LMFDB - Embedded newform 3997.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3997,2,Mod(1,3997)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3997, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3997.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3997 = 7 \cdot 571 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3997.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:	$31.9162056879$
Analytic rank:	$1$
Dimension:	$73$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	3997.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50126 q^{2} -0.776277 q^{3} +4.25628 q^{4} +3.25093 q^{5} +1.94167 q^{6} -1.00000 q^{7} -5.64353 q^{8} -2.39739 q^{9} +O(q^{10})$	\(q-2.50126 q^{2} -0.776277 q^{3} +4.25628 q^{4} +3.25093 q^{5} +1.94167 q^{6} -1.00000 q^{7} -5.64353 q^{8} -2.39739 q^{9} -8.13141 q^{10} +5.04145 q^{11} -3.30405 q^{12} -5.52713 q^{13} +2.50126 q^{14} -2.52362 q^{15} +5.60335 q^{16} -6.87634 q^{17} +5.99650 q^{18} -0.997175 q^{19} +13.8369 q^{20} +0.776277 q^{21} -12.6100 q^{22} +4.96241 q^{23} +4.38094 q^{24} +5.56854 q^{25} +13.8248 q^{26} +4.18987 q^{27} -4.25628 q^{28} -2.00137 q^{29} +6.31222 q^{30} -0.946519 q^{31} -2.72835 q^{32} -3.91356 q^{33} +17.1995 q^{34} -3.25093 q^{35} -10.2040 q^{36} -3.78926 q^{37} +2.49419 q^{38} +4.29058 q^{39} -18.3467 q^{40} +7.87522 q^{41} -1.94167 q^{42} +11.1969 q^{43} +21.4578 q^{44} -7.79376 q^{45} -12.4123 q^{46} -3.88998 q^{47} -4.34975 q^{48} +1.00000 q^{49} -13.9283 q^{50} +5.33794 q^{51} -23.5250 q^{52} +10.7237 q^{53} -10.4799 q^{54} +16.3894 q^{55} +5.64353 q^{56} +0.774084 q^{57} +5.00593 q^{58} -0.0493753 q^{59} -10.7412 q^{60} -6.11444 q^{61} +2.36749 q^{62} +2.39739 q^{63} -4.38240 q^{64} -17.9683 q^{65} +9.78882 q^{66} -1.63213 q^{67} -29.2676 q^{68} -3.85220 q^{69} +8.13141 q^{70} +12.3539 q^{71} +13.5298 q^{72} -5.28800 q^{73} +9.47792 q^{74} -4.32273 q^{75} -4.24426 q^{76} -5.04145 q^{77} -10.7318 q^{78} +0.795173 q^{79} +18.2161 q^{80} +3.93968 q^{81} -19.6979 q^{82} -14.2623 q^{83} +3.30405 q^{84} -22.3545 q^{85} -28.0064 q^{86} +1.55362 q^{87} -28.4516 q^{88} -13.5814 q^{89} +19.4942 q^{90} +5.52713 q^{91} +21.1214 q^{92} +0.734761 q^{93} +9.72983 q^{94} -3.24175 q^{95} +2.11796 q^{96} -15.2307 q^{97} -2.50126 q^{98} -12.0864 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9}+O(q^{10}) $ 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9	\( 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9} - 8 q^{10} - 24 q^{11} - 33 q^{12} - 24 q^{13} + 12 q^{14} - 9 q^{15} + 76 q^{16} - 31 q^{17} - 42 q^{18} - 31 q^{19} - 54 q^{20} + 18 q^{21} - 4 q^{22} - 30 q^{23} - 6 q^{24} + 71 q^{25} - 22 q^{26} - 78 q^{27} - 74 q^{28} + 39 q^{29} - 13 q^{30} - 30 q^{31} - 100 q^{32} - 51 q^{33} - 10 q^{34} + 22 q^{35} + 97 q^{36} - 22 q^{37} - 61 q^{38} - 4 q^{39} - 15 q^{40} - 49 q^{41} + 7 q^{42} - 50 q^{43} - 21 q^{44} - 86 q^{45} + 17 q^{46} - 73 q^{47} - 70 q^{48} + 73 q^{49} - 50 q^{50} - 47 q^{51} - 82 q^{52} + 4 q^{53} - 8 q^{54} - 34 q^{55} + 39 q^{56} - 27 q^{57} - 27 q^{58} - 118 q^{59} - 24 q^{60} - 25 q^{61} - 61 q^{62} - 81 q^{63} + 77 q^{64} - 32 q^{65} - 46 q^{66} - 71 q^{67} - 92 q^{68} - 21 q^{69} + 8 q^{70} + 2 q^{71} - 141 q^{72} - 54 q^{73} - 12 q^{74} - 66 q^{75} - 36 q^{76} + 24 q^{77} - 28 q^{78} - 6 q^{79} - 133 q^{80} + 101 q^{81} - 38 q^{82} - 214 q^{83} + 33 q^{84} - 21 q^{85} - 34 q^{86} - 115 q^{87} + 3 q^{88} - 60 q^{89} - 67 q^{90} + 24 q^{91} - 99 q^{92} - 15 q^{93} - 11 q^{94} + q^{95} - 43 q^{96} - 48 q^{97} - 12 q^{98} - 55 q^{99}+O(q^{100}) \) 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9 - 8 * q^10 - 24 * q^11 - 33 * q^12 - 24 * q^13 + 12 * q^14 - 9 * q^15 + 76 * q^16 - 31 * q^17 - 42 * q^18 - 31 * q^19 - 54 * q^20 + 18 * q^21 - 4 * q^22 - 30 * q^23 - 6 * q^24 + 71 * q^25 - 22 * q^26 - 78 * q^27 - 74 * q^28 + 39 * q^29 - 13 * q^30 - 30 * q^31 - 100 * q^32 - 51 * q^33 - 10 * q^34 + 22 * q^35 + 97 * q^36 - 22 * q^37 - 61 * q^38 - 4 * q^39 - 15 * q^40 - 49 * q^41 + 7 * q^42 - 50 * q^43 - 21 * q^44 - 86 * q^45 + 17 * q^46 - 73 * q^47 - 70 * q^48 + 73 * q^49 - 50 * q^50 - 47 * q^51 - 82 * q^52 + 4 * q^53 - 8 * q^54 - 34 * q^55 + 39 * q^56 - 27 * q^57 - 27 * q^58 - 118 * q^59 - 24 * q^60 - 25 * q^61 - 61 * q^62 - 81 * q^63 + 77 * q^64 - 32 * q^65 - 46 * q^66 - 71 * q^67 - 92 * q^68 - 21 * q^69 + 8 * q^70 + 2 * q^71 - 141 * q^72 - 54 * q^73 - 12 * q^74 - 66 * q^75 - 36 * q^76 + 24 * q^77 - 28 * q^78 - 6 * q^79 - 133 * q^80 + 101 * q^81 - 38 * q^82 - 214 * q^83 + 33 * q^84 - 21 * q^85 - 34 * q^86 - 115 * q^87 + 3 * q^88 - 60 * q^89 - 67 * q^90 + 24 * q^91 - 99 * q^92 - 15 * q^93 - 11 * q^94 + q^95 - 43 * q^96 - 48 * q^97 - 12 * q^98 - 55 * 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 $

$2$	−2.50126	−1.76865	−0.884327	−	0.466867i	$-0.845383\pi$
$2$	−2.50126	−1.76865	−0.884327	+	0.466867i	$0.845383\pi$
$3$	−0.776277	−0.448184	−0.224092	−	0.974568i	$-0.571942\pi$
$3$	−0.776277	−0.448184	−0.224092	+	0.974568i	$0.571942\pi$
$4$	4.25628	2.12814
$5$	3.25093	1.45386	0.726930	−	0.686712i	$-0.240946\pi$
$5$	3.25093	1.45386	0.726930	+	0.686712i	$0.240946\pi$
$6$	1.94167	0.792682
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−5.64353	−1.99529
$9$	−2.39739	−0.799132
$10$	−8.13141	−2.57138
$11$	5.04145	1.52006	0.760028	−	0.649891i	$-0.225185\pi$
$11$	5.04145	1.52006	0.760028	+	0.649891i	$0.225185\pi$
$12$	−3.30405	−0.953797
$13$	−5.52713	−1.53295	−0.766475	−	0.642274i	$-0.777991\pi$
$13$	−5.52713	−1.53295	−0.766475	+	0.642274i	$0.777991\pi$
$14$	2.50126	0.668489
$15$	−2.52362	−0.651596
$16$	5.60335	1.40084
$17$	−6.87634	−1.66776	−0.833879	−	0.551947i	$-0.813885\pi$
$17$	−6.87634	−1.66776	−0.833879	+	0.551947i	$0.813885\pi$
$18$	5.99650	1.41339
$19$	−0.997175	−0.228768	−0.114384	−	0.993437i	$-0.536489\pi$
$19$	−0.997175	−0.228768	−0.114384	+	0.993437i	$0.536489\pi$
$20$	13.8369	3.09402
$21$	0.776277	0.169397
$22$	−12.6100	−2.68845
$23$	4.96241	1.03473	0.517367	−	0.855764i	$-0.326912\pi$
$23$	4.96241	1.03473	0.517367	+	0.855764i	$0.326912\pi$
$24$	4.38094	0.894256
$25$	5.56854	1.11371
$26$	13.8248	2.71126
$27$	4.18987	0.806341
$28$	−4.25628	−0.804361
