LMFDB - Embedded newform 8007.2.a.e.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83962 q^{2} +1.00000 q^{3} +1.38420 q^{4} +0.446319 q^{5} -1.83962 q^{6} +3.21050 q^{7} +1.13283 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.83962 q^{2} +1.00000 q^{3} +1.38420 q^{4} +0.446319 q^{5} -1.83962 q^{6} +3.21050 q^{7} +1.13283 q^{8} +1.00000 q^{9} -0.821058 q^{10} +0.755063 q^{11} +1.38420 q^{12} -6.00546 q^{13} -5.90609 q^{14} +0.446319 q^{15} -4.85239 q^{16} -1.00000 q^{17} -1.83962 q^{18} -3.47410 q^{19} +0.617796 q^{20} +3.21050 q^{21} -1.38903 q^{22} -1.33575 q^{23} +1.13283 q^{24} -4.80080 q^{25} +11.0478 q^{26} +1.00000 q^{27} +4.44398 q^{28} -5.99876 q^{29} -0.821058 q^{30} +0.937935 q^{31} +6.66089 q^{32} +0.755063 q^{33} +1.83962 q^{34} +1.43291 q^{35} +1.38420 q^{36} +7.61442 q^{37} +6.39102 q^{38} -6.00546 q^{39} +0.505605 q^{40} +4.83292 q^{41} -5.90609 q^{42} +11.5470 q^{43} +1.04516 q^{44} +0.446319 q^{45} +2.45728 q^{46} +4.65967 q^{47} -4.85239 q^{48} +3.30728 q^{49} +8.83165 q^{50} -1.00000 q^{51} -8.31277 q^{52} -0.522651 q^{53} -1.83962 q^{54} +0.336999 q^{55} +3.63695 q^{56} -3.47410 q^{57} +11.0354 q^{58} -3.59680 q^{59} +0.617796 q^{60} -3.89238 q^{61} -1.72544 q^{62} +3.21050 q^{63} -2.54873 q^{64} -2.68035 q^{65} -1.38903 q^{66} -4.41034 q^{67} -1.38420 q^{68} -1.33575 q^{69} -2.63600 q^{70} -10.1666 q^{71} +1.13283 q^{72} +15.6550 q^{73} -14.0076 q^{74} -4.80080 q^{75} -4.80885 q^{76} +2.42413 q^{77} +11.0478 q^{78} -15.3608 q^{79} -2.16571 q^{80} +1.00000 q^{81} -8.89074 q^{82} +3.70090 q^{83} +4.44398 q^{84} -0.446319 q^{85} -21.2422 q^{86} -5.99876 q^{87} +0.855361 q^{88} +1.35034 q^{89} -0.821058 q^{90} -19.2805 q^{91} -1.84895 q^{92} +0.937935 q^{93} -8.57202 q^{94} -1.55056 q^{95} +6.66089 q^{96} -12.1618 q^{97} -6.08414 q^{98} +0.755063 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) $ 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.83962	−1.30081	−0.650404	−	0.759588i	$-0.725400\pi$
$2$	−1.83962	−1.30081	−0.650404	+	0.759588i	$0.725400\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.38420	0.692102
$5$	0.446319	0.199600	0.0998000	−	0.995008i	$-0.468180\pi$
$5$	0.446319	0.199600	0.0998000	+	0.995008i	$0.468180\pi$
$6$	−1.83962	−0.751022
$7$	3.21050	1.21345	0.606727	−	0.794911i	$-0.292482\pi$
$7$	3.21050	1.21345	0.606727	+	0.794911i	$0.292482\pi$
$8$	1.13283	0.400517
$9$	1.00000	0.333333
$10$	−0.821058	−0.259641
$11$	0.755063	0.227660	0.113830	−	0.993500i	$-0.463688\pi$
$11$	0.755063	0.227660	0.113830	+	0.993500i	$0.463688\pi$
$12$	1.38420	0.399585
$13$	−6.00546	−1.66561	−0.832807	−	0.553563i	$-0.813268\pi$
$13$	−6.00546	−1.66561	−0.832807	+	0.553563i	$0.813268\pi$
$14$	−5.90609	−1.57847
$15$	0.446319	0.115239
$16$	−4.85239	−1.21310
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.83962	−0.433603
$19$	−3.47410	−0.797012	−0.398506	−	0.917166i	$-0.630471\pi$
$19$	−3.47410	−0.797012	−0.398506	+	0.917166i	$0.630471\pi$
$20$	0.617796	0.138143
$21$	3.21050	0.700588
$22$	−1.38903	−0.296142
$23$	−1.33575	−0.278524	−0.139262	−	0.990256i	$-0.544473\pi$
$23$	−1.33575	−0.278524	−0.139262	+	0.990256i	$0.544473\pi$
$24$	1.13283	0.231238
$25$	−4.80080	−0.960160
$26$	11.0478	2.16664
$27$	1.00000	0.192450
$28$	4.44398	0.839833
$29$	−5.99876	−1.11394	−0.556971	−	0.830532i	$-0.688037\pi$
$29$	−5.99876	−1.11394	−0.556971	+	0.830532i	$0.688037\pi$
$30$	−0.821058	−0.149904
$31$	0.937935	0.168458	0.0842290	−	0.996446i	$-0.473157\pi$
$31$	0.937935	0.168458	0.0842290	+	0.996446i	$0.473157\pi$
$32$	6.66089	1.17749
$33$	0.755063	0.131440
$34$	1.83962	0.315492
$35$	1.43291	0.242205
$36$	1.38420	0.230701
$37$	7.61442	1.25180	0.625901	−	0.779902i	$-0.284731\pi$
$37$	7.61442	1.25180	0.625901	+	0.779902i	$0.284731\pi$
$38$	6.39102	1.03676
$39$	−6.00546	−0.961643
$40$	0.505605	0.0799432
$41$	4.83292	0.754776	0.377388	−	0.926055i	$-0.376822\pi$
$41$	4.83292	0.754776	0.377388	+	0.926055i	$0.376822\pi$
$42$	−5.90609	−0.911330
$43$	11.5470	1.76091	0.880454	−	0.474131i	$-0.157238\pi$
$43$	11.5470	1.76091	0.880454	+	0.474131i	$0.157238\pi$
$44$	1.04516	0.157564
$45$	0.446319	0.0665333
$46$	2.45728	0.362306
$47$	4.65967	0.679682	0.339841	−	0.940483i	$-0.389627\pi$
$47$	4.65967	0.679682	0.339841	+	0.940483i	$0.389627\pi$
$48$	−4.85239	−0.700382
$49$	3.30728	0.472469
$50$	8.83165	1.24898
$51$	−1.00000	−0.140028
$52$	−8.31277	−1.15277
$53$	−0.522651	−0.0717917	−0.0358958	−	0.999356i	$-0.511428\pi$
$53$	−0.522651	−0.0717917	−0.0358958	+	0.999356i	$0.511428\pi$
$54$	−1.83962	−0.250341
$55$	0.336999	0.0454410
$56$	3.63695	0.486008
$57$	−3.47410	−0.460155
$58$	11.0354	1.44902
$59$	−3.59680	−0.468264	−0.234132	−	0.972205i	$-0.575225\pi$
$59$	−3.59680	−0.468264	−0.234132	+	0.972205i	$0.575225\pi$
$60$	0.617796	0.0797572
$61$	−3.89238	−0.498369	−0.249184	−	0.968456i	$-0.580163\pi$
$61$	−3.89238	−0.498369	−0.249184	+	0.968456i	$0.580163\pi$
$62$	−1.72544	−0.219132
$63$	3.21050	0.404484
$64$	−2.54873	−0.318591
$65$	−2.68035	−0.332457
$66$	−1.38903	−0.170978
$67$	−4.41034	−0.538809	−0.269404	−	0.963027i	$-0.586827\pi$
$67$	−4.41034	−0.538809	−0.269404	+	0.963027i	$0.586827\pi$
$68$	−1.38420	−0.167859
$69$	−1.33575	−0.160806
$70$	−2.63600	−0.315063
$71$	−10.1666	−1.20655	−0.603275	−	0.797533i	$-0.706138\pi$
$71$	−10.1666	−1.20655	−0.603275	+	0.797533i	$0.706138\pi$
$72$	1.13283	0.133506
$73$	15.6550	1.83228	0.916142	−	0.400854i	$-0.131287\pi$
$73$	15.6550	1.83228	0.916142	+	0.400854i	$0.131287\pi$
$74$	−14.0076	−1.62836
$75$	−4.80080	−0.554349
