LMFDB - Embedded newform 528.8.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,8,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	528.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:	$164.939293456$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{97}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 24 $ x^2 - x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.42443$ of defining polynomial
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000 q^{3} -303.826 q^{5} +829.478 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -303.826 q^{5} +829.478 q^{7} +729.000 q^{9} -1331.00 q^{11} -6061.62 q^{13} -8203.30 q^{15} -2468.81 q^{17} -42140.2 q^{19} +22395.9 q^{21} +37118.2 q^{23} +14185.2 q^{25} +19683.0 q^{27} -59769.3 q^{29} +21301.7 q^{31} -35937.0 q^{33} -252017. q^{35} +330635. q^{37} -163664. q^{39} +816862. q^{41} +958652. q^{43} -221489. q^{45} -38915.5 q^{47} -135509. q^{49} -66657.8 q^{51} -1.47350e6 q^{53} +404392. q^{55} -1.13779e6 q^{57} -1.59205e6 q^{59} -3.08178e6 q^{61} +604689. q^{63} +1.84168e6 q^{65} +1.59464e6 q^{67} +1.00219e6 q^{69} +2.69496e6 q^{71} -4.48213e6 q^{73} +383002. q^{75} -1.10404e6 q^{77} +1.41084e6 q^{79} +531441. q^{81} -653174. q^{83} +750088. q^{85} -1.61377e6 q^{87} +2.32673e6 q^{89} -5.02798e6 q^{91} +575146. q^{93} +1.28033e7 q^{95} -2.16642e6 q^{97} -970299. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 54 q^{3} - 194 q^{5} + 418 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q + 54 * q^3 - 194 * q^5 + 418 * q^7 + 1458 * q^9	$ 2 q + 54 q^{3} - 194 q^{5} + 418 q^{7} + 1458 q^{9} - 2662 q^{11} - 13246 q^{13} - 5238 q^{15} - 10256 q^{17} - 14196 q^{19} + 11286 q^{21} + 13666 q^{23} - 51878 q^{25} + 39366 q^{27} + 15312 q^{29} + 48040 q^{31} - 71874 q^{33} - 297208 q^{35} + 274092 q^{37} - 357642 q^{39} + 755836 q^{41} + 1704096 q^{43} - 141426 q^{45} + 1182094 q^{47} - 789738 q^{49} - 276912 q^{51} - 2156394 q^{53} + 258214 q^{55} - 383292 q^{57} - 927332 q^{59} - 1061994 q^{61} + 304722 q^{63} + 1052644 q^{65} + 3259952 q^{67} + 368982 q^{69} + 5495514 q^{71} - 5450812 q^{73} - 1400706 q^{75} - 556358 q^{77} + 1536590 q^{79} + 1062882 q^{81} - 8850888 q^{83} - 105148 q^{85} + 413424 q^{87} + 6810132 q^{89} - 2071760 q^{91} + 1297080 q^{93} + 15872304 q^{95} + 9897376 q^{97} - 1940598 q^{99}+O(q^{100}) $ 2 * q + 54 * q^3 - 194 * q^5 + 418 * q^7 + 1458 * q^9 - 2662 * q^11 - 13246 * q^13 - 5238 * q^15 - 10256 * q^17 - 14196 * q^19 + 11286 * q^21 + 13666 * q^23 - 51878 * q^25 + 39366 * q^27 + 15312 * q^29 + 48040 * q^31 - 71874 * q^33 - 297208 * q^35 + 274092 * q^37 - 357642 * q^39 + 755836 * q^41 + 1704096 * q^43 - 141426 * q^45 + 1182094 * q^47 - 789738 * q^49 - 276912 * q^51 - 2156394 * q^53 + 258214 * q^55 - 383292 * q^57 - 927332 * q^59 - 1061994 * q^61 + 304722 * q^63 + 1052644 * q^65 + 3259952 * q^67 + 368982 * q^69 + 5495514 * q^71 - 5450812 * q^73 - 1400706 * q^75 - 556358 * q^77 + 1536590 * q^79 + 1062882 * q^81 - 8850888 * q^83 - 105148 * q^85 + 413424 * q^87 + 6810132 * q^89 - 2071760 * q^91 + 1297080 * q^93 + 15872304 * q^95 + 9897376 * q^97 - 1940598 * 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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	−303.826	−1.08700	−0.543500	−	0.839409i	$-0.682901\pi$
$5$	−303.826	−1.08700	−0.543500	+	0.839409i	$0.682901\pi$
$6$	0	0
$7$	829.478	0.914033	0.457016	−	0.889458i	$-0.348918\pi$
$7$	829.478	0.914033	0.457016	+	0.889458i	$0.348918\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−1331.00	−0.301511
$11$	−1331.00	−0.301511
$12$	0	0
$13$	−6061.62	−0.765221	−0.382610	−	0.923910i	$-0.624975\pi$
$13$	−6061.62	−0.765221	−0.382610	+	0.923910i	$0.624975\pi$
$14$	0	0
$15$	−8203.30	−0.627580
$16$	0	0
$17$	−2468.81	−0.121875	−0.0609377	−	0.998142i	$-0.519409\pi$
$17$	−2468.81	−0.121875	−0.0609377	+	0.998142i	$0.519409\pi$
$18$	0	0
$19$	−42140.2	−1.40948	−0.704741	−	0.709465i	$-0.748937\pi$
$19$	−42140.2	−1.40948	−0.704741	+	0.709465i	$0.748937\pi$
$20$	0	0
$21$	22395.9	0.527717
$22$	0	0
$23$	37118.2	0.636121	0.318061	−	0.948070i	$-0.396968\pi$
$23$	37118.2	0.636121	0.318061	+	0.948070i	$0.396968\pi$
$24$	0	0
$25$	14185.2	0.181571
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−59769.3	−0.455077	−0.227539	−	0.973769i	$-0.573068\pi$
$29$	−59769.3	−0.455077	−0.227539	+	0.973769i	$0.573068\pi$
$30$	0	0
$31$	21301.7	0.128425	0.0642124	−	0.997936i	$-0.479546\pi$
$31$	21301.7	0.128425	0.0642124	+	0.997936i	$0.479546\pi$
$32$	0	0
$33$	−35937.0	−0.174078
$34$	0	0
$35$	−252017.	−0.993554
$36$	0	0
$37$	330635.	1.07311	0.536553	−	0.843866i	$-0.319726\pi$
$37$	330635.	1.07311	0.536553	+	0.843866i	$0.319726\pi$
$38$	0	0
$39$	−163664.	−0.441800
$40$	0	0
$41$	816862.	1.85099	0.925497	−	0.378754i	$-0.123647\pi$
$41$	816862.	1.85099	0.925497	+	0.378754i	$0.123647\pi$
$42$	0	0
$43$	958652.	1.83874	0.919372	−	0.393389i	$-0.128697\pi$
$43$	958652.	1.83874	0.919372	+	0.393389i	$0.128697\pi$
$44$	0	0
$45$	−221489.	−0.362334
$46$	0	0
$47$	−38915.5	−0.0546739	−0.0273369	−	0.999626i	$-0.508703\pi$
