LMFDB - Embedded newform 1502.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1502,2,Mod(1,1502)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1502, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1502.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1502 = 2 \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1502.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:	$11.9935303836$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} + \cdots + 4222 $ x^19 - 6x^18 - 23x^17 + 190x^16 + 128x^15 - 2394x^14 + 749x^13 + 15539x^12 - 11338x^11 - 56744x^10 + 50183x^9 + 120237x^8 - 102992x^7 - 148539x^6 + 99519x^5 + 104388x^4 - 41081x^3 - 35733x^2 + 5542x + 4222
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.94953$ of defining polynomial
Character	$\chi$	$=$	1502.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.94953 q^{3} +1.00000 q^{4} +2.00911 q^{5} +1.94953 q^{6} +0.0633867 q^{7} -1.00000 q^{8} +0.800679 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.94953 q^{3} +1.00000 q^{4} +2.00911 q^{5} +1.94953 q^{6} +0.0633867 q^{7} -1.00000 q^{8} +0.800679 q^{9} -2.00911 q^{10} +5.29902 q^{11} -1.94953 q^{12} +5.54482 q^{13} -0.0633867 q^{14} -3.91683 q^{15} +1.00000 q^{16} +3.37027 q^{17} -0.800679 q^{18} -4.39789 q^{19} +2.00911 q^{20} -0.123574 q^{21} -5.29902 q^{22} -4.01286 q^{23} +1.94953 q^{24} -0.963477 q^{25} -5.54482 q^{26} +4.28765 q^{27} +0.0633867 q^{28} +6.55340 q^{29} +3.91683 q^{30} -0.0684522 q^{31} -1.00000 q^{32} -10.3306 q^{33} -3.37027 q^{34} +0.127351 q^{35} +0.800679 q^{36} +2.26303 q^{37} +4.39789 q^{38} -10.8098 q^{39} -2.00911 q^{40} -1.65633 q^{41} +0.123574 q^{42} +8.33738 q^{43} +5.29902 q^{44} +1.60865 q^{45} +4.01286 q^{46} -9.34693 q^{47} -1.94953 q^{48} -6.99598 q^{49} +0.963477 q^{50} -6.57046 q^{51} +5.54482 q^{52} +0.485846 q^{53} -4.28765 q^{54} +10.6463 q^{55} -0.0633867 q^{56} +8.57383 q^{57} -6.55340 q^{58} -5.70498 q^{59} -3.91683 q^{60} -0.634911 q^{61} +0.0684522 q^{62} +0.0507524 q^{63} +1.00000 q^{64} +11.1401 q^{65} +10.3306 q^{66} -3.13919 q^{67} +3.37027 q^{68} +7.82320 q^{69} -0.127351 q^{70} +2.59120 q^{71} -0.800679 q^{72} +11.7757 q^{73} -2.26303 q^{74} +1.87833 q^{75} -4.39789 q^{76} +0.335887 q^{77} +10.8098 q^{78} -15.5510 q^{79} +2.00911 q^{80} -10.7609 q^{81} +1.65633 q^{82} +16.2184 q^{83} -0.123574 q^{84} +6.77125 q^{85} -8.33738 q^{86} -12.7761 q^{87} -5.29902 q^{88} +0.587608 q^{89} -1.60865 q^{90} +0.351468 q^{91} -4.01286 q^{92} +0.133450 q^{93} +9.34693 q^{94} -8.83584 q^{95} +1.94953 q^{96} +4.24165 q^{97} +6.99598 q^{98} +4.24281 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) $ 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.94953	−1.12556	−0.562782	−	0.826606i	$-0.690269\pi$
$3$	−1.94953	−1.12556	−0.562782	+	0.826606i	$0.690269\pi$
$4$	1.00000	0.500000
$5$	2.00911	0.898501	0.449251	−	0.893406i	$-0.351691\pi$
$5$	2.00911	0.898501	0.449251	+	0.893406i	$0.351691\pi$
$6$	1.94953	0.795894
$7$	0.0633867	0.0239579	0.0119790	−	0.999928i	$-0.496187\pi$
$7$	0.0633867	0.0239579	0.0119790	+	0.999928i	$0.496187\pi$
$8$	−1.00000	−0.353553
$9$	0.800679	0.266893
$10$	−2.00911	−0.635336
$11$	5.29902	1.59771	0.798857	−	0.601521i	$-0.205438\pi$
$11$	5.29902	1.59771	0.798857	+	0.601521i	$0.205438\pi$
$12$	−1.94953	−0.562782
$13$	5.54482	1.53786	0.768928	−	0.639336i	$-0.220791\pi$
$13$	5.54482	1.53786	0.768928	+	0.639336i	$0.220791\pi$
$14$	−0.0633867	−0.0169408
$15$	−3.91683	−1.01132
$16$	1.00000	0.250000
$17$	3.37027	0.817411	0.408705	−	0.912666i	$-0.365980\pi$
$17$	3.37027	0.817411	0.408705	+	0.912666i	$0.365980\pi$
$18$	−0.800679	−0.188722
$19$	−4.39789	−1.00895	−0.504473	−	0.863428i	$-0.668313\pi$
$19$	−4.39789	−1.00895	−0.504473	+	0.863428i	$0.668313\pi$
$20$	2.00911	0.449251
$21$	−0.123574	−0.0269662
$22$	−5.29902	−1.12975
$23$	−4.01286	−0.836739	−0.418369	−	0.908277i	$-0.637398\pi$
$23$	−4.01286	−0.836739	−0.418369	+	0.908277i	$0.637398\pi$
$24$	1.94953	0.397947
$25$	−0.963477	−0.192695
$26$	−5.54482	−1.08743
$27$	4.28765	0.825158
$28$	0.0633867	0.0119790
$29$	6.55340	1.21694	0.608468	−	0.793578i	$-0.291784\pi$
$29$	6.55340	1.21694	0.608468	+	0.793578i	$0.291784\pi$
$30$	3.91683	0.715111
$31$	−0.0684522	−0.0122944	−0.00614719	−	0.999981i	$-0.501957\pi$
$31$	−0.0684522	−0.0122944	−0.00614719	+	0.999981i	$0.501957\pi$
$32$	−1.00000	−0.176777
$33$	−10.3306	−1.79833
$34$	−3.37027	−0.577997
$35$	0.127351	0.0215262
$36$	0.800679	0.133446
$37$	2.26303	0.372040	0.186020	−	0.982546i	$-0.440441\pi$
$37$	2.26303	0.372040	0.186020	+	0.982546i	$0.440441\pi$
$38$	4.39789	0.713432
$39$	−10.8098	−1.73095
$40$	−2.00911	−0.317668
$41$	−1.65633	−0.258675	−0.129337	−	0.991601i	$-0.541285\pi$
$41$	−1.65633	−0.258675	−0.129337	+	0.991601i	$0.541285\pi$
$42$	0.123574	0.0190679
$43$	8.33738	1.27144	0.635719	−	0.771920i	$-0.280704\pi$
$43$	8.33738	1.27144	0.635719	+	0.771920i	$0.280704\pi$
$44$	5.29902	0.798857
$45$	1.60865	0.239804
$46$	4.01286	0.591664
$47$	−9.34693	−1.36339	−0.681695	−	0.731637i	$-0.738757\pi$
$47$	−9.34693	−1.36339	−0.681695	+	0.731637i	$0.738757\pi$
$48$	−1.94953	−0.281391
$49$	−6.99598	−0.999426
$50$	0.963477	0.136256
$51$	−6.57046	−0.920048
$52$	5.54482	0.768928
$53$	0.485846	0.0667360	0.0333680	−	0.999443i	$-0.489377\pi$
$53$	0.485846	0.0667360	0.0333680	+	0.999443i	$0.489377\pi$
$54$	−4.28765	−0.583475
$55$	10.6463	1.43555
$56$	−0.0633867	−0.00847040
$57$	8.57383	1.13563
$58$	−6.55340	−0.860504
$59$	−5.70498	−0.742726	−0.371363	−	0.928488i	$-0.621109\pi$
$59$	−5.70498	−0.742726	−0.371363	+	0.928488i	$0.621109\pi$
$60$	−3.91683	−0.505660
$61$	−0.634911	−0.0812920	−0.0406460	−	0.999174i	$-0.512942\pi$
