LMFDB - Embedded newform 1682.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1682 = 2 \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.32836640625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} - 18x^{6} + 17x^{5} + 95x^{4} - 77x^{3} - 128x^{2} + 51x + 31$ x^8 - x^7 - 18x^6 + 17x^5 + 95x^4 - 77x^3 - 128x^2 + 51x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.88972$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.88972 q^{3} +1.00000 q^{4} -2.81297 q^{5} +2.88972 q^{6} +3.85101 q^{7} -1.00000 q^{8} +5.35050 q^{9} +2.81297 q^{10} -2.90591 q^{11} -2.88972 q^{12} -0.364636 q^{13} -3.85101 q^{14} +8.12870 q^{15} +1.00000 q^{16} +0.925336 q^{17} -5.35050 q^{18} +5.15058 q^{19} -2.81297 q^{20} -11.1284 q^{21} +2.90591 q^{22} -2.53872 q^{23} +2.88972 q^{24} +2.91279 q^{25} +0.364636 q^{26} -6.79231 q^{27} +3.85101 q^{28} -8.12870 q^{30} -8.03616 q^{31} -1.00000 q^{32} +8.39729 q^{33} -0.925336 q^{34} -10.8328 q^{35} +5.35050 q^{36} +3.62923 q^{37} -5.15058 q^{38} +1.05370 q^{39} +2.81297 q^{40} -4.99001 q^{41} +11.1284 q^{42} -8.55107 q^{43} -2.90591 q^{44} -15.0508 q^{45} +2.53872 q^{46} +7.50775 q^{47} -2.88972 q^{48} +7.83031 q^{49} -2.91279 q^{50} -2.67397 q^{51} -0.364636 q^{52} -9.16362 q^{53} +6.79231 q^{54} +8.17425 q^{55} -3.85101 q^{56} -14.8838 q^{57} -4.66605 q^{59} +8.12870 q^{60} -6.85219 q^{61} +8.03616 q^{62} +20.6049 q^{63} +1.00000 q^{64} +1.02571 q^{65} -8.39729 q^{66} -1.40270 q^{67} +0.925336 q^{68} +7.33619 q^{69} +10.8328 q^{70} -4.31712 q^{71} -5.35050 q^{72} +12.5974 q^{73} -3.62923 q^{74} -8.41716 q^{75} +5.15058 q^{76} -11.1907 q^{77} -1.05370 q^{78} +12.0164 q^{79} -2.81297 q^{80} +3.57638 q^{81} +4.99001 q^{82} +14.5760 q^{83} -11.1284 q^{84} -2.60294 q^{85} +8.55107 q^{86} +2.90591 q^{88} +11.4600 q^{89} +15.0508 q^{90} -1.40422 q^{91} -2.53872 q^{92} +23.2223 q^{93} -7.50775 q^{94} -14.4884 q^{95} +2.88972 q^{96} -12.4026 q^{97} -7.83031 q^{98} -15.5481 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} + q^{6} + 7 q^{7} - 8 q^{8} + 13 q^{9} - 5 q^{10} + 7 q^{11} - q^{12} + 13 q^{13} - 7 q^{14} + 20 q^{15} + 8 q^{16} - 9 q^{17} - 13 q^{18} + 13 q^{19} + 5 q^{20}+ \cdots + 2 q^{99}+O(q^{100})$ 8 * q - 8 * q^2 - q^3 + 8 * q^4 + 5 * q^5 + q^6 + 7 * q^7 - 8 * q^8 + 13 * q^9 - 5 * q^10 + 7 * q^11 - q^12 + 13 * q^13 - 7 * q^14 + 20 * q^15 + 8 * q^16 - 9 * q^17 - 13 * q^18 + 13 * q^19 + 5 * q^20 - 34 * q^21 - 7 * q^22 + 12 * q^23 + q^24 + 45 * q^25 - 13 * q^26 + 2 * q^27 + 7 * q^28 - 20 * q^30 - 15 * q^31 - 8 * q^32 - 4 * q^33 + 9 * q^34 + 13 * q^36 - 4 * q^37 - 13 * q^38 + 9 * q^39 - 5 * q^40 - 8 * q^41 + 34 * q^42 - 12 * q^43 + 7 * q^44 + 30 * q^45 - 12 * q^46 + 13 * q^47 - q^48 + 27 * q^49 - 45 * q^50 - 42 * q^51 + 13 * q^52 + 4 * q^53 - 2 * q^54 + 35 * q^55 - 7 * q^56 - 11 * q^57 + 8 * q^59 + 20 * q^60 - 9 * q^61 + 15 * q^62 + 12 * q^63 + 8 * q^64 - 15 * q^65 + 4 * q^66 + 34 * q^67 - 9 * q^68 - 4 * q^69 - 11 * q^71 - 13 * q^72 + 13 * q^73 + 4 * q^74 + 15 * q^75 + 13 * q^76 + 23 * q^77 - 9 * q^78 + 8 * q^79 + 5 * q^80 + 12 * q^81 + 8 * q^82 + 10 * q^83 - 34 * q^84 + 15 * q^85 + 12 * q^86 - 7 * q^88 - 6 * q^89 - 30 * q^90 + 12 * q^91 + 12 * q^92 + 15 * q^93 - 13 * q^94 + 15 * q^95 + q^96 + 11 * q^97 - 27 * q^98 + 2 * q^99

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
$3$	−2.88972	−1.66838	−0.834191	−	0.551475i	$-0.814065\pi$
$3$	−2.88972	−1.66838	−0.834191	+	0.551475i	$0.814065\pi$
$4$	1.00000	0.500000
$5$	−2.81297	−1.25800	−0.628999	−	0.777406i	$-0.716535\pi$
$5$	−2.81297	−1.25800	−0.628999	+	0.777406i	$0.716535\pi$
$6$	2.88972	1.17972
$7$	3.85101	1.45555	0.727773	−	0.685818i	$-0.240555\pi$
$7$	3.85101	1.45555	0.727773	+	0.685818i	$0.240555\pi$
$8$	−1.00000	−0.353553
$9$	5.35050	1.78350
$10$	2.81297	0.889539
$11$	−2.90591	−0.876166	−0.438083	−	0.898934i	$-0.644342\pi$
$11$	−2.90591	−0.876166	−0.438083	+	0.898934i	$0.644342\pi$
$12$	−2.88972	−0.834191
$13$	−0.364636	−0.101132	−0.0505660	−	0.998721i	$-0.516103\pi$
$13$	−0.364636	−0.101132	−0.0505660	+	0.998721i	$0.516103\pi$
$14$	−3.85101	−1.02923
$15$	8.12870	2.09882
$16$	1.00000	0.250000
$17$	0.925336	0.224427	0.112214	−	0.993684i	$-0.464206\pi$
$17$	0.925336	0.224427	0.112214	+	0.993684i	$0.464206\pi$
$18$	−5.35050	−1.26113
$19$	5.15058	1.18162	0.590812	−	0.806809i	$-0.298807\pi$
$19$	5.15058	1.18162	0.590812	+	0.806809i	$0.298807\pi$
$20$	−2.81297	−0.628999
$21$	−11.1284	−2.42841
$22$	2.90591	0.619543
$23$	−2.53872	−0.529359	−0.264680	−	0.964336i	$-0.585266\pi$
$23$	−2.53872	−0.529359	−0.264680	+	0.964336i	$0.585266\pi$
$24$	2.88972	0.589862
$25$	2.91279	0.582558
$26$	0.364636	0.0715111
$27$	−6.79231	−1.30718
$28$	3.85101	0.727773
$29$	0	0
$29$	0	0
$30$	−8.12870	−1.48409
$31$	−8.03616	−1.44334	−0.721669	−	0.692239i	$-0.756625\pi$
$31$	−8.03616	−1.44334	−0.721669	+	0.692239i	$0.756625\pi$
$32$	−1.00000	−0.176777
$33$	8.39729	1.46178
$34$	−0.925336	−0.158694
$35$	−10.8328	−1.83107
$36$	5.35050	0.891751
$37$	3.62923	0.596642	0.298321	−	0.954466i	$-0.403573\pi$
$37$	3.62923	0.596642	0.298321	+	0.954466i	$0.403573\pi$
$38$	−5.15058	−0.835535
$39$	1.05370	0.168727
$40$	2.81297	0.444769
$41$	−4.99001	−0.779308	−0.389654	−	0.920961i	$-0.627405\pi$
$41$	−4.99001	−0.779308	−0.389654	+	0.920961i	$0.627405\pi$
$42$	11.1284	1.71714
$43$	−8.55107	−1.30403	−0.652013	−	0.758208i	$-0.726075\pi$
$43$	−8.55107	−1.30403	−0.652013	+	0.758208i	$0.726075\pi$