$76$	−4.80885	−0.551613
$77$	2.42413	0.276255
$78$	11.0478	1.25091
$79$	−15.3608	−1.72822	−0.864112	−	0.503300i	$-0.832119\pi$
$79$	−15.3608	−1.72822	−0.864112	+	0.503300i	$0.832119\pi$
$80$	−2.16571	−0.242134
$81$	1.00000	0.111111
$82$	−8.89074	−0.981818
$83$	3.70090	0.406226	0.203113	−	0.979155i	$-0.434894\pi$
$83$	3.70090	0.406226	0.203113	+	0.979155i	$0.434894\pi$
$84$	4.44398	0.484878
$85$	−0.446319	−0.0484101
$86$	−21.2422	−2.29060
$87$	−5.99876	−0.643135
$88$	0.855361	0.0911817
$89$	1.35034	0.143135	0.0715677	−	0.997436i	$-0.477200\pi$
$89$	1.35034	0.143135	0.0715677	+	0.997436i	$0.477200\pi$
$90$	−0.821058	−0.0865471
$91$	−19.2805	−2.02114
$92$	−1.84895	−0.192767
$93$	0.937935	0.0972593
$94$	−8.57202	−0.884136
$95$	−1.55056	−0.159084
$96$	6.66089	0.679824
$97$	−12.1618	−1.23484	−0.617421	−	0.786633i	$-0.711823\pi$
$97$	−12.1618	−1.23484	−0.617421	+	0.786633i	$0.711823\pi$
$98$	−6.08414	−0.614591
$99$	0.755063	0.0758867
$100$	−6.64528	−0.664528
$101$	−9.20932	−0.916362	−0.458181	−	0.888859i	$-0.651499\pi$
$101$	−9.20932	−0.916362	−0.458181	+	0.888859i	$0.651499\pi$
$102$	1.83962	0.182150
$103$	12.9331	1.27434	0.637170	−	0.770724i	$-0.280105\pi$
$103$	12.9331	1.27434	0.637170	+	0.770724i	$0.280105\pi$
$104$	−6.80318	−0.667106
$105$	1.43291	0.139837
$106$	0.961480	0.0933872
$107$	2.40569	0.232567	0.116284	−	0.993216i	$-0.462902\pi$
$107$	2.40569	0.232567	0.116284	+	0.993216i	$0.462902\pi$
$108$	1.38420	0.133195
$109$	−4.36744	−0.418325	−0.209163	−	0.977881i	$-0.567074\pi$
$109$	−4.36744	−0.418325	−0.209163	+	0.977881i	$0.567074\pi$
$110$	−0.619951	−0.0591100
$111$	7.61442	0.722729
$112$	−15.5786	−1.47204
$113$	−3.50544	−0.329764	−0.164882	−	0.986313i	$-0.552724\pi$
$113$	−3.50544	−0.329764	−0.164882	+	0.986313i	$0.552724\pi$
$114$	6.39102	0.598574
$115$	−0.596173	−0.0555934
$116$	−8.30350	−0.770961
$117$	−6.00546	−0.555205
$118$	6.61675	0.609122
$119$	−3.21050	−0.294306
$120$	0.505605	0.0461552
$121$	−10.4299	−0.948171
$122$	7.16051	0.648282
$123$	4.83292	0.435770
$124$	1.29829	0.116590
$125$	−4.37428	−0.391248
$126$	−5.90609	−0.526157
$127$	8.25531	0.732541	0.366270	−	0.930508i	$-0.380635\pi$
$127$	8.25531	0.732541	0.366270	+	0.930508i	$0.380635\pi$
$128$	−8.63308	−0.763064
$129$	11.5470	1.01666
$130$	4.93083	0.432462
$131$	−22.4186	−1.95872	−0.979360	−	0.202122i	$-0.935216\pi$
$131$	−22.4186	−1.95872	−0.979360	+	0.202122i	$0.935216\pi$
$132$	1.04516	0.0909696
$133$	−11.1536	−0.967137
$134$	8.11335	0.700887
$135$	0.446319	0.0384130
$136$	−1.13283	−0.0971396
$137$	−18.1655	−1.55198	−0.775992	−	0.630743i	$-0.782750\pi$
$137$	−18.1655	−1.55198	−0.775992	+	0.630743i	$0.782750\pi$
$138$	2.45728	0.209178
$139$	3.44808	0.292462	0.146231	−	0.989250i	$-0.453286\pi$
$139$	3.44808	0.292462	0.146231	+	0.989250i	$0.453286\pi$
$140$	1.98343	0.167631
$141$	4.65967	0.392415
$142$	18.7026	1.56949
$143$	−4.53450	−0.379194
$144$	−4.85239	−0.404366
$145$	−2.67736	−0.222343
$146$	−28.7993	−2.38345
$147$	3.30728	0.272780
$148$	10.5399	0.866375
$149$	−8.39683	−0.687895	−0.343947	−	0.938989i	$-0.611764\pi$
$149$	−8.39683	−0.687895	−0.343947	+	0.938989i	$0.611764\pi$
$150$	8.83165	0.721101
$151$	4.71303	0.383541	0.191771	−	0.981440i	$-0.438577\pi$
$151$	4.71303	0.383541	0.191771	+	0.981440i	$0.438577\pi$
$152$	−3.93557	−0.319217
$153$	−1.00000	−0.0808452
$154$	−4.45947	−0.359355
$155$	0.418618	0.0336242
$156$	−8.31277	−0.665554
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	28.2580	2.24809
$159$	−0.522651	−0.0414489
$160$	2.97288	0.235027
$161$	−4.28843	−0.337976
$162$	−1.83962	−0.144534
$163$	−2.89363	−0.226646	−0.113323	−	0.993558i	$-0.536150\pi$
$163$	−2.89363	−0.226646	−0.113323	+	0.993558i	$0.536150\pi$
$164$	6.68975	0.522381
$165$	0.336999	0.0262354
$166$	−6.80824	−0.528422
$167$	−19.4708	−1.50670	−0.753349	−	0.657621i	$-0.771563\pi$
$167$	−19.4708	−1.50670	−0.753349	+	0.657621i	$0.771563\pi$
$168$	3.63695	0.280597
$169$	23.0655	1.77427
$170$	0.821058	0.0629723
$171$	−3.47410	−0.265671
$172$	15.9835	1.21873
$173$	7.43832	0.565525	0.282763	−	0.959190i	$-0.408749\pi$
$173$	7.43832	0.565525	0.282763	+	0.959190i	$0.408749\pi$
$174$	11.0354	0.836595
$175$	−15.4129	−1.16511
$176$	−3.66386	−0.276174
$177$	−3.59680	−0.270352
$178$	−2.48411	−0.186192
$179$	17.2445	1.28892	0.644458	−	0.764639i	$-0.277083\pi$
$179$	17.2445	1.28892	0.644458	+	0.764639i	$0.277083\pi$
$180$	0.617796	0.0460478
$181$	1.93638	0.143930	0.0719649	−	0.997407i	$-0.477073\pi$
$181$	1.93638	0.143930	0.0719649	+	0.997407i	$0.477073\pi$
$182$	35.4688	2.62912
$183$	−3.89238	−0.287733
$184$	−1.51319	−0.111554
$185$	3.39846	0.249860
$186$	−1.72544	−0.126516
$187$	−0.755063	−0.0552157
$188$	6.44992	0.470409
$189$	3.21050	0.233529
$190$	2.85243	0.206937
$191$	−17.5547	−1.27022	−0.635108	−	0.772423i	$-0.719045\pi$
$191$	−17.5547	−1.27022	−0.635108	+	0.772423i	$0.719045\pi$
$192$	−2.54873	−0.183938
$193$	−17.4943	−1.25926	−0.629632	−	0.776894i	$-0.716794\pi$
$193$	−17.4943	−1.25926	−0.629632	+	0.776894i	$0.716794\pi$
$194$	22.3731	1.60629
$195$	−2.68035	−0.191944
$196$	4.57795	0.326996
$197$	0.781602	0.0556868	0.0278434	−	0.999612i	$-0.491136\pi$
$197$	0.781602	0.0556868	0.0278434	+	0.999612i	$0.491136\pi$
$198$	−1.38903	−0.0987141
$199$	3.72043	0.263734	0.131867	−	0.991267i	$-0.457903\pi$
$199$	3.72043	0.263734	0.131867	+	0.991267i	$0.457903\pi$
$200$	−5.43850	−0.384560
$201$	−4.41034	−0.311081
$202$	16.9417	1.19201
$203$	−19.2590	−1.35172
$204$	−1.38420	−0.0969136
$205$	2.15703	0.150653
$206$	−23.7921	−1.65767