$47$	−38915.5	−0.0546739	−0.0273369	+	0.999626i	$0.508703\pi$
$48$	0	0
$49$	−135509.	−0.164544
$50$	0	0
$51$	−66657.8	−0.0703648
$52$	0	0
$53$	−1.47350e6	−1.35952	−0.679759	−	0.733436i	$-0.737916\pi$
$53$	−1.47350e6	−1.35952	−0.679759	+	0.733436i	$0.737916\pi$
$54$	0	0
$55$	404392.	0.327743
$56$	0	0
$57$	−1.13779e6	−0.813764
$58$	0	0
$59$	−1.59205e6	−1.00919	−0.504597	−	0.863355i	$-0.668359\pi$
$59$	−1.59205e6	−1.00919	−0.504597	+	0.863355i	$0.668359\pi$
$60$	0	0
$61$	−3.08178e6	−1.73839	−0.869196	−	0.494468i	$-0.835363\pi$
$61$	−3.08178e6	−1.73839	−0.869196	+	0.494468i	$0.835363\pi$
$62$	0	0
$63$	604689.	0.304678
$64$	0	0
$65$	1.84168e6	0.831795
$66$	0	0
$67$	1.59464e6	0.647739	0.323870	−	0.946102i	$-0.395016\pi$
$67$	1.59464e6	0.647739	0.323870	+	0.946102i	$0.395016\pi$
$68$	0	0
$69$	1.00219e6	0.367265
$70$	0	0
$71$	2.69496e6	0.893609	0.446805	−	0.894632i	$-0.352562\pi$
$71$	2.69496e6	0.893609	0.446805	+	0.894632i	$0.352562\pi$
$72$	0	0
$73$	−4.48213e6	−1.34851	−0.674255	−	0.738499i	$-0.735535\pi$
$73$	−4.48213e6	−1.34851	−0.674255	+	0.738499i	$0.735535\pi$
$74$	0	0
$75$	383002.	0.104830
$76$	0	0
$77$	−1.10404e6	−0.275591
$78$	0	0
$79$	1.41084e6	0.321947	0.160973	−	0.986959i	$-0.448537\pi$
$79$	1.41084e6	0.321947	0.160973	+	0.986959i	$0.448537\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−653174.	−0.125388	−0.0626939	−	0.998033i	$-0.519969\pi$
$83$	−653174.	−0.125388	−0.0626939	+	0.998033i	$0.519969\pi$
$84$	0	0
$85$	750088.	0.132479
$86$	0	0
$87$	−1.61377e6	−0.262739
$88$	0	0
$89$	2.32673e6	0.349850	0.174925	−	0.984582i	$-0.444032\pi$
$89$	2.32673e6	0.349850	0.174925	+	0.984582i	$0.444032\pi$
$90$	0	0
$91$	−5.02798e6	−0.699437
$92$	0	0
$93$	575146.	0.0741460
$94$	0	0
$95$	1.28033e7	1.53211
$96$	0	0
$97$	−2.16642e6	−0.241014	−0.120507	−	0.992712i	$-0.538452\pi$
$97$	−2.16642e6	−0.241014	−0.120507	+	0.992712i	$0.538452\pi$
$98$	0	0
$99$	−970299.	−0.100504
$100$	0	0
$101$	6.34311e6	0.612600	0.306300	−	0.951935i	$-0.400909\pi$
$101$	6.34311e6	0.612600	0.306300	+	0.951935i	$0.400909\pi$
$102$	0	0
$103$	−5.53067e6	−0.498709	−0.249355	−	0.968412i	$-0.580218\pi$
$103$	−5.53067e6	−0.498709	−0.249355	+	0.968412i	$0.580218\pi$
$104$	0	0
$105$	−6.80446e6	−0.573629
$106$	0	0
$107$	1.24836e7	0.985141	0.492570	−	0.870273i	$-0.336057\pi$
$107$	1.24836e7	0.985141	0.492570	+	0.870273i	$0.336057\pi$
$108$	0	0
$109$	1.15577e7	0.854831	0.427416	−	0.904055i	$-0.359424\pi$
$109$	1.15577e7	0.854831	0.427416	+	0.904055i	$0.359424\pi$
$110$	0	0
$111$	8.92715e6	0.619559
$112$	0	0
$113$	2.91312e7	1.89926	0.949629	−	0.313378i	$-0.101461\pi$
$113$	2.91312e7	1.89926	0.949629	+	0.313378i	$0.101461\pi$
$114$	0	0
$115$	−1.12775e7	−0.691464
$116$	0	0
$117$	−4.41892e6	−0.255074
$118$	0	0
$119$	−2.04782e6	−0.111398
$120$	0	0
$121$	1.77156e6	0.0909091
$122$	0	0
$123$	2.20553e7	1.06867
$124$	0	0
$125$	1.94266e7	0.889633
$126$	0	0
$127$	4.29589e7	1.86097	0.930487	−	0.366326i	$-0.119384\pi$
$127$	4.29589e7	1.86097	0.930487	+	0.366326i	$0.119384\pi$
$128$	0	0
$129$	2.58836e7	1.06160
$130$	0	0
$131$	−3.87619e7	−1.50645	−0.753226	−	0.657761i	$-0.771504\pi$
$131$	−3.87619e7	−1.50645	−0.753226	+	0.657761i	$0.771504\pi$
$132$	0	0
$133$	−3.49544e7	−1.28831
$134$	0	0
$135$	−5.98021e6	−0.209193
$136$	0	0
$137$	2.16108e7	0.718040	0.359020	−	0.933330i	$-0.383111\pi$
$137$	2.16108e7	0.718040	0.359020	+	0.933330i	$0.383111\pi$
$138$	0	0
$139$	1.06241e7	0.335537	0.167769	−	0.985826i	$-0.446344\pi$
$139$	1.06241e7	0.335537	0.167769	+	0.985826i	$0.446344\pi$
$140$	0	0
$141$	−1.05072e6	−0.0315660
$142$	0	0
$143$	8.06801e6	0.230723
$144$	0	0
$145$	1.81595e7	0.494669
$146$	0	0
$147$	−3.65875e6	−0.0949996
$148$	0	0
$149$	6.36303e7	1.57584	0.787920	−	0.615777i	$-0.211158\pi$
$149$	6.36303e7	1.57584	0.787920	+	0.615777i	$0.211158\pi$
$150$	0	0
$151$	1.42528e7	0.336885	0.168443	−	0.985711i	$-0.446126\pi$
$151$	1.42528e7	0.336885	0.168443	+	0.985711i	$0.446126\pi$
$152$	0	0
$153$	−1.79976e6	−0.0406251
$154$	0	0
$155$	−6.47202e6	−0.139598
$156$	0	0
$157$	−1.42133e7	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$157$	−1.42133e7	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$158$	0	0
$159$	−3.97845e7	−0.784918
$160$	0	0
$161$	3.07888e7	0.581436
$162$	0	0
$163$	−8.04737e7	−1.45545	−0.727725	−	0.685869i	$-0.759422\pi$
$163$	−8.04737e7	−1.45545	−0.727725	+	0.685869i	$0.759422\pi$
$164$	0	0
$165$	1.09186e7	0.189223
$166$	0	0
$167$	−7.01073e6	−0.116481	−0.0582406	−	0.998303i	$-0.518549\pi$
$167$	−7.01073e6	−0.116481	−0.0582406	+	0.998303i	$0.518549\pi$
$168$	0	0
$169$	−2.60053e7	−0.414438
$170$	0	0
$171$	−3.07202e7	−0.469827
$172$	0	0
$173$	−1.81447e7	−0.266433	−0.133216	−	0.991087i	$-0.542531\pi$
$173$	−1.81447e7	−0.266433	−0.133216	+	0.991087i	$0.542531\pi$
$174$	0	0
$175$	1.17664e7	0.165962
$176$	0	0
$177$	−4.29853e7	−0.582658
$178$	0	0
$179$	1.16784e8	1.52195	0.760973	−	0.648784i	$-0.224722\pi$
$179$	1.16784e8	1.52195	0.760973	+	0.648784i	$0.224722\pi$
$180$	0	0
$181$	−9.11199e7	−1.14219	−0.571095	−	0.820884i	$-0.693481\pi$
$181$	−9.11199e7	−1.14219	−0.571095	+	0.820884i	$0.693481\pi$
$182$	0	0
