LMFDB - Embedded newform 1850.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.27307$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.27307 q^{3} +1.00000 q^{4} -2.27307 q^{6} -1.27307 q^{7} +1.00000 q^{8} +2.16686 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.27307 q^{3} +1.00000 q^{4} -2.27307 q^{6} -1.27307 q^{7} +1.00000 q^{8} +2.16686 q^{9} -4.43993 q^{11} -2.27307 q^{12} +5.27307 q^{13} -1.27307 q^{14} +1.00000 q^{16} +0.273073 q^{17} +2.16686 q^{18} -5.71301 q^{19} +2.89379 q^{21} -4.43993 q^{22} +6.54615 q^{23} -2.27307 q^{24} +5.27307 q^{26} +1.89379 q^{27} -1.27307 q^{28} +0.546146 q^{29} -5.98608 q^{31} +1.00000 q^{32} +10.0923 q^{33} +0.273073 q^{34} +2.16686 q^{36} +1.00000 q^{37} -5.71301 q^{38} -11.9861 q^{39} +11.8799 q^{41} +2.89379 q^{42} +10.3793 q^{43} -4.43993 q^{44} +6.54615 q^{46} -2.27307 q^{48} -5.37929 q^{49} -0.620715 q^{51} +5.27307 q^{52} -2.33372 q^{53} +1.89379 q^{54} -1.27307 q^{56} +12.9861 q^{57} +0.546146 q^{58} +2.37929 q^{59} +11.8192 q^{61} -5.98608 q^{62} -2.75857 q^{63} +1.00000 q^{64} +10.0923 q^{66} -7.15294 q^{67} +0.273073 q^{68} -14.8799 q^{69} +5.60679 q^{71} +2.16686 q^{72} +9.71301 q^{73} +1.00000 q^{74} -5.71301 q^{76} +5.65236 q^{77} -11.9861 q^{78} +11.4260 q^{79} -10.8053 q^{81} +11.8799 q^{82} -3.89379 q^{83} +2.89379 q^{84} +10.3793 q^{86} -1.24143 q^{87} -4.43993 q^{88} +13.8659 q^{89} -6.71301 q^{91} +6.54615 q^{92} +13.6068 q^{93} -2.27307 q^{96} -0.879866 q^{97} -5.37929 q^{98} -9.62071 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 5 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{18} - q^{19} + 16 q^{21} - 5 q^{22} + 4 q^{23} + q^{24} + 8 q^{26} + 13 q^{27} + 4 q^{28} - 14 q^{29} + 6 q^{31} + 3 q^{32} - q^{33} - 7 q^{34} + 6 q^{36} + 3 q^{37} - q^{38} - 12 q^{39} + 19 q^{41} + 16 q^{42} + 16 q^{43} - 5 q^{44} + 4 q^{46} + q^{48} - q^{49} - 17 q^{51} + 8 q^{52} - 6 q^{53} + 13 q^{54} + 4 q^{56} + 15 q^{57} - 14 q^{58} - 8 q^{59} + 12 q^{61} + 6 q^{62} + 22 q^{63} + 3 q^{64} - q^{66} + 3 q^{67} - 7 q^{68} - 28 q^{69} + 8 q^{71} + 6 q^{72} + 13 q^{73} + 3 q^{74} - q^{76} - 6 q^{77} - 12 q^{78} + 2 q^{79} + 15 q^{81} + 19 q^{82} - 19 q^{83} + 16 q^{84} + 16 q^{86} - 34 q^{87} - 5 q^{88} + q^{89} - 4 q^{91} + 4 q^{92} + 32 q^{93} + q^{96} + 14 q^{97} - q^{98} - 44 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9 - 5 * q^11 + q^12 + 8 * q^13 + 4 * q^14 + 3 * q^16 - 7 * q^17 + 6 * q^18 - q^19 + 16 * q^21 - 5 * q^22 + 4 * q^23 + q^24 + 8 * q^26 + 13 * q^27 + 4 * q^28 - 14 * q^29 + 6 * q^31 + 3 * q^32 - q^33 - 7 * q^34 + 6 * q^36 + 3 * q^37 - q^38 - 12 * q^39 + 19 * q^41 + 16 * q^42 + 16 * q^43 - 5 * q^44 + 4 * q^46 + q^48 - q^49 - 17 * q^51 + 8 * q^52 - 6 * q^53 + 13 * q^54 + 4 * q^56 + 15 * q^57 - 14 * q^58 - 8 * q^59 + 12 * q^61 + 6 * q^62 + 22 * q^63 + 3 * q^64 - q^66 + 3 * q^67 - 7 * q^68 - 28 * q^69 + 8 * q^71 + 6 * q^72 + 13 * q^73 + 3 * q^74 - q^76 - 6 * q^77 - 12 * q^78 + 2 * q^79 + 15 * q^81 + 19 * q^82 - 19 * q^83 + 16 * q^84 + 16 * q^86 - 34 * q^87 - 5 * q^88 + q^89 - 4 * q^91 + 4 * q^92 + 32 * q^93 + q^96 + 14 * q^97 - q^98 - 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.27307	−1.31236	−0.656180	−	0.754605i	$-0.727829\pi$
$3$	−2.27307	−1.31236	−0.656180	+	0.754605i	$0.727829\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.27307	−0.927978
$7$	−1.27307	−0.481176	−0.240588	−	0.970627i	$-0.577340\pi$
$7$	−1.27307	−0.481176	−0.240588	+	0.970627i	$0.577340\pi$
$8$	1.00000	0.353553
$9$	2.16686	0.722287
$10$	0	0
$11$	−4.43993	−1.33869	−0.669345	−	0.742952i	$-0.733425\pi$
$11$	−4.43993	−1.33869	−0.669345	+	0.742952i	$0.733425\pi$
$12$	−2.27307	−0.656180
$13$	5.27307	1.46249	0.731244	−	0.682116i	$-0.238940\pi$
$13$	5.27307	1.46249	0.731244	+	0.682116i	$0.238940\pi$
$14$	−1.27307	−0.340243
$15$	0	0
$16$	1.00000	0.250000
$17$	0.273073	0.0662299	0.0331149	−	0.999452i	$-0.489457\pi$
$17$	0.273073	0.0662299	0.0331149	+	0.999452i	$0.489457\pi$
$18$	2.16686	0.510734
$19$	−5.71301	−1.31065	−0.655327	−	0.755346i	$-0.727469\pi$
$19$	−5.71301	−1.31065	−0.655327	+	0.755346i	$0.727469\pi$
$20$	0	0
$21$	2.89379	0.631476
$22$	−4.43993	−0.946597
$23$	6.54615	1.36497	0.682483	−	0.730902i	$-0.260900\pi$
$23$	6.54615	1.36497	0.682483	+	0.730902i	$0.260900\pi$
$24$	−2.27307	−0.463989
$25$	0	0
$26$	5.27307	1.03413
$27$	1.89379	0.364460
$28$	−1.27307	−0.240588
$29$	0.546146	0.101417	0.0507084	−	0.998714i	$-0.483852\pi$
$29$	0.546146	0.101417	0.0507084	+	0.998714i	$0.483852\pi$
$30$	0	0
$31$	−5.98608	−1.07513	−0.537566	−	0.843222i	$-0.680656\pi$
$31$	−5.98608	−1.07513	−0.537566	+	0.843222i	$0.680656\pi$
$32$	1.00000	0.176777
$33$	10.0923	1.75684
$34$	0.273073	0.0468316
$35$	0	0
$36$	2.16686	0.361143
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−5.71301	−0.926772
$39$	−11.9861	−1.91931
$40$	0	0
$41$	11.8799	1.85532	0.927662	−	0.373422i	$-0.121816\pi$
$41$	11.8799	1.85532	0.927662	+	0.373422i	$0.121816\pi$
$42$	2.89379	0.446521
$43$	10.3793	1.58283	0.791413	−	0.611282i	$-0.209346\pi$
$43$	10.3793	1.58283	0.791413	+	0.611282i	$0.209346\pi$
$44$	−4.43993	−0.669345
$45$	0	0
$46$	6.54615	0.965177
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.27307	−0.328090
$49$	−5.37929	−0.768469
$50$	0	0
$51$	−0.620715	−0.0869174
$52$	5.27307	0.731244
$53$	−2.33372	−0.320561	−0.160281	−	0.987071i	$-0.551240\pi$
$53$	−2.33372	−0.320561	−0.160281	+	0.987071i	$0.551240\pi$
$54$	1.89379	0.257712
