LMFDB - Embedded newform 5775.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.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:	$46.1136071673$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} +0.539189 q^{6} -1.00000 q^{7} -2.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} +0.539189 q^{6} -1.00000 q^{7} -2.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.70928 q^{12} +3.17009 q^{13} -0.539189 q^{14} +2.34017 q^{16} -2.70928 q^{17} +0.539189 q^{18} -3.34017 q^{19} -1.00000 q^{21} -0.539189 q^{22} +1.63090 q^{23} -2.00000 q^{24} +1.70928 q^{26} +1.00000 q^{27} +1.70928 q^{28} +1.24846 q^{29} +5.21953 q^{31} +5.26180 q^{32} -1.00000 q^{33} -1.46081 q^{34} -1.70928 q^{36} -3.17009 q^{37} -1.80098 q^{38} +3.17009 q^{39} -0.829914 q^{41} -0.539189 q^{42} -1.24846 q^{43} +1.70928 q^{44} +0.879362 q^{46} +4.87936 q^{47} +2.34017 q^{48} +1.00000 q^{49} -2.70928 q^{51} -5.41855 q^{52} -5.92162 q^{53} +0.539189 q^{54} +2.00000 q^{56} -3.34017 q^{57} +0.673158 q^{58} -4.46081 q^{59} -7.44748 q^{61} +2.81432 q^{62} -1.00000 q^{63} -1.84324 q^{64} -0.539189 q^{66} -2.00000 q^{67} +4.63090 q^{68} +1.63090 q^{69} -11.6670 q^{71} -2.00000 q^{72} +6.92162 q^{73} -1.70928 q^{74} +5.70928 q^{76} +1.00000 q^{77} +1.70928 q^{78} +9.21953 q^{79} +1.00000 q^{81} -0.447480 q^{82} -15.0494 q^{83} +1.70928 q^{84} -0.673158 q^{86} +1.24846 q^{87} +2.00000 q^{88} -11.4319 q^{89} -3.17009 q^{91} -2.78765 q^{92} +5.21953 q^{93} +2.63090 q^{94} +5.26180 q^{96} -4.32684 q^{97} +0.539189 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 2 q^{4} - 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^7 - 6 * q^8 + 3 * q^9	$ 3 q + 3 q^{3} + 2 q^{4} - 3 q^{7} - 6 q^{8} + 3 q^{9} - 3 q^{11} + 2 q^{12} + 4 q^{13} - 4 q^{16} - q^{17} + q^{19} - 3 q^{21} + q^{23} - 6 q^{24} - 2 q^{26} + 3 q^{27} - 2 q^{28} - 5 q^{29} - 8 q^{31} + 8 q^{32} - 3 q^{33} - 6 q^{34} + 2 q^{36} - 4 q^{37} + 4 q^{38} + 4 q^{39} - 8 q^{41} + 5 q^{43} - 2 q^{44} - 10 q^{46} + 2 q^{47} - 4 q^{48} + 3 q^{49} - q^{51} - 2 q^{52} - 21 q^{53} + 6 q^{56} + q^{57} + 14 q^{58} - 15 q^{59} - 23 q^{61} - 3 q^{63} - 12 q^{64} - 6 q^{67} + 10 q^{68} + q^{69} - 12 q^{71} - 6 q^{72} + 24 q^{73} + 2 q^{74} + 10 q^{76} + 3 q^{77} - 2 q^{78} + 4 q^{79} + 3 q^{81} - 2 q^{82} - 27 q^{83} - 2 q^{84} - 14 q^{86} - 5 q^{87} + 6 q^{88} - 21 q^{89} - 4 q^{91} + 2 q^{92} - 8 q^{93} + 4 q^{94} + 8 q^{96} - q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^7 - 6 * q^8 + 3 * q^9 - 3 * q^11 + 2 * q^12 + 4 * q^13 - 4 * q^16 - q^17 + q^19 - 3 * q^21 + q^23 - 6 * q^24 - 2 * q^26 + 3 * q^27 - 2 * q^28 - 5 * q^29 - 8 * q^31 + 8 * q^32 - 3 * q^33 - 6 * q^34 + 2 * q^36 - 4 * q^37 + 4 * q^38 + 4 * q^39 - 8 * q^41 + 5 * q^43 - 2 * q^44 - 10 * q^46 + 2 * q^47 - 4 * q^48 + 3 * q^49 - q^51 - 2 * q^52 - 21 * q^53 + 6 * q^56 + q^57 + 14 * q^58 - 15 * q^59 - 23 * q^61 - 3 * q^63 - 12 * q^64 - 6 * q^67 + 10 * q^68 + q^69 - 12 * q^71 - 6 * q^72 + 24 * q^73 + 2 * q^74 + 10 * q^76 + 3 * q^77 - 2 * q^78 + 4 * q^79 + 3 * q^81 - 2 * q^82 - 27 * q^83 - 2 * q^84 - 14 * q^86 - 5 * q^87 + 6 * q^88 - 21 * q^89 - 4 * q^91 + 2 * q^92 - 8 * q^93 + 4 * q^94 + 8 * q^96 - q^97 - 3 * 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 $

$2$	0.539189	0.381264	0.190632	−	0.981662i	$-0.438946\pi$
$2$	0.539189	0.381264	0.190632	+	0.981662i	$0.438946\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.70928	−0.854638
$5$	0	0
$5$	0	0
$6$	0.539189	0.220123
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.00000	−0.707107
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.70928	−0.493425
$13$	3.17009	0.879224	0.439612	−	0.898188i	$-0.355116\pi$
$13$	3.17009	0.879224	0.439612	+	0.898188i	$0.355116\pi$
$14$	−0.539189	−0.144104
$15$	0	0
$16$	2.34017	0.585043
$17$	−2.70928	−0.657096	−0.328548	−	0.944487i	$-0.606559\pi$
$17$	−2.70928	−0.657096	−0.328548	+	0.944487i	$0.606559\pi$
$18$	0.539189	0.127088
$19$	−3.34017	−0.766288	−0.383144	−	0.923689i	$-0.625159\pi$
$19$	−3.34017	−0.766288	−0.383144	+	0.923689i	$0.625159\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−0.539189	−0.114955
$23$	1.63090	0.340066	0.170033	−	0.985438i	$-0.445613\pi$
$23$	1.63090	0.340066	0.170033	+	0.985438i	$0.445613\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	1.70928	0.335216
$27$	1.00000	0.192450
$28$	1.70928	0.323023
$29$	1.24846	0.231834	0.115917	−	0.993259i	$-0.463019\pi$
$29$	1.24846	0.231834	0.115917	+	0.993259i	$0.463019\pi$
$30$	0	0
$31$	5.21953	0.937456	0.468728	−	0.883343i	$-0.344712\pi$
$31$	5.21953	0.937456	0.468728	+	0.883343i	$0.344712\pi$
$32$	5.26180	0.930163
$33$	−1.00000	−0.174078
$34$	−1.46081	−0.250527
$35$	0	0
$36$	−1.70928	−0.284879
$37$	−3.17009	−0.521159	−0.260580	−	0.965452i	$-0.583914\pi$
$37$	−3.17009	−0.521159	−0.260580	+	0.965452i	$0.583914\pi$
$38$	−1.80098	−0.292158
$39$	3.17009	0.507620
$40$	0	0
$41$	−0.829914	−0.129611	−0.0648054	−	0.997898i	$-0.520643\pi$
$41$	−0.829914	−0.129611	−0.0648054	+	0.997898i	$0.520643\pi$
$42$	−0.539189	−0.0831986
$43$	−1.24846	−0.190389	−0.0951945	−	0.995459i	$-0.530347\pi$
$43$	−1.24846	−0.190389	−0.0951945	+	0.995459i	$0.530347\pi$
$44$	1.70928	0.257683
$45$	0	0
$46$	0.879362	0.129655
$47$	4.87936	0.711728	0.355864	−	0.934538i	$-0.384187\pi$
$47$	4.87936	0.711728	0.355864	+	0.934538i	$0.384187\pi$
$48$	2.34017	0.337775
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.70928	−0.379374
$52$	−5.41855	−0.751418
$53$	−5.92162	−0.813397	−0.406699	−	0.913562i	$-0.633320\pi$
$53$	−5.92162	−0.813397	−0.406699	+	0.913562i	$0.633320\pi$
