LMFDB - Embedded newform 6422.2.a.bg.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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 22x^{5} + 3x^{4} - 28x^{3} + 7x^{2} + 6x - 2 $ x^8 - 4x^7 - 4x^6 + 22x^5 + 3x^4 - 28x^3 + 7x^2 + 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.533995$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.533995 q^{3} +1.00000 q^{4} -2.79724 q^{5} +0.533995 q^{6} +0.904890 q^{7} -1.00000 q^{8} -2.71485 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.533995 q^{3} +1.00000 q^{4} -2.79724 q^{5} +0.533995 q^{6} +0.904890 q^{7} -1.00000 q^{8} -2.71485 q^{9} +2.79724 q^{10} +2.41304 q^{11} -0.533995 q^{12} -0.904890 q^{14} +1.49371 q^{15} +1.00000 q^{16} +0.336478 q^{17} +2.71485 q^{18} +1.00000 q^{19} -2.79724 q^{20} -0.483206 q^{21} -2.41304 q^{22} -8.62179 q^{23} +0.533995 q^{24} +2.82455 q^{25} +3.05170 q^{27} +0.904890 q^{28} +7.46588 q^{29} -1.49371 q^{30} +7.75451 q^{31} -1.00000 q^{32} -1.28855 q^{33} -0.336478 q^{34} -2.53119 q^{35} -2.71485 q^{36} -11.0929 q^{37} -1.00000 q^{38} +2.79724 q^{40} +5.16998 q^{41} +0.483206 q^{42} -4.41024 q^{43} +2.41304 q^{44} +7.59409 q^{45} +8.62179 q^{46} -3.95332 q^{47} -0.533995 q^{48} -6.18117 q^{49} -2.82455 q^{50} -0.179678 q^{51} +6.74316 q^{53} -3.05170 q^{54} -6.74985 q^{55} -0.904890 q^{56} -0.533995 q^{57} -7.46588 q^{58} +12.8470 q^{59} +1.49371 q^{60} -2.89826 q^{61} -7.75451 q^{62} -2.45664 q^{63} +1.00000 q^{64} +1.28855 q^{66} +11.7900 q^{67} +0.336478 q^{68} +4.60399 q^{69} +2.53119 q^{70} +16.2329 q^{71} +2.71485 q^{72} -11.6519 q^{73} +11.0929 q^{74} -1.50830 q^{75} +1.00000 q^{76} +2.18354 q^{77} -5.00620 q^{79} -2.79724 q^{80} +6.51496 q^{81} -5.16998 q^{82} -14.4781 q^{83} -0.483206 q^{84} -0.941211 q^{85} +4.41024 q^{86} -3.98674 q^{87} -2.41304 q^{88} +15.0155 q^{89} -7.59409 q^{90} -8.62179 q^{92} -4.14087 q^{93} +3.95332 q^{94} -2.79724 q^{95} +0.533995 q^{96} -6.79232 q^{97} +6.18117 q^{98} -6.55104 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8} - 2 q^{10} + 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} + 8 q^{19} + 2 q^{20} - 12 q^{21} - 10 q^{22} - 8 q^{23} + 4 q^{24} - 14 q^{25} - 22 q^{27} + 2 q^{28} + 8 q^{29} + 12 q^{31} - 8 q^{32} - 4 q^{33} - 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} - 2 q^{41} + 12 q^{42} - 16 q^{43} + 10 q^{44} + 12 q^{45} + 8 q^{46} - 12 q^{47} - 4 q^{48} - 14 q^{49} + 14 q^{50} - 22 q^{51} - 24 q^{53} + 22 q^{54} - 10 q^{55} - 2 q^{56} - 4 q^{57} - 8 q^{58} + 4 q^{59} - 26 q^{61} - 12 q^{62} + 12 q^{63} + 8 q^{64} + 4 q^{66} - 10 q^{67} + 2 q^{68} + 6 q^{69} + 10 q^{70} + 36 q^{71} - 20 q^{73} + 6 q^{75} + 8 q^{76} - 12 q^{77} - 22 q^{79} + 2 q^{80} + 36 q^{81} + 2 q^{82} - 18 q^{83} - 12 q^{84} - 26 q^{85} + 16 q^{86} - 38 q^{87} - 10 q^{88} + 18 q^{89} - 12 q^{90} - 8 q^{92} - 12 q^{93} + 12 q^{94} + 2 q^{95} + 4 q^{96} + 8 q^{97} + 14 q^{98} + 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8 - 2 * q^10 + 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 + 8 * q^19 + 2 * q^20 - 12 * q^21 - 10 * q^22 - 8 * q^23 + 4 * q^24 - 14 * q^25 - 22 * q^27 + 2 * q^28 + 8 * q^29 + 12 * q^31 - 8 * q^32 - 4 * q^33 - 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 - 2 * q^41 + 12 * q^42 - 16 * q^43 + 10 * q^44 + 12 * q^45 + 8 * q^46 - 12 * q^47 - 4 * q^48 - 14 * q^49 + 14 * q^50 - 22 * q^51 - 24 * q^53 + 22 * q^54 - 10 * q^55 - 2 * q^56 - 4 * q^57 - 8 * q^58 + 4 * q^59 - 26 * q^61 - 12 * q^62 + 12 * q^63 + 8 * q^64 + 4 * q^66 - 10 * q^67 + 2 * q^68 + 6 * q^69 + 10 * q^70 + 36 * q^71 - 20 * q^73 + 6 * q^75 + 8 * q^76 - 12 * q^77 - 22 * q^79 + 2 * q^80 + 36 * q^81 + 2 * q^82 - 18 * q^83 - 12 * q^84 - 26 * q^85 + 16 * q^86 - 38 * q^87 - 10 * q^88 + 18 * q^89 - 12 * q^90 - 8 * q^92 - 12 * q^93 + 12 * q^94 + 2 * q^95 + 4 * q^96 + 8 * q^97 + 14 * q^98 + 8 * 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
$3$	−0.533995	−0.308302	−0.154151	−	0.988047i	$-0.549264\pi$
$3$	−0.533995	−0.308302	−0.154151	+	0.988047i	$0.549264\pi$
$4$	1.00000	0.500000
$5$	−2.79724	−1.25096	−0.625482	−	0.780239i	$-0.715098\pi$
$5$	−2.79724	−1.25096	−0.625482	+	0.780239i	$0.715098\pi$
$6$	0.533995	0.218002
$7$	0.904890	0.342016	0.171008	−	0.985270i	$-0.445298\pi$
$7$	0.904890	0.342016	0.171008	+	0.985270i	$0.445298\pi$
$8$	−1.00000	−0.353553
$9$	−2.71485	−0.904950
$10$	2.79724	0.884565
$11$	2.41304	0.727559	0.363779	−	0.931485i	$-0.381486\pi$
$11$	2.41304	0.727559	0.363779	+	0.931485i	$0.381486\pi$
$12$	−0.533995	−0.154151
$13$	0	0
$13$	0	0
$14$	−0.904890	−0.241842
$15$	1.49371	0.385675
$16$	1.00000	0.250000
$17$	0.336478	0.0816080	0.0408040	−	0.999167i	$-0.487008\pi$
$17$	0.336478	0.0816080	0.0408040	+	0.999167i	$0.487008\pi$
$18$	2.71485	0.639896
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.79724	−0.625482
$21$	−0.483206	−0.105444
$22$	−2.41304	−0.514462
$23$	−8.62179	−1.79777	−0.898884	−	0.438187i	$-0.855621\pi$
$23$	−8.62179	−1.79777	−0.898884	+	0.438187i	$0.855621\pi$
$24$	0.533995	0.109001
$25$	2.82455	0.564910
$26$	0	0
$27$	3.05170	0.587300
$28$	0.904890	0.171008
$29$	7.46588	1.38638	0.693190	−	0.720755i	$-0.256205\pi$
$29$	7.46588	1.38638	0.693190	+	0.720755i	$0.256205\pi$
$30$	−1.49371	−0.272713
$31$	7.75451	1.39275	0.696376	−	0.717677i	$-0.254795\pi$
$31$	7.75451	1.39275	0.696376	+	0.717677i	$0.254795\pi$
$32$	−1.00000	−0.176777
$33$	−1.28855	−0.224308
$34$	−0.336478	−0.0577056
$35$	−2.53119	−0.427850
$36$	−2.71485	−0.452475
$37$	−11.0929	−1.82366	−0.911832	−	0.410563i	$-0.865332\pi$
$37$	−11.0929	−1.82366	−0.911832	+	0.410563i	$0.865332\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	2.79724	0.442282
$41$	5.16998	0.807415	0.403708	−	0.914888i	$-0.367721\pi$
$41$	5.16998	0.807415	0.403708	+	0.914888i	$0.367721\pi$
$42$	0.483206	0.0745604
$43$	−4.41024	−0.672555	−0.336278	−	0.941763i	$-0.609168\pi$
$43$	−4.41024	−0.672555	−0.336278	+	0.941763i	$0.609168\pi$
