LMFDB - Embedded newform 2013.2.a.h.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,2,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2013.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:	$16.0738859269$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.0561655$ of defining polynomial
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0561655 q^{2} -1.00000 q^{3} -1.99685 q^{4} -2.87016 q^{5} -0.0561655 q^{6} +3.53988 q^{7} -0.224485 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.0561655 q^{2} -1.00000 q^{3} -1.99685 q^{4} -2.87016 q^{5} -0.0561655 q^{6} +3.53988 q^{7} -0.224485 q^{8} +1.00000 q^{9} -0.161204 q^{10} -1.00000 q^{11} +1.99685 q^{12} +1.66374 q^{13} +0.198819 q^{14} +2.87016 q^{15} +3.98108 q^{16} -6.51228 q^{17} +0.0561655 q^{18} -4.27158 q^{19} +5.73126 q^{20} -3.53988 q^{21} -0.0561655 q^{22} -8.02418 q^{23} +0.224485 q^{24} +3.23780 q^{25} +0.0934449 q^{26} -1.00000 q^{27} -7.06858 q^{28} +3.26557 q^{29} +0.161204 q^{30} +6.97737 q^{31} +0.672570 q^{32} +1.00000 q^{33} -0.365766 q^{34} -10.1600 q^{35} -1.99685 q^{36} +0.826353 q^{37} -0.239916 q^{38} -1.66374 q^{39} +0.644307 q^{40} -11.9366 q^{41} -0.198819 q^{42} -2.21299 q^{43} +1.99685 q^{44} -2.87016 q^{45} -0.450683 q^{46} +10.4001 q^{47} -3.98108 q^{48} +5.53072 q^{49} +0.181853 q^{50} +6.51228 q^{51} -3.32224 q^{52} +8.11158 q^{53} -0.0561655 q^{54} +2.87016 q^{55} -0.794649 q^{56} +4.27158 q^{57} +0.183413 q^{58} +2.00484 q^{59} -5.73126 q^{60} +1.00000 q^{61} +0.391888 q^{62} +3.53988 q^{63} -7.92439 q^{64} -4.77520 q^{65} +0.0561655 q^{66} +10.0237 q^{67} +13.0040 q^{68} +8.02418 q^{69} -0.570642 q^{70} -2.13305 q^{71} -0.224485 q^{72} +2.54693 q^{73} +0.0464125 q^{74} -3.23780 q^{75} +8.52969 q^{76} -3.53988 q^{77} -0.0934449 q^{78} -6.81997 q^{79} -11.4263 q^{80} +1.00000 q^{81} -0.670428 q^{82} +0.562991 q^{83} +7.06858 q^{84} +18.6913 q^{85} -0.124294 q^{86} -3.26557 q^{87} +0.224485 q^{88} +7.23252 q^{89} -0.161204 q^{90} +5.88944 q^{91} +16.0231 q^{92} -6.97737 q^{93} +0.584128 q^{94} +12.2601 q^{95} -0.672570 q^{96} +11.1816 q^{97} +0.310636 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - q^2 - 14 * q^3 + 15 * q^4 + q^5 + q^6 + 9 * q^7 + 14 * q^9	\( 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9} + 6 q^{10} - 14 q^{11} - 15 q^{12} + q^{13} - 7 q^{14} - q^{15} + 17 q^{16} - 9 q^{17} - q^{18} + 22 q^{19} + 23 q^{20} - 9 q^{21} + q^{22} + q^{23} + 25 q^{25} + 4 q^{26} - 14 q^{27} + 37 q^{28} - 6 q^{29} - 6 q^{30} + 9 q^{31} + 4 q^{32} + 14 q^{33} + 8 q^{34} + 18 q^{35} + 15 q^{36} + 18 q^{37} + 8 q^{38} - q^{39} + 16 q^{40} - 25 q^{41} + 7 q^{42} + 25 q^{43} - 15 q^{44} + q^{45} + 20 q^{46} + 36 q^{47} - 17 q^{48} + 25 q^{49} + 2 q^{50} + 9 q^{51} - 13 q^{52} + q^{54} - q^{55} - 40 q^{56} - 22 q^{57} + 33 q^{58} + 17 q^{59} - 23 q^{60} + 14 q^{61} - 13 q^{62} + 9 q^{63} - 6 q^{64} - 61 q^{65} - q^{66} + 22 q^{67} + 66 q^{68} - q^{69} + 44 q^{70} - 13 q^{71} + 20 q^{73} - 12 q^{74} - 25 q^{75} + 49 q^{76} - 9 q^{77} - 4 q^{78} + 31 q^{79} + 88 q^{80} + 14 q^{81} + 2 q^{82} + 32 q^{83} - 37 q^{84} + 2 q^{85} - 14 q^{86} + 6 q^{87} - 21 q^{89} + 6 q^{90} + 45 q^{91} - 14 q^{92} - 9 q^{93} - 31 q^{94} + 23 q^{95} - 4 q^{96} + 37 q^{97} - 38 q^{98} - 14 q^{99}+O(q^{100}) \) 14 * q - q^2 - 14 * q^3 + 15 * q^4 + q^5 + q^6 + 9 * q^7 + 14 * q^9 + 6 * q^10 - 14 * q^11 - 15 * q^12 + q^13 - 7 * q^14 - q^15 + 17 * q^16 - 9 * q^17 - q^18 + 22 * q^19 + 23 * q^20 - 9 * q^21 + q^22 + q^23 + 25 * q^25 + 4 * q^26 - 14 * q^27 + 37 * q^28 - 6 * q^29 - 6 * q^30 + 9 * q^31 + 4 * q^32 + 14 * q^33 + 8 * q^34 + 18 * q^35 + 15 * q^36 + 18 * q^37 + 8 * q^38 - q^39 + 16 * q^40 - 25 * q^41 + 7 * q^42 + 25 * q^43 - 15 * q^44 + q^45 + 20 * q^46 + 36 * q^47 - 17 * q^48 + 25 * q^49 + 2 * q^50 + 9 * q^51 - 13 * q^52 + q^54 - q^55 - 40 * q^56 - 22 * q^57 + 33 * q^58 + 17 * q^59 - 23 * q^60 + 14 * q^61 - 13 * q^62 + 9 * q^63 - 6 * q^64 - 61 * q^65 - q^66 + 22 * q^67 + 66 * q^68 - q^69 + 44 * q^70 - 13 * q^71 + 20 * q^73 - 12 * q^74 - 25 * q^75 + 49 * q^76 - 9 * q^77 - 4 * q^78 + 31 * q^79 + 88 * q^80 + 14 * q^81 + 2 * q^82 + 32 * q^83 - 37 * q^84 + 2 * q^85 - 14 * q^86 + 6 * q^87 - 21 * q^89 + 6 * q^90 + 45 * q^91 - 14 * q^92 - 9 * q^93 - 31 * q^94 + 23 * q^95 - 4 * q^96 + 37 * q^97 - 38 * q^98 - 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.0561655	0.0397150	0.0198575	−	0.999803i	$-0.493679\pi$
$2$	0.0561655	0.0397150	0.0198575	+	0.999803i	$0.493679\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.99685	−0.998423
$5$	−2.87016	−1.28357	−0.641786	−	0.766883i	$-0.721806\pi$
$5$	−2.87016	−1.28357	−0.641786	+	0.766883i	$0.721806\pi$
$6$	−0.0561655	−0.0229295
$7$	3.53988	1.33795	0.668974	−	0.743286i	$-0.266734\pi$
$7$	3.53988	1.33795	0.668974	+	0.743286i	$0.266734\pi$
$8$	−0.224485	−0.0793674
$9$	1.00000	0.333333
$10$	−0.161204	−0.0509771
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.99685	0.576440
$13$	1.66374	0.461439	0.230719	−	0.973020i	$-0.425892\pi$
$13$	1.66374	0.461439	0.230719	+	0.973020i	$0.425892\pi$
$14$	0.198819	0.0531366
$15$	2.87016	0.741071
$16$	3.98108	0.995271
$17$	−6.51228	−1.57946	−0.789730	−	0.613455i	$-0.789779\pi$
$17$	−6.51228	−1.57946	−0.789730	+	0.613455i	$0.789779\pi$
$18$	0.0561655	0.0132383
$19$	−4.27158	−0.979968	−0.489984	−	0.871731i	$-0.662997\pi$
$19$	−4.27158	−0.979968	−0.489984	+	0.871731i	$0.662997\pi$
$20$	5.73126	1.28155
$21$	−3.53988	−0.772464
$22$	−0.0561655	−0.0119745
$23$	−8.02418	−1.67316	−0.836579	−	0.547846i	$-0.815448\pi$
$23$	−8.02418	−1.67316	−0.836579	+	0.547846i	$0.815448\pi$
$24$	0.224485	0.0458228
$25$	3.23780	0.647559
$26$	0.0934449	0.0183261
$27$	−1.00000	−0.192450
$28$	−7.06858	−1.33584
$29$	3.26557	0.606402	0.303201	−	0.952927i	$-0.401945\pi$
$29$	3.26557	0.606402	0.303201	+	0.952927i	$0.401945\pi$
$30$	0.161204	0.0294317
$31$	6.97737	1.25317	0.626587	−	0.779352i	$-0.284451\pi$
$31$	6.97737	1.25317	0.626587	+	0.779352i	$0.284451\pi$
$32$	0.672570	0.118895
$33$	1.00000	0.174078
$34$	−0.365766	−0.0627283
$35$	−10.1600	−1.71735
$36$	−1.99685	−0.332808
$37$	0.826353	0.135852	0.0679258	−	0.997690i	$-0.478362\pi$
