LMFDB - Embedded newform 1850.2.a.bd.1.4

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:	$5$
Coefficient field:	5.5.1791440.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} + 13x - 4 $ x^5 - 9x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.09441$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} +3.20984 q^{7} -1.00000 q^{8} -1.80226 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} +3.20984 q^{7} -1.00000 q^{8} -1.80226 q^{9} +3.82327 q^{11} +1.09441 q^{12} -0.147332 q^{13} -3.20984 q^{14} +1.00000 q^{16} -0.978989 q^{17} +1.80226 q^{18} +2.67594 q^{19} +3.51289 q^{21} -3.82327 q^{22} +2.33616 q^{23} -1.09441 q^{24} +0.147332 q^{26} -5.25565 q^{27} +3.20984 q^{28} +6.30425 q^{29} -3.62372 q^{31} -1.00000 q^{32} +4.18424 q^{33} +0.978989 q^{34} -1.80226 q^{36} -1.00000 q^{37} -2.67594 q^{38} -0.161242 q^{39} +11.8265 q^{41} -3.51289 q^{42} +4.53390 q^{43} +3.82327 q^{44} -2.33616 q^{46} -6.23085 q^{47} +1.09441 q^{48} +3.30305 q^{49} -1.07142 q^{51} -0.147332 q^{52} -11.2978 q^{53} +5.25565 q^{54} -3.20984 q^{56} +2.92858 q^{57} -6.30425 q^{58} +6.92858 q^{59} +10.4885 q^{61} +3.62372 q^{62} -5.78496 q^{63} +1.00000 q^{64} -4.18424 q^{66} -2.80936 q^{67} -0.978989 q^{68} +2.55672 q^{69} -12.3189 q^{71} +1.80226 q^{72} +13.9966 q^{73} +1.00000 q^{74} +2.67594 q^{76} +12.2721 q^{77} +0.161242 q^{78} +15.6057 q^{79} -0.345071 q^{81} -11.8265 q^{82} -13.5371 q^{83} +3.51289 q^{84} -4.53390 q^{86} +6.89945 q^{87} -3.82327 q^{88} -6.46929 q^{89} -0.472913 q^{91} +2.33616 q^{92} -3.96585 q^{93} +6.23085 q^{94} -1.09441 q^{96} +3.07063 q^{97} -3.30305 q^{98} -6.89054 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.09441	0.631859	0.315930	−	0.948783i	$-0.397684\pi$
$3$	1.09441	0.631859	0.315930	+	0.948783i	$0.397684\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.09441	−0.446792
$7$	3.20984	1.21320	0.606602	−	0.795006i	$-0.292532\pi$
$7$	3.20984	1.21320	0.606602	+	0.795006i	$0.292532\pi$
$8$	−1.00000	−0.353553
$9$	−1.80226	−0.600754
$10$	0	0
$11$	3.82327	1.15276	0.576380	−	0.817182i	$-0.304465\pi$
$11$	3.82327	1.15276	0.576380	+	0.817182i	$0.304465\pi$
$12$	1.09441	0.315930
$13$	−0.147332	−0.0408626	−0.0204313	−	0.999791i	$-0.506504\pi$
$13$	−0.147332	−0.0408626	−0.0204313	+	0.999791i	$0.506504\pi$
$14$	−3.20984	−0.857865
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.978989	−0.237440	−0.118720	−	0.992928i	$-0.537879\pi$
$17$	−0.978989	−0.237440	−0.118720	+	0.992928i	$0.537879\pi$
$18$	1.80226	0.424797
$19$	2.67594	0.613903	0.306951	−	0.951725i	$-0.400691\pi$
$19$	2.67594	0.613903	0.306951	+	0.951725i	$0.400691\pi$
$20$	0	0
$21$	3.51289	0.766574
$22$	−3.82327	−0.815124
$23$	2.33616	0.487122	0.243561	−	0.969886i	$-0.421684\pi$
$23$	2.33616	0.487122	0.243561	+	0.969886i	$0.421684\pi$
$24$	−1.09441	−0.223396
$25$	0	0
$26$	0.147332	0.0288942
$27$	−5.25565	−1.01145
$28$	3.20984	0.606602
$29$	6.30425	1.17067	0.585335	−	0.810792i	$-0.300963\pi$
$29$	6.30425	1.17067	0.585335	+	0.810792i	$0.300963\pi$
$30$	0	0
$31$	−3.62372	−0.650839	−0.325420	−	0.945570i	$-0.605506\pi$
$31$	−3.62372	−0.650839	−0.325420	+	0.945570i	$0.605506\pi$
$32$	−1.00000	−0.176777
$33$	4.18424	0.728382
$34$	0.978989	0.167895
$35$	0	0
$36$	−1.80226	−0.300377
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−2.67594	−0.434095
$39$	−0.161242	−0.0258194
$40$	0	0
$41$	11.8265	1.84698	0.923491	−	0.383620i	$-0.125323\pi$
$41$	11.8265	1.84698	0.923491	+	0.383620i	$0.125323\pi$
$42$	−3.51289	−0.542050
$43$	4.53390	0.691413	0.345706	−	0.938343i	$-0.387639\pi$
$43$	4.53390	0.691413	0.345706	+	0.938343i	$0.387639\pi$
$44$	3.82327	0.576380
$45$	0	0
$46$	−2.33616	−0.344448
$47$	−6.23085	−0.908863	−0.454431	−	0.890782i	$-0.650157\pi$
$47$	−6.23085	−0.908863	−0.454431	+	0.890782i	$0.650157\pi$
$48$	1.09441	0.157965
$49$	3.30305	0.471864
$50$	0	0
$51$	−1.07142	−0.150028
$52$	−0.147332	−0.0204313
$53$	−11.2978	−1.55188	−0.775939	−	0.630807i	$-0.782724\pi$
$53$	−11.2978	−1.55188	−0.775939	+	0.630807i	$0.782724\pi$
$54$	5.25565	0.715204
$55$	0	0
$56$	−3.20984	−0.428932
$57$	2.92858	0.387900
$58$	−6.30425	−0.827788
$59$	6.92858	0.902025	0.451012	−	0.892518i	$-0.351063\pi$
$59$	6.92858	0.902025	0.451012	+	0.892518i	$0.351063\pi$
$60$	0	0
$61$	10.4885	1.34291	0.671457	−	0.741044i	$-0.265669\pi$
$61$	10.4885	1.34291	0.671457	+	0.741044i	$0.265669\pi$
$62$	3.62372	0.460213
$63$	−5.78496	−0.728837
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.18424	−0.515044
$67$	−2.80936	−0.343218	−0.171609	−	0.985165i	$-0.554897\pi$
$67$	−2.80936	−0.343218	−0.171609	+	0.985165i	$0.554897\pi$
$68$	−0.978989	−0.118720
$69$	2.55672	0.307793
$70$	0	0
$71$	−12.3189	−1.46198	−0.730990	−	0.682388i	$-0.760941\pi$
$71$	−12.3189	−1.46198	−0.730990	+	0.682388i	$0.760941\pi$
$72$	1.80226	0.212399
$73$	13.9966	1.63818	0.819090	−	0.573665i	$-0.194479\pi$
$73$	13.9966	1.63818	0.819090	+	0.573665i	$0.194479\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.67594	0.306951
$77$	12.2721	1.39853
$78$	0.161242	0.0182571
$79$	15.6057	1.75578	0.877890	−	0.478861i	$-0.158950\pi$
$79$	15.6057	1.75578	0.877890	+	0.478861i	$0.158950\pi$
