LMFDB - Embedded newform 6422.2.a.bl.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 $ x^9 - x^8 - 12x^7 + 14x^6 + 35x^5 - 35x^4 - 28x^3 + 10x^2 + 4*x - 1
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			$1.89407$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.953982 q^{3} +1.00000 q^{4} -3.75463 q^{5} -0.953982 q^{6} -3.81192 q^{7} +1.00000 q^{8} -2.08992 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.953982 q^{3} +1.00000 q^{4} -3.75463 q^{5} -0.953982 q^{6} -3.81192 q^{7} +1.00000 q^{8} -2.08992 q^{9} -3.75463 q^{10} +5.51745 q^{11} -0.953982 q^{12} -3.81192 q^{14} +3.58185 q^{15} +1.00000 q^{16} +4.01406 q^{17} -2.08992 q^{18} -1.00000 q^{19} -3.75463 q^{20} +3.63650 q^{21} +5.51745 q^{22} -8.48080 q^{23} -0.953982 q^{24} +9.09726 q^{25} +4.85569 q^{27} -3.81192 q^{28} +3.65007 q^{29} +3.58185 q^{30} +10.5573 q^{31} +1.00000 q^{32} -5.26355 q^{33} +4.01406 q^{34} +14.3124 q^{35} -2.08992 q^{36} +0.902026 q^{37} -1.00000 q^{38} -3.75463 q^{40} -4.14463 q^{41} +3.63650 q^{42} -5.76706 q^{43} +5.51745 q^{44} +7.84688 q^{45} -8.48080 q^{46} +6.37763 q^{47} -0.953982 q^{48} +7.53075 q^{49} +9.09726 q^{50} -3.82934 q^{51} +10.7754 q^{53} +4.85569 q^{54} -20.7160 q^{55} -3.81192 q^{56} +0.953982 q^{57} +3.65007 q^{58} -0.808242 q^{59} +3.58185 q^{60} -1.66708 q^{61} +10.5573 q^{62} +7.96661 q^{63} +1.00000 q^{64} -5.26355 q^{66} -11.7663 q^{67} +4.01406 q^{68} +8.09053 q^{69} +14.3124 q^{70} -5.59901 q^{71} -2.08992 q^{72} -2.82357 q^{73} +0.902026 q^{74} -8.67862 q^{75} -1.00000 q^{76} -21.0321 q^{77} +12.2482 q^{79} -3.75463 q^{80} +1.63752 q^{81} -4.14463 q^{82} -13.2017 q^{83} +3.63650 q^{84} -15.0713 q^{85} -5.76706 q^{86} -3.48210 q^{87} +5.51745 q^{88} -11.6108 q^{89} +7.84688 q^{90} -8.48080 q^{92} -10.0714 q^{93} +6.37763 q^{94} +3.75463 q^{95} -0.953982 q^{96} -11.4819 q^{97} +7.53075 q^{98} -11.5310 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * 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
$3$	−0.953982	−0.550782	−0.275391	−	0.961332i	$-0.588807\pi$
$3$	−0.953982	−0.550782	−0.275391	+	0.961332i	$0.588807\pi$
$4$	1.00000	0.500000
$5$	−3.75463	−1.67912	−0.839561	−	0.543265i	$-0.817188\pi$
$5$	−3.75463	−1.67912	−0.839561	+	0.543265i	$0.817188\pi$
$6$	−0.953982	−0.389461
$7$	−3.81192	−1.44077	−0.720386	−	0.693574i	$-0.756035\pi$
$7$	−3.81192	−1.44077	−0.720386	+	0.693574i	$0.756035\pi$
$8$	1.00000	0.353553
$9$	−2.08992	−0.696640
$10$	−3.75463	−1.18732
$11$	5.51745	1.66358	0.831788	−	0.555094i	$-0.187318\pi$
$11$	5.51745	1.66358	0.831788	+	0.555094i	$0.187318\pi$
$12$	−0.953982	−0.275391
$13$	0	0
$13$	0	0
$14$	−3.81192	−1.01878
$15$	3.58185	0.924830
$16$	1.00000	0.250000
$17$	4.01406	0.973553	0.486777	−	0.873527i	$-0.338173\pi$
$17$	4.01406	0.973553	0.486777	+	0.873527i	$0.338173\pi$
$18$	−2.08992	−0.492599
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.75463	−0.839561
$21$	3.63650	0.793550
$22$	5.51745	1.17633
$23$	−8.48080	−1.76837	−0.884185	−	0.467137i	$-0.845286\pi$
$23$	−8.48080	−1.76837	−0.884185	+	0.467137i	$0.845286\pi$
$24$	−0.953982	−0.194731
$25$	9.09726	1.81945
$26$	0	0
$27$	4.85569	0.934478
$28$	−3.81192	−0.720386
$29$	3.65007	0.677801	0.338901	−	0.940822i	$-0.389945\pi$
$29$	3.65007	0.677801	0.338901	+	0.940822i	$0.389945\pi$
$30$	3.58185	0.653953
$31$	10.5573	1.89614	0.948072	−	0.318057i	$-0.103030\pi$
$31$	10.5573	1.89614	0.948072	+	0.318057i	$0.103030\pi$
$32$	1.00000	0.176777
$33$	−5.26355	−0.916266
$34$	4.01406	0.688406
$35$	14.3124	2.41923
$36$	−2.08992	−0.348320
$37$	0.902026	0.148292	0.0741461	−	0.997247i	$-0.476377\pi$
$37$	0.902026	0.148292	0.0741461	+	0.997247i	$0.476377\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−3.75463	−0.593659
$41$	−4.14463	−0.647282	−0.323641	−	0.946180i	$-0.604907\pi$
$41$	−4.14463	−0.647282	−0.323641	+	0.946180i	$0.604907\pi$
$42$	3.63650	0.561125
$43$	−5.76706	−0.879469	−0.439734	−	0.898128i	$-0.644927\pi$
$43$	−5.76706	−0.879469	−0.439734	+	0.898128i	$0.644927\pi$
$44$	5.51745	0.831788
$45$	7.84688	1.16974
$46$	−8.48080	−1.25043
$47$	6.37763	0.930272	0.465136	−	0.885239i	$-0.346005\pi$
$47$	6.37763	0.930272	0.465136	+	0.885239i	$0.346005\pi$
$48$	−0.953982	−0.137695
$49$	7.53075	1.07582
$50$	9.09726	1.28655
$51$	−3.82934	−0.536215
$52$	0	0
$53$	10.7754	1.48012	0.740058	−	0.672543i	$-0.234798\pi$
$53$	10.7754	1.48012	0.740058	+	0.672543i	$0.234798\pi$
$54$	4.85569	0.660776
$55$	−20.7160	−2.79335
$56$	−3.81192	−0.509390
$57$	0.953982	0.126358
$58$	3.65007	0.479278
$59$	−0.808242	−0.105224	−0.0526121	−	0.998615i	$-0.516755\pi$
$59$	−0.808242	−0.105224	−0.0526121	+	0.998615i	$0.516755\pi$
$60$	3.58185	0.462415
$61$	−1.66708	−0.213447	−0.106724	−	0.994289i	$-0.534036\pi$
$61$	−1.66708	−0.213447	−0.106724	+	0.994289i	$0.534036\pi$
$62$	10.5573	1.34078
$63$	7.96661	1.00370
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.26355	−0.647898
$67$	−11.7663	−1.43749	−0.718744	−	0.695275i	$-0.755283\pi$
$67$	−11.7663	−1.43749	−0.718744	+	0.695275i	$0.755283\pi$
$68$	4.01406	0.486777
$69$	8.09053	0.973986
$70$	14.3124	1.71065
$71$	−5.59901	−0.664481	−0.332240	−	0.943195i	$-0.607804\pi$
$71$	−5.59901	−0.664481	−0.332240	+	0.943195i	$0.607804\pi$
$72$	−2.08992	−0.246299
$73$	−2.82357	−0.330473	−0.165237	−	0.986254i	$-0.552839\pi$
$73$	−2.82357	−0.330473	−0.165237	+	0.986254i	$0.552839\pi$
$74$	0.902026	0.104858
