LMFDB - Embedded newform 8.22.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,22,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	8.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:	$22.3581875430$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{358549}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 89637 $ x^2 - x - 89637
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$299.895$ of defining polynomial
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-167684. q^{3} +3.35342e6 q^{5} +7.07779e8 q^{7} +1.76574e10 q^{9} +O(q^{10})$	$q-167684. q^{3} +3.35342e6 q^{5} +7.07779e8 q^{7} +1.76574e10 q^{9} +8.75119e10 q^{11} -7.41212e11 q^{13} -5.62314e11 q^{15} -6.82853e12 q^{17} +5.17730e13 q^{19} -1.18683e14 q^{21} -3.13428e14 q^{23} -4.65592e14 q^{25} -1.20683e15 q^{27} +1.46400e15 q^{29} -6.42150e15 q^{31} -1.46743e16 q^{33} +2.37348e15 q^{35} -6.93987e15 q^{37} +1.24289e17 q^{39} +3.25339e16 q^{41} +4.37536e16 q^{43} +5.92128e16 q^{45} -5.34426e16 q^{47} -5.75946e16 q^{49} +1.14503e18 q^{51} -1.19249e18 q^{53} +2.93464e17 q^{55} -8.68148e18 q^{57} -4.48047e17 q^{59} -9.05611e17 q^{61} +1.24976e19 q^{63} -2.48560e18 q^{65} -6.06820e17 q^{67} +5.25568e19 q^{69} -2.18611e19 q^{71} -6.65250e19 q^{73} +7.80721e19 q^{75} +6.19391e19 q^{77} -1.81938e18 q^{79} +1.76631e19 q^{81} -2.58951e20 q^{83} -2.28989e19 q^{85} -2.45490e20 q^{87} -1.80968e20 q^{89} -5.24614e20 q^{91} +1.07678e21 q^{93} +1.73617e20 q^{95} +4.09632e20 q^{97} +1.54524e21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9}+O(q^{10}) $ 2 * q - 105432 * q^3 + 2108140 * q^5 + 444771792 * q^7 + 11072347578 * q^9	$ 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9} + 53806403320 q^{11} - 490366676932 q^{13} - 639834721680 q^{15} - 6593864672092 q^{17} + 19302397925320 q^{19} - 135055584824256 q^{21} - 409737865776272 q^{23} - 940878149007650 q^{25} - 22\!\cdots\!56 q^{27}+ \cdots + 17\!\cdots\!04 q^{99}+O(q^{100}) $ 2 * q - 105432 * q^3 + 2108140 * q^5 + 444771792 * q^7 + 11072347578 * q^9 + 53806403320 * q^11 - 490366676932 * q^13 - 639834721680 * q^15 - 6593864672092 * q^17 + 19302397925320 * q^19 - 135055584824256 * q^21 - 409737865776272 * q^23 - 940878149007650 * q^25 - 2267939450073456 * q^27 - 2404787522145060 * q^29 - 8689907170559168 * q^31 - 16772541164782752 * q^33 + 2701000503537120 * q^35 - 2186204096251860 * q^37 + 139904623071296688 * q^39 + 68178038573558676 * q^41 + 264529652266004024 * q^43 + 67413140092548540 * q^45 + 426494411558622432 * q^47 - 546967577640131534 * q^49 + 1159640456555021520 * q^51 - 3055980275589518132 * q^53 + 335437371694825040 * q^55 - 10702822880368260192 * q^57 + 783424997522814424 * q^59 - 7177279049078597092 * q^61 + 14229493501717343376 * q^63 - 2797969869756700760 * q^65 + 16674123174011538088 * q^67 + 46561318352256083136 * q^69 + 9448263149848716368 * q^71 - 11586140334503007532 * q^73 + 48484754640473875800 * q^75 + 70803926825553273024 * q^77 - 85280702218715897824 * q^79 + 20489906394726573618 * q^81 - 381814622040086245816 * q^83 - 23191160664932407400 * q^85 - 486328037598631978704 * q^87 - 59742932430695979660 * q^89 - 590588534777956016544 * q^91 + 935567489996727394560 * q^93 + 214051632007883873840 * q^95 + 783394660926711950788 * q^97 + 1767190682286719892504 * 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
$3$	−167684.	−1.63952	−0.819761	−	0.572705i	$-0.805894\pi$
$3$	−167684.	−1.63952	−0.819761	+	0.572705i	$0.805894\pi$
$4$	0	0
$5$	3.35342e6	0.153569	0.0767844	−	0.997048i	$-0.475535\pi$
$5$	3.35342e6	0.153569	0.0767844	+	0.997048i	$0.475535\pi$
$6$	0	0
$7$	7.07779e8	0.947040	0.473520	−	0.880783i	$-0.342983\pi$
$7$	7.07779e8	0.947040	0.473520	+	0.880783i	$0.342983\pi$
$8$	0	0
$9$	1.76574e10	1.68803
$10$	0	0
$11$	8.75119e10	1.01729	0.508644	−	0.860977i	$-0.330147\pi$
$11$	8.75119e10	1.01729	0.508644	+	0.860977i	$0.330147\pi$
$12$	0	0
$13$	−7.41212e11	−1.49120	−0.745602	−	0.666391i	$-0.767838\pi$
$13$	−7.41212e11	−1.49120	−0.745602	+	0.666391i	$0.767838\pi$
$14$	0	0
$15$	−5.62314e11	−0.251780
$16$	0	0
$17$	−6.82853e12	−0.821511	−0.410756	−	0.911746i	$-0.634735\pi$
$17$	−6.82853e12	−0.821511	−0.410756	+	0.911746i	$0.634735\pi$
$18$	0	0
$19$	5.17730e13	1.93727	0.968635	−	0.248487i	$-0.0799334\pi$
$19$	5.17730e13	1.93727	0.968635	+	0.248487i	$0.0799334\pi$
$20$	0	0
$21$	−1.18683e14	−1.55269
$22$	0	0
$23$	−3.13428e14	−1.57760	−0.788798	−	0.614653i	$-0.789296\pi$
$23$	−3.13428e14	−1.57760	−0.788798	+	0.614653i	$0.789296\pi$
$24$	0	0
$25$	−4.65592e14	−0.976417
$26$	0	0
$27$	−1.20683e15	−1.12805
$28$	0	0
$29$	1.46400e15	0.646195	0.323097	−	0.946366i	$-0.395276\pi$
$29$	1.46400e15	0.646195	0.323097	+	0.946366i	$0.395276\pi$
$30$	0	0
$31$	−6.42150e15	−1.40714	−0.703572	−	0.710624i	$-0.748413\pi$
$31$	−6.42150e15	−1.40714	−0.703572	+	0.710624i	$0.748413\pi$
$32$	0	0
$33$	−1.46743e16	−1.66787
$34$	0	0
$35$	2.37348e15	0.145436
$36$	0	0
$37$	−6.93987e15	−0.237265	−0.118632	−	0.992938i	$-0.537851\pi$
$37$	−6.93987e15	−0.237265	−0.118632	+	0.992938i	$0.537851\pi$
$38$	0	0
$39$	1.24289e17	2.44486
$40$	0	0
$41$	3.25339e16	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$41$	3.25339e16	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$42$	0	0