$55$	0	0
$56$	−1.27307	−0.170122
$57$	12.9861	1.72005
$58$	0.546146	0.0717124
$59$	2.37929	0.309757	0.154878	−	0.987934i	$-0.450501\pi$
$59$	2.37929	0.309757	0.154878	+	0.987934i	$0.450501\pi$
$60$	0	0
$61$	11.8192	1.51330	0.756648	−	0.653823i	$-0.226836\pi$
$61$	11.8192	1.51330	0.756648	+	0.653823i	$0.226836\pi$
$62$	−5.98608	−0.760233
$63$	−2.75857	−0.347547
$64$	1.00000	0.125000
$65$	0	0
$66$	10.0923	1.24228
$67$	−7.15294	−0.873871	−0.436935	−	0.899493i	$-0.643936\pi$
$67$	−7.15294	−0.873871	−0.436935	+	0.899493i	$0.643936\pi$
$68$	0.273073	0.0331149
$69$	−14.8799	−1.79133
$70$	0	0
$71$	5.60679	0.665404	0.332702	−	0.943032i	$-0.392040\pi$
$71$	5.60679	0.665404	0.332702	+	0.943032i	$0.392040\pi$
$72$	2.16686	0.255367
$73$	9.71301	1.13682	0.568411	−	0.822745i	$-0.307558\pi$
$73$	9.71301	1.13682	0.568411	+	0.822745i	$0.307558\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−5.71301	−0.655327
$77$	5.65236	0.644146
$78$	−11.9861	−1.35716
$79$	11.4260	1.28553	0.642763	−	0.766065i	$-0.277788\pi$
$79$	11.4260	1.28553	0.642763	+	0.766065i	$0.277788\pi$
$80$	0	0
$81$	−10.8053	−1.20059
$82$	11.8799	1.31191
$83$	−3.89379	−0.427399	−0.213699	−	0.976899i	$-0.568551\pi$
$83$	−3.89379	−0.427399	−0.213699	+	0.976899i	$0.568551\pi$
$84$	2.89379	0.315738
$85$	0	0
$86$	10.3793	1.11923
$87$	−1.24143	−0.133095
$88$	−4.43993	−0.473298
$89$	13.8659	1.46979	0.734894	−	0.678182i	$-0.237232\pi$
$89$	13.8659	1.46979	0.734894	+	0.678182i	$0.237232\pi$
$90$	0	0
$91$	−6.71301	−0.703714
$92$	6.54615	0.682483
$93$	13.6068	1.41096
$94$	0	0
$95$	0	0
$96$	−2.27307	−0.231995
$97$	−0.879866	−0.0893369	−0.0446684	−	0.999002i	$-0.514223\pi$
$97$	−0.879866	−0.0893369	−0.0446684	+	0.999002i	$0.514223\pi$
$98$	−5.37929	−0.543390
$99$	−9.62071	−0.966918
$100$	0	0
$101$	3.66628	0.364808	0.182404	−	0.983224i	$-0.441612\pi$
$101$	3.66628	0.364808	0.182404	+	0.983224i	$0.441612\pi$
$102$	−0.620715	−0.0614599
$103$	12.3198	1.21391	0.606953	−	0.794738i	$-0.292392\pi$
$103$	12.3198	1.21391	0.606953	+	0.794738i	$0.292392\pi$
$104$	5.27307	0.517067
$105$	0	0
$106$	−2.33372	−0.226671
$107$	−7.36536	−0.712037	−0.356018	−	0.934479i	$-0.615866\pi$
$107$	−7.36536	−0.712037	−0.356018	+	0.934479i	$0.615866\pi$
$108$	1.89379	0.182230
$109$	4.33372	0.415095	0.207548	−	0.978225i	$-0.433452\pi$
$109$	4.33372	0.415095	0.207548	+	0.978225i	$0.433452\pi$
$110$	0	0
$111$	−2.27307	−0.215751
$112$	−1.27307	−0.120294
$113$	−2.10621	−0.198136	−0.0990679	−	0.995081i	$-0.531586\pi$
$113$	−2.10621	−0.198136	−0.0990679	+	0.995081i	$0.531586\pi$
$114$	12.9861	1.21626
$115$	0	0
$116$	0.546146	0.0507084
$117$	11.4260	1.05634
$118$	2.37929	0.219031
$119$	−0.347642	−0.0318683
$120$	0	0
$121$	8.71301	0.792091
$122$	11.8192	1.07006
$123$	−27.0038	−2.43485
$124$	−5.98608	−0.537566
$125$	0	0
$126$	−2.75857	−0.245753
$127$	13.6068	1.20741	0.603704	−	0.797209i	$-0.293691\pi$
$127$	13.6068	1.20741	0.603704	+	0.797209i	$0.293691\pi$
$128$	1.00000	0.0883883
$129$	−23.5929	−2.07724
$130$	0	0
$131$	4.16686	0.364060	0.182030	−	0.983293i	$-0.441733\pi$
$131$	4.16686	0.364060	0.182030	+	0.983293i	$0.441733\pi$
$132$	10.0923	0.878421
$133$	7.27307	0.630655
$134$	−7.15294	−0.617920
$135$	0	0
$136$	0.273073	0.0234158
$137$	−1.21243	−0.103584	−0.0517922	−	0.998658i	$-0.516493\pi$
$137$	−1.21243	−0.103584	−0.0517922	+	0.998658i	$0.516493\pi$
$138$	−14.8799	−1.26666
$139$	7.53222	0.638875	0.319437	−	0.947607i	$-0.396506\pi$
$139$	7.53222	0.638875	0.319437	+	0.947607i	$0.396506\pi$
$140$	0	0
$141$	0	0
$142$	5.60679	0.470512
$143$	−23.4121	−1.95782
$144$	2.16686	0.180572
$145$	0	0
$146$	9.71301	0.803854
$147$	12.2275	1.00851
$148$	1.00000	0.0821995
$149$	−13.9861	−1.14578	−0.572892	−	0.819631i	$-0.694179\pi$
$149$	−13.9861	−1.14578	−0.572892	+	0.819631i	$0.694179\pi$
$150$	0	0
$151$	13.7597	1.11975	0.559876	−	0.828577i	$-0.310849\pi$
$151$	13.7597	1.11975	0.559876	+	0.828577i	$0.310849\pi$
$152$	−5.71301	−0.463386
$153$	0.591711	0.0478370
$154$	5.65236	0.455480
$155$	0	0
$156$	−11.9861	−0.959654
$157$	−2.56007	−0.204316	−0.102158	−	0.994768i	$-0.532575\pi$
$157$	−2.56007	−0.204316	−0.102158	+	0.994768i	$0.532575\pi$
$158$	11.4260	0.909005
$159$	5.30472	0.420691
$160$	0	0
$161$	−8.33372	−0.656789
$162$	−10.8053	−0.848944
$163$	21.7130	1.70069	0.850347	−	0.526222i	$-0.176392\pi$
$163$	21.7130	1.70069	0.850347	+	0.526222i	$0.176392\pi$
$164$	11.8799	0.927662
$165$	0	0
$166$	−3.89379	−0.302217
$167$	−16.8659	−1.30513	−0.652563	−	0.757734i	$-0.726306\pi$
$167$	−16.8659	−1.30513	−0.652563	+	0.757734i	$0.726306\pi$
$168$	2.89379	0.223261
$169$	14.8053	1.13887
$170$	0	0
$171$	−12.3793	−0.946668
$172$	10.3793	0.791413
$173$	−10.7586	−0.817959	−0.408980	−	0.912544i	$-0.634115\pi$
$173$	−10.7586	−0.817959	−0.408980	+	0.912544i	$0.634115\pi$
$174$	−1.24143	−0.0941125
$175$	0	0
$176$	−4.43993	−0.334673
$177$	−5.40829	−0.406512
$178$	13.8659	1.03930
$179$	1.66744	0.124630	0.0623152	−	0.998057i	$-0.480152\pi$
$179$	1.66744	0.124630	0.0623152	+	0.998057i	$0.480152\pi$
$180$	0	0
$181$	5.97216	0.443907	0.221953	−	0.975057i	$-0.428757\pi$
$181$	5.97216	0.443907	0.221953	+	0.975057i	$0.428757\pi$
$182$	−6.71301	−0.497601
$183$	−26.8659	−1.98599
$184$	6.54615	0.482588
$185$	0	0
$186$	13.6068	0.997698
$187$	−1.21243	−0.0886613
$188$	0	0
$189$	−2.41093	−0.175369
$190$	0	0
$191$	−24.8520	−1.79823	−0.899115	−	0.437713i	$-0.855789\pi$
$191$	−24.8520	−1.79823	−0.899115	+	0.437713i	$0.855789\pi$
