LMFDB - Embedded newform 507.4.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.73419$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73419 q^{2} -3.00000 q^{3} -0.524213 q^{4} -21.1246 q^{5} +8.20257 q^{6} -25.8618 q^{7} +23.3068 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.73419 q^{2} -3.00000 q^{3} -0.524213 q^{4} -21.1246 q^{5} +8.20257 q^{6} -25.8618 q^{7} +23.3068 q^{8} +9.00000 q^{9} +57.7586 q^{10} -6.96892 q^{11} +1.57264 q^{12} +70.7111 q^{14} +63.3738 q^{15} -59.5315 q^{16} +122.879 q^{17} -24.6077 q^{18} +43.1340 q^{19} +11.0738 q^{20} +77.5854 q^{21} +19.0543 q^{22} -75.5492 q^{23} -69.9204 q^{24} +321.248 q^{25} -27.0000 q^{27} +13.5571 q^{28} -163.764 q^{29} -173.276 q^{30} +139.421 q^{31} -23.6841 q^{32} +20.9068 q^{33} -335.975 q^{34} +546.320 q^{35} -4.71792 q^{36} +2.80616 q^{37} -117.936 q^{38} -492.347 q^{40} -300.555 q^{41} -212.133 q^{42} +363.145 q^{43} +3.65320 q^{44} -190.121 q^{45} +206.566 q^{46} -41.2660 q^{47} +178.594 q^{48} +325.834 q^{49} -878.354 q^{50} -368.638 q^{51} -125.763 q^{53} +73.8231 q^{54} +147.216 q^{55} -602.756 q^{56} -129.402 q^{57} +447.762 q^{58} -407.311 q^{59} -33.2214 q^{60} +536.710 q^{61} -381.204 q^{62} -232.756 q^{63} +541.009 q^{64} -57.1630 q^{66} +340.155 q^{67} -64.4150 q^{68} +226.648 q^{69} -1493.74 q^{70} +514.831 q^{71} +209.761 q^{72} -491.231 q^{73} -7.67257 q^{74} -963.745 q^{75} -22.6114 q^{76} +180.229 q^{77} +762.869 q^{79} +1257.58 q^{80} +81.0000 q^{81} +821.774 q^{82} +345.948 q^{83} -40.6713 q^{84} -2595.78 q^{85} -992.907 q^{86} +491.292 q^{87} -162.423 q^{88} +362.482 q^{89} +519.828 q^{90} +39.6039 q^{92} -418.264 q^{93} +112.829 q^{94} -911.188 q^{95} +71.0523 q^{96} -276.297 q^{97} -890.890 q^{98} -62.7203 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9	\( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9} + 198 q^{10} - 37 q^{11} - 96 q^{12} + 98 q^{14} + 123 q^{15} + 32 q^{16} - 134 q^{17} - 72 q^{18} + 72 q^{19} - 356 q^{20} + 3 q^{21} + 274 q^{22} + 226 q^{23} + 333 q^{24} + 612 q^{25} - 243 q^{27} - 132 q^{28} - 547 q^{29} - 594 q^{30} + 521 q^{31} - 721 q^{32} + 111 q^{33} + 100 q^{34} + 138 q^{35} + 288 q^{36} - 584 q^{37} - 416 q^{38} + 1342 q^{40} - 482 q^{41} - 294 q^{42} + 158 q^{43} - 1453 q^{44} - 369 q^{45} - 1537 q^{46} - 1500 q^{47} - 96 q^{48} + 642 q^{49} - 2777 q^{50} + 402 q^{51} + 1399 q^{53} + 216 q^{54} - 1408 q^{55} - 616 q^{56} - 216 q^{57} - 1455 q^{58} - 1541 q^{59} + 1068 q^{60} + 2092 q^{61} - 293 q^{62} - 9 q^{63} + 2481 q^{64} - 822 q^{66} - 252 q^{67} - 1579 q^{68} - 678 q^{69} - 2492 q^{70} - 2352 q^{71} - 999 q^{72} - 903 q^{73} + 1037 q^{74} - 1836 q^{75} + 485 q^{76} - 1686 q^{77} - 115 q^{79} - 5701 q^{80} + 729 q^{81} - 5147 q^{82} - 1207 q^{83} + 396 q^{84} - 4308 q^{85} - 5691 q^{86} + 1641 q^{87} - 484 q^{88} - 2336 q^{89} + 1782 q^{90} + 2087 q^{92} - 1563 q^{93} - 468 q^{94} - 222 q^{95} + 2163 q^{96} - 2155 q^{97} - 5593 q^{98} - 333 q^{99}+O(q^{100}) \) 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9 + 198 * q^10 - 37 * q^11 - 96 * q^12 + 98 * q^14 + 123 * q^15 + 32 * q^16 - 134 * q^17 - 72 * q^18 + 72 * q^19 - 356 * q^20 + 3 * q^21 + 274 * q^22 + 226 * q^23 + 333 * q^24 + 612 * q^25 - 243 * q^27 - 132 * q^28 - 547 * q^29 - 594 * q^30 + 521 * q^31 - 721 * q^32 + 111 * q^33 + 100 * q^34 + 138 * q^35 + 288 * q^36 - 584 * q^37 - 416 * q^38 + 1342 * q^40 - 482 * q^41 - 294 * q^42 + 158 * q^43 - 1453 * q^44 - 369 * q^45 - 1537 * q^46 - 1500 * q^47 - 96 * q^48 + 642 * q^49 - 2777 * q^50 + 402 * q^51 + 1399 * q^53 + 216 * q^54 - 1408 * q^55 - 616 * q^56 - 216 * q^57 - 1455 * q^58 - 1541 * q^59 + 1068 * q^60 + 2092 * q^61 - 293 * q^62 - 9 * q^63 + 2481 * q^64 - 822 * q^66 - 252 * q^67 - 1579 * q^68 - 678 * q^69 - 2492 * q^70 - 2352 * q^71 - 999 * q^72 - 903 * q^73 + 1037 * q^74 - 1836 * q^75 + 485 * q^76 - 1686 * q^77 - 115 * q^79 - 5701 * q^80 + 729 * q^81 - 5147 * q^82 - 1207 * q^83 + 396 * q^84 - 4308 * q^85 - 5691 * q^86 + 1641 * q^87 - 484 * q^88 - 2336 * q^89 + 1782 * q^90 + 2087 * q^92 - 1563 * q^93 - 468 * q^94 - 222 * q^95 + 2163 * q^96 - 2155 * q^97 - 5593 * q^98 - 333 * 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 $

$2$	−2.73419	−0.966682	−0.483341	−	0.875432i	$-0.660577\pi$
$2$	−2.73419	−0.966682	−0.483341	+	0.875432i	$0.660577\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−0.524213	−0.0655266
$5$	−21.1246	−1.88944	−0.944721	−	0.327877i	$-0.893667\pi$
$5$	−21.1246	−1.88944	−0.944721	+	0.327877i	$0.893667\pi$
$6$	8.20257	0.558114
$7$	−25.8618	−1.39641	−0.698203	−	0.715899i	$-0.746017\pi$
$7$	−25.8618	−1.39641	−0.698203	+	0.715899i	$0.746017\pi$
$8$	23.3068	1.03003
$9$	9.00000	0.333333
$10$	57.7586	1.82649
$11$	−6.96892	−0.191019	−0.0955095	−	0.995429i	$-0.530448\pi$
$11$	−6.96892	−0.191019	−0.0955095	+	0.995429i	$0.530448\pi$
$12$	1.57264	0.0378318
$13$	0	0
$13$	0	0
$14$	70.7111	1.34988
$15$	63.3738	1.09087
$16$	−59.5315	−0.930180
$17$	122.879	1.75310	0.876548	−	0.481315i	$-0.159841\pi$
$17$	122.879	1.75310	0.876548	+	0.481315i	$0.159841\pi$
$18$	−24.6077	−0.322227
$19$	43.1340	0.520822	0.260411	−	0.965498i	$-0.416142\pi$
$19$	43.1340	0.520822	0.260411	+	0.965498i	$0.416142\pi$
$20$	11.0738	0.123809
$21$	77.5854	0.806216
$22$	19.0543	0.184655
$23$	−75.5492	−0.684918	−0.342459	−	0.939533i	$-0.611260\pi$
$23$	−75.5492	−0.684918	−0.342459	+	0.939533i	$0.611260\pi$
$24$	−69.9204	−0.594685
$25$	321.248	2.56999
$26$	0	0
$27$	−27.0000	−0.192450
$28$	13.5571	0.0915018
$29$	−163.764	−1.04863	−0.524314	−	0.851525i	$-0.675678\pi$
$29$	−163.764	−1.04863	−0.524314	+	0.851525i	$0.675678\pi$
$30$	−173.276	−1.05452
$31$	139.421	0.807767	0.403884	−	0.914810i	$-0.367660\pi$
$31$	139.421	0.807767	0.403884	+	0.914810i	$0.367660\pi$
$32$	−23.6841	−0.130837
$33$	20.9068	0.110285
$34$	−335.975	−1.69469
$35$	546.320	2.63843
$36$	−4.71792	−0.0218422
$37$	2.80616	0.0124684	0.00623419	−	0.999981i	$-0.498016\pi$
$37$	2.80616	0.0124684	0.00623419	+	0.999981i	$0.498016\pi$
