LMFDB - Embedded newform 6018.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6018,2,Mod(1,6018)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6018, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6018.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6018 = 2 \cdot 3 \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6018.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:	$48.0539719364$
Analytic rank:	$0$
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} - 22x^{7} + 20x^{6} + 129x^{5} - 106x^{4} - 126x^{3} + 48x^{2} + 24x - 8 $ x^9 - x^8 - 22x^7 + 20x^6 + 129x^5 - 106x^4 - 126x^3 + 48x^2 + 24*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.354931$ of defining polynomial
Character	$\chi$	$=$	6018.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.354931 q^{5} -1.00000 q^{6} +1.61788 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.354931 q^{5} -1.00000 q^{6} +1.61788 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.354931 q^{10} +0.127157 q^{11} +1.00000 q^{12} +5.66458 q^{13} -1.61788 q^{14} +0.354931 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +2.49911 q^{19} +0.354931 q^{20} +1.61788 q^{21} -0.127157 q^{22} +6.45730 q^{23} -1.00000 q^{24} -4.87402 q^{25} -5.66458 q^{26} +1.00000 q^{27} +1.61788 q^{28} +4.75367 q^{29} -0.354931 q^{30} +1.34348 q^{31} -1.00000 q^{32} +0.127157 q^{33} -1.00000 q^{34} +0.574235 q^{35} +1.00000 q^{36} -2.13363 q^{37} -2.49911 q^{38} +5.66458 q^{39} -0.354931 q^{40} +9.25649 q^{41} -1.61788 q^{42} -8.24988 q^{43} +0.127157 q^{44} +0.354931 q^{45} -6.45730 q^{46} +12.1871 q^{47} +1.00000 q^{48} -4.38247 q^{49} +4.87402 q^{50} +1.00000 q^{51} +5.66458 q^{52} -3.16949 q^{53} -1.00000 q^{54} +0.0451320 q^{55} -1.61788 q^{56} +2.49911 q^{57} -4.75367 q^{58} +1.00000 q^{59} +0.354931 q^{60} -11.2691 q^{61} -1.34348 q^{62} +1.61788 q^{63} +1.00000 q^{64} +2.01054 q^{65} -0.127157 q^{66} -1.41366 q^{67} +1.00000 q^{68} +6.45730 q^{69} -0.574235 q^{70} -3.75070 q^{71} -1.00000 q^{72} +11.1255 q^{73} +2.13363 q^{74} -4.87402 q^{75} +2.49911 q^{76} +0.205725 q^{77} -5.66458 q^{78} -5.60546 q^{79} +0.354931 q^{80} +1.00000 q^{81} -9.25649 q^{82} -9.91629 q^{83} +1.61788 q^{84} +0.354931 q^{85} +8.24988 q^{86} +4.75367 q^{87} -0.127157 q^{88} +9.25235 q^{89} -0.354931 q^{90} +9.16461 q^{91} +6.45730 q^{92} +1.34348 q^{93} -12.1871 q^{94} +0.887011 q^{95} -1.00000 q^{96} +3.62535 q^{97} +4.38247 q^{98} +0.127157 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + 9 q^{3} + 9 q^{4} + q^{5} - 9 q^{6} - 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^2 + 9 * q^3 + 9 * q^4 + q^5 - 9 * q^6 - 9 * q^8 + 9 * q^9	$ 9 q - 9 q^{2} + 9 q^{3} + 9 q^{4} + q^{5} - 9 q^{6} - 9 q^{8} + 9 q^{9} - q^{10} + 6 q^{11} + 9 q^{12} + 2 q^{13} + q^{15} + 9 q^{16} + 9 q^{17} - 9 q^{18} - 5 q^{19} + q^{20} - 6 q^{22} + 15 q^{23} - 9 q^{24} - 2 q^{26} + 9 q^{27} + 11 q^{29} - q^{30} - 5 q^{31} - 9 q^{32} + 6 q^{33} - 9 q^{34} + 22 q^{35} + 9 q^{36} + 9 q^{37} + 5 q^{38} + 2 q^{39} - q^{40} + q^{41} + 4 q^{43} + 6 q^{44} + q^{45} - 15 q^{46} + 14 q^{47} + 9 q^{48} - q^{49} + 9 q^{51} + 2 q^{52} + 4 q^{53} - 9 q^{54} + 4 q^{55} - 5 q^{57} - 11 q^{58} + 9 q^{59} + q^{60} + 10 q^{61} + 5 q^{62} + 9 q^{64} + 8 q^{65} - 6 q^{66} - q^{67} + 9 q^{68} + 15 q^{69} - 22 q^{70} + 14 q^{71} - 9 q^{72} - q^{73} - 9 q^{74} - 5 q^{76} + 30 q^{77} - 2 q^{78} + 4 q^{79} + q^{80} + 9 q^{81} - q^{82} + 22 q^{83} + q^{85} - 4 q^{86} + 11 q^{87} - 6 q^{88} + 22 q^{89} - q^{90} - 3 q^{91} + 15 q^{92} - 5 q^{93} - 14 q^{94} + 43 q^{95} - 9 q^{96} - 15 q^{97} + q^{98} + 6 q^{99}+O(q^{100}) $ 9 * q - 9 * q^2 + 9 * q^3 + 9 * q^4 + q^5 - 9 * q^6 - 9 * q^8 + 9 * q^9 - q^10 + 6 * q^11 + 9 * q^12 + 2 * q^13 + q^15 + 9 * q^16 + 9 * q^17 - 9 * q^18 - 5 * q^19 + q^20 - 6 * q^22 + 15 * q^23 - 9 * q^24 - 2 * q^26 + 9 * q^27 + 11 * q^29 - q^30 - 5 * q^31 - 9 * q^32 + 6 * q^33 - 9 * q^34 + 22 * q^35 + 9 * q^36 + 9 * q^37 + 5 * q^38 + 2 * q^39 - q^40 + q^41 + 4 * q^43 + 6 * q^44 + q^45 - 15 * q^46 + 14 * q^47 + 9 * q^48 - q^49 + 9 * q^51 + 2 * q^52 + 4 * q^53 - 9 * q^54 + 4 * q^55 - 5 * q^57 - 11 * q^58 + 9 * q^59 + q^60 + 10 * q^61 + 5 * q^62 + 9 * q^64 + 8 * q^65 - 6 * q^66 - q^67 + 9 * q^68 + 15 * q^69 - 22 * q^70 + 14 * q^71 - 9 * q^72 - q^73 - 9 * q^74 - 5 * q^76 + 30 * q^77 - 2 * q^78 + 4 * q^79 + q^80 + 9 * q^81 - q^82 + 22 * q^83 + q^85 - 4 * q^86 + 11 * q^87 - 6 * q^88 + 22 * q^89 - q^90 - 3 * q^91 + 15 * q^92 - 5 * q^93 - 14 * q^94 + 43 * q^95 - 9 * q^96 - 15 * q^97 + q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0.354931	0.158730	0.0793650	−	0.996846i	$-0.474711\pi$
$5$	0.354931	0.158730	0.0793650	+	0.996846i	$0.474711\pi$
$6$	−1.00000	−0.408248
$7$	1.61788	0.611501	0.305750	−	0.952112i	$-0.401093\pi$
$7$	1.61788	0.611501	0.305750	+	0.952112i	$0.401093\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−0.354931	−0.112239
$11$	0.127157	0.0383393	0.0191697	−	0.999816i	$-0.493898\pi$
$11$	0.127157	0.0383393	0.0191697	+	0.999816i	$0.493898\pi$
$12$	1.00000	0.288675
$13$	5.66458	1.57107	0.785536	−	0.618815i	$-0.212387\pi$
$13$	5.66458	1.57107	0.785536	+	0.618815i	$0.212387\pi$
$14$	−1.61788	−0.432396
$15$	0.354931	0.0916428
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	2.49911	0.573335	0.286667	−	0.958030i	$-0.407453\pi$
$19$	2.49911	0.573335	0.286667	+	0.958030i	$0.407453\pi$
$20$	0.354931	0.0793650
$21$	1.61788	0.353050
$22$	−0.127157	−0.0271100
$23$	6.45730	1.34644	0.673220	−	0.739442i	$-0.264910\pi$
$23$	6.45730	1.34644	0.673220	+	0.739442i	$0.264910\pi$
$24$	−1.00000	−0.204124
$25$	−4.87402	−0.974805
$26$	−5.66458	−1.11092
$27$	1.00000	0.192450
$28$	1.61788	0.305750
$29$	4.75367	0.882734	0.441367	−	0.897327i	$-0.354494\pi$
$29$	4.75367	0.882734	0.441367	+	0.897327i	$0.354494\pi$
$30$	−0.354931	−0.0648012
$31$	1.34348	0.241296	0.120648	−	0.992695i	$-0.461503\pi$
$31$	1.34348	0.241296	0.120648	+	0.992695i	$0.461503\pi$
$32$	−1.00000	−0.176777
$33$	0.127157	0.0221352
$34$	−1.00000	−0.171499
$35$	0.574235	0.0970635
$36$	1.00000	0.166667
$37$	−2.13363	−0.350766	−0.175383	−	0.984500i	$-0.556116\pi$