$54$	0.539189	0.0733743
$55$	0	0
$56$	2.00000	0.267261
$57$	−3.34017	−0.442417
$58$	0.673158	0.0883900
$59$	−4.46081	−0.580748	−0.290374	−	0.956913i	$-0.593780\pi$
$59$	−4.46081	−0.580748	−0.290374	+	0.956913i	$0.593780\pi$
$60$	0	0
$61$	−7.44748	−0.953552	−0.476776	−	0.879025i	$-0.658195\pi$
$61$	−7.44748	−0.953552	−0.476776	+	0.879025i	$0.658195\pi$
$62$	2.81432	0.357418
$63$	−1.00000	−0.125988
$64$	−1.84324	−0.230406
$65$	0	0
$66$	−0.539189	−0.0663696
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	4.63090	0.561579
$69$	1.63090	0.196337
$70$	0	0
$71$	−11.6670	−1.38462	−0.692310	−	0.721600i	$-0.743407\pi$
$71$	−11.6670	−1.38462	−0.692310	+	0.721600i	$0.743407\pi$
$72$	−2.00000	−0.235702
$73$	6.92162	0.810115	0.405057	−	0.914291i	$-0.367252\pi$
$73$	6.92162	0.810115	0.405057	+	0.914291i	$0.367252\pi$
$74$	−1.70928	−0.198699
$75$	0	0
$76$	5.70928	0.654899
$77$	1.00000	0.113961
$78$	1.70928	0.193537
$79$	9.21953	1.03728	0.518639	−	0.854993i	$-0.326439\pi$
$79$	9.21953	1.03728	0.518639	+	0.854993i	$0.326439\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.447480	−0.0494159
$83$	−15.0494	−1.65189	−0.825946	−	0.563750i	$-0.809358\pi$
$83$	−15.0494	−1.65189	−0.825946	+	0.563750i	$0.809358\pi$
$84$	1.70928	0.186497
$85$	0	0
$86$	−0.673158	−0.0725885
$87$	1.24846	0.133849
$88$	2.00000	0.213201
$89$	−11.4319	−1.21178	−0.605889	−	0.795550i	$-0.707182\pi$
$89$	−11.4319	−1.21178	−0.605889	+	0.795550i	$0.707182\pi$
$90$	0	0
$91$	−3.17009	−0.332315
$92$	−2.78765	−0.290633
$93$	5.21953	0.541241
$94$	2.63090	0.271356
$95$	0	0
$96$	5.26180	0.537030
$97$	−4.32684	−0.439324	−0.219662	−	0.975576i	$-0.570495\pi$
$97$	−4.32684	−0.439324	−0.219662	+	0.975576i	$0.570495\pi$
$98$	0.539189	0.0544663
$99$	−1.00000	−0.100504
$100$	0	0
$101$	2.68035	0.266704	0.133352	−	0.991069i	$-0.457426\pi$
$101$	2.68035	0.266704	0.133352	+	0.991069i	$0.457426\pi$
$102$	−1.46081	−0.144642
$103$	−9.00719	−0.887505	−0.443752	−	0.896149i	$-0.646353\pi$
$103$	−9.00719	−0.887505	−0.443752	+	0.896149i	$0.646353\pi$
$104$	−6.34017	−0.621705
$105$	0	0
$106$	−3.19287	−0.310119
$107$	−9.80098	−0.947497	−0.473748	−	0.880660i	$-0.657099\pi$
$107$	−9.80098	−0.947497	−0.473748	+	0.880660i	$0.657099\pi$
$108$	−1.70928	−0.164475
$109$	−12.2979	−1.17793	−0.588963	−	0.808160i	$-0.700464\pi$
$109$	−12.2979	−1.17793	−0.588963	+	0.808160i	$0.700464\pi$
$110$	0	0
$111$	−3.17009	−0.300891
$112$	−2.34017	−0.221126
$113$	−12.8660	−1.21033	−0.605167	−	0.796098i	$-0.706894\pi$
$113$	−12.8660	−1.21033	−0.605167	+	0.796098i	$0.706894\pi$
$114$	−1.80098	−0.168678
$115$	0	0
$116$	−2.13397	−0.198134
$117$	3.17009	0.293075
$118$	−2.40522	−0.221418
$119$	2.70928	0.248359
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.01560	−0.363555
$123$	−0.829914	−0.0748308
$124$	−8.92162	−0.801185
$125$	0	0
$126$	−0.539189	−0.0480348
$127$	−7.14116	−0.633675	−0.316838	−	0.948480i	$-0.602621\pi$
$127$	−7.14116	−0.633675	−0.316838	+	0.948480i	$0.602621\pi$
$128$	−11.5174	−1.01801
$129$	−1.24846	−0.109921
$130$	0	0
$131$	−2.09171	−0.182753	−0.0913767	−	0.995816i	$-0.529127\pi$
$131$	−2.09171	−0.182753	−0.0913767	+	0.995816i	$0.529127\pi$
$132$	1.70928	0.148773
$133$	3.34017	0.289630
$134$	−1.07838	−0.0931576
$135$	0	0
$136$	5.41855	0.464637
$137$	11.2846	0.964107	0.482053	−	0.876142i	$-0.339891\pi$
$137$	11.2846	0.964107	0.482053	+	0.876142i	$0.339891\pi$
$138$	0.879362	0.0748563
$139$	19.9916	1.69566	0.847832	−	0.530265i	$-0.177907\pi$
$139$	19.9916	1.69566	0.847832	+	0.530265i	$0.177907\pi$
$140$	0	0
$141$	4.87936	0.410916
$142$	−6.29072	−0.527906
$143$	−3.17009	−0.265096
$144$	2.34017	0.195014
$145$	0	0
$146$	3.73206	0.308868
$147$	1.00000	0.0824786
$148$	5.41855	0.445402
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−3.26180	−0.265441	−0.132721	−	0.991153i	$-0.542371\pi$
$151$	−3.26180	−0.265441	−0.132721	+	0.991153i	$0.542371\pi$
$152$	6.68035	0.541848
$153$	−2.70928	−0.219032
$154$	0.539189	0.0434491
$155$	0	0
$156$	−5.41855	−0.433831
$157$	−8.06278	−0.643480	−0.321740	−	0.946828i	$-0.604268\pi$
$157$	−8.06278	−0.643480	−0.321740	+	0.946828i	$0.604268\pi$
$158$	4.97107	0.395477
$159$	−5.92162	−0.469615
$160$	0	0
$161$	−1.63090	−0.128533
$162$	0.539189	0.0423627
$163$	3.17009	0.248300	0.124150	−	0.992263i	$-0.460380\pi$
$163$	3.17009	0.248300	0.124150	+	0.992263i	$0.460380\pi$
$164$	1.41855	0.110770
$165$	0	0
$166$	−8.11450	−0.629807
$167$	4.20620	0.325486	0.162743	−	0.986669i	$-0.447966\pi$
$167$	4.20620	0.325486	0.162743	+	0.986669i	$0.447966\pi$
$168$	2.00000	0.154303
$169$	−2.95055	−0.226966
$170$	0	0
$171$	−3.34017	−0.255429
$172$	2.13397	0.162714
$173$	2.65756	0.202051	0.101025	−	0.994884i	$-0.467788\pi$
$173$	2.65756	0.202051	0.101025	+	0.994884i	$0.467788\pi$
$174$	0.673158	0.0510320
$175$	0	0
$176$	−2.34017	−0.176397
$177$	−4.46081	−0.335295
$178$	−6.16394	−0.462007
$179$	19.3679	1.44762	0.723812	−	0.689998i	$-0.242388\pi$
$179$	19.3679	1.44762	0.723812	+	0.689998i	$0.242388\pi$
$180$	0	0
$181$	−10.7070	−0.795846	−0.397923	−	0.917419i	$-0.630269\pi$
$181$	−10.7070	−0.795846	−0.397923	+	0.917419i	$0.630269\pi$
$182$	−1.70928	−0.126700
$183$	−7.44748	−0.550534
$184$	−3.26180	−0.240463
$185$	0	0
$186$	2.81432	0.206356
$187$	2.70928	0.198122
$188$	−8.34017	−0.608270
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−18.6225	−1.34748	−0.673738	−	0.738970i	$-0.735312\pi$
$191$	−18.6225	−1.34748	−0.673738	+	0.738970i	$0.735312\pi$
$192$	−1.84324	−0.133025
$193$	14.7792	1.06383	0.531917	−	0.846797i	$-0.321472\pi$
