LMFDB - Embedded newform 765.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [765,2,Mod(1,765)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(765, 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("765.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	765.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:	$6.10855575463$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	765.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66908 q^{2} +5.12398 q^{4} -1.00000 q^{5} +0.454904 q^{7} +8.33816 q^{8} -2.66908 q^{10} -1.54510 q^{11} +2.00000 q^{13} +1.21417 q^{14} +12.0072 q^{16} +1.00000 q^{17} +3.54510 q^{19} -5.12398 q^{20} -4.12398 q^{22} +1.00000 q^{25} +5.33816 q^{26} +2.33092 q^{28} -9.79306 q^{29} -0.909808 q^{31} +15.3719 q^{32} +2.66908 q^{34} -0.454904 q^{35} -7.79306 q^{37} +9.46214 q^{38} -8.33816 q^{40} +6.70287 q^{41} -8.67632 q^{43} -7.91705 q^{44} -12.0410 q^{47} -6.79306 q^{49} +2.66908 q^{50} +10.2480 q^{52} -0.883254 q^{53} +1.54510 q^{55} +3.79306 q^{56} -26.1385 q^{58} +3.09019 q^{59} +9.58612 q^{61} -2.42835 q^{62} +17.0145 q^{64} -2.00000 q^{65} +15.7665 q^{67} +5.12398 q^{68} -1.21417 q^{70} -13.5861 q^{71} +1.11675 q^{73} -20.8003 q^{74} +18.1650 q^{76} -0.702870 q^{77} +8.00000 q^{79} -12.0072 q^{80} +17.8905 q^{82} +5.81962 q^{83} -1.00000 q^{85} -23.1578 q^{86} -12.8833 q^{88} -13.7665 q^{89} +0.909808 q^{91} -32.1385 q^{94} -3.54510 q^{95} +15.5861 q^{97} -18.1312 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{5} + 9 q^{8} - 6 q^{11} + 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{17} + 12 q^{19} - 6 q^{20} - 3 q^{22} + 3 q^{25} + 15 q^{28} - 12 q^{29} + 18 q^{32} - 6 q^{37} + 3 q^{38} - 9 q^{40}+ \cdots - 21 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^5 + 9 * q^8 - 6 * q^11 + 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^17 + 12 * q^19 - 6 * q^20 - 3 * q^22 + 3 * q^25 + 15 * q^28 - 12 * q^29 + 18 * q^32 - 6 * q^37 + 3 * q^38 - 9 * q^40 + 6 * q^43 + 3 * q^44 - 3 * q^49 + 12 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 21 * q^58 + 12 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 - 6 * q^65 + 18 * q^67 + 6 * q^68 + 3 * q^70 - 6 * q^71 + 18 * q^73 - 21 * q^74 + 9 * q^76 + 18 * q^77 + 24 * q^79 - 12 * q^80 + 15 * q^82 + 12 * q^83 - 3 * q^85 - 48 * q^86 - 24 * q^88 - 12 * q^89 - 39 * q^94 - 12 * q^95 + 12 * q^97 - 21 * q^98

Coefficient data

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

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

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

$2$	2.66908	1.88732	0.943662	−	0.330911i	$-0.107356\pi$
$2$	2.66908	1.88732	0.943662	+	0.330911i	$0.107356\pi$
$3$	0	0
$3$	0	0
$4$	5.12398	2.56199
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.454904	0.171938	0.0859688	−	0.996298i	$-0.472601\pi$