$192$	−2.27307	−0.164045
$193$	−7.65352	−0.550912	−0.275456	−	0.961314i	$-0.588829\pi$
$193$	−7.65352	−0.550912	−0.275456	+	0.961314i	$0.588829\pi$
$194$	−0.879866	−0.0631707
$195$	0	0
$196$	−5.37929	−0.384235
$197$	14.5322	1.03538	0.517689	−	0.855569i	$-0.326792\pi$
$197$	14.5322	1.03538	0.517689	+	0.855569i	$0.326792\pi$
$198$	−9.62071	−0.683714
$199$	−12.8799	−0.913030	−0.456515	−	0.889716i	$-0.650902\pi$
$199$	−12.8799	−0.913030	−0.456515	+	0.889716i	$0.650902\pi$
$200$	0	0
$201$	16.2592	1.14683
$202$	3.66628	0.257959
$203$	−0.695283	−0.0487993
$204$	−0.620715	−0.0434587
$205$	0	0
$206$	12.3198	0.858361
$207$	14.1846	0.985897
$208$	5.27307	0.365622
$209$	25.3654	1.75456
$210$	0	0
$211$	−19.1529	−1.31854	−0.659271	−	0.751905i	$-0.729135\pi$
$211$	−19.1529	−1.31854	−0.659271	+	0.751905i	$0.729135\pi$
$212$	−2.33372	−0.160281
$213$	−12.7446	−0.873249
$214$	−7.36536	−0.503486
$215$	0	0
$216$	1.89379	0.128856
$217$	7.62071	0.517328
$218$	4.33372	0.293517
$219$	−22.0784	−1.49192
$220$	0	0
$221$	1.43993	0.0968604
$222$	−2.27307	−0.152559
$223$	−2.75857	−0.184728	−0.0923638	−	0.995725i	$-0.529442\pi$
$223$	−2.75857	−0.184728	−0.0923638	+	0.995725i	$0.529442\pi$
$224$	−1.27307	−0.0850608
$225$	0	0
$226$	−2.10621	−0.140103
$227$	17.2592	1.14553	0.572765	−	0.819720i	$-0.305871\pi$
$227$	17.2592	1.14553	0.572765	+	0.819720i	$0.305871\pi$
$228$	12.9861	0.860024
$229$	−22.3059	−1.47401	−0.737007	−	0.675885i	$-0.763762\pi$
$229$	−22.3059	−1.47401	−0.737007	+	0.675885i	$0.763762\pi$
$230$	0	0
$231$	−12.8482	−0.845351
$232$	0.546146	0.0358562
$233$	24.3514	1.59532	0.797658	−	0.603110i	$-0.206072\pi$
$233$	24.3514	1.59532	0.797658	+	0.603110i	$0.206072\pi$
$234$	11.4260	0.746942
$235$	0	0
$236$	2.37929	0.154878
$237$	−25.9722	−1.68707
$238$	−0.347642	−0.0225343
$239$	1.78757	0.115629	0.0578143	−	0.998327i	$-0.481587\pi$
$239$	1.78757	0.115629	0.0578143	+	0.998327i	$0.481587\pi$
$240$	0	0
$241$	−3.72693	−0.240072	−0.120036	−	0.992770i	$-0.538301\pi$
$241$	−3.72693	−0.240072	−0.120036	+	0.992770i	$0.538301\pi$
$242$	8.71301	0.560093
$243$	18.8799	1.21114
$244$	11.8192	0.756648
$245$	0	0
$246$	−27.0038	−1.72170
$247$	−30.1251	−1.91681
$248$	−5.98608	−0.380116
$249$	8.85086	0.560901
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	−2.75857	−0.173774
$253$	−29.0644	−1.82727
$254$	13.6068	0.853766
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.42601	0.213709	0.106854	−	0.994275i	$-0.465922\pi$
$257$	3.42601	0.213709	0.106854	+	0.994275i	$0.465922\pi$
$258$	−23.5929	−1.46883
$259$	−1.27307	−0.0791049
$260$	0	0
$261$	1.18342	0.0732519
$262$	4.16686	0.257429
$263$	−2.57399	−0.158719	−0.0793595	−	0.996846i	$-0.525287\pi$
$263$	−2.57399	−0.158719	−0.0793595	+	0.996846i	$0.525287\pi$
$264$	10.0923	0.621138
$265$	0	0
$266$	7.27307	0.445941
$267$	−31.5183	−1.92889
$268$	−7.15294	−0.436935
$269$	−27.7319	−1.69084	−0.845422	−	0.534100i	$-0.820651\pi$
$269$	−27.7319	−1.69084	−0.845422	+	0.534100i	$0.820651\pi$
$270$	0	0
$271$	26.4588	1.60726	0.803629	−	0.595130i	$-0.202899\pi$
$271$	26.4588	1.60726	0.803629	+	0.595130i	$0.202899\pi$
$272$	0.273073	0.0165575
$273$	15.2592	0.923526
$274$	−1.21243	−0.0732453
$275$	0	0
$276$	−14.8799	−0.895663
$277$	−0.573988	−0.0344876	−0.0172438	−	0.999851i	$-0.505489\pi$
$277$	−0.573988	−0.0344876	−0.0172438	+	0.999851i	$0.505489\pi$
$278$	7.53222	0.451753
$279$	−12.9710	−0.776553
$280$	0	0
$281$	−29.2592	−1.74545	−0.872727	−	0.488208i	$-0.837651\pi$
$281$	−29.2592	−1.74545	−0.872727	+	0.488208i	$0.837651\pi$
$282$	0	0
$283$	21.0177	1.24937	0.624687	−	0.780875i	$-0.285227\pi$
$283$	21.0177	1.24937	0.624687	+	0.780875i	$0.285227\pi$
$284$	5.60679	0.332702
$285$	0	0
$286$	−23.4121	−1.38439
$287$	−15.1239	−0.892738
$288$	2.16686	0.127683
$289$	−16.9254	−0.995614
$290$	0	0
$291$	2.00000	0.117242
$292$	9.71301	0.568411
$293$	17.9582	1.04913	0.524566	−	0.851370i	$-0.324228\pi$
$293$	17.9582	1.04913	0.524566	+	0.851370i	$0.324228\pi$
$294$	12.2275	0.713123
$295$	0	0
$296$	1.00000	0.0581238
$297$	−8.40829	−0.487898
$298$	−13.9861	−0.810192
$299$	34.5183	1.99625
$300$	0	0
$301$	−13.2136	−0.761618
$302$	13.7597	0.791784
$303$	−8.33372	−0.478760
$304$	−5.71301	−0.327663
$305$	0	0
$306$	0.591711	0.0338259
$307$	23.1985	1.32401	0.662004	−	0.749500i	$-0.269706\pi$
$307$	23.1985	1.32401	0.662004	+	0.749500i	$0.269706\pi$
$308$	5.65236	0.322073
$309$	−28.0038	−1.59308
$310$	0	0
$311$	−6.98492	−0.396078	−0.198039	−	0.980194i	$-0.563457\pi$
$311$	−6.98492	−0.396078	−0.198039	+	0.980194i	$0.563457\pi$
$312$	−11.9861	−0.678578
$313$	24.1669	1.36599	0.682996	−	0.730422i	$-0.260677\pi$
$313$	24.1669	1.36599	0.682996	+	0.730422i	$0.260677\pi$
$314$	−2.56007	−0.144473
$315$	0	0
$316$	11.4260	0.642763
$317$	−32.2920	−1.81370	−0.906848	−	0.421457i	$-0.861519\pi$
$317$	−32.2920	−1.81370	−0.906848	+	0.421457i	$0.861519\pi$
$318$	5.30472	0.297474
$319$	−2.42485	−0.135766
$320$	0	0
$321$	16.7420	0.934448
$322$	−8.33372	−0.464420
$323$	−1.56007	−0.0868044
$324$	−10.8053	−0.600294
$325$	0	0
$326$	21.7130	1.20257
$327$	−9.85086	−0.544754
$328$	11.8799	0.655956
$329$	0	0
$330$	0	0
$331$	−27.9254	−1.53492	−0.767460	−	0.641097i	$-0.778480\pi$
$331$	−27.9254	−1.53492	−0.767460	+	0.641097i	$0.778480\pi$
$332$	−3.89379	−0.213699
$333$	2.16686	0.118743
$334$	−16.8659	−0.922863
$335$	0	0
$336$	2.89379	0.157869
$337$	−8.33256	−0.453903	−0.226952	−	0.973906i	$-0.572876\pi$