$43$	4.37536e16	0.308741	0.154371	−	0.988013i	$-0.450665\pi$
$43$	4.37536e16	0.308741	0.154371	+	0.988013i	$0.450665\pi$
$44$	0	0
$45$	5.92128e16	0.259230
$46$	0	0
$47$	−5.34426e16	−0.148204	−0.0741020	−	0.997251i	$-0.523609\pi$
$47$	−5.34426e16	−0.148204	−0.0741020	+	0.997251i	$0.523609\pi$
$48$	0	0
$49$	−5.75946e16	−0.103115
$50$	0	0
$51$	1.14503e18	1.34689
$52$	0	0
$53$	−1.19249e18	−0.936611	−0.468306	−	0.883567i	$-0.655135\pi$
$53$	−1.19249e18	−0.936611	−0.468306	+	0.883567i	$0.655135\pi$
$54$	0	0
$55$	2.93464e17	0.156224
$56$	0	0
$57$	−8.68148e18	−3.17620
$58$	0	0
$59$	−4.48047e17	−0.114124	−0.0570620	−	0.998371i	$-0.518173\pi$
$59$	−4.48047e17	−0.114124	−0.0570620	+	0.998371i	$0.518173\pi$
$60$	0	0
$61$	−9.05611e17	−0.162547	−0.0812734	−	0.996692i	$-0.525899\pi$
$61$	−9.05611e17	−0.162547	−0.0812734	+	0.996692i	$0.525899\pi$
$62$	0	0
$63$	1.24976e19	1.59864
$64$	0	0
$65$	−2.48560e18	−0.229003
$66$	0	0
$67$	−6.06820e17	−0.0406700	−0.0203350	−	0.999793i	$-0.506473\pi$
$67$	−6.06820e17	−0.0406700	−0.0203350	+	0.999793i	$0.506473\pi$
$68$	0	0
$69$	5.25568e19	2.58650
$70$	0	0
$71$	−2.18611e19	−0.797003	−0.398501	−	0.917168i	$-0.630470\pi$
$71$	−2.18611e19	−0.797003	−0.398501	+	0.917168i	$0.630470\pi$
$72$	0	0
$73$	−6.65250e19	−1.81174	−0.905868	−	0.423559i	$-0.860781\pi$
$73$	−6.65250e19	−1.81174	−0.905868	+	0.423559i	$0.860781\pi$
$74$	0	0
$75$	7.80721e19	1.60086
$76$	0	0
$77$	6.19391e19	0.963413
$78$	0	0
$79$	−1.81938e18	−0.0216191	−0.0108096	−	0.999942i	$-0.503441\pi$
$79$	−1.81938e18	−0.0216191	−0.0108096	+	0.999942i	$0.503441\pi$
$80$	0	0
$81$	1.76631e19	0.161426
$82$	0	0
$83$	−2.58951e20	−1.83188	−0.915940	−	0.401316i	$-0.868553\pi$
$83$	−2.58951e20	−1.83188	−0.915940	+	0.401316i	$0.868553\pi$
$84$	0	0
$85$	−2.28989e19	−0.126159
$86$	0	0
$87$	−2.45490e20	−1.05945
$88$	0	0
$89$	−1.80968e20	−0.615187	−0.307594	−	0.951518i	$-0.599524\pi$
$89$	−1.80968e20	−0.615187	−0.307594	+	0.951518i	$0.599524\pi$
$90$	0	0
$91$	−5.24614e20	−1.41223
$92$	0	0
$93$	1.07678e21	2.30704
$94$	0	0
$95$	1.73617e20	0.297504
$96$	0	0
$97$	4.09632e20	0.564016	0.282008	−	0.959412i	$-0.409000\pi$
$97$	4.09632e20	0.564016	0.282008	+	0.959412i	$0.409000\pi$
$98$	0	0
$99$	1.54524e21	1.71722
$100$	0	0
$101$	−1.49956e21	−1.35080	−0.675398	−	0.737454i	$-0.736028\pi$
$101$	−1.49956e21	−1.35080	−0.675398	+	0.737454i	$0.736028\pi$
$102$	0	0
$103$	5.14179e20	0.376984	0.188492	−	0.982075i	$-0.439640\pi$
$103$	5.14179e20	0.376984	0.188492	+	0.982075i	$0.439640\pi$
$104$	0	0
$105$	−3.97994e20	−0.238445
$106$	0	0
$107$	1.11062e21	0.545801	0.272900	−	0.962042i	$-0.412017\pi$
$107$	1.11062e21	0.545801	0.272900	+	0.962042i	$0.412017\pi$
$108$	0	0
$109$	−1.75592e21	−0.710437	−0.355219	−	0.934783i	$-0.615594\pi$
$109$	−1.75592e21	−0.710437	−0.355219	+	0.934783i	$0.615594\pi$
$110$	0	0
$111$	1.16370e21	0.389001
$112$	0	0
$113$	5.38798e21	1.49315	0.746573	−	0.665303i	$-0.231698\pi$
$113$	5.38798e21	1.49315	0.746573	+	0.665303i	$0.231698\pi$
$114$	0	0
$115$	−1.05106e21	−0.242270
$116$	0	0
$117$	−1.30879e22	−2.51721
$118$	0	0
$119$	−4.83309e21	−0.778004
$120$	0	0
$121$	2.58090e20	0.0348758
$122$	0	0
$123$	−5.45540e21	−0.620616
$124$	0	0
$125$	−3.16036e21	−0.303516
$126$	0	0
$127$	1.11104e22	0.903212	0.451606	−	0.892217i	$-0.350851\pi$
$127$	1.11104e22	0.903212	0.451606	+	0.892217i	$0.350851\pi$
$128$	0	0
$129$	−7.33676e21	−0.506189
$130$	0	0
$131$	2.53513e22	1.48817	0.744084	−	0.668086i	$-0.232886\pi$
$131$	2.53513e22	1.48817	0.744084	+	0.668086i	$0.232886\pi$
$132$	0	0
$133$	3.66438e22	1.83467
$134$	0	0
$135$	−4.04702e21	−0.173233
$136$	0	0
$137$	1.99626e22	0.732235	0.366117	−	0.930569i	$-0.380687\pi$
$137$	1.99626e22	0.732235	0.366117	+	0.930569i	$0.380687\pi$
$138$	0	0
$139$	−2.94117e22	−0.926540	−0.463270	−	0.886217i	$-0.653324\pi$
$139$	−2.94117e22	−0.926540	−0.463270	+	0.886217i	$0.653324\pi$
$140$	0	0
$141$	8.96144e21	0.242984
$142$	0	0
$143$	−6.48649e22	−1.51699
$144$	0	0
$145$	4.90942e21	0.0992354
$146$	0	0
$147$	9.65767e21	0.169060
$148$	0	0
$149$	1.21478e23	1.84520	0.922598	−	0.385762i	$-0.126061\pi$
$149$	1.21478e23	1.84520	0.922598	+	0.385762i	$0.126061\pi$
$150$	0	0
$151$	−5.95169e22	−0.785928	−0.392964	−	0.919554i	$-0.628550\pi$
$151$	−5.95169e22	−0.785928	−0.392964	+	0.919554i	$0.628550\pi$
$152$	0	0
$153$	−1.20574e23	−1.38674
$154$	0	0
$155$	−2.15340e22	−0.216093
$156$	0	0
$157$	−1.28748e23	−1.12926	−0.564630	−	0.825344i	$-0.690981\pi$
$157$	−1.28748e23	−1.12926	−0.564630	+	0.825344i	$0.690981\pi$
$158$	0	0
$159$	1.99962e23	1.53560
$160$	0	0
$161$	−2.21838e23	−1.49405
$162$	0	0
$163$	−5.66875e22	−0.335365	−0.167682	−	0.985841i	$-0.553628\pi$
$163$	−5.66875e22	−0.335365	−0.167682	+	0.985841i	$0.553628\pi$
$164$	0	0
$165$	−4.92092e22	−0.256133
$166$	0	0
$167$	−1.99516e23	−0.915071	−0.457535	−	0.889191i	$-0.651268\pi$
$167$	−1.99516e23	−0.915071	−0.457535	+	0.889191i	$0.651268\pi$
$168$	0	0
$169$	3.02331e23	1.22369
$170$	0	0
$171$	9.14178e23	3.27018
$172$	0	0
$173$	3.56107e23	1.12745	0.563723	−	0.825964i	$-0.309368\pi$
$173$	3.56107e23	1.12745	0.563723	+	0.825964i	$0.309368\pi$
$174$	0	0
$175$	−3.29536e23	−0.924706
$176$	0	0
$177$	7.51301e22	0.187109
$178$	0	0