$7$	0.454904	0.171938	0.0859688	+	0.996298i	$0.472601\pi$
$8$	8.33816	2.94798
$9$	0	0
$10$	−2.66908	−0.844037
$11$	−1.54510	−0.465864	−0.232932	−	0.972493i	$-0.574832\pi$
$11$	−1.54510	−0.465864	−0.232932	+	0.972493i	$0.574832\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.21417	0.324502
$15$	0	0
$16$	12.0072	3.00181
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.54510	0.813301	0.406650	−	0.913584i	$-0.366697\pi$
$19$	3.54510	0.813301	0.406650	+	0.913584i	$0.366697\pi$
$20$	−5.12398	−1.14576
$21$	0	0
$22$	−4.12398	−0.879236
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.33816	1.04690
$27$	0	0
$28$	2.33092	0.440503
$29$	−9.79306	−1.81853	−0.909263	−	0.416222i	$-0.863354\pi$
$29$	−9.79306	−1.81853	−0.909263	+	0.416222i	$0.863354\pi$
$30$	0	0
$31$	−0.909808	−0.163406	−0.0817032	−	0.996657i	$-0.526036\pi$
$31$	−0.909808	−0.163406	−0.0817032	+	0.996657i	$0.526036\pi$
$32$	15.3719	2.71740
$33$	0	0
$34$	2.66908	0.457743
$35$	−0.454904	−0.0768928
$36$	0	0
$37$	−7.79306	−1.28117	−0.640586	−	0.767887i	$-0.721308\pi$
$37$	−7.79306	−1.28117	−0.640586	+	0.767887i	$0.721308\pi$
$38$	9.46214	1.53496
$39$	0	0
$40$	−8.33816	−1.31838
$41$	6.70287	1.04681	0.523406	−	0.852083i	$-0.324661\pi$
$41$	6.70287	1.04681	0.523406	+	0.852083i	$0.324661\pi$
$42$	0	0
$43$	−8.67632	−1.32313	−0.661563	−	0.749890i	$-0.730107\pi$
$43$	−8.67632	−1.32313	−0.661563	+	0.749890i	$0.730107\pi$
$44$	−7.91705	−1.19354
$45$	0	0
$46$	0	0
$47$	−12.0410	−1.75636	−0.878182	−	0.478326i	$-0.841244\pi$
$47$	−12.0410	−1.75636	−0.878182	+	0.478326i	$0.841244\pi$
$48$	0	0
$49$	−6.79306	−0.970437
$50$	2.66908	0.377465
$51$	0	0
$52$	10.2480	1.42114
$53$	−0.883254	−0.121324	−0.0606621	−	0.998158i	$-0.519321\pi$
$53$	−0.883254	−0.121324	−0.0606621	+	0.998158i	$0.519321\pi$
$54$	0	0
$55$	1.54510	0.208341
$56$	3.79306	0.506869
$57$	0	0
$58$	−26.1385	−3.43215
$59$	3.09019	0.402309	0.201154	−	0.979560i	$-0.435531\pi$
$59$	3.09019	0.402309	0.201154	+	0.979560i	$0.435531\pi$
$60$	0	0
$61$	9.58612	1.22738	0.613689	−	0.789548i	$-0.289685\pi$
$61$	9.58612	1.22738	0.613689	+	0.789548i	$0.289685\pi$
$62$	−2.42835	−0.308401
$63$	0	0
$64$	17.0145	2.12681
$65$	−2.00000	−0.248069
$66$	0	0
$67$	15.7665	1.92619	0.963093	−	0.269170i	$-0.0867492\pi$
$67$	15.7665	1.92619	0.963093	+	0.269170i	$0.0867492\pi$
$68$	5.12398	0.621374
$69$	0	0
$70$	−1.21417	−0.145122
$71$	−13.5861	−1.61238	−0.806188	−	0.591659i	$-0.798473\pi$
$71$	−13.5861	−1.61238	−0.806188	+	0.591659i	$0.798473\pi$
$72$	0	0
$73$	1.11675	0.130705	0.0653526	−	0.997862i	$-0.479183\pi$
$73$	1.11675	0.130705	0.0653526	+	0.997862i	$0.479183\pi$
$74$	−20.8003	−2.41799
$75$	0	0
$76$	18.1650	2.08367
$77$	−0.702870	−0.0800995
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−12.0072	−1.34245
$81$	0	0
$82$	17.8905	1.97567