$337$	−8.33256	−0.453903	−0.226952	+	0.973906i	$0.572876\pi$
$338$	14.8053	0.805302
$339$	4.78757	0.260025
$340$	0	0
$341$	26.5778	1.43927
$342$	−12.3793	−0.669395
$343$	15.7597	0.850946
$344$	10.3793	0.559614
$345$	0	0
$346$	−10.7586	−0.578384
$347$	−1.99884	−0.107303	−0.0536516	−	0.998560i	$-0.517086\pi$
$347$	−1.99884	−0.107303	−0.0536516	+	0.998560i	$0.517086\pi$
$348$	−1.24143	−0.0665476
$349$	−21.9582	−1.17540	−0.587699	−	0.809080i	$-0.699966\pi$
$349$	−21.9582	−1.17540	−0.587699	+	0.809080i	$0.699966\pi$
$350$	0	0
$351$	9.98608	0.533017
$352$	−4.43993	−0.236649
$353$	11.7597	0.625907	0.312954	−	0.949768i	$-0.398682\pi$
$353$	11.7597	0.625907	0.312954	+	0.949768i	$0.398682\pi$
$354$	−5.40829	−0.287447
$355$	0	0
$356$	13.8659	0.734894
$357$	0.790215	0.0418226
$358$	1.66744	0.0881270
$359$	2.57399	0.135850	0.0679249	−	0.997690i	$-0.478362\pi$
$359$	2.57399	0.135850	0.0679249	+	0.997690i	$0.478362\pi$
$360$	0	0
$361$	13.6384	0.717812
$362$	5.97216	0.313890
$363$	−19.8053	−1.03951
$364$	−6.71301	−0.351857
$365$	0	0
$366$	−26.8659	−1.40431
$367$	−12.4867	−0.651798	−0.325899	−	0.945405i	$-0.605667\pi$
$367$	−12.4867	−0.651798	−0.325899	+	0.945405i	$0.605667\pi$
$368$	6.54615	0.341241
$369$	25.7420	1.34008
$370$	0	0
$371$	2.97100	0.154246
$372$	13.6068	0.705479
$373$	−14.8659	−0.769729	−0.384865	−	0.922973i	$-0.625752\pi$
$373$	−14.8659	−0.769729	−0.384865	+	0.922973i	$0.625752\pi$
$374$	−1.21243	−0.0626930
$375$	0	0
$376$	0	0
$377$	2.87987	0.148321
$378$	−2.41093	−0.124005
$379$	−2.39437	−0.122990	−0.0614952	−	0.998107i	$-0.519587\pi$
$379$	−2.39437	−0.122990	−0.0614952	+	0.998107i	$0.519587\pi$
$380$	0	0
$381$	−30.9292	−1.58455
$382$	−24.8520	−1.27154
$383$	−14.7725	−0.754839	−0.377420	−	0.926042i	$-0.623189\pi$
$383$	−14.7725	−0.754839	−0.377420	+	0.926042i	$0.623189\pi$
$384$	−2.27307	−0.115997
$385$	0	0
$386$	−7.65352	−0.389554
$387$	22.4905	1.14325
$388$	−0.879866	−0.0446684
$389$	−12.6991	−0.643869	−0.321935	−	0.946762i	$-0.604333\pi$
$389$	−12.6991	−0.643869	−0.321935	+	0.946762i	$0.604333\pi$
$390$	0	0
$391$	1.78757	0.0904015
$392$	−5.37929	−0.271695
$393$	−9.47158	−0.477778
$394$	14.5322	0.732123
$395$	0	0
$396$	−9.62071	−0.483459
$397$	−4.98492	−0.250186	−0.125093	−	0.992145i	$-0.539923\pi$
$397$	−4.98492	−0.250186	−0.125093	+	0.992145i	$0.539923\pi$
$398$	−12.8799	−0.645609
$399$	−16.5322	−0.827646
$400$	0	0
$401$	27.1985	1.35823	0.679114	−	0.734033i	$-0.262364\pi$
$401$	27.1985	1.35823	0.679114	+	0.734033i	$0.262364\pi$
$402$	16.2592	0.810933
$403$	−31.5650	−1.57237
$404$	3.66628	0.182404
$405$	0	0
$406$	−0.695283	−0.0345063
$407$	−4.43993	−0.220079
$408$	−0.620715	−0.0307299
$409$	35.5499	1.75783	0.878916	−	0.476977i	$-0.158267\pi$
$409$	35.5499	1.75783	0.878916	+	0.476977i	$0.158267\pi$
$410$	0	0
$411$	2.75593	0.135940
$412$	12.3198	0.606953
$413$	−3.02900	−0.149048
$414$	14.1846	0.697134
$415$	0	0
$416$	5.27307	0.258534
$417$	−17.1213	−0.838433
$418$	25.3654	1.24066
$419$	−29.5322	−1.44274	−0.721372	−	0.692548i	$-0.756488\pi$
$419$	−29.5322	−1.44274	−0.721372	+	0.692548i	$0.756488\pi$
$420$	0	0
$421$	−30.4905	−1.48601	−0.743007	−	0.669284i	$-0.766601\pi$
$421$	−30.4905	−1.48601	−0.743007	+	0.669284i	$0.766601\pi$
$422$	−19.1529	−0.932350
$423$	0	0
$424$	−2.33372	−0.113335
$425$	0	0
$426$	−12.7446	−0.617480
$427$	−15.0467	−0.728162
$428$	−7.36536	−0.356018
$429$	53.2174	2.56936
$430$	0	0
$431$	21.9722	1.05836	0.529181	−	0.848509i	$-0.322499\pi$
$431$	21.9722	1.05836	0.529181	+	0.848509i	$0.322499\pi$
$432$	1.89379	0.0911149
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	7.62071	0.365806
$435$	0	0
$436$	4.33372	0.207548
$437$	−37.3982	−1.78900
$438$	−22.0784	−1.05495
$439$	29.5171	1.40878	0.704388	−	0.709815i	$-0.251221\pi$
$439$	29.5171	1.40878	0.704388	+	0.709815i	$0.251221\pi$
$440$	0	0
$441$	−11.6562	−0.555055
$442$	1.43993	0.0684906
$443$	−28.9861	−1.37717	−0.688585	−	0.725156i	$-0.741768\pi$
$443$	−28.9861	−1.37717	−0.688585	+	0.725156i	$0.741768\pi$
$444$	−2.27307	−0.107875
$445$	0	0
$446$	−2.75857	−0.130622
$447$	31.7914	1.50368
$448$	−1.27307	−0.0601470
$449$	36.6245	1.72842	0.864209	−	0.503133i	$-0.167819\pi$
$449$	36.6245	1.72842	0.864209	+	0.503133i	$0.167819\pi$
$450$	0	0
$451$	−52.7458	−2.48370
$452$	−2.10621	−0.0990679
$453$	−31.2769	−1.46952
$454$	17.2592	0.810012
$455$	0	0
$456$	12.9861	0.608129
$457$	−19.7446	−0.923616	−0.461808	−	0.886980i	$-0.652799\pi$
$457$	−19.7446	−0.923616	−0.461808	+	0.886980i	$0.652799\pi$
$458$	−22.3059	−1.04229
$459$	0.517142	0.0241381
$460$	0	0
$461$	34.9733	1.62887	0.814435	−	0.580255i	$-0.197047\pi$
$461$	34.9733	1.62887	0.814435	+	0.580255i	$0.197047\pi$
$462$	−12.8482	−0.597753
$463$	24.1707	1.12331	0.561653	−	0.827373i	$-0.310166\pi$
$463$	24.1707	1.12331	0.561653	+	0.827373i	$0.310166\pi$
$464$	0.546146	0.0253542
$465$	0	0
$466$	24.3514	1.12806
$467$	−7.63844	−0.353465	−0.176732	−	0.984259i	$-0.556553\pi$
$467$	−7.63844	−0.353465	−0.176732	+	0.984259i	$0.556553\pi$
$468$	11.4260	0.528168
$469$	9.10621	0.420486
$470$	0	0
$471$	5.81922	0.268135
$472$	2.37929	0.109515
$473$	−46.0833	−2.11891
$474$	−25.9722	−1.19294
$475$	0	0
$476$	−0.347642	−0.0159341
$477$	−5.05685	−0.231537
$478$	1.78757	0.0817618
$479$	−4.01392	−0.183401	−0.0917004	−	0.995787i	$-0.529230\pi$
$479$	−4.01392	−0.183401	−0.0917004	+	0.995787i	$0.529230\pi$
$480$	0	0
$481$	5.27307	0.240431
$482$	−3.72693	−0.169757
$483$	18.9432	0.861943