$179$	1.56436e23	0.346243	0.173122	−	0.984900i	$-0.444615\pi$
$179$	1.56436e23	0.346243	0.173122	+	0.984900i	$0.444615\pi$
$180$	0	0
$181$	−7.11852e23	−1.40205	−0.701027	−	0.713135i	$-0.747275\pi$
$181$	−7.11852e23	−1.40205	−0.701027	+	0.713135i	$0.747275\pi$
$182$	0	0
$183$	1.51856e23	0.266499
$184$	0	0
$185$	−2.32723e22	−0.0364365
$186$	0	0
$187$	−5.97578e23	−0.835714
$188$	0	0
$189$	−8.54171e23	−1.06831
$190$	0	0
$191$	−1.15222e23	−0.129029	−0.0645143	−	0.997917i	$-0.520550\pi$
$191$	−1.15222e23	−0.129029	−0.0645143	+	0.997917i	$0.520550\pi$
$192$	0	0
$193$	9.86701e23	0.990453	0.495227	−	0.868764i	$-0.335085\pi$
$193$	9.86701e23	0.990453	0.495227	+	0.868764i	$0.335085\pi$
$194$	0	0
$195$	4.16794e23	0.375455
$196$	0	0
$197$	−4.06615e23	−0.329070	−0.164535	−	0.986371i	$-0.552612\pi$
$197$	−4.06615e23	−0.329070	−0.164535	+	0.986371i	$0.552612\pi$
$198$	0	0
$199$	−1.80332e24	−1.31255	−0.656276	−	0.754521i	$-0.727869\pi$
$199$	−1.80332e24	−1.31255	−0.656276	+	0.754521i	$0.727869\pi$
$200$	0	0
$201$	1.01754e23	0.0666795
$202$	0	0
$203$	1.03619e24	0.611972
$204$	0	0
$205$	1.09100e23	0.0581311
$206$	0	0
$207$	−5.53434e24	−2.66304
$208$	0	0
$209$	4.53075e24	1.97076
$210$	0	0
$211$	3.74550e23	0.147416	0.0737080	−	0.997280i	$-0.476517\pi$
$211$	3.74550e23	0.147416	0.0737080	+	0.997280i	$0.476517\pi$
$212$	0	0
$213$	3.66575e24	1.30670
$214$	0	0
$215$	1.46724e23	0.0474131
$216$	0	0
$217$	−4.54500e24	−1.33262
$218$	0	0
$219$	1.11552e25	2.97038
$220$	0	0
$221$	5.06139e24	1.22504
$222$	0	0
$223$	3.09034e24	0.680465	0.340232	−	0.940341i	$-0.389494\pi$
$223$	3.09034e24	0.680465	0.340232	+	0.940341i	$0.389494\pi$
$224$	0	0
$225$	−8.22116e24	−1.64823
$226$	0	0
$227$	−9.67101e23	−0.176685	−0.0883426	−	0.996090i	$-0.528157\pi$
$227$	−9.67101e23	−0.176685	−0.0883426	+	0.996090i	$0.528157\pi$
$228$	0	0
$229$	−3.37848e24	−0.562922	−0.281461	−	0.959573i	$-0.590819\pi$
$229$	−3.37848e24	−0.562922	−0.281461	+	0.959573i	$0.590819\pi$
$230$	0	0
$231$	−1.03862e25	−1.57954
$232$	0	0
$233$	−8.86566e24	−1.23161	−0.615806	−	0.787898i	$-0.711169\pi$
$233$	−8.86566e24	−1.23161	−0.615806	+	0.787898i	$0.711169\pi$
$234$	0	0
$235$	−1.79215e23	−0.0227595
$236$	0	0
$237$	3.05079e23	0.0354450
$238$	0	0
$239$	9.61276e24	1.02252	0.511258	−	0.859427i	$-0.329180\pi$
$239$	9.61276e24	1.02252	0.511258	+	0.859427i	$0.329180\pi$
$240$	0	0
$241$	−1.04750e25	−1.02088	−0.510442	−	0.859912i	$-0.670518\pi$
$241$	−1.04750e25	−1.02088	−0.510442	+	0.859912i	$0.670518\pi$
$242$	0	0
$243$	9.66209e24	0.863386
$244$	0	0
$245$	−1.93139e23	−0.0158353
$246$	0	0
$247$	−3.83747e25	−2.88887
$248$	0	0
$249$	4.34218e25	3.00341
$250$	0	0
$251$	2.60451e25	1.65635	0.828174	−	0.560471i	$-0.189380\pi$
$251$	2.60451e25	1.65635	0.828174	+	0.560471i	$0.189380\pi$
$252$	0	0
$253$	−2.74287e25	−1.60487
$254$	0	0
$255$	3.83978e24	0.206840
$256$	0	0
$257$	−1.83013e25	−0.908208	−0.454104	−	0.890949i	$-0.650041\pi$
$257$	−1.83013e25	−0.908208	−0.454104	+	0.890949i	$0.650041\pi$
$258$	0	0
$259$	−4.91189e24	−0.224699
$260$	0	0
$261$	2.58506e25	1.09080
$262$	0	0
$263$	7.36593e24	0.286874	0.143437	−	0.989659i	$-0.454184\pi$
$263$	7.36593e24	0.286874	0.143437	+	0.989659i	$0.454184\pi$
$264$	0	0
$265$	−3.99893e24	−0.143834
$266$	0	0
$267$	3.03454e25	1.00861
$268$	0	0
$269$	2.31544e25	0.711599	0.355800	−	0.934562i	$-0.384209\pi$
$269$	2.31544e25	0.711599	0.355800	+	0.934562i	$0.384209\pi$
$270$	0	0
$271$	1.17274e25	0.333445	0.166722	−	0.986004i	$-0.446682\pi$
$271$	1.17274e25	0.333445	0.166722	+	0.986004i	$0.446682\pi$
$272$	0	0
$273$	8.79692e25	2.31538
$274$	0	0
$275$	−4.07448e25	−0.993297
$276$	0	0
$277$	−5.46772e25	−1.23529	−0.617644	−	0.786458i	$-0.711913\pi$
$277$	−5.46772e25	−1.23529	−0.617644	+	0.786458i	$0.711913\pi$
$278$	0	0
$279$	−1.13387e26	−2.37531
$280$	0	0
$281$	6.73811e24	0.130955	0.0654775	−	0.997854i	$-0.479143\pi$
$281$	6.73811e24	0.130955	0.0654775	+	0.997854i	$0.479143\pi$
$282$	0	0
$283$	−3.72254e25	−0.671555	−0.335778	−	0.941941i	$-0.608999\pi$
$283$	−3.72254e25	−0.671555	−0.335778	+	0.941941i	$0.608999\pi$
$284$	0	0
$285$	−2.91127e25	−0.487765
$286$	0	0
$287$	2.30268e25	0.358487
$288$	0	0
$289$	−2.24631e25	−0.325119
$290$	0	0
$291$	−6.86886e25	−0.924716
$292$	0	0
$293$	6.42074e24	0.0804406	0.0402203	−	0.999191i	$-0.487194\pi$
$293$	6.42074e24	0.0804406	0.0402203	+	0.999191i	$0.487194\pi$
$294$	0	0
$295$	−1.50249e24	−0.0175259
$296$	0	0
$297$	−1.05612e26	−1.14755
$298$	0	0
$299$	2.32317e26	2.35252
$300$	0	0
$301$	3.09679e25	0.292390
$302$	0	0
$303$	2.51452e26	2.21466
$304$	0	0
$305$	−3.03690e24	−0.0249621
$306$	0	0
$307$	−7.52469e25	−0.577478	−0.288739	−	0.957408i	$-0.593236\pi$
$307$	−7.52469e25	−0.577478	−0.288739	+	0.957408i	$0.593236\pi$
$308$	0	0
$309$	−8.62194e25	−0.618074
$310$	0	0
$311$	−2.48646e25	−0.166570	−0.0832850	−	0.996526i	$-0.526541\pi$
$311$	−2.48646e25	−0.166570	−0.0832850	+	0.996526i	$0.526541\pi$
$312$	0	0
$313$	−1.07554e26	−0.673615	−0.336808	−	0.941574i	$-0.609347\pi$
$313$	−1.07554e26	−0.673615	−0.336808	+	0.941574i	$0.609347\pi$
$314$	0	0
$315$	4.19096e25	0.245501
$316$	0	0
$317$	2.82324e26	1.54748	0.773742	−	0.633501i	$-0.218383\pi$
$317$	2.82324e26	1.54748	0.773742	+	0.633501i	$0.218383\pi$
$318$	0	0
$319$	1.28118e26	0.657366