$38$	−117.936	−0.503469
$39$	0	0
$40$	−492.347	−1.94617
$41$	−300.555	−1.14485	−0.572425	−	0.819957i	$-0.693997\pi$
$41$	−300.555	−1.14485	−0.572425	+	0.819957i	$0.693997\pi$
$42$	−212.133	−0.779354
$43$	363.145	1.28789	0.643943	−	0.765073i	$-0.277297\pi$
$43$	363.145	1.28789	0.643943	+	0.765073i	$0.277297\pi$
$44$	3.65320	0.0125168
$45$	−190.121	−0.629814
$46$	206.566	0.662097
$47$	−41.2660	−0.128070	−0.0640348	−	0.997948i	$-0.520397\pi$
$47$	−41.2660	−0.128070	−0.0640348	+	0.997948i	$0.520397\pi$
$48$	178.594	0.537039
$49$	325.834	0.949952
$50$	−878.354	−2.48436
$51$	−368.638	−1.01215
$52$	0	0
$53$	−125.763	−0.325941	−0.162971	−	0.986631i	$-0.552108\pi$
$53$	−125.763	−0.325941	−0.162971	+	0.986631i	$0.552108\pi$
$54$	73.8231	0.186038
$55$	147.216	0.360919
$56$	−602.756	−1.43833
$57$	−129.402	−0.300697
$58$	447.762	1.01369
$59$	−407.311	−0.898768	−0.449384	−	0.893339i	$-0.648357\pi$
$59$	−407.311	−0.898768	−0.449384	+	0.893339i	$0.648357\pi$
$60$	−33.2214	−0.0714810
$61$	536.710	1.12654	0.563268	−	0.826274i	$-0.309544\pi$
$61$	536.710	1.12654	0.563268	+	0.826274i	$0.309544\pi$
$62$	−381.204	−0.780854
$63$	−232.756	−0.465469
$64$	541.009	1.05666
$65$	0	0
$66$	−57.1630	−0.106610
$67$	340.155	0.620247	0.310124	−	0.950696i	$-0.399630\pi$
$67$	340.155	0.620247	0.310124	+	0.950696i	$0.399630\pi$
$68$	−64.4150	−0.114874
$69$	226.648	0.395437
$70$	−1493.74	−2.55052
$71$	514.831	0.860552	0.430276	−	0.902697i	$-0.358416\pi$
$71$	514.831	0.860552	0.430276	+	0.902697i	$0.358416\pi$
$72$	209.761	0.343342
$73$	−491.231	−0.787592	−0.393796	−	0.919198i	$-0.628838\pi$
$73$	−491.231	−0.787592	−0.393796	+	0.919198i	$0.628838\pi$
$74$	−7.67257	−0.0120529
$75$	−963.745	−1.48378
$76$	−22.6114	−0.0341277
$77$	180.229	0.266740
$78$	0	0
$79$	762.869	1.08645	0.543225	−	0.839587i	$-0.317203\pi$
$79$	762.869	1.08645	0.543225	+	0.839587i	$0.317203\pi$
$80$	1257.58	1.75752
$81$	81.0000	0.111111
$82$	821.774	1.10670
$83$	345.948	0.457503	0.228752	−	0.973485i	$-0.426536\pi$
$83$	345.948	0.457503	0.228752	+	0.973485i	$0.426536\pi$
$84$	−40.6713	−0.0528286
$85$	−2595.78	−3.31237
$86$	−992.907	−1.24498
$87$	491.292	0.605426
$88$	−162.423	−0.196754
$89$	362.482	0.431720	0.215860	−	0.976424i	$-0.430745\pi$
$89$	362.482	0.431720	0.215860	+	0.976424i	$0.430745\pi$
$90$	519.828	0.608829
$91$	0	0
$92$	39.6039	0.0448803
$93$	−418.264	−0.466365
$94$	112.829	0.123803
$95$	−911.188	−0.984062
$96$	71.0523	0.0755390
$97$	−276.297	−0.289213	−0.144607	−	0.989489i	$-0.546192\pi$
$97$	−276.297	−0.289213	−0.144607	+	0.989489i	$0.546192\pi$
$98$	−890.890	−0.918301
$99$	−62.7203	−0.0636730
$100$	−168.403	−0.168403
$101$	313.657	0.309010	0.154505	−	0.987992i	$-0.450622\pi$
$101$	313.657	0.309010	0.154505	+	0.987992i	$0.450622\pi$
$102$	1007.93	0.978427
$103$	507.209	0.485212	0.242606	−	0.970125i	$-0.421998\pi$
$103$	507.209	0.485212	0.242606	+	0.970125i	$0.421998\pi$
$104$	0	0
$105$	−1638.96	−1.52330
$106$	343.860	0.315082
$107$	−1262.50	−1.14066	−0.570330	−	0.821416i	$-0.693185\pi$
$107$	−1262.50	−1.14066	−0.570330	+	0.821416i	$0.693185\pi$
$108$	14.1537	0.0126106
$109$	−469.356	−0.412442	−0.206221	−	0.978505i	$-0.566117\pi$
$109$	−469.356	−0.412442	−0.206221	+	0.978505i	$0.566117\pi$
$110$	−402.515	−0.348894
$111$	−8.41848	−0.00719862
$112$	1539.59	1.29891
$113$	−1110.69	−0.924649	−0.462325	−	0.886711i	$-0.652985\pi$
$113$	−1110.69	−0.924649	−0.462325	+	0.886711i	$0.652985\pi$
$114$	353.809	0.290678
$115$	1595.95	1.29411
$116$	85.8472	0.0687131
$117$	0	0
$118$	1113.66	0.868823
$119$	−3177.88	−2.44803
$120$	1477.04	1.12362
$121$	−1282.43	−0.963512
$122$	−1467.47	−1.08900
$123$	901.665	0.660979
$124$	−73.0864	−0.0529303
$125$	−4145.67	−2.96640
$126$	636.400	0.449960
$127$	−2547.15	−1.77971	−0.889853	−	0.456248i	$-0.849193\pi$
$127$	−2547.15	−1.77971	−0.889853	+	0.456248i	$0.849193\pi$
$128$	−1289.75	−0.890614
$129$	−1089.44	−0.743561
$130$	0	0
$131$	−701.027	−0.467550	−0.233775	−	0.972291i	$-0.575108\pi$
$131$	−701.027	−0.467550	−0.233775	+	0.972291i	$0.575108\pi$
$132$	−10.9596	−0.00722659
$133$	−1115.52	−0.727279
$134$	−930.048	−0.599581
$135$	570.364	0.363623
$136$	2863.93	1.80573
$137$	2576.33	1.60665	0.803325	−	0.595541i	$-0.203062\pi$
$137$	2576.33	1.60665	0.803325	+	0.595541i	$0.203062\pi$
$138$	−619.698	−0.382262
$139$	233.539	0.142508	0.0712538	−	0.997458i	$-0.477300\pi$
$139$	233.539	0.142508	0.0712538	+	0.997458i	$0.477300\pi$
$140$	−286.388	−0.172887
$141$	123.798	0.0739410
$142$	−1407.64	−0.831880
$143$	0	0
$144$	−535.783	−0.310060
$145$	3459.45	1.98132
$146$	1343.12	0.761350
$147$	−977.501	−0.548455
$148$	−1.47102	−0.000817010 0
$149$	1595.97	0.877495	0.438748	−	0.898610i	$-0.355422\pi$
$149$	1595.97	0.877495	0.438748	+	0.898610i	$0.355422\pi$
$150$	2635.06	1.43435
$151$	3494.13	1.88310	0.941552	−	0.336869i	$-0.109368\pi$
$151$	3494.13	1.88310	0.941552	+	0.336869i	$0.109368\pi$
$152$	1005.32	0.536459
$153$	1105.91	0.584365
$154$	−492.780	−0.257853
$155$	−2945.22	−1.52623
$156$	0	0
$157$	−2203.71	−1.12022	−0.560112	−	0.828417i	$-0.689242\pi$
$157$	−2203.71	−1.12022	−0.560112	+	0.828417i	$0.689242\pi$
$158$	−2085.83	−1.05025
$159$	377.289	0.188182
$160$	500.317	0.247210
$161$	1953.84	0.956424
$162$	−221.469	−0.107409
$163$	758.697	0.364575	0.182288	−	0.983245i	$-0.441650\pi$
$163$	758.697	0.364575	0.182288	+	0.983245i	$0.441650\pi$
$164$	157.555	0.0750181
$165$	−441.647	−0.208377
$166$	−945.888	−0.442260
$167$	−855.272	−0.396305	−0.198152	−	0.980171i	$-0.563494\pi$
$167$	−855.272	−0.396305	−0.198152	+	0.980171i	$0.563494\pi$
$168$	1808.27	0.830422
$169$	0	0
$170$	7097.34	3.20201
$171$	388.206	0.173607
$172$	−190.365	−0.0843908
$173$	−76.6923	−0.0337041	−0.0168521	−	0.999858i	$-0.505364\pi$
$173$	−76.6923	−0.0337041	−0.0168521	+	0.999858i	$0.505364\pi$
$174$	−1343.29	−0.585254
$175$	−8308.07	−3.58875
$176$	414.870	0.177682
$177$	1221.93	0.518904
$178$	−991.095	−0.417335
$179$	3533.09	1.47528	0.737642	−	0.675193i	$-0.235939\pi$
$179$	3533.09	1.47528	0.737642	+	0.675193i	$0.235939\pi$