$37$	−2.13363	−0.350766	−0.175383	+	0.984500i	$0.556116\pi$
$38$	−2.49911	−0.405409
$39$	5.66458	0.907059
$40$	−0.354931	−0.0561195
$41$	9.25649	1.44562	0.722811	−	0.691046i	$-0.242850\pi$
$41$	9.25649	1.44562	0.722811	+	0.691046i	$0.242850\pi$
$42$	−1.61788	−0.249644
$43$	−8.24988	−1.25809	−0.629047	−	0.777367i	$-0.716555\pi$
$43$	−8.24988	−1.25809	−0.629047	+	0.777367i	$0.716555\pi$
$44$	0.127157	0.0191697
$45$	0.354931	0.0529100
$46$	−6.45730	−0.952077
$47$	12.1871	1.77767	0.888836	−	0.458226i	$-0.151515\pi$
$47$	12.1871	1.77767	0.888836	+	0.458226i	$0.151515\pi$
$48$	1.00000	0.144338
$49$	−4.38247	−0.626067
$50$	4.87402	0.689291
$51$	1.00000	0.140028
$52$	5.66458	0.785536
$53$	−3.16949	−0.435363	−0.217682	−	0.976020i	$-0.569849\pi$
$53$	−3.16949	−0.435363	−0.217682	+	0.976020i	$0.569849\pi$
$54$	−1.00000	−0.136083
$55$	0.0451320	0.00608560
$56$	−1.61788	−0.216198
$57$	2.49911	0.331015
$58$	−4.75367	−0.624187
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0.354931	0.0458214
$61$	−11.2691	−1.44286	−0.721432	−	0.692486i	$-0.756516\pi$
$61$	−11.2691	−1.44286	−0.721432	+	0.692486i	$0.756516\pi$
$62$	−1.34348	−0.170622
$63$	1.61788	0.203834
$64$	1.00000	0.125000
$65$	2.01054	0.249376
$66$	−0.127157	−0.0156520
$67$	−1.41366	−0.172706	−0.0863531	−	0.996265i	$-0.527521\pi$
$67$	−1.41366	−0.172706	−0.0863531	+	0.996265i	$0.527521\pi$
$68$	1.00000	0.121268
$69$	6.45730	0.777368
$70$	−0.574235	−0.0686342
$71$	−3.75070	−0.445127	−0.222563	−	0.974918i	$-0.571442\pi$
$71$	−3.75070	−0.445127	−0.222563	+	0.974918i	$0.571442\pi$
$72$	−1.00000	−0.117851
$73$	11.1255	1.30214	0.651068	−	0.759019i	$-0.274321\pi$
$73$	11.1255	1.30214	0.651068	+	0.759019i	$0.274321\pi$
$74$	2.13363	0.248029
$75$	−4.87402	−0.562804
$76$	2.49911	0.286667
$77$	0.205725	0.0234445
$78$	−5.66458	−0.641388
$79$	−5.60546	−0.630664	−0.315332	−	0.948981i	$-0.602116\pi$
$79$	−5.60546	−0.630664	−0.315332	+	0.948981i	$0.602116\pi$
$80$	0.354931	0.0396825
$81$	1.00000	0.111111
$82$	−9.25649	−1.02221
$83$	−9.91629	−1.08845	−0.544227	−	0.838938i	$-0.683177\pi$
$83$	−9.91629	−1.08845	−0.544227	+	0.838938i	$0.683177\pi$
$84$	1.61788	0.176525
$85$	0.354931	0.0384977
$86$	8.24988	0.889607
$87$	4.75367	0.509647
$88$	−0.127157	−0.0135550
$89$	9.25235	0.980747	0.490373	−	0.871512i	$-0.336860\pi$
$89$	9.25235	0.980747	0.490373	+	0.871512i	$0.336860\pi$
$90$	−0.354931	−0.0374130
$91$	9.16461	0.960712
$92$	6.45730	0.673220
$93$	1.34348	0.139313
$94$	−12.1871	−1.25700
$95$	0.887011	0.0910054
$96$	−1.00000	−0.102062
$97$	3.62535	0.368099	0.184049	−	0.982917i	$-0.441079\pi$
$97$	3.62535	0.368099	0.184049	+	0.982917i	$0.441079\pi$
$98$	4.38247	0.442696
$99$	0.127157	0.0127798
$100$	−4.87402	−0.487402
$101$	5.87625	0.584709	0.292354	−	0.956310i	$-0.405561\pi$
$101$	5.87625	0.584709	0.292354	+	0.956310i	$0.405561\pi$
$102$	−1.00000	−0.0990148
$103$	−15.7621	−1.55309	−0.776543	−	0.630064i	$-0.783029\pi$
$103$	−15.7621	−1.55309	−0.776543	+	0.630064i	$0.783029\pi$
$104$	−5.66458	−0.555458
$105$	0.574235	0.0560396
$106$	3.16949	0.307848
$107$	11.8127	1.14198	0.570988	−	0.820958i	$-0.306560\pi$
$107$	11.8127	1.14198	0.570988	+	0.820958i	$0.306560\pi$
$108$	1.00000	0.0962250
$109$	0.371196	0.0355541	0.0177770	−	0.999842i	$-0.494341\pi$
$109$	0.371196	0.0355541	0.0177770	+	0.999842i	$0.494341\pi$
$110$	−0.0451320	−0.00430317
$111$	−2.13363	−0.202515
$112$	1.61788	0.152875
$113$	−2.97215	−0.279597	−0.139798	−	0.990180i	$-0.544645\pi$
$113$	−2.97215	−0.279597	−0.139798	+	0.990180i	$0.544645\pi$
$114$	−2.49911	−0.234063
$115$	2.29190	0.213721
$116$	4.75367	0.441367
$117$	5.66458	0.523691
$118$	−1.00000	−0.0920575
$119$	1.61788	0.148311
$120$	−0.354931	−0.0324006
$121$	−10.9838	−0.998530
$122$	11.2691	1.02026
$123$	9.25649	0.834630
$124$	1.34348	0.120648
$125$	−3.50460	−0.313461
$126$	−1.61788	−0.144132
$127$	−19.5881	−1.73816	−0.869081	−	0.494670i	$-0.835289\pi$
$127$	−19.5881	−1.73816	−0.869081	+	0.494670i	$0.835289\pi$
$128$	−1.00000	−0.0883883
$129$	−8.24988	−0.726361
$130$	−2.01054	−0.176336
$131$	1.66060	0.145088	0.0725438	−	0.997365i	$-0.476888\pi$
$131$	1.66060	0.145088	0.0725438	+	0.997365i	$0.476888\pi$
$132$	0.127157	0.0110676
$133$	4.04325	0.350595
$134$	1.41366	0.122122
$135$	0.354931	0.0305476
$136$	−1.00000	−0.0857493
$137$	3.24592	0.277318	0.138659	−	0.990340i	$-0.455721\pi$
$137$	3.24592	0.277318	0.138659	+	0.990340i	$0.455721\pi$
$138$	−6.45730	−0.549682
$139$	−15.0025	−1.27249	−0.636247	−	0.771486i	$-0.719514\pi$
$139$	−15.0025	−1.27249	−0.636247	+	0.771486i	$0.719514\pi$
$140$	0.574235	0.0485317
$141$	12.1871	1.02634
$142$	3.75070	0.314752
$143$	0.720292	0.0602338
$144$	1.00000	0.0833333
$145$	1.68722	0.140116
$146$	−11.1255	−0.920749
$147$	−4.38247	−0.361460
$148$	−2.13363	−0.175383
$149$	−6.05926	−0.496394	−0.248197	−	0.968710i	$-0.579838\pi$
$149$	−6.05926	−0.496394	−0.248197	+	0.968710i	$0.579838\pi$
$150$	4.87402	0.397962
$151$	20.6175	1.67782	0.838912	−	0.544267i	$-0.183192\pi$
$151$	20.6175	1.67782	0.838912	+	0.544267i	$0.183192\pi$
$152$	−2.49911	−0.202704
$153$	1.00000	0.0808452
$154$	−0.205725	−0.0165778
$155$	0.476843	0.0383010
$156$	5.66458	0.453530
$157$	−18.0803	−1.44296	−0.721482	−	0.692434i	$-0.756539\pi$
$157$	−18.0803	−1.44296	−0.721482	+	0.692434i	$0.756539\pi$
$158$	5.60546	0.445947
$159$	−3.16949	−0.251357
$160$	−0.354931	−0.0280598
$161$	10.4471	0.823349
$162$	−1.00000	−0.0785674
$163$	14.6221	1.14529	0.572644	−	0.819804i	$-0.305918\pi$
$163$	14.6221	1.14529	0.572644	+	0.819804i	$0.305918\pi$
$164$	9.25649	0.722811
$165$	0.0451320	0.00351352
$166$	9.91629	0.769653
$167$	11.7148	0.906516	0.453258	−	0.891379i	$-0.350262\pi$
$167$	11.7148	0.906516	0.453258	+	0.891379i	$0.350262\pi$
$168$	−1.61788	−0.124822
$169$	19.0875	1.46827
$170$	−0.354931	−0.0272220
$171$	2.49911	0.191112
$172$	−8.24988	−0.629047
$173$	−13.6384	−1.03691	−0.518453	−	0.855106i	$-0.673492\pi$
$173$	−13.6384	−1.03691	−0.518453	+	0.855106i	$0.673492\pi$
$174$	−4.75367	−0.360375
$175$	−7.88558	−0.596094
$176$	0.127157	0.00958483
$177$	1.00000	0.0751646
$178$	−9.25235	−0.693493
$179$	17.3260	1.29501	0.647504	−	0.762062i	$-0.275813\pi$