$193$	14.7792	1.06383	0.531917	+	0.846797i	$0.321472\pi$
$194$	−2.33299	−0.167499
$195$	0	0
$196$	−1.70928	−0.122091
$197$	−19.0928	−1.36030	−0.680151	−	0.733072i	$-0.738086\pi$
$197$	−19.0928	−1.36030	−0.680151	+	0.733072i	$0.738086\pi$
$198$	−0.539189	−0.0383185
$199$	−24.4391	−1.73244	−0.866220	−	0.499663i	$-0.833457\pi$
$199$	−24.4391	−1.73244	−0.866220	+	0.499663i	$0.833457\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	1.44521	0.101685
$203$	−1.24846	−0.0876250
$204$	4.63090	0.324228
$205$	0	0
$206$	−4.85658	−0.338374
$207$	1.63090	0.113355
$208$	7.41855	0.514384
$209$	3.34017	0.231045
$210$	0	0
$211$	−22.2557	−1.53214	−0.766071	−	0.642756i	$-0.777791\pi$
$211$	−22.2557	−1.53214	−0.766071	+	0.642756i	$0.777791\pi$
$212$	10.1217	0.695160
$213$	−11.6670	−0.799411
$214$	−5.28458	−0.361247
$215$	0	0
$216$	−2.00000	−0.136083
$217$	−5.21953	−0.354325
$218$	−6.63090	−0.449101
$219$	6.92162	0.467720
$220$	0	0
$221$	−8.58864	−0.577734
$222$	−1.70928	−0.114719
$223$	−24.1845	−1.61951	−0.809756	−	0.586767i	$-0.800400\pi$
$223$	−24.1845	−1.61951	−0.809756	+	0.586767i	$0.800400\pi$
$224$	−5.26180	−0.351568
$225$	0	0
$226$	−6.93722	−0.461457
$227$	25.9360	1.72143	0.860716	−	0.509085i	$-0.170016\pi$
$227$	25.9360	1.72143	0.860716	+	0.509085i	$0.170016\pi$
$228$	5.70928	0.378106
$229$	0.937221	0.0619333	0.0309666	−	0.999520i	$-0.490141\pi$
$229$	0.937221	0.0619333	0.0309666	+	0.999520i	$0.490141\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	−2.49693	−0.163931
$233$	17.4752	1.14484	0.572419	−	0.819961i	$-0.306005\pi$
$233$	17.4752	1.14484	0.572419	+	0.819961i	$0.306005\pi$
$234$	1.70928	0.111739
$235$	0	0
$236$	7.62475	0.496329
$237$	9.21953	0.598873
$238$	1.46081	0.0946903
$239$	13.0423	0.843634	0.421817	−	0.906681i	$-0.361393\pi$
$239$	13.0423	0.843634	0.421817	+	0.906681i	$0.361393\pi$
$240$	0	0
$241$	−18.6042	−1.19840	−0.599202	−	0.800598i	$-0.704515\pi$
$241$	−18.6042	−1.19840	−0.599202	+	0.800598i	$0.704515\pi$
$242$	0.539189	0.0346604
$243$	1.00000	0.0641500
$244$	12.7298	0.814942
$245$	0	0
$246$	−0.447480	−0.0285303
$247$	−10.5886	−0.673739
$248$	−10.4391	−0.662882
$249$	−15.0494	−0.953720
$250$	0	0
$251$	−2.21008	−0.139499	−0.0697495	−	0.997565i	$-0.522220\pi$
$251$	−2.21008	−0.139499	−0.0697495	+	0.997565i	$0.522220\pi$
$252$	1.70928	0.107674
$253$	−1.63090	−0.102534
$254$	−3.85043	−0.241598
$255$	0	0
$256$	−2.52359	−0.157724
$257$	−1.20516	−0.0751758	−0.0375879	−	0.999293i	$-0.511967\pi$
$257$	−1.20516	−0.0751758	−0.0375879	+	0.999293i	$0.511967\pi$
$258$	−0.673158	−0.0419090
$259$	3.17009	0.196980
$260$	0	0
$261$	1.24846	0.0772780
$262$	−1.12783	−0.0696773
$263$	−11.8432	−0.730286	−0.365143	−	0.930951i	$-0.618980\pi$
$263$	−11.8432	−0.730286	−0.365143	+	0.930951i	$0.618980\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	1.80098	0.110425
$267$	−11.4319	−0.699620
$268$	3.41855	0.208821
$269$	−23.5802	−1.43771	−0.718856	−	0.695159i	$-0.755334\pi$
$269$	−23.5802	−1.43771	−0.718856	+	0.695159i	$0.755334\pi$
$270$	0	0
$271$	−9.99773	−0.607319	−0.303660	−	0.952781i	$-0.598209\pi$
$271$	−9.99773	−0.607319	−0.303660	+	0.952781i	$0.598209\pi$
$272$	−6.34017	−0.384429
$273$	−3.17009	−0.191862
$274$	6.08452	0.367579
$275$	0	0
$276$	−2.78765	−0.167797
$277$	29.8660	1.79448	0.897238	−	0.441547i	$-0.145570\pi$
$277$	29.8660	1.79448	0.897238	+	0.441547i	$0.145570\pi$
$278$	10.7792	0.646496
$279$	5.21953	0.312485
$280$	0	0
$281$	−17.8660	−1.06580	−0.532899	−	0.846179i	$-0.678897\pi$
$281$	−17.8660	−1.06580	−0.532899	+	0.846179i	$0.678897\pi$
$282$	2.63090	0.156668
$283$	−4.56198	−0.271181	−0.135591	−	0.990765i	$-0.543293\pi$
$283$	−4.56198	−0.271181	−0.135591	+	0.990765i	$0.543293\pi$
$284$	19.9421	1.18335
$285$	0	0
$286$	−1.70928	−0.101072
$287$	0.829914	0.0489882
$288$	5.26180	0.310054
$289$	−9.65983	−0.568225
$290$	0	0
$291$	−4.32684	−0.253644
$292$	−11.8310	−0.692354
$293$	−15.0144	−0.877149	−0.438575	−	0.898695i	$-0.644517\pi$
$293$	−15.0144	−0.877149	−0.438575	+	0.898695i	$0.644517\pi$
$294$	0.539189	0.0314461
$295$	0	0
$296$	6.34017	0.368515
$297$	−1.00000	−0.0580259
$298$	5.39189	0.312344
$299$	5.17009	0.298994
$300$	0	0
$301$	1.24846	0.0719603
$302$	−1.75872	−0.101203
$303$	2.68035	0.153982
$304$	−7.81658	−0.448312
$305$	0	0
$306$	−1.46081	−0.0835090
$307$	20.6537	1.17877	0.589384	−	0.807853i	$-0.299371\pi$
$307$	20.6537	1.17877	0.589384	+	0.807853i	$0.299371\pi$
$308$	−1.70928	−0.0973950
$309$	−9.00719	−0.512401
$310$	0	0
$311$	−23.1506	−1.31275	−0.656375	−	0.754434i	$-0.727911\pi$
$311$	−23.1506	−1.31275	−0.656375	+	0.754434i	$0.727911\pi$
$312$	−6.34017	−0.358942
$313$	25.9516	1.46687	0.733435	−	0.679759i	$-0.237916\pi$
$313$	25.9516	1.46687	0.733435	+	0.679759i	$0.237916\pi$
$314$	−4.34736	−0.245336
$315$	0	0
$316$	−15.7587	−0.886497
$317$	−7.06997	−0.397089	−0.198544	−	0.980092i	$-0.563621\pi$
$317$	−7.06997	−0.397089	−0.198544	+	0.980092i	$0.563621\pi$
$318$	−3.19287	−0.179047
$319$	−1.24846	−0.0699006
$320$	0	0
$321$	−9.80098	−0.547038
$322$	−0.879362	−0.0490049
$323$	9.04945	0.503525
$324$	−1.70928	−0.0949597
$325$	0	0
$326$	1.70928	0.0946680
$327$	−12.2979	−0.680076
$328$	1.65983	0.0916486
$329$	−4.87936	−0.269008
$330$	0	0
$331$	1.26794	0.0696922	0.0348461	−	0.999393i	$-0.488906\pi$
$331$	1.26794	0.0696922	0.0348461	+	0.999393i	$0.488906\pi$
$332$	25.7237	1.41177
$333$	−3.17009	−0.173720
$334$	2.26794	0.124096
$335$	0	0
$336$	−2.34017	−0.127667
$337$	−2.51026	−0.136743	−0.0683713	−	0.997660i	$-0.521780\pi$
$337$	−2.51026	−0.136743	−0.0683713	+	0.997660i	$0.521780\pi$
