LMFDB - Embedded newform 531.8.a.e.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,8,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("531.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.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:	$165.876448532$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.80127$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.80127 q^{2} -113.550 q^{4} +87.8737 q^{5} +1306.06 q^{7} +918.197 q^{8} +O(q^{10})$	\(q-3.80127 q^{2} -113.550 q^{4} +87.8737 q^{5} +1306.06 q^{7} +918.197 q^{8} -334.031 q^{10} +3799.92 q^{11} +8976.87 q^{13} -4964.66 q^{14} +11044.1 q^{16} +4357.97 q^{17} -57085.5 q^{19} -9978.10 q^{20} -14444.5 q^{22} -96669.6 q^{23} -70403.2 q^{25} -34123.5 q^{26} -148303. q^{28} -22753.4 q^{29} -53358.6 q^{31} -159511. q^{32} -16565.8 q^{34} +114768. q^{35} +319784. q^{37} +216997. q^{38} +80685.4 q^{40} -476338. q^{41} +707558. q^{43} -431483. q^{44} +367467. q^{46} -946330. q^{47} +882238. q^{49} +267621. q^{50} -1.01933e6 q^{52} -799926. q^{53} +333913. q^{55} +1.19922e6 q^{56} +86491.9 q^{58} -205379. q^{59} +539847. q^{61} +202830. q^{62} -807306. q^{64} +788832. q^{65} +2.90460e6 q^{67} -494849. q^{68} -436264. q^{70} +523802. q^{71} -2.11143e6 q^{73} -1.21558e6 q^{74} +6.48208e6 q^{76} +4.96291e6 q^{77} -3.80223e6 q^{79} +970490. q^{80} +1.81069e6 q^{82} -1.88881e6 q^{83} +382951. q^{85} -2.68962e6 q^{86} +3.48908e6 q^{88} -6.48074e6 q^{89} +1.17243e7 q^{91} +1.09769e7 q^{92} +3.59725e6 q^{94} -5.01631e6 q^{95} +2.84275e6 q^{97} -3.35362e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) $ 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * q^98

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 $

$2$	−3.80127	−0.335988	−0.167994	−	0.985788i	$-0.553729\pi$
$2$	−3.80127	−0.335988	−0.167994	+	0.985788i	$0.553729\pi$
$3$	0	0
$3$	0	0
$4$	−113.550	−0.887112
$5$	87.8737	0.314387	0.157193	−	0.987568i	$-0.449755\pi$
$5$	87.8737	0.314387	0.157193	+	0.987568i	$0.449755\pi$
$6$	0	0
$7$	1306.06	1.43919	0.719596	−	0.694393i	$-0.244327\pi$
$7$	1306.06	1.43919	0.719596	+	0.694393i	$0.244327\pi$
$8$	918.197	0.634046
$9$	0	0
$10$	−334.031	−0.105630
$11$	3799.92	0.860796	0.430398	−	0.902639i	$-0.358373\pi$
$11$	3799.92	0.860796	0.430398	+	0.902639i	$0.358373\pi$
$12$	0	0
$13$	8976.87	1.13324	0.566622	−	0.823978i	$-0.308250\pi$
$13$	8976.87	1.13324	0.566622	+	0.823978i	$0.308250\pi$
$14$	−4964.66	−0.483550
$15$	0	0
$16$	11044.1	0.674081
$17$	4357.97	0.215136	0.107568	−	0.994198i	$-0.465694\pi$
$17$	4357.97	0.215136	0.107568	+	0.994198i	$0.465694\pi$
$18$	0	0
$19$	−57085.5	−1.90936	−0.954680	−	0.297633i	$-0.903803\pi$
$19$	−57085.5	−1.90936	−0.954680	+	0.297633i	$0.903803\pi$
$20$	−9978.10	−0.278896
$21$	0	0
$22$	−14444.5	−0.289217
$23$	−96669.6	−1.65670	−0.828348	−	0.560214i	$-0.810719\pi$
$23$	−96669.6	−1.65670	−0.828348	+	0.560214i	$0.810719\pi$
$24$	0	0
$25$	−70403.2	−0.901161
$26$	−34123.5	−0.380756
$27$	0	0
$28$	−148303.	−1.27672
$29$	−22753.4	−0.173242	−0.0866212	−	0.996241i	$-0.527607\pi$
$29$	−22753.4	−0.173242	−0.0866212	+	0.996241i	$0.527607\pi$
$30$	0	0
$31$	−53358.6	−0.321691	−0.160845	−	0.986980i	$-0.551422\pi$
$31$	−53358.6	−0.321691	−0.160845	+	0.986980i	$0.551422\pi$
$32$	−159511.	−0.860529
$33$	0	0
$34$	−16565.8	−0.0722830
$35$	114768.	0.452462
$36$	0	0
$37$	319784.	1.03789	0.518944	−	0.854808i	$-0.326325\pi$
$37$	319784.	1.03789	0.518944	+	0.854808i	$0.326325\pi$
$38$	216997.	0.641522
$39$	0	0
$40$	80685.4	0.199336
$41$	−476338.	−1.07937	−0.539687	−	0.841866i	$-0.681457\pi$
$41$	−476338.	−1.07937	−0.539687	+	0.841866i	$0.681457\pi$
$42$	0	0
$43$	707558.	1.35713	0.678567	−	0.734539i	$-0.262601\pi$
$43$	707558.	1.35713	0.678567	+	0.734539i	$0.262601\pi$
$44$	−431483.	−0.763623
$45$	0	0
$46$	367467.	0.556629
$47$	−946330.	−1.32954	−0.664768	−	0.747050i	$-0.731470\pi$
$47$	−946330.	−1.32954	−0.664768	+	0.747050i	$0.731470\pi$
$48$	0	0
$49$	882238.	1.07127
$50$	267621.	0.302779
$51$	0	0
$52$	−1.01933e6	−1.00531
$53$	−799926.	−0.738047	−0.369024	−	0.929420i	$-0.620308\pi$
$53$	−799926.	−0.738047	−0.369024	+	0.929420i	$0.620308\pi$
$54$	0	0
$55$	333913.	0.270623
$56$	1.19922e6	0.912514
$57$	0	0
$58$	86491.9	0.0582073
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	539847.	0.304520	0.152260	−	0.988340i	$-0.451345\pi$
$61$	539847.	0.304520	0.152260	+	0.988340i	$0.451345\pi$
$62$	202830.	0.108084
$63$	0	0
$64$	−807306.	−0.384954
$65$	788832.	0.356277
$66$	0	0
$67$	2.90460e6	1.17985	0.589923	−	0.807460i	$-0.299158\pi$
$67$	2.90460e6	1.17985	0.589923	+	0.807460i	$0.299158\pi$
$68$	−494849.	−0.190850
$69$	0	0
$70$	−436264.	−0.152022
$71$	523802.	0.173685	0.0868426	−	0.996222i	$-0.472322\pi$
$71$	523802.	0.173685	0.0868426	+	0.996222i	$0.472322\pi$
$72$	0	0
$73$	−2.11143e6	−0.635252	−0.317626	−	0.948216i	$-0.602886\pi$
$73$	−2.11143e6	−0.635252	−0.317626	+	0.948216i	$0.602886\pi$
$74$	−1.21558e6	−0.348718
$75$	0	0
$76$	6.48208e6	1.69382
$77$	4.96291e6	1.23885
$78$	0	0
$79$	−3.80223e6	−0.867648	−0.433824	−	0.900998i	$-0.642836\pi$