$180$	99.6641	0.0412696
$181$	2352.77	0.966189	0.483095	−	0.875568i	$-0.339513\pi$
$181$	2352.77	0.966189	0.483095	+	0.875568i	$0.339513\pi$
$182$	0	0
$183$	−1610.13	−0.650406
$184$	−1760.81	−0.705482
$185$	−59.2790	−0.0235583
$186$	1143.61	0.450826
$187$	−856.337	−0.334875
$188$	21.6322	0.00839197
$189$	698.269	0.268739
$190$	2491.36	0.951275
$191$	−1532.91	−0.580719	−0.290360	−	0.956918i	$-0.593775\pi$
$191$	−1532.91	−0.580719	−0.290360	+	0.956918i	$0.593775\pi$
$192$	−1623.03	−0.610062
$193$	4611.82	1.72003	0.860015	−	0.510269i	$-0.170454\pi$
$193$	4611.82	1.72003	0.860015	+	0.510269i	$0.170454\pi$
$194$	755.447	0.279577
$195$	0	0
$196$	−170.806	−0.0622471
$197$	−1914.37	−0.692353	−0.346176	−	0.938169i	$-0.612520\pi$
$197$	−1914.37	−0.692353	−0.346176	+	0.938169i	$0.612520\pi$
$198$	171.489	0.0615515
$199$	2304.94	0.821070	0.410535	−	0.911845i	$-0.365342\pi$
$199$	2304.94	0.821070	0.410535	+	0.911845i	$0.365342\pi$
$200$	7487.27	2.64715
$201$	−1020.47	−0.358100
$202$	−857.596	−0.298714
$203$	4235.24	1.46431
$204$	193.245	0.0663228
$205$	6349.10	2.16312
$206$	−1386.81	−0.469045
$207$	−679.943	−0.228306
$208$	0	0
$209$	−300.597	−0.0994868
$210$	4481.23	1.47254
$211$	−3500.56	−1.14213	−0.571064	−	0.820906i	$-0.693469\pi$
$211$	−3500.56	−1.14213	−0.571064	+	0.820906i	$0.693469\pi$
$212$	65.9266	0.0213578
$213$	−1544.49	−0.496840
$214$	3451.92	1.10266
$215$	−7671.29	−2.43339
$216$	−629.284	−0.198228
$217$	−3605.69	−1.12797
$218$	1283.31	0.398700
$219$	1473.69	0.454716
$220$	−77.1723	−0.0236498
$221$	0	0
$222$	23.0177	0.00695877
$223$	1255.99	0.377163	0.188582	−	0.982058i	$-0.439611\pi$
$223$	1255.99	0.377163	0.188582	+	0.982058i	$0.439611\pi$
$224$	612.514	0.182702
$225$	2891.24	0.856662
$226$	3036.85	0.893842
$227$	−3408.52	−0.996615	−0.498307	−	0.867000i	$-0.666045\pi$
$227$	−3408.52	−0.996615	−0.498307	+	0.867000i	$0.666045\pi$
$228$	67.8342	0.0197036
$229$	−4135.13	−1.19326	−0.596631	−	0.802515i	$-0.703495\pi$
$229$	−4135.13	−1.19326	−0.596631	+	0.802515i	$0.703495\pi$
$230$	−4363.62	−1.25099
$231$	−540.687	−0.154003
$232$	−3816.82	−1.08011
$233$	3788.88	1.06531	0.532656	−	0.846332i	$-0.321194\pi$
$233$	3788.88	1.06531	0.532656	+	0.846332i	$0.321194\pi$
$234$	0	0
$235$	871.728	0.241980
$236$	213.518	0.0588933
$237$	−2288.61	−0.627262
$238$	8688.93	2.36647
$239$	1543.75	0.417810	0.208905	−	0.977936i	$-0.433010\pi$
$239$	1543.75	0.417810	0.208905	+	0.977936i	$0.433010\pi$
$240$	−3772.74	−1.01470
$241$	6295.40	1.68266	0.841332	−	0.540518i	$-0.181772\pi$
$241$	6295.40	1.68266	0.841332	+	0.540518i	$0.181772\pi$
$242$	3506.42	0.931409
$243$	−243.000	−0.0641500
$244$	−281.350	−0.0738181
$245$	−6883.10	−1.79488
$246$	−2465.32	−0.638956
$247$	0	0
$248$	3249.46	0.832021
$249$	−1037.84	−0.264140
$250$	11335.0	2.86756
$251$	2241.93	0.563781	0.281891	−	0.959447i	$-0.409038\pi$
$251$	2241.93	0.563781	0.281891	+	0.959447i	$0.409038\pi$
$252$	122.014	0.0305006
$253$	526.497	0.130832
$254$	6964.38	1.72041
$255$	7787.33	1.91240
$256$	−801.658	−0.195717
$257$	3603.80	0.874703	0.437351	−	0.899291i	$-0.355917\pi$
$257$	3603.80	0.874703	0.437351	+	0.899291i	$0.355917\pi$
$258$	2978.72	0.718787
$259$	−72.5724	−0.0174109
$260$	0	0
$261$	−1473.88	−0.349543
$262$	1916.74	0.451972
$263$	−2131.06	−0.499646	−0.249823	−	0.968292i	$-0.580372\pi$
$263$	−2131.06	−0.499646	−0.249823	+	0.968292i	$0.580372\pi$
$264$	487.270	0.113596
$265$	2656.69	0.615847
$266$	3050.05	0.703047
$267$	−1087.45	−0.249253
$268$	−178.314	−0.0406427
$269$	−2505.79	−0.567958	−0.283979	−	0.958830i	$-0.591655\pi$
$269$	−2505.79	−0.567958	−0.283979	+	0.958830i	$0.591655\pi$
$270$	−1559.48	−0.351508
$271$	−230.220	−0.0516046	−0.0258023	−	0.999667i	$-0.508214\pi$
$271$	−230.220	−0.0516046	−0.0258023	+	0.999667i	$0.508214\pi$
$272$	−7315.19	−1.63069
$273$	0	0
$274$	−7044.18	−1.55312
$275$	−2238.75	−0.490916
$276$	−118.812	−0.0259117
$277$	−7750.38	−1.68114	−0.840569	−	0.541705i	$-0.817779\pi$
$277$	−7750.38	−1.68114	−0.840569	+	0.541705i	$0.817779\pi$
$278$	−638.541	−0.137760
$279$	1254.79	0.269256
$280$	12733.0	2.71765
$281$	−5179.21	−1.09952	−0.549761	−	0.835322i	$-0.685281\pi$
$281$	−5179.21	−1.09952	−0.549761	+	0.835322i	$0.685281\pi$
$282$	−338.487	−0.0714774
$283$	3799.93	0.798171	0.399085	−	0.916914i	$-0.369328\pi$
$283$	3799.93	0.798171	0.399085	+	0.916914i	$0.369328\pi$
$284$	−269.881	−0.0563891
$285$	2733.56	0.568148
$286$	0	0
$287$	7772.90	1.59868
$288$	−213.157	−0.0436125
$289$	10186.3	2.07334
$290$	−9458.78	−1.91531
$291$	828.890	0.166977
$292$	257.509	0.0516082
$293$	−5840.85	−1.16459	−0.582297	−	0.812976i	$-0.697846\pi$
$293$	−5840.85	−1.16459	−0.582297	+	0.812976i	$0.697846\pi$
$294$	2672.67	0.530181
$295$	8604.27	1.69817
$296$	65.4026	0.0128427
$297$	188.161	0.0367616
$298$	−4363.68	−0.848258
$299$	0	0
$300$	505.208	0.0972273
$301$	−9391.59	−1.79841
$302$	−9553.62	−1.82036
$303$	−940.970	−0.178407
$304$	−2567.83	−0.484458
$305$	−11337.8	−2.12852
$306$	−3023.78	−0.564895
$307$	1686.29	0.313490	0.156745	−	0.987639i	$-0.449900\pi$
$307$	1686.29	0.313490	0.156745	+	0.987639i	$0.449900\pi$
$308$	−94.4783	−0.0174786
$309$	−1521.63	−0.280137
$310$	8052.78	1.47538
$311$	−10601.1	−1.93290	−0.966449	−	0.256860i	$-0.917312\pi$
$311$	−10601.1	−1.93290	−0.966449	+	0.256860i	$0.917312\pi$
$312$	0	0
$313$	−5889.99	−1.06365	−0.531824	−	0.846855i	$-0.678493\pi$
$313$	−5889.99	−1.06365	−0.531824	+	0.846855i	$0.678493\pi$
$314$	6025.35	1.08290
$315$	4916.88	0.879476
$316$	−399.906	−0.0711913
$317$	−2432.98	−0.431071	−0.215536	−	0.976496i	$-0.569150\pi$
$317$	−2432.98	−0.431071	−0.215536	+	0.976496i	$0.569150\pi$
$318$	−1031.58	−0.181912
$319$	1141.26	0.200308
$320$	−11428.6	−1.99649
$321$	3787.50	0.658560
$322$	−5342.17	−0.924557
$323$	5300.28	0.913050
$324$	−42.4612	−0.00728073
$325$	0	0
$326$	−2074.42	−0.352428
$327$	1408.07	0.238123
$328$	−7004.98	−1.17922
$329$	1067.21	0.178837
$330$	1207.55	0.201434
$331$	−757.534	−0.125794	−0.0628971	−	0.998020i	$-0.520034\pi$