$484$	8.71301	0.396046
$485$	0	0
$486$	18.8799	0.856408
$487$	−43.3842	−1.96593	−0.982964	−	0.183798i	$-0.941161\pi$
$487$	−43.3842	−1.96593	−0.982964	+	0.183798i	$0.941161\pi$
$488$	11.8192	0.535031
$489$	−49.3552	−2.23192
$490$	0	0
$491$	−20.7307	−0.935565	−0.467782	−	0.883844i	$-0.654947\pi$
$491$	−20.7307	−0.935565	−0.467782	+	0.883844i	$0.654947\pi$
$492$	−27.0038	−1.21743
$493$	0.149138	0.00671682
$494$	−30.1251	−1.35539
$495$	0	0
$496$	−5.98608	−0.268783
$497$	−7.13786	−0.320177
$498$	8.85086	0.396617
$499$	5.92659	0.265311	0.132655	−	0.991162i	$-0.457650\pi$
$499$	5.92659	0.265311	0.132655	+	0.991162i	$0.457650\pi$
$500$	0	0
$501$	38.3375	1.71279
$502$	−9.00000	−0.401690
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−2.75857	−0.122877
$505$	0	0
$506$	−29.0644	−1.29207
$507$	−33.6535	−1.49461
$508$	13.6068	0.603704
$509$	−9.07837	−0.402392	−0.201196	−	0.979551i	$-0.564483\pi$
$509$	−9.07837	−0.402392	−0.201196	+	0.979551i	$0.564483\pi$
$510$	0	0
$511$	−12.3654	−0.547012
$512$	1.00000	0.0441942
$513$	−10.8192	−0.477680
$514$	3.42601	0.151115
$515$	0	0
$516$	−23.5929	−1.03862
$517$	0	0
$518$	−1.27307	−0.0559356
$519$	24.4550	1.07346
$520$	0	0
$521$	−7.16686	−0.313986	−0.156993	−	0.987600i	$-0.550180\pi$
$521$	−7.16686	−0.313986	−0.156993	+	0.987600i	$0.550180\pi$
$522$	1.18342	0.0517970
$523$	29.3970	1.28544	0.642721	−	0.766101i	$-0.277806\pi$
$523$	29.3970	1.28544	0.642721	+	0.766101i	$0.277806\pi$
$524$	4.16686	0.182030
$525$	0	0
$526$	−2.57399	−0.112231
$527$	−1.63464	−0.0712058
$528$	10.0923	0.439211
$529$	19.8520	0.863131
$530$	0	0
$531$	5.15558	0.223733
$532$	7.27307	0.315328
$533$	62.6434	2.71339
$534$	−31.5183	−1.36393
$535$	0	0
$536$	−7.15294	−0.308960
$537$	−3.79021	−0.163560
$538$	−27.7319	−1.19561
$539$	23.8837	1.02874
$540$	0	0
$541$	26.0038	1.11799	0.558995	−	0.829171i	$-0.311187\pi$
$541$	26.0038	1.11799	0.558995	+	0.829171i	$0.311187\pi$
$542$	26.4588	1.13650
$543$	−13.5751	−0.582565
$544$	0.273073	0.0117079
$545$	0	0
$546$	15.2592	0.653031
$547$	8.77746	0.375297	0.187648	−	0.982236i	$-0.439913\pi$
$547$	8.77746	0.375297	0.187648	+	0.982236i	$0.439913\pi$
$548$	−1.21243	−0.0517922
$549$	25.6106	1.09303
$550$	0	0
$551$	−3.12013	−0.132922
$552$	−14.8799	−0.633329
$553$	−14.5461	−0.618565
$554$	−0.573988	−0.0243864
$555$	0	0
$556$	7.53222	0.319437
$557$	21.6663	0.918030	0.459015	−	0.888429i	$-0.348202\pi$
$557$	21.6663	0.918030	0.459015	+	0.888429i	$0.348202\pi$
$558$	−12.9710	−0.549106
$559$	54.7307	2.31486
$560$	0	0
$561$	2.75593	0.116355
$562$	−29.2592	−1.23422
$563$	−19.4437	−0.819456	−0.409728	−	0.912208i	$-0.634376\pi$
$563$	−19.4437	−0.819456	−0.409728	+	0.912208i	$0.634376\pi$
$564$	0	0
$565$	0	0
$566$	21.0177	0.883441
$567$	13.7559	0.577695
$568$	5.60679	0.235256
$569$	21.7446	0.911583	0.455792	−	0.890087i	$-0.349356\pi$
$569$	21.7446	0.911583	0.455792	+	0.890087i	$0.349356\pi$
$570$	0	0
$571$	1.54498	0.0646556	0.0323278	−	0.999477i	$-0.489708\pi$
$571$	1.54498	0.0646556	0.0323278	+	0.999477i	$0.489708\pi$
$572$	−23.4121	−0.978909
$573$	56.4905	2.35992
$574$	−15.1239	−0.631261
$575$	0	0
$576$	2.16686	0.0902858
$577$	37.8837	1.57712	0.788559	−	0.614959i	$-0.210828\pi$
$577$	37.8837	1.57712	0.788559	+	0.614959i	$0.210828\pi$
$578$	−16.9254	−0.704005
$579$	17.3970	0.722995
$580$	0	0
$581$	4.95708	0.205654
$582$	2.00000	0.0829027
$583$	10.3616	0.429132
$584$	9.71301	0.401927
$585$	0	0
$586$	17.9582	0.741848
$587$	38.8053	1.60167	0.800833	−	0.598888i	$-0.204390\pi$
$587$	38.8053	1.60167	0.800833	+	0.598888i	$0.204390\pi$
$588$	12.2275	0.504254
$589$	34.1985	1.40912
$590$	0	0
$591$	−33.0328	−1.35879
$592$	1.00000	0.0410997
$593$	−22.0923	−0.907222	−0.453611	−	0.891200i	$-0.649864\pi$
$593$	−22.0923	−0.907222	−0.453611	+	0.891200i	$0.649864\pi$
$594$	−8.40829	−0.344996
$595$	0	0
$596$	−13.9861	−0.572892
$597$	29.2769	1.19822
$598$	34.5183	1.41156
$599$	−17.8471	−0.729211	−0.364606	−	0.931162i	$-0.618796\pi$
$599$	−17.8471	−0.729211	−0.364606	+	0.931162i	$0.618796\pi$
$600$	0	0
$601$	46.1100	1.88087	0.940433	−	0.339978i	$-0.110420\pi$
$601$	46.1100	1.88087	0.940433	+	0.339978i	$0.110420\pi$
$602$	−13.2136	−0.538546
$603$	−15.4994	−0.631185
$604$	13.7597	0.559876
$605$	0	0
$606$	−8.33372	−0.338534
$607$	47.3842	1.92327	0.961634	−	0.274337i	$-0.0884583\pi$
$607$	47.3842	1.92327	0.961634	+	0.274337i	$0.0884583\pi$
$608$	−5.71301	−0.231693
$609$	1.58043	0.0640422
$610$	0	0
$611$	0	0
$612$	0.591711	0.0239185
$613$	15.7876	0.637654	0.318827	−	0.947813i	$-0.396711\pi$
$613$	15.7876	0.637654	0.318827	+	0.947813i	$0.396711\pi$
$614$	23.1985	0.936215
$615$	0	0
$616$	5.65236	0.227740
$617$	26.1390	1.05232	0.526159	−	0.850386i	$-0.323632\pi$
$617$	26.1390	1.05232	0.526159	+	0.850386i	$0.323632\pi$
$618$	−28.0038	−1.12648
$619$	31.3047	1.25824	0.629121	−	0.777307i	$-0.283415\pi$
$619$	31.3047	1.25824	0.629121	+	0.777307i	$0.283415\pi$
$620$	0	0
$621$	12.3970	0.497475
$622$	−6.98492	−0.280070
$623$	−17.6524	−0.707227
$624$	−11.9861	−0.479827
$625$	0	0
$626$	24.1669	0.965902
$627$	−57.6573	−2.30261
$628$	−2.56007	−0.102158
$629$	0.273073	0.0108881
$630$	0	0
$631$	−15.3047	−0.609271	−0.304636	−	0.952469i	$-0.598535\pi$
$631$	−15.3047	−0.609271	−0.304636	+	0.952469i	$0.598535\pi$
$632$	11.4260	0.454502
$633$	43.5360	1.73040
$634$	−32.2920	−1.28248
$635$	0	0
$636$	5.30472	0.210346
$637$	−28.3654	−1.12388
$638$	−2.42485	−0.0960008
$639$	12.1491	0.480612