$183$	−8.32081e7	−1.00366
$184$	0	0
$185$	−1.00456e8	−1.16647
$186$	0	0
$187$	3.28598e6	0.0367468
$188$	0	0
$189$	1.63266e7	0.175906
$190$	0	0
$191$	1.82885e8	1.89916	0.949581	−	0.313522i	$-0.101509\pi$
$191$	1.82885e8	1.89916	0.949581	+	0.313522i	$0.101509\pi$
$192$	0	0
$193$	1.32994e8	1.33163	0.665814	−	0.746118i	$-0.268084\pi$
$193$	1.32994e8	1.33163	0.665814	+	0.746118i	$0.268084\pi$
$194$	0	0
$195$	4.97253e7	0.480237
$196$	0	0
$197$	−2.82339e7	−0.263111	−0.131555	−	0.991309i	$-0.541997\pi$
$197$	−2.82339e7	−0.263111	−0.131555	+	0.991309i	$0.541997\pi$
$198$	0	0
$199$	1.99071e8	1.79070	0.895350	−	0.445362i	$-0.146925\pi$
$199$	1.99071e8	1.79070	0.895350	+	0.445362i	$0.146925\pi$
$200$	0	0
$201$	4.30552e7	0.373973
$202$	0	0
$203$	−4.95773e7	−0.415955
$204$	0	0
$205$	−2.48184e8	−2.01203
$206$	0	0
$207$	2.70592e7	0.212040
$208$	0	0
$209$	5.60887e7	0.424975
$210$	0	0
$211$	−3.80016e7	−0.278493	−0.139246	−	0.990258i	$-0.544468\pi$
$211$	−3.80016e7	−0.278493	−0.139246	+	0.990258i	$0.544468\pi$
$212$	0	0
$213$	7.27638e7	0.515926
$214$	0	0
$215$	−2.91263e8	−1.99872
$216$	0	0
$217$	1.76693e7	0.117384
$218$	0	0
$219$	−1.21017e8	−0.778563
$220$	0	0
$221$	1.49650e7	0.0932616
$222$	0	0
$223$	−1.14601e8	−0.692022	−0.346011	−	0.938230i	$-0.612464\pi$
$223$	−1.14601e8	−0.692022	−0.346011	+	0.938230i	$0.612464\pi$
$224$	0	0
$225$	1.03410e7	0.0605237
$226$	0	0
$227$	9.29427e7	0.527381	0.263691	−	0.964607i	$-0.415060\pi$
$227$	9.29427e7	0.527381	0.263691	+	0.964607i	$0.415060\pi$
$228$	0	0
$229$	3.51772e8	1.93569	0.967847	−	0.251541i	$-0.0809374\pi$
$229$	3.51772e8	1.93569	0.967847	+	0.251541i	$0.0809374\pi$
$230$	0	0
$231$	−2.98090e7	−0.159113
$232$	0	0
$233$	1.90315e8	0.985661	0.492831	−	0.870125i	$-0.335962\pi$
$233$	1.90315e8	0.985661	0.492831	+	0.870125i	$0.335962\pi$
$234$	0	0
$235$	1.18235e7	0.0594306
$236$	0	0
$237$	3.80928e7	0.185876
$238$	0	0
$239$	2.00720e8	0.951038	0.475519	−	0.879705i	$-0.342260\pi$
$239$	2.00720e8	0.951038	0.475519	+	0.879705i	$0.342260\pi$
$240$	0	0
$241$	−2.08229e8	−0.958258	−0.479129	−	0.877745i	$-0.659047\pi$
$241$	−2.08229e8	−0.958258	−0.479129	+	0.877745i	$0.659047\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	4.11712e7	0.178860
$246$	0	0
$247$	2.55438e8	1.07856
$248$	0	0
$249$	−1.76357e7	−0.0723927
$250$	0	0
$251$	4.78103e7	0.190837	0.0954186	−	0.995437i	$-0.469581\pi$
$251$	4.78103e7	0.190837	0.0954186	+	0.995437i	$0.469581\pi$
$252$	0	0
$253$	−4.94044e7	−0.191798
$254$	0	0
$255$	2.02524e7	0.0764866
$256$	0	0
$257$	−2.02303e8	−0.743425	−0.371712	−	0.928348i	$-0.621229\pi$
$257$	−2.02303e8	−0.743425	−0.371712	+	0.928348i	$0.621229\pi$
$258$	0	0
$259$	2.74255e8	0.980855
$260$	0	0
$261$	−4.35718e7	−0.151692
$262$	0	0
$263$	4.06426e8	1.37764	0.688822	−	0.724931i	$-0.258128\pi$
$263$	4.06426e8	1.37764	0.688822	+	0.724931i	$0.258128\pi$
$264$	0	0
$265$	4.47688e8	1.47780
$266$	0	0
$267$	6.28218e7	0.201986
$268$	0	0
$269$	5.29696e8	1.65918	0.829590	−	0.558373i	$-0.188574\pi$
$269$	5.29696e8	1.65918	0.829590	+	0.558373i	$0.188574\pi$
$270$	0	0
$271$	4.79694e8	1.46410	0.732051	−	0.681249i	$-0.238563\pi$
$271$	4.79694e8	1.46410	0.732051	+	0.681249i	$0.238563\pi$
$272$	0	0
$273$	−1.35755e8	−0.403820
$274$	0	0
$275$	−1.88806e7	−0.0547458
$276$	0	0
$277$	5.74020e8	1.62274	0.811369	−	0.584535i	$-0.198723\pi$
$277$	5.74020e8	1.62274	0.811369	+	0.584535i	$0.198723\pi$
$278$	0	0
$279$	1.55290e7	0.0428082
$280$	0	0
$281$	−5.08869e8	−1.36815	−0.684075	−	0.729412i	$-0.739794\pi$
$281$	−5.08869e8	−1.36815	−0.684075	+	0.729412i	$0.739794\pi$
$282$	0	0
$283$	−1.55996e8	−0.409130	−0.204565	−	0.978853i	$-0.565578\pi$
$283$	−1.55996e8	−0.409130	−0.204565	+	0.978853i	$0.565578\pi$
$284$	0	0
$285$	3.45689e8	0.884563
$286$	0	0
$287$	6.77569e8	1.69187
$288$	0	0
$289$	−4.04244e8	−0.985146
$290$	0	0
$291$	−5.84934e7	−0.139149
$292$	0	0
$293$	3.31026e8	0.768821	0.384411	−	0.923162i	$-0.374405\pi$
$293$	3.31026e8	0.768821	0.384411	+	0.923162i	$0.374405\pi$
$294$	0	0
$295$	4.83706e8	1.09699
$296$	0	0
$297$	−2.61981e7	−0.0580259
$298$	0	0
$299$	−2.24996e8	−0.486773
$300$	0	0
$301$	7.95181e8	1.68067
$302$	0	0
$303$	1.71264e8	0.353685
$304$	0	0
$305$	9.36326e8	1.88963
$306$	0	0
$307$	1.59688e8	0.314983	0.157491	−	0.987520i	$-0.449659\pi$
$307$	1.59688e8	0.314983	0.157491	+	0.987520i	$0.449659\pi$
$308$	0	0
$309$	−1.49328e8	−0.287930
$310$	0	0
$311$	−3.51157e8	−0.661973	−0.330986	−	0.943636i	$-0.607381\pi$
$311$	−3.51157e8	−0.661973	−0.330986	+	0.943636i	$0.607381\pi$
$312$	0	0
$313$	−5.46772e8	−1.00786	−0.503931	−	0.863744i	$-0.668114\pi$
$313$	−5.46772e8	−1.00786	−0.503931	+	0.863744i	$0.668114\pi$
$314$	0	0
$315$	−1.83720e8	−0.331185
$316$	0	0
$317$	−6.18068e8	−1.08975	−0.544877	−	0.838516i	$-0.683424\pi$
$317$	−6.18068e8	−1.08975	−0.544877	+	0.838516i	$0.683424\pi$
$318$	0	0
$319$	7.95529e7	0.137211
$320$	0	0
$321$	3.37058e8	0.568771
$322$	0	0
$323$	1.04036e8	0.171781
$324$	0	0
$325$	−8.59855e7	−0.138942
$326$	0	0
$327$	3.12059e8	0.493537
$328$	0	0
$329$	−3.22795e7	−0.0499737
$330$	0	0
$331$	6.84846e8	1.03799	0.518997	−	0.854776i	$-0.326306\pi$