$207$	−1.33575	−0.0928413
$208$	29.1408	2.02055
$209$	−2.62316	−0.181448
$210$	−2.63600	−0.181901
$211$	−2.95112	−0.203164	−0.101582	−	0.994827i	$-0.532390\pi$
$211$	−2.95112	−0.203164	−0.101582	+	0.994827i	$0.532390\pi$
$212$	−0.723455	−0.0496871
$213$	−10.1666	−0.696602
$214$	−4.42556	−0.302525
$215$	5.15367	0.351477
$216$	1.13283	0.0770795
$217$	3.01123	0.204416
$218$	8.03444	0.544161
$219$	15.6550	1.05787
$220$	0.466475	0.0314498
$221$	6.00546	0.403971
$222$	−14.0076	−0.940131
$223$	5.81311	0.389275	0.194637	−	0.980875i	$-0.437647\pi$
$223$	5.81311	0.389275	0.194637	+	0.980875i	$0.437647\pi$
$224$	21.3847	1.42883
$225$	−4.80080	−0.320053
$226$	6.44867	0.428959
$227$	−2.81084	−0.186562	−0.0932812	−	0.995640i	$-0.529736\pi$
$227$	−2.81084	−0.186562	−0.0932812	+	0.995640i	$0.529736\pi$
$228$	−4.80885	−0.318474
$229$	19.7529	1.30531	0.652655	−	0.757655i	$-0.273655\pi$
$229$	19.7529	1.30531	0.652655	+	0.757655i	$0.273655\pi$
$230$	1.09673	0.0723163
$231$	2.42413	0.159496
$232$	−6.79559	−0.446153
$233$	−8.79173	−0.575965	−0.287983	−	0.957636i	$-0.592985\pi$
$233$	−8.79173	−0.575965	−0.287983	+	0.957636i	$0.592985\pi$
$234$	11.0478	0.722215
$235$	2.07970	0.135665
$236$	−4.97871	−0.324086
$237$	−15.3608	−0.997790
$238$	5.90609	0.382835
$239$	−11.7069	−0.757258	−0.378629	−	0.925549i	$-0.623604\pi$
$239$	−11.7069	−0.757258	−0.378629	+	0.925549i	$0.623604\pi$
$240$	−2.16571	−0.139796
$241$	21.6664	1.39565	0.697826	−	0.716267i	$-0.254151\pi$
$241$	21.6664	1.39565	0.697826	+	0.716267i	$0.254151\pi$
$242$	19.1870	1.23339
$243$	1.00000	0.0641500
$244$	−5.38785	−0.344922
$245$	1.47610	0.0943048
$246$	−8.89074	−0.566853
$247$	20.8635	1.32752
$248$	1.06252	0.0674703
$249$	3.70090	0.234535
$250$	8.04702	0.508938
$251$	−16.4457	−1.03804	−0.519022	−	0.854761i	$-0.673704\pi$
$251$	−16.4457	−1.03804	−0.519022	+	0.854761i	$0.673704\pi$
$252$	4.44398	0.279944
$253$	−1.00858	−0.0634088
$254$	−15.1866	−0.952895
$255$	−0.446319	−0.0279496
$256$	20.9791	1.31119
$257$	−2.86778	−0.178887	−0.0894437	−	0.995992i	$-0.528509\pi$
$257$	−2.86778	−0.178887	−0.0894437	+	0.995992i	$0.528509\pi$
$258$	−21.2422	−1.32248
$259$	24.4461	1.51900
$260$	−3.71015	−0.230094
$261$	−5.99876	−0.371314
$262$	41.2417	2.54792
$263$	12.7348	0.785263	0.392631	−	0.919696i	$-0.371565\pi$
$263$	12.7348	0.785263	0.392631	+	0.919696i	$0.371565\pi$
$264$	0.855361	0.0526438
$265$	−0.233269	−0.0143296
$266$	20.5183	1.25806
$267$	1.35034	0.0826392
$268$	−6.10481	−0.372910
$269$	−22.2195	−1.35475	−0.677373	−	0.735640i	$-0.736882\pi$
$269$	−22.2195	−1.35475	−0.677373	+	0.735640i	$0.736882\pi$
$270$	−0.821058	−0.0499680
$271$	6.43735	0.391041	0.195520	−	0.980700i	$-0.437360\pi$
$271$	6.43735	0.391041	0.195520	+	0.980700i	$0.437360\pi$
$272$	4.85239	0.294219
$273$	−19.2805	−1.16691
$274$	33.4176	2.01883
$275$	−3.62491	−0.218590
$276$	−1.84895	−0.111294
$277$	12.7060	0.763430	0.381715	−	0.924280i	$-0.375334\pi$
$277$	12.7060	0.763430	0.381715	+	0.924280i	$0.375334\pi$
$278$	−6.34315	−0.380437
$279$	0.937935	0.0561527
$280$	1.62324	0.0970073
$281$	−20.0757	−1.19761	−0.598807	−	0.800893i	$-0.704358\pi$
$281$	−20.0757	−1.19761	−0.598807	+	0.800893i	$0.704358\pi$
$282$	−8.57202	−0.510456
$283$	−17.1401	−1.01887	−0.509437	−	0.860508i	$-0.670146\pi$
$283$	−17.1401	−1.01887	−0.509437	+	0.860508i	$0.670146\pi$
$284$	−14.0726	−0.835056
$285$	−1.55056	−0.0918470
$286$	8.34176	0.493259
$287$	15.5161	0.915885
$288$	6.66089	0.392496
$289$	1.00000	0.0588235
$290$	4.92533	0.289225
$291$	−12.1618	−0.712937
$292$	21.6698	1.26813
$293$	14.7028	0.858945	0.429473	−	0.903080i	$-0.358699\pi$
$293$	14.7028	0.858945	0.429473	+	0.903080i	$0.358699\pi$
$294$	−6.08414	−0.354834
$295$	−1.60532	−0.0934655
$296$	8.62586	0.501368
$297$	0.755063	0.0438132
$298$	15.4470	0.894819
$299$	8.02181	0.463913
$300$	−6.64528	−0.383665
$301$	37.0717	2.13678
$302$	−8.67019	−0.498913
$303$	−9.20932	−0.529062
$304$	16.8577	0.966853
$305$	−1.73725	−0.0994744
$306$	1.83962	0.105164
$307$	−12.8210	−0.731736	−0.365868	−	0.930667i	$-0.619228\pi$
$307$	−12.8210	−0.731736	−0.365868	+	0.930667i	$0.619228\pi$
$308$	3.35549	0.191197
$309$	12.9331	0.735740
$310$	−0.770098	−0.0437387
$311$	−10.9232	−0.619399	−0.309699	−	0.950834i	$-0.600228\pi$
$311$	−10.9232	−0.619399	−0.309699	+	0.950834i	$0.600228\pi$
$312$	−6.80318	−0.385154
$313$	−19.8697	−1.12310	−0.561552	−	0.827442i	$-0.689795\pi$
$313$	−19.8697	−1.12310	−0.561552	+	0.827442i	$0.689795\pi$
$314$	−1.83962	−0.103816
$315$	1.43291	0.0807351
$316$	−21.2624	−1.19611
$317$	8.79359	0.493897	0.246948	−	0.969029i	$-0.420572\pi$
$317$	8.79359	0.493897	0.246948	+	0.969029i	$0.420572\pi$
$318$	0.961480	0.0539171
$319$	−4.52945	−0.253600
$320$	−1.13755	−0.0635907
$321$	2.40569	0.134273
$322$	7.88909	0.439642
$323$	3.47410	0.193304
$324$	1.38420	0.0769002
$325$	28.8310	1.59926
$326$	5.32318	0.294824
$327$	−4.36744	−0.241520
$328$	5.47489	0.302300
$329$	14.9598	0.824762
$330$	−0.619951	−0.0341272
$331$	−18.6818	−1.02684	−0.513422	−	0.858136i	$-0.671622\pi$
$331$	−18.6818	−1.02684	−0.513422	+	0.858136i	$0.671622\pi$
$332$	5.12279	0.281150
$333$	7.61442	0.417268
$334$	35.8189	1.95992
$335$	−1.96842	−0.107546
$336$	−15.5786	−0.849881
$337$	−10.3404	−0.563276	−0.281638	−	0.959521i	$-0.590878\pi$
$337$	−10.3404	−0.563276	−0.281638	+	0.959521i	$0.590878\pi$
$338$	−42.4318	−2.30799
$339$	−3.50544	−0.190389
$340$	−0.617796	−0.0335047
$341$	0.708200	0.0383512
$342$	6.39102	0.345587
$343$	−11.8555	−0.640135
$344$	13.0809	0.705273
$345$	−0.596173	−0.0320969
