LMFDB - Embedded newform 8043.2.a.n.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.765993 q^{2} +1.00000 q^{3} -1.41325 q^{4} -3.34403 q^{5} -0.765993 q^{6} +1.00000 q^{7} +2.61453 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.765993 q^{2} +1.00000 q^{3} -1.41325 q^{4} -3.34403 q^{5} -0.765993 q^{6} +1.00000 q^{7} +2.61453 q^{8} +1.00000 q^{9} +2.56151 q^{10} -2.72240 q^{11} -1.41325 q^{12} +2.91478 q^{13} -0.765993 q^{14} -3.34403 q^{15} +0.823799 q^{16} -1.11556 q^{17} -0.765993 q^{18} -1.90424 q^{19} +4.72597 q^{20} +1.00000 q^{21} +2.08534 q^{22} -3.99381 q^{23} +2.61453 q^{24} +6.18255 q^{25} -2.23270 q^{26} +1.00000 q^{27} -1.41325 q^{28} -5.21312 q^{29} +2.56151 q^{30} +4.59365 q^{31} -5.86008 q^{32} -2.72240 q^{33} +0.854513 q^{34} -3.34403 q^{35} -1.41325 q^{36} +10.2502 q^{37} +1.45863 q^{38} +2.91478 q^{39} -8.74307 q^{40} -4.17097 q^{41} -0.765993 q^{42} -2.49906 q^{43} +3.84745 q^{44} -3.34403 q^{45} +3.05923 q^{46} +7.33669 q^{47} +0.823799 q^{48} +1.00000 q^{49} -4.73579 q^{50} -1.11556 q^{51} -4.11933 q^{52} +9.56467 q^{53} -0.765993 q^{54} +9.10380 q^{55} +2.61453 q^{56} -1.90424 q^{57} +3.99321 q^{58} -1.70771 q^{59} +4.72597 q^{60} +6.64870 q^{61} -3.51870 q^{62} +1.00000 q^{63} +2.84118 q^{64} -9.74713 q^{65} +2.08534 q^{66} -5.44386 q^{67} +1.57657 q^{68} -3.99381 q^{69} +2.56151 q^{70} -7.55180 q^{71} +2.61453 q^{72} -9.55859 q^{73} -7.85159 q^{74} +6.18255 q^{75} +2.69118 q^{76} -2.72240 q^{77} -2.23270 q^{78} +6.37889 q^{79} -2.75481 q^{80} +1.00000 q^{81} +3.19493 q^{82} -2.15517 q^{83} -1.41325 q^{84} +3.73048 q^{85} +1.91426 q^{86} -5.21312 q^{87} -7.11780 q^{88} +8.52274 q^{89} +2.56151 q^{90} +2.91478 q^{91} +5.64427 q^{92} +4.59365 q^{93} -5.61985 q^{94} +6.36784 q^{95} -5.86008 q^{96} -3.69646 q^{97} -0.765993 q^{98} -2.72240 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * 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 $

$2$	−0.765993	−0.541639	−0.270819	−	0.962630i	$-0.587295\pi$
$2$	−0.765993	−0.541639	−0.270819	+	0.962630i	$0.587295\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.41325	−0.706627
$5$	−3.34403	−1.49550	−0.747748	−	0.663982i	$-0.768865\pi$
$5$	−3.34403	−1.49550	−0.747748	+	0.663982i	$0.768865\pi$
$6$	−0.765993	−0.312715
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.61453	0.924376
$9$	1.00000	0.333333
$10$	2.56151	0.810019
$11$	−2.72240	−0.820835	−0.410418	−	0.911898i	$-0.634617\pi$
$11$	−2.72240	−0.820835	−0.410418	+	0.911898i	$0.634617\pi$
$12$	−1.41325	−0.407972
$13$	2.91478	0.808415	0.404208	−	0.914667i	$-0.367547\pi$
$13$	2.91478	0.808415	0.404208	+	0.914667i	$0.367547\pi$
$14$	−0.765993	−0.204720
$15$	−3.34403	−0.863426
$16$	0.823799	0.205950
$17$	−1.11556	−0.270564	−0.135282	−	0.990807i	$-0.543194\pi$
$17$	−1.11556	−0.270564	−0.135282	+	0.990807i	$0.543194\pi$
$18$	−0.765993	−0.180546
$19$	−1.90424	−0.436863	−0.218431	−	0.975852i	$-0.570094\pi$
$19$	−1.90424	−0.436863	−0.218431	+	0.975852i	$0.570094\pi$
$20$	4.72597	1.05676
$21$	1.00000	0.218218
$22$	2.08534	0.444596
$23$	−3.99381	−0.832766	−0.416383	−	0.909189i	$-0.636702\pi$
$23$	−3.99381	−0.832766	−0.416383	+	0.909189i	$0.636702\pi$
$24$	2.61453	0.533689
$25$	6.18255	1.23651
$26$	−2.23270	−0.437869
$27$	1.00000	0.192450
$28$	−1.41325	−0.267080
$29$	−5.21312	−0.968052	−0.484026	−	0.875054i	$-0.660826\pi$
$29$	−5.21312	−0.968052	−0.484026	+	0.875054i	$0.660826\pi$
$30$	2.56151	0.467665
$31$	4.59365	0.825044	0.412522	−	0.910948i	$-0.364648\pi$
$31$	4.59365	0.825044	0.412522	+	0.910948i	$0.364648\pi$
$32$	−5.86008	−1.03593
$33$	−2.72240	−0.473909
$34$	0.854513	0.146548
$35$	−3.34403	−0.565245
$36$	−1.41325	−0.235542
$37$	10.2502	1.68512	0.842562	−	0.538600i	$-0.181046\pi$
$37$	10.2502	1.68512	0.842562	+	0.538600i	$0.181046\pi$
$38$	1.45863	0.236622
$39$	2.91478	0.466739
$40$	−8.74307	−1.38240
$41$	−4.17097	−0.651396	−0.325698	−	0.945474i	$-0.605599\pi$
$41$	−4.17097	−0.651396	−0.325698	+	0.945474i	$0.605599\pi$
$42$	−0.765993	−0.118195
$43$	−2.49906	−0.381104	−0.190552	−	0.981677i	$-0.561028\pi$
$43$	−2.49906	−0.381104	−0.190552	+	0.981677i	$0.561028\pi$
$44$	3.84745	0.580025
$45$	−3.34403	−0.498499
$46$	3.05923	0.451058
$47$	7.33669	1.07017	0.535083	−	0.844800i	$-0.320280\pi$
$47$	7.33669	1.07017	0.535083	+	0.844800i	$0.320280\pi$
$48$	0.823799	0.118905
$49$	1.00000	0.142857
$50$	−4.73579	−0.669742
$51$	−1.11556	−0.156210
$52$	−4.11933	−0.571248
$53$	9.56467	1.31381	0.656904	−	0.753974i	$-0.271866\pi$
$53$	9.56467	1.31381	0.656904	+	0.753974i	$0.271866\pi$
$54$	−0.765993	−0.104238
$55$	9.10380	1.22756
$56$	2.61453	0.349381
$57$	−1.90424	−0.252223
$58$	3.99321	0.524334
$59$	−1.70771	−0.222325	−0.111162	−	0.993802i	$-0.535457\pi$
$59$	−1.70771	−0.222325	−0.111162	+	0.993802i	$0.535457\pi$
$60$	4.72597	0.610120
$61$	6.64870	0.851279	0.425640	−	0.904893i	$-0.360049\pi$
$61$	6.64870	0.851279	0.425640	+	0.904893i	$0.360049\pi$
$62$	−3.51870	−0.446876
$63$	1.00000	0.125988
$64$	2.84118	0.355148
$65$	−9.74713	−1.20898
$66$	2.08534	0.256688
$67$	−5.44386	−0.665073	−0.332537	−	0.943090i	$-0.607904\pi$
$67$	−5.44386	−0.665073	−0.332537	+	0.943090i	$0.607904\pi$
$68$	1.57657	0.191188
$69$	−3.99381	−0.480798
$70$	2.56151	0.306158
$71$	−7.55180	−0.896234	−0.448117	−	0.893975i	$-0.647905\pi$
$71$	−7.55180	−0.896234	−0.448117	+	0.893975i	$0.647905\pi$
$72$	2.61453	0.308125
$73$	−9.55859	−1.11875	−0.559374	−	0.828915i	$-0.688959\pi$
$73$	−9.55859	−1.11875	−0.559374	+	0.828915i	$0.688959\pi$
$74$	−7.85159	−0.912728
$75$	6.18255	0.713900