$75$	−8.67862	−1.00212
$76$	−1.00000	−0.114708
$77$	−21.0321	−2.39683
$78$	0	0
$79$	12.2482	1.37803	0.689014	−	0.724748i	$-0.258044\pi$
$79$	12.2482	1.37803	0.689014	+	0.724748i	$0.258044\pi$
$80$	−3.75463	−0.419781
$81$	1.63752	0.181947
$82$	−4.14463	−0.457698
$83$	−13.2017	−1.44907	−0.724537	−	0.689236i	$-0.757946\pi$
$83$	−13.2017	−1.44907	−0.724537	+	0.689236i	$0.757946\pi$
$84$	3.63650	0.396775
$85$	−15.0713	−1.63471
$86$	−5.76706	−0.621878
$87$	−3.48210	−0.373320
$88$	5.51745	0.588163
$89$	−11.6108	−1.23075	−0.615373	−	0.788236i	$-0.710994\pi$
$89$	−11.6108	−1.23075	−0.615373	+	0.788236i	$0.710994\pi$
$90$	7.84688	0.827133
$91$	0	0
$92$	−8.48080	−0.884185
$93$	−10.0714	−1.04436
$94$	6.37763	0.657802
$95$	3.75463	0.385217
$96$	−0.953982	−0.0973653
$97$	−11.4819	−1.16581	−0.582907	−	0.812539i	$-0.698085\pi$
$97$	−11.4819	−1.16581	−0.582907	+	0.812539i	$0.698085\pi$
$98$	7.53075	0.760721
$99$	−11.5310	−1.15891
$100$	9.09726	0.909726
$101$	14.2302	1.41596	0.707979	−	0.706233i	$-0.249607\pi$
$101$	14.2302	1.41596	0.707979	+	0.706233i	$0.249607\pi$
$102$	−3.82934	−0.379161
$103$	8.73011	0.860203	0.430102	−	0.902780i	$-0.358478\pi$
$103$	8.73011	0.860203	0.430102	+	0.902780i	$0.358478\pi$
$104$	0	0
$105$	−13.6537	−1.33247
$106$	10.7754	1.04660
$107$	−0.275928	−0.0266750	−0.0133375	−	0.999911i	$-0.504246\pi$
$107$	−0.275928	−0.0266750	−0.0133375	+	0.999911i	$0.504246\pi$
$108$	4.85569	0.467239
$109$	−7.56862	−0.724942	−0.362471	−	0.931995i	$-0.618067\pi$
$109$	−7.56862	−0.724942	−0.362471	+	0.931995i	$0.618067\pi$
$110$	−20.7160	−1.97519
$111$	−0.860516	−0.0816766
$112$	−3.81192	−0.360193
$113$	−9.72329	−0.914690	−0.457345	−	0.889289i	$-0.651200\pi$
$113$	−9.72329	−0.914690	−0.457345	+	0.889289i	$0.651200\pi$
$114$	0.953982	0.0893486
$115$	31.8423	2.96931
$116$	3.65007	0.338901
$117$	0	0
$118$	−0.808242	−0.0744047
$119$	−15.3013	−1.40267
$120$	3.58185	0.326977
$121$	19.4423	1.76748
$122$	−1.66708	−0.150930
$123$	3.95390	0.356511
$124$	10.5573	0.948072
$125$	−15.3837	−1.37596
$126$	7.96661	0.709722
$127$	10.7228	0.951493	0.475747	−	0.879582i	$-0.342178\pi$
$127$	10.7228	0.951493	0.475747	+	0.879582i	$0.342178\pi$
$128$	1.00000	0.0883883
$129$	5.50167	0.484395
$130$	0	0
$131$	−12.8060	−1.11886	−0.559431	−	0.828877i	$-0.688980\pi$
$131$	−12.8060	−1.11886	−0.559431	+	0.828877i	$0.688980\pi$
$132$	−5.26355	−0.458133
$133$	3.81192	0.330536
$134$	−11.7663	−1.01646
$135$	−18.2313	−1.56910
$136$	4.01406	0.344203
$137$	−7.37387	−0.629993	−0.314996	−	0.949093i	$-0.602003\pi$
$137$	−7.37387	−0.629993	−0.314996	+	0.949093i	$0.602003\pi$
$138$	8.09053	0.688712
$139$	8.31102	0.704931	0.352466	−	0.935825i	$-0.385343\pi$
$139$	8.31102	0.704931	0.352466	+	0.935825i	$0.385343\pi$
$140$	14.3124	1.20962
$141$	−6.08414	−0.512377
$142$	−5.59901	−0.469859
$143$	0	0
$144$	−2.08992	−0.174160
$145$	−13.7047	−1.13811
$146$	−2.82357	−0.233680
$147$	−7.18420	−0.592543
$148$	0.902026	0.0741461
$149$	−5.21031	−0.426845	−0.213423	−	0.976960i	$-0.568461\pi$
$149$	−5.21031	−0.426845	−0.213423	+	0.976960i	$0.568461\pi$
$150$	−8.67862	−0.708606
$151$	−19.3003	−1.57064	−0.785318	−	0.619092i	$-0.787501\pi$
$151$	−19.3003	−1.57064	−0.785318	+	0.619092i	$0.787501\pi$
$152$	−1.00000	−0.0811107
$153$	−8.38907	−0.678216
$154$	−21.0321	−1.69482
$155$	−39.6387	−3.18386
$156$	0	0
$157$	−7.27011	−0.580218	−0.290109	−	0.956994i	$-0.593692\pi$
$157$	−7.27011	−0.580218	−0.290109	+	0.956994i	$0.593692\pi$
$158$	12.2482	0.974413
$159$	−10.2795	−0.815221
$160$	−3.75463	−0.296830
$161$	32.3282	2.54782
$162$	1.63752	0.128656
$163$	−17.0956	−1.33903	−0.669515	−	0.742798i	$-0.733498\pi$
$163$	−17.0956	−1.33903	−0.669515	+	0.742798i	$0.733498\pi$
$164$	−4.14463	−0.323641
$165$	19.7627	1.53852
$166$	−13.2017	−1.02465
$167$	13.6416	1.05562	0.527809	−	0.849363i	$-0.323013\pi$
$167$	13.6416	1.05562	0.527809	+	0.849363i	$0.323013\pi$
$168$	3.63650	0.280562
$169$	0	0
$170$	−15.0713	−1.15592
$171$	2.08992	0.159820
$172$	−5.76706	−0.439734
$173$	−9.20312	−0.699700	−0.349850	−	0.936806i	$-0.613767\pi$
$173$	−9.20312	−0.699700	−0.349850	+	0.936806i	$0.613767\pi$
$174$	−3.48210	−0.263977
$175$	−34.6781	−2.62141
$176$	5.51745	0.415894
$177$	0.771048	0.0579555
$178$	−11.6108	−0.870268
$179$	−2.00310	−0.149718	−0.0748592	−	0.997194i	$-0.523851\pi$
$179$	−2.00310	−0.149718	−0.0748592	+	0.997194i	$0.523851\pi$
$180$	7.84688	0.584872
$181$	10.3661	0.770505	0.385253	−	0.922811i	$-0.374114\pi$
$181$	10.3661	0.770505	0.385253	+	0.922811i	$0.374114\pi$
$182$	0	0
$183$	1.59036	0.117563
$184$	−8.48080	−0.625213
$185$	−3.38678	−0.249001
$186$	−10.0714	−0.738474
$187$	22.1474	1.61958
$188$	6.37763	0.465136
$189$	−18.5095	−1.34637
$190$	3.75463	0.272390
$191$	−0.0436664	−0.00315959	−0.00157979	−	0.999999i	$-0.500503\pi$
$191$	−0.0436664	−0.00315959	−0.00157979	+	0.999999i	$0.500503\pi$
$192$	−0.953982	−0.0688477
$193$	−1.89921	−0.136708	−0.0683541	−	0.997661i	$-0.521775\pi$
$193$	−1.89921	−0.136708	−0.0683541	+	0.997661i	$0.521775\pi$
$194$	−11.4819	−0.824355
$195$	0	0
$196$	7.53075	0.537911
$197$	−0.327292	−0.0233186	−0.0116593	−	0.999932i	$-0.503711\pi$
$197$	−0.327292	−0.0233186	−0.0116593	+	0.999932i	$0.503711\pi$
$198$	−11.5310	−0.819475
$199$	19.3957	1.37493	0.687464	−	0.726218i	$-0.258724\pi$
$199$	19.3957	1.37493	0.687464	+	0.726218i	$0.258724\pi$
$200$	9.09726	0.643273
$201$	11.2249	0.791742