$346$	−13.6837	−0.735640
$347$	−11.9471	−0.641356	−0.320678	−	0.947188i	$-0.603911\pi$
$347$	−11.9471	−0.641356	−0.320678	+	0.947188i	$0.603911\pi$
$348$	−8.30350	−0.445114
$349$	−24.1205	−1.29114	−0.645571	−	0.763700i	$-0.723380\pi$
$349$	−24.1205	−1.29114	−0.645571	+	0.763700i	$0.723380\pi$
$350$	28.3540	1.51558
$351$	−6.00546	−0.320548
$352$	5.02939	0.268067
$353$	−30.4099	−1.61855	−0.809277	−	0.587427i	$-0.800141\pi$
$353$	−30.4099	−1.61855	−0.809277	+	0.587427i	$0.800141\pi$
$354$	6.61675	0.351677
$355$	−4.53754	−0.240828
$356$	1.86914	0.0990642
$357$	−3.21050	−0.169917
$358$	−31.7234	−1.67663
$359$	−1.91066	−0.100841	−0.0504204	−	0.998728i	$-0.516056\pi$
$359$	−1.91066	−0.100841	−0.0504204	+	0.998728i	$0.516056\pi$
$360$	0.505605	0.0266477
$361$	−6.93065	−0.364771
$362$	−3.56220	−0.187225
$363$	−10.4299	−0.547427
$364$	−26.6881	−1.39884
$365$	6.98714	0.365724
$366$	7.16051	0.374286
$367$	−18.2460	−0.952436	−0.476218	−	0.879327i	$-0.657993\pi$
$367$	−18.2460	−0.952436	−0.476218	+	0.879327i	$0.657993\pi$
$368$	6.48160	0.337877
$369$	4.83292	0.251592
$370$	−6.25188	−0.325020
$371$	−1.67797	−0.0871158
$372$	1.29829	0.0673133
$373$	23.9197	1.23851	0.619257	−	0.785189i	$-0.287434\pi$
$373$	23.9197	1.23851	0.619257	+	0.785189i	$0.287434\pi$
$374$	1.38903	0.0718250
$375$	−4.37428	−0.225887
$376$	5.27862	0.272224
$377$	36.0253	1.85540
$378$	−5.90609	−0.303777
$379$	−27.7164	−1.42370	−0.711848	−	0.702333i	$-0.752142\pi$
$379$	−27.7164	−1.42370	−0.711848	+	0.702333i	$0.752142\pi$
$380$	−2.14628	−0.110102
$381$	8.25531	0.422933
$382$	32.2941	1.65231
$383$	−22.7856	−1.16429	−0.582146	−	0.813084i	$-0.697787\pi$
$383$	−22.7856	−1.16429	−0.582146	+	0.813084i	$0.697787\pi$
$384$	−8.63308	−0.440555
$385$	1.08193	0.0551405
$386$	32.1828	1.63806
$387$	11.5470	0.586970
$388$	−16.8344	−0.854636
$389$	−8.42203	−0.427014	−0.213507	−	0.976942i	$-0.568489\pi$
$389$	−8.42203	−0.427014	−0.213507	+	0.976942i	$0.568489\pi$
$390$	4.93083	0.249682
$391$	1.33575	0.0675520
$392$	3.74660	0.189232
$393$	−22.4186	−1.13087
$394$	−1.43785	−0.0724379
$395$	−6.85581	−0.344953
$396$	1.04516	0.0525213
$397$	23.1340	1.16106	0.580532	−	0.814237i	$-0.302845\pi$
$397$	23.1340	1.16106	0.580532	+	0.814237i	$0.302845\pi$
$398$	−6.84418	−0.343068
$399$	−11.1536	−0.558377
$400$	23.2953	1.16477
$401$	7.12378	0.355745	0.177872	−	0.984054i	$-0.443079\pi$
$401$	7.12378	0.355745	0.177872	+	0.984054i	$0.443079\pi$
$402$	8.11335	0.404657
$403$	−5.63273	−0.280586
$404$	−12.7476	−0.634215
$405$	0.446319	0.0221778
$406$	35.4292	1.75832
$407$	5.74937	0.284986
$408$	−1.13283	−0.0560836
$409$	−4.09474	−0.202472	−0.101236	−	0.994862i	$-0.532280\pi$
$409$	−4.09474	−0.202472	−0.101236	+	0.994862i	$0.532280\pi$
$410$	−3.96811	−0.195971
$411$	−18.1655	−0.896038
$412$	17.9021	0.881972
$413$	−11.5475	−0.568217
$414$	2.45728	0.120769
$415$	1.65178	0.0810827
$416$	−40.0017	−1.96124
$417$	3.44808	0.168853
$418$	4.82562	0.236029
$419$	−4.09700	−0.200152	−0.100076	−	0.994980i	$-0.531909\pi$
$419$	−4.09700	−0.200152	−0.100076	+	0.994980i	$0.531909\pi$
$420$	1.98343	0.0967816
$421$	16.7932	0.818453	0.409226	−	0.912433i	$-0.365799\pi$
$421$	16.7932	0.818453	0.409226	+	0.912433i	$0.365799\pi$
$422$	5.42895	0.264277
$423$	4.65967	0.226561
$424$	−0.592076	−0.0287538
$425$	4.80080	0.232873
$426$	18.7026	0.906146
$427$	−12.4965	−0.604747
$428$	3.32997	0.160960
$429$	−4.53450	−0.218928
$430$	−9.48080	−0.457205
$431$	−38.3833	−1.84886	−0.924430	−	0.381351i	$-0.875459\pi$
$431$	−38.3833	−1.84886	−0.924430	+	0.381351i	$0.875459\pi$
$432$	−4.85239	−0.233461
$433$	21.1848	1.01808	0.509038	−	0.860744i	$-0.330001\pi$
$433$	21.1848	1.01808	0.509038	+	0.860744i	$0.330001\pi$
$434$	−5.53953	−0.265906
$435$	−2.67736	−0.128370
$436$	−6.04543	−0.289524
$437$	4.64054	0.221987
$438$	−28.7993	−1.37609
$439$	−5.09873	−0.243349	−0.121675	−	0.992570i	$-0.538826\pi$
$439$	−5.09873	−0.243349	−0.121675	+	0.992570i	$0.538826\pi$
$440$	0.381764	0.0181999
$441$	3.30728	0.157490
$442$	−11.0478	−0.525488
$443$	9.28038	0.440924	0.220462	−	0.975396i	$-0.429243\pi$
$443$	9.28038	0.440924	0.220462	+	0.975396i	$0.429243\pi$
$444$	10.5399	0.500202
$445$	0.602681	0.0285698
$446$	−10.6939	−0.506372
$447$	−8.39683	−0.397156
$448$	−8.18267	−0.386595
$449$	26.1201	1.23269	0.616343	−	0.787478i	$-0.288614\pi$
$449$	26.1201	1.23269	0.616343	+	0.787478i	$0.288614\pi$
$450$	8.83165	0.416328
$451$	3.64916	0.171832
$452$	−4.85224	−0.228230
$453$	4.71303	0.221438
$454$	5.17089	0.242682
$455$	−8.60525	−0.403421
$456$	−3.93557	−0.184300
$457$	−31.8956	−1.49201	−0.746006	−	0.665940i	$-0.768031\pi$
$457$	−31.8956	−1.49201	−0.746006	+	0.665940i	$0.768031\pi$
$458$	−36.3379	−1.69796
$459$	−1.00000	−0.0466760
$460$	−0.825224	−0.0384763
$461$	−17.8924	−0.833334	−0.416667	−	0.909059i	$-0.636802\pi$
$461$	−17.8924	−0.833334	−0.416667	+	0.909059i	$0.636802\pi$
$462$	−4.45947	−0.207474
$463$	−15.5257	−0.721538	−0.360769	−	0.932655i	$-0.617486\pi$
$463$	−15.5257	−0.721538	−0.360769	+	0.932655i	$0.617486\pi$
$464$	29.1083	1.35132
$465$	0.418618	0.0194130
$466$	16.1734	0.749220
$467$	−28.7219	−1.32909	−0.664545	−	0.747248i	$-0.731375\pi$
$467$	−28.7219	−1.32909	−0.664545	+	0.747248i	$0.731375\pi$
$468$	−8.31277	−0.384258
$469$	−14.1594	−0.653819
$470$	−3.82586	−0.176474
$471$	1.00000	0.0460776
$472$	−4.07458	−0.187548
$473$	8.71876	0.400889
$474$	28.2580	1.29793
$475$	16.6784	0.765259
$476$	−4.44398	−0.203689
$477$	−0.522651	−0.0239306
$478$	21.5363	0.985047
$479$	4.37172	0.199749	0.0998744	−	0.995000i	$-0.468156\pi$