$179$	17.3260	1.29501	0.647504	+	0.762062i	$0.275813\pi$
$180$	0.354931	0.0264550
$181$	9.77124	0.726291	0.363145	−	0.931733i	$-0.381703\pi$
$181$	9.77124	0.726291	0.363145	+	0.931733i	$0.381703\pi$
$182$	−9.16461	−0.679326
$183$	−11.2691	−0.833038
$184$	−6.45730	−0.476039
$185$	−0.757291	−0.0556771
$186$	−1.34348	−0.0985088
$187$	0.127157	0.00929865
$188$	12.1871	0.888836
$189$	1.61788	0.117683
$190$	−0.887011	−0.0643505
$191$	−8.67311	−0.627564	−0.313782	−	0.949495i	$-0.601596\pi$
$191$	−8.67311	−0.627564	−0.313782	+	0.949495i	$0.601596\pi$
$192$	1.00000	0.0721688
$193$	3.65793	0.263304	0.131652	−	0.991296i	$-0.457972\pi$
$193$	3.65793	0.263304	0.131652	+	0.991296i	$0.457972\pi$
$194$	−3.62535	−0.260285
$195$	2.01054	0.143977
$196$	−4.38247	−0.313034
$197$	7.39726	0.527033	0.263516	−	0.964655i	$-0.415118\pi$
$197$	7.39726	0.527033	0.263516	+	0.964655i	$0.415118\pi$
$198$	−0.127157	−0.00903666
$199$	22.3060	1.58123	0.790614	−	0.612315i	$-0.209762\pi$
$199$	22.3060	1.58123	0.790614	+	0.612315i	$0.209762\pi$
$200$	4.87402	0.344646
$201$	−1.41366	−0.0997119
$202$	−5.87625	−0.413452
$203$	7.69086	0.539792
$204$	1.00000	0.0700140
$205$	3.28542	0.229463
$206$	15.7621	1.09820
$207$	6.45730	0.448814
$208$	5.66458	0.392768
$209$	0.317779	0.0219813
$210$	−0.574235	−0.0396260
$211$	5.07524	0.349394	0.174697	−	0.984622i	$-0.444105\pi$
$211$	5.07524	0.349394	0.174697	+	0.984622i	$0.444105\pi$
$212$	−3.16949	−0.217682
$213$	−3.75070	−0.256994
$214$	−11.8127	−0.807499
$215$	−2.92814	−0.199697
$216$	−1.00000	−0.0680414
$217$	2.17359	0.147553
$218$	−0.371196	−0.0251405
$219$	11.1255	0.751789
$220$	0.0451320	0.00304280
$221$	5.66458	0.381041
$222$	2.13363	0.143200
$223$	23.1987	1.55350	0.776750	−	0.629810i	$-0.216867\pi$
$223$	23.1987	1.55350	0.776750	+	0.629810i	$0.216867\pi$
$224$	−1.61788	−0.108099
$225$	−4.87402	−0.324935
$226$	2.97215	0.197705
$227$	−23.5651	−1.56407	−0.782035	−	0.623234i	$-0.785818\pi$
$227$	−23.5651	−1.56407	−0.782035	+	0.623234i	$0.785818\pi$
$228$	2.49911	0.165507
$229$	−18.3178	−1.21048	−0.605238	−	0.796044i	$-0.706922\pi$
$229$	−18.3178	−1.21048	−0.605238	+	0.796044i	$0.706922\pi$
$230$	−2.29190	−0.151123
$231$	0.205725	0.0135357
$232$	−4.75367	−0.312094
$233$	12.5344	0.821159	0.410579	−	0.911825i	$-0.365326\pi$
$233$	12.5344	0.821159	0.410579	+	0.911825i	$0.365326\pi$
$234$	−5.66458	−0.370305
$235$	4.32558	0.282170
$236$	1.00000	0.0650945
$237$	−5.60546	−0.364114
$238$	−1.61788	−0.104871
$239$	−26.3711	−1.70581	−0.852904	−	0.522068i	$-0.825161\pi$
$239$	−26.3711	−1.70581	−0.852904	+	0.522068i	$0.825161\pi$
$240$	0.354931	0.0229107
$241$	23.6126	1.52102	0.760511	−	0.649325i	$-0.224948\pi$
$241$	23.6126	1.52102	0.760511	+	0.649325i	$0.224948\pi$
$242$	10.9838	0.706067
$243$	1.00000	0.0641500
$244$	−11.2691	−0.721432
$245$	−1.55547	−0.0993756
$246$	−9.25649	−0.590172
$247$	14.1564	0.900751
$248$	−1.34348	−0.0853111
$249$	−9.91629	−0.628419
$250$	3.50460	0.221650
$251$	7.29061	0.460179	0.230090	−	0.973169i	$-0.426098\pi$
$251$	7.29061	0.460179	0.230090	+	0.973169i	$0.426098\pi$
$252$	1.61788	0.101917
$253$	0.821092	0.0516216
$254$	19.5881	1.22907
$255$	0.354931	0.0222266
$256$	1.00000	0.0625000
$257$	−16.5447	−1.03203	−0.516014	−	0.856580i	$-0.672585\pi$
$257$	−16.5447	−1.03203	−0.516014	+	0.856580i	$0.672585\pi$
$258$	8.24988	0.513615
$259$	−3.45195	−0.214494
$260$	2.01054	0.124688
$261$	4.75367	0.294245
$262$	−1.66060	−0.102592
$263$	−15.3604	−0.947165	−0.473582	−	0.880750i	$-0.657039\pi$
$263$	−15.3604	−0.947165	−0.473582	+	0.880750i	$0.657039\pi$
$264$	−0.127157	−0.00782598
$265$	−1.12495	−0.0691052
$266$	−4.04325	−0.247908
$267$	9.25235	0.566234
$268$	−1.41366	−0.0863531
$269$	−21.6836	−1.32207	−0.661035	−	0.750355i	$-0.729883\pi$
$269$	−21.6836	−1.32207	−0.661035	+	0.750355i	$0.729883\pi$
$270$	−0.354931	−0.0216004
$271$	6.48269	0.393796	0.196898	−	0.980424i	$-0.436913\pi$
$271$	6.48269	0.393796	0.196898	+	0.980424i	$0.436913\pi$
$272$	1.00000	0.0606339
$273$	9.16461	0.554667
$274$	−3.24592	−0.196093
$275$	−0.619767	−0.0373733
$276$	6.45730	0.388684
$277$	−11.2225	−0.674295	−0.337147	−	0.941452i	$-0.609462\pi$
$277$	−11.2225	−0.674295	−0.337147	+	0.941452i	$0.609462\pi$
$278$	15.0025	0.899789
$279$	1.34348	0.0804321
$280$	−0.574235	−0.0343171
$281$	−3.40072	−0.202870	−0.101435	−	0.994842i	$-0.532343\pi$
$281$	−3.40072	−0.202870	−0.101435	+	0.994842i	$0.532343\pi$
$282$	−12.1871	−0.725731
$283$	7.41484	0.440767	0.220383	−	0.975413i	$-0.429269\pi$
$283$	7.41484	0.440767	0.220383	+	0.975413i	$0.429269\pi$
$284$	−3.75070	−0.222563
$285$	0.887011	0.0525420
$286$	−0.720292	−0.0425918
$287$	14.9759	0.883998
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−1.68722	−0.0990772
$291$	3.62535	0.212522
$292$	11.1255	0.651068
$293$	−0.298302	−0.0174270	−0.00871351	−	0.999962i	$-0.502774\pi$
$293$	−0.298302	−0.0174270	−0.00871351	+	0.999962i	$0.502774\pi$
$294$	4.38247	0.255591
$295$	0.354931	0.0206649
$296$	2.13363	0.124015
$297$	0.127157	0.00737840
$298$	6.05926	0.351003
$299$	36.5779	2.11536
$300$	−4.87402	−0.281402
$301$	−13.3473	−0.769326
$302$	−20.6175	−1.18640
$303$	5.87625	0.337582
$304$	2.49911	0.143334
$305$	−3.99976	−0.229026
$306$	−1.00000	−0.0571662
$307$	5.73215	0.327151	0.163576	−	0.986531i	$-0.447697\pi$
$307$	5.73215	0.327151	0.163576	+	0.986531i	$0.447697\pi$
$308$	0.205725	0.0117223
$309$	−15.7621	−0.896675
$310$	−0.476843	−0.0270829
$311$	−7.23600	−0.410316	−0.205158	−	0.978729i	$-0.565771\pi$
$311$	−7.23600	−0.410316	−0.205158	+	0.978729i	$0.565771\pi$
$312$	−5.66458	−0.320694
$313$	21.3597	1.20732	0.603661	−	0.797241i	$-0.293708\pi$
$313$	21.3597	1.20732	0.603661	+	0.797241i	$0.293708\pi$
$314$	18.0803	1.02033
$315$	0.574235	0.0323545
$316$	−5.60546	−0.315332
$317$	−1.36565	−0.0767023	−0.0383512	−	0.999264i	$-0.512211\pi$
$317$	−1.36565	−0.0767023	−0.0383512	+	0.999264i	$0.512211\pi$
$318$	3.16949	0.177736
$319$	0.604463	0.0338434
$320$	0.354931	0.0198412
$321$	11.8127	0.659320
$322$	−10.4471	−0.582196
$323$	2.49911	0.139054
$324$	1.00000	0.0555556
$325$	−27.6093	−1.53149