$61$	−0.634911	−0.0812920	−0.0406460	+	0.999174i	$0.512942\pi$
$62$	0.0684522	0.00869344
$63$	0.0507524	0.00639420
$64$	1.00000	0.125000
$65$	11.1401	1.38177
$66$	10.3306	1.27161
$67$	−3.13919	−0.383513	−0.191757	−	0.981443i	$-0.561418\pi$
$67$	−3.13919	−0.383513	−0.191757	+	0.981443i	$0.561418\pi$
$68$	3.37027	0.408705
$69$	7.82320	0.941802
$70$	−0.127351	−0.0152213
$71$	2.59120	0.307519	0.153759	−	0.988108i	$-0.450862\pi$
$71$	2.59120	0.307519	0.153759	+	0.988108i	$0.450862\pi$
$72$	−0.800679	−0.0943609
$73$	11.7757	1.37824	0.689121	−	0.724646i	$-0.257997\pi$
$73$	11.7757	1.37824	0.689121	+	0.724646i	$0.257997\pi$
$74$	−2.26303	−0.263072
$75$	1.87833	0.216891
$76$	−4.39789	−0.504473
$77$	0.335887	0.0382779
$78$	10.8098	1.22397
$79$	−15.5510	−1.74962	−0.874812	−	0.484462i	$-0.839015\pi$
$79$	−15.5510	−1.74962	−0.874812	+	0.484462i	$0.839015\pi$
$80$	2.00911	0.224625
$81$	−10.7609	−1.19566
$82$	1.65633	0.182911
$83$	16.2184	1.78021	0.890103	−	0.455760i	$-0.150633\pi$
$83$	16.2184	1.78021	0.890103	+	0.455760i	$0.150633\pi$
$84$	−0.123574	−0.0134831
$85$	6.77125	0.734445
$86$	−8.33738	−0.899043
$87$	−12.7761	−1.36974
$88$	−5.29902	−0.564877
$89$	0.587608	0.0622863	0.0311431	−	0.999515i	$-0.490085\pi$
$89$	0.587608	0.0622863	0.0311431	+	0.999515i	$0.490085\pi$
$90$	−1.60865	−0.169567
$91$	0.351468	0.0368438
$92$	−4.01286	−0.418369
$93$	0.133450	0.0138381
$94$	9.34693	0.964062
$95$	−8.83584	−0.906539
$96$	1.94953	0.198973
$97$	4.24165	0.430674	0.215337	−	0.976540i	$-0.430915\pi$
$97$	4.24165	0.430674	0.215337	+	0.976540i	$0.430915\pi$
$98$	6.99598	0.706701
$99$	4.24281	0.426419
$100$	−0.963477	−0.0963477
$101$	10.1553	1.01049	0.505246	−	0.862976i	$-0.331402\pi$
$101$	10.1553	1.01049	0.505246	+	0.862976i	$0.331402\pi$
$102$	6.57046	0.650572
$103$	6.03535	0.594681	0.297340	−	0.954772i	$-0.403900\pi$
$103$	6.03535	0.594681	0.297340	+	0.954772i	$0.403900\pi$
$104$	−5.54482	−0.543714
$105$	−0.248275	−0.0242291
$106$	−0.485846	−0.0471895
$107$	6.48560	0.626987	0.313493	−	0.949590i	$-0.398501\pi$
$107$	6.48560	0.626987	0.313493	+	0.949590i	$0.398501\pi$
$108$	4.28765	0.412579
$109$	14.3170	1.37132	0.685660	−	0.727922i	$-0.259514\pi$
$109$	14.3170	1.37132	0.685660	+	0.727922i	$0.259514\pi$
$110$	−10.6463	−1.01509
$111$	−4.41186	−0.418755
$112$	0.0633867	0.00598948
$113$	−5.21323	−0.490419	−0.245210	−	0.969470i	$-0.578857\pi$
$113$	−5.21323	−0.490419	−0.245210	+	0.969470i	$0.578857\pi$
$114$	−8.57383	−0.803013
$115$	−8.06227	−0.751811
$116$	6.55340	0.608468
$117$	4.43962	0.410443
$118$	5.70498	0.525186
$119$	0.213630	0.0195835
$120$	3.91683	0.357556
$121$	17.0796	1.55269
$122$	0.634911	0.0574821
$123$	3.22907	0.291155
$124$	−0.0684522	−0.00614719
$125$	−11.9813	−1.07164
$126$	−0.0507524	−0.00452138
$127$	19.4274	1.72390	0.861951	−	0.506991i	$-0.169242\pi$
$127$	19.4274	1.72390	0.861951	+	0.506991i	$0.169242\pi$
$128$	−1.00000	−0.0883883
$129$	−16.2540	−1.43108
$130$	−11.1401	−0.977055
$131$	−5.96335	−0.521020	−0.260510	−	0.965471i	$-0.583891\pi$
$131$	−5.96335	−0.521020	−0.260510	+	0.965471i	$0.583891\pi$
$132$	−10.3306	−0.899164
$133$	−0.278768	−0.0241722
$134$	3.13919	0.271185
$135$	8.61436	0.741406
$136$	−3.37027	−0.288998
$137$	−3.95926	−0.338262	−0.169131	−	0.985594i	$-0.554096\pi$
$137$	−3.95926	−0.338262	−0.169131	+	0.985594i	$0.554096\pi$
$138$	−7.82320	−0.665955
$139$	3.66226	0.310629	0.155314	−	0.987865i	$-0.450361\pi$
$139$	3.66226	0.310629	0.155314	+	0.987865i	$0.450361\pi$
$140$	0.127351	0.0107631
$141$	18.2221	1.53458
$142$	−2.59120	−0.217449
$143$	29.3821	2.45705
$144$	0.800679	0.0667232
$145$	13.1665	1.09342
$146$	−11.7757	−0.974565
$147$	13.6389	1.12492
$148$	2.26303	0.186020
$149$	12.5991	1.03216	0.516080	−	0.856541i	$-0.327391\pi$
$149$	12.5991	1.03216	0.516080	+	0.856541i	$0.327391\pi$
$150$	−1.87833	−0.153365
$151$	8.28741	0.674420	0.337210	−	0.941430i	$-0.390517\pi$
$151$	8.28741	0.674420	0.337210	+	0.941430i	$0.390517\pi$
$152$	4.39789	0.356716
$153$	2.69851	0.218161
$154$	−0.335887	−0.0270666
$155$	−0.137528	−0.0110465
$156$	−10.8098	−0.865477
$157$	18.2518	1.45665	0.728327	−	0.685230i	$-0.240298\pi$
$157$	18.2518	1.45665	0.728327	+	0.685230i	$0.240298\pi$
$158$	15.5510	1.23717
$159$	−0.947172	−0.0751156
$160$	−2.00911	−0.158834
$161$	−0.254362	−0.0200465
$162$	10.7609	0.845460
$163$	9.58504	0.750759	0.375379	−	0.926871i	$-0.377512\pi$
$163$	9.58504	0.750759	0.375379	+	0.926871i	$0.377512\pi$
$164$	−1.65633	−0.129337
$165$	−20.7553	−1.61580
$166$	−16.2184	−1.25880
$167$	4.38854	0.339596	0.169798	−	0.985479i	$-0.445689\pi$
$167$	4.38854	0.339596	0.169798	+	0.985479i	$0.445689\pi$
$168$	0.123574	0.00953397
$169$	17.7450	1.36500
$170$	−6.77125	−0.519331
$171$	−3.52130	−0.269280
$172$	8.33738	0.635719
$173$	3.03467	0.230721	0.115361	−	0.993324i	$-0.463198\pi$
$173$	3.03467	0.230721	0.115361	+	0.993324i	$0.463198\pi$
$174$	12.7761	0.968551
$175$	−0.0610716	−0.00461658
$176$	5.29902	0.399429
$177$	11.1221	0.835985
$178$	−0.587608	−0.0440430
$179$	−25.1408	−1.87911	−0.939557	−	0.342392i	$-0.888763\pi$
$179$	−25.1408	−1.87911	−0.939557	+	0.342392i	$0.888763\pi$
$180$	1.60865	0.119902
$181$	11.9144	0.885588	0.442794	−	0.896623i	$-0.353987\pi$
$181$	11.9144	0.885588	0.442794	+	0.896623i	$0.353987\pi$
$182$	−0.351468	−0.0260525
$183$	1.23778	0.0914993
$184$	4.01286	0.295832
$185$	4.54668	0.334279
$186$	−0.133450	−0.00978502
$187$	17.8591	1.30599
$188$	−9.34693	−0.681695
$189$	0.271780	0.0197691
$190$	8.83584	0.641020
$191$	−11.3433	−0.820770	−0.410385	−	0.911912i	$-0.634606\pi$
$191$	−11.3433	−0.820770	−0.410385	+	0.911912i	$0.634606\pi$
$192$	−1.94953	−0.140695
$193$	−1.10008	−0.0791855	−0.0395927	−	0.999216i	$-0.512606\pi$
