LMFDB - Embedded newform 1122.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1122,2,Mod(1,1122)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1122, 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("1122.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1122 = 2 \cdot 3 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1122.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:	$8.95921510679$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 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			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1122.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +0.763932 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +0.763932 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} -0.763932 q^{14} -1.23607 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.47214 q^{19} +1.23607 q^{20} -0.763932 q^{21} +1.00000 q^{22} +3.23607 q^{23} +1.00000 q^{24} -3.47214 q^{25} -1.00000 q^{27} +0.763932 q^{28} -3.23607 q^{29} +1.23607 q^{30} -1.23607 q^{31} -1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +0.944272 q^{35} +1.00000 q^{36} +9.70820 q^{37} -6.47214 q^{38} -1.23607 q^{40} -4.47214 q^{41} +0.763932 q^{42} +8.94427 q^{43} -1.00000 q^{44} +1.23607 q^{45} -3.23607 q^{46} -2.00000 q^{47} -1.00000 q^{48} -6.41641 q^{49} +3.47214 q^{50} -1.00000 q^{51} -4.94427 q^{53} +1.00000 q^{54} -1.23607 q^{55} -0.763932 q^{56} -6.47214 q^{57} +3.23607 q^{58} +1.52786 q^{59} -1.23607 q^{60} +5.23607 q^{61} +1.23607 q^{62} +0.763932 q^{63} +1.00000 q^{64} -1.00000 q^{66} +1.00000 q^{68} -3.23607 q^{69} -0.944272 q^{70} +8.76393 q^{71} -1.00000 q^{72} +7.52786 q^{73} -9.70820 q^{74} +3.47214 q^{75} +6.47214 q^{76} -0.763932 q^{77} +12.1803 q^{79} +1.23607 q^{80} +1.00000 q^{81} +4.47214 q^{82} +16.0000 q^{83} -0.763932 q^{84} +1.23607 q^{85} -8.94427 q^{86} +3.23607 q^{87} +1.00000 q^{88} +10.9443 q^{89} -1.23607 q^{90} +3.23607 q^{92} +1.23607 q^{93} +2.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +10.9443 q^{97} +6.41641 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} - 6 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 4 q^{19} - 2 q^{20} - 6 q^{21} + 2 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{27} + 6 q^{28} - 2 q^{29} - 2 q^{30} + 2 q^{31} - 2 q^{32} + 2 q^{33} - 2 q^{34} - 16 q^{35} + 2 q^{36} + 6 q^{37} - 4 q^{38} + 2 q^{40} + 6 q^{42} - 2 q^{44} - 2 q^{45} - 2 q^{46} - 4 q^{47} - 2 q^{48} + 14 q^{49} - 2 q^{50} - 2 q^{51} + 8 q^{53} + 2 q^{54} + 2 q^{55} - 6 q^{56} - 4 q^{57} + 2 q^{58} + 12 q^{59} + 2 q^{60} + 6 q^{61} - 2 q^{62} + 6 q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{68} - 2 q^{69} + 16 q^{70} + 22 q^{71} - 2 q^{72} + 24 q^{73} - 6 q^{74} - 2 q^{75} + 4 q^{76} - 6 q^{77} + 2 q^{79} - 2 q^{80} + 2 q^{81} + 32 q^{83} - 6 q^{84} - 2 q^{85} + 2 q^{87} + 2 q^{88} + 4 q^{89} + 2 q^{90} + 2 q^{92} - 2 q^{93} + 4 q^{94} + 16 q^{95} + 2 q^{96} + 4 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 - 2 * q^12 - 6 * q^14 + 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 4 * q^19 - 2 * q^20 - 6 * q^21 + 2 * q^22 + 2 * q^23 + 2 * q^24 + 2 * q^25 - 2 * q^27 + 6 * q^28 - 2 * q^29 - 2 * q^30 + 2 * q^31 - 2 * q^32 + 2 * q^33 - 2 * q^34 - 16 * q^35 + 2 * q^36 + 6 * q^37 - 4 * q^38 + 2 * q^40 + 6 * q^42 - 2 * q^44 - 2 * q^45 - 2 * q^46 - 4 * q^47 - 2 * q^48 + 14 * q^49 - 2 * q^50 - 2 * q^51 + 8 * q^53 + 2 * q^54 + 2 * q^55 - 6 * q^56 - 4 * q^57 + 2 * q^58 + 12 * q^59 + 2 * q^60 + 6 * q^61 - 2 * q^62 + 6 * q^63 + 2 * q^64 - 2 * q^66 + 2 * q^68 - 2 * q^69 + 16 * q^70 + 22 * q^71 - 2 * q^72 + 24 * q^73 - 6 * q^74 - 2 * q^75 + 4 * q^76 - 6 * q^77 + 2 * q^79 - 2 * q^80 + 2 * q^81 + 32 * q^83 - 6 * q^84 - 2 * q^85 + 2 * q^87 + 2 * q^88 + 4 * q^89 + 2 * q^90 + 2 * q^92 - 2 * q^93 + 4 * q^94 + 16 * q^95 + 2 * q^96 + 4 * q^97 - 14 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	1.00000	0.408248
$7$	0.763932	0.288739	0.144370	−	0.989524i	$-0.453885\pi$
$7$	0.763932	0.288739	0.144370	+	0.989524i	$0.453885\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.23607	−0.390879
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−0.763932	−0.204169
$15$	−1.23607	−0.319151
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	1.23607	0.276393
$21$	−0.763932	−0.166704
$22$	1.00000	0.213201
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	1.00000	0.204124
$25$	−3.47214	−0.694427
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0.763932	0.144370
$29$	−3.23607	−0.600923	−0.300461	−	0.953794i	$-0.597141\pi$
$29$	−3.23607	−0.600923	−0.300461	+	0.953794i	$0.597141\pi$
$30$	1.23607	0.225674
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−1.00000	−0.171499
$35$	0.944272	0.159611
$36$	1.00000	0.166667
$37$	9.70820	1.59602	0.798009	−	0.602645i	$-0.205886\pi$
$37$	9.70820	1.59602	0.798009	+	0.602645i	$0.205886\pi$
$38$	−6.47214	−1.04992
$39$	0	0
$40$	−1.23607	−0.195440
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0.763932	0.117877
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	−1.00000	−0.150756
$45$	1.23607	0.184262
$46$	−3.23607	−0.477132
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−1.00000	−0.144338
$49$	−6.41641	−0.916630
$50$	3.47214	0.491034
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−4.94427	−0.679148	−0.339574	−	0.940579i	$-0.610283\pi$
$53$	−4.94427	−0.679148	−0.339574	+	0.940579i	$0.610283\pi$
$54$	1.00000	0.136083
$55$	−1.23607	−0.166671
$56$	−0.763932	−0.102085
$57$	−6.47214	−0.857255
$58$	3.23607	0.424917
$59$	1.52786	0.198911	0.0994555	−	0.995042i	$-0.468290\pi$