$326$	−14.6221	−0.809842
$327$	0.371196	0.0205272
$328$	−9.25649	−0.511104
$329$	19.7173	1.08705
$330$	−0.0451320	−0.00248443
$331$	−31.4699	−1.72974	−0.864871	−	0.501994i	$-0.832600\pi$
$331$	−31.4699	−1.72974	−0.864871	+	0.501994i	$0.832600\pi$
$332$	−9.91629	−0.544227
$333$	−2.13363	−0.116922
$334$	−11.7148	−0.641004
$335$	−0.501752	−0.0274136
$336$	1.61788	0.0882625
$337$	−8.76848	−0.477649	−0.238825	−	0.971063i	$-0.576762\pi$
$337$	−8.76848	−0.477649	−0.238825	+	0.971063i	$0.576762\pi$
$338$	−19.0875	−1.03822
$339$	−2.97215	−0.161425
$340$	0.354931	0.0192488
$341$	0.170833	0.00925114
$342$	−2.49911	−0.135136
$343$	−18.4155	−0.994341
$344$	8.24988	0.444804
$345$	2.29190	0.123392
$346$	13.6384	0.733203
$347$	−2.05202	−0.110158	−0.0550791	−	0.998482i	$-0.517541\pi$
$347$	−2.05202	−0.110158	−0.0550791	+	0.998482i	$0.517541\pi$
$348$	4.75367	0.254823
$349$	10.2241	0.547282	0.273641	−	0.961832i	$-0.411772\pi$
$349$	10.2241	0.547282	0.273641	+	0.961832i	$0.411772\pi$
$350$	7.88558	0.421502
$351$	5.66458	0.302353
$352$	−0.127157	−0.00677750
$353$	11.3052	0.601715	0.300858	−	0.953669i	$-0.402727\pi$
$353$	11.3052	0.601715	0.300858	+	0.953669i	$0.402727\pi$
$354$	−1.00000	−0.0531494
$355$	−1.33124	−0.0706549
$356$	9.25235	0.490373
$357$	1.61788	0.0856272
$358$	−17.3260	−0.915708
$359$	26.5169	1.39951	0.699755	−	0.714383i	$-0.253293\pi$
$359$	26.5169	1.39951	0.699755	+	0.714383i	$0.253293\pi$
$360$	−0.354931	−0.0187065
$361$	−12.7545	−0.671287
$362$	−9.77124	−0.513565
$363$	−10.9838	−0.576502
$364$	9.16461	0.480356
$365$	3.94877	0.206688
$366$	11.2691	0.589047
$367$	11.4511	0.597742	0.298871	−	0.954294i	$-0.403390\pi$
$367$	11.4511	0.597742	0.298871	+	0.954294i	$0.403390\pi$
$368$	6.45730	0.336610
$369$	9.25649	0.481874
$370$	0.757291	0.0393697
$371$	−5.12785	−0.266225
$372$	1.34348	0.0696563
$373$	−13.7494	−0.711916	−0.355958	−	0.934502i	$-0.615845\pi$
$373$	−13.7494	−0.711916	−0.355958	+	0.934502i	$0.615845\pi$
$374$	−0.127157	−0.00657514
$375$	−3.50460	−0.180977
$376$	−12.1871	−0.628502
$377$	26.9276	1.38684
$378$	−1.61788	−0.0832147
$379$	1.22316	0.0628294	0.0314147	−	0.999506i	$-0.489999\pi$
$379$	1.22316	0.0628294	0.0314147	+	0.999506i	$0.489999\pi$
$380$	0.887011	0.0455027
$381$	−19.5881	−1.00353
$382$	8.67311	0.443755
$383$	24.0623	1.22953	0.614763	−	0.788712i	$-0.289252\pi$
$383$	24.0623	1.22953	0.614763	+	0.788712i	$0.289252\pi$
$384$	−1.00000	−0.0510310
$385$	0.0730181	0.00372135
$386$	−3.65793	−0.186184
$387$	−8.24988	−0.419365
$388$	3.62535	0.184049
$389$	−6.59857	−0.334561	−0.167280	−	0.985909i	$-0.553498\pi$
$389$	−6.59857	−0.334561	−0.167280	+	0.985909i	$0.553498\pi$
$390$	−2.01054	−0.101807
$391$	6.45730	0.326560
$392$	4.38247	0.221348
$393$	1.66060	0.0837663
$394$	−7.39726	−0.372668
$395$	−1.98955	−0.100105
$396$	0.127157	0.00638989
$397$	15.3985	0.772829	0.386415	−	0.922325i	$-0.373713\pi$
$397$	15.3985	0.772829	0.386415	+	0.922325i	$0.373713\pi$
$398$	−22.3060	−1.11810
$399$	4.04325	0.202416
$400$	−4.87402	−0.243701
$401$	7.36831	0.367956	0.183978	−	0.982930i	$-0.441102\pi$
$401$	7.36831	0.367956	0.183978	+	0.982930i	$0.441102\pi$
$402$	1.41366	0.0705070
$403$	7.61026	0.379094
$404$	5.87625	0.292354
$405$	0.354931	0.0176367
$406$	−7.69086	−0.381691
$407$	−0.271306	−0.0134481
$408$	−1.00000	−0.0495074
$409$	22.5962	1.11731	0.558654	−	0.829401i	$-0.311318\pi$
$409$	22.5962	1.11731	0.558654	+	0.829401i	$0.311318\pi$
$410$	−3.28542	−0.162255
$411$	3.24592	0.160110
$412$	−15.7621	−0.776543
$413$	1.61788	0.0796106
$414$	−6.45730	−0.317359
$415$	−3.51960	−0.172770
$416$	−5.66458	−0.277729
$417$	−15.0025	−0.734674
$418$	−0.317779	−0.0155431
$419$	26.0654	1.27338	0.636688	−	0.771121i	$-0.280304\pi$
$419$	26.0654	1.27338	0.636688	+	0.771121i	$0.280304\pi$
$420$	0.574235	0.0280198
$421$	2.24486	0.109408	0.0547039	−	0.998503i	$-0.482579\pi$
$421$	2.24486	0.109408	0.0547039	+	0.998503i	$0.482579\pi$
$422$	−5.07524	−0.247059
$423$	12.1871	0.592557
$424$	3.16949	0.153924
$425$	−4.87402	−0.236425
$426$	3.75070	0.181722
$427$	−18.2321	−0.882312
$428$	11.8127	0.570988
$429$	0.720292	0.0347760
$430$	2.92814	0.141207
$431$	−13.5984	−0.655013	−0.327506	−	0.944849i	$-0.606208\pi$
$431$	−13.5984	−0.655013	−0.327506	+	0.944849i	$0.606208\pi$
$432$	1.00000	0.0481125
$433$	−14.3600	−0.690096	−0.345048	−	0.938585i	$-0.612137\pi$
$433$	−14.3600	−0.690096	−0.345048	+	0.938585i	$0.612137\pi$
$434$	−2.17359	−0.104336
$435$	1.68722	0.0808962
$436$	0.371196	0.0177770
$437$	16.1375	0.771961
$438$	−11.1255	−0.531595
$439$	28.6254	1.36621	0.683107	−	0.730319i	$-0.260628\pi$
$439$	28.6254	1.36621	0.683107	+	0.730319i	$0.260628\pi$
$440$	−0.0451320	−0.00215158
$441$	−4.38247	−0.208689
$442$	−5.66458	−0.269437
$443$	21.1015	1.00256	0.501281	−	0.865285i	$-0.332862\pi$
$443$	21.1015	1.00256	0.501281	+	0.865285i	$0.332862\pi$
$444$	−2.13363	−0.101258
$445$	3.28394	0.155674
$446$	−23.1987	−1.09849
$447$	−6.05926	−0.286593
$448$	1.61788	0.0764376
$449$	−37.2005	−1.75560	−0.877800	−	0.479028i	$-0.840989\pi$
$449$	−37.2005	−1.75560	−0.877800	+	0.479028i	$0.840989\pi$
$450$	4.87402	0.229764
$451$	1.17703	0.0554241
$452$	−2.97215	−0.139798
$453$	20.6175	0.968692
$454$	23.5651	1.10596
$455$	3.25280	0.152494
$456$	−2.49911	−0.117031
$457$	−25.0328	−1.17098	−0.585492	−	0.810678i	$-0.699099\pi$
$457$	−25.0328	−1.17098	−0.585492	+	0.810678i	$0.699099\pi$
$458$	18.3178	0.855936
$459$	1.00000	0.0466760
$460$	2.29190	0.106860
$461$	20.2175	0.941623	0.470811	−	0.882234i	$-0.343961\pi$
$461$	20.2175	0.941623	0.470811	+	0.882234i	$0.343961\pi$
$462$	−0.205725	−0.00957118
$463$	−19.1835	−0.891534	−0.445767	−	0.895149i	$-0.647069\pi$
$463$	−19.1835	−0.891534	−0.445767	+	0.895149i	$0.647069\pi$
$464$	4.75367	0.220684
$465$	0.476843	0.0221131
$466$	−12.5344	−0.580647
$467$	4.45626	0.206211	0.103105	−	0.994670i	$-0.467122\pi$
$467$	4.45626	0.206211	0.103105	+	0.994670i	$0.467122\pi$
$468$	5.66458	0.261845
$469$	−2.28713	−0.105610
$470$	−4.32558	−0.199524
$471$	−18.0803	−0.833095
$472$	−1.00000	−0.0460287
$473$	−1.04903	−0.0482345
$474$	5.60546	0.257467
$475$	−12.1807	−0.558889
$476$	1.61788	0.0741553