$193$	−1.10008	−0.0791855	−0.0395927	+	0.999216i	$0.512606\pi$
$194$	−4.24165	−0.304532
$195$	−21.7181	−1.55526
$196$	−6.99598	−0.499713
$197$	−20.4151	−1.45452	−0.727259	−	0.686363i	$-0.759206\pi$
$197$	−20.4151	−1.45452	−0.727259	+	0.686363i	$0.759206\pi$
$198$	−4.24281	−0.301524
$199$	−4.28997	−0.304108	−0.152054	−	0.988372i	$-0.548589\pi$
$199$	−4.28997	−0.304108	−0.152054	+	0.988372i	$0.548589\pi$
$200$	0.963477	0.0681281
$201$	6.11996	0.431669
$202$	−10.1553	−0.714525
$203$	0.415398	0.0291552
$204$	−6.57046	−0.460024
$205$	−3.32774	−0.232420
$206$	−6.03535	−0.420503
$207$	−3.21301	−0.223320
$208$	5.54482	0.384464
$209$	−23.3045	−1.61201
$210$	0.248275	0.0171326
$211$	20.1989	1.39055	0.695274	−	0.718745i	$-0.255283\pi$
$211$	20.1989	1.39055	0.695274	+	0.718745i	$0.255283\pi$
$212$	0.485846	0.0333680
$213$	−5.05163	−0.346132
$214$	−6.48560	−0.443346
$215$	16.7507	1.14239
$216$	−4.28765	−0.291738
$217$	−0.00433896	−0.000294548 0
$218$	−14.3170	−0.969670
$219$	−22.9571	−1.55130
$220$	10.6463	0.717774
$221$	18.6875	1.25706
$222$	4.41186	0.296105
$223$	−24.1584	−1.61776	−0.808882	−	0.587972i	$-0.799927\pi$
$223$	−24.1584	−1.61776	−0.808882	+	0.587972i	$0.799927\pi$
$224$	−0.0633867	−0.00423520
$225$	−0.771436	−0.0514290
$226$	5.21323	0.346779
$227$	−3.84036	−0.254894	−0.127447	−	0.991845i	$-0.540678\pi$
$227$	−3.84036	−0.254894	−0.127447	+	0.991845i	$0.540678\pi$
$228$	8.57383	0.567816
$229$	0.856250	0.0565826	0.0282913	−	0.999600i	$-0.490993\pi$
$229$	0.856250	0.0565826	0.0282913	+	0.999600i	$0.490993\pi$
$230$	8.06227	0.531610
$231$	−0.654823	−0.0430842
$232$	−6.55340	−0.430252
$233$	1.19018	0.0779715	0.0389857	−	0.999240i	$-0.487587\pi$
$233$	1.19018	0.0779715	0.0389857	+	0.999240i	$0.487587\pi$
$234$	−4.43962	−0.290227
$235$	−18.7790	−1.22501
$236$	−5.70498	−0.371363
$237$	30.3172	1.96931
$238$	−0.213630	−0.0138476
$239$	−12.0156	−0.777225	−0.388613	−	0.921401i	$-0.627046\pi$
$239$	−12.0156	−0.777225	−0.388613	+	0.921401i	$0.627046\pi$
$240$	−3.91683	−0.252830
$241$	−5.11342	−0.329384	−0.164692	−	0.986345i	$-0.552663\pi$
$241$	−5.11342	−0.329384	−0.164692	+	0.986345i	$0.552663\pi$
$242$	−17.0796	−1.09792
$243$	8.11588	0.520634
$244$	−0.634911	−0.0406460
$245$	−14.0557	−0.897986
$246$	−3.22907	−0.205878
$247$	−24.3855	−1.55161
$248$	0.0684522	0.00434672
$249$	−31.6184	−2.00373
$250$	11.9813	0.757763
$251$	−2.86144	−0.180612	−0.0903061	−	0.995914i	$-0.528785\pi$
$251$	−2.86144	−0.180612	−0.0903061	+	0.995914i	$0.528785\pi$
$252$	0.0507524	0.00319710
$253$	−21.2642	−1.33687
$254$	−19.4274	−1.21898
$255$	−13.2008	−0.826664
$256$	1.00000	0.0625000
$257$	22.7601	1.41974	0.709868	−	0.704334i	$-0.248754\pi$
$257$	22.7601	1.41974	0.709868	+	0.704334i	$0.248754\pi$
$258$	16.2540	1.01193
$259$	0.143446	0.00891331
$260$	11.1401	0.690883
$261$	5.24717	0.324792
$262$	5.96335	0.368417
$263$	15.0405	0.927438	0.463719	−	0.885982i	$-0.346515\pi$
$263$	15.0405	0.927438	0.463719	+	0.885982i	$0.346515\pi$
$264$	10.3306	0.635805
$265$	0.976117	0.0599624
$266$	0.278768	0.0170923
$267$	−1.14556	−0.0701071
$268$	−3.13919	−0.191757
$269$	−10.4723	−0.638510	−0.319255	−	0.947669i	$-0.603433\pi$
$269$	−10.4723	−0.638510	−0.319255	+	0.947669i	$0.603433\pi$
$270$	−8.61436	−0.524253
$271$	−17.2409	−1.04731	−0.523656	−	0.851930i	$-0.675432\pi$
$271$	−17.2409	−1.04731	−0.523656	+	0.851930i	$0.675432\pi$
$272$	3.37027	0.204353
$273$	−0.685197	−0.0414700
$274$	3.95926	0.239188
$275$	−5.10548	−0.307872
$276$	7.82320	0.470901
$277$	−18.7864	−1.12877	−0.564384	−	0.825513i	$-0.690886\pi$
$277$	−18.7864	−1.12877	−0.564384	+	0.825513i	$0.690886\pi$
$278$	−3.66226	−0.219648
$279$	−0.0548082	−0.00328128
$280$	−0.127351	−0.00761067
$281$	9.73379	0.580669	0.290335	−	0.956925i	$-0.406233\pi$
$281$	9.73379	0.580669	0.290335	+	0.956925i	$0.406233\pi$
$282$	−18.2221	−1.08511
$283$	15.7302	0.935060	0.467530	−	0.883977i	$-0.345144\pi$
$283$	15.7302	0.935060	0.467530	+	0.883977i	$0.345144\pi$
$284$	2.59120	0.153759
$285$	17.2258	1.02037
$286$	−29.3821	−1.73740
$287$	−0.104989	−0.00619731
$288$	−0.800679	−0.0471805
$289$	−5.64127	−0.331839
$290$	−13.1665	−0.773164
$291$	−8.26923	−0.484751
$292$	11.7757	0.689121
$293$	−5.16792	−0.301913	−0.150957	−	0.988540i	$-0.548235\pi$
$293$	−5.16792	−0.301913	−0.150957	+	0.988540i	$0.548235\pi$
$294$	−13.6389	−0.795437
$295$	−11.4619	−0.667340
$296$	−2.26303	−0.131536
$297$	22.7203	1.31837
$298$	−12.5991	−0.729847
$299$	−22.2506	−1.28678
$300$	1.87833	0.108445
$301$	0.528479	0.0304610
$302$	−8.28741	−0.476887
$303$	−19.7981	−1.13737
$304$	−4.39789	−0.252236
$305$	−1.27561	−0.0730409
$306$	−2.69851	−0.154263
$307$	−31.4445	−1.79463	−0.897317	−	0.441386i	$-0.854487\pi$
$307$	−31.4445	−1.79463	−0.897317	+	0.441386i	$0.854487\pi$
$308$	0.335887	0.0191390
$309$	−11.7661	−0.669351
$310$	0.137528	0.00781107
$311$	−26.9175	−1.52635	−0.763175	−	0.646192i	$-0.776361\pi$
$311$	−26.9175	−1.52635	−0.763175	+	0.646192i	$0.776361\pi$
$312$	10.8098	0.611985
$313$	8.55726	0.483685	0.241842	−	0.970316i	$-0.422248\pi$
$313$	8.55726	0.483685	0.241842	+	0.970316i	$0.422248\pi$
$314$	−18.2518	−1.03001
$315$	0.101967	0.00574520
$316$	−15.5510	−0.874812
$317$	−6.76917	−0.380194	−0.190097	−	0.981765i	$-0.560880\pi$
$317$	−6.76917	−0.380194	−0.190097	+	0.981765i	$0.560880\pi$
$318$	0.947172	0.0531148
$319$	34.7266	1.94432
$320$	2.00911	0.112313
$321$	−12.6439	−0.705713
$322$	0.254362	0.0141750
$323$	−14.8221	−0.824723
$324$	−10.7609	−0.597831
$325$	−5.34230	−0.296338
$326$	−9.58504	−0.530866
$327$	−27.9115	−1.54351
$328$	1.65633	0.0914554
$329$	−0.592471	−0.0326640
$330$	20.7553	1.14254
$331$	18.4295	1.01298	0.506489	−	0.862246i	$-0.330943\pi$