$202$	14.2302	1.00123
$203$	−13.9138	−0.976556
$204$	−3.82934	−0.268108
$205$	15.5616	1.08687
$206$	8.73011	0.608256
$207$	17.7242	1.23192
$208$	0	0
$209$	−5.51745	−0.381650
$210$	−13.6537	−0.942197
$211$	−6.51849	−0.448751	−0.224376	−	0.974503i	$-0.572034\pi$
$211$	−6.51849	−0.448751	−0.224376	+	0.974503i	$0.572034\pi$
$212$	10.7754	0.740058
$213$	5.34135	0.365984
$214$	−0.275928	−0.0188621
$215$	21.6532	1.47674
$216$	4.85569	0.330388
$217$	−40.2435	−2.73191
$218$	−7.56862	−0.512612
$219$	2.69363	0.182019
$220$	−20.7160	−1.39667
$221$	0	0
$222$	−0.860516	−0.0577541
$223$	−22.1419	−1.48273	−0.741366	−	0.671101i	$-0.765822\pi$
$223$	−22.1419	−1.48273	−0.741366	+	0.671101i	$0.765822\pi$
$224$	−3.81192	−0.254695
$225$	−19.0125	−1.26750
$226$	−9.72329	−0.646784
$227$	24.7975	1.64587	0.822934	−	0.568137i	$-0.192336\pi$
$227$	24.7975	1.64587	0.822934	+	0.568137i	$0.192336\pi$
$228$	0.953982	0.0631790
$229$	12.8014	0.845943	0.422972	−	0.906143i	$-0.360987\pi$
$229$	12.8014	0.845943	0.422972	+	0.906143i	$0.360987\pi$
$230$	31.8423	2.09962
$231$	20.0642	1.32013
$232$	3.65007	0.239639
$233$	18.6566	1.22224	0.611118	−	0.791540i	$-0.290720\pi$
$233$	18.6566	1.22224	0.611118	+	0.791540i	$0.290720\pi$
$234$	0	0
$235$	−23.9456	−1.56204
$236$	−0.808242	−0.0526121
$237$	−11.6845	−0.758992
$238$	−15.3013	−0.991836
$239$	−19.8498	−1.28397	−0.641987	−	0.766715i	$-0.721890\pi$
$239$	−19.8498	−1.28397	−0.641987	+	0.766715i	$0.721890\pi$
$240$	3.58185	0.231207
$241$	−5.31705	−0.342501	−0.171251	−	0.985228i	$-0.554781\pi$
$241$	−5.31705	−0.342501	−0.171251	+	0.985228i	$0.554781\pi$
$242$	19.4423	1.24980
$243$	−16.1292	−1.03469
$244$	−1.66708	−0.106724
$245$	−28.2752	−1.80644
$246$	3.95390	0.252091
$247$	0	0
$248$	10.5573	0.670388
$249$	12.5942	0.798123
$250$	−15.3837	−0.972951
$251$	−0.904204	−0.0570728	−0.0285364	−	0.999593i	$-0.509085\pi$
$251$	−0.904204	−0.0570728	−0.0285364	+	0.999593i	$0.509085\pi$
$252$	7.96661	0.501849
$253$	−46.7925	−2.94182
$254$	10.7228	0.672807
$255$	14.3778	0.900371
$256$	1.00000	0.0625000
$257$	−12.4257	−0.775091	−0.387546	−	0.921851i	$-0.626677\pi$
$257$	−12.4257	−0.775091	−0.387546	+	0.921851i	$0.626677\pi$
$258$	5.50167	0.342519
$259$	−3.43845	−0.213655
$260$	0	0
$261$	−7.62835	−0.472183
$262$	−12.8060	−0.791155
$263$	−18.3088	−1.12897	−0.564483	−	0.825444i	$-0.690925\pi$
$263$	−18.3088	−1.12897	−0.564483	+	0.825444i	$0.690925\pi$
$264$	−5.26355	−0.323949
$265$	−40.4577	−2.48530
$266$	3.81192	0.233724
$267$	11.0765	0.677872
$268$	−11.7663	−0.718744
$269$	3.02898	0.184680	0.0923400	−	0.995728i	$-0.470565\pi$
$269$	3.02898	0.184680	0.0923400	+	0.995728i	$0.470565\pi$
$270$	−18.2313	−1.10952
$271$	−1.86245	−0.113136	−0.0565679	−	0.998399i	$-0.518016\pi$
$271$	−1.86245	−0.113136	−0.0565679	+	0.998399i	$0.518016\pi$
$272$	4.01406	0.243388
$273$	0	0
$274$	−7.37387	−0.445472
$275$	50.1937	3.02680
$276$	8.09053	0.486993
$277$	−28.3273	−1.70202	−0.851011	−	0.525148i	$-0.824010\pi$
$277$	−28.3273	−1.70202	−0.851011	+	0.525148i	$0.824010\pi$
$278$	8.31102	0.498462
$279$	−22.0639	−1.32093
$280$	14.3124	0.855327
$281$	−8.89920	−0.530882	−0.265441	−	0.964127i	$-0.585518\pi$
$281$	−8.89920	−0.530882	−0.265441	+	0.964127i	$0.585518\pi$
$282$	−6.08414	−0.362305
$283$	8.51060	0.505903	0.252951	−	0.967479i	$-0.418599\pi$
$283$	8.51060	0.505903	0.252951	+	0.967479i	$0.418599\pi$
$284$	−5.59901	−0.332240
$285$	−3.58185	−0.212170
$286$	0	0
$287$	15.7990	0.932586
$288$	−2.08992	−0.123150
$289$	−0.887304	−0.0521943
$290$	−13.7047	−0.804766
$291$	10.9536	0.642109
$292$	−2.82357	−0.165237
$293$	−1.27549	−0.0745151	−0.0372575	−	0.999306i	$-0.511862\pi$
$293$	−1.27549	−0.0745151	−0.0372575	+	0.999306i	$0.511862\pi$
$294$	−7.18420	−0.418991
$295$	3.03465	0.176684
$296$	0.902026	0.0524292
$297$	26.7910	1.55457
$298$	−5.21031	−0.301825
$299$	0	0
$300$	−8.67862	−0.501060
$301$	21.9836	1.26711
$302$	−19.3003	−1.11061
$303$	−13.5754	−0.779884
$304$	−1.00000	−0.0573539
$305$	6.25926	0.358404
$306$	−8.38907	−0.479571
$307$	26.2199	1.49645	0.748226	−	0.663444i	$-0.230906\pi$
$307$	26.2199	1.49645	0.748226	+	0.663444i	$0.230906\pi$
$308$	−21.0321	−1.19842
$309$	−8.32836	−0.473784
$310$	−39.6387	−2.25133
$311$	−11.5880	−0.657094	−0.328547	−	0.944488i	$-0.606559\pi$
$311$	−11.5880	−0.657094	−0.328547	+	0.944488i	$0.606559\pi$
$312$	0	0
$313$	−27.7945	−1.57104	−0.785520	−	0.618836i	$-0.787604\pi$
$313$	−27.7945	−1.57104	−0.785520	+	0.618836i	$0.787604\pi$
$314$	−7.27011	−0.410276
$315$	−29.9117	−1.68533
$316$	12.2482	0.689014
$317$	−22.0123	−1.23633	−0.618166	−	0.786048i	$-0.712124\pi$
$317$	−22.0123	−1.23633	−0.618166	+	0.786048i	$0.712124\pi$
$318$	−10.2795	−0.576448
$319$	20.1391	1.12757
$320$	−3.75463	−0.209890
$321$	0.263231	0.0146921
$322$	32.3282	1.80158
$323$	−4.01406	−0.223348
$324$	1.63752	0.0909733
$325$	0	0
$326$	−17.0956	−0.946837
$327$	7.22033	0.399285
$328$	−4.14463	−0.228849
$329$	−24.3110	−1.34031
$330$	19.7627	1.08790
$331$	−34.3346	−1.88720	−0.943600	−	0.331088i	$-0.892584\pi$
$331$	−34.3346	−1.88720	−0.943600	+	0.331088i	$0.892584\pi$
$332$	−13.2017	−0.724537
$333$	−1.88516	−0.103306
$334$	13.6416	0.746435
$335$	44.1783	2.41372
$336$	3.63650	0.198388
$337$	8.71841	0.474922	0.237461	−	0.971397i	$-0.423685\pi$
$337$	8.71841	0.474922	0.237461	+	0.971397i	$0.423685\pi$
$338$	0	0
$339$	9.27584	0.503794
$340$	−15.0713	−0.817357