$29$	−2.00137	−0.371645	−0.185822	−	0.982583i	$-0.559495\pi$
$29$	−2.00137	−0.371645	−0.185822	+	0.982583i	$0.559495\pi$
$30$	6.31222	1.15245
$31$	−0.946519	−0.170000	−0.0850000	−	0.996381i	$-0.527089\pi$
$31$	−0.946519	−0.170000	−0.0850000	+	0.996381i	$0.527089\pi$
$32$	−2.72835	−0.482309
$33$	−3.91356	−0.681264
$34$	17.1995	2.94969
$35$	−3.25093	−0.549507
$36$	−10.2040	−1.70066
$37$	−3.78926	−0.622951	−0.311476	−	0.950254i	$-0.600823\pi$
$37$	−3.78926	−0.622951	−0.311476	+	0.950254i	$0.600823\pi$
$38$	2.49419	0.404611
$39$	4.29058	0.687043
$40$	−18.3467	−2.90087
$41$	7.87522	1.22990	0.614951	−	0.788565i	$-0.289176\pi$
$41$	7.87522	1.22990	0.614951	+	0.788565i	$0.289176\pi$
$42$	−1.94167	−0.299606
$43$	11.1969	1.70752	0.853759	−	0.520668i	$-0.174317\pi$
$43$	11.1969	1.70752	0.853759	+	0.520668i	$0.174317\pi$
$44$	21.4578	3.23489
$45$	−7.79376	−1.16183
$46$	−12.4123	−1.83009
$47$	−3.88998	−0.567412	−0.283706	−	0.958911i	$-0.591564\pi$
$47$	−3.88998	−0.567412	−0.283706	+	0.958911i	$0.591564\pi$
$48$	−4.34975	−0.627832
$49$	1.00000	0.142857
$50$	−13.9283	−1.96977
$51$	5.33794	0.747462
$52$	−23.5250	−3.26233
$53$	10.7237	1.47301	0.736503	−	0.676434i	$-0.236476\pi$
$53$	10.7237	1.47301	0.736503	+	0.676434i	$0.236476\pi$
$54$	−10.4799	−1.42614
$55$	16.3894	2.20995
$56$	5.64353	0.754148
$57$	0.774084	0.102530
$58$	5.00593	0.657311
$59$	−0.0493753	−0.00642811	−0.00321406	−	0.999995i	$-0.501023\pi$
$59$	−0.0493753	−0.00642811	−0.00321406	+	0.999995i	$0.501023\pi$
$60$	−10.7412	−1.38669
$61$	−6.11444	−0.782873	−0.391437	−	0.920205i	$-0.628022\pi$
$61$	−6.11444	−0.782873	−0.391437	+	0.920205i	$0.628022\pi$
$62$	2.36749	0.300671
$63$	2.39739	0.302043
$64$	−4.38240	−0.547800
$65$	−17.9683	−2.22870
$66$	9.78882	1.20492
$67$	−1.63213	−0.199396	−0.0996982	−	0.995018i	$-0.531788\pi$
$67$	−1.63213	−0.199396	−0.0996982	+	0.995018i	$0.531788\pi$
$68$	−29.2676	−3.54922
$69$	−3.85220	−0.463751
$70$	8.13141	0.971889
$71$	12.3539	1.46613	0.733067	−	0.680156i	$-0.238088\pi$
$71$	12.3539	1.46613	0.733067	+	0.680156i	$0.238088\pi$
$72$	13.5298	1.59450
$73$	−5.28800	−0.618914	−0.309457	−	0.950913i	$-0.600147\pi$
$73$	−5.28800	−0.618914	−0.309457	+	0.950913i	$0.600147\pi$
$74$	9.47792	1.10179
$75$	−4.32273	−0.499146
$76$	−4.24426	−0.486849
$77$	−5.04145	−0.574527
$78$	−10.7318	−1.21514
$79$	0.795173	0.0894639	0.0447320	−	0.998999i	$-0.485757\pi$
$79$	0.795173	0.0894639	0.0447320	+	0.998999i	$0.485757\pi$
$80$	18.2161	2.03662
$81$	3.93968	0.437743
$82$	−19.6979	−2.17527
$83$	−14.2623	−1.56549	−0.782743	−	0.622345i	$-0.786180\pi$
$83$	−14.2623	−1.56549	−0.782743	+	0.622345i	$0.786180\pi$
$84$	3.30405	0.360501
$85$	−22.3545	−2.42469
$86$	−28.0064	−3.02001
$87$	1.55362	0.166565
$88$	−28.4516	−3.03295
$89$	−13.5814	−1.43962	−0.719812	−	0.694169i	$-0.755772\pi$
$89$	−13.5814	−1.43962	−0.719812	+	0.694169i	$0.755772\pi$
$90$	19.4942	2.05487
$91$	5.52713	0.579401
$92$	21.1214	2.20206
$93$	0.734761	0.0761912
$94$	9.72983	1.00356
$95$	−3.24175	−0.332596
$96$	2.11796	0.216163
$97$	−15.2307	−1.54645	−0.773223	−	0.634135i	$-0.781356\pi$
$97$	−15.2307	−1.54645	−0.773223	+	0.634135i	$0.781356\pi$
$98$	−2.50126	−0.252665
$99$	−12.0864	−1.21472
$100$	23.7013	2.37013
$101$	−11.0280	−1.09732	−0.548661	−	0.836045i	$-0.684862\pi$
$101$	−11.0280	−1.09732	−0.548661	+	0.836045i	$0.684862\pi$
$102$	−13.3516	−1.32200
$103$	−2.19213	−0.215997	−0.107998	−	0.994151i	$-0.534444\pi$
$103$	−2.19213	−0.215997	−0.107998	+	0.994151i	$0.534444\pi$
$104$	31.1925	3.05868
$105$	2.52362	0.246280
$106$	−26.8226	−2.60524
$107$	−18.0728	−1.74716	−0.873581	−	0.486678i	$-0.838208\pi$
$107$	−18.0728	−1.74716	−0.873581	+	0.486678i	$0.838208\pi$
$108$	17.8333	1.71601
$109$	−6.16636	−0.590630	−0.295315	−	0.955400i	$-0.595425\pi$
$109$	−6.16636	−0.590630	−0.295315	+	0.955400i	$0.595425\pi$
$110$	−40.9941	−3.90863
$111$	2.94152	0.279196
$112$	−5.60335	−0.529467
$113$	17.9455	1.68817	0.844086	−	0.536208i	$-0.180144\pi$
$113$	17.9455	1.68817	0.844086	+	0.536208i	$0.180144\pi$
$114$	−1.93618	−0.181340
$115$	16.1324	1.50436
$116$	−8.51838	−0.790912
$117$	13.2507	1.22503
$118$	0.123500	0.0113691
$119$	6.87634	0.630353
$120$	14.2421	1.30012
$121$	14.4163	1.31057
$122$	15.2938	1.38463
$123$	−6.11335	−0.551222
$124$	−4.02865	−0.361783
$125$	1.84830	0.165317
$126$	−5.99650	−0.534210
$127$	−17.1977	−1.52605	−0.763026	−	0.646368i	$-0.776287\pi$
$127$	−17.1977	−1.52605	−0.763026	+	0.646368i	$0.776287\pi$
$128$	16.4182	1.45118
$129$	−8.69193	−0.765282
$130$	44.9433	3.94179
$131$	19.2550	1.68232	0.841159	−	0.540788i	$-0.181874\pi$
$131$	19.2550	1.68232	0.841159	+	0.540788i	$0.181874\pi$
$132$	−16.6572	−1.44982
$133$	0.997175	0.0864661
$134$	4.08237	0.352663
$135$	13.6210	1.17231
$136$	38.8068	3.32766
$137$	−1.18069	−0.100873	−0.0504366	−	0.998727i	$-0.516061\pi$
$137$	−1.18069	−0.100873	−0.0504366	+	0.998727i	$0.516061\pi$
$138$	9.63534	0.820215
$139$	0.210433	0.0178487	0.00892436	−	0.999960i	$-0.497159\pi$
$139$	0.210433	0.0178487	0.00892436	+	0.999960i	$0.497159\pi$
$140$	−13.8369	−1.16943
$141$	3.01970	0.254305
$142$	−30.9002	−2.59309
$143$	−27.8648	−2.33017
$144$	−13.4334	−1.11945
$145$	−6.50631	−0.540319
$146$	13.2266	1.09465
$147$	−0.776277	−0.0640262
$148$	−16.1282	−1.32573
$149$	6.70669	0.549434	0.274717	−	0.961525i	$-0.411416\pi$
$149$	6.70669	0.549434	0.274717	+	0.961525i	$0.411416\pi$
$150$	10.8123	0.882817
$151$	10.7788	0.877169	0.438585	−	0.898690i	$-0.355480\pi$
$151$	10.7788	0.877169	0.438585	+	0.898690i	$0.355480\pi$
$152$	5.62759	0.456458
$153$	16.4853	1.33276
$154$	12.6100	1.01614
$155$	−3.07707	−0.247156
$156$	18.2619	1.46212
$157$	0.297042	0.0237065	0.0118533	−	0.999930i	$-0.496227\pi$
$157$	0.297042	0.0237065	0.0118533	+	0.999930i	$0.496227\pi$
$158$	−1.98893	−0.158231
$159$	−8.32452	−0.660177
$160$	−8.86968	−0.701210
$161$	−4.96241	−0.391093
$162$	−9.85416	−0.774216
$163$	10.9206	0.855367	0.427684	−	0.903928i	$-0.359330\pi$
$163$	10.9206	0.855367	0.427684	+	0.903928i	$0.359330\pi$
$164$	33.5191	2.61740
$165$	−12.7227	−0.990462
$166$	35.6735	2.76880
$167$	−24.6201	−1.90516	−0.952582	−	0.304284i	$-0.901583\pi$
$167$	−24.6201	−1.90516	−0.952582	+	0.304284i	$0.901583\pi$