$331$	18.4295	1.01298	0.506489	+	0.862246i	$0.330943\pi$
$332$	16.2184	0.890103
$333$	1.81196	0.0992950
$334$	−4.38854	−0.240130
$335$	−6.30698	−0.344587
$336$	−0.123574	−0.00674154
$337$	1.17947	0.0642498	0.0321249	−	0.999484i	$-0.489773\pi$
$337$	1.17947	0.0642498	0.0321249	+	0.999484i	$0.489773\pi$
$338$	−17.7450	−0.965200
$339$	10.1634	0.551998
$340$	6.77125	0.367222
$341$	−0.362730	−0.0196429
$342$	3.52130	0.190410
$343$	−0.887159	−0.0479021
$344$	−8.33738	−0.449521
$345$	15.7177	0.846211
$346$	−3.03467	−0.163145
$347$	−31.6275	−1.69786	−0.848928	−	0.528509i	$-0.822751\pi$
$347$	−31.6275	−1.69786	−0.848928	+	0.528509i	$0.822751\pi$
$348$	−12.7761	−0.684869
$349$	15.4394	0.826452	0.413226	−	0.910629i	$-0.364402\pi$
$349$	15.4394	0.826452	0.413226	+	0.910629i	$0.364402\pi$
$350$	0.0610716	0.00326441
$351$	23.7742	1.26897
$352$	−5.29902	−0.282439
$353$	19.0055	1.01156	0.505781	−	0.862662i	$-0.331204\pi$
$353$	19.0055	1.01156	0.505781	+	0.862662i	$0.331204\pi$
$354$	−11.1221	−0.591131
$355$	5.20600	0.276306
$356$	0.587608	0.0311431
$357$	−0.416479	−0.0220424
$358$	25.1408	1.32873
$359$	−30.1771	−1.59269	−0.796344	−	0.604844i	$-0.793235\pi$
$359$	−30.1771	−1.59269	−0.796344	+	0.604844i	$0.793235\pi$
$360$	−1.60865	−0.0847834
$361$	0.341435	0.0179702
$362$	−11.9144	−0.626205
$363$	−33.2973	−1.74765
$364$	0.351468	0.0184219
$365$	23.6587	1.23835
$366$	−1.23778	−0.0646997
$367$	−19.8634	−1.03686	−0.518430	−	0.855120i	$-0.673483\pi$
$367$	−19.8634	−1.03686	−0.518430	+	0.855120i	$0.673483\pi$
$368$	−4.01286	−0.209185
$369$	−1.32619	−0.0690385
$370$	−4.54668	−0.236371
$371$	0.0307961	0.00159886
$372$	0.133450	0.00691905
$373$	6.33701	0.328118	0.164059	−	0.986451i	$-0.447541\pi$
$373$	6.33701	0.328118	0.164059	+	0.986451i	$0.447541\pi$
$374$	−17.8591	−0.923474
$375$	23.3579	1.20620
$376$	9.34693	0.482031
$377$	36.3374	1.87147
$378$	−0.271780	−0.0139788
$379$	37.5876	1.93074	0.965371	−	0.260879i	$-0.0840125\pi$
$379$	37.5876	1.93074	0.965371	+	0.260879i	$0.0840125\pi$
$380$	−8.83584	−0.453269
$381$	−37.8744	−1.94036
$382$	11.3433	0.580372
$383$	33.7385	1.72396	0.861980	−	0.506943i	$-0.169225\pi$
$383$	33.7385	1.72396	0.861980	+	0.506943i	$0.169225\pi$
$384$	1.94953	0.0994867
$385$	0.674834	0.0343927
$386$	1.10008	0.0559926
$387$	6.67556	0.339338
$388$	4.24165	0.215337
$389$	12.9403	0.656098	0.328049	−	0.944661i	$-0.393609\pi$
$389$	12.9403	0.656098	0.328049	+	0.944661i	$0.393609\pi$
$390$	21.7181	1.09974
$391$	−13.5244	−0.683959
$392$	6.99598	0.353350
$393$	11.6257	0.586441
$394$	20.4151	1.02850
$395$	−31.2437	−1.57204
$396$	4.24281	0.213209
$397$	22.5317	1.13083	0.565416	−	0.824806i	$-0.308716\pi$
$397$	22.5317	1.13083	0.565416	+	0.824806i	$0.308716\pi$
$398$	4.28997	0.215037
$399$	0.543467	0.0272074
$400$	−0.963477	−0.0481738
$401$	1.11468	0.0556644	0.0278322	−	0.999613i	$-0.491140\pi$
$401$	1.11468	0.0556644	0.0278322	+	0.999613i	$0.491140\pi$
$402$	−6.11996	−0.305236
$403$	−0.379555	−0.0189070
$404$	10.1553	0.505246
$405$	−21.6199	−1.07430
$406$	−0.415398	−0.0206159
$407$	11.9919	0.594414
$408$	6.57046	0.325286
$409$	13.2976	0.657522	0.328761	−	0.944413i	$-0.393369\pi$
$409$	13.2976	0.657522	0.328761	+	0.944413i	$0.393369\pi$
$410$	3.32774	0.164346
$411$	7.71871	0.380736
$412$	6.03535	0.297340
$413$	−0.361620	−0.0177942
$414$	3.21301	0.157911
$415$	32.5846	1.59952
$416$	−5.54482	−0.271857
$417$	−7.13969	−0.349632
$418$	23.3045	1.13986
$419$	−31.4431	−1.53609	−0.768047	−	0.640394i	$-0.778771\pi$
$419$	−31.4431	−1.53609	−0.768047	+	0.640394i	$0.778771\pi$
$420$	−0.248275	−0.0121146
$421$	2.74645	0.133854	0.0669270	−	0.997758i	$-0.478681\pi$
$421$	2.74645	0.133854	0.0669270	+	0.997758i	$0.478681\pi$
$422$	−20.1989	−0.983266
$423$	−7.48389	−0.363879
$424$	−0.485846	−0.0235947
$425$	−3.24718	−0.157511
$426$	5.05163	0.244752
$427$	−0.0402449	−0.00194759
$428$	6.48560	0.313493
$429$	−57.2814	−2.76557
$430$	−16.7507	−0.807791
$431$	34.5057	1.66208	0.831040	−	0.556213i	$-0.187746\pi$
$431$	34.5057	1.66208	0.831040	+	0.556213i	$0.187746\pi$
$432$	4.28765	0.206290
$433$	−14.9572	−0.718798	−0.359399	−	0.933184i	$-0.617018\pi$
$433$	−14.9572	−0.718798	−0.359399	+	0.933184i	$0.617018\pi$
$434$	0.00433896	0.000208277 0
$435$	−25.6685	−1.23071
$436$	14.3170	0.685660
$437$	17.6481	0.844223
$438$	22.9571	1.09693
$439$	−4.45586	−0.212667	−0.106333	−	0.994331i	$-0.533911\pi$
$439$	−4.45586	−0.212667	−0.106333	+	0.994331i	$0.533911\pi$
$440$	−10.6463	−0.507543
$441$	−5.60153	−0.266740
$442$	−18.6875	−0.888875
$443$	−9.06519	−0.430700	−0.215350	−	0.976537i	$-0.569089\pi$
$443$	−9.06519	−0.430700	−0.215350	+	0.976537i	$0.569089\pi$
$444$	−4.41186	−0.209378
$445$	1.18057	0.0559643
$446$	24.1584	1.14393
$447$	−24.5624	−1.16176
$448$	0.0633867	0.00299474
$449$	−21.1122	−0.996345	−0.498173	−	0.867078i	$-0.665995\pi$
$449$	−21.1122	−0.996345	−0.498173	+	0.867078i	$0.665995\pi$
$450$	0.771436	0.0363658
$451$	−8.77691	−0.413289
$452$	−5.21323	−0.245210
$453$	−16.1566	−0.759102
$454$	3.84036	0.180237
$455$	0.706137	0.0331042
$456$	−8.57383	−0.401506
$457$	−1.86568	−0.0872727	−0.0436363	−	0.999047i	$-0.513894\pi$
$457$	−1.86568	−0.0872727	−0.0436363	+	0.999047i	$0.513894\pi$
$458$	−0.856250	−0.0400099
$459$	14.4505	0.674494
$460$	−8.06227	−0.375905
$461$	−21.2495	−0.989686	−0.494843	−	0.868982i	$-0.664775\pi$
$461$	−21.2495	−0.989686	−0.494843	+	0.868982i	$0.664775\pi$
$462$	0.654823	0.0304651
$463$	18.2977	0.850365	0.425182	−	0.905108i	$-0.360210\pi$
$463$	18.2977	0.850365	0.425182	+	0.905108i	$0.360210\pi$
$464$	6.55340	0.304234
$465$	0.268115	0.0124336
$466$	−1.19018	−0.0551342
$467$	−22.8970	−1.05955	−0.529774	−	0.848139i	$-0.677723\pi$