$80$	0	0
$81$	−0.345071	−0.0383412
$82$	−11.8265	−1.30601
$83$	−13.5371	−1.48589	−0.742944	−	0.669354i	$-0.766571\pi$
$83$	−13.5371	−1.48589	−0.742944	+	0.669354i	$0.766571\pi$
$84$	3.51289	0.383287
$85$	0	0
$86$	−4.53390	−0.488903
$87$	6.89945	0.739699
$88$	−3.82327	−0.407562
$89$	−6.46929	−0.685743	−0.342872	−	0.939382i	$-0.611400\pi$
$89$	−6.46929	−0.685743	−0.342872	+	0.939382i	$0.611400\pi$
$90$	0	0
$91$	−0.472913	−0.0495747
$92$	2.33616	0.243561
$93$	−3.96585	−0.411239
$94$	6.23085	0.642663
$95$	0	0
$96$	−1.09441	−0.111698
$97$	3.07063	0.311775	0.155887	−	0.987775i	$-0.450176\pi$
$97$	3.07063	0.311775	0.155887	+	0.987775i	$0.450176\pi$
$98$	−3.30305	−0.333658
$99$	−6.89054	−0.692525
$100$	0	0
$101$	−6.57513	−0.654250	−0.327125	−	0.944981i	$-0.606080\pi$
$101$	−6.57513	−0.654250	−0.327125	+	0.944981i	$0.606080\pi$
$102$	1.07142	0.106086
$103$	2.52180	0.248480	0.124240	−	0.992252i	$-0.460351\pi$
$103$	2.52180	0.248480	0.124240	+	0.992252i	$0.460351\pi$
$104$	0.147332	0.0144471
$105$	0	0
$106$	11.2978	1.09734
$107$	−5.70291	−0.551321	−0.275661	−	0.961255i	$-0.588897\pi$
$107$	−5.70291	−0.551321	−0.275661	+	0.961255i	$0.588897\pi$
$108$	−5.25565	−0.505726
$109$	11.9318	1.14286	0.571428	−	0.820652i	$-0.306390\pi$
$109$	11.9318	1.14286	0.571428	+	0.820652i	$0.306390\pi$
$110$	0	0
$111$	−1.09441	−0.103877
$112$	3.20984	0.303301
$113$	2.18044	0.205119	0.102559	−	0.994727i	$-0.467297\pi$
$113$	2.18044	0.205119	0.102559	+	0.994727i	$0.467297\pi$
$114$	−2.92858	−0.274287
$115$	0	0
$116$	6.30425	0.585335
$117$	0.265531	0.0245484
$118$	−6.92858	−0.637828
$119$	−3.14239	−0.288063
$120$	0	0
$121$	3.61741	0.328856
$122$	−10.4885	−0.949583
$123$	12.9430	1.16703
$124$	−3.62372	−0.325420
$125$	0	0
$126$	5.78496	0.515365
$127$	5.39390	0.478631	0.239316	−	0.970942i	$-0.423077\pi$
$127$	5.39390	0.478631	0.239316	+	0.970942i	$0.423077\pi$
$128$	−1.00000	−0.0883883
$129$	4.96195	0.436876
$130$	0	0
$131$	−8.97060	−0.783765	−0.391883	−	0.920015i	$-0.628176\pi$
$131$	−8.97060	−0.783765	−0.391883	+	0.920015i	$0.628176\pi$
$132$	4.18424	0.364191
$133$	8.58933	0.744789
$134$	2.80936	0.242692
$135$	0	0
$136$	0.978989	0.0839476
$137$	2.70352	0.230978	0.115489	−	0.993309i	$-0.463157\pi$
$137$	2.70352	0.230978	0.115489	+	0.993309i	$0.463157\pi$
$138$	−2.55672	−0.217642
$139$	4.30358	0.365025	0.182512	−	0.983204i	$-0.441577\pi$
$139$	4.30358	0.365025	0.182512	+	0.983204i	$0.441577\pi$
$140$	0	0
$141$	−6.81912	−0.574273
$142$	12.3189	1.03378
$143$	−0.563292	−0.0471048
$144$	−1.80226	−0.150188
$145$	0	0
$146$	−13.9966	−1.15837
$147$	3.61490	0.298152
$148$	−1.00000	−0.0821995
$149$	21.6949	1.77732	0.888659	−	0.458568i	$-0.151638\pi$
$149$	21.6949	1.77732	0.888659	+	0.458568i	$0.151638\pi$
$150$	0	0
$151$	2.06744	0.168246	0.0841230	−	0.996455i	$-0.473191\pi$
$151$	2.06744	0.168246	0.0841230	+	0.996455i	$0.473191\pi$
$152$	−2.67594	−0.217047
$153$	1.76439	0.142643
$154$	−12.2721	−0.988912
$155$	0	0
$156$	−0.161242	−0.0129097
$157$	−18.5439	−1.47996	−0.739982	−	0.672627i	$-0.765166\pi$
$157$	−18.5439	−1.47996	−0.739982	+	0.672627i	$0.765166\pi$
$158$	−15.6057	−1.24152
$159$	−12.3645	−0.980569
$160$	0	0
$161$	7.49868	0.590979
$162$	0.345071	0.0271113
$163$	21.6240	1.69372	0.846860	−	0.531817i	$-0.178490\pi$
$163$	21.6240	1.69372	0.846860	+	0.531817i	$0.178490\pi$
$164$	11.8265	0.923491
$165$	0	0
$166$	13.5371	1.05068
$167$	11.7729	0.911012	0.455506	−	0.890233i	$-0.349458\pi$
$167$	11.7729	0.911012	0.455506	+	0.890233i	$0.349458\pi$
$168$	−3.51289	−0.271025
$169$	−12.9783	−0.998330
$170$	0	0
$171$	−4.82274	−0.368804
$172$	4.53390	0.345706
$173$	−14.0158	−1.06560	−0.532801	−	0.846240i	$-0.678861\pi$
$173$	−14.0158	−1.06560	−0.532801	+	0.846240i	$0.678861\pi$
$174$	−6.89945	−0.523046
$175$	0	0
$176$	3.82327	0.288190
$177$	7.58273	0.569953
$178$	6.46929	0.484894
$179$	−12.4160	−0.928019	−0.464009	−	0.885830i	$-0.653590\pi$
$179$	−12.4160	−0.928019	−0.464009	+	0.885830i	$0.653590\pi$
$180$	0	0
$181$	8.19748	0.609314	0.304657	−	0.952462i	$-0.401458\pi$
$181$	8.19748	0.609314	0.304657	+	0.952462i	$0.401458\pi$
$182$	0.472913	0.0350546
$183$	11.4787	0.848532
$184$	−2.33616	−0.172224
$185$	0	0
$186$	3.96585	0.290790
$187$	−3.74294	−0.273711
$188$	−6.23085	−0.454431
$189$	−16.8698	−1.22710
$190$	0	0
$191$	8.00137	0.578959	0.289479	−	0.957184i	$-0.406518\pi$
$191$	8.00137	0.578959	0.289479	+	0.957184i	$0.406518\pi$
$192$	1.09441	0.0789824
$193$	14.0420	1.01077	0.505383	−	0.862895i	$-0.331351\pi$
$193$	14.0420	1.01077	0.505383	+	0.862895i	$0.331351\pi$
$194$	−3.07063	−0.220458
$195$	0	0
$196$	3.30305	0.235932
$197$	−3.36608	−0.239823	−0.119912	−	0.992785i	$-0.538261\pi$
$197$	−3.36608	−0.239823	−0.119912	+	0.992785i	$0.538261\pi$
$198$	6.89054	0.489689
$199$	−14.8890	−1.05545	−0.527725	−	0.849415i	$-0.676955\pi$
$199$	−14.8890	−1.05545	−0.527725	+	0.849415i	$0.676955\pi$
$200$	0	0
$201$	−3.07460	−0.216866
$202$	6.57513	0.462624
$203$	20.2356	1.42026
$204$	−1.07142	−0.0750142
$205$	0	0
$206$	−2.52180	−0.175702
$207$	−4.21037	−0.292641
$208$	−0.147332	−0.0102157
$209$	10.2308	0.707683
$210$	0	0