$168$	−4.38094	−0.337997
$169$	17.5492	1.34994
$170$	55.9143	4.28843
$171$	2.39062	0.182815
$172$	47.6573	3.63384
$173$	−5.91502	−0.449710	−0.224855	−	0.974392i	$-0.572191\pi$
$173$	−5.91502	−0.449710	−0.224855	+	0.974392i	$0.572191\pi$
$174$	−3.88599	−0.294596
$175$	−5.56854	−0.420942
$176$	28.2490	2.12935
$177$	0.0383289	0.00288097
$178$	33.9705	2.54620
$179$	−14.5113	−1.08463	−0.542314	−	0.840176i	$-0.682452\pi$
$179$	−14.5113	−1.08463	−0.542314	+	0.840176i	$0.682452\pi$
$180$	−33.1724	−2.47253
$181$	−14.5752	−1.08336	−0.541682	−	0.840584i	$-0.682212\pi$
$181$	−14.5752	−1.08336	−0.541682	+	0.840584i	$0.682212\pi$
$182$	−13.8248	−1.02476
$183$	4.74649	0.350871
$184$	−28.0055	−2.06459
$185$	−12.3186	−0.905684
$186$	−1.83782	−0.134756
$187$	−34.6668	−2.53508
$188$	−16.5568	−1.20753
$189$	−4.18987	−0.304768
$190$	8.10844	0.588248
$191$	−8.92613	−0.645872	−0.322936	−	0.946421i	$-0.604670\pi$
$191$	−8.92613	−0.645872	−0.322936	+	0.946421i	$0.604670\pi$
$192$	3.40195	0.245515
$193$	−4.14811	−0.298587	−0.149294	−	0.988793i	$-0.547700\pi$
$193$	−4.14811	−0.298587	−0.149294	+	0.988793i	$0.547700\pi$
$194$	38.0959	2.73513
$195$	13.9484	0.998864
$196$	4.25628	0.304020
$197$	−24.8781	−1.77249	−0.886247	−	0.463212i	$-0.846697\pi$
$197$	−24.8781	−1.77249	−0.886247	+	0.463212i	$0.846697\pi$
$198$	30.2311	2.14843
$199$	3.33653	0.236521	0.118260	−	0.992983i	$-0.462268\pi$
$199$	3.33653	0.236521	0.118260	+	0.992983i	$0.462268\pi$
$200$	−31.4262	−2.22217
$201$	1.26698	0.0893662
$202$	27.5837	1.94078
$203$	2.00137	0.140468
$204$	22.7198	1.59070
$205$	25.6018	1.78811
$206$	5.48307	0.382024
$207$	−11.8969	−0.826888
$208$	−30.9704	−2.14741
$209$	−5.02721	−0.347740
$210$	−6.31222	−0.435585
$211$	−2.17022	−0.149404	−0.0747022	−	0.997206i	$-0.523801\pi$
$211$	−2.17022	−0.149404	−0.0747022	+	0.997206i	$0.523801\pi$
$212$	45.6429	3.13476
$213$	−9.59002	−0.657097
$214$	45.2047	3.09013
$215$	36.4005	2.48249
$216$	−23.6457	−1.60888
$217$	0.946519	0.0642539
$218$	15.4236	1.04462
$219$	4.10495	0.277387
$220$	69.7579	4.70308
$221$	38.0065	2.55659
$222$	−7.35749	−0.493802
$223$	−5.32691	−0.356716	−0.178358	−	0.983966i	$-0.557079\pi$
$223$	−5.32691	−0.356716	−0.178358	+	0.983966i	$0.557079\pi$
$224$	2.72835	0.182296
$225$	−13.3500	−0.890000
$226$	−44.8863	−2.98579
$227$	−1.95506	−0.129762	−0.0648808	−	0.997893i	$-0.520667\pi$
$227$	−1.95506	−0.129762	−0.0648808	+	0.997893i	$0.520667\pi$
$228$	3.29472	0.218198
$229$	25.5971	1.69150	0.845752	−	0.533577i	$-0.179152\pi$
$229$	25.5971	1.69150	0.845752	+	0.533577i	$0.179152\pi$
$230$	−40.3514	−2.66069
$231$	3.91356	0.257494
$232$	11.2948	0.741538
$233$	−8.19142	−0.536638	−0.268319	−	0.963330i	$-0.586468\pi$
$233$	−8.19142	−0.536638	−0.268319	+	0.963330i	$0.586468\pi$
$234$	−33.1434	−2.16665
$235$	−12.6461	−0.824937
$236$	−0.210155	−0.0136799
$237$	−0.617274	−0.0400963
$238$	−17.1995	−1.11488
$239$	10.4038	0.672963	0.336482	−	0.941690i	$-0.390763\pi$
$239$	10.4038	0.672963	0.336482	+	0.941690i	$0.390763\pi$
$240$	−14.1407	−0.912780
$241$	16.4304	1.05838	0.529189	−	0.848504i	$-0.322496\pi$
$241$	16.4304	1.05838	0.529189	+	0.848504i	$0.322496\pi$
$242$	−36.0587	−2.31794
$243$	−15.6279	−1.00253
$244$	−26.0247	−1.66606
$245$	3.25093	0.207694
$246$	15.2910	0.974922
$247$	5.51152	0.350690
$248$	5.34171	0.339199
$249$	11.0715	0.701625
$250$	−4.62306	−0.292388
$251$	−24.9228	−1.57312	−0.786558	−	0.617517i	$-0.788139\pi$
$251$	−24.9228	−1.57312	−0.786558	+	0.617517i	$0.788139\pi$
$252$	10.2040	0.642790
$253$	25.0178	1.57285
$254$	43.0159	2.69906
$255$	17.3533	1.08670
$256$	−32.3013	−2.01883
$257$	−17.5099	−1.09224	−0.546118	−	0.837708i	$-0.683895\pi$
$257$	−17.5099	−1.09224	−0.546118	+	0.837708i	$0.683895\pi$
$258$	21.7407	1.35352
$259$	3.78926	0.235453
$260$	−76.4782	−4.74297
$261$	4.79807	0.296993
$262$	−48.1617	−2.97544
$263$	0.559141	0.0344781	0.0172390	−	0.999851i	$-0.494512\pi$
$263$	0.559141	0.0344781	0.0172390	+	0.999851i	$0.494512\pi$
$264$	22.0863	1.35932
$265$	34.8618	2.14155
$266$	−2.49419	−0.152929
$267$	10.5429	0.645216
$268$	−6.94680	−0.424343
$269$	−14.7368	−0.898516	−0.449258	−	0.893402i	$-0.648312\pi$
$269$	−14.7368	−0.898516	−0.449258	+	0.893402i	$0.648312\pi$
$270$	−34.0695	−2.07341
$271$	−17.4348	−1.05909	−0.529543	−	0.848283i	$-0.677637\pi$
$271$	−17.4348	−1.05909	−0.529543	+	0.848283i	$0.677637\pi$
$272$	−38.5306	−2.33626
$273$	−4.29058	−0.259678
$274$	2.95321	0.178410
$275$	28.0736	1.69290
$276$	−16.3960	−0.986926
$277$	25.6459	1.54091	0.770456	−	0.637493i	$-0.220028\pi$
$277$	25.6459	1.54091	0.770456	+	0.637493i	$0.220028\pi$
$278$	−0.526347	−0.0315682
$279$	2.26918	0.135852
$280$	18.3467	1.09643
$281$	−3.35489	−0.200136	−0.100068	−	0.994981i	$-0.531906\pi$
$281$	−3.35489	−0.200136	−0.100068	+	0.994981i	$0.531906\pi$
$282$	−7.55304	−0.449777
$283$	−22.0834	−1.31272	−0.656360	−	0.754447i	$-0.727905\pi$
$283$	−22.0834	−1.31272	−0.656360	+	0.754447i	$0.727905\pi$
$284$	52.5815	3.12014
$285$	2.51649	0.149064
$286$	69.6969	4.12127
$287$	−7.87522	−0.464860
$288$	6.54093	0.385428
$289$	30.2841	1.78142
$290$	16.2739	0.955638
$291$	11.8233	0.693091
$292$	−22.5072	−1.31714
$293$	−32.5271	−1.90026	−0.950128	−	0.311860i	$-0.899048\pi$
$293$	−32.5271	−1.90026	−0.950128	+	0.311860i	$0.899048\pi$
$294$	1.94167	0.113240
$295$	−0.160515	−0.00934557
$296$	21.3848	1.24297
$297$	21.1230	1.22568
$298$	−16.7751	−0.971758
$299$	−27.4279	−1.58620
$300$	−18.3987	−1.06225
$301$	−11.1969	−0.645381
$302$	−26.9606	−1.55141
$303$	8.56074	0.491802
$304$	−5.58752	−0.320466
$305$	−19.8776	−1.13819
$306$	−41.2340	−2.35719
$307$	9.78513	0.558467	0.279233	−	0.960223i	$-0.409920\pi$
$307$	9.78513	0.558467	0.279233	+	0.960223i	$0.409920\pi$
$308$	−21.4578	−1.22267
$309$	1.70170	0.0968062
$310$	7.69653	0.437134
$311$	16.4733	0.934113	0.467056	−	0.884228i	$-0.345315\pi$
$311$	16.4733	0.934113	0.467056	+	0.884228i	$0.345315\pi$
$312$	−24.2140	−1.37085
$313$	−17.5560	−0.992325	−0.496162	−	0.868230i	$-0.665258\pi$
$313$	−17.5560	−0.992325	−0.496162	+	0.868230i	$0.665258\pi$
$314$	−0.742978	−0.0419287
$315$	7.79376	0.439129
$316$	3.38448	0.190392
$317$	1.30897	0.0735189	0.0367595	−	0.999324i	$-0.488296\pi$