$83$	5.81962	0.638786	0.319393	−	0.947622i	$-0.396521\pi$
$83$	5.81962	0.638786	0.319393	+	0.947622i	$0.396521\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−23.1578	−2.49717
$87$	0	0
$88$	−12.8833	−1.37336
$89$	−13.7665	−1.45925	−0.729623	−	0.683849i	$-0.760305\pi$
$89$	−13.7665	−1.45925	−0.729623	+	0.683849i	$0.760305\pi$
$90$	0	0
$91$	0.909808	0.0953738
$92$	0	0
$93$	0	0
$94$	−32.1385	−3.31483
$95$	−3.54510	−0.363719
$96$	0	0
$97$	15.5861	1.58253	0.791266	−	0.611473i	$-0.209423\pi$
$97$	15.5861	1.58253	0.791266	+	0.611473i	$0.209423\pi$
$98$	−18.1312	−1.83153
$99$	0	0
$100$	5.12398	0.512398
$101$	5.81962	0.579073	0.289537	−	0.957167i	$-0.406499\pi$
$101$	5.81962	0.579073	0.289537	+	0.957167i	$0.406499\pi$
$102$	0	0
$103$	−5.58612	−0.550417	−0.275209	−	0.961385i	$-0.588747\pi$
$103$	−5.58612	−0.550417	−0.275209	+	0.961385i	$0.588747\pi$
$104$	16.6763	1.63525
$105$	0	0
$106$	−2.35748	−0.228978
$107$	8.90981	0.861344	0.430672	−	0.902509i	$-0.358277\pi$
$107$	8.90981	0.861344	0.430672	+	0.902509i	$0.358277\pi$
$108$	0	0
$109$	−11.7665	−1.12703	−0.563514	−	0.826107i	$-0.690551\pi$
$109$	−11.7665	−1.12703	−0.563514	+	0.826107i	$0.690551\pi$
$110$	4.12398	0.393206
$111$	0	0
$112$	5.46214	0.516124
$113$	−13.5861	−1.27808	−0.639038	−	0.769176i	$-0.720667\pi$
$113$	−13.5861	−1.27808	−0.639038	+	0.769176i	$0.720667\pi$
$114$	0	0
$115$	0	0
$116$	−50.1795	−4.65905
$117$	0	0
$118$	8.24797	0.759287
$119$	0.454904	0.0417010
$120$	0	0
$121$	−8.61268	−0.782971
$122$	25.5861	2.31646
$123$	0	0
$124$	−4.66184	−0.418646
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.7665	1.39905	0.699526	−	0.714607i	$-0.253395\pi$
$127$	15.7665	1.39905	0.699526	+	0.714607i	$0.253395\pi$
$128$	14.6691	1.29658
$129$	0	0
$130$	−5.33816	−0.468187
$131$	7.76651	0.678563	0.339281	−	0.940685i	$-0.389816\pi$
$131$	7.76651	0.678563	0.339281	+	0.940685i	$0.389816\pi$
$132$	0	0
$133$	1.61268	0.139837
$134$	42.0821	3.63534
$135$	0	0
$136$	8.33816	0.714991
$137$	−3.61268	−0.308652	−0.154326	−	0.988020i	$-0.549321\pi$
$137$	−3.61268	−0.308652	−0.154326	+	0.988020i	$0.549321\pi$
$138$	0	0
$139$	4.90981	0.416444	0.208222	−	0.978082i	$-0.433232\pi$
$139$	4.90981	0.416444	0.208222	+	0.978082i	$0.433232\pi$
$140$	−2.33092	−0.196999
$141$	0	0
$142$	−36.2624	−3.04308
$143$	−3.09019	−0.258415
$144$	0	0
$145$	9.79306	0.813270
$146$	2.98068	0.246683
$147$	0	0
$148$	−39.9315	−3.28235
$149$	13.7665	1.12780	0.563898	−	0.825844i	$-0.309301\pi$
$149$	13.7665	1.12780	0.563898	+	0.825844i	$0.309301\pi$
$150$	0	0
$151$	8.04103	0.654370	0.327185	−	0.944960i	$-0.393900\pi$
$151$	8.04103	0.654370	0.327185	+	0.944960i	$0.393900\pi$
$152$	29.5596	2.39760
$153$	0	0
$154$	−1.87602	−0.151174
$155$	0.909808	0.0730776
$156$	0	0
$157$	17.1722	1.37049	0.685247	−	0.728310i	$-0.259694\pi$