$338$	−1.59090	−0.0865338
$339$	−12.8660	−0.698787
$340$	0	0
$341$	−5.21953	−0.282654
$342$	−1.80098	−0.0973861
$343$	−1.00000	−0.0539949
$344$	2.49693	0.134625
$345$	0	0
$346$	1.43293	0.0770346
$347$	−1.68649	−0.0905355	−0.0452677	−	0.998975i	$-0.514414\pi$
$347$	−1.68649	−0.0905355	−0.0452677	+	0.998975i	$0.514414\pi$
$348$	−2.13397	−0.114393
$349$	−33.5113	−1.79382	−0.896909	−	0.442214i	$-0.854193\pi$
$349$	−33.5113	−1.79382	−0.896909	+	0.442214i	$0.854193\pi$
$350$	0	0
$351$	3.17009	0.169207
$352$	−5.26180	−0.280455
$353$	−1.86991	−0.0995251	−0.0497625	−	0.998761i	$-0.515846\pi$
$353$	−1.86991	−0.0995251	−0.0497625	+	0.998761i	$0.515846\pi$
$354$	−2.40522	−0.127836
$355$	0	0
$356$	19.5402	1.03563
$357$	2.70928	0.143390
$358$	10.4429	0.551927
$359$	1.98667	0.104852	0.0524262	−	0.998625i	$-0.483305\pi$
$359$	1.98667	0.104852	0.0524262	+	0.998625i	$0.483305\pi$
$360$	0	0
$361$	−7.84324	−0.412802
$362$	−5.77310	−0.303427
$363$	1.00000	0.0524864
$364$	5.41855	0.284009
$365$	0	0
$366$	−4.01560	−0.209899
$367$	29.0300	1.51535	0.757676	−	0.652631i	$-0.226335\pi$
$367$	29.0300	1.51535	0.757676	+	0.652631i	$0.226335\pi$
$368$	3.81658	0.198953
$369$	−0.829914	−0.0432036
$370$	0	0
$371$	5.92162	0.307435
$372$	−8.92162	−0.462565
$373$	21.6381	1.12038	0.560189	−	0.828365i	$-0.310729\pi$
$373$	21.6381	1.12038	0.560189	+	0.828365i	$0.310729\pi$
$374$	1.46081	0.0755367
$375$	0	0
$376$	−9.75872	−0.503268
$377$	3.95774	0.203834
$378$	−0.539189	−0.0277329
$379$	11.4163	0.586415	0.293208	−	0.956049i	$-0.405277\pi$
$379$	11.4163	0.586415	0.293208	+	0.956049i	$0.405277\pi$
$380$	0	0
$381$	−7.14116	−0.365853
$382$	−10.0410	−0.513744
$383$	17.8010	0.909588	0.454794	−	0.890597i	$-0.349713\pi$
$383$	17.8010	0.909588	0.454794	+	0.890597i	$0.349713\pi$
$384$	−11.5174	−0.587747
$385$	0	0
$386$	7.96880	0.405601
$387$	−1.24846	−0.0634630
$388$	7.39576	0.375463
$389$	17.7659	0.900767	0.450384	−	0.892835i	$-0.351287\pi$
$389$	17.7659	0.900767	0.450384	+	0.892835i	$0.351287\pi$
$390$	0	0
$391$	−4.41855	−0.223456
$392$	−2.00000	−0.101015
$393$	−2.09171	−0.105513
$394$	−10.2946	−0.518634
$395$	0	0
$396$	1.70928	0.0858943
$397$	−12.2907	−0.616854	−0.308427	−	0.951248i	$-0.599803\pi$
$397$	−12.2907	−0.616854	−0.308427	+	0.951248i	$0.599803\pi$
$398$	−13.1773	−0.660517
$399$	3.34017	0.167218
$400$	0	0
$401$	8.79872	0.439387	0.219693	−	0.975569i	$-0.429494\pi$
$401$	8.79872	0.439387	0.219693	+	0.975569i	$0.429494\pi$
$402$	−1.07838	−0.0537846
$403$	16.5464	0.824234
$404$	−4.58145	−0.227936
$405$	0	0
$406$	−0.673158	−0.0334083
$407$	3.17009	0.157135
$408$	5.41855	0.268258
$409$	7.65142	0.378338	0.189169	−	0.981945i	$-0.439421\pi$
$409$	7.65142	0.378338	0.189169	+	0.981945i	$0.439421\pi$
$410$	0	0
$411$	11.2846	0.556627
$412$	15.3958	0.758495
$413$	4.46081	0.219502
$414$	0.879362	0.0432183
$415$	0	0
$416$	16.6803	0.817821
$417$	19.9916	0.978992
$418$	1.80098	0.0880890
$419$	−35.8710	−1.75241	−0.876205	−	0.481938i	$-0.839933\pi$
$419$	−35.8710	−1.75241	−0.876205	+	0.481938i	$0.839933\pi$
$420$	0	0
$421$	−10.2351	−0.498830	−0.249415	−	0.968397i	$-0.580238\pi$
$421$	−10.2351	−0.498830	−0.249415	+	0.968397i	$0.580238\pi$
$422$	−12.0000	−0.584151
$423$	4.87936	0.237243
$424$	11.8432	0.575159
$425$	0	0
$426$	−6.29072	−0.304787
$427$	7.44748	0.360409
$428$	16.7526	0.809767
$429$	−3.17009	−0.153053
$430$	0	0
$431$	−29.9565	−1.44295	−0.721477	−	0.692438i	$-0.756537\pi$
$431$	−29.9565	−1.44295	−0.721477	+	0.692438i	$0.756537\pi$
$432$	2.34017	0.112592
$433$	−31.8264	−1.52948	−0.764740	−	0.644339i	$-0.777133\pi$
$433$	−31.8264	−1.52948	−0.764740	+	0.644339i	$0.777133\pi$
$434$	−2.81432	−0.135091
$435$	0	0
$436$	21.0205	1.00670
$437$	−5.44748	−0.260588
$438$	3.73206	0.178325
$439$	−13.7792	−0.657647	−0.328824	−	0.944391i	$-0.606652\pi$
$439$	−13.7792	−0.657647	−0.328824	+	0.944391i	$0.606652\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−4.63090	−0.220269
$443$	0.573039	0.0272259	0.0136129	−	0.999907i	$-0.495667\pi$
$443$	0.573039	0.0272259	0.0136129	+	0.999907i	$0.495667\pi$
$444$	5.41855	0.257153
$445$	0	0
$446$	−13.0400	−0.617462
$447$	10.0000	0.472984
$448$	1.84324	0.0870851
$449$	−14.2751	−0.673685	−0.336842	−	0.941561i	$-0.609359\pi$
$449$	−14.2751	−0.673685	−0.336842	+	0.941561i	$0.609359\pi$
$450$	0	0
$451$	0.829914	0.0390791
$452$	21.9916	1.03440
$453$	−3.26180	−0.153253
$454$	13.9844	0.656320
$455$	0	0
$456$	6.68035	0.312836
$457$	−12.9350	−0.605072	−0.302536	−	0.953138i	$-0.597833\pi$
$457$	−12.9350	−0.605072	−0.302536	+	0.953138i	$0.597833\pi$
$458$	0.505339	0.0236129
$459$	−2.70928	−0.126458
$460$	0	0
$461$	22.7070	1.05757	0.528785	−	0.848756i	$-0.322648\pi$
$461$	22.7070	1.05757	0.528785	+	0.848756i	$0.322648\pi$
$462$	0.539189	0.0250853
$463$	4.29914	0.199798	0.0998989	−	0.994998i	$-0.468148\pi$
$463$	4.29914	0.199798	0.0998989	+	0.994998i	$0.468148\pi$
$464$	2.92162	0.135633
$465$	0	0
$466$	9.42243	0.436485
$467$	24.8371	1.14932	0.574662	−	0.818391i	$-0.305134\pi$
$467$	24.8371	1.14932	0.574662	+	0.818391i	$0.305134\pi$
$468$	−5.41855	−0.250473
$469$	2.00000	0.0923514
$470$	0	0
$471$	−8.06278	−0.371513
$472$	8.92162	0.410651
$473$	1.24846	0.0574044
$474$	4.97107	0.228329
$475$	0	0
$476$	−4.63090	−0.212257
$477$	−5.92162	−0.271132
$478$	7.03224	0.321647
$479$	−19.9350	−0.910851	−0.455426	−	0.890274i	$-0.650513\pi$
$479$	−19.9350	−0.910851	−0.455426	+	0.890274i	$0.650513\pi$
$480$	0	0
$481$	−10.0494	−0.458215
$482$	−10.0312	−0.456908
$483$	−1.63090	−0.0742084
$484$	−1.70928	−0.0776943