$320$	0	0
$321$	−1.86232e26	−0.894853
$322$	0	0
$323$	−3.53533e26	−1.59149
$324$	0	0
$325$	3.45102e26	1.45604
$326$	0	0
$327$	2.94438e26	1.16478
$328$	0	0
$329$	−3.78255e25	−0.140355
$330$	0	0
$331$	−8.70208e25	−0.302991	−0.151495	−	0.988458i	$-0.548409\pi$
$331$	−8.70208e25	−0.302991	−0.151495	+	0.988458i	$0.548409\pi$
$332$	0	0
$333$	−1.22540e26	−0.400511
$334$	0	0
$335$	−2.03492e24	−0.00624565
$336$	0	0
$337$	1.22441e26	0.353030	0.176515	−	0.984298i	$-0.443518\pi$
$337$	1.22441e26	0.353030	0.176515	+	0.984298i	$0.443518\pi$
$338$	0	0
$339$	−9.03476e26	−2.44805
$340$	0	0
$341$	−5.61958e26	−1.43147
$342$	0	0
$343$	−4.36091e26	−1.04469
$344$	0	0
$345$	1.76245e26	0.397206
$346$	0	0
$347$	3.10647e26	0.658882	0.329441	−	0.944176i	$-0.393140\pi$
$347$	3.10647e26	0.658882	0.329441	+	0.944176i	$0.393140\pi$
$348$	0	0
$349$	−3.62929e26	−0.724695	−0.362347	−	0.932043i	$-0.618025\pi$
$349$	−3.62929e26	−0.724695	−0.362347	+	0.932043i	$0.618025\pi$
$350$	0	0
$351$	8.94519e26	1.68215
$352$	0	0
$353$	−9.02974e25	−0.159971	−0.0799854	−	0.996796i	$-0.525487\pi$
$353$	−9.02974e25	−0.159971	−0.0799854	+	0.996796i	$0.525487\pi$
$354$	0	0
$355$	−7.33096e25	−0.122395
$356$	0	0
$357$	8.10430e26	1.27556
$358$	0	0
$359$	−5.84199e26	−0.867100	−0.433550	−	0.901130i	$-0.642739\pi$
$359$	−5.84199e26	−0.867100	−0.433550	+	0.901130i	$0.642739\pi$
$360$	0	0
$361$	1.96623e27	2.75302
$362$	0	0
$363$	−4.32774e25	−0.0571796
$364$	0	0
$365$	−2.23087e26	−0.278226
$366$	0	0
$367$	7.68446e26	0.904940	0.452470	−	0.891780i	$-0.350543\pi$
$367$	7.68446e26	0.904940	0.452470	+	0.891780i	$0.350543\pi$
$368$	0	0
$369$	5.74465e26	0.638979
$370$	0	0
$371$	−8.44022e26	−0.887008
$372$	0	0
$373$	−6.12745e26	−0.608607	−0.304304	−	0.952575i	$-0.598424\pi$
$373$	−6.12745e26	−0.608607	−0.304304	+	0.952575i	$0.598424\pi$
$374$	0	0
$375$	5.29941e26	0.497621
$376$	0	0
$377$	−1.08514e27	−0.963609
$378$	0	0
$379$	−1.02134e27	−0.857945	−0.428972	−	0.903318i	$-0.641124\pi$
$379$	−1.02134e27	−0.857945	−0.428972	+	0.903318i	$0.641124\pi$
$380$	0	0
$381$	−1.86303e27	−1.48084
$382$	0	0
$383$	2.10822e27	1.58609	0.793045	−	0.609163i	$-0.208494\pi$
$383$	2.10822e27	1.58609	0.793045	+	0.609163i	$0.208494\pi$
$384$	0	0
$385$	2.07708e26	0.147950
$386$	0	0
$387$	7.72576e26	0.521166
$388$	0	0
$389$	5.16494e26	0.330061	0.165030	−	0.986288i	$-0.447228\pi$
$389$	5.16494e26	0.330061	0.165030	+	0.986288i	$0.447228\pi$
$390$	0	0
$391$	2.14025e27	1.29601
$392$	0	0
$393$	−4.25100e27	−2.43989
$394$	0	0
$395$	−6.10113e24	−0.00332002
$396$	0	0
$397$	2.07736e27	1.07204	0.536019	−	0.844206i	$-0.319927\pi$
$397$	2.07736e27	1.07204	0.536019	+	0.844206i	$0.319927\pi$
$398$	0	0
$399$	−6.14457e27	−3.00799
$400$	0	0
$401$	−3.03566e27	−1.41006	−0.705029	−	0.709179i	$-0.749066\pi$
$401$	−3.03566e27	−1.41006	−0.705029	+	0.709179i	$0.749066\pi$
$402$	0	0
$403$	4.75969e27	2.09834
$404$	0	0
$405$	5.92318e25	0.0247900
$406$	0	0
$407$	−6.07321e26	−0.241367
$408$	0	0
$409$	1.38404e27	0.522462	0.261231	−	0.965276i	$-0.415872\pi$
$409$	1.38404e27	0.522462	0.261231	+	0.965276i	$0.415872\pi$
$410$	0	0
$411$	−3.34739e27	−1.20052
$412$	0	0
$413$	−3.17118e26	−0.108080
$414$	0	0
$415$	−8.68370e26	−0.281320
$416$	0	0
$417$	4.93185e27	1.51908
$418$	0	0
$419$	2.05547e27	0.602093	0.301046	−	0.953610i	$-0.402664\pi$
$419$	2.05547e27	0.602093	0.301046	+	0.953610i	$0.402664\pi$
$420$	0	0
$421$	4.61061e27	1.28468	0.642341	−	0.766419i	$-0.277963\pi$
$421$	4.61061e27	1.28468	0.642341	+	0.766419i	$0.277963\pi$
$422$	0	0
$423$	−9.43659e26	−0.250173
$424$	0	0
$425$	3.17931e27	0.802137
$426$	0	0
$427$	−6.40972e26	−0.153938
$428$	0	0
$429$	1.08768e28	2.48713
$430$	0	0
$431$	5.49501e27	1.19662	0.598312	−	0.801264i	$-0.295839\pi$
$431$	5.49501e27	1.19662	0.598312	+	0.801264i	$0.295839\pi$
$432$	0	0
$433$	−5.65719e27	−1.17349	−0.586743	−	0.809773i	$-0.699590\pi$
$433$	−5.65719e27	−1.17349	−0.586743	+	0.809773i	$0.699590\pi$
$434$	0	0
$435$	−8.23230e26	−0.162699
$436$	0	0
$437$	−1.62271e28	−3.05623
$438$	0	0
$439$	−2.13975e27	−0.384136	−0.192068	−	0.981382i	$-0.561519\pi$
$439$	−2.13975e27	−0.384136	−0.192068	+	0.981382i	$0.561519\pi$
$440$	0	0
$441$	−1.01697e27	−0.174062
$442$	0	0
$443$	5.22577e27	0.852926	0.426463	−	0.904505i	$-0.359759\pi$
$443$	5.22577e27	0.852926	0.426463	+	0.904505i	$0.359759\pi$
$444$	0	0
$445$	−6.06863e26	−0.0944736
$446$	0	0
$447$	−2.03699e28	−3.02524
$448$	0	0
$449$	3.58499e27	0.508044	0.254022	−	0.967198i	$-0.418246\pi$
$449$	3.58499e27	0.508044	0.254022	+	0.967198i	$0.418246\pi$
$450$	0	0
$451$	2.84710e27	0.385079
$452$	0	0
$453$	9.98001e27	1.28855
$454$	0	0
$455$	−1.75925e27	−0.216875
$456$	0	0
$457$	1.08470e27	0.127700	0.0638498	−	0.997960i	$-0.479662\pi$
$457$	1.08470e27	0.127700	0.0638498	+	0.997960i	$0.479662\pi$
$458$	0	0
$459$	8.24089e27	0.926704
$460$	0	0
$461$	−1.61325e28	−1.73317	−0.866585	−	0.499029i	$-0.833690\pi$
$461$	−1.61325e28	−1.73317	−0.866585	+	0.499029i	$0.833690\pi$
$462$	0	0
$463$	−1.26030e28	−1.29382	−0.646909	−	0.762567i	$-0.723939\pi$
$463$	−1.26030e28	−1.29382	−0.646909	+	0.762567i	$0.723939\pi$
$464$	0	0
$465$	3.61090e27	0.354290
$466$	0	0
$467$	1.01065e28	0.947923	0.473961	−	0.880546i	$-0.342824\pi$