$331$	6.84846e8	1.03799	0.518997	+	0.854776i	$0.326306\pi$
$332$	0	0
$333$	2.41033e8	0.357702
$334$	0	0
$335$	−4.84493e8	−0.704093
$336$	0	0
$337$	1.19934e6	0.00170701	0.000853507	−	1.00000i	$-0.499728\pi$
$337$	1.19934e6	0.00170701	0.000853507		1.00000i	$0.499728\pi$
$338$	0	0
$339$	7.86542e8	1.09654
$340$	0	0
$341$	−2.83526e7	−0.0387215
$342$	0	0
$343$	−7.95513e8	−1.06443
$344$	0	0
$345$	−3.04492e8	−0.399217
$346$	0	0
$347$	2.08559e8	0.267964	0.133982	−	0.990984i	$-0.457224\pi$
$347$	2.08559e8	0.267964	0.133982	+	0.990984i	$0.457224\pi$
$348$	0	0
$349$	1.13428e7	0.0142834	0.00714172	−	0.999974i	$-0.497727\pi$
$349$	1.13428e7	0.0142834	0.00714172	+	0.999974i	$0.497727\pi$
$350$	0	0
$351$	−1.19311e8	−0.147267
$352$	0	0
$353$	5.47130e8	0.662032	0.331016	−	0.943625i	$-0.392609\pi$
$353$	5.47130e8	0.662032	0.331016	+	0.943625i	$0.392609\pi$
$354$	0	0
$355$	−8.18798e8	−0.971354
$356$	0	0
$357$	−5.52912e7	−0.0643157
$358$	0	0
$359$	−1.04623e9	−1.19343	−0.596715	−	0.802453i	$-0.703528\pi$
$359$	−1.04623e9	−1.19343	−0.596715	+	0.802453i	$0.703528\pi$
$360$	0	0
$361$	8.81928e8	0.986638
$362$	0	0
$363$	4.78321e7	0.0524864
$364$	0	0
$365$	1.36179e9	1.46583
$366$	0	0
$367$	−4.80538e8	−0.507453	−0.253727	−	0.967276i	$-0.581656\pi$
$367$	−4.80538e8	−0.507453	−0.253727	+	0.967276i	$0.581656\pi$
$368$	0	0
$369$	5.95492e8	0.616998
$370$	0	0
$371$	−1.22224e9	−1.24264
$372$	0	0
$373$	−1.45793e9	−1.45464	−0.727320	−	0.686298i	$-0.759235\pi$
$373$	−1.45793e9	−1.45464	−0.727320	+	0.686298i	$0.759235\pi$
$374$	0	0
$375$	5.24517e8	0.513630
$376$	0	0
$377$	3.62298e8	0.348234
$378$	0	0
$379$	1.16963e8	0.110359	0.0551797	−	0.998476i	$-0.482427\pi$
$379$	1.16963e8	0.110359	0.0551797	+	0.998476i	$0.482427\pi$
$380$	0	0
$381$	1.15989e9	1.07443
$382$	0	0
$383$	1.54022e9	1.40084	0.700418	−	0.713732i	$-0.252997\pi$
$383$	1.54022e9	1.40084	0.700418	+	0.713732i	$0.252997\pi$
$384$	0	0
$385$	3.35435e8	0.299568
$386$	0	0
$387$	6.98857e8	0.612915
$388$	0	0
$389$	6.82704e8	0.588043	0.294022	−	0.955799i	$-0.405006\pi$
$389$	6.82704e8	0.588043	0.294022	+	0.955799i	$0.405006\pi$
$390$	0	0
$391$	−9.16378e7	−0.0775276
$392$	0	0
$393$	−1.04657e9	−0.869751
$394$	0	0
$395$	−4.28651e8	−0.349957
$396$	0	0
$397$	−4.39437e8	−0.352476	−0.176238	−	0.984348i	$-0.556393\pi$
$397$	−4.39437e8	−0.352476	−0.176238	+	0.984348i	$0.556393\pi$
$398$	0	0
$399$	−9.43769e8	−0.743807
$400$	0	0
$401$	−1.92220e9	−1.48865	−0.744326	−	0.667816i	$-0.767229\pi$
$401$	−1.92220e9	−1.48865	−0.744326	+	0.667816i	$0.767229\pi$
$402$	0	0
$403$	−1.29123e8	−0.0982732
$404$	0	0
$405$	−1.61466e8	−0.120778
$406$	0	0
$407$	−4.40075e8	−0.323554
$408$	0	0
$409$	−5.70106e8	−0.412026	−0.206013	−	0.978549i	$-0.566049\pi$
$409$	−5.70106e8	−0.412026	−0.206013	+	0.978549i	$0.566049\pi$
$410$	0	0
$411$	5.83492e8	0.414561
$412$	0	0
$413$	−1.32057e9	−0.922436
$414$	0	0
$415$	1.98451e8	0.136297
$416$	0	0
$417$	2.86851e8	0.193723
$418$	0	0
$419$	1.12807e9	0.749179	0.374589	−	0.927191i	$-0.377784\pi$
$419$	1.12807e9	0.749179	0.374589	+	0.927191i	$0.377784\pi$
$420$	0	0
$421$	2.91413e9	1.90336	0.951681	−	0.307089i	$-0.0993549\pi$
$421$	2.91413e9	1.90336	0.951681	+	0.307089i	$0.0993549\pi$
$422$	0	0
$423$	−2.83694e7	−0.0182246
$424$	0	0
$425$	−3.50207e7	−0.0221291
$426$	0	0
$427$	−2.55627e9	−1.58895
$428$	0	0
$429$	2.17836e8	0.133208
$430$	0	0
$431$	−8.77990e8	−0.528225	−0.264113	−	0.964492i	$-0.585079\pi$
$431$	−8.77990e8	−0.528225	−0.264113	+	0.964492i	$0.585079\pi$
$432$	0	0
$433$	−7.45894e8	−0.441539	−0.220770	−	0.975326i	$-0.570857\pi$
$433$	−7.45894e8	−0.441539	−0.220770	+	0.975326i	$0.570857\pi$
$434$	0	0
$435$	4.90305e8	0.285597
$436$	0	0
$437$	−1.56417e9	−0.896601
$438$	0	0
$439$	−3.31044e8	−0.186749	−0.0933747	−	0.995631i	$-0.529765\pi$
$439$	−3.31044e8	−0.186749	−0.0933747	+	0.995631i	$0.529765\pi$
$440$	0	0
$441$	−9.87862e7	−0.0548480
$442$	0	0
$443$	1.92657e9	1.05286	0.526432	−	0.850217i	$-0.323530\pi$
$443$	1.92657e9	1.05286	0.526432	+	0.850217i	$0.323530\pi$
$444$	0	0
$445$	−7.06922e8	−0.380287
$446$	0	0
$447$	1.71802e9	0.909812
$448$	0	0
$449$	3.35149e8	0.174733	0.0873667	−	0.996176i	$-0.472155\pi$
$449$	3.35149e8	0.174733	0.0873667	+	0.996176i	$0.472155\pi$
$450$	0	0
$451$	−1.08724e9	−0.558096
$452$	0	0
$453$	3.84826e8	0.194501
$454$	0	0
$455$	1.52763e9	0.760288
$456$	0	0
$457$	−9.81385e8	−0.480986	−0.240493	−	0.970651i	$-0.577309\pi$
$457$	−9.81385e8	−0.480986	−0.240493	+	0.970651i	$0.577309\pi$
$458$	0	0
$459$	−4.85936e7	−0.0234549
$460$	0	0
$461$	−7.99148e8	−0.379904	−0.189952	−	0.981793i	$-0.560833\pi$
$461$	−7.99148e8	−0.379904	−0.189952	+	0.981793i	$0.560833\pi$
$462$	0	0
$463$	1.01685e9	0.476128	0.238064	−	0.971249i	$-0.423487\pi$
$463$	1.01685e9	0.476128	0.238064	+	0.971249i	$0.423487\pi$
$464$	0	0
$465$	−1.74744e8	−0.0805968
$466$	0	0
$467$	2.83923e9	1.29000	0.645002	−	0.764181i	$-0.276857\pi$
$467$	2.83923e9	1.29000	0.645002	+	0.764181i	$0.276857\pi$
$468$	0	0
$469$	1.32272e9	0.592055
$470$	0	0
$471$	−3.83758e8	−0.169233
$472$	0	0
$473$	−1.27597e9	−0.554402
$474$	0	0
$475$	−5.97770e8	−0.255921
$476$	0	0