$479$	4.37172	0.199749	0.0998744	+	0.995000i	$0.468156\pi$
$480$	2.97288	0.135693
$481$	−45.7281	−2.08502
$482$	−39.8579	−1.81548
$483$	−4.28843	−0.195130
$484$	−14.4371	−0.656230
$485$	−5.42804	−0.246475
$486$	−1.83962	−0.0834469
$487$	−13.5242	−0.612839	−0.306419	−	0.951897i	$-0.599131\pi$
$487$	−13.5242	−0.612839	−0.306419	+	0.951897i	$0.599131\pi$
$488$	−4.40942	−0.199605
$489$	−2.89363	−0.130854
$490$	−2.71547	−0.122672
$491$	33.9148	1.53055	0.765276	−	0.643702i	$-0.222602\pi$
$491$	33.9148	1.53055	0.765276	+	0.643702i	$0.222602\pi$
$492$	6.68975	0.301597
$493$	5.99876	0.270171
$494$	−38.3810	−1.72684
$495$	0.336999	0.0151470
$496$	−4.55122	−0.204356
$497$	−32.6398	−1.46409
$498$	−6.80824	−0.305085
$499$	13.9621	0.625030	0.312515	−	0.949913i	$-0.398829\pi$
$499$	13.9621	0.625030	0.312515	+	0.949913i	$0.398829\pi$
$500$	−6.05490	−0.270783
$501$	−19.4708	−0.869892
$502$	30.2539	1.35030
$503$	−1.05698	−0.0471283	−0.0235642	−	0.999722i	$-0.507501\pi$
$503$	−1.05698	−0.0471283	−0.0235642	+	0.999722i	$0.507501\pi$
$504$	3.63695	0.162003
$505$	−4.11030	−0.182906
$506$	1.85540	0.0824827
$507$	23.0655	1.02438
$508$	11.4270	0.506993
$509$	11.0058	0.487823	0.243911	−	0.969798i	$-0.421569\pi$
$509$	11.0058	0.487823	0.243911	+	0.969798i	$0.421569\pi$
$510$	0.821058	0.0363571
$511$	50.2604	2.22339
$512$	−21.3273	−0.942543
$513$	−3.47410	−0.153385
$514$	5.27563	0.232698
$515$	5.77230	0.254358
$516$	15.9835	0.703633
$517$	3.51834	0.154737
$518$	−44.9715	−1.97593
$519$	7.43832	0.326506
$520$	−3.03639	−0.133154
$521$	−17.9608	−0.786876	−0.393438	−	0.919351i	$-0.628714\pi$
$521$	−17.9608	−0.786876	−0.393438	+	0.919351i	$0.628714\pi$
$522$	11.0354	0.483008
$523$	−0.352744	−0.0154244	−0.00771220	−	0.999970i	$-0.502455\pi$
$523$	−0.352744	−0.0154244	−0.00771220	+	0.999970i	$0.502455\pi$
$524$	−31.0319	−1.35563
$525$	−15.4129	−0.672676
$526$	−23.4272	−1.02148
$527$	−0.937935	−0.0408571
$528$	−3.66386	−0.159449
$529$	−21.2158	−0.922424
$530$	0.429127	0.0186401
$531$	−3.59680	−0.156088
$532$	−15.4388	−0.669357
$533$	−29.0239	−1.25717
$534$	−2.48411	−0.107498
$535$	1.07371	0.0464204
$536$	−4.99618	−0.215802
$537$	17.2445	0.744156
$538$	40.8754	1.76226
$539$	2.49721	0.107562
$540$	0.617796	0.0265857
$541$	−0.711043	−0.0305701	−0.0152851	−	0.999883i	$-0.504866\pi$
$541$	−0.711043	−0.0305701	−0.0152851	+	0.999883i	$0.504866\pi$
$542$	−11.8423	−0.508669
$543$	1.93638	0.0830979
$544$	−6.66089	−0.285583
$545$	−1.94927	−0.0834977
$546$	35.4688	1.51792
$547$	−19.5158	−0.834436	−0.417218	−	0.908807i	$-0.636995\pi$
$547$	−19.5158	−0.834436	−0.417218	+	0.908807i	$0.636995\pi$
$548$	−25.1447	−1.07413
$549$	−3.89238	−0.166123
$550$	6.66845	0.284344
$551$	20.8403	0.887826
$552$	−1.51319	−0.0644055
$553$	−49.3157	−2.09712
$554$	−23.3742	−0.993076
$555$	3.39846	0.144257
$556$	4.77284	0.202413
$557$	−7.48203	−0.317024	−0.158512	−	0.987357i	$-0.550670\pi$
$557$	−7.48203	−0.317024	−0.158512	+	0.987357i	$0.550670\pi$
$558$	−1.72544	−0.0730438
$559$	−69.3453	−2.93299
$560$	−6.95301	−0.293818
$561$	−0.755063	−0.0318788
$562$	36.9316	1.55787
$563$	13.2808	0.559721	0.279860	−	0.960041i	$-0.409712\pi$
$563$	13.2808	0.559721	0.279860	+	0.960041i	$0.409712\pi$
$564$	6.44992	0.271591
$565$	−1.56454	−0.0658208
$566$	31.5313	1.32536
$567$	3.21050	0.134828
$568$	−11.5170	−0.483244
$569$	34.2610	1.43630	0.718148	−	0.695891i	$-0.244990\pi$
$569$	34.2610	1.43630	0.718148	+	0.695891i	$0.244990\pi$
$570$	2.85243	0.119475
$571$	26.5105	1.10943	0.554714	−	0.832041i	$-0.312828\pi$
$571$	26.5105	1.10943	0.554714	+	0.832041i	$0.312828\pi$
$572$	−6.27667	−0.262441
$573$	−17.5547	−0.733360
$574$	−28.5437	−1.19139
$575$	6.41269	0.267428
$576$	−2.54873	−0.106197
$577$	27.5838	1.14833	0.574165	−	0.818740i	$-0.305327\pi$
$577$	27.5838	1.14833	0.574165	+	0.818740i	$0.305327\pi$
$578$	−1.83962	−0.0765181
$579$	−17.4943	−0.727036
$580$	−3.70601	−0.153884
$581$	11.8817	0.492936
$582$	22.3731	0.927394
$583$	−0.394635	−0.0163441
$584$	17.7345	0.733860
$585$	−2.68035	−0.110819
$586$	−27.0475	−1.11732
$587$	−40.4654	−1.67019	−0.835093	−	0.550109i	$-0.814586\pi$
$587$	−40.4654	−1.67019	−0.835093	+	0.550109i	$0.814586\pi$
$588$	4.57795	0.188791
$589$	−3.25847	−0.134263
$590$	2.95318	0.121581
$591$	0.781602	0.0321508
$592$	−36.9481	−1.51856
$593$	24.0408	0.987239	0.493620	−	0.869678i	$-0.335674\pi$
$593$	24.0408	0.987239	0.493620	+	0.869678i	$0.335674\pi$
$594$	−1.38903	−0.0569926
$595$	−1.43291	−0.0587434
$596$	−11.6229	−0.476093
$597$	3.72043	0.152267
$598$	−14.7571	−0.603462
$599$	22.8014	0.931639	0.465819	−	0.884880i	$-0.345760\pi$
$599$	22.8014	0.931639	0.465819	+	0.884880i	$0.345760\pi$
$600$	−5.43850	−0.222026
$601$	−11.5020	−0.469175	−0.234587	−	0.972095i	$-0.575374\pi$
$601$	−11.5020	−0.469175	−0.234587	+	0.972095i	$0.575374\pi$
$602$	−68.1979	−2.77954
$603$	−4.41034	−0.179603
$604$	6.52379	0.265449
$605$	−4.65505	−0.189255
$606$	16.9417	0.688208
$607$	26.0331	1.05665	0.528326	−	0.849042i	$-0.322820\pi$
$607$	26.0331	1.05665	0.528326	+	0.849042i	$0.322820\pi$
$608$	−23.1406	−0.938474
$609$	−19.2590	−0.780414
$610$	3.19587	0.129397
$611$	−27.9834	−1.13209
$612$	−1.38420	−0.0559531
$613$	42.7072	1.72493	0.862464	−	0.506119i	$-0.168920\pi$
$613$	42.7072	1.72493	0.862464	+	0.506119i	$0.168920\pi$
$614$	23.5859	0.951848
$615$	2.15703	0.0869797
$616$	2.74613	0.110645
$617$	12.5604	0.505663	0.252831	−	0.967510i	$-0.418638\pi$
$617$	12.5604	0.505663	0.252831	+	0.967510i	$0.418638\pi$
$618$	−23.7921	−0.957057