$211$	−9.80544	−0.675035	−0.337517	−	0.941319i	$-0.609587\pi$
$211$	−9.80544	−0.675035	−0.337517	+	0.941319i	$0.609587\pi$
$212$	−11.2978	−0.775939
$213$	−13.4819	−0.923766
$214$	5.70291	0.389843
$215$	0	0
$216$	5.25565	0.357602
$217$	−11.6316	−0.789601
$218$	−11.9318	−0.808121
$219$	15.3181	1.03510
$220$	0	0
$221$	0.144237	0.00970241
$222$	1.09441	0.0734522
$223$	15.4447	1.03425	0.517125	−	0.855910i	$-0.327002\pi$
$223$	15.4447	1.03425	0.517125	+	0.855910i	$0.327002\pi$
$224$	−3.20984	−0.214466
$225$	0	0
$226$	−2.18044	−0.145041
$227$	−6.46090	−0.428825	−0.214413	−	0.976743i	$-0.568784\pi$
$227$	−6.46090	−0.428825	−0.214413	+	0.976743i	$0.568784\pi$
$228$	2.92858	0.193950
$229$	7.97458	0.526975	0.263488	−	0.964663i	$-0.415127\pi$
$229$	7.97458	0.526975	0.263488	+	0.964663i	$0.415127\pi$
$230$	0	0
$231$	13.4307	0.883676
$232$	−6.30425	−0.413894
$233$	−25.2907	−1.65685	−0.828423	−	0.560103i	$-0.810762\pi$
$233$	−25.2907	−1.65685	−0.828423	+	0.560103i	$0.810762\pi$
$234$	−0.265531	−0.0173583
$235$	0	0
$236$	6.92858	0.451012
$237$	17.0791	1.10941
$238$	3.14239	0.203691
$239$	5.11481	0.330850	0.165425	−	0.986222i	$-0.447100\pi$
$239$	5.11481	0.330850	0.165425	+	0.986222i	$0.447100\pi$
$240$	0	0
$241$	16.4415	1.05909	0.529544	−	0.848282i	$-0.322363\pi$
$241$	16.4415	1.05909	0.529544	+	0.848282i	$0.322363\pi$
$242$	−3.61741	−0.232536
$243$	15.3893	0.987225
$244$	10.4885	0.671457
$245$	0	0
$246$	−12.9430	−0.825217
$247$	−0.394252	−0.0250857
$248$	3.62372	0.230107
$249$	−14.8152	−0.938872
$250$	0	0
$251$	6.35586	0.401178	0.200589	−	0.979675i	$-0.435714\pi$
$251$	6.35586	0.401178	0.200589	+	0.979675i	$0.435714\pi$
$252$	−5.78496	−0.364418
$253$	8.93177	0.561535
$254$	−5.39390	−0.338444
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.45772	0.340443	0.170222	−	0.985406i	$-0.445552\pi$
$257$	5.45772	0.340443	0.170222	+	0.985406i	$0.445552\pi$
$258$	−4.96195	−0.308918
$259$	−3.20984	−0.199450
$260$	0	0
$261$	−11.3619	−0.703284
$262$	8.97060	0.554206
$263$	14.8882	0.918044	0.459022	−	0.888425i	$-0.348200\pi$
$263$	14.8882	0.918044	0.459022	+	0.888425i	$0.348200\pi$
$264$	−4.18424	−0.257522
$265$	0	0
$266$	−8.58933	−0.526646
$267$	−7.08007	−0.433293
$268$	−2.80936	−0.171609
$269$	15.5573	0.948546	0.474273	−	0.880378i	$-0.342711\pi$
$269$	15.5573	0.948546	0.474273	+	0.880378i	$0.342711\pi$
$270$	0	0
$271$	0.462920	0.0281204	0.0140602	−	0.999901i	$-0.495524\pi$
$271$	0.462920	0.0281204	0.0140602	+	0.999901i	$0.495524\pi$
$272$	−0.978989	−0.0593599
$273$	−0.517562	−0.0313242
$274$	−2.70352	−0.163326
$275$	0	0
$276$	2.55672	0.153896
$277$	−9.12112	−0.548035	−0.274018	−	0.961725i	$-0.588353\pi$
$277$	−9.12112	−0.548035	−0.274018	+	0.961725i	$0.588353\pi$
$278$	−4.30358	−0.258111
$279$	6.53089	0.390994
$280$	0	0
$281$	−24.8734	−1.48382	−0.741912	−	0.670497i	$-0.766081\pi$
$281$	−24.8734	−1.48382	−0.741912	+	0.670497i	$0.766081\pi$
$282$	6.81912	0.406073
$283$	−19.7259	−1.17258	−0.586292	−	0.810100i	$-0.699413\pi$
$283$	−19.7259	−1.17258	−0.586292	+	0.810100i	$0.699413\pi$
$284$	−12.3189	−0.730990
$285$	0	0
$286$	0.563292	0.0333081
$287$	37.9610	2.24077
$288$	1.80226	0.106199
$289$	−16.0416	−0.943622
$290$	0	0
$291$	3.36053	0.196998
$292$	13.9966	0.819090
$293$	2.65528	0.155123	0.0775615	−	0.996988i	$-0.475287\pi$
$293$	2.65528	0.155123	0.0775615	+	0.996988i	$0.475287\pi$
$294$	−3.61490	−0.210825
$295$	0	0
$296$	1.00000	0.0581238
$297$	−20.0938	−1.16596
$298$	−21.6949	−1.25675
$299$	−0.344191	−0.0199051
$300$	0	0
$301$	14.5531	0.838825
$302$	−2.06744	−0.118968
$303$	−7.19590	−0.413394
$304$	2.67594	0.153476
$305$	0	0
$306$	−1.76439	−0.100864
$307$	25.3545	1.44706	0.723528	−	0.690295i	$-0.242519\pi$
$307$	25.3545	1.44706	0.723528	+	0.690295i	$0.242519\pi$
$308$	12.2721	0.699267
$309$	2.75989	0.157005
$310$	0	0
$311$	11.1234	0.630748	0.315374	−	0.948967i	$-0.397870\pi$
$311$	11.1234	0.630748	0.315374	+	0.948967i	$0.397870\pi$
$312$	0.161242	0.00912855
$313$	−3.49109	−0.197328	−0.0986640	−	0.995121i	$-0.531457\pi$
$313$	−3.49109	−0.197328	−0.0986640	+	0.995121i	$0.531457\pi$
$314$	18.5439	1.04649
$315$	0	0
$316$	15.6057	0.877890
$317$	−18.3656	−1.03152	−0.515759	−	0.856734i	$-0.672490\pi$
$317$	−18.3656	−1.03152	−0.515759	+	0.856734i	$0.672490\pi$
$318$	12.3645	0.693367
$319$	24.1029	1.34950
$320$	0	0
$321$	−6.24134	−0.348357
$322$	−7.49868	−0.417885
$323$	−2.61972	−0.145765
$324$	−0.345071	−0.0191706
$325$	0	0
$326$	−21.6240	−1.19764
$327$	13.0583	0.722124
$328$	−11.8265	−0.653007
$329$	−20.0000	−1.10264
$330$	0	0
$331$	−21.9365	−1.20574	−0.602870	−	0.797839i	$-0.705976\pi$
$331$	−21.9365	−1.20574	−0.602870	+	0.797839i	$0.705976\pi$
$332$	−13.5371	−0.742944
$333$	1.80226	0.0987633
$334$	−11.7729	−0.644183
$335$	0	0
$336$	3.51289	0.191644
$337$	8.65472	0.471453	0.235726	−	0.971819i	$-0.424253\pi$
$337$	8.65472	0.471453	0.235726	+	0.971819i	$0.424253\pi$
$338$	12.9783	0.705926
$339$	2.38630	0.129606
$340$	0	0
$341$	−13.8545	−0.750262
$342$	4.82274	0.260784
$343$	−11.8666	−0.640737
$344$	−4.53390	−0.244451
$345$	0	0
$346$	14.0158	0.753495
$347$	21.3265	1.14486	0.572432	−	0.819952i	$-0.306000\pi$