$331$	−757.534	−0.125794	−0.0628971	+	0.998020i	$0.520034\pi$
$332$	−181.351	−0.0299786
$333$	25.2554	0.00415612
$334$	2338.47	0.383101
$335$	−7185.64	−1.17192
$336$	−4618.78	−0.749926
$337$	−889.307	−0.143750	−0.0718748	−	0.997414i	$-0.522898\pi$
$337$	−889.307	−0.143750	−0.0718748	+	0.997414i	$0.522898\pi$
$338$	0	0
$339$	3332.08	0.533847
$340$	1360.74	0.217048
$341$	−971.615	−0.154299
$342$	−1061.43	−0.167823
$343$	443.956	0.0698874
$344$	8463.75	1.32655
$345$	−4787.84	−0.747156
$346$	209.691	0.0325811
$347$	−9644.48	−1.49205	−0.746027	−	0.665916i	$-0.768041\pi$
$347$	−9644.48	−1.49205	−0.746027	+	0.665916i	$0.768041\pi$
$348$	−257.542	−0.0396715
$349$	1807.89	0.277289	0.138645	−	0.990342i	$-0.455725\pi$
$349$	1807.89	0.277289	0.138645	+	0.990342i	$0.455725\pi$
$350$	22715.8	3.46918
$351$	0	0
$352$	165.053	0.0249924
$353$	−10681.8	−1.61058	−0.805289	−	0.592883i	$-0.797990\pi$
$353$	−10681.8	−1.61058	−0.805289	+	0.592883i	$0.797990\pi$
$354$	−3340.99	−0.501615
$355$	−10875.6	−1.62596
$356$	−190.018	−0.0282891
$357$	9533.65	1.41337
$358$	−9660.14	−1.42613
$359$	8195.95	1.20492	0.602459	−	0.798149i	$-0.294188\pi$
$359$	8195.95	1.20492	0.602459	+	0.798149i	$0.294188\pi$
$360$	−4431.12	−0.648724
$361$	−4998.46	−0.728745
$362$	−6432.92	−0.933997
$363$	3847.30	0.556284
$364$	0	0
$365$	10377.0	1.48811
$366$	4402.40	0.628736
$367$	−2555.29	−0.363447	−0.181724	−	0.983350i	$-0.558168\pi$
$367$	−2555.29	−0.363447	−0.181724	+	0.983350i	$0.558168\pi$
$368$	4497.56	0.637096
$369$	−2705.00	−0.381616
$370$	162.080	0.0227733
$371$	3252.46	0.455147
$372$	219.259	0.0305593
$373$	−600.757	−0.0833941	−0.0416971	−	0.999130i	$-0.513276\pi$
$373$	−600.757	−0.0833941	−0.0416971	+	0.999130i	$0.513276\pi$
$374$	2341.39	0.323717
$375$	12437.0	1.71265
$376$	−961.779	−0.131915
$377$	0	0
$378$	−1909.20	−0.259785
$379$	863.612	0.117047	0.0585234	−	0.998286i	$-0.481361\pi$
$379$	863.612	0.117047	0.0585234	+	0.998286i	$0.481361\pi$
$380$	477.656	0.0644822
$381$	7641.44	1.02751
$382$	4191.26	0.561371
$383$	9089.38	1.21265	0.606326	−	0.795216i	$-0.292643\pi$
$383$	9089.38	1.21265	0.606326	+	0.795216i	$0.292643\pi$
$384$	3869.24	0.514196
$385$	−3807.26	−0.503990
$386$	−12609.6	−1.66272
$387$	3268.31	0.429295
$388$	144.838	0.0189511
$389$	−1513.16	−0.197224	−0.0986120	−	0.995126i	$-0.531440\pi$
$389$	−1513.16	−0.197224	−0.0986120	+	0.995126i	$0.531440\pi$
$390$	0	0
$391$	−9283.44	−1.20073
$392$	7594.14	0.978474
$393$	2103.08	0.269940
$394$	5234.26	0.669285
$395$	−16115.3	−2.05278
$396$	32.8788	0.00417228
$397$	−7572.77	−0.957346	−0.478673	−	0.877993i	$-0.658882\pi$
$397$	−7572.77	−0.957346	−0.478673	+	0.877993i	$0.658882\pi$
$398$	−6302.15	−0.793714
$399$	3346.57	0.419895
$400$	−19124.4	−2.39055
$401$	11075.5	1.37926	0.689630	−	0.724162i	$-0.257773\pi$
$401$	11075.5	1.37926	0.689630	+	0.724162i	$0.257773\pi$
$402$	2790.14	0.346168
$403$	0	0
$404$	−164.423	−0.0202484
$405$	−1711.09	−0.209938
$406$	−11579.9	−1.41552
$407$	−19.5559	−0.00238170
$408$	−8591.78	−1.04254
$409$	11082.8	1.33988	0.669940	−	0.742415i	$-0.266320\pi$
$409$	11082.8	1.33988	0.669940	+	0.742415i	$0.266320\pi$
$410$	−17359.6	−2.09105
$411$	−7729.00	−0.927600
$412$	−265.886	−0.0317943
$413$	10533.8	1.25505
$414$	1859.09	0.220699
$415$	−7308.02	−0.864425
$416$	0	0
$417$	−700.618	−0.0822768
$418$	821.890	0.0961721
$419$	13035.6	1.51988	0.759940	−	0.649993i	$-0.225228\pi$
$419$	13035.6	1.51988	0.759940	+	0.649993i	$0.225228\pi$
$420$	859.164	0.0998165
$421$	−14664.0	−1.69757	−0.848787	−	0.528735i	$-0.822666\pi$
$421$	−14664.0	−1.69757	−0.848787	+	0.528735i	$0.822666\pi$
$422$	9571.20	1.10407
$423$	−371.394	−0.0426899
$424$	−2931.14	−0.335728
$425$	39474.8	4.50543
$426$	4222.93	0.480286
$427$	−13880.3	−1.57310
$428$	661.820	0.0747436
$429$	0	0
$430$	20974.8	2.35231
$431$	1454.87	0.162596	0.0812979	−	0.996690i	$-0.474093\pi$
$431$	1454.87	0.162596	0.0812979	+	0.996690i	$0.474093\pi$
$432$	1607.35	0.179013
$433$	−8982.19	−0.996897	−0.498449	−	0.866919i	$-0.666097\pi$
$433$	−8982.19	−0.996897	−0.498449	+	0.866919i	$0.666097\pi$
$434$	9858.62	1.09039
$435$	−10378.3	−1.14392
$436$	246.042	0.0270259
$437$	−3258.74	−0.356720
$438$	−4029.35	−0.439566
$439$	3348.05	0.363995	0.181997	−	0.983299i	$-0.441744\pi$
$439$	3348.05	0.363995	0.181997	+	0.983299i	$0.441744\pi$
$440$	3431.13	0.371756
$441$	2932.50	0.316651
$442$	0	0
$443$	−15671.4	−1.68075	−0.840375	−	0.542006i	$-0.817665\pi$
$443$	−15671.4	−1.68075	−0.840375	+	0.542006i	$0.817665\pi$
$444$	4.41307	0.000471701 0
$445$	−7657.29	−0.815709
$446$	−3434.12	−0.364597
$447$	−4787.90	−0.506622
$448$	−13991.5	−1.47552
$449$	−12265.8	−1.28922	−0.644611	−	0.764511i	$-0.722981\pi$
$449$	−12265.8	−1.28922	−0.644611	+	0.764511i	$0.722981\pi$
$450$	−7905.18	−0.828120
$451$	2094.54	0.218688
$452$	582.240	0.0605891
$453$	−10482.4	−1.08721
$454$	9319.54	0.963409
$455$	0	0
$456$	−3015.95	−0.309725
$457$	8382.54	0.858027	0.429014	−	0.903298i	$-0.358861\pi$
$457$	8382.54	0.858027	0.429014	+	0.903298i	$0.358861\pi$
$458$	11306.2	1.15351
$459$	−3317.74	−0.337383
$460$	−836.616	−0.0847987
$461$	−2308.36	−0.233212	−0.116606	−	0.993178i	$-0.537202\pi$
$461$	−2308.36	−0.233212	−0.116606	+	0.993178i	$0.537202\pi$
$462$	1478.34	0.148871
$463$	2330.53	0.233928	0.116964	−	0.993136i	$-0.462684\pi$
$463$	2330.53	0.233928	0.116964	+	0.993136i	$0.462684\pi$
$464$	9749.12	0.975413
$465$	8835.65	0.881169
$466$	−10359.5	−1.02982
$467$	3437.89	0.340657	0.170328	−	0.985387i	$-0.445517\pi$
$467$	3437.89	0.340657	0.170328	+	0.985387i	$0.445517\pi$
$468$	0	0
$469$	−8797.03	−0.866117
$470$	−2383.47	−0.233918
$471$	6611.12	0.646761
$472$	−9493.11	−0.925754
$473$	−2530.73	−0.246011
$474$	6257.48	0.606362
$475$	13856.7	1.33851
$476$	1665.89	0.160411
$477$	−1131.87	−0.108647
$478$	−4220.89	−0.403889
$479$	−15820.0	−1.50905	−0.754525	−	0.656272i	$-0.772133\pi$
$479$	−15820.0	−1.50905	−0.754525	+	0.656272i	$0.772133\pi$
$480$	−1500.95	−0.142727
$481$	0	0
$482$	−17212.8	−1.62660
$483$	−5861.52	−0.552191