$467$	1.01065e28	0.947923	0.473961	+	0.880546i	$0.342824\pi$
$468$	0	0
$469$	−4.29495e26	−0.0385162
$470$	0	0
$471$	2.15889e28	1.85145
$472$	0	0
$473$	3.82896e27	0.314079
$474$	0	0
$475$	−2.41051e28	−1.89158
$476$	0	0
$477$	−2.10564e28	−1.58103
$478$	0	0
$479$	−7.85064e27	−0.564134	−0.282067	−	0.959395i	$-0.591020\pi$
$479$	−7.85064e27	−0.564134	−0.282067	+	0.959395i	$0.591020\pi$
$480$	0	0
$481$	5.14391e27	0.353810
$482$	0	0
$483$	3.71986e28	2.44952
$484$	0	0
$485$	1.37367e27	0.0866152
$486$	0	0
$487$	1.35819e28	0.820172	0.410086	−	0.912047i	$-0.365499\pi$
$487$	1.35819e28	0.820172	0.410086	+	0.912047i	$0.365499\pi$
$488$	0	0
$489$	9.50556e27	0.549838
$490$	0	0
$491$	3.02246e28	1.67496	0.837482	−	0.546465i	$-0.184027\pi$
$491$	3.02246e28	1.67496	0.837482	+	0.546465i	$0.184027\pi$
$492$	0	0
$493$	−9.99700e27	−0.530856
$494$	0	0
$495$	5.18183e27	0.263711
$496$	0	0
$497$	−1.54728e28	−0.754793
$498$	0	0
$499$	−1.52172e28	−0.711669	−0.355835	−	0.934549i	$-0.615803\pi$
$499$	−1.52172e28	−0.711669	−0.355835	+	0.934549i	$0.615803\pi$
$500$	0	0
$501$	3.34556e28	1.50028
$502$	0	0
$503$	9.52959e27	0.409836	0.204918	−	0.978779i	$-0.434307\pi$
$503$	9.52959e27	0.409836	0.204918	+	0.978779i	$0.434307\pi$
$504$	0	0
$505$	−5.02866e27	−0.207440
$506$	0	0
$507$	−5.06959e28	−2.00627
$508$	0	0
$509$	3.87204e28	1.47029	0.735146	−	0.677909i	$-0.237114\pi$
$509$	3.87204e28	1.47029	0.735146	+	0.677909i	$0.237114\pi$
$510$	0	0
$511$	−4.70850e28	−1.71579
$512$	0	0
$513$	−6.24813e28	−2.18533
$514$	0	0
$515$	1.72426e27	0.0578931
$516$	0	0
$517$	−4.67686e27	−0.150766
$518$	0	0
$519$	−5.97132e28	−1.84847
$520$	0	0
$521$	8.03085e26	0.0238762	0.0119381	−	0.999929i	$-0.496200\pi$
$521$	8.03085e26	0.0238762	0.0119381	+	0.999929i	$0.496200\pi$
$522$	0	0
$523$	−1.91554e28	−0.547047	−0.273523	−	0.961865i	$-0.588189\pi$
$523$	−1.91554e28	−0.547047	−0.273523	+	0.961865i	$0.588189\pi$
$524$	0	0
$525$	5.52578e28	1.51608
$526$	0	0
$527$	4.38494e28	1.15598
$528$	0	0
$529$	5.87656e28	1.48881
$530$	0	0
$531$	−7.91136e27	−0.192645
$532$	0	0
$533$	−2.41145e28	−0.564472
$534$	0	0
$535$	3.72436e27	0.0838180
$536$	0	0
$537$	−2.62318e28	−0.567673
$538$	0	0
$539$	−5.04021e27	−0.104898
$540$	0	0
$541$	−3.68313e27	−0.0737303	−0.0368651	−	0.999320i	$-0.511737\pi$
$541$	−3.68313e27	−0.0737303	−0.0368651	+	0.999320i	$0.511737\pi$
$542$	0	0
$543$	1.19366e29	2.29870
$544$	0	0
$545$	−5.88833e27	−0.109101
$546$	0	0
$547$	−4.83617e28	−0.862253	−0.431127	−	0.902291i	$-0.641884\pi$
$547$	−4.83617e28	−0.862253	−0.431127	+	0.902291i	$0.641884\pi$
$548$	0	0
$549$	−1.59908e28	−0.274385
$550$	0	0
$551$	7.57959e28	1.25185
$552$	0	0
$553$	−1.28772e27	−0.0204742
$554$	0	0
$555$	3.90238e27	0.0597384
$556$	0	0
$557$	−6.17549e28	−0.910316	−0.455158	−	0.890411i	$-0.650417\pi$
$557$	−6.17549e28	−0.910316	−0.455158	+	0.890411i	$0.650417\pi$
$558$	0	0
$559$	−3.24307e28	−0.460397
$560$	0	0
$561$	1.00204e29	1.37017
$562$	0	0
$563$	−3.51350e28	−0.462809	−0.231405	−	0.972858i	$-0.574332\pi$
$563$	−3.51350e28	−0.462809	−0.231405	+	0.972858i	$0.574332\pi$
$564$	0	0
$565$	1.80682e28	0.229301
$566$	0	0
$567$	1.25016e28	0.152877
$568$	0	0
$569$	1.45054e28	0.170943	0.0854716	−	0.996341i	$-0.472760\pi$
$569$	1.45054e28	0.170943	0.0854716	+	0.996341i	$0.472760\pi$
$570$	0	0
$571$	6.07413e28	0.689930	0.344965	−	0.938616i	$-0.387891\pi$
$571$	6.07413e28	0.689930	0.344965	+	0.938616i	$0.387891\pi$
$572$	0	0
$573$	1.93209e28	0.211545
$574$	0	0
$575$	1.45930e29	1.54039
$576$	0	0
$577$	2.71266e28	0.276089	0.138044	−	0.990426i	$-0.455918\pi$
$577$	2.71266e28	0.276089	0.138044	+	0.990426i	$0.455918\pi$
$578$	0	0
$579$	−1.65454e29	−1.62387
$580$	0	0
$581$	−1.83280e29	−1.73486
$582$	0	0
$583$	−1.04357e29	−0.952804
$584$	0	0
$585$	−4.38893e28	−0.386564
$586$	0	0
$587$	2.51262e28	0.213514	0.106757	−	0.994285i	$-0.465953\pi$
$587$	2.51262e28	0.213514	0.106757	+	0.994285i	$0.465953\pi$
$588$	0	0
$589$	−3.32460e29	−2.72602
$590$	0	0
$591$	6.81827e28	0.539517
$592$	0	0
$593$	−1.95530e29	−1.49328	−0.746638	−	0.665231i	$-0.768333\pi$
$593$	−1.95530e29	−1.49328	−0.746638	+	0.665231i	$0.768333\pi$
$594$	0	0
$595$	−1.62074e28	−0.119477
$596$	0	0
$597$	3.02388e29	2.15196
$598$	0	0
$599$	−1.11818e29	−0.768299	−0.384149	−	0.923271i	$-0.625505\pi$
$599$	−1.11818e29	−0.768299	−0.384149	+	0.923271i	$0.625505\pi$
$600$	0	0
$601$	2.32726e29	1.54405	0.772027	−	0.635590i	$-0.219243\pi$
$601$	2.32726e29	1.54405	0.772027	+	0.635590i	$0.219243\pi$
$602$	0	0
$603$	−1.07149e28	−0.0686525
$604$	0	0
$605$	8.65483e26	0.00535583
$606$	0	0
$607$	2.39690e29	1.43274	0.716371	−	0.697720i	$-0.245802\pi$
$607$	2.39690e29	1.43274	0.716371	+	0.697720i	$0.245802\pi$
$608$	0	0
$609$	−1.73752e29	−1.00334
$610$	0	0
$611$	3.96123e28	0.221002
$612$	0	0
$613$	1.89238e29	1.02017	0.510086	−	0.860123i	$-0.329613\pi$
$613$	1.89238e29	1.02017	0.510086	+	0.860123i	$0.329613\pi$
$614$	0	0
$615$	−1.82943e28	−0.0953072
$616$	0	0
$617$	1.13109e29	0.569510	0.284755	−	0.958600i	$-0.408088\pi$
$617$	1.13109e29	0.569510	0.284755	+	0.958600i	$0.408088\pi$
$618$	0	0
$619$	1.89565e29	0.922585	0.461293	−	0.887248i	$-0.347386\pi$
$619$	1.89565e29	0.922585	0.461293	+	0.887248i	$0.347386\pi$
$620$	0	0
$621$	3.78255e29	1.77960
$622$	0	0
$623$	−1.28086e29	−0.582607