$44$	−2.90591	−0.438083
$45$	−15.0508	−2.24364
$46$	2.53872	0.374313
$47$	7.50775	1.09512	0.547559	−	0.836767i	$-0.315557\pi$
$47$	7.50775	1.09512	0.547559	+	0.836767i	$0.315557\pi$
$48$	−2.88972	−0.417096
$49$	7.83031	1.11862
$50$	−2.91279	−0.411931
$51$	−2.67397	−0.374430
$52$	−0.364636	−0.0505660
$53$	−9.16362	−1.25872	−0.629360	−	0.777114i	$-0.716683\pi$
$53$	−9.16362	−1.25872	−0.629360	+	0.777114i	$0.716683\pi$
$54$	6.79231	0.924316
$55$	8.17425	1.10222
$56$	−3.85101	−0.514613
$57$	−14.8838	−1.97140
$58$	0	0
$59$	−4.66605	−0.607468	−0.303734	−	0.952757i	$-0.598233\pi$
$59$	−4.66605	−0.607468	−0.303734	+	0.952757i	$0.598233\pi$
$60$	8.12870	1.04941
$61$	−6.85219	−0.877334	−0.438667	−	0.898650i	$-0.644549\pi$
$61$	−6.85219	−0.877334	−0.438667	+	0.898650i	$0.644549\pi$
$62$	8.03616	1.02059
$63$	20.6049	2.59597
$64$	1.00000	0.125000
$65$	1.02571	0.127224
$66$	−8.39729	−1.03364
$67$	−1.40270	−0.171367	−0.0856834	−	0.996322i	$-0.527307\pi$
$67$	−1.40270	−0.171367	−0.0856834	+	0.996322i	$0.527307\pi$
$68$	0.925336	0.112214
$69$	7.33619	0.883174
$70$	10.8328	1.29477
$71$	−4.31712	−0.512348	−0.256174	−	0.966631i	$-0.582462\pi$
$71$	−4.31712	−0.512348	−0.256174	+	0.966631i	$0.582462\pi$
$72$	−5.35050	−0.630563
$73$	12.5974	1.47442	0.737208	−	0.675666i	$-0.236144\pi$
$73$	12.5974	1.47442	0.737208	+	0.675666i	$0.236144\pi$
$74$	−3.62923	−0.421889
$75$	−8.41716	−0.971930
$76$	5.15058	0.590812
$77$	−11.1907	−1.27530
$78$	−1.05370	−0.119308
$79$	12.0164	1.35195	0.675976	−	0.736923i	$-0.263722\pi$
$79$	12.0164	1.35195	0.675976	+	0.736923i	$0.263722\pi$
$80$	−2.81297	−0.314499
$81$	3.57638	0.397376
$82$	4.99001	0.551054
$83$	14.5760	1.59992	0.799960	−	0.600053i	$-0.204854\pi$
$83$	14.5760	1.59992	0.799960	+	0.600053i	$0.204854\pi$
$84$	−11.1284	−1.21420
$85$	−2.60294	−0.282329
$86$	8.55107	0.922085
$87$	0	0
$88$	2.90591	0.309772
$89$	11.4600	1.21475	0.607376	−	0.794414i	$-0.292222\pi$
$89$	11.4600	1.21475	0.607376	+	0.794414i	$0.292222\pi$
$90$	15.0508	1.58649
$91$	−1.40422	−0.147202
$92$	−2.53872	−0.264680
$93$	23.2223	2.40804
$94$	−7.50775	−0.774365
$95$	−14.4884	−1.48648
$96$	2.88972	0.294931
$97$	−12.4026	−1.25930	−0.629648	−	0.776880i	$-0.716801\pi$
$97$	−12.4026	−1.25930	−0.629648	+	0.776880i	$0.716801\pi$
$98$	−7.83031	−0.790981
$99$	−15.5481	−1.56264
$100$	2.91279	0.291279
$101$	7.17961	0.714397	0.357199	−	0.934028i	$-0.383732\pi$
$101$	7.17961	0.714397	0.357199	+	0.934028i	$0.383732\pi$
$102$	2.67397	0.264762
$103$	16.0746	1.58387	0.791937	−	0.610603i	$-0.209073\pi$
$103$	16.0746	1.58387	0.791937	+	0.610603i	$0.209073\pi$
$104$	0.364636	0.0357555
$105$	31.3037	3.05493
$106$	9.16362	0.890049
$107$	11.7282	1.13381	0.566906	−	0.823783i	$-0.308140\pi$
$107$	11.7282	1.13381	0.566906	+	0.823783i	$0.308140\pi$
$108$	−6.79231	−0.653590
$109$	0.0291942	0.00279630	0.00139815	−	0.999999i	$-0.499555\pi$
$109$	0.0291942	0.00279630	0.00139815	+	0.999999i	$0.499555\pi$
$110$	−8.17425	−0.779384
$111$	−10.4875	−0.995427
$112$	3.85101	0.363887
$113$	1.64135	0.154405	0.0772026	−	0.997015i	$-0.475401\pi$
$113$	1.64135	0.154405	0.0772026	+	0.997015i	$0.475401\pi$
$114$	14.8838	1.39399
$115$	7.14133	0.665932
$116$	0	0
$117$	−1.95099	−0.180369
$118$	4.66605	0.429544
$119$	3.56348	0.326664
$120$	−8.12870	−0.742046
$121$	−2.55566	−0.232333
$122$	6.85219	0.620368
$123$	14.4197	1.30018
$124$	−8.03616	−0.721669
$125$	5.87126	0.525141
$126$	−20.6049	−1.83563
$127$	16.2246	1.43970	0.719849	−	0.694130i	$-0.244211\pi$
$127$	16.2246	1.43970	0.719849	+	0.694130i	$0.244211\pi$
$128$	−1.00000	−0.0883883
$129$	24.7102	2.17561
$130$	−1.02571	−0.0899608
$131$	−13.3119	−1.16306	−0.581532	−	0.813524i	$-0.697546\pi$
$131$	−13.3119	−1.16306	−0.581532	+	0.813524i	$0.697546\pi$
$132$	8.39729	0.730890
$133$	19.8350	1.71991
$134$	1.40270	0.121175
$135$	19.1065	1.64443
$136$	−0.925336	−0.0793469
$137$	19.6985	1.68296	0.841479	−	0.540290i	$-0.181685\pi$
$137$	19.6985	1.68296	0.841479	+	0.540290i	$0.181685\pi$
$138$	−7.33619	−0.624498
$139$	7.91712	0.671522	0.335761	−	0.941947i	$-0.391007\pi$
$139$	7.91712	0.671522	0.335761	+	0.941947i	$0.391007\pi$
$140$	−10.8328	−0.915537
$141$	−21.6953	−1.82707
$142$	4.31712	0.362285
$143$	1.05960	0.0886084
$144$	5.35050	0.445875
$145$	0	0
$146$	−12.5974	−1.04257
$147$	−22.6274	−1.86628
$148$	3.62923	0.298321
$149$	−6.66087	−0.545680	−0.272840	−	0.962059i	$-0.587963\pi$
$149$	−6.66087	−0.545680	−0.272840	+	0.962059i	$0.587963\pi$
$150$	8.41716	0.687258
$151$	10.8305	0.881376	0.440688	−	0.897660i	$-0.354734\pi$
$151$	10.8305	0.881376	0.440688	+	0.897660i	$0.354734\pi$
$152$	−5.15058	−0.417768
$153$	4.95102	0.400266
$154$	11.1907	0.901774
$155$	22.6055	1.81572
$156$	1.05370	0.0843634
$157$	14.8837	1.18785	0.593924	−	0.804521i	$-0.297578\pi$
$157$	14.8837	1.18785	0.593924	+	0.804521i	$0.297578\pi$
$158$	−12.0164	−0.955975
$159$	26.4803	2.10003
$160$	2.81297	0.222385
$161$	−9.77664	−0.770507
$162$	−3.57638	−0.280987
$163$	−12.4608	−0.976006	−0.488003	−	0.872842i	$-0.662274\pi$
$163$	−12.4608	−0.976006	−0.488003	+	0.872842i	$0.662274\pi$
$164$	−4.99001	−0.389654
$165$	−23.6213	−1.83892
$166$	−14.5760	−1.13131
$167$	−4.31880	−0.334199	−0.167099	−	0.985940i	$-0.553440\pi$
$167$	−4.31880	−0.334199	−0.167099	+	0.985940i	$0.553440\pi$
$168$	11.1284	0.858572
$169$	−12.8670	−0.989772
$170$	2.60294	0.199636
$171$	27.5582	2.10743
$172$	−8.55107	−0.652013
$173$	−21.3247	−1.62129	−0.810644	−	0.585540i	$-0.800883\pi$