$467$	−22.8970	−1.05955	−0.529774	+	0.848139i	$0.677723\pi$
$468$	4.43962	0.205221
$469$	−0.198983	−0.00918818
$470$	18.7790	0.866211
$471$	−35.5825	−1.63956
$472$	5.70498	0.262593
$473$	44.1799	2.03140
$474$	−30.3172	−1.39251
$475$	4.23727	0.194419
$476$	0.213630	0.00979173
$477$	0.389006	0.0178114
$478$	12.0156	0.549581
$479$	29.7797	1.36067	0.680334	−	0.732902i	$-0.261835\pi$
$479$	29.7797	1.36067	0.680334	+	0.732902i	$0.261835\pi$
$480$	3.91683	0.178778
$481$	12.5481	0.572144
$482$	5.11342	0.232910
$483$	0.495887	0.0225636
$484$	17.0796	0.776346
$485$	8.52193	0.386961
$486$	−8.11588	−0.368144
$487$	32.6593	1.47993	0.739966	−	0.672644i	$-0.234842\pi$
$487$	32.6593	1.47993	0.739966	+	0.672644i	$0.234842\pi$
$488$	0.634911	0.0287410
$489$	−18.6864	−0.845026
$490$	14.0557	0.634972
$491$	−0.358739	−0.0161897	−0.00809483	−	0.999967i	$-0.502577\pi$
$491$	−0.358739	−0.0161897	−0.00809483	+	0.999967i	$0.502577\pi$
$492$	3.22907	0.145578
$493$	22.0867	0.994737
$494$	24.3855	1.09716
$495$	8.52428	0.383138
$496$	−0.0684522	−0.00307359
$497$	0.164247	0.00736751
$498$	31.6184	1.41685
$499$	−37.1941	−1.66504	−0.832518	−	0.553998i	$-0.813101\pi$
$499$	−37.1941	−1.66504	−0.832518	+	0.553998i	$0.813101\pi$
$500$	−11.9813	−0.535819
$501$	−8.55561	−0.382236
$502$	2.86144	0.127712
$503$	−15.2039	−0.677910	−0.338955	−	0.940803i	$-0.610073\pi$
$503$	−15.2039	−0.677910	−0.338955	+	0.940803i	$0.610073\pi$
$504$	−0.0507524	−0.00226069
$505$	20.4031	0.907928
$506$	21.2642	0.945309
$507$	−34.5944	−1.53639
$508$	19.4274	0.861951
$509$	−5.75050	−0.254886	−0.127443	−	0.991846i	$-0.540677\pi$
$509$	−5.75050	−0.254886	−0.127443	+	0.991846i	$0.540677\pi$
$510$	13.2008	0.584540
$511$	0.746423	0.0330198
$512$	−1.00000	−0.0441942
$513$	−18.8566	−0.832540
$514$	−22.7601	−1.00391
$515$	12.1257	0.534321
$516$	−16.2540	−0.715542
$517$	−49.5296	−2.17831
$518$	−0.143446	−0.00630266
$519$	−5.91618	−0.259692
$520$	−11.1401	−0.488528
$521$	−31.5704	−1.38312	−0.691561	−	0.722318i	$-0.743077\pi$
$521$	−31.5704	−1.38312	−0.691561	+	0.722318i	$0.743077\pi$
$522$	−5.24717	−0.229662
$523$	41.3285	1.80717	0.903586	−	0.428408i	$-0.140925\pi$
$523$	41.3285	1.80717	0.903586	+	0.428408i	$0.140925\pi$
$524$	−5.96335	−0.260510
$525$	0.119061	0.00519625
$526$	−15.0405	−0.655798
$527$	−0.230703	−0.0100496
$528$	−10.3306	−0.449582
$529$	−6.89698	−0.299869
$530$	−0.976117	−0.0423998
$531$	−4.56786	−0.198228
$532$	−0.278768	−0.0120861
$533$	−9.18403	−0.397805
$534$	1.14556	0.0495732
$535$	13.0303	0.563348
$536$	3.13919	0.135592
$537$	49.0129	2.11506
$538$	10.4723	0.451495
$539$	−37.0718	−1.59680
$540$	8.61436	0.370703
$541$	24.0134	1.03241	0.516207	−	0.856464i	$-0.327343\pi$
$541$	24.0134	1.03241	0.516207	+	0.856464i	$0.327343\pi$
$542$	17.2409	0.740561
$543$	−23.2274	−0.996785
$544$	−3.37027	−0.144499
$545$	28.7644	1.23213
$546$	0.685197	0.0293237
$547$	13.0258	0.556945	0.278472	−	0.960444i	$-0.410172\pi$
$547$	13.0258	0.556945	0.278472	+	0.960444i	$0.410172\pi$
$548$	−3.95926	−0.169131
$549$	−0.508359	−0.0216963
$550$	5.10548	0.217699
$551$	−28.8211	−1.22782
$552$	−7.82320	−0.332977
$553$	−0.985726	−0.0419173
$554$	18.7864	0.798159
$555$	−8.86391	−0.376252
$556$	3.66226	0.155314
$557$	43.8246	1.85691	0.928455	−	0.371445i	$-0.121138\pi$
$557$	43.8246	1.85691	0.928455	+	0.371445i	$0.121138\pi$
$558$	0.0548082	0.00232022
$559$	46.2292	1.95529
$560$	0.127351	0.00538155
$561$	−34.8170	−1.46997
$562$	−9.73379	−0.410595
$563$	0.532450	0.0224401	0.0112200	−	0.999937i	$-0.496428\pi$
$563$	0.532450	0.0224401	0.0112200	+	0.999937i	$0.496428\pi$
$564$	18.2221	0.767291
$565$	−10.4739	−0.440642
$566$	−15.7302	−0.661188
$567$	−0.682101	−0.0286455
$568$	−2.59120	−0.108724
$569$	−35.5026	−1.48834	−0.744172	−	0.667988i	$-0.767156\pi$
$569$	−35.5026	−1.48834	−0.744172	+	0.667988i	$0.767156\pi$
$570$	−17.2258	−0.721508
$571$	−19.7227	−0.825368	−0.412684	−	0.910874i	$-0.635409\pi$
$571$	−19.7227	−0.825368	−0.412684	+	0.910874i	$0.635409\pi$
$572$	29.3821	1.22853
$573$	22.1141	0.923829
$574$	0.104989	0.00438216
$575$	3.86630	0.161236
$576$	0.800679	0.0333616
$577$	−11.5767	−0.481946	−0.240973	−	0.970532i	$-0.577466\pi$
$577$	−11.5767	−0.481946	−0.240973	+	0.970532i	$0.577466\pi$
$578$	5.64127	0.234646
$579$	2.14464	0.0891283
$580$	13.1665	0.546709
$581$	1.02803	0.0426500
$582$	8.26923	0.342771
$583$	2.57451	0.106625
$584$	−11.7757	−0.487282
$585$	8.91968	0.368783
$586$	5.16792	0.213485
$587$	−20.4216	−0.842890	−0.421445	−	0.906854i	$-0.638477\pi$
$587$	−20.4216	−0.842890	−0.421445	+	0.906854i	$0.638477\pi$
$588$	13.6389	0.562459
$589$	0.301045	0.0124044
$590$	11.4619	0.471881
$591$	39.8000	1.63715
$592$	2.26303	0.0930101
$593$	−23.7000	−0.973241	−0.486620	−	0.873613i	$-0.661771\pi$
$593$	−23.7000	−0.973241	−0.486620	+	0.873613i	$0.661771\pi$
$594$	−22.7203	−0.932227
$595$	0.429207	0.0175958
$596$	12.5991	0.516080
$597$	8.36344	0.342293
$598$	22.2506	0.909893
$599$	26.4967	1.08263	0.541314	−	0.840821i	$-0.317927\pi$
$599$	26.4967	1.08263	0.541314	+	0.840821i	$0.317927\pi$
$600$	−1.87833	−0.0766825
$601$	9.92290	0.404763	0.202382	−	0.979307i	$-0.435132\pi$
$601$	9.92290	0.404763	0.202382	+	0.979307i	$0.435132\pi$
$602$	−0.528479	−0.0215392
$603$	−2.51348	−0.102357
$604$	8.28741	0.337210
$605$	34.3148	1.39510
$606$	19.7981	0.804243
$607$	−3.25958	−0.132302	−0.0661510	−	0.997810i	$-0.521072\pi$
$607$	−3.25958	−0.132302	−0.0661510	+	0.997810i	$0.521072\pi$
$608$	4.39789	0.178358
$609$	−0.809833	−0.0328161
$610$	1.27561	0.0516477
$611$	−51.8270	−2.09670
$612$	2.69851	0.109081
$613$	−20.3488	−0.821881	−0.410941	−	0.911662i	$-0.634800\pi$