$44$	2.41304	0.363779
$45$	7.59409	1.13206
$46$	8.62179	1.27121
$47$	−3.95332	−0.576650	−0.288325	−	0.957533i	$-0.593098\pi$
$47$	−3.95332	−0.576650	−0.288325	+	0.957533i	$0.593098\pi$
$48$	−0.533995	−0.0770755
$49$	−6.18117	−0.883025
$50$	−2.82455	−0.399452
$51$	−0.179678	−0.0251599
$52$	0	0
$53$	6.74316	0.926245	0.463122	−	0.886294i	$-0.346729\pi$
$53$	6.74316	0.926245	0.463122	+	0.886294i	$0.346729\pi$
$54$	−3.05170	−0.415284
$55$	−6.74985	−0.910150
$56$	−0.904890	−0.120921
$57$	−0.533995	−0.0707293
$58$	−7.46588	−0.980318
$59$	12.8470	1.67254	0.836271	−	0.548317i	$-0.184731\pi$
$59$	12.8470	1.67254	0.836271	+	0.548317i	$0.184731\pi$
$60$	1.49371	0.192837
$61$	−2.89826	−0.371084	−0.185542	−	0.982636i	$-0.559404\pi$
$61$	−2.89826	−0.371084	−0.185542	+	0.982636i	$0.559404\pi$
$62$	−7.75451	−0.984824
$63$	−2.45664	−0.309508
$64$	1.00000	0.125000
$65$	0	0
$66$	1.28855	0.158610
$67$	11.7900	1.44038	0.720190	−	0.693776i	$-0.244054\pi$
$67$	11.7900	1.44038	0.720190	+	0.693776i	$0.244054\pi$
$68$	0.336478	0.0408040
$69$	4.60399	0.554255
$70$	2.53119	0.302536
$71$	16.2329	1.92648	0.963242	−	0.268633i	$-0.0865720\pi$
$71$	16.2329	1.92648	0.963242	+	0.268633i	$0.0865720\pi$
$72$	2.71485	0.319948
$73$	−11.6519	−1.36375	−0.681874	−	0.731470i	$-0.738835\pi$
$73$	−11.6519	−1.36375	−0.681874	+	0.731470i	$0.738835\pi$
$74$	11.0929	1.28953
$75$	−1.50830	−0.174163
$76$	1.00000	0.114708
$77$	2.18354	0.248837
$78$	0	0
$79$	−5.00620	−0.563242	−0.281621	−	0.959526i	$-0.590872\pi$
$79$	−5.00620	−0.563242	−0.281621	+	0.959526i	$0.590872\pi$
$80$	−2.79724	−0.312741
$81$	6.51496	0.723884
$82$	−5.16998	−0.570929
$83$	−14.4781	−1.58918	−0.794590	−	0.607146i	$-0.792314\pi$
$83$	−14.4781	−1.58918	−0.794590	+	0.607146i	$0.792314\pi$
$84$	−0.483206	−0.0527221
$85$	−0.941211	−0.102089
$86$	4.41024	0.475568
$87$	−3.98674	−0.427423
$88$	−2.41304	−0.257231
$89$	15.0155	1.59164	0.795822	−	0.605530i	$-0.207039\pi$
$89$	15.0155	1.59164	0.795822	+	0.605530i	$0.207039\pi$
$90$	−7.59409	−0.800487
$91$	0	0
$92$	−8.62179	−0.898884
$93$	−4.14087	−0.429388
$94$	3.95332	0.407753
$95$	−2.79724	−0.286991
$96$	0.533995	0.0545006
$97$	−6.79232	−0.689655	−0.344828	−	0.938666i	$-0.612063\pi$
$97$	−6.79232	−0.689655	−0.344828	+	0.938666i	$0.612063\pi$
$98$	6.18117	0.624393
$99$	−6.55104	−0.658404
$100$	2.82455	0.282455
$101$	−1.28544	−0.127906	−0.0639531	−	0.997953i	$-0.520371\pi$
$101$	−1.28544	−0.127906	−0.0639531	+	0.997953i	$0.520371\pi$
$102$	0.179678	0.0177907
$103$	−8.97969	−0.884795	−0.442397	−	0.896819i	$-0.645872\pi$
$103$	−8.97969	−0.884795	−0.442397	+	0.896819i	$0.645872\pi$
$104$	0	0
$105$	1.35164	0.131907
$106$	−6.74316	−0.654954
$107$	−10.4109	−1.00646	−0.503230	−	0.864152i	$-0.667855\pi$
$107$	−10.4109	−1.00646	−0.503230	+	0.864152i	$0.667855\pi$
$108$	3.05170	0.293650
$109$	19.6093	1.87823	0.939117	−	0.343598i	$-0.111646\pi$
$109$	19.6093	1.87823	0.939117	+	0.343598i	$0.111646\pi$
$110$	6.74985	0.643573
$111$	5.92356	0.562239
$112$	0.904890	0.0855041
$113$	4.53745	0.426847	0.213424	−	0.976960i	$-0.431539\pi$
$113$	4.53745	0.426847	0.213424	+	0.976960i	$0.431539\pi$
$114$	0.533995	0.0500132
$115$	24.1172	2.24894
$116$	7.46588	0.693190
$117$	0	0
$118$	−12.8470	−1.18267
$119$	0.304476	0.0279113
$120$	−1.49371	−0.136357
$121$	−5.17724	−0.470658
$122$	2.89826	0.262396
$123$	−2.76074	−0.248928
$124$	7.75451	0.696376
$125$	6.08525	0.544281
$126$	2.45664	0.218855
$127$	6.15251	0.545947	0.272973	−	0.962022i	$-0.411993\pi$
$127$	6.15251	0.545947	0.272973	+	0.962022i	$0.411993\pi$
$128$	−1.00000	−0.0883883
$129$	2.35504	0.207350
$130$	0	0
$131$	3.52588	0.308058	0.154029	−	0.988066i	$-0.450775\pi$
$131$	3.52588	0.308058	0.154029	+	0.988066i	$0.450775\pi$
$132$	−1.28855	−0.112154
$133$	0.904890	0.0784639
$134$	−11.7900	−1.01850
$135$	−8.53634	−0.734691
$136$	−0.336478	−0.0288528
$137$	17.1629	1.46633	0.733165	−	0.680051i	$-0.238042\pi$
$137$	17.1629	1.46633	0.733165	+	0.680051i	$0.238042\pi$
$138$	−4.60399	−0.391918
$139$	−1.05024	−0.0890800	−0.0445400	−	0.999008i	$-0.514182\pi$
$139$	−1.05024	−0.0890800	−0.0445400	+	0.999008i	$0.514182\pi$
$140$	−2.53119	−0.213925
$141$	2.11105	0.177782
$142$	−16.2329	−1.36223
$143$	0	0
$144$	−2.71485	−0.226237
$145$	−20.8839	−1.73431
$146$	11.6519	0.964315
$147$	3.30071	0.272238
$148$	−11.0929	−0.911832
$149$	11.6327	0.952989	0.476495	−	0.879177i	$-0.341907\pi$
$149$	11.6327	0.952989	0.476495	+	0.879177i	$0.341907\pi$
$150$	1.50830	0.123152
$151$	−20.7774	−1.69084	−0.845419	−	0.534103i	$-0.820649\pi$
$151$	−20.7774	−1.69084	−0.845419	+	0.534103i	$0.820649\pi$
$152$	−1.00000	−0.0811107
$153$	−0.913488	−0.0738511
$154$	−2.18354	−0.175954
$155$	−21.6912	−1.74228
$156$	0	0
$157$	−10.3488	−0.825924	−0.412962	−	0.910748i	$-0.635506\pi$
$157$	−10.3488	−0.825924	−0.412962	+	0.910748i	$0.635506\pi$
$158$	5.00620	0.398272
$159$	−3.60081	−0.285563
$160$	2.79724	0.221141
$161$	−7.80177	−0.614866
$162$	−6.51496	−0.511863
$163$	1.12803	0.0883540	0.0441770	−	0.999024i	$-0.485933\pi$
$163$	1.12803	0.0883540	0.0441770	+	0.999024i	$0.485933\pi$
$164$	5.16998	0.403708
$165$	3.60439	0.280601
$166$	14.4781	1.12372
$167$	−8.11258	−0.627770	−0.313885	−	0.949461i	$-0.601631\pi$
$167$	−8.11258	−0.627770	−0.313885	+	0.949461i	$0.601631\pi$
$168$	0.483206	0.0372802
$169$	0	0
$170$	0.941211	0.0721876
$171$	−2.71485	−0.207610
$172$	−4.41024	−0.336278
$173$	13.6484	1.03767	0.518833	−	0.854876i	$-0.326367\pi$
$173$	13.6484	1.03767	0.518833	+	0.854876i	$0.326367\pi$
$174$	3.98674	0.302234
$175$	2.55591	0.193209
$176$	2.41304	0.181890
$177$	−6.86025	−0.515648
$178$	−15.0155	−1.12546
$179$	−5.33366	−0.398656	−0.199328	−	0.979933i	$-0.563876\pi$
$179$	−5.33366	−0.398656	−0.199328	+	0.979933i	$0.563876\pi$
$180$	7.59409	0.566030
$181$	−24.3388	−1.80909	−0.904545	−	0.426378i	$-0.859789\pi$
$181$	−24.3388	−1.80909	−0.904545	+	0.426378i	$0.859789\pi$
$182$	0	0