$173$	−21.3247	−1.62129	−0.810644	+	0.585540i	$0.800883\pi$
$174$	0	0
$175$	11.2172	0.847940
$176$	−2.90591	−0.219042
$177$	13.4836	1.01349
$178$	−11.4600	−0.858960
$179$	3.51602	0.262799	0.131400	−	0.991329i	$-0.458053\pi$
$179$	3.51602	0.262799	0.131400	+	0.991329i	$0.458053\pi$
$180$	−15.0508	−1.12182
$181$	−4.81159	−0.357643	−0.178821	−	0.983882i	$-0.557228\pi$
$181$	−4.81159	−0.357643	−0.178821	+	0.983882i	$0.557228\pi$
$182$	1.40422	0.104088
$183$	19.8009	1.46373
$184$	2.53872	0.187157
$185$	−10.2089	−0.750574
$186$	−23.2223	−1.70274
$187$	−2.68895	−0.196635
$188$	7.50775	0.547559
$189$	−26.1573	−1.90266
$190$	14.4884	1.05110
$191$	16.5999	1.20113	0.600563	−	0.799577i	$-0.294943\pi$
$191$	16.5999	1.20113	0.600563	+	0.799577i	$0.294943\pi$
$192$	−2.88972	−0.208548
$193$	−7.73601	−0.556850	−0.278425	−	0.960458i	$-0.589812\pi$
$193$	−7.73601	−0.556850	−0.278425	+	0.960458i	$0.589812\pi$
$194$	12.4026	0.890457
$195$	−2.96402	−0.212258
$196$	7.83031	0.559308
$197$	−7.22310	−0.514624	−0.257312	−	0.966328i	$-0.582837\pi$
$197$	−7.22310	−0.514624	−0.257312	+	0.966328i	$0.582837\pi$
$198$	15.5481	1.10496
$199$	11.3438	0.804142	0.402071	−	0.915609i	$-0.368291\pi$
$199$	11.3438	0.804142	0.402071	+	0.915609i	$0.368291\pi$
$200$	−2.91279	−0.205965
$201$	4.05341	0.285906
$202$	−7.17961	−0.505155
$203$	0	0
$204$	−2.67397	−0.187215
$205$	14.0367	0.980368
$206$	−16.0746	−1.11997
$207$	−13.5834	−0.944113
$208$	−0.364636	−0.0252830
$209$	−14.9672	−1.03530
$210$	−31.3037	−2.16016
$211$	−0.202325	−0.0139286	−0.00696431	−	0.999976i	$-0.502217\pi$
$211$	−0.202325	−0.0139286	−0.00696431	+	0.999976i	$0.502217\pi$
$212$	−9.16362	−0.629360
$213$	12.4753	0.854793
$214$	−11.7282	−0.801725
$215$	24.0539	1.64046
$216$	6.79231	0.462158
$217$	−30.9474	−2.10085
$218$	−0.0291942	−0.00197728
$219$	−36.4031	−2.45989
$220$	8.17425	0.551108
$221$	−0.337411	−0.0226967
$222$	10.4875	0.703873
$223$	−6.99226	−0.468236	−0.234118	−	0.972208i	$-0.575220\pi$
$223$	−6.99226	−0.468236	−0.234118	+	0.972208i	$0.575220\pi$
$224$	−3.85101	−0.257307
$225$	15.5849	1.03899
$226$	−1.64135	−0.109181
$227$	25.5874	1.69830	0.849148	−	0.528155i	$-0.177116\pi$
$227$	25.5874	1.69830	0.849148	+	0.528155i	$0.177116\pi$
$228$	−14.8838	−0.985701
$229$	−8.03460	−0.530941	−0.265471	−	0.964119i	$-0.585527\pi$
$229$	−8.03460	−0.530941	−0.265471	+	0.964119i	$0.585527\pi$
$230$	−7.14133	−0.470885
$231$	32.3381	2.12769
$232$	0	0
$233$	−12.3568	−0.809520	−0.404760	−	0.914423i	$-0.632645\pi$
$233$	−12.3568	−0.809520	−0.404760	+	0.914423i	$0.632645\pi$
$234$	1.95099	0.127540
$235$	−21.1190	−1.37765
$236$	−4.66605	−0.303734
$237$	−34.7241	−2.25557
$238$	−3.56348	−0.230986
$239$	0.451586	0.0292107	0.0146053	−	0.999893i	$-0.495351\pi$
$239$	0.451586	0.0292107	0.0146053	+	0.999893i	$0.495351\pi$
$240$	8.12870	0.524705
$241$	−15.7861	−1.01688	−0.508438	−	0.861099i	$-0.669777\pi$
$241$	−15.7861	−1.01688	−0.508438	+	0.861099i	$0.669777\pi$
$242$	2.55566	0.164284
$243$	10.0422	0.644205
$244$	−6.85219	−0.438667
$245$	−22.0264	−1.40722
$246$	−14.4197	−0.919370
$247$	−1.87809	−0.119500
$248$	8.03616	0.510297
$249$	−42.1205	−2.66928
$250$	−5.87126	−0.371331
$251$	7.80780	0.492824	0.246412	−	0.969165i	$-0.420748\pi$
$251$	7.80780	0.492824	0.246412	+	0.969165i	$0.420748\pi$
$252$	20.6049	1.29798
$253$	7.37729	0.463807
$254$	−16.2246	−1.01802
$255$	7.52178	0.471032
$256$	1.00000	0.0625000
$257$	7.42710	0.463290	0.231645	−	0.972800i	$-0.425589\pi$
$257$	7.42710	0.463290	0.231645	+	0.972800i	$0.425589\pi$
$258$	−24.7102	−1.53839
$259$	13.9762	0.868440
$260$	1.02571	0.0636119
$261$	0	0
$262$	13.3119	0.822410
$263$	15.8103	0.974907	0.487454	−	0.873149i	$-0.337926\pi$
$263$	15.8103	0.974907	0.487454	+	0.873149i	$0.337926\pi$
$264$	−8.39729	−0.516818
$265$	25.7770	1.58347
$266$	−19.8350	−1.21616
$267$	−33.1161	−2.02667
$268$	−1.40270	−0.0856834
$269$	15.7747	0.961798	0.480899	−	0.876776i	$-0.340310\pi$
$269$	15.7747	0.961798	0.480899	+	0.876776i	$0.340310\pi$
$270$	−19.1065	−1.16279
$271$	5.72904	0.348014	0.174007	−	0.984744i	$-0.444328\pi$
$271$	5.72904	0.348014	0.174007	+	0.984744i	$0.444328\pi$
$272$	0.925336	0.0561068
$273$	4.05781	0.245590
$274$	−19.6985	−1.19003
$275$	−8.46432	−0.510418
$276$	7.33619	0.441587
$277$	12.1246	0.728500	0.364250	−	0.931301i	$-0.381325\pi$
$277$	12.1246	0.728500	0.364250	+	0.931301i	$0.381325\pi$
$278$	−7.91712	−0.474838
$279$	−42.9975	−2.57419
$280$	10.8328	0.647383
$281$	−3.65322	−0.217933	−0.108967	−	0.994045i	$-0.534754\pi$
$281$	−3.65322	−0.217933	−0.108967	+	0.994045i	$0.534754\pi$
$282$	21.6953	1.29194
$283$	28.3178	1.68332	0.841658	−	0.540011i	$-0.181580\pi$
$283$	28.3178	1.68332	0.841658	+	0.540011i	$0.181580\pi$
$284$	−4.31712	−0.256174
$285$	41.8676	2.48002
$286$	−1.05960	−0.0626556
$287$	−19.2166	−1.13432
$288$	−5.35050	−0.315281
$289$	−16.1438	−0.949633
$290$	0	0
$291$	35.8402	2.10099
$292$	12.5974	0.737208
$293$	0.660825	0.0386058	0.0193029	−	0.999814i	$-0.493855\pi$
$293$	0.660825	0.0386058	0.0193029	+	0.999814i	$0.493855\pi$
$294$	22.6274	1.31966
$295$	13.1254	0.764193
$296$	−3.62923	−0.210945
$297$	19.7379	1.14531
$298$	6.66087	0.385854
$299$	0.925709	0.0535351
$300$	−8.41716	−0.485965
$301$	−32.9303	−1.89807
$302$	−10.8305	−0.623227
$303$	−20.7471	−1.19189
$304$	5.15058	0.295406
$305$	19.2750	1.10368
$306$	−4.95102	−0.283031
$307$	14.1116	0.805390	0.402695	−	0.915334i	$-0.368074\pi$