$59$	1.52786	0.198911	0.0994555	+	0.995042i	$0.468290\pi$
$60$	−1.23607	−0.159576
$61$	5.23607	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$61$	5.23607	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$62$	1.23607	0.156981
$63$	0.763932	0.0962464
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	1.00000	0.121268
$69$	−3.23607	−0.389577
$70$	−0.944272	−0.112862
$71$	8.76393	1.04009	0.520044	−	0.854140i	$-0.325916\pi$
$71$	8.76393	1.04009	0.520044	+	0.854140i	$0.325916\pi$
$72$	−1.00000	−0.117851
$73$	7.52786	0.881070	0.440535	−	0.897735i	$-0.354789\pi$
$73$	7.52786	0.881070	0.440535	+	0.897735i	$0.354789\pi$
$74$	−9.70820	−1.12856
$75$	3.47214	0.400928
$76$	6.47214	0.742405
$77$	−0.763932	−0.0870581
$78$	0	0
$79$	12.1803	1.37040	0.685198	−	0.728357i	$-0.259716\pi$
$79$	12.1803	1.37040	0.685198	+	0.728357i	$0.259716\pi$
$80$	1.23607	0.138197
$81$	1.00000	0.111111
$82$	4.47214	0.493865
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	−0.763932	−0.0833518
$85$	1.23607	0.134070
$86$	−8.94427	−0.964486
$87$	3.23607	0.346943
$88$	1.00000	0.106600
$89$	10.9443	1.16009	0.580045	−	0.814584i	$-0.303035\pi$
$89$	10.9443	1.16009	0.580045	+	0.814584i	$0.303035\pi$
$90$	−1.23607	−0.130293
$91$	0	0
$92$	3.23607	0.337383
$93$	1.23607	0.128174
$94$	2.00000	0.206284
$95$	8.00000	0.820783
$96$	1.00000	0.102062
$97$	10.9443	1.11122	0.555611	−	0.831442i	$-0.312484\pi$
$97$	10.9443	1.11122	0.555611	+	0.831442i	$0.312484\pi$
$98$	6.41641	0.648155
$99$	−1.00000	−0.100504
$100$	−3.47214	−0.347214
$101$	−12.4721	−1.24102	−0.620512	−	0.784197i	$-0.713075\pi$
$101$	−12.4721	−1.24102	−0.620512	+	0.784197i	$0.713075\pi$
$102$	1.00000	0.0990148
$103$	1.52786	0.150545	0.0752725	−	0.997163i	$-0.476017\pi$
$103$	1.52786	0.150545	0.0752725	+	0.997163i	$0.476017\pi$
$104$	0	0
$105$	−0.944272	−0.0921515
$106$	4.94427	0.480230
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	3.70820	0.355182	0.177591	−	0.984104i	$-0.443170\pi$
$109$	3.70820	0.355182	0.177591	+	0.984104i	$0.443170\pi$
$110$	1.23607	0.117854
$111$	−9.70820	−0.921462
$112$	0.763932	0.0721848
$113$	−9.41641	−0.885821	−0.442911	−	0.896566i	$-0.646054\pi$
$113$	−9.41641	−0.885821	−0.442911	+	0.896566i	$0.646054\pi$
$114$	6.47214	0.606171
$115$	4.00000	0.373002
$116$	−3.23607	−0.300461
$117$	0	0
$118$	−1.52786	−0.140651
$119$	0.763932	0.0700295
$120$	1.23607	0.112837
$121$	1.00000	0.0909091
$122$	−5.23607	−0.474051
$123$	4.47214	0.403239
$124$	−1.23607	−0.111002
$125$	−10.4721	−0.936656
$126$	−0.763932	−0.0680565
$127$	16.4721	1.46167	0.730833	−	0.682556i	$-0.239132\pi$
$127$	16.4721	1.46167	0.730833	+	0.682556i	$0.239132\pi$
$128$	−1.00000	−0.0883883
$129$	−8.94427	−0.787499
$130$	0	0
$131$	−11.4164	−0.997456	−0.498728	−	0.866758i	$-0.666199\pi$
$131$	−11.4164	−0.997456	−0.498728	+	0.866758i	$0.666199\pi$
$132$	1.00000	0.0870388
$133$	4.94427	0.428723
$134$	0	0
$135$	−1.23607	−0.106384
$136$	−1.00000	−0.0857493
$137$	2.94427	0.251546	0.125773	−	0.992059i	$-0.459859\pi$
$137$	2.94427	0.251546	0.125773	+	0.992059i	$0.459859\pi$
$138$	3.23607	0.275472
$139$	10.4721	0.888235	0.444117	−	0.895969i	$-0.353517\pi$
$139$	10.4721	0.888235	0.444117	+	0.895969i	$0.353517\pi$
$140$	0.944272	0.0798055
$141$	2.00000	0.168430
$142$	−8.76393	−0.735453
$143$	0	0
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−7.52786	−0.623010
$147$	6.41641	0.529216
$148$	9.70820	0.798009
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−3.47214	−0.283499
$151$	3.52786	0.287094	0.143547	−	0.989644i	$-0.454149\pi$
$151$	3.52786	0.287094	0.143547	+	0.989644i	$0.454149\pi$
$152$	−6.47214	−0.524960
$153$	1.00000	0.0808452
$154$	0.763932	0.0615594
$155$	−1.52786	−0.122721
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−12.1803	−0.969016
$159$	4.94427	0.392106
$160$	−1.23607	−0.0977198
$161$	2.47214	0.194832
$162$	−1.00000	−0.0785674
$163$	−0.944272	−0.0739611	−0.0369805	−	0.999316i	$-0.511774\pi$
$163$	−0.944272	−0.0739611	−0.0369805	+	0.999316i	$0.511774\pi$
$164$	−4.47214	−0.349215
$165$	1.23607	0.0962278
$166$	−16.0000	−1.24184
$167$	12.6525	0.979078	0.489539	−	0.871981i	$-0.337165\pi$
$167$	12.6525	0.979078	0.489539	+	0.871981i	$0.337165\pi$
$168$	0.763932	0.0589386
$169$	−13.0000	−1.00000
$170$	−1.23607	−0.0948021
$171$	6.47214	0.494937
$172$	8.94427	0.681994
$173$	17.7082	1.34633	0.673165	−	0.739492i	$-0.264934\pi$
$173$	17.7082	1.34633	0.673165	+	0.739492i	$0.264934\pi$
$174$	−3.23607	−0.245326
$175$	−2.65248	−0.200508
$176$	−1.00000	−0.0753778
$177$	−1.52786	−0.114841
$178$	−10.9443	−0.820308
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	1.23607	0.0921311
$181$	0.763932	0.0567826	0.0283913	−	0.999597i	$-0.490962\pi$
$181$	0.763932	0.0567826	0.0283913	+	0.999597i	$0.490962\pi$
$182$	0	0
$183$	−5.23607	−0.387061
$184$	−3.23607	−0.238566
$185$	12.0000	0.882258
$186$	−1.23607	−0.0906329
$187$	−1.00000	−0.0731272
$188$	−2.00000	−0.145865
$189$	−0.763932	−0.0555679
$190$	−8.00000	−0.580381
$191$	10.9443	0.791900	0.395950	−	0.918272i	$-0.370415\pi$
$191$	10.9443	0.791900	0.395950	+	0.918272i	$0.370415\pi$
$192$	−1.00000	−0.0721688
$193$	8.47214	0.609838	0.304919	−	0.952378i	$-0.401371\pi$
$193$	8.47214	0.609838	0.304919	+	0.952378i	$0.401371\pi$
$194$	−10.9443	−0.785753
$195$	0	0
$196$	−6.41641	−0.458315
$197$	−18.6525	−1.32893	−0.664467	−	0.747318i	$-0.731341\pi$
$197$	−18.6525	−1.32893	−0.664467	+	0.747318i	$0.731341\pi$
$198$	1.00000	0.0710669
$199$	−12.6525	−0.896910	−0.448455	−	0.893805i	$-0.648026\pi$
$199$	−12.6525	−0.896910	−0.448455	+	0.893805i	$0.648026\pi$
$200$	3.47214	0.245517
$201$	0	0
$202$	12.4721	0.877536
$203$	−2.47214	−0.173510