$76$	2.69118	0.308699
$77$	−2.72240	−0.310247
$78$	−2.23270	−0.252804
$79$	6.37889	0.717681	0.358841	−	0.933399i	$-0.383172\pi$
$79$	6.37889	0.717681	0.358841	+	0.933399i	$0.383172\pi$
$80$	−2.75481	−0.307997
$81$	1.00000	0.111111
$82$	3.19493	0.352821
$83$	−2.15517	−0.236560	−0.118280	−	0.992980i	$-0.537738\pi$
$83$	−2.15517	−0.236560	−0.118280	+	0.992980i	$0.537738\pi$
$84$	−1.41325	−0.154199
$85$	3.73048	0.404627
$86$	1.91426	0.206420
$87$	−5.21312	−0.558905
$88$	−7.11780	−0.758760
$89$	8.52274	0.903409	0.451704	−	0.892168i	$-0.350816\pi$
$89$	8.52274	0.903409	0.451704	+	0.892168i	$0.350816\pi$
$90$	2.56151	0.270006
$91$	2.91478	0.305552
$92$	5.64427	0.588455
$93$	4.59365	0.476339
$94$	−5.61985	−0.579643
$95$	6.36784	0.653327
$96$	−5.86008	−0.598092
$97$	−3.69646	−0.375319	−0.187660	−	0.982234i	$-0.560090\pi$
$97$	−3.69646	−0.375319	−0.187660	+	0.982234i	$0.560090\pi$
$98$	−0.765993	−0.0773770
$99$	−2.72240	−0.273612
$100$	−8.73753	−0.873753
$101$	−15.8518	−1.57732	−0.788658	−	0.614832i	$-0.789224\pi$
$101$	−15.8518	−1.57732	−0.788658	+	0.614832i	$0.789224\pi$
$102$	0.854513	0.0846094
$103$	17.2065	1.69540	0.847702	−	0.530473i	$-0.177986\pi$
$103$	17.2065	1.69540	0.847702	+	0.530473i	$0.177986\pi$
$104$	7.62079	0.747279
$105$	−3.34403	−0.326344
$106$	−7.32647	−0.711610
$107$	−0.855698	−0.0827235	−0.0413617	−	0.999144i	$-0.513170\pi$
$107$	−0.855698	−0.0827235	−0.0413617	+	0.999144i	$0.513170\pi$
$108$	−1.41325	−0.135991
$109$	11.5522	1.10650	0.553250	−	0.833015i	$-0.313387\pi$
$109$	11.5522	1.10650	0.553250	+	0.833015i	$0.313387\pi$
$110$	−6.97345	−0.664892
$111$	10.2502	0.972907
$112$	0.823799	0.0778417
$113$	−1.82776	−0.171941	−0.0859706	−	0.996298i	$-0.527399\pi$
$113$	−1.82776	−0.171941	−0.0859706	+	0.996298i	$0.527399\pi$
$114$	1.45863	0.136614
$115$	13.3554	1.24540
$116$	7.36746	0.684052
$117$	2.91478	0.269472
$118$	1.30809	0.120420
$119$	−1.11556	−0.102263
$120$	−8.74307	−0.798130
$121$	−3.58852	−0.326229
$122$	−5.09286	−0.461086
$123$	−4.17097	−0.376084
$124$	−6.49200	−0.582999
$125$	−3.95450	−0.353701
$126$	−0.765993	−0.0682401
$127$	−3.43917	−0.305177	−0.152588	−	0.988290i	$-0.548761\pi$
$127$	−3.43917	−0.305177	−0.152588	+	0.988290i	$0.548761\pi$
$128$	9.54384	0.843564
$129$	−2.49906	−0.220030
$130$	7.46623	0.654832
$131$	−1.81779	−0.158821	−0.0794106	−	0.996842i	$-0.525304\pi$
$131$	−1.81779	−0.158821	−0.0794106	+	0.996842i	$0.525304\pi$
$132$	3.84745	0.334877
$133$	−1.90424	−0.165119
$134$	4.16996	0.360229
$135$	−3.34403	−0.287809
$136$	−2.91667	−0.250102
$137$	10.0217	0.856208	0.428104	−	0.903729i	$-0.359182\pi$
$137$	10.0217	0.856208	0.428104	+	0.903729i	$0.359182\pi$
$138$	3.05923	0.260419
$139$	−14.7821	−1.25380	−0.626901	−	0.779099i	$-0.715677\pi$
$139$	−14.7821	−1.25380	−0.626901	+	0.779099i	$0.715677\pi$
$140$	4.72597	0.399417
$141$	7.33669	0.617860
$142$	5.78462	0.485435
$143$	−7.93521	−0.663576
$144$	0.823799	0.0686499
$145$	17.4328	1.44772
$146$	7.32182	0.605958
$147$	1.00000	0.0824786
$148$	−14.4862	−1.19075
$149$	15.0025	1.22906	0.614528	−	0.788895i	$-0.289347\pi$
$149$	15.0025	1.22906	0.614528	+	0.788895i	$0.289347\pi$
$150$	−4.73579	−0.386676
$151$	−8.83358	−0.718867	−0.359433	−	0.933171i	$-0.617030\pi$
$151$	−8.83358	−0.718867	−0.359433	+	0.933171i	$0.617030\pi$
$152$	−4.97869	−0.403825
$153$	−1.11556	−0.0901879
$154$	2.08534	0.168042
$155$	−15.3613	−1.23385
$156$	−4.11933	−0.329810
$157$	−4.62020	−0.368732	−0.184366	−	0.982858i	$-0.559023\pi$
$157$	−4.62020	−0.368732	−0.184366	+	0.982858i	$0.559023\pi$
$158$	−4.88619	−0.388724
$159$	9.56467	0.758528
$160$	19.5963	1.54922
$161$	−3.99381	−0.314756
$162$	−0.765993	−0.0601821
$163$	2.59561	0.203304	0.101652	−	0.994820i	$-0.467587\pi$
$163$	2.59561	0.203304	0.101652	+	0.994820i	$0.467587\pi$
$164$	5.89464	0.460294
$165$	9.10380	0.708730
$166$	1.65084	0.128130
$167$	5.68511	0.439927	0.219964	−	0.975508i	$-0.429406\pi$
$167$	5.68511	0.439927	0.219964	+	0.975508i	$0.429406\pi$
$168$	2.61453	0.201715
$169$	−4.50404	−0.346465
$170$	−2.85752	−0.219162
$171$	−1.90424	−0.145621
$172$	3.53181	0.269298
$173$	−5.62630	−0.427759	−0.213880	−	0.976860i	$-0.568610\pi$
$173$	−5.62630	−0.427759	−0.213880	+	0.976860i	$0.568610\pi$
$174$	3.99321	0.302725
$175$	6.18255	0.467357
$176$	−2.24271	−0.169051
$177$	−1.70771	−0.128359
$178$	−6.52836	−0.489321
$179$	−13.6527	−1.02045	−0.510227	−	0.860040i	$-0.670439\pi$
$179$	−13.6527	−1.02045	−0.510227	+	0.860040i	$0.670439\pi$
$180$	4.72597	0.352253
$181$	−4.79393	−0.356330	−0.178165	−	0.984001i	$-0.557016\pi$
$181$	−4.79393	−0.356330	−0.178165	+	0.984001i	$0.557016\pi$
$182$	−2.23270	−0.165499
$183$	6.64870	0.491486
$184$	−10.4419	−0.769789
$185$	−34.2770	−2.52010
$186$	−3.51870	−0.258004
$187$	3.03701	0.222088
$188$	−10.3686	−0.756208
$189$	1.00000	0.0727393
$190$	−4.87772	−0.353867
$191$	26.0660	1.88607	0.943035	−	0.332694i	$-0.107958\pi$
$191$	26.0660	1.88607	0.943035	+	0.332694i	$0.107958\pi$
$192$	2.84118	0.205045
$193$	−7.73723	−0.556938	−0.278469	−	0.960445i	$-0.589827\pi$
$193$	−7.73723	−0.556938	−0.278469	+	0.960445i	$0.589827\pi$
$194$	2.83147	0.203287
$195$	−9.74713	−0.698006
$196$	−1.41325	−0.100947
$197$	6.86453	0.489078	0.244539	−	0.969639i	$-0.421363\pi$
$197$	6.86453	0.489078	0.244539	+	0.969639i	$0.421363\pi$
$198$	2.08534	0.148199
$199$	6.00494	0.425679	0.212840	−	0.977087i	$-0.431729\pi$
$199$	6.00494	0.425679	0.212840	+	0.977087i	$0.431729\pi$
$200$	16.1645	1.14300
$201$	−5.44386	−0.383980
$202$	12.1424	0.854336