$307$	14.1116	0.805390	0.402695	+	0.915334i	$0.368074\pi$
$308$	−11.1907	−0.637650
$309$	−46.4510	−2.64251
$310$	−22.6055	−1.28390
$311$	15.8616	0.899429	0.449714	−	0.893172i	$-0.351526\pi$
$311$	15.8616	0.899429	0.449714	+	0.893172i	$0.351526\pi$
$312$	−1.05370	−0.0596539
$313$	23.2523	1.31430	0.657149	−	0.753761i	$-0.271762\pi$
$313$	23.2523	1.31430	0.657149	+	0.753761i	$0.271762\pi$
$314$	−14.8837	−0.839935
$315$	−57.9608	−3.26572
$316$	12.0164	0.675976
$317$	15.7881	0.886748	0.443374	−	0.896337i	$-0.353781\pi$
$317$	15.7881	0.886748	0.443374	+	0.896337i	$0.353781\pi$
$318$	−26.4803	−1.48494
$319$	0	0
$320$	−2.81297	−0.157250
$321$	−33.8914	−1.89163
$322$	9.77664	0.544831
$323$	4.76602	0.265189
$324$	3.57638	0.198688
$325$	−1.06211	−0.0589152
$326$	12.4608	0.690140
$327$	−0.0843632	−0.00466530
$328$	4.99001	0.275527
$329$	28.9124	1.59399
$330$	23.6213	1.30031
$331$	3.87620	0.213055	0.106528	−	0.994310i	$-0.466027\pi$
$331$	3.87620	0.213055	0.106528	+	0.994310i	$0.466027\pi$
$332$	14.5760	0.799960
$333$	19.4182	1.06411
$334$	4.31880	0.236314
$335$	3.94575	0.215579
$336$	−11.1284	−0.607102
$337$	13.8500	0.754459	0.377230	−	0.926120i	$-0.376877\pi$
$337$	13.8500	0.754459	0.377230	+	0.926120i	$0.376877\pi$
$338$	12.8670	0.699875
$339$	−4.74305	−0.257607
$340$	−2.60294	−0.141164
$341$	23.3524	1.26460
$342$	−27.5582	−1.49018
$343$	3.19755	0.172651
$344$	8.55107	0.461043
$345$	−20.6365	−1.11103
$346$	21.3247	1.14642
$347$	24.8212	1.33247	0.666235	−	0.745742i	$-0.267905\pi$
$347$	24.8212	1.33247	0.666235	+	0.745742i	$0.267905\pi$
$348$	0	0
$349$	19.3592	1.03627	0.518136	−	0.855298i	$-0.326626\pi$
$349$	19.3592	1.03627	0.518136	+	0.855298i	$0.326626\pi$
$350$	−11.2172	−0.599584
$351$	2.47672	0.132198
$352$	2.90591	0.154886
$353$	34.2123	1.82094	0.910468	−	0.413579i	$-0.135722\pi$
$353$	34.2123	1.82094	0.910468	+	0.413579i	$0.135722\pi$
$354$	−13.4836	−0.716644
$355$	12.1439	0.644533
$356$	11.4600	0.607376
$357$	−10.2975	−0.545001
$358$	−3.51602	−0.185827
$359$	5.64481	0.297922	0.148961	−	0.988843i	$-0.452407\pi$
$359$	5.64481	0.297922	0.148961	+	0.988843i	$0.452407\pi$
$360$	15.0508	0.793247
$361$	7.52852	0.396238
$362$	4.81159	0.252892
$363$	7.38515	0.387620
$364$	−1.40422	−0.0736011
$365$	−35.4361	−1.85481
$366$	−19.8009	−1.03501
$367$	−28.8967	−1.50840	−0.754199	−	0.656646i	$-0.771974\pi$
$367$	−28.8967	−1.50840	−0.754199	+	0.656646i	$0.771974\pi$
$368$	−2.53872	−0.132340
$369$	−26.6991	−1.38990
$370$	10.2089	0.530736
$371$	−35.2892	−1.83213
$372$	23.2223	1.20402
$373$	−14.9847	−0.775877	−0.387939	−	0.921685i	$-0.626813\pi$
$373$	−14.9847	−0.775877	−0.387939	+	0.921685i	$0.626813\pi$
$374$	2.68895	0.139042
$375$	−16.9663	−0.876136
$376$	−7.50775	−0.387182
$377$	0	0
$378$	26.1573	1.34539
$379$	−22.4281	−1.15206	−0.576028	−	0.817430i	$-0.695398\pi$
$379$	−22.4281	−1.15206	−0.576028	+	0.817430i	$0.695398\pi$
$380$	−14.4884	−0.743241
$381$	−46.8846	−2.40197
$382$	−16.5999	−0.849325
$383$	−3.74411	−0.191315	−0.0956574	−	0.995414i	$-0.530495\pi$
$383$	−3.74411	−0.191315	−0.0956574	+	0.995414i	$0.530495\pi$
$384$	2.88972	0.147466
$385$	31.4791	1.60433
$386$	7.73601	0.393752
$387$	−45.7525	−2.32573
$388$	−12.4026	−0.629648
$389$	−4.62048	−0.234268	−0.117134	−	0.993116i	$-0.537371\pi$
$389$	−4.62048	−0.234268	−0.117134	+	0.993116i	$0.537371\pi$
$390$	2.96402	0.150089
$391$	−2.34917	−0.118802
$392$	−7.83031	−0.395491
$393$	38.4676	1.94044
$394$	7.22310	0.363894
$395$	−33.8018	−1.70075
$396$	−15.5481	−0.781322
$397$	17.0364	0.855035	0.427517	−	0.904007i	$-0.359388\pi$
$397$	17.0364	0.855035	0.427517	+	0.904007i	$0.359388\pi$
$398$	−11.3438	−0.568614
$399$	−57.3176	−2.86947
$400$	2.91279	0.145640
$401$	−35.9180	−1.79366	−0.896830	−	0.442374i	$-0.854136\pi$
$401$	−35.9180	−1.79366	−0.896830	+	0.442374i	$0.854136\pi$
$402$	−4.05341	−0.202166
$403$	2.93028	0.145968
$404$	7.17961	0.357199
$405$	−10.0602	−0.499898
$406$	0	0
$407$	−10.5462	−0.522757
$408$	2.67397	0.132381
$409$	9.10554	0.450240	0.225120	−	0.974331i	$-0.427723\pi$
$409$	9.10554	0.450240	0.225120	+	0.974331i	$0.427723\pi$
$410$	−14.0367	−0.693225
$411$	−56.9233	−2.80782
$412$	16.0746	0.791937
$413$	−17.9690	−0.884197
$414$	13.5834	0.667588
$415$	−41.0017	−2.01270
$416$	0.364636	0.0178778
$417$	−22.8783	−1.12036
$418$	14.9672	0.732068
$419$	1.84770	0.0902661	0.0451331	−	0.998981i	$-0.485629\pi$
$419$	1.84770	0.0902661	0.0451331	+	0.998981i	$0.485629\pi$
$420$	31.3037	1.52747
$421$	−35.2513	−1.71805	−0.859023	−	0.511938i	$-0.828928\pi$
$421$	−35.2513	−1.71805	−0.859023	+	0.511938i	$0.828928\pi$
$422$	0.202325	0.00984902
$423$	40.1702	1.95314
$424$	9.16362	0.445025
$425$	2.69531	0.130742
$426$	−12.4753	−0.604430
$427$	−26.3879	−1.27700
$428$	11.7282	0.566906
$429$	−3.06196	−0.147833
$430$	−24.0539	−1.15998
$431$	6.75999	0.325617	0.162809	−	0.986658i	$-0.447945\pi$
$431$	6.75999	0.325617	0.162809	+	0.986658i	$0.447945\pi$
$432$	−6.79231	−0.326795
$433$	2.34317	0.112606	0.0563029	−	0.998414i	$-0.482069\pi$
$433$	2.34317	0.112606	0.0563029	+	0.998414i	$0.482069\pi$
$434$	30.9474	1.48552
$435$	0	0
$436$	0.0291942	0.00139815
$437$	−13.0759	−0.625504
$438$	36.4031	1.73941
$439$	27.2219	1.29923	0.649616	−	0.760263i	$-0.274930\pi$
$439$	27.2219	1.29923	0.649616	+	0.760263i	$0.274930\pi$
$440$	−8.17425	−0.389692
$441$	41.8961	1.99505
$442$	0.337411	0.0160490
$443$	−3.70498	−0.176029	−0.0880145	−	0.996119i	$-0.528052\pi$