$183$	1.54765	0.114406
$184$	8.62179	0.635607
$185$	31.0296	2.28134
$186$	4.14087	0.303623
$187$	0.811936	0.0593746
$188$	−3.95332	−0.288325
$189$	2.76145	0.200866
$190$	2.79724	0.202933
$191$	6.74347	0.487940	0.243970	−	0.969783i	$-0.421550\pi$
$191$	6.74347	0.487940	0.243970	+	0.969783i	$0.421550\pi$
$192$	−0.533995	−0.0385377
$193$	−3.38300	−0.243514	−0.121757	−	0.992560i	$-0.538853\pi$
$193$	−3.38300	−0.243514	−0.121757	+	0.992560i	$0.538853\pi$
$194$	6.79232	0.487660
$195$	0	0
$196$	−6.18117	−0.441512
$197$	−4.72792	−0.336851	−0.168425	−	0.985714i	$-0.553868\pi$
$197$	−4.72792	−0.336851	−0.168425	+	0.985714i	$0.553868\pi$
$198$	6.55104	0.465562
$199$	6.76173	0.479326	0.239663	−	0.970856i	$-0.422963\pi$
$199$	6.76173	0.479326	0.239663	+	0.970856i	$0.422963\pi$
$200$	−2.82455	−0.199726
$201$	−6.29581	−0.444072
$202$	1.28544	0.0904433
$203$	6.75580	0.474164
$204$	−0.179678	−0.0125800
$205$	−14.4617	−1.01005
$206$	8.97969	0.625644
$207$	23.4069	1.62689
$208$	0	0
$209$	2.41304	0.166913
$210$	−1.35164	−0.0932723
$211$	0.0337822	0.00232566	0.00116283	−	0.999999i	$-0.499630\pi$
$211$	0.0337822	0.00232566	0.00116283	+	0.999999i	$0.499630\pi$
$212$	6.74316	0.463122
$213$	−8.66826	−0.593939
$214$	10.4109	0.711675
$215$	12.3365	0.841342
$216$	−3.05170	−0.207642
$217$	7.01698	0.476344
$218$	−19.6093	−1.32811
$219$	6.22204	0.420446
$220$	−6.74985	−0.455075
$221$	0	0
$222$	−5.92356	−0.397563
$223$	3.69513	0.247444	0.123722	−	0.992317i	$-0.460517\pi$
$223$	3.69513	0.247444	0.123722	+	0.992317i	$0.460517\pi$
$224$	−0.904890	−0.0604605
$225$	−7.66824	−0.511216
$226$	−4.53745	−0.301827
$227$	−10.6674	−0.708017	−0.354008	−	0.935242i	$-0.615182\pi$
$227$	−10.6674	−0.708017	−0.354008	+	0.935242i	$0.615182\pi$
$228$	−0.533995	−0.0353647
$229$	7.72323	0.510365	0.255183	−	0.966893i	$-0.417864\pi$
$229$	7.72323	0.510365	0.255183	+	0.966893i	$0.417864\pi$
$230$	−24.1172	−1.59024
$231$	−1.16600	−0.0767169
$232$	−7.46588	−0.490159
$233$	3.32091	0.217560	0.108780	−	0.994066i	$-0.465306\pi$
$233$	3.32091	0.217560	0.108780	+	0.994066i	$0.465306\pi$
$234$	0	0
$235$	11.0584	0.721369
$236$	12.8470	0.836271
$237$	2.67328	0.173648
$238$	−0.304476	−0.0197362
$239$	−10.8271	−0.700348	−0.350174	−	0.936685i	$-0.613878\pi$
$239$	−10.8271	−0.700348	−0.350174	+	0.936685i	$0.613878\pi$
$240$	1.49371	0.0964187
$241$	−11.9629	−0.770601	−0.385301	−	0.922791i	$-0.625902\pi$
$241$	−11.9629	−0.770601	−0.385301	+	0.922791i	$0.625902\pi$
$242$	5.17724	0.332805
$243$	−12.6341	−0.810475
$244$	−2.89826	−0.185542
$245$	17.2902	1.10463
$246$	2.76074	0.176019
$247$	0	0
$248$	−7.75451	−0.492412
$249$	7.73124	0.489948
$250$	−6.08525	−0.384865
$251$	6.11624	0.386054	0.193027	−	0.981193i	$-0.438170\pi$
$251$	6.11624	0.386054	0.193027	+	0.981193i	$0.438170\pi$
$252$	−2.45664	−0.154754
$253$	−20.8047	−1.30798
$254$	−6.15251	−0.386043
$255$	0.502602	0.0314741
$256$	1.00000	0.0625000
$257$	0.742012	0.0462854	0.0231427	−	0.999732i	$-0.492633\pi$
$257$	0.742012	0.0462854	0.0231427	+	0.999732i	$0.492633\pi$
$258$	−2.35504	−0.146619
$259$	−10.0379	−0.623723
$260$	0	0
$261$	−20.2687	−1.25460
$262$	−3.52588	−0.217830
$263$	−8.16537	−0.503498	−0.251749	−	0.967793i	$-0.581006\pi$
$263$	−8.16537	−0.503498	−0.251749	+	0.967793i	$0.581006\pi$
$264$	1.28855	0.0793048
$265$	−18.8622	−1.15870
$266$	−0.904890	−0.0554824
$267$	−8.01822	−0.490707
$268$	11.7900	0.720190
$269$	−14.4846	−0.883139	−0.441569	−	0.897227i	$-0.645578\pi$
$269$	−14.4846	−0.883139	−0.441569	+	0.897227i	$0.645578\pi$
$270$	8.53634	0.519505
$271$	−29.9304	−1.81814	−0.909070	−	0.416642i	$-0.863207\pi$
$271$	−29.9304	−1.81814	−0.909070	+	0.416642i	$0.863207\pi$
$272$	0.336478	0.0204020
$273$	0	0
$274$	−17.1629	−1.03685
$275$	6.81576	0.411006
$276$	4.60399	0.277128
$277$	−0.553968	−0.0332847	−0.0166424	−	0.999862i	$-0.505298\pi$
$277$	−0.553968	−0.0332847	−0.0166424	+	0.999862i	$0.505298\pi$
$278$	1.05024	0.0629891
$279$	−21.0523	−1.26037
$280$	2.53119	0.151268
$281$	−28.5562	−1.70352	−0.851760	−	0.523933i	$-0.824464\pi$
$281$	−28.5562	−1.70352	−0.851760	+	0.523933i	$0.824464\pi$
$282$	−2.11105	−0.125711
$283$	−14.1226	−0.839504	−0.419752	−	0.907639i	$-0.637883\pi$
$283$	−14.1226	−0.839504	−0.419752	+	0.907639i	$0.637883\pi$
$284$	16.2329	0.963242
$285$	1.49371	0.0884798
$286$	0	0
$287$	4.67826	0.276149
$288$	2.71485	0.159974
$289$	−16.8868	−0.993340
$290$	20.8839	1.22634
$291$	3.62706	0.212622
$292$	−11.6519	−0.681874
$293$	−12.6001	−0.736104	−0.368052	−	0.929805i	$-0.619975\pi$
$293$	−12.6001	−0.736104	−0.368052	+	0.929805i	$0.619975\pi$
$294$	−3.30071	−0.192502
$295$	−35.9362	−2.09229
$296$	11.0929	0.644763
$297$	7.36387	0.427295
$298$	−11.6327	−0.673865
$299$	0	0
$300$	−1.50830	−0.0870815
$301$	−3.99078	−0.230025
$302$	20.7774	1.19560
$303$	0.686419	0.0394337
$304$	1.00000	0.0573539
$305$	8.10713	0.464213
$306$	0.913488	0.0522206
$307$	−8.82083	−0.503431	−0.251716	−	0.967801i	$-0.580995\pi$
$307$	−8.82083	−0.503431	−0.251716	+	0.967801i	$0.580995\pi$
$308$	2.18354	0.124418
$309$	4.79510	0.272784
$310$	21.6912	1.23198
$311$	−27.7921	−1.57595	−0.787973	−	0.615710i	$-0.788869\pi$
$311$	−27.7921	−1.57595	−0.787973	+	0.615710i	$0.788869\pi$
$312$	0	0
$313$	28.2286	1.59557	0.797786	−	0.602940i	$-0.206004\pi$
$313$	28.2286	1.59557	0.797786	+	0.602940i	$0.206004\pi$
$314$	10.3488	0.584017
$315$	6.87181	0.387183
$316$	−5.00620	−0.281621
$317$	−3.84232	−0.215806	−0.107903	−	0.994161i	$-0.534414\pi$
$317$	−3.84232	−0.215806	−0.107903	+	0.994161i	$0.534414\pi$
$318$	3.60081	0.201924
$319$	18.0155	1.00867
$320$	−2.79724	−0.156370
$321$	5.55937	0.310294
$322$	7.80177	0.434776
$323$	0.336478	0.0187222
$324$	6.51496	0.361942
$325$	0	0
$326$	−1.12803	−0.0624757
$327$	−10.4713	−0.579063
$328$	−5.16998	−0.285464
$329$	−3.57732	−0.197224
$330$	−3.60439	−0.198415
$331$	10.9197	0.600203	0.300102	−	0.953907i	$-0.402979\pi$