$341$	58.2493	3.15438
$342$	2.08992	0.113010
$343$	−2.02319	−0.109242
$344$	−5.76706	−0.310939
$345$	−30.3770	−1.63544
$346$	−9.20312	−0.494763
$347$	−13.4880	−0.724075	−0.362038	−	0.932163i	$-0.617919\pi$
$347$	−13.4880	−0.724075	−0.362038	+	0.932163i	$0.617919\pi$
$348$	−3.48210	−0.186660
$349$	27.0934	1.45027	0.725137	−	0.688605i	$-0.241776\pi$
$349$	27.0934	1.45027	0.725137	+	0.688605i	$0.241776\pi$
$350$	−34.6781	−1.85362
$351$	0	0
$352$	5.51745	0.294081
$353$	−2.67238	−0.142236	−0.0711182	−	0.997468i	$-0.522657\pi$
$353$	−2.67238	−0.142236	−0.0711182	+	0.997468i	$0.522657\pi$
$354$	0.771048	0.0409808
$355$	21.0222	1.11574
$356$	−11.6108	−0.615373
$357$	14.5972	0.772563
$358$	−2.00310	−0.105867
$359$	11.6976	0.617376	0.308688	−	0.951163i	$-0.400110\pi$
$359$	11.6976	0.617376	0.308688	+	0.951163i	$0.400110\pi$
$360$	7.84688	0.413567
$361$	1.00000	0.0526316
$362$	10.3661	0.544829
$363$	−18.5476	−0.973497
$364$	0	0
$365$	10.6015	0.554905
$366$	1.59036	0.0831295
$367$	14.7357	0.769197	0.384598	−	0.923084i	$-0.374340\pi$
$367$	14.7357	0.769197	0.384598	+	0.923084i	$0.374340\pi$
$368$	−8.48080	−0.442093
$369$	8.66194	0.450923
$370$	−3.38678	−0.176070
$371$	−41.0750	−2.13251
$372$	−10.0714	−0.522180
$373$	−23.1411	−1.19820	−0.599100	−	0.800674i	$-0.704475\pi$
$373$	−23.1411	−1.19820	−0.599100	+	0.800674i	$0.704475\pi$
$374$	22.1474	1.14522
$375$	14.6758	0.757854
$376$	6.37763	0.328901
$377$	0	0
$378$	−18.5095	−0.952027
$379$	6.05005	0.310770	0.155385	−	0.987854i	$-0.450338\pi$
$379$	6.05005	0.310770	0.155385	+	0.987854i	$0.450338\pi$
$380$	3.75463	0.192609
$381$	−10.2293	−0.524065
$382$	−0.0436664	−0.00223417
$383$	1.22283	0.0624836	0.0312418	−	0.999512i	$-0.490054\pi$
$383$	1.22283	0.0624836	0.0312418	+	0.999512i	$0.490054\pi$
$384$	−0.953982	−0.0486827
$385$	78.9678	4.02457
$386$	−1.89921	−0.0966673
$387$	12.0527	0.612673
$388$	−11.4819	−0.582907
$389$	−25.7113	−1.30362	−0.651808	−	0.758384i	$-0.725989\pi$
$389$	−25.7113	−1.30362	−0.651808	+	0.758384i	$0.725989\pi$
$390$	0	0
$391$	−34.0425	−1.72160
$392$	7.53075	0.380360
$393$	12.2167	0.616249
$394$	−0.327292	−0.0164888
$395$	−45.9874	−2.31388
$396$	−11.5310	−0.579456
$397$	−1.98500	−0.0996244	−0.0498122	−	0.998759i	$-0.515862\pi$
$397$	−1.98500	−0.0996244	−0.0498122	+	0.998759i	$0.515862\pi$
$398$	19.3957	0.972221
$399$	−3.63650	−0.182053
$400$	9.09726	0.454863
$401$	23.6454	1.18080	0.590398	−	0.807112i	$-0.298971\pi$
$401$	23.6454	1.18080	0.590398	+	0.807112i	$0.298971\pi$
$402$	11.2249	0.559846
$403$	0	0
$404$	14.2302	0.707979
$405$	−6.14828	−0.305511
$406$	−13.9138	−0.690530
$407$	4.97689	0.246695
$408$	−3.82934	−0.189581
$409$	−25.9461	−1.28295	−0.641477	−	0.767143i	$-0.721678\pi$
$409$	−25.9461	−1.28295	−0.641477	+	0.767143i	$0.721678\pi$
$410$	15.5616	0.768531
$411$	7.03454	0.346988
$412$	8.73011	0.430102
$413$	3.08096	0.151604
$414$	17.7242	0.871097
$415$	49.5675	2.43317
$416$	0	0
$417$	−7.92856	−0.388263
$418$	−5.51745	−0.269868
$419$	−13.8045	−0.674393	−0.337197	−	0.941434i	$-0.609479\pi$
$419$	−13.8045	−0.674393	−0.337197	+	0.941434i	$0.609479\pi$
$420$	−13.6537	−0.666234
$421$	3.00420	0.146416	0.0732079	−	0.997317i	$-0.476676\pi$
$421$	3.00420	0.146416	0.0732079	+	0.997317i	$0.476676\pi$
$422$	−6.51849	−0.317315
$423$	−13.3287	−0.648065
$424$	10.7754	0.523300
$425$	36.5170	1.77133
$426$	5.34135	0.258789
$427$	6.35477	0.307529
$428$	−0.275928	−0.0133375
$429$	0	0
$430$	21.6532	1.04421
$431$	10.6452	0.512759	0.256380	−	0.966576i	$-0.417470\pi$
$431$	10.6452	0.512759	0.256380	+	0.966576i	$0.417470\pi$
$432$	4.85569	0.233619
$433$	12.1047	0.581713	0.290857	−	0.956767i	$-0.406060\pi$
$433$	12.1047	0.581713	0.290857	+	0.956767i	$0.406060\pi$
$434$	−40.2435	−1.93175
$435$	13.0740	0.626851
$436$	−7.56862	−0.362471
$437$	8.48080	0.405692
$438$	2.69363	0.128707
$439$	−2.42866	−0.115914	−0.0579569	−	0.998319i	$-0.518459\pi$
$439$	−2.42866	−0.115914	−0.0579569	+	0.998319i	$0.518459\pi$
$440$	−20.7160	−0.987597
$441$	−15.7387	−0.749460
$442$	0	0
$443$	−19.1495	−0.909821	−0.454910	−	0.890537i	$-0.650329\pi$
$443$	−19.1495	−0.909821	−0.454910	+	0.890537i	$0.650329\pi$
$444$	−0.860516	−0.0408383
$445$	43.5944	2.06657
$446$	−22.1419	−1.04845
$447$	4.97054	0.235099
$448$	−3.81192	−0.180096
$449$	7.24041	0.341696	0.170848	−	0.985297i	$-0.445349\pi$
$449$	7.24041	0.341696	0.170848	+	0.985297i	$0.445349\pi$
$450$	−19.0125	−0.896260
$451$	−22.8678	−1.07680
$452$	−9.72329	−0.457345
$453$	18.4121	0.865078
$454$	24.7975	1.16380
$455$	0	0
$456$	0.953982	0.0446743
$457$	16.6298	0.777911	0.388955	−	0.921257i	$-0.372836\pi$
$457$	16.6298	0.777911	0.388955	+	0.921257i	$0.372836\pi$
$458$	12.8014	0.598172
$459$	19.4910	0.909764
$460$	31.8423	1.48465
$461$	26.9244	1.25399	0.626996	−	0.779022i	$-0.284284\pi$
$461$	26.9244	1.25399	0.626996	+	0.779022i	$0.284284\pi$
$462$	20.0642	0.933473
$463$	−31.5292	−1.46529	−0.732643	−	0.680613i	$-0.761714\pi$
$463$	−31.5292	−1.46529	−0.732643	+	0.680613i	$0.761714\pi$
$464$	3.65007	0.169450
$465$	37.8146	1.75361
$466$	18.6566	0.864251
$467$	31.9897	1.48031	0.740154	−	0.672437i	$-0.234753\pi$
$467$	31.9897	1.48031	0.740154	+	0.672437i	$0.234753\pi$
$468$	0	0
$469$	44.8524	2.07109
$470$	−23.9456	−1.10453
$471$	6.93555	0.319573
$472$	−0.808242	−0.0372024
$473$	−31.8195	−1.46306
$474$	−11.6845	−0.536689
$475$	−9.09726	−0.417411
$476$	−15.3013	−0.701334