$443$	−3.70498	−0.176029	−0.0880145	+	0.996119i	$0.528052\pi$
$444$	−10.4875	−0.497713
$445$	−32.2365	−1.52816
$446$	6.99226	0.331093
$447$	19.2481	0.910403
$448$	3.85101	0.181943
$449$	20.6490	0.974485	0.487243	−	0.873267i	$-0.338003\pi$
$449$	20.6490	0.974485	0.487243	+	0.873267i	$0.338003\pi$
$450$	−15.5849	−0.734679
$451$	14.5005	0.682804
$452$	1.64135	0.0772026
$453$	−31.2973	−1.47047
$454$	−25.5874	−1.20088
$455$	3.95003	0.185180
$456$	14.8838	0.696996
$457$	0.0396615	0.00185529	0.000927644	−	1.00000i	$-0.499705\pi$
$457$	0.0396615	0.00185529	0.000927644		1.00000i	$0.499705\pi$
$458$	8.03460	0.375432
$459$	−6.28517	−0.293367
$460$	7.14133	0.332966
$461$	−29.6008	−1.37864	−0.689322	−	0.724455i	$-0.742092\pi$
$461$	−29.6008	−1.37864	−0.689322	+	0.724455i	$0.742092\pi$
$462$	−32.3381	−1.50450
$463$	−22.9957	−1.06870	−0.534351	−	0.845263i	$-0.679444\pi$
$463$	−22.9957	−1.06870	−0.534351	+	0.845263i	$0.679444\pi$
$464$	0	0
$465$	−65.3236	−3.02931
$466$	12.3568	0.572417
$467$	14.4157	0.667080	0.333540	−	0.942736i	$-0.391757\pi$
$467$	14.4157	0.667080	0.333540	+	0.942736i	$0.391757\pi$
$468$	−1.95099	−0.0901845
$469$	−5.40181	−0.249432
$470$	21.1190	0.974149
$471$	−43.0098	−1.98179
$472$	4.66605	0.214772
$473$	24.8487	1.14254
$474$	34.7241	1.59493
$475$	15.0026	0.688365
$476$	3.56348	0.163332
$477$	−49.0300	−2.24493
$478$	−0.451586	−0.0206551
$479$	3.99470	0.182522	0.0912612	−	0.995827i	$-0.470910\pi$
$479$	3.99470	0.182522	0.0912612	+	0.995827i	$0.470910\pi$
$480$	−8.12870	−0.371023
$481$	−1.32335	−0.0603395
$482$	15.7861	0.719039
$483$	28.2518	1.28550
$484$	−2.55566	−0.116166
$485$	34.8882	1.58419
$486$	−10.0422	−0.455522
$487$	18.6434	0.844815	0.422407	−	0.906406i	$-0.361185\pi$
$487$	18.6434	0.844815	0.422407	+	0.906406i	$0.361185\pi$
$488$	6.85219	0.310184
$489$	36.0083	1.62835
$490$	22.0264	0.995052
$491$	3.44975	0.155685	0.0778426	−	0.996966i	$-0.475197\pi$
$491$	3.44975	0.155685	0.0778426	+	0.996966i	$0.475197\pi$
$492$	14.4197	0.650092
$493$	0	0
$494$	1.87809	0.0844993
$495$	43.7363	1.96580
$496$	−8.03616	−0.360834
$497$	−16.6253	−0.745747
$498$	42.1205	1.88747
$499$	6.02903	0.269896	0.134948	−	0.990853i	$-0.456913\pi$
$499$	6.02903	0.269896	0.134948	+	0.990853i	$0.456913\pi$
$500$	5.87126	0.262571
$501$	12.4801	0.557571
$502$	−7.80780	−0.348479
$503$	−5.96356	−0.265902	−0.132951	−	0.991123i	$-0.542445\pi$
$503$	−5.96356	−0.265902	−0.132951	+	0.991123i	$0.542445\pi$
$504$	−20.6049	−0.917814
$505$	−20.1960	−0.898710
$506$	−7.37729	−0.327961
$507$	37.1822	1.65132
$508$	16.2246	0.719849
$509$	4.71837	0.209138	0.104569	−	0.994518i	$-0.466654\pi$
$509$	4.71837	0.209138	0.104569	+	0.994518i	$0.466654\pi$
$510$	−7.52178	−0.333070
$511$	48.5128	2.14608
$512$	−1.00000	−0.0441942
$513$	−34.9844	−1.54460
$514$	−7.42710	−0.327596
$515$	−45.2172	−1.99251
$516$	24.7102	1.08781
$517$	−21.8169	−0.959505
$518$	−13.9762	−0.614080
$519$	61.6225	2.70493
$520$	−1.02571	−0.0449804
$521$	27.8336	1.21941	0.609707	−	0.792627i	$-0.291287\pi$
$521$	27.8336	1.21941	0.609707	+	0.792627i	$0.291287\pi$
$522$	0	0
$523$	29.4055	1.28581	0.642906	−	0.765945i	$-0.277728\pi$
$523$	29.4055	1.28581	0.642906	+	0.765945i	$0.277728\pi$
$524$	−13.3119	−0.581532
$525$	−32.4146	−1.41469
$526$	−15.8103	−0.689364
$527$	−7.43615	−0.323924
$528$	8.39729	0.365445
$529$	−16.5549	−0.719779
$530$	−25.7770	−1.11968
$531$	−24.9657	−1.08342
$532$	19.8350	0.859955
$533$	1.81954	0.0788130
$534$	33.1161	1.43307
$535$	−32.9911	−1.42633
$536$	1.40270	0.0605873
$537$	−10.1603	−0.438450
$538$	−15.7747	−0.680094
$539$	−22.7542	−0.980094
$540$	19.1065	0.822215
$541$	27.5775	1.18565	0.592825	−	0.805331i	$-0.298013\pi$
$541$	27.5775	1.18565	0.592825	+	0.805331i	$0.298013\pi$
$542$	−5.72904	−0.246083
$543$	13.9042	0.596685
$544$	−0.925336	−0.0396735
$545$	−0.0821224	−0.00351774
$546$	−4.05781	−0.173658
$547$	−3.86399	−0.165212	−0.0826061	−	0.996582i	$-0.526324\pi$
$547$	−3.86399	−0.165212	−0.0826061	+	0.996582i	$0.526324\pi$
$548$	19.6985	0.841479
$549$	−36.6627	−1.56473
$550$	8.46432	0.360920
$551$	0	0
$552$	−7.33619	−0.312249
$553$	46.2754	1.96783
$554$	−12.1246	−0.515127
$555$	29.5009	1.25224
$556$	7.91712	0.335761
$557$	18.0890	0.766454	0.383227	−	0.923654i	$-0.374813\pi$
$557$	18.0890	0.766454	0.383227	+	0.923654i	$0.374813\pi$
$558$	42.9975	1.82023
$559$	3.11803	0.131879
$560$	−10.8328	−0.457769
$561$	7.77032	0.328063
$562$	3.65322	0.154102
$563$	−10.0817	−0.424893	−0.212447	−	0.977173i	$-0.568143\pi$
$563$	−10.0817	−0.424893	−0.212447	+	0.977173i	$0.568143\pi$
$564$	−21.6953	−0.913537
$565$	−4.61707	−0.194241
$566$	−28.3178	−1.19028
$567$	13.7727	0.578399
$568$	4.31712	0.181142
$569$	8.42654	0.353259	0.176630	−	0.984277i	$-0.443481\pi$
$569$	8.42654	0.353259	0.176630	+	0.984277i	$0.443481\pi$
$570$	−41.8676	−1.75364
$571$	25.6175	1.07206	0.536029	−	0.844199i	$-0.319924\pi$
$571$	25.6175	1.07206	0.536029	+	0.844199i	$0.319924\pi$
$572$	1.05960	0.0443042
$573$	−47.9691	−2.00394
$574$	19.2166	0.802085
$575$	−7.39475	−0.308382
$576$	5.35050	0.222938
$577$	14.2757	0.594307	0.297154	−	0.954830i	$-0.403963\pi$
$577$	14.2757	0.594307	0.297154	+	0.954830i	$0.403963\pi$
$578$	16.1438	0.671492
$579$	22.3549	0.929039
$580$	0	0
$581$	56.1323	2.32876
$582$	−35.8402	−1.48562
$583$	26.6287	1.10285
$584$	−12.5974	−0.521285
$585$	5.48807	0.226904
$586$	−0.660825	−0.0272984
$587$	3.16614	0.130681	0.0653403	−	0.997863i	$-0.479187\pi$