$203$	−5.21312	−0.365889
$204$	1.57657	0.110382
$205$	13.9479	0.974160
$206$	−13.1800	−0.918296
$207$	−3.99381	−0.277589
$208$	2.40119	0.166493
$209$	5.18411	0.358592
$210$	2.56151	0.176761
$211$	11.2540	0.774755	0.387378	−	0.921921i	$-0.373381\pi$
$211$	11.2540	0.774755	0.387378	+	0.921921i	$0.373381\pi$
$212$	−13.5173	−0.928373
$213$	−7.55180	−0.517441
$214$	0.655459	0.0448062
$215$	8.35695	0.569939
$216$	2.61453	0.177896
$217$	4.59365	0.311837
$218$	−8.84890	−0.599323
$219$	−9.55859	−0.645910
$220$	−12.8660	−0.867425
$221$	−3.25162	−0.218728
$222$	−7.85159	−0.526964
$223$	12.0377	0.806101	0.403051	−	0.915178i	$-0.367950\pi$
$223$	12.0377	0.806101	0.403051	+	0.915178i	$0.367950\pi$
$224$	−5.86008	−0.391543
$225$	6.18255	0.412170
$226$	1.40005	0.0931300
$227$	−18.9482	−1.25763	−0.628817	−	0.777553i	$-0.716461\pi$
$227$	−18.9482	−1.25763	−0.628817	+	0.777553i	$0.716461\pi$
$228$	2.69118	0.178228
$229$	−12.0721	−0.797745	−0.398873	−	0.917006i	$-0.630598\pi$
$229$	−12.0721	−0.797745	−0.398873	+	0.917006i	$0.630598\pi$
$230$	−10.2302	−0.674557
$231$	−2.72240	−0.179121
$232$	−13.6298	−0.894843
$233$	0.809941	0.0530610	0.0265305	−	0.999648i	$-0.491554\pi$
$233$	0.809941	0.0530610	0.0265305	+	0.999648i	$0.491554\pi$
$234$	−2.23270	−0.145956
$235$	−24.5341	−1.60043
$236$	2.41343	0.157101
$237$	6.37889	0.414353
$238$	0.854513	0.0553899
$239$	−23.8789	−1.54460	−0.772299	−	0.635259i	$-0.780893\pi$
$239$	−23.8789	−1.54460	−0.772299	+	0.635259i	$0.780893\pi$
$240$	−2.75481	−0.177822
$241$	−3.93525	−0.253492	−0.126746	−	0.991935i	$-0.540453\pi$
$241$	−3.93525	−0.253492	−0.126746	+	0.991935i	$0.540453\pi$
$242$	2.74878	0.176699
$243$	1.00000	0.0641500
$244$	−9.39631	−0.601537
$245$	−3.34403	−0.213642
$246$	3.19493	0.203701
$247$	−5.55045	−0.353166
$248$	12.0102	0.762651
$249$	−2.15517	−0.136578
$250$	3.02912	0.191578
$251$	0.104105	0.00657107	0.00328553	−	0.999995i	$-0.498954\pi$
$251$	0.104105	0.00657107	0.00328553	+	0.999995i	$0.498954\pi$
$252$	−1.41325	−0.0890267
$253$	10.8727	0.683564
$254$	2.63438	0.165296
$255$	3.73048	0.233612
$256$	−12.9929	−0.812055
$257$	25.9757	1.62032	0.810161	−	0.586207i	$-0.199380\pi$
$257$	25.9757	1.62032	0.810161	+	0.586207i	$0.199380\pi$
$258$	1.91426	0.119177
$259$	10.2502	0.636917
$260$	13.7752	0.854300
$261$	−5.21312	−0.322684
$262$	1.39242	0.0860237
$263$	27.2621	1.68105	0.840527	−	0.541770i	$-0.182246\pi$
$263$	27.2621	1.68105	0.840527	+	0.541770i	$0.182246\pi$
$264$	−7.11780	−0.438070
$265$	−31.9846	−1.96480
$266$	1.45863	0.0894346
$267$	8.52274	0.521583
$268$	7.69356	0.469959
$269$	−2.56855	−0.156607	−0.0783037	−	0.996930i	$-0.524950\pi$
$269$	−2.56855	−0.156607	−0.0783037	+	0.996930i	$0.524950\pi$
$270$	2.56151	0.155888
$271$	11.4403	0.694950	0.347475	−	0.937689i	$-0.387039\pi$
$271$	11.4403	0.694950	0.347475	+	0.937689i	$0.387039\pi$
$272$	−0.918999	−0.0557225
$273$	2.91478	0.176411
$274$	−7.67652	−0.463756
$275$	−16.8314	−1.01497
$276$	5.64427	0.339745
$277$	−3.06005	−0.183861	−0.0919304	−	0.995765i	$-0.529304\pi$
$277$	−3.06005	−0.183861	−0.0919304	+	0.995765i	$0.529304\pi$
$278$	11.3230	0.679108
$279$	4.59365	0.275015
$280$	−8.74307	−0.522498
$281$	4.30953	0.257085	0.128542	−	0.991704i	$-0.458970\pi$
$281$	4.30953	0.257085	0.128542	+	0.991704i	$0.458970\pi$
$282$	−5.61985	−0.334657
$283$	−8.72909	−0.518890	−0.259445	−	0.965758i	$-0.583540\pi$
$283$	−8.72909	−0.518890	−0.259445	+	0.965758i	$0.583540\pi$
$284$	10.6726	0.633303
$285$	6.36784	0.377198
$286$	6.07832	0.359418
$287$	−4.17097	−0.246204
$288$	−5.86008	−0.345309
$289$	−15.7555	−0.926795
$290$	−13.3534	−0.784140
$291$	−3.69646	−0.216691
$292$	13.5087	0.790539
$293$	−25.8810	−1.51198	−0.755992	−	0.654581i	$-0.772845\pi$
$293$	−25.8810	−1.51198	−0.755992	+	0.654581i	$0.772845\pi$
$294$	−0.765993	−0.0446736
$295$	5.71064	0.332486
$296$	26.7995	1.55769
$297$	−2.72240	−0.157970
$298$	−11.4918	−0.665704
$299$	−11.6411	−0.673221
$300$	−8.73753	−0.504461
$301$	−2.49906	−0.144044
$302$	6.76646	0.389366
$303$	−15.8518	−0.910664
$304$	−1.56871	−0.0899717
$305$	−22.2335	−1.27309
$306$	0.854513	0.0488493
$307$	18.3792	1.04895	0.524477	−	0.851424i	$-0.324261\pi$
$307$	18.3792	1.04895	0.524477	+	0.851424i	$0.324261\pi$
$308$	3.84745	0.219229
$309$	17.2065	0.978842
$310$	11.7667	0.668302
$311$	−14.3119	−0.811553	−0.405777	−	0.913972i	$-0.632999\pi$
$311$	−14.3119	−0.811553	−0.405777	+	0.913972i	$0.632999\pi$
$312$	7.62079	0.431442
$313$	−3.05933	−0.172923	−0.0864617	−	0.996255i	$-0.527556\pi$
$313$	−3.05933	−0.172923	−0.0864617	+	0.996255i	$0.527556\pi$
$314$	3.53904	0.199720
$315$	−3.34403	−0.188415
$316$	−9.01500	−0.507133
$317$	−17.9555	−1.00848	−0.504241	−	0.863563i	$-0.668228\pi$
$317$	−17.9555	−1.00848	−0.504241	+	0.863563i	$0.668228\pi$
$318$	−7.32647	−0.410848
$319$	14.1922	0.794611
$320$	−9.50101	−0.531123
$321$	−0.855698	−0.0477604
$322$	3.05923	0.170484
$323$	2.12430	0.118199
$324$	−1.41325	−0.0785142
$325$	18.0208	0.999614
$326$	−1.98822	−0.110117
$327$	11.5522	0.638838
$328$	−10.9051	−0.602134
$329$	7.33669	0.404485
$330$	−6.97345	−0.383876
$331$	0.465717	0.0255981	0.0127991	−	0.999918i	$-0.495926\pi$
$331$	0.465717	0.0255981	0.0127991	+	0.999918i	$0.495926\pi$
$332$	3.04580	0.167160
$333$	10.2502	0.561708
$334$	−4.35476	−0.238282
$335$	18.2044	0.994615
$336$	0.823799	0.0449419
$337$	2.74717	0.149648	0.0748238	−	0.997197i	$-0.476161\pi$
$337$	2.74717	0.149648	0.0748238	+	0.997197i	$0.476161\pi$
$338$	3.45006	0.187659
$339$	−1.82776	−0.0992703
$340$	−5.27212	−0.285921
$341$	−12.5058	−0.677225