$204$	−1.00000	−0.0700140
$205$	−5.52786	−0.386083
$206$	−1.52786	−0.106451
$207$	3.23607	0.224922
$208$	0	0
$209$	−6.47214	−0.447687
$210$	0.944272	0.0651610
$211$	−21.8885	−1.50687	−0.753435	−	0.657523i	$-0.771604\pi$
$211$	−21.8885	−1.50687	−0.753435	+	0.657523i	$0.771604\pi$
$212$	−4.94427	−0.339574
$213$	−8.76393	−0.600495
$214$	−4.00000	−0.273434
$215$	11.0557	0.753994
$216$	1.00000	0.0680414
$217$	−0.944272	−0.0641014
$218$	−3.70820	−0.251151
$219$	−7.52786	−0.508686
$220$	−1.23607	−0.0833357
$221$	0	0
$222$	9.70820	0.651572
$223$	2.47214	0.165546	0.0827732	−	0.996568i	$-0.473622\pi$
$223$	2.47214	0.165546	0.0827732	+	0.996568i	$0.473622\pi$
$224$	−0.763932	−0.0510424
$225$	−3.47214	−0.231476
$226$	9.41641	0.626370
$227$	−24.3607	−1.61688	−0.808438	−	0.588582i	$-0.799686\pi$
$227$	−24.3607	−1.61688	−0.808438	+	0.588582i	$0.799686\pi$
$228$	−6.47214	−0.428628
$229$	−28.8328	−1.90533	−0.952663	−	0.304028i	$-0.901668\pi$
$229$	−28.8328	−1.90533	−0.952663	+	0.304028i	$0.901668\pi$
$230$	−4.00000	−0.263752
$231$	0.763932	0.0502630
$232$	3.23607	0.212458
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	0	0
$235$	−2.47214	−0.161264
$236$	1.52786	0.0994555
$237$	−12.1803	−0.791198
$238$	−0.763932	−0.0495184
$239$	−9.52786	−0.616306	−0.308153	−	0.951337i	$-0.599711\pi$
$239$	−9.52786	−0.616306	−0.308153	+	0.951337i	$0.599711\pi$
$240$	−1.23607	−0.0797878
$241$	26.3607	1.69804	0.849020	−	0.528360i	$-0.177193\pi$
$241$	26.3607	1.69804	0.849020	+	0.528360i	$0.177193\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	5.23607	0.335205
$245$	−7.93112	−0.506700
$246$	−4.47214	−0.285133
$247$	0	0
$248$	1.23607	0.0784904
$249$	−16.0000	−1.01396
$250$	10.4721	0.662316
$251$	−3.05573	−0.192876	−0.0964379	−	0.995339i	$-0.530745\pi$
$251$	−3.05573	−0.192876	−0.0964379	+	0.995339i	$0.530745\pi$
$252$	0.763932	0.0481232
$253$	−3.23607	−0.203450
$254$	−16.4721	−1.03355
$255$	−1.23607	−0.0774056
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	8.94427	0.556846
$259$	7.41641	0.460833
$260$	0	0
$261$	−3.23607	−0.200308
$262$	11.4164	0.705308
$263$	31.4164	1.93722	0.968609	−	0.248588i	$-0.0799665\pi$
$263$	31.4164	1.93722	0.968609	+	0.248588i	$0.0799665\pi$
$264$	−1.00000	−0.0615457
$265$	−6.11146	−0.375424
$266$	−4.94427	−0.303153
$267$	−10.9443	−0.669779
$268$	0	0
$269$	1.23607	0.0753644	0.0376822	−	0.999290i	$-0.488003\pi$
$269$	1.23607	0.0753644	0.0376822	+	0.999290i	$0.488003\pi$
$270$	1.23607	0.0752247
$271$	−24.4721	−1.48658	−0.743288	−	0.668971i	$-0.766735\pi$
$271$	−24.4721	−1.48658	−0.743288	+	0.668971i	$0.766735\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−2.94427	−0.177870
$275$	3.47214	0.209378
$276$	−3.23607	−0.194788
$277$	−5.23607	−0.314605	−0.157302	−	0.987550i	$-0.550280\pi$
$277$	−5.23607	−0.314605	−0.157302	+	0.987550i	$0.550280\pi$
$278$	−10.4721	−0.628077
$279$	−1.23607	−0.0740015
$280$	−0.944272	−0.0564310
$281$	−1.05573	−0.0629795	−0.0314897	−	0.999504i	$-0.510025\pi$
$281$	−1.05573	−0.0629795	−0.0314897	+	0.999504i	$0.510025\pi$
$282$	−2.00000	−0.119098
$283$	−0.944272	−0.0561311	−0.0280656	−	0.999606i	$-0.508935\pi$
$283$	−0.944272	−0.0561311	−0.0280656	+	0.999606i	$0.508935\pi$
$284$	8.76393	0.520044
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−3.41641	−0.201664
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	4.00000	0.234888
$291$	−10.9443	−0.641565
$292$	7.52786	0.440535
$293$	−10.9443	−0.639371	−0.319686	−	0.947524i	$-0.603577\pi$
$293$	−10.9443	−0.639371	−0.319686	+	0.947524i	$0.603577\pi$
$294$	−6.41641	−0.374213
$295$	1.88854	0.109955
$296$	−9.70820	−0.564278
$297$	1.00000	0.0580259
$298$	−2.00000	−0.115857
$299$	0	0
$300$	3.47214	0.200464
$301$	6.83282	0.393837
$302$	−3.52786	−0.203006
$303$	12.4721	0.716505
$304$	6.47214	0.371202
$305$	6.47214	0.370593
$306$	−1.00000	−0.0571662
$307$	3.05573	0.174400	0.0871998	−	0.996191i	$-0.472208\pi$
$307$	3.05573	0.174400	0.0871998	+	0.996191i	$0.472208\pi$
$308$	−0.763932	−0.0435291
$309$	−1.52786	−0.0869171
$310$	1.52786	0.0867768
$311$	−20.7639	−1.17741	−0.588707	−	0.808346i	$-0.700363\pi$
$311$	−20.7639	−1.17741	−0.588707	+	0.808346i	$0.700363\pi$
$312$	0	0
$313$	−21.4164	−1.21053	−0.605263	−	0.796025i	$-0.706932\pi$
$313$	−21.4164	−1.21053	−0.605263	+	0.796025i	$0.706932\pi$
$314$	6.00000	0.338600
$315$	0.944272	0.0532037
$316$	12.1803	0.685198
$317$	12.6525	0.710634	0.355317	−	0.934746i	$-0.384373\pi$
$317$	12.6525	0.710634	0.355317	+	0.934746i	$0.384373\pi$
$318$	−4.94427	−0.277261
$319$	3.23607	0.181185
$320$	1.23607	0.0690983
$321$	−4.00000	−0.223258
$322$	−2.47214	−0.137767
$323$	6.47214	0.360119
$324$	1.00000	0.0555556
$325$	0	0
$326$	0.944272	0.0522984
$327$	−3.70820	−0.205064
$328$	4.47214	0.246932
$329$	−1.52786	−0.0842339
$330$	−1.23607	−0.0680433
$331$	16.9443	0.931341	0.465671	−	0.884958i	$-0.345813\pi$
$331$	16.9443	0.931341	0.465671	+	0.884958i	$0.345813\pi$
$332$	16.0000	0.878114
$333$	9.70820	0.532006
$334$	−12.6525	−0.692313
$335$	0	0
$336$	−0.763932	−0.0416759
$337$	−26.9443	−1.46775	−0.733874	−	0.679286i	$-0.762290\pi$
$337$	−26.9443	−1.46775	−0.733874	+	0.679286i	$0.762290\pi$
$338$	13.0000	0.707107
$339$	9.41641	0.511429
$340$	1.23607	0.0670352
$341$	1.23607	0.0669368
$342$	−6.47214	−0.349973
$343$	−10.2492	−0.553406
$344$	−8.94427	−0.482243
$345$	−4.00000	−0.215353
$346$	−17.7082	−0.951999
$347$	21.8885	1.17504	0.587519	−	0.809210i	$-0.300105\pi$
$347$	21.8885	1.17504	0.587519	+	0.809210i	$0.300105\pi$
$348$	3.23607	0.173471
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	2.65248	0.141781
$351$	0	0
$352$	1.00000	0.0533002
