LMFDB - Embedded newform 3850.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.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:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.117688.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 1 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.97382$ of defining polynomial
Character	$\chi$	$=$	3850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.27510 q^{3} +1.00000 q^{4} -3.27510 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.72630 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.27510 q^{3} +1.00000 q^{4} -3.27510 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.72630 q^{9} +1.00000 q^{11} -3.27510 q^{12} +6.46446 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.38123 q^{17} +7.72630 q^{18} +6.62018 q^{19} +3.27510 q^{21} +1.00000 q^{22} -5.53694 q^{23} -3.27510 q^{24} +6.46446 q^{26} -15.4791 q^{27} -1.00000 q^{28} -1.01326 q^{29} -2.72630 q^{31} +1.00000 q^{32} -3.27510 q^{33} +3.38123 q^{34} +7.72630 q^{36} +2.15571 q^{37} +6.62018 q^{38} -21.1718 q^{39} +2.81064 q^{41} +3.27510 q^{42} -12.4190 q^{43} +1.00000 q^{44} -5.53694 q^{46} +2.82139 q^{47} -3.27510 q^{48} +1.00000 q^{49} -11.0739 q^{51} +6.46446 q^{52} -8.39450 q^{53} -15.4791 q^{54} -1.00000 q^{56} -21.6818 q^{57} -1.01326 q^{58} +11.4308 q^{59} +5.64307 q^{61} -2.72630 q^{62} -7.72630 q^{63} +1.00000 q^{64} -3.27510 q^{66} +3.16897 q^{67} +3.38123 q^{68} +18.1341 q^{69} +4.85755 q^{71} +7.72630 q^{72} +0.831025 q^{73} +2.15571 q^{74} +6.62018 q^{76} -1.00000 q^{77} -21.1718 q^{78} -10.5610 q^{79} +27.5169 q^{81} +2.81064 q^{82} +4.81205 q^{83} +3.27510 q^{84} -12.4190 q^{86} +3.31854 q^{87} +1.00000 q^{88} +8.02653 q^{89} -6.46446 q^{91} -5.53694 q^{92} +8.92893 q^{93} +2.82139 q^{94} -3.27510 q^{96} -11.3961 q^{97} +1.00000 q^{98} +7.72630 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{7} + 5 q^{8} + 13 q^{9}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^7 + 5 * q^8 + 13 * q^9	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{7} + 5 q^{8} + 13 q^{9} + 5 q^{11} + 10 q^{13} - 5 q^{14} + 5 q^{16} + 2 q^{17} + 13 q^{18} + 6 q^{19} + 5 q^{22} - 8 q^{23} + 10 q^{26} - 5 q^{28} + 8 q^{29} + 12 q^{31} + 5 q^{32} + 2 q^{34} + 13 q^{36} + 6 q^{37} + 6 q^{38} - 4 q^{39} + 20 q^{41} - 12 q^{43} + 5 q^{44} - 8 q^{46} + 10 q^{47} + 5 q^{49} - 16 q^{51} + 10 q^{52} - 14 q^{53} - 5 q^{56} - 12 q^{57} + 8 q^{58} + 36 q^{59} + 10 q^{61} + 12 q^{62} - 13 q^{63} + 5 q^{64} - 2 q^{67} + 2 q^{68} + 24 q^{69} + 16 q^{71} + 13 q^{72} + 22 q^{73} + 6 q^{74} + 6 q^{76} - 5 q^{77} - 4 q^{78} - 10 q^{79} + 25 q^{81} + 20 q^{82} - 12 q^{83} - 12 q^{86} + 32 q^{87} + 5 q^{88} + 14 q^{89} - 10 q^{91} - 8 q^{92} + 10 q^{94} + 2 q^{97} + 5 q^{98} + 13 q^{99}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^7 + 5 * q^8 + 13 * q^9 + 5 * q^11 + 10 * q^13 - 5 * q^14 + 5 * q^16 + 2 * q^17 + 13 * q^18 + 6 * q^19 + 5 * q^22 - 8 * q^23 + 10 * q^26 - 5 * q^28 + 8 * q^29 + 12 * q^31 + 5 * q^32 + 2 * q^34 + 13 * q^36 + 6 * q^37 + 6 * q^38 - 4 * q^39 + 20 * q^41 - 12 * q^43 + 5 * q^44 - 8 * q^46 + 10 * q^47 + 5 * q^49 - 16 * q^51 + 10 * q^52 - 14 * q^53 - 5 * q^56 - 12 * q^57 + 8 * q^58 + 36 * q^59 + 10 * q^61 + 12 * q^62 - 13 * q^63 + 5 * q^64 - 2 * q^67 + 2 * q^68 + 24 * q^69 + 16 * q^71 + 13 * q^72 + 22 * q^73 + 6 * q^74 + 6 * q^76 - 5 * q^77 - 4 * q^78 - 10 * q^79 + 25 * q^81 + 20 * q^82 - 12 * q^83 - 12 * q^86 + 32 * q^87 + 5 * q^88 + 14 * q^89 - 10 * q^91 - 8 * q^92 + 10 * q^94 + 2 * q^97 + 5 * q^98 + 13 * 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
$3$	−3.27510	−1.89088	−0.945441	−	0.325793i	$-0.894369\pi$
$3$	−3.27510	−1.89088	−0.945441	+	0.325793i	$0.894369\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.27510	−1.33706
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	7.72630	2.57543
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−3.27510	−0.945441
$13$	6.46446	1.79292	0.896460	−	0.443125i	$-0.146130\pi$
$13$	6.46446	1.79292	0.896460	+	0.443125i	$0.146130\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.38123	0.820069	0.410035	−	0.912070i	$-0.365517\pi$
$17$	3.38123	0.820069	0.410035	+	0.912070i	$0.365517\pi$
$18$	7.72630	1.82111
$19$	6.62018	1.51877	0.759386	−	0.650640i	$-0.225499\pi$
$19$	6.62018	1.51877	0.759386	+	0.650640i	$0.225499\pi$
$20$	0	0
$21$	3.27510	0.714686
$22$	1.00000	0.213201
$23$	−5.53694	−1.15453	−0.577266	−	0.816556i	$-0.695881\pi$
$23$	−5.53694	−1.15453	−0.577266	+	0.816556i	$0.695881\pi$
$24$	−3.27510	−0.668528
$25$	0	0
$26$	6.46446	1.26779
$27$	−15.4791	−2.97896
$28$	−1.00000	−0.188982
$29$	−1.01326	−0.188158	−0.0940792	−	0.995565i	$-0.529991\pi$
$29$	−1.01326	−0.188158	−0.0940792	+	0.995565i	$0.529991\pi$
$30$	0	0
$31$	−2.72630	−0.489659	−0.244829	−	0.969566i	$-0.578732\pi$
$31$	−2.72630	−0.489659	−0.244829	+	0.969566i	$0.578732\pi$
$32$	1.00000	0.176777
$33$	−3.27510	−0.570122
$34$	3.38123	0.579877
$35$	0	0
$36$	7.72630	1.28772
$37$	2.15571	0.354397	0.177198	−	0.984175i	$-0.443297\pi$
$37$	2.15571	0.354397	0.177198	+	0.984175i	$0.443297\pi$
$38$	6.62018	1.07393
$39$	−21.1718	−3.39020
$40$	0	0
$41$	2.81064	0.438948	0.219474	−	0.975618i	$-0.429566\pi$
$41$	2.81064	0.438948	0.219474	+	0.975618i	$0.429566\pi$
$42$	3.27510	0.505359
$43$	−12.4190	−1.89387	−0.946937	−	0.321420i	$-0.895840\pi$
$43$	−12.4190	−1.89387	−0.946937	+	0.321420i	$0.895840\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−5.53694	−0.816378
$47$	2.82139	0.411542	0.205771	−	0.978600i	$-0.434030\pi$