$477$	−22.5197	−1.03111
$478$	−19.8498	−0.907907
$479$	−1.52205	−0.0695443	−0.0347722	−	0.999395i	$-0.511071\pi$
$479$	−1.52205	−0.0695443	−0.0347722	+	0.999395i	$0.511071\pi$
$480$	3.58185	0.163488
$481$	0	0
$482$	−5.31705	−0.242185
$483$	−30.8405	−1.40329
$484$	19.4423	0.883741
$485$	43.1104	1.95754
$486$	−16.1292	−0.731637
$487$	14.6946	0.665876	0.332938	−	0.942949i	$-0.391960\pi$
$487$	14.6946	0.665876	0.332938	+	0.942949i	$0.391960\pi$
$488$	−1.66708	−0.0754651
$489$	16.3089	0.737513
$490$	−28.2752	−1.27734
$491$	−29.9345	−1.35092	−0.675462	−	0.737395i	$-0.736056\pi$
$491$	−29.9345	−1.35092	−0.675462	+	0.737395i	$0.736056\pi$
$492$	3.95390	0.178256
$493$	14.6516	0.659875
$494$	0	0
$495$	43.2948	1.94596
$496$	10.5573	0.474036
$497$	21.3430	0.957364
$498$	12.5942	0.564358
$499$	−17.1820	−0.769174	−0.384587	−	0.923089i	$-0.625656\pi$
$499$	−17.1820	−0.769174	−0.384587	+	0.923089i	$0.625656\pi$
$500$	−15.3837	−0.687980
$501$	−13.0138	−0.581415
$502$	−0.904204	−0.0403566
$503$	−0.323693	−0.0144327	−0.00721637	−	0.999974i	$-0.502297\pi$
$503$	−0.323693	−0.0144327	−0.00721637	+	0.999974i	$0.502297\pi$
$504$	7.96661	0.354861
$505$	−53.4292	−2.37757
$506$	−46.7925	−2.08018
$507$	0	0
$508$	10.7228	0.475747
$509$	−30.4293	−1.34876	−0.674378	−	0.738386i	$-0.735588\pi$
$509$	−30.4293	−1.34876	−0.674378	+	0.738386i	$0.735588\pi$
$510$	14.3778	0.636658
$511$	10.7632	0.476137
$512$	1.00000	0.0441942
$513$	−4.85569	−0.214384
$514$	−12.4257	−0.548072
$515$	−32.7783	−1.44439
$516$	5.50167	0.242198
$517$	35.1883	1.54758
$518$	−3.43845	−0.151077
$519$	8.77960	0.385382
$520$	0	0
$521$	10.6247	0.465478	0.232739	−	0.972539i	$-0.425231\pi$
$521$	10.6247	0.465478	0.232739	+	0.972539i	$0.425231\pi$
$522$	−7.62835	−0.333884
$523$	11.6523	0.509518	0.254759	−	0.967005i	$-0.418004\pi$
$523$	11.6523	0.509518	0.254759	+	0.967005i	$0.418004\pi$
$524$	−12.8060	−0.559431
$525$	33.0822	1.44383
$526$	−18.3088	−0.798300
$527$	42.3776	1.84600
$528$	−5.26355	−0.229067
$529$	48.9240	2.12713
$530$	−40.4577	−1.75737
$531$	1.68916	0.0733033
$532$	3.81192	0.165268
$533$	0	0
$534$	11.0765	0.479328
$535$	1.03601	0.0447906
$536$	−11.7663	−0.508229
$537$	1.91092	0.0824622
$538$	3.02898	0.130589
$539$	41.5506	1.78971
$540$	−18.2313	−0.784551
$541$	10.1419	0.436035	0.218017	−	0.975945i	$-0.430041\pi$
$541$	10.1419	0.436035	0.218017	+	0.975945i	$0.430041\pi$
$542$	−1.86245	−0.0799990
$543$	−9.88905	−0.424380
$544$	4.01406	0.172102
$545$	28.4174	1.21727
$546$	0	0
$547$	−12.9662	−0.554396	−0.277198	−	0.960813i	$-0.589406\pi$
$547$	−12.9662	−0.554396	−0.277198	+	0.960813i	$0.589406\pi$
$548$	−7.37387	−0.314996
$549$	3.48406	0.148696
$550$	50.1937	2.14027
$551$	−3.65007	−0.155498
$552$	8.09053	0.344356
$553$	−46.6891	−1.98542
$554$	−28.3273	−1.20351
$555$	3.23092	0.137145
$556$	8.31102	0.352466
$557$	20.5235	0.869610	0.434805	−	0.900525i	$-0.356817\pi$
$557$	20.5235	0.869610	0.434805	+	0.900525i	$0.356817\pi$
$558$	−22.0639	−0.934038
$559$	0	0
$560$	14.3124	0.604808
$561$	−21.1282	−0.892034
$562$	−8.89920	−0.375390
$563$	−21.1454	−0.891172	−0.445586	−	0.895239i	$-0.647005\pi$
$563$	−21.1454	−0.891172	−0.445586	+	0.895239i	$0.647005\pi$
$564$	−6.08414	−0.256188
$565$	36.5074	1.53588
$566$	8.51060	0.357727
$567$	−6.24210	−0.262143
$568$	−5.59901	−0.234929
$569$	22.8679	0.958671	0.479335	−	0.877632i	$-0.340878\pi$
$569$	22.8679	0.958671	0.479335	+	0.877632i	$0.340878\pi$
$570$	−3.58185	−0.150027
$571$	31.1432	1.30330	0.651651	−	0.758519i	$-0.274077\pi$
$571$	31.1432	1.30330	0.651651	+	0.758519i	$0.274077\pi$
$572$	0	0
$573$	0.0416569	0.00174024
$574$	15.7990	0.659438
$575$	−77.1521	−3.21746
$576$	−2.08992	−0.0870800
$577$	−3.48756	−0.145189	−0.0725944	−	0.997362i	$-0.523128\pi$
$577$	−3.48756	−0.145189	−0.0725944	+	0.997362i	$0.523128\pi$
$578$	−0.887304	−0.0369070
$579$	1.81181	0.0752963
$580$	−13.7047	−0.569056
$581$	50.3238	2.08778
$582$	10.9536	0.454039
$583$	59.4528	2.46228
$584$	−2.82357	−0.116840
$585$	0	0
$586$	−1.27549	−0.0526901
$587$	−35.0364	−1.44611	−0.723053	−	0.690793i	$-0.757262\pi$
$587$	−35.0364	−1.44611	−0.723053	+	0.690793i	$0.757262\pi$
$588$	−7.18420	−0.296271
$589$	−10.5573	−0.435005
$590$	3.03465	0.124935
$591$	0.312231	0.0128435
$592$	0.902026	0.0370730
$593$	35.8979	1.47415	0.737074	−	0.675812i	$-0.236207\pi$
$593$	35.8979	1.47415	0.737074	+	0.675812i	$0.236207\pi$
$594$	26.7910	1.09925
$595$	57.4507	2.35525
$596$	−5.21031	−0.213423
$597$	−18.5032	−0.757285
$598$	0	0
$599$	21.2466	0.868114	0.434057	−	0.900885i	$-0.357082\pi$
$599$	21.2466	0.868114	0.434057	+	0.900885i	$0.357082\pi$
$600$	−8.67862	−0.354303
$601$	−13.3478	−0.544469	−0.272235	−	0.962231i	$-0.587763\pi$
$601$	−13.3478	−0.544469	−0.272235	+	0.962231i	$0.587763\pi$
$602$	21.9836	0.895984
$603$	24.5907	1.00141
$604$	−19.3003	−0.785318
$605$	−72.9987	−2.96782
$606$	−13.5754	−0.551461
$607$	−1.12716	−0.0457499	−0.0228749	−	0.999738i	$-0.507282\pi$
$607$	−1.12716	−0.0457499	−0.0228749	+	0.999738i	$0.507282\pi$
$608$	−1.00000	−0.0405554
$609$	13.2735	0.537869
$610$	6.25926	0.253430
$611$	0	0
$612$	−8.38907	−0.339108
$613$	−19.1157	−0.772075	−0.386038	−	0.922483i	$-0.626156\pi$
$613$	−19.1157	−0.772075	−0.386038	+	0.922483i	$0.626156\pi$
$614$	26.2199	1.05815
$615$	−14.8454	−0.598626
$616$	−21.0321	−0.847408
$617$	25.4631	1.02511	0.512553	−	0.858656i	$-0.328700\pi$