$624$	0	0
$625$	2.11413e29	0.929806
$626$	0	0
$627$	−7.59733e29	−3.23111
$628$	0	0
$629$	4.73891e28	0.194916
$630$	0	0
$631$	1.39864e29	0.556415	0.278208	−	0.960521i	$-0.410260\pi$
$631$	1.39864e29	0.556415	0.278208	+	0.960521i	$0.410260\pi$
$632$	0	0
$633$	−6.28059e28	−0.241692
$634$	0	0
$635$	3.72578e28	0.138705
$636$	0	0
$637$	4.26898e28	0.153766
$638$	0	0
$639$	−3.86011e29	−1.34537
$640$	0	0
$641$	−7.73898e28	−0.261020	−0.130510	−	0.991447i	$-0.541662\pi$
$641$	−7.73898e28	−0.261020	−0.130510	+	0.991447i	$0.541662\pi$
$642$	0	0
$643$	2.02401e29	0.660691	0.330345	−	0.943860i	$-0.392835\pi$
$643$	2.02401e29	0.660691	0.330345	+	0.943860i	$0.392835\pi$
$644$	0	0
$645$	−2.46032e28	−0.0777348
$646$	0	0
$647$	−6.36949e28	−0.194809	−0.0974046	−	0.995245i	$-0.531054\pi$
$647$	−6.36949e28	−0.194809	−0.0974046	+	0.995245i	$0.531054\pi$
$648$	0	0
$649$	−3.92094e28	−0.116097
$650$	0	0
$651$	7.62122e29	2.18486
$652$	0	0
$653$	−1.78655e29	−0.495938	−0.247969	−	0.968768i	$-0.579763\pi$
$653$	−1.78655e29	−0.495938	−0.247969	+	0.968768i	$0.579763\pi$
$654$	0	0
$655$	8.50137e28	0.228536
$656$	0	0
$657$	−1.17466e30	−3.05827
$658$	0	0
$659$	5.55707e29	1.40136	0.700678	−	0.713477i	$-0.252881\pi$
$659$	5.55707e29	1.40136	0.700678	+	0.713477i	$0.252881\pi$
$660$	0	0
$661$	2.72969e29	0.666804	0.333402	−	0.942785i	$-0.391803\pi$
$661$	2.72969e29	0.666804	0.333402	+	0.942785i	$0.391803\pi$
$662$	0	0
$663$	−8.48712e29	−2.00848
$664$	0	0
$665$	1.22882e29	0.281749
$666$	0	0
$667$	−4.58860e29	−1.01943
$668$	0	0
$669$	−5.18200e29	−1.11564
$670$	0	0
$671$	−7.92518e28	−0.165357
$672$	0	0
$673$	3.87928e29	0.784499	0.392250	−	0.919859i	$-0.371697\pi$
$673$	3.87928e29	0.784499	0.392250	+	0.919859i	$0.371697\pi$
$674$	0	0
$675$	5.61892e29	1.10145
$676$	0	0
$677$	−7.66988e29	−1.45750	−0.728748	−	0.684782i	$-0.759898\pi$
$677$	−7.66988e29	−1.45750	−0.728748	+	0.684782i	$0.759898\pi$
$678$	0	0
$679$	2.89929e29	0.534145
$680$	0	0
$681$	1.62167e29	0.289680
$682$	0	0
$683$	−5.57644e29	−0.965917	−0.482959	−	0.875643i	$-0.660438\pi$
$683$	−5.57644e29	−0.965917	−0.482959	+	0.875643i	$0.660438\pi$
$684$	0	0
$685$	6.69429e28	0.112448
$686$	0	0
$687$	5.66515e29	0.922923
$688$	0	0
$689$	8.83890e29	1.39668
$690$	0	0
$691$	6.52122e29	0.999561	0.499781	−	0.866152i	$-0.333414\pi$
$691$	6.52122e29	0.999561	0.499781	+	0.866152i	$0.333414\pi$
$692$	0	0
$693$	1.09369e30	1.62627
$694$	0	0
$695$	−9.86297e28	−0.142288
$696$	0	0
$697$	−2.22158e29	−0.310970
$698$	0	0
$699$	1.48663e30	2.01925
$700$	0	0
$701$	1.49391e29	0.196917	0.0984586	−	0.995141i	$-0.468609\pi$
$701$	1.49391e29	0.196917	0.0984586	+	0.995141i	$0.468609\pi$
$702$	0	0
$703$	−3.59297e29	−0.459646
$704$	0	0
$705$	3.00515e28	0.0373147
$706$	0	0
$707$	−1.06136e30	−1.27926
$708$	0	0
$709$	−1.42826e30	−1.67117	−0.835585	−	0.549361i	$-0.814871\pi$
$709$	−1.42826e30	−1.67117	−0.835585	+	0.549361i	$0.814871\pi$
$710$	0	0
$711$	−3.21255e28	−0.0364938
$712$	0	0
$713$	2.01268e30	2.21990
$714$	0	0
$715$	−2.17519e29	−0.232962
$716$	0	0
$717$	−1.61190e30	−1.67644
$718$	0	0
$719$	−1.67477e30	−1.69162	−0.845810	−	0.533484i	$-0.820882\pi$
$719$	−1.67477e30	−1.69162	−0.845810	+	0.533484i	$0.820882\pi$
$720$	0	0
$721$	3.63925e29	0.357019
$722$	0	0
$723$	1.75649e30	1.67376
$724$	0	0
$725$	−6.81628e29	−0.630955
$726$	0	0
$727$	−1.28121e30	−1.15215	−0.576073	−	0.817398i	$-0.695416\pi$
$727$	−1.28121e30	−1.15215	−0.576073	+	0.817398i	$0.695416\pi$
$728$	0	0
$729$	−1.80494e30	−1.57697
$730$	0	0
$731$	−2.98773e29	−0.253634
$732$	0	0
$733$	−3.55290e29	−0.293083	−0.146542	−	0.989205i	$-0.546814\pi$
$733$	−3.55290e29	−0.293083	−0.146542	+	0.989205i	$0.546814\pi$
$734$	0	0
$735$	3.23862e28	0.0259623
$736$	0	0
$737$	−5.31040e28	−0.0413732
$738$	0	0
$739$	3.02616e29	0.229153	0.114576	−	0.993414i	$-0.463449\pi$
$739$	3.02616e29	0.229153	0.114576	+	0.993414i	$0.463449\pi$
$740$	0	0
$741$	6.43482e30	4.73636
$742$	0	0
$743$	3.53318e29	0.252803	0.126402	−	0.991979i	$-0.459657\pi$
$743$	3.53318e29	0.252803	0.126402	+	0.991979i	$0.459657\pi$
$744$	0	0
$745$	4.07368e29	0.283365
$746$	0	0
$747$	−4.57240e30	−3.09228
$748$	0	0
$749$	7.86071e29	0.516895
$750$	0	0
$751$	−1.66286e30	−1.06325	−0.531625	−	0.846980i	$-0.678419\pi$
$751$	−1.66286e30	−1.06325	−0.531625	+	0.846980i	$0.678419\pi$
$752$	0	0
$753$	−4.36733e30	−2.71562
$754$	0	0
$755$	−1.99585e29	−0.120694
$756$	0	0
$757$	−1.26493e30	−0.743979	−0.371989	−	0.928237i	$-0.621324\pi$
$757$	−1.26493e30	−0.743979	−0.371989	+	0.928237i	$0.621324\pi$
$758$	0	0
$759$	4.59934e30	2.63122
$760$	0	0
$761$	3.19527e30	1.77815	0.889075	−	0.457762i	$-0.151349\pi$
$761$	3.19527e30	1.77815	0.889075	+	0.457762i	$0.151349\pi$
$762$	0	0
$763$	−1.24280e30	−0.672812
$764$	0	0
$765$	−4.04337e29	−0.212960
$766$	0	0
$767$	3.32098e29	0.170182
$768$	0	0
$769$	−2.81246e30	−1.40236	−0.701182	−	0.712983i	$-0.747344\pi$
$769$	−2.81246e30	−1.40236	−0.701182	+	0.712983i	$0.747344\pi$
$770$	0	0
$771$	3.06884e30	1.48903
$772$	0	0
$773$	2.43052e30	1.14766	0.573831	−	0.818974i	$-0.305457\pi$
$773$	2.43052e30	1.14766	0.573831	+	0.818974i	$0.305457\pi$
$774$	0	0
$775$	2.98980e30	1.37396
$776$	0	0
$777$	8.23644e29	0.368399
$778$	0	0
$779$	1.68438e30	0.733323
$780$	0	0