$317$	1.30897	0.0735189	0.0367595	+	0.999324i	$0.488296\pi$
$318$	20.8218	1.16763
$319$	−10.0898	−0.564921
$320$	−14.2469	−0.796424
$321$	14.0295	0.783050
$322$	12.4123	0.691708
$323$	6.85692	0.381529
$324$	16.7684	0.931577
$325$	−30.7781	−1.70726
$326$	−27.3152	−1.51285
$327$	4.78680	0.264711
$328$	−44.4440	−2.45401
$329$	3.88998	0.214462
$330$	31.8228	1.75179
$331$	−10.8948	−0.598833	−0.299417	−	0.954122i	$-0.596792\pi$
$331$	−10.8948	−0.598833	−0.299417	+	0.954122i	$0.596792\pi$
$332$	−60.7041	−3.33157
$333$	9.08436	0.497820
$334$	61.5812	3.36958
$335$	−5.30594	−0.289895
$336$	4.34975	0.237298
$337$	−21.2372	−1.15686	−0.578432	−	0.815730i	$-0.696335\pi$
$337$	−21.2372	−1.15686	−0.578432	+	0.815730i	$0.696335\pi$
$338$	−43.8950	−2.38757
$339$	−13.9307	−0.756611
$340$	−95.1470	−5.16007
$341$	−4.77183	−0.258409
$342$	−5.97956	−0.323337
$343$	−1.00000	−0.0539949
$344$	−63.1903	−3.40699
$345$	−12.5232	−0.674228
$346$	14.7950	0.795382
$347$	26.9717	1.44791	0.723957	−	0.689845i	$-0.242321\pi$
$347$	26.9717	1.44791	0.723957	+	0.689845i	$0.242321\pi$
$348$	6.61262	0.354474
$349$	32.5997	1.74502	0.872510	−	0.488596i	$-0.162491\pi$
$349$	32.5997	1.74502	0.872510	+	0.488596i	$0.162491\pi$
$350$	13.9283	0.744502
$351$	−23.1580	−1.23608
$352$	−13.7549	−0.733136
$353$	−24.8148	−1.32076	−0.660380	−	0.750932i	$-0.729605\pi$
$353$	−24.8148	−1.32076	−0.660380	+	0.750932i	$0.729605\pi$
$354$	−0.0958703	−0.00509545
$355$	40.1616	2.13155
$356$	−57.8062	−3.06372
$357$	−5.33794	−0.282514
$358$	36.2965	1.91833
$359$	−20.2553	−1.06903	−0.534517	−	0.845158i	$-0.679506\pi$
$359$	−20.2553	−1.06903	−0.534517	+	0.845158i	$0.679506\pi$
$360$	43.9843	2.31818
$361$	−18.0056	−0.947665
$362$	36.4562	1.91610
$363$	−11.1910	−0.587375
$364$	23.5250	1.23305
$365$	−17.1909	−0.899814
$366$	−11.8722	−0.620569
$367$	9.80169	0.511644	0.255822	−	0.966724i	$-0.417654\pi$
$367$	9.80169	0.511644	0.255822	+	0.966724i	$0.417654\pi$
$368$	27.8061	1.44949
$369$	−18.8800	−0.982854
$370$	30.8120	1.60184
$371$	−10.7237	−0.556744
$372$	3.12735	0.162145
$373$	12.7955	0.662528	0.331264	−	0.943538i	$-0.392525\pi$
$373$	12.7955	0.662528	0.331264	+	0.943538i	$0.392525\pi$
$374$	86.7104	4.48369
$375$	−1.43479	−0.0740922
$376$	21.9532	1.13215
$377$	11.0618	0.569713
$378$	10.4799	0.539030
$379$	−28.3713	−1.45733	−0.728667	−	0.684868i	$-0.759860\pi$
$379$	−28.3713	−1.45733	−0.728667	+	0.684868i	$0.759860\pi$
$380$	−13.7978	−0.707811
$381$	13.3502	0.683951
$382$	22.3265	1.14232
$383$	5.96591	0.304844	0.152422	−	0.988316i	$-0.451293\pi$
$383$	5.96591	0.304844	0.152422	+	0.988316i	$0.451293\pi$
$384$	−12.7451	−0.650394
$385$	−16.3894	−0.835282
$386$	10.3755	0.528098
$387$	−26.8435	−1.36453
$388$	−64.8262	−3.29105
$389$	−13.3980	−0.679303	−0.339652	−	0.940551i	$-0.610309\pi$
$389$	−13.3980	−0.679303	−0.339652	+	0.940551i	$0.610309\pi$
$390$	−34.8885	−1.76665
$391$	−34.1232	−1.72569
$392$	−5.64353	−0.285041
$393$	−14.9472	−0.753987
$394$	62.2266	3.13493
$395$	2.58505	0.130068
$396$	−51.4429	−2.58510
$397$	11.9775	0.601134	0.300567	−	0.953761i	$-0.402824\pi$
$397$	11.9775	0.601134	0.300567	+	0.953761i	$0.402824\pi$
$398$	−8.34552	−0.418323
$399$	−0.774084	−0.0387527
$400$	31.2025	1.56012
$401$	7.10422	0.354768	0.177384	−	0.984142i	$-0.443237\pi$
$401$	7.10422	0.354768	0.177384	+	0.984142i	$0.443237\pi$
$402$	−3.16905	−0.158058
$403$	5.23154	0.260601
$404$	−46.9380	−2.33525
$405$	12.8076	0.636417
$406$	−5.00593	−0.248440
$407$	−19.1034	−0.946920
$408$	−30.1248	−1.49140
$409$	16.6484	0.823209	0.411604	−	0.911363i	$-0.364969\pi$
$409$	16.6484	0.823209	0.411604	+	0.911363i	$0.364969\pi$
$410$	−64.0366	−3.16254
$411$	0.916542	0.0452097
$412$	−9.33030	−0.459671
$413$	0.0493753	0.00242960
$414$	29.7571	1.46248
$415$	−46.3656	−2.27600
$416$	15.0800	0.739356
$417$	−0.163354	−0.00799950
$418$	12.5743	0.615031
$419$	−24.9707	−1.21990	−0.609949	−	0.792441i	$-0.708810\pi$
$419$	−24.9707	−1.21990	−0.609949	+	0.792441i	$0.708810\pi$
$420$	10.7412	0.524119
$421$	1.51207	0.0736939	0.0368470	−	0.999321i	$-0.488269\pi$
$421$	1.51207	0.0736939	0.0368470	+	0.999321i	$0.488269\pi$
$422$	5.42828	0.264245
$423$	9.32582	0.453437
$424$	−60.5192	−2.93907
$425$	−38.2912	−1.85740
$426$	23.9871	1.16218
$427$	6.11444	0.295898
$428$	−76.9228	−3.71821
$429$	21.6308	1.04434
$430$	−91.0469	−4.39067
$431$	−24.5897	−1.18444	−0.592221	−	0.805776i	$-0.701749\pi$
$431$	−24.5897	−1.18444	−0.592221	+	0.805776i	$0.701749\pi$
$432$	23.4773	1.12955
$433$	20.9589	1.00722	0.503610	−	0.863931i	$-0.332005\pi$
$433$	20.9589	1.00722	0.503610	+	0.863931i	$0.332005\pi$
$434$	−2.36749	−0.113643
$435$	5.05069	0.242162
$436$	−26.2457	−1.25694
$437$	−4.94839	−0.236714
$438$	−10.2675	−0.490602
$439$	6.04445	0.288486	0.144243	−	0.989542i	$-0.453925\pi$
$439$	6.04445	0.288486	0.144243	+	0.989542i	$0.453925\pi$
$440$	−92.4941	−4.40948
$441$	−2.39739	−0.114162
$442$	−95.0638	−4.52173
$443$	−0.767557	−0.0364677	−0.0182339	−	0.999834i	$-0.505804\pi$
$443$	−0.767557	−0.0364677	−0.0182339	+	0.999834i	$0.505804\pi$
$444$	12.5199	0.594169
$445$	−44.1521	−2.09301
$446$	13.3240	0.630907
$447$	−5.20625	−0.246247
$448$	4.38240	0.207049
$449$	−23.8747	−1.12672	−0.563359	−	0.826212i	$-0.690491\pi$
$449$	−23.8747	−1.12672	−0.563359	+	0.826212i	$0.690491\pi$
$450$	33.3918	1.57410
$451$	39.7026	1.86952
$452$	76.3811	3.59266
$453$	−8.36736	−0.393133
$454$	4.89010	0.229504
$455$	17.9683	0.842368
$456$	−4.36856	−0.204577
$457$	−23.3509	−1.09231	−0.546154	−	0.837685i	$-0.683909\pi$
$457$	−23.3509	−1.09231	−0.546154	+	0.837685i	$0.683909\pi$
$458$	−64.0249	−2.99169
$459$	−28.8110	−1.34478
$460$	68.6642	3.20148
$461$	−31.5718	−1.47045	−0.735223	−	0.677826i	$-0.762922\pi$
$461$	−31.5718	−1.47045	−0.735223	+	0.677826i	$0.762922\pi$
$462$	−9.78882	−0.455417
$463$	−9.52471	−0.442651	−0.221325	−	0.975200i	$-0.571038\pi$
$463$	−9.52471	−0.442651	−0.221325	+	0.975200i	$0.571038\pi$
$464$	−11.2144	−0.520614
$465$	2.38866	0.110771
$466$	20.4888	0.949127
$467$	4.86555	0.225151	0.112575	−	0.993643i	$-0.464090\pi$
$467$	4.86555	0.225151	0.112575	+	0.993643i	$0.464090\pi$
$468$	56.3987	2.60703
$469$	1.63213	0.0753648
$470$	31.6310	1.45903