$617$	25.4631	1.02511	0.512553	+	0.858656i	$0.328700\pi$
$618$	−8.32836	−0.335016
$619$	10.0267	0.403007	0.201504	−	0.979488i	$-0.435417\pi$
$619$	10.0267	0.403007	0.201504	+	0.979488i	$0.435417\pi$
$620$	−39.6387	−1.59193
$621$	−41.1802	−1.65250
$622$	−11.5880	−0.464636
$623$	44.2596	1.77322
$624$	0	0
$625$	12.2738	0.490954
$626$	−27.7945	−1.11089
$627$	5.26355	0.210206
$628$	−7.27011	−0.290109
$629$	3.62079	0.144370
$630$	−29.9117	−1.19171
$631$	29.6985	1.18228	0.591139	−	0.806569i	$-0.298678\pi$
$631$	29.6985	1.18228	0.591139	+	0.806569i	$0.298678\pi$
$632$	12.2482	0.487206
$633$	6.21852	0.247164
$634$	−22.0123	−0.874219
$635$	−40.2601	−1.59767
$636$	−10.2795	−0.407610
$637$	0	0
$638$	20.1391	0.797315
$639$	11.7015	0.462904
$640$	−3.75463	−0.148415
$641$	−22.8183	−0.901269	−0.450635	−	0.892709i	$-0.648802\pi$
$641$	−22.8183	−0.901269	−0.450635	+	0.892709i	$0.648802\pi$
$642$	0.263231	0.0103889
$643$	−8.64376	−0.340877	−0.170438	−	0.985368i	$-0.554518\pi$
$643$	−8.64376	−0.340877	−0.170438	+	0.985368i	$0.554518\pi$
$644$	32.3282	1.27391
$645$	−20.6567	−0.813359
$646$	−4.01406	−0.157931
$647$	21.5338	0.846583	0.423291	−	0.905994i	$-0.360875\pi$
$647$	21.5338	0.846583	0.423291	+	0.905994i	$0.360875\pi$
$648$	1.63752	0.0643278
$649$	−4.45944	−0.175048
$650$	0	0
$651$	38.3916	1.50468
$652$	−17.0956	−0.669515
$653$	16.4492	0.643707	0.321854	−	0.946789i	$-0.395694\pi$
$653$	16.4492	0.643707	0.321854	+	0.946789i	$0.395694\pi$
$654$	7.22033	0.282337
$655$	48.0817	1.87871
$656$	−4.14463	−0.161821
$657$	5.90103	0.230221
$658$	−24.3110	−0.947742
$659$	14.5074	0.565129	0.282565	−	0.959248i	$-0.408815\pi$
$659$	14.5074	0.565129	0.282565	+	0.959248i	$0.408815\pi$
$660$	19.7627	0.769262
$661$	−44.0413	−1.71301	−0.856504	−	0.516141i	$-0.827368\pi$
$661$	−44.0413	−1.71301	−0.856504	+	0.516141i	$0.827368\pi$
$662$	−34.3346	−1.33445
$663$	0	0
$664$	−13.2017	−0.512325
$665$	−14.3124	−0.555010
$666$	−1.88516	−0.0730485
$667$	−30.9555	−1.19860
$668$	13.6416	0.527809
$669$	21.1230	0.816662
$670$	44.1783	1.70676
$671$	−9.19802	−0.355086
$672$	3.63650	0.140281
$673$	−43.5078	−1.67710	−0.838552	−	0.544822i	$-0.816597\pi$
$673$	−43.5078	−1.67710	−0.838552	+	0.544822i	$0.816597\pi$
$674$	8.71841	0.335821
$675$	44.1735	1.70024
$676$	0	0
$677$	−18.5667	−0.713575	−0.356787	−	0.934186i	$-0.616128\pi$
$677$	−18.5667	−0.713575	−0.356787	+	0.934186i	$0.616128\pi$
$678$	9.27584	0.356236
$679$	43.7682	1.67967
$680$	−15.0713	−0.577959
$681$	−23.6564	−0.906514
$682$	58.2493	2.23048
$683$	−45.8023	−1.75258	−0.876289	−	0.481786i	$-0.839988\pi$
$683$	−45.8023	−1.75258	−0.876289	+	0.481786i	$0.839988\pi$
$684$	2.08992	0.0799101
$685$	27.6862	1.05783
$686$	−2.02319	−0.0772458
$687$	−12.2123	−0.465930
$688$	−5.76706	−0.219867
$689$	0	0
$690$	−30.3770	−1.15643
$691$	38.1283	1.45047	0.725235	−	0.688502i	$-0.241731\pi$
$691$	38.1283	1.45047	0.725235	+	0.688502i	$0.241731\pi$
$692$	−9.20312	−0.349850
$693$	43.9554	1.66973
$694$	−13.4880	−0.511999
$695$	−31.2048	−1.18367
$696$	−3.48210	−0.131989
$697$	−16.6368	−0.630164
$698$	27.0934	1.02550
$699$	−17.7981	−0.673185
$700$	−34.6781	−1.31071
$701$	−4.10370	−0.154995	−0.0774973	−	0.996993i	$-0.524693\pi$
$701$	−4.10370	−0.154995	−0.0774973	+	0.996993i	$0.524693\pi$
$702$	0	0
$703$	−0.902026	−0.0340206
$704$	5.51745	0.207947
$705$	22.8437	0.860343
$706$	−2.67238	−0.100576
$707$	−54.2444	−2.04007
$708$	0.771048	0.0289778
$709$	17.4783	0.656412	0.328206	−	0.944606i	$-0.393556\pi$
$709$	17.4783	0.656412	0.328206	+	0.944606i	$0.393556\pi$
$710$	21.0222	0.788950
$711$	−25.5977	−0.959989
$712$	−11.6108	−0.435134
$713$	−89.5342	−3.35308
$714$	14.5972	0.546285
$715$	0	0
$716$	−2.00310	−0.0748592
$717$	18.9363	0.707189
$718$	11.6976	0.436551
$719$	−37.3236	−1.39193	−0.695967	−	0.718073i	$-0.745024\pi$
$719$	−37.3236	−1.39193	−0.695967	+	0.718073i	$0.745024\pi$
$720$	7.84688	0.292436
$721$	−33.2785	−1.23936
$722$	1.00000	0.0372161
$723$	5.07237	0.188643
$724$	10.3661	0.385253
$725$	33.2056	1.23323
$726$	−18.5476	−0.688366
$727$	−35.6394	−1.32179	−0.660897	−	0.750477i	$-0.729824\pi$
$727$	−35.6394	−1.32179	−0.660897	+	0.750477i	$0.729824\pi$
$728$	0	0
$729$	10.4744	0.387942
$730$	10.6015	0.392377
$731$	−23.1493	−0.856209
$732$	1.59036	0.0587814
$733$	8.27236	0.305547	0.152773	−	0.988261i	$-0.451180\pi$
$733$	8.27236	0.305547	0.152773	+	0.988261i	$0.451180\pi$
$734$	14.7357	0.543904
$735$	26.9740	0.994952
$736$	−8.48080	−0.312607
$737$	−64.9203	−2.39137
$738$	8.66194	0.318850
$739$	−36.1762	−1.33076	−0.665382	−	0.746503i	$-0.731731\pi$
$739$	−36.1762	−1.33076	−0.665382	+	0.746503i	$0.731731\pi$
$740$	−3.38678	−0.124500
$741$	0	0
$742$	−41.0750	−1.50791
$743$	−31.0975	−1.14086	−0.570429	−	0.821347i	$-0.693223\pi$
$743$	−31.0975	−1.14086	−0.570429	+	0.821347i	$0.693223\pi$
$744$	−10.0714	−0.369237
$745$	19.5628	0.716726
$746$	−23.1411	−0.847256
$747$	27.5905	1.00948
$748$	22.1474	0.809789
$749$	1.05182	0.0384326
$750$	14.6758	0.535883
$751$	−18.9389	−0.691092	−0.345546	−	0.938402i	$-0.612306\pi$
$751$	−18.9389	−0.691092	−0.345546	+	0.938402i	$0.612306\pi$
$752$	6.37763	0.232568
$753$	0.862594	0.0314347
$754$	0	0
$755$	72.4655	2.63729
$756$	−18.5095	−0.673184
$757$	19.1608	0.696412	0.348206	−	0.937418i	$-0.386791\pi$
$757$	19.1608	0.696412	0.348206	+	0.937418i	$0.386791\pi$
$758$	6.05005	0.219748
$759$	44.6391	1.62030
$760$	3.75463	0.136195