$157$	17.1722	1.37049	0.685247	+	0.728310i	$0.259694\pi$
$158$	21.3526	1.69872
$159$	0	0
$160$	−15.3719	−1.21526
$161$	0	0
$162$	0	0
$163$	8.40180	0.658079	0.329040	−	0.944316i	$-0.393275\pi$
$163$	8.40180	0.658079	0.329040	+	0.944316i	$0.393275\pi$
$164$	34.3454	2.68192
$165$	0	0
$166$	15.5330	1.20560
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−2.66908	−0.204709
$171$	0	0
$172$	−44.4573	−3.38984
$173$	13.9469	1.06036	0.530181	−	0.847884i	$-0.322124\pi$
$173$	13.9469	1.06036	0.530181	+	0.847884i	$0.322124\pi$
$174$	0	0
$175$	0.454904	0.0343875
$176$	−18.5523	−1.39843
$177$	0	0
$178$	−36.7439	−2.75407
$179$	−21.3526	−1.59597	−0.797985	−	0.602677i	$-0.794101\pi$
$179$	−21.3526	−1.59597	−0.797985	+	0.602677i	$0.794101\pi$
$180$	0	0
$181$	−11.7665	−0.874598	−0.437299	−	0.899316i	$-0.644065\pi$
$181$	−11.7665	−0.874598	−0.437299	+	0.899316i	$0.644065\pi$
$182$	2.42835	0.180001
$183$	0	0
$184$	0	0
$185$	7.79306	0.572957
$186$	0	0
$187$	−1.54510	−0.112989
$188$	−61.6980	−4.49979
$189$	0	0
$190$	−9.46214	−0.686456
$191$	15.1722	1.09783	0.548913	−	0.835880i	$-0.315042\pi$
$191$	15.1722	1.09783	0.548913	+	0.835880i	$0.315042\pi$
$192$	0	0
$193$	−11.5861	−0.833987	−0.416994	−	0.908909i	$-0.636916\pi$
$193$	−11.5861	−0.833987	−0.416994	+	0.908909i	$0.636916\pi$
$194$	41.6006	2.98675
$195$	0	0
$196$	−34.8075	−2.48625
$197$	−0.180384	−0.0128518	−0.00642590	−	0.999979i	$-0.502045\pi$
$197$	−0.180384	−0.0128518	−0.00642590	+	0.999979i	$0.502045\pi$
$198$	0	0
$199$	−22.2624	−1.57814	−0.789071	−	0.614302i	$-0.789438\pi$
$199$	−22.2624	−1.57814	−0.789071	+	0.614302i	$0.789438\pi$
$200$	8.33816	0.589597
$201$	0	0
$202$	15.5330	1.09290
$203$	−4.45490	−0.312673
$204$	0	0
$205$	−6.70287	−0.468149
$206$	−14.9098	−1.03882
$207$	0	0
$208$	24.0145	1.66510
$209$	−5.47751	−0.378888
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−4.52578	−0.310832
$213$	0	0
$214$	23.7810	1.62563
$215$	8.67632	0.591720
$216$	0	0
$217$	−0.413875	−0.0280957
$218$	−31.4057	−2.12707
$219$	0	0
$220$	7.91705	0.533767
$221$	2.00000	0.134535
$222$	0	0
$223$	21.2254	1.42136	0.710678	−	0.703518i	$-0.248389\pi$
$223$	21.2254	1.42136	0.710678	+	0.703518i	$0.248389\pi$
$224$	6.99276	0.467224
$225$	0	0
$226$	−36.2624	−2.41214
$227$	0.360767	0.0239450	0.0119725	−	0.999928i	$-0.496189\pi$
$227$	0.360767	0.0239450	0.0119725	+	0.999928i	$0.496189\pi$
$228$	0	0
$229$	8.88325	0.587022	0.293511	−	0.955956i	$-0.405176\pi$
$229$	8.88325	0.587022	0.293511	+	0.955956i	$0.405176\pi$
$230$	0	0
$231$	0	0
$232$	−81.6561	−5.36099
$233$	19.7665	1.29495	0.647474	−	0.762088i	$-0.275826\pi$
$233$	19.7665	1.29495	0.647474	+	0.762088i	$0.275826\pi$
$234$	0	0
$235$	12.0410	0.785470
$236$	15.8341	1.03071
$237$	0	0
$238$	1.21417	0.0787033
$239$	18.2624	1.18130	0.590650	−	0.806928i	$-0.298872\pi$