$79$	−3.80223e6	−0.867648	−0.433824	+	0.900998i	$0.642836\pi$
$80$	970490.	0.211922
$81$	0	0
$82$	1.81069e6	0.362656
$83$	−1.88881e6	−0.362589	−0.181294	−	0.983429i	$-0.558029\pi$
$83$	−1.88881e6	−0.362589	−0.181294	+	0.983429i	$0.558029\pi$
$84$	0	0
$85$	382951.	0.0676359
$86$	−2.68962e6	−0.455980
$87$	0	0
$88$	3.48908e6	0.545785
$89$	−6.48074e6	−0.974450	−0.487225	−	0.873277i	$-0.661991\pi$
$89$	−6.48074e6	−0.974450	−0.487225	+	0.873277i	$0.661991\pi$
$90$	0	0
$91$	1.17243e7	1.63095
$92$	1.09769e7	1.46967
$93$	0	0
$94$	3.59725e6	0.446708
$95$	−5.01631e6	−0.600278
$96$	0	0
$97$	2.84275e6	0.316256	0.158128	−	0.987419i	$-0.449454\pi$
$97$	2.84275e6	0.316256	0.158128	+	0.987419i	$0.449454\pi$
$98$	−3.35362e6	−0.359934
$99$	0	0
$100$	7.99431e6	0.799431
$101$	−1.41130e6	−0.136299	−0.0681497	−	0.997675i	$-0.521710\pi$
$101$	−1.41130e6	−0.136299	−0.0681497	+	0.997675i	$0.521710\pi$
$102$	0	0
$103$	1.05972e7	0.955564	0.477782	−	0.878478i	$-0.341441\pi$
$103$	1.05972e7	0.955564	0.477782	+	0.878478i	$0.341441\pi$
$104$	8.24254e6	0.718529
$105$	0	0
$106$	3.04073e6	0.247975
$107$	−1.84524e7	−1.45616	−0.728082	−	0.685490i	$-0.759588\pi$
$107$	−1.84524e7	−1.45616	−0.728082	+	0.685490i	$0.759588\pi$
$108$	0	0
$109$	−1.18614e7	−0.877289	−0.438645	−	0.898661i	$-0.644541\pi$
$109$	−1.18614e7	−0.877289	−0.438645	+	0.898661i	$0.644541\pi$
$110$	−1.26929e6	−0.0909259
$111$	0	0
$112$	1.44243e7	0.970131
$113$	−1.57496e7	−1.02682	−0.513412	−	0.858142i	$-0.671619\pi$
$113$	−1.57496e7	−1.02682	−0.513412	+	0.858142i	$0.671619\pi$
$114$	0	0
$115$	−8.49472e6	−0.520843
$116$	2.58366e6	0.153685
$117$	0	0
$118$	780700.	0.0437419
$119$	5.69175e6	0.309622
$120$	0	0
$121$	−5.04777e6	−0.259030
$122$	−2.05210e6	−0.102315
$123$	0	0
$124$	6.05889e6	0.285376
$125$	−1.30517e7	−0.597700
$126$	0	0
$127$	−1.74843e7	−0.757417	−0.378708	−	0.925516i	$-0.623632\pi$
$127$	−1.74843e7	−0.757417	−0.378708	+	0.925516i	$0.623632\pi$
$128$	2.34862e7	0.989869
$129$	0	0
$130$	−2.99856e6	−0.119705
$131$	−3.83994e7	−1.49236	−0.746182	−	0.665742i	$-0.768115\pi$
$131$	−3.83994e7	−1.49236	−0.746182	+	0.665742i	$0.768115\pi$
$132$	0	0
$133$	−7.45568e7	−2.74794
$134$	−1.10412e7	−0.396413
$135$	0	0
$136$	4.00148e6	0.136406
$137$	4.77564e7	1.58675	0.793376	−	0.608731i	$-0.208321\pi$
$137$	4.77564e7	1.58675	0.793376	+	0.608731i	$0.208321\pi$
$138$	0	0
$139$	−9.01121e6	−0.284598	−0.142299	−	0.989824i	$-0.545449\pi$
$139$	−9.01121e6	−0.284598	−0.142299	+	0.989824i	$0.545449\pi$
$140$	−1.30319e7	−0.401385
$141$	0	0
$142$	−1.99111e6	−0.0583561
$143$	3.41114e7	0.975492
$144$	0	0
$145$	−1.99943e6	−0.0544651
$146$	8.02609e6	0.213437
$147$	0	0
$148$	−3.63116e7	−0.920724
$149$	1.73883e7	0.430631	0.215315	−	0.976545i	$-0.430922\pi$
$149$	1.73883e7	0.430631	0.215315	+	0.976545i	$0.430922\pi$
$150$	0	0
$151$	1.46672e7	0.346680	0.173340	−	0.984862i	$-0.444544\pi$
$151$	1.46672e7	0.346680	0.173340	+	0.984862i	$0.444544\pi$
$152$	−5.24157e7	−1.21062
$153$	0	0
$154$	−1.88653e7	−0.416238
$155$	−4.68882e6	−0.101135
$156$	0	0
$157$	4.11300e7	0.848222	0.424111	−	0.905610i	$-0.360587\pi$
$157$	4.11300e7	0.848222	0.424111	+	0.905610i	$0.360587\pi$
$158$	1.44533e7	0.291519
$159$	0	0
$160$	−1.40168e7	−0.270539
$161$	−1.26256e8	−2.38430
$162$	0	0
$163$	−1.02265e6	−0.0184957	−0.00924783	−	0.999957i	$-0.502944\pi$
$163$	−1.02265e6	−0.0184957	−0.00924783	+	0.999957i	$0.502944\pi$
$164$	5.40884e7	0.957526
$165$	0	0
$166$	7.17986e6	0.121825
$167$	4.54084e6	0.0754447	0.0377224	−	0.999288i	$-0.487990\pi$
$167$	4.54084e6	0.0754447	0.0377224	+	0.999288i	$0.487990\pi$
$168$	0	0
$169$	1.78358e7	0.284242
$170$	−1.45570e6	−0.0227248
$171$	0	0
$172$	−8.03435e7	−1.20393
$173$	−1.45643e7	−0.213860	−0.106930	−	0.994267i	$-0.534102\pi$
$173$	−1.45643e7	−0.213860	−0.106930	+	0.994267i	$0.534102\pi$
$174$	0	0
$175$	−9.19505e7	−1.29694
$176$	4.19669e7	0.580246
$177$	0	0
$178$	2.46350e7	0.327403
$179$	7.59717e7	0.990071	0.495036	−	0.868873i	$-0.335155\pi$
$179$	7.59717e7	0.990071	0.495036	+	0.868873i	$0.335155\pi$
$180$	0	0
$181$	−1.08062e8	−1.35456	−0.677281	−	0.735724i	$-0.736842\pi$
$181$	−1.08062e8	−1.35456	−0.677281	+	0.735724i	$0.736842\pi$
$182$	−4.45672e7	−0.547981
$183$	0	0
$184$	−8.87618e7	−1.05042
$185$	2.81006e7	0.326298
$186$	0	0
$187$	1.65599e7	0.185188
$188$	1.07456e8	1.17945
$189$	0	0
$190$	1.90683e7	0.201686
$191$	1.32489e8	1.37583	0.687914	−	0.725792i	$-0.258527\pi$
$191$	1.32489e8	1.37583	0.687914	+	0.725792i	$0.258527\pi$
$192$	0	0
$193$	1.01108e8	1.01236	0.506178	−	0.862429i	$-0.331058\pi$
$193$	1.01108e8	1.01236	0.506178	+	0.862429i	$0.331058\pi$
$194$	−1.08061e7	−0.106258
$195$	0	0
$196$	−1.00178e8	−0.950338
$197$	−1.00275e7	−0.0934464	−0.0467232	−	0.998908i	$-0.514878\pi$
$197$	−1.00275e7	−0.0934464	−0.0467232	+	0.998908i	$0.514878\pi$
$198$	0	0
$199$	6.35124e7	0.571312	0.285656	−	0.958332i	$-0.407789\pi$
$199$	6.35124e7	0.571312	0.285656	+	0.958332i	$0.407789\pi$
$200$	−6.46440e7	−0.571378
$201$	0	0
$202$	5.36472e6	0.0457949
$203$	−2.97173e7	−0.249329
$204$	0	0