$353$	−3.88854	−0.206966	−0.103483	−	0.994631i	$-0.532999\pi$
$353$	−3.88854	−0.206966	−0.103483	+	0.994631i	$0.532999\pi$
$354$	1.52786	0.0812051
$355$	10.8328	0.574946
$356$	10.9443	0.580045
$357$	−0.763932	−0.0404316
$358$	24.0000	1.26844
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	−1.23607	−0.0651465
$361$	22.8885	1.20466
$362$	−0.763932	−0.0401514
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	9.30495	0.487043
$366$	5.23607	0.273694
$367$	−32.0689	−1.67398	−0.836991	−	0.547217i	$-0.815687\pi$
$367$	−32.0689	−1.67398	−0.836991	+	0.547217i	$0.815687\pi$
$368$	3.23607	0.168692
$369$	−4.47214	−0.232810
$370$	−12.0000	−0.623850
$371$	−3.77709	−0.196097
$372$	1.23607	0.0640871
$373$	−30.8328	−1.59646	−0.798231	−	0.602351i	$-0.794231\pi$
$373$	−30.8328	−1.59646	−0.798231	+	0.602351i	$0.794231\pi$
$374$	1.00000	0.0517088
$375$	10.4721	0.540779
$376$	2.00000	0.103142
$377$	0	0
$378$	0.763932	0.0392924
$379$	26.8328	1.37831	0.689155	−	0.724614i	$-0.257982\pi$
$379$	26.8328	1.37831	0.689155	+	0.724614i	$0.257982\pi$
$380$	8.00000	0.410391
$381$	−16.4721	−0.843893
$382$	−10.9443	−0.559958
$383$	6.94427	0.354836	0.177418	−	0.984136i	$-0.443226\pi$
$383$	6.94427	0.354836	0.177418	+	0.984136i	$0.443226\pi$
$384$	1.00000	0.0510310
$385$	−0.944272	−0.0481246
$386$	−8.47214	−0.431220
$387$	8.94427	0.454663
$388$	10.9443	0.555611
$389$	−2.47214	−0.125342	−0.0626711	−	0.998034i	$-0.519962\pi$
$389$	−2.47214	−0.125342	−0.0626711	+	0.998034i	$0.519962\pi$
$390$	0	0
$391$	3.23607	0.163655
$392$	6.41641	0.324078
$393$	11.4164	0.575882
$394$	18.6525	0.939698
$395$	15.0557	0.757536
$396$	−1.00000	−0.0502519
$397$	−15.2361	−0.764676	−0.382338	−	0.924022i	$-0.624881\pi$
$397$	−15.2361	−0.764676	−0.382338	+	0.924022i	$0.624881\pi$
$398$	12.6525	0.634211
$399$	−4.94427	−0.247523
$400$	−3.47214	−0.173607
$401$	17.4164	0.869734	0.434867	−	0.900495i	$-0.356795\pi$
$401$	17.4164	0.869734	0.434867	+	0.900495i	$0.356795\pi$
$402$	0	0
$403$	0	0
$404$	−12.4721	−0.620512
$405$	1.23607	0.0614207
$406$	2.47214	0.122690
$407$	−9.70820	−0.481218
$408$	1.00000	0.0495074
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	5.52786	0.273002
$411$	−2.94427	−0.145230
$412$	1.52786	0.0752725
$413$	1.16718	0.0574334
$414$	−3.23607	−0.159044
$415$	19.7771	0.970819
$416$	0	0
$417$	−10.4721	−0.512823
$418$	6.47214	0.316563
$419$	−7.41641	−0.362315	−0.181158	−	0.983454i	$-0.557984\pi$
$419$	−7.41641	−0.362315	−0.181158	+	0.983454i	$0.557984\pi$
$420$	−0.944272	−0.0460758
$421$	−33.4164	−1.62862	−0.814308	−	0.580433i	$-0.802883\pi$
$421$	−33.4164	−1.62862	−0.814308	+	0.580433i	$0.802883\pi$
$422$	21.8885	1.06552
$423$	−2.00000	−0.0972433
$424$	4.94427	0.240115
$425$	−3.47214	−0.168423
$426$	8.76393	0.424614
$427$	4.00000	0.193574
$428$	4.00000	0.193347
$429$	0	0
$430$	−11.0557	−0.533155
$431$	9.81966	0.472996	0.236498	−	0.971632i	$-0.424000\pi$
$431$	9.81966	0.472996	0.236498	+	0.971632i	$0.424000\pi$
$432$	−1.00000	−0.0481125
$433$	−2.58359	−0.124160	−0.0620798	−	0.998071i	$-0.519773\pi$
$433$	−2.58359	−0.124160	−0.0620798	+	0.998071i	$0.519773\pi$
$434$	0.944272	0.0453265
$435$	4.00000	0.191785
$436$	3.70820	0.177591
$437$	20.9443	1.00190
$438$	7.52786	0.359695
$439$	36.7639	1.75465	0.877323	−	0.479900i	$-0.159327\pi$
$439$	36.7639	1.75465	0.877323	+	0.479900i	$0.159327\pi$
$440$	1.23607	0.0589272
$441$	−6.41641	−0.305543
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	−9.70820	−0.460731
$445$	13.5279	0.641282
$446$	−2.47214	−0.117059
$447$	−2.00000	−0.0945968
$448$	0.763932	0.0360924
$449$	−13.4164	−0.633159	−0.316580	−	0.948566i	$-0.602534\pi$
$449$	−13.4164	−0.633159	−0.316580	+	0.948566i	$0.602534\pi$
$450$	3.47214	0.163678
$451$	4.47214	0.210585
$452$	−9.41641	−0.442911
$453$	−3.52786	−0.165754
$454$	24.3607	1.14330
$455$	0	0
$456$	6.47214	0.303086
$457$	0.111456	0.00521370	0.00260685	−	0.999997i	$-0.499170\pi$
$457$	0.111456	0.00521370	0.00260685	+	0.999997i	$0.499170\pi$
$458$	28.8328	1.34727
$459$	−1.00000	−0.0466760
$460$	4.00000	0.186501
$461$	14.3607	0.668844	0.334422	−	0.942424i	$-0.391459\pi$
$461$	14.3607	0.668844	0.334422	+	0.942424i	$0.391459\pi$
$462$	−0.763932	−0.0355413
$463$	−14.4721	−0.672577	−0.336289	−	0.941759i	$-0.609172\pi$
$463$	−14.4721	−0.672577	−0.336289	+	0.941759i	$0.609172\pi$
$464$	−3.23607	−0.150231
$465$	1.52786	0.0708530
$466$	−2.94427	−0.136391
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	0	0
$469$	0	0
$470$	2.47214	0.114031
$471$	6.00000	0.276465
$472$	−1.52786	−0.0703256
$473$	−8.94427	−0.411258
$474$	12.1803	0.559462
$475$	−22.4721	−1.03109
$476$	0.763932	0.0350148
$477$	−4.94427	−0.226383
$478$	9.52786	0.435794
$479$	12.2918	0.561626	0.280813	−	0.959762i	$-0.409396\pi$
$479$	12.2918	0.561626	0.280813	+	0.959762i	$0.409396\pi$
$480$	1.23607	0.0564185
$481$	0	0
$482$	−26.3607	−1.20070
$483$	−2.47214	−0.112486
$484$	1.00000	0.0454545
$485$	13.5279	0.614269
$486$	1.00000	0.0453609
$487$	5.81966	0.263714	0.131857	−	0.991269i	$-0.457906\pi$
$487$	5.81966	0.263714	0.131857	+	0.991269i	$0.457906\pi$
$488$	−5.23607	−0.237026
$489$	0.944272	0.0427015
$490$	7.93112	0.358291
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	4.47214	0.201619
$493$	−3.23607	−0.145745
$494$	0	0
$495$	−1.23607	−0.0555571
$496$	−1.23607	−0.0555011
$497$	6.69505	0.300314
$498$	16.0000	0.716977
$499$	−30.4721	−1.36412	−0.682060	−	0.731296i	$-0.738916\pi$
$499$	−30.4721	−1.36412	−0.682060	+	0.731296i	$0.738916\pi$
$500$	−10.4721	−0.468328
$501$	−12.6525	−0.565271
$502$	3.05573	0.136384
$503$	13.8197	0.616188	0.308094	−	0.951356i	$-0.400309\pi$