$484$	672.268	0.0631357
$485$	5836.65	0.546451
$486$	664.408	0.0620127
$487$	−14504.1	−1.34957	−0.674787	−	0.738012i	$-0.735765\pi$
$487$	−14504.1	−1.34957	−0.674787	+	0.738012i	$0.735765\pi$
$488$	12509.0	1.16036
$489$	−2276.09	−0.210488
$490$	18819.7	1.73508
$491$	−17513.3	−1.60970	−0.804850	−	0.593479i	$-0.797754\pi$
$491$	−17513.3	−1.60970	−0.804850	+	0.593479i	$0.797754\pi$
$492$	−472.665	−0.0433117
$493$	−20123.2	−1.83835
$494$	0	0
$495$	1324.94	0.120306
$496$	−8299.95	−0.751369
$497$	−13314.5	−1.20168
$498$	2837.66	0.255339
$499$	3856.26	0.345952	0.172976	−	0.984926i	$-0.444662\pi$
$499$	3856.26	0.345952	0.172976	+	0.984926i	$0.444662\pi$
$500$	2173.21	0.194378
$501$	2565.82	0.228807
$502$	−6129.85	−0.544997
$503$	−1822.27	−0.161533	−0.0807666	−	0.996733i	$-0.525737\pi$
$503$	−1822.27	−0.161533	−0.0807666	+	0.996733i	$0.525737\pi$
$504$	−5424.81	−0.479445
$505$	−6625.87	−0.583856
$506$	−1439.54	−0.126473
$507$	0	0
$508$	1335.25	0.116618
$509$	−3738.01	−0.325510	−0.162755	−	0.986667i	$-0.552038\pi$
$509$	−3738.01	−0.325510	−0.162755	+	0.986667i	$0.552038\pi$
$510$	−21292.0	−1.84868
$511$	12704.1	1.09980
$512$	12509.9	1.07981
$513$	−1164.62	−0.100232
$514$	−9853.46	−0.845559
$515$	−10714.6	−0.916779
$516$	571.096	0.0487231
$517$	287.580	0.0244637
$518$	198.427	0.0168308
$519$	230.077	0.0194591
$520$	0	0
$521$	17735.5	1.49138	0.745690	−	0.666294i	$-0.232120\pi$
$521$	17735.5	1.49138	0.745690	+	0.666294i	$0.232120\pi$
$522$	4029.86	0.337897
$523$	−6099.99	−0.510007	−0.255004	−	0.966940i	$-0.582077\pi$
$523$	−6099.99	−0.510007	−0.255004	+	0.966940i	$0.582077\pi$
$524$	367.487	0.0306370
$525$	24924.2	2.07196
$526$	5826.72	0.482998
$527$	17132.0	1.41609
$528$	−1244.61	−0.102585
$529$	−6459.31	−0.530888
$530$	−7263.90	−0.595328
$531$	−3665.80	−0.299589
$532$	584.772	0.0476561
$533$	0	0
$534$	2973.28	0.240949
$535$	26669.8	2.15521
$536$	7927.93	0.638870
$537$	−10599.3	−0.851755
$538$	6851.30	0.549035
$539$	−2270.71	−0.181459
$540$	−298.992	−0.0238270
$541$	4453.47	0.353918	0.176959	−	0.984218i	$-0.443374\pi$
$541$	4453.47	0.353918	0.176959	+	0.984218i	$0.443374\pi$
$542$	629.464	0.0498852
$543$	−7058.32	−0.557830
$544$	−2910.29	−0.229371
$545$	9914.95	0.779284
$546$	0	0
$547$	−17915.1	−1.40035	−0.700176	−	0.713970i	$-0.746895\pi$
$547$	−17915.1	−1.40035	−0.700176	+	0.713970i	$0.746895\pi$
$548$	−1350.55	−0.105278
$549$	4830.39	0.375512
$550$	6121.18	0.474560
$551$	−7063.79	−0.546148
$552$	5282.43	0.407310
$553$	−19729.2	−1.51712
$554$	21191.0	1.62513
$555$	177.837	0.0136014
$556$	−122.424	−0.00933804
$557$	−3127.12	−0.237882	−0.118941	−	0.992901i	$-0.537950\pi$
$557$	−3127.12	−0.237882	−0.118941	+	0.992901i	$0.537950\pi$
$558$	−3430.83	−0.260285
$559$	0	0
$560$	−32523.3	−2.45421
$561$	2569.01	0.193340
$562$	14160.9	1.06289
$563$	12978.0	0.971507	0.485753	−	0.874096i	$-0.338545\pi$
$563$	12978.0	0.971507	0.485753	+	0.874096i	$0.338545\pi$
$564$	−64.8966	−0.00484510
$565$	23463.0	1.74707
$566$	−10389.7	−0.771577
$567$	−2094.81	−0.155156
$568$	11999.1	0.886390
$569$	1275.65	0.0939857	0.0469928	−	0.998895i	$-0.485036\pi$
$569$	1275.65	0.0939857	0.0469928	+	0.998895i	$0.485036\pi$
$570$	−7474.08	−0.549219
$571$	26406.4	1.93533	0.967663	−	0.252245i	$-0.0811689\pi$
$571$	26406.4	1.93533	0.967663	+	0.252245i	$0.0811689\pi$
$572$	0	0
$573$	4598.73	0.335278
$574$	−21252.6	−1.54541
$575$	−24270.1	−1.76023
$576$	4869.08	0.352219
$577$	−14705.2	−1.06098	−0.530489	−	0.847692i	$-0.677992\pi$
$577$	−14705.2	−1.06098	−0.530489	+	0.847692i	$0.677992\pi$
$578$	−27851.4	−2.00426
$579$	−13835.4	−0.993060
$580$	−1813.49	−0.129829
$581$	−8946.85	−0.638860
$582$	−2266.34	−0.161414
$583$	876.433	0.0622610
$584$	−11449.0	−0.811239
$585$	0	0
$586$	15970.0	1.12579
$587$	18627.3	1.30976	0.654881	−	0.755732i	$-0.272719\pi$
$587$	18627.3	1.30976	0.654881	+	0.755732i	$0.272719\pi$
$588$	512.418	0.0359384
$589$	6013.79	0.420703
$590$	−23525.7	−1.64159
$591$	5743.12	0.399730
$592$	−167.055	−0.0115978
$593$	4495.16	0.311288	0.155644	−	0.987813i	$-0.450255\pi$
$593$	4495.16	0.311288	0.155644	+	0.987813i	$0.450255\pi$
$594$	−514.467	−0.0355368
$595$	67131.5	4.62542
$596$	−836.627	−0.0574993
$597$	−6914.83	−0.474045
$598$	0	0
$599$	3262.10	0.222514	0.111257	−	0.993792i	$-0.464512\pi$
$599$	3262.10	0.222514	0.111257	+	0.993792i	$0.464512\pi$
$600$	−22461.8	−1.52833
$601$	−5618.35	−0.381327	−0.190663	−	0.981655i	$-0.561064\pi$
$601$	−5618.35	−0.381327	−0.190663	+	0.981655i	$0.561064\pi$
$602$	25678.4	1.73849
$603$	3061.40	0.206749
$604$	−1831.67	−0.123393
$605$	27090.9	1.82050
$606$	2572.79	0.172463
$607$	−19129.2	−1.27913	−0.639563	−	0.768738i	$-0.720885\pi$
$607$	−19129.2	−1.27913	−0.639563	+	0.768738i	$0.720885\pi$
$608$	−1021.59	−0.0681430
$609$	−12705.7	−0.845421
$610$	30999.6	2.05760
$611$	0	0
$612$	−579.735	−0.0382915
$613$	−16054.6	−1.05781	−0.528907	−	0.848679i	$-0.677398\pi$
$613$	−16054.6	−1.05781	−0.528907	+	0.848679i	$0.677398\pi$
$614$	−4610.62	−0.303045
$615$	−19047.3	−1.24888
$616$	4200.56	0.274749
$617$	−18669.4	−1.21816	−0.609079	−	0.793110i	$-0.708461\pi$
$617$	−18669.4	−1.21816	−0.609079	+	0.793110i	$0.708461\pi$
$618$	4160.42	0.270803
$619$	2488.87	0.161609	0.0808045	−	0.996730i	$-0.474251\pi$
$619$	2488.87	0.161609	0.0808045	+	0.996730i	$0.474251\pi$
$620$	1543.92	0.100009
$621$	2039.83	0.131812
$622$	28985.3	1.86850
$623$	−9374.45	−0.602856
$624$	0	0
$625$	47419.5	3.03485
$626$	16104.4	1.02821
$627$	901.792	0.0574388
$628$	1155.21	0.0734044
$629$	344.819	0.0218582
$630$	−13443.7	−0.850173
$631$	−3595.22	−0.226820	−0.113410	−	0.993548i	$-0.536177\pi$
$631$	−3595.22	−0.226820	−0.113410	+	0.993548i	$0.536177\pi$
$632$	17780.0	1.11907
$633$	10501.7	0.659407
$634$	6652.22	0.416709
$635$	53807.4	3.36265
$636$	−197.780	−0.0123309
$637$	0	0
$638$	−3120.42	−0.193634
$639$	4633.48	0.286851
$640$	27245.4	1.68276
$641$	−4048.14	−0.249441	−0.124721	−	0.992192i	$-0.539803\pi$
$641$	−4048.14	−0.249441	−0.124721	+	0.992192i	$0.539803\pi$