$239$	18.2624	1.18130	0.590650	+	0.806928i	$0.298872\pi$
$240$	0	0
$241$	15.7665	1.01561	0.507805	−	0.861472i	$-0.330457\pi$
$241$	15.7665	1.01561	0.507805	+	0.861472i	$0.330457\pi$
$242$	−22.9879	−1.47772
$243$	0	0
$244$	49.1191	3.14453
$245$	6.79306	0.433993
$246$	0	0
$247$	7.09019	0.451138
$248$	−7.58612	−0.481719
$249$	0	0
$250$	−2.66908	−0.168807
$251$	−8.90981	−0.562382	−0.281191	−	0.959652i	$-0.590729\pi$
$251$	−8.90981	−0.562382	−0.281191	+	0.959652i	$0.590729\pi$
$252$	0	0
$253$	0	0
$254$	42.0821	2.64046
$255$	0	0
$256$	5.12398	0.320249
$257$	21.1722	1.32069	0.660344	−	0.750963i	$-0.270410\pi$
$257$	21.1722	1.32069	0.660344	+	0.750963i	$0.270410\pi$
$258$	0	0
$259$	−3.54510	−0.220282
$260$	−10.2480	−0.635552
$261$	0	0
$262$	20.7294	1.28067
$263$	−3.13122	−0.193079	−0.0965397	−	0.995329i	$-0.530777\pi$
$263$	−3.13122	−0.193079	−0.0965397	+	0.995329i	$0.530777\pi$
$264$	0	0
$265$	0.883254	0.0542579
$266$	4.30437	0.263918
$267$	0	0
$268$	80.7873	4.93487
$269$	3.97345	0.242265	0.121133	−	0.992636i	$-0.461347\pi$
$269$	3.97345	0.242265	0.121133	+	0.992636i	$0.461347\pi$
$270$	0	0
$271$	−25.3526	−1.54006	−0.770031	−	0.638006i	$-0.779759\pi$
$271$	−25.3526	−1.54006	−0.770031	+	0.638006i	$0.779759\pi$
$272$	12.0072	0.728046
$273$	0	0
$274$	−9.64252	−0.582526
$275$	−1.54510	−0.0931728
$276$	0	0
$277$	−11.5861	−0.696143	−0.348071	−	0.937468i	$-0.613163\pi$
$277$	−11.5861	−0.696143	−0.348071	+	0.937468i	$0.613163\pi$
$278$	13.1047	0.785966
$279$	0	0
$280$	−3.79306	−0.226679
$281$	6.18038	0.368691	0.184345	−	0.982862i	$-0.440984\pi$
$281$	6.18038	0.368691	0.184345	+	0.982862i	$0.440984\pi$
$282$	0	0
$283$	1.77859	0.105726	0.0528630	−	0.998602i	$-0.483165\pi$
$283$	1.77859	0.105726	0.0528630	+	0.998602i	$0.483165\pi$
$284$	−69.6151	−4.13089
$285$	0	0
$286$	−8.24797	−0.487712
$287$	3.04916	0.179986
$288$	0	0
$289$	1.00000	0.0588235
$290$	26.1385	1.53490
$291$	0	0
$292$	5.72219	0.334866
$293$	20.8301	1.21691	0.608455	−	0.793588i	$-0.291790\pi$
$293$	20.8301	1.21691	0.608455	+	0.793588i	$0.291790\pi$
$294$	0	0
$295$	−3.09019	−0.179918
$296$	−64.9798	−3.77687
$297$	0	0
$298$	36.7439	2.12852
$299$	0	0
$300$	0	0
$301$	−3.94689	−0.227495
$302$	21.4621	1.23501
$303$	0	0
$304$	42.5668	2.44137
$305$	−9.58612	−0.548900
$306$	0	0
$307$	0.233492	0.0133261	0.00666304	−	0.999978i	$-0.497879\pi$
$307$	0.233492	0.0133261	0.00666304	+	0.999978i	$0.497879\pi$
$308$	−3.60150	−0.205214
$309$	0	0
$310$	2.42835	0.137921
$311$	2.95084	0.167327	0.0836633	−	0.996494i	$-0.473338\pi$
$311$	2.95084	0.167327	0.0836633	+	0.996494i	$0.473338\pi$
$312$	0	0
$313$	−16.3421	−0.923710	−0.461855	−	0.886955i	$-0.652816\pi$
$313$	−16.3421	−0.923710	−0.461855	+	0.886955i	$0.652816\pi$
$314$	45.8341	2.58657
$315$	0	0
$316$	40.9919	2.30597
$317$	−15.3526	−0.862290	−0.431145	−	0.902283i	$-0.641890\pi$