$613$	−20.3488	−0.821881	−0.410941	+	0.911662i	$0.634800\pi$
$614$	31.4445	1.26900
$615$	6.48755	0.261603
$616$	−0.335887	−0.0135333
$617$	−35.3844	−1.42452	−0.712260	−	0.701915i	$-0.752328\pi$
$617$	−35.3844	−1.42452	−0.712260	+	0.701915i	$0.752328\pi$
$618$	11.7661	0.473303
$619$	27.6607	1.11178	0.555890	−	0.831256i	$-0.312378\pi$
$619$	27.6607	1.11178	0.555890	+	0.831256i	$0.312378\pi$
$620$	−0.137528	−0.00552326
$621$	−17.2057	−0.690442
$622$	26.9175	1.07929
$623$	0.0372465	0.00149225
$624$	−10.8098	−0.432738
$625$	−19.2543	−0.770173
$626$	−8.55726	−0.342017
$627$	45.4329	1.81442
$628$	18.2518	0.728327
$629$	7.62704	0.304110
$630$	−0.101967	−0.00406247
$631$	0.501236	0.0199539	0.00997694	−	0.999950i	$-0.496824\pi$
$631$	0.501236	0.0199539	0.00997694	+	0.999950i	$0.496824\pi$
$632$	15.5510	0.618586
$633$	−39.3784	−1.56515
$634$	6.76917	0.268838
$635$	39.0318	1.54893
$636$	−0.947172	−0.0375578
$637$	−38.7914	−1.53697
$638$	−34.7266	−1.37484
$639$	2.07472	0.0820746
$640$	−2.00911	−0.0794170
$641$	−27.3092	−1.07865	−0.539324	−	0.842099i	$-0.681320\pi$
$641$	−27.3092	−1.07865	−0.539324	+	0.842099i	$0.681320\pi$
$642$	12.6439	0.499015
$643$	30.2129	1.19148	0.595741	−	0.803177i	$-0.296858\pi$
$643$	30.2129	1.19148	0.595741	+	0.803177i	$0.296858\pi$
$644$	−0.254362	−0.0100233
$645$	−32.6561	−1.28583
$646$	14.8221	0.583167
$647$	26.0165	1.02281	0.511407	−	0.859339i	$-0.329125\pi$
$647$	26.0165	1.02281	0.511407	+	0.859339i	$0.329125\pi$
$648$	10.7609	0.422730
$649$	−30.2308	−1.18666
$650$	5.34230	0.209542
$651$	0.00845894	0.000331532 0
$652$	9.58504	0.375379
$653$	36.7644	1.43870	0.719351	−	0.694647i	$-0.244439\pi$
$653$	36.7644	1.43870	0.719351	+	0.694647i	$0.244439\pi$
$654$	27.9115	1.09142
$655$	−11.9810	−0.468137
$656$	−1.65633	−0.0646687
$657$	9.42856	0.367843
$658$	0.592471	0.0230969
$659$	14.9274	0.581491	0.290745	−	0.956800i	$-0.406097\pi$
$659$	14.9274	0.581491	0.290745	+	0.956800i	$0.406097\pi$
$660$	−20.7553	−0.807900
$661$	−45.0594	−1.75261	−0.876303	−	0.481760i	$-0.839998\pi$
$661$	−45.0594	−1.75261	−0.876303	+	0.481760i	$0.839998\pi$
$662$	−18.4295	−0.716284
$663$	−36.4320	−1.41490
$664$	−16.2184	−0.629398
$665$	−0.560075	−0.0217188
$666$	−1.81196	−0.0702122
$667$	−26.2979	−1.01826
$668$	4.38854	0.169798
$669$	47.0975	1.82089
$670$	6.30698	0.243660
$671$	−3.36440	−0.129881
$672$	0.123574	0.00476699
$673$	−21.0214	−0.810317	−0.405158	−	0.914246i	$-0.632784\pi$
$673$	−21.0214	−0.810317	−0.405158	+	0.914246i	$0.632784\pi$
$674$	−1.17947	−0.0454314
$675$	−4.13105	−0.159004
$676$	17.7450	0.682500
$677$	3.71665	0.142842	0.0714212	−	0.997446i	$-0.477247\pi$
$677$	3.71665	0.142842	0.0714212	+	0.997446i	$0.477247\pi$
$678$	−10.1634	−0.390322
$679$	0.268864	0.0103180
$680$	−6.77125	−0.259665
$681$	7.48691	0.286899
$682$	0.362730	0.0138896
$683$	−10.3433	−0.395773	−0.197887	−	0.980225i	$-0.563408\pi$
$683$	−10.3433	−0.395773	−0.197887	+	0.980225i	$0.563408\pi$
$684$	−3.52130	−0.134640
$685$	−7.95459	−0.303929
$686$	0.887159	0.0338719
$687$	−1.66929	−0.0636873
$688$	8.33738	0.317860
$689$	2.69392	0.102630
$690$	−15.7177	−0.598361
$691$	−23.1198	−0.879520	−0.439760	−	0.898115i	$-0.644937\pi$
$691$	−23.1198	−0.879520	−0.439760	+	0.898115i	$0.644937\pi$
$692$	3.03467	0.115361
$693$	0.268938	0.0102161
$694$	31.6275	1.20057
$695$	7.35788	0.279100
$696$	12.7761	0.484276
$697$	−5.58227	−0.211444
$698$	−15.4394	−0.584389
$699$	−2.32030	−0.0877618
$700$	−0.0610716	−0.00230829
$701$	−19.4155	−0.733314	−0.366657	−	0.930356i	$-0.619498\pi$
$701$	−19.4155	−0.733314	−0.366657	+	0.930356i	$0.619498\pi$
$702$	−23.7742	−0.897300
$703$	−9.95257	−0.375368
$704$	5.29902	0.199714
$705$	36.6103	1.37882
$706$	−19.0055	−0.715283
$707$	0.643712	0.0242093
$708$	11.1221	0.417992
$709$	8.49293	0.318959	0.159479	−	0.987201i	$-0.449018\pi$
$709$	8.49293	0.318959	0.159479	+	0.987201i	$0.449018\pi$
$710$	−5.20600	−0.195378
$711$	−12.4514	−0.466962
$712$	−0.587608	−0.0220215
$713$	0.274689	0.0102872
$714$	0.416479	0.0155863
$715$	59.0318	2.20767
$716$	−25.1408	−0.939557
$717$	23.4248	0.874816
$718$	30.1771	1.12620
$719$	−44.1043	−1.64481	−0.822406	−	0.568901i	$-0.807369\pi$
$719$	−44.1043	−1.64481	−0.822406	+	0.568901i	$0.807369\pi$
$720$	1.60865	0.0599509
$721$	0.382561	0.0142473
$722$	−0.341435	−0.0127069
$723$	9.96877	0.370743
$724$	11.9144	0.442794
$725$	−6.31405	−0.234498
$726$	33.2973	1.23578
$727$	43.4656	1.61205	0.806024	−	0.591883i	$-0.201615\pi$
$727$	43.4656	1.61205	0.806024	+	0.591883i	$0.201615\pi$
$728$	−0.351468	−0.0130263
$729$	16.4607	0.609655
$730$	−23.6587	−0.875648
$731$	28.0992	1.03929
$732$	1.23778	0.0457496
$733$	−25.1495	−0.928918	−0.464459	−	0.885595i	$-0.653751\pi$
$733$	−25.1495	−0.928918	−0.464459	+	0.885595i	$0.653751\pi$
$734$	19.8634	0.733171
$735$	27.4020	1.01074
$736$	4.01286	0.147916
$737$	−16.6346	−0.612745
$738$	1.32619	0.0488176
$739$	38.2385	1.40663	0.703313	−	0.710881i	$-0.251703\pi$
$739$	38.2385	1.40663	0.703313	+	0.710881i	$0.251703\pi$
$740$	4.54668	0.167139
$741$	47.5403	1.74644
$742$	−0.0307961	−0.00113056
$743$	−14.4289	−0.529346	−0.264673	−	0.964338i	$-0.585264\pi$
$743$	−14.4289	−0.529346	−0.264673	+	0.964338i	$0.585264\pi$
$744$	−0.133450	−0.00489251
$745$	25.3130	0.927397
$746$	−6.33701	−0.232015
$747$	12.9858	0.475124
$748$	17.8591	0.652995
$749$	0.411101	0.0150213
$750$	−23.3579	−0.852910
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	−9.34693	−0.340847
$753$	5.57846	0.203291
$754$	−36.3374	−1.32333
$755$	16.6503	0.605967
$756$	0.271780	0.00988454
$757$	−15.5755	−0.566103	−0.283051	−	0.959105i	$-0.591347\pi$
$757$	−15.5755	−0.566103	−0.283051	+	0.959105i	$0.591347\pi$
$758$	−37.5876	−1.36524