$347$	21.3265	1.14486	0.572432	+	0.819952i	$0.306000\pi$
$348$	6.89945	0.369849
$349$	−1.16305	−0.0622569	−0.0311285	−	0.999515i	$-0.509910\pi$
$349$	−1.16305	−0.0622569	−0.0311285	+	0.999515i	$0.509910\pi$
$350$	0	0
$351$	0.774328	0.0413306
$352$	−3.82327	−0.203781
$353$	0.470079	0.0250198	0.0125099	−	0.999922i	$-0.496018\pi$
$353$	0.470079	0.0250198	0.0125099	+	0.999922i	$0.496018\pi$
$354$	−7.58273	−0.403017
$355$	0	0
$356$	−6.46929	−0.342872
$357$	−3.43908	−0.182015
$358$	12.4160	0.656208
$359$	−33.3086	−1.75796	−0.878981	−	0.476856i	$-0.841776\pi$
$359$	−33.3086	−1.75796	−0.878981	+	0.476856i	$0.841776\pi$
$360$	0	0
$361$	−11.8393	−0.623123
$362$	−8.19748	−0.430850
$363$	3.95894	0.207790
$364$	−0.472913	−0.0247874
$365$	0	0
$366$	−11.4787	−0.600003
$367$	11.1884	0.584029	0.292014	−	0.956414i	$-0.405674\pi$
$367$	11.1884	0.584029	0.292014	+	0.956414i	$0.405674\pi$
$368$	2.33616	0.121781
$369$	−21.3144	−1.10958
$370$	0	0
$371$	−36.2642	−1.88275
$372$	−3.96585	−0.205620
$373$	−23.1634	−1.19936	−0.599678	−	0.800242i	$-0.704705\pi$
$373$	−23.1634	−1.19936	−0.599678	+	0.800242i	$0.704705\pi$
$374$	3.74294	0.193543
$375$	0	0
$376$	6.23085	0.321331
$377$	−0.928820	−0.0478366
$378$	16.8698	0.867688
$379$	−26.7721	−1.37519	−0.687595	−	0.726095i	$-0.741333\pi$
$379$	−26.7721	−1.37519	−0.687595	+	0.726095i	$0.741333\pi$
$380$	0	0
$381$	5.90315	0.302428
$382$	−8.00137	−0.409386
$383$	−22.3189	−1.14044	−0.570220	−	0.821492i	$-0.693142\pi$
$383$	−22.3189	−1.14044	−0.570220	+	0.821492i	$0.693142\pi$
$384$	−1.09441	−0.0558490
$385$	0	0
$386$	−14.0420	−0.714720
$387$	−8.17127	−0.415369
$388$	3.07063	0.155887
$389$	−24.3011	−1.23211	−0.616057	−	0.787701i	$-0.711271\pi$
$389$	−24.3011	−1.23211	−0.616057	+	0.787701i	$0.711271\pi$
$390$	0	0
$391$	−2.28707	−0.115662
$392$	−3.30305	−0.166829
$393$	−9.81754	−0.495229
$394$	3.36608	0.169581
$395$	0	0
$396$	−6.89054	−0.346262
$397$	−28.1888	−1.41476	−0.707378	−	0.706835i	$-0.750122\pi$
$397$	−28.1888	−1.41476	−0.707378	+	0.706835i	$0.750122\pi$
$398$	14.8890	0.746316
$399$	9.40027	0.470602
$400$	0	0
$401$	−15.9528	−0.796644	−0.398322	−	0.917246i	$-0.630407\pi$
$401$	−15.9528	−0.796644	−0.398322	+	0.917246i	$0.630407\pi$
$402$	3.07460	0.153347
$403$	0.533891	0.0265950
$404$	−6.57513	−0.327125
$405$	0	0
$406$	−20.2356	−1.00428
$407$	−3.82327	−0.189513
$408$	1.07142	0.0530431
$409$	31.6014	1.56259	0.781294	−	0.624164i	$-0.214560\pi$
$409$	31.6014	1.56259	0.781294	+	0.624164i	$0.214560\pi$
$410$	0	0
$411$	2.95877	0.145945
$412$	2.52180	0.124240
$413$	22.2396	1.09434
$414$	4.21037	0.206928
$415$	0	0
$416$	0.147332	0.00722356
$417$	4.70989	0.230644
$418$	−10.2308	−0.500407
$419$	−18.4266	−0.900197	−0.450098	−	0.892979i	$-0.648611\pi$
$419$	−18.4266	−0.900197	−0.450098	+	0.892979i	$0.648611\pi$
$420$	0	0
$421$	−19.1570	−0.933655	−0.466828	−	0.884348i	$-0.654603\pi$
$421$	−19.1570	−0.933655	−0.466828	+	0.884348i	$0.654603\pi$
$422$	9.80544	0.477322
$423$	11.2296	0.546003
$424$	11.2978	0.548672
$425$	0	0
$426$	13.4819	0.653201
$427$	33.6663	1.62923
$428$	−5.70291	−0.275661
$429$	−0.616473	−0.0297636
$430$	0	0
$431$	−10.3947	−0.500694	−0.250347	−	0.968156i	$-0.580545\pi$
$431$	−10.3947	−0.500694	−0.250347	+	0.968156i	$0.580545\pi$
$432$	−5.25565	−0.252863
$433$	−3.36622	−0.161770	−0.0808852	−	0.996723i	$-0.525775\pi$
$433$	−3.36622	−0.161770	−0.0808852	+	0.996723i	$0.525775\pi$
$434$	11.6316	0.558332
$435$	0	0
$436$	11.9318	0.571428
$437$	6.25142	0.299046
$438$	−15.3181	−0.731926
$439$	−32.5045	−1.55135	−0.775677	−	0.631130i	$-0.782591\pi$
$439$	−32.5045	−1.55135	−0.775677	+	0.631130i	$0.782591\pi$
$440$	0	0
$441$	−5.95296	−0.283474
$442$	−0.144237	−0.00686064
$443$	23.5479	1.11879	0.559397	−	0.828900i	$-0.311033\pi$
$443$	23.5479	1.11879	0.559397	+	0.828900i	$0.311033\pi$
$444$	−1.09441	−0.0519385
$445$	0	0
$446$	−15.4447	−0.731325
$447$	23.7432	1.12302
$448$	3.20984	0.151651
$449$	12.5629	0.592878	0.296439	−	0.955052i	$-0.404201\pi$
$449$	12.5629	0.592878	0.296439	+	0.955052i	$0.404201\pi$
$450$	0	0
$451$	45.2158	2.12913
$452$	2.18044	0.102559
$453$	2.26263	0.106308
$454$	6.46090	0.303225
$455$	0	0
$456$	−2.92858	−0.137143
$457$	−6.53390	−0.305643	−0.152821	−	0.988254i	$-0.548836\pi$
$457$	−6.53390	−0.305643	−0.152821	+	0.988254i	$0.548836\pi$
$458$	−7.97458	−0.372628
$459$	5.14523	0.240159
$460$	0	0
$461$	24.7616	1.15326	0.576631	−	0.817005i	$-0.304367\pi$
$461$	24.7616	1.15326	0.576631	+	0.817005i	$0.304367\pi$
$462$	−13.4307	−0.624853
$463$	−25.9239	−1.20479	−0.602393	−	0.798200i	$-0.705786\pi$
$463$	−25.9239	−1.20479	−0.602393	+	0.798200i	$0.705786\pi$
$464$	6.30425	0.292667
$465$	0	0
$466$	25.2907	1.17157
$467$	−28.8182	−1.33355	−0.666773	−	0.745261i	$-0.732325\pi$
$467$	−28.8182	−1.33355	−0.666773	+	0.745261i	$0.732325\pi$
$468$	0.265531	0.0122742
$469$	−9.01759	−0.416394
$470$	0	0
$471$	−20.2947	−0.935129
$472$	−6.92858	−0.318914
$473$	17.3343	0.797033
$474$	−17.0791	−0.784469
$475$	0	0
$476$	−3.14239	−0.144031
$477$	20.3617	0.932297
$478$	−5.11481	−0.233946
$479$	−41.7873	−1.90931	−0.954655	−	0.297714i	$-0.903776\pi$
