338
$861$	0	0
$862$	−39.6086	−1.34908
$863$	−2.21429	−0.0753753	−0.0376876	−	0.999290i	$-0.511999\pi$
$863$	−2.21429	−0.0753753	−0.0376876	+	0.999290i	$0.511999\pi$
$864$	0	0
$865$	−85.1076	−2.89375
$866$	17.5235	0.595472
$867$	0	0
$868$	1.72311	0.0584863
$869$	−73.5093	−2.49363
$870$	0	0
$871$	−12.8123	−0.434127
$872$	30.7586	1.04162
$873$	0	0
$874$	−18.6518	−0.630907
$875$	43.0428	1.45511
$876$	0	0
$877$	−34.7091	−1.17204	−0.586021	−	0.810296i	$-0.699306\pi$
$877$	−34.7091	−1.17204	−0.586021	+	0.810296i	$0.699306\pi$
$878$	21.9349	0.740269
$879$	0	0
$880$	58.1234	1.95934
$881$	−0.174959	−0.00589451	−0.00294725	−	0.999996i	$-0.500938\pi$
$881$	−0.174959	−0.00589451	−0.00294725	+	0.999996i	$0.500938\pi$
$882$	0	0
$883$	47.3602	1.59380	0.796899	−	0.604113i	$-0.206473\pi$
$883$	47.3602	1.59380	0.796899	+	0.604113i	$0.206473\pi$
$884$	−3.41998	−0.115026
$885$	0	0
$886$	−2.61668	−0.0879090
$887$	−40.2851	−1.35264	−0.676321	−	0.736607i	$-0.736427\pi$
$887$	−40.2851	−1.35264	−0.676321	+	0.736607i	$0.736427\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	30.2518	1.01404
$891$	0	0
$892$	−11.8341	−0.396235
$893$	13.5788	0.454398
$894$	0	0
$895$	−31.1894	−1.04255
$896$	−3.69860	−0.123561
$897$	0	0
$898$	−15.2929	−0.510332
$899$	11.0893	0.369850
$900$	0	0
$901$	25.5961	0.852731
$902$	4.52223	0.150574
$903$	0	0
$904$	−25.6474	−0.853020
$905$	68.1916	2.26677
$906$	0	0
$907$	−16.1755	−0.537100	−0.268550	−	0.963266i	$-0.586544\pi$
$907$	−16.1755	−0.537100	−0.268550	+	0.963266i	$0.586544\pi$
$908$	7.59259	0.251969
$909$	0	0
$910$	−12.4904	−0.414054
$911$	5.34893	0.177218	0.0886090	−	0.996066i	$-0.471758\pi$
$911$	5.34893	0.177218	0.0886090	+	0.996066i	$0.471758\pi$
$912$	0	0
$913$	31.7320	1.05017
$914$	−42.7070	−1.41262
$915$	0	0
$916$	15.3822	0.508242
$917$	−9.14569	−0.302017
$918$	0	0
$919$	−26.5607	−0.876156	−0.438078	−	0.898937i	$-0.644341\pi$
$919$	−26.5607	−0.876156	−0.438078	+	0.898937i	$0.644341\pi$
$920$	78.3440	2.58292
$921$	0	0
$922$	−48.6371	−1.60178
$923$	−20.9634	−0.690020
$924$	0	0
$925$	−52.7265	−1.73364
$926$	−46.2282	−1.51915
$927$	0	0
$928$	12.9051	0.423630
$929$	−42.5323	−1.39544	−0.697719	−	0.716371i	$-0.745802\pi$
$929$	−42.5323	−1.39544	−0.697719	+	0.716371i	$0.745802\pi$
$930$	0	0
$931$	−2.75497	−0.0902906
$932$	−6.02152	−0.197241
$933$	0	0
$934$	32.4545	1.06194
$935$	56.7983	1.85750
$936$	0	0
$937$	8.57209	0.280038	0.140019	−	0.990149i	$-0.455284\pi$
$937$	8.57209	0.280038	0.140019	+	0.990149i	$0.455284\pi$
$938$	−6.34047	−0.207024
$939$	0	0
$940$	−13.2848	−0.433301
$941$	−30.0271	−0.978857	−0.489428	−	0.872044i	$-0.662795\pi$
$941$	−30.0271	−0.978857	−0.489428	+	0.872044i	$0.662795\pi$
$942$	0	0
$943$	4.05663	0.132102
$944$	21.0305	0.684484
$945$	0	0
$946$	58.0555	1.88755
$947$	−27.6070	−0.897108	−0.448554	−	0.893756i	$-0.648061\pi$
$947$	−27.6070	−0.897108	−0.448554	+	0.893756i	$0.648061\pi$
$948$	0	0
$949$	34.7797	1.12900
$950$	−47.7870	−1.55041
$951$	0	0
$952$	−7.26627	−0.235501
$953$	−56.3567	−1.82557	−0.912786	−	0.408437i	$-0.866074\pi$