$37$	0.826353	0.135852	0.0679258	+	0.997690i	$0.478362\pi$
$38$	−0.239916	−0.0389194
$39$	−1.66374	−0.266412
$40$	0.644307	0.101874
$41$	−11.9366	−1.86419	−0.932095	−	0.362214i	$-0.882021\pi$
$41$	−11.9366	−1.86419	−0.932095	+	0.362214i	$0.882021\pi$
$42$	−0.198819	−0.0306784
$43$	−2.21299	−0.337477	−0.168739	−	0.985661i	$-0.553969\pi$
$43$	−2.21299	−0.337477	−0.168739	+	0.985661i	$0.553969\pi$
$44$	1.99685	0.301036
$45$	−2.87016	−0.427858
$46$	−0.450683	−0.0664495
$47$	10.4001	1.51701	0.758506	−	0.651666i	$-0.225930\pi$
$47$	10.4001	1.51701	0.758506	+	0.651666i	$0.225930\pi$
$48$	−3.98108	−0.574620
$49$	5.53072	0.790103
$50$	0.181853	0.0257178
$51$	6.51228	0.911902
$52$	−3.32224	−0.460711
$53$	8.11158	1.11421	0.557106	−	0.830442i	$-0.311912\pi$
$53$	8.11158	1.11421	0.557106	+	0.830442i	$0.311912\pi$
$54$	−0.0561655	−0.00764316
$55$	2.87016	0.387012
$56$	−0.794649	−0.106189
$57$	4.27158	0.565785
$58$	0.183413	0.0240833
$59$	2.00484	0.261007	0.130504	−	0.991448i	$-0.458341\pi$
$59$	2.00484	0.261007	0.130504	+	0.991448i	$0.458341\pi$
$60$	−5.73126	−0.739902
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	0.391888	0.0497698
$63$	3.53988	0.445982
$64$	−7.92439	−0.990549
$65$	−4.77520	−0.592291
$66$	0.0561655	0.00691350
$67$	10.0237	1.22459	0.612293	−	0.790631i	$-0.290247\pi$
$67$	10.0237	1.22459	0.612293	+	0.790631i	$0.290247\pi$
$68$	13.0040	1.57697
$69$	8.02418	0.965998
$70$	−0.570642	−0.0682047
$71$	−2.13305	−0.253146	−0.126573	−	0.991957i	$-0.540398\pi$
$71$	−2.13305	−0.253146	−0.126573	+	0.991957i	$0.540398\pi$
$72$	−0.224485	−0.0264558
$73$	2.54693	0.298096	0.149048	−	0.988830i	$-0.452379\pi$
$73$	2.54693	0.298096	0.149048	+	0.988830i	$0.452379\pi$
$74$	0.0464125	0.00539535
$75$	−3.23780	−0.373868
$76$	8.52969	0.978422
$77$	−3.53988	−0.403406
$78$	−0.0934449	−0.0105806
$79$	−6.81997	−0.767306	−0.383653	−	0.923477i	$-0.625334\pi$
$79$	−6.81997	−0.767306	−0.383653	+	0.923477i	$0.625334\pi$
$80$	−11.4263	−1.27750
$81$	1.00000	0.111111
$82$	−0.670428	−0.0740364
$83$	0.562991	0.0617963	0.0308982	−	0.999523i	$-0.490163\pi$
$83$	0.562991	0.0617963	0.0308982	+	0.999523i	$0.490163\pi$
$84$	7.06858	0.771246
$85$	18.6913	2.02735
$86$	−0.124294	−0.0134029
$87$	−3.26557	−0.350106
$88$	0.224485	0.0239302
$89$	7.23252	0.766646	0.383323	−	0.923614i	$-0.374780\pi$
$89$	7.23252	0.766646	0.383323	+	0.923614i	$0.374780\pi$
$90$	−0.161204	−0.0169924
$91$	5.88944	0.617381
$92$	16.0231	1.67052
$93$	−6.97737	−0.723520
$94$	0.584128	0.0602482
$95$	12.2601	1.25786
$96$	−0.672570	−0.0686438
$97$	11.1816	1.13532	0.567660	−	0.823263i	$-0.307849\pi$
$97$	11.1816	1.13532	0.567660	+	0.823263i	$0.307849\pi$
$98$	0.310636	0.0313789
$99$	−1.00000	−0.100504
$100$	−6.46538	−0.646538
$101$	10.8938	1.08398	0.541989	−	0.840386i	$-0.317671\pi$
$101$	10.8938	1.08398	0.541989	+	0.840386i	$0.317671\pi$
$102$	0.365766	0.0362162
$103$	−1.11625	−0.109988	−0.0549938	−	0.998487i	$-0.517514\pi$
$103$	−1.11625	−0.109988	−0.0549938	+	0.998487i	$0.517514\pi$
$104$	−0.373485	−0.0366232
$105$	10.1600	0.991514
$106$	0.455591	0.0442509
$107$	−2.80022	−0.270707	−0.135354	−	0.990797i	$-0.543217\pi$
$107$	−2.80022	−0.270707	−0.135354	+	0.990797i	$0.543217\pi$
$108$	1.99685	0.192147
$109$	10.6923	1.02414	0.512069	−	0.858944i	$-0.328879\pi$
$109$	10.6923	1.02414	0.512069	+	0.858944i	$0.328879\pi$
$110$	0.161204	0.0153702
$111$	−0.826353	−0.0784339
$112$	14.0925	1.33162
$113$	13.9443	1.31177	0.655885	−	0.754861i	$-0.272296\pi$
$113$	13.9443	1.31177	0.655885	+	0.754861i	$0.272296\pi$
$114$	0.239916	0.0224702
$115$	23.0307	2.14762
$116$	−6.52085	−0.605446
$117$	1.66374	0.153813
$118$	0.112603	0.0103659
$119$	−23.0527	−2.11323
$120$	−0.644307	−0.0588169
$121$	1.00000	0.0909091
$122$	0.0561655	0.00508499
$123$	11.9366	1.07629
$124$	−13.9327	−1.25120
$125$	5.05780	0.452383
$126$	0.198819	0.0177122
$127$	−15.3492	−1.36202	−0.681012	−	0.732272i	$-0.738460\pi$
$127$	−15.3492	−1.36202	−0.681012	+	0.732272i	$0.738460\pi$
$128$	−1.79022	−0.158234
$129$	2.21299	0.194843
$130$	−0.268202	−0.0235228
$131$	17.1289	1.49656	0.748280	−	0.663383i	$-0.230880\pi$
$131$	17.1289	1.49656	0.748280	+	0.663383i	$0.230880\pi$
$132$	−1.99685	−0.173803
$133$	−15.1209	−1.31115
$134$	0.562985	0.0486345
$135$	2.87016	0.247024
$136$	1.46191	0.125358
$137$	−1.15929	−0.0990451	−0.0495225	−	0.998773i	$-0.515770\pi$
$137$	−1.15929	−0.0990451	−0.0495225	+	0.998773i	$0.515770\pi$
$138$	0.450683	0.0383646
$139$	7.66907	0.650482	0.325241	−	0.945631i	$-0.394555\pi$
$139$	7.66907	0.650482	0.325241	+	0.945631i	$0.394555\pi$
$140$	20.2879	1.71464
$141$	−10.4001	−0.875847
$142$	−0.119804	−0.0100537
$143$	−1.66374	−0.139129
$144$	3.98108	0.331757
$145$	−9.37271	−0.778361
$146$	0.143050	0.0118389
$147$	−5.53072	−0.456166
$148$	−1.65010	−0.135637
$149$	−1.19140	−0.0976031	−0.0488015	−	0.998808i	$-0.515540\pi$
$149$	−1.19140	−0.0976031	−0.0488015	+	0.998808i	$0.515540\pi$
$150$	−0.181853	−0.0148482
$151$	11.2781	0.917798	0.458899	−	0.888488i	$-0.348244\pi$
$151$	11.2781	0.917798	0.458899	+	0.888488i	$0.348244\pi$
$152$	0.958906	0.0777775
$153$	−6.51228	−0.526487
$154$	−0.198819	−0.0160213
$155$	−20.0262	−1.60854
$156$	3.32224	0.265992
$157$	17.0614	1.36165	0.680825	−	0.732446i	$-0.261621\pi$
$157$	17.0614	1.36165	0.680825	+	0.732446i	$0.261621\pi$
$158$	−0.383047	−0.0304736
$159$	−8.11158	−0.643290
$160$	−1.93038	−0.152610
$161$	−28.4046	−2.23860
$162$	0.0561655	0.00441278
$163$	−18.8321	−1.47504	−0.737521	−	0.675325i	$-0.764004\pi$
$163$	−18.8321	−1.47504	−0.737521	+	0.675325i	$0.764004\pi$
$164$	23.8356	1.86125
$165$	−2.87016	−0.223441
$166$	0.0316207	0.00245424
$167$	20.7776	1.60782	0.803911	−	0.594749i	$-0.202749\pi$
$167$	20.7776	1.60782	0.803911	+	0.594749i	$0.202749\pi$
$168$	0.794649	0.0613085
$169$	−10.2320	−0.787074
$170$	1.04980	0.0805163
$171$	−4.27158	−0.326656
$172$	4.41899	0.336945
$173$	8.58089	0.652393	0.326197	−	0.945302i	$-0.394233\pi$
$173$	8.58089	0.652393	0.326197	+	0.945302i	$0.394233\pi$
$174$	−0.183413	−0.0139045
$175$	11.4614	0.866400
$176$	−3.98108	−0.300085
$177$	−2.00484	−0.150693
$178$	0.406218	0.0304474
$179$	1.62970	0.121810	0.0609049	−	0.998144i	$-0.480601\pi$