$317$	−15.3526	−0.862290	−0.431145	+	0.902283i	$0.641890\pi$
$318$	0	0
$319$	15.1312	0.847186
$320$	−17.0145	−0.951138
$321$	0	0
$322$	0	0
$323$	3.54510	0.197254
$324$	0	0
$325$	2.00000	0.110940
$326$	22.4251	1.24201
$327$	0	0
$328$	55.8896	3.08599
$329$	−5.47751	−0.301985
$330$	0	0
$331$	3.18433	0.175027	0.0875133	−	0.996163i	$-0.472108\pi$
$331$	3.18433	0.175027	0.0875133	+	0.996163i	$0.472108\pi$
$332$	29.8196	1.63656
$333$	0	0
$334$	−32.0289	−1.75255
$335$	−15.7665	−0.861416
$336$	0	0
$337$	4.64976	0.253289	0.126644	−	0.991948i	$-0.459579\pi$
$337$	4.64976	0.253289	0.126644	+	0.991948i	$0.459579\pi$
$338$	−24.0217	−1.30661
$339$	0	0
$340$	−5.12398	−0.277887
$341$	1.40574	0.0761251
$342$	0	0
$343$	−6.27452	−0.338792
$344$	−72.3445	−3.90055
$345$	0	0
$346$	37.2254	2.00125
$347$	20.9919	1.12690	0.563451	−	0.826149i	$-0.309473\pi$
$347$	20.9919	1.12690	0.563451	+	0.826149i	$0.309473\pi$
$348$	0	0
$349$	−18.6498	−0.998299	−0.499149	−	0.866516i	$-0.666354\pi$
$349$	−18.6498	−0.998299	−0.499149	+	0.866516i	$0.666354\pi$
$350$	1.21417	0.0649004
$351$	0	0
$352$	−23.7511	−1.26594
$353$	−19.1457	−1.01902	−0.509511	−	0.860464i	$-0.670174\pi$
$353$	−19.1457	−1.01902	−0.509511	+	0.860464i	$0.670174\pi$
$354$	0	0
$355$	13.5861	0.721077
$356$	−70.5394	−3.73858
$357$	0	0
$358$	−56.9919	−3.01211
$359$	3.09019	0.163094	0.0815470	−	0.996669i	$-0.474014\pi$
$359$	3.09019	0.163094	0.0815470	+	0.996669i	$0.474014\pi$
$360$	0	0
$361$	−6.43229	−0.338542
$362$	−31.4057	−1.65065
$363$	0	0
$364$	4.66184	0.244347
$365$	−1.11675	−0.0584532
$366$	0	0
$367$	35.5330	1.85481	0.927404	−	0.374061i	$-0.122035\pi$
$367$	35.5330	1.85481	0.927404	+	0.374061i	$0.122035\pi$
$368$	0	0
$369$	0	0
$370$	20.8003	1.08136
$371$	−0.401796	−0.0208602
$372$	0	0
$373$	−23.7665	−1.23058	−0.615292	−	0.788300i	$-0.710962\pi$
$373$	−23.7665	−1.23058	−0.615292	+	0.788300i	$0.710962\pi$
$374$	−4.12398	−0.213246
$375$	0	0
$376$	−100.400	−5.17773
$377$	−19.5861	−1.00874
$378$	0	0
$379$	−10.2624	−0.527146	−0.263573	−	0.964639i	$-0.584901\pi$
$379$	−10.2624	−0.527146	−0.263573	+	0.964639i	$0.584901\pi$
$380$	−18.1650	−0.931845
$381$	0	0
$382$	40.4959	2.07195
$383$	36.0410	1.84161	0.920805	−	0.390023i	$-0.127533\pi$
$383$	36.0410	1.84161	0.920805	+	0.390023i	$0.127533\pi$
$384$	0	0
$385$	0.702870	0.0358216
$386$	−30.9243	−1.57400
$387$	0	0
$388$	79.8630	4.05443
$389$	−19.2254	−0.974764	−0.487382	−	0.873189i	$-0.662048\pi$
$389$	−19.2254	−0.974764	−0.487382	+	0.873189i	$0.662048\pi$
$390$	0	0
$391$	0	0
$392$	−56.6416	−2.86083
$393$	0	0
$394$	−0.481458	−0.0242555
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−1.61268	−0.0809380	−0.0404690	−	0.999181i	$-0.512885\pi$
$397$	−1.61268	−0.0809380	−0.0404690	+	0.999181i	$0.512885\pi$
$398$	−59.4202	−2.97847
$399$	0	0
$400$	12.0072	0.600362