$761$	5.91488	0.214414	0.107207	−	0.994237i	$-0.465809\pi$
$761$	5.91488	0.214414	0.107207	+	0.994237i	$0.465809\pi$
$762$	−10.2293	−0.370570
$763$	28.8510	1.04448
$764$	−0.0436664	−0.00157979
$765$	31.4979	1.13881
$766$	1.22283	0.0441826
$767$	0	0
$768$	−0.953982	−0.0344238
$769$	17.8765	0.644645	0.322322	−	0.946630i	$-0.395537\pi$
$769$	17.8765	0.644645	0.322322	+	0.946630i	$0.395537\pi$
$770$	78.9678	2.84580
$771$	11.8538	0.426906
$772$	−1.89921	−0.0683541
$773$	17.1602	0.617208	0.308604	−	0.951191i	$-0.400138\pi$
$773$	17.1602	0.617208	0.308604	+	0.951191i	$0.400138\pi$
$774$	12.0527	0.433225
$775$	96.0423	3.44994
$776$	−11.4819	−0.412177
$777$	3.28022	0.117677
$778$	−25.7113	−0.921796
$779$	4.14463	0.148497
$780$	0	0
$781$	−30.8923	−1.10541
$782$	−34.0425	−1.21736
$783$	17.7236	0.633390
$784$	7.53075	0.268955
$785$	27.2966	0.974257
$786$	12.2167	0.435754
$787$	−34.4755	−1.22892	−0.614460	−	0.788948i	$-0.710626\pi$
$787$	−34.4755	−1.22892	−0.614460	+	0.788948i	$0.710626\pi$
$788$	−0.327292	−0.0116593
$789$	17.4662	0.621814
$790$	−45.9874	−1.63616
$791$	37.0644	1.31786
$792$	−11.5310	−0.409737
$793$	0	0
$794$	−1.98500	−0.0704451
$795$	38.5959	1.36886
$796$	19.3957	0.687464
$797$	41.7815	1.47998	0.739988	−	0.672620i	$-0.234831\pi$
$797$	41.7815	1.47998	0.739988	+	0.672620i	$0.234831\pi$
$798$	−3.63650	−0.128731
$799$	25.6002	0.905669
$800$	9.09726	0.321637
$801$	24.2657	0.857386
$802$	23.6454	0.834949
$803$	−15.5789	−0.549767
$804$	11.2249	0.395871
$805$	−121.380	−4.27810
$806$	0	0
$807$	−2.88959	−0.101718
$808$	14.2302	0.500617
$809$	17.9219	0.630102	0.315051	−	0.949075i	$-0.397978\pi$
$809$	17.9219	0.630102	0.315051	+	0.949075i	$0.397978\pi$
$810$	−6.14828	−0.216029
$811$	−20.9475	−0.735566	−0.367783	−	0.929912i	$-0.619883\pi$
$811$	−20.9475	−0.735566	−0.367783	+	0.929912i	$0.619883\pi$
$812$	−13.9138	−0.488278
$813$	1.77674	0.0623131
$814$	4.97689	0.174440
$815$	64.1877	2.24840
$816$	−3.82934	−0.134054
$817$	5.76706	0.201764
$818$	−25.9461	−0.907185
$819$	0	0
$820$	15.5616	0.543433
$821$	31.6371	1.10414	0.552071	−	0.833797i	$-0.313838\pi$
$821$	31.6371	1.10414	0.552071	+	0.833797i	$0.313838\pi$
$822$	7.03454	0.245358
$823$	6.33227	0.220729	0.110365	−	0.993891i	$-0.464798\pi$
$823$	6.33227	0.220729	0.110365	+	0.993891i	$0.464798\pi$
$824$	8.73011	0.304128
$825$	−47.8839	−1.66710
$826$	3.08096	0.107200
$827$	32.5610	1.13226	0.566129	−	0.824317i	$-0.308440\pi$
$827$	32.5610	1.13226	0.566129	+	0.824317i	$0.308440\pi$
$828$	17.7242	0.615958
$829$	−36.8637	−1.28033	−0.640164	−	0.768238i	$-0.721134\pi$
$829$	−36.8637	−1.28033	−0.640164	+	0.768238i	$0.721134\pi$
$830$	49.5675	1.72051
$831$	27.0237	0.937442
$832$	0	0
$833$	30.2289	1.04737
$834$	−7.92856	−0.274543
$835$	−51.2192	−1.77251
$836$	−5.51745	−0.190825
$837$	51.2629	1.77190
$838$	−13.8045	−0.476868
$839$	35.7390	1.23385	0.616924	−	0.787023i	$-0.288379\pi$
$839$	35.7390	1.23385	0.616924	+	0.787023i	$0.288379\pi$
$840$	−13.6537	−0.471099
$841$	−15.6770	−0.540586
$842$	3.00420	0.103532
$843$	8.48967	0.292400
$844$	−6.51849	−0.224376
$845$	0	0
$846$	−13.3287	−0.458251
$847$	−74.1126	−2.54654
$848$	10.7754	0.370029
$849$	−8.11896	−0.278642
$850$	36.5170	1.25252
$851$	−7.64991	−0.262235
$852$	5.34135	0.182992
$853$	−49.0109	−1.67810	−0.839050	−	0.544054i	$-0.816889\pi$
$853$	−49.0109	−1.67810	−0.839050	+	0.544054i	$0.816889\pi$
$854$	6.35477	0.217456
$855$	−7.84688	−0.268358
$856$	−0.275928	−0.00943104
$857$	−36.6260	−1.25112	−0.625561	−	0.780175i	$-0.715130\pi$
$857$	−36.6260	−1.25112	−0.625561	+	0.780175i	$0.715130\pi$
$858$	0	0
$859$	−21.3043	−0.726892	−0.363446	−	0.931615i	$-0.618400\pi$
$859$	−21.3043	−0.726892	−0.363446	+	0.931615i	$0.618400\pi$
$860$	21.6532	0.738368
$861$	−15.0720	−0.513651
$862$	10.6452	0.362576
$863$	−29.4691	−1.00314	−0.501569	−	0.865118i	$-0.667244\pi$
$863$	−29.4691	−1.00314	−0.501569	+	0.865118i	$0.667244\pi$
$864$	4.85569	0.165194
$865$	34.5543	1.17488
$866$	12.1047	0.411334
$867$	0.846471	0.0287477
$868$	−40.2435	−1.36595
$869$	67.5788	2.29245
$870$	13.0740	0.443250
$871$	0	0
$872$	−7.56862	−0.256306
$873$	23.9963	0.812152
$874$	8.48080	0.286867
$875$	58.6415	1.98244
$876$	2.69363	0.0910093
$877$	−34.8576	−1.17706	−0.588528	−	0.808477i	$-0.700292\pi$
$877$	−34.8576	−1.17706	−0.588528	+	0.808477i	$0.700292\pi$
$878$	−2.42866	−0.0819634
$879$	1.21680	0.0410415
$880$	−20.7160	−0.698337
$881$	−3.13769	−0.105712	−0.0528558	−	0.998602i	$-0.516832\pi$
$881$	−3.13769	−0.105712	−0.0528558	+	0.998602i	$0.516832\pi$
$882$	−15.7387	−0.529948
$883$	1.83112	0.0616221	0.0308110	−	0.999525i	$-0.490191\pi$
$883$	1.83112	0.0616221	0.0308110	+	0.999525i	$0.490191\pi$
$884$	0	0
$885$	−2.89500	−0.0973144
$886$	−19.1495	−0.643340
$887$	−48.4704	−1.62748	−0.813739	−	0.581231i	$-0.802571\pi$
$887$	−48.4704	−1.62748	−0.813739	+	0.581231i	$0.802571\pi$
$888$	−0.860516	−0.0288770
$889$	−40.8744	−1.37088
$890$	43.5944	1.46129
$891$	9.03494	0.302682
$892$	−22.1419	−0.741366
$893$	−6.37763	−0.213419
$894$	4.97054	0.166240
$895$	7.52089	0.251396
$896$	−3.81192	−0.127347
$897$	0	0
$898$	7.24041	0.241616
$899$	38.5348	1.28521
$900$	−19.0125	−0.633751
$901$	43.2532	1.44097
$902$	−22.8678	−0.761415
$903$	−20.9719	−0.697903
$904$	−9.72329	−0.323392
$905$	−38.9208	−1.29377
$906$	18.4121	0.611702
$907$	−21.2069	−0.704165	−0.352082	−	0.935969i	$-0.614526\pi$