$781$	−1.91311e30	−0.810782
$782$	0	0
$783$	−1.76681e30	−0.728939
$784$	0	0
$785$	−4.31746e29	−0.173419
$786$	0	0
$787$	2.79697e30	1.09384	0.546919	−	0.837185i	$-0.315800\pi$
$787$	2.79697e30	1.09384	0.546919	+	0.837185i	$0.315800\pi$
$788$	0	0
$789$	−1.23514e30	−0.470337
$790$	0	0
$791$	3.81350e30	1.41407
$792$	0	0
$793$	6.71250e29	0.242390
$794$	0	0
$795$	6.70556e29	0.235820
$796$	0	0
$797$	−2.37782e30	−0.814456	−0.407228	−	0.913327i	$-0.633504\pi$
$797$	−2.37782e30	−0.814456	−0.407228	+	0.913327i	$0.633504\pi$
$798$	0	0
$799$	3.64934e29	0.121751
$800$	0	0
$801$	−3.19544e30	−1.03846
$802$	0	0
$803$	−5.82173e30	−1.84306
$804$	0	0
$805$	−7.43916e29	−0.229439
$806$	0	0
$807$	−3.88262e30	−1.16668
$808$	0	0
$809$	6.13505e29	0.179622	0.0898108	−	0.995959i	$-0.471374\pi$
$809$	6.13505e29	0.179622	0.0898108	+	0.995959i	$0.471374\pi$
$810$	0	0
$811$	−4.24348e30	−1.21061	−0.605303	−	0.795995i	$-0.706948\pi$
$811$	−4.24348e30	−1.21061	−0.605303	+	0.795995i	$0.706948\pi$
$812$	0	0
$813$	−1.96649e30	−0.546690
$814$	0	0
$815$	−1.90097e29	−0.0515016
$816$	0	0
$817$	2.26525e30	0.598116
$818$	0	0
$819$	−9.26335e30	−2.38389
$820$	0	0
$821$	2.33090e30	0.584683	0.292342	−	0.956314i	$-0.405566\pi$
$821$	2.33090e30	0.584683	0.292342	+	0.956314i	$0.405566\pi$
$822$	0	0
$823$	4.92628e30	1.20454	0.602270	−	0.798292i	$-0.294263\pi$
$823$	4.92628e30	1.20454	0.602270	+	0.798292i	$0.294263\pi$
$824$	0	0
$825$	6.83224e30	1.62853
$826$	0	0
$827$	3.48099e30	0.808899	0.404450	−	0.914560i	$-0.367463\pi$
$827$	3.48099e30	0.808899	0.404450	+	0.914560i	$0.367463\pi$
$828$	0	0
$829$	−7.46934e30	−1.69223	−0.846116	−	0.532999i	$-0.821065\pi$
$829$	−7.46934e30	−1.69223	−0.846116	+	0.532999i	$0.821065\pi$
$830$	0	0
$831$	9.16847e30	2.02528
$832$	0	0
$833$	3.93286e29	0.0847103
$834$	0	0
$835$	−6.69061e29	−0.140526
$836$	0	0
$837$	7.74968e30	1.58733
$838$	0	0
$839$	−7.84188e30	−1.56646	−0.783231	−	0.621731i	$-0.786430\pi$
$839$	−7.84188e30	−1.56646	−0.783231	+	0.621731i	$0.786430\pi$
$840$	0	0
$841$	−2.98953e30	−0.582432
$842$	0	0
$843$	−1.12987e30	−0.214704
$844$	0	0
$845$	1.01384e30	0.187921
$846$	0	0
$847$	1.82670e29	0.0330288
$848$	0	0
$849$	6.24209e30	1.10103
$850$	0	0
$851$	2.17515e30	0.374308
$852$	0	0
$853$	−1.56244e30	−0.262324	−0.131162	−	0.991361i	$-0.541871\pi$
$853$	−1.56244e30	−0.262324	−0.131162	+	0.991361i	$0.541871\pi$
$854$	0	0
$855$	3.06562e30	0.502198
$856$	0	0
$857$	3.39912e29	0.0543336	0.0271668	−	0.999631i	$-0.491351\pi$
$857$	3.39912e29	0.0543336	0.0271668	+	0.999631i	$0.491351\pi$
$858$	0	0
$859$	−3.06417e30	−0.477953	−0.238976	−	0.971025i	$-0.576812\pi$
$859$	−3.06417e30	−0.477953	−0.238976	+	0.971025i	$0.576812\pi$
$860$	0	0
$861$	−3.86122e30	−0.587748
$862$	0	0
$863$	1.16441e30	0.172979	0.0864896	−	0.996253i	$-0.472435\pi$
$863$	1.16441e30	0.172979	0.0864896	+	0.996253i	$0.472435\pi$
$864$	0	0
$865$	1.19418e30	0.173141
$866$	0	0
$867$	3.76670e30	0.533041
$868$	0	0
$869$	−1.59217e29	−0.0219929
$870$	0	0
$871$	4.49783e29	0.0606474
$872$	0	0
$873$	7.23306e30	0.952078
$874$	0	0
$875$	−2.23684e30	−0.287442
$876$	0	0
$877$	1.22709e31	1.53950	0.769752	−	0.638343i	$-0.220380\pi$
$877$	1.22709e31	1.53950	0.769752	+	0.638343i	$0.220380\pi$
$878$	0	0
$879$	−1.07665e30	−0.131884
$880$	0	0
$881$	−7.15175e30	−0.855392	−0.427696	−	0.903923i	$-0.640675\pi$
$881$	−7.15175e30	−0.855392	−0.427696	+	0.903923i	$0.640675\pi$
$882$	0	0
$883$	4.43259e30	0.517691	0.258845	−	0.965919i	$-0.416658\pi$
$883$	4.43259e30	0.517691	0.258845	+	0.965919i	$0.416658\pi$
$884$	0	0
$885$	2.51943e29	0.0287341
$886$	0	0
$887$	−1.71592e31	−1.91117	−0.955586	−	0.294712i	$-0.904776\pi$
$887$	−1.71592e31	−1.91117	−0.955586	+	0.294712i	$0.904776\pi$
$888$	0	0
$889$	7.86369e30	0.855378
$890$	0	0
$891$	1.54573e30	0.164217
$892$	0	0
$893$	−2.76688e30	−0.287111
$894$	0	0
$895$	5.24597e29	0.0531722
$896$	0	0
$897$	−3.89557e31	−3.85701
$898$	0	0
$899$	−9.40110e30	−0.909289
$900$	0	0
$901$	8.14298e30	0.769437
$902$	0	0
$903$	−5.19280e30	−0.479381
$904$	0	0
$905$	−2.38714e30	−0.215312
$906$	0	0
$907$	1.99778e31	1.76064	0.880322	−	0.474377i	$-0.157327\pi$
$907$	1.99778e31	1.76064	0.880322	+	0.474377i	$0.157327\pi$
$908$	0	0
$909$	−2.64784e31	−2.28019
$910$	0	0
$911$	1.13275e31	0.953219	0.476610	−	0.879115i	$-0.341866\pi$
$911$	1.13275e31	0.953219	0.476610	+	0.879115i	$0.341866\pi$
$912$	0	0
$913$	−2.26613e31	−1.86355
$914$	0	0
$915$	5.09238e29	0.0409260
$916$	0	0
$917$	1.79431e31	1.40935
$918$	0	0
$919$	4.22159e30	0.324089	0.162044	−	0.986783i	$-0.448191\pi$
$919$	4.22159e30	0.324089	0.162044	+	0.986783i	$0.448191\pi$
$920$	0	0
$921$	1.26177e31	0.946789
$922$	0	0
$923$	1.62037e31	1.18849
$924$	0	0
$925$	3.23114e30	0.231669
$926$	0	0
$927$	9.07908e30	0.636363
$928$	0	0
$929$	7.45447e30	0.510801	0.255400	−	0.966835i	$-0.417793\pi$
$929$	7.45447e30	0.510801	0.255400	+	0.966835i	$0.417793\pi$
$930$	0	0
$931$	−2.98184e30	−0.199762
$932$	0	0
$933$	4.16938e30	0.273095
$934$	0	0
$935$	−2.00393e30	−0.128340
$936$	0	0
$937$	2.86894e31	1.79662	0.898308	−	0.439366i	$-0.144797\pi$
$937$	2.86894e31	1.79662	0.898308	+	0.439366i	$0.144797\pi$
$938$	0	0
$939$	1.80351e31	1.10441
$940$	0	0
$941$	−1.96096e31	−1.17430	−0.587148	−	0.809480i	$-0.699749\pi$