$342$	1.45863	0.0788739
$343$	1.00000	0.0539949
$344$	−6.53387	−0.352283
$345$	13.3554	0.719032
$346$	4.30970	0.231691
$347$	5.91613	0.317594	0.158797	−	0.987311i	$-0.449238\pi$
$347$	5.91613	0.317594	0.158797	+	0.987311i	$0.449238\pi$
$348$	7.36746	0.394937
$349$	−15.0601	−0.806148	−0.403074	−	0.915167i	$-0.632058\pi$
$349$	−15.0601	−0.806148	−0.403074	+	0.915167i	$0.632058\pi$
$350$	−4.73579	−0.253139
$351$	2.91478	0.155580
$352$	15.9535	0.850325
$353$	−17.6636	−0.940137	−0.470068	−	0.882630i	$-0.655771\pi$
$353$	−17.6636	−0.940137	−0.470068	+	0.882630i	$0.655771\pi$
$354$	1.30809	0.0695244
$355$	25.2535	1.34031
$356$	−12.0448	−0.638373
$357$	−1.11556	−0.0590418
$358$	10.4579	0.552718
$359$	−14.3528	−0.757513	−0.378756	−	0.925496i	$-0.623648\pi$
$359$	−14.3528	−0.757513	−0.378756	+	0.925496i	$0.623648\pi$
$360$	−8.74307	−0.460800
$361$	−15.3739	−0.809151
$362$	3.67212	0.193002
$363$	−3.58852	−0.188349
$364$	−4.11933	−0.215912
$365$	31.9643	1.67309
$366$	−5.09286	−0.266208
$367$	−20.2214	−1.05555	−0.527773	−	0.849385i	$-0.676973\pi$
$367$	−20.2214	−1.05555	−0.527773	+	0.849385i	$0.676973\pi$
$368$	−3.29009	−0.171508
$369$	−4.17097	−0.217132
$370$	26.2560	1.36498
$371$	9.56467	0.496573
$372$	−6.49200	−0.336595
$373$	−25.3811	−1.31418	−0.657092	−	0.753810i	$-0.728214\pi$
$373$	−25.3811	−1.31418	−0.657092	+	0.753810i	$0.728214\pi$
$374$	−2.32633	−0.120292
$375$	−3.95450	−0.204210
$376$	19.1820	0.989235
$377$	−15.1951	−0.782588
$378$	−0.765993	−0.0393984
$379$	20.7301	1.06483	0.532417	−	0.846482i	$-0.321284\pi$
$379$	20.7301	1.06483	0.532417	+	0.846482i	$0.321284\pi$
$380$	−8.99938	−0.461659
$381$	−3.43917	−0.176194
$382$	−19.9664	−1.02157
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	9.54384	0.487032
$385$	9.10380	0.463973
$386$	5.92666	0.301659
$387$	−2.49906	−0.127035
$388$	5.22405	0.265211
$389$	−37.9472	−1.92400	−0.961999	−	0.273052i	$-0.911967\pi$
$389$	−37.9472	−1.92400	−0.961999	+	0.273052i	$0.911967\pi$
$390$	7.46623	0.378067
$391$	4.45534	0.225316
$392$	2.61453	0.132054
$393$	−1.81779	−0.0916955
$394$	−5.25818	−0.264903
$395$	−21.3312	−1.07329
$396$	3.84745	0.193342
$397$	−29.1285	−1.46192	−0.730958	−	0.682422i	$-0.760927\pi$
$397$	−29.1285	−1.46192	−0.730958	+	0.682422i	$0.760927\pi$
$398$	−4.59975	−0.230564
$399$	−1.90424	−0.0953312
$400$	5.09318	0.254659
$401$	−31.3392	−1.56500	−0.782502	−	0.622649i	$-0.786057\pi$
$401$	−31.3392	−1.56500	−0.782502	+	0.622649i	$0.786057\pi$
$402$	4.16996	0.207979
$403$	13.3895	0.666978
$404$	22.4027	1.11458
$405$	−3.34403	−0.166166
$406$	3.99321	0.198180
$407$	−27.9052	−1.38321
$408$	−2.91667	−0.144397
$409$	−33.5866	−1.66075	−0.830375	−	0.557206i	$-0.811874\pi$
$409$	−33.5866	−1.66075	−0.830375	+	0.557206i	$0.811874\pi$
$410$	−10.6840	−0.527643
$411$	10.0217	0.494332
$412$	−24.3171	−1.19802
$413$	−1.70771	−0.0840309
$414$	3.05923	0.150353
$415$	7.20695	0.353775
$416$	−17.0809	−0.837458
$417$	−14.7821	−0.723883
$418$	−3.97099	−0.194227
$419$	−1.41080	−0.0689219	−0.0344609	−	0.999406i	$-0.510971\pi$
$419$	−1.41080	−0.0689219	−0.0344609	+	0.999406i	$0.510971\pi$
$420$	4.72597	0.230604
$421$	−13.0345	−0.635264	−0.317632	−	0.948214i	$-0.602888\pi$
$421$	−13.0345	−0.635264	−0.317632	+	0.948214i	$0.602888\pi$
$422$	−8.62046	−0.419637
$423$	7.33669	0.356722
$424$	25.0071	1.21445
$425$	−6.89703	−0.334555
$426$	5.78462	0.280266
$427$	6.64870	0.321753
$428$	1.20932	0.0584547
$429$	−7.93521	−0.383116
$430$	−6.40136	−0.308701
$431$	−40.7525	−1.96298	−0.981488	−	0.191521i	$-0.938658\pi$
$431$	−40.7525	−1.96298	−0.981488	+	0.191521i	$0.938658\pi$
$432$	0.823799	0.0396350
$433$	−4.78511	−0.229958	−0.114979	−	0.993368i	$-0.536680\pi$
$433$	−4.78511	−0.229958	−0.114979	+	0.993368i	$0.536680\pi$
$434$	−3.51870	−0.168903
$435$	17.4328	0.835841
$436$	−16.3262	−0.781883
$437$	7.60517	0.363804
$438$	7.32182	0.349850
$439$	−26.7334	−1.27591	−0.637957	−	0.770072i	$-0.720220\pi$
$439$	−26.7334	−1.27591	−0.637957	+	0.770072i	$0.720220\pi$
$440$	23.8022	1.13472
$441$	1.00000	0.0476190
$442$	2.49072	0.118471
$443$	−7.43419	−0.353209	−0.176605	−	0.984282i	$-0.556511\pi$
$443$	−7.43419	−0.353209	−0.176605	+	0.984282i	$0.556511\pi$
$444$	−14.4862	−0.687482
$445$	−28.5003	−1.35104
$446$	−9.22076	−0.436616
$447$	15.0025	0.709595
$448$	2.84118	0.134233
$449$	−13.9203	−0.656939	−0.328470	−	0.944515i	$-0.606533\pi$
$449$	−13.9203	−0.656939	−0.328470	+	0.944515i	$0.606533\pi$
$450$	−4.73579	−0.223247
$451$	11.3551	0.534689
$452$	2.58309	0.121498
$453$	−8.83358	−0.415038
$454$	14.5142	0.681184
$455$	−9.74713	−0.456953
$456$	−4.97869	−0.233149
$457$	−14.8036	−0.692485	−0.346242	−	0.938145i	$-0.612542\pi$
$457$	−14.8036	−0.692485	−0.346242	+	0.938145i	$0.612542\pi$
$458$	9.24712	0.432090
$459$	−1.11556	−0.0520700
$460$	−18.8746	−0.880033
$461$	−25.4815	−1.18679	−0.593395	−	0.804911i	$-0.702213\pi$
$461$	−25.4815	−1.18679	−0.593395	+	0.804911i	$0.702213\pi$
$462$	2.08534	0.0970189
$463$	−33.0485	−1.53589	−0.767946	−	0.640514i	$-0.778721\pi$
$463$	−33.0485	−1.53589	−0.767946	+	0.640514i	$0.778721\pi$
$464$	−4.29456	−0.199370
$465$	−15.3613	−0.712364
$466$	−0.620409	−0.0287399
$467$	−16.9693	−0.785246	−0.392623	−	0.919700i	$-0.628432\pi$
$467$	−16.9693	−0.785246	−0.392623	+	0.919700i	$0.628432\pi$
$468$	−4.11933	−0.190416
$469$	−5.44386	−0.251374
$470$	18.7930	0.866855
$471$	−4.62020	−0.212888
$472$	−4.46486	−0.205512
$473$	6.80346	0.312823
$474$	−4.88619	−0.224430
$475$	−11.7731	−0.540185
$476$	1.57657	0.0722622
$477$	9.56467	0.437936