$640$	0	0
$641$	−12.8520	−0.507624	−0.253812	−	0.967254i	$-0.581685\pi$
$641$	−12.8520	−0.507624	−0.253812	+	0.967254i	$0.581685\pi$
$642$	16.7420	0.660754
$643$	−0.139018	−0.00548233	−0.00274117	−	0.999996i	$-0.500873\pi$
$643$	−0.139018	−0.00548233	−0.00274117	+	0.999996i	$0.500873\pi$
$644$	−8.33372	−0.328395
$645$	0	0
$646$	−1.56007	−0.0613800
$647$	46.8659	1.84249	0.921245	−	0.388982i	$-0.127173\pi$
$647$	46.8659	1.84249	0.921245	+	0.388982i	$0.127173\pi$
$648$	−10.8053	−0.424472
$649$	−10.5639	−0.414668
$650$	0	0
$651$	−17.3224	−0.678920
$652$	21.7130	0.850347
$653$	−7.48550	−0.292930	−0.146465	−	0.989216i	$-0.546790\pi$
$653$	−7.48550	−0.292930	−0.146465	+	0.989216i	$0.546790\pi$
$654$	−9.85086	−0.385199
$655$	0	0
$656$	11.8799	0.463831
$657$	21.0467	0.821111
$658$	0	0
$659$	−22.7914	−0.887826	−0.443913	−	0.896070i	$-0.646410\pi$
$659$	−22.7914	−0.887826	−0.443913	+	0.896070i	$0.646410\pi$
$660$	0	0
$661$	16.4249	0.638853	0.319426	−	0.947611i	$-0.396510\pi$
$661$	16.4249	0.638853	0.319426	+	0.947611i	$0.396510\pi$
$662$	−27.9254	−1.08535
$663$	−3.27307	−0.127116
$664$	−3.89379	−0.151108
$665$	0	0
$666$	2.16686	0.0839641
$667$	3.57515	0.138430
$668$	−16.8659	−0.652563
$669$	6.27043	0.242429
$670$	0	0
$671$	−52.4765	−2.02583
$672$	2.89379	0.111630
$673$	51.2035	1.97375	0.986874	−	0.161490i	$-0.0516300\pi$
$673$	51.2035	1.97375	0.986874	+	0.161490i	$0.0516300\pi$
$674$	−8.33256	−0.320958
$675$	0	0
$676$	14.8053	0.569435
$677$	−1.00116	−0.0384778	−0.0192389	−	0.999815i	$-0.506124\pi$
$677$	−1.00116	−0.0384778	−0.0192389	+	0.999815i	$0.506124\pi$
$678$	4.78757	0.183866
$679$	1.12013	0.0429868
$680$	0	0
$681$	−39.2313	−1.50335
$682$	26.5778	1.01772
$683$	28.3526	1.08488	0.542441	−	0.840094i	$-0.317500\pi$
$683$	28.3526	1.08488	0.542441	+	0.840094i	$0.317500\pi$
$684$	−12.3793	−0.473334
$685$	0	0
$686$	15.7597	0.601709
$687$	50.7029	1.93444
$688$	10.3793	0.395707
$689$	−12.3059	−0.468817
$690$	0	0
$691$	26.3842	1.00370	0.501852	−	0.864953i	$-0.332652\pi$
$691$	26.3842	1.00370	0.501852	+	0.864953i	$0.332652\pi$
$692$	−10.7586	−0.408980
$693$	12.2479	0.465258
$694$	−1.99884	−0.0758749
$695$	0	0
$696$	−1.24143	−0.0470562
$697$	3.24407	0.122878
$698$	−21.9582	−0.831132
$699$	−55.3526	−2.09363
$700$	0	0
$701$	40.5801	1.53269	0.766345	−	0.642429i	$-0.222073\pi$
$701$	40.5801	1.53269	0.766345	+	0.642429i	$0.222073\pi$
$702$	9.98608	0.376900
$703$	−5.71301	−0.215470
$704$	−4.43993	−0.167336
$705$	0	0
$706$	11.7597	0.442583
$707$	−4.66744	−0.175537
$708$	−5.40829	−0.203256
$709$	14.5461	0.546292	0.273146	−	0.961973i	$-0.411936\pi$
$709$	14.5461	0.546292	0.273146	+	0.961973i	$0.411936\pi$
$710$	0	0
$711$	24.7586	0.928519
$712$	13.8659	0.519648
$713$	−39.1857	−1.46752
$714$	0.790215	0.0295730
$715$	0	0
$716$	1.66744	0.0623152
$717$	−4.06329	−0.151746
$718$	2.57399	0.0960604
$719$	21.6701	0.808158	0.404079	−	0.914724i	$-0.367592\pi$
$719$	21.6701	0.808158	0.404079	+	0.914724i	$0.367592\pi$
$720$	0	0
$721$	−15.6840	−0.584103
$722$	13.6384	0.507570
$723$	8.47158	0.315061
$724$	5.97216	0.221953
$725$	0	0
$726$	−19.8053	−0.735044
$727$	28.8799	1.07109	0.535547	−	0.844505i	$-0.320105\pi$
$727$	28.8799	1.07109	0.535547	+	0.844505i	$0.320105\pi$
$728$	−6.71301	−0.248801
$729$	−10.4994	−0.388867
$730$	0	0
$731$	2.83430	0.104830
$732$	−26.8659	−0.992994
$733$	−21.9165	−0.809503	−0.404752	−	0.914427i	$-0.632642\pi$
$733$	−21.9165	−0.809503	−0.404752	+	0.914427i	$0.632642\pi$
$734$	−12.4867	−0.460891
$735$	0	0
$736$	6.54615	0.241294
$737$	31.7586	1.16984
$738$	25.7420	0.947576
$739$	−29.7040	−1.09268	−0.546341	−	0.837563i	$-0.683980\pi$
$739$	−29.7040	−1.09268	−0.546341	+	0.837563i	$0.683980\pi$
$740$	0	0
$741$	68.4765	2.51555
$742$	2.97100	0.109069
$743$	−39.5171	−1.44974	−0.724872	−	0.688884i	$-0.758101\pi$
$743$	−39.5171	−1.44974	−0.724872	+	0.688884i	$0.758101\pi$
$744$	13.6068	0.498849
$745$	0	0
$746$	−14.8659	−0.544281
$747$	−8.43729	−0.308704
$748$	−1.21243	−0.0443307
$749$	9.37665	0.342615
$750$	0	0
$751$	−11.6384	−0.424693	−0.212346	−	0.977194i	$-0.568110\pi$
$751$	−11.6384	−0.424693	−0.212346	+	0.977194i	$0.568110\pi$
$752$	0	0
$753$	20.4577	0.745518
$754$	2.87987	0.104879
$755$	0	0
$756$	−2.41093	−0.0876847
$757$	21.0923	0.766612	0.383306	−	0.923621i	$-0.374785\pi$
$757$	21.0923	0.766612	0.383306	+	0.923621i	$0.374785\pi$
$758$	−2.39437	−0.0869674
$759$	66.0656	2.39803
$760$	0	0
$761$	−16.8988	−0.612579	−0.306290	−	0.951938i	$-0.599088\pi$
$761$	−16.8988	−0.612579	−0.306290	+	0.951938i	$0.599088\pi$
$762$	−30.9292	−1.12045
$763$	−5.51714	−0.199734
$764$	−24.8520	−0.899115
$765$	0	0
$766$	−14.7725	−0.533752
$767$	12.5461	0.453015
$768$	−2.27307	−0.0820225
$769$	28.6524	1.03323	0.516615	−	0.856218i	$-0.327192\pi$
$769$	28.6524	1.03323	0.516615	+	0.856218i	$0.327192\pi$
$770$	0	0
$771$	−7.78757	−0.280463
$772$	−7.65352	−0.275456
$773$	−10.3198	−0.371177	−0.185589	−	0.982628i	$-0.559419\pi$
$773$	−10.3198	−0.371177	−0.185589	+	0.982628i	$0.559419\pi$
$774$	22.4905	0.808403
$775$	0	0
$776$	−0.879866	−0.0315854
$777$	2.89379	0.103814
$778$	−12.6991	−0.455284
$779$	−67.8697	−2.43169
$780$	0	0
$781$	−24.8938	−0.890770
$782$	1.78757	0.0639235
$783$	1.03428	0.0369623
$784$	−5.37929	−0.192117
$785$	0	0
$786$	−9.47158	−0.337840
$787$	16.3337	0.582234	0.291117	−	0.956687i	$-0.405973\pi$
$787$	16.3337	0.582234	0.291117	+	0.956687i	$0.405973\pi$
$788$	14.5322	0.517689
$789$	5.85086	0.208296