$477$	−1.07418e9	−0.453172
$478$	0	0
$479$	−3.24833e9	−1.35047	−0.675236	−	0.737602i	$-0.735958\pi$
$479$	−3.24833e9	−1.35047	−0.675236	+	0.737602i	$0.735958\pi$
$480$	0	0
$481$	−2.00418e9	−0.821163
$482$	0	0
$483$	8.31297e8	0.335692
$484$	0	0
$485$	6.58215e8	0.261982
$486$	0	0
$487$	6.55732e8	0.257262	0.128631	−	0.991693i	$-0.458942\pi$
$487$	6.55732e8	0.257262	0.128631	+	0.991693i	$0.458942\pi$
$488$	0	0
$489$	−2.17279e9	−0.840304
$490$	0	0
$491$	−9.37028e8	−0.357246	−0.178623	−	0.983918i	$-0.557164\pi$
$491$	−9.37028e8	−0.357246	−0.178623	+	0.983918i	$0.557164\pi$
$492$	0	0
$493$	1.47559e8	0.0554627
$494$	0	0
$495$	2.94802e8	0.109248
$496$	0	0
$497$	2.23541e9	0.816788
$498$	0	0
$499$	5.07602e9	1.82882	0.914411	−	0.404786i	$-0.132654\pi$
$499$	5.07602e9	1.82882	0.914411	+	0.404786i	$0.132654\pi$
$500$	0	0
$501$	−1.89290e8	−0.0672504
$502$	0	0
$503$	3.30955e9	1.15953	0.579763	−	0.814785i	$-0.303145\pi$
$503$	3.30955e9	1.15953	0.579763	+	0.814785i	$0.303145\pi$
$504$	0	0
$505$	−1.92720e9	−0.665897
$506$	0	0
$507$	−7.02144e8	−0.239276
$508$	0	0
$509$	−5.09686e8	−0.171313	−0.0856566	−	0.996325i	$-0.527299\pi$
$509$	−5.09686e8	−0.171313	−0.0856566	+	0.996325i	$0.527299\pi$
$510$	0	0
$511$	−3.71783e9	−1.23258
$512$	0	0
$513$	−8.29446e8	−0.271255
$514$	0	0
$515$	1.68036e9	0.542097
$516$	0	0
$517$	5.17965e7	0.0164848
$518$	0	0
$519$	−4.89906e8	−0.153825
$520$	0	0
$521$	4.24284e9	1.31439	0.657196	−	0.753720i	$-0.271742\pi$
$521$	4.24284e9	1.31439	0.657196	+	0.753720i	$0.271742\pi$
$522$	0	0
$523$	−4.77407e9	−1.45926	−0.729630	−	0.683843i	$-0.760308\pi$
$523$	−4.77407e9	−1.45926	−0.729630	+	0.683843i	$0.760308\pi$
$524$	0	0
$525$	3.17691e8	0.0958182
$526$	0	0
$527$	−5.25899e7	−0.0156518
$528$	0	0
$529$	−2.02706e9	−0.595350
$530$	0	0
$531$	−1.16060e9	−0.336398
$532$	0	0
$533$	−4.95150e9	−1.41642
$534$	0	0
$535$	−3.79286e9	−1.07085
$536$	0	0
$537$	3.15318e9	0.878696
$538$	0	0
$539$	1.80363e8	0.0496119
$540$	0	0
$541$	3.10457e9	0.842967	0.421483	−	0.906836i	$-0.361510\pi$
$541$	3.10457e9	0.842967	0.421483	+	0.906836i	$0.361510\pi$
$542$	0	0
$543$	−2.46024e9	−0.659443
$544$	0	0
$545$	−3.51154e9	−0.929202
$546$	0	0
$547$	−5.81117e9	−1.51813	−0.759063	−	0.651017i	$-0.774342\pi$
$547$	−5.81117e9	−1.51813	−0.759063	+	0.651017i	$0.774342\pi$
$548$	0	0
$549$	−2.24662e9	−0.579464
$550$	0	0
$551$	2.51869e9	0.641423
$552$	0	0
$553$	1.17026e9	0.294270
$554$	0	0
$555$	−2.71230e9	−0.673461
$556$	0	0
$557$	−4.69891e8	−0.115214	−0.0576068	−	0.998339i	$-0.518347\pi$
$557$	−4.69891e8	−0.115214	−0.0576068	+	0.998339i	$0.518347\pi$
$558$	0	0
$559$	−5.81098e9	−1.40705
$560$	0	0
$561$	8.87216e7	0.0212158
$562$	0	0
$563$	−4.00223e9	−0.945199	−0.472599	−	0.881277i	$-0.656684\pi$
$563$	−4.00223e9	−0.945199	−0.472599	+	0.881277i	$0.656684\pi$
$564$	0	0
$565$	−8.85082e9	−2.06449
$566$	0	0
$567$	4.40819e8	0.101559
$568$	0	0
$569$	1.36018e9	0.309530	0.154765	−	0.987951i	$-0.450538\pi$
$569$	1.36018e9	0.309530	0.154765	+	0.987951i	$0.450538\pi$
$570$	0	0
$571$	−5.92967e9	−1.33292	−0.666461	−	0.745540i	$-0.732192\pi$
$571$	−5.92967e9	−1.33292	−0.666461	+	0.745540i	$0.732192\pi$
$572$	0	0
$573$	4.93790e9	1.09648
$574$	0	0
$575$	5.26531e8	0.115501
$576$	0	0
$577$	−5.08690e9	−1.10240	−0.551199	−	0.834374i	$-0.685829\pi$
$577$	−5.08690e9	−1.10240	−0.551199	+	0.834374i	$0.685829\pi$
$578$	0	0
$579$	3.59085e9	0.768816
$580$	0	0
$581$	−5.41793e8	−0.114609
$582$	0	0
$583$	1.96123e9	0.409910
$584$	0	0
$585$	1.34258e9	0.277265
$586$	0	0
$587$	3.07480e9	0.627456	0.313728	−	0.949513i	$-0.398422\pi$
$587$	3.07480e9	0.627456	0.313728	+	0.949513i	$0.398422\pi$
$588$	0	0
$589$	−8.97659e8	−0.181012
$590$	0	0
$591$	−7.62315e8	−0.151907
$592$	0	0
$593$	4.96104e9	0.976970	0.488485	−	0.872572i	$-0.337550\pi$
$593$	4.96104e9	0.976970	0.488485	+	0.872572i	$0.337550\pi$
$594$	0	0
$595$	6.22182e8	0.121090
$596$	0	0
$597$	5.37493e9	1.03386
$598$	0	0
$599$	−4.34257e9	−0.825568	−0.412784	−	0.910829i	$-0.635443\pi$
$599$	−4.34257e9	−0.825568	−0.412784	+	0.910829i	$0.635443\pi$
$600$	0	0
$601$	1.77668e9	0.333848	0.166924	−	0.985970i	$-0.446617\pi$
$601$	1.77668e9	0.333848	0.166924	+	0.985970i	$0.446617\pi$
$602$	0	0
$603$	1.16249e9	0.215913
$604$	0	0
$605$	−5.38246e8	−0.0988183
$606$	0	0
$607$	8.76418e9	1.59056	0.795282	−	0.606240i	$-0.207323\pi$
$607$	8.76418e9	1.59056	0.795282	+	0.606240i	$0.207323\pi$
$608$	0	0
$609$	−1.33859e9	−0.240152
$610$	0	0
$611$	2.35891e8	0.0418376
$612$	0	0
$613$	−1.13075e9	−0.198269	−0.0991346	−	0.995074i	$-0.531607\pi$
$613$	−1.13075e9	−0.198269	−0.0991346	+	0.995074i	$0.531607\pi$
$614$	0	0
$615$	−6.70097e9	−1.16165
$616$	0	0
$617$	2.36299e7	0.00405008	0.00202504	−	0.999998i	$-0.499355\pi$
$617$	2.36299e7	0.00405008	0.00202504	+	0.999998i	$0.499355\pi$
$618$	0	0
$619$	−5.90610e9	−1.00088	−0.500442	−	0.865770i	$-0.666829\pi$
$619$	−5.90610e9	−1.00088	−0.500442	+	0.865770i	$0.666829\pi$
$620$	0	0
$621$	7.30598e8	0.122422
$622$	0	0
$623$	1.92997e9	0.319774
$624$	0	0
$625$	−7.01052e9	−1.14860
$626$	0	0
$627$	1.51439e9	0.245359
$628$	0	0
$629$	−8.16275e8	−0.130785
$630$	0	0