$478$	18.2911	0.836615
$479$	−7.07547	−0.323286	−0.161643	−	0.986849i	$-0.551679\pi$
$479$	−7.07547	−0.323286	−0.161643	+	0.986849i	$0.551679\pi$
$480$	19.5963	0.894445
$481$	29.8771	1.36228
$482$	3.01437	0.137301
$483$	−3.99381	−0.181724
$484$	5.07150	0.230523
$485$	12.3611	0.561289
$486$	−0.765993	−0.0347461
$487$	35.0243	1.58710	0.793552	−	0.608502i	$-0.208229\pi$
$487$	35.0243	1.58710	0.793552	+	0.608502i	$0.208229\pi$
$488$	17.3832	0.786902
$489$	2.59561	0.117377
$490$	2.56151	0.115717
$491$	−4.43307	−0.200061	−0.100031	−	0.994984i	$-0.531894\pi$
$491$	−4.43307	−0.200061	−0.100031	+	0.994984i	$0.531894\pi$
$492$	5.89464	0.265751
$493$	5.81556	0.261920
$494$	4.25160	0.191289
$495$	9.10380	0.409186
$496$	3.78424	0.169918
$497$	−7.55180	−0.338744
$498$	1.65084	0.0739760
$499$	−36.9526	−1.65422	−0.827112	−	0.562037i	$-0.810018\pi$
$499$	−36.9526	−1.65422	−0.827112	+	0.562037i	$0.810018\pi$
$500$	5.58872	0.249935
$501$	5.68511	0.253992
$502$	−0.0797439	−0.00355915
$503$	2.34185	0.104418	0.0522089	−	0.998636i	$-0.483374\pi$
$503$	2.34185	0.104418	0.0522089	+	0.998636i	$0.483374\pi$
$504$	2.61453	0.116460
$505$	53.0091	2.35887
$506$	−8.32845	−0.370245
$507$	−4.50404	−0.200031
$508$	4.86042	0.215646
$509$	−16.6895	−0.739750	−0.369875	−	0.929082i	$-0.620599\pi$
$509$	−16.6895	−0.739750	−0.369875	+	0.929082i	$0.620599\pi$
$510$	−2.85752	−0.126533
$511$	−9.55859	−0.422847
$512$	−9.13522	−0.403724
$513$	−1.90424	−0.0840743
$514$	−19.8972	−0.877629
$515$	−57.5390	−2.53547
$516$	3.53181	0.155479
$517$	−19.9734	−0.878430
$518$	−7.85159	−0.344979
$519$	−5.62630	−0.246967
$520$	−25.4842	−1.11755
$521$	−20.1068	−0.880896	−0.440448	−	0.897778i	$-0.645180\pi$
$521$	−20.1068	−0.880896	−0.440448	+	0.897778i	$0.645180\pi$
$522$	3.99321	0.174778
$523$	−18.3024	−0.800310	−0.400155	−	0.916447i	$-0.631044\pi$
$523$	−18.3024	−0.800310	−0.400155	+	0.916447i	$0.631044\pi$
$524$	2.56900	0.112227
$525$	6.18255	0.269829
$526$	−20.8826	−0.910524
$527$	−5.12451	−0.223227
$528$	−2.24271	−0.0976015
$529$	−7.04951	−0.306500
$530$	24.5000	1.06421
$531$	−1.70771	−0.0741083
$532$	2.69118	0.116677
$533$	−12.1575	−0.526598
$534$	−6.52836	−0.282510
$535$	2.86148	0.123713
$536$	−14.2331	−0.614777
$537$	−13.6527	−0.589159
$538$	1.96749	0.0848247
$539$	−2.72240	−0.117262
$540$	4.72597	0.203373
$541$	37.9543	1.63178	0.815892	−	0.578204i	$-0.196246\pi$
$541$	37.9543	1.63178	0.815892	+	0.578204i	$0.196246\pi$
$542$	−8.76321	−0.376412
$543$	−4.79393	−0.205727
$544$	6.53729	0.280284
$545$	−38.6309	−1.65477
$546$	−2.23270	−0.0955509
$547$	−29.6532	−1.26788	−0.633940	−	0.773382i	$-0.718563\pi$
$547$	−29.6532	−1.26788	−0.633940	+	0.773382i	$0.718563\pi$
$548$	−14.1632	−0.605020
$549$	6.64870	0.283760
$550$	12.8927	0.549748
$551$	9.92703	0.422906
$552$	−10.4419	−0.444438
$553$	6.37889	0.271258
$554$	2.34398	0.0995861
$555$	−34.2770	−1.45498
$556$	20.8909	0.885971
$557$	14.6256	0.619707	0.309853	−	0.950784i	$-0.399720\pi$
$557$	14.6256	0.619707	0.309853	+	0.950784i	$0.399720\pi$
$558$	−3.51870	−0.148959
$559$	−7.28423	−0.308090
$560$	−2.75481	−0.116412
$561$	3.03701	0.128223
$562$	−3.30107	−0.139247
$563$	7.20376	0.303602	0.151801	−	0.988411i	$-0.451493\pi$
$563$	7.20376	0.303602	0.151801	+	0.988411i	$0.451493\pi$
$564$	−10.3686	−0.436597
$565$	6.11209	0.257137
$566$	6.68642	0.281051
$567$	1.00000	0.0419961
$568$	−19.7444	−0.828457
$569$	37.2261	1.56060	0.780300	−	0.625405i	$-0.215066\pi$
$569$	37.2261	1.56060	0.780300	+	0.625405i	$0.215066\pi$
$570$	−4.87772	−0.204305
$571$	29.1772	1.22103	0.610514	−	0.792006i	$-0.290963\pi$
$571$	29.1772	1.22103	0.610514	+	0.792006i	$0.290963\pi$
$572$	11.2145	0.468901
$573$	26.0660	1.08892
$574$	3.19493	0.133354
$575$	−24.6919	−1.02972
$576$	2.84118	0.118383
$577$	25.3983	1.05735	0.528673	−	0.848826i	$-0.322690\pi$
$577$	25.3983	1.05735	0.528673	+	0.848826i	$0.322690\pi$
$578$	12.0686	0.501988
$579$	−7.73723	−0.321548
$580$	−24.6370	−1.02300
$581$	−2.15517	−0.0894114
$582$	2.83147	0.117368
$583$	−26.0389	−1.07842
$584$	−24.9912	−1.03414
$585$	−9.74713	−0.402994
$586$	19.8247	0.818949
$587$	1.32661	0.0547550	0.0273775	−	0.999625i	$-0.491284\pi$
$587$	1.32661	0.0547550	0.0273775	+	0.999625i	$0.491284\pi$
$588$	−1.41325	−0.0582816
$589$	−8.74741	−0.360431
$590$	−4.37431	−0.180087
$591$	6.86453	0.282369
$592$	8.44411	0.347051
$593$	32.9756	1.35415	0.677073	−	0.735916i	$-0.263248\pi$
$593$	32.9756	1.35415	0.677073	+	0.735916i	$0.263248\pi$
$594$	2.08534	0.0855626
$595$	3.73048	0.152935
$596$	−21.2024	−0.868484
$597$	6.00494	0.245766
$598$	8.91698	0.364643
$599$	−34.8093	−1.42227	−0.711135	−	0.703055i	$-0.751819\pi$
$599$	−34.8093	−1.42227	−0.711135	+	0.703055i	$0.751819\pi$
$600$	16.1645	0.659912
$601$	22.3549	0.911875	0.455937	−	0.890012i	$-0.349304\pi$
$601$	22.3549	0.911875	0.455937	+	0.890012i	$0.349304\pi$
$602$	1.91426	0.0780196
$603$	−5.44386	−0.221691
$604$	12.4841	0.507971
$605$	12.0001	0.487875
$606$	12.1424	0.493251
$607$	31.5739	1.28155	0.640774	−	0.767730i	$-0.278614\pi$
$607$	31.5739	1.28155	0.640774	+	0.767730i	$0.278614\pi$
$608$	11.1590	0.452557
$609$	−5.21312	−0.211246
$610$	17.0307	0.689552
$611$	21.3848	0.865138
$612$	1.57657	0.0637292
$613$	−6.43451	−0.259887	−0.129944	−	0.991521i	$-0.541480\pi$
$613$	−6.43451	−0.259887	−0.129944	+	0.991521i	$0.541480\pi$
$614$	−14.0783	−0.568155
$615$	13.9479	0.562432
$616$	−7.11780	−0.286784
$617$	−10.6606	−0.429178	−0.214589	−	0.976704i	$-0.568841\pi$
$617$	−10.6606	−0.429178	−0.214589	+	0.976704i	$0.568841\pi$