$47$	2.82139	0.411542	0.205771	+	0.978600i	$0.434030\pi$
$48$	−3.27510	−0.472720
$49$	1.00000	0.142857
$50$	0	0
$51$	−11.0739	−1.55065
$52$	6.46446	0.896460
$53$	−8.39450	−1.15307	−0.576536	−	0.817071i	$-0.695596\pi$
$53$	−8.39450	−1.15307	−0.576536	+	0.817071i	$0.695596\pi$
$54$	−15.4791	−2.10644
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−21.6818	−2.87182
$58$	−1.01326	−0.133048
$59$	11.4308	1.48817	0.744083	−	0.668088i	$-0.232887\pi$
$59$	11.4308	1.48817	0.744083	+	0.668088i	$0.232887\pi$
$60$	0	0
$61$	5.64307	0.722521	0.361261	−	0.932465i	$-0.382346\pi$
$61$	5.64307	0.722521	0.361261	+	0.932465i	$0.382346\pi$
$62$	−2.72630	−0.346241
$63$	−7.72630	−0.973423
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.27510	−0.403137
$67$	3.16897	0.387152	0.193576	−	0.981085i	$-0.437991\pi$
$67$	3.16897	0.387152	0.193576	+	0.981085i	$0.437991\pi$
$68$	3.38123	0.410035
$69$	18.1341	2.18308
$70$	0	0
$71$	4.85755	0.576485	0.288243	−	0.957557i	$-0.406929\pi$
$71$	4.85755	0.576485	0.288243	+	0.957557i	$0.406929\pi$
$72$	7.72630	0.910554
$73$	0.831025	0.0972641	0.0486321	−	0.998817i	$-0.484514\pi$
$73$	0.831025	0.0972641	0.0486321	+	0.998817i	$0.484514\pi$
$74$	2.15571	0.250596
$75$	0	0
$76$	6.62018	0.759386
$77$	−1.00000	−0.113961
$78$	−21.1718	−2.39723
$79$	−10.5610	−1.18820	−0.594100	−	0.804391i	$-0.702492\pi$
$79$	−10.5610	−1.18820	−0.594100	+	0.804391i	$0.702492\pi$
$80$	0	0
$81$	27.5169	3.05743
$82$	2.81064	0.310383
$83$	4.81205	0.528191	0.264095	−	0.964497i	$-0.414927\pi$
$83$	4.81205	0.528191	0.264095	+	0.964497i	$0.414927\pi$
$84$	3.27510	0.357343
$85$	0	0
$86$	−12.4190	−1.33917
$87$	3.31854	0.355785
$88$	1.00000	0.106600
$89$	8.02653	0.850810	0.425405	−	0.905003i	$-0.360132\pi$
$89$	8.02653	0.850810	0.425405	+	0.905003i	$0.360132\pi$
$90$	0	0
$91$	−6.46446	−0.677660
$92$	−5.53694	−0.577266
$93$	8.92893	0.925887
$94$	2.82139	0.291004
$95$	0	0
$96$	−3.27510	−0.334264
$97$	−11.3961	−1.15709	−0.578547	−	0.815649i	$-0.696380\pi$
$97$	−11.3961	−1.15709	−0.578547	+	0.815649i	$0.696380\pi$
$98$	1.00000	0.101015
$99$	7.72630	0.776523
$100$	0	0
$101$	−1.47159	−0.146428	−0.0732141	−	0.997316i	$-0.523326\pi$
$101$	−1.47159	−0.146428	−0.0732141	+	0.997316i	$0.523326\pi$
$102$	−11.0739	−1.09648
$103$	5.49003	0.540949	0.270474	−	0.962727i	$-0.412819\pi$
$103$	5.49003	0.540949	0.270474	+	0.962727i	$0.412819\pi$
$104$	6.46446	0.633893
$105$	0	0
$106$	−8.39450	−0.815346
$107$	10.1341	0.979697	0.489848	−	0.871808i	$-0.337052\pi$
$107$	10.1341	0.979697	0.489848	+	0.871808i	$0.337052\pi$
$108$	−15.4791	−1.48948
$109$	11.5635	1.10758	0.553790	−	0.832656i	$-0.313181\pi$
$109$	11.5635	1.10758	0.553790	+	0.832656i	$0.313181\pi$
$110$	0	0
$111$	−7.06018	−0.670122
$112$	−1.00000	−0.0944911
$113$	−2.47883	−0.233189	−0.116594	−	0.993180i	$-0.537198\pi$
$113$	−2.47883	−0.233189	−0.116594	+	0.993180i	$0.537198\pi$
$114$	−21.6818	−2.03068
$115$	0	0
$116$	−1.01326	−0.0940792
$117$	49.9464	4.61755
$118$	11.4308	1.05229
$119$	−3.38123	−0.309957
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.64307	0.510900
$123$	−9.20514	−0.829999
$124$	−2.72630	−0.244829
$125$	0	0
$126$	−7.72630	−0.688314
$127$	9.78648	0.868410	0.434205	−	0.900814i	$-0.357029\pi$
$127$	9.78648	0.868410	0.434205	+	0.900814i	$0.357029\pi$
$128$	1.00000	0.0883883
$129$	40.6734	3.58109
$130$	0	0
$131$	8.64670	0.755466	0.377733	−	0.925915i	$-0.376704\pi$
$131$	8.64670	0.755466	0.377733	+	0.925915i	$0.376704\pi$
$132$	−3.27510	−0.285061
$133$	−6.62018	−0.574042
$134$	3.16897	0.273758
$135$	0	0
$136$	3.38123	0.289938
$137$	2.95264	0.252261	0.126131	−	0.992014i	$-0.459744\pi$
$137$	2.95264	0.252261	0.126131	+	0.992014i	$0.459744\pi$
$138$	18.1341	1.54367
$139$	−17.9063	−1.51879	−0.759397	−	0.650627i	$-0.774506\pi$
$139$	−17.9063	−1.51879	−0.759397	+	0.650627i	$0.774506\pi$
$140$	0	0
$141$	−9.24035	−0.778178
$142$	4.85755	0.407637
$143$	6.46446	0.540586
$144$	7.72630	0.643859
$145$	0	0
$146$	0.831025	0.0687761
$147$	−3.27510	−0.270126
$148$	2.15571	0.177198
$149$	10.9867	0.900069	0.450034	−	0.893011i	$-0.351412\pi$
$149$	10.9867	0.900069	0.450034	+	0.893011i	$0.351412\pi$
$150$	0	0
$151$	−7.15320	−0.582119	−0.291060	−	0.956705i	$-0.594008\pi$
$151$	−7.15320	−0.582119	−0.291060	+	0.956705i	$0.594008\pi$
$152$	6.62018	0.536967
$153$	26.1244	2.11204
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−21.1718	−1.69510
$157$	11.8905	0.948969	0.474484	−	0.880264i	$-0.342635\pi$
$157$	11.8905	0.948969	0.474484	+	0.880264i	$0.342635\pi$
$158$	−10.5610	−0.840185
$159$	27.4928	2.18032
$160$	0	0
$161$	5.53694	0.436372
$162$	27.5169	2.16193
$163$	−6.47381	−0.507068	−0.253534	−	0.967326i	$-0.581593\pi$
$163$	−6.47381	−0.507068	−0.253534	+	0.967326i	$0.581593\pi$
$164$	2.81064	0.219474
$165$	0	0
$166$	4.81205	0.373487
$167$	13.6241	1.05426	0.527132	−	0.849783i	$-0.323267\pi$
$167$	13.6241	1.05426	0.527132	+	0.849783i	$0.323267\pi$
$168$	3.27510	0.252680
$169$	28.7893	2.21456
$170$	0	0
$171$	51.1495	3.91150
$172$	−12.4190	−0.946937
$173$	3.05209	0.232046	0.116023	−	0.993247i	$-0.462985\pi$
$173$	3.05209	0.232046	0.116023	+	0.993247i	$0.462985\pi$
$174$	3.31854	0.251578
$175$	0	0
$176$	1.00000	0.0753778
$177$	−37.4371	−2.81394
$178$	8.02653	0.601614
$179$	−17.4417	−1.30365	−0.651827	−	0.758367i	$-0.725997\pi$
$179$	−17.4417	−1.30365	−0.651827	+	0.758367i	$0.725997\pi$
$180$	0	0
$181$	−24.9813	−1.85685	−0.928424	−	0.371523i	$-0.878836\pi$
$181$	−24.9813	−1.85685	−0.928424	+	0.371523i	$0.878836\pi$
$182$	−6.46446	−0.479178
$183$	−18.4816	−1.36620