$179$	1.62970	0.121810	0.0609049	+	0.998144i	$0.480601\pi$
$180$	5.73126	0.427183
$181$	−6.80075	−0.505496	−0.252748	−	0.967532i	$-0.581334\pi$
$181$	−6.80075	−0.505496	−0.252748	+	0.967532i	$0.581334\pi$
$182$	0.330783	0.0245193
$183$	−1.00000	−0.0739221
$184$	1.80131	0.132794
$185$	−2.37176	−0.174375
$186$	−0.391888	−0.0287346
$187$	6.51228	0.476225
$188$	−20.7674	−1.51462
$189$	−3.53988	−0.257488
$190$	0.688595	0.0499559
$191$	21.1056	1.52714	0.763572	−	0.645723i	$-0.223444\pi$
$191$	21.1056	1.52714	0.763572	+	0.645723i	$0.223444\pi$
$192$	7.92439	0.571894
$193$	−15.3449	−1.10455	−0.552276	−	0.833661i	$-0.686241\pi$
$193$	−15.3449	−1.10455	−0.552276	+	0.833661i	$0.686241\pi$
$194$	0.628021	0.0450893
$195$	4.77520	0.341959
$196$	−11.0440	−0.788856
$197$	8.01939	0.571358	0.285679	−	0.958325i	$-0.407781\pi$
$197$	8.01939	0.571358	0.285679	+	0.958325i	$0.407781\pi$
$198$	−0.0561655	−0.00399151
$199$	18.9435	1.34287	0.671435	−	0.741063i	$-0.265678\pi$
$199$	18.9435	1.34287	0.671435	+	0.741063i	$0.265678\pi$
$200$	−0.726836	−0.0513951
$201$	−10.0237	−0.707016
$202$	0.611859	0.0430502
$203$	11.5597	0.811334
$204$	−13.0040	−0.910463
$205$	34.2600	2.39282
$206$	−0.0626949	−0.00436816
$207$	−8.02418	−0.557719
$208$	6.62349	0.459257
$209$	4.27158	0.295471
$210$	0.570642	0.0393780
$211$	−14.2888	−0.983682	−0.491841	−	0.870685i	$-0.663676\pi$
$211$	−14.2888	−0.983682	−0.491841	+	0.870685i	$0.663676\pi$
$212$	−16.1976	−1.11245
$213$	2.13305	0.146154
$214$	−0.157276	−0.0107511
$215$	6.35162	0.433177
$216$	0.224485	0.0152743
$217$	24.6990	1.67668
$218$	0.600540	0.0406737
$219$	−2.54693	−0.172106
$220$	−5.73126	−0.386401
$221$	−10.8348	−0.728824
$222$	−0.0464125	−0.00311501
$223$	−2.84596	−0.190580	−0.0952899	−	0.995450i	$-0.530378\pi$
$223$	−2.84596	−0.190580	−0.0952899	+	0.995450i	$0.530378\pi$
$224$	2.38081	0.159075
$225$	3.23780	0.215853
$226$	0.783189	0.0520970
$227$	−19.4833	−1.29315	−0.646576	−	0.762849i	$-0.723800\pi$
$227$	−19.4833	−1.29315	−0.646576	+	0.762849i	$0.723800\pi$
$228$	−8.52969	−0.564892
$229$	−11.7161	−0.774219	−0.387109	−	0.922034i	$-0.626526\pi$
$229$	−11.7161	−0.774219	−0.387109	+	0.922034i	$0.626526\pi$
$230$	1.29353	0.0852928
$231$	3.53988	0.232907
$232$	−0.733072	−0.0481286
$233$	−27.8004	−1.82126	−0.910632	−	0.413219i	$-0.864404\pi$
$233$	−27.8004	−1.82126	−0.910632	+	0.413219i	$0.864404\pi$
$234$	0.0934449	0.00610869
$235$	−29.8499	−1.94720
$236$	−4.00335	−0.260596
$237$	6.81997	0.443004
$238$	−1.29476	−0.0839271
$239$	1.27569	0.0825176	0.0412588	−	0.999148i	$-0.486863\pi$
$239$	1.27569	0.0825176	0.0412588	+	0.999148i	$0.486863\pi$
$240$	11.4263	0.737566
$241$	14.9945	0.965878	0.482939	−	0.875654i	$-0.339569\pi$
$241$	14.9945	0.965878	0.482939	+	0.875654i	$0.339569\pi$
$242$	0.0561655	0.00361046
$243$	−1.00000	−0.0641500
$244$	−1.99685	−0.127835
$245$	−15.8740	−1.01415
$246$	0.670428	0.0427449
$247$	−7.10681	−0.452195
$248$	−1.56632	−0.0994611
$249$	−0.562991	−0.0356781
$250$	0.284074	0.0179664
$251$	−25.6541	−1.61927	−0.809637	−	0.586931i	$-0.800336\pi$
$251$	−25.6541	−1.61927	−0.809637	+	0.586931i	$0.800336\pi$
$252$	−7.06858	−0.445279
$253$	8.02418	0.504476
$254$	−0.862098	−0.0540929
$255$	−18.6913	−1.17049
$256$	15.7482	0.984264
$257$	−23.5239	−1.46738	−0.733690	−	0.679485i	$-0.762203\pi$
$257$	−23.5239	−1.46738	−0.733690	+	0.679485i	$0.762203\pi$
$258$	0.124294	0.00773818
$259$	2.92519	0.181762
$260$	9.53533	0.591356
$261$	3.26557	0.202134
$262$	0.962055	0.0594360
$263$	−7.55266	−0.465717	−0.232859	−	0.972511i	$-0.574808\pi$
$263$	−7.55266	−0.465717	−0.232859	+	0.972511i	$0.574808\pi$
$264$	−0.224485	−0.0138161
$265$	−23.2815	−1.43017
$266$	−0.849271	−0.0520722
$267$	−7.23252	−0.442623
$268$	−20.0157	−1.22266
$269$	25.3390	1.54495	0.772474	−	0.635046i	$-0.219019\pi$
$269$	25.3390	1.54495	0.772474	+	0.635046i	$0.219019\pi$
$270$	0.161204	0.00981055
$271$	−24.0494	−1.46090	−0.730449	−	0.682967i	$-0.760689\pi$
$271$	−24.0494	−1.46090	−0.730449	+	0.682967i	$0.760689\pi$
$272$	−25.9259	−1.57199
$273$	−5.88944	−0.356445
$274$	−0.0651123	−0.00393358
$275$	−3.23780	−0.195246
$276$	−16.0231	−0.964475
$277$	25.9274	1.55782	0.778912	−	0.627133i	$-0.215772\pi$
$277$	25.9274	1.55782	0.778912	+	0.627133i	$0.215772\pi$
$278$	0.430738	0.0258339
$279$	6.97737	0.417724
$280$	2.28077	0.136302
$281$	−16.8801	−1.00698	−0.503491	−	0.864000i	$-0.667951\pi$
$281$	−16.8801	−1.00698	−0.503491	+	0.864000i	$0.667951\pi$
$282$	−0.584128	−0.0347843
$283$	14.2351	0.846191	0.423095	−	0.906085i	$-0.360944\pi$
$283$	14.2351	0.846191	0.423095	+	0.906085i	$0.360944\pi$
$284$	4.25937	0.252747
$285$	−12.2601	−0.726226
$286$	−0.0934449	−0.00552552
$287$	−42.2542	−2.49419
$288$	0.672570	0.0396315
$289$	25.4098	1.49469
$290$	−0.526423	−0.0309126
$291$	−11.1816	−0.655477
$292$	−5.08583	−0.297626
$293$	25.1762	1.47081	0.735406	−	0.677627i	$-0.236992\pi$
$293$	25.1762	1.47081	0.735406	+	0.677627i	$0.236992\pi$
$294$	−0.310636	−0.0181166
$295$	−5.75419	−0.335022
$296$	−0.185504	−0.0107822
$297$	1.00000	0.0580259
$298$	−0.0669155	−0.00387631
$299$	−13.3502	−0.772060
$300$	6.46538	0.373279
$301$	−7.83369	−0.451527
$302$	0.633440	0.0364504
$303$	−10.8938	−0.625835
$304$	−17.0055	−0.975333
$305$	−2.87016	−0.164345
$306$	−0.365766	−0.0209094
$307$	20.2369	1.15498	0.577490	−	0.816398i	$-0.304032\pi$
$307$	20.2369	1.15498	0.577490	+	0.816398i	$0.304032\pi$
$308$	7.06858	0.402770
$309$	1.11625	0.0635014
$310$	−1.12478	−0.0638832
$311$	20.5089	1.16295	0.581477	−	0.813563i	$-0.302475\pi$
$311$	20.5089	1.16295	0.581477	+	0.813563i	$0.302475\pi$
$312$	0.373485	0.0211444
$313$	−10.5734	−0.597644	−0.298822	−	0.954309i	$-0.596594\pi$
$313$	−10.5734	−0.597644	−0.298822	+	0.954309i	$0.596594\pi$
$314$	0.958264	0.0540780
$315$	−10.1600	−0.572451
$316$	13.6184	0.766096
$317$	20.3836	1.14486	0.572429	−	0.819955i	$-0.306001\pi$
$317$	20.3836	1.14486	0.572429	+	0.819955i	$0.306001\pi$
$318$	−0.455591	−0.0255483
$319$	−3.26557	−0.182837
$320$	22.7442	1.27144
$321$	2.80022	0.156293
$322$	−1.59536	−0.0889059
$323$	27.8177	1.54782
$324$	−1.99685	−0.110936
$325$	5.38686	0.298809
$326$	−1.05771	−0.0585813