$631$	3.87609e9	0.614173	0.307087	−	0.951682i	$-0.400646\pi$
$631$	3.87609e9	0.614173	0.307087	+	0.951682i	$0.400646\pi$
$632$	0	0
$633$	−1.02604e9	−0.160788
$634$	0	0
$635$	−1.30520e10	−2.02288
$636$	0	0
$637$	8.21404e8	0.125913
$638$	0	0
$639$	1.96462e9	0.297870
$640$	0	0
$641$	4.60264e9	0.690246	0.345123	−	0.938558i	$-0.387837\pi$
$641$	4.60264e9	0.690246	0.345123	+	0.938558i	$0.387837\pi$
$642$	0	0
$643$	−1.10814e10	−1.64383	−0.821917	−	0.569607i	$-0.807095\pi$
$643$	−1.10814e10	−1.64383	−0.821917	+	0.569607i	$0.807095\pi$
$644$	0	0
$645$	−7.86411e9	−1.15396
$646$	0	0
$647$	−3.95562e9	−0.574183	−0.287091	−	0.957903i	$-0.592688\pi$
$647$	−3.95562e9	−0.574183	−0.287091	+	0.957903i	$0.592688\pi$
$648$	0	0
$649$	2.11902e9	0.304283
$650$	0	0
$651$	4.77071e8	0.0677719
$652$	0	0
$653$	−3.72327e9	−0.523274	−0.261637	−	0.965166i	$-0.584262\pi$
$653$	−3.72327e9	−0.523274	−0.261637	+	0.965166i	$0.584262\pi$
$654$	0	0
$655$	1.17769e10	1.63752
$656$	0	0
$657$	−3.26747e9	−0.449503
$658$	0	0
$659$	2.45031e9	0.333520	0.166760	−	0.985998i	$-0.446670\pi$
$659$	2.45031e9	0.333520	0.166760	+	0.985998i	$0.446670\pi$
$660$	0	0
$661$	−4.82392e9	−0.649673	−0.324836	−	0.945770i	$-0.605309\pi$
$661$	−4.82392e9	−0.649673	−0.324836	+	0.945770i	$0.605309\pi$
$662$	0	0
$663$	4.04054e8	0.0538446
$664$	0	0
$665$	1.06201e10	1.40040
$666$	0	0
$667$	−2.21853e9	−0.289484
$668$	0	0
$669$	−3.09422e9	−0.399539
$670$	0	0
$671$	4.10185e9	0.524145
$672$	0	0
$673$	−2.17377e9	−0.274891	−0.137446	−	0.990509i	$-0.543889\pi$
$673$	−2.17377e9	−0.274891	−0.137446	+	0.990509i	$0.543889\pi$
$674$	0	0
$675$	2.79208e8	0.0349434
$676$	0	0
$677$	−4.71648e9	−0.584194	−0.292097	−	0.956389i	$-0.594353\pi$
$677$	−4.71648e9	−0.584194	−0.292097	+	0.956389i	$0.594353\pi$
$678$	0	0
$679$	−1.79700e9	−0.220295
$680$	0	0
$681$	2.50945e9	0.304484
$682$	0	0
$683$	6.72757e9	0.807953	0.403976	−	0.914769i	$-0.367628\pi$
$683$	6.72757e9	0.807953	0.403976	+	0.914769i	$0.367628\pi$
$684$	0	0
$685$	−6.56592e9	−0.780510
$686$	0	0
$687$	9.49783e9	1.11757
$688$	0	0
$689$	8.93179e9	1.04033
$690$	0	0
$691$	4.98106e9	0.574313	0.287157	−	0.957884i	$-0.407290\pi$
$691$	4.98106e9	0.574313	0.287157	+	0.957884i	$0.407290\pi$
$692$	0	0
$693$	−8.04842e8	−0.0918637
$694$	0	0
$695$	−3.22788e9	−0.364729
$696$	0	0
$697$	−2.01668e9	−0.225591
$698$	0	0
$699$	5.13851e9	0.569072
$700$	0	0
$701$	4.43468e9	0.486238	0.243119	−	0.969996i	$-0.421829\pi$
$701$	4.43468e9	0.486238	0.243119	+	0.969996i	$0.421829\pi$
$702$	0	0
$703$	−1.39330e10	−1.51252
$704$	0	0
$705$	3.19236e8	0.0343123
$706$	0	0
$707$	5.26147e9	0.559937
$708$	0	0
$709$	−4.96472e9	−0.523158	−0.261579	−	0.965182i	$-0.584243\pi$
$709$	−4.96472e9	−0.523158	−0.261579	+	0.965182i	$0.584243\pi$
$710$	0	0
$711$	1.02851e9	0.107316
$712$	0	0
$713$	7.90682e8	0.0816937
$714$	0	0
$715$	−2.45127e9	−0.250796
$716$	0	0
$717$	5.41944e9	0.549082
$718$	0	0
$719$	−7.62145e9	−0.764692	−0.382346	−	0.924019i	$-0.624884\pi$
$719$	−7.62145e9	−0.764692	−0.382346	+	0.924019i	$0.624884\pi$
$720$	0	0
$721$	−4.58757e9	−0.455837
$722$	0	0
$723$	−5.62219e9	−0.553250
$724$	0	0
$725$	−8.47842e8	−0.0826289
$726$	0	0
$727$	9.67350e9	0.933712	0.466856	−	0.884333i	$-0.345387\pi$
$727$	9.67350e9	0.933712	0.466856	+	0.884333i	$0.345387\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−2.36673e9	−0.224098
$732$	0	0
$733$	6.37101e9	0.597509	0.298755	−	0.954330i	$-0.403429\pi$
$733$	6.37101e9	0.597509	0.298755	+	0.954330i	$0.403429\pi$
$734$	0	0
$735$	1.11162e9	0.103265
$736$	0	0
$737$	−2.12246e9	−0.195301
$738$	0	0
$739$	−1.16634e10	−1.06309	−0.531544	−	0.847031i	$-0.678388\pi$
$739$	−1.16634e10	−1.06309	−0.531544	+	0.847031i	$0.678388\pi$
$740$	0	0
$741$	6.89682e9	0.622709
$742$	0	0
$743$	1.87781e9	0.167955	0.0839773	−	0.996468i	$-0.473238\pi$
$743$	1.87781e9	0.167955	0.0839773	+	0.996468i	$0.473238\pi$
$744$	0	0
$745$	−1.93326e10	−1.71294
$746$	0	0
$747$	−4.76164e8	−0.0417960
$748$	0	0
$749$	1.03549e10	0.900451
$750$	0	0
$751$	1.89594e10	1.63337	0.816684	−	0.577085i	$-0.195810\pi$
$751$	1.89594e10	1.63337	0.816684	+	0.577085i	$0.195810\pi$
$752$	0	0
$753$	1.29088e9	0.110180
$754$	0	0
$755$	−4.33038e9	−0.366194
$756$	0	0
$757$	2.13638e9	0.178996	0.0894980	−	0.995987i	$-0.471474\pi$
$757$	2.13638e9	0.178996	0.0894980	+	0.995987i	$0.471474\pi$
$758$	0	0
$759$	−1.33392e9	−0.110735
$760$	0	0
$761$	−8.14245e9	−0.669744	−0.334872	−	0.942264i	$-0.608693\pi$
$761$	−8.14245e9	−0.669744	−0.334872	+	0.942264i	$0.608693\pi$
$762$	0	0
$763$	9.58690e9	0.781344
$764$	0	0
$765$	5.46814e8	0.0441596
$766$	0	0
$767$	9.65039e9	0.772256
$768$	0	0
$769$	3.93260e9	0.311844	0.155922	−	0.987769i	$-0.450165\pi$
$769$	3.93260e9	0.311844	0.155922	+	0.987769i	$0.450165\pi$
$770$	0	0
$771$	−5.46219e9	−0.429216
$772$	0	0
$773$	−2.04769e10	−1.59455	−0.797273	−	0.603619i	$-0.793725\pi$
$773$	−2.04769e10	−1.59455	−0.797273	+	0.603619i	$0.793725\pi$
$774$	0	0
$775$	3.02170e8	0.0233182
$776$	0	0
$777$	7.40487e9	0.566297
$778$	0	0
$779$	−3.44228e10	−2.60894
$780$	0	0
$781$	−3.58699e9	−0.269433
$782$	0	0
$783$	−1.17644e9	−0.0875796
$784$	0	0
$785$	4.31836e9	0.318621