$184$	−5.53694	−0.408189
$185$	0	0
$186$	8.92893	0.654701
$187$	3.38123	0.247260
$188$	2.82139	0.205771
$189$	15.4791	1.12594
$190$	0	0
$191$	18.6624	1.35036	0.675180	−	0.737653i	$-0.264066\pi$
$191$	18.6624	1.35036	0.675180	+	0.737653i	$0.264066\pi$
$192$	−3.27510	−0.236360
$193$	4.82641	0.347413	0.173706	−	0.984797i	$-0.444426\pi$
$193$	4.82641	0.347413	0.173706	+	0.984797i	$0.444426\pi$
$194$	−11.3961	−0.818190
$195$	0	0
$196$	1.00000	0.0714286
$197$	−13.1677	−0.938161	−0.469080	−	0.883155i	$-0.655415\pi$
$197$	−13.1677	−0.938161	−0.469080	+	0.883155i	$0.655415\pi$
$198$	7.72630	0.549085
$199$	−17.6955	−1.25440	−0.627199	−	0.778859i	$-0.715799\pi$
$199$	−17.6955	−1.25440	−0.627199	+	0.778859i	$0.715799\pi$
$200$	0	0
$201$	−10.3787	−0.732059
$202$	−1.47159	−0.103540
$203$	1.01326	0.0711172
$204$	−11.0739	−0.775327
$205$	0	0
$206$	5.49003	0.382509
$207$	−42.7801	−2.97342
$208$	6.46446	0.448230
$209$	6.62018	0.457927
$210$	0	0
$211$	11.2669	0.775644	0.387822	−	0.921734i	$-0.373228\pi$
$211$	11.2669	0.775644	0.387822	+	0.921734i	$0.373228\pi$
$212$	−8.39450	−0.576536
$213$	−15.9090	−1.09007
$214$	10.1341	0.692750
$215$	0	0
$216$	−15.4791	−1.05322
$217$	2.72630	0.185074
$218$	11.5635	0.783177
$219$	−2.72169	−0.183915
$220$	0	0
$221$	21.8579	1.47032
$222$	−7.06018	−0.473848
$223$	−24.2768	−1.62570	−0.812848	−	0.582476i	$-0.802084\pi$
$223$	−24.2768	−1.62570	−0.812848	+	0.582476i	$0.802084\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−2.47883	−0.164889
$227$	19.9074	1.32130	0.660652	−	0.750693i	$-0.270280\pi$
$227$	19.9074	1.32130	0.660652	+	0.750693i	$0.270280\pi$
$228$	−21.6818	−1.43591
$229$	−10.9196	−0.721586	−0.360793	−	0.932646i	$-0.617494\pi$
$229$	−10.9196	−0.721586	−0.360793	+	0.932646i	$0.617494\pi$
$230$	0	0
$231$	3.27510	0.215486
$232$	−1.01326	−0.0665240
$233$	6.86037	0.449438	0.224719	−	0.974424i	$-0.427854\pi$
$233$	6.86037	0.449438	0.224719	+	0.974424i	$0.427854\pi$
$234$	49.9464	3.26510
$235$	0	0
$236$	11.4308	0.744083
$237$	34.5882	2.24675
$238$	−3.38123	−0.219173
$239$	19.0161	1.23005	0.615024	−	0.788508i	$-0.289146\pi$
$239$	19.0161	1.23005	0.615024	+	0.788508i	$0.289146\pi$
$240$	0	0
$241$	−9.59461	−0.618043	−0.309022	−	0.951055i	$-0.600002\pi$
$241$	−9.59461	−0.618043	−0.309022	+	0.951055i	$0.600002\pi$
$242$	1.00000	0.0642824
$243$	−43.6832	−2.80228
$244$	5.64307	0.361261
$245$	0	0
$246$	−9.20514	−0.586898
$247$	42.7959	2.72304
$248$	−2.72630	−0.173120
$249$	−15.7600	−0.998746
$250$	0	0
$251$	−22.9823	−1.45063	−0.725314	−	0.688418i	$-0.758305\pi$
$251$	−22.9823	−1.45063	−0.725314	+	0.688418i	$0.758305\pi$
$252$	−7.72630	−0.486711
$253$	−5.53694	−0.348105
$254$	9.78648	0.614058
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.18067	0.0736484	0.0368242	−	0.999322i	$-0.488276\pi$
$257$	1.18067	0.0736484	0.0368242	+	0.999322i	$0.488276\pi$
$258$	40.6734	2.53221
$259$	−2.15571	−0.133949
$260$	0	0
$261$	−7.82878	−0.484590
$262$	8.64670	0.534195
$263$	26.9368	1.66099	0.830496	−	0.557024i	$-0.188057\pi$
$263$	26.9368	1.66099	0.830496	+	0.557024i	$0.188057\pi$
$264$	−3.27510	−0.201569
$265$	0	0
$266$	−6.62018	−0.405909
$267$	−26.2877	−1.60878
$268$	3.16897	0.193576
$269$	−20.4439	−1.24649	−0.623244	−	0.782027i	$-0.714186\pi$
$269$	−20.4439	−1.24649	−0.623244	+	0.782027i	$0.714186\pi$
$270$	0	0
$271$	25.6269	1.55672	0.778362	−	0.627816i	$-0.216051\pi$
$271$	25.6269	1.55672	0.778362	+	0.627816i	$0.216051\pi$
$272$	3.38123	0.205017
$273$	21.1718	1.28138
$274$	2.95264	0.178376
$275$	0	0
$276$	18.1341	1.09154
$277$	16.3408	0.981821	0.490911	−	0.871210i	$-0.336664\pi$
$277$	16.3408	0.981821	0.490911	+	0.871210i	$0.336664\pi$
$278$	−17.9063	−1.07395
$279$	−21.0643	−1.26108
$280$	0	0
$281$	4.37872	0.261213	0.130606	−	0.991434i	$-0.458308\pi$
$281$	4.37872	0.261213	0.130606	+	0.991434i	$0.458308\pi$
$282$	−9.24035	−0.550255
$283$	15.1500	0.900574	0.450287	−	0.892884i	$-0.351322\pi$
$283$	15.1500	0.900574	0.450287	+	0.892884i	$0.351322\pi$
$284$	4.85755	0.288243
$285$	0	0
$286$	6.46446	0.382252
$287$	−2.81064	−0.165907
$288$	7.72630	0.455277
$289$	−5.56727	−0.327486
$290$	0	0
$291$	37.3233	2.18793
$292$	0.831025	0.0486321
$293$	−0.630927	−0.0368592	−0.0184296	−	0.999830i	$-0.505867\pi$
$293$	−0.630927	−0.0368592	−0.0184296	+	0.999830i	$0.505867\pi$
$294$	−3.27510	−0.191008
$295$	0	0
$296$	2.15571	0.125298
$297$	−15.4791	−0.898191
$298$	10.9867	0.636445
$299$	−35.7934	−2.06998
$300$	0	0
$301$	12.4190	0.715817
$302$	−7.15320	−0.411620
$303$	4.81960	0.276879
$304$	6.62018	0.379693
$305$	0	0
$306$	26.1244	1.49343
$307$	17.5287	1.00042	0.500208	−	0.865905i	$-0.333257\pi$
$307$	17.5287	1.00042	0.500208	+	0.865905i	$0.333257\pi$
$308$	−1.00000	−0.0569803
$309$	−17.9804	−1.02287
$310$	0	0
$311$	29.0181	1.64547	0.822734	−	0.568427i	$-0.192448\pi$
$311$	29.0181	1.64547	0.822734	+	0.568427i	$0.192448\pi$
$312$	−21.1718	−1.19862
$313$	26.9475	1.52316	0.761582	−	0.648069i	$-0.224423\pi$
$313$	26.9475	1.52316	0.761582	+	0.648069i	$0.224423\pi$
$314$	11.8905	0.671022
$315$	0	0
$316$	−10.5610	−0.594100
$317$	16.7087	0.938456	0.469228	−	0.883077i	$-0.344532\pi$
$317$	16.7087	0.938456	0.469228	+	0.883077i	$0.344532\pi$
$318$	27.4928	1.54172
$319$	−1.01326	−0.0567319
$320$	0	0
$321$	−33.1901	−1.85249
$322$	5.53694	0.308562
$323$	22.3844	1.24550
$324$	27.5169	1.52871
$325$	0	0
$326$	−6.47381	−0.358551
$327$	−37.8716	−2.09430
$328$	2.81064	0.155192
$329$	−2.82139	−0.155548
$330$	0	0
$331$	−31.0222	−1.70514	−0.852568	−	0.522617i	$-0.824956\pi$
$331$	−31.0222	−1.70514	−0.852568	+	0.522617i	$0.824956\pi$