$503$	13.8197	0.616188	0.308094	+	0.951356i	$0.400309\pi$
$504$	−0.763932	−0.0340282
$505$	−15.4164	−0.686021
$506$	3.23607	0.143861
$507$	13.0000	0.577350
$508$	16.4721	0.730833
$509$	−0.583592	−0.0258673	−0.0129336	−	0.999916i	$-0.504117\pi$
$509$	−0.583592	−0.0258673	−0.0129336	+	0.999916i	$0.504117\pi$
$510$	1.23607	0.0547340
$511$	5.75078	0.254399
$512$	−1.00000	−0.0441942
$513$	−6.47214	−0.285752
$514$	−2.00000	−0.0882162
$515$	1.88854	0.0832192
$516$	−8.94427	−0.393750
$517$	2.00000	0.0879599
$518$	−7.41641	−0.325858
$519$	−17.7082	−0.777304
$520$	0	0
$521$	44.4721	1.94836	0.974180	−	0.225773i	$-0.0724909\pi$
$521$	44.4721	1.94836	0.974180	+	0.225773i	$0.0724909\pi$
$522$	3.23607	0.141639
$523$	−15.4164	−0.674112	−0.337056	−	0.941485i	$-0.609431\pi$
$523$	−15.4164	−0.674112	−0.337056	+	0.941485i	$0.609431\pi$
$524$	−11.4164	−0.498728
$525$	2.65248	0.115764
$526$	−31.4164	−1.36982
$527$	−1.23607	−0.0538440
$528$	1.00000	0.0435194
$529$	−12.5279	−0.544690
$530$	6.11146	0.265465
$531$	1.52786	0.0663037
$532$	4.94427	0.214361
$533$	0	0
$534$	10.9443	0.473605
$535$	4.94427	0.213760
$536$	0	0
$537$	24.0000	1.03568
$538$	−1.23607	−0.0532907
$539$	6.41641	0.276374
$540$	−1.23607	−0.0531919
$541$	11.1246	0.478284	0.239142	−	0.970985i	$-0.423134\pi$
$541$	11.1246	0.478284	0.239142	+	0.970985i	$0.423134\pi$
$542$	24.4721	1.05117
$543$	−0.763932	−0.0327835
$544$	−1.00000	−0.0428746
$545$	4.58359	0.196340
$546$	0	0
$547$	10.4721	0.447756	0.223878	−	0.974617i	$-0.428128\pi$
$547$	10.4721	0.447756	0.223878	+	0.974617i	$0.428128\pi$
$548$	2.94427	0.125773
$549$	5.23607	0.223470
$550$	−3.47214	−0.148052
$551$	−20.9443	−0.892256
$552$	3.23607	0.137736
$553$	9.30495	0.395687
$554$	5.23607	0.222459
$555$	−12.0000	−0.509372
$556$	10.4721	0.444117
$557$	17.4164	0.737957	0.368978	−	0.929438i	$-0.379708\pi$
$557$	17.4164	0.737957	0.368978	+	0.929438i	$0.379708\pi$
$558$	1.23607	0.0523269
$559$	0	0
$560$	0.944272	0.0399028
$561$	1.00000	0.0422200
$562$	1.05573	0.0445332
$563$	12.9443	0.545536	0.272768	−	0.962080i	$-0.412061\pi$
$563$	12.9443	0.545536	0.272768	+	0.962080i	$0.412061\pi$
$564$	2.00000	0.0842152
$565$	−11.6393	−0.489670
$566$	0.944272	0.0396907
$567$	0.763932	0.0320821
$568$	−8.76393	−0.367726
$569$	−28.4721	−1.19361	−0.596807	−	0.802385i	$-0.703564\pi$
$569$	−28.4721	−1.19361	−0.596807	+	0.802385i	$0.703564\pi$
$570$	8.00000	0.335083
$571$	10.4721	0.438245	0.219123	−	0.975697i	$-0.429681\pi$
$571$	10.4721	0.438245	0.219123	+	0.975697i	$0.429681\pi$
$572$	0	0
$573$	−10.9443	−0.457204
$574$	3.41641	0.142598
$575$	−11.2361	−0.468576
$576$	1.00000	0.0416667
$577$	28.8328	1.20033	0.600163	−	0.799878i	$-0.295102\pi$
$577$	28.8328	1.20033	0.600163	+	0.799878i	$0.295102\pi$
$578$	−1.00000	−0.0415945
$579$	−8.47214	−0.352090
$580$	−4.00000	−0.166091
$581$	12.2229	0.507092
$582$	10.9443	0.453655
$583$	4.94427	0.204771
$584$	−7.52786	−0.311505
$585$	0	0
$586$	10.9443	0.452104
$587$	−25.8885	−1.06853	−0.534267	−	0.845316i	$-0.679412\pi$
$587$	−25.8885	−1.06853	−0.534267	+	0.845316i	$0.679412\pi$
$588$	6.41641	0.264608
$589$	−8.00000	−0.329634
$590$	−1.88854	−0.0777501
$591$	18.6525	0.767260
$592$	9.70820	0.399005
$593$	24.4721	1.00495	0.502475	−	0.864592i	$-0.332423\pi$
$593$	24.4721	1.00495	0.502475	+	0.864592i	$0.332423\pi$
$594$	−1.00000	−0.0410305
$595$	0.944272	0.0387114
$596$	2.00000	0.0819232
$597$	12.6525	0.517831
$598$	0	0
$599$	−39.3050	−1.60596	−0.802978	−	0.596008i	$-0.796753\pi$
$599$	−39.3050	−1.60596	−0.802978	+	0.596008i	$0.796753\pi$
$600$	−3.47214	−0.141749
$601$	43.3050	1.76645	0.883223	−	0.468953i	$-0.155369\pi$
$601$	43.3050	1.76645	0.883223	+	0.468953i	$0.155369\pi$
$602$	−6.83282	−0.278485
$603$	0	0
$604$	3.52786	0.143547
$605$	1.23607	0.0502533
$606$	−12.4721	−0.506646
$607$	2.29180	0.0930211	0.0465106	−	0.998918i	$-0.485190\pi$
$607$	2.29180	0.0930211	0.0465106	+	0.998918i	$0.485190\pi$
$608$	−6.47214	−0.262480
$609$	2.47214	0.100176
$610$	−6.47214	−0.262049
$611$	0	0
$612$	1.00000	0.0404226
$613$	17.5279	0.707944	0.353972	−	0.935256i	$-0.384831\pi$
$613$	17.5279	0.707944	0.353972	+	0.935256i	$0.384831\pi$
$614$	−3.05573	−0.123319
$615$	5.52786	0.222905
$616$	0.763932	0.0307797
$617$	−0.111456	−0.00448706	−0.00224353	−	0.999997i	$-0.500714\pi$
$617$	−0.111456	−0.00448706	−0.00224353	+	0.999997i	$0.500714\pi$
$618$	1.52786	0.0614597
$619$	−37.3050	−1.49941	−0.749706	−	0.661771i	$-0.769805\pi$
$619$	−37.3050	−1.49941	−0.749706	+	0.661771i	$0.769805\pi$
$620$	−1.52786	−0.0613605
$621$	−3.23607	−0.129859
$622$	20.7639	0.832558
$623$	8.36068	0.334964
$624$	0	0
$625$	4.41641	0.176656
$626$	21.4164	0.855972
$627$	6.47214	0.258472
$628$	−6.00000	−0.239426
$629$	9.70820	0.387091
$630$	−0.944272	−0.0376207
$631$	12.3607	0.492071	0.246035	−	0.969261i	$-0.420872\pi$
$631$	12.3607	0.492071	0.246035	+	0.969261i	$0.420872\pi$
$632$	−12.1803	−0.484508
$633$	21.8885	0.869992
$634$	−12.6525	−0.502494
$635$	20.3607	0.807989
$636$	4.94427	0.196053
$637$	0	0
$638$	−3.23607	−0.128117
$639$	8.76393	0.346696
$640$	−1.23607	−0.0488599
$641$	−36.8328	−1.45481	−0.727404	−	0.686209i	$-0.759274\pi$
$641$	−36.8328	−1.45481	−0.727404	+	0.686209i	$0.759274\pi$
$642$	4.00000	0.157867
$643$	17.5279	0.691231	0.345616	−	0.938376i	$-0.387670\pi$
$643$	17.5279	0.691231	0.345616	+	0.938376i	$0.387670\pi$
$644$	2.47214	0.0974158
$645$	−11.0557	−0.435319
$646$	−6.47214	−0.254643
$647$	37.7771	1.48517	0.742585	−	0.669752i	$-0.233599\pi$
$647$	37.7771	1.48517	0.742585	+	0.669752i	$0.233599\pi$
$648$	−1.00000	−0.0392837
$649$	−1.52786	−0.0599739
$650$	0	0
$651$	0.944272	0.0370089