$479$	−41.7873	−1.90931	−0.954655	+	0.297714i	$0.903776\pi$
$480$	0	0
$481$	0.147332	0.00671778
$482$	−16.4415	−0.748888
$483$	8.20665	0.373416
$484$	3.61741	0.164428
$485$	0	0
$486$	−15.3893	−0.698073
$487$	−9.34463	−0.423446	−0.211723	−	0.977330i	$-0.567907\pi$
$487$	−9.34463	−0.423446	−0.211723	+	0.977330i	$0.567907\pi$
$488$	−10.4885	−0.474791
$489$	23.6655	1.07019
$490$	0	0
$491$	−10.7558	−0.485404	−0.242702	−	0.970101i	$-0.578034\pi$
$491$	−10.7558	−0.485404	−0.242702	+	0.970101i	$0.578034\pi$
$492$	12.9430	0.583516
$493$	−6.17179	−0.277963
$494$	0.394252	0.0177383
$495$	0	0
$496$	−3.62372	−0.162710
$497$	−39.5415	−1.77368
$498$	14.8152	0.663883
$499$	29.0702	1.30136	0.650680	−	0.759352i	$-0.274484\pi$
$499$	29.0702	1.30136	0.650680	+	0.759352i	$0.274484\pi$
$500$	0	0
$501$	12.8844	0.575631
$502$	−6.35586	−0.283676
$503$	−36.2683	−1.61712	−0.808561	−	0.588412i	$-0.799753\pi$
$503$	−36.2683	−1.61712	−0.808561	+	0.588412i	$0.799753\pi$
$504$	5.78496	0.257683
$505$	0	0
$506$	−8.93177	−0.397065
$507$	−14.2036	−0.630804
$508$	5.39390	0.239316
$509$	−22.2170	−0.984751	−0.492375	−	0.870383i	$-0.663871\pi$
$509$	−22.2170	−0.984751	−0.492375	+	0.870383i	$0.663871\pi$
$510$	0	0
$511$	44.9268	1.98745
$512$	−1.00000	−0.0441942
$513$	−14.0638	−0.620933
$514$	−5.45772	−0.240730
$515$	0	0
$516$	4.96195	0.218438
$517$	−23.8222	−1.04770
$518$	3.20984	0.141032
$519$	−15.3391	−0.673311
$520$	0	0
$521$	41.2732	1.80821	0.904107	−	0.427306i	$-0.140537\pi$
$521$	41.2732	1.80821	0.904107	+	0.427306i	$0.140537\pi$
$522$	11.3619	0.497297
$523$	−11.0354	−0.482545	−0.241273	−	0.970457i	$-0.577565\pi$
$523$	−11.0354	−0.482545	−0.241273	+	0.970457i	$0.577565\pi$
$524$	−8.97060	−0.391883
$525$	0	0
$526$	−14.8882	−0.649155
$527$	3.54758	0.154535
$528$	4.18424	0.182096
$529$	−17.5424	−0.762712
$530$	0	0
$531$	−12.4871	−0.541895
$532$	8.58933	0.372395
$533$	−1.74242	−0.0754726
$534$	7.08007	0.306385
$535$	0	0
$536$	2.80936	0.121346
$537$	−13.5883	−0.586377
$538$	−15.5573	−0.670723
$539$	12.6285	0.543946
$540$	0	0
$541$	−32.2338	−1.38584	−0.692920	−	0.721014i	$-0.743676\pi$
$541$	−32.2338	−1.38584	−0.692920	+	0.721014i	$0.743676\pi$
$542$	−0.462920	−0.0198841
$543$	8.97142	0.385001
$544$	0.978989	0.0419738
$545$	0	0
$546$	0.517562	0.0221496
$547$	−15.0234	−0.642355	−0.321177	−	0.947019i	$-0.604079\pi$
$547$	−15.0234	−0.642355	−0.321177	+	0.947019i	$0.604079\pi$
$548$	2.70352	0.115489
$549$	−18.9030	−0.806760
$550$	0	0
$551$	16.8698	0.718677
$552$	−2.55672	−0.108821
$553$	50.0918	2.13012
$554$	9.12112	0.387519
$555$	0	0
$556$	4.30358	0.182512
$557$	−39.1275	−1.65788	−0.828942	−	0.559334i	$-0.811057\pi$
$557$	−39.1275	−1.65788	−0.828942	+	0.559334i	$0.811057\pi$
$558$	−6.53089	−0.276475
$559$	−0.667989	−0.0282529
$560$	0	0
$561$	−4.09632	−0.172947
$562$	24.8734	1.04922
$563$	27.1970	1.14622	0.573109	−	0.819479i	$-0.305737\pi$
$563$	27.1970	1.14622	0.573109	+	0.819479i	$0.305737\pi$
$564$	−6.81912	−0.287137
$565$	0	0
$566$	19.7259	0.829142
$567$	−1.10762	−0.0465157
$568$	12.3189	0.516888
$569$	−33.4499	−1.40229	−0.701146	−	0.713018i	$-0.747328\pi$
$569$	−33.4499	−1.40229	−0.701146	+	0.713018i	$0.747328\pi$
$570$	0	0
$571$	28.9145	1.21003	0.605017	−	0.796213i	$-0.293166\pi$
$571$	28.9145	1.21003	0.605017	+	0.796213i	$0.293166\pi$
$572$	−0.563292	−0.0235524
$573$	8.75680	0.365821
$574$	−37.9610	−1.58446
$575$	0	0
$576$	−1.80226	−0.0750942
$577$	−33.5613	−1.39717	−0.698587	−	0.715525i	$-0.746188\pi$
$577$	−33.5613	−1.39717	−0.698587	+	0.715525i	$0.746188\pi$
$578$	16.0416	0.667242
$579$	15.3678	0.638663
$580$	0	0
$581$	−43.4518	−1.80268
$582$	−3.36053	−0.139299
$583$	−43.1948	−1.78894
$584$	−13.9966	−0.579184
$585$	0	0
$586$	−2.65528	−0.109689
$587$	4.83581	0.199595	0.0997976	−	0.995008i	$-0.468180\pi$
$587$	4.83581	0.199595	0.0997976	+	0.995008i	$0.468180\pi$
$588$	3.61490	0.149076
$589$	−9.69686	−0.399552
$590$	0	0
$591$	−3.68388	−0.151535
$592$	−1.00000	−0.0410997
$593$	−43.1680	−1.77270	−0.886348	−	0.463020i	$-0.846766\pi$
$593$	−43.1680	−1.77270	−0.886348	+	0.463020i	$0.846766\pi$
$594$	20.0938	0.824459
$595$	0	0
$596$	21.6949	0.888659
$597$	−16.2947	−0.666896
$598$	0.344191	0.0140750
$599$	−38.7714	−1.58416	−0.792078	−	0.610420i	$-0.791001\pi$
$599$	−38.7714	−1.58416	−0.792078	+	0.610420i	$0.791001\pi$
$600$	0	0
$601$	1.56064	0.0636597	0.0318299	−	0.999493i	$-0.489867\pi$
$601$	1.56064	0.0636597	0.0318299	+	0.999493i	$0.489867\pi$
$602$	−14.5531	−0.593139
$603$	5.06320	0.206190
$604$	2.06744	0.0841230
$605$	0	0
$606$	7.19590	0.292314
$607$	−21.0494	−0.854370	−0.427185	−	0.904164i	$-0.640495\pi$
$607$	−21.0494	−0.854370	−0.427185	+	0.904164i	$0.640495\pi$
$608$	−2.67594	−0.108524
$609$	22.1461	0.897405
$610$	0	0
$611$	0.918005	0.0371385
$612$	1.76439	0.0713214
$613$	5.28786	0.213574	0.106787	−	0.994282i	$-0.465944\pi$
$613$	5.28786	0.213574	0.106787	+	0.994282i	$0.465944\pi$
$614$	−25.3545	−1.02322
$615$	0	0
$616$	−12.2721	−0.494456
$617$	34.8545	1.40319	0.701595	−	0.712576i	$-0.252472\pi$
$617$	34.8545	1.40319	0.701595	+	0.712576i	$0.252472\pi$
$618$	−2.75989	−0.111019
$619$	−7.30122	−0.293461	−0.146730	−	0.989177i	$-0.546875\pi$