$205$	−4.18576e7	−0.339341
$206$	−4.02827e7	−0.321058
$207$	0	0
$208$	9.91418e7	0.763898
$209$	−2.16920e8	−1.64357
$210$	0	0
$211$	2.91443e7	0.213582	0.106791	−	0.994281i	$-0.465942\pi$
$211$	2.91443e7	0.213582	0.106791	+	0.994281i	$0.465942\pi$
$212$	9.08319e7	0.654731
$213$	0	0
$214$	7.01426e7	0.489253
$215$	6.21758e7	0.426665
$216$	0	0
$217$	−6.96893e7	−0.462975
$218$	4.50883e7	0.294758
$219$	0	0
$220$	−3.79160e7	−0.240073
$221$	3.91210e7	0.243802
$222$	0	0
$223$	−2.58995e8	−1.56396	−0.781979	−	0.623305i	$-0.785789\pi$
$223$	−2.58995e8	−1.56396	−0.781979	+	0.623305i	$0.785789\pi$
$224$	−2.08330e8	−1.23847
$225$	0	0
$226$	5.98686e7	0.345000
$227$	2.65429e8	1.50612	0.753058	−	0.657955i	$-0.228578\pi$
$227$	2.65429e8	1.50612	0.753058	+	0.657955i	$0.228578\pi$
$228$	0	0
$229$	1.83433e8	1.00938	0.504688	−	0.863302i	$-0.331607\pi$
$229$	1.83433e8	1.00938	0.504688	+	0.863302i	$0.331607\pi$
$230$	3.22907e7	0.174997
$231$	0	0
$232$	−2.08921e7	−0.109844
$233$	−3.14430e8	−1.62846	−0.814232	−	0.580540i	$-0.802841\pi$
$233$	−3.14430e8	−1.62846	−0.814232	+	0.580540i	$0.802841\pi$
$234$	0	0
$235$	−8.31576e7	−0.417989
$236$	2.33209e7	0.115492
$237$	0	0
$238$	−2.16359e7	−0.104029
$239$	3.42402e7	0.162235	0.0811173	−	0.996705i	$-0.474151\pi$
$239$	3.42402e7	0.162235	0.0811173	+	0.996705i	$0.474151\pi$
$240$	0	0
$241$	−1.98928e8	−0.915452	−0.457726	−	0.889093i	$-0.651336\pi$
$241$	−1.98928e8	−0.915452	−0.457726	+	0.889093i	$0.651336\pi$
$242$	1.91879e7	0.0870309
$243$	0	0
$244$	−6.12999e7	−0.270144
$245$	7.75255e7	0.336793
$246$	0	0
$247$	−5.12449e8	−2.16377
$248$	−4.89938e7	−0.203967
$249$	0	0
$250$	4.96131e7	0.200820
$251$	4.06923e8	1.62425	0.812127	−	0.583481i	$-0.198310\pi$
$251$	4.06923e8	1.62425	0.812127	+	0.583481i	$0.198310\pi$
$252$	0	0
$253$	−3.67337e8	−1.42608
$254$	6.64624e7	0.254483
$255$	0	0
$256$	1.40580e7	0.0523700
$257$	−1.04442e8	−0.383804	−0.191902	−	0.981414i	$-0.561466\pi$
$257$	−1.04442e8	−0.383804	−0.191902	+	0.981414i	$0.561466\pi$
$258$	0	0
$259$	4.17656e8	1.49372
$260$	−8.95721e7	−0.316058
$261$	0	0
$262$	1.45966e8	0.501415
$263$	−2.20493e8	−0.747393	−0.373696	−	0.927551i	$-0.621910\pi$
$263$	−2.20493e8	−0.747393	−0.373696	+	0.927551i	$0.621910\pi$
$264$	0	0
$265$	−7.02925e7	−0.232032
$266$	2.83410e8	0.923272
$267$	0	0
$268$	−3.29819e8	−1.04666
$269$	−3.53895e8	−1.10852	−0.554258	−	0.832345i	$-0.686998\pi$
$269$	−3.53895e8	−1.10852	−0.554258	+	0.832345i	$0.686998\pi$
$270$	0	0
$271$	−4.86543e8	−1.48501	−0.742504	−	0.669842i	$-0.766362\pi$
$271$	−4.86543e8	−1.48501	−0.742504	+	0.669842i	$0.766362\pi$
$272$	4.81300e7	0.145019
$273$	0	0
$274$	−1.81535e8	−0.533129
$275$	−2.67527e8	−0.775716
$276$	0	0
$277$	−4.24477e8	−1.19998	−0.599992	−	0.800006i	$-0.704829\pi$
$277$	−4.24477e8	−1.19998	−0.599992	+	0.800006i	$0.704829\pi$
$278$	3.42540e7	0.0956213
$279$	0	0
$280$	1.05380e8	0.286882
$281$	−5.85274e8	−1.57357	−0.786787	−	0.617225i	$-0.788257\pi$
$281$	−5.85274e8	−1.57357	−0.786787	+	0.617225i	$0.788257\pi$
$282$	0	0
$283$	1.23529e8	0.323979	0.161989	−	0.986793i	$-0.448209\pi$
$283$	1.23529e8	0.323979	0.161989	+	0.986793i	$0.448209\pi$
$284$	−5.94779e7	−0.154078
$285$	0	0
$286$	−1.29667e8	−0.327753
$287$	−6.22124e8	−1.55343
$288$	0	0
$289$	−3.91347e8	−0.953716
$290$	7.60036e6	0.0182996
$291$	0	0
$292$	2.39753e8	0.563540
$293$	3.34796e8	0.777577	0.388788	−	0.921327i	$-0.372894\pi$
$293$	3.34796e8	0.777577	0.388788	+	0.921327i	$0.372894\pi$
$294$	0	0
$295$	−1.80474e7	−0.0409297
$296$	2.93625e8	0.658069
$297$	0	0
$298$	−6.60975e7	−0.144687
$299$	−8.67791e8	−1.87744
$300$	0	0
$301$	9.24110e8	1.95317
$302$	−5.57541e7	−0.116480
$303$	0	0
$304$	−6.30460e8	−1.28706
$305$	4.74384e7	0.0957372
$306$	0	0
$307$	5.54710e8	1.09416	0.547081	−	0.837080i	$-0.315739\pi$
$307$	5.54710e8	1.09416	0.547081	+	0.837080i	$0.315739\pi$
$308$	−5.63540e8	−1.09900
$309$	0	0
$310$	1.78235e7	0.0339802
$311$	4.01552e8	0.756973	0.378486	−	0.925607i	$-0.376445\pi$
$311$	4.01552e8	0.756973	0.378486	+	0.925607i	$0.376445\pi$
$312$	0	0
$313$	7.36811e8	1.35816	0.679080	−	0.734064i	$-0.262379\pi$
$313$	7.36811e8	1.35816	0.679080	+	0.734064i	$0.262379\pi$
$314$	−1.56346e8	−0.284992
$315$	0	0
$316$	4.31745e8	0.769701
$317$	2.52797e7	0.0445722	0.0222861	−	0.999752i	$-0.492906\pi$
$317$	2.52797e7	0.0445722	0.0222861	+	0.999752i	$0.492906\pi$
$318$	0	0
$319$	−8.64613e7	−0.149126
$320$	−7.09410e7	−0.121024
$321$	0	0
$322$	4.79932e8	0.801096
$323$	−2.48777e8	−0.410772
$324$	0	0
$325$	−6.32001e8	−1.02124
$326$	3.88736e6	0.00621431
$327$	0	0
$328$	−4.37372e8	−0.684373
$329$	−1.23596e9	−1.91346
$330$	0	0
$331$	6.96332e8	1.05540	0.527702	−	0.849430i	$-0.323054\pi$
$331$	6.96332e8	1.05540	0.527702	+	0.849430i	$0.323054\pi$
$332$	2.14475e8	0.321657
$333$	0	0
$334$	−1.72609e7	−0.0253485
$335$	2.55238e8	0.370928
$336$	0	0
$337$	−7.32502e8	−1.04257	−0.521284	−	0.853384i	$-0.674547\pi$
$337$	−7.32502e8	−1.04257	−0.521284	+	0.853384i	$0.674547\pi$
$338$	−6.77984e7	−0.0955017
$339$	0	0
$340$	−4.34843e7	−0.0600006
$341$	−2.02759e8	−0.276910