$953$	−56.3567	−1.82557	−0.912786	+	0.408437i	$0.866074\pi$
$954$	0	0
$955$	−34.4606	−1.11512
$956$	11.1892	0.361885
$957$	0	0
$958$	2.65893	0.0859060
$959$	13.9083	0.449121
$960$	0	0
$961$	−22.9493	−0.740299
$962$	10.0956	0.325495
$963$	0	0
$964$	−5.38746	−0.173518
$965$	109.817	3.53513
$966$	0	0
$967$	−10.0649	−0.323664	−0.161832	−	0.986818i	$-0.551740\pi$
$967$	−10.0649	−0.323664	−0.161832	+	0.986818i	$0.551740\pi$
$968$	−56.5147	−1.81645
$969$	0	0
$970$	−72.5827	−2.33049
$971$	−47.6227	−1.52829	−0.764143	−	0.645047i	$-0.776838\pi$
$971$	−47.6227	−1.52829	−0.764143	+	0.645047i	$0.776838\pi$
$972$	0	0
$973$	17.0334	0.546067
$974$	5.11437	0.163875
$975$	0	0
$976$	−37.4021	−1.19721
$977$	2.84827	0.0911243	0.0455621	−	0.998962i	$-0.485492\pi$
$977$	2.84827	0.0911243	0.0455621	+	0.998962i	$0.485492\pi$
$978$	0	0
$979$	−31.2997	−1.00034
$980$	2.69531	0.0860986
$981$	0	0
$982$	4.16600	0.132942
$983$	37.5274	1.19694	0.598470	−	0.801146i	$-0.295776\pi$
$983$	37.5274	1.19694	0.598470	+	0.801146i	$0.295776\pi$
$984$	0	0
$985$	−46.9543	−1.49609
$986$	−10.8921	−0.346874
$987$	0	0
$988$	−3.98978	−0.126932
$989$	52.0782	1.65599
$990$	0	0
$991$	−42.7222	−1.35711	−0.678557	−	0.734547i	$-0.737395\pi$
$991$	−42.7222	−1.35711	−0.678557	+	0.734547i	$0.737395\pi$
$992$	9.36893	0.297464
$993$	0	0
$994$	−10.3743	−0.329052
$995$	−3.60851	−0.114397
$996$	0	0
$997$	3.76493	0.119237	0.0596183	−	0.998221i	$-0.481012\pi$
$997$	3.76493	0.119237	0.0596183	+	0.998221i	$0.481012\pi$
$998$	−23.0374	−0.729238
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.23	yes	32
3.2	odd	2	inner	8001.2.a.z.1.10	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.10	✓	32	3.2	odd	2	inner
8001.2.a.z.1.23	yes	32	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18013 q^{2} -0.607291 q^{4} -4.43826 q^{5} -1.00000 q^{7} -3.07694 q^{8} +O(q^{10})\)	\(q+1.18013 q^{2} -0.607291 q^{4} -4.43826 q^{5} -1.00000 q^{7} -3.07694 q^{8} -5.23773 q^{10} +5.41915 q^{11} -2.38471 q^{13} -1.18013 q^{14} -2.41662 q^{16} -2.36152 q^{17} -2.75497 q^{19} +2.69531 q^{20} +6.39530 q^{22} +5.73685 q^{23} +14.6981 q^{25} -2.81427 q^{26} +0.607291 q^{28} +3.90830 q^{29} +2.83738 q^{31} +3.30196 q^{32} -2.78691 q^{34} +4.43826 q^{35} -3.58729 q^{37} -3.25123 q^{38} +13.6563 q^{40} +0.707118 q^{41} +9.07783 q^{43} -3.29100 q^{44} +6.77024 q^{46} -4.92884 q^{47} +1.00000 q^{49} +17.3457 q^{50} +1.44821 q^{52} -10.8388 q^{53} -24.0516 q^{55} +3.07694 q^{56} +4.61231 q^{58} -8.70245 q^{59} +15.4770 q^{61} +3.34848 q^{62} +8.72998 q^{64} +10.5839 q^{65} +5.37268 q^{67} +1.43413 q^{68} +5.23773 q^{70} +8.79079 q^{71} -14.5845 q^{73} -4.23348 q^{74} +1.67307 q^{76} -5.41915 q^{77} -13.5647 q^{79} +10.7256 q^{80} +0.834492 q^{82} +5.85553 q^{83} +10.4810 q^{85} +10.7130 q^{86} -16.6744 q^{88} -5.77576 q^{89} +2.38471 q^{91} -3.48394 q^{92} -5.81668 q^{94} +12.2273 q^{95} +13.8577 q^{97} +1.18013 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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