$790$	0	0
$791$	2.68136	0.0953383
$792$	−9.62071	−0.341857
$793$	62.3236	2.21318
$794$	−4.98492	−0.176908
$795$	0	0
$796$	−12.8799	−0.456515
$797$	2.96719	0.105103	0.0525517	−	0.998618i	$-0.483265\pi$
$797$	2.96719	0.105103	0.0525517	+	0.998618i	$0.483265\pi$
$798$	−16.5322	−0.585234
$799$	0	0
$800$	0	0
$801$	30.0456	1.06161
$802$	27.1985	0.960413
$803$	−43.1251	−1.52185
$804$	16.2592	0.573416
$805$	0	0
$806$	−31.5650	−1.11183
$807$	63.0366	2.21899
$808$	3.66628	0.128979
$809$	−7.87870	−0.277001	−0.138500	−	0.990362i	$-0.544228\pi$
$809$	−7.87870	−0.277001	−0.138500	+	0.990362i	$0.544228\pi$
$810$	0	0
$811$	−48.6396	−1.70797	−0.853984	−	0.520300i	$-0.825820\pi$
$811$	−48.6396	−1.70797	−0.853984	+	0.520300i	$0.825820\pi$
$812$	−0.695283	−0.0243997
$813$	−60.1428	−2.10930
$814$	−4.43993	−0.155620
$815$	0	0
$816$	−0.620715	−0.0217294
$817$	−59.2969	−2.07454
$818$	35.5499	1.24297
$819$	−14.5461	−0.508283
$820$	0	0
$821$	−7.68020	−0.268041	−0.134020	−	0.990979i	$-0.542789\pi$
$821$	−7.68020	−0.268041	−0.134020	+	0.990979i	$0.542789\pi$
$822$	2.75593	0.0961241
$823$	−26.3654	−0.919039	−0.459519	−	0.888168i	$-0.651978\pi$
$823$	−26.3654	−0.919039	−0.459519	+	0.888168i	$0.651978\pi$
$824$	12.3198	0.429181
$825$	0	0
$826$	−3.02900	−0.105393
$827$	−17.8343	−0.620159	−0.310080	−	0.950711i	$-0.600356\pi$
$827$	−17.8343	−0.620159	−0.310080	+	0.950711i	$0.600356\pi$
$828$	14.1846	0.492948
$829$	−5.17962	−0.179896	−0.0899478	−	0.995946i	$-0.528670\pi$
$829$	−5.17962	−0.179896	−0.0899478	+	0.995946i	$0.528670\pi$
$830$	0	0
$831$	1.30472	0.0452601
$832$	5.27307	0.182811
$833$	−1.46894	−0.0508956
$834$	−17.1213	−0.592862
$835$	0	0
$836$	25.3654	0.877279
$837$	−11.3364	−0.391842
$838$	−29.5322	−1.02017
$839$	−6.15294	−0.212423	−0.106212	−	0.994344i	$-0.533872\pi$
$839$	−6.15294	−0.212423	−0.106212	+	0.994344i	$0.533872\pi$
$840$	0	0
$841$	−28.7017	−0.989715
$842$	−30.4905	−1.05077
$843$	66.5082	2.29066
$844$	−19.1529	−0.659271
$845$	0	0
$846$	0	0
$847$	−11.0923	−0.381136
$848$	−2.33372	−0.0801403
$849$	−47.7748	−1.63963
$850$	0	0
$851$	6.54615	0.224399
$852$	−12.7446	−0.436625
$853$	−30.8837	−1.05744	−0.528718	−	0.848797i	$-0.677327\pi$
$853$	−30.8837	−1.05744	−0.528718	+	0.848797i	$0.677327\pi$
$854$	−15.0467	−0.514888
$855$	0	0
$856$	−7.36536	−0.251743
$857$	15.1529	0.517615	0.258807	−	0.965929i	$-0.416671\pi$
$857$	15.1529	0.517615	0.258807	+	0.965929i	$0.416671\pi$
$858$	53.2174	1.81681
$859$	−7.28583	−0.248589	−0.124295	−	0.992245i	$-0.539667\pi$
$859$	−7.28583	−0.248589	−0.124295	+	0.992245i	$0.539667\pi$
$860$	0	0
$861$	34.3778	1.17159
$862$	21.9722	0.748375
$863$	−51.7357	−1.76110	−0.880552	−	0.473950i	$-0.842828\pi$
$863$	−51.7357	−1.76110	−0.880552	+	0.473950i	$0.842828\pi$
$864$	1.89379	0.0644280
$865$	0	0
$866$	5.00000	0.169907
$867$	38.4727	1.30660
$868$	7.62071	0.258664
$869$	−50.7307	−1.72092
$870$	0	0
$871$	−37.7180	−1.27802
$872$	4.33372	0.146758
$873$	−1.90655	−0.0645268
$874$	−37.3982	−1.26501
$875$	0	0
$876$	−22.0784	−0.745959
$877$	−50.4109	−1.70226	−0.851128	−	0.524958i	$-0.824081\pi$
$877$	−50.4109	−1.70226	−0.851128	+	0.524958i	$0.824081\pi$
$878$	29.5171	0.996155
$879$	−40.8204	−1.37684
$880$	0	0
$881$	−45.0822	−1.51886	−0.759428	−	0.650591i	$-0.774521\pi$
$881$	−45.0822	−1.51886	−0.759428	+	0.650591i	$0.774521\pi$
$882$	−11.6562	−0.392483
$883$	−17.9988	−0.605709	−0.302855	−	0.953037i	$-0.597940\pi$
$883$	−17.9988	−0.605709	−0.302855	+	0.953037i	$0.597940\pi$
$884$	1.43993	0.0484302
$885$	0	0
$886$	−28.9861	−0.973806
$887$	−38.2502	−1.28432	−0.642158	−	0.766572i	$-0.721961\pi$
$887$	−38.2502	−1.28432	−0.642158	+	0.766572i	$0.721961\pi$
$888$	−2.27307	−0.0762793
$889$	−17.3224	−0.580976
$890$	0	0
$891$	47.9748	1.60722
$892$	−2.75857	−0.0923638
$893$	0	0
$894$	31.7914	1.06326
$895$	0	0
$896$	−1.27307	−0.0425304
$897$	−78.4626	−2.61979
$898$	36.6245	1.22218
$899$	−3.26927	−0.109036
$900$	0	0
$901$	−0.637276	−0.0212307
$902$	−52.7458	−1.75624
$903$	30.0354	0.999517
$904$	−2.10621	−0.0700516
$905$	0	0
$906$	−31.2769	−1.03910
$907$	10.0734	0.334482	0.167241	−	0.985916i	$-0.446514\pi$
$907$	10.0734	0.334482	0.167241	+	0.985916i	$0.446514\pi$
$908$	17.2592	0.572765
$909$	7.94432	0.263496
$910$	0	0
$911$	−5.56123	−0.184252	−0.0921259	−	0.995747i	$-0.529366\pi$
$911$	−5.56123	−0.184252	−0.0921259	+	0.995747i	$0.529366\pi$
$912$	12.9861	0.430012
$913$	17.2882	0.572154
$914$	−19.7446	−0.653095
$915$	0	0
$916$	−22.3059	−0.737007
$917$	−5.30472	−0.175177
$918$	0.517142	0.0170682
$919$	−3.75973	−0.124022	−0.0620111	−	0.998075i	$-0.519751\pi$
$919$	−3.75973	−0.124022	−0.0620111	+	0.998075i	$0.519751\pi$
$920$	0	0
$921$	−52.7319	−1.73757
$922$	34.9733	1.15178
$923$	29.5650	0.973145
$924$	−12.8482	−0.422675
$925$	0	0
$926$	24.1707	0.794297
$927$	26.6953	0.876788
$928$	0.546146	0.0179281
$929$	3.40597	0.111746	0.0558731	−	0.998438i	$-0.482206\pi$
$929$	3.40597	0.111746	0.0558731	+	0.998438i	$0.482206\pi$
$930$	0	0
$931$	30.7319	1.00720
$932$	24.3514	0.797658
$933$	15.8772	0.519797
$934$	−7.63844	−0.249937
$935$	0	0
$936$	11.4260	0.373471
$937$	34.8053	1.13704	0.568520	−	0.822670i	$-0.307516\pi$
$937$	34.8053	1.13704	0.568520	+	0.822670i	$0.307516\pi$
$938$	9.10621	0.297328
$939$	−54.9330	−1.79267
$940$	0	0
$941$	−1.63844	−0.0534115	−0.0267058	−	0.999643i	$-0.508502\pi$
$941$	−1.63844	−0.0534115	−0.0267058	+	0.999643i	$0.508502\pi$
$942$	5.81922	0.189600
$943$	77.7673	2.53245