$342$	0	0
$343$	7.66587e7	0.102573
$344$	6.49678e8	0.860485
$345$	0	0
$346$	5.53629e7	0.0718543
$347$	6.58184e8	0.845657	0.422828	−	0.906210i	$-0.361037\pi$
$347$	6.58184e8	0.845657	0.422828	+	0.906210i	$0.361037\pi$
$348$	0	0
$349$	−3.82277e7	−0.0481381	−0.0240691	−	0.999710i	$-0.507662\pi$
$349$	−3.82277e7	−0.0481381	−0.0240691	+	0.999710i	$0.507662\pi$
$350$	3.49528e8	0.435757
$351$	0	0
$352$	−6.06129e8	−0.740740
$353$	−8.56374e8	−1.03622	−0.518110	−	0.855314i	$-0.673364\pi$
$353$	−8.56374e8	−1.03622	−0.518110	+	0.855314i	$0.673364\pi$
$354$	0	0
$355$	4.60284e7	0.0546043
$356$	7.35890e8	0.864446
$357$	0	0
$358$	−2.88789e8	−0.332652
$359$	−1.50693e9	−1.71895	−0.859475	−	0.511177i	$-0.829210\pi$
$359$	−1.50693e9	−1.71895	−0.859475	+	0.511177i	$0.829210\pi$
$360$	0	0
$361$	2.36488e9	2.64566
$362$	4.10773e8	0.455116
$363$	0	0
$364$	−1.33130e9	−1.44684
$365$	−1.85539e8	−0.199715
$366$	0	0
$367$	−7.69693e8	−0.812805	−0.406402	−	0.913694i	$-0.633217\pi$
$367$	−7.69693e8	−0.812805	−0.406402	+	0.913694i	$0.633217\pi$
$368$	−1.06763e9	−1.11675
$369$	0	0
$370$	−1.06818e8	−0.109632
$371$	−1.04475e9	−1.06219
$372$	0	0
$373$	−8.69970e8	−0.868008	−0.434004	−	0.900911i	$-0.642900\pi$
$373$	−8.69970e8	−0.868008	−0.434004	+	0.900911i	$0.642900\pi$
$374$	−6.29488e7	−0.0622209
$375$	0	0
$376$	−8.68918e8	−0.842988
$377$	−2.04255e8	−0.196326
$378$	0	0
$379$	−6.86916e8	−0.648136	−0.324068	−	0.946034i	$-0.605051\pi$
$379$	−6.86916e8	−0.648136	−0.324068	+	0.946034i	$0.605051\pi$
$380$	5.69604e8	0.532514
$381$	0	0
$382$	−5.03627e8	−0.462261
$383$	−6.92766e8	−0.630073	−0.315037	−	0.949080i	$-0.602017\pi$
$383$	−6.92766e8	−0.630073	−0.315037	+	0.949080i	$0.602017\pi$
$384$	0	0
$385$	4.36109e8	0.389478
$386$	−3.84337e8	−0.340139
$387$	0	0
$388$	−3.22796e8	−0.280554
$389$	1.80412e7	0.0155397	0.00776985	−	0.999970i	$-0.497527\pi$
$389$	1.80412e7	0.0155397	0.00776985	+	0.999970i	$0.497527\pi$
$390$	0	0
$391$	−4.21283e8	−0.356415
$392$	8.10068e8	0.679236
$393$	0	0
$394$	3.81174e7	0.0313968
$395$	−3.34116e8	−0.272777
$396$	0	0
$397$	6.47110e8	0.519053	0.259526	−	0.965736i	$-0.416434\pi$
$397$	6.47110e8	0.519053	0.259526	+	0.965736i	$0.416434\pi$
$398$	−2.41428e8	−0.191954
$399$	0	0
$400$	−7.77543e8	−0.607455
$401$	2.37807e8	0.184171	0.0920853	−	0.995751i	$-0.470647\pi$
$401$	2.37807e8	0.184171	0.0920853	+	0.995751i	$0.470647\pi$
$402$	0	0
$403$	−4.78994e8	−0.364554
$404$	1.60253e8	0.120913
$405$	0	0
$406$	1.12963e8	0.0837714
$407$	1.21515e9	0.893410
$408$	0	0
$409$	1.17282e9	0.847618	0.423809	−	0.905752i	$-0.360693\pi$
$409$	1.17282e9	0.847618	0.423809	+	0.905752i	$0.360693\pi$
$410$	1.59112e8	0.114014
$411$	0	0
$412$	−1.20331e9	−0.847693
$413$	−2.68236e8	−0.187367
$414$	0	0
$415$	−1.65977e8	−0.113993
$416$	−1.43191e9	−0.975189
$417$	0	0
$418$	8.24572e8	0.552219
$419$	1.40480e9	0.932962	0.466481	−	0.884531i	$-0.345522\pi$
$419$	1.40480e9	0.932962	0.466481	+	0.884531i	$0.345522\pi$
$420$	0	0
$421$	−8.16663e8	−0.533403	−0.266702	−	0.963779i	$-0.585934\pi$
$421$	−8.16663e8	−0.533403	−0.266702	+	0.963779i	$0.585934\pi$
$422$	−1.10785e8	−0.0717609
$423$	0	0
$424$	−7.34490e8	−0.467956
$425$	−3.06815e8	−0.193872
$426$	0	0
$427$	7.05070e8	0.438263
$428$	2.09528e9	1.29178
$429$	0	0
$430$	−2.36347e8	−0.143354
$431$	−2.46225e9	−1.48137	−0.740683	−	0.671855i	$-0.765498\pi$
$431$	−2.46225e9	−1.48137	−0.740683	+	0.671855i	$0.765498\pi$
$432$	0	0
$433$	−1.20496e9	−0.713289	−0.356645	−	0.934240i	$-0.616079\pi$
$433$	−1.20496e9	−0.713289	−0.356645	+	0.934240i	$0.616079\pi$
$434$	2.64908e8	0.155554
$435$	0	0
$436$	1.34687e9	0.778254
$437$	5.51843e9	3.16323
$438$	0	0
$439$	7.43862e8	0.419630	0.209815	−	0.977741i	$-0.432714\pi$
$439$	7.43862e8	0.419630	0.209815	+	0.977741i	$0.432714\pi$
$440$	3.06598e8	0.171587
$441$	0	0
$442$	−1.48709e8	−0.0819143
$443$	−5.65900e8	−0.309262	−0.154631	−	0.987972i	$-0.549419\pi$
$443$	−5.65900e8	−0.309262	−0.154631	+	0.987972i	$0.549419\pi$
$444$	0	0
$445$	−5.69487e8	−0.306354
$446$	9.84510e8	0.525470
$447$	0	0
$448$	−1.05439e9	−0.554022
$449$	1.60627e9	0.837443	0.418722	−	0.908115i	$-0.362478\pi$
$449$	1.60627e9	0.837443	0.418722	+	0.908115i	$0.362478\pi$
$450$	0	0
$451$	−1.81005e9	−0.929121
$452$	1.78838e9	0.910909
$453$	0	0
$454$	−1.00897e9	−0.506036
$455$	1.03026e9	0.512750
$456$	0	0
$457$	−1.42935e9	−0.700538	−0.350269	−	0.936649i	$-0.613910\pi$
$457$	−1.42935e9	−0.700538	−0.350269	+	0.936649i	$0.613910\pi$
$458$	−6.97277e8	−0.339138
$459$	0	0
$460$	9.64579e8	0.462046
$461$	−6.78035e8	−0.322328	−0.161164	−	0.986928i	$-0.551525\pi$
$461$	−6.78035e8	−0.322328	−0.161164	+	0.986928i	$0.551525\pi$
$462$	0	0
$463$	7.42705e7	0.0347763	0.0173881	−	0.999849i	$-0.494465\pi$
$463$	7.42705e7	0.0347763	0.0173881	+	0.999849i	$0.494465\pi$
$464$	−2.51292e8	−0.116779
$465$	0	0
$466$	1.19523e9	0.547143
$467$	8.65991e8	0.393464	0.196732	−	0.980457i	$-0.436967\pi$
$467$	8.65991e8	0.393464	0.196732	+	0.980457i	$0.436967\pi$
$468$	0	0
$469$	3.79357e9	1.69802
$470$	3.16104e8	0.140439
$471$	0	0
$472$	−1.88578e8	−0.0825458