$759$	41.4553	1.50473
$760$	8.83584	0.320510
$761$	14.6261	0.530196	0.265098	−	0.964221i	$-0.414596\pi$
$761$	14.6261	0.530196	0.265098	+	0.964221i	$0.414596\pi$
$762$	37.8744	1.37204
$763$	0.907507	0.0328540
$764$	−11.3433	−0.410385
$765$	5.42159	0.196018
$766$	−33.7385	−1.21902
$767$	−31.6331	−1.14220
$768$	−1.94953	−0.0703477
$769$	25.1599	0.907288	0.453644	−	0.891183i	$-0.350124\pi$
$769$	25.1599	0.907288	0.453644	+	0.891183i	$0.350124\pi$
$770$	−0.674834	−0.0243193
$771$	−44.3716	−1.59800
$772$	−1.10008	−0.0395927
$773$	−5.83890	−0.210011	−0.105005	−	0.994472i	$-0.533486\pi$
$773$	−5.83890	−0.210011	−0.105005	+	0.994472i	$0.533486\pi$
$774$	−6.67556	−0.239948
$775$	0.0659521	0.00236907
$776$	−4.24165	−0.152266
$777$	−0.279653	−0.0100325
$778$	−12.9403	−0.463931
$779$	7.28435	0.260989
$780$	−21.7181	−0.777632
$781$	13.7308	0.491327
$782$	13.5244	0.483632
$783$	28.0987	1.00416
$784$	−6.99598	−0.249857
$785$	36.6699	1.30881
$786$	−11.6257	−0.414676
$787$	17.0776	0.608752	0.304376	−	0.952552i	$-0.401552\pi$
$787$	17.0776	0.608752	0.304376	+	0.952552i	$0.401552\pi$
$788$	−20.4151	−0.727259
$789$	−29.3220	−1.04389
$790$	31.2437	1.11160
$791$	−0.330449	−0.0117494
$792$	−4.24281	−0.150762
$793$	−3.52046	−0.125015
$794$	−22.5317	−0.799619
$795$	−1.90297	−0.0674915
$796$	−4.28997	−0.152054
$797$	−2.74057	−0.0970759	−0.0485380	−	0.998821i	$-0.515456\pi$
$797$	−2.74057	−0.0970759	−0.0485380	+	0.998821i	$0.515456\pi$
$798$	−0.543467	−0.0192385
$799$	−31.5017	−1.11445
$800$	0.963477	0.0340641
$801$	0.470485	0.0166238
$802$	−1.11468	−0.0393607
$803$	62.3997	2.20204
$804$	6.11996	0.215834
$805$	−0.511041	−0.0180118
$806$	0.379555	0.0133693
$807$	20.4162	0.718683
$808$	−10.1553	−0.357263
$809$	−52.2517	−1.83707	−0.918537	−	0.395336i	$-0.870628\pi$
$809$	−52.2517	−1.83707	−0.918537	+	0.395336i	$0.870628\pi$
$810$	21.6199	0.759647
$811$	−51.5452	−1.81000	−0.904998	−	0.425417i	$-0.860128\pi$
$811$	−51.5452	−1.81000	−0.904998	+	0.425417i	$0.860128\pi$
$812$	0.415398	0.0145776
$813$	33.6117	1.17882
$814$	−11.9919	−0.420315
$815$	19.2574	0.674558
$816$	−6.57046	−0.230012
$817$	−36.6669	−1.28281
$818$	−13.2976	−0.464938
$819$	0.281413	0.00983335
$820$	−3.32774	−0.116210
$821$	4.26496	0.148848	0.0744240	−	0.997227i	$-0.476288\pi$
$821$	4.26496	0.148848	0.0744240	+	0.997227i	$0.476288\pi$
$822$	−7.71871	−0.269221
$823$	−30.6151	−1.06717	−0.533587	−	0.845745i	$-0.679156\pi$
$823$	−30.6151	−1.06717	−0.533587	+	0.845745i	$0.679156\pi$
$824$	−6.03535	−0.210251
$825$	9.95331	0.346530
$826$	0.361620	0.0125824
$827$	−24.1246	−0.838893	−0.419447	−	0.907780i	$-0.637776\pi$
$827$	−24.1246	−0.838893	−0.419447	+	0.907780i	$0.637776\pi$
$828$	−3.21301	−0.111660
$829$	8.85440	0.307526	0.153763	−	0.988108i	$-0.450861\pi$
$829$	8.85440	0.307526	0.153763	+	0.988108i	$0.450861\pi$
$830$	−32.5846	−1.13103
$831$	36.6248	1.27050
$832$	5.54482	0.192232
$833$	−23.5784	−0.816942
$834$	7.13969	0.247227
$835$	8.81706	0.305127
$836$	−23.3045	−0.806003
$837$	−0.293499	−0.0101448
$838$	31.4431	1.08618
$839$	−47.0708	−1.62506	−0.812532	−	0.582916i	$-0.801912\pi$
$839$	−47.0708	−1.62506	−0.812532	+	0.582916i	$0.801912\pi$
$840$	0.248275	0.00856629
$841$	13.9471	0.480933
$842$	−2.74645	−0.0946491
$843$	−18.9763	−0.653580
$844$	20.1989	0.695274
$845$	35.6516	1.22645
$846$	7.48389	0.257301
$847$	1.08262	0.0371993
$848$	0.485846	0.0166840
$849$	−30.6664	−1.05247
$850$	3.24718	0.111377
$851$	−9.08123	−0.311301
$852$	−5.05163	−0.173066
$853$	−19.2290	−0.658387	−0.329193	−	0.944263i	$-0.606777\pi$
$853$	−19.2290	−0.658387	−0.329193	+	0.944263i	$0.606777\pi$
$854$	0.0402449	0.00137715
$855$	−7.07467	−0.241949
$856$	−6.48560	−0.221673
$857$	11.4945	0.392644	0.196322	−	0.980539i	$-0.437100\pi$
$857$	11.4945	0.392644	0.196322	+	0.980539i	$0.437100\pi$
$858$	57.2814	1.95555
$859$	29.2792	0.998994	0.499497	−	0.866316i	$-0.333518\pi$
$859$	29.2792	0.998994	0.499497	+	0.866316i	$0.333518\pi$
$860$	16.7507	0.571194
$861$	0.204680	0.00697547
$862$	−34.5057	−1.17527
$863$	14.5647	0.495789	0.247895	−	0.968787i	$-0.420261\pi$
$863$	14.5647	0.495789	0.247895	+	0.968787i	$0.420261\pi$
$864$	−4.28765	−0.145869
$865$	6.09698	0.207304
$866$	14.9572	0.508267
$867$	10.9978	0.373506
$868$	−0.00433896	−0.000147274 0
$869$	−82.4050	−2.79540
$870$	25.6685	0.870245
$871$	−17.4062	−0.589788
$872$	−14.3170	−0.484835
$873$	3.39620	0.114944
$874$	−17.6481	−0.596956
$875$	−0.759454	−0.0256742
$876$	−22.9571	−0.775650
$877$	−8.83880	−0.298465	−0.149233	−	0.988802i	$-0.547680\pi$
$877$	−8.83880	−0.298465	−0.149233	+	0.988802i	$0.547680\pi$
$878$	4.45586	0.150378
$879$	10.0750	0.339822
$880$	10.6463	0.358887
$881$	51.1790	1.72426	0.862132	−	0.506683i	$-0.169129\pi$
$881$	51.1790	1.72426	0.862132	+	0.506683i	$0.169129\pi$
$882$	5.60153	0.188613
$883$	−1.20889	−0.0406823	−0.0203411	−	0.999793i	$-0.506475\pi$
$883$	−1.20889	−0.0406823	−0.0203411	+	0.999793i	$0.506475\pi$
$884$	18.6875	0.628530
$885$	22.3454	0.751133
$886$	9.06519	0.304551
$887$	28.8487	0.968646	0.484323	−	0.874889i	$-0.339066\pi$
$887$	28.8487	0.968646	0.484323	+	0.874889i	$0.339066\pi$
$888$	4.41186	0.148052
$889$	1.23144	0.0413011
$890$	−1.18057	−0.0395727
$891$	−57.0225	−1.91033
$892$	−24.1584	−0.808882
$893$	41.1068	1.37559
$894$	24.5624	0.821489
$895$	−50.5107	−1.68839
$896$	−0.0633867	−0.00211760
$897$	43.3782	1.44836
$898$	21.1122	0.704523
$899$	−0.448595	−0.0149615
$900$	−0.771436	−0.0257145
$901$	1.63743	0.0545508
$902$	8.77691	0.292239
$903$	−1.03029	−0.0342858
$904$	5.21323	0.173389
$905$	23.9373	0.795702
$906$	16.1566	0.536766
$907$	2.32723	0.0772746	0.0386373	−	0.999253i	$-0.487698\pi$
$907$	2.32723	0.0772746	0.0386373	+	0.999253i	$0.487698\pi$