$471$	−0.230587	−0.0106249
$472$	0.278651	0.0128259
$473$	56.4489	2.59552
$474$	1.54396	0.0709164
$475$	−5.55281	−0.254781
$476$	29.2676	1.34148
$477$	−25.7088	−1.17713
$478$	−26.0225	−1.19024
$479$	22.3716	1.02219	0.511093	−	0.859525i	$-0.329241\pi$
$479$	22.3716	1.02219	0.511093	+	0.859525i	$0.329241\pi$
$480$	6.88532	0.314271
$481$	20.9438	0.954953
$482$	−41.0967	−1.87190
$483$	3.85220	0.175281
$484$	61.3596	2.78907
$485$	−49.5140	−2.24831
$486$	39.0894	1.77313
$487$	19.2983	0.874491	0.437246	−	0.899342i	$-0.355954\pi$
$487$	19.2983	0.874491	0.437246	+	0.899342i	$0.355954\pi$
$488$	34.5070	1.56206
$489$	−8.47741	−0.383362
$490$	−8.13141	−0.367339
$491$	4.24034	0.191364	0.0956819	−	0.995412i	$-0.469497\pi$
$491$	4.24034	0.191364	0.0956819	+	0.995412i	$0.469497\pi$
$492$	−26.0201	−1.17308
$493$	13.7621	0.619813
$494$	−13.7857	−0.620249
$495$	−39.2919	−1.76604
$496$	−5.30368	−0.238142
$497$	−12.3539	−0.554147
$498$	−27.6925	−1.24093
$499$	10.8547	0.485922	0.242961	−	0.970036i	$-0.421881\pi$
$499$	10.8547	0.485922	0.242961	+	0.970036i	$0.421881\pi$
$500$	7.86686	0.351817
$501$	19.1120	0.853863
$502$	62.3384	2.78230
$503$	−8.48799	−0.378461	−0.189230	−	0.981933i	$-0.560599\pi$
$503$	−8.48799	−0.378461	−0.189230	+	0.981933i	$0.560599\pi$
$504$	−13.5298	−0.602664
$505$	−35.8511	−1.59535
$506$	−62.5758	−2.78183
$507$	−13.6230	−0.605019
$508$	−73.1983	−3.24765
$509$	−3.26431	−0.144688	−0.0723440	−	0.997380i	$-0.523048\pi$
$509$	−3.26431	−0.144688	−0.0723440	+	0.997380i	$0.523048\pi$
$510$	−43.4050	−1.92201
$511$	5.28800	0.233928
$512$	47.9574	2.11944
$513$	−4.17804	−0.184465
$514$	43.7967	1.93179
$515$	−7.12645	−0.314029
$516$	−36.9953	−1.62863
$517$	−19.6112	−0.862498
$518$	−9.47792	−0.416436
$519$	4.59169	0.201553
$520$	101.405	4.44689
$521$	4.79007	0.209857	0.104928	−	0.994480i	$-0.466539\pi$
$521$	4.79007	0.209857	0.104928	+	0.994480i	$0.466539\pi$
$522$	−12.0012	−0.525278
$523$	−11.3129	−0.494677	−0.247338	−	0.968929i	$-0.579556\pi$
$523$	−11.3129	−0.494677	−0.247338	+	0.968929i	$0.579556\pi$
$524$	81.9547	3.58021
$525$	4.32273	0.188659
$526$	−1.39855	−0.0609798
$527$	6.50859	0.283519
$528$	−21.9291	−0.954340
$529$	1.62550	0.0706738
$530$	−87.1984	−3.78765
$531$	0.118372	0.00513691
$532$	4.24426	0.184012
$533$	−43.5274	−1.88538
$534$	−26.3705	−1.14116
$535$	−58.7534	−2.54013
$536$	9.21097	0.397853
$537$	11.2648	0.486112
$538$	36.8604	1.58917
$539$	5.04145	0.217151
$540$	57.9747	2.49483
$541$	−19.5038	−0.838534	−0.419267	−	0.907863i	$-0.637713\pi$
$541$	−19.5038	−0.838534	−0.419267	+	0.907863i	$0.637713\pi$
$542$	43.6088	1.87316
$543$	11.3144	0.485545
$544$	18.7611	0.804375
$545$	−20.0464	−0.858693
$546$	10.7318	0.459281
$547$	−33.9246	−1.45051	−0.725255	−	0.688481i	$-0.758278\pi$
$547$	−33.9246	−1.45051	−0.725255	+	0.688481i	$0.758278\pi$
$548$	−5.02534	−0.214672
$549$	14.6587	0.625619
$550$	−70.2191	−2.99415
$551$	1.99571	0.0850203
$552$	21.7400	0.925316
$553$	−0.795173	−0.0338142
$554$	−64.1469	−2.72534
$555$	9.56266	0.405913
$556$	0.895663	0.0379846
$557$	27.7904	1.17752	0.588758	−	0.808309i	$-0.299617\pi$
$557$	27.7904	1.17752	0.588758	+	0.808309i	$0.299617\pi$
$558$	−5.67580	−0.240276
$559$	−61.8870	−2.61754
$560$	−18.2161	−0.769771
$561$	26.9110	1.13618
$562$	8.39145	0.353972
$563$	15.0675	0.635020	0.317510	−	0.948255i	$-0.397153\pi$
$563$	15.0675	0.635020	0.317510	+	0.948255i	$0.397153\pi$
$564$	12.8527	0.541196
$565$	58.3396	2.45437
$566$	55.2362	2.32175
$567$	−3.93968	−0.165451
$568$	−69.7194	−2.92536
$569$	−29.8968	−1.25334	−0.626669	−	0.779285i	$-0.715582\pi$
$569$	−29.8968	−1.25334	−0.626669	+	0.779285i	$0.715582\pi$
$570$	−6.29439	−0.263643
$571$	−1.00000	−0.0418487
$571$	−1.00000	−0.0418487
$572$	−118.600	−4.95893
$573$	6.92915	0.289469
$574$	19.6979	0.822176
$575$	27.6334	1.15239
$576$	10.5063	0.437764
$577$	46.2871	1.92696	0.963478	−	0.267788i	$-0.0862927\pi$
$577$	46.2871	1.92696	0.963478	+	0.267788i	$0.0862927\pi$
$578$	−75.7483	−3.15071
$579$	3.22008	0.133822
$580$	−27.6927	−1.14987
$581$	14.2623	0.591698
$582$	−29.5730	−1.22584
$583$	54.0628	2.23905
$584$	29.8430	1.23491
$585$	43.0771	1.78102
$586$	81.3587	3.36090
$587$	3.85736	0.159210	0.0796052	−	0.996826i	$-0.474634\pi$
$587$	3.85736	0.159210	0.0796052	+	0.996826i	$0.474634\pi$
$588$	−3.30405	−0.136257
$589$	0.943846	0.0388905
$590$	0.401490	0.0165291
$591$	19.3123	0.794403
$592$	−21.2326	−0.872653
$593$	−14.4651	−0.594009	−0.297005	−	0.954876i	$-0.595988\pi$
$593$	−14.4651	−0.594009	−0.297005	+	0.954876i	$0.595988\pi$
$594$	−52.8341	−2.16781
$595$	22.3545	0.916445
$596$	28.5455	1.16927
$597$	−2.59007	−0.106005
$598$	68.6041	2.80543
$599$	−7.55614	−0.308735	−0.154368	−	0.988013i	$-0.549334\pi$
$599$	−7.55614	−0.308735	−0.154368	+	0.988013i	$0.549334\pi$
$600$	24.3955	0.995940
$601$	−32.1569	−1.31171	−0.655853	−	0.754889i	$-0.727691\pi$
$601$	−32.1569	−1.31171	−0.655853	+	0.754889i	$0.727691\pi$
$602$	28.0064	1.14146
$603$	3.91286	0.159344
$604$	45.8777	1.86674
$605$	46.8662	1.90538
$606$	−21.4126	−0.869827
$607$	14.7444	0.598456	0.299228	−	0.954182i	$-0.403271\pi$
$607$	14.7444	0.598456	0.299228	+	0.954182i	$0.403271\pi$
$608$	2.72064	0.110337
$609$	−1.55362	−0.0629557
$610$	49.7189	2.01306
$611$	21.5004	0.869814
$612$	70.1661	2.83629
$613$	−25.2959	−1.02169	−0.510847	−	0.859672i	$-0.670668\pi$
$613$	−25.2959	−1.02169	−0.510847	+	0.859672i	$0.670668\pi$
$614$	−24.4751	−0.987735
$615$	−19.8741	−0.801400
$616$	28.4516	1.14635
$617$	12.6943	0.511051	0.255526	−	0.966802i	$-0.417751\pi$
$617$	12.6943	0.511051	0.255526	+	0.966802i	$0.417751\pi$
$618$	−4.25638	−0.171217
$619$	13.7642	0.553230	0.276615	−	0.960981i	$-0.410787\pi$
$619$	13.7642	0.553230	0.276615	+	0.960981i	$0.410787\pi$
$620$	−13.0969	−0.525983
$621$	20.7919	0.834348
$622$	−41.2038	−1.65212
$623$	13.5814	0.544127
$624$	24.0416	0.962436
$625$	−21.8340	−0.873362
$626$	43.9121	1.75508
$627$	3.90251	0.155851
$628$	1.26429	0.0504508
$629$	26.0563	1.03893
$630$	−19.4942	−0.776667
$631$	−48.5249	−1.93175	−0.965874	−	0.259013i	$-0.916603\pi$
$631$	−48.5249	−1.93175	−0.965874	+	0.259013i	$0.916603\pi$
$632$	−4.48758	−0.178506
$633$	1.68469	0.0669606
$634$	−3.27406	−0.130030