$642$	−10355.8	−0.636618
$643$	20768.9	1.27379	0.636894	−	0.770951i	$-0.280219\pi$
$643$	20768.9	1.27379	0.636894	+	0.770951i	$0.280219\pi$
$644$	−1024.23	−0.0626712
$645$	23013.9	1.40492
$646$	−14492.0	−0.882629
$647$	14788.6	0.898610	0.449305	−	0.893378i	$-0.351672\pi$
$647$	14788.6	0.898610	0.449305	+	0.893378i	$0.351672\pi$
$648$	1887.85	0.114447
$649$	2838.52	0.171682
$650$	0	0
$651$	10817.1	0.651235
$652$	−397.719	−0.0238894
$653$	13026.1	0.780628	0.390314	−	0.920682i	$-0.372367\pi$
$653$	13026.1	0.780628	0.390314	+	0.920682i	$0.372367\pi$
$654$	−3849.92	−0.230189
$655$	14808.9	0.883408
$656$	17892.5	1.06492
$657$	−4421.07	−0.262531
$658$	−2917.97	−0.172879
$659$	1633.07	0.0965330	0.0482665	−	0.998834i	$-0.484630\pi$
$659$	1633.07	0.0965330	0.0482665	+	0.998834i	$0.484630\pi$
$660$	231.517	0.0136542
$661$	−14205.6	−0.835908	−0.417954	−	0.908468i	$-0.637253\pi$
$661$	−14205.6	−0.835908	−0.417954	+	0.908468i	$0.637253\pi$
$662$	2071.24	0.121603
$663$	0	0
$664$	8062.95	0.471240
$665$	23565.0	1.37415
$666$	−69.0531	−0.00401765
$667$	12372.2	0.718224
$668$	448.345	0.0259685
$669$	−3767.97	−0.217755
$670$	19646.9	1.13287
$671$	−3740.29	−0.215190
$672$	−1837.54	−0.105483
$673$	−27400.1	−1.56939	−0.784693	−	0.619885i	$-0.787179\pi$
$673$	−27400.1	−1.56939	−0.784693	+	0.619885i	$0.787179\pi$
$674$	2431.53	0.138960
$675$	−8673.71	−0.494594
$676$	0	0
$677$	−2463.68	−0.139862	−0.0699311	−	0.997552i	$-0.522278\pi$
$677$	−2463.68	−0.139862	−0.0699311	+	0.997552i	$0.522278\pi$
$678$	−9110.55	−0.516060
$679$	7145.53	0.403859
$680$	−60499.3	−3.41182
$681$	10225.6	0.575396
$682$	2656.58	0.149158
$683$	21663.1	1.21364	0.606820	−	0.794839i	$-0.292445\pi$
$683$	21663.1	1.21364	0.606820	+	0.794839i	$0.292445\pi$
$684$	−203.502	−0.0113759
$685$	−54424.0	−3.03567
$686$	−1213.86	−0.0675589
$687$	12405.4	0.688931
$688$	−21618.6	−1.19797
$689$	0	0
$690$	13090.9	0.722262
$691$	3787.41	0.208509	0.104254	−	0.994551i	$-0.466754\pi$
$691$	3787.41	0.208509	0.104254	+	0.994551i	$0.466754\pi$
$692$	40.2031	0.00220852
$693$	1622.06	0.0889134
$694$	26369.8	1.44234
$695$	−4933.43	−0.269260
$696$	11450.4	0.623604
$697$	−36932.0	−2.00703
$698$	−4943.10	−0.268051
$699$	−11366.6	−0.615058
$700$	4355.20	0.235158
$701$	−12593.4	−0.678527	−0.339263	−	0.940691i	$-0.610178\pi$
$701$	−12593.4	−0.678527	−0.339263	+	0.940691i	$0.610178\pi$
$702$	0	0
$703$	121.041	0.00649380
$704$	−3770.25	−0.201842
$705$	−2615.18	−0.139707
$706$	29206.0	1.55692
$707$	−8111.73	−0.431504
$708$	−640.553	−0.0340020
$709$	−22849.8	−1.21036	−0.605178	−	0.796090i	$-0.706898\pi$
$709$	−22849.8	−1.21036	−0.605178	+	0.796090i	$0.706898\pi$
$710$	29735.9	1.57179
$711$	6865.82	0.362150
$712$	8448.30	0.444682
$713$	−10533.2	−0.553254
$714$	−26066.8	−1.36628
$715$	0	0
$716$	−1852.09	−0.0966703
$717$	−4631.24	−0.241223
$718$	−22409.3	−1.16477
$719$	7269.04	0.377037	0.188518	−	0.982070i	$-0.439632\pi$
$719$	7269.04	0.377037	0.188518	+	0.982070i	$0.439632\pi$
$720$	11318.2	0.585840
$721$	−13117.4	−0.677553
$722$	13666.7	0.704464
$723$	−18886.2	−0.971487
$724$	−1233.35	−0.0633111
$725$	−52608.9	−2.69496
$726$	−10519.3	−0.537749
$727$	21380.1	1.09071	0.545355	−	0.838205i	$-0.316395\pi$
$727$	21380.1	1.09071	0.545355	+	0.838205i	$0.316395\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−28372.8	−1.43853
$731$	44623.0	2.25779
$732$	844.051	0.0426189
$733$	−5430.11	−0.273623	−0.136811	−	0.990597i	$-0.543685\pi$
$733$	−5430.11	−0.273623	−0.136811	+	0.990597i	$0.543685\pi$
$734$	6986.65	0.351338
$735$	20649.3	1.03627
$736$	1789.32	0.0896128
$737$	−2370.51	−0.118479
$738$	7395.97	0.368902
$739$	30838.6	1.53507	0.767534	−	0.641008i	$-0.221483\pi$
$739$	30838.6	1.53507	0.767534	+	0.641008i	$0.221483\pi$
$740$	31.0748	0.00154369
$741$	0	0
$742$	−8892.85	−0.439982
$743$	−31665.8	−1.56353	−0.781767	−	0.623571i	$-0.785681\pi$
$743$	−31665.8	−1.56353	−0.781767	+	0.623571i	$0.785681\pi$
$744$	−9748.39	−0.480367
$745$	−33714.2	−1.65798
$746$	1642.58	0.0806156
$747$	3113.53	0.152501
$748$	448.903	0.0219432
$749$	32650.6	1.59283
$750$	−34005.1	−1.65559
$751$	−23714.1	−1.15225	−0.576124	−	0.817362i	$-0.695436\pi$
$751$	−23714.1	−1.15225	−0.576124	+	0.817362i	$0.695436\pi$
$752$	2456.63	0.119128
$753$	−6725.78	−0.325499
$754$	0	0
$755$	−73812.2	−3.55801
$756$	−366.042	−0.0176095
$757$	−19213.1	−0.922474	−0.461237	−	0.887277i	$-0.652594\pi$
$757$	−19213.1	−0.922474	−0.461237	+	0.887277i	$0.652594\pi$
$758$	−2361.28	−0.113147
$759$	−1579.49	−0.0755360
$760$	−21236.9	−1.01361
$761$	11406.5	0.543345	0.271673	−	0.962390i	$-0.412423\pi$
$761$	11406.5	0.543345	0.271673	+	0.962390i	$0.412423\pi$
$762$	−20893.1	−0.993279
$763$	12138.4	0.575936
$764$	803.570	0.0380526
$765$	−23362.0	−1.10412
$766$	−24852.1	−1.17225
$767$	0	0
$768$	2404.97	0.112997
$769$	−6384.27	−0.299379	−0.149690	−	0.988733i	$-0.547827\pi$
$769$	−6384.27	−0.299379	−0.149690	+	0.988733i	$0.547827\pi$
$770$	10409.8	0.487198
$771$	−10811.4	−0.505010
$772$	−2417.57	−0.112708
$773$	−1561.96	−0.0726776	−0.0363388	−	0.999340i	$-0.511570\pi$
$773$	−1561.96	−0.0726776	−0.0363388	+	0.999340i	$0.511570\pi$
$774$	−8936.16	−0.414992
$775$	44788.8	2.07595
$776$	−6439.59	−0.297897
$777$	217.717	0.0100522
$778$	4137.26	0.190653
$779$	−12964.1	−0.596262
$780$	0	0
$781$	−3587.82	−0.164382
$782$	25382.7	1.16072
$783$	4421.63	0.201809
$784$	−19397.4	−0.883626
$785$	46552.4	2.11660
$786$	−5750.22	−0.260946
$787$	11134.6	0.504326	0.252163	−	0.967685i	$-0.418858\pi$
$787$	11134.6	0.504326	0.252163	+	0.967685i	$0.418858\pi$
$788$	1003.54	0.0453675
$789$	6393.18	0.288471
$790$	44062.3	1.98439
$791$	28724.6	1.29119
$792$	−1461.81	−0.0655848
$793$	0	0
$794$	20705.4	0.925448
$795$	−7970.08	−0.355559
$796$	−1208.28	−0.0538020
$797$	13798.3	0.613250	0.306625	−	0.951830i	$-0.400800\pi$
$797$	13798.3	0.613250	0.306625	+	0.951830i	$0.400800\pi$
$798$	−9150.15	−0.405905
$799$	−5070.74	−0.224518
$800$	−7608.48	−0.336251
$801$	3262.34	0.143907
$802$	−30282.5	−1.33331
$803$	3423.35	0.150445