$786$	0	0
$787$	−2.33821e10	−1.70991	−0.854954	−	0.518704i	$-0.826415\pi$
$787$	−2.33821e10	−1.70991	−0.854954	+	0.518704i	$0.826415\pi$
$788$	0	0
$789$	1.09735e10	0.795383
$790$	0	0
$791$	2.41637e10	1.73598
$792$	0	0
$793$	1.86806e10	1.33025
$794$	0	0
$795$	1.20876e10	0.853206
$796$	0	0
$797$	5.55295e9	0.388526	0.194263	−	0.980950i	$-0.437769\pi$
$797$	5.55295e9	0.388526	0.194263	+	0.980950i	$0.437769\pi$
$798$	0	0
$799$	9.60749e7	0.00666340
$800$	0	0
$801$	1.69619e9	0.116617
$802$	0	0
$803$	5.96571e9	0.406591
$804$	0	0
$805$	−9.35443e9	−0.632021
$806$	0	0
$807$	1.43018e10	0.957928
$808$	0	0
$809$	−1.91933e10	−1.27447	−0.637237	−	0.770668i	$-0.719923\pi$
$809$	−1.91933e10	−1.27447	−0.637237	+	0.770668i	$0.719923\pi$
$810$	0	0
$811$	1.30956e9	0.0862091	0.0431045	−	0.999071i	$-0.486275\pi$
$811$	1.30956e9	0.0862091	0.0431045	+	0.999071i	$0.486275\pi$
$812$	0	0
$813$	1.29517e10	0.845300
$814$	0	0
$815$	2.44500e10	1.58208
$816$	0	0
$817$	−4.03978e10	−2.59168
$818$	0	0
$819$	−3.66539e9	−0.233146
$820$	0	0
$821$	1.43007e10	0.901896	0.450948	−	0.892550i	$-0.351086\pi$
$821$	1.43007e10	0.901896	0.450948	+	0.892550i	$0.351086\pi$
$822$	0	0
$823$	−2.08892e10	−1.30624	−0.653120	−	0.757254i	$-0.726540\pi$
$823$	−2.08892e10	−1.30624	−0.653120	+	0.757254i	$0.726540\pi$
$824$	0	0
$825$	−5.09775e8	−0.0316075
$826$	0	0
$827$	−1.31770e10	−0.810116	−0.405058	−	0.914291i	$-0.632749\pi$
$827$	−1.31770e10	−0.810116	−0.405058	+	0.914291i	$0.632749\pi$
$828$	0	0
$829$	−1.77646e10	−1.08296	−0.541481	−	0.840713i	$-0.682136\pi$
$829$	−1.77646e10	−1.08296	−0.541481	+	0.840713i	$0.682136\pi$
$830$	0	0
$831$	1.54986e10	0.936888
$832$	0	0
$833$	3.34546e8	0.0200539
$834$	0	0
$835$	2.13004e9	0.126615
$836$	0	0
$837$	4.19282e8	0.0247153
$838$	0	0
$839$	−1.05490e10	−0.616659	−0.308329	−	0.951280i	$-0.599770\pi$
$839$	−1.05490e10	−0.616659	−0.308329	+	0.951280i	$0.599770\pi$
$840$	0	0
$841$	−1.36775e10	−0.792905
$842$	0	0
$843$	−1.37394e10	−0.789901
$844$	0	0
$845$	7.90110e9	0.450494
$846$	0	0
$847$	1.46947e9	0.0830939
$848$	0	0
$849$	−4.21190e9	−0.236212
$850$	0	0
$851$	1.22726e10	0.682626
$852$	0	0
$853$	6.93413e8	0.0382534	0.0191267	−	0.999817i	$-0.493911\pi$
$853$	6.93413e8	0.0382534	0.0191267	+	0.999817i	$0.493911\pi$
$854$	0	0
$855$	9.33361e9	0.510703
$856$	0	0
$857$	2.42332e10	1.31516	0.657578	−	0.753387i	$-0.271581\pi$
$857$	2.42332e10	1.31516	0.657578	+	0.753387i	$0.271581\pi$
$858$	0	0
$859$	−1.50069e10	−0.807823	−0.403911	−	0.914798i	$-0.632350\pi$
$859$	−1.50069e10	−0.807823	−0.403911	+	0.914798i	$0.632350\pi$
$860$	0	0
$861$	1.82944e10	0.976801
$862$	0	0
$863$	−4.48430e9	−0.237496	−0.118748	−	0.992924i	$-0.537888\pi$
$863$	−4.48430e9	−0.237496	−0.118748	+	0.992924i	$0.537888\pi$
$864$	0	0
$865$	5.51282e9	0.289613
$866$	0	0
$867$	−1.09146e10	−0.568775
$868$	0	0
$869$	−1.87783e9	−0.0970706
$870$	0	0
$871$	−9.66608e9	−0.495664
$872$	0	0
$873$	−1.57932e9	−0.0803380
$874$	0	0
$875$	1.61139e10	0.813154
$876$	0	0
$877$	−2.15250e10	−1.07757	−0.538784	−	0.842444i	$-0.681116\pi$
$877$	−2.15250e10	−1.07757	−0.538784	+	0.842444i	$0.681116\pi$
$878$	0	0
$879$	8.93770e9	0.443879
$880$	0	0
$881$	1.89298e10	0.932676	0.466338	−	0.884607i	$-0.345573\pi$
$881$	1.89298e10	0.932676	0.466338	+	0.884607i	$0.345573\pi$
$882$	0	0
$883$	−7.30415e9	−0.357032	−0.178516	−	0.983937i	$-0.557130\pi$
$883$	−7.30415e9	−0.357032	−0.178516	+	0.983937i	$0.557130\pi$
$884$	0	0
$885$	1.30601e10	0.633350
$886$	0	0
$887$	2.12379e10	1.02183	0.510915	−	0.859631i	$-0.329307\pi$
$887$	2.12379e10	1.02183	0.510915	+	0.859631i	$0.329307\pi$
$888$	0	0
$889$	3.56335e10	1.70099
$890$	0	0
$891$	−7.07348e8	−0.0335013
$892$	0	0
$893$	1.63991e9	0.0770618
$894$	0	0
$895$	−3.54821e10	−1.65436
$896$	0	0
$897$	−6.07490e9	−0.281039
$898$	0	0
$899$	−1.27319e9	−0.0584432
$900$	0	0
$901$	3.63779e9	0.165692
$902$	0	0
$903$	2.14699e10	0.970337
$904$	0	0
$905$	2.76846e10	1.24156
$906$	0	0
$907$	3.80530e10	1.69341	0.846707	−	0.532059i	$-0.178581\pi$
$907$	3.80530e10	1.69341	0.846707	+	0.532059i	$0.178581\pi$
$908$	0	0
$909$	4.62413e9	0.204200
$910$	0	0
$911$	3.43756e10	1.50638	0.753192	−	0.657800i	$-0.228513\pi$
$911$	3.43756e10	1.50638	0.753192	+	0.657800i	$0.228513\pi$
$912$	0	0
$913$	8.69374e8	0.0378059
$914$	0	0
$915$	2.52808e10	1.09098
$916$	0	0
$917$	−3.21521e10	−1.37695
$918$	0	0
$919$	9.00499e9	0.382718	0.191359	−	0.981520i	$-0.438711\pi$
$919$	9.00499e9	0.382718	0.191359	+	0.981520i	$0.438711\pi$
$920$	0	0
$921$	4.31156e9	0.181855
$922$	0	0
$923$	−1.63358e10	−0.683808
$924$	0	0
$925$	4.69014e9	0.194845
$926$	0	0
$927$	−4.03186e9	−0.166236
$928$	0	0
$929$	−1.39486e10	−0.570788	−0.285394	−	0.958410i	$-0.592125\pi$
$929$	−1.39486e10	−0.570788	−0.285394	+	0.958410i	$0.592125\pi$
$930$	0	0
$931$	5.71039e9	0.231922
$932$	0	0
$933$	−9.48124e9	−0.382190
$934$	0	0
$935$	−9.98367e8	−0.0399438
$936$	0	0
$937$	3.54170e10	1.40645	0.703223	−	0.710970i	$-0.251744\pi$
$937$	3.54170e10	1.40645	0.703223	+	0.710970i	$0.251744\pi$
$938$	0	0
$939$	−1.47629e10	−0.581890
$940$	0	0
$941$	2.25006e10	0.880299	0.440150	−	0.897924i	$-0.354925\pi$