$618$	−13.1800	−0.530179
$619$	17.9684	0.722213	0.361106	−	0.932525i	$-0.382399\pi$
$619$	17.9684	0.722213	0.361106	+	0.932525i	$0.382399\pi$
$620$	21.7095	0.871873
$621$	−3.99381	−0.160266
$622$	10.9628	0.439569
$623$	8.52274	0.341456
$624$	2.40119	0.0961247
$625$	−17.6888	−0.707552
$626$	2.34342	0.0936620
$627$	5.18411	0.207033
$628$	6.52952	0.260556
$629$	−11.4347	−0.455933
$630$	2.56151	0.102053
$631$	−16.7146	−0.665399	−0.332700	−	0.943033i	$-0.607960\pi$
$631$	−16.7146	−0.665399	−0.332700	+	0.943033i	$0.607960\pi$
$632$	16.6778	0.663407
$633$	11.2540	0.447305
$634$	13.7538	0.546233
$635$	11.5007	0.456391
$636$	−13.5173	−0.535996
$637$	2.91478	0.115488
$638$	−10.8711	−0.430392
$639$	−7.55180	−0.298745
$640$	−31.9149	−1.26155
$641$	−26.0301	−1.02813	−0.514063	−	0.857752i	$-0.671860\pi$
$641$	−26.0301	−1.02813	−0.514063	+	0.857752i	$0.671860\pi$
$642$	0.655459	0.0258689
$643$	23.0988	0.910928	0.455464	−	0.890254i	$-0.349473\pi$
$643$	23.0988	0.910928	0.455464	+	0.890254i	$0.349473\pi$
$644$	5.64427	0.222415
$645$	8.35695	0.329055
$646$	−1.62720	−0.0640213
$647$	29.0549	1.14227	0.571133	−	0.820857i	$-0.306504\pi$
$647$	29.0549	1.14227	0.571133	+	0.820857i	$0.306504\pi$
$648$	2.61453	0.102708
$649$	4.64907	0.182492
$650$	−13.8038	−0.541430
$651$	4.59365	0.180039
$652$	−3.66825	−0.143660
$653$	0.457687	0.0179107	0.00895533	−	0.999960i	$-0.497149\pi$
$653$	0.457687	0.0179107	0.00895533	+	0.999960i	$0.497149\pi$
$654$	−8.84890	−0.346019
$655$	6.07876	0.237517
$656$	−3.43604	−0.134155
$657$	−9.55859	−0.372916
$658$	−5.61985	−0.219085
$659$	0.872249	0.0339780	0.0169890	−	0.999856i	$-0.494592\pi$
$659$	0.872249	0.0339780	0.0169890	+	0.999856i	$0.494592\pi$
$660$	−12.8660	−0.500808
$661$	−24.0266	−0.934526	−0.467263	−	0.884118i	$-0.654760\pi$
$661$	−24.0266	−0.934526	−0.467263	+	0.884118i	$0.654760\pi$
$662$	−0.356736	−0.0138649
$663$	−3.25162	−0.126283
$664$	−5.63475	−0.218671
$665$	6.36784	0.246934
$666$	−7.85159	−0.304243
$667$	20.8202	0.806161
$668$	−8.03452	−0.310865
$669$	12.0377	0.465403
$670$	−13.9445	−0.538722
$671$	−18.1004	−0.698760
$672$	−5.86008	−0.226058
$673$	−5.07236	−0.195525	−0.0977626	−	0.995210i	$-0.531169\pi$
$673$	−5.07236	−0.195525	−0.0977626	+	0.995210i	$0.531169\pi$
$674$	−2.10431	−0.0810550
$675$	6.18255	0.237967
$676$	6.36536	0.244821
$677$	7.18299	0.276065	0.138032	−	0.990428i	$-0.455922\pi$
$677$	7.18299	0.276065	0.138032	+	0.990428i	$0.455922\pi$
$678$	1.40005	0.0537686
$679$	−3.69646	−0.141857
$680$	9.75344	0.374027
$681$	−18.9482	−0.726096
$682$	9.57933	0.366811
$683$	7.55259	0.288992	0.144496	−	0.989505i	$-0.453844\pi$
$683$	7.55259	0.288992	0.144496	+	0.989505i	$0.453844\pi$
$684$	2.69118	0.102900
$685$	−33.5128	−1.28046
$686$	−0.765993	−0.0292457
$687$	−12.0721	−0.460578
$688$	−2.05873	−0.0784882
$689$	27.8789	1.06210
$690$	−10.2302	−0.389455
$691$	−13.6534	−0.519401	−0.259700	−	0.965689i	$-0.583624\pi$
$691$	−13.6534	−0.519401	−0.259700	+	0.965689i	$0.583624\pi$
$692$	7.95139	0.302267
$693$	−2.72240	−0.103416
$694$	−4.53171	−0.172022
$695$	49.4319	1.87506
$696$	−13.6298	−0.516638
$697$	4.65298	0.176244
$698$	11.5359	0.436641
$699$	0.809941	0.0306348
$700$	−8.73753	−0.330247
$701$	−22.5684	−0.852396	−0.426198	−	0.904630i	$-0.640147\pi$
$701$	−22.5684	−0.852396	−0.426198	+	0.904630i	$0.640147\pi$
$702$	−2.23270	−0.0842680
$703$	−19.5189	−0.736168
$704$	−7.73485	−0.291518
$705$	−24.5341	−0.924008
$706$	13.5302	0.509215
$707$	−15.8518	−0.596170
$708$	2.41343	0.0907022
$709$	−7.92761	−0.297728	−0.148864	−	0.988858i	$-0.547562\pi$
$709$	−7.92761	−0.297728	−0.148864	+	0.988858i	$0.547562\pi$
$710$	−19.3440	−0.725966
$711$	6.37889	0.239227
$712$	22.2829	0.835089
$713$	−18.3462	−0.687069
$714$	0.854513	0.0319794
$715$	26.5356	0.992376
$716$	19.2948	0.721081
$717$	−23.8789	−0.891774
$718$	10.9942	0.410298
$719$	35.2036	1.31287	0.656436	−	0.754382i	$-0.272063\pi$
$719$	35.2036	1.31287	0.656436	+	0.754382i	$0.272063\pi$
$720$	−2.75481	−0.102666
$721$	17.2065	0.640802
$722$	11.7763	0.438268
$723$	−3.93525	−0.146353
$724$	6.77505	0.251793
$725$	−32.2304	−1.19701
$726$	2.74878	0.102017
$727$	−27.2422	−1.01036	−0.505179	−	0.863014i	$-0.668574\pi$
$727$	−27.2422	−1.01036	−0.505179	+	0.863014i	$0.668574\pi$
$728$	7.62079	0.282445
$729$	1.00000	0.0370370
$730$	−24.4844	−0.906208
$731$	2.78786	0.103113
$732$	−9.39631	−0.347298
$733$	−39.1428	−1.44577	−0.722887	−	0.690967i	$-0.757185\pi$
$733$	−39.1428	−1.44577	−0.722887	+	0.690967i	$0.757185\pi$
$734$	15.4894	0.571725
$735$	−3.34403	−0.123347
$736$	23.4040	0.862684
$737$	14.8204	0.545916
$738$	3.19493	0.117607
$739$	27.8083	1.02294	0.511472	−	0.859300i	$-0.329100\pi$
$739$	27.8083	1.02294	0.511472	+	0.859300i	$0.329100\pi$
$740$	48.4422	1.78077
$741$	−5.55045	−0.203901
$742$	−7.32647	−0.268963
$743$	−4.27593	−0.156869	−0.0784343	−	0.996919i	$-0.524992\pi$
$743$	−4.27593	−0.156869	−0.0784343	+	0.996919i	$0.524992\pi$
$744$	12.0102	0.440317
$745$	−50.1690	−1.83805
$746$	19.4418	0.711813
$747$	−2.15517	−0.0788534
$748$	−4.29207	−0.156934
$749$	−0.855698	−0.0312665
$750$	3.02912	0.110608
$751$	−18.2691	−0.666648	−0.333324	−	0.942812i	$-0.608170\pi$
$751$	−18.2691	−0.666648	−0.333324	+	0.942812i	$0.608170\pi$
$752$	6.04395	0.220400
$753$	0.104105	0.00379381
$754$	11.6393	0.423880
$755$	29.5398	1.07506
$756$	−1.41325	−0.0513996
$757$	−43.8729	−1.59459	−0.797294	−	0.603591i	$-0.793736\pi$
$757$	−43.8729	−1.59459	−0.797294	+	0.603591i	$0.793736\pi$
$758$	−15.8791	−0.576755
$759$	10.8727	0.394656
$760$	16.6489	0.603919