$327$	−10.6923	−0.591287
$328$	2.67960	0.147956
$329$	36.8151	2.02968
$330$	−0.161204	−0.00887398
$331$	11.9595	0.657353	0.328676	−	0.944443i	$-0.393397\pi$
$331$	11.9595	0.657353	0.328676	+	0.944443i	$0.393397\pi$
$332$	−1.12421	−0.0616989
$333$	0.826353	0.0452838
$334$	1.16699	0.0638547
$335$	−28.7695	−1.57185
$336$	−14.0925	−0.768811
$337$	−17.6438	−0.961117	−0.480559	−	0.876963i	$-0.659566\pi$
$337$	−17.6438	−0.961117	−0.480559	+	0.876963i	$0.659566\pi$
$338$	−0.574684	−0.0312587
$339$	−13.9443	−0.757351
$340$	−37.3236	−2.02415
$341$	−6.97737	−0.377846
$342$	−0.239916	−0.0129731
$343$	−5.20108	−0.280832
$344$	0.496782	0.0267847
$345$	−23.0307	−1.23993
$346$	0.481950	0.0259098
$347$	0.648489	0.0348127	0.0174063	−	0.999848i	$-0.494459\pi$
$347$	0.648489	0.0348127	0.0174063	+	0.999848i	$0.494459\pi$
$348$	6.52085	0.349554
$349$	−23.3381	−1.24926	−0.624631	−	0.780920i	$-0.714751\pi$
$349$	−23.3381	−1.24926	−0.624631	+	0.780920i	$0.714751\pi$
$350$	0.643735	0.0344091
$351$	−1.66374	−0.0888040
$352$	−0.672570	−0.0358481
$353$	−35.7398	−1.90224	−0.951119	−	0.308825i	$-0.900064\pi$
$353$	−35.7398	−1.90224	−0.951119	+	0.308825i	$0.900064\pi$
$354$	−0.112603	−0.00598476
$355$	6.12219	0.324932
$356$	−14.4422	−0.765437
$357$	23.0527	1.22008
$358$	0.0915332	0.00483768
$359$	10.9472	0.577770	0.288885	−	0.957364i	$-0.406715\pi$
$359$	10.9472	0.577770	0.288885	+	0.957364i	$0.406715\pi$
$360$	0.644307	0.0339580
$361$	−0.753598	−0.0396631
$362$	−0.381968	−0.0200758
$363$	−1.00000	−0.0524864
$364$	−11.7603	−0.616407
$365$	−7.31009	−0.382628
$366$	−0.0561655	−0.00293582
$367$	17.0313	0.889025	0.444513	−	0.895773i	$-0.353377\pi$
$367$	17.0313	0.889025	0.444513	+	0.895773i	$0.353377\pi$
$368$	−31.9449	−1.66525
$369$	−11.9366	−0.621397
$370$	−0.133211	−0.00692532
$371$	28.7140	1.49076
$372$	13.9327	0.722379
$373$	26.7579	1.38547	0.692735	−	0.721192i	$-0.256406\pi$
$373$	26.7579	1.38547	0.692735	+	0.721192i	$0.256406\pi$
$374$	0.365766	0.0189133
$375$	−5.05780	−0.261184
$376$	−2.33467	−0.120401
$377$	5.43307	0.279818
$378$	−0.198819	−0.0102261
$379$	7.77600	0.399426	0.199713	−	0.979854i	$-0.435999\pi$
$379$	7.77600	0.399426	0.199713	+	0.979854i	$0.435999\pi$
$380$	−24.4815	−1.25588
$381$	15.3492	0.786365
$382$	1.18540	0.0606506
$383$	−28.3894	−1.45063	−0.725316	−	0.688416i	$-0.758306\pi$
$383$	−28.3894	−1.45063	−0.725316	+	0.688416i	$0.758306\pi$
$384$	1.79022	0.0913566
$385$	10.1600	0.517801
$386$	−0.861857	−0.0438673
$387$	−2.21299	−0.112492
$388$	−22.3279	−1.13353
$389$	32.7737	1.66169	0.830846	−	0.556503i	$-0.187857\pi$
$389$	32.7737	1.66169	0.830846	+	0.556503i	$0.187857\pi$
$390$	0.268202	0.0135809
$391$	52.2557	2.64269
$392$	−1.24156	−0.0627084
$393$	−17.1289	−0.864040
$394$	0.450413	0.0226915
$395$	19.5744	0.984893
$396$	1.99685	0.100345
$397$	23.4981	1.17934	0.589668	−	0.807646i	$-0.299259\pi$
$397$	23.4981	1.17934	0.589668	+	0.807646i	$0.299259\pi$
$398$	1.06397	0.0533321
$399$	15.1209	0.756990
$400$	12.8899	0.644497
$401$	−5.67692	−0.283492	−0.141746	−	0.989903i	$-0.545272\pi$
$401$	−5.67692	−0.283492	−0.141746	+	0.989903i	$0.545272\pi$
$402$	−0.562985	−0.0280791
$403$	11.6086	0.578263
$404$	−21.7533	−1.08227
$405$	−2.87016	−0.142619
$406$	0.649258	0.0322221
$407$	−0.826353	−0.0409608
$408$	−1.46191	−0.0723753
$409$	15.8544	0.783950	0.391975	−	0.919976i	$-0.371792\pi$
$409$	15.8544	0.783950	0.391975	+	0.919976i	$0.371792\pi$
$410$	1.92423	0.0950311
$411$	1.15929	0.0571837
$412$	2.22898	0.109814
$413$	7.09687	0.349214
$414$	−0.450683	−0.0221498
$415$	−1.61587	−0.0793201
$416$	1.11898	0.0548626
$417$	−7.66907	−0.375556
$418$	0.239916	0.0117347
$419$	9.40019	0.459229	0.229615	−	0.973282i	$-0.426253\pi$
$419$	9.40019	0.459229	0.229615	+	0.973282i	$0.426253\pi$
$420$	−20.2879	−0.989950
$421$	19.3416	0.942650	0.471325	−	0.881960i	$-0.343776\pi$
$421$	19.3416	0.942650	0.471325	+	0.881960i	$0.343776\pi$
$422$	−0.802538	−0.0390669
$423$	10.4001	0.505671
$424$	−1.82093	−0.0884321
$425$	−21.0854	−1.02279
$426$	0.119804	0.00580452
$427$	3.53988	0.171307
$428$	5.59160	0.270280
$429$	1.66374	0.0803262
$430$	0.356742	0.0172036
$431$	22.0619	1.06268	0.531342	−	0.847158i	$-0.321688\pi$
$431$	22.0619	1.06268	0.531342	+	0.847158i	$0.321688\pi$
$432$	−3.98108	−0.191540
$433$	16.4389	0.790005	0.395002	−	0.918680i	$-0.370744\pi$
$433$	16.4389	0.790005	0.395002	+	0.918680i	$0.370744\pi$
$434$	1.38723	0.0665894
$435$	9.37271	0.449387
$436$	−21.3509	−1.02252
$437$	34.2760	1.63964
$438$	−0.143050	−0.00683518
$439$	29.1315	1.39037	0.695185	−	0.718831i	$-0.255322\pi$
$439$	29.1315	1.39037	0.695185	+	0.718831i	$0.255322\pi$
$440$	−0.644307	−0.0307161
$441$	5.53072	0.263368
$442$	−0.608540	−0.0289453
$443$	−39.8835	−1.89492	−0.947461	−	0.319871i	$-0.896360\pi$
$443$	−39.8835	−1.89492	−0.947461	+	0.319871i	$0.896360\pi$
$444$	1.65010	0.0783102
$445$	−20.7585	−0.984046
$446$	−0.159845	−0.00756888
$447$	1.19140	0.0563512
$448$	−28.0514	−1.32530
$449$	−6.39145	−0.301631	−0.150816	−	0.988562i	$-0.548190\pi$
$449$	−6.39145	−0.301631	−0.150816	+	0.988562i	$0.548190\pi$
$450$	0.181853	0.00857261
$451$	11.9366	0.562074
$452$	−27.8446	−1.30970
$453$	−11.2781	−0.529891
$454$	−1.09429	−0.0513576
$455$	−16.9036	−0.792453
$456$	−0.958906	−0.0449049
$457$	−6.55699	−0.306723	−0.153362	−	0.988170i	$-0.549010\pi$
$457$	−6.55699	−0.306723	−0.153362	+	0.988170i	$0.549010\pi$
$458$	−0.658038	−0.0307481
$459$	6.51228	0.303967
$460$	−45.9887	−2.14423
$461$	−7.90935	−0.368375	−0.184188	−	0.982891i	$-0.558965\pi$
$461$	−7.90935	−0.368375	−0.184188	+	0.982891i	$0.558965\pi$
$462$	0.198819	0.00924990
$463$	29.9602	1.39237	0.696184	−	0.717864i	$-0.254880\pi$
$463$	29.9602	1.39237	0.696184	+	0.717864i	$0.254880\pi$
$464$	13.0005	0.603534
$465$	20.0262	0.928691
$466$	−1.56142	−0.0723315
$467$	11.4395	0.529355	0.264678	−	0.964337i	$-0.414734\pi$
$467$	11.4395	0.529355	0.264678	+	0.964337i	$0.414734\pi$
$468$	−3.32224	−0.153570
$469$	35.4826	1.63843
$470$	−1.67654	−0.0773329
$471$	−17.0614	−0.786149
$472$	−0.450055	−0.0207155
$473$	2.21299	0.101753
$474$	0.383047	0.0175939
$475$	−13.8305	−0.634587
$476$	46.0326	2.10990
$477$	8.11158	0.371404