$941$	2.25006e10	0.880299	0.440150	+	0.897924i	$0.354925\pi$
$942$	0	0
$943$	3.03205e10	1.17746
$944$	0	0
$945$	−4.96045e9	−0.191210
$946$	0	0
$947$	−1.92828e10	−0.737810	−0.368905	−	0.929467i	$-0.620267\pi$
$947$	−1.92828e10	−0.737810	−0.368905	+	0.929467i	$0.620267\pi$
$948$	0	0
$949$	2.71689e10	1.03191
$950$	0	0
$951$	−1.66878e10	−0.629170
$952$	0	0
$953$	1.08950e10	0.407759	0.203879	−	0.978996i	$-0.434645\pi$
$953$	1.08950e10	0.407759	0.203879	+	0.978996i	$0.434645\pi$
$954$	0	0
$955$	−5.55653e10	−2.06439
$956$	0	0
$957$	2.14793e9	0.0792188
$958$	0	0
$959$	1.79257e10	0.656312
$960$	0	0
$961$	−2.70589e10	−0.983507
$962$	0	0
$963$	9.10058e9	0.328380
$964$	0	0
$965$	−4.04072e10	−1.44748
$966$	0	0
$967$	1.35695e10	0.482584	0.241292	−	0.970453i	$-0.422429\pi$
$967$	1.35695e10	0.482584	0.241292	+	0.970453i	$0.422429\pi$
$968$	0	0
$969$	2.80898e9	0.0991779
$970$	0	0
$971$	4.33843e9	0.152078	0.0760388	−	0.997105i	$-0.475773\pi$
$971$	4.33843e9	0.152078	0.0760388	+	0.997105i	$0.475773\pi$
$972$	0	0
$973$	8.81247e9	0.306692
$974$	0	0
$975$	−2.32161e9	−0.0802182
$976$	0	0
$977$	−3.90981e10	−1.34130	−0.670648	−	0.741776i	$-0.733984\pi$
$977$	−3.90981e10	−1.34130	−0.670648	+	0.741776i	$0.733984\pi$
$978$	0	0
$979$	−3.09688e9	−0.105484
$980$	0	0
$981$	8.42560e9	0.284944
$982$	0	0
$983$	3.05704e10	1.02651	0.513257	−	0.858235i	$-0.328439\pi$
$983$	3.05704e10	1.02651	0.513257	+	0.858235i	$0.328439\pi$
$984$	0	0
$985$	8.57819e9	0.286002
$986$	0	0
$987$	−8.71548e8	−0.0288523
$988$	0	0
$989$	3.55835e10	1.16966
$990$	0	0
$991$	−3.59751e10	−1.17421	−0.587103	−	0.809512i	$-0.699732\pi$
$991$	−3.59751e10	−1.17421	−0.587103	+	0.809512i	$0.699732\pi$
$992$	0	0
$993$	1.84908e10	0.599286
$994$	0	0
$995$	−6.04831e10	−1.94649
$996$	0	0
$997$	2.97463e10	0.950604	0.475302	−	0.879823i	$-0.342339\pi$
$997$	2.97463e10	0.950604	0.475302	+	0.879823i	$0.342339\pi$
$998$	0	0
$999$	6.50789e9	0.206520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.8.a.h.1.1		2
4.3	odd	2		33.8.a.c.1.2	✓	2
12.11	even	2		99.8.a.b.1.1		2
44.43	even	2		363.8.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.a.c.1.2	✓	2	4.3	odd	2
99.8.a.b.1.1		2	12.11	even	2
363.8.a.c.1.1		2	44.43	even	2
528.8.a.h.1.1		2	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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -303.826 q^{5} +829.478 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -303.826 q^{5} +829.478 q^{7} +729.000 q^{9} -1331.00 q^{11} -6061.62 q^{13} -8203.30 q^{15} -2468.81 q^{17} -42140.2 q^{19} +22395.9 q^{21} +37118.2 q^{23} +14185.2 q^{25} +19683.0 q^{27} -59769.3 q^{29} +21301.7 q^{31} -35937.0 q^{33} -252017. q^{35} +330635. q^{37} -163664. q^{39} +816862. q^{41} +958652. q^{43} -221489. q^{45} -38915.5 q^{47} -135509. q^{49} -66657.8 q^{51} -1.47350e6 q^{53} +404392. q^{55} -1.13779e6 q^{57} -1.59205e6 q^{59} -3.08178e6 q^{61} +604689. q^{63} +1.84168e6 q^{65} +1.59464e6 q^{67} +1.00219e6 q^{69} +2.69496e6 q^{71} -4.48213e6 q^{73} +383002. q^{75} -1.10404e6 q^{77} +1.41084e6 q^{79} +531441. q^{81} -653174. q^{83} +750088. q^{85} -1.61377e6 q^{87} +2.32673e6 q^{89} -5.02798e6 q^{91} +575146. q^{93} +1.28033e7 q^{95} -2.16642e6 q^{97} -970299. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 54 q^{3} - 194 q^{5} + 418 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q + 54 * q^3 - 194 * q^5 + 418 * q^7 + 1458 * q^9	\( 2 q + 54 q^{3} - 194 q^{5} + 418 q^{7} + 1458 q^{9} - 2662 q^{11} - 13246 q^{13} - 5238 q^{15} - 10256 q^{17} - 14196 q^{19} + 11286 q^{21} + 13666 q^{23} - 51878 q^{25} + 39366 q^{27} + 15312 q^{29} + 48040 q^{31} - 71874 q^{33} - 297208 q^{35} + 274092 q^{37} - 357642 q^{39} + 755836 q^{41} + 1704096 q^{43} - 141426 q^{45} + 1182094 q^{47} - 789738 q^{49} - 276912 q^{51} - 2156394 q^{53} + 258214 q^{55} - 383292 q^{57} - 927332 q^{59} - 1061994 q^{61} + 304722 q^{63} + 1052644 q^{65} + 3259952 q^{67} + 368982 q^{69} + 5495514 q^{71} - 5450812 q^{73} - 1400706 q^{75} - 556358 q^{77} + 1536590 q^{79} + 1062882 q^{81} - 8850888 q^{83} - 105148 q^{85} + 413424 q^{87} + 6810132 q^{89} - 2071760 q^{91} + 1297080 q^{93} + 15872304 q^{95} + 9897376 q^{97} - 1940598 q^{99}+O(q^{100}) \) 2 * q + 54 * q^3 - 194 * q^5 + 418 * q^7 + 1458 * q^9 - 2662 * q^11 - 13246 * q^13 - 5238 * q^15 - 10256 * q^17 - 14196 * q^19 + 11286 * q^21 + 13666 * q^23 - 51878 * q^25 + 39366 * q^27 + 15312 * q^29 + 48040 * q^31 - 71874 * q^33 - 297208 * q^35 + 274092 * q^37 - 357642 * q^39 + 755836 * q^41 + 1704096 * q^43 - 141426 * q^45 + 1182094 * q^47 - 789738 * q^49 - 276912 * q^51 - 2156394 * q^53 + 258214 * q^55 - 383292 * q^57 - 927332 * q^59 - 1061994 * q^61 + 304722 * q^63 + 1052644 * q^65 + 3259952 * q^67 + 368982 * q^69 + 5495514 * q^71 - 5450812 * q^73 - 1400706 * q^75 - 556358 * q^77 + 1536590 * q^79 + 1062882 * q^81 - 8850888 * q^83 - 105148 * q^85 + 413424 * q^87 + 6810132 * q^89 - 2071760 * q^91 + 1297080 * q^93 + 15872304 * q^95 + 9897376 * q^97 - 1940598 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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