$619$	−7.30122	−0.293461	−0.146730	+	0.989177i	$0.546875\pi$
$620$	0	0
$621$	−12.2780	−0.492701
$622$	−11.1234	−0.446006
$623$	−20.7654	−0.831946
$624$	−0.161242	−0.00645486
$625$	0	0
$626$	3.49109	0.139532
$627$	11.1968	0.447156
$628$	−18.5439	−0.739982
$629$	0.978989	0.0390348
$630$	0	0
$631$	24.2752	0.966381	0.483190	−	0.875515i	$-0.339478\pi$
$631$	24.2752	0.966381	0.483190	+	0.875515i	$0.339478\pi$
$632$	−15.6057	−0.620762
$633$	−10.7312	−0.426527
$634$	18.3656	0.729393
$635$	0	0
$636$	−12.3645	−0.490285
$637$	−0.486646	−0.0192816
$638$	−24.1029	−0.954241
$639$	22.2018	0.878290
$640$	0	0
$641$	−8.22087	−0.324705	−0.162353	−	0.986733i	$-0.551908\pi$
$641$	−8.22087	−0.324705	−0.162353	+	0.986733i	$0.551908\pi$
$642$	6.24134	0.246326
$643$	−29.7758	−1.17424	−0.587121	−	0.809499i	$-0.699739\pi$
$643$	−29.7758	−1.17424	−0.587121	+	0.809499i	$0.699739\pi$
$644$	7.49868	0.295489
$645$	0	0
$646$	2.61972	0.103071
$647$	35.6356	1.40098	0.700490	−	0.713662i	$-0.252965\pi$
$647$	35.6356	1.40098	0.700490	+	0.713662i	$0.252965\pi$
$648$	0.345071	0.0135557
$649$	26.4899	1.03982
$650$	0	0
$651$	−12.7297	−0.498917
$652$	21.6240	0.846860
$653$	−37.6128	−1.47190	−0.735951	−	0.677035i	$-0.763264\pi$
$653$	−37.6128	−1.47190	−0.735951	+	0.677035i	$0.763264\pi$
$654$	−13.0583	−0.510619
$655$	0	0
$656$	11.8265	0.461746
$657$	−25.2256	−0.984142
$658$	20.0000	0.779681
$659$	45.0553	1.75510	0.877552	−	0.479481i	$-0.159175\pi$
$659$	45.0553	1.75510	0.877552	+	0.479481i	$0.159175\pi$
$660$	0	0
$661$	7.32365	0.284857	0.142429	−	0.989805i	$-0.454509\pi$
$661$	7.32365	0.284857	0.142429	+	0.989805i	$0.454509\pi$
$662$	21.9365	0.852587
$663$	0.157854	0.00613056
$664$	13.5371	0.525341
$665$	0	0
$666$	−1.80226	−0.0698362
$667$	14.7277	0.570260
$668$	11.7729	0.455506
$669$	16.9028	0.653501
$670$	0	0
$671$	40.1003	1.54806
$672$	−3.51289	−0.135512
$673$	8.58807	0.331046	0.165523	−	0.986206i	$-0.447069\pi$
$673$	8.58807	0.331046	0.165523	+	0.986206i	$0.447069\pi$
$674$	−8.65472	−0.333368
$675$	0	0
$676$	−12.9783	−0.499165
$677$	9.13523	0.351096	0.175548	−	0.984471i	$-0.443830\pi$
$677$	9.13523	0.351096	0.175548	+	0.984471i	$0.443830\pi$
$678$	−2.38630	−0.0916454
$679$	9.85621	0.378247
$680$	0	0
$681$	−7.07089	−0.270957
$682$	13.8545	0.530515
$683$	40.7414	1.55893	0.779463	−	0.626449i	$-0.215492\pi$
$683$	40.7414	1.55893	0.779463	+	0.626449i	$0.215492\pi$
$684$	−4.82274	−0.184402
$685$	0	0
$686$	11.8666	0.453069
$687$	8.72748	0.332974
$688$	4.53390	0.172853
$689$	1.66454	0.0634139
$690$	0	0
$691$	−28.7543	−1.09387	−0.546933	−	0.837176i	$-0.684205\pi$
$691$	−28.7543	−1.09387	−0.546933	+	0.837176i	$0.684205\pi$
$692$	−14.0158	−0.532801
$693$	−22.1175	−0.840174
$694$	−21.3265	−0.809541
$695$	0	0
$696$	−6.89945	−0.261523
$697$	−11.5780	−0.438547
$698$	1.16305	0.0440223
$699$	−27.6784	−1.04689
$700$	0	0
$701$	−7.87020	−0.297253	−0.148627	−	0.988893i	$-0.547485\pi$
$701$	−7.87020	−0.297253	−0.148627	+	0.988893i	$0.547485\pi$
$702$	−0.774328	−0.0292251
$703$	−2.67594	−0.100925
$704$	3.82327	0.144095
$705$	0	0
$706$	−0.470079	−0.0176917
$707$	−21.1051	−0.793738
$708$	7.58273	0.284976
$709$	33.0458	1.24106	0.620530	−	0.784182i	$-0.286917\pi$
$709$	33.0458	1.24106	0.620530	+	0.784182i	$0.286917\pi$
$710$	0	0
$711$	−28.1256	−1.05479
$712$	6.46929	0.242447
$713$	−8.46558	−0.317039
$714$	3.43908	0.128704
$715$	0	0
$716$	−12.4160	−0.464009
$717$	5.59771	0.209050
$718$	33.3086	1.24307
$719$	4.94138	0.184282	0.0921412	−	0.995746i	$-0.470629\pi$
$719$	4.94138	0.184282	0.0921412	+	0.995746i	$0.470629\pi$
$720$	0	0
$721$	8.09456	0.301457
$722$	11.8393	0.440615
$723$	17.9937	0.669195
$724$	8.19748	0.304657
$725$	0	0
$726$	−3.95894	−0.146930
$727$	−42.5178	−1.57690	−0.788448	−	0.615101i	$-0.789115\pi$
$727$	−42.5178	−1.57690	−0.788448	+	0.615101i	$0.789115\pi$
$728$	0.472913	0.0175273
$729$	17.8775	0.662129
$730$	0	0
$731$	−4.43863	−0.164169
$732$	11.4787	0.424266
$733$	−10.2114	−0.377167	−0.188584	−	0.982057i	$-0.560390\pi$
$733$	−10.2114	−0.377167	−0.188584	+	0.982057i	$0.560390\pi$
$734$	−11.1884	−0.412971
$735$	0	0
$736$	−2.33616	−0.0861119
$737$	−10.7410	−0.395648
$738$	21.3144	0.784593
$739$	25.5528	0.939975	0.469988	−	0.882673i	$-0.344258\pi$
$739$	25.5528	0.939975	0.469988	+	0.882673i	$0.344258\pi$
$740$	0	0
$741$	−0.431475	−0.0158506
$742$	36.2642	1.33130
$743$	−44.2852	−1.62467	−0.812334	−	0.583193i	$-0.801803\pi$
$743$	−44.2852	−1.62467	−0.812334	+	0.583193i	$0.801803\pi$
$744$	3.96585	0.145395
$745$	0	0
$746$	23.1634	0.848072
$747$	24.3974	0.892652
$748$	−3.74294	−0.136855
$749$	−18.3054	−0.668865
$750$	0	0
$751$	41.2897	1.50668	0.753342	−	0.657629i	$-0.228440\pi$
$751$	41.2897	1.50668	0.753342	+	0.657629i	$0.228440\pi$
$752$	−6.23085	−0.227216
$753$	6.95593	0.253488
$754$	0.928820	0.0338256
$755$	0	0
$756$	−16.8698	−0.613548
$757$	−6.57802	−0.239082	−0.119541	−	0.992829i	$-0.538142\pi$
$757$	−6.57802	−0.239082	−0.119541	+	0.992829i	$0.538142\pi$
$758$	26.7721	0.972406
$759$	9.77504	0.354811
$760$	0	0
$761$	20.3166	0.736475	0.368237	−	0.929732i	$-0.379961\pi$
$761$	20.3166	0.736475	0.368237	+	0.929732i	$0.379961\pi$