$332$	4.81205	0.264095
$333$	16.6557	0.912726
$334$	13.6241	0.745477
$335$	0	0
$336$	3.27510	0.178672
$337$	−14.1958	−0.773295	−0.386647	−	0.922228i	$-0.626367\pi$
$337$	−14.1958	−0.773295	−0.386647	+	0.922228i	$0.626367\pi$
$338$	28.7893	1.56593
$339$	8.11843	0.440933
$340$	0	0
$341$	−2.72630	−0.147638
$342$	51.1495	2.76585
$343$	−1.00000	−0.0539949
$344$	−12.4190	−0.669585
$345$	0	0
$346$	3.05209	0.164082
$347$	−5.38584	−0.289127	−0.144564	−	0.989496i	$-0.546178\pi$
$347$	−5.38584	−0.289127	−0.144564	+	0.989496i	$0.546178\pi$
$348$	3.31854	0.177893
$349$	−22.9557	−1.22879	−0.614396	−	0.788997i	$-0.710600\pi$
$349$	−22.9557	−1.22879	−0.614396	+	0.788997i	$0.710600\pi$
$350$	0	0
$351$	−100.064	−5.34104
$352$	1.00000	0.0533002
$353$	−2.80195	−0.149133	−0.0745664	−	0.997216i	$-0.523757\pi$
$353$	−2.80195	−0.149133	−0.0745664	+	0.997216i	$0.523757\pi$
$354$	−37.4371	−1.98976
$355$	0	0
$356$	8.02653	0.425405
$357$	11.0739	0.586092
$358$	−17.4417	−0.921823
$359$	9.53192	0.503076	0.251538	−	0.967847i	$-0.419064\pi$
$359$	9.53192	0.503076	0.251538	+	0.967847i	$0.419064\pi$
$360$	0	0
$361$	24.8267	1.30667
$362$	−24.9813	−1.31299
$363$	−3.27510	−0.171898
$364$	−6.46446	−0.338830
$365$	0	0
$366$	−18.4816	−0.966051
$367$	11.8716	0.619691	0.309845	−	0.950787i	$-0.399723\pi$
$367$	11.8716	0.619691	0.309845	+	0.950787i	$0.399723\pi$
$368$	−5.53694	−0.288633
$369$	21.7159	1.13048
$370$	0	0
$371$	8.39450	0.435821
$372$	8.92893	0.462943
$373$	12.3379	0.638835	0.319417	−	0.947614i	$-0.396513\pi$
$373$	12.3379	0.638835	0.319417	+	0.947614i	$0.396513\pi$
$374$	3.38123	0.174839
$375$	0	0
$376$	2.82139	0.145502
$377$	−6.55021	−0.337353
$378$	15.4791	0.796161
$379$	−23.2669	−1.19514	−0.597570	−	0.801817i	$-0.703867\pi$
$379$	−23.2669	−1.19514	−0.597570	+	0.801817i	$0.703867\pi$
$380$	0	0
$381$	−32.0517	−1.64206
$382$	18.6624	0.954849
$383$	9.41519	0.481094	0.240547	−	0.970638i	$-0.422673\pi$
$383$	9.41519	0.481094	0.240547	+	0.970638i	$0.422673\pi$
$384$	−3.27510	−0.167132
$385$	0	0
$386$	4.82641	0.245658
$387$	−95.9526	−4.87755
$388$	−11.3961	−0.578547
$389$	24.7245	1.25358	0.626791	−	0.779187i	$-0.284368\pi$
$389$	24.7245	1.25358	0.626791	+	0.779187i	$0.284368\pi$
$390$	0	0
$391$	−18.7217	−0.946797
$392$	1.00000	0.0505076
$393$	−28.3188	−1.42850
$394$	−13.1677	−0.663380
$395$	0	0
$396$	7.72630	0.388261
$397$	−0.894317	−0.0448845	−0.0224423	−	0.999748i	$-0.507144\pi$
$397$	−0.894317	−0.0448845	−0.0224423	+	0.999748i	$0.507144\pi$
$398$	−17.6955	−0.886994
$399$	21.6818	1.08545
$400$	0	0
$401$	3.60662	0.180106	0.0900531	−	0.995937i	$-0.471296\pi$
$401$	3.60662	0.180106	0.0900531	+	0.995937i	$0.471296\pi$
$402$	−10.3787	−0.517644
$403$	−17.6241	−0.877919
$404$	−1.47159	−0.0732141
$405$	0	0
$406$	1.01326	0.0502874
$407$	2.15571	0.106855
$408$	−11.0739	−0.548239
$409$	24.4591	1.20942	0.604711	−	0.796445i	$-0.293289\pi$
$409$	24.4591	1.20942	0.604711	+	0.796445i	$0.293289\pi$
$410$	0	0
$411$	−9.67020	−0.476996
$412$	5.49003	0.270474
$413$	−11.4308	−0.562474
$414$	−42.7801	−2.10253
$415$	0	0
$416$	6.46446	0.316946
$417$	58.6451	2.87186
$418$	6.62018	0.323803
$419$	4.70912	0.230056	0.115028	−	0.993362i	$-0.463304\pi$
$419$	4.70912	0.230056	0.115028	+	0.993362i	$0.463304\pi$
$420$	0	0
$421$	13.1731	0.642015	0.321008	−	0.947077i	$-0.395978\pi$
$421$	13.1731	0.642015	0.321008	+	0.947077i	$0.395978\pi$
$422$	11.2669	0.548463
$423$	21.7989	1.05990
$424$	−8.39450	−0.407673
$425$	0	0
$426$	−15.9090	−0.770793
$427$	−5.64307	−0.273087
$428$	10.1341	0.489848
$429$	−21.1718	−1.02218
$430$	0	0
$431$	−6.96302	−0.335397	−0.167699	−	0.985838i	$-0.553634\pi$
$431$	−6.96302	−0.335397	−0.167699	+	0.985838i	$0.553634\pi$
$432$	−15.4791	−0.744740
$433$	3.53318	0.169794	0.0848969	−	0.996390i	$-0.472944\pi$
$433$	3.53318	0.169794	0.0848969	+	0.996390i	$0.472944\pi$
$434$	2.72630	0.130867
$435$	0	0
$436$	11.5635	0.553790
$437$	−36.6555	−1.75347
$438$	−2.72169	−0.130048
$439$	32.6519	1.55839	0.779194	−	0.626782i	$-0.215628\pi$
$439$	32.6519	1.55839	0.779194	+	0.626782i	$0.215628\pi$
$440$	0	0
$441$	7.72630	0.367919
$442$	21.8579	1.03967
$443$	−40.9084	−1.94362	−0.971808	−	0.235772i	$-0.924238\pi$
$443$	−40.9084	−1.94362	−0.971808	+	0.235772i	$0.924238\pi$
$444$	−7.06018	−0.335061
$445$	0	0
$446$	−24.2768	−1.14954
$447$	−35.9827	−1.70192
$448$	−1.00000	−0.0472456
$449$	−29.6135	−1.39755	−0.698774	−	0.715342i	$-0.746271\pi$
$449$	−29.6135	−1.39755	−0.698774	+	0.715342i	$0.746271\pi$
$450$	0	0
$451$	2.81064	0.132348
$452$	−2.47883	−0.116594
$453$	23.4275	1.10072
$454$	19.9074	0.934302
$455$	0	0
$456$	−21.6818	−1.01534
$457$	−15.4719	−0.723744	−0.361872	−	0.932228i	$-0.617862\pi$
$457$	−15.4719	−0.723744	−0.361872	+	0.932228i	$0.617862\pi$
$458$	−10.9196	−0.510239
$459$	−52.3386	−2.44295
$460$	0	0
$461$	0.231236	0.0107697	0.00538486	−	0.999986i	$-0.498286\pi$
$461$	0.231236	0.0107697	0.00538486	+	0.999986i	$0.498286\pi$
$462$	3.27510	0.152372
$463$	0.741666	0.0344682	0.0172341	−	0.999851i	$-0.494514\pi$
$463$	0.741666	0.0344682	0.0172341	+	0.999851i	$0.494514\pi$
$464$	−1.01326	−0.0470396
$465$	0	0
$466$	6.86037	0.317800
$467$	−35.1500	−1.62655	−0.813274	−	0.581880i	$-0.802317\pi$
$467$	−35.1500	−1.62655	−0.813274	+	0.581880i	$0.802317\pi$
$468$	49.9464	2.30877
$469$	−3.16897	−0.146330
$470$	0	0
$471$	−38.9428	−1.79439
$472$	11.4308	0.526146
$473$	−12.4190	−0.571024
$474$	34.5882	1.58869
$475$	0	0
$476$	−3.38123	−0.154979
$477$	−64.8584	−2.96966
$478$	19.0161	0.869775
$479$	7.82851	0.357694	0.178847	−	0.983877i	$-0.442763\pi$