$941$	−1.96096e31	−1.17430	−0.587148	+	0.809480i	$0.699749\pi$
$942$	0	0
$943$	−1.01970e31	−0.597174
$944$	0	0
$945$	−2.86440e30	−0.164059
$946$	0	0
$947$	−6.20608e30	−0.347650	−0.173825	−	0.984777i	$-0.555613\pi$
$947$	−6.20608e30	−0.347650	−0.173825	+	0.984777i	$0.555613\pi$
$948$	0	0
$949$	4.93092e31	2.70167
$950$	0	0
$951$	−4.73412e31	−2.53714
$952$	0	0
$953$	−2.74365e31	−1.43831	−0.719156	−	0.694849i	$-0.755471\pi$
$953$	−2.74365e31	−1.43831	−0.719156	+	0.694849i	$0.755471\pi$
$954$	0	0
$955$	−3.86389e29	−0.0198148
$956$	0	0
$957$	−2.14833e31	−1.07777
$958$	0	0
$959$	1.41291e31	0.693456
$960$	0	0
$961$	2.04101e31	0.980054
$962$	0	0
$963$	1.96106e31	0.921331
$964$	0	0
$965$	3.30883e30	0.152103
$966$	0	0
$967$	−2.94115e31	−1.32294	−0.661469	−	0.749973i	$-0.730067\pi$
$967$	−2.94115e31	−1.32294	−0.661469	+	0.749973i	$0.730067\pi$
$968$	0	0
$969$	5.92817e31	2.60928
$970$	0	0
$971$	−3.73435e30	−0.160847	−0.0804236	−	0.996761i	$-0.525627\pi$
$971$	−3.73435e30	−0.160847	−0.0804236	+	0.996761i	$0.525627\pi$
$972$	0	0
$973$	−2.08170e31	−0.877470
$974$	0	0
$975$	−5.78680e31	−2.38721
$976$	0	0
$977$	4.55570e30	0.183934	0.0919670	−	0.995762i	$-0.470685\pi$
$977$	4.55570e30	0.183934	0.0919670	+	0.995762i	$0.470685\pi$
$978$	0	0
$979$	−1.58369e31	−0.625823
$980$	0	0
$981$	−3.10050e31	−1.19924
$982$	0	0
$983$	4.81915e31	1.82456	0.912280	−	0.409567i	$-0.134320\pi$
$983$	4.81915e31	1.82456	0.912280	+	0.409567i	$0.134320\pi$
$984$	0	0
$985$	−1.36355e30	−0.0505349
$986$	0	0
$987$	6.34272e30	0.230115
$988$	0	0
$989$	−1.37136e31	−0.487069
$990$	0	0
$991$	2.54898e31	0.886324	0.443162	−	0.896441i	$-0.353857\pi$
$991$	2.54898e31	0.886324	0.443162	+	0.896441i	$0.353857\pi$
$992$	0	0
$993$	1.45920e31	0.496760
$994$	0	0
$995$	−6.04731e30	−0.201567
$996$	0	0
$997$	−4.55000e31	−1.48495	−0.742476	−	0.669873i	$-0.766349\pi$
$997$	−4.55000e31	−1.48495	−0.742476	+	0.669873i	$0.766349\pi$
$998$	0	0
$999$	8.37526e30	0.267646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.22.a.a.1.1	✓	2
3.2	odd	2		72.22.a.b.1.1		2
4.3	odd	2		16.22.a.e.1.2		2
8.3	odd	2		64.22.a.h.1.1		2
8.5	even	2		64.22.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.a.a.1.1	✓	2	1.1	even	1	trivial
16.22.a.e.1.2		2	4.3	odd	2
64.22.a.h.1.1		2	8.3	odd	2
64.22.a.k.1.2		2	8.5	even	2
72.22.a.b.1.1		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q-167684. q^{3} +3.35342e6 q^{5} +7.07779e8 q^{7} +1.76574e10 q^{9} +O(q^{10})\)	\(q-167684. q^{3} +3.35342e6 q^{5} +7.07779e8 q^{7} +1.76574e10 q^{9} +8.75119e10 q^{11} -7.41212e11 q^{13} -5.62314e11 q^{15} -6.82853e12 q^{17} +5.17730e13 q^{19} -1.18683e14 q^{21} -3.13428e14 q^{23} -4.65592e14 q^{25} -1.20683e15 q^{27} +1.46400e15 q^{29} -6.42150e15 q^{31} -1.46743e16 q^{33} +2.37348e15 q^{35} -6.93987e15 q^{37} +1.24289e17 q^{39} +3.25339e16 q^{41} +4.37536e16 q^{43} +5.92128e16 q^{45} -5.34426e16 q^{47} -5.75946e16 q^{49} +1.14503e18 q^{51} -1.19249e18 q^{53} +2.93464e17 q^{55} -8.68148e18 q^{57} -4.48047e17 q^{59} -9.05611e17 q^{61} +1.24976e19 q^{63} -2.48560e18 q^{65} -6.06820e17 q^{67} +5.25568e19 q^{69} -2.18611e19 q^{71} -6.65250e19 q^{73} +7.80721e19 q^{75} +6.19391e19 q^{77} -1.81938e18 q^{79} +1.76631e19 q^{81} -2.58951e20 q^{83} -2.28989e19 q^{85} -2.45490e20 q^{87} -1.80968e20 q^{89} -5.24614e20 q^{91} +1.07678e21 q^{93} +1.73617e20 q^{95} +4.09632e20 q^{97} +1.54524e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9}+O(q^{10}) \) 2 * q - 105432 * q^3 + 2108140 * q^5 + 444771792 * q^7 + 11072347578 * q^9	\( 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9} + 53806403320 q^{11} - 490366676932 q^{13} - 639834721680 q^{15} - 6593864672092 q^{17} + 19302397925320 q^{19} - 135055584824256 q^{21} - 409737865776272 q^{23} - 940878149007650 q^{25} - 22\!\cdots\!56 q^{27}+ \cdots + 17\!\cdots\!04 q^{99}+O(q^{100}) \) 2 * q - 105432 * q^3 + 2108140 * q^5 + 444771792 * q^7 + 11072347578 * q^9 + 53806403320 * q^11 - 490366676932 * q^13 - 639834721680 * q^15 - 6593864672092 * q^17 + 19302397925320 * q^19 - 135055584824256 * q^21 - 409737865776272 * q^23 - 940878149007650 * q^25 - 2267939450073456 * q^27 - 2404787522145060 * q^29 - 8689907170559168 * q^31 - 16772541164782752 * q^33 + 2701000503537120 * q^35 - 2186204096251860 * q^37 + 139904623071296688 * q^39 + 68178038573558676 * q^41 + 264529652266004024 * q^43 + 67413140092548540 * q^45 + 426494411558622432 * q^47 - 546967577640131534 * q^49 + 1159640456555021520 * q^51 - 3055980275589518132 * q^53 + 335437371694825040 * q^55 - 10702822880368260192 * q^57 + 783424997522814424 * q^59 - 7177279049078597092 * q^61 + 14229493501717343376 * q^63 - 2797969869756700760 * q^65 + 16674123174011538088 * q^67 + 46561318352256083136 * q^69 + 9448263149848716368 * q^71 - 11586140334503007532 * q^73 + 48484754640473875800 * q^75 + 70803926825553273024 * q^77 - 85280702218715897824 * q^79 + 20489906394726573618 * q^81 - 381814622040086245816 * q^83 - 23191160664932407400 * q^85 - 486328037598631978704 * q^87 - 59742932430695979660 * q^89 - 590588534777956016544 * q^91 + 935567489996727394560 * q^93 + 214051632007883873840 * q^95 + 783394660926711950788 * q^97 + 1767190682286719892504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more