$331$	10.9197	0.600203	0.300102	+	0.953907i	$0.402979\pi$
$332$	−14.4781	−0.794590
$333$	30.1156	1.65032
$334$	8.11258	0.443901
$335$	−32.9795	−1.80186
$336$	−0.483206	−0.0263611
$337$	−5.79951	−0.315920	−0.157960	−	0.987446i	$-0.550492\pi$
$337$	−5.79951	−0.315920	−0.157960	+	0.987446i	$0.550492\pi$
$338$	0	0
$339$	−2.42297	−0.131598
$340$	−0.941211	−0.0510443
$341$	18.7119	1.01331
$342$	2.71485	0.146802
$343$	−11.9275	−0.644025
$344$	4.41024	0.237784
$345$	−12.8785	−0.693353
$346$	−13.6484	−0.733741
$347$	11.3297	0.608212	0.304106	−	0.952638i	$-0.401642\pi$
$347$	11.3297	0.608212	0.304106	+	0.952638i	$0.401642\pi$
$348$	−3.98674	−0.213712
$349$	32.3965	1.73415	0.867073	−	0.498181i	$-0.165998\pi$
$349$	32.3965	1.73415	0.867073	+	0.498181i	$0.165998\pi$
$350$	−2.55591	−0.136619
$351$	0	0
$352$	−2.41304	−0.128615
$353$	10.7881	0.574192	0.287096	−	0.957902i	$-0.407310\pi$
$353$	10.7881	0.574192	0.287096	+	0.957902i	$0.407310\pi$
$354$	6.86025	0.364618
$355$	−45.4072	−2.40996
$356$	15.0155	0.795822
$357$	−0.162589	−0.00860510
$358$	5.33366	0.281893
$359$	−34.3183	−1.81125	−0.905624	−	0.424081i	$-0.860597\pi$
$359$	−34.3183	−1.81125	−0.905624	+	0.424081i	$0.860597\pi$
$360$	−7.59409	−0.400244
$361$	1.00000	0.0526316
$362$	24.3388	1.27922
$363$	2.76462	0.145105
$364$	0	0
$365$	32.5931	1.70600
$366$	−1.54765	−0.0808972
$367$	23.5526	1.22944	0.614718	−	0.788747i	$-0.289270\pi$
$367$	23.5526	1.22944	0.614718	+	0.788747i	$0.289270\pi$
$368$	−8.62179	−0.449442
$369$	−14.0357	−0.730671
$370$	−31.0296	−1.61315
$371$	6.10182	0.316791
$372$	−4.14087	−0.214694
$373$	−25.7122	−1.33133	−0.665663	−	0.746252i	$-0.731851\pi$
$373$	−25.7122	−1.33133	−0.665663	+	0.746252i	$0.731851\pi$
$374$	−0.811936	−0.0419842
$375$	−3.24949	−0.167803
$376$	3.95332	0.203877
$377$	0	0
$378$	−2.76145	−0.142034
$379$	−12.3452	−0.634131	−0.317065	−	0.948404i	$-0.602697\pi$
$379$	−12.3452	−0.634131	−0.317065	+	0.948404i	$0.602697\pi$
$380$	−2.79724	−0.143495
$381$	−3.28541	−0.168316
$382$	−6.74347	−0.345026
$383$	11.5251	0.588904	0.294452	−	0.955666i	$-0.404863\pi$
$383$	11.5251	0.588904	0.294452	+	0.955666i	$0.404863\pi$
$384$	0.533995	0.0272503
$385$	−6.10787	−0.311286
$386$	3.38300	0.172190
$387$	11.9731	0.608629
$388$	−6.79232	−0.344828
$389$	−13.2871	−0.673683	−0.336842	−	0.941561i	$-0.609359\pi$
$389$	−13.2871	−0.673683	−0.336842	+	0.941561i	$0.609359\pi$
$390$	0	0
$391$	−2.90105	−0.146712
$392$	6.18117	0.312196
$393$	−1.88280	−0.0949748
$394$	4.72792	0.238189
$395$	14.0035	0.704595
$396$	−6.55104	−0.329202
$397$	2.23448	0.112145	0.0560727	−	0.998427i	$-0.482142\pi$
$397$	2.23448	0.112145	0.0560727	+	0.998427i	$0.482142\pi$
$398$	−6.76173	−0.338935
$399$	−0.483206	−0.0241906
$400$	2.82455	0.141228
$401$	4.31218	0.215340	0.107670	−	0.994187i	$-0.465661\pi$
$401$	4.31218	0.215340	0.107670	+	0.994187i	$0.465661\pi$
$402$	6.29581	0.314007
$403$	0	0
$404$	−1.28544	−0.0639531
$405$	−18.2239	−0.905553
$406$	−6.75580	−0.335285
$407$	−26.7677	−1.32682
$408$	0.179678	0.00889537
$409$	23.1334	1.14387	0.571936	−	0.820298i	$-0.306192\pi$
$409$	23.1334	1.14387	0.571936	+	0.820298i	$0.306192\pi$
$410$	14.4617	0.714211
$411$	−9.16492	−0.452072
$412$	−8.97969	−0.442397
$413$	11.6252	0.572036
$414$	−23.4069	−1.15038
$415$	40.4988	1.98801
$416$	0	0
$417$	0.560822	0.0274635
$418$	−2.41304	−0.118026
$419$	−9.27300	−0.453016	−0.226508	−	0.974009i	$-0.572731\pi$
$419$	−9.27300	−0.453016	−0.226508	+	0.974009i	$0.572731\pi$
$420$	1.35164	0.0659535
$421$	−19.3226	−0.941728	−0.470864	−	0.882206i	$-0.656058\pi$
$421$	−19.3226	−0.941728	−0.470864	+	0.882206i	$0.656058\pi$
$422$	−0.0337822	−0.00164449
$423$	10.7327	0.521840
$424$	−6.74316	−0.327477
$425$	0.950401	0.0461012
$426$	8.66826	0.419978
$427$	−2.62261	−0.126917
$428$	−10.4109	−0.503230
$429$	0	0
$430$	−12.3365	−0.594919
$431$	−28.0060	−1.34900	−0.674500	−	0.738275i	$-0.735641\pi$
$431$	−28.0060	−1.34900	−0.674500	+	0.738275i	$0.735641\pi$
$432$	3.05170	0.146825
$433$	28.6014	1.37450	0.687248	−	0.726423i	$-0.258818\pi$
$433$	28.6014	1.37450	0.687248	+	0.726423i	$0.258818\pi$
$434$	−7.01698	−0.336826
$435$	11.1519	0.534691
$436$	19.6093	0.939117
$437$	−8.62179	−0.412436
$438$	−6.22204	−0.297300
$439$	−18.2630	−0.871644	−0.435822	−	0.900033i	$-0.643542\pi$
$439$	−18.2630	−0.871644	−0.435822	+	0.900033i	$0.643542\pi$
$440$	6.74985	0.321787
$441$	16.7810	0.799093
$442$	0	0
$443$	−21.0754	−1.00132	−0.500660	−	0.865644i	$-0.666909\pi$
$443$	−21.0754	−1.00132	−0.500660	+	0.865644i	$0.666909\pi$
$444$	5.92356	0.281120
$445$	−42.0021	−1.99109
$446$	−3.69513	−0.174970
$447$	−6.21181	−0.293808
$448$	0.904890	0.0427520
$449$	14.3812	0.678691	0.339345	−	0.940662i	$-0.389795\pi$
$449$	14.3812	0.678691	0.339345	+	0.940662i	$0.389795\pi$
$450$	7.66824	0.361484
$451$	12.4754	0.587442
$452$	4.53745	0.213424
$453$	11.0950	0.521289
$454$	10.6674	0.500644
$455$	0	0
$456$	0.533995	0.0250066
$457$	−21.2580	−0.994407	−0.497203	−	0.867634i	$-0.665640\pi$
$457$	−21.2580	−0.994407	−0.497203	+	0.867634i	$0.665640\pi$
$458$	−7.72323	−0.360883
$459$	1.02683	0.0479284
$460$	24.1172	1.12447
$461$	−30.5002	−1.42053	−0.710267	−	0.703932i	$-0.751426\pi$
$461$	−30.5002	−1.42053	−0.710267	+	0.703932i	$0.751426\pi$
$462$	1.16600	0.0542471
$463$	0.544292	0.0252954	0.0126477	−	0.999920i	$-0.495974\pi$
$463$	0.544292	0.0252954	0.0126477	+	0.999920i	$0.495974\pi$
$464$	7.46588	0.346595
$465$	11.5830	0.537149
$466$	−3.32091	−0.153838
$467$	−3.93860	−0.182257	−0.0911284	−	0.995839i	$-0.529047\pi$
$467$	−3.93860	−0.182257	−0.0911284	+	0.995839i	$0.529047\pi$
$468$	0	0
$469$	10.6687	0.492634
$470$	−11.0584	−0.510085
$471$	5.52621	0.254634
$472$	−12.8470	−0.591333
$473$	−10.6421	−0.489324
$474$	−2.67328	−0.122788
$475$	2.82455	0.129599
$476$	0.304476	0.0139556
$477$	−18.3067	−0.838205
$478$	10.8271	0.495221
$479$	15.4244	0.704757	0.352378	−	0.935858i	$-0.385373\pi$
$479$	15.4244	0.704757	0.352378	+	0.935858i	$0.385373\pi$