$907$	−21.2069	−0.704165	−0.352082	+	0.935969i	$0.614526\pi$
$908$	24.7975	0.822934
$909$	−29.7400	−0.986413
$910$	0	0
$911$	−46.5216	−1.54133	−0.770664	−	0.637241i	$-0.780075\pi$
$911$	−46.5216	−1.54133	−0.770664	+	0.637241i	$0.780075\pi$
$912$	0.953982	0.0315895
$913$	−72.8397	−2.41064
$914$	16.6298	0.550066
$915$	−5.97122	−0.197402
$916$	12.8014	0.422972
$917$	48.8154	1.61203
$918$	19.4910	0.643300
$919$	−10.7055	−0.353141	−0.176571	−	0.984288i	$-0.556500\pi$
$919$	−10.7055	−0.353141	−0.176571	+	0.984288i	$0.556500\pi$
$920$	31.8423	1.04981
$921$	−25.0133	−0.824218
$922$	26.9244	0.886706
$923$	0	0
$924$	20.0642	0.660065
$925$	8.20597	0.269811
$926$	−31.5292	−1.03611
$927$	−18.2452	−0.599252
$928$	3.65007	0.119819
$929$	13.1905	0.432768	0.216384	−	0.976308i	$-0.430574\pi$
$929$	13.1905	0.432768	0.216384	+	0.976308i	$0.430574\pi$
$930$	37.8146	1.23999
$931$	−7.53075	−0.246810
$932$	18.6566	0.611118
$933$	11.0547	0.361915
$934$	31.9897	1.04674
$935$	−83.1554	−2.71947
$936$	0	0
$937$	−2.26181	−0.0738901	−0.0369450	−	0.999317i	$-0.511763\pi$
$937$	−2.26181	−0.0738901	−0.0369450	+	0.999317i	$0.511763\pi$
$938$	44.8524	1.46448
$939$	26.5155	0.865300
$940$	−23.9456	−0.781020
$941$	−34.0381	−1.10961	−0.554806	−	0.831980i	$-0.687207\pi$
$941$	−34.0381	−1.10961	−0.554806	+	0.831980i	$0.687207\pi$
$942$	6.93555	0.225972
$943$	35.1498	1.14463
$944$	−0.808242	−0.0263060
$945$	69.4964	2.26072
$946$	−31.8195	−1.03454
$947$	−29.2672	−0.951057	−0.475529	−	0.879700i	$-0.657743\pi$
$947$	−29.2672	−0.951057	−0.475529	+	0.879700i	$0.657743\pi$
$948$	−11.6845	−0.379496
$949$	0	0
$950$	−9.09726	−0.295154
$951$	20.9993	0.680949
$952$	−15.3013	−0.495918
$953$	−40.0117	−1.29611	−0.648053	−	0.761595i	$-0.724416\pi$
$953$	−40.0117	−1.29611	−0.648053	+	0.761595i	$0.724416\pi$
$954$	−22.5197	−0.729103
$955$	0.163951	0.00530534
$956$	−19.8498	−0.641987
$957$	−19.2123	−0.621047
$958$	−1.52205	−0.0491753
$959$	28.1086	0.907675
$960$	3.58185	0.115604
$961$	80.4561	2.59536
$962$	0	0
$963$	0.576668	0.0185829
$964$	−5.31705	−0.171251
$965$	7.13084	0.229550
$966$	−30.8405	−0.992276
$967$	−56.1954	−1.80712	−0.903561	−	0.428459i	$-0.859057\pi$
$967$	−56.1954	−1.80712	−0.903561	+	0.428459i	$0.859057\pi$
$968$	19.4423	0.624899
$969$	3.82934	0.123016
$970$	43.1104	1.38419
$971$	−32.1777	−1.03263	−0.516317	−	0.856398i	$-0.672697\pi$
$971$	−32.1777	−1.03263	−0.516317	+	0.856398i	$0.672697\pi$
$972$	−16.1292	−0.517345
$973$	−31.6810	−1.01564
$974$	14.6946	0.470845
$975$	0	0
$976$	−1.66708	−0.0533619
$977$	45.7738	1.46444	0.732218	−	0.681071i	$-0.238485\pi$
$977$	45.7738	1.46444	0.732218	+	0.681071i	$0.238485\pi$
$978$	16.3089	0.521501
$979$	−64.0622	−2.04744
$980$	−28.2752	−0.903218
$981$	15.8178	0.505024
$982$	−29.9345	−0.955247
$983$	38.0408	1.21331	0.606657	−	0.794964i	$-0.292510\pi$
$983$	38.0408	1.21331	0.606657	+	0.794964i	$0.292510\pi$
$984$	3.95390	0.126046
$985$	1.22886	0.0391548
$986$	14.6516	0.466602
$987$	23.1923	0.738218
$988$	0	0
$989$	48.9093	1.55523
$990$	43.2948	1.37600
$991$	39.4205	1.25223	0.626117	−	0.779729i	$-0.284643\pi$
$991$	39.4205	1.25223	0.626117	+	0.779729i	$0.284643\pi$
$992$	10.5573	0.335194
$993$	32.7546	1.03943
$994$	21.3430	0.676959
$995$	−72.8239	−2.30867
$996$	12.5942	0.399061
$997$	31.1740	0.987289	0.493645	−	0.869664i	$-0.335664\pi$
$997$	31.1740	0.987289	0.493645	+	0.869664i	$0.335664\pi$
$998$	−17.1820	−0.543888
$999$	4.37996	0.138576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bl.1.4	yes	9
13.12	even	2		6422.2.a.bj.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bj.1.4	✓	9	13.12	even	2
6422.2.a.bl.1.4	yes	9	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.953982 q^{3} +1.00000 q^{4} -3.75463 q^{5} -0.953982 q^{6} -3.81192 q^{7} +1.00000 q^{8} -2.08992 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.953982 q^{3} +1.00000 q^{4} -3.75463 q^{5} -0.953982 q^{6} -3.81192 q^{7} +1.00000 q^{8} -2.08992 q^{9} -3.75463 q^{10} +5.51745 q^{11} -0.953982 q^{12} -3.81192 q^{14} +3.58185 q^{15} +1.00000 q^{16} +4.01406 q^{17} -2.08992 q^{18} -1.00000 q^{19} -3.75463 q^{20} +3.63650 q^{21} +5.51745 q^{22} -8.48080 q^{23} -0.953982 q^{24} +9.09726 q^{25} +4.85569 q^{27} -3.81192 q^{28} +3.65007 q^{29} +3.58185 q^{30} +10.5573 q^{31} +1.00000 q^{32} -5.26355 q^{33} +4.01406 q^{34} +14.3124 q^{35} -2.08992 q^{36} +0.902026 q^{37} -1.00000 q^{38} -3.75463 q^{40} -4.14463 q^{41} +3.63650 q^{42} -5.76706 q^{43} +5.51745 q^{44} +7.84688 q^{45} -8.48080 q^{46} +6.37763 q^{47} -0.953982 q^{48} +7.53075 q^{49} +9.09726 q^{50} -3.82934 q^{51} +10.7754 q^{53} +4.85569 q^{54} -20.7160 q^{55} -3.81192 q^{56} +0.953982 q^{57} +3.65007 q^{58} -0.808242 q^{59} +3.58185 q^{60} -1.66708 q^{61} +10.5573 q^{62} +7.96661 q^{63} +1.00000 q^{64} -5.26355 q^{66} -11.7663 q^{67} +4.01406 q^{68} +8.09053 q^{69} +14.3124 q^{70} -5.59901 q^{71} -2.08992 q^{72} -2.82357 q^{73} +0.902026 q^{74} -8.67862 q^{75} -1.00000 q^{76} -21.0321 q^{77} +12.2482 q^{79} -3.75463 q^{80} +1.63752 q^{81} -4.14463 q^{82} -13.2017 q^{83} +3.63650 q^{84} -15.0713 q^{85} -5.76706 q^{86} -3.48210 q^{87} +5.51745 q^{88} -11.6108 q^{89} +7.84688 q^{90} -8.48080 q^{92} -10.0714 q^{93} +6.37763 q^{94} +3.75463 q^{95} -0.953982 q^{96} -11.4819 q^{97} +7.53075 q^{98} -11.5310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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