$479$	7.82851	0.357694	0.178847	+	0.983877i	$0.442763\pi$
$480$	0	0
$481$	13.9355	0.635405
$482$	−9.59461	−0.437022
$483$	−18.1341	−0.825128
$484$	1.00000	0.0454545
$485$	0	0
$486$	−43.6832	−1.98151
$487$	38.4687	1.74318	0.871591	−	0.490233i	$-0.163088\pi$
$487$	38.4687	1.74318	0.871591	+	0.490233i	$0.163088\pi$
$488$	5.64307	0.255450
$489$	21.2024	0.958805
$490$	0	0
$491$	7.04234	0.317816	0.158908	−	0.987293i	$-0.449203\pi$
$491$	7.04234	0.317816	0.158908	+	0.987293i	$0.449203\pi$
$492$	−9.20514	−0.415000
$493$	−3.42608	−0.154303
$494$	42.7959	1.92548
$495$	0	0
$496$	−2.72630	−0.122415
$497$	−4.85755	−0.217891
$498$	−15.7600	−0.706220
$499$	−10.7080	−0.479355	−0.239678	−	0.970853i	$-0.577042\pi$
$499$	−10.7080	−0.479355	−0.239678	+	0.970853i	$0.577042\pi$
$500$	0	0
$501$	−44.6203	−1.99349
$502$	−22.9823	−1.02575
$503$	1.57235	0.0701078	0.0350539	−	0.999385i	$-0.488840\pi$
$503$	1.57235	0.0701078	0.0350539	+	0.999385i	$0.488840\pi$
$504$	−7.72630	−0.344157
$505$	0	0
$506$	−5.53694	−0.246147
$507$	−94.2879	−4.18747
$508$	9.78648	0.434205
$509$	18.1484	0.804415	0.402208	−	0.915549i	$-0.368243\pi$
$509$	18.1484	0.804415	0.402208	+	0.915549i	$0.368243\pi$
$510$	0	0
$511$	−0.831025	−0.0367624
$512$	1.00000	0.0441942
$513$	−102.475	−4.52436
$514$	1.18067	0.0520773
$515$	0	0
$516$	40.6734	1.79055
$517$	2.82139	0.124085
$518$	−2.15571	−0.0947165
$519$	−9.99592	−0.438772
$520$	0	0
$521$	8.77193	0.384305	0.192153	−	0.981365i	$-0.438453\pi$
$521$	8.77193	0.384305	0.192153	+	0.981365i	$0.438453\pi$
$522$	−7.82878	−0.342657
$523$	15.0998	0.660266	0.330133	−	0.943934i	$-0.392906\pi$
$523$	15.0998	0.660266	0.330133	+	0.943934i	$0.392906\pi$
$524$	8.64670	0.377733
$525$	0	0
$526$	26.9368	1.17450
$527$	−9.21827	−0.401554
$528$	−3.27510	−0.142531
$529$	7.65774	0.332945
$530$	0	0
$531$	88.3179	3.83267
$532$	−6.62018	−0.287021
$533$	18.1693	0.786999
$534$	−26.2877	−1.13758
$535$	0	0
$536$	3.16897	0.136879
$537$	57.1234	2.46506
$538$	−20.4439	−0.881401
$539$	1.00000	0.0430730
$540$	0	0
$541$	17.8143	0.765896	0.382948	−	0.923770i	$-0.374909\pi$
$541$	17.8143	0.765896	0.382948	+	0.923770i	$0.374909\pi$
$542$	25.6269	1.10077
$543$	81.8164	3.51108
$544$	3.38123	0.144969
$545$	0	0
$546$	21.1718	0.906069
$547$	39.7962	1.70156	0.850781	−	0.525520i	$-0.176129\pi$
$547$	39.7962	1.70156	0.850781	+	0.525520i	$0.176129\pi$
$548$	2.95264	0.126131
$549$	43.6001	1.86081
$550$	0	0
$551$	−6.70798	−0.285770
$552$	18.1341	0.771837
$553$	10.5610	0.449098
$554$	16.3408	0.694253
$555$	0	0
$556$	−17.9063	−0.759397
$557$	5.61626	0.237968	0.118984	−	0.992896i	$-0.462036\pi$
$557$	5.61626	0.237968	0.118984	+	0.992896i	$0.462036\pi$
$558$	−21.0643	−0.891721
$559$	−80.2819	−3.39556
$560$	0	0
$561$	−11.0739	−0.467540
$562$	4.37872	0.184705
$563$	−27.4346	−1.15623	−0.578115	−	0.815955i	$-0.696211\pi$
$563$	−27.4346	−1.15623	−0.578115	+	0.815955i	$0.696211\pi$
$564$	−9.24035	−0.389089
$565$	0	0
$566$	15.1500	0.636802
$567$	−27.5169	−1.15560
$568$	4.85755	0.203818
$569$	6.02245	0.252474	0.126237	−	0.992000i	$-0.459710\pi$
$569$	6.02245	0.252474	0.126237	+	0.992000i	$0.459710\pi$
$570$	0	0
$571$	8.38374	0.350849	0.175424	−	0.984493i	$-0.443870\pi$
$571$	8.38374	0.350849	0.175424	+	0.984493i	$0.443870\pi$
$572$	6.46446	0.270293
$573$	−61.1211	−2.55337
$574$	−2.81064	−0.117314
$575$	0	0
$576$	7.72630	0.321929
$577$	1.12762	0.0469434	0.0234717	−	0.999725i	$-0.492528\pi$
$577$	1.12762	0.0469434	0.0234717	+	0.999725i	$0.492528\pi$
$578$	−5.56727	−0.231568
$579$	−15.8070	−0.656917
$580$	0	0
$581$	−4.81205	−0.199637
$582$	37.3233	1.54710
$583$	−8.39450	−0.347665
$584$	0.831025	0.0343881
$585$	0	0
$586$	−0.630927	−0.0260634
$587$	11.3187	0.467172	0.233586	−	0.972336i	$-0.424954\pi$
$587$	11.3187	0.467172	0.233586	+	0.972336i	$0.424954\pi$
$588$	−3.27510	−0.135063
$589$	−18.0486	−0.743680
$590$	0	0
$591$	43.1256	1.77395
$592$	2.15571	0.0885992
$593$	19.0319	0.781545	0.390772	−	0.920487i	$-0.372208\pi$
$593$	19.0319	0.781545	0.390772	+	0.920487i	$0.372208\pi$
$594$	−15.4791	−0.635117
$595$	0	0
$596$	10.9867	0.450034
$597$	57.9545	2.37192
$598$	−35.7934	−1.46370
$599$	−12.8576	−0.525345	−0.262673	−	0.964885i	$-0.584604\pi$
$599$	−12.8576	−0.525345	−0.262673	+	0.964885i	$0.584604\pi$
$600$	0	0
$601$	3.16565	0.129129	0.0645647	−	0.997914i	$-0.479434\pi$
$601$	3.16565	0.129129	0.0645647	+	0.997914i	$0.479434\pi$
$602$	12.4190	0.506159
$603$	24.4845	0.997084
$604$	−7.15320	−0.291060
$605$	0	0
$606$	4.81960	0.195783
$607$	2.18017	0.0884905	0.0442453	−	0.999021i	$-0.485912\pi$
$607$	2.18017	0.0884905	0.0442453	+	0.999021i	$0.485912\pi$
$608$	6.62018	0.268484
$609$	−3.31854	−0.134474
$610$	0	0
$611$	18.2388	0.737862
$612$	26.1244	1.05602
$613$	−32.4835	−1.31200	−0.655998	−	0.754762i	$-0.727752\pi$
$613$	−32.4835	−1.31200	−0.655998	+	0.754762i	$0.727752\pi$
$614$	17.5287	0.707401
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−8.85755	−0.356592	−0.178296	−	0.983977i	$-0.557058\pi$
$617$	−8.85755	−0.356592	−0.178296	+	0.983977i	$0.557058\pi$
$618$	−17.9804	−0.723279
$619$	28.4997	1.14550	0.572749	−	0.819731i	$-0.305877\pi$
$619$	28.4997	1.14550	0.572749	+	0.819731i	$0.305877\pi$
$620$	0	0
$621$	85.7071	3.43931
$622$	29.0181	1.16352
$623$	−8.02653	−0.321576
$624$	−21.1718	−0.847550
$625$	0	0
$626$	26.9475	1.07704
$627$	−21.6818	−0.865886
$628$	11.8905	0.474484
$629$	7.28896	0.290630
$630$	0	0
$631$	−44.8847	−1.78683	−0.893416	−	0.449231i	$-0.851698\pi$
$631$	−44.8847	−1.78683	−0.893416	+	0.449231i	$0.851698\pi$
$632$	−10.5610	−0.420092