$652$	−0.944272	−0.0369805
$653$	−9.59675	−0.375550	−0.187775	−	0.982212i	$-0.560128\pi$
$653$	−9.59675	−0.375550	−0.187775	+	0.982212i	$0.560128\pi$
$654$	3.70820	0.145002
$655$	−14.1115	−0.551380
$656$	−4.47214	−0.174608
$657$	7.52786	0.293690
$658$	1.52786	0.0595623
$659$	−41.8885	−1.63175	−0.815873	−	0.578231i	$-0.803743\pi$
$659$	−41.8885	−1.63175	−0.815873	+	0.578231i	$0.803743\pi$
$660$	1.23607	0.0481139
$661$	−50.3607	−1.95880	−0.979402	−	0.201921i	$-0.935281\pi$
$661$	−50.3607	−1.95880	−0.979402	+	0.201921i	$0.935281\pi$
$662$	−16.9443	−0.658558
$663$	0	0
$664$	−16.0000	−0.620920
$665$	6.11146	0.236992
$666$	−9.70820	−0.376185
$667$	−10.4721	−0.405483
$668$	12.6525	0.489539
$669$	−2.47214	−0.0955783
$670$	0	0
$671$	−5.23607	−0.202136
$672$	0.763932	0.0294693
$673$	−31.5279	−1.21531	−0.607655	−	0.794201i	$-0.707890\pi$
$673$	−31.5279	−1.21531	−0.607655	+	0.794201i	$0.707890\pi$
$674$	26.9443	1.03785
$675$	3.47214	0.133643
$676$	−13.0000	−0.500000
$677$	30.0689	1.15564	0.577821	−	0.816164i	$-0.303903\pi$
$677$	30.0689	1.15564	0.577821	+	0.816164i	$0.303903\pi$
$678$	−9.41641	−0.361635
$679$	8.36068	0.320853
$680$	−1.23607	−0.0474010
$681$	24.3607	0.933503
$682$	−1.23607	−0.0473315
$683$	36.3607	1.39130	0.695651	−	0.718380i	$-0.255116\pi$
$683$	36.3607	1.39130	0.695651	+	0.718380i	$0.255116\pi$
$684$	6.47214	0.247468
$685$	3.63932	0.139051
$686$	10.2492	0.391317
$687$	28.8328	1.10004
$688$	8.94427	0.340997
$689$	0	0
$690$	4.00000	0.152277
$691$	2.83282	0.107765	0.0538827	−	0.998547i	$-0.482840\pi$
$691$	2.83282	0.107765	0.0538827	+	0.998547i	$0.482840\pi$
$692$	17.7082	0.673165
$693$	−0.763932	−0.0290194
$694$	−21.8885	−0.830878
$695$	12.9443	0.491004
$696$	−3.23607	−0.122663
$697$	−4.47214	−0.169394
$698$	16.0000	0.605609
$699$	−2.94427	−0.111363
$700$	−2.65248	−0.100254
$701$	−3.52786	−0.133246	−0.0666228	−	0.997778i	$-0.521222\pi$
$701$	−3.52786	−0.133246	−0.0666228	+	0.997778i	$0.521222\pi$
$702$	0	0
$703$	62.8328	2.36978
$704$	−1.00000	−0.0376889
$705$	2.47214	0.0931060
$706$	3.88854	0.146347
$707$	−9.52786	−0.358332
$708$	−1.52786	−0.0574206
$709$	22.2918	0.837186	0.418593	−	0.908174i	$-0.362523\pi$
$709$	22.2918	0.837186	0.418593	+	0.908174i	$0.362523\pi$
$710$	−10.8328	−0.406548
$711$	12.1803	0.456798
$712$	−10.9443	−0.410154
$713$	−4.00000	−0.149801
$714$	0.763932	0.0285894
$715$	0	0
$716$	−24.0000	−0.896922
$717$	9.52786	0.355825
$718$	12.0000	0.447836
$719$	−5.34752	−0.199429	−0.0997145	−	0.995016i	$-0.531793\pi$
$719$	−5.34752	−0.199429	−0.0997145	+	0.995016i	$0.531793\pi$
$720$	1.23607	0.0460655
$721$	1.16718	0.0434682
$722$	−22.8885	−0.851823
$723$	−26.3607	−0.980364
$724$	0.763932	0.0283913
$725$	11.2361	0.417297
$726$	1.00000	0.0371135
$727$	−16.9443	−0.628428	−0.314214	−	0.949352i	$-0.601741\pi$
$727$	−16.9443	−0.628428	−0.314214	+	0.949352i	$0.601741\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−9.30495	−0.344392
$731$	8.94427	0.330816
$732$	−5.23607	−0.193531
$733$	−36.9443	−1.36457	−0.682284	−	0.731087i	$-0.739013\pi$
$733$	−36.9443	−1.36457	−0.682284	+	0.731087i	$0.739013\pi$
$734$	32.0689	1.18368
$735$	7.93112	0.292544
$736$	−3.23607	−0.119283
$737$	0	0
$738$	4.47214	0.164622
$739$	−47.4164	−1.74424	−0.872120	−	0.489291i	$-0.837255\pi$
$739$	−47.4164	−1.74424	−0.872120	+	0.489291i	$0.837255\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	3.77709	0.138661
$743$	−32.2918	−1.18467	−0.592336	−	0.805691i	$-0.701794\pi$
$743$	−32.2918	−1.18467	−0.592336	+	0.805691i	$0.701794\pi$
$744$	−1.23607	−0.0453165
$745$	2.47214	0.0905721
$746$	30.8328	1.12887
$747$	16.0000	0.585409
$748$	−1.00000	−0.0365636
$749$	3.05573	0.111654
$750$	−10.4721	−0.382388
$751$	−0.652476	−0.0238092	−0.0119046	−	0.999929i	$-0.503789\pi$
$751$	−0.652476	−0.0238092	−0.0119046	+	0.999929i	$0.503789\pi$
$752$	−2.00000	−0.0729325
$753$	3.05573	0.111357
$754$	0	0
$755$	4.36068	0.158701
$756$	−0.763932	−0.0277839
$757$	−29.7771	−1.08227	−0.541133	−	0.840937i	$-0.682005\pi$
$757$	−29.7771	−1.08227	−0.541133	+	0.840937i	$0.682005\pi$
$758$	−26.8328	−0.974612
$759$	3.23607	0.117462
$760$	−8.00000	−0.290191
$761$	−32.2492	−1.16903	−0.584517	−	0.811382i	$-0.698716\pi$
$761$	−32.2492	−1.16903	−0.584517	+	0.811382i	$0.698716\pi$
$762$	16.4721	0.596723
$763$	2.83282	0.102555
$764$	10.9443	0.395950
$765$	1.23607	0.0446901
$766$	−6.94427	−0.250907
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	10.9443	0.394661	0.197330	−	0.980337i	$-0.436773\pi$
$769$	10.9443	0.394661	0.197330	+	0.980337i	$0.436773\pi$
$770$	0.944272	0.0340292
$771$	−2.00000	−0.0720282
$772$	8.47214	0.304919
$773$	−10.4721	−0.376657	−0.188328	−	0.982106i	$-0.560307\pi$
$773$	−10.4721	−0.376657	−0.188328	+	0.982106i	$0.560307\pi$
$774$	−8.94427	−0.321495
$775$	4.29180	0.154166
$776$	−10.9443	−0.392876
$777$	−7.41641	−0.266062
$778$	2.47214	0.0886304
$779$	−28.9443	−1.03704
$780$	0	0
$781$	−8.76393	−0.313598
$782$	−3.23607	−0.115722
$783$	3.23607	0.115648
$784$	−6.41641	−0.229157
$785$	−7.41641	−0.264703
$786$	−11.4164	−0.407210
$787$	−7.05573	−0.251510	−0.125755	−	0.992061i	$-0.540135\pi$
$787$	−7.05573	−0.251510	−0.125755	+	0.992061i	$0.540135\pi$
$788$	−18.6525	−0.664467
$789$	−31.4164	−1.11845
$790$	−15.0557	−0.535659
$791$	−7.19350	−0.255771
$792$	1.00000	0.0355335
$793$	0	0
$794$	15.2361	0.540708
$795$	6.11146	0.216751
$796$	−12.6525	−0.448455
$797$	33.8885	1.20039	0.600197	−	0.799852i	$-0.295089\pi$
$797$	33.8885	1.20039	0.600197	+	0.799852i	$0.295089\pi$
$798$	4.94427	0.175025
$799$	−2.00000	−0.0707549
$800$	3.47214	0.122759
$801$	10.9443	0.386697
$802$	−17.4164	−0.614995