$480$	−1.49371	−0.0681783
$481$	0	0
$482$	11.9629	0.544897
$483$	4.16611	0.189564
$484$	−5.17724	−0.235329
$485$	18.9997	0.862734
$486$	12.6341	0.573092
$487$	9.45313	0.428362	0.214181	−	0.976794i	$-0.431292\pi$
$487$	9.45313	0.428362	0.214181	+	0.976794i	$0.431292\pi$
$488$	2.89826	0.131198
$489$	−0.602361	−0.0272397
$490$	−17.2902	−0.781093
$491$	16.2955	0.735403	0.367702	−	0.929944i	$-0.380145\pi$
$491$	16.2955	0.735403	0.367702	+	0.929944i	$0.380145\pi$
$492$	−2.76074	−0.124464
$493$	2.51211	0.113140
$494$	0	0
$495$	18.3248	0.823640
$496$	7.75451	0.348188
$497$	14.6889	0.658889
$498$	−7.73124	−0.346445
$499$	−39.8915	−1.78579	−0.892895	−	0.450265i	$-0.851329\pi$
$499$	−39.8915	−1.78579	−0.892895	+	0.450265i	$0.851329\pi$
$500$	6.08525	0.272141
$501$	4.33207	0.193543
$502$	−6.11624	−0.272981
$503$	−5.95636	−0.265581	−0.132791	−	0.991144i	$-0.542394\pi$
$503$	−5.95636	−0.265581	−0.132791	+	0.991144i	$0.542394\pi$
$504$	2.45664	0.109427
$505$	3.59569	0.160006
$506$	20.8047	0.924883
$507$	0	0
$508$	6.15251	0.272973
$509$	6.55633	0.290604	0.145302	−	0.989387i	$-0.453585\pi$
$509$	6.55633	0.290604	0.145302	+	0.989387i	$0.453585\pi$
$510$	−0.502602	−0.0222556
$511$	−10.5437	−0.466424
$512$	−1.00000	−0.0441942
$513$	3.05170	0.134736
$514$	−0.742012	−0.0327287
$515$	25.1183	1.10685
$516$	2.35504	0.103675
$517$	−9.53951	−0.419547
$518$	10.0379	0.441039
$519$	−7.28816	−0.319915
$520$	0	0
$521$	−43.4867	−1.90519	−0.952594	−	0.304244i	$-0.901596\pi$
$521$	−43.4867	−1.90519	−0.952594	+	0.304244i	$0.901596\pi$
$522$	20.2687	0.887139
$523$	1.81240	0.0792505	0.0396253	−	0.999215i	$-0.487384\pi$
$523$	1.81240	0.0792505	0.0396253	+	0.999215i	$0.487384\pi$
$524$	3.52588	0.154029
$525$	−1.36484	−0.0595666
$526$	8.16537	0.356027
$527$	2.60923	0.113660
$528$	−1.28855	−0.0560770
$529$	51.3353	2.23197
$530$	18.8622	0.819324
$531$	−34.8778	−1.51357
$532$	0.904890	0.0392320
$533$	0	0
$534$	8.01822	0.346982
$535$	29.1218	1.25905
$536$	−11.7900	−0.509252
$537$	2.84814	0.122906
$538$	14.4846	0.624473
$539$	−14.9154	−0.642453
$540$	−8.53634	−0.367345
$541$	22.3828	0.962311	0.481156	−	0.876635i	$-0.340217\pi$
$541$	22.3828	0.962311	0.481156	+	0.876635i	$0.340217\pi$
$542$	29.9304	1.28562
$543$	12.9968	0.557746
$544$	−0.336478	−0.0144264
$545$	−54.8520	−2.34960
$546$	0	0
$547$	7.51807	0.321449	0.160725	−	0.986999i	$-0.448617\pi$
$547$	7.51807	0.321449	0.160725	+	0.986999i	$0.448617\pi$
$548$	17.1629	0.733165
$549$	7.86834	0.335812
$550$	−6.81576	−0.290625
$551$	7.46588	0.318057
$552$	−4.60399	−0.195959
$553$	−4.53006	−0.192638
$554$	0.553968	0.0235359
$555$	−16.5696	−0.703341
$556$	−1.05024	−0.0445400
$557$	−33.7940	−1.43190	−0.715948	−	0.698154i	$-0.754005\pi$
$557$	−33.7940	−1.43190	−0.715948	+	0.698154i	$0.754005\pi$
$558$	21.0523	0.891217
$559$	0	0
$560$	−2.53119	−0.106962
$561$	−0.433569	−0.0183053
$562$	28.5562	1.20457
$563$	−33.4641	−1.41034	−0.705172	−	0.709036i	$-0.749130\pi$
$563$	−33.4641	−1.41034	−0.705172	+	0.709036i	$0.749130\pi$
$564$	2.11105	0.0888912
$565$	−12.6923	−0.533971
$566$	14.1226	0.593619
$567$	5.89532	0.247580
$568$	−16.2329	−0.681115
$569$	−20.1683	−0.845499	−0.422749	−	0.906247i	$-0.638935\pi$
$569$	−20.1683	−0.845499	−0.422749	+	0.906247i	$0.638935\pi$
$570$	−1.49371	−0.0625647
$571$	−16.5864	−0.694121	−0.347061	−	0.937843i	$-0.612820\pi$
$571$	−16.5864	−0.694121	−0.347061	+	0.937843i	$0.612820\pi$
$572$	0	0
$573$	−3.60098	−0.150433
$574$	−4.67826	−0.195267
$575$	−24.3527	−1.01558
$576$	−2.71485	−0.113119
$577$	0.178768	0.00744219	0.00372109	−	0.999993i	$-0.498816\pi$
$577$	0.178768	0.00744219	0.00372109	+	0.999993i	$0.498816\pi$
$578$	16.8868	0.702398
$579$	1.80651	0.0750758
$580$	−20.8839	−0.867155
$581$	−13.1011	−0.543526
$582$	−3.62706	−0.150346
$583$	16.2715	0.673898
$584$	11.6519	0.482158
$585$	0	0
$586$	12.6001	0.520504
$587$	−10.5370	−0.434907	−0.217454	−	0.976071i	$-0.569775\pi$
$587$	−10.5370	−0.434907	−0.217454	+	0.976071i	$0.569775\pi$
$588$	3.30071	0.136119
$589$	7.75451	0.319519
$590$	35.9362	1.47947
$591$	2.52469	0.103852
$592$	−11.0929	−0.455916
$593$	23.9731	0.984457	0.492228	−	0.870466i	$-0.336182\pi$
$593$	23.9731	0.984457	0.492228	+	0.870466i	$0.336182\pi$
$594$	−7.36387	−0.302143
$595$	−0.851692	−0.0349160
$596$	11.6327	0.476495
$597$	−3.61073	−0.147777
$598$	0	0
$599$	9.27316	0.378891	0.189446	−	0.981891i	$-0.439331\pi$
$599$	9.27316	0.378891	0.189446	+	0.981891i	$0.439331\pi$
$600$	1.50830	0.0615759
$601$	−20.9989	−0.856564	−0.428282	−	0.903645i	$-0.640881\pi$
$601$	−20.9989	−0.856564	−0.428282	+	0.903645i	$0.640881\pi$
$602$	3.99078	0.162652
$603$	−32.0081	−1.30347
$604$	−20.7774	−0.845419
$605$	14.4820	0.588776
$606$	−0.686419	−0.0278838
$607$	4.92557	0.199923	0.0999614	−	0.994991i	$-0.468128\pi$
$607$	4.92557	0.199923	0.0999614	+	0.994991i	$0.468128\pi$
$608$	−1.00000	−0.0405554
$609$	−3.60756	−0.146186
$610$	−8.10713	−0.328248
$611$	0	0
$612$	−0.913488	−0.0369256
$613$	8.08312	0.326474	0.163237	−	0.986587i	$-0.447806\pi$
$613$	8.08312	0.326474	0.163237	+	0.986587i	$0.447806\pi$
$614$	8.82083	0.355980
$615$	7.72246	0.311400
$616$	−2.18354	−0.0879772
$617$	−0.745927	−0.0300299	−0.0150150	−	0.999887i	$-0.504780\pi$
$617$	−0.745927	−0.0300299	−0.0150150	+	0.999887i	$0.504780\pi$
$618$	−4.79510	−0.192887
$619$	39.0106	1.56797	0.783984	−	0.620781i	$-0.213184\pi$
$619$	39.0106	1.56797	0.783984	+	0.620781i	$0.213184\pi$
$620$	−21.6912	−0.871141
$621$	−26.3111	−1.05583
$622$	27.7921	1.11436
$623$	13.5874	0.544368
$624$	0	0
$625$	−31.1447	−1.24579
$626$	−28.2286	−1.12824
$627$	−1.28855	−0.0514597
$628$	−10.3488	−0.412962
$629$	−3.73253	−0.148826
$630$	−6.87181	−0.273780
$631$	−6.95623	−0.276923	−0.138462	−	0.990368i	$-0.544216\pi$
$631$	−6.95623	−0.276923	−0.138462	+	0.990368i	$0.544216\pi$
$632$	5.00620	0.199136
$633$	−0.0180395	−0.000717005 0
$634$	3.84232	0.152598
$635$	−17.2100	−0.682960