$477$	−3.16949	−0.145121
$478$	26.3711	1.20619
$479$	−7.04558	−0.321921	−0.160960	−	0.986961i	$-0.551459\pi$
$479$	−7.04558	−0.321921	−0.160960	+	0.986961i	$0.551459\pi$
$480$	−0.354931	−0.0162003
$481$	−12.0861	−0.551080
$482$	−23.6126	−1.07553
$483$	10.4471	0.475361
$484$	−10.9838	−0.499265
$485$	1.28675	0.0584283
$486$	−1.00000	−0.0453609
$487$	−3.14277	−0.142413	−0.0712064	−	0.997462i	$-0.522685\pi$
$487$	−3.14277	−0.142413	−0.0712064	+	0.997462i	$0.522685\pi$
$488$	11.2691	0.510129
$489$	14.6221	0.661233
$490$	1.55547	0.0702692
$491$	12.6814	0.572305	0.286153	−	0.958184i	$-0.407624\pi$
$491$	12.6814	0.572305	0.286153	+	0.958184i	$0.407624\pi$
$492$	9.25649	0.417315
$493$	4.75367	0.214094
$494$	−14.1564	−0.636927
$495$	0.0451320	0.00202853
$496$	1.34348	0.0603241
$497$	−6.06818	−0.272195
$498$	9.91629	0.444360
$499$	−30.0798	−1.34656	−0.673279	−	0.739389i	$-0.735115\pi$
$499$	−30.0798	−1.34656	−0.673279	+	0.739389i	$0.735115\pi$
$500$	−3.50460	−0.156730
$501$	11.7148	0.523378
$502$	−7.29061	−0.325396
$503$	−38.7158	−1.72625	−0.863127	−	0.504987i	$-0.831497\pi$
$503$	−38.7158	−1.72625	−0.863127	+	0.504987i	$0.831497\pi$
$504$	−1.61788	−0.0720660
$505$	2.08566	0.0928108
$506$	−0.821092	−0.0365020
$507$	19.0875	0.847706
$508$	−19.5881	−0.869081
$509$	10.7607	0.476961	0.238481	−	0.971147i	$-0.423351\pi$
$509$	10.7607	0.476961	0.238481	+	0.971147i	$0.423351\pi$
$510$	−0.354931	−0.0157166
$511$	17.9996	0.796257
$512$	−1.00000	−0.0441942
$513$	2.49911	0.110338
$514$	16.5447	0.729754
$515$	−5.59446	−0.246521
$516$	−8.24988	−0.363181
$517$	1.54968	0.0681547
$518$	3.45195	0.151670
$519$	−13.6384	−0.598658
$520$	−2.01054	−0.0881678
$521$	15.0286	0.658416	0.329208	−	0.944257i	$-0.393218\pi$
$521$	15.0286	0.658416	0.329208	+	0.944257i	$0.393218\pi$
$522$	−4.75367	−0.208062
$523$	37.6964	1.64835	0.824174	−	0.566337i	$-0.191640\pi$
$523$	37.6964	1.64835	0.824174	+	0.566337i	$0.191640\pi$
$524$	1.66060	0.0725438
$525$	−7.88558	−0.344155
$526$	15.3604	0.669747
$527$	1.34348	0.0585230
$528$	0.127157	0.00553380
$529$	18.6968	0.812903
$530$	1.12495	0.0488647
$531$	1.00000	0.0433963
$532$	4.04325	0.175297
$533$	52.4342	2.27118
$534$	−9.25235	−0.400388
$535$	4.19269	0.181266
$536$	1.41366	0.0610608
$537$	17.3260	0.747673
$538$	21.6836	0.934845
$539$	−0.557262	−0.0240030
$540$	0.354931	0.0152738
$541$	34.5295	1.48454	0.742271	−	0.670100i	$-0.233749\pi$
$541$	34.5295	1.48454	0.742271	+	0.670100i	$0.233749\pi$
$542$	−6.48269	−0.278456
$543$	9.77124	0.419324
$544$	−1.00000	−0.0428746
$545$	0.131749	0.00564350
$546$	−9.16461	−0.392209
$547$	−20.8524	−0.891585	−0.445793	−	0.895136i	$-0.647078\pi$
$547$	−20.8524	−0.891585	−0.445793	+	0.895136i	$0.647078\pi$
$548$	3.24592	0.138659
$549$	−11.2691	−0.480954
$550$	0.619767	0.0264269
$551$	11.8799	0.506102
$552$	−6.45730	−0.274841
$553$	−9.06896	−0.385651
$554$	11.2225	0.476798
$555$	−0.757291	−0.0321452
$556$	−15.0025	−0.636247
$557$	−17.1281	−0.725741	−0.362870	−	0.931840i	$-0.618203\pi$
$557$	−17.1281	−0.725741	−0.362870	+	0.931840i	$0.618203\pi$
$558$	−1.34348	−0.0568741
$559$	−46.7321	−1.97656
$560$	0.574235	0.0242659
$561$	0.127157	0.00536858
$562$	3.40072	0.143451
$563$	−8.72374	−0.367662	−0.183831	−	0.982958i	$-0.558850\pi$
$563$	−8.72374	−0.367662	−0.183831	+	0.982958i	$0.558850\pi$
$564$	12.1871	0.513170
$565$	−1.05491	−0.0443804
$566$	−7.41484	−0.311669
$567$	1.61788	0.0679445
$568$	3.75070	0.157376
$569$	5.02192	0.210530	0.105265	−	0.994444i	$-0.466431\pi$
$569$	5.02192	0.210530	0.105265	+	0.994444i	$0.466431\pi$
$570$	−0.887011	−0.0371528
$571$	−15.8701	−0.664144	−0.332072	−	0.943254i	$-0.607748\pi$
$571$	−15.8701	−0.664144	−0.332072	+	0.943254i	$0.607748\pi$
$572$	0.720292	0.0301169
$573$	−8.67311	−0.362324
$574$	−14.9759	−0.625081
$575$	−31.4731	−1.31252
$576$	1.00000	0.0416667
$577$	29.9558	1.24708	0.623538	−	0.781793i	$-0.285695\pi$
$577$	29.9558	1.24708	0.623538	+	0.781793i	$0.285695\pi$
$578$	−1.00000	−0.0415945
$579$	3.65793	0.152018
$580$	1.68722	0.0700582
$581$	−16.0434	−0.665591
$582$	−3.62535	−0.150276
$583$	−0.403023	−0.0166915
$584$	−11.1255	−0.460375
$585$	2.01054	0.0831254
$586$	0.298302	0.0123228
$587$	28.3142	1.16865	0.584327	−	0.811518i	$-0.301359\pi$
$587$	28.3142	1.16865	0.584327	+	0.811518i	$0.301359\pi$
$588$	−4.38247	−0.180730
$589$	3.35750	0.138344
$590$	−0.354931	−0.0146123
$591$	7.39726	0.304282
$592$	−2.13363	−0.0876916
$593$	−0.937389	−0.0384940	−0.0192470	−	0.999815i	$-0.506127\pi$
$593$	−0.937389	−0.0384940	−0.0192470	+	0.999815i	$0.506127\pi$
$594$	−0.127157	−0.00521732
$595$	0.574235	0.0235413
$596$	−6.05926	−0.248197
$597$	22.3060	0.912922
$598$	−36.5779	−1.49578
$599$	−4.86161	−0.198640	−0.0993200	−	0.995056i	$-0.531667\pi$
$599$	−4.86161	−0.198640	−0.0993200	+	0.995056i	$0.531667\pi$
$600$	4.87402	0.198981
$601$	−17.3569	−0.708003	−0.354002	−	0.935245i	$-0.615179\pi$
$601$	−17.3569	−0.708003	−0.354002	+	0.935245i	$0.615179\pi$
$602$	13.3473	0.543995
$603$	−1.41366	−0.0575687
$604$	20.6175	0.838912
$605$	−3.89850	−0.158497
$606$	−5.87625	−0.238706
$607$	−1.07001	−0.0434303	−0.0217152	−	0.999764i	$-0.506913\pi$
$607$	−1.07001	−0.0434303	−0.0217152	+	0.999764i	$0.506913\pi$
$608$	−2.49911	−0.101352
$609$	7.69086	0.311649
$610$	3.99976	0.161946
$611$	69.0349	2.79285
$612$	1.00000	0.0404226
$613$	−23.7859	−0.960705	−0.480352	−	0.877076i	$-0.659491\pi$
$613$	−23.7859	−0.960705	−0.480352	+	0.877076i	$0.659491\pi$
$614$	−5.73215	−0.231331
$615$	3.28542	0.132481
$616$	−0.205725	−0.00828889
$617$	−22.7869	−0.917367	−0.458684	−	0.888600i	$-0.651679\pi$
$617$	−22.7869	−0.917367	−0.458684	+	0.888600i	$0.651679\pi$
$618$	15.7621	0.634045
$619$	4.22673	0.169887	0.0849433	−	0.996386i	$-0.472929\pi$
$619$	4.22673	0.169887	0.0849433	+	0.996386i	$0.472929\pi$
$620$	0.476843	0.0191505
$621$	6.45730	0.259123
$622$	7.23600	0.290137
$623$	14.9692	0.599727
$624$	5.66458	0.226765
$625$	23.1262	0.925049
$626$	−21.3597	−0.853706
$627$	0.317779	0.0126909
$628$	−18.0803	−0.721482
$629$	−2.13363	−0.0850734
$630$	−0.574235	−0.0228781
$631$	−46.0405	−1.83284	−0.916421	−	0.400215i	$-0.868935\pi$