$485$	0	0
$486$	0.539189	0.0244581
$487$	34.9048	1.58169	0.790844	−	0.612018i	$-0.209642\pi$
$487$	34.9048	1.58169	0.790844	+	0.612018i	$0.209642\pi$
$488$	14.8950	0.674263
$489$	3.17009	0.143356
$490$	0	0
$491$	28.6853	1.29455	0.647274	−	0.762257i	$-0.275909\pi$
$491$	28.6853	1.29455	0.647274	+	0.762257i	$0.275909\pi$
$492$	1.41855	0.0639532
$493$	−3.38243	−0.152337
$494$	−5.70928	−0.256872
$495$	0	0
$496$	12.2146	0.548452
$497$	11.6670	0.523337
$498$	−8.11450	−0.363619
$499$	15.4040	0.689578	0.344789	−	0.938680i	$-0.387951\pi$
$499$	15.4040	0.689578	0.344789	+	0.938680i	$0.387951\pi$
$500$	0	0
$501$	4.20620	0.187919
$502$	−1.19165	−0.0531860
$503$	5.49693	0.245096	0.122548	−	0.992463i	$-0.460893\pi$
$503$	5.49693	0.245096	0.122548	+	0.992463i	$0.460893\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	−0.879362	−0.0390924
$507$	−2.95055	−0.131039
$508$	12.2062	0.541563
$509$	24.8381	1.10093	0.550466	−	0.834858i	$-0.314450\pi$
$509$	24.8381	1.10093	0.550466	+	0.834858i	$0.314450\pi$
$510$	0	0
$511$	−6.92162	−0.306195
$512$	21.6742	0.957873
$513$	−3.34017	−0.147472
$514$	−0.649808	−0.0286618
$515$	0	0
$516$	2.13397	0.0939428
$517$	−4.87936	−0.214594
$518$	1.70928	0.0751012
$519$	2.65756	0.116654
$520$	0	0
$521$	−13.0300	−0.570854	−0.285427	−	0.958401i	$-0.592135\pi$
$521$	−13.0300	−0.570854	−0.285427	+	0.958401i	$0.592135\pi$
$522$	0.673158	0.0294633
$523$	21.6020	0.944588	0.472294	−	0.881441i	$-0.343426\pi$
$523$	21.6020	0.944588	0.472294	+	0.881441i	$0.343426\pi$
$524$	3.57531	0.156188
$525$	0	0
$526$	−6.38575	−0.278432
$527$	−14.1412	−0.615998
$528$	−2.34017	−0.101843
$529$	−20.3402	−0.884355
$530$	0	0
$531$	−4.46081	−0.193583
$532$	−5.70928	−0.247528
$533$	−2.63090	−0.113957
$534$	−6.16394	−0.266740
$535$	0	0
$536$	4.00000	0.172774
$537$	19.3679	0.835786
$538$	−12.7142	−0.548148
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−27.5285	−1.18354	−0.591772	−	0.806106i	$-0.701571\pi$
$541$	−27.5285	−1.18354	−0.591772	+	0.806106i	$0.701571\pi$
$542$	−5.39067	−0.231549
$543$	−10.7070	−0.459482
$544$	−14.2557	−0.611206
$545$	0	0
$546$	−1.70928	−0.0731502
$547$	37.8625	1.61889	0.809443	−	0.587199i	$-0.199769\pi$
$547$	37.8625	1.61889	0.809443	+	0.587199i	$0.199769\pi$
$548$	−19.2885	−0.823962
$549$	−7.44748	−0.317851
$550$	0	0
$551$	−4.17009	−0.177652
$552$	−3.26180	−0.138831
$553$	−9.21953	−0.392055
$554$	16.1034	0.684169
$555$	0	0
$556$	−34.1711	−1.44918
$557$	−5.86991	−0.248716	−0.124358	−	0.992237i	$-0.539687\pi$
$557$	−5.86991	−0.248716	−0.124358	+	0.992237i	$0.539687\pi$
$558$	2.81432	0.119139
$559$	−3.95774	−0.167395
$560$	0	0
$561$	2.70928	0.114386
$562$	−9.63317	−0.406351
$563$	−10.4163	−0.438994	−0.219497	−	0.975613i	$-0.570442\pi$
$563$	−10.4163	−0.438994	−0.219497	+	0.975613i	$0.570442\pi$
$564$	−8.34017	−0.351185
$565$	0	0
$566$	−2.45977	−0.103392
$567$	−1.00000	−0.0419961
$568$	23.3340	0.979074
$569$	−22.9532	−0.962248	−0.481124	−	0.876652i	$-0.659771\pi$
$569$	−22.9532	−0.962248	−0.481124	+	0.876652i	$0.659771\pi$
$570$	0	0
$571$	37.8154	1.58252	0.791262	−	0.611478i	$-0.209425\pi$
$571$	37.8154	1.58252	0.791262	+	0.611478i	$0.209425\pi$
$572$	5.41855	0.226561
$573$	−18.6225	−0.777966
$574$	0.447480	0.0186775
$575$	0	0
$576$	−1.84324	−0.0768019
$577$	32.7070	1.36161	0.680805	−	0.732464i	$-0.261630\pi$
$577$	32.7070	1.36161	0.680805	+	0.732464i	$0.261630\pi$
$578$	−5.20847	−0.216644
$579$	14.7792	0.614204
$580$	0	0
$581$	15.0494	0.624356
$582$	−2.33299	−0.0967053
$583$	5.92162	0.245249
$584$	−13.8432	−0.572838
$585$	0	0
$586$	−8.09558	−0.334426
$587$	−36.2979	−1.49818	−0.749088	−	0.662471i	$-0.769508\pi$
$587$	−36.2979	−1.49818	−0.749088	+	0.662471i	$0.769508\pi$
$588$	−1.70928	−0.0704893
$589$	−17.4341	−0.718362
$590$	0	0
$591$	−19.0928	−0.785371
$592$	−7.41855	−0.304901
$593$	2.78765	0.114475	0.0572376	−	0.998361i	$-0.481771\pi$
$593$	2.78765	0.114475	0.0572376	+	0.998361i	$0.481771\pi$
$594$	−0.539189	−0.0221232
$595$	0	0
$596$	−17.0928	−0.700146
$597$	−24.4391	−1.00022
$598$	2.78765	0.113996
$599$	−18.9627	−0.774793	−0.387397	−	0.921913i	$-0.626626\pi$
$599$	−18.9627	−0.774793	−0.387397	+	0.921913i	$0.626626\pi$
$600$	0	0
$601$	19.5464	0.797313	0.398657	−	0.917100i	$-0.369477\pi$
$601$	19.5464	0.797313	0.398657	+	0.917100i	$0.369477\pi$
$602$	0.673158	0.0274359
$603$	−2.00000	−0.0814463
$604$	5.57531	0.226856
$605$	0	0
$606$	1.44521	0.0587078
$607$	−10.3935	−0.421859	−0.210930	−	0.977501i	$-0.567649\pi$
$607$	−10.3935	−0.421859	−0.210930	+	0.977501i	$0.567649\pi$
$608$	−17.5753	−0.712773
$609$	−1.24846	−0.0505903
$610$	0	0
$611$	15.4680	0.625768
$612$	4.63090	0.187193
$613$	−1.65529	−0.0668566	−0.0334283	−	0.999441i	$-0.510643\pi$
$613$	−1.65529	−0.0668566	−0.0334283	+	0.999441i	$0.510643\pi$
$614$	11.1362	0.449422
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−28.4931	−1.14709	−0.573544	−	0.819175i	$-0.694432\pi$
$617$	−28.4931	−1.14709	−0.573544	+	0.819175i	$0.694432\pi$
$618$	−4.85658	−0.195360
$619$	25.8843	1.04038	0.520189	−	0.854051i	$-0.325862\pi$
$619$	25.8843	1.04038	0.520189	+	0.854051i	$0.325862\pi$
$620$	0	0
$621$	1.63090	0.0654457
$622$	−12.4826	−0.500505
$623$	11.4319	0.458009
$624$	7.41855	0.296980
$625$	0	0
$626$	13.9928	0.559265
$627$	3.34017	0.133394
$628$	13.7815	0.549942
$629$	8.58864	0.342451
$630$	0	0
$631$	39.3318	1.56577	0.782886	−	0.622165i	$-0.213747\pi$
$631$	39.3318	1.56577	0.782886	+	0.622165i	$0.213747\pi$
$632$	−18.4391	−0.733467
$633$	−22.2557	−0.884583
$634$	−3.81205	−0.151396
$635$	0	0
$636$	10.1217	0.401351