$636$	−3.60081	−0.142782
$637$	0	0
$638$	−18.0155	−0.713239
$639$	−44.0698	−1.74337
$640$	2.79724	0.110571
$641$	23.5324	0.929474	0.464737	−	0.885449i	$-0.346149\pi$
$641$	23.5324	0.929474	0.464737	+	0.885449i	$0.346149\pi$
$642$	−5.55937	−0.219411
$643$	14.6652	0.578340	0.289170	−	0.957278i	$-0.406621\pi$
$643$	14.6652	0.578340	0.289170	+	0.957278i	$0.406621\pi$
$644$	−7.80177	−0.307433
$645$	−6.58763	−0.259387
$646$	−0.336478	−0.0132386
$647$	−34.1919	−1.34422	−0.672111	−	0.740450i	$-0.734612\pi$
$647$	−34.1919	−1.34422	−0.672111	+	0.740450i	$0.734612\pi$
$648$	−6.51496	−0.255932
$649$	31.0004	1.21687
$650$	0	0
$651$	−3.74703	−0.146858
$652$	1.12803	0.0441770
$653$	−43.9586	−1.72023	−0.860117	−	0.510096i	$-0.829610\pi$
$653$	−43.9586	−1.72023	−0.860117	+	0.510096i	$0.829610\pi$
$654$	10.4713	0.409459
$655$	−9.86274	−0.385369
$656$	5.16998	0.201854
$657$	31.6331	1.23412
$658$	3.57732	0.139458
$659$	44.5256	1.73447	0.867235	−	0.497898i	$-0.165895\pi$
$659$	44.5256	1.73447	0.867235	+	0.497898i	$0.165895\pi$
$660$	3.60439	0.140300
$661$	−41.5515	−1.61617	−0.808083	−	0.589069i	$-0.799495\pi$
$661$	−41.5515	−1.61617	−0.808083	+	0.589069i	$0.799495\pi$
$662$	−10.9197	−0.424408
$663$	0	0
$664$	14.4781	0.561860
$665$	−2.53119	−0.0981555
$666$	−30.1156	−1.16696
$667$	−64.3693	−2.49239
$668$	−8.11258	−0.313885
$669$	−1.97318	−0.0762875
$670$	32.9795	1.27411
$671$	−6.99361	−0.269985
$672$	0.483206	0.0186401
$673$	3.89935	0.150309	0.0751545	−	0.997172i	$-0.476055\pi$
$673$	3.89935	0.150309	0.0751545	+	0.997172i	$0.476055\pi$
$674$	5.79951	0.223389
$675$	8.61968	0.331772
$676$	0	0
$677$	11.1816	0.429746	0.214873	−	0.976642i	$-0.431066\pi$
$677$	11.1816	0.429746	0.214873	+	0.976642i	$0.431066\pi$
$678$	2.42297	0.0930537
$679$	−6.14630	−0.235873
$680$	0.941211	0.0360938
$681$	5.69631	0.218283
$682$	−18.7119	−0.716518
$683$	−31.2557	−1.19596	−0.597982	−	0.801509i	$-0.704031\pi$
$683$	−31.2557	−1.19596	−0.597982	+	0.801509i	$0.704031\pi$
$684$	−2.71485	−0.103805
$685$	−48.0089	−1.83432
$686$	11.9275	0.455395
$687$	−4.12416	−0.157347
$688$	−4.41024	−0.168139
$689$	0	0
$690$	12.8785	0.490275
$691$	−7.50328	−0.285438	−0.142719	−	0.989763i	$-0.545585\pi$
$691$	−7.50328	−0.285438	−0.142719	+	0.989763i	$0.545585\pi$
$692$	13.6484	0.518833
$693$	−5.92797	−0.225185
$694$	−11.3297	−0.430071
$695$	2.93777	0.111436
$696$	3.98674	0.151117
$697$	1.73959	0.0658915
$698$	−32.3965	−1.22623
$699$	−1.77335	−0.0670741
$700$	2.55591	0.0966043
$701$	3.98975	0.150691	0.0753454	−	0.997157i	$-0.475994\pi$
$701$	3.98975	0.150691	0.0753454	+	0.997157i	$0.475994\pi$
$702$	0	0
$703$	−11.0929	−0.418377
$704$	2.41304	0.0909449
$705$	−5.90511	−0.222399
$706$	−10.7881	−0.406015
$707$	−1.16318	−0.0437460
$708$	−6.86025	−0.257824
$709$	−33.6512	−1.26380	−0.631899	−	0.775051i	$-0.717724\pi$
$709$	−33.6512	−1.26380	−0.631899	+	0.775051i	$0.717724\pi$
$710$	45.4072	1.70410
$711$	13.5911	0.509705
$712$	−15.0155	−0.562731
$713$	−66.8578	−2.50384
$714$	0.162589	0.00608472
$715$	0	0
$716$	−5.33366	−0.199328
$717$	5.78162	0.215919
$718$	34.3183	1.28075
$719$	−12.6836	−0.473017	−0.236508	−	0.971629i	$-0.576003\pi$
$719$	−12.6836	−0.473017	−0.236508	+	0.971629i	$0.576003\pi$
$720$	7.59409	0.283015
$721$	−8.12563	−0.302614
$722$	−1.00000	−0.0372161
$723$	6.38815	0.237578
$724$	−24.3388	−0.904545
$725$	21.0878	0.783180
$726$	−2.76462	−0.102605
$727$	−21.0014	−0.778898	−0.389449	−	0.921048i	$-0.627335\pi$
$727$	−21.0014	−0.778898	−0.389449	+	0.921048i	$0.627335\pi$
$728$	0	0
$729$	−12.7984	−0.474013
$730$	−32.5931	−1.20632
$731$	−1.48395	−0.0548859
$732$	1.54765	0.0572030
$733$	−33.8656	−1.25085	−0.625427	−	0.780283i	$-0.715075\pi$
$733$	−33.8656	−1.25085	−0.625427	+	0.780283i	$0.715075\pi$
$734$	−23.5526	−0.869343
$735$	−9.23289	−0.340560
$736$	8.62179	0.317803
$737$	28.4498	1.04796
$738$	14.0357	0.516662
$739$	−40.6197	−1.49422	−0.747110	−	0.664700i	$-0.768559\pi$
$739$	−40.6197	−1.49422	−0.747110	+	0.664700i	$0.768559\pi$
$740$	31.0296	1.14067
$741$	0	0
$742$	−6.10182	−0.224005
$743$	32.2364	1.18264	0.591319	−	0.806438i	$-0.298607\pi$
$743$	32.2364	1.18264	0.591319	+	0.806438i	$0.298607\pi$
$744$	4.14087	0.151812
$745$	−32.5395	−1.19215
$746$	25.7122	0.941390
$747$	39.3059	1.43813
$748$	0.811936	0.0296873
$749$	−9.42073	−0.344226
$750$	3.24949	0.118655
$751$	8.09625	0.295436	0.147718	−	0.989030i	$-0.452807\pi$
$751$	8.09625	0.295436	0.147718	+	0.989030i	$0.452807\pi$
$752$	−3.95332	−0.144163
$753$	−3.26604	−0.119021
$754$	0	0
$755$	58.1193	2.11518
$756$	2.76145	0.100433
$757$	−4.36063	−0.158490	−0.0792448	−	0.996855i	$-0.525251\pi$
$757$	−4.36063	−0.158490	−0.0792448	+	0.996855i	$0.525251\pi$
$758$	12.3452	0.448398
$759$	11.1096	0.403253
$760$	2.79724	0.101467
$761$	3.83887	0.139159	0.0695794	−	0.997576i	$-0.477834\pi$
$761$	3.83887	0.139159	0.0695794	+	0.997576i	$0.477834\pi$
$762$	3.28541	0.119018
$763$	17.7443	0.642386
$764$	6.74347	0.243970
$765$	2.55525	0.0923851
$766$	−11.5251	−0.416418
$767$	0	0
$768$	−0.533995	−0.0192689
$769$	−22.5787	−0.814208	−0.407104	−	0.913382i	$-0.633461\pi$
$769$	−22.5787	−0.814208	−0.407104	+	0.913382i	$0.633461\pi$
$770$	6.10787	0.220112
$771$	−0.396230	−0.0142699
$772$	−3.38300	−0.121757
$773$	−12.7949	−0.460199	−0.230100	−	0.973167i	$-0.573905\pi$
$773$	−12.7949	−0.460199	−0.230100	+	0.973167i	$0.573905\pi$
$774$	−11.9731	−0.430366
$775$	21.9030	0.786780
$776$	6.79232	0.243830
$777$	5.36017	0.192295
$778$	13.2871	0.476366
$779$	5.16998	0.185234
$780$	0	0
$781$	39.1705	1.40163
$782$	2.90105	0.103741
$783$	22.7836	0.814220
$784$	−6.18117	−0.220756
$785$	28.9481	1.03320
$786$	1.88280	0.0671574
$787$	8.39256	0.299162	0.149581	−	0.988749i	$-0.452207\pi$
$787$	8.39256	0.299162	0.149581	+	0.988749i	$0.452207\pi$
$788$	−4.72792	−0.168425
$789$	4.36026	0.155229
$790$	−14.0035	−0.498224
$791$	4.10589	0.145989
$792$	6.55104	0.232781
$793$	0	0