$619$	2.29891	0.0924010	0.0462005	−	0.998932i	$-0.485289\pi$
$619$	2.29891	0.0924010	0.0462005	+	0.998932i	$0.485289\pi$
$620$	0.579453	0.0232714
$621$	−1.33575	−0.0536020
$622$	20.0946	0.805719
$623$	4.33525	0.173688
$624$	29.1408	1.16657
$625$	22.0517	0.882067
$626$	36.5527	1.46094
$627$	−2.62316	−0.104759
$628$	1.38420	0.0552357
$629$	−7.61442	−0.303607
$630$	−2.63600	−0.105021
$631$	7.25565	0.288843	0.144421	−	0.989516i	$-0.453868\pi$
$631$	7.25565	0.288843	0.144421	+	0.989516i	$0.453868\pi$
$632$	−17.4012	−0.692183
$633$	−2.95112	−0.117297
$634$	−16.1769	−0.642465
$635$	3.68451	0.146215
$636$	−0.723455	−0.0286869
$637$	−19.8617	−0.786951
$638$	8.33246	0.329885
$639$	−10.1666	−0.402184
$640$	−3.85311	−0.152308
$641$	22.8705	0.903332	0.451666	−	0.892187i	$-0.350830\pi$
$641$	22.8705	0.903332	0.451666	+	0.892187i	$0.350830\pi$
$642$	−4.42556	−0.174663
$643$	27.5324	1.08577	0.542887	−	0.839806i	$-0.317331\pi$
$643$	27.5324	1.08577	0.542887	+	0.839806i	$0.317331\pi$
$644$	−5.93606	−0.233914
$645$	5.15367	0.202926
$646$	−6.39102	−0.251451
$647$	−13.7759	−0.541586	−0.270793	−	0.962638i	$-0.587286\pi$
$647$	−13.7759	−0.541586	−0.270793	+	0.962638i	$0.587286\pi$
$648$	1.13283	0.0445019
$649$	−2.71582	−0.106605
$650$	−53.0381	−2.08032
$651$	3.01123	0.118020
$652$	−4.00537	−0.156862
$653$	31.2456	1.22273	0.611367	−	0.791347i	$-0.290620\pi$
$653$	31.2456	1.22273	0.611367	+	0.791347i	$0.290620\pi$
$654$	8.03444	0.314171
$655$	−10.0058	−0.390961
$656$	−23.4512	−0.915616
$657$	15.6550	0.610761
$658$	−27.5204	−1.07286
$659$	−23.7045	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$659$	−23.7045	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$660$	0.466475	0.0181575
$661$	−19.8230	−0.771026	−0.385513	−	0.922702i	$-0.625976\pi$
$661$	−19.8230	−0.771026	−0.385513	+	0.922702i	$0.625976\pi$
$662$	34.3674	1.33573
$663$	6.00546	0.233233
$664$	4.19250	0.162700
$665$	−4.97805	−0.193041
$666$	−14.0076	−0.542785
$667$	8.01287	0.310260
$668$	−26.9516	−1.04279
$669$	5.81311	0.224748
$670$	3.62114	0.139897
$671$	−2.93900	−0.113459
$672$	21.3847	0.824934
$673$	41.4024	1.59594	0.797972	−	0.602694i	$-0.205906\pi$
$673$	41.4024	1.59594	0.797972	+	0.602694i	$0.205906\pi$
$674$	19.0224	0.732714
$675$	−4.80080	−0.184783
$676$	31.9274	1.22798
$677$	40.4080	1.55301	0.776503	−	0.630114i	$-0.216992\pi$
$677$	40.4080	1.55301	0.776503	+	0.630114i	$0.216992\pi$
$678$	6.44867	0.247660
$679$	−39.0454	−1.49842
$680$	−0.505605	−0.0193891
$681$	−2.81084	−0.107712
$682$	−1.30282	−0.0498875
$683$	−45.2385	−1.73100	−0.865501	−	0.500907i	$-0.833000\pi$
$683$	−45.2385	−1.73100	−0.865501	+	0.500907i	$0.833000\pi$
$684$	−4.80885	−0.183871
$685$	−8.10761	−0.309776
$686$	21.8095	0.832692
$687$	19.7529	0.753621
$688$	−56.0308	−2.13615
$689$	3.13876	0.119577
$690$	1.09673	0.0417518
$691$	−46.2740	−1.76034	−0.880172	−	0.474655i	$-0.842573\pi$
$691$	−46.2740	−1.76034	−0.880172	+	0.474655i	$0.842573\pi$
$692$	10.2962	0.391401
$693$	2.42413	0.0920850
$694$	21.9782	0.834280
$695$	1.53894	0.0583754
$696$	−6.79559	−0.257586
$697$	−4.83292	−0.183060
$698$	44.3726	1.67953
$699$	−8.79173	−0.332534
$700$	−21.3346	−0.806374
$701$	8.49922	0.321011	0.160505	−	0.987035i	$-0.448688\pi$
$701$	8.49922	0.321011	0.160505	+	0.987035i	$0.448688\pi$
$702$	11.0478	0.416971
$703$	−26.4532	−0.997702
$704$	−1.92445	−0.0725304
$705$	2.07970	0.0783260
$706$	55.9426	2.10543
$707$	−29.5665	−1.11196
$708$	−4.97871	−0.187111
$709$	−12.5160	−0.470048	−0.235024	−	0.971990i	$-0.575517\pi$
$709$	−12.5160	−0.470048	−0.235024	+	0.971990i	$0.575517\pi$
$710$	8.34735	0.313270
$711$	−15.3608	−0.576075
$712$	1.52970	0.0573281
$713$	−1.25285	−0.0469196
$714$	5.90609	0.221030
$715$	−2.02383	−0.0756871
$716$	23.8699	0.892061
$717$	−11.7069	−0.437203
$718$	3.51489	0.131175
$719$	13.9248	0.519306	0.259653	−	0.965702i	$-0.416392\pi$
$719$	13.9248	0.519306	0.259653	+	0.965702i	$0.416392\pi$
$720$	−2.16571	−0.0807114
$721$	41.5218	1.54635
$722$	12.7498	0.474497
$723$	21.6664	0.805780
$724$	2.68034	0.0996140
$725$	28.7988	1.06956
$726$	19.1870	0.712097
$727$	20.0022	0.741839	0.370919	−	0.928665i	$-0.379043\pi$
$727$	20.0022	0.741839	0.370919	+	0.928665i	$0.379043\pi$
$728$	−21.8416	−0.809503
$729$	1.00000	0.0370370
$730$	−12.8537	−0.475736
$731$	−11.5470	−0.427083
$732$	−5.38785	−0.199141
$733$	−34.0639	−1.25818	−0.629090	−	0.777332i	$-0.716572\pi$
$733$	−34.0639	−1.25818	−0.629090	+	0.777332i	$0.716572\pi$
$734$	33.5658	1.23894
$735$	1.47610	0.0544469
$736$	−8.89731	−0.327959
$737$	−3.33009	−0.122665
$738$	−8.89074	−0.327273
$739$	−23.0416	−0.847600	−0.423800	−	0.905756i	$-0.639304\pi$
$739$	−23.0416	−0.847600	−0.423800	+	0.905756i	$0.639304\pi$
$740$	4.70416	0.172928
$741$	20.8635	0.766441
$742$	3.08683	0.113321
$743$	33.6432	1.23425	0.617126	−	0.786865i	$-0.288297\pi$
$743$	33.6432	1.23425	0.617126	+	0.786865i	$0.288297\pi$
$744$	1.06252	0.0389540
$745$	−3.74766	−0.137304
$746$	−44.0031	−1.61107
$747$	3.70090	0.135409
$748$	−1.04516	−0.0382149
$749$	7.72347	0.282209
$750$	8.04702	0.293836
$751$	−8.22992	−0.300314	−0.150157	−	0.988662i	$-0.547978\pi$
$751$	−8.22992	−0.300314	−0.150157	+	0.988662i	$0.547978\pi$
$752$	−22.6105	−0.824520
$753$	−16.4457	−0.599315
$754$	−66.2729	−2.41352
$755$	2.10352	0.0765548
$756$	4.44398	0.161626
$757$	36.0285	1.30948	0.654738	−	0.755856i	$-0.272779\pi$
$757$	36.0285	1.30948	0.654738	+	0.755856i	$0.272779\pi$
$758$	50.9877	1.85196
$759$	−1.00858	−0.0366091
$760$	−1.75652	−0.0637157
$761$	−9.75040	−0.353452	−0.176726	−	0.984260i	$-0.556551\pi$