$635$	−55.9086	−2.21867
$636$	−35.4315	−1.40495
$637$	−5.52713	−0.218993
$638$	25.2372	0.999149
$639$	−29.6171	−1.17163
$640$	53.3744	2.10981
$641$	−1.75878	−0.0694674	−0.0347337	−	0.999397i	$-0.511058\pi$
$641$	−1.75878	−0.0694674	−0.0347337	+	0.999397i	$0.511058\pi$
$642$	−35.0913	−1.38494
$643$	7.14419	0.281739	0.140870	−	0.990028i	$-0.455010\pi$
$643$	7.14419	0.281739	0.140870	+	0.990028i	$0.455010\pi$
$644$	−21.1214	−0.832299
$645$	−28.2569	−1.11261
$646$	−17.1509	−0.674793
$647$	27.7557	1.09119	0.545595	−	0.838049i	$-0.316304\pi$
$647$	27.7557	1.09119	0.545595	+	0.838049i	$0.316304\pi$
$648$	−22.2337	−0.873423
$649$	−0.248923	−0.00977108
$650$	76.9838	3.01955
$651$	−0.734761	−0.0287976
$652$	46.4811	1.82034
$653$	25.5980	1.00173	0.500864	−	0.865526i	$-0.333016\pi$
$653$	25.5980	1.00173	0.500864	+	0.865526i	$0.333016\pi$
$654$	−11.9730	−0.468182
$655$	62.5967	2.44586
$656$	44.1276	1.72289
$657$	12.6774	0.494594
$658$	−9.72983	−0.379308
$659$	−4.17933	−0.162803	−0.0814017	−	0.996681i	$-0.525940\pi$
$659$	−4.17933	−0.162803	−0.0814017	+	0.996681i	$0.525940\pi$
$660$	−54.1514	−2.10784
$661$	−10.5499	−0.410344	−0.205172	−	0.978726i	$-0.565775\pi$
$661$	−10.5499	−0.410344	−0.205172	+	0.978726i	$0.565775\pi$
$662$	27.2507	1.05913
$663$	−29.5035	−1.14582
$664$	80.4895	3.12360
$665$	3.24175	0.125710
$666$	−22.7223	−0.880471
$667$	−9.93161	−0.384553
$668$	−104.790	−4.05445
$669$	4.13515	0.159874
$670$	13.2715	0.512723
$671$	−30.8256	−1.19001
$672$	−2.11796	−0.0817019
$673$	34.1454	1.31621	0.658103	−	0.752928i	$-0.271359\pi$
$673$	34.1454	1.31621	0.658103	+	0.752928i	$0.271359\pi$
$674$	53.1197	2.04609
$675$	23.3315	0.898029
$676$	74.6942	2.87285
$677$	46.9886	1.80592	0.902960	−	0.429726i	$-0.141390\pi$
$677$	46.9886	1.80592	0.902960	+	0.429726i	$0.141390\pi$
$678$	34.8442	1.33818
$679$	15.2307	0.584501
$680$	126.158	4.83795
$681$	1.51766	0.0581570
$682$	11.9356	0.457037
$683$	12.9134	0.494118	0.247059	−	0.969000i	$-0.420536\pi$
$683$	12.9134	0.494118	0.247059	+	0.969000i	$0.420536\pi$
$684$	10.1752	0.389057
$685$	−3.83834	−0.146655
$686$	2.50126	0.0954984
$687$	−19.8704	−0.758104
$688$	62.7404	2.39196
$689$	−59.2710	−2.25805
$690$	31.3238	1.19248
$691$	28.0207	1.06596	0.532978	−	0.846129i	$-0.321073\pi$
$691$	28.0207	1.06596	0.532978	+	0.846129i	$0.321073\pi$
$692$	−25.1760	−0.957046
$693$	12.0864	0.459123
$694$	−67.4630	−2.56086
$695$	0.684104	0.0259495
$696$	−8.76787	−0.332345
$697$	−54.1527	−2.05118
$698$	−81.5401	−3.08634
$699$	6.35881	0.240512
$700$	−23.7013	−0.895824
$701$	18.1247	0.684561	0.342280	−	0.939598i	$-0.388801\pi$
$701$	18.1247	0.684561	0.342280	+	0.939598i	$0.388801\pi$
$702$	57.9240	2.18620
$703$	3.77856	0.142511
$704$	−22.0937	−0.832686
$705$	9.81683	0.369723
$706$	62.0682	2.33597
$707$	11.0280	0.414749
$708$	0.163138	0.00613111
$709$	39.8523	1.49669	0.748343	−	0.663312i	$-0.230850\pi$
$709$	39.8523	1.49669	0.748343	+	0.663312i	$0.230850\pi$
$710$	−100.454	−3.76998
$711$	−1.90634	−0.0714934
$712$	76.6469	2.87247
$713$	−4.69702	−0.175905
$714$	13.3516	0.499670
$715$	−90.5864	−3.38774
$716$	−61.7642	−2.30824
$717$	−8.07620	−0.301611
$718$	50.6636	1.89075
$719$	34.0127	1.26846	0.634231	−	0.773144i	$-0.281317\pi$
$719$	34.0127	1.26846	0.634231	+	0.773144i	$0.281317\pi$
$720$	−43.6712	−1.62753
$721$	2.19213	0.0816391
$722$	45.0367	1.67609
$723$	−12.7546	−0.474347
$724$	−62.0359	−2.30555
$725$	−11.1447	−0.413904
$726$	27.9916	1.03886
$727$	−13.9119	−0.515964	−0.257982	−	0.966150i	$-0.583058\pi$
$727$	−13.9119	−0.515964	−0.257982	+	0.966150i	$0.583058\pi$
$728$	−31.1925	−1.15607
$729$	0.312520	0.0115748
$730$	42.9989	1.59146
$731$	−76.9941	−2.84773
$732$	20.2024	0.746702
$733$	−8.85185	−0.326951	−0.163475	−	0.986547i	$-0.552270\pi$
$733$	−8.85185	−0.326951	−0.163475	+	0.986547i	$0.552270\pi$
$734$	−24.5165	−0.904921
$735$	−2.52362	−0.0930852
$736$	−13.5392	−0.499061
$737$	−8.22831	−0.303094
$738$	47.2237	1.73833
$739$	−48.1531	−1.77134	−0.885670	−	0.464315i	$-0.846300\pi$
$739$	−48.1531	−1.77134	−0.885670	+	0.464315i	$0.846300\pi$
$740$	−52.4315	−1.92742
$741$	−4.27846	−0.157173
$742$	26.8226	0.984688
$743$	−30.9535	−1.13557	−0.567787	−	0.823175i	$-0.692200\pi$
$743$	−30.9535	−1.13557	−0.567787	+	0.823175i	$0.692200\pi$
$744$	−4.14664	−0.152023
$745$	21.8030	0.798799
$746$	−32.0049	−1.17178
$747$	34.1923	1.25103
$748$	−147.551	−5.39501
$749$	18.0728	0.660365
$750$	3.58877	0.131043
$751$	−15.0447	−0.548991	−0.274495	−	0.961588i	$-0.588511\pi$
$751$	−15.0447	−0.548991	−0.274495	+	0.961588i	$0.588511\pi$
$752$	−21.7969	−0.794852
$753$	19.3470	0.705044
$754$	−27.6684	−1.00763
$755$	35.0412	1.27528
$756$	−17.8333	−0.648589
$757$	−3.79790	−0.138037	−0.0690184	−	0.997615i	$-0.521987\pi$
$757$	−3.79790	−0.138037	−0.0690184	+	0.997615i	$0.521987\pi$
$758$	70.9638	2.57752
$759$	−19.4207	−0.704927
$760$	18.2949	0.663625
$761$	38.0989	1.38108	0.690542	−	0.723292i	$-0.257372\pi$
$761$	38.0989	1.38108	0.690542	+	0.723292i	$0.257372\pi$
$762$	−33.3923	−1.20967
$763$	6.16636	0.223237
$764$	−37.9921	−1.37451
$765$	53.5926	1.93764
$766$	−14.9223	−0.539163
$767$	0.272904	0.00985397
$768$	25.0747	0.904807
$769$	16.8151	0.606368	0.303184	−	0.952932i	$-0.401950\pi$
$769$	16.8151	0.606368	0.303184	+	0.952932i	$0.401950\pi$
$770$	40.9941	1.47732
$771$	13.5925	0.489522
$772$	−17.6555	−0.635436
$773$	6.59870	0.237339	0.118669	−	0.992934i	$-0.462137\pi$
$773$	6.59870	0.237339	0.118669	+	0.992934i	$0.462137\pi$
$774$	67.1425	2.41339
$775$	−5.27074	−0.189330
$776$	85.9550	3.08560
$777$	−2.94152	−0.105526
$778$	33.5117	1.20145
$779$	−7.85298	−0.281362
$780$	59.3682	2.12572
$781$	62.2815	2.22861
$782$	85.3509	3.05214
$783$	−8.38547	−0.299672
$784$	5.60335	0.200120
$785$	0.965663	0.0344660
$786$	37.3868	1.33354
$787$	−3.42063	−0.121932	−0.0609661	−	0.998140i	$-0.519418\pi$
$787$	−3.42063	−0.121932	−0.0609661	+	0.998140i	$0.519418\pi$
$788$	−105.888	−3.77212
$789$	−0.434048	−0.0154525
$790$	−6.46587	−0.230045
$791$	−17.9455	−0.638069
$792$	68.2097	2.42373
$793$	33.7953	1.20011
$794$	−29.9588	−1.06320
$795$	−27.0624	−0.959806
$796$	14.2012	0.503349
$797$	−0.702483	−0.0248832	−0.0124416	−	0.999923i	$-0.503960\pi$