$794$	−2.23448	−0.0792987
$795$	10.0723	0.357229
$796$	6.76173	0.239663
$797$	−23.2451	−0.823385	−0.411692	−	0.911323i	$-0.635062\pi$
$797$	−23.2451	−0.823385	−0.411692	+	0.911323i	$0.635062\pi$
$798$	0.483206	0.0171053
$799$	−1.33021	−0.0470593
$800$	−2.82455	−0.0998630
$801$	−40.7649	−1.44036
$802$	−4.31218	−0.152268
$803$	−28.1164	−0.992207
$804$	−6.29581	−0.222036
$805$	21.8234	0.769175
$806$	0	0
$807$	7.73467	0.272273
$808$	1.28544	0.0452217
$809$	14.7407	0.518255	0.259128	−	0.965843i	$-0.416565\pi$
$809$	14.7407	0.518255	0.259128	+	0.965843i	$0.416565\pi$
$810$	18.2239	0.640323
$811$	−12.9564	−0.454961	−0.227481	−	0.973783i	$-0.573049\pi$
$811$	−12.9564	−0.454961	−0.227481	+	0.973783i	$0.573049\pi$
$812$	6.75580	0.237082
$813$	15.9827	0.560536
$814$	26.7677	0.938206
$815$	−3.15537	−0.110528
$816$	−0.179678	−0.00628998
$817$	−4.41024	−0.154295
$818$	−23.1334	−0.808840
$819$	0	0
$820$	−14.4617	−0.505024
$821$	8.53578	0.297901	0.148950	−	0.988845i	$-0.452411\pi$
$821$	8.53578	0.297901	0.148950	+	0.988845i	$0.452411\pi$
$822$	9.16492	0.319663
$823$	−10.6260	−0.370399	−0.185200	−	0.982701i	$-0.559293\pi$
$823$	−10.6260	−0.370399	−0.185200	+	0.982701i	$0.559293\pi$
$824$	8.97969	0.312822
$825$	−3.63958	−0.126714
$826$	−11.6252	−0.404491
$827$	1.34654	0.0468236	0.0234118	−	0.999726i	$-0.492547\pi$
$827$	1.34654	0.0468236	0.0234118	+	0.999726i	$0.492547\pi$
$828$	23.4069	0.813445
$829$	25.4619	0.884329	0.442164	−	0.896934i	$-0.354211\pi$
$829$	25.4619	0.884329	0.442164	+	0.896934i	$0.354211\pi$
$830$	−40.4988	−1.40573
$831$	0.295816	0.0102617
$832$	0	0
$833$	−2.07983	−0.0720619
$834$	−0.560822	−0.0194197
$835$	22.6928	0.785318
$836$	2.41304	0.0834567
$837$	23.6644	0.817963
$838$	9.27300	0.320330
$839$	49.5051	1.70910	0.854552	−	0.519365i	$-0.173832\pi$
$839$	49.5051	1.70910	0.854552	+	0.519365i	$0.173832\pi$
$840$	−1.35164	−0.0466362
$841$	26.7394	0.922048
$842$	19.3226	0.665902
$843$	15.2489	0.525198
$844$	0.0337822	0.00116283
$845$	0	0
$846$	−10.7327	−0.368996
$847$	−4.68483	−0.160973
$848$	6.74316	0.231561
$849$	7.54142	0.258821
$850$	−0.950401	−0.0325985
$851$	95.6408	3.27853
$852$	−8.66826	−0.296970
$853$	2.27962	0.0780526	0.0390263	−	0.999238i	$-0.487574\pi$
$853$	2.27962	0.0780526	0.0390263	+	0.999238i	$0.487574\pi$
$854$	2.62261	0.0897437
$855$	7.59409	0.259712
$856$	10.4109	0.355838
$857$	−7.47740	−0.255423	−0.127712	−	0.991811i	$-0.540763\pi$
$857$	−7.47740	−0.255423	−0.127712	+	0.991811i	$0.540763\pi$
$858$	0	0
$859$	−48.9070	−1.66869	−0.834343	−	0.551246i	$-0.814152\pi$
$859$	−48.9070	−1.66869	−0.834343	+	0.551246i	$0.814152\pi$
$860$	12.3365	0.420671
$861$	−2.49817	−0.0851374
$862$	28.0060	0.953887
$863$	28.9776	0.986409	0.493205	−	0.869913i	$-0.335825\pi$
$863$	28.9776	0.986409	0.493205	+	0.869913i	$0.335825\pi$
$864$	−3.05170	−0.103821
$865$	−38.1778	−1.29808
$866$	−28.6014	−0.971916
$867$	9.01745	0.306249
$868$	7.01698	0.238172
$869$	−12.0802	−0.409791
$870$	−11.1519	−0.378084
$871$	0	0
$872$	−19.6093	−0.664056
$873$	18.4401	0.624103
$874$	8.62179	0.291636
$875$	5.50648	0.186153
$876$	6.22204	0.210223
$877$	−28.1620	−0.950963	−0.475482	−	0.879726i	$-0.657726\pi$
$877$	−28.1620	−0.950963	−0.475482	+	0.879726i	$0.657726\pi$
$878$	18.2630	0.616346
$879$	6.72837	0.226942
$880$	−6.74985	−0.227537
$881$	−19.8174	−0.667664	−0.333832	−	0.942633i	$-0.608342\pi$
$881$	−19.8174	−0.667664	−0.333832	+	0.942633i	$0.608342\pi$
$882$	−16.7810	−0.565044
$883$	17.2065	0.579045	0.289522	−	0.957171i	$-0.406504\pi$
$883$	17.2065	0.579045	0.289522	+	0.957171i	$0.406504\pi$
$884$	0	0
$885$	19.1898	0.645057
$886$	21.0754	0.708041
$887$	16.2904	0.546977	0.273488	−	0.961875i	$-0.411822\pi$
$887$	16.2904	0.546977	0.273488	+	0.961875i	$0.411822\pi$
$888$	−5.92356	−0.198782
$889$	5.56734	0.186723
$890$	42.0021	1.40791
$891$	15.7209	0.526668
$892$	3.69513	0.123722
$893$	−3.95332	−0.132293
$894$	6.21181	0.207754
$895$	14.9195	0.498704
$896$	−0.904890	−0.0302303
$897$	0	0
$898$	−14.3812	−0.479907
$899$	57.8943	1.93088
$900$	−7.66824	−0.255608
$901$	2.26893	0.0755890
$902$	−12.4754	−0.415384
$903$	2.13106	0.0709171
$904$	−4.53745	−0.150913
$905$	68.0815	2.26311
$906$	−11.0950	−0.368607
$907$	18.9021	0.627633	0.313817	−	0.949484i	$-0.398392\pi$
$907$	18.9021	0.627633	0.313817	+	0.949484i	$0.398392\pi$
$908$	−10.6674	−0.354008
$909$	3.48978	0.115749
$910$	0	0
$911$	−21.4108	−0.709371	−0.354685	−	0.934986i	$-0.615412\pi$
$911$	−21.4108	−0.709371	−0.354685	+	0.934986i	$0.615412\pi$
$912$	−0.533995	−0.0176823
$913$	−34.9363	−1.15622
$914$	21.2580	0.703152
$915$	−4.32916	−0.143118
$916$	7.72323	0.255183
$917$	3.19054	0.105361
$918$	−1.02683	−0.0338905
$919$	−17.2404	−0.568707	−0.284353	−	0.958720i	$-0.591779\pi$
$919$	−17.2404	−0.568707	−0.284353	+	0.958720i	$0.591779\pi$
$920$	−24.1172	−0.795121
$921$	4.71028	0.155209
$922$	30.5002	1.00447
$923$	0	0
$924$	−1.16600	−0.0383585
$925$	−31.3325	−1.03021
$926$	−0.544292	−0.0178865
$927$	24.3785	0.800695
$928$	−7.46588	−0.245080
$929$	−45.7801	−1.50200	−0.750999	−	0.660304i	$-0.770428\pi$
$929$	−45.7801	−1.50200	−0.750999	+	0.660304i	$0.770428\pi$
$930$	−11.5830	−0.379822
$931$	−6.18117	−0.202580
$932$	3.32091	0.108780
$933$	14.8408	0.485867
$934$	3.93860	0.128875
$935$	−2.27118	−0.0742755
$936$	0	0
$937$	21.2790	0.695155	0.347578	−	0.937651i	$-0.387004\pi$
$937$	21.2790	0.695155	0.347578	+	0.937651i	$0.387004\pi$
$938$	−10.6687	−0.348345
$939$	−15.0739	−0.491918
$940$	11.0584	0.360684
$941$	28.3916	0.925540	0.462770	−	0.886478i	$-0.346856\pi$
$941$	28.3916	0.925540	0.462770	+	0.886478i	$0.346856\pi$
$942$	−5.52621	−0.180054
$943$	−44.5745	−1.45155
$944$	12.8470	0.418135
$945$	−7.72444	−0.251276
$946$	10.6421	0.346004
$947$	13.9684	0.453913	0.226957	−	0.973905i	$-0.427122\pi$
$947$	13.9684	0.453913	0.226957	+	0.973905i	$0.427122\pi$
$948$	2.67328	0.0868242
$949$	0	0
$950$	−2.82455	−0.0916406
$951$	2.05178	0.0665335