$637$	3.17009	0.125603
$638$	−0.673158	−0.0266506
$639$	−11.6670	−0.461540
$640$	0	0
$641$	−25.3028	−0.999402	−0.499701	−	0.866198i	$-0.666557\pi$
$641$	−25.3028	−0.999402	−0.499701	+	0.866198i	$0.666557\pi$
$642$	−5.28458	−0.208566
$643$	42.6053	1.68019	0.840094	−	0.542441i	$-0.182500\pi$
$643$	42.6053	1.68019	0.840094	+	0.542441i	$0.182500\pi$
$644$	2.78765	0.109849
$645$	0	0
$646$	4.87936	0.191976
$647$	−34.6369	−1.36172	−0.680858	−	0.732416i	$-0.738393\pi$
$647$	−34.6369	−1.36172	−0.680858	+	0.732416i	$0.738393\pi$
$648$	−2.00000	−0.0785674
$649$	4.46081	0.175102
$650$	0	0
$651$	−5.21953	−0.204570
$652$	−5.41855	−0.212207
$653$	−1.94828	−0.0762423	−0.0381211	−	0.999273i	$-0.512137\pi$
$653$	−1.94828	−0.0762423	−0.0381211	+	0.999273i	$0.512137\pi$
$654$	−6.63090	−0.259289
$655$	0	0
$656$	−1.94214	−0.0758279
$657$	6.92162	0.270038
$658$	−2.63090	−0.102563
$659$	−0.506384	−0.0197259	−0.00986296	−	0.999951i	$-0.503140\pi$
$659$	−0.506384	−0.0197259	−0.00986296	+	0.999951i	$0.503140\pi$
$660$	0	0
$661$	23.0316	0.895825	0.447912	−	0.894077i	$-0.352168\pi$
$661$	23.0316	0.895825	0.447912	+	0.894077i	$0.352168\pi$
$662$	0.683658	0.0265711
$663$	−8.58864	−0.333555
$664$	30.0989	1.16806
$665$	0	0
$666$	−1.70928	−0.0662331
$667$	2.03612	0.0788388
$668$	−7.18956	−0.278172
$669$	−24.1845	−0.935025
$670$	0	0
$671$	7.44748	0.287507
$672$	−5.26180	−0.202978
$673$	17.4280	0.671800	0.335900	−	0.941898i	$-0.390960\pi$
$673$	17.4280	0.671800	0.335900	+	0.941898i	$0.390960\pi$
$674$	−1.35350	−0.0521350
$675$	0	0
$676$	5.04331	0.193973
$677$	−13.0183	−0.500332	−0.250166	−	0.968203i	$-0.580485\pi$
$677$	−13.0183	−0.500332	−0.250166	+	0.968203i	$0.580485\pi$
$678$	−6.93722	−0.266422
$679$	4.32684	0.166049
$680$	0	0
$681$	25.9360	0.993870
$682$	−2.81432	−0.107766
$683$	−25.4764	−0.974828	−0.487414	−	0.873171i	$-0.662060\pi$
$683$	−25.4764	−0.974828	−0.487414	+	0.873171i	$0.662060\pi$
$684$	5.70928	0.218300
$685$	0	0
$686$	−0.539189	−0.0205863
$687$	0.937221	0.0357572
$688$	−2.92162	−0.111386
$689$	−18.7721	−0.715158
$690$	0	0
$691$	34.6635	1.31866	0.659331	−	0.751853i	$-0.270839\pi$
$691$	34.6635	1.31866	0.659331	+	0.751853i	$0.270839\pi$
$692$	−4.54250	−0.172680
$693$	1.00000	0.0379869
$694$	−0.909336	−0.0345179
$695$	0	0
$696$	−2.49693	−0.0946458
$697$	2.24846	0.0851667
$698$	−18.0689	−0.683919
$699$	17.4752	0.660972
$700$	0	0
$701$	15.4007	0.581676	0.290838	−	0.956772i	$-0.406066\pi$
$701$	15.4007	0.581676	0.290838	+	0.956772i	$0.406066\pi$
$702$	1.70928	0.0645124
$703$	10.5886	0.399358
$704$	1.84324	0.0694699
$705$	0	0
$706$	−1.00823	−0.0379453
$707$	−2.68035	−0.100805
$708$	7.62475	0.286556
$709$	−30.2123	−1.13465	−0.567324	−	0.823495i	$-0.692021\pi$
$709$	−30.2123	−1.13465	−0.567324	+	0.823495i	$0.692021\pi$
$710$	0	0
$711$	9.21953	0.345760
$712$	22.8638	0.856856
$713$	8.51253	0.318797
$714$	1.46081	0.0546695
$715$	0	0
$716$	−33.1050	−1.23719
$717$	13.0423	0.487072
$718$	1.07119	0.0399764
$719$	35.7514	1.33330	0.666650	−	0.745371i	$-0.267727\pi$
$719$	35.7514	1.33330	0.666650	+	0.745371i	$0.267727\pi$
$720$	0	0
$721$	9.00719	0.335445
$722$	−4.22899	−0.157387
$723$	−18.6042	−0.691899
$724$	18.3012	0.680160
$725$	0	0
$726$	0.539189	0.0200112
$727$	22.6319	0.839372	0.419686	−	0.907669i	$-0.362140\pi$
$727$	22.6319	0.839372	0.419686	+	0.907669i	$0.362140\pi$
$728$	6.34017	0.234982
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.38243	0.125104
$732$	12.7298	0.470507
$733$	27.8987	1.03046	0.515230	−	0.857052i	$-0.327706\pi$
$733$	27.8987	1.03046	0.515230	+	0.857052i	$0.327706\pi$
$734$	15.6526	0.577749
$735$	0	0
$736$	8.58145	0.316316
$737$	2.00000	0.0736709
$738$	−0.447480	−0.0164720
$739$	3.60650	0.132667	0.0663337	−	0.997797i	$-0.478870\pi$
$739$	3.60650	0.132667	0.0663337	+	0.997797i	$0.478870\pi$
$740$	0	0
$741$	−10.5886	−0.388983
$742$	3.19287	0.117214
$743$	−7.83096	−0.287290	−0.143645	−	0.989629i	$-0.545882\pi$
$743$	−7.83096	−0.287290	−0.143645	+	0.989629i	$0.545882\pi$
$744$	−10.4391	−0.382715
$745$	0	0
$746$	11.6670	0.427160
$747$	−15.0494	−0.550631
$748$	−4.63090	−0.169322
$749$	9.80098	0.358120
$750$	0	0
$751$	−19.6803	−0.718146	−0.359073	−	0.933309i	$-0.616907\pi$
$751$	−19.6803	−0.718146	−0.359073	+	0.933309i	$0.616907\pi$
$752$	11.4186	0.416392
$753$	−2.21008	−0.0805398
$754$	2.13397	0.0777146
$755$	0	0
$756$	1.70928	0.0621657
$757$	8.11327	0.294882	0.147441	−	0.989071i	$-0.452896\pi$
$757$	8.11327	0.294882	0.147441	+	0.989071i	$0.452896\pi$
$758$	6.15553	0.223579
$759$	−1.63090	−0.0591978
$760$	0	0
$761$	12.2967	0.445755	0.222877	−	0.974846i	$-0.428455\pi$
$761$	12.2967	0.445755	0.222877	+	0.974846i	$0.428455\pi$
$762$	−3.85043	−0.139486
$763$	12.2979	0.445214
$764$	31.8310	1.15160
$765$	0	0
$766$	9.59809	0.346793
$767$	−14.1412	−0.510608
$768$	−2.52359	−0.0910622
$769$	−28.3028	−1.02063	−0.510313	−	0.859989i	$-0.670470\pi$
$769$	−28.3028	−1.02063	−0.510313	+	0.859989i	$0.670470\pi$
$770$	0	0
$771$	−1.20516	−0.0434027
$772$	−25.2618	−0.909192
$773$	54.9914	1.97790	0.988952	−	0.148237i	$-0.0473600\pi$
$773$	54.9914	1.97790	0.988952	+	0.148237i	$0.0473600\pi$
$774$	−0.673158	−0.0241962
$775$	0	0
$776$	8.65368	0.310649
$777$	3.17009	0.113726
$778$	9.57918	0.343430
$779$	2.77205	0.0993192
$780$	0	0
$781$	11.6670	0.417479
$782$	−2.38243	−0.0851956
$783$	1.24846	0.0446165
$784$	2.34017	0.0835776
$785$	0	0
$786$	−1.12783	−0.0402282
$787$	−11.3028	−0.402902	−0.201451	−	0.979499i	$-0.564566\pi$
$787$	−11.3028	−0.402902	−0.201451	+	0.979499i	$0.564566\pi$
$788$	32.6348	1.16257