$473$	2.68867e9	1.16821
$474$	0	0
$475$	4.01900e9	1.72064
$476$	−6.46301e8	−0.274669
$477$	0	0
$478$	−1.30156e8	−0.0545088
$479$	−3.87827e9	−1.61237	−0.806183	−	0.591666i	$-0.798470\pi$
$479$	−3.87827e9	−1.61237	−0.806183	+	0.591666i	$0.798470\pi$
$480$	0	0
$481$	2.87066e9	1.17618
$482$	7.56177e8	0.307581
$483$	0	0
$484$	5.73176e8	0.229789
$485$	2.49803e8	0.0994266
$486$	0	0
$487$	3.18862e9	1.25098	0.625492	−	0.780231i	$-0.284898\pi$
$487$	3.18862e9	1.25098	0.625492	+	0.780231i	$0.284898\pi$
$488$	4.95686e8	0.193080
$489$	0	0
$490$	−2.94695e8	−0.113158
$491$	−1.59045e9	−0.606364	−0.303182	−	0.952933i	$-0.598049\pi$
$491$	−1.59045e9	−0.606364	−0.303182	+	0.952933i	$0.598049\pi$
$492$	0	0
$493$	−9.91588e7	−0.0372707
$494$	1.94796e9	0.727000
$495$	0	0
$496$	−5.89300e8	−0.216846
$497$	6.84114e8	0.249966
$498$	0	0
$499$	−3.98538e9	−1.43588	−0.717940	−	0.696105i	$-0.754915\pi$
$499$	−3.98538e9	−1.43588	−0.717940	+	0.696105i	$0.754915\pi$
$500$	1.48203e9	0.530227
$501$	0	0
$502$	−1.54682e9	−0.545729
$503$	−4.29501e9	−1.50479	−0.752396	−	0.658711i	$-0.771102\pi$
$503$	−4.29501e9	−1.50479	−0.752396	+	0.658711i	$0.771102\pi$
$504$	0	0
$505$	−1.24016e8	−0.0428507
$506$	1.39635e9	0.479144
$507$	0	0
$508$	1.98535e9	0.671914
$509$	2.62335e9	0.881746	0.440873	−	0.897570i	$-0.354669\pi$
$509$	2.62335e9	0.881746	0.440873	+	0.897570i	$0.354669\pi$
$510$	0	0
$511$	−2.75764e9	−0.914248
$512$	−3.05967e9	−1.00746
$513$	0	0
$514$	3.97013e8	0.128954
$515$	9.31213e8	0.300417
$516$	0	0
$517$	−3.59598e9	−1.14446
$518$	−1.58762e9	−0.501871
$519$	0	0
$520$	7.24303e8	0.225896
$521$	4.65644e9	1.44252	0.721260	−	0.692664i	$-0.243563\pi$
$521$	4.65644e9	1.44252	0.721260	+	0.692664i	$0.243563\pi$
$522$	0	0
$523$	−5.16451e9	−1.57860	−0.789301	−	0.614006i	$-0.789557\pi$
$523$	−5.16451e9	−1.57860	−0.789301	+	0.614006i	$0.789557\pi$
$524$	4.36026e9	1.32389
$525$	0	0
$526$	8.38151e8	0.251115
$527$	−2.32535e8	−0.0692073
$528$	0	0
$529$	5.94019e9	1.74464
$530$	2.67200e8	0.0779599
$531$	0	0
$532$	8.46595e9	2.43773
$533$	−4.27603e9	−1.22319
$534$	0	0
$535$	−1.62148e9	−0.457799
$536$	2.66700e9	0.748076
$537$	0	0
$538$	1.34525e9	0.372447
$539$	3.35243e9	0.922146
$540$	0	0
$541$	7.10793e9	1.92998	0.964990	−	0.262288i	$-0.0844772\pi$
$541$	7.10793e9	1.92998	0.964990	+	0.262288i	$0.0844772\pi$
$542$	1.84948e9	0.498944
$543$	0	0
$544$	−6.95144e8	−0.185131
$545$	−1.04230e9	−0.275808
$546$	0	0
$547$	4.92835e9	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$547$	4.92835e9	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$548$	−5.42275e9	−1.40763
$549$	0	0
$550$	1.01694e9	0.260631
$551$	1.29889e9	0.330782
$552$	0	0
$553$	−4.96592e9	−1.24871
$554$	1.61355e9	0.403179
$555$	0	0
$556$	1.02323e9	0.252470
$557$	5.04922e9	1.23803	0.619015	−	0.785379i	$-0.287532\pi$
$557$	5.04922e9	1.23803	0.619015	+	0.785379i	$0.287532\pi$
$558$	0	0
$559$	6.35166e9	1.53796
$560$	1.26751e9	0.304996
$561$	0	0
$562$	2.22478e9	0.528701
$563$	1.36397e9	0.322125	0.161063	−	0.986944i	$-0.448508\pi$
$563$	1.36397e9	0.322125	0.161063	+	0.986944i	$0.448508\pi$
$564$	0	0
$565$	−1.38398e9	−0.322820
$566$	−4.69566e8	−0.108853
$567$	0	0
$568$	4.80953e8	0.110124
$569$	1.81850e9	0.413829	0.206915	−	0.978359i	$-0.433658\pi$
$569$	1.81850e9	0.413829	0.206915	+	0.978359i	$0.433658\pi$
$570$	0	0
$571$	−9.20531e8	−0.206925	−0.103462	−	0.994633i	$-0.532992\pi$
$571$	−9.20531e8	−0.206925	−0.103462	+	0.994633i	$0.532992\pi$
$572$	−3.87336e9	−0.865371
$573$	0	0
$574$	2.36486e9	0.521932
$575$	6.80585e9	1.49295
$576$	0	0
$577$	−6.37038e9	−1.38054	−0.690271	−	0.723551i	$-0.742509\pi$
$577$	−6.37038e9	−1.38054	−0.690271	+	0.723551i	$0.742509\pi$
$578$	1.48761e9	0.320437
$579$	0	0
$580$	2.27036e8	0.0483167
$581$	−2.46689e9	−0.521834
$582$	0	0
$583$	−3.03966e9	−0.635308
$584$	−1.93871e9	−0.402779
$585$	0	0
$586$	−1.27265e9	−0.261256
$587$	−4.67433e9	−0.953863	−0.476931	−	0.878941i	$-0.658251\pi$
$587$	−4.67433e9	−0.953863	−0.476931	+	0.878941i	$0.658251\pi$
$588$	0	0
$589$	3.04600e9	0.614224
$590$	6.86030e7	0.0137519
$591$	0	0
$592$	3.53174e9	0.699620
$593$	−5.82162e9	−1.14644	−0.573221	−	0.819401i	$-0.694306\pi$
$593$	−5.82162e9	−1.14644	−0.573221	+	0.819401i	$0.694306\pi$
$594$	0	0
$595$	5.00156e8	0.0973410
$596$	−1.97445e9	−0.382018
$597$	0	0
$598$	3.29870e9	0.630796
$599$	−6.17800e9	−1.17450	−0.587251	−	0.809405i	$-0.699790\pi$
$599$	−6.17800e9	−1.17450	−0.587251	+	0.809405i	$0.699790\pi$
$600$	0	0
$601$	8.32412e9	1.56415	0.782074	−	0.623186i	$-0.214162\pi$
$601$	8.32412e9	1.56415	0.782074	+	0.623186i	$0.214162\pi$
$602$	−3.51279e9	−0.656242
$603$	0	0
$604$	−1.66547e9	−0.307544
$605$	−4.43566e8	−0.0814356
$606$	0	0
$607$	−6.60638e9	−1.19896	−0.599478	−	0.800391i	$-0.704625\pi$
$607$	−6.60638e9	−1.19896	−0.599478	+	0.800391i	$0.704625\pi$
$608$	9.10576e9	1.64306
$609$	0	0
$610$	−1.80326e8	−0.0321665
$611$	−8.49509e9	−1.50669
$612$	0	0
$613$	−5.00900e9	−0.878293	−0.439146	−	0.898415i	$-0.644719\pi$
$613$	−5.00900e9	−0.878293	−0.439146	+	0.898415i	$0.644719\pi$
$614$	−2.10860e9	−0.367625