$478$	0.0716499	0.00327719
$479$	6.05533	0.276675	0.138338	−	0.990385i	$-0.455824\pi$
$479$	6.05533	0.276675	0.138338	+	0.990385i	$0.455824\pi$
$480$	1.93038	0.0881094
$481$	1.37484	0.0626872
$482$	0.842172	0.0383599
$483$	28.4046	1.29245
$484$	−1.99685	−0.0907657
$485$	−32.0930	−1.45727
$486$	−0.0561655	−0.00254772
$487$	5.74022	0.260114	0.130057	−	0.991507i	$-0.458484\pi$
$487$	5.74022	0.260114	0.130057	+	0.991507i	$0.458484\pi$
$488$	−0.224485	−0.0101620
$489$	18.8321	0.851616
$490$	−0.891573	−0.0402772
$491$	5.78303	0.260984	0.130492	−	0.991449i	$-0.458344\pi$
$491$	5.78303	0.260984	0.130492	+	0.991449i	$0.458344\pi$
$492$	−23.8356	−1.07459
$493$	−21.2663	−0.957788
$494$	−0.399158	−0.0179590
$495$	2.87016	0.129004
$496$	27.7775	1.24725
$497$	−7.55073	−0.338697
$498$	−0.0316207	−0.00141696
$499$	−9.90735	−0.443514	−0.221757	−	0.975102i	$-0.571179\pi$
$499$	−9.90735	−0.443514	−0.221757	+	0.975102i	$0.571179\pi$
$500$	−10.0996	−0.451670
$501$	−20.7776	−0.928277
$502$	−1.44088	−0.0643095
$503$	33.8356	1.50865	0.754327	−	0.656499i	$-0.227963\pi$
$503$	33.8356	1.50865	0.754327	+	0.656499i	$0.227963\pi$
$504$	−0.794649	−0.0353965
$505$	−31.2670	−1.39136
$506$	0.450683	0.0200353
$507$	10.2320	0.454417
$508$	30.6501	1.35988
$509$	−18.3516	−0.813422	−0.406711	−	0.913557i	$-0.633324\pi$
$509$	−18.3516	−0.813422	−0.406711	+	0.913557i	$0.633324\pi$
$510$	−1.04980	−0.0464861
$511$	9.01582	0.398836
$512$	4.46494	0.197324
$513$	4.27158	0.188595
$514$	−1.32123	−0.0582770
$515$	3.20382	0.141177
$516$	−4.41899	−0.194535
$517$	−10.4001	−0.457396
$518$	0.164295	0.00721869
$519$	−8.58089	−0.376659
$520$	1.07196	0.0470086
$521$	14.9139	0.653389	0.326694	−	0.945130i	$-0.394065\pi$
$521$	14.9139	0.653389	0.326694	+	0.945130i	$0.394065\pi$
$522$	0.183413	0.00802776
$523$	12.6492	0.553112	0.276556	−	0.960998i	$-0.410807\pi$
$523$	12.6492	0.553112	0.276556	+	0.960998i	$0.410807\pi$
$524$	−34.2038	−1.49420
$525$	−11.4614	−0.500216
$526$	−0.424199	−0.0184960
$527$	−45.4386	−1.97934
$528$	3.98108	0.173254
$529$	41.3875	1.79946
$530$	−1.30762	−0.0567993
$531$	2.00484	0.0870024
$532$	30.1940	1.30908
$533$	−19.8595	−0.860210
$534$	−0.406218	−0.0175788
$535$	8.03706	0.347472
$536$	−2.25016	−0.0971923
$537$	−1.62970	−0.0703269
$538$	1.42318	0.0613577
$539$	−5.53072	−0.238225
$540$	−5.73126	−0.246634
$541$	16.0561	0.690305	0.345153	−	0.938547i	$-0.387827\pi$
$541$	16.0561	0.690305	0.345153	+	0.938547i	$0.387827\pi$
$542$	−1.35075	−0.0580196
$543$	6.80075	0.291848
$544$	−4.37996	−0.187789
$545$	−30.6886	−1.31456
$546$	−0.330783	−0.0141562
$547$	7.91664	0.338491	0.169246	−	0.985574i	$-0.445867\pi$
$547$	7.91664	0.338491	0.169246	+	0.985574i	$0.445867\pi$
$548$	2.31493	0.0988889
$549$	1.00000	0.0426790
$550$	−0.181853	−0.00775422
$551$	−13.9492	−0.594254
$552$	−1.80131	−0.0766688
$553$	−24.1418	−1.02662
$554$	1.45622	0.0618691
$555$	2.37176	0.100676
$556$	−15.3140	−0.649456
$557$	−30.0118	−1.27164	−0.635821	−	0.771837i	$-0.719338\pi$
$557$	−30.0118	−1.27164	−0.635821	+	0.771837i	$0.719338\pi$
$558$	0.391888	0.0165899
$559$	−3.68184	−0.155725
$560$	−40.4478	−1.70923
$561$	−6.51228	−0.274949
$562$	−0.948080	−0.0399923
$563$	−0.239936	−0.0101121	−0.00505604	−	0.999987i	$-0.501609\pi$
$563$	−0.239936	−0.0101121	−0.00505604	+	0.999987i	$0.501609\pi$
$564$	20.7674	0.874466
$565$	−40.0223	−1.68375
$566$	0.799524	0.0336065
$567$	3.53988	0.148661
$568$	0.478838	0.0200916
$569$	−7.76917	−0.325701	−0.162850	−	0.986651i	$-0.552069\pi$
$569$	−7.76917	−0.325701	−0.162850	+	0.986651i	$0.552069\pi$
$570$	−0.688595	−0.0288421
$571$	−46.4165	−1.94247	−0.971235	−	0.238124i	$-0.923468\pi$
$571$	−46.4165	−1.94247	−0.971235	+	0.238124i	$0.923468\pi$
$572$	3.32224	0.138910
$573$	−21.1056	−0.881697
$574$	−2.37323	−0.0990567
$575$	−25.9807	−1.08347
$576$	−7.92439	−0.330183
$577$	41.0776	1.71008	0.855042	−	0.518559i	$-0.173531\pi$
$577$	41.0776	1.71008	0.855042	+	0.518559i	$0.173531\pi$
$578$	1.42715	0.0593618
$579$	15.3449	0.637714
$580$	18.7159	0.777133
$581$	1.99292	0.0826802
$582$	−0.628021	−0.0260323
$583$	−8.11158	−0.335947
$584$	−0.571748	−0.0236591
$585$	−4.77520	−0.197430
$586$	1.41404	0.0584133
$587$	32.5698	1.34430	0.672150	−	0.740415i	$-0.265371\pi$
$587$	32.5698	1.34430	0.672150	+	0.740415i	$0.265371\pi$
$588$	11.0440	0.455446
$589$	−29.8044	−1.22807
$590$	−0.323187	−0.0133054
$591$	−8.01939	−0.329874
$592$	3.28978	0.135209
$593$	−37.9151	−1.55699	−0.778493	−	0.627653i	$-0.784016\pi$
$593$	−37.9151	−1.55699	−0.778493	+	0.627653i	$0.784016\pi$
$594$	0.0561655	0.00230450
$595$	66.1647	2.71249
$596$	2.37904	0.0974491
$597$	−18.9435	−0.775307
$598$	−0.749820	−0.0306624
$599$	−6.61867	−0.270432	−0.135216	−	0.990816i	$-0.543173\pi$
$599$	−6.61867	−0.270432	−0.135216	+	0.990816i	$0.543173\pi$
$600$	0.726836	0.0296730
$601$	28.1603	1.14868	0.574341	−	0.818616i	$-0.305258\pi$
$601$	28.1603	1.14868	0.574341	+	0.818616i	$0.305258\pi$
$602$	−0.439984	−0.0179324
$603$	10.0237	0.408196
$604$	−22.5206	−0.916351
$605$	−2.87016	−0.116688
$606$	−0.611859	−0.0248551
$607$	−43.2487	−1.75541	−0.877705	−	0.479201i	$-0.840927\pi$
$607$	−43.2487	−1.75541	−0.877705	+	0.479201i	$0.840927\pi$
$608$	−2.87293	−0.116513
$609$	−11.5597	−0.468424
$610$	−0.161204	−0.00652695
$611$	17.3031	0.700009
$612$	13.0040	0.525656
$613$	−28.6880	−1.15870	−0.579349	−	0.815079i	$-0.696693\pi$
$613$	−28.6880	−1.15870	−0.579349	+	0.815079i	$0.696693\pi$
$614$	1.13661	0.0458700
$615$	−34.2600	−1.38150
$616$	0.794649	0.0320173
$617$	−0.710736	−0.0286132	−0.0143066	−	0.999898i	$-0.504554\pi$
$617$	−0.710736	−0.0286132	−0.0143066	+	0.999898i	$0.504554\pi$
$618$	0.0626949	0.00252196
$619$	−47.7932	−1.92097	−0.960486	−	0.278329i	$-0.910220\pi$
$619$	−47.7932	−1.92097	−0.960486	+	0.278329i	$0.910220\pi$
$620$	39.9891	1.60600
$621$	8.02418	0.321999
$622$	1.15189	0.0461867
$623$	25.6022	1.02573
$624$	−6.62349	−0.265152
$625$	−30.7057	−1.22823
$626$	−0.593861	−0.0237355
$627$	−4.27158	−0.170591
$628$	−34.0690	−1.35950
$629$	−5.38144	−0.214572
$630$	−0.570642	−0.0227349
$631$	−20.5702	−0.818889	−0.409444	−	0.912335i	$-0.634277\pi$
$631$	−20.5702	−0.818889	−0.409444	+	0.912335i	$0.634277\pi$