$631$	−46.0405	−1.83284	−0.916421	+	0.400215i	$0.868935\pi$
$632$	5.60546	0.222973
$633$	5.07524	0.201723
$634$	1.36565	0.0542367
$635$	−6.95242	−0.275898
$636$	−3.16949	−0.125678
$637$	−24.8249	−0.983597
$638$	−0.604463	−0.0239309
$639$	−3.75070	−0.148376
$640$	−0.354931	−0.0140299
$641$	−28.9345	−1.14284	−0.571422	−	0.820656i	$-0.693608\pi$
$641$	−28.9345	−1.14284	−0.571422	+	0.820656i	$0.693608\pi$
$642$	−11.8127	−0.466210
$643$	8.91520	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$643$	8.91520	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$644$	10.4471	0.411675
$645$	−2.92814	−0.115295
$646$	−2.49911	−0.0983261
$647$	24.3491	0.957261	0.478631	−	0.878016i	$-0.341133\pi$
$647$	24.3491	0.957261	0.478631	+	0.878016i	$0.341133\pi$
$648$	−1.00000	−0.0392837
$649$	0.127157	0.00499135
$650$	27.6093	1.08293
$651$	2.17359	0.0851897
$652$	14.6221	0.572644
$653$	2.72155	0.106503	0.0532513	−	0.998581i	$-0.483042\pi$
$653$	2.72155	0.106503	0.0532513	+	0.998581i	$0.483042\pi$
$654$	−0.371196	−0.0145149
$655$	0.589399	0.0230297
$656$	9.25649	0.361405
$657$	11.1255	0.434045
$658$	−19.7173	−0.768658
$659$	18.5167	0.721308	0.360654	−	0.932700i	$-0.382554\pi$
$659$	18.5167	0.721308	0.360654	+	0.932700i	$0.382554\pi$
$660$	0.0451320	0.00175676
$661$	37.1560	1.44520	0.722601	−	0.691266i	$-0.242947\pi$
$661$	37.1560	1.44520	0.722601	+	0.691266i	$0.242947\pi$
$662$	31.4699	1.22311
$663$	5.66458	0.219994
$664$	9.91629	0.384827
$665$	1.43508	0.0556499
$666$	2.13363	0.0826765
$667$	30.6959	1.18855
$668$	11.7148	0.453258
$669$	23.1987	0.896913
$670$	0.501752	0.0193844
$671$	−1.43295	−0.0553184
$672$	−1.61788	−0.0624110
$673$	42.6863	1.64544	0.822718	−	0.568450i	$-0.192457\pi$
$673$	42.6863	1.64544	0.822718	+	0.568450i	$0.192457\pi$
$674$	8.76848	0.337749
$675$	−4.87402	−0.187601
$676$	19.0875	0.734135
$677$	−10.2934	−0.395607	−0.197804	−	0.980242i	$-0.563381\pi$
$677$	−10.2934	−0.395607	−0.197804	+	0.980242i	$0.563381\pi$
$678$	2.97215	0.114145
$679$	5.86538	0.225093
$680$	−0.354931	−0.0136110
$681$	−23.5651	−0.903016
$682$	−0.170833	−0.00654154
$683$	−0.984890	−0.0376858	−0.0188429	−	0.999822i	$-0.505998\pi$
$683$	−0.984890	−0.0376858	−0.0188429	+	0.999822i	$0.505998\pi$
$684$	2.49911	0.0955558
$685$	1.15208	0.0440187
$686$	18.4155	0.703105
$687$	−18.3178	−0.698869
$688$	−8.24988	−0.314524
$689$	−17.9538	−0.683987
$690$	−2.29190	−0.0872510
$691$	13.4154	0.510344	0.255172	−	0.966896i	$-0.417868\pi$
$691$	13.4154	0.510344	0.255172	+	0.966896i	$0.417868\pi$
$692$	−13.6384	−0.518453
$693$	0.205725	0.00781484
$694$	2.05202	0.0778935
$695$	−5.32484	−0.201983
$696$	−4.75367	−0.180187
$697$	9.25649	0.350615
$698$	−10.2241	−0.386987
$699$	12.5344	0.474096
$700$	−7.88558	−0.298047
$701$	11.3964	0.430435	0.215217	−	0.976566i	$-0.430954\pi$
$701$	11.3964	0.430435	0.215217	+	0.976566i	$0.430954\pi$
$702$	−5.66458	−0.213796
$703$	−5.33217	−0.201107
$704$	0.127157	0.00479241
$705$	4.32558	0.162911
$706$	−11.3052	−0.425477
$707$	9.50706	0.357550
$708$	1.00000	0.0375823
$709$	−35.7253	−1.34169	−0.670846	−	0.741597i	$-0.734069\pi$
$709$	−35.7253	−1.34169	−0.670846	+	0.741597i	$0.734069\pi$
$710$	1.33124	0.0499606
$711$	−5.60546	−0.210221
$712$	−9.25235	−0.346746
$713$	8.67527	0.324891
$714$	−1.61788	−0.0605476
$715$	0.255654	0.00956092
$716$	17.3260	0.647504
$717$	−26.3711	−0.984849
$718$	−26.5169	−0.989602
$719$	−7.98098	−0.297640	−0.148820	−	0.988864i	$-0.547548\pi$
$719$	−7.98098	−0.297640	−0.148820	+	0.988864i	$0.547548\pi$
$720$	0.354931	0.0132275
$721$	−25.5012	−0.949714
$722$	12.7545	0.474672
$723$	23.6126	0.878163
$724$	9.77124	0.363145
$725$	−23.1695	−0.860494
$726$	10.9838	0.407648
$727$	−37.6999	−1.39821	−0.699107	−	0.715017i	$-0.746419\pi$
$727$	−37.6999	−1.39821	−0.699107	+	0.715017i	$0.746419\pi$
$728$	−9.16461	−0.339663
$729$	1.00000	0.0370370
$730$	−3.94877	−0.146150
$731$	−8.24988	−0.305133
$732$	−11.2691	−0.416519
$733$	−30.0246	−1.10898	−0.554492	−	0.832189i	$-0.687087\pi$
$733$	−30.0246	−1.10898	−0.554492	+	0.832189i	$0.687087\pi$
$734$	−11.4511	−0.422667
$735$	−1.55547	−0.0573745
$736$	−6.45730	−0.238019
$737$	−0.179757	−0.00662143
$738$	−9.25649	−0.340736
$739$	−5.56116	−0.204571	−0.102285	−	0.994755i	$-0.532615\pi$
$739$	−5.56116	−0.204571	−0.102285	+	0.994755i	$0.532615\pi$
$740$	−0.757291	−0.0278386
$741$	14.1564	0.520049
$742$	5.12785	0.188249
$743$	−6.54665	−0.240173	−0.120087	−	0.992763i	$-0.538317\pi$
$743$	−6.54665	−0.240173	−0.120087	+	0.992763i	$0.538317\pi$
$744$	−1.34348	−0.0492544
$745$	−2.15062	−0.0787926
$746$	13.7494	0.503400
$747$	−9.91629	−0.362818
$748$	0.127157	0.00464932
$749$	19.1115	0.698319
$750$	3.50460	0.127970
$751$	44.5978	1.62740	0.813698	−	0.581288i	$-0.197451\pi$
$751$	44.5978	1.62740	0.813698	+	0.581288i	$0.197451\pi$
$752$	12.1871	0.444418
$753$	7.29061	0.265685
$754$	−26.9276	−0.980644
$755$	7.31777	0.266321
$756$	1.61788	0.0588417
$757$	29.8558	1.08513	0.542565	−	0.840014i	$-0.317453\pi$
$757$	29.8558	1.08513	0.542565	+	0.840014i	$0.317453\pi$
$758$	−1.22316	−0.0444271
$759$	0.821092	0.0298038
$760$	−0.887011	−0.0321753
$761$	17.2153	0.624053	0.312027	−	0.950073i	$-0.398992\pi$
$761$	17.2153	0.624053	0.312027	+	0.950073i	$0.398992\pi$
$762$	19.5881	0.709602
$763$	0.600549	0.0217413
$764$	−8.67311	−0.313782
$765$	0.354931	0.0128326
$766$	−24.0623	−0.869406
$767$	5.66458	0.204536
$768$	1.00000	0.0360844
$769$	17.9451	0.647115	0.323558	−	0.946208i	$-0.395121\pi$
$769$	17.9451	0.647115	0.323558	+	0.946208i	$0.395121\pi$
$770$	−0.0730181	−0.00263139
$771$	−16.5447	−0.595841
$772$	3.65793	0.131652
$773$	51.2855	1.84461	0.922306	−	0.386461i	$-0.126303\pi$
$773$	51.2855	1.84461	0.922306	+	0.386461i	$0.126303\pi$
$774$	8.24988	0.296536
$775$	−6.54816	−0.235217
$776$	−3.62535	−0.130143
$777$	−3.45195	−0.123838
$778$	6.59857	0.236570
$779$	23.1330	0.828825
$780$	2.01054	0.0719887
$781$	−0.476929	−0.0170658
$782$	−6.45730	−0.230913
$783$	4.75367	0.169882
$784$	−4.38247	−0.156517
$785$	−6.41725	−0.229041
$786$	−1.66060	−0.0592317
$787$	−34.5643	−1.23209	−0.616043	−	0.787713i	$-0.711265\pi$
$787$	−34.5643	−1.23209	−0.616043	+	0.787713i	$0.711265\pi$