$633$	−36.9002	−1.46665
$634$	16.7087	0.663589
$635$	0	0
$636$	27.4928	1.09016
$637$	6.46446	0.256131
$638$	−1.01326	−0.0401155
$639$	37.5309	1.48470
$640$	0	0
$641$	38.3641	1.51529	0.757646	−	0.652666i	$-0.226350\pi$
$641$	38.3641	1.51529	0.757646	+	0.652666i	$0.226350\pi$
$642$	−33.1901	−1.30991
$643$	7.61526	0.300317	0.150158	−	0.988662i	$-0.452022\pi$
$643$	7.61526	0.300317	0.150158	+	0.988662i	$0.452022\pi$
$644$	5.53694	0.218186
$645$	0	0
$646$	22.3844	0.880701
$647$	−30.7250	−1.20793	−0.603963	−	0.797012i	$-0.706413\pi$
$647$	−30.7250	−1.20793	−0.603963	+	0.797012i	$0.706413\pi$
$648$	27.5169	1.08096
$649$	11.4308	0.448699
$650$	0	0
$651$	−8.92893	−0.349952
$652$	−6.47381	−0.253534
$653$	−37.5653	−1.47005	−0.735023	−	0.678043i	$-0.762829\pi$
$653$	−37.5653	−1.47005	−0.735023	+	0.678043i	$0.762829\pi$
$654$	−37.8716	−1.48090
$655$	0	0
$656$	2.81064	0.109737
$657$	6.42075	0.250497
$658$	−2.82139	−0.109989
$659$	−39.9380	−1.55576	−0.777882	−	0.628410i	$-0.783706\pi$
$659$	−39.9380	−1.55576	−0.777882	+	0.628410i	$0.783706\pi$
$660$	0	0
$661$	−36.3398	−1.41345	−0.706727	−	0.707486i	$-0.749829\pi$
$661$	−36.3398	−1.41345	−0.706727	+	0.707486i	$0.749829\pi$
$662$	−31.0222	−1.20571
$663$	−71.5867	−2.78020
$664$	4.81205	0.186744
$665$	0	0
$666$	16.6557	0.645394
$667$	5.61038	0.217235
$668$	13.6241	0.527132
$669$	79.5091	3.07400
$670$	0	0
$671$	5.64307	0.217848
$672$	3.27510	0.126340
$673$	−33.8212	−1.30371	−0.651855	−	0.758344i	$-0.726009\pi$
$673$	−33.8212	−1.30371	−0.651855	+	0.758344i	$0.726009\pi$
$674$	−14.1958	−0.546802
$675$	0	0
$676$	28.7893	1.10728
$677$	47.8344	1.83843	0.919213	−	0.393760i	$-0.128826\pi$
$677$	47.8344	1.83843	0.919213	+	0.393760i	$0.128826\pi$
$678$	8.11843	0.311786
$679$	11.3961	0.437341
$680$	0	0
$681$	−65.1989	−2.49843
$682$	−2.72630	−0.104396
$683$	−44.6775	−1.70954	−0.854768	−	0.519011i	$-0.826300\pi$
$683$	−44.6775	−1.70954	−0.854768	+	0.519011i	$0.826300\pi$
$684$	51.1495	1.95575
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	35.7628	1.36443
$688$	−12.4190	−0.473468
$689$	−54.2659	−2.06737
$690$	0	0
$691$	−9.08364	−0.345558	−0.172779	−	0.984961i	$-0.555275\pi$
$691$	−9.08364	−0.345558	−0.172779	+	0.984961i	$0.555275\pi$
$692$	3.05209	0.116023
$693$	−7.72630	−0.293498
$694$	−5.38584	−0.204444
$695$	0	0
$696$	3.31854	0.125789
$697$	9.50343	0.359968
$698$	−22.9557	−0.868888
$699$	−22.4684	−0.849834
$700$	0	0
$701$	21.5344	0.813344	0.406672	−	0.913574i	$-0.366689\pi$
$701$	21.5344	0.813344	0.406672	+	0.913574i	$0.366689\pi$
$702$	−100.064	−3.77668
$703$	14.2712	0.538248
$704$	1.00000	0.0376889
$705$	0	0
$706$	−2.80195	−0.105453
$707$	1.47159	0.0553447
$708$	−37.4371	−1.40697
$709$	−0.0726367	−0.00272793	−0.00136396	−	0.999999i	$-0.500434\pi$
$709$	−0.0726367	−0.00272793	−0.00136396	+	0.999999i	$0.500434\pi$
$710$	0	0
$711$	−81.5972	−3.06013
$712$	8.02653	0.300807
$713$	15.0954	0.565327
$714$	11.0739	0.414430
$715$	0	0
$716$	−17.4417	−0.651827
$717$	−62.2796	−2.32588
$718$	9.53192	0.355728
$719$	−14.8626	−0.554281	−0.277140	−	0.960829i	$-0.589387\pi$
$719$	−14.8626	−0.554281	−0.277140	+	0.960829i	$0.589387\pi$
$720$	0	0
$721$	−5.49003	−0.204459
$722$	24.8267	0.923955
$723$	31.4233	1.16865
$724$	−24.9813	−0.928424
$725$	0	0
$726$	−3.27510	−0.121550
$727$	14.3413	0.531889	0.265945	−	0.963988i	$-0.414316\pi$
$727$	14.3413	0.531889	0.265945	+	0.963988i	$0.414316\pi$
$728$	−6.46446	−0.239589
$729$	60.5163	2.24135
$730$	0	0
$731$	−41.9914	−1.55311
$732$	−18.4816	−0.683101
$733$	−38.0104	−1.40394	−0.701972	−	0.712204i	$-0.747697\pi$
$733$	−38.0104	−1.40394	−0.701972	+	0.712204i	$0.747697\pi$
$734$	11.8716	0.438188
$735$	0	0
$736$	−5.53694	−0.204094
$737$	3.16897	0.116731
$738$	21.7159	0.799372
$739$	8.83792	0.325108	0.162554	−	0.986700i	$-0.448027\pi$
$739$	8.83792	0.325108	0.162554	+	0.986700i	$0.448027\pi$
$740$	0	0
$741$	−140.161	−5.14894
$742$	8.39450	0.308172
$743$	32.2049	1.18148	0.590742	−	0.806861i	$-0.298835\pi$
$743$	32.2049	1.18148	0.590742	+	0.806861i	$0.298835\pi$
$744$	8.92893	0.327350
$745$	0	0
$746$	12.3379	0.451724
$747$	37.1793	1.36032
$748$	3.38123	0.123630
$749$	−10.1341	−0.370291
$750$	0	0
$751$	−37.1493	−1.35560	−0.677799	−	0.735247i	$-0.737066\pi$
$751$	−37.1493	−1.35560	−0.677799	+	0.735247i	$0.737066\pi$
$752$	2.82139	0.102886
$753$	75.2693	2.74297
$754$	−6.55021	−0.238544
$755$	0	0
$756$	15.4791	0.562971
$757$	−3.34714	−0.121654	−0.0608269	−	0.998148i	$-0.519374\pi$
$757$	−3.34714	−0.121654	−0.0608269	+	0.998148i	$0.519374\pi$
$758$	−23.2669	−0.845091
$759$	18.1341	0.658225
$760$	0	0
$761$	24.7682	0.897848	0.448924	−	0.893570i	$-0.351807\pi$
$761$	24.7682	0.897848	0.448924	+	0.893570i	$0.351807\pi$
$762$	−32.0517	−1.16111
$763$	−11.5635	−0.418626
$764$	18.6624	0.675180
$765$	0	0
$766$	9.41519	0.340184
$767$	73.8941	2.66816
$768$	−3.27510	−0.118180
$769$	−13.2016	−0.476063	−0.238032	−	0.971257i	$-0.576502\pi$
$769$	−13.2016	−0.476063	−0.238032	+	0.971257i	$0.576502\pi$
$770$	0	0
$771$	−3.86683	−0.139260
$772$	4.82641	0.173706
$773$	5.00683	0.180083	0.0900416	−	0.995938i	$-0.471300\pi$
$773$	5.00683	0.180083	0.0900416	+	0.995938i	$0.471300\pi$
$774$	−95.9526	−3.44895
$775$	0	0
$776$	−11.3961	−0.409095
$777$	7.06018	0.253282
$778$	24.7245	0.886417
$779$	18.6069	0.666662
$780$	0	0
$781$	4.85755	0.173817
$782$	−18.7217	−0.669486
$783$	15.6844	0.560516
$784$	1.00000	0.0357143
$785$	0	0
$786$	−28.3188	−1.01010
$787$	−20.5280	−0.731746	−0.365873	−	0.930665i	$-0.619230\pi$
$787$	−20.5280	−0.731746	−0.365873	+	0.930665i	$0.619230\pi$