$632$	1.53098	0.0608991
$633$	14.2888	0.567929
$634$	1.14486	0.0454680
$635$	44.0547	1.74826
$636$	16.1976	0.642276
$637$	9.20169	0.364584
$638$	−0.183413	−0.00726138
$639$	−2.13305	−0.0843822
$640$	5.13820	0.203105
$641$	−13.0911	−0.517069	−0.258535	−	0.966002i	$-0.583240\pi$
$641$	−13.0911	−0.517069	−0.258535	+	0.966002i	$0.583240\pi$
$642$	0.157276	0.00620717
$643$	23.2585	0.917226	0.458613	−	0.888636i	$-0.348346\pi$
$643$	23.2585	0.917226	0.458613	+	0.888636i	$0.348346\pi$
$644$	56.7196	2.23507
$645$	−6.35162	−0.250095
$646$	1.56240	0.0614717
$647$	30.4427	1.19683	0.598413	−	0.801188i	$-0.295798\pi$
$647$	30.4427	1.19683	0.598413	+	0.801188i	$0.295798\pi$
$648$	−0.224485	−0.00881860
$649$	−2.00484	−0.0786967
$650$	0.302556	0.0118672
$651$	−24.6990	−0.968032
$652$	37.6047	1.47271
$653$	−28.5204	−1.11609	−0.558045	−	0.829811i	$-0.688448\pi$
$653$	−28.5204	−1.11609	−0.558045	+	0.829811i	$0.688448\pi$
$654$	−0.600540	−0.0234830
$655$	−49.1627	−1.92094
$656$	−47.5207	−1.85537
$657$	2.54693	0.0993653
$658$	2.06774	0.0806089
$659$	9.78396	0.381129	0.190564	−	0.981675i	$-0.438968\pi$
$659$	9.78396	0.381129	0.190564	+	0.981675i	$0.438968\pi$
$660$	5.73126	0.223089
$661$	−26.0822	−1.01448	−0.507239	−	0.861805i	$-0.669334\pi$
$661$	−26.0822	−1.01448	−0.507239	+	0.861805i	$0.669334\pi$
$662$	0.671711	0.0261068
$663$	10.8348	0.420787
$664$	−0.126383	−0.00490462
$665$	43.3992	1.68295
$666$	0.0464125	0.00179845
$667$	−26.2036	−1.01461
$668$	−41.4897	−1.60529
$669$	2.84596	0.110031
$670$	−1.61586	−0.0624259
$671$	−1.00000	−0.0386046
$672$	−2.38081	−0.0918418
$673$	−6.37530	−0.245750	−0.122875	−	0.992422i	$-0.539211\pi$
$673$	−6.37530	−0.245750	−0.122875	+	0.992422i	$0.539211\pi$
$674$	−0.990972	−0.0381708
$675$	−3.23780	−0.124623
$676$	20.4316	0.785833
$677$	−22.2919	−0.856747	−0.428374	−	0.903602i	$-0.640913\pi$
$677$	−22.2919	−0.856747	−0.428374	+	0.903602i	$0.640913\pi$
$678$	−0.783189	−0.0300782
$679$	39.5815	1.51900
$680$	−4.19591	−0.160906
$681$	19.4833	0.746602
$682$	−0.391888	−0.0150062
$683$	24.9233	0.953662	0.476831	−	0.878995i	$-0.341785\pi$
$683$	24.9233	0.953662	0.476831	+	0.878995i	$0.341785\pi$
$684$	8.52969	0.326141
$685$	3.32735	0.127132
$686$	−0.292121	−0.0111532
$687$	11.7161	0.446995
$688$	−8.81008	−0.335881
$689$	13.4956	0.514141
$690$	−1.29353	−0.0492438
$691$	−0.490891	−0.0186744	−0.00933718	−	0.999956i	$-0.502972\pi$
$691$	−0.490891	−0.0186744	−0.00933718	+	0.999956i	$0.502972\pi$
$692$	−17.1347	−0.651364
$693$	−3.53988	−0.134469
$694$	0.0364227	0.00138259
$695$	−22.0114	−0.834941
$696$	0.733072	0.0277870
$697$	77.7347	2.94441
$698$	−1.31080	−0.0496145
$699$	27.8004	1.05151
$700$	−22.8866	−0.865033
$701$	−12.9149	−0.487791	−0.243895	−	0.969802i	$-0.578425\pi$
$701$	−12.9149	−0.487791	−0.243895	+	0.969802i	$0.578425\pi$
$702$	−0.0934449	−0.00352685
$703$	−3.52983	−0.133130
$704$	7.92439	0.298662
$705$	29.8499	1.12421
$706$	−2.00735	−0.0755474
$707$	38.5628	1.45031
$708$	4.00335	0.150455
$709$	−21.6649	−0.813642	−0.406821	−	0.913508i	$-0.633363\pi$
$709$	−21.6649	−0.813642	−0.406821	+	0.913508i	$0.633363\pi$
$710$	0.343856	0.0129047
$711$	−6.81997	−0.255769
$712$	−1.62359	−0.0608467
$713$	−55.9877	−2.09676
$714$	1.29476	0.0484554
$715$	4.77520	0.178582
$716$	−3.25427	−0.121618
$717$	−1.27569	−0.0476416
$718$	0.614854	0.0229462
$719$	3.05414	0.113900	0.0569502	−	0.998377i	$-0.481862\pi$
$719$	3.05414	0.113900	0.0569502	+	0.998377i	$0.481862\pi$
$720$	−11.4263	−0.425834
$721$	−3.95140	−0.147158
$722$	−0.0423262	−0.00157522
$723$	−14.9945	−0.557650
$724$	13.5800	0.504698
$725$	10.5733	0.392681
$726$	−0.0561655	−0.00208450
$727$	−11.0808	−0.410964	−0.205482	−	0.978661i	$-0.565876\pi$
$727$	−11.0808	−0.410964	−0.205482	+	0.978661i	$0.565876\pi$
$728$	−1.32209	−0.0489999
$729$	1.00000	0.0370370
$730$	−0.410575	−0.0151961
$731$	14.4116	0.533032
$732$	1.99685	0.0738055
$733$	−10.7441	−0.396843	−0.198421	−	0.980117i	$-0.563581\pi$
$733$	−10.7441	−0.396843	−0.198421	+	0.980117i	$0.563581\pi$
$734$	0.956571	0.0353077
$735$	15.8740	0.585522
$736$	−5.39682	−0.198929
$737$	−10.0237	−0.369227
$738$	−0.670428	−0.0246788
$739$	−25.1340	−0.924569	−0.462284	−	0.886732i	$-0.652970\pi$
$739$	−25.1340	−0.924569	−0.462284	+	0.886732i	$0.652970\pi$
$740$	4.73604	0.174100
$741$	7.10681	0.261075
$742$	1.61274	0.0592054
$743$	−25.0852	−0.920285	−0.460143	−	0.887845i	$-0.652202\pi$
$743$	−25.0852	−0.920285	−0.460143	+	0.887845i	$0.652202\pi$
$744$	1.56632	0.0574239
$745$	3.41950	0.125281
$746$	1.50287	0.0550240
$747$	0.562991	0.0205988
$748$	−13.0040	−0.475474
$749$	−9.91242	−0.362192
$750$	−0.284074	−0.0103729
$751$	40.8602	1.49101	0.745505	−	0.666500i	$-0.232209\pi$
$751$	40.8602	1.49101	0.745505	+	0.666500i	$0.232209\pi$
$752$	41.4037	1.50984
$753$	25.6541	0.934888
$754$	0.305151	0.0111130
$755$	−32.3699	−1.17806
$756$	7.06858	0.257082
$757$	−23.1061	−0.839807	−0.419903	−	0.907569i	$-0.637936\pi$
$757$	−23.1061	−0.839807	−0.419903	+	0.907569i	$0.637936\pi$
$758$	0.436743	0.0158632
$759$	−8.02418	−0.291259
$760$	−2.75221	−0.0998331
$761$	−24.9673	−0.905066	−0.452533	−	0.891748i	$-0.649479\pi$
$761$	−24.9673	−0.905066	−0.452533	+	0.891748i	$0.649479\pi$
$762$	0.862098	0.0312305
$763$	37.8495	1.37024
$764$	−42.1445	−1.52474
$765$	18.6913	0.675784
$766$	−1.59451	−0.0576119
$767$	3.33553	0.120439
$768$	−15.7482	−0.568265
$769$	−39.7456	−1.43326	−0.716632	−	0.697452i	$-0.754317\pi$
$769$	−39.7456	−1.43326	−0.716632	+	0.697452i	$0.754317\pi$
$770$	0.570642	0.0205645
$771$	23.5239	0.847192
$772$	30.6415	1.10281
$773$	47.6555	1.71405	0.857025	−	0.515276i	$-0.172310\pi$
$773$	47.6555	1.71405	0.857025	+	0.515276i	$0.172310\pi$
$774$	−0.124294	−0.00446764
$775$	22.5913	0.811504
$776$	−2.51010	−0.0901074
$777$	−2.92519	−0.104940
$778$	1.84075	0.0659941
$779$	50.9883	1.82685
$780$	−9.53533	−0.341420
$781$	2.13305	0.0763265
$782$	2.93497	0.104954
$783$	−3.26557	−0.116702
$784$	22.0182	0.786366
$785$	−48.9690	−1.74778
$786$	−0.962055	−0.0343154
$787$	25.6637	0.914811	0.457405	−	0.889258i	$-0.348779\pi$
$787$	25.6637	0.914811	0.457405	+	0.889258i	$0.348779\pi$
$788$	−16.0135	−0.570457
$789$	7.55266	0.268882