$615$	0	0
$616$	4.55693e9	0.785488
$617$	−8.30278e9	−1.42307	−0.711534	−	0.702652i	$-0.751999\pi$
$617$	−8.30278e9	−1.42307	−0.711534	+	0.702652i	$0.751999\pi$
$618$	0	0
$619$	−4.13223e9	−0.700272	−0.350136	−	0.936699i	$-0.613865\pi$
$619$	−4.13223e9	−0.700272	−0.350136	+	0.936699i	$0.613865\pi$
$620$	5.32418e8	0.0897184
$621$	0	0
$622$	−1.52641e9	−0.254333
$623$	−8.46420e9	−1.40242
$624$	0	0
$625$	4.35335e9	0.713252
$626$	−2.80082e9	−0.456325
$627$	0	0
$628$	−4.67032e9	−0.752468
$629$	1.39361e9	0.223287
$630$	0	0
$631$	8.48866e8	0.134504	0.0672522	−	0.997736i	$-0.478577\pi$
$631$	8.48866e8	0.134504	0.0672522	+	0.997736i	$0.478577\pi$
$632$	−3.49120e9	−0.550129
$633$	0	0
$634$	−9.60948e7	−0.0149757
$635$	−1.53641e9	−0.238122
$636$	0	0
$637$	7.91974e9	1.21401
$638$	3.28662e8	0.0501046
$639$	0	0
$640$	2.06382e9	0.311202
$641$	−4.40906e9	−0.661216	−0.330608	−	0.943768i	$-0.607254\pi$
$641$	−4.40906e9	−0.661216	−0.330608	+	0.943768i	$0.607254\pi$
$642$	0	0
$643$	−1.11714e10	−1.65717	−0.828586	−	0.559862i	$-0.810854\pi$
$643$	−1.11714e10	−1.65717	−0.828586	+	0.559862i	$0.810854\pi$
$644$	1.43364e10	2.11514
$645$	0	0
$646$	9.45667e8	0.138014
$647$	1.09926e10	1.59564	0.797818	−	0.602899i	$-0.205988\pi$
$647$	1.09926e10	1.59564	0.797818	+	0.602899i	$0.205988\pi$
$648$	0	0
$649$	−7.80424e8	−0.112066
$650$	2.40240e9	0.343122
$651$	0	0
$652$	1.16122e8	0.0164077
$653$	5.52901e9	0.777053	0.388527	−	0.921437i	$-0.372984\pi$
$653$	5.52901e9	0.777053	0.388527	+	0.921437i	$0.372984\pi$
$654$	0	0
$655$	−3.37430e9	−0.469179
$656$	−5.26075e9	−0.727585
$657$	0	0
$658$	4.69821e9	0.642898
$659$	8.17686e9	1.11298	0.556491	−	0.830854i	$-0.312148\pi$
$659$	8.17686e9	1.11298	0.556491	+	0.830854i	$0.312148\pi$
$660$	0	0
$661$	1.00210e10	1.34960	0.674801	−	0.737999i	$-0.264229\pi$
$661$	1.00210e10	1.34960	0.674801	+	0.737999i	$0.264229\pi$
$662$	−2.64694e9	−0.354603
$663$	0	0
$664$	−1.73430e9	−0.229898
$665$	−6.55158e9	−0.863914
$666$	0	0
$667$	2.19957e9	0.287010
$668$	−5.15614e8	−0.0669279
$669$	0	0
$670$	−9.70229e8	−0.124627
$671$	2.05138e9	0.262130
$672$	0	0
$673$	−1.41401e10	−1.78814	−0.894068	−	0.447931i	$-0.852161\pi$
$673$	−1.41401e10	−1.78814	−0.894068	+	0.447931i	$0.852161\pi$
$674$	2.78443e9	0.350290
$675$	0	0
$676$	−2.02526e9	−0.252154
$677$	−9.91668e9	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$677$	−9.91668e9	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$678$	0	0
$679$	3.71280e9	0.455152
$680$	3.51625e8	0.0428843
$681$	0	0
$682$	7.70740e8	0.0930384
$683$	1.13006e10	1.35715	0.678575	−	0.734531i	$-0.262598\pi$
$683$	1.13006e10	1.35715	0.678575	+	0.734531i	$0.262598\pi$
$684$	0	0
$685$	4.19653e9	0.498854
$686$	−2.91400e8	−0.0344632
$687$	0	0
$688$	7.81437e9	0.914817
$689$	−7.18083e9	−0.836388
$690$	0	0
$691$	6.57667e9	0.758286	0.379143	−	0.925338i	$-0.376219\pi$
$691$	6.57667e9	0.758286	0.379143	+	0.925338i	$0.376219\pi$
$692$	1.65379e9	0.189718
$693$	0	0
$694$	−2.50193e9	−0.284130
$695$	−7.91849e8	−0.0894737
$696$	0	0
$697$	−2.07587e9	−0.232212
$698$	1.45314e8	0.0161738
$699$	0	0
$700$	1.04410e10	1.15053
$701$	1.12915e10	1.23805	0.619027	−	0.785370i	$-0.287527\pi$
$701$	1.12915e10	1.23805	0.619027	+	0.785370i	$0.287527\pi$
$702$	0	0
$703$	−1.82550e10	−1.98170
$704$	−3.06770e9	−0.331366
$705$	0	0
$706$	3.25530e9	0.348157
$707$	−1.84323e9	−0.196161
$708$	0	0
$709$	4.04329e9	0.426062	0.213031	−	0.977045i	$-0.431666\pi$
$709$	4.04329e9	0.426062	0.213031	+	0.977045i	$0.431666\pi$
$710$	−1.74966e8	−0.0183464
$711$	0	0
$712$	−5.95060e9	−0.617846
$713$	5.15816e9	0.532944
$714$	0	0
$715$	2.99750e9	0.306682
$716$	−8.62662e9	−0.878305
$717$	0	0
$718$	5.72825e9	0.577546
$719$	1.82139e10	1.82748	0.913739	−	0.406302i	$-0.133182\pi$
$719$	1.82139e10	1.82748	0.913739	+	0.406302i	$0.133182\pi$
$720$	0	0
$721$	1.38405e10	1.37524
$722$	−8.98954e9	−0.888908
$723$	0	0
$724$	1.22705e10	1.20165
$725$	1.60192e9	0.156119
$726$	0	0
$727$	8.36470e9	0.807384	0.403692	−	0.914895i	$-0.367727\pi$
$727$	8.36470e9	0.807384	0.403692	+	0.914895i	$0.367727\pi$
$728$	1.07652e10	1.03410
$729$	0	0
$730$	7.05283e8	0.0671016
$731$	3.08352e9	0.291968
$732$	0	0
$733$	1.25663e10	1.17853	0.589267	−	0.807939i	$-0.299417\pi$
$733$	1.25663e10	1.17853	0.589267	+	0.807939i	$0.299417\pi$
$734$	2.92581e9	0.273092
$735$	0	0
$736$	1.54199e10	1.42563
$737$	1.10373e10	1.01561
$738$	0	0
$739$	−1.15684e10	−1.05443	−0.527214	−	0.849733i	$-0.676763\pi$
$739$	−1.15684e10	−1.05443	−0.527214	+	0.849733i	$0.676763\pi$
$740$	−3.19084e9	−0.289463
$741$	0	0
$742$	3.97136e9	0.356883
$743$	−1.87018e10	−1.67272	−0.836360	−	0.548181i	$-0.815321\pi$
$743$	−1.87018e10	−1.67272	−0.836360	+	0.548181i	$0.815321\pi$
$744$	0	0
$745$	1.52797e9	0.135385
$746$	3.30699e9	0.291640
$747$	0	0
$748$	−1.88039e9	−0.164283
$749$	−2.40999e10	−2.09570
$750$	0	0
$751$	−1.25529e10	−1.08145	−0.540723	−	0.841201i	$-0.681849\pi$
$751$	−1.25529e10	−1.08145	−0.540723	+	0.841201i	$0.681849\pi$
$752$	−1.04514e10	−0.896215
$753$	0	0
$754$	7.76427e8	0.0659631
$755$	1.28886e9	0.108992
$756$	0	0