$797$	−0.702483	−0.0248832	−0.0124416	+	0.999923i	$0.503960\pi$
$798$	1.93618	0.0685401
$799$	26.7488	0.946306
$800$	−15.1929	−0.537152
$801$	32.5599	1.15045
$802$	−17.7695	−0.627461
$803$	−26.6592	−0.940784
$804$	5.39264	0.190184
$805$	−16.1324	−0.568594
$806$	−13.0854	−0.460914
$807$	11.4398	0.402700
$808$	62.2366	2.18947
$809$	43.4985	1.52933	0.764664	−	0.644430i	$-0.222905\pi$
$809$	43.4985	1.52933	0.764664	+	0.644430i	$0.222905\pi$
$810$	−32.0352	−1.12560
$811$	49.7957	1.74856	0.874282	−	0.485418i	$-0.161333\pi$
$811$	49.7957	1.74856	0.874282	+	0.485418i	$0.161333\pi$
$812$	8.51838	0.298937
$813$	13.5342	0.474665
$814$	47.7825	1.67477
$815$	35.5021	1.24358
$816$	29.9104	1.04707
$817$	−11.1653	−0.390625
$818$	−41.6418	−1.45597
$819$	−13.2507	−0.463017
$820$	108.968	3.80534
$821$	10.4276	0.363925	0.181962	−	0.983306i	$-0.441755\pi$
$821$	10.4276	0.363925	0.181962	+	0.983306i	$0.441755\pi$
$822$	−2.29251	−0.0799603
$823$	−42.4306	−1.47904	−0.739519	−	0.673136i	$-0.764947\pi$
$823$	−42.4306	−1.47904	−0.739519	+	0.673136i	$0.764947\pi$
$824$	12.3713	0.430976
$825$	−21.7928	−0.758729
$826$	−0.123500	−0.00429712
$827$	−12.8783	−0.447822	−0.223911	−	0.974610i	$-0.571883\pi$
$827$	−12.8783	−0.447822	−0.223911	+	0.974610i	$0.571883\pi$
$828$	−50.6363	−1.75973
$829$	−0.445376	−0.0154686	−0.00773428	−	0.999970i	$-0.502462\pi$
$829$	−0.445376	−0.0154686	−0.00773428	+	0.999970i	$0.502462\pi$
$830$	115.972	4.02545
$831$	−19.9083	−0.690612
$832$	24.2221	0.839750
$833$	−6.87634	−0.238251
$834$	0.408591	0.0141484
$835$	−80.0383	−2.76984
$836$	−21.3972	−0.740038
$837$	−3.96579	−0.137078
$838$	62.4580	2.15758
$839$	18.2408	0.629743	0.314871	−	0.949134i	$-0.398039\pi$
$839$	18.2408	0.629743	0.314871	+	0.949134i	$0.398039\pi$
$840$	−14.2421	−0.491400
$841$	−24.9945	−0.861880
$842$	−3.78208	−0.130339
$843$	2.60433	0.0896977
$844$	−9.23707	−0.317953
$845$	57.0511	1.96262
$846$	−23.3262	−0.801973
$847$	−14.4163	−0.495348
$848$	60.0884	2.06344
$849$	17.1428	0.588340
$850$	95.7761	3.28509
$851$	−18.8039	−0.644588
$852$	−40.8178	−1.39839
$853$	45.1001	1.54420	0.772099	−	0.635502i	$-0.219207\pi$
$853$	45.1001	1.54420	0.772099	+	0.635502i	$0.219207\pi$
$854$	−15.2938	−0.523342
$855$	7.77175	0.265788
$856$	101.994	3.48609
$857$	−13.5937	−0.464351	−0.232175	−	0.972674i	$-0.574584\pi$
$857$	−13.5937	−0.464351	−0.232175	+	0.972674i	$0.574584\pi$
$858$	−54.1041	−1.84708
$859$	43.1103	1.47090	0.735452	−	0.677577i	$-0.236970\pi$
$859$	43.1103	1.47090	0.735452	+	0.677577i	$0.236970\pi$
$860$	154.931	5.28309
$861$	6.11335	0.208342
$862$	61.5050	2.09487
$863$	−54.8951	−1.86865	−0.934326	−	0.356420i	$-0.883997\pi$
$863$	−54.8951	−1.86865	−0.934326	+	0.356420i	$0.883997\pi$
$864$	−11.4314	−0.388905
$865$	−19.2293	−0.653816
$866$	−52.4235	−1.78142
$867$	−23.5088	−0.798402
$868$	4.02865	0.136741
$869$	4.00883	0.135990
$870$	−12.6331	−0.428301
$871$	9.02100	0.305665
$872$	34.8000	1.17848
$873$	36.5140	1.23581
$874$	12.3772	0.418665
$875$	−1.84830	−0.0624838
$876$	17.4718	0.590318
$877$	29.6069	0.999753	0.499876	−	0.866097i	$-0.333379\pi$
$877$	29.6069	0.999753	0.499876	+	0.866097i	$0.333379\pi$
$878$	−15.1187	−0.510231
$879$	25.2501	0.851664
$880$	91.8356	3.09578
$881$	26.1310	0.880374	0.440187	−	0.897906i	$-0.354912\pi$
$881$	26.1310	0.880374	0.440187	+	0.897906i	$0.354912\pi$
$882$	5.99650	0.201913
$883$	48.9766	1.64819	0.824097	−	0.566448i	$-0.191683\pi$
$883$	48.9766	1.64819	0.824097	+	0.566448i	$0.191683\pi$
$884$	161.766	5.44078
$885$	0.124604	0.00418853
$886$	1.91986	0.0644988
$887$	−37.8971	−1.27246	−0.636230	−	0.771500i	$-0.719507\pi$
$887$	−37.8971	−1.27246	−0.636230	+	0.771500i	$0.719507\pi$
$888$	−16.6005	−0.557077
$889$	17.1977	0.576793
$890$	110.436	3.70181
$891$	19.8617	0.665393
$892$	−22.6728	−0.759141
$893$	3.87899	0.129806
$894$	13.0222	0.435526
$895$	−47.1753	−1.57690
$896$	−16.4182	−0.548493
$897$	21.2916	0.710907
$898$	59.7168	1.99277
$899$	1.89433	0.0631796
$900$	−56.8213	−1.89404
$901$	−73.7395	−2.45662
$902$	−99.3062	−3.30654
$903$	8.69193	0.289249
$904$	−101.276	−3.36839
$905$	−47.3828	−1.57506
$906$	20.9289	0.695316
$907$	−13.8259	−0.459083	−0.229541	−	0.973299i	$-0.573723\pi$
$907$	−13.8259	−0.459083	−0.229541	+	0.973299i	$0.573723\pi$
$908$	−8.32127	−0.276151
$909$	26.4383	0.876905
$910$	−44.9433	−1.48986
$911$	15.5776	0.516110	0.258055	−	0.966130i	$-0.416918\pi$
$911$	15.5776	0.516110	0.258055	+	0.966130i	$0.416918\pi$
$912$	4.33746	0.143628
$913$	−71.9025	−2.37963
$914$	58.4065	1.93192
$915$	15.4305	0.510117
$916$	108.948	3.59975
$917$	−19.2550	−0.635857
$918$	72.0637	2.37845
$919$	−56.4865	−1.86332	−0.931658	−	0.363336i	$-0.881638\pi$
$919$	−56.4865	−1.86332	−0.931658	+	0.363336i	$0.881638\pi$
$920$	−91.0439	−3.00163
$921$	−7.59596	−0.250296
$922$	78.9691	2.60071
$923$	−68.2815	−2.24751
$924$	16.6572	0.547982
$925$	−21.1007	−0.693786
$926$	23.8237	0.782896
$927$	5.25539	0.172610
$928$	5.46043	0.179248
$929$	−13.3857	−0.439172	−0.219586	−	0.975593i	$-0.570471\pi$
$929$	−13.3857	−0.439172	−0.219586	+	0.975593i	$0.570471\pi$
$930$	−5.97464	−0.195916
$931$	−0.997175	−0.0326811
$932$	−34.8650	−1.14204
$933$	−12.7878	−0.418654
$934$	−12.1700	−0.398214
$935$	−112.699	−3.68566
$936$	−74.7808	−2.44429
$937$	−18.7350	−0.612046	−0.306023	−	0.952024i	$-0.598998\pi$
$937$	−18.7350	−0.612046	−0.306023	+	0.952024i	$0.598998\pi$
$938$	−4.08237	−0.133294
$939$	13.6283	0.444744
$940$	−53.8251	−1.75558
$941$	25.2482	0.823067	0.411533	−	0.911395i	$-0.364993\pi$
$941$	25.2482	0.823067	0.411533	+	0.911395i	$0.364993\pi$
$942$	0.576757	0.0187918
$943$	39.0801	1.27262
$944$	−0.276667	−0.00900474
$945$	−13.6210	−0.443090
$946$	−141.193	−4.59058
$947$	47.0892	1.53019	0.765097	−	0.643915i	$-0.222691\pi$
$947$	47.0892	1.53019	0.765097	+	0.643915i	$0.222691\pi$
$948$	−2.62729	−0.0853304
$949$	29.2275	0.948765
$950$	13.8890	0.450619
$951$	−1.01612	−0.0329500
$952$	−38.8068	−1.25774
$953$	−49.7583	−1.61183	−0.805915	−	0.592031i	$-0.798326\pi$
$953$	−49.7583	−1.61183	−0.805915	+	0.592031i	$0.798326\pi$
$954$	64.3043	2.08193
$955$	−29.0182	−0.939007
$956$	44.2813	1.43216
$957$	7.83248	0.253188
$958$	−55.9572	−1.80789