$761$	12.8643	0.466331	0.233165	−	0.972437i	$-0.425092\pi$
$761$	12.8643	0.466331	0.233165	+	0.972437i	$0.425092\pi$
$762$	2.63438	0.0954335
$763$	11.5522	0.418218
$764$	−36.8379	−1.33275
$765$	3.73048	0.134876
$766$	0.765993	0.0276764
$767$	−4.97760	−0.179731
$768$	−12.9929	−0.468840
$769$	25.0880	0.904697	0.452348	−	0.891841i	$-0.350586\pi$
$769$	25.0880	0.904697	0.452348	+	0.891841i	$0.350586\pi$
$770$	−6.97345	−0.251306
$771$	25.9757	0.935493
$772$	10.9347	0.393548
$773$	52.2073	1.87776	0.938882	−	0.344238i	$-0.111863\pi$
$773$	52.2073	1.87776	0.938882	+	0.344238i	$0.111863\pi$
$774$	1.91426	0.0688068
$775$	28.4005	1.02018
$776$	−9.66451	−0.346936
$777$	10.2502	0.367724
$778$	29.0673	1.04211
$779$	7.94253	0.284570
$780$	13.7752	0.493231
$781$	20.5590	0.735660
$782$	−3.41276	−0.122040
$783$	−5.21312	−0.186302
$784$	0.823799	0.0294214
$785$	15.4501	0.551438
$786$	1.39242	0.0496658
$787$	21.0615	0.750762	0.375381	−	0.926871i	$-0.377512\pi$
$787$	21.0615	0.750762	0.375381	+	0.926871i	$0.377512\pi$
$788$	−9.70134	−0.345596
$789$	27.2621	0.970557
$790$	16.3396	0.581336
$791$	−1.82776	−0.0649876
$792$	−7.11780	−0.252920
$793$	19.3795	0.688187
$794$	22.3122	0.791831
$795$	−31.9846	−1.13438
$796$	−8.48652	−0.300797
$797$	46.6748	1.65331	0.826653	−	0.562712i	$-0.190242\pi$
$797$	46.6748	1.65331	0.826653	+	0.562712i	$0.190242\pi$
$798$	1.45863	0.0516351
$799$	−8.18453	−0.289548
$800$	−36.2303	−1.28093
$801$	8.52274	0.301136
$802$	24.0056	0.847666
$803$	26.0223	0.918309
$804$	7.69356	0.271331
$805$	13.3554	0.470717
$806$	−10.2563	−0.361261
$807$	−2.56855	−0.0904173
$808$	−41.4451	−1.45803
$809$	12.9073	0.453795	0.226898	−	0.973919i	$-0.427142\pi$
$809$	12.9073	0.453795	0.226898	+	0.973919i	$0.427142\pi$
$810$	2.56151	0.0900021
$811$	33.1270	1.16325	0.581623	−	0.813458i	$-0.302418\pi$
$811$	33.1270	1.16325	0.581623	+	0.813458i	$0.302418\pi$
$812$	7.36746	0.258547
$813$	11.4403	0.401230
$814$	21.3752	0.749200
$815$	−8.67979	−0.304040
$816$	−0.918999	−0.0321714
$817$	4.75882	0.166490
$818$	25.7271	0.899526
$819$	2.91478	0.101851
$820$	−19.7119	−0.688368
$821$	30.3962	1.06083	0.530417	−	0.847737i	$-0.322036\pi$
$821$	30.3962	1.06083	0.530417	+	0.847737i	$0.322036\pi$
$822$	−7.67652	−0.267749
$823$	−41.4479	−1.44478	−0.722392	−	0.691484i	$-0.756957\pi$
$823$	−41.4479	−1.44478	−0.722392	+	0.691484i	$0.756957\pi$
$824$	44.9868	1.56719
$825$	−16.8314	−0.585994
$826$	1.30809	0.0455144
$827$	−36.2788	−1.26154	−0.630769	−	0.775971i	$-0.717261\pi$
$827$	−36.2788	−1.26154	−0.630769	+	0.775971i	$0.717261\pi$
$828$	5.64427	0.196152
$829$	23.3217	0.809996	0.404998	−	0.914318i	$-0.367272\pi$
$829$	23.3217	0.809996	0.404998	+	0.914318i	$0.367272\pi$
$830$	−5.52047	−0.191618
$831$	−3.06005	−0.106152
$832$	8.28144	0.287107
$833$	−1.11556	−0.0386520
$834$	11.3230	0.392083
$835$	−19.0112	−0.657910
$836$	−7.32647	−0.253391
$837$	4.59365	0.158780
$838$	1.08066	0.0373308
$839$	16.7491	0.578242	0.289121	−	0.957293i	$-0.406637\pi$
$839$	16.7491	0.578242	0.289121	+	0.957293i	$0.406637\pi$
$840$	−8.74307	−0.301665
$841$	−1.82341	−0.0628760
$842$	9.98436	0.344084
$843$	4.30953	0.148428
$844$	−15.9047	−0.547463
$845$	15.0617	0.518137
$846$	−5.61985	−0.193214
$847$	−3.58852	−0.123303
$848$	7.87936	0.270578
$849$	−8.72909	−0.299582
$850$	5.28307	0.181208
$851$	−40.9373	−1.40331
$852$	10.6726	0.365638
$853$	32.9290	1.12747	0.563734	−	0.825957i	$-0.309364\pi$
$853$	32.9290	1.12747	0.563734	+	0.825957i	$0.309364\pi$
$854$	−5.09286	−0.174274
$855$	6.36784	0.217776
$856$	−2.23725	−0.0764675
$857$	36.4654	1.24563	0.622817	−	0.782368i	$-0.285988\pi$
$857$	36.4654	1.24563	0.622817	+	0.782368i	$0.285988\pi$
$858$	6.07832	0.207510
$859$	0.777602	0.0265314	0.0132657	−	0.999912i	$-0.495777\pi$
$859$	0.777602	0.0265314	0.0132657	+	0.999912i	$0.495777\pi$
$860$	−11.8105	−0.402735
$861$	−4.17097	−0.142146
$862$	31.2161	1.06322
$863$	−54.6021	−1.85868	−0.929339	−	0.369228i	$-0.879622\pi$
$863$	−54.6021	−1.85868	−0.929339	+	0.369228i	$0.879622\pi$
$864$	−5.86008	−0.199364
$865$	18.8145	0.639713
$866$	3.66536	0.124554
$867$	−15.7555	−0.535086
$868$	−6.49200	−0.220353
$869$	−17.3659	−0.589098
$870$	−13.3534	−0.452724
$871$	−15.8677	−0.537655
$872$	30.2036	1.02282
$873$	−3.69646	−0.125106
$874$	−5.82550	−0.197051
$875$	−3.95450	−0.133687
$876$	13.5087	0.456418
$877$	27.5082	0.928884	0.464442	−	0.885603i	$-0.346255\pi$
$877$	27.5082	0.928884	0.464442	+	0.885603i	$0.346255\pi$
$878$	20.4776	0.691084
$879$	−25.8810	−0.872945
$880$	7.49970	0.252815
$881$	−15.1975	−0.512017	−0.256008	−	0.966675i	$-0.582407\pi$
$881$	−15.1975	−0.512017	−0.256008	+	0.966675i	$0.582407\pi$
$882$	−0.765993	−0.0257923
$883$	−26.8553	−0.903753	−0.451876	−	0.892081i	$-0.649245\pi$
$883$	−26.8553	−0.903753	−0.451876	+	0.892081i	$0.649245\pi$
$884$	4.59537	0.154559
$885$	5.71064	0.191961
$886$	5.69454	0.191312
$887$	46.7584	1.56999	0.784996	−	0.619500i	$-0.212665\pi$
$887$	46.7584	1.56999	0.784996	+	0.619500i	$0.212665\pi$
$888$	26.7995	0.899331
$889$	−3.43917	−0.115346
$890$	21.8310	0.731778
$891$	−2.72240	−0.0912039
$892$	−17.0123	−0.569613
$893$	−13.9708	−0.467515
$894$	−11.4918	−0.384344
$895$	45.6552	1.52609
$896$	9.54384	0.318837
$897$	−11.6411	−0.388684
$898$	10.6628	0.355824
$899$	−23.9472	−0.798685
$900$	−8.73753	−0.291251
$901$	−10.6700	−0.355469
$902$	−8.69789	−0.289608
$903$	−2.49906	−0.0831636
$904$	−4.77873	−0.158938
$905$	16.0311	0.532891
$906$	6.76646	0.224801
$907$	42.5169	1.41175	0.705876	−	0.708335i	$-0.250554\pi$