$757$	−9.74904e9	−0.816820	−0.408410	−	0.912799i	$-0.633917\pi$
$757$	−9.74904e9	−0.816820	−0.408410	+	0.912799i	$0.633917\pi$
$758$	2.61115e9	0.217766
$759$	0	0
$760$	−4.60597e9	−0.380604
$761$	−1.96148e10	−1.61338	−0.806692	−	0.590972i	$-0.798745\pi$
$761$	−1.96148e10	−1.61338	−0.806692	+	0.590972i	$0.798745\pi$
$762$	0	0
$763$	−1.54916e10	−1.26259
$764$	−1.50442e10	−1.22051
$765$	0	0
$766$	2.63339e9	0.211697
$767$	−1.84366e9	−0.147536
$768$	0	0
$769$	1.59863e9	0.126767	0.0633834	−	0.997989i	$-0.479811\pi$
$769$	1.59863e9	0.126767	0.0633834	+	0.997989i	$0.479811\pi$
$770$	−1.65777e9	−0.130860
$771$	0	0
$772$	−1.14808e10	−0.898074
$773$	2.19525e10	1.70945	0.854725	−	0.519081i	$-0.173726\pi$
$773$	2.19525e10	1.70945	0.854725	+	0.519081i	$0.173726\pi$
$774$	0	0
$775$	3.75662e9	0.289895
$776$	2.61021e9	0.200521
$777$	0	0
$778$	−6.85795e7	−0.00522115
$779$	2.71920e10	2.06091
$780$	0	0
$781$	1.99041e9	0.149508
$782$	1.60141e9	0.119751
$783$	0	0
$784$	9.74356e9	0.722123
$785$	3.61424e9	0.266670
$786$	0	0
$787$	2.94002e9	0.215000	0.107500	−	0.994205i	$-0.465715\pi$
$787$	2.94002e9	0.215000	0.107500	+	0.994205i	$0.465715\pi$
$788$	1.13863e9	0.0828975
$789$	0	0
$790$	1.27006e9	0.0916496
$791$	−2.05699e10	−1.47780
$792$	0	0
$793$	4.84614e9	0.345096
$794$	−2.45984e9	−0.174395
$795$	0	0
$796$	−7.21186e9	−0.506818
$797$	2.52071e10	1.76367	0.881836	−	0.471556i	$-0.156307\pi$
$797$	2.52071e10	1.76367	0.881836	+	0.471556i	$0.156307\pi$
$798$	0	0
$799$	−4.12408e9	−0.286031
$800$	1.12301e10	0.775475
$801$	0	0
$802$	−9.03969e8	−0.0618790
$803$	−8.02325e9	−0.546822
$804$	0	0
$805$	−1.10946e10	−0.749592
$806$	1.82078e9	0.122486
$807$	0	0
$808$	−1.29585e9	−0.0864201
$809$	4.67981e9	0.310748	0.155374	−	0.987856i	$-0.450342\pi$
$809$	4.67981e9	0.310748	0.155374	+	0.987856i	$0.450342\pi$
$810$	0	0
$811$	1.13390e10	0.746451	0.373225	−	0.927741i	$-0.378252\pi$
$811$	1.13390e10	0.746451	0.373225	+	0.927741i	$0.378252\pi$
$812$	3.37441e9	0.221183
$813$	0	0
$814$	−4.61912e9	−0.300175
$815$	−8.98639e7	−0.00581479
$816$	0	0
$817$	−4.03913e10	−2.59126
$818$	−4.45820e9	−0.284789
$819$	0	0
$820$	4.75295e9	0.301033
$821$	2.40573e10	1.51721	0.758606	−	0.651549i	$-0.225881\pi$
$821$	2.40573e10	1.51721	0.758606	+	0.651549i	$0.225881\pi$
$822$	0	0
$823$	3.05338e10	1.90933	0.954666	−	0.297681i	$-0.0962131\pi$
$823$	3.05338e10	1.90933	0.954666	+	0.297681i	$0.0962131\pi$
$824$	9.73030e9	0.605872
$825$	0	0
$826$	1.01964e9	0.0629529
$827$	2.87404e10	1.76695	0.883473	−	0.468482i	$-0.155199\pi$
$827$	2.87404e10	1.76695	0.883473	+	0.468482i	$0.155199\pi$
$828$	0	0
$829$	−1.64553e10	−1.00315	−0.501575	−	0.865114i	$-0.667246\pi$
$829$	−1.64553e10	−1.00315	−0.501575	+	0.865114i	$0.667246\pi$
$830$	6.30921e8	0.0383003
$831$	0	0
$832$	−7.24709e9	−0.436246
$833$	3.84477e9	0.230469
$834$	0	0
$835$	3.99021e8	0.0237188
$836$	2.46314e10	1.45803
$837$	0	0
$838$	−5.34000e9	−0.313464
$839$	−1.71860e10	−1.00463	−0.502316	−	0.864684i	$-0.667519\pi$
$839$	−1.71860e10	−1.00463	−0.502316	+	0.864684i	$0.667519\pi$
$840$	0	0
$841$	−1.67322e10	−0.969987
$842$	3.10435e9	0.179217
$843$	0	0
$844$	−3.30934e9	−0.189471
$845$	1.56729e9	0.0893618
$846$	0	0
$847$	−6.59266e9	−0.372794
$848$	−8.83449e9	−0.497503
$849$	0	0
$850$	1.16629e9	0.0651386
$851$	−3.09134e10	−1.71946
$852$	0	0
$853$	−2.96393e9	−0.163511	−0.0817555	−	0.996652i	$-0.526053\pi$
$853$	−2.96393e9	−0.163511	−0.0817555	+	0.996652i	$0.526053\pi$
$854$	−2.68016e9	−0.147251
$855$	0	0
$856$	−1.69430e10	−0.923276
$857$	7.20996e9	0.391291	0.195646	−	0.980675i	$-0.437320\pi$
$857$	7.20996e9	0.391291	0.195646	+	0.980675i	$0.437320\pi$
$858$	0	0
$859$	−2.08163e10	−1.12054	−0.560269	−	0.828311i	$-0.689302\pi$
$859$	−2.08163e10	−1.12054	−0.560269	+	0.828311i	$0.689302\pi$
$860$	−7.06008e9	−0.378499
$861$	0	0
$862$	9.35968e9	0.497720
$863$	−1.95918e9	−0.103761	−0.0518807	−	0.998653i	$-0.516522\pi$
$863$	−1.95918e9	−0.103761	−0.0518807	+	0.998653i	$0.516522\pi$
$864$	0	0
$865$	−1.27982e9	−0.0672347
$866$	4.58038e9	0.239656
$867$	0	0
$868$	7.91325e9	0.410711
$869$	−1.44482e10	−0.746868
$870$	0	0
$871$	2.60743e10	1.33705
$872$	−1.08911e10	−0.556242
$873$	0	0
$874$	−2.09770e10	−1.06281
$875$	−1.70463e10	−0.860204
$876$	0	0
$877$	−3.94405e10	−1.97444	−0.987220	−	0.159366i	$-0.949055\pi$
$877$	−3.94405e10	−1.97444	−0.987220	+	0.159366i	$0.949055\pi$
$878$	−2.82762e9	−0.140991
$879$	0	0
$880$	3.68778e9	0.182422
$881$	−2.19999e10	−1.08394	−0.541970	−	0.840398i	$-0.682321\pi$
$881$	−2.19999e10	−1.08394	−0.541970	+	0.840398i	$0.682321\pi$
$882$	0	0
$883$	−8.61343e8	−0.0421031	−0.0210515	−	0.999778i	$-0.506701\pi$
$883$	−8.61343e8	−0.0421031	−0.0210515	+	0.999778i	$0.506701\pi$
$884$	−4.44220e9	−0.216279
$885$	0	0
$886$	2.15114e9	0.103908
$887$	−4.62675e9	−0.222609	−0.111305	−	0.993786i	$-0.535503\pi$
$887$	−4.62675e9	−0.222609	−0.111305	+	0.993786i	$0.535503\pi$
$888$	0	0
$889$	−2.28354e10	−1.09007
$890$	2.16477e9	0.102931
$891$	0	0
$892$	2.94090e10	1.38741
$893$	5.40217e10	2.53857
$894$	0	0
$895$	6.67592e9	0.311265