$587$	3.16614	0.130681	0.0653403	+	0.997863i	$0.479187\pi$
$588$	−22.6274	−0.933140
$589$	−41.3909	−1.70548
$590$	−13.1254	−0.540366
$591$	20.8728	0.858591
$592$	3.62923	0.149160
$593$	32.1045	1.31837	0.659186	−	0.751980i	$-0.270901\pi$
$593$	32.1045	1.31837	0.659186	+	0.751980i	$0.270901\pi$
$594$	−19.7379	−0.809854
$595$	−10.0240	−0.410943
$596$	−6.66087	−0.272840
$597$	−32.7805	−1.34162
$598$	−0.925709	−0.0378550
$599$	3.87056	0.158147	0.0790735	−	0.996869i	$-0.474804\pi$
$599$	3.87056	0.158147	0.0790735	+	0.996869i	$0.474804\pi$
$600$	8.41716	0.343629
$601$	−27.7877	−1.13349	−0.566743	−	0.823895i	$-0.691797\pi$
$601$	−27.7877	−1.13349	−0.566743	+	0.823895i	$0.691797\pi$
$602$	32.9303	1.34214
$603$	−7.50514	−0.305633
$604$	10.8305	0.440688
$605$	7.18899	0.292274
$606$	20.7471	0.842792
$607$	25.3730	1.02986	0.514929	−	0.857233i	$-0.327818\pi$
$607$	25.3730	1.02986	0.514929	+	0.857233i	$0.327818\pi$
$608$	−5.15058	−0.208884
$609$	0	0
$610$	−19.2750	−0.780422
$611$	−2.73760	−0.110751
$612$	4.95102	0.200133
$613$	−31.3639	−1.26678	−0.633388	−	0.773835i	$-0.718336\pi$
$613$	−31.3639	−1.26678	−0.633388	+	0.773835i	$0.718336\pi$
$614$	−14.1116	−0.569497
$615$	−40.5623	−1.63563
$616$	11.1907	0.450887
$617$	−42.0303	−1.69208	−0.846039	−	0.533121i	$-0.821019\pi$
$617$	−42.0303	−1.69208	−0.846039	+	0.533121i	$0.821019\pi$
$618$	46.4510	1.86853
$619$	−30.2687	−1.21660	−0.608300	−	0.793707i	$-0.708148\pi$
$619$	−30.2687	−1.21660	−0.608300	+	0.793707i	$0.708148\pi$
$620$	22.6055	0.907858
$621$	17.2437	0.691968
$622$	−15.8616	−0.635992
$623$	44.1324	1.76813
$624$	1.05370	0.0421817
$625$	−31.0796	−1.24318
$626$	−23.2523	−0.929349
$627$	43.2510	1.72728
$628$	14.8837	0.593924
$629$	3.35826	0.133902
$630$	57.9608	2.30922
$631$	37.0307	1.47417	0.737085	−	0.675800i	$-0.236202\pi$
$631$	37.0307	1.47417	0.737085	+	0.675800i	$0.236202\pi$
$632$	−12.0164	−0.477987
$633$	0.584663	0.0232383
$634$	−15.7881	−0.627025
$635$	−45.6392	−1.81114
$636$	26.4803	1.05001
$637$	−2.85522	−0.113128
$638$	0	0
$639$	−23.0988	−0.913774
$640$	2.81297	0.111192
$641$	−28.4320	−1.12300	−0.561499	−	0.827477i	$-0.689775\pi$
$641$	−28.4320	−1.12300	−0.561499	+	0.827477i	$0.689775\pi$
$642$	33.8914	1.33759
$643$	36.5756	1.44240	0.721201	−	0.692726i	$-0.243590\pi$
$643$	36.5756	1.44240	0.721201	+	0.692726i	$0.243590\pi$
$644$	−9.77664	−0.385253
$645$	−69.5091	−2.73692
$646$	−4.76602	−0.187517
$647$	12.7395	0.500840	0.250420	−	0.968137i	$-0.419431\pi$
$647$	12.7395	0.500840	0.250420	+	0.968137i	$0.419431\pi$
$648$	−3.57638	−0.140494
$649$	13.5591	0.532243
$650$	1.06211	0.0416594
$651$	89.4294	3.50501
$652$	−12.4608	−0.488003
$653$	42.2415	1.65304	0.826518	−	0.562910i	$-0.190318\pi$
$653$	42.2415	1.65304	0.826518	+	0.562910i	$0.190318\pi$
$654$	0.0843632	0.00329886
$655$	37.4459	1.46313
$656$	−4.99001	−0.194827
$657$	67.4025	2.62962
$658$	−28.9124	−1.12712
$659$	−29.8537	−1.16293	−0.581467	−	0.813570i	$-0.697521\pi$
$659$	−29.8537	−1.16293	−0.581467	+	0.813570i	$0.697521\pi$
$660$	−23.6213	−0.919458
$661$	−20.9321	−0.814164	−0.407082	−	0.913392i	$-0.633454\pi$
$661$	−20.9321	−0.814164	−0.407082	+	0.913392i	$0.633454\pi$
$662$	−3.87620	−0.150653
$663$	0.975025	0.0378668
$664$	−14.5760	−0.565657
$665$	−55.7952	−2.16364
$666$	−19.4182	−0.752440
$667$	0	0
$668$	−4.31880	−0.167099
$669$	20.2057	0.781197
$670$	−3.94575	−0.152437
$671$	19.9119	0.768690
$672$	11.1284	0.429286
$673$	8.30904	0.320290	0.160145	−	0.987094i	$-0.448804\pi$
$673$	8.30904	0.320290	0.160145	+	0.987094i	$0.448804\pi$
$674$	−13.8500	−0.533483
$675$	−19.7846	−0.761508
$676$	−12.8670	−0.494886
$677$	−23.9829	−0.921738	−0.460869	−	0.887468i	$-0.652462\pi$
$677$	−23.9829	−0.921738	−0.460869	+	0.887468i	$0.652462\pi$
$678$	4.74305	0.182156
$679$	−47.7627	−1.83297
$680$	2.60294	0.0998182
$681$	−73.9405	−2.83341
$682$	−23.3524	−0.894210
$683$	36.5532	1.39867	0.699334	−	0.714795i	$-0.253480\pi$
$683$	36.5532	1.39867	0.699334	+	0.714795i	$0.253480\pi$
$684$	27.5582	1.05371
$685$	−55.4113	−2.11716
$686$	−3.19755	−0.122083
$687$	23.2178	0.885813
$688$	−8.55107	−0.326006
$689$	3.34139	0.127297
$690$	20.6365	0.785617
$691$	34.4772	1.31158	0.655788	−	0.754945i	$-0.272337\pi$
$691$	34.4772	1.31158	0.655788	+	0.754945i	$0.272337\pi$
$692$	−21.3247	−0.810644
$693$	−59.8760	−2.27450
$694$	−24.8212	−0.942198
$695$	−22.2706	−0.844773
$696$	0	0
$697$	−4.61744	−0.174898
$698$	−19.3592	−0.732756
$699$	35.7077	1.35059
$700$	11.2172	0.423970
$701$	36.8109	1.39033	0.695165	−	0.718850i	$-0.255331\pi$
$701$	36.8109	1.39033	0.695165	+	0.718850i	$0.255331\pi$
$702$	−2.47672	−0.0934779
$703$	18.6927	0.705007
$704$	−2.90591	−0.109521
$705$	61.0282	2.29846
$706$	−34.2123	−1.28760
$707$	27.6488	1.03984
$708$	13.4836	0.506744
$709$	20.6501	0.775529	0.387765	−	0.921758i	$-0.373247\pi$
$709$	20.6501	0.775529	0.387765	+	0.921758i	$0.373247\pi$
$710$	−12.1439	−0.455753
$711$	64.2939	2.41121
$712$	−11.4600	−0.429480
$713$	20.4015	0.764044
$714$	10.2975	0.385374
$715$	−2.98063	−0.111469
$716$	3.51602	0.131400
$717$	−1.30496	−0.0487346
$718$	−5.64481	−0.210662
$719$	−9.39581	−0.350404	−0.175202	−	0.984532i	$-0.556058\pi$
$719$	−9.39581	−0.350404	−0.175202	+	0.984532i	$0.556058\pi$
$720$	−15.0508	−0.560910
$721$	61.9034	2.30540
$722$	−7.52852	−0.280182
$723$	45.6176	1.69654
$724$	−4.81159	−0.178821
$725$	0	0
$726$	−7.38515	−0.274089
$727$	−0.457584	−0.0169708	−0.00848542	−	0.999964i	$-0.502701\pi$