$790$	1.09940	0.0391151
$791$	49.3611	1.75508
$792$	0.224485	0.00797673
$793$	1.66374	0.0590812
$794$	1.31978	0.0468374
$795$	23.2815	0.825710
$796$	−37.8273	−1.34075
$797$	8.44244	0.299047	0.149523	−	0.988758i	$-0.452226\pi$
$797$	8.44244	0.299047	0.149523	+	0.988758i	$0.452226\pi$
$798$	0.849271	0.0300639
$799$	−67.7284	−2.39606
$800$	2.17764	0.0769913
$801$	7.23252	0.255549
$802$	−0.318848	−0.0112589
$803$	−2.54693	−0.0898793
$804$	20.0157	0.705900
$805$	81.5257	2.87340
$806$	0.652000	0.0229657
$807$	−25.3390	−0.891976
$808$	−2.44550	−0.0860325
$809$	14.1868	0.498780	0.249390	−	0.968403i	$-0.419770\pi$
$809$	14.1868	0.498780	0.249390	+	0.968403i	$0.419770\pi$
$810$	−0.161204	−0.00566413
$811$	−2.52616	−0.0887054	−0.0443527	−	0.999016i	$-0.514123\pi$
$811$	−2.52616	−0.0887054	−0.0443527	+	0.999016i	$0.514123\pi$
$812$	−23.0830	−0.810054
$813$	24.0494	0.843450
$814$	−0.0464125	−0.00162676
$815$	54.0510	1.89332
$816$	25.9259	0.907589
$817$	9.45295	0.330717
$818$	0.890471	0.0311346
$819$	5.88944	0.205794
$820$	−68.4120	−2.38905
$821$	14.9783	0.522746	0.261373	−	0.965238i	$-0.415825\pi$
$821$	14.9783	0.522746	0.261373	+	0.965238i	$0.415825\pi$
$822$	0.0651123	0.00227105
$823$	1.14435	0.0398894	0.0199447	−	0.999801i	$-0.493651\pi$
$823$	1.14435	0.0398894	0.0199447	+	0.999801i	$0.493651\pi$
$824$	0.250582	0.00872944
$825$	3.23780	0.112726
$826$	0.398599	0.0138690
$827$	40.5209	1.40905	0.704525	−	0.709679i	$-0.251160\pi$
$827$	40.5209	1.40905	0.704525	+	0.709679i	$0.251160\pi$
$828$	16.0231	0.556840
$829$	−41.9288	−1.45625	−0.728124	−	0.685446i	$-0.759607\pi$
$829$	−41.9288	−1.45625	−0.728124	+	0.685446i	$0.759607\pi$
$830$	−0.0907564	−0.00315020
$831$	−25.9274	−0.899411
$832$	−13.1841	−0.457078
$833$	−36.0176	−1.24794
$834$	−0.430738	−0.0149152
$835$	−59.6351	−2.06376
$836$	−8.52969	−0.295005
$837$	−6.97737	−0.241173
$838$	0.527967	0.0182383
$839$	−17.9705	−0.620412	−0.310206	−	0.950669i	$-0.600398\pi$
$839$	−17.9705	−0.620412	−0.310206	+	0.950669i	$0.600398\pi$
$840$	−2.28077	−0.0786939
$841$	−18.3360	−0.632277
$842$	1.08633	0.0374374
$843$	16.8801	0.581382
$844$	28.5325	0.982130
$845$	29.3673	1.01027
$846$	0.584128	0.0200827
$847$	3.53988	0.121632
$848$	32.2929	1.10894
$849$	−14.2351	−0.488548
$850$	−1.18427	−0.0406203
$851$	−6.63081	−0.227301
$852$	−4.25937	−0.145924
$853$	−29.2979	−1.00314	−0.501571	−	0.865117i	$-0.667244\pi$
$853$	−29.2979	−1.00314	−0.501571	+	0.865117i	$0.667244\pi$
$854$	0.198819	0.00680345
$855$	12.2601	0.419287
$856$	0.628606	0.0214853
$857$	13.4529	0.459543	0.229772	−	0.973245i	$-0.426202\pi$
$857$	13.4529	0.459543	0.229772	+	0.973245i	$0.426202\pi$
$858$	0.0934449	0.00319016
$859$	11.4298	0.389981	0.194990	−	0.980805i	$-0.437532\pi$
$859$	11.4298	0.389981	0.194990	+	0.980805i	$0.437532\pi$
$860$	−12.6832	−0.432493
$861$	42.2542	1.44002
$862$	1.23912	0.0422045
$863$	22.5690	0.768259	0.384129	−	0.923279i	$-0.374502\pi$
$863$	22.5690	0.768259	0.384129	+	0.923279i	$0.374502\pi$
$864$	−0.672570	−0.0228813
$865$	−24.6285	−0.837394
$866$	0.923301	0.0313751
$867$	−25.4098	−0.862962
$868$	−49.3202	−1.67404
$869$	6.81997	0.231352
$870$	0.526423	0.0178474
$871$	16.6768	0.565072
$872$	−2.40027	−0.0812833
$873$	11.1816	0.378440
$874$	1.92513	0.0651184
$875$	17.9040	0.605265
$876$	5.08583	0.171834
$877$	42.1546	1.42346	0.711729	−	0.702454i	$-0.247912\pi$
$877$	42.1546	1.42346	0.711729	+	0.702454i	$0.247912\pi$
$878$	1.63619	0.0552186
$879$	−25.1762	−0.849173
$880$	11.4263	0.385181
$881$	4.79100	0.161413	0.0807065	−	0.996738i	$-0.474282\pi$
$881$	4.79100	0.161413	0.0807065	+	0.996738i	$0.474282\pi$
$882$	0.310636	0.0104596
$883$	35.8055	1.20495	0.602476	−	0.798137i	$-0.294181\pi$
$883$	35.8055	1.20495	0.602476	+	0.798137i	$0.294181\pi$
$884$	21.6353	0.727675
$885$	5.75419	0.193425
$886$	−2.24008	−0.0752569
$887$	17.9056	0.601210	0.300605	−	0.953749i	$-0.402811\pi$
$887$	17.9056	0.601210	0.300605	+	0.953749i	$0.402811\pi$
$888$	0.185504	0.00622510
$889$	−54.3344	−1.82232
$890$	−1.16591	−0.0390814
$891$	−1.00000	−0.0335013
$892$	5.68295	0.190279
$893$	−44.4249	−1.48662
$894$	0.0669155	0.00223799
$895$	−4.67751	−0.156352
$896$	−6.33714	−0.211709
$897$	13.3502	0.445749
$898$	−0.358979	−0.0119793
$899$	22.7851	0.759927
$900$	−6.46538	−0.215513
$901$	−52.8249	−1.75985
$902$	0.670428	0.0223228
$903$	7.83369	0.260689
$904$	−3.13029	−0.104112
$905$	19.5192	0.648841
$906$	−0.633440	−0.0210446
$907$	16.6708	0.553544	0.276772	−	0.960936i	$-0.410735\pi$
$907$	16.6708	0.553544	0.276772	+	0.960936i	$0.410735\pi$
$908$	38.9052	1.29111
$909$	10.8938	0.361326
$910$	−0.949400	−0.0314723
$911$	−36.4248	−1.20681	−0.603403	−	0.797436i	$-0.706189\pi$
$911$	−36.4248	−1.20681	−0.603403	+	0.797436i	$0.706189\pi$
$912$	17.0055	0.563109
$913$	−0.562991	−0.0186323
$914$	−0.368277	−0.0121815
$915$	2.87016	0.0948844
$916$	23.3952	0.772998
$917$	60.6342	2.00232
$918$	0.365766	0.0120721
$919$	4.21597	0.139072	0.0695359	−	0.997579i	$-0.477848\pi$
$919$	4.21597	0.139072	0.0695359	+	0.997579i	$0.477848\pi$
$920$	−5.17004	−0.170451
$921$	−20.2369	−0.666827
$922$	−0.444233	−0.0146300
$923$	−3.54884	−0.116812
$924$	−7.06858	−0.232539
$925$	2.67556	0.0879719
$926$	1.68273	0.0552979
$927$	−1.11625	−0.0366626
$928$	2.19633	0.0720979
$929$	28.4286	0.932713	0.466356	−	0.884597i	$-0.345566\pi$
$929$	28.4286	0.932713	0.466356	+	0.884597i	$0.345566\pi$
$930$	1.12478	0.0368830
$931$	−23.6249	−0.774275
$932$	55.5131	1.81839
$933$	−20.5089	−0.671431
$934$	0.642503	0.0210234
$935$	−18.6913	−0.611270
$936$	−0.373485	−0.0122077
$937$	−20.5654	−0.671843	−0.335921	−	0.941890i	$-0.609048\pi$
$937$	−20.5654	−0.671843	−0.335921	+	0.941890i	$0.609048\pi$
$938$	1.99290	0.0650704
$939$	10.5734	0.345050
$940$	59.6057	1.94412
$941$	−7.87253	−0.256637	−0.128318	−	0.991733i	$-0.540958\pi$
$941$	−7.87253	−0.256637	−0.128318	+	0.991733i	$0.540958\pi$
$942$	−0.958264	−0.0312219
$943$	95.7818	3.11908
$944$	7.98142	0.259773
$945$	10.1600	0.330505
$946$	0.124294	0.00404113
$947$	51.6536	1.67852	0.839258	−	0.543734i	$-0.182990\pi$
$947$	51.6536	1.67852	0.839258	+	0.543734i	$0.182990\pi$
$948$	−13.6184	−0.442306
$949$	4.23744	0.137553