$803$	−7.52786	−0.265653
$804$	0	0
$805$	3.05573	0.107700
$806$	0	0
$807$	−1.23607	−0.0435117
$808$	12.4721	0.438768
$809$	15.3050	0.538093	0.269047	−	0.963127i	$-0.413291\pi$
$809$	15.3050	0.538093	0.269047	+	0.963127i	$0.413291\pi$
$810$	−1.23607	−0.0434310
$811$	29.8885	1.04953	0.524764	−	0.851248i	$-0.324153\pi$
$811$	29.8885	1.04953	0.524764	+	0.851248i	$0.324153\pi$
$812$	−2.47214	−0.0867550
$813$	24.4721	0.858275
$814$	9.70820	0.340272
$815$	−1.16718	−0.0408847
$816$	−1.00000	−0.0350070
$817$	57.8885	2.02526
$818$	14.0000	0.489499
$819$	0	0
$820$	−5.52786	−0.193041
$821$	10.2918	0.359186	0.179593	−	0.983741i	$-0.442522\pi$
$821$	10.2918	0.359186	0.179593	+	0.983741i	$0.442522\pi$
$822$	2.94427	0.102693
$823$	9.81966	0.342292	0.171146	−	0.985246i	$-0.445253\pi$
$823$	9.81966	0.342292	0.171146	+	0.985246i	$0.445253\pi$
$824$	−1.52786	−0.0532257
$825$	−3.47214	−0.120884
$826$	−1.16718	−0.0406115
$827$	−5.30495	−0.184471	−0.0922356	−	0.995737i	$-0.529401\pi$
$827$	−5.30495	−0.184471	−0.0922356	+	0.995737i	$0.529401\pi$
$828$	3.23607	0.112461
$829$	−29.7771	−1.03420	−0.517101	−	0.855925i	$-0.672989\pi$
$829$	−29.7771	−1.03420	−0.517101	+	0.855925i	$0.672989\pi$
$830$	−19.7771	−0.686473
$831$	5.23607	0.181637
$832$	0	0
$833$	−6.41641	−0.222315
$834$	10.4721	0.362620
$835$	15.6393	0.541221
$836$	−6.47214	−0.223844
$837$	1.23607	0.0427248
$838$	7.41641	0.256196
$839$	−6.29180	−0.217217	−0.108608	−	0.994085i	$-0.534639\pi$
$839$	−6.29180	−0.217217	−0.108608	+	0.994085i	$0.534639\pi$
$840$	0.944272	0.0325805
$841$	−18.5279	−0.638892
$842$	33.4164	1.15161
$843$	1.05573	0.0363612
$844$	−21.8885	−0.753435
$845$	−16.0689	−0.552786
$846$	2.00000	0.0687614
$847$	0.763932	0.0262490
$848$	−4.94427	−0.169787
$849$	0.944272	0.0324073
$850$	3.47214	0.119093
$851$	31.4164	1.07694
$852$	−8.76393	−0.300247
$853$	30.5410	1.04570	0.522852	−	0.852423i	$-0.324868\pi$
$853$	30.5410	1.04570	0.522852	+	0.852423i	$0.324868\pi$
$854$	−4.00000	−0.136877
$855$	8.00000	0.273594
$856$	−4.00000	−0.136717
$857$	−49.4164	−1.68803	−0.844016	−	0.536318i	$-0.819815\pi$
$857$	−49.4164	−1.68803	−0.844016	+	0.536318i	$0.819815\pi$
$858$	0	0
$859$	−10.1115	−0.344998	−0.172499	−	0.985010i	$-0.555184\pi$
$859$	−10.1115	−0.344998	−0.172499	+	0.985010i	$0.555184\pi$
$860$	11.0557	0.376997
$861$	3.41641	0.116431
$862$	−9.81966	−0.334459
$863$	13.4164	0.456700	0.228350	−	0.973579i	$-0.426667\pi$
$863$	13.4164	0.456700	0.228350	+	0.973579i	$0.426667\pi$
$864$	1.00000	0.0340207
$865$	21.8885	0.744233
$866$	2.58359	0.0877940
$867$	−1.00000	−0.0339618
$868$	−0.944272	−0.0320507
$869$	−12.1803	−0.413190
$870$	−4.00000	−0.135613
$871$	0	0
$872$	−3.70820	−0.125576
$873$	10.9443	0.370407
$874$	−20.9443	−0.708451
$875$	−8.00000	−0.270449
$876$	−7.52786	−0.254343
$877$	10.5410	0.355945	0.177972	−	0.984035i	$-0.443046\pi$
$877$	10.5410	0.355945	0.177972	+	0.984035i	$0.443046\pi$
$878$	−36.7639	−1.24072
$879$	10.9443	0.369141
$880$	−1.23607	−0.0416678
$881$	−5.05573	−0.170332	−0.0851659	−	0.996367i	$-0.527142\pi$
$881$	−5.05573	−0.170332	−0.0851659	+	0.996367i	$0.527142\pi$
$882$	6.41641	0.216052
$883$	−1.16718	−0.0392789	−0.0196394	−	0.999807i	$-0.506252\pi$
$883$	−1.16718	−0.0392789	−0.0196394	+	0.999807i	$0.506252\pi$
$884$	0	0
$885$	−1.88854	−0.0634827
$886$	16.0000	0.537531
$887$	47.1246	1.58229	0.791145	−	0.611629i	$-0.209485\pi$
$887$	47.1246	1.58229	0.791145	+	0.611629i	$0.209485\pi$
$888$	9.70820	0.325786
$889$	12.5836	0.422040
$890$	−13.5279	−0.453455
$891$	−1.00000	−0.0335013
$892$	2.47214	0.0827732
$893$	−12.9443	−0.433164
$894$	2.00000	0.0668900
$895$	−29.6656	−0.991613
$896$	−0.763932	−0.0255212
$897$	0	0
$898$	13.4164	0.447711
$899$	4.00000	0.133407
$900$	−3.47214	−0.115738
$901$	−4.94427	−0.164718
$902$	−4.47214	−0.148906
$903$	−6.83282	−0.227382
$904$	9.41641	0.313185
$905$	0.944272	0.0313887
$906$	3.52786	0.117205
$907$	−18.8328	−0.625333	−0.312667	−	0.949863i	$-0.601222\pi$
$907$	−18.8328	−0.625333	−0.312667	+	0.949863i	$0.601222\pi$
$908$	−24.3607	−0.808438
$909$	−12.4721	−0.413675
$910$	0	0
$911$	−2.29180	−0.0759306	−0.0379653	−	0.999279i	$-0.512088\pi$
$911$	−2.29180	−0.0759306	−0.0379653	+	0.999279i	$0.512088\pi$
$912$	−6.47214	−0.214314
$913$	−16.0000	−0.529523
$914$	−0.111456	−0.00368664
$915$	−6.47214	−0.213962
$916$	−28.8328	−0.952663
$917$	−8.72136	−0.288005
$918$	1.00000	0.0330049
$919$	−0.472136	−0.0155743	−0.00778716	−	0.999970i	$-0.502479\pi$
$919$	−0.472136	−0.0155743	−0.00778716	+	0.999970i	$0.502479\pi$
$920$	−4.00000	−0.131876
$921$	−3.05573	−0.100690
$922$	−14.3607	−0.472944
$923$	0	0
$924$	0.763932	0.0251315
$925$	−33.7082	−1.10832
$926$	14.4721	0.475584
$927$	1.52786	0.0501816
$928$	3.23607	0.106229
$929$	16.8328	0.552267	0.276133	−	0.961119i	$-0.410947\pi$
$929$	16.8328	0.552267	0.276133	+	0.961119i	$0.410947\pi$
$930$	−1.52786	−0.0501006
$931$	−41.5279	−1.36102
$932$	2.94427	0.0964428
$933$	20.7639	0.679781
$934$	−8.94427	−0.292666
$935$	−1.23607	−0.0404237
$936$	0	0
$937$	−11.8885	−0.388382	−0.194191	−	0.980964i	$-0.562208\pi$
$937$	−11.8885	−0.388382	−0.194191	+	0.980964i	$0.562208\pi$
$938$	0	0
$939$	21.4164	0.698898
$940$	−2.47214	−0.0806322
$941$	10.2918	0.335503	0.167751	−	0.985829i	$-0.446349\pi$
$941$	10.2918	0.335503	0.167751	+	0.985829i	$0.446349\pi$
$942$	−6.00000	−0.195491
$943$	−14.4721	−0.471278
$944$	1.52786	0.0497277
$945$	−0.944272	−0.0307172
$946$	8.94427	0.290803
$947$	−39.0557	−1.26914	−0.634570	−	0.772865i	$-0.718823\pi$
$947$	−39.0557	−1.26914	−0.634570	+	0.772865i	$0.718823\pi$
$948$	−12.1803	−0.395599
$949$	0	0
$950$	22.4721	0.729092