$761$	−9.75040	−0.353452	−0.176726	+	0.984260i	$0.556551\pi$
$762$	−15.1866	−0.550154
$763$	−14.0217	−0.507618
$764$	−24.2993	−0.879119
$765$	−0.446319	−0.0161367
$766$	41.9169	1.51452
$767$	21.6005	0.779947
$768$	20.9791	0.757016
$769$	9.07378	0.327209	0.163604	−	0.986526i	$-0.447688\pi$
$769$	9.07378	0.327209	0.163604	+	0.986526i	$0.447688\pi$
$770$	−1.99035	−0.0717272
$771$	−2.86778	−0.103281
$772$	−24.2156	−0.871538
$773$	−10.3566	−0.372499	−0.186250	−	0.982502i	$-0.559633\pi$
$773$	−10.3566	−0.372499	−0.186250	+	0.982502i	$0.559633\pi$
$774$	−21.2422	−0.763535
$775$	−4.50284	−0.161747
$776$	−13.7773	−0.494575
$777$	24.4461	0.876997
$778$	15.4933	0.555463
$779$	−16.7900	−0.601566
$780$	−3.71015	−0.132845
$781$	−7.67641	−0.274684
$782$	−2.45728	−0.0878722
$783$	−5.99876	−0.214378
$784$	−16.0482	−0.573150
$785$	0.446319	0.0159298
$786$	41.2417	1.47104
$787$	47.6846	1.69977	0.849886	−	0.526967i	$-0.176671\pi$
$787$	47.6846	1.69977	0.849886	+	0.526967i	$0.176671\pi$
$788$	1.08190	0.0385409
$789$	12.7348	0.453372
$790$	12.6121	0.448718
$791$	−11.2542	−0.400153
$792$	0.855361	0.0303939
$793$	23.3755	0.830090
$794$	−42.5579	−1.51032
$795$	−0.233269	−0.00827321
$796$	5.14983	0.182531
$797$	−43.3536	−1.53566	−0.767831	−	0.640652i	$-0.778664\pi$
$797$	−43.3536	−1.53566	−0.767831	+	0.640652i	$0.778664\pi$
$798$	20.5183	0.726341
$799$	−4.65967	−0.164847
$800$	−31.9776	−1.13058
$801$	1.35034	0.0477118
$802$	−13.1051	−0.462755
$803$	11.8205	0.417138
$804$	−6.10481	−0.215300
$805$	−1.91401	−0.0674600
$806$	10.3621	0.364989
$807$	−22.2195	−0.782163
$808$	−10.4326	−0.367018
$809$	27.1159	0.953345	0.476672	−	0.879081i	$-0.341843\pi$
$809$	27.1159	0.953345	0.476672	+	0.879081i	$0.341843\pi$
$810$	−0.821058	−0.0288490
$811$	46.4477	1.63100	0.815500	−	0.578758i	$-0.196462\pi$
$811$	46.4477	1.63100	0.815500	+	0.578758i	$0.196462\pi$
$812$	−26.6584	−0.935525
$813$	6.43735	0.225768
$814$	−10.5767	−0.370712
$815$	−1.29148	−0.0452386
$816$	4.85239	0.169868
$817$	−40.1156	−1.40347
$818$	7.53277	0.263377
$819$	−19.2805	−0.673715
$820$	2.98576	0.104267
$821$	−34.2620	−1.19575	−0.597876	−	0.801589i	$-0.703988\pi$
$821$	−34.2620	−1.19575	−0.597876	+	0.801589i	$0.703988\pi$
$822$	33.4176	1.16557
$823$	−21.4199	−0.746651	−0.373326	−	0.927700i	$-0.621783\pi$
$823$	−21.4199	−0.746651	−0.373326	+	0.927700i	$0.621783\pi$
$824$	14.6511	0.510394
$825$	−3.62491	−0.126203
$826$	21.2431	0.739141
$827$	24.3985	0.848419	0.424210	−	0.905564i	$-0.360552\pi$
$827$	24.3985	0.848419	0.424210	+	0.905564i	$0.360552\pi$
$828$	−1.84895	−0.0642556
$829$	55.4033	1.92424	0.962119	−	0.272631i	$-0.0878937\pi$
$829$	55.4033	1.92424	0.962119	+	0.272631i	$0.0878937\pi$
$830$	−3.03865	−0.105473
$831$	12.7060	0.440767
$832$	15.3063	0.530649
$833$	−3.30728	−0.114590
$834$	−6.34315	−0.219645
$835$	−8.69020	−0.300737
$836$	−3.63099	−0.125580
$837$	0.937935	0.0324198
$838$	7.53692	0.260359
$839$	−35.7078	−1.23277	−0.616384	−	0.787445i	$-0.711403\pi$
$839$	−35.7078	−1.23277	−0.616384	+	0.787445i	$0.711403\pi$
$840$	1.62324	0.0560072
$841$	6.98513	0.240867
$842$	−30.8932	−1.06465
$843$	−20.0757	−0.691443
$844$	−4.08495	−0.140610
$845$	10.2946	0.354144
$846$	−8.57202	−0.294712
$847$	−33.4851	−1.15056
$848$	2.53611	0.0870903
$849$	−17.1401	−0.588247
$850$	−8.83165	−0.302923
$851$	−10.1710	−0.348657
$852$	−14.0726	−0.482120
$853$	−38.9845	−1.33480	−0.667402	−	0.744697i	$-0.732594\pi$
$853$	−38.9845	−1.33480	−0.667402	+	0.744697i	$0.732594\pi$
$854$	22.9888	0.786660
$855$	−1.55056	−0.0530279
$856$	2.72525	0.0931471
$857$	−7.36037	−0.251425	−0.125713	−	0.992067i	$-0.540122\pi$
$857$	−7.36037	−0.251425	−0.125713	+	0.992067i	$0.540122\pi$
$858$	8.34176	0.284783
$859$	7.23709	0.246926	0.123463	−	0.992349i	$-0.460600\pi$
$859$	7.23709	0.246926	0.123463	+	0.992349i	$0.460600\pi$
$860$	7.13372	0.243258
$861$	15.5161	0.528786
$862$	70.6108	2.40501
$863$	44.5640	1.51697	0.758487	−	0.651688i	$-0.225939\pi$
$863$	44.5640	1.51697	0.758487	+	0.651688i	$0.225939\pi$
$864$	6.66089	0.226608
$865$	3.31987	0.112879
$866$	−38.9720	−1.32432
$867$	1.00000	0.0339618
$868$	4.16816	0.141477
$869$	−11.5984	−0.393448
$870$	4.92533	0.166984
$871$	26.4861	0.897448
$872$	−4.94758	−0.167546
$873$	−12.1618	−0.411614
$874$	−8.53683	−0.288763
$875$	−14.0436	−0.474761
$876$	21.6698	0.732153
$877$	58.5585	1.97738	0.988690	−	0.149974i	$-0.0479190\pi$
$877$	58.5585	1.97738	0.988690	+	0.149974i	$0.0479190\pi$
$878$	9.37973	0.316551
$879$	14.7028	0.495912
$880$	−1.63525	−0.0551243
$881$	−14.2452	−0.479932	−0.239966	−	0.970781i	$-0.577136\pi$
$881$	−14.2452	−0.479932	−0.239966	+	0.970781i	$0.577136\pi$
$882$	−6.08414	−0.204864
$883$	36.4077	1.22522	0.612609	−	0.790386i	$-0.290120\pi$
$883$	36.4077	1.22522	0.612609	+	0.790386i	$0.290120\pi$
$884$	8.31277	0.279589
$885$	−1.60532	−0.0539623
$886$	−17.0724	−0.573558
$887$	1.35943	0.0456453	0.0228226	−	0.999740i	$-0.492735\pi$
$887$	1.35943	0.0456453	0.0228226	+	0.999740i	$0.492735\pi$
$888$	8.62586	0.289465
$889$	26.5036	0.888904
$890$	−1.10870	−0.0371638
$891$	0.755063	0.0252956
$892$	8.04653	0.269418
$893$	−16.1881	−0.541715
$894$	15.4470	0.516624
$895$	7.69656	0.257268
$896$	−27.7165	−0.925943
$897$	8.02181	0.267841
$898$	−48.0511	−1.60349
$899$	−5.62645	−0.187652
$900$	−6.64528	−0.221509
$901$	0.522651	0.0174120
$902$	−6.71308	−0.223521
$903$	37.0717	1.23367
$904$	−3.97107	−0.132076
$905$	0.864242	0.0287284
$906$	−8.67019	−0.288048
$907$	17.4641	0.579887	0.289943	−	0.957044i	$-0.406364\pi$
$907$	17.4641	0.579887	0.289943	+	0.957044i	$0.406364\pi$