$959$	1.18069	0.0381265
$960$	11.0595	0.356944
$961$	−30.1041	−0.971100
$962$	−52.3857	−1.68898
$963$	43.3276	1.39621
$964$	69.9325	2.25237
$965$	−13.4852	−0.434104
$966$	−9.63534	−0.310012
$967$	46.3788	1.49144	0.745721	−	0.666258i	$-0.232105\pi$
$967$	46.3788	1.49144	0.745721	+	0.666258i	$0.232105\pi$
$968$	−81.3585	−2.61496
$969$	−5.32287	−0.170995
$970$	123.847	3.97649
$971$	−54.5480	−1.75053	−0.875265	−	0.483643i	$-0.839313\pi$
$971$	−54.5480	−1.75053	−0.875265	+	0.483643i	$0.839313\pi$
$972$	−66.5167	−2.13352
$973$	−0.210433	−0.00674618
$974$	−48.2701	−1.54667
$975$	23.8923	0.765166
$976$	−34.2613	−1.09668
$977$	−34.0600	−1.08968	−0.544838	−	0.838541i	$-0.683409\pi$
$977$	−34.0600	−1.08968	−0.544838	+	0.838541i	$0.683409\pi$
$978$	21.2042	0.678034
$979$	−68.4699	−2.18831
$980$	13.8369	0.442002
$981$	14.7832	0.471991
$982$	−10.6062	−0.338456
$983$	14.4864	0.462045	0.231022	−	0.972948i	$-0.425793\pi$
$983$	14.4864	0.462045	0.231022	+	0.972948i	$0.425793\pi$
$984$	34.5009	1.09985
$985$	−80.8771	−2.57696
$986$	−34.4225	−1.09624
$987$	−3.01970	−0.0961181
$988$	23.4586	0.746316
$989$	55.5638	1.76683
$990$	98.2790	3.12351
$991$	17.7119	0.562639	0.281319	−	0.959614i	$-0.409228\pi$
$991$	17.7119	0.562639	0.281319	+	0.959614i	$0.409228\pi$
$992$	2.58244	0.0819925
$993$	8.45739	0.268387
$994$	30.9002	0.980094
$995$	10.8468	0.343868
$996$	47.1232	1.49316
$997$	−14.9485	−0.473423	−0.236711	−	0.971580i	$-0.576070\pi$
$997$	−14.9485	−0.473423	−0.236711	+	0.971580i	$0.576070\pi$
$998$	−27.1503	−0.859429
$999$	−15.8765	−0.502311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3997.2.a.e.1.7	✓	73

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3997.2.a.e.1.7	✓	73	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:	\( N \)	\(=\)	\( 3997 = 7 \cdot 571 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3997.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50126 q^{2} -0.776277 q^{3} +4.25628 q^{4} +3.25093 q^{5} +1.94167 q^{6} -1.00000 q^{7} -5.64353 q^{8} -2.39739 q^{9} +O(q^{10})\)	\(q-2.50126 q^{2} -0.776277 q^{3} +4.25628 q^{4} +3.25093 q^{5} +1.94167 q^{6} -1.00000 q^{7} -5.64353 q^{8} -2.39739 q^{9} -8.13141 q^{10} +5.04145 q^{11} -3.30405 q^{12} -5.52713 q^{13} +2.50126 q^{14} -2.52362 q^{15} +5.60335 q^{16} -6.87634 q^{17} +5.99650 q^{18} -0.997175 q^{19} +13.8369 q^{20} +0.776277 q^{21} -12.6100 q^{22} +4.96241 q^{23} +4.38094 q^{24} +5.56854 q^{25} +13.8248 q^{26} +4.18987 q^{27} -4.25628 q^{28} -2.00137 q^{29} +6.31222 q^{30} -0.946519 q^{31} -2.72835 q^{32} -3.91356 q^{33} +17.1995 q^{34} -3.25093 q^{35} -10.2040 q^{36} -3.78926 q^{37} +2.49419 q^{38} +4.29058 q^{39} -18.3467 q^{40} +7.87522 q^{41} -1.94167 q^{42} +11.1969 q^{43} +21.4578 q^{44} -7.79376 q^{45} -12.4123 q^{46} -3.88998 q^{47} -4.34975 q^{48} +1.00000 q^{49} -13.9283 q^{50} +5.33794 q^{51} -23.5250 q^{52} +10.7237 q^{53} -10.4799 q^{54} +16.3894 q^{55} +5.64353 q^{56} +0.774084 q^{57} +5.00593 q^{58} -0.0493753 q^{59} -10.7412 q^{60} -6.11444 q^{61} +2.36749 q^{62} +2.39739 q^{63} -4.38240 q^{64} -17.9683 q^{65} +9.78882 q^{66} -1.63213 q^{67} -29.2676 q^{68} -3.85220 q^{69} +8.13141 q^{70} +12.3539 q^{71} +13.5298 q^{72} -5.28800 q^{73} +9.47792 q^{74} -4.32273 q^{75} -4.24426 q^{76} -5.04145 q^{77} -10.7318 q^{78} +0.795173 q^{79} +18.2161 q^{80} +3.93968 q^{81} -19.6979 q^{82} -14.2623 q^{83} +3.30405 q^{84} -22.3545 q^{85} -28.0064 q^{86} +1.55362 q^{87} -28.4516 q^{88} -13.5814 q^{89} +19.4942 q^{90} +5.52713 q^{91} +21.1214 q^{92} +0.734761 q^{93} +9.72983 q^{94} -3.24175 q^{95} +2.11796 q^{96} -15.2307 q^{97} -2.50126 q^{98} -12.0864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9}+O(q^{10}) \) 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9	\( 73 q - 12 q^{2} - 18 q^{3} + 74 q^{4} - 22 q^{5} - 7 q^{6} - 73 q^{7} - 39 q^{8} + 81 q^{9} - 8 q^{10} - 24 q^{11} - 33 q^{12} - 24 q^{13} + 12 q^{14} - 9 q^{15} + 76 q^{16} - 31 q^{17} - 42 q^{18} - 31 q^{19} - 54 q^{20} + 18 q^{21} - 4 q^{22} - 30 q^{23} - 6 q^{24} + 71 q^{25} - 22 q^{26} - 78 q^{27} - 74 q^{28} + 39 q^{29} - 13 q^{30} - 30 q^{31} - 100 q^{32} - 51 q^{33} - 10 q^{34} + 22 q^{35} + 97 q^{36} - 22 q^{37} - 61 q^{38} - 4 q^{39} - 15 q^{40} - 49 q^{41} + 7 q^{42} - 50 q^{43} - 21 q^{44} - 86 q^{45} + 17 q^{46} - 73 q^{47} - 70 q^{48} + 73 q^{49} - 50 q^{50} - 47 q^{51} - 82 q^{52} + 4 q^{53} - 8 q^{54} - 34 q^{55} + 39 q^{56} - 27 q^{57} - 27 q^{58} - 118 q^{59} - 24 q^{60} - 25 q^{61} - 61 q^{62} - 81 q^{63} + 77 q^{64} - 32 q^{65} - 46 q^{66} - 71 q^{67} - 92 q^{68} - 21 q^{69} + 8 q^{70} + 2 q^{71} - 141 q^{72} - 54 q^{73} - 12 q^{74} - 66 q^{75} - 36 q^{76} + 24 q^{77} - 28 q^{78} - 6 q^{79} - 133 q^{80} + 101 q^{81} - 38 q^{82} - 214 q^{83} + 33 q^{84} - 21 q^{85} - 34 q^{86} - 115 q^{87} + 3 q^{88} - 60 q^{89} - 67 q^{90} + 24 q^{91} - 99 q^{92} - 15 q^{93} - 11 q^{94} + q^{95} - 43 q^{96} - 48 q^{97} - 12 q^{98} - 55 q^{99}+O(q^{100}) \) 73 * q - 12 * q^2 - 18 * q^3 + 74 * q^4 - 22 * q^5 - 7 * q^6 - 73 * q^7 - 39 * q^8 + 81 * q^9 - 8 * q^10 - 24 * q^11 - 33 * q^12 - 24 * q^13 + 12 * q^14 - 9 * q^15 + 76 * q^16 - 31 * q^17 - 42 * q^18 - 31 * q^19 - 54 * q^20 + 18 * q^21 - 4 * q^22 - 30 * q^23 - 6 * q^24 + 71 * q^25 - 22 * q^26 - 78 * q^27 - 74 * q^28 + 39 * q^29 - 13 * q^30 - 30 * q^31 - 100 * q^32 - 51 * q^33 - 10 * q^34 + 22 * q^35 + 97 * q^36 - 22 * q^37 - 61 * q^38 - 4 * q^39 - 15 * q^40 - 49 * q^41 + 7 * q^42 - 50 * q^43 - 21 * q^44 - 86 * q^45 + 17 * q^46 - 73 * q^47 - 70 * q^48 + 73 * q^49 - 50 * q^50 - 47 * q^51 - 82 * q^52 + 4 * q^53 - 8 * q^54 - 34 * q^55 + 39 * q^56 - 27 * q^57 - 27 * q^58 - 118 * q^59 - 24 * q^60 - 25 * q^61 - 61 * q^62 - 81 * q^63 + 77 * q^64 - 32 * q^65 - 46 * q^66 - 71 * q^67 - 92 * q^68 - 21 * q^69 + 8 * q^70 + 2 * q^71 - 141 * q^72 - 54 * q^73 - 12 * q^74 - 66 * q^75 - 36 * q^76 + 24 * q^77 - 28 * q^78 - 6 * q^79 - 133 * q^80 + 101 * q^81 - 38 * q^82 - 214 * q^83 + 33 * q^84 - 21 * q^85 - 34 * q^86 - 115 * q^87 + 3 * q^88 - 60 * q^89 - 67 * q^90 + 24 * q^91 - 99 * q^92 - 15 * q^93 - 11 * q^94 + q^95 - 43 * q^96 - 48 * q^97 - 12 * q^98 - 55 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more