$951$	−12.6525	−0.410285
$952$	−0.763932	−0.0247592
$953$	−24.8328	−0.804414	−0.402207	−	0.915549i	$-0.631757\pi$
$953$	−24.8328	−0.804414	−0.402207	+	0.915549i	$0.631757\pi$
$954$	4.94427	0.160077
$955$	13.5279	0.437751
$956$	−9.52786	−0.308153
$957$	−3.23607	−0.104607
$958$	−12.2918	−0.397130
$959$	2.24922	0.0726312
$960$	−1.23607	−0.0398939
$961$	−29.4721	−0.950714
$962$	0	0
$963$	4.00000	0.128898
$964$	26.3607	0.849020
$965$	10.4721	0.337110
$966$	2.47214	0.0795397
$967$	33.7771	1.08620	0.543099	−	0.839669i	$-0.317251\pi$
$967$	33.7771	1.08620	0.543099	+	0.839669i	$0.317251\pi$
$968$	−1.00000	−0.0321412
$969$	−6.47214	−0.207915
$970$	−13.5279	−0.434354
$971$	−24.3607	−0.781771	−0.390886	−	0.920439i	$-0.627831\pi$
$971$	−24.3607	−0.781771	−0.390886	+	0.920439i	$0.627831\pi$
$972$	−1.00000	−0.0320750
$973$	8.00000	0.256468
$974$	−5.81966	−0.186474
$975$	0	0
$976$	5.23607	0.167602
$977$	36.8328	1.17839	0.589193	−	0.807992i	$-0.299446\pi$
$977$	36.8328	1.17839	0.589193	+	0.807992i	$0.299446\pi$
$978$	−0.944272	−0.0301945
$979$	−10.9443	−0.349780
$980$	−7.93112	−0.253350
$981$	3.70820	0.118394
$982$	−12.0000	−0.382935
$983$	−23.2361	−0.741115	−0.370558	−	0.928809i	$-0.620833\pi$
$983$	−23.2361	−0.741115	−0.370558	+	0.928809i	$0.620833\pi$
$984$	−4.47214	−0.142566
$985$	−23.0557	−0.734617
$986$	3.23607	0.103057
$987$	1.52786	0.0486324
$988$	0	0
$989$	28.9443	0.920374
$990$	1.23607	0.0392848
$991$	33.2361	1.05578	0.527889	−	0.849313i	$-0.322984\pi$
$991$	33.2361	1.05578	0.527889	+	0.849313i	$0.322984\pi$
$992$	1.23607	0.0392452
$993$	−16.9443	−0.537710
$994$	−6.69505	−0.212354
$995$	−15.6393	−0.495800
$996$	−16.0000	−0.506979
$997$	7.34752	0.232698	0.116349	−	0.993208i	$-0.462881\pi$
$997$	7.34752	0.232698	0.116349	+	0.993208i	$0.462881\pi$
$998$	30.4721	0.964579
$999$	−9.70820	−0.307154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1122.2.a.o.1.2	✓	2
3.2	odd	2		3366.2.a.y.1.1		2
4.3	odd	2		8976.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1122.2.a.o.1.2	✓	2	1.1	even	1	trivial
3366.2.a.y.1.1		2	3.2	odd	2
8976.2.a.bo.1.2		2	4.3	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:	\( N \)	\(=\)	\( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1122.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +0.763932 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +0.763932 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} -0.763932 q^{14} -1.23607 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.47214 q^{19} +1.23607 q^{20} -0.763932 q^{21} +1.00000 q^{22} +3.23607 q^{23} +1.00000 q^{24} -3.47214 q^{25} -1.00000 q^{27} +0.763932 q^{28} -3.23607 q^{29} +1.23607 q^{30} -1.23607 q^{31} -1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +0.944272 q^{35} +1.00000 q^{36} +9.70820 q^{37} -6.47214 q^{38} -1.23607 q^{40} -4.47214 q^{41} +0.763932 q^{42} +8.94427 q^{43} -1.00000 q^{44} +1.23607 q^{45} -3.23607 q^{46} -2.00000 q^{47} -1.00000 q^{48} -6.41641 q^{49} +3.47214 q^{50} -1.00000 q^{51} -4.94427 q^{53} +1.00000 q^{54} -1.23607 q^{55} -0.763932 q^{56} -6.47214 q^{57} +3.23607 q^{58} +1.52786 q^{59} -1.23607 q^{60} +5.23607 q^{61} +1.23607 q^{62} +0.763932 q^{63} +1.00000 q^{64} -1.00000 q^{66} +1.00000 q^{68} -3.23607 q^{69} -0.944272 q^{70} +8.76393 q^{71} -1.00000 q^{72} +7.52786 q^{73} -9.70820 q^{74} +3.47214 q^{75} +6.47214 q^{76} -0.763932 q^{77} +12.1803 q^{79} +1.23607 q^{80} +1.00000 q^{81} +4.47214 q^{82} +16.0000 q^{83} -0.763932 q^{84} +1.23607 q^{85} -8.94427 q^{86} +3.23607 q^{87} +1.00000 q^{88} +10.9443 q^{89} -1.23607 q^{90} +3.23607 q^{92} +1.23607 q^{93} +2.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +10.9443 q^{97} +6.41641 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} - 2 q^{12} - 6 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 4 q^{19} - 2 q^{20} - 6 q^{21} + 2 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{27} + 6 q^{28} - 2 q^{29} - 2 q^{30} + 2 q^{31} - 2 q^{32} + 2 q^{33} - 2 q^{34} - 16 q^{35} + 2 q^{36} + 6 q^{37} - 4 q^{38} + 2 q^{40} + 6 q^{42} - 2 q^{44} - 2 q^{45} - 2 q^{46} - 4 q^{47} - 2 q^{48} + 14 q^{49} - 2 q^{50} - 2 q^{51} + 8 q^{53} + 2 q^{54} + 2 q^{55} - 6 q^{56} - 4 q^{57} + 2 q^{58} + 12 q^{59} + 2 q^{60} + 6 q^{61} - 2 q^{62} + 6 q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{68} - 2 q^{69} + 16 q^{70} + 22 q^{71} - 2 q^{72} + 24 q^{73} - 6 q^{74} - 2 q^{75} + 4 q^{76} - 6 q^{77} + 2 q^{79} - 2 q^{80} + 2 q^{81} + 32 q^{83} - 6 q^{84} - 2 q^{85} + 2 q^{87} + 2 q^{88} + 4 q^{89} + 2 q^{90} + 2 q^{92} - 2 q^{93} + 4 q^{94} + 16 q^{95} + 2 q^{96} + 4 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 - 2 * q^11 - 2 * q^12 - 6 * q^14 + 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 4 * q^19 - 2 * q^20 - 6 * q^21 + 2 * q^22 + 2 * q^23 + 2 * q^24 + 2 * q^25 - 2 * q^27 + 6 * q^28 - 2 * q^29 - 2 * q^30 + 2 * q^31 - 2 * q^32 + 2 * q^33 - 2 * q^34 - 16 * q^35 + 2 * q^36 + 6 * q^37 - 4 * q^38 + 2 * q^40 + 6 * q^42 - 2 * q^44 - 2 * q^45 - 2 * q^46 - 4 * q^47 - 2 * q^48 + 14 * q^49 - 2 * q^50 - 2 * q^51 + 8 * q^53 + 2 * q^54 + 2 * q^55 - 6 * q^56 - 4 * q^57 + 2 * q^58 + 12 * q^59 + 2 * q^60 + 6 * q^61 - 2 * q^62 + 6 * q^63 + 2 * q^64 - 2 * q^66 + 2 * q^68 - 2 * q^69 + 16 * q^70 + 22 * q^71 - 2 * q^72 + 24 * q^73 - 6 * q^74 - 2 * q^75 + 4 * q^76 - 6 * q^77 + 2 * q^79 - 2 * q^80 + 2 * q^81 + 32 * q^83 - 6 * q^84 - 2 * q^85 + 2 * q^87 + 2 * q^88 + 4 * q^89 + 2 * q^90 + 2 * q^92 - 2 * q^93 + 4 * q^94 + 16 * q^95 + 2 * q^96 + 4 * q^97 - 14 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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