$908$	−3.89078	−0.129120
$909$	−9.20932	−0.305454
$910$	15.8304	0.524773
$911$	32.8539	1.08850	0.544249	−	0.838923i	$-0.316815\pi$
$911$	32.8539	1.08850	0.544249	+	0.838923i	$0.316815\pi$
$912$	16.8577	0.558213
$913$	2.79441	0.0924815
$914$	58.6757	1.94082
$915$	−1.73725	−0.0574316
$916$	27.3421	0.903407
$917$	−71.9748	−2.37682
$918$	1.83962	0.0607165
$919$	−22.6488	−0.747114	−0.373557	−	0.927607i	$-0.621862\pi$
$919$	−22.6488	−0.747114	−0.373557	+	0.927607i	$0.621862\pi$
$920$	−0.675364	−0.0222661
$921$	−12.8210	−0.422468
$922$	32.9153	1.08401
$923$	61.0549	2.00965
$924$	3.35549	0.110387
$925$	−36.5553	−1.20193
$926$	28.5613	0.938583
$927$	12.9331	0.424780
$928$	−39.9571	−1.31165
$929$	−4.60508	−0.151088	−0.0755438	−	0.997142i	$-0.524069\pi$
$929$	−4.60508	−0.151088	−0.0755438	+	0.997142i	$0.524069\pi$
$930$	−0.770098	−0.0252525
$931$	−11.4898	−0.376563
$932$	−12.1695	−0.398626
$933$	−10.9232	−0.357610
$934$	52.8374	1.72889
$935$	−0.336999	−0.0110211
$936$	−6.80318	−0.222369
$937$	−37.9397	−1.23943	−0.619717	−	0.784825i	$-0.712753\pi$
$937$	−37.9397	−1.23943	−0.619717	+	0.784825i	$0.712753\pi$
$938$	26.0479	0.850493
$939$	−19.8697	−0.648424
$940$	2.87872	0.0938936
$941$	−39.5174	−1.28823	−0.644116	−	0.764928i	$-0.722774\pi$
$941$	−39.5174	−1.28823	−0.644116	+	0.764928i	$0.722774\pi$
$942$	−1.83962	−0.0599381
$943$	−6.45560	−0.210223
$944$	17.4531	0.568050
$945$	1.43291	0.0466124
$946$	−16.0392	−0.521479
$947$	46.1968	1.50119	0.750597	−	0.660760i	$-0.229766\pi$
$947$	46.1968	1.50119	0.750597	+	0.660760i	$0.229766\pi$
$948$	−21.2624	−0.690572
$949$	−94.0157	−3.05188
$950$	−30.6820	−0.995455
$951$	8.79359	0.285152
$952$	−3.63695	−0.117874
$953$	−29.7051	−0.962242	−0.481121	−	0.876654i	$-0.659770\pi$
$953$	−29.7051	−0.962242	−0.481121	+	0.876654i	$0.659770\pi$
$954$	0.961480	0.0311291
$955$	−7.83502	−0.253535
$956$	−16.2048	−0.524099
$957$	−4.52945	−0.146416
$958$	−8.04230	−0.259835
$959$	−58.3203	−1.88326
$960$	−1.13755	−0.0367141
$961$	−30.1203	−0.971622
$962$	84.1223	2.71221
$963$	2.40569	0.0775224
$964$	29.9906	0.965933
$965$	−7.80802	−0.251349
$966$	7.88909	0.253827
$967$	−9.07931	−0.291971	−0.145986	−	0.989287i	$-0.546635\pi$
$967$	−9.07931	−0.291971	−0.145986	+	0.989287i	$0.546635\pi$
$968$	−11.8153	−0.379758
$969$	3.47410	0.111604
$970$	9.98553	0.320616
$971$	13.8983	0.446018	0.223009	−	0.974816i	$-0.428412\pi$
$971$	13.8983	0.446018	0.223009	+	0.974816i	$0.428412\pi$
$972$	1.38420	0.0443983
$973$	11.0700	0.354889
$974$	24.8793	0.797185
$975$	28.8310	0.923331
$976$	18.8874	0.604570
$977$	−23.1988	−0.742196	−0.371098	−	0.928594i	$-0.621019\pi$
$977$	−23.1988	−0.742196	−0.371098	+	0.928594i	$0.621019\pi$
$978$	5.32318	0.170216
$979$	1.01959	0.0325862
$980$	2.04323	0.0652685
$981$	−4.36744	−0.139442
$982$	−62.3903	−1.99095
$983$	−4.95426	−0.158016	−0.0790082	−	0.996874i	$-0.525175\pi$
$983$	−4.95426	−0.158016	−0.0790082	+	0.996874i	$0.525175\pi$
$984$	5.47489	0.174533
$985$	0.348844	0.0111151
$986$	−11.0354	−0.351440
$987$	14.9598	0.476177
$988$	28.8794	0.918775
$989$	−15.4240	−0.490455
$990$	−0.619951	−0.0197033
$991$	−30.3631	−0.964516	−0.482258	−	0.876029i	$-0.660183\pi$
$991$	−30.3631	−0.964516	−0.482258	+	0.876029i	$0.660183\pi$
$992$	6.24748	0.198358
$993$	−18.6818	−0.592848
$994$	60.0448	1.90450
$995$	1.66050	0.0526414
$996$	5.12279	0.162322
$997$	−33.7430	−1.06865	−0.534326	−	0.845278i	$-0.679435\pi$
$997$	−33.7430	−1.06865	−0.534326	+	0.845278i	$0.679435\pi$
$998$	−25.6850	−0.813044
$999$	7.61442	0.240910

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.e.1.10	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.e.1.10	✓	46	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83962 q^{2} +1.00000 q^{3} +1.38420 q^{4} +0.446319 q^{5} -1.83962 q^{6} +3.21050 q^{7} +1.13283 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.83962 q^{2} +1.00000 q^{3} +1.38420 q^{4} +0.446319 q^{5} -1.83962 q^{6} +3.21050 q^{7} +1.13283 q^{8} +1.00000 q^{9} -0.821058 q^{10} +0.755063 q^{11} +1.38420 q^{12} -6.00546 q^{13} -5.90609 q^{14} +0.446319 q^{15} -4.85239 q^{16} -1.00000 q^{17} -1.83962 q^{18} -3.47410 q^{19} +0.617796 q^{20} +3.21050 q^{21} -1.38903 q^{22} -1.33575 q^{23} +1.13283 q^{24} -4.80080 q^{25} +11.0478 q^{26} +1.00000 q^{27} +4.44398 q^{28} -5.99876 q^{29} -0.821058 q^{30} +0.937935 q^{31} +6.66089 q^{32} +0.755063 q^{33} +1.83962 q^{34} +1.43291 q^{35} +1.38420 q^{36} +7.61442 q^{37} +6.39102 q^{38} -6.00546 q^{39} +0.505605 q^{40} +4.83292 q^{41} -5.90609 q^{42} +11.5470 q^{43} +1.04516 q^{44} +0.446319 q^{45} +2.45728 q^{46} +4.65967 q^{47} -4.85239 q^{48} +3.30728 q^{49} +8.83165 q^{50} -1.00000 q^{51} -8.31277 q^{52} -0.522651 q^{53} -1.83962 q^{54} +0.336999 q^{55} +3.63695 q^{56} -3.47410 q^{57} +11.0354 q^{58} -3.59680 q^{59} +0.617796 q^{60} -3.89238 q^{61} -1.72544 q^{62} +3.21050 q^{63} -2.54873 q^{64} -2.68035 q^{65} -1.38903 q^{66} -4.41034 q^{67} -1.38420 q^{68} -1.33575 q^{69} -2.63600 q^{70} -10.1666 q^{71} +1.13283 q^{72} +15.6550 q^{73} -14.0076 q^{74} -4.80080 q^{75} -4.80885 q^{76} +2.42413 q^{77} +11.0478 q^{78} -15.3608 q^{79} -2.16571 q^{80} +1.00000 q^{81} -8.89074 q^{82} +3.70090 q^{83} +4.44398 q^{84} -0.446319 q^{85} -21.2422 q^{86} -5.99876 q^{87} +0.855361 q^{88} +1.35034 q^{89} -0.821058 q^{90} -19.2805 q^{91} -1.84895 q^{92} +0.937935 q^{93} -8.57202 q^{94} -1.55056 q^{95} +6.66089 q^{96} -12.1618 q^{97} -6.08414 q^{98} +0.755063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more