$804$	534.941	0.0234651
$805$	−41274.1	−1.80711
$806$	0	0
$807$	7517.37	0.327911
$808$	7310.33	0.318288
$809$	8271.72	0.359479	0.179739	−	0.983714i	$-0.442475\pi$
$809$	8271.72	0.359479	0.179739	+	0.983714i	$0.442475\pi$
$810$	4678.45	0.202943
$811$	31496.6	1.36374	0.681872	−	0.731472i	$-0.261166\pi$
$811$	31496.6	1.36374	0.681872	+	0.731472i	$0.261166\pi$
$812$	−2220.16	−0.0959514
$813$	690.659	0.0297939
$814$	53.4695	0.00230234
$815$	−16027.2	−0.688843
$816$	21945.6	0.941482
$817$	15663.9	0.670759
$818$	−30302.6	−1.29524
$819$	0	0
$820$	−3328.28	−0.141742
$821$	−12125.3	−0.515441	−0.257721	−	0.966219i	$-0.582971\pi$
$821$	−12125.3	−0.515441	−0.257721	+	0.966219i	$0.582971\pi$
$822$	21132.5	0.896693
$823$	6237.45	0.264184	0.132092	−	0.991237i	$-0.457831\pi$
$823$	6237.45	0.264184	0.132092	+	0.991237i	$0.457831\pi$
$824$	11821.4	0.499780
$825$	6716.26	0.283431
$826$	−28801.4	−1.21323
$827$	−31790.0	−1.33669	−0.668347	−	0.743850i	$-0.732998\pi$
$827$	−31790.0	−1.33669	−0.668347	+	0.743850i	$0.732998\pi$
$828$	356.435	0.0149601
$829$	−23138.6	−0.969404	−0.484702	−	0.874679i	$-0.661072\pi$
$829$	−23138.6	−0.969404	−0.484702	+	0.874679i	$0.661072\pi$
$830$	19981.5	0.835624
$831$	23251.1	0.970605
$832$	0	0
$833$	40038.2	1.66536
$834$	1915.62	0.0795355
$835$	18067.3	0.748795
$836$	157.577	0.00651904
$837$	−3764.37	−0.155455
$838$	−35641.7	−1.46924
$839$	−3357.69	−0.138165	−0.0690825	−	0.997611i	$-0.522007\pi$
$839$	−3357.69	−0.138165	−0.0690825	+	0.997611i	$0.522007\pi$
$840$	−38198.9	−1.56903
$841$	2429.66	0.0996211
$842$	40094.1	1.64101
$843$	15537.6	0.634809
$844$	1835.04	0.0748397
$845$	0	0
$846$	1015.46	0.0412675
$847$	33166.1	1.34545
$848$	7486.87	0.303184
$849$	−11399.8	−0.460824
$850$	−107932.	−4.35532
$851$	−212.003	−0.00853981
$852$	809.643	0.0325562
$853$	16262.6	0.652778	0.326389	−	0.945236i	$-0.394168\pi$
$853$	16262.6	0.652778	0.326389	+	0.945236i	$0.394168\pi$
$854$	37951.4	1.52069
$855$	−8200.69	−0.328021
$856$	−29424.9	−1.17491
$857$	33293.4	1.32705	0.663524	−	0.748155i	$-0.269060\pi$
$857$	33293.4	1.32705	0.663524	+	0.748155i	$0.269060\pi$
$858$	0	0
$859$	26170.5	1.03950	0.519748	−	0.854320i	$-0.326026\pi$
$859$	26170.5	1.03950	0.519748	+	0.854320i	$0.326026\pi$
$860$	4021.39	0.159451
$861$	−23318.7	−0.922996
$862$	−3977.90	−0.157178
$863$	−37186.2	−1.46678	−0.733391	−	0.679807i	$-0.762064\pi$
$863$	−37186.2	−1.46678	−0.733391	+	0.679807i	$0.762064\pi$
$864$	639.471	0.0251797
$865$	1620.09	0.0636819
$866$	24559.0	0.963682
$867$	−30559.0	−1.19705
$868$	1890.15	0.0739122
$869$	−5316.37	−0.207532
$870$	28376.4	1.10580
$871$	0	0
$872$	−10939.2	−0.424825
$873$	−2486.67	−0.0964043
$874$	8910.01	0.344835
$875$	107215.	4.14230
$876$	−772.528	−0.0297960
$877$	19008.1	0.731880	0.365940	−	0.930638i	$-0.380748\pi$
$877$	19008.1	0.731880	0.365940	+	0.930638i	$0.380748\pi$
$878$	−9154.20	−0.351867
$879$	17522.5	0.672379
$880$	−8763.97	−0.335720
$881$	−11259.4	−0.430578	−0.215289	−	0.976550i	$-0.569069\pi$
$881$	−11259.4	−0.430578	−0.215289	+	0.976550i	$0.569069\pi$
$882$	−8018.01	−0.306100
$883$	34604.5	1.31884	0.659419	−	0.751776i	$-0.270802\pi$
$883$	34604.5	1.31884	0.659419	+	0.751776i	$0.270802\pi$
$884$	0	0
$885$	−25812.8	−0.980439
$886$	42848.6	1.62475
$887$	36352.6	1.37610	0.688050	−	0.725663i	$-0.258467\pi$
$887$	36352.6	1.37610	0.688050	+	0.725663i	$0.258467\pi$
$888$	−196.208	−0.00741476
$889$	65873.8	2.48519
$890$	20936.5	0.788531
$891$	−564.483	−0.0212243
$892$	−658.407	−0.0247142
$893$	−1779.97	−0.0667014
$894$	13091.0	0.489742
$895$	−74635.1	−2.78746
$896$	33355.2	1.24366
$897$	0	0
$898$	33537.1	1.24627
$899$	−22832.2	−0.847048
$900$	−1515.62	−0.0561342
$901$	−15453.7	−0.571406
$902$	−5726.88	−0.211402
$903$	28174.8	1.03831
$904$	−25886.7	−0.952412
$905$	−49701.4	−1.82556
$906$	28660.9	1.05099
$907$	−36358.6	−1.33106	−0.665528	−	0.746373i	$-0.731794\pi$
$907$	−36358.6	−1.33106	−0.665528	+	0.746373i	$0.731794\pi$
$908$	1786.79	0.0653048
$909$	2822.91	0.103003
$910$	0	0
$911$	16789.4	0.610603	0.305301	−	0.952256i	$-0.401243\pi$
$911$	16789.4	0.610603	0.305301	+	0.952256i	$0.401243\pi$
$912$	7703.49	0.279702
$913$	−2410.89	−0.0873918
$914$	−22919.4	−0.829439
$915$	34013.4	1.22890
$916$	2167.69	0.0781905
$917$	18129.8	0.652890
$918$	9071.34	0.326142
$919$	42322.8	1.51915	0.759575	−	0.650419i	$-0.225407\pi$
$919$	42322.8	1.51915	0.759575	+	0.650419i	$0.225407\pi$
$920$	37196.4	1.33297
$921$	−5058.86	−0.180993
$922$	6311.48	0.225442
$923$	0	0
$924$	283.435	0.0100913
$925$	901.474	0.0320436
$926$	−6372.11	−0.226134
$927$	4564.88	0.161737
$928$	3878.60	0.137200
$929$	−15413.1	−0.544337	−0.272168	−	0.962250i	$-0.587741\pi$
$929$	−15413.1	−0.544337	−0.272168	+	0.962250i	$0.587741\pi$
$930$	−24158.3	−0.851810
$931$	14054.5	0.494756
$932$	−1986.18	−0.0698063
$933$	31803.2	1.11596
$934$	−9399.85	−0.329307
$935$	18089.8	0.632726
$936$	0	0
$937$	45360.4	1.58149	0.790747	−	0.612143i	$-0.209692\pi$
$937$	45360.4	1.58149	0.790747	+	0.612143i	$0.209692\pi$
$938$	24052.7	0.837260
$939$	17670.0	0.614098
$940$	−456.971	−0.0158561
$941$	−37604.6	−1.30274	−0.651368	−	0.758762i	$-0.725805\pi$
$941$	−37604.6	−1.30274	−0.651368	+	0.758762i	$0.725805\pi$
$942$	−18076.1	−0.625212
$943$	22706.7	0.784127
$944$	24247.8	0.836016
$945$	−14750.6	−0.507766
$946$	6919.49	0.237814
$947$	−28270.9	−0.970095	−0.485048	−	0.874488i	$-0.661198\pi$
$947$	−28270.9	−0.970095	−0.485048	+	0.874488i	$0.661198\pi$
$948$	1199.72	0.0411023
$949$	0	0
$950$	−37886.9	−1.29391
$951$	7298.93	0.248879
$952$	−74066.3	−2.52154
$953$	21396.1	0.727268	0.363634	−	0.931542i	$-0.381536\pi$
$953$	21396.1	0.727268	0.363634	+	0.931542i	$0.381536\pi$
$954$	3094.74	0.105027
$955$	32382.1	1.09723
$956$	−809.252	−0.0273777
$957$	−3423.78	−0.115648
$958$	43254.9	1.45877
$959$	−66628.7	−2.24354
$960$	34285.8	1.15268
$961$	−10352.7	−0.347512
$962$	0	0
$963$	−11362.5	−0.380220
$964$	−3300.13	−0.110259
$965$	−97422.7	−3.24990
$966$	16026.5	0.533793
$967$	898.300	0.0298732	0.0149366	−	0.999888i	$-0.495245\pi$