$788$	7.39726	0.263516
$789$	−15.3604	−0.546846
$790$	1.98955	0.0707851
$791$	−4.80858	−0.170973
$792$	−0.127157	−0.00451833
$793$	−63.8349	−2.26684
$794$	−15.3985	−0.546473
$795$	−1.12495	−0.0398979
$796$	22.3060	0.790614
$797$	−6.90883	−0.244723	−0.122362	−	0.992486i	$-0.539047\pi$
$797$	−6.90883	−0.244723	−0.122362	+	0.992486i	$0.539047\pi$
$798$	−4.04325	−0.143130
$799$	12.1871	0.431149
$800$	4.87402	0.172323
$801$	9.25235	0.326916
$802$	−7.36831	−0.260184
$803$	1.41468	0.0499230
$804$	−1.41366	−0.0498560
$805$	3.70801	0.130690
$806$	−7.61026	−0.268060
$807$	−21.6836	−0.763298
$808$	−5.87625	−0.206726
$809$	7.28774	0.256223	0.128112	−	0.991760i	$-0.459108\pi$
$809$	7.28774	0.256223	0.128112	+	0.991760i	$0.459108\pi$
$810$	−0.354931	−0.0124710
$811$	−8.20573	−0.288142	−0.144071	−	0.989567i	$-0.546019\pi$
$811$	−8.20573	−0.288142	−0.144071	+	0.989567i	$0.546019\pi$
$812$	7.69086	0.269896
$813$	6.48269	0.227358
$814$	0.271306	0.00950927
$815$	5.18983	0.181792
$816$	1.00000	0.0350070
$817$	−20.6173	−0.721309
$818$	−22.5962	−0.790056
$819$	9.16461	0.320237
$820$	3.28542	0.114732
$821$	31.6418	1.10431	0.552154	−	0.833742i	$-0.313806\pi$
$821$	31.6418	1.10431	0.552154	+	0.833742i	$0.313806\pi$
$822$	−3.24592	−0.113215
$823$	−24.6205	−0.858217	−0.429108	−	0.903253i	$-0.641172\pi$
$823$	−24.6205	−0.858217	−0.429108	+	0.903253i	$0.641172\pi$
$824$	15.7621	0.549099
$825$	−0.619767	−0.0215775
$826$	−1.61788	−0.0562932
$827$	−22.7318	−0.790463	−0.395232	−	0.918582i	$-0.629336\pi$
$827$	−22.7318	−0.790463	−0.395232	+	0.918582i	$0.629336\pi$
$828$	6.45730	0.224407
$829$	36.5642	1.26993	0.634963	−	0.772542i	$-0.281015\pi$
$829$	36.5642	1.26993	0.634963	+	0.772542i	$0.281015\pi$
$830$	3.51960	0.122167
$831$	−11.2225	−0.389304
$832$	5.66458	0.196384
$833$	−4.38247	−0.151844
$834$	15.0025	0.519493
$835$	4.15794	0.143891
$836$	0.317779	0.0109906
$837$	1.34348	0.0464375
$838$	−26.0654	−0.900413
$839$	37.8352	1.30621	0.653107	−	0.757265i	$-0.273465\pi$
$839$	37.8352	1.30621	0.653107	+	0.757265i	$0.273465\pi$
$840$	−0.574235	−0.0198130
$841$	−6.40263	−0.220780
$842$	−2.24486	−0.0773630
$843$	−3.40072	−0.117127
$844$	5.07524	0.174697
$845$	6.77475	0.233058
$846$	−12.1871	−0.419001
$847$	−17.7705	−0.610602
$848$	−3.16949	−0.108841
$849$	7.41484	0.254477
$850$	4.87402	0.167178
$851$	−13.7775	−0.472286
$852$	−3.75070	−0.128497
$853$	33.1489	1.13500	0.567498	−	0.823375i	$-0.307911\pi$
$853$	33.1489	1.13500	0.567498	+	0.823375i	$0.307911\pi$
$854$	18.2321	0.623889
$855$	0.887011	0.0303351
$856$	−11.8127	−0.403750
$857$	17.8207	0.608743	0.304371	−	0.952553i	$-0.401554\pi$
$857$	17.8207	0.608743	0.304371	+	0.952553i	$0.401554\pi$
$858$	−0.720292	−0.0245904
$859$	36.3808	1.24130	0.620648	−	0.784089i	$-0.286870\pi$
$859$	36.3808	1.24130	0.620648	+	0.784089i	$0.286870\pi$
$860$	−2.92814	−0.0998486
$861$	14.9759	0.510377
$862$	13.5984	0.463164
$863$	41.6535	1.41790	0.708950	−	0.705259i	$-0.249169\pi$
$863$	41.6535	1.41790	0.708950	+	0.705259i	$0.249169\pi$
$864$	−1.00000	−0.0340207
$865$	−4.84068	−0.164588
$866$	14.3600	0.487972
$867$	1.00000	0.0339618
$868$	2.17359	0.0737764
$869$	−0.712775	−0.0241792
$870$	−1.68722	−0.0572023
$871$	−8.00780	−0.271334
$872$	−0.371196	−0.0125703
$873$	3.62535	0.122700
$874$	−16.1375	−0.545859
$875$	−5.67001	−0.191681
$876$	11.1255	0.375894
$877$	−55.6117	−1.87787	−0.938936	−	0.344090i	$-0.888187\pi$
$877$	−55.6117	−1.87787	−0.938936	+	0.344090i	$0.888187\pi$
$878$	−28.6254	−0.966059
$879$	−0.298302	−0.0100615
$880$	0.0451320	0.00152140
$881$	−29.2473	−0.985368	−0.492684	−	0.870208i	$-0.663984\pi$
$881$	−29.2473	−0.985368	−0.492684	+	0.870208i	$0.663984\pi$
$882$	4.38247	0.147565
$883$	−11.8646	−0.399275	−0.199638	−	0.979870i	$-0.563976\pi$
$883$	−11.8646	−0.399275	−0.199638	+	0.979870i	$0.563976\pi$
$884$	5.66458	0.190521
$885$	0.354931	0.0119309
$886$	−21.1015	−0.708918
$887$	42.6031	1.43047	0.715237	−	0.698882i	$-0.246319\pi$
$887$	42.6031	1.43047	0.715237	+	0.698882i	$0.246319\pi$
$888$	2.13363	0.0715999
$889$	−31.6912	−1.06289
$890$	−3.28394	−0.110078
$891$	0.127157	0.00425992
$892$	23.1987	0.776750
$893$	30.4569	1.01920
$894$	6.05926	0.202652
$895$	6.14954	0.205556
$896$	−1.61788	−0.0540495
$897$	36.5779	1.22130
$898$	37.2005	1.24140
$899$	6.38646	0.213001
$900$	−4.87402	−0.162467
$901$	−3.16949	−0.105591
$902$	−1.17703	−0.0391908
$903$	−13.3473	−0.444170
$904$	2.97215	0.0988523
$905$	3.46812	0.115284
$906$	−20.6175	−0.684969
$907$	37.3946	1.24167	0.620833	−	0.783943i	$-0.286794\pi$
$907$	37.3946	1.24167	0.620833	+	0.783943i	$0.286794\pi$
$908$	−23.5651	−0.782035
$909$	5.87625	0.194903
$910$	−3.25280	−0.107829
$911$	23.3144	0.772441	0.386220	−	0.922407i	$-0.373780\pi$
$911$	23.3144	0.772441	0.386220	+	0.922407i	$0.373780\pi$
$912$	2.49911	0.0827537
$913$	−1.26093	−0.0417306
$914$	25.0328	0.828011
$915$	−3.99976	−0.132228
$916$	−18.3178	−0.605238
$917$	2.68665	0.0887211
$918$	−1.00000	−0.0330049
$919$	14.0411	0.463172	0.231586	−	0.972814i	$-0.425609\pi$
$919$	14.0411	0.463172	0.231586	+	0.972814i	$0.425609\pi$
$920$	−2.29190	−0.0755616
$921$	5.73215	0.188881
$922$	−20.2175	−0.665828
$923$	−21.2462	−0.699326
$924$	0.205725	0.00676785
$925$	10.3994	0.341929
$926$	19.1835	0.630410
$927$	−15.7621	−0.517696
$928$	−4.75367	−0.156047
$929$	−42.4328	−1.39217	−0.696087	−	0.717957i	$-0.745077\pi$
$929$	−42.4328	−1.39217	−0.696087	+	0.717957i	$0.745077\pi$
$930$	−0.476843	−0.0156363
$931$	−10.9523	−0.358946
$932$	12.5344	0.410579
$933$	−7.23600	−0.236896
$934$	−4.45626	−0.145813
$935$	0.0451320	0.00147597
$936$	−5.66458	−0.185153
$937$	−9.23948	−0.301841	−0.150920	−	0.988546i	$-0.548224\pi$
$937$	−9.23948	−0.301841	−0.150920	+	0.988546i	$0.548224\pi$
$938$	2.28713	0.0746775
$939$	21.3597	0.697048
$940$	4.32558	0.141085
$941$	33.6927	1.09835	0.549175	−	0.835707i	$-0.314942\pi$
$941$	33.6927	1.09835	0.549175	+	0.835707i	$0.314942\pi$
$942$	18.0803	0.589087
$943$	59.7720	1.94644
$944$	1.00000	0.0325472
$945$	0.574235	0.0186799
$946$	1.04903	0.0341069
$947$	−50.5724	−1.64338	−0.821691	−	0.569933i	$-0.806969\pi$