$762$	−5.90315	−0.213849
$763$	38.2990	1.38652
$764$	8.00137	0.289479
$765$	0	0
$766$	22.3189	0.806413
$767$	−1.02080	−0.0368591
$768$	1.09441	0.0394912
$769$	25.2435	0.910303	0.455151	−	0.890414i	$-0.349585\pi$
$769$	25.2435	0.910303	0.455151	+	0.890414i	$0.349585\pi$
$770$	0	0
$771$	5.97300	0.215112
$772$	14.0420	0.505383
$773$	48.9781	1.76162	0.880810	−	0.473470i	$-0.156999\pi$
$773$	48.9781	1.76162	0.880810	+	0.473470i	$0.156999\pi$
$774$	8.17127	0.293710
$775$	0	0
$776$	−3.07063	−0.110229
$777$	−3.51289	−0.126024
$778$	24.3011	0.871237
$779$	31.6469	1.13387
$780$	0	0
$781$	−47.0984	−1.68531
$782$	2.28707	0.0817855
$783$	−33.1330	−1.18408
$784$	3.30305	0.117966
$785$	0	0
$786$	9.81754	0.350180
$787$	3.78073	0.134768	0.0673842	−	0.997727i	$-0.478535\pi$
$787$	3.78073	0.134768	0.0673842	+	0.997727i	$0.478535\pi$
$788$	−3.36608	−0.119912
$789$	16.2938	0.580075
$790$	0	0
$791$	6.99886	0.248851
$792$	6.89054	0.244845
$793$	−1.54529	−0.0548750
$794$	28.1888	1.00038
$795$	0	0
$796$	−14.8890	−0.527725
$797$	29.5487	1.04667	0.523334	−	0.852128i	$-0.324688\pi$
$797$	29.5487	1.04667	0.523334	+	0.852128i	$0.324688\pi$
$798$	−9.40027	−0.332766
$799$	6.09993	0.215800
$800$	0	0
$801$	11.6593	0.411963
$802$	15.9528	0.563312
$803$	53.5129	1.88843
$804$	−3.07460	−0.108433
$805$	0	0
$806$	−0.533891	−0.0188055
$807$	17.0261	0.599348
$808$	6.57513	0.231312
$809$	47.4231	1.66731	0.833654	−	0.552287i	$-0.186245\pi$
$809$	47.4231	1.66731	0.833654	+	0.552287i	$0.186245\pi$
$810$	0	0
$811$	25.6962	0.902317	0.451159	−	0.892444i	$-0.351011\pi$
$811$	25.6962	0.902317	0.451159	+	0.892444i	$0.351011\pi$
$812$	20.2356	0.710131
$813$	0.506626	0.0177681
$814$	3.82327	0.134006
$815$	0	0
$816$	−1.07142	−0.0375071
$817$	12.1324	0.424460
$818$	−31.6014	−1.10492
$819$	0.852312	0.0297822
$820$	0	0
$821$	0.286013	0.00998192	0.00499096	−	0.999988i	$-0.498411\pi$
$821$	0.286013	0.00998192	0.00499096	+	0.999988i	$0.498411\pi$
$822$	−2.95877	−0.103199
$823$	−44.6093	−1.55498	−0.777491	−	0.628893i	$-0.783508\pi$
$823$	−44.6093	−1.55498	−0.777491	+	0.628893i	$0.783508\pi$
$824$	−2.52180	−0.0878510
$825$	0	0
$826$	−22.2396	−0.773815
$827$	55.1277	1.91698	0.958488	−	0.285131i	$-0.0920373\pi$
$827$	55.1277	1.91698	0.958488	+	0.285131i	$0.0920373\pi$
$828$	−4.21037	−0.146320
$829$	−17.2029	−0.597482	−0.298741	−	0.954334i	$-0.596567\pi$
$829$	−17.2029	−0.597482	−0.298741	+	0.954334i	$0.596567\pi$
$830$	0	0
$831$	−9.98227	−0.346281
$832$	−0.147332	−0.00510783
$833$	−3.23365	−0.112039
$834$	−4.70989	−0.163090
$835$	0	0
$836$	10.2308	0.353841
$837$	19.0450	0.658292
$838$	18.4266	0.636535
$839$	27.3527	0.944320	0.472160	−	0.881513i	$-0.343474\pi$
$839$	27.3527	0.944320	0.472160	+	0.881513i	$0.343474\pi$
$840$	0	0
$841$	10.7436	0.370467
$842$	19.1570	0.660194
$843$	−27.2218	−0.937568
$844$	−9.80544	−0.337517
$845$	0	0
$846$	−11.2296	−0.386082
$847$	11.6113	0.398969
$848$	−11.2978	−0.387970
$849$	−21.5883	−0.740908
$850$	0	0
$851$	−2.33616	−0.0800824
$852$	−13.4819	−0.461883
$853$	51.8565	1.77553	0.887766	−	0.460294i	$-0.152256\pi$
$853$	51.8565	1.77553	0.887766	+	0.460294i	$0.152256\pi$
$854$	−33.6663	−1.15204
$855$	0	0
$856$	5.70291	0.194921
$857$	4.63553	0.158347	0.0791733	−	0.996861i	$-0.474772\pi$
$857$	4.63553	0.158347	0.0791733	+	0.996861i	$0.474772\pi$
$858$	0.616473	0.0210461
$859$	43.1006	1.47057	0.735287	−	0.677755i	$-0.237047\pi$
$859$	43.1006	1.47057	0.735287	+	0.677755i	$0.237047\pi$
$860$	0	0
$861$	41.5450	1.41585
$862$	10.3947	0.354044
$863$	−11.2463	−0.382829	−0.191414	−	0.981509i	$-0.561307\pi$
$863$	−11.2463	−0.382829	−0.191414	+	0.981509i	$0.561307\pi$
$864$	5.25565	0.178801
$865$	0	0
$866$	3.36622	0.114389
$867$	−17.5561	−0.596237
$868$	−11.6316	−0.394801
$869$	59.6649	2.02399
$870$	0	0
$871$	0.413910	0.0140248
$872$	−11.9318	−0.404061
$873$	−5.53407	−0.187300
$874$	−6.25142	−0.211457
$875$	0	0
$876$	15.3181	0.517550
$877$	17.9444	0.605939	0.302969	−	0.953000i	$-0.402022\pi$
$877$	17.9444	0.605939	0.302969	+	0.953000i	$0.402022\pi$
$878$	32.5045	1.09697
$879$	2.90597	0.0980160
$880$	0	0
$881$	−16.6495	−0.560935	−0.280468	−	0.959864i	$-0.590489\pi$
$881$	−16.6495	−0.560935	−0.280468	+	0.959864i	$0.590489\pi$
$882$	5.95296	0.200446
$883$	−28.7758	−0.968383	−0.484191	−	0.874962i	$-0.660886\pi$
$883$	−28.7758	−0.968383	−0.484191	+	0.874962i	$0.660886\pi$
$884$	0.144237	0.00485121
$885$	0	0
$886$	−23.5479	−0.791107
$887$	23.6957	0.795625	0.397812	−	0.917467i	$-0.369770\pi$
$887$	23.6957	0.795625	0.397812	+	0.917467i	$0.369770\pi$
$888$	1.09441	0.0367261
$889$	17.3135	0.580678
$890$	0	0
$891$	−1.31930	−0.0441982
$892$	15.4447	0.517125
$893$	−16.6734	−0.557953
$894$	−23.7432	−0.794092
$895$	0	0
$896$	−3.20984	−0.107233
$897$	−0.376687	−0.0125772
$898$	−12.5629	−0.419228
$899$	−22.8448	−0.761918
$900$	0	0
$901$	11.0605	0.368478
$902$	−45.2158	−1.50552
$903$	15.9271	0.530019
$904$	−2.18044	−0.0725204
$905$	0	0
$906$	−2.26263	−0.0751710
$907$	5.56548	0.184799	0.0923994	−	0.995722i	$-0.470546\pi$
$907$	5.56548	0.184799	0.0923994	+	0.995722i	$0.470546\pi$
$908$	−6.46090	−0.214413
$909$	11.8501	0.393043