$967$	898.300	0.0298732	0.0149366	+	0.999888i	$0.495245\pi$
$968$	−29889.4	−0.992441
$969$	−15900.8	−0.527150
$970$	−15958.5	−0.528244
$971$	3749.51	0.123921	0.0619607	−	0.998079i	$-0.480265\pi$
$971$	3749.51	0.123921	0.0619607	+	0.998079i	$0.480265\pi$
$972$	127.384	0.00420353
$973$	−6039.76	−0.198999
$974$	39656.9	1.30461
$975$	0	0
$976$	−31951.2	−1.04788
$977$	2717.19	0.0889771	0.0444885	−	0.999010i	$-0.485834\pi$
$977$	2717.19	0.0889771	0.0444885	+	0.999010i	$0.485834\pi$
$978$	6223.26	0.203474
$979$	−2526.11	−0.0824666
$980$	3608.21	0.117612
$981$	−4224.20	−0.137481
$982$	47884.6	1.55607
$983$	4179.98	0.135626	0.0678132	−	0.997698i	$-0.478398\pi$
$983$	4179.98	0.135626	0.0678132	+	0.997698i	$0.478398\pi$
$984$	21014.9	0.680825
$985$	40440.4	1.30816
$986$	55020.7	1.77709
$987$	−3201.64	−0.103252
$988$	0	0
$989$	−27435.3	−0.882096
$990$	−3622.64	−0.116298
$991$	−15297.9	−0.490367	−0.245184	−	0.969477i	$-0.578848\pi$
$991$	−15297.9	−0.490367	−0.245184	+	0.969477i	$0.578848\pi$
$992$	−3302.07	−0.105686
$993$	2272.60	0.0726273
$994$	36404.3	1.16164
$995$	−48691.0	−1.55136
$996$	544.052	0.0173082
$997$	9164.26	0.291108	0.145554	−	0.989350i	$-0.453504\pi$
$997$	9164.26	0.291108	0.145554	+	0.989350i	$0.453504\pi$
$998$	−10543.7	−0.334425
$999$	−75.7663	−0.00239954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.n.1.4	✓	9
3.2	odd	2		1521.4.a.bj.1.6		9
13.5	odd	4		507.4.b.j.337.13		18
13.8	odd	4		507.4.b.j.337.6		18
13.12	even	2		507.4.a.q.1.6	yes	9
39.38	odd	2		1521.4.a.be.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.4	✓	9	1.1	even	1	trivial
507.4.a.q.1.6	yes	9	13.12	even	2
507.4.b.j.337.6		18	13.8	odd	4
507.4.b.j.337.13		18	13.5	odd	4
1521.4.a.be.1.4		9	39.38	odd	2
1521.4.a.bj.1.6		9	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73419 q^{2} -3.00000 q^{3} -0.524213 q^{4} -21.1246 q^{5} +8.20257 q^{6} -25.8618 q^{7} +23.3068 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.73419 q^{2} -3.00000 q^{3} -0.524213 q^{4} -21.1246 q^{5} +8.20257 q^{6} -25.8618 q^{7} +23.3068 q^{8} +9.00000 q^{9} +57.7586 q^{10} -6.96892 q^{11} +1.57264 q^{12} +70.7111 q^{14} +63.3738 q^{15} -59.5315 q^{16} +122.879 q^{17} -24.6077 q^{18} +43.1340 q^{19} +11.0738 q^{20} +77.5854 q^{21} +19.0543 q^{22} -75.5492 q^{23} -69.9204 q^{24} +321.248 q^{25} -27.0000 q^{27} +13.5571 q^{28} -163.764 q^{29} -173.276 q^{30} +139.421 q^{31} -23.6841 q^{32} +20.9068 q^{33} -335.975 q^{34} +546.320 q^{35} -4.71792 q^{36} +2.80616 q^{37} -117.936 q^{38} -492.347 q^{40} -300.555 q^{41} -212.133 q^{42} +363.145 q^{43} +3.65320 q^{44} -190.121 q^{45} +206.566 q^{46} -41.2660 q^{47} +178.594 q^{48} +325.834 q^{49} -878.354 q^{50} -368.638 q^{51} -125.763 q^{53} +73.8231 q^{54} +147.216 q^{55} -602.756 q^{56} -129.402 q^{57} +447.762 q^{58} -407.311 q^{59} -33.2214 q^{60} +536.710 q^{61} -381.204 q^{62} -232.756 q^{63} +541.009 q^{64} -57.1630 q^{66} +340.155 q^{67} -64.4150 q^{68} +226.648 q^{69} -1493.74 q^{70} +514.831 q^{71} +209.761 q^{72} -491.231 q^{73} -7.67257 q^{74} -963.745 q^{75} -22.6114 q^{76} +180.229 q^{77} +762.869 q^{79} +1257.58 q^{80} +81.0000 q^{81} +821.774 q^{82} +345.948 q^{83} -40.6713 q^{84} -2595.78 q^{85} -992.907 q^{86} +491.292 q^{87} -162.423 q^{88} +362.482 q^{89} +519.828 q^{90} +39.6039 q^{92} -418.264 q^{93} +112.829 q^{94} -911.188 q^{95} +71.0523 q^{96} -276.297 q^{97} -890.890 q^{98} -62.7203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9	\( 9 q - 8 q^{2} - 27 q^{3} + 32 q^{4} - 41 q^{5} + 24 q^{6} - q^{7} - 111 q^{8} + 81 q^{9} + 198 q^{10} - 37 q^{11} - 96 q^{12} + 98 q^{14} + 123 q^{15} + 32 q^{16} - 134 q^{17} - 72 q^{18} + 72 q^{19} - 356 q^{20} + 3 q^{21} + 274 q^{22} + 226 q^{23} + 333 q^{24} + 612 q^{25} - 243 q^{27} - 132 q^{28} - 547 q^{29} - 594 q^{30} + 521 q^{31} - 721 q^{32} + 111 q^{33} + 100 q^{34} + 138 q^{35} + 288 q^{36} - 584 q^{37} - 416 q^{38} + 1342 q^{40} - 482 q^{41} - 294 q^{42} + 158 q^{43} - 1453 q^{44} - 369 q^{45} - 1537 q^{46} - 1500 q^{47} - 96 q^{48} + 642 q^{49} - 2777 q^{50} + 402 q^{51} + 1399 q^{53} + 216 q^{54} - 1408 q^{55} - 616 q^{56} - 216 q^{57} - 1455 q^{58} - 1541 q^{59} + 1068 q^{60} + 2092 q^{61} - 293 q^{62} - 9 q^{63} + 2481 q^{64} - 822 q^{66} - 252 q^{67} - 1579 q^{68} - 678 q^{69} - 2492 q^{70} - 2352 q^{71} - 999 q^{72} - 903 q^{73} + 1037 q^{74} - 1836 q^{75} + 485 q^{76} - 1686 q^{77} - 115 q^{79} - 5701 q^{80} + 729 q^{81} - 5147 q^{82} - 1207 q^{83} + 396 q^{84} - 4308 q^{85} - 5691 q^{86} + 1641 q^{87} - 484 q^{88} - 2336 q^{89} + 1782 q^{90} + 2087 q^{92} - 1563 q^{93} - 468 q^{94} - 222 q^{95} + 2163 q^{96} - 2155 q^{97} - 5593 q^{98} - 333 q^{99}+O(q^{100}) \) 9 * q - 8 * q^2 - 27 * q^3 + 32 * q^4 - 41 * q^5 + 24 * q^6 - q^7 - 111 * q^8 + 81 * q^9 + 198 * q^10 - 37 * q^11 - 96 * q^12 + 98 * q^14 + 123 * q^15 + 32 * q^16 - 134 * q^17 - 72 * q^18 + 72 * q^19 - 356 * q^20 + 3 * q^21 + 274 * q^22 + 226 * q^23 + 333 * q^24 + 612 * q^25 - 243 * q^27 - 132 * q^28 - 547 * q^29 - 594 * q^30 + 521 * q^31 - 721 * q^32 + 111 * q^33 + 100 * q^34 + 138 * q^35 + 288 * q^36 - 584 * q^37 - 416 * q^38 + 1342 * q^40 - 482 * q^41 - 294 * q^42 + 158 * q^43 - 1453 * q^44 - 369 * q^45 - 1537 * q^46 - 1500 * q^47 - 96 * q^48 + 642 * q^49 - 2777 * q^50 + 402 * q^51 + 1399 * q^53 + 216 * q^54 - 1408 * q^55 - 616 * q^56 - 216 * q^57 - 1455 * q^58 - 1541 * q^59 + 1068 * q^60 + 2092 * q^61 - 293 * q^62 - 9 * q^63 + 2481 * q^64 - 822 * q^66 - 252 * q^67 - 1579 * q^68 - 678 * q^69 - 2492 * q^70 - 2352 * q^71 - 999 * q^72 - 903 * q^73 + 1037 * q^74 - 1836 * q^75 + 485 * q^76 - 1686 * q^77 - 115 * q^79 - 5701 * q^80 + 729 * q^81 - 5147 * q^82 - 1207 * q^83 + 396 * q^84 - 4308 * q^85 - 5691 * q^86 + 1641 * q^87 - 484 * q^88 - 2336 * q^89 + 1782 * q^90 + 2087 * q^92 - 1563 * q^93 - 468 * q^94 - 222 * q^95 + 2163 * q^96 - 2155 * q^97 - 5593 * q^98 - 333 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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