$727$	−0.457584	−0.0169708	−0.00848542	+	0.999964i	$0.502701\pi$
$728$	1.40422	0.0520439
$729$	−39.7482	−1.47216
$730$	35.4361	1.31155
$731$	−7.91261	−0.292659
$732$	19.8009	0.731864
$733$	19.2200	0.709907	0.354953	−	0.934884i	$-0.384497\pi$
$733$	19.2200	0.709907	0.354953	+	0.934884i	$0.384497\pi$
$734$	28.8967	1.06660
$735$	63.6503	2.34778
$736$	2.53872	0.0935783
$737$	4.07612	0.150146
$738$	26.6991	0.982806
$739$	−43.7081	−1.60783	−0.803914	−	0.594746i	$-0.797253\pi$
$739$	−43.7081	−1.60783	−0.803914	+	0.594746i	$0.797253\pi$
$740$	−10.2089	−0.375287
$741$	5.42716	0.199372
$742$	35.2892	1.29551
$743$	−33.9116	−1.24410	−0.622048	−	0.782979i	$-0.713699\pi$
$743$	−33.9116	−1.24410	−0.622048	+	0.782979i	$0.713699\pi$
$744$	−23.2223	−0.851371
$745$	18.7368	0.686464
$746$	14.9847	0.548628
$747$	77.9888	2.85346
$748$	−2.68895	−0.0983177
$749$	45.1656	1.65031
$750$	16.9663	0.619522
$751$	−9.81273	−0.358072	−0.179036	−	0.983843i	$-0.557298\pi$
$751$	−9.81273	−0.358072	−0.179036	+	0.983843i	$0.557298\pi$
$752$	7.50775	0.273779
$753$	−22.5624	−0.822220
$754$	0	0
$755$	−30.4660	−1.10877
$756$	−26.1573	−0.951331
$757$	7.61721	0.276852	0.138426	−	0.990373i	$-0.455796\pi$
$757$	7.61721	0.276852	0.138426	+	0.990373i	$0.455796\pi$
$758$	22.4281	0.814627
$759$	−21.3183	−0.773807
$760$	14.4884	0.525551
$761$	35.5488	1.28864	0.644321	−	0.764755i	$-0.277140\pi$
$761$	35.5488	1.28864	0.644321	+	0.764755i	$0.277140\pi$
$762$	46.8846	1.69845
$763$	0.112427	0.00407014
$764$	16.5999	0.600563
$765$	−13.9270	−0.503534
$766$	3.74411	0.135280
$767$	1.70141	0.0614344
$768$	−2.88972	−0.104274
$769$	−29.8344	−1.07586	−0.537928	−	0.842991i	$-0.680793\pi$
$769$	−29.8344	−1.07586	−0.537928	+	0.842991i	$0.680793\pi$
$770$	−31.4791	−1.13443
$771$	−21.4623	−0.772945
$772$	−7.73601	−0.278425
$773$	−33.0140	−1.18743	−0.593715	−	0.804675i	$-0.702339\pi$
$773$	−33.0140	−1.18743	−0.593715	+	0.804675i	$0.702339\pi$
$774$	45.7525	1.64454
$775$	−23.4077	−0.840828
$776$	12.4026	0.445229
$777$	−40.3874	−1.44889
$778$	4.62048	0.165652
$779$	−25.7015	−0.920850
$780$	−2.96402	−0.106129
$781$	12.5452	0.448902
$782$	2.34917	0.0840060
$783$	0	0
$784$	7.83031	0.279654
$785$	−41.8674	−1.49431
$786$	−38.4676	−1.37210
$787$	−37.9408	−1.35245	−0.676223	−	0.736697i	$-0.736384\pi$
$787$	−37.9408	−1.35245	−0.676223	+	0.736697i	$0.736384\pi$
$788$	−7.22310	−0.257312
$789$	−45.6875	−1.62652
$790$	33.8018	1.20261
$791$	6.32086	0.224744
$792$	15.5481	0.552478
$793$	2.49856	0.0887264
$794$	−17.0364	−0.604601
$795$	−74.4883	−2.64183
$796$	11.3438	0.402071
$797$	0.0808665	0.00286444	0.00143222	−	0.999999i	$-0.499544\pi$
$797$	0.0808665	0.00286444	0.00143222	+	0.999999i	$0.499544\pi$
$798$	57.3176	2.02902
$799$	6.94719	0.245774
$800$	−2.91279	−0.102983
$801$	61.3165	2.16651
$802$	35.9180	1.26831
$803$	−36.6070	−1.29183
$804$	4.05341	0.142953
$805$	27.5014	0.969296
$806$	−2.93028	−0.103215
$807$	−45.5844	−1.60465
$808$	−7.17961	−0.252578
$809$	43.1159	1.51587	0.757936	−	0.652328i	$-0.226208\pi$
$809$	43.1159	1.51587	0.757936	+	0.652328i	$0.226208\pi$
$810$	10.0602	0.353481
$811$	−5.32713	−0.187061	−0.0935305	−	0.995616i	$-0.529815\pi$
$811$	−5.32713	−0.187061	−0.0935305	+	0.995616i	$0.529815\pi$
$812$	0	0
$813$	−16.5553	−0.580621
$814$	10.5462	0.369645
$815$	35.0519	1.22781
$816$	−2.67397	−0.0936075
$817$	−44.0430	−1.54087
$818$	−9.10554	−0.318368
$819$	−7.51329	−0.262535
$820$	14.0367	0.490184
$821$	39.3983	1.37501	0.687506	−	0.726179i	$-0.258706\pi$
$821$	39.3983	1.37501	0.687506	+	0.726179i	$0.258706\pi$
$822$	56.9233	1.98543
$823$	−52.6721	−1.83603	−0.918017	−	0.396541i	$-0.870210\pi$
$823$	−52.6721	−1.83603	−0.918017	+	0.396541i	$0.870210\pi$
$824$	−16.0746	−0.559984
$825$	24.4595	0.851572
$826$	17.9690	0.625222
$827$	44.2163	1.53755	0.768777	−	0.639517i	$-0.220866\pi$
$827$	44.2163	1.53755	0.768777	+	0.639517i	$0.220866\pi$
$828$	−13.5834	−0.472056
$829$	9.10353	0.316179	0.158089	−	0.987425i	$-0.449467\pi$
$829$	9.10353	0.316179	0.158089	+	0.987425i	$0.449467\pi$
$830$	41.0017	1.42319
$831$	−35.0369	−1.21542
$832$	−0.364636	−0.0126415
$833$	7.24567	0.251048
$834$	22.8783	0.792211
$835$	12.1486	0.420421
$836$	−14.9672	−0.517650
$837$	54.5841	1.88670
$838$	−1.84770	−0.0638278
$839$	46.4865	1.60489	0.802446	−	0.596725i	$-0.203532\pi$
$839$	46.4865	1.60489	0.802446	+	0.596725i	$0.203532\pi$
$840$	−31.3037	−1.08008
$841$	0	0
$842$	35.2513	1.21484
$843$	10.5568	0.363596
$844$	−0.202325	−0.00696431
$845$	36.1946	1.24513
$846$	−40.1702	−1.38108
$847$	−9.84188	−0.338171
$848$	−9.16362	−0.314680
$849$	−81.8305	−2.80842
$850$	−2.69531	−0.0924484
$851$	−9.21359	−0.315838
$852$	12.4753	0.427396
$853$	36.3433	1.24437	0.622185	−	0.782870i	$-0.286245\pi$
$853$	36.3433	1.24437	0.622185	+	0.782870i	$0.286245\pi$
$854$	26.3879	0.902975
$855$	−77.5204	−2.65114
$856$	−11.7282	−0.400863
$857$	−33.1429	−1.13214	−0.566069	−	0.824358i	$-0.691537\pi$
$857$	−33.1429	−1.13214	−0.566069	+	0.824358i	$0.691537\pi$
$858$	3.06196	0.104534
$859$	33.6381	1.14772	0.573858	−	0.818955i	$-0.305446\pi$
$859$	33.6381	1.14772	0.573858	+	0.818955i	$0.305446\pi$
$860$	24.0539	0.820230
$861$	55.5307	1.89248
$862$	−6.75999	−0.230246
$863$	2.79233	0.0950519	0.0475259	−	0.998870i	$-0.484866\pi$
$863$	2.79233	0.0950519	0.0475259	+	0.998870i	$0.484866\pi$
$864$	6.79231	0.231079
$865$	59.9857	2.03958
$866$	−2.34317	−0.0796243
$867$	46.6510	1.58435
$868$	−30.9474	−1.05042