$910$	0	0
$911$	−17.2514	−0.571565	−0.285782	−	0.958295i	$-0.592253\pi$
$911$	−17.2514	−0.571565	−0.285782	+	0.958295i	$0.592253\pi$
$912$	2.92858	0.0969751
$913$	−51.7559	−1.71287
$914$	6.53390	0.216122
$915$	0	0
$916$	7.97458	0.263488
$917$	−28.7942	−0.950867
$918$	−5.14523	−0.169818
$919$	−8.34007	−0.275114	−0.137557	−	0.990494i	$-0.543925\pi$
$919$	−8.34007	−0.275114	−0.137557	+	0.990494i	$0.543925\pi$
$920$	0	0
$921$	27.7483	0.914336
$922$	−24.7616	−0.815480
$923$	1.81497	0.0597403
$924$	13.4307	0.441838
$925$	0	0
$926$	25.9239	0.851913
$927$	−4.54494	−0.149275
$928$	−6.30425	−0.206947
$929$	−48.2529	−1.58313	−0.791564	−	0.611087i	$-0.790733\pi$
$929$	−48.2529	−1.58313	−0.791564	+	0.611087i	$0.790733\pi$
$930$	0	0
$931$	8.83876	0.289679
$932$	−25.2907	−0.828423
$933$	12.1736	0.398544
$934$	28.8182	0.942959
$935$	0	0
$936$	−0.265531	−0.00867916
$937$	30.2000	0.986592	0.493296	−	0.869862i	$-0.335792\pi$
$937$	30.2000	0.986592	0.493296	+	0.869862i	$0.335792\pi$
$938$	9.01759	0.294435
$939$	−3.82069	−0.124684
$940$	0	0
$941$	14.9206	0.486399	0.243199	−	0.969976i	$-0.421803\pi$
$941$	14.9206	0.486399	0.243199	+	0.969976i	$0.421803\pi$
$942$	20.2947	0.661236
$943$	27.6285	0.899707
$944$	6.92858	0.225506
$945$	0	0
$946$	−17.3343	−0.563587
$947$	48.1091	1.56334	0.781668	−	0.623694i	$-0.214369\pi$
$947$	48.1091	1.56334	0.781668	+	0.623694i	$0.214369\pi$
$948$	17.0791	0.554703
$949$	−2.06215	−0.0669403
$950$	0	0
$951$	−20.0996	−0.651774
$952$	3.14239	0.101846
$953$	8.67149	0.280897	0.140449	−	0.990088i	$-0.455146\pi$
$953$	8.67149	0.280897	0.140449	+	0.990088i	$0.455146\pi$
$954$	−20.3617	−0.659234
$955$	0	0
$956$	5.11481	0.165425
$957$	26.3785	0.852695
$958$	41.7873	1.35009
$959$	8.67787	0.280223
$960$	0	0
$961$	−17.8686	−0.576408
$962$	−0.147332	−0.00475018
$963$	10.2781	0.331208
$964$	16.4415	0.529544
$965$	0	0
$966$	−8.20665	−0.264045
$967$	18.5944	0.597955	0.298978	−	0.954260i	$-0.403354\pi$
$967$	18.5944	0.597955	0.298978	+	0.954260i	$0.403354\pi$
$968$	−3.61741	−0.116268
$969$	−2.86705	−0.0921029
$970$	0	0
$971$	−32.6679	−1.04836	−0.524182	−	0.851606i	$-0.675629\pi$
$971$	−32.6679	−1.04836	−0.524182	+	0.851606i	$0.675629\pi$
$972$	15.3893	0.493612
$973$	13.8138	0.442850
$974$	9.34463	0.299421
$975$	0	0
$976$	10.4885	0.335728
$977$	−28.1856	−0.901737	−0.450868	−	0.892590i	$-0.648886\pi$
$977$	−28.1856	−0.901737	−0.450868	+	0.892590i	$0.648886\pi$
$978$	−23.6655	−0.756740
$979$	−24.7338	−0.790497
$980$	0	0
$981$	−21.5042	−0.686575
$982$	10.7558	0.343232
$983$	25.3845	0.809639	0.404819	−	0.914397i	$-0.367334\pi$
$983$	25.3845	0.809639	0.404819	+	0.914397i	$0.367334\pi$
$984$	−12.9430	−0.412608
$985$	0	0
$986$	6.17179	0.196550
$987$	−21.8883	−0.696711
$988$	−0.394252	−0.0125428
$989$	10.5919	0.336803
$990$	0	0
$991$	−21.1083	−0.670527	−0.335264	−	0.942124i	$-0.608825\pi$
$991$	−21.1083	−0.670527	−0.335264	+	0.942124i	$0.608825\pi$
$992$	3.62372	0.115053
$993$	−24.0076	−0.761858
$994$	39.5415	1.25418
$995$	0	0
$996$	−14.8152	−0.469436
$997$	−39.0978	−1.23824	−0.619120	−	0.785296i	$-0.712511\pi$
$997$	−39.0978	−1.23824	−0.619120	+	0.785296i	$0.712511\pi$
$998$	−29.0702	−0.920201
$999$	5.25565	0.166282

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bd.1.4		5
5.2	odd	4		370.2.b.d.149.2	✓	10
5.3	odd	4		370.2.b.d.149.9	yes	10
5.4	even	2		1850.2.a.be.1.2		5
15.2	even	4		3330.2.d.p.1999.7		10
15.8	even	4		3330.2.d.p.1999.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.d.149.2	✓	10	5.2	odd	4
370.2.b.d.149.9	yes	10	5.3	odd	4
1850.2.a.bd.1.4		5	1.1	even	1	trivial
1850.2.a.be.1.2		5	5.4	even	2
3330.2.d.p.1999.2		10	15.8	even	4
3330.2.d.p.1999.7		10	15.2	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} +3.20984 q^{7} -1.00000 q^{8} -1.80226 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.09441 q^{3} +1.00000 q^{4} -1.09441 q^{6} +3.20984 q^{7} -1.00000 q^{8} -1.80226 q^{9} +3.82327 q^{11} +1.09441 q^{12} -0.147332 q^{13} -3.20984 q^{14} +1.00000 q^{16} -0.978989 q^{17} +1.80226 q^{18} +2.67594 q^{19} +3.51289 q^{21} -3.82327 q^{22} +2.33616 q^{23} -1.09441 q^{24} +0.147332 q^{26} -5.25565 q^{27} +3.20984 q^{28} +6.30425 q^{29} -3.62372 q^{31} -1.00000 q^{32} +4.18424 q^{33} +0.978989 q^{34} -1.80226 q^{36} -1.00000 q^{37} -2.67594 q^{38} -0.161242 q^{39} +11.8265 q^{41} -3.51289 q^{42} +4.53390 q^{43} +3.82327 q^{44} -2.33616 q^{46} -6.23085 q^{47} +1.09441 q^{48} +3.30305 q^{49} -1.07142 q^{51} -0.147332 q^{52} -11.2978 q^{53} +5.25565 q^{54} -3.20984 q^{56} +2.92858 q^{57} -6.30425 q^{58} +6.92858 q^{59} +10.4885 q^{61} +3.62372 q^{62} -5.78496 q^{63} +1.00000 q^{64} -4.18424 q^{66} -2.80936 q^{67} -0.978989 q^{68} +2.55672 q^{69} -12.3189 q^{71} +1.80226 q^{72} +13.9966 q^{73} +1.00000 q^{74} +2.67594 q^{76} +12.2721 q^{77} +0.161242 q^{78} +15.6057 q^{79} -0.345071 q^{81} -11.8265 q^{82} -13.5371 q^{83} +3.51289 q^{84} -4.53390 q^{86} +6.89945 q^{87} -3.82327 q^{88} -6.46929 q^{89} -0.472913 q^{91} +2.33616 q^{92} -3.96585 q^{93} +6.23085 q^{94} -1.09441 q^{96} +3.07063 q^{97} -3.30305 q^{98} -6.89054 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more