$789$	−11.8432	−0.421631
$790$	0	0
$791$	12.8660	0.457463
$792$	2.00000	0.0710669
$793$	−23.6092	−0.838386
$794$	−6.62702	−0.235184
$795$	0	0
$796$	41.7731	1.48061
$797$	−52.5380	−1.86099	−0.930495	−	0.366304i	$-0.880623\pi$
$797$	−52.5380	−1.86099	−0.930495	+	0.366304i	$0.880623\pi$
$798$	1.80098	0.0637541
$799$	−13.2195	−0.467674
$800$	0	0
$801$	−11.4319	−0.403926
$802$	4.74417	0.167522
$803$	−6.92162	−0.244259
$804$	3.41855	0.120563
$805$	0	0
$806$	8.92162	0.314251
$807$	−23.5802	−0.830063
$808$	−5.36069	−0.188588
$809$	21.6163	0.759990	0.379995	−	0.924989i	$-0.375926\pi$
$809$	21.6163	0.759990	0.379995	+	0.924989i	$0.375926\pi$
$810$	0	0
$811$	27.9611	0.981845	0.490923	−	0.871203i	$-0.336660\pi$
$811$	27.9611	0.981845	0.490923	+	0.871203i	$0.336660\pi$
$812$	2.13397	0.0748876
$813$	−9.99773	−0.350636
$814$	1.70928	0.0599101
$815$	0	0
$816$	−6.34017	−0.221950
$817$	4.17009	0.145893
$818$	4.12556	0.144247
$819$	−3.17009	−0.110772
$820$	0	0
$821$	25.0710	0.874984	0.437492	−	0.899222i	$-0.355867\pi$
$821$	25.0710	0.874984	0.437492	+	0.899222i	$0.355867\pi$
$822$	6.08452	0.212222
$823$	51.6502	1.80041	0.900206	−	0.435464i	$-0.143416\pi$
$823$	51.6502	1.80041	0.900206	+	0.435464i	$0.143416\pi$
$824$	18.0144	0.627561
$825$	0	0
$826$	2.40522	0.0836883
$827$	−45.2039	−1.57189	−0.785947	−	0.618293i	$-0.787824\pi$
$827$	−45.2039	−1.57189	−0.785947	+	0.618293i	$0.787824\pi$
$828$	−2.78765	−0.0968776
$829$	7.88428	0.273832	0.136916	−	0.990583i	$-0.456281\pi$
$829$	7.88428	0.273832	0.136916	+	0.990583i	$0.456281\pi$
$830$	0	0
$831$	29.8660	1.03604
$832$	−5.84324	−0.202578
$833$	−2.70928	−0.0938708
$834$	10.7792	0.373255
$835$	0	0
$836$	−5.70928	−0.197459
$837$	5.21953	0.180414
$838$	−19.3412	−0.668131
$839$	27.4729	0.948471	0.474235	−	0.880398i	$-0.342725\pi$
$839$	27.4729	0.948471	0.474235	+	0.880398i	$0.342725\pi$
$840$	0	0
$841$	−27.4413	−0.946253
$842$	−5.51867	−0.190186
$843$	−17.8660	−0.615339
$844$	38.0410	1.30943
$845$	0	0
$846$	2.63090	0.0904521
$847$	−1.00000	−0.0343604
$848$	−13.8576	−0.475873
$849$	−4.56198	−0.156567
$850$	0	0
$851$	−5.17009	−0.177228
$852$	19.9421	0.683206
$853$	2.85658	0.0978073	0.0489036	−	0.998804i	$-0.484427\pi$
$853$	2.85658	0.0978073	0.0489036	+	0.998804i	$0.484427\pi$
$854$	4.01560	0.137411
$855$	0	0
$856$	19.6020	0.669981
$857$	−20.4079	−0.697120	−0.348560	−	0.937287i	$-0.613329\pi$
$857$	−20.4079	−0.697120	−0.348560	+	0.937287i	$0.613329\pi$
$858$	−1.70928	−0.0583537
$859$	−34.7514	−1.18570	−0.592851	−	0.805313i	$-0.701998\pi$
$859$	−34.7514	−1.18570	−0.592851	+	0.805313i	$0.701998\pi$
$860$	0	0
$861$	0.829914	0.0282834
$862$	−16.1522	−0.550147
$863$	50.6619	1.72455	0.862276	−	0.506439i	$-0.169038\pi$
$863$	50.6619	1.72455	0.862276	+	0.506439i	$0.169038\pi$
$864$	5.26180	0.179010
$865$	0	0
$866$	−17.1605	−0.583136
$867$	−9.65983	−0.328065
$868$	8.92162	0.302820
$869$	−9.21953	−0.312751
$870$	0	0
$871$	−6.34017	−0.214829
$872$	24.5958	0.832920
$873$	−4.32684	−0.146441
$874$	−2.93722	−0.0993530
$875$	0	0
$876$	−11.8310	−0.399731
$877$	37.0027	1.24949	0.624745	−	0.780829i	$-0.285203\pi$
$877$	37.0027	1.24949	0.624745	+	0.780829i	$0.285203\pi$
$878$	−7.42961	−0.250737
$879$	−15.0144	−0.506422
$880$	0	0
$881$	−12.0445	−0.405790	−0.202895	−	0.979200i	$-0.565035\pi$
$881$	−12.0445	−0.405790	−0.202895	+	0.979200i	$0.565035\pi$
$882$	0.539189	0.0181554
$883$	−52.7358	−1.77470	−0.887350	−	0.461097i	$-0.847456\pi$
$883$	−52.7358	−1.77470	−0.887350	+	0.461097i	$0.847456\pi$
$884$	14.6803	0.493753
$885$	0	0
$886$	0.308976	0.0103803
$887$	14.1362	0.474648	0.237324	−	0.971431i	$-0.423730\pi$
$887$	14.1362	0.474648	0.237324	+	0.971431i	$0.423730\pi$
$888$	6.34017	0.212762
$889$	7.14116	0.239507
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	41.3379	1.38410
$893$	−16.2979	−0.545389
$894$	5.39189	0.180332
$895$	0	0
$896$	11.5174	0.384771
$897$	5.17009	0.172624
$898$	−7.69699	−0.256852
$899$	6.51640	0.217334
$900$	0	0
$901$	16.0433	0.534480
$902$	0.447480	0.0148995
$903$	1.24846	0.0415463
$904$	25.7321	0.855836
$905$	0	0
$906$	−1.75872	−0.0584297
$907$	6.47745	0.215080	0.107540	−	0.994201i	$-0.465703\pi$
$907$	6.47745	0.215080	0.107540	+	0.994201i	$0.465703\pi$
$908$	−44.3318	−1.47120
$909$	2.68035	0.0889015
$910$	0	0
$911$	22.5380	0.746716	0.373358	−	0.927687i	$-0.378206\pi$
$911$	22.5380	0.746716	0.373358	+	0.927687i	$0.378206\pi$
$912$	−7.81658	−0.258833
$913$	15.0494	0.498064
$914$	−6.97438	−0.230692
$915$	0	0
$916$	−1.60197	−0.0529305
$917$	2.09171	0.0690743
$918$	−1.46081	−0.0482140
$919$	−21.1773	−0.698574	−0.349287	−	0.937016i	$-0.613576\pi$
$919$	−21.1773	−0.698574	−0.349287	+	0.937016i	$0.613576\pi$
$920$	0	0
$921$	20.6537	0.680562
$922$	12.2434	0.403214
$923$	−36.9854	−1.21739
$924$	−1.70928	−0.0562310
$925$	0	0
$926$	2.31805	0.0761757
$927$	−9.00719	−0.295835
$928$	6.56916	0.215643
$929$	24.4352	0.801693	0.400846	−	0.916145i	$-0.368716\pi$
$929$	24.4352	0.801693	0.400846	+	0.916145i	$0.368716\pi$
$930$	0	0
$931$	−3.34017	−0.109470
$932$	−29.8699	−0.978421
$933$	−23.1506	−0.757917
$934$	13.3919	0.438196
$935$	0	0
$936$	−6.34017	−0.207235
$937$	18.3063	0.598042	0.299021	−	0.954247i	$-0.403340\pi$
$937$	18.3063	0.598042	0.299021	+	0.954247i	$0.403340\pi$
$938$	1.07838	0.0352103
$939$	25.9516	0.846898
$940$	0	0
$941$	−26.7598	−0.872344	−0.436172	−	0.899863i	$-0.643666\pi$
$941$	−26.7598	−0.872344	−0.436172	+	0.899863i	$0.643666\pi$
$942$	−4.34736	−0.141645
$943$	−1.35350	−0.0440762
$944$	−10.4391	−0.339763
$945$	0	0
$946$	0.673158	0.0218863