$907$	42.5169	1.41175	0.705876	+	0.708335i	$0.250554\pi$
$908$	26.7786	0.888679
$909$	−15.8518	−0.525772
$910$	7.46623	0.247503
$911$	9.17051	0.303833	0.151916	−	0.988393i	$-0.451456\pi$
$911$	9.17051	0.303833	0.151916	+	0.988393i	$0.451456\pi$
$912$	−1.56871	−0.0519452
$913$	5.86723	0.194177
$914$	11.3395	0.375076
$915$	−22.2335	−0.735016
$916$	17.0609	0.563709
$917$	−1.81779	−0.0600288
$918$	0.854513	0.0282031
$919$	−18.5593	−0.612215	−0.306108	−	0.951997i	$-0.599027\pi$
$919$	−18.5593	−0.612215	−0.306108	+	0.951997i	$0.599027\pi$
$920$	34.9181	1.15122
$921$	18.3792	0.605614
$922$	19.5186	0.642812
$923$	−22.0119	−0.724529
$924$	3.84745	0.126572
$925$	63.3725	2.08367
$926$	25.3149	0.831899
$927$	17.2065	0.565134
$928$	30.5493	1.00283
$929$	27.0075	0.886086	0.443043	−	0.896500i	$-0.353899\pi$
$929$	27.0075	0.886086	0.443043	+	0.896500i	$0.353899\pi$
$930$	11.7667	0.385844
$931$	−1.90424	−0.0624089
$932$	−1.14465	−0.0374943
$933$	−14.3119	−0.468551
$934$	12.9984	0.425320
$935$	−10.1559	−0.332132
$936$	7.62079	0.249093
$937$	−14.8285	−0.484427	−0.242213	−	0.970223i	$-0.577873\pi$
$937$	−14.8285	−0.484427	−0.242213	+	0.970223i	$0.577873\pi$
$938$	4.16996	0.136154
$939$	−3.05933	−0.0998374
$940$	34.6730	1.13091
$941$	−19.7521	−0.643899	−0.321949	−	0.946757i	$-0.604338\pi$
$941$	−19.7521	−0.643899	−0.321949	+	0.946757i	$0.604338\pi$
$942$	3.53904	0.115308
$943$	16.6580	0.542460
$944$	−1.40681	−0.0457877
$945$	−3.34403	−0.108781
$946$	−5.21140	−0.169437
$947$	19.2629	0.625959	0.312979	−	0.949760i	$-0.398673\pi$
$947$	19.2629	0.625959	0.312979	+	0.949760i	$0.398673\pi$
$948$	−9.01500	−0.292794
$949$	−27.8612	−0.904414
$950$	9.01809	0.292585
$951$	−17.9555	−0.582248
$952$	−2.91667	−0.0945299
$953$	37.2132	1.20545	0.602726	−	0.797948i	$-0.294081\pi$
$953$	37.2132	1.20545	0.602726	+	0.797948i	$0.294081\pi$
$954$	−7.32647	−0.237203
$955$	−87.1656	−2.82061
$956$	33.7470	1.09146
$957$	14.1922	0.458769
$958$	5.41976	0.175104
$959$	10.0217	0.323616
$960$	−9.50101	−0.306644
$961$	−9.89837	−0.319302
$962$	−22.8857	−0.737864
$963$	−0.855698	−0.0275745
$964$	5.56151	0.179124
$965$	25.8735	0.832899
$966$	3.05923	0.0984290
$967$	−21.2242	−0.682524	−0.341262	−	0.939968i	$-0.610854\pi$
$967$	−21.2242	−0.682524	−0.341262	+	0.939968i	$0.610854\pi$
$968$	−9.38230	−0.301559
$969$	2.12430	0.0682423
$970$	−9.46851	−0.304016
$971$	−41.3070	−1.32560	−0.662802	−	0.748795i	$-0.730633\pi$
$971$	−41.3070	−1.32560	−0.662802	+	0.748795i	$0.730633\pi$
$972$	−1.41325	−0.0453302
$973$	−14.7821	−0.473893
$974$	−26.8284	−0.859637
$975$	18.0208	0.577128
$976$	5.47719	0.175321
$977$	28.8581	0.923251	0.461626	−	0.887075i	$-0.347266\pi$
$977$	28.8581	0.923251	0.461626	+	0.887075i	$0.347266\pi$
$978$	−1.98822	−0.0635761
$979$	−23.2023	−0.741550
$980$	4.72597	0.150966
$981$	11.5522	0.368833
$982$	3.39570	0.108361
$983$	−59.1821	−1.88761	−0.943807	−	0.330496i	$-0.892784\pi$
$983$	−59.1821	−1.88761	−0.943807	+	0.330496i	$0.892784\pi$
$984$	−10.9051	−0.347642
$985$	−22.9552	−0.731414
$986$	−4.45468	−0.141866
$987$	7.33669	0.233529
$988$	7.84420	0.249557
$989$	9.98078	0.317370
$990$	−6.97345	−0.221631
$991$	5.80589	0.184430	0.0922150	−	0.995739i	$-0.470605\pi$
$991$	5.80589	0.184430	0.0922150	+	0.995739i	$0.470605\pi$
$992$	−26.9192	−0.854685
$993$	0.465717	0.0147791
$994$	5.78462	0.183477
$995$	−20.0807	−0.636602
$996$	3.04580	0.0965099
$997$	29.1828	0.924230	0.462115	−	0.886820i	$-0.347091\pi$
$997$	29.1828	0.924230	0.462115	+	0.886820i	$0.347091\pi$
$998$	28.3054	0.895992
$999$	10.2502	0.324302

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.n.1.17	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.n.1.17	✓	40	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.765993 q^{2} +1.00000 q^{3} -1.41325 q^{4} -3.34403 q^{5} -0.765993 q^{6} +1.00000 q^{7} +2.61453 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.765993 q^{2} +1.00000 q^{3} -1.41325 q^{4} -3.34403 q^{5} -0.765993 q^{6} +1.00000 q^{7} +2.61453 q^{8} +1.00000 q^{9} +2.56151 q^{10} -2.72240 q^{11} -1.41325 q^{12} +2.91478 q^{13} -0.765993 q^{14} -3.34403 q^{15} +0.823799 q^{16} -1.11556 q^{17} -0.765993 q^{18} -1.90424 q^{19} +4.72597 q^{20} +1.00000 q^{21} +2.08534 q^{22} -3.99381 q^{23} +2.61453 q^{24} +6.18255 q^{25} -2.23270 q^{26} +1.00000 q^{27} -1.41325 q^{28} -5.21312 q^{29} +2.56151 q^{30} +4.59365 q^{31} -5.86008 q^{32} -2.72240 q^{33} +0.854513 q^{34} -3.34403 q^{35} -1.41325 q^{36} +10.2502 q^{37} +1.45863 q^{38} +2.91478 q^{39} -8.74307 q^{40} -4.17097 q^{41} -0.765993 q^{42} -2.49906 q^{43} +3.84745 q^{44} -3.34403 q^{45} +3.05923 q^{46} +7.33669 q^{47} +0.823799 q^{48} +1.00000 q^{49} -4.73579 q^{50} -1.11556 q^{51} -4.11933 q^{52} +9.56467 q^{53} -0.765993 q^{54} +9.10380 q^{55} +2.61453 q^{56} -1.90424 q^{57} +3.99321 q^{58} -1.70771 q^{59} +4.72597 q^{60} +6.64870 q^{61} -3.51870 q^{62} +1.00000 q^{63} +2.84118 q^{64} -9.74713 q^{65} +2.08534 q^{66} -5.44386 q^{67} +1.57657 q^{68} -3.99381 q^{69} +2.56151 q^{70} -7.55180 q^{71} +2.61453 q^{72} -9.55859 q^{73} -7.85159 q^{74} +6.18255 q^{75} +2.69118 q^{76} -2.72240 q^{77} -2.23270 q^{78} +6.37889 q^{79} -2.75481 q^{80} +1.00000 q^{81} +3.19493 q^{82} -2.15517 q^{83} -1.41325 q^{84} +3.73048 q^{85} +1.91426 q^{86} -5.21312 q^{87} -7.11780 q^{88} +8.52274 q^{89} +2.56151 q^{90} +2.91478 q^{91} +5.64427 q^{92} +4.59365 q^{93} -5.61985 q^{94} +6.36784 q^{95} -5.86008 q^{96} -3.69646 q^{97} -0.765993 q^{98} -2.72240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more