$950$	−0.776798	−0.0252026
$951$	−20.3836	−0.660984
$952$	5.17498	0.167722
$953$	−50.9942	−1.65186	−0.825932	−	0.563769i	$-0.809351\pi$
$953$	−50.9942	−1.65186	−0.825932	+	0.563769i	$0.809351\pi$
$954$	0.455591	0.0147503
$955$	−60.5762	−1.96020
$956$	−2.54736	−0.0823875
$957$	3.26557	0.105561
$958$	0.340101	0.0109882
$959$	−4.10375	−0.132517
$960$	−22.7442	−0.734067
$961$	17.6838	0.570444
$962$	0.0772185	0.00248962
$963$	−2.80022	−0.0902357
$964$	−29.9416	−0.964355
$965$	44.0424	1.41777
$966$	1.59536	0.0513299
$967$	31.7512	1.02105	0.510524	−	0.859863i	$-0.329451\pi$
$967$	31.7512	1.02105	0.510524	+	0.859863i	$0.329451\pi$
$968$	−0.224485	−0.00721522
$969$	−27.8177	−0.893634
$970$	−1.80252	−0.0578754
$971$	27.2104	0.873222	0.436611	−	0.899650i	$-0.356179\pi$
$971$	27.2104	0.873222	0.436611	+	0.899650i	$0.356179\pi$
$972$	1.99685	0.0640488
$973$	27.1476	0.870311
$974$	0.322402	0.0103304
$975$	−5.38686	−0.172517
$976$	3.98108	0.127431
$977$	49.0474	1.56916	0.784582	−	0.620025i	$-0.212877\pi$
$977$	49.0474	1.56916	0.784582	+	0.620025i	$0.212877\pi$
$978$	1.05771	0.0338219
$979$	−7.23252	−0.231152
$980$	31.6980	1.01255
$981$	10.6923	0.341380
$982$	0.324807	0.0103650
$983$	29.9573	0.955490	0.477745	−	0.878499i	$-0.341454\pi$
$983$	29.9573	0.955490	0.477745	+	0.878499i	$0.341454\pi$
$984$	−2.67960	−0.0854224
$985$	−23.0169	−0.733380
$986$	−1.19444	−0.0380386
$987$	−36.8151	−1.17184
$988$	14.1912	0.451482
$989$	17.7574	0.564653
$990$	0.161204	0.00512339
$991$	−45.8678	−1.45704	−0.728519	−	0.685026i	$-0.759791\pi$
$991$	−45.8678	−1.45704	−0.728519	+	0.685026i	$0.759791\pi$
$992$	4.69277	0.148996
$993$	−11.9595	−0.379523
$994$	−0.424091	−0.0134513
$995$	−54.3709	−1.72367
$996$	1.12421	0.0356219
$997$	−17.7103	−0.560892	−0.280446	−	0.959870i	$-0.590482\pi$
$997$	−17.7103	−0.560892	−0.280446	+	0.959870i	$0.590482\pi$
$998$	−0.556452	−0.0176142
$999$	−0.826353	−0.0261446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.2.a.h.1.8	✓	14
3.2	odd	2		6039.2.a.j.1.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.8	✓	14	1.1	even	1	trivial
6039.2.a.j.1.7		14	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0561655 q^{2} -1.00000 q^{3} -1.99685 q^{4} -2.87016 q^{5} -0.0561655 q^{6} +3.53988 q^{7} -0.224485 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.0561655 q^{2} -1.00000 q^{3} -1.99685 q^{4} -2.87016 q^{5} -0.0561655 q^{6} +3.53988 q^{7} -0.224485 q^{8} +1.00000 q^{9} -0.161204 q^{10} -1.00000 q^{11} +1.99685 q^{12} +1.66374 q^{13} +0.198819 q^{14} +2.87016 q^{15} +3.98108 q^{16} -6.51228 q^{17} +0.0561655 q^{18} -4.27158 q^{19} +5.73126 q^{20} -3.53988 q^{21} -0.0561655 q^{22} -8.02418 q^{23} +0.224485 q^{24} +3.23780 q^{25} +0.0934449 q^{26} -1.00000 q^{27} -7.06858 q^{28} +3.26557 q^{29} +0.161204 q^{30} +6.97737 q^{31} +0.672570 q^{32} +1.00000 q^{33} -0.365766 q^{34} -10.1600 q^{35} -1.99685 q^{36} +0.826353 q^{37} -0.239916 q^{38} -1.66374 q^{39} +0.644307 q^{40} -11.9366 q^{41} -0.198819 q^{42} -2.21299 q^{43} +1.99685 q^{44} -2.87016 q^{45} -0.450683 q^{46} +10.4001 q^{47} -3.98108 q^{48} +5.53072 q^{49} +0.181853 q^{50} +6.51228 q^{51} -3.32224 q^{52} +8.11158 q^{53} -0.0561655 q^{54} +2.87016 q^{55} -0.794649 q^{56} +4.27158 q^{57} +0.183413 q^{58} +2.00484 q^{59} -5.73126 q^{60} +1.00000 q^{61} +0.391888 q^{62} +3.53988 q^{63} -7.92439 q^{64} -4.77520 q^{65} +0.0561655 q^{66} +10.0237 q^{67} +13.0040 q^{68} +8.02418 q^{69} -0.570642 q^{70} -2.13305 q^{71} -0.224485 q^{72} +2.54693 q^{73} +0.0464125 q^{74} -3.23780 q^{75} +8.52969 q^{76} -3.53988 q^{77} -0.0934449 q^{78} -6.81997 q^{79} -11.4263 q^{80} +1.00000 q^{81} -0.670428 q^{82} +0.562991 q^{83} +7.06858 q^{84} +18.6913 q^{85} -0.124294 q^{86} -3.26557 q^{87} +0.224485 q^{88} +7.23252 q^{89} -0.161204 q^{90} +5.88944 q^{91} +16.0231 q^{92} -6.97737 q^{93} +0.584128 q^{94} +12.2601 q^{95} -0.672570 q^{96} +11.1816 q^{97} +0.310636 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - q^2 - 14 * q^3 + 15 * q^4 + q^5 + q^6 + 9 * q^7 + 14 * q^9	\( 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9} + 6 q^{10} - 14 q^{11} - 15 q^{12} + q^{13} - 7 q^{14} - q^{15} + 17 q^{16} - 9 q^{17} - q^{18} + 22 q^{19} + 23 q^{20} - 9 q^{21} + q^{22} + q^{23} + 25 q^{25} + 4 q^{26} - 14 q^{27} + 37 q^{28} - 6 q^{29} - 6 q^{30} + 9 q^{31} + 4 q^{32} + 14 q^{33} + 8 q^{34} + 18 q^{35} + 15 q^{36} + 18 q^{37} + 8 q^{38} - q^{39} + 16 q^{40} - 25 q^{41} + 7 q^{42} + 25 q^{43} - 15 q^{44} + q^{45} + 20 q^{46} + 36 q^{47} - 17 q^{48} + 25 q^{49} + 2 q^{50} + 9 q^{51} - 13 q^{52} + q^{54} - q^{55} - 40 q^{56} - 22 q^{57} + 33 q^{58} + 17 q^{59} - 23 q^{60} + 14 q^{61} - 13 q^{62} + 9 q^{63} - 6 q^{64} - 61 q^{65} - q^{66} + 22 q^{67} + 66 q^{68} - q^{69} + 44 q^{70} - 13 q^{71} + 20 q^{73} - 12 q^{74} - 25 q^{75} + 49 q^{76} - 9 q^{77} - 4 q^{78} + 31 q^{79} + 88 q^{80} + 14 q^{81} + 2 q^{82} + 32 q^{83} - 37 q^{84} + 2 q^{85} - 14 q^{86} + 6 q^{87} - 21 q^{89} + 6 q^{90} + 45 q^{91} - 14 q^{92} - 9 q^{93} - 31 q^{94} + 23 q^{95} - 4 q^{96} + 37 q^{97} - 38 q^{98} - 14 q^{99}+O(q^{100}) \) 14 * q - q^2 - 14 * q^3 + 15 * q^4 + q^5 + q^6 + 9 * q^7 + 14 * q^9 + 6 * q^10 - 14 * q^11 - 15 * q^12 + q^13 - 7 * q^14 - q^15 + 17 * q^16 - 9 * q^17 - q^18 + 22 * q^19 + 23 * q^20 - 9 * q^21 + q^22 + q^23 + 25 * q^25 + 4 * q^26 - 14 * q^27 + 37 * q^28 - 6 * q^29 - 6 * q^30 + 9 * q^31 + 4 * q^32 + 14 * q^33 + 8 * q^34 + 18 * q^35 + 15 * q^36 + 18 * q^37 + 8 * q^38 - q^39 + 16 * q^40 - 25 * q^41 + 7 * q^42 + 25 * q^43 - 15 * q^44 + q^45 + 20 * q^46 + 36 * q^47 - 17 * q^48 + 25 * q^49 + 2 * q^50 + 9 * q^51 - 13 * q^52 + q^54 - q^55 - 40 * q^56 - 22 * q^57 + 33 * q^58 + 17 * q^59 - 23 * q^60 + 14 * q^61 - 13 * q^62 + 9 * q^63 - 6 * q^64 - 61 * q^65 - q^66 + 22 * q^67 + 66 * q^68 - q^69 + 44 * q^70 - 13 * q^71 + 20 * q^73 - 12 * q^74 - 25 * q^75 + 49 * q^76 - 9 * q^77 - 4 * q^78 + 31 * q^79 + 88 * q^80 + 14 * q^81 + 2 * q^82 + 32 * q^83 - 37 * q^84 + 2 * q^85 - 14 * q^86 + 6 * q^87 - 21 * q^89 + 6 * q^90 + 45 * q^91 - 14 * q^92 - 9 * q^93 - 31 * q^94 + 23 * q^95 - 4 * q^96 + 37 * q^97 - 38 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more