$896$	3.06743e10	1.42461
$897$	0	0
$898$	−6.10584e9	−0.281370
$899$	1.21409e9	0.0557305
$900$	0	0
$901$	−3.48605e9	−0.158781
$902$	6.88047e9	0.312173
$903$	0	0
$904$	−1.44613e10	−0.651054
$905$	−9.49583e9	−0.425856
$906$	0	0
$907$	−5.22593e9	−0.232562	−0.116281	−	0.993216i	$-0.537097\pi$
$907$	−5.22593e9	−0.232562	−0.116281	+	0.993216i	$0.537097\pi$
$908$	−3.01396e10	−1.33609
$909$	0	0
$910$	−3.91628e9	−0.172278
$911$	3.60178e9	0.157835	0.0789175	−	0.996881i	$-0.474854\pi$
$911$	3.60178e9	0.157835	0.0789175	+	0.996881i	$0.474854\pi$
$912$	0	0
$913$	−7.17732e9	−0.312115
$914$	5.43334e9	0.235372
$915$	0	0
$916$	−2.08289e10	−0.895430
$917$	−5.01517e10	−2.14780
$918$	0	0
$919$	8.64702e9	0.367504	0.183752	−	0.982973i	$-0.441176\pi$
$919$	8.64702e9	0.367504	0.183752	+	0.982973i	$0.441176\pi$
$920$	−7.79983e9	−0.330239
$921$	0	0
$922$	2.57739e9	0.108298
$923$	4.70210e9	0.196828
$924$	0	0
$925$	−2.25138e10	−0.935305
$926$	−2.82322e8	−0.0116844
$927$	0	0
$928$	3.62942e9	0.149080
$929$	2.75239e9	0.112630	0.0563151	−	0.998413i	$-0.482065\pi$
$929$	2.75239e9	0.112630	0.0563151	+	0.998413i	$0.482065\pi$
$930$	0	0
$931$	−5.03630e10	−2.04544
$932$	3.57036e10	1.44463
$933$	0	0
$934$	−3.29186e9	−0.132199
$935$	1.45518e9	0.0582207
$936$	0	0
$937$	2.00871e10	0.797678	0.398839	−	0.917021i	$-0.369413\pi$
$937$	2.00871e10	0.797678	0.398839	+	0.917021i	$0.369413\pi$
$938$	−1.44204e10	−0.570515
$939$	0	0
$940$	9.44258e9	0.370803
$941$	5.18639e9	0.202909	0.101455	−	0.994840i	$-0.467650\pi$
$941$	5.18639e9	0.202909	0.101455	+	0.994840i	$0.467650\pi$
$942$	0	0
$943$	4.60474e10	1.78819
$944$	−2.26823e9	−0.0877578
$945$	0	0
$946$	−1.02203e10	−0.392506
$947$	−3.37384e10	−1.29092	−0.645460	−	0.763794i	$-0.723334\pi$
$947$	−3.37384e10	−1.29092	−0.645460	+	0.763794i	$0.723334\pi$
$948$	0	0
$949$	−1.89540e10	−0.719895
$950$	−1.52773e10	−0.578114
$951$	0	0
$952$	5.22615e9	0.196315
$953$	−3.47736e10	−1.30144	−0.650720	−	0.759318i	$-0.725533\pi$
$953$	−3.47736e10	−1.30144	−0.650720	+	0.759318i	$0.725533\pi$
$954$	0	0
$955$	1.16423e10	0.432542
$956$	−3.88799e9	−0.143920
$957$	0	0
$958$	1.47423e10	0.541735
$959$	6.23725e10	2.28364
$960$	0	0
$961$	−2.46655e10	−0.896515
$962$	−1.09121e10	−0.395182
$963$	0	0
$964$	2.25883e10	0.812109
$965$	8.88470e9	0.318271
$966$	0	0
$967$	−4.82322e10	−1.71532	−0.857659	−	0.514219i	$-0.828082\pi$
$967$	−4.82322e10	−1.71532	−0.857659	+	0.514219i	$0.828082\pi$
$968$	−4.63484e9	−0.164237
$969$	0	0
$970$	−9.49569e8	−0.0334061
$971$	−6.11947e9	−0.214509	−0.107255	−	0.994232i	$-0.534206\pi$
$971$	−6.11947e9	−0.214509	−0.107255	+	0.994232i	$0.534206\pi$
$972$	0	0
$973$	−1.17691e10	−0.409591
$974$	−1.21208e10	−0.420315
$975$	0	0
$976$	5.96215e9	0.205271
$977$	−4.14878e10	−1.42328	−0.711638	−	0.702546i	$-0.752046\pi$
$977$	−4.14878e10	−1.42328	−0.711638	+	0.702546i	$0.752046\pi$
$978$	0	0
$979$	−2.46263e10	−0.838803
$980$	−8.80305e9	−0.298774
$981$	0	0
$982$	6.04571e9	0.203731
$983$	−4.34606e10	−1.45935	−0.729674	−	0.683795i	$-0.760328\pi$
$983$	−4.34606e10	−1.45935	−0.729674	+	0.683795i	$0.760328\pi$
$984$	0	0
$985$	−8.81158e8	−0.0293783
$986$	3.76929e8	0.0125225
$987$	0	0
$988$	5.81888e10	1.91951
$989$	−6.83994e10	−2.24836
$990$	0	0
$991$	2.63085e10	0.858694	0.429347	−	0.903140i	$-0.358744\pi$
$991$	2.63085e10	0.858694	0.429347	+	0.903140i	$0.358744\pi$
$992$	8.51129e9	0.276824
$993$	0	0
$994$	−2.60050e9	−0.0839855
$995$	5.58108e9	0.179613
$996$	0	0
$997$	−4.90773e10	−1.56837	−0.784183	−	0.620530i	$-0.786918\pi$
$997$	−4.90773e10	−1.56837	−0.784183	+	0.620530i	$0.786918\pi$
$998$	1.51495e10	0.482438
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.10		18
3.2	odd	2		177.8.a.d.1.9	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.9	✓	18	3.2	odd	2
531.8.a.e.1.10		18	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-3.80127 q^{2} -113.550 q^{4} +87.8737 q^{5} +1306.06 q^{7} +918.197 q^{8} +O(q^{10})\)	\(q-3.80127 q^{2} -113.550 q^{4} +87.8737 q^{5} +1306.06 q^{7} +918.197 q^{8} -334.031 q^{10} +3799.92 q^{11} +8976.87 q^{13} -4964.66 q^{14} +11044.1 q^{16} +4357.97 q^{17} -57085.5 q^{19} -9978.10 q^{20} -14444.5 q^{22} -96669.6 q^{23} -70403.2 q^{25} -34123.5 q^{26} -148303. q^{28} -22753.4 q^{29} -53358.6 q^{31} -159511. q^{32} -16565.8 q^{34} +114768. q^{35} +319784. q^{37} +216997. q^{38} +80685.4 q^{40} -476338. q^{41} +707558. q^{43} -431483. q^{44} +367467. q^{46} -946330. q^{47} +882238. q^{49} +267621. q^{50} -1.01933e6 q^{52} -799926. q^{53} +333913. q^{55} +1.19922e6 q^{56} +86491.9 q^{58} -205379. q^{59} +539847. q^{61} +202830. q^{62} -807306. q^{64} +788832. q^{65} +2.90460e6 q^{67} -494849. q^{68} -436264. q^{70} +523802. q^{71} -2.11143e6 q^{73} -1.21558e6 q^{74} +6.48208e6 q^{76} +4.96291e6 q^{77} -3.80223e6 q^{79} +970490. q^{80} +1.81069e6 q^{82} -1.88881e6 q^{83} +382951. q^{85} -2.68962e6 q^{86} +3.48908e6 q^{88} -6.48074e6 q^{89} +1.17243e7 q^{91} +1.09769e7 q^{92} +3.59725e6 q^{94} -5.01631e6 q^{95} +2.84275e6 q^{97} -3.35362e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more