$947$	−50.5724	−1.64338	−0.821691	+	0.569933i	$0.806969\pi$
$948$	−5.60546	−0.182057
$949$	63.0211	2.04575
$950$	12.1807	0.395194
$951$	−1.36565	−0.0442841
$952$	−1.61788	−0.0524357
$953$	−0.961187	−0.0311359	−0.0155680	−	0.999879i	$-0.504956\pi$
$953$	−0.961187	−0.0311359	−0.0155680	+	0.999879i	$0.504956\pi$
$954$	3.16949	0.102616
$955$	−3.07835	−0.0996132
$956$	−26.3711	−0.852904
$957$	0.604463	0.0195395
$958$	7.04558	0.227632
$959$	5.25151	0.169580
$960$	0.354931	0.0114553
$961$	−29.1951	−0.941776
$962$	12.0861	0.389672
$963$	11.8127	0.380659
$964$	23.6126	0.760511
$965$	1.29831	0.0417942
$966$	−10.4471	−0.336131
$967$	16.9793	0.546017	0.273008	−	0.962012i	$-0.411981\pi$
$967$	16.9793	0.546017	0.273008	+	0.962012i	$0.411981\pi$
$968$	10.9838	0.353034
$969$	2.49911	0.0802829
$970$	−1.28675	−0.0413150
$971$	−4.71113	−0.151188	−0.0755938	−	0.997139i	$-0.524085\pi$
$971$	−4.71113	−0.151188	−0.0755938	+	0.997139i	$0.524085\pi$
$972$	1.00000	0.0320750
$973$	−24.2722	−0.778130
$974$	3.14277	0.100701
$975$	−27.6093	−0.884206
$976$	−11.2691	−0.360716
$977$	16.6279	0.531972	0.265986	−	0.963977i	$-0.414302\pi$
$977$	16.6279	0.531972	0.265986	+	0.963977i	$0.414302\pi$
$978$	−14.6221	−0.467562
$979$	1.17650	0.0376012
$980$	−1.55547	−0.0496878
$981$	0.371196	0.0118514
$982$	−12.6814	−0.404681
$983$	13.0975	0.417744	0.208872	−	0.977943i	$-0.433021\pi$
$983$	13.0975	0.417744	0.208872	+	0.977943i	$0.433021\pi$
$984$	−9.25649	−0.295086
$985$	2.62552	0.0836559
$986$	−4.75367	−0.151388
$987$	19.7173	0.627607
$988$	14.1564	0.450375
$989$	−53.2720	−1.69395
$990$	−0.0451320	−0.00143439
$991$	−10.4780	−0.332844	−0.166422	−	0.986055i	$-0.553221\pi$
$991$	−10.4780	−0.332844	−0.166422	+	0.986055i	$0.553221\pi$
$992$	−1.34348	−0.0426556
$993$	−31.4699	−0.998667
$994$	6.06818	0.192471
$995$	7.91708	0.250988
$996$	−9.91629	−0.314210
$997$	−42.4839	−1.34548	−0.672740	−	0.739879i	$-0.734883\pi$
$997$	−42.4839	−1.34548	−0.672740	+	0.739879i	$0.734883\pi$
$998$	30.0798	0.952160
$999$	−2.13363	−0.0675050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6018.2.a.w.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6018.2.a.w.1.5	✓	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:	\( N \)	\(=\)	\( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6018.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.354931 q^{5} -1.00000 q^{6} +1.61788 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.354931 q^{5} -1.00000 q^{6} +1.61788 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.354931 q^{10} +0.127157 q^{11} +1.00000 q^{12} +5.66458 q^{13} -1.61788 q^{14} +0.354931 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +2.49911 q^{19} +0.354931 q^{20} +1.61788 q^{21} -0.127157 q^{22} +6.45730 q^{23} -1.00000 q^{24} -4.87402 q^{25} -5.66458 q^{26} +1.00000 q^{27} +1.61788 q^{28} +4.75367 q^{29} -0.354931 q^{30} +1.34348 q^{31} -1.00000 q^{32} +0.127157 q^{33} -1.00000 q^{34} +0.574235 q^{35} +1.00000 q^{36} -2.13363 q^{37} -2.49911 q^{38} +5.66458 q^{39} -0.354931 q^{40} +9.25649 q^{41} -1.61788 q^{42} -8.24988 q^{43} +0.127157 q^{44} +0.354931 q^{45} -6.45730 q^{46} +12.1871 q^{47} +1.00000 q^{48} -4.38247 q^{49} +4.87402 q^{50} +1.00000 q^{51} +5.66458 q^{52} -3.16949 q^{53} -1.00000 q^{54} +0.0451320 q^{55} -1.61788 q^{56} +2.49911 q^{57} -4.75367 q^{58} +1.00000 q^{59} +0.354931 q^{60} -11.2691 q^{61} -1.34348 q^{62} +1.61788 q^{63} +1.00000 q^{64} +2.01054 q^{65} -0.127157 q^{66} -1.41366 q^{67} +1.00000 q^{68} +6.45730 q^{69} -0.574235 q^{70} -3.75070 q^{71} -1.00000 q^{72} +11.1255 q^{73} +2.13363 q^{74} -4.87402 q^{75} +2.49911 q^{76} +0.205725 q^{77} -5.66458 q^{78} -5.60546 q^{79} +0.354931 q^{80} +1.00000 q^{81} -9.25649 q^{82} -9.91629 q^{83} +1.61788 q^{84} +0.354931 q^{85} +8.24988 q^{86} +4.75367 q^{87} -0.127157 q^{88} +9.25235 q^{89} -0.354931 q^{90} +9.16461 q^{91} +6.45730 q^{92} +1.34348 q^{93} -12.1871 q^{94} +0.887011 q^{95} -1.00000 q^{96} +3.62535 q^{97} +4.38247 q^{98} +0.127157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + 9 q^{3} + 9 q^{4} + q^{5} - 9 q^{6} - 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^2 + 9 * q^3 + 9 * q^4 + q^5 - 9 * q^6 - 9 * q^8 + 9 * q^9	\( 9 q - 9 q^{2} + 9 q^{3} + 9 q^{4} + q^{5} - 9 q^{6} - 9 q^{8} + 9 q^{9} - q^{10} + 6 q^{11} + 9 q^{12} + 2 q^{13} + q^{15} + 9 q^{16} + 9 q^{17} - 9 q^{18} - 5 q^{19} + q^{20} - 6 q^{22} + 15 q^{23} - 9 q^{24} - 2 q^{26} + 9 q^{27} + 11 q^{29} - q^{30} - 5 q^{31} - 9 q^{32} + 6 q^{33} - 9 q^{34} + 22 q^{35} + 9 q^{36} + 9 q^{37} + 5 q^{38} + 2 q^{39} - q^{40} + q^{41} + 4 q^{43} + 6 q^{44} + q^{45} - 15 q^{46} + 14 q^{47} + 9 q^{48} - q^{49} + 9 q^{51} + 2 q^{52} + 4 q^{53} - 9 q^{54} + 4 q^{55} - 5 q^{57} - 11 q^{58} + 9 q^{59} + q^{60} + 10 q^{61} + 5 q^{62} + 9 q^{64} + 8 q^{65} - 6 q^{66} - q^{67} + 9 q^{68} + 15 q^{69} - 22 q^{70} + 14 q^{71} - 9 q^{72} - q^{73} - 9 q^{74} - 5 q^{76} + 30 q^{77} - 2 q^{78} + 4 q^{79} + q^{80} + 9 q^{81} - q^{82} + 22 q^{83} + q^{85} - 4 q^{86} + 11 q^{87} - 6 q^{88} + 22 q^{89} - q^{90} - 3 q^{91} + 15 q^{92} - 5 q^{93} - 14 q^{94} + 43 q^{95} - 9 q^{96} - 15 q^{97} + q^{98} + 6 q^{99}+O(q^{100}) \) 9 * q - 9 * q^2 + 9 * q^3 + 9 * q^4 + q^5 - 9 * q^6 - 9 * q^8 + 9 * q^9 - q^10 + 6 * q^11 + 9 * q^12 + 2 * q^13 + q^15 + 9 * q^16 + 9 * q^17 - 9 * q^18 - 5 * q^19 + q^20 - 6 * q^22 + 15 * q^23 - 9 * q^24 - 2 * q^26 + 9 * q^27 + 11 * q^29 - q^30 - 5 * q^31 - 9 * q^32 + 6 * q^33 - 9 * q^34 + 22 * q^35 + 9 * q^36 + 9 * q^37 + 5 * q^38 + 2 * q^39 - q^40 + q^41 + 4 * q^43 + 6 * q^44 + q^45 - 15 * q^46 + 14 * q^47 + 9 * q^48 - q^49 + 9 * q^51 + 2 * q^52 + 4 * q^53 - 9 * q^54 + 4 * q^55 - 5 * q^57 - 11 * q^58 + 9 * q^59 + q^60 + 10 * q^61 + 5 * q^62 + 9 * q^64 + 8 * q^65 - 6 * q^66 - q^67 + 9 * q^68 + 15 * q^69 - 22 * q^70 + 14 * q^71 - 9 * q^72 - q^73 - 9 * q^74 - 5 * q^76 + 30 * q^77 - 2 * q^78 + 4 * q^79 + q^80 + 9 * q^81 - q^82 + 22 * q^83 + q^85 - 4 * q^86 + 11 * q^87 - 6 * q^88 + 22 * q^89 - q^90 - 3 * q^91 + 15 * q^92 - 5 * q^93 - 14 * q^94 + 43 * q^95 - 9 * q^96 - 15 * q^97 + q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more