$908$	−3.84036	−0.127447
$909$	8.13114	0.269693
$910$	−0.706137	−0.0234082
$911$	37.4724	1.24152	0.620758	−	0.784002i	$-0.286825\pi$
$911$	37.4724	1.24152	0.620758	+	0.784002i	$0.286825\pi$
$912$	8.57383	0.283908
$913$	85.9418	2.84426
$914$	1.86568	0.0617111
$915$	2.48683	0.0822122
$916$	0.856250	0.0282913
$917$	−0.377997	−0.0124826
$918$	−14.4505	−0.476939
$919$	−9.13638	−0.301382	−0.150691	−	0.988581i	$-0.548150\pi$
$919$	−9.13638	−0.301382	−0.150691	+	0.988581i	$0.548150\pi$
$920$	8.06227	0.265805
$921$	61.3022	2.01997
$922$	21.2495	0.699814
$923$	14.3677	0.472919
$924$	−0.654823	−0.0215421
$925$	−2.18038	−0.0716905
$926$	−18.2977	−0.601299
$927$	4.83238	0.158716
$928$	−6.55340	−0.215126
$929$	−28.0094	−0.918960	−0.459480	−	0.888188i	$-0.651964\pi$
$929$	−28.0094	−0.918960	−0.459480	+	0.888188i	$0.651964\pi$
$930$	−0.268115	−0.00879185
$931$	30.7676	1.00837
$932$	1.19018	0.0389857
$933$	52.4765	1.71800
$934$	22.8970	0.749213
$935$	35.8810	1.17343
$936$	−4.43962	−0.145113
$937$	−10.2976	−0.336408	−0.168204	−	0.985752i	$-0.553797\pi$
$937$	−10.2976	−0.336408	−0.168204	+	0.985752i	$0.553797\pi$
$938$	0.198983	0.00649702
$939$	−16.6827	−0.544418
$940$	−18.7790	−0.612504
$941$	−12.6945	−0.413828	−0.206914	−	0.978359i	$-0.566342\pi$
$941$	−12.6945	−0.413828	−0.206914	+	0.978359i	$0.566342\pi$
$942$	35.5825	1.15934
$943$	6.64661	0.216443
$944$	−5.70498	−0.185681
$945$	0.546036	0.0177625
$946$	−44.1799	−1.43641
$947$	−15.2924	−0.496938	−0.248469	−	0.968640i	$-0.579927\pi$
$947$	−15.2924	−0.496938	−0.248469	+	0.968640i	$0.579927\pi$
$948$	30.3172	0.984656
$949$	65.2942	2.11954
$950$	−4.23727	−0.137475
$951$	13.1967	0.427933
$952$	−0.213630	−0.00692380
$953$	55.4730	1.79695	0.898473	−	0.439029i	$-0.144677\pi$
$953$	55.4730	1.79695	0.898473	+	0.439029i	$0.144677\pi$
$954$	−0.389006	−0.0125945
$955$	−22.7899	−0.737463
$956$	−12.0156	−0.388613
$957$	−67.7006	−2.18845
$958$	−29.7797	−0.962138
$959$	−0.250964	−0.00810406
$960$	−3.91683	−0.126415
$961$	−30.9953	−0.999849
$962$	−12.5481	−0.404567
$963$	5.19288	0.167338
$964$	−5.11342	−0.164692
$965$	−2.21018	−0.0711483
$966$	−0.495887	−0.0159549
$967$	−29.1390	−0.937047	−0.468524	−	0.883451i	$-0.655214\pi$
$967$	−29.1390	−0.937047	−0.468524	+	0.883451i	$0.655214\pi$
$968$	−17.0796	−0.548959
$969$	28.8961	0.928278
$970$	−8.52193	−0.273623
$971$	44.2510	1.42008	0.710041	−	0.704161i	$-0.248677\pi$
$971$	44.2510	1.42008	0.710041	+	0.704161i	$0.248677\pi$
$972$	8.11588	0.260317
$973$	0.232138	0.00744201
$974$	−32.6593	−1.04647
$975$	10.4150	0.333547
$976$	−0.634911	−0.0203230
$977$	−14.1207	−0.451762	−0.225881	−	0.974155i	$-0.572526\pi$
$977$	−14.1207	−0.451762	−0.225881	+	0.974155i	$0.572526\pi$
$978$	18.6864	0.597524
$979$	3.11374	0.0995157
$980$	−14.0557	−0.448993
$981$	11.4633	0.365996
$982$	0.358739	0.0114478
$983$	−13.1391	−0.419073	−0.209536	−	0.977801i	$-0.567195\pi$
$983$	−13.1391	−0.419073	−0.209536	+	0.977801i	$0.567195\pi$
$984$	−3.22907	−0.102939
$985$	−41.0163	−1.30689
$986$	−22.0867	−0.703385
$987$	1.15504	0.0367654
$988$	−24.3855	−0.775806
$989$	−33.4567	−1.06386
$990$	−8.52428	−0.270919
$991$	14.9736	0.475651	0.237826	−	0.971308i	$-0.423565\pi$
$991$	14.9736	0.475651	0.237826	+	0.971308i	$0.423565\pi$
$992$	0.0684522	0.00217336
$993$	−35.9290	−1.14017
$994$	−0.164247	−0.00520961
$995$	−8.61903	−0.273242
$996$	−31.6184	−1.00187
$997$	29.2340	0.925850	0.462925	−	0.886398i	$-0.346800\pi$
$997$	29.2340	0.925850	0.462925	+	0.886398i	$0.346800\pi$
$998$	37.1941	1.17736
$999$	9.70310	0.306992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1502.2.a.h.1.4	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1502.2.a.h.1.4	✓	19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1502 = 2 \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1502.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.94953 q^{3} +1.00000 q^{4} +2.00911 q^{5} +1.94953 q^{6} +0.0633867 q^{7} -1.00000 q^{8} +0.800679 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.94953 q^{3} +1.00000 q^{4} +2.00911 q^{5} +1.94953 q^{6} +0.0633867 q^{7} -1.00000 q^{8} +0.800679 q^{9} -2.00911 q^{10} +5.29902 q^{11} -1.94953 q^{12} +5.54482 q^{13} -0.0633867 q^{14} -3.91683 q^{15} +1.00000 q^{16} +3.37027 q^{17} -0.800679 q^{18} -4.39789 q^{19} +2.00911 q^{20} -0.123574 q^{21} -5.29902 q^{22} -4.01286 q^{23} +1.94953 q^{24} -0.963477 q^{25} -5.54482 q^{26} +4.28765 q^{27} +0.0633867 q^{28} +6.55340 q^{29} +3.91683 q^{30} -0.0684522 q^{31} -1.00000 q^{32} -10.3306 q^{33} -3.37027 q^{34} +0.127351 q^{35} +0.800679 q^{36} +2.26303 q^{37} +4.39789 q^{38} -10.8098 q^{39} -2.00911 q^{40} -1.65633 q^{41} +0.123574 q^{42} +8.33738 q^{43} +5.29902 q^{44} +1.60865 q^{45} +4.01286 q^{46} -9.34693 q^{47} -1.94953 q^{48} -6.99598 q^{49} +0.963477 q^{50} -6.57046 q^{51} +5.54482 q^{52} +0.485846 q^{53} -4.28765 q^{54} +10.6463 q^{55} -0.0633867 q^{56} +8.57383 q^{57} -6.55340 q^{58} -5.70498 q^{59} -3.91683 q^{60} -0.634911 q^{61} +0.0684522 q^{62} +0.0507524 q^{63} +1.00000 q^{64} +11.1401 q^{65} +10.3306 q^{66} -3.13919 q^{67} +3.37027 q^{68} +7.82320 q^{69} -0.127351 q^{70} +2.59120 q^{71} -0.800679 q^{72} +11.7757 q^{73} -2.26303 q^{74} +1.87833 q^{75} -4.39789 q^{76} +0.335887 q^{77} +10.8098 q^{78} -15.5510 q^{79} +2.00911 q^{80} -10.7609 q^{81} +1.65633 q^{82} +16.2184 q^{83} -0.123574 q^{84} +6.77125 q^{85} -8.33738 q^{86} -12.7761 q^{87} -5.29902 q^{88} +0.587608 q^{89} -1.60865 q^{90} +0.351468 q^{91} -4.01286 q^{92} +0.133450 q^{93} +9.34693 q^{94} -8.83584 q^{95} +1.94953 q^{96} +4.24165 q^{97} +6.99598 q^{98} +4.24281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more