$788$	−13.1677	−0.469080
$789$	−88.2207	−3.14074
$790$	0	0
$791$	2.47883	0.0881371
$792$	7.72630	0.274542
$793$	36.4794	1.29542
$794$	−0.894317	−0.0317381
$795$	0	0
$796$	−17.6955	−0.627199
$797$	−4.72846	−0.167491	−0.0837454	−	0.996487i	$-0.526688\pi$
$797$	−4.72846	−0.167491	−0.0837454	+	0.996487i	$0.526688\pi$
$798$	21.6818	0.767526
$799$	9.53978	0.337493
$800$	0	0
$801$	62.0154	2.19121
$802$	3.60662	0.127354
$803$	0.831025	0.0293262
$804$	−10.3787	−0.366029
$805$	0	0
$806$	−17.6241	−0.620782
$807$	66.9560	2.35696
$808$	−1.47159	−0.0517702
$809$	−31.1911	−1.09662	−0.548310	−	0.836275i	$-0.684728\pi$
$809$	−31.1911	−1.09662	−0.548310	+	0.836275i	$0.684728\pi$
$810$	0	0
$811$	3.74209	0.131403	0.0657013	−	0.997839i	$-0.479072\pi$
$811$	3.74209	0.131403	0.0657013	+	0.997839i	$0.479072\pi$
$812$	1.01326	0.0355586
$813$	−83.9308	−2.94358
$814$	2.15571	0.0755576
$815$	0	0
$816$	−11.0739	−0.387664
$817$	−82.2157	−2.87636
$818$	24.4591	0.855191
$819$	−49.9464	−1.74527
$820$	0	0
$821$	16.1553	0.563825	0.281913	−	0.959440i	$-0.409031\pi$
$821$	16.1553	0.563825	0.281913	+	0.959440i	$0.409031\pi$
$822$	−9.67020	−0.337287
$823$	1.62701	0.0567139	0.0283570	−	0.999598i	$-0.490972\pi$
$823$	1.62701	0.0567139	0.0283570	+	0.999598i	$0.490972\pi$
$824$	5.49003	0.191254
$825$	0	0
$826$	−11.4308	−0.397729
$827$	−28.5580	−0.993061	−0.496530	−	0.868019i	$-0.665393\pi$
$827$	−28.5580	−0.993061	−0.496530	+	0.868019i	$0.665393\pi$
$828$	−42.7801	−1.48671
$829$	18.7101	0.649830	0.324915	−	0.945743i	$-0.394664\pi$
$829$	18.7101	0.649830	0.324915	+	0.945743i	$0.394664\pi$
$830$	0	0
$831$	−53.5177	−1.85651
$832$	6.46446	0.224115
$833$	3.38123	0.117153
$834$	58.6451	2.03071
$835$	0	0
$836$	6.62018	0.228964
$837$	42.2008	1.45867
$838$	4.70912	0.162674
$839$	36.4414	1.25810	0.629049	−	0.777366i	$-0.283445\pi$
$839$	36.4414	1.25810	0.629049	+	0.777366i	$0.283445\pi$
$840$	0	0
$841$	−27.9733	−0.964596
$842$	13.1731	0.453973
$843$	−14.3408	−0.493922
$844$	11.2669	0.387822
$845$	0	0
$846$	21.7989	0.749463
$847$	−1.00000	−0.0343604
$848$	−8.39450	−0.288268
$849$	−49.6178	−1.70288
$850$	0	0
$851$	−11.9361	−0.409163
$852$	−15.9090	−0.545033
$853$	25.9043	0.886945	0.443472	−	0.896288i	$-0.353746\pi$
$853$	25.9043	0.886945	0.443472	+	0.896288i	$0.353746\pi$
$854$	−5.64307	−0.193102
$855$	0	0
$856$	10.1341	0.346375
$857$	−15.6452	−0.534431	−0.267215	−	0.963637i	$-0.586104\pi$
$857$	−15.6452	−0.534431	−0.267215	+	0.963637i	$0.586104\pi$
$858$	−21.1718	−0.722793
$859$	−31.7666	−1.08386	−0.541930	−	0.840423i	$-0.682306\pi$
$859$	−31.7666	−1.08386	−0.541930	+	0.840423i	$0.682306\pi$
$860$	0	0
$861$	9.20514	0.313710
$862$	−6.96302	−0.237162
$863$	−7.07738	−0.240917	−0.120458	−	0.992718i	$-0.538436\pi$
$863$	−7.07738	−0.240917	−0.120458	+	0.992718i	$0.538436\pi$
$864$	−15.4791	−0.526611
$865$	0	0
$866$	3.53318	0.120062
$867$	18.2334	0.619238
$868$	2.72630	0.0925368
$869$	−10.5610	−0.358256
$870$	0	0
$871$	20.4857	0.694132
$872$	11.5635	0.391589
$873$	−88.0494	−2.98002
$874$	−36.6555	−1.23989
$875$	0	0
$876$	−2.72169	−0.0919575
$877$	13.7101	0.462957	0.231478	−	0.972840i	$-0.425644\pi$
$877$	13.7101	0.462957	0.231478	+	0.972840i	$0.425644\pi$
$878$	32.6519	1.10195
$879$	2.06635	0.0696963
$880$	0	0
$881$	−15.9858	−0.538574	−0.269287	−	0.963060i	$-0.586788\pi$
$881$	−15.9858	−0.538574	−0.269287	+	0.963060i	$0.586788\pi$
$882$	7.72630	0.260158
$883$	44.0546	1.48256	0.741278	−	0.671198i	$-0.234220\pi$
$883$	44.0546	1.48256	0.741278	+	0.671198i	$0.234220\pi$
$884$	21.8579	0.735159
$885$	0	0
$886$	−40.9084	−1.37434
$887$	51.9962	1.74586	0.872930	−	0.487845i	$-0.162217\pi$
$887$	51.9962	1.74586	0.872930	+	0.487845i	$0.162217\pi$
$888$	−7.06018	−0.236924
$889$	−9.78648	−0.328228
$890$	0	0
$891$	27.5169	0.921850
$892$	−24.2768	−0.812848
$893$	18.6781	0.625039
$894$	−35.9827	−1.20344
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	117.227	3.91410
$898$	−29.6135	−0.988216
$899$	2.76247	0.0921334
$900$	0	0
$901$	−28.3837	−0.945600
$902$	2.81064	0.0935841
$903$	−40.6734	−1.35353
$904$	−2.47883	−0.0824447
$905$	0	0
$906$	23.4275	0.778326
$907$	−25.8158	−0.857200	−0.428600	−	0.903494i	$-0.640993\pi$
$907$	−25.8158	−0.857200	−0.428600	+	0.903494i	$0.640993\pi$
$908$	19.9074	0.660652
$909$	−11.3699	−0.377116
$910$	0	0
$911$	56.2763	1.86452	0.932258	−	0.361794i	$-0.117836\pi$
$911$	56.2763	1.86452	0.932258	+	0.361794i	$0.117836\pi$
$912$	−21.6818	−0.717955
$913$	4.81205	0.159256
$914$	−15.4719	−0.511764
$915$	0	0
$916$	−10.9196	−0.360793
$917$	−8.64670	−0.285539
$918$	−52.3386	−1.72743
$919$	−21.0335	−0.693831	−0.346916	−	0.937896i	$-0.612771\pi$
$919$	−21.0335	−0.693831	−0.346916	+	0.937896i	$0.612771\pi$
$920$	0	0
$921$	−57.4084	−1.89167
$922$	0.231236	0.00761534
$923$	31.4015	1.03359
$924$	3.27510	0.107743
$925$	0	0
$926$	0.741666	0.0243727
$927$	42.4176	1.39318
$928$	−1.01326	−0.0332620
$929$	−17.0767	−0.560268	−0.280134	−	0.959961i	$-0.590379\pi$
$929$	−17.0767	−0.560268	−0.280134	+	0.959961i	$0.590379\pi$
$930$	0	0
$931$	6.62018	0.216967
$932$	6.86037	0.224719
$933$	−95.0374	−3.11139
$934$	−35.1500	−1.15014
$935$	0	0
$936$	49.9464	1.63255
$937$	−4.00689	−0.130899	−0.0654497	−	0.997856i	$-0.520848\pi$
$937$	−4.00689	−0.130899	−0.0654497	+	0.997856i	$0.520848\pi$
$938$	−3.16897	−0.103471
$939$	−88.2559	−2.88012
$940$	0	0
$941$	−14.4810	−0.472067	−0.236033	−	0.971745i	$-0.575847\pi$
$941$	−14.4810	−0.472067	−0.236033	+	0.971745i	$0.575847\pi$
$942$	−38.9428	−1.26882
$943$	−15.5624	−0.506780
$944$	11.4308	0.372041
$945$	0	0