$952$	−0.304476	−0.00986812
$953$	40.0033	1.29583	0.647917	−	0.761711i	$-0.275640\pi$
$953$	40.0033	1.29583	0.647917	+	0.761711i	$0.275640\pi$
$954$	18.3067	0.592700
$955$	−18.8631	−0.610396
$956$	−10.8271	−0.350174
$957$	−9.62016	−0.310976
$958$	−15.4244	−0.498338
$959$	15.5306	0.501508
$960$	1.49371	0.0482093
$961$	29.1325	0.939757
$962$	0	0
$963$	28.2641	0.910796
$964$	−11.9629	−0.385301
$965$	9.46307	0.304627
$966$	−4.16611	−0.134042
$967$	52.9273	1.70203	0.851013	−	0.525145i	$-0.175989\pi$
$967$	52.9273	1.70203	0.851013	+	0.525145i	$0.175989\pi$
$968$	5.17724	0.166403
$969$	−0.179678	−0.00577208
$970$	−18.9997	−0.610045
$971$	2.36683	0.0759553	0.0379776	−	0.999279i	$-0.487908\pi$
$971$	2.36683	0.0759553	0.0379776	+	0.999279i	$0.487908\pi$
$972$	−12.6341	−0.405237
$973$	−0.950350	−0.0304668
$974$	−9.45313	−0.302898
$975$	0	0
$976$	−2.89826	−0.0927710
$977$	−40.9698	−1.31074	−0.655370	−	0.755308i	$-0.727487\pi$
$977$	−40.9698	−1.31074	−0.655370	+	0.755308i	$0.727487\pi$
$978$	0.602361	0.0192614
$979$	36.2331	1.15802
$980$	17.2902	0.552316
$981$	−53.2364	−1.69971
$982$	−16.2955	−0.520009
$983$	−54.7742	−1.74703	−0.873513	−	0.486801i	$-0.838164\pi$
$983$	−54.7742	−1.74703	−0.873513	+	0.486801i	$0.838164\pi$
$984$	2.76074	0.0880093
$985$	13.2251	0.421388
$986$	−2.51211	−0.0800018
$987$	1.91027	0.0608045
$988$	0	0
$989$	38.0242	1.20910
$990$	−18.3248	−0.582401
$991$	−27.3301	−0.868169	−0.434084	−	0.900872i	$-0.642928\pi$
$991$	−27.3301	−0.868169	−0.434084	+	0.900872i	$0.642928\pi$
$992$	−7.75451	−0.246206
$993$	−5.83108	−0.185044
$994$	−14.6889	−0.465905
$995$	−18.9142	−0.599620
$996$	7.73124	0.244974
$997$	−22.5276	−0.713455	−0.356728	−	0.934208i	$-0.616108\pi$
$997$	−22.5276	−0.713455	−0.356728	+	0.934208i	$0.616108\pi$
$998$	39.8915	1.26274
$999$	−33.8522	−1.07104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bg.1.4		8
13.2	odd	12		494.2.m.a.381.3	yes	16
13.7	odd	12		494.2.m.a.153.3	✓	16
13.12	even	2		6422.2.a.bh.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.m.a.153.3	✓	16	13.7	odd	12
494.2.m.a.381.3	yes	16	13.2	odd	12
6422.2.a.bg.1.4		8	1.1	even	1	trivial
6422.2.a.bh.1.4		8	13.12	even	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.533995 q^{3} +1.00000 q^{4} -2.79724 q^{5} +0.533995 q^{6} +0.904890 q^{7} -1.00000 q^{8} -2.71485 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.533995 q^{3} +1.00000 q^{4} -2.79724 q^{5} +0.533995 q^{6} +0.904890 q^{7} -1.00000 q^{8} -2.71485 q^{9} +2.79724 q^{10} +2.41304 q^{11} -0.533995 q^{12} -0.904890 q^{14} +1.49371 q^{15} +1.00000 q^{16} +0.336478 q^{17} +2.71485 q^{18} +1.00000 q^{19} -2.79724 q^{20} -0.483206 q^{21} -2.41304 q^{22} -8.62179 q^{23} +0.533995 q^{24} +2.82455 q^{25} +3.05170 q^{27} +0.904890 q^{28} +7.46588 q^{29} -1.49371 q^{30} +7.75451 q^{31} -1.00000 q^{32} -1.28855 q^{33} -0.336478 q^{34} -2.53119 q^{35} -2.71485 q^{36} -11.0929 q^{37} -1.00000 q^{38} +2.79724 q^{40} +5.16998 q^{41} +0.483206 q^{42} -4.41024 q^{43} +2.41304 q^{44} +7.59409 q^{45} +8.62179 q^{46} -3.95332 q^{47} -0.533995 q^{48} -6.18117 q^{49} -2.82455 q^{50} -0.179678 q^{51} +6.74316 q^{53} -3.05170 q^{54} -6.74985 q^{55} -0.904890 q^{56} -0.533995 q^{57} -7.46588 q^{58} +12.8470 q^{59} +1.49371 q^{60} -2.89826 q^{61} -7.75451 q^{62} -2.45664 q^{63} +1.00000 q^{64} +1.28855 q^{66} +11.7900 q^{67} +0.336478 q^{68} +4.60399 q^{69} +2.53119 q^{70} +16.2329 q^{71} +2.71485 q^{72} -11.6519 q^{73} +11.0929 q^{74} -1.50830 q^{75} +1.00000 q^{76} +2.18354 q^{77} -5.00620 q^{79} -2.79724 q^{80} +6.51496 q^{81} -5.16998 q^{82} -14.4781 q^{83} -0.483206 q^{84} -0.941211 q^{85} +4.41024 q^{86} -3.98674 q^{87} -2.41304 q^{88} +15.0155 q^{89} -7.59409 q^{90} -8.62179 q^{92} -4.14087 q^{93} +3.95332 q^{94} -2.79724 q^{95} +0.533995 q^{96} -6.79232 q^{97} +6.18117 q^{98} -6.55104 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8} - 2 q^{10} + 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} + 8 q^{19} + 2 q^{20} - 12 q^{21} - 10 q^{22} - 8 q^{23} + 4 q^{24} - 14 q^{25} - 22 q^{27} + 2 q^{28} + 8 q^{29} + 12 q^{31} - 8 q^{32} - 4 q^{33} - 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} - 2 q^{41} + 12 q^{42} - 16 q^{43} + 10 q^{44} + 12 q^{45} + 8 q^{46} - 12 q^{47} - 4 q^{48} - 14 q^{49} + 14 q^{50} - 22 q^{51} - 24 q^{53} + 22 q^{54} - 10 q^{55} - 2 q^{56} - 4 q^{57} - 8 q^{58} + 4 q^{59} - 26 q^{61} - 12 q^{62} + 12 q^{63} + 8 q^{64} + 4 q^{66} - 10 q^{67} + 2 q^{68} + 6 q^{69} + 10 q^{70} + 36 q^{71} - 20 q^{73} + 6 q^{75} + 8 q^{76} - 12 q^{77} - 22 q^{79} + 2 q^{80} + 36 q^{81} + 2 q^{82} - 18 q^{83} - 12 q^{84} - 26 q^{85} + 16 q^{86} - 38 q^{87} - 10 q^{88} + 18 q^{89} - 12 q^{90} - 8 q^{92} - 12 q^{93} + 12 q^{94} + 2 q^{95} + 4 q^{96} + 8 q^{97} + 14 q^{98} + 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^5 + 4 * q^6 + 2 * q^7 - 8 * q^8 - 2 * q^10 + 10 * q^11 - 4 * q^12 - 2 * q^14 + 8 * q^16 + 2 * q^17 + 8 * q^19 + 2 * q^20 - 12 * q^21 - 10 * q^22 - 8 * q^23 + 4 * q^24 - 14 * q^25 - 22 * q^27 + 2 * q^28 + 8 * q^29 + 12 * q^31 - 8 * q^32 - 4 * q^33 - 2 * q^34 - 10 * q^35 - 8 * q^38 - 2 * q^40 - 2 * q^41 + 12 * q^42 - 16 * q^43 + 10 * q^44 + 12 * q^45 + 8 * q^46 - 12 * q^47 - 4 * q^48 - 14 * q^49 + 14 * q^50 - 22 * q^51 - 24 * q^53 + 22 * q^54 - 10 * q^55 - 2 * q^56 - 4 * q^57 - 8 * q^58 + 4 * q^59 - 26 * q^61 - 12 * q^62 + 12 * q^63 + 8 * q^64 + 4 * q^66 - 10 * q^67 + 2 * q^68 + 6 * q^69 + 10 * q^70 + 36 * q^71 - 20 * q^73 + 6 * q^75 + 8 * q^76 - 12 * q^77 - 22 * q^79 + 2 * q^80 + 36 * q^81 + 2 * q^82 - 18 * q^83 - 12 * q^84 - 26 * q^85 + 16 * q^86 - 38 * q^87 - 10 * q^88 + 18 * q^89 - 12 * q^90 - 8 * q^92 - 12 * q^93 + 12 * q^94 + 2 * q^95 + 4 * q^96 + 8 * q^97 + 14 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

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Groups

Database

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