$944$	2.37929	0.0774391
$945$	0	0
$946$	−46.0833	−1.49830
$947$	30.8520	1.00256	0.501278	−	0.865286i	$-0.332863\pi$
$947$	30.8520	1.00256	0.501278	+	0.865286i	$0.332863\pi$
$948$	−25.9722	−0.843536
$949$	51.2174	1.66259
$950$	0	0
$951$	73.4020	2.38022
$952$	−0.347642	−0.0112671
$953$	−34.0467	−1.10288	−0.551441	−	0.834214i	$-0.685922\pi$
$953$	−34.0467	−1.10288	−0.551441	+	0.834214i	$0.685922\pi$
$954$	−5.05685	−0.163721
$955$	0	0
$956$	1.78757	0.0578143
$957$	5.51186	0.178173
$958$	−4.01392	−0.129684
$959$	1.54351	0.0498424
$960$	0	0
$961$	4.83314	0.155908
$962$	5.27307	0.170011
$963$	−15.9597	−0.514295
$964$	−3.72693	−0.120036
$965$	0	0
$966$	18.9432	0.609486
$967$	58.2757	1.87402	0.937010	−	0.349302i	$-0.113581\pi$
$967$	58.2757	1.87402	0.937010	+	0.349302i	$0.113581\pi$
$968$	8.71301	0.280047
$969$	3.54615	0.113919
$970$	0	0
$971$	57.6611	1.85043	0.925217	−	0.379439i	$-0.123883\pi$
$971$	57.6611	1.85043	0.925217	+	0.379439i	$0.123883\pi$
$972$	18.8799	0.605572
$973$	−9.58907	−0.307411
$974$	−43.3842	−1.39012
$975$	0	0
$976$	11.8192	0.378324
$977$	−3.78261	−0.121016	−0.0605082	−	0.998168i	$-0.519272\pi$
$977$	−3.78261	−0.121016	−0.0605082	+	0.998168i	$0.519272\pi$
$978$	−49.3552	−1.57821
$979$	−61.5639	−1.96759
$980$	0	0
$981$	9.39057	0.299818
$982$	−20.7307	−0.661544
$983$	41.3047	1.31742	0.658708	−	0.752399i	$-0.271103\pi$
$983$	41.3047	1.31742	0.658708	+	0.752399i	$0.271103\pi$
$984$	−27.0038	−0.860850
$985$	0	0
$986$	0.149138	0.00474951
$987$	0	0
$988$	−30.1251	−0.958407
$989$	67.9443	2.16050
$990$	0	0
$991$	−9.89263	−0.314250	−0.157125	−	0.987579i	$-0.550222\pi$
$991$	−9.89263	−0.314250	−0.157125	+	0.987579i	$0.550222\pi$
$992$	−5.98608	−0.190058
$993$	63.4765	2.01437
$994$	−7.13786	−0.226399
$995$	0	0
$996$	8.85086	0.280450
$997$	−44.5221	−1.41003	−0.705015	−	0.709193i	$-0.749060\pi$
$997$	−44.5221	−1.41003	−0.705015	+	0.709193i	$0.749060\pi$
$998$	5.92659	0.187603
$999$	1.89379	0.0599168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bc.1.1	yes	3
5.2	odd	4		1850.2.b.p.149.6		6
5.3	odd	4		1850.2.b.p.149.1		6
5.4	even	2		1850.2.a.y.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.y.1.3	✓	3	5.4	even	2
1850.2.a.bc.1.1	yes	3	1.1	even	1	trivial
1850.2.b.p.149.1		6	5.3	odd	4
1850.2.b.p.149.6		6	5.2	odd	4

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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.27307 q^{3} +1.00000 q^{4} -2.27307 q^{6} -1.27307 q^{7} +1.00000 q^{8} +2.16686 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.27307 q^{3} +1.00000 q^{4} -2.27307 q^{6} -1.27307 q^{7} +1.00000 q^{8} +2.16686 q^{9} -4.43993 q^{11} -2.27307 q^{12} +5.27307 q^{13} -1.27307 q^{14} +1.00000 q^{16} +0.273073 q^{17} +2.16686 q^{18} -5.71301 q^{19} +2.89379 q^{21} -4.43993 q^{22} +6.54615 q^{23} -2.27307 q^{24} +5.27307 q^{26} +1.89379 q^{27} -1.27307 q^{28} +0.546146 q^{29} -5.98608 q^{31} +1.00000 q^{32} +10.0923 q^{33} +0.273073 q^{34} +2.16686 q^{36} +1.00000 q^{37} -5.71301 q^{38} -11.9861 q^{39} +11.8799 q^{41} +2.89379 q^{42} +10.3793 q^{43} -4.43993 q^{44} +6.54615 q^{46} -2.27307 q^{48} -5.37929 q^{49} -0.620715 q^{51} +5.27307 q^{52} -2.33372 q^{53} +1.89379 q^{54} -1.27307 q^{56} +12.9861 q^{57} +0.546146 q^{58} +2.37929 q^{59} +11.8192 q^{61} -5.98608 q^{62} -2.75857 q^{63} +1.00000 q^{64} +10.0923 q^{66} -7.15294 q^{67} +0.273073 q^{68} -14.8799 q^{69} +5.60679 q^{71} +2.16686 q^{72} +9.71301 q^{73} +1.00000 q^{74} -5.71301 q^{76} +5.65236 q^{77} -11.9861 q^{78} +11.4260 q^{79} -10.8053 q^{81} +11.8799 q^{82} -3.89379 q^{83} +2.89379 q^{84} +10.3793 q^{86} -1.24143 q^{87} -4.43993 q^{88} +13.8659 q^{89} -6.71301 q^{91} +6.54615 q^{92} +13.6068 q^{93} -2.27307 q^{96} -0.879866 q^{97} -5.37929 q^{98} -9.62071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 5 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{18} - q^{19} + 16 q^{21} - 5 q^{22} + 4 q^{23} + q^{24} + 8 q^{26} + 13 q^{27} + 4 q^{28} - 14 q^{29} + 6 q^{31} + 3 q^{32} - q^{33} - 7 q^{34} + 6 q^{36} + 3 q^{37} - q^{38} - 12 q^{39} + 19 q^{41} + 16 q^{42} + 16 q^{43} - 5 q^{44} + 4 q^{46} + q^{48} - q^{49} - 17 q^{51} + 8 q^{52} - 6 q^{53} + 13 q^{54} + 4 q^{56} + 15 q^{57} - 14 q^{58} - 8 q^{59} + 12 q^{61} + 6 q^{62} + 22 q^{63} + 3 q^{64} - q^{66} + 3 q^{67} - 7 q^{68} - 28 q^{69} + 8 q^{71} + 6 q^{72} + 13 q^{73} + 3 q^{74} - q^{76} - 6 q^{77} - 12 q^{78} + 2 q^{79} + 15 q^{81} + 19 q^{82} - 19 q^{83} + 16 q^{84} + 16 q^{86} - 34 q^{87} - 5 q^{88} + q^{89} - 4 q^{91} + 4 q^{92} + 32 q^{93} + q^{96} + 14 q^{97} - q^{98} - 44 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9 - 5 * q^11 + q^12 + 8 * q^13 + 4 * q^14 + 3 * q^16 - 7 * q^17 + 6 * q^18 - q^19 + 16 * q^21 - 5 * q^22 + 4 * q^23 + q^24 + 8 * q^26 + 13 * q^27 + 4 * q^28 - 14 * q^29 + 6 * q^31 + 3 * q^32 - q^33 - 7 * q^34 + 6 * q^36 + 3 * q^37 - q^38 - 12 * q^39 + 19 * q^41 + 16 * q^42 + 16 * q^43 - 5 * q^44 + 4 * q^46 + q^48 - q^49 - 17 * q^51 + 8 * q^52 - 6 * q^53 + 13 * q^54 + 4 * q^56 + 15 * q^57 - 14 * q^58 - 8 * q^59 + 12 * q^61 + 6 * q^62 + 22 * q^63 + 3 * q^64 - q^66 + 3 * q^67 - 7 * q^68 - 28 * q^69 + 8 * q^71 + 6 * q^72 + 13 * q^73 + 3 * q^74 - q^76 - 6 * q^77 - 12 * q^78 + 2 * q^79 + 15 * q^81 + 19 * q^82 - 19 * q^83 + 16 * q^84 + 16 * q^86 - 34 * q^87 - 5 * q^88 + q^89 - 4 * q^91 + 4 * q^92 + 32 * q^93 + q^96 + 14 * q^97 - q^98 - 44 * q^99

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