$869$	−34.9187	−1.18454
$870$	0	0
$871$	0.511475	0.0173307
$872$	−0.0291942	−0.000988641 0
$873$	−66.3603	−2.24596
$874$	13.0759	0.442298
$875$	22.6103	0.764367
$876$	−36.4031	−1.22995
$877$	35.6070	1.20236	0.601181	−	0.799113i	$-0.294697\pi$
$877$	35.6070	1.20236	0.601181	+	0.799113i	$0.294697\pi$
$878$	−27.2219	−0.918696
$879$	−1.90960	−0.0644093
$880$	8.17425	0.275554
$881$	−31.0805	−1.04713	−0.523565	−	0.851986i	$-0.675398\pi$
$881$	−31.0805	−1.04713	−0.523565	+	0.851986i	$0.675398\pi$
$882$	−41.8961	−1.41072
$883$	−27.2833	−0.918156	−0.459078	−	0.888396i	$-0.651820\pi$
$883$	−27.2833	−0.918156	−0.459078	+	0.888396i	$0.651820\pi$
$884$	−0.337411	−0.0113484
$885$	−37.9289	−1.27497
$886$	3.70498	0.124471
$887$	4.60352	0.154571	0.0772855	−	0.997009i	$-0.475375\pi$
$887$	4.60352	0.154571	0.0772855	+	0.997009i	$0.475375\pi$
$888$	10.4875	0.351936
$889$	62.4811	2.09555
$890$	32.2365	1.08057
$891$	−10.3927	−0.348167
$892$	−6.99226	−0.234118
$893$	38.6693	1.29402
$894$	−19.2481	−0.643752
$895$	−9.89045	−0.330601
$896$	−3.85101	−0.128653
$897$	−2.67504	−0.0893171
$898$	−20.6490	−0.689065
$899$	0	0
$900$	15.5849	0.519496
$901$	−8.47943	−0.282491
$902$	−14.5005	−0.482815
$903$	95.1594	3.16671
$904$	−1.64135	−0.0545905
$905$	13.5348	0.449914
$906$	31.2973	1.03978
$907$	−17.7136	−0.588169	−0.294085	−	0.955779i	$-0.595015\pi$
$907$	−17.7136	−0.588169	−0.294085	+	0.955779i	$0.595015\pi$
$908$	25.5874	0.849148
$909$	38.4145	1.27413
$910$	−3.95003	−0.130942
$911$	−46.6546	−1.54573	−0.772867	−	0.634567i	$-0.781178\pi$
$911$	−46.6546	−1.54573	−0.772867	+	0.634567i	$0.781178\pi$
$912$	−14.8838	−0.492851
$913$	−42.3565	−1.40180
$914$	−0.0396615	−0.00131189
$915$	−55.6994	−1.84137
$916$	−8.03460	−0.265471
$917$	−51.2642	−1.69289
$918$	6.28517	0.207441
$919$	−37.6676	−1.24254	−0.621270	−	0.783596i	$-0.713383\pi$
$919$	−37.6676	−1.24254	−0.621270	+	0.783596i	$0.713383\pi$
$920$	−7.14133	−0.235443
$921$	−40.7785	−1.34370
$922$	29.6008	0.974849
$923$	1.57418	0.0518148
$924$	32.3381	1.06385
$925$	10.5712	0.347578
$926$	22.9957	0.755687
$927$	86.0070	2.82484
$928$	0	0
$929$	−57.6558	−1.89162	−0.945812	−	0.324714i	$-0.894732\pi$
$929$	−57.6558	−1.89162	−0.945812	+	0.324714i	$0.894732\pi$
$930$	65.3236	2.14204
$931$	40.3307	1.32178
$932$	−12.3568	−0.404760
$933$	−45.8357	−1.50059
$934$	−14.4157	−0.471697
$935$	7.56393	0.247367
$936$	1.95099	0.0637701
$937$	51.6641	1.68779	0.843897	−	0.536506i	$-0.180256\pi$
$937$	51.6641	1.68779	0.843897	+	0.536506i	$0.180256\pi$
$938$	5.40181	0.176375
$939$	−67.1927	−2.19275
$940$	−21.1190	−0.688827
$941$	21.5672	0.703070	0.351535	−	0.936175i	$-0.385660\pi$
$941$	21.5672	0.703070	0.351535	+	0.936175i	$0.385660\pi$
$942$	43.0098	1.40133
$943$	12.6682	0.412534
$944$	−4.66605	−0.151867
$945$	73.5796	2.39354
$946$	−24.8487	−0.807900
$947$	5.74054	0.186543	0.0932713	−	0.995641i	$-0.470268\pi$
$947$	5.74054	0.186543	0.0932713	+	0.995641i	$0.470268\pi$
$948$	−34.7241	−1.12779
$949$	−4.59348	−0.149111
$950$	−15.0026	−0.486748
$951$	−45.6232	−1.47943
$952$	−3.56348	−0.115493
$953$	−26.6321	−0.862697	−0.431348	−	0.902185i	$-0.641962\pi$
$953$	−26.6321	−0.862697	−0.431348	+	0.902185i	$0.641962\pi$
$954$	49.0300	1.58740
$955$	−46.6950	−1.51101
$956$	0.451586	0.0146053
$957$	0	0
$958$	−3.99470	−0.129063
$959$	75.8593	2.44962
$960$	8.12870	0.262353
$961$	33.5799	1.08322
$962$	1.32335	0.0426665
$963$	62.7520	2.02215
$964$	−15.7861	−0.508438
$965$	21.7611	0.700516
$966$	−28.2518	−0.908986
$967$	9.34598	0.300546	0.150273	−	0.988645i	$-0.451985\pi$
$967$	9.34598	0.300546	0.150273	+	0.988645i	$0.451985\pi$
$968$	2.55566	0.0821420
$969$	−13.7725	−0.442436
$970$	−34.8882	−1.12019
$971$	13.4411	0.431346	0.215673	−	0.976466i	$-0.430805\pi$
$971$	13.4411	0.431346	0.215673	+	0.976466i	$0.430805\pi$
$972$	10.0422	0.322103
$973$	30.4890	0.977431
$974$	−18.6434	−0.597374
$975$	3.06920	0.0982931
$976$	−6.85219	−0.219333
$977$	−8.50896	−0.272226	−0.136113	−	0.990693i	$-0.543461\pi$
$977$	−8.50896	−0.272226	−0.136113	+	0.990693i	$0.543461\pi$
$978$	−36.0083	−1.15142
$979$	−33.3016	−1.06433
$980$	−22.0264	−0.703608
$981$	0.156204	0.00498720
$982$	−3.44975	−0.110086
$983$	−41.6581	−1.32869	−0.664343	−	0.747428i	$-0.731289\pi$
$983$	−41.6581	−1.32869	−0.664343	+	0.747428i	$0.731289\pi$
$984$	−14.4197	−0.459685
$985$	20.3183	0.647396
$986$	0	0
$987$	−83.5490	−2.65939
$988$	−1.87809	−0.0597500
$989$	21.7087	0.690298
$990$	−43.7363	−1.39003
$991$	29.5715	0.939369	0.469684	−	0.882834i	$-0.344368\pi$
$991$	29.5715	0.939369	0.469684	+	0.882834i	$0.344368\pi$
$992$	8.03616	0.255148
$993$	−11.2012	−0.355458
$994$	16.6253	0.527323
$995$	−31.9098	−1.01161
$996$	−42.1205	−1.33464
$997$	47.3436	1.49939	0.749694	−	0.661784i	$-0.230201\pi$
$997$	47.3436	1.49939	0.749694	+	0.661784i	$0.230201\pi$
$998$	−6.02903	−0.190846
$999$	−24.6508	−0.779918

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.u.1.1	✓	8
29.12	odd	4		1682.2.b.k.1681.1		16
29.17	odd	4		1682.2.b.k.1681.16		16
29.28	even	2		1682.2.a.v.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1682.2.a.u.1.1	✓	8	1.1	even	1	trivial
1682.2.a.v.1.8	yes	8	29.28	even	2
1682.2.b.k.1681.1		16	29.12	odd	4
1682.2.b.k.1681.16		16	29.17	odd	4

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1682.2.a.u.1.1

Level	$1682$
Weight	$2$
Character	1682.1
Self dual	yes
Analytic conductor	$13.431$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$1$