$946$	−12.4190	−0.403775
$947$	34.1301	1.10908	0.554539	−	0.832158i	$-0.312895\pi$
$947$	34.1301	1.10908	0.554539	+	0.832158i	$0.312895\pi$
$948$	34.5882	1.12337
$949$	5.37213	0.174387
$950$	0	0
$951$	−54.7228	−1.77451
$952$	−3.38123	−0.109586
$953$	6.31885	0.204688	0.102344	−	0.994749i	$-0.467366\pi$
$953$	6.31885	0.204688	0.102344	+	0.994749i	$0.467366\pi$
$954$	−64.8584	−2.09987
$955$	0	0
$956$	19.0161	0.615024
$957$	3.31854	0.107273
$958$	7.82851	0.252928
$959$	−2.95264	−0.0953457
$960$	0	0
$961$	−23.5673	−0.760234
$962$	13.9355	0.449299
$963$	78.2989	2.52315
$964$	−9.59461	−0.309022
$965$	0	0
$966$	−18.1341	−0.583454
$967$	−32.6668	−1.05049	−0.525247	−	0.850950i	$-0.676027\pi$
$967$	−32.6668	−1.05049	−0.525247	+	0.850950i	$0.676027\pi$
$968$	1.00000	0.0321412
$969$	−73.3111	−2.35509
$970$	0	0
$971$	−32.9801	−1.05838	−0.529190	−	0.848503i	$-0.677504\pi$
$971$	−32.9801	−1.05838	−0.529190	+	0.848503i	$0.677504\pi$
$972$	−43.6832	−1.40114
$973$	17.9063	0.574050
$974$	38.4687	1.23262
$975$	0	0
$976$	5.64307	0.180630
$977$	5.50566	0.176142	0.0880709	−	0.996114i	$-0.471930\pi$
$977$	5.50566	0.176142	0.0880709	+	0.996114i	$0.471930\pi$
$978$	21.2024	0.677978
$979$	8.02653	0.256529
$980$	0	0
$981$	89.3429	2.85250
$982$	7.04234	0.224730
$983$	−15.5437	−0.495767	−0.247883	−	0.968790i	$-0.579735\pi$
$983$	−15.5437	−0.495767	−0.247883	+	0.968790i	$0.579735\pi$
$984$	−9.20514	−0.293449
$985$	0	0
$986$	−3.42608	−0.109109
$987$	9.24035	0.294124
$988$	42.7959	1.36152
$989$	68.7631	2.18654
$990$	0	0
$991$	−2.47470	−0.0786115	−0.0393057	−	0.999227i	$-0.512515\pi$
$991$	−2.47470	−0.0786115	−0.0393057	+	0.999227i	$0.512515\pi$
$992$	−2.72630	−0.0865602
$993$	101.601	3.22421
$994$	−4.85755	−0.154072
$995$	0	0
$996$	−15.7600	−0.499373
$997$	−13.9735	−0.442544	−0.221272	−	0.975212i	$-0.571021\pi$
$997$	−13.9735	−0.442544	−0.221272	+	0.975212i	$0.571021\pi$
$998$	−10.7080	−0.338955
$999$	−33.3685	−1.05573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.a.ca.1.1		5
5.2	odd	4		770.2.c.f.309.10	yes	10
5.3	odd	4		770.2.c.f.309.1	✓	10
5.4	even	2		3850.2.a.bz.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.c.f.309.1	✓	10	5.3	odd	4
770.2.c.f.309.10	yes	10	5.2	odd	4
3850.2.a.bz.1.5		5	5.4	even	2
3850.2.a.ca.1.1		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.27510 q^{3} +1.00000 q^{4} -3.27510 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.72630 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.27510 q^{3} +1.00000 q^{4} -3.27510 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.72630 q^{9} +1.00000 q^{11} -3.27510 q^{12} +6.46446 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.38123 q^{17} +7.72630 q^{18} +6.62018 q^{19} +3.27510 q^{21} +1.00000 q^{22} -5.53694 q^{23} -3.27510 q^{24} +6.46446 q^{26} -15.4791 q^{27} -1.00000 q^{28} -1.01326 q^{29} -2.72630 q^{31} +1.00000 q^{32} -3.27510 q^{33} +3.38123 q^{34} +7.72630 q^{36} +2.15571 q^{37} +6.62018 q^{38} -21.1718 q^{39} +2.81064 q^{41} +3.27510 q^{42} -12.4190 q^{43} +1.00000 q^{44} -5.53694 q^{46} +2.82139 q^{47} -3.27510 q^{48} +1.00000 q^{49} -11.0739 q^{51} +6.46446 q^{52} -8.39450 q^{53} -15.4791 q^{54} -1.00000 q^{56} -21.6818 q^{57} -1.01326 q^{58} +11.4308 q^{59} +5.64307 q^{61} -2.72630 q^{62} -7.72630 q^{63} +1.00000 q^{64} -3.27510 q^{66} +3.16897 q^{67} +3.38123 q^{68} +18.1341 q^{69} +4.85755 q^{71} +7.72630 q^{72} +0.831025 q^{73} +2.15571 q^{74} +6.62018 q^{76} -1.00000 q^{77} -21.1718 q^{78} -10.5610 q^{79} +27.5169 q^{81} +2.81064 q^{82} +4.81205 q^{83} +3.27510 q^{84} -12.4190 q^{86} +3.31854 q^{87} +1.00000 q^{88} +8.02653 q^{89} -6.46446 q^{91} -5.53694 q^{92} +8.92893 q^{93} +2.82139 q^{94} -3.27510 q^{96} -11.3961 q^{97} +1.00000 q^{98} +7.72630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{7} + 5 q^{8} + 13 q^{9}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^7 + 5 * q^8 + 13 * q^9	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{7} + 5 q^{8} + 13 q^{9} + 5 q^{11} + 10 q^{13} - 5 q^{14} + 5 q^{16} + 2 q^{17} + 13 q^{18} + 6 q^{19} + 5 q^{22} - 8 q^{23} + 10 q^{26} - 5 q^{28} + 8 q^{29} + 12 q^{31} + 5 q^{32} + 2 q^{34} + 13 q^{36} + 6 q^{37} + 6 q^{38} - 4 q^{39} + 20 q^{41} - 12 q^{43} + 5 q^{44} - 8 q^{46} + 10 q^{47} + 5 q^{49} - 16 q^{51} + 10 q^{52} - 14 q^{53} - 5 q^{56} - 12 q^{57} + 8 q^{58} + 36 q^{59} + 10 q^{61} + 12 q^{62} - 13 q^{63} + 5 q^{64} - 2 q^{67} + 2 q^{68} + 24 q^{69} + 16 q^{71} + 13 q^{72} + 22 q^{73} + 6 q^{74} + 6 q^{76} - 5 q^{77} - 4 q^{78} - 10 q^{79} + 25 q^{81} + 20 q^{82} - 12 q^{83} - 12 q^{86} + 32 q^{87} + 5 q^{88} + 14 q^{89} - 10 q^{91} - 8 q^{92} + 10 q^{94} + 2 q^{97} + 5 q^{98} + 13 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^7 + 5 * q^8 + 13 * q^9 + 5 * q^11 + 10 * q^13 - 5 * q^14 + 5 * q^16 + 2 * q^17 + 13 * q^18 + 6 * q^19 + 5 * q^22 - 8 * q^23 + 10 * q^26 - 5 * q^28 + 8 * q^29 + 12 * q^31 + 5 * q^32 + 2 * q^34 + 13 * q^36 + 6 * q^37 + 6 * q^38 - 4 * q^39 + 20 * q^41 - 12 * q^43 + 5 * q^44 - 8 * q^46 + 10 * q^47 + 5 * q^49 - 16 * q^51 + 10 * q^52 - 14 * q^53 - 5 * q^56 - 12 * q^57 + 8 * q^58 + 36 * q^59 + 10 * q^61 + 12 * q^62 - 13 * q^63 + 5 * q^64 - 2 * q^67 + 2 * q^68 + 24 * q^69 + 16 * q^71 + 13 * q^72 + 22 * q^73 + 6 * q^74 + 6 * q^76 - 5 * q^77 - 4 * q^78 - 10 * q^79 + 25 * q^81 + 20 * q^82 - 12 * q^83 - 12 * q^86 + 32 * q^87 + 5 * q^88 + 14 * q^89 - 10 * q^91 - 8 * q^92 + 10 * q^94 + 2 * q^97 + 5 * q^98 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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