$947$	−54.8948	−1.78384	−0.891920	−	0.452192i	$-0.850642\pi$
$947$	−54.8948	−1.78384	−0.891920	+	0.452192i	$0.850642\pi$
$948$	−15.7587	−0.511820
$949$	21.9421	0.712272
$950$	0	0
$951$	−7.06997	−0.229259
$952$	−5.41855	−0.175616
$953$	26.5646	0.860513	0.430256	−	0.902707i	$-0.358423\pi$
$953$	26.5646	0.860513	0.430256	+	0.902707i	$0.358423\pi$
$954$	−3.19287	−0.103373
$955$	0	0
$956$	−22.2928	−0.721001
$957$	−1.24846	−0.0403571
$958$	−10.7487	−0.347275
$959$	−11.2846	−0.364398
$960$	0	0
$961$	−3.75646	−0.121176
$962$	−5.41855	−0.174701
$963$	−9.80098	−0.315832
$964$	31.7998	1.02420
$965$	0	0
$966$	−0.879362	−0.0282930
$967$	29.4196	0.946070	0.473035	−	0.881044i	$-0.343158\pi$
$967$	29.4196	0.946070	0.473035	+	0.881044i	$0.343158\pi$
$968$	−2.00000	−0.0642824
$969$	9.04945	0.290710
$970$	0	0
$971$	−38.4668	−1.23446	−0.617229	−	0.786784i	$-0.711745\pi$
$971$	−38.4668	−1.23446	−0.617229	+	0.786784i	$0.711745\pi$
$972$	−1.70928	−0.0548250
$973$	−19.9916	−0.640901
$974$	18.8203	0.603041
$975$	0	0
$976$	−17.4284	−0.557869
$977$	−14.7464	−0.471780	−0.235890	−	0.971780i	$-0.575801\pi$
$977$	−14.7464	−0.471780	−0.235890	+	0.971780i	$0.575801\pi$
$978$	1.70928	0.0546566
$979$	11.4319	0.365365
$980$	0	0
$981$	−12.2979	−0.392642
$982$	15.4668	0.493565
$983$	47.7152	1.52188	0.760940	−	0.648822i	$-0.224738\pi$
$983$	47.7152	1.52188	0.760940	+	0.648822i	$0.224738\pi$
$984$	1.65983	0.0529134
$985$	0	0
$986$	−1.82377	−0.0580807
$987$	−4.87936	−0.155312
$988$	18.0989	0.575803
$989$	−2.03612	−0.0647448
$990$	0	0
$991$	21.9893	0.698514	0.349257	−	0.937027i	$-0.386434\pi$
$991$	21.9893	0.698514	0.349257	+	0.937027i	$0.386434\pi$
$992$	27.4641	0.871987
$993$	1.26794	0.0402368
$994$	6.29072	0.199530
$995$	0	0
$996$	25.7237	0.815085
$997$	26.4657	0.838178	0.419089	−	0.907945i	$-0.362350\pi$
$997$	26.4657	0.838178	0.419089	+	0.907945i	$0.362350\pi$
$998$	8.30566	0.262911
$999$	−3.17009	−0.100297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bu.1.2		3
5.2	odd	4		1155.2.c.c.694.4	yes	6
5.3	odd	4		1155.2.c.c.694.3	✓	6
5.4	even	2		5775.2.a.bt.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.c.c.694.3	✓	6	5.3	odd	4
1155.2.c.c.694.4	yes	6	5.2	odd	4
5775.2.a.bt.1.2		3	5.4	even	2
5775.2.a.bu.1.2		3	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:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q+0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} +0.539189 q^{6} -1.00000 q^{7} -2.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} +0.539189 q^{6} -1.00000 q^{7} -2.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.70928 q^{12} +3.17009 q^{13} -0.539189 q^{14} +2.34017 q^{16} -2.70928 q^{17} +0.539189 q^{18} -3.34017 q^{19} -1.00000 q^{21} -0.539189 q^{22} +1.63090 q^{23} -2.00000 q^{24} +1.70928 q^{26} +1.00000 q^{27} +1.70928 q^{28} +1.24846 q^{29} +5.21953 q^{31} +5.26180 q^{32} -1.00000 q^{33} -1.46081 q^{34} -1.70928 q^{36} -3.17009 q^{37} -1.80098 q^{38} +3.17009 q^{39} -0.829914 q^{41} -0.539189 q^{42} -1.24846 q^{43} +1.70928 q^{44} +0.879362 q^{46} +4.87936 q^{47} +2.34017 q^{48} +1.00000 q^{49} -2.70928 q^{51} -5.41855 q^{52} -5.92162 q^{53} +0.539189 q^{54} +2.00000 q^{56} -3.34017 q^{57} +0.673158 q^{58} -4.46081 q^{59} -7.44748 q^{61} +2.81432 q^{62} -1.00000 q^{63} -1.84324 q^{64} -0.539189 q^{66} -2.00000 q^{67} +4.63090 q^{68} +1.63090 q^{69} -11.6670 q^{71} -2.00000 q^{72} +6.92162 q^{73} -1.70928 q^{74} +5.70928 q^{76} +1.00000 q^{77} +1.70928 q^{78} +9.21953 q^{79} +1.00000 q^{81} -0.447480 q^{82} -15.0494 q^{83} +1.70928 q^{84} -0.673158 q^{86} +1.24846 q^{87} +2.00000 q^{88} -11.4319 q^{89} -3.17009 q^{91} -2.78765 q^{92} +5.21953 q^{93} +2.63090 q^{94} +5.26180 q^{96} -4.32684 q^{97} +0.539189 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 2 q^{4} - 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^7 - 6 * q^8 + 3 * q^9	\( 3 q + 3 q^{3} + 2 q^{4} - 3 q^{7} - 6 q^{8} + 3 q^{9} - 3 q^{11} + 2 q^{12} + 4 q^{13} - 4 q^{16} - q^{17} + q^{19} - 3 q^{21} + q^{23} - 6 q^{24} - 2 q^{26} + 3 q^{27} - 2 q^{28} - 5 q^{29} - 8 q^{31} + 8 q^{32} - 3 q^{33} - 6 q^{34} + 2 q^{36} - 4 q^{37} + 4 q^{38} + 4 q^{39} - 8 q^{41} + 5 q^{43} - 2 q^{44} - 10 q^{46} + 2 q^{47} - 4 q^{48} + 3 q^{49} - q^{51} - 2 q^{52} - 21 q^{53} + 6 q^{56} + q^{57} + 14 q^{58} - 15 q^{59} - 23 q^{61} - 3 q^{63} - 12 q^{64} - 6 q^{67} + 10 q^{68} + q^{69} - 12 q^{71} - 6 q^{72} + 24 q^{73} + 2 q^{74} + 10 q^{76} + 3 q^{77} - 2 q^{78} + 4 q^{79} + 3 q^{81} - 2 q^{82} - 27 q^{83} - 2 q^{84} - 14 q^{86} - 5 q^{87} + 6 q^{88} - 21 q^{89} - 4 q^{91} + 2 q^{92} - 8 q^{93} + 4 q^{94} + 8 q^{96} - q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^7 - 6 * q^8 + 3 * q^9 - 3 * q^11 + 2 * q^12 + 4 * q^13 - 4 * q^16 - q^17 + q^19 - 3 * q^21 + q^23 - 6 * q^24 - 2 * q^26 + 3 * q^27 - 2 * q^28 - 5 * q^29 - 8 * q^31 + 8 * q^32 - 3 * q^33 - 6 * q^34 + 2 * q^36 - 4 * q^37 + 4 * q^38 + 4 * q^39 - 8 * q^41 + 5 * q^43 - 2 * q^44 - 10 * q^46 + 2 * q^47 - 4 * q^48 + 3 * q^49 - q^51 - 2 * q^52 - 21 * q^53 + 6 * q^56 + q^57 + 14 * q^58 - 15 * q^59 - 23 * q^61 - 3 * q^63 - 12 * q^64 - 6 * q^67 + 10 * q^68 + q^69 - 12 * q^71 - 6 * q^72 + 24 * q^73 + 2 * q^74 + 10 * q^76 + 3 * q^77 - 2 * q^78 + 4 * q^79 + 3 * q^81 - 2 * q^82 - 27 * q^83 - 2 * q^84 - 14 * q^86 - 5 * q^87 + 6 * q^88 - 21 * q^89 - 4 * q^91 + 2 * q^92 - 8 * q^93 + 4 * q^94 + 8 * q^96 - q^97 - 3 * q^99

Introduction

L-functions

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Database

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