$401$	20.4694	1.02219	0.511096	−	0.859524i	$-0.329240\pi$
$401$	20.4694	1.02219	0.511096	+	0.859524i	$0.329240\pi$
$402$	0	0
$403$	−1.81962	−0.0906415
$404$	29.8196	1.48358
$405$	0	0
$406$	−11.8905	−0.590115
$407$	12.0410	0.596852
$408$	0	0
$409$	−36.9122	−1.82519	−0.912595	−	0.408864i	$-0.865925\pi$
$409$	−36.9122	−1.82519	−0.912595	+	0.408864i	$0.865925\pi$
$410$	−17.8905	−0.883548
$411$	0	0
$412$	−28.6232	−1.41016
$413$	1.40574	0.0691720
$414$	0	0
$415$	−5.81962	−0.285674
$416$	30.7439	1.50734
$417$	0	0
$418$	−14.6199	−0.715083
$419$	1.90586	0.0931075	0.0465538	−	0.998916i	$-0.485176\pi$
$419$	1.90586	0.0931075	0.0465538	+	0.998916i	$0.485176\pi$
$420$	0	0
$421$	−16.5225	−0.805257	−0.402628	−	0.915364i	$-0.631903\pi$
$421$	−16.5225	−0.805257	−0.402628	+	0.915364i	$0.631903\pi$
$422$	53.3816	2.59857
$423$	0	0
$424$	−7.36471	−0.357662
$425$	1.00000	0.0485071
$426$	0	0
$427$	4.36077	0.211032
$428$	45.6537	2.20676
$429$	0	0
$430$	23.1578	1.11677
$431$	18.4018	0.886383	0.443192	−	0.896427i	$-0.353846\pi$
$431$	18.4018	0.886383	0.443192	+	0.896427i	$0.353846\pi$
$432$	0	0
$433$	−2.05311	−0.0986661	−0.0493330	−	0.998782i	$-0.515710\pi$
$433$	−2.05311	−0.0986661	−0.0493330	+	0.998782i	$0.515710\pi$
$434$	−1.10467	−0.0530257
$435$	0	0
$436$	−60.2914	−2.88743
$437$	0	0
$438$	0	0
$439$	−9.81962	−0.468665	−0.234332	−	0.972157i	$-0.575290\pi$
$439$	−9.81962	−0.468665	−0.234332	+	0.972157i	$0.575290\pi$
$440$	12.8833	0.614185
$441$	0	0
$442$	5.33816	0.253910
$443$	−0.360767	−0.0171406	−0.00857029	−	0.999963i	$-0.502728\pi$
$443$	−0.360767	−0.0171406	−0.00857029	+	0.999963i	$0.502728\pi$
$444$	0	0
$445$	13.7665	0.652595
$446$	56.6522	2.68256
$447$	0	0
$448$	7.73995	0.365678
$449$	9.53302	0.449891	0.224945	−	0.974371i	$-0.427780\pi$
$449$	9.53302	0.449891	0.224945	+	0.974371i	$0.427780\pi$
$450$	0	0
$451$	−10.3566	−0.487672
$452$	−69.6151	−3.27442
$453$	0	0
$454$	0.962916	0.0451919
$455$	−0.909808	−0.0426525
$456$	0	0
$457$	3.76651	0.176190	0.0880949	−	0.996112i	$-0.471922\pi$
$457$	3.76651	0.176190	0.0880949	+	0.996112i	$0.471922\pi$
$458$	23.7101	1.10790
$459$	0	0
$460$	0	0
$461$	−9.35263	−0.435596	−0.217798	−	0.975994i	$-0.569887\pi$
$461$	−9.35263	−0.435596	−0.217798	+	0.975994i	$0.569887\pi$
$462$	0	0
$463$	15.4057	0.715965	0.357983	−	0.933728i	$-0.383465\pi$
$463$	15.4057	0.715965	0.357983	+	0.933728i	$0.383465\pi$
$464$	−117.588	−5.45887
$465$	0	0
$466$	52.7584	2.44398
$467$	28.8977	1.33723	0.668614	−	0.743610i	$-0.266888\pi$
$467$	28.8977	1.33723	0.668614	+	0.743610i	$0.266888\pi$
$468$	0	0
$469$	7.17225	0.331184
$470$	32.1385	1.48244
$471$	0	0
$472$	25.7665	1.18600
$473$	13.4057	0.616397
$474$	0	0
$475$	3.54510	0.162660
$476$	2.33092	0.106838
$477$	0	0
$478$	48.7439	2.22949
$479$	−31.4057	−1.43496	−0.717482	−	0.696577i	$-0.754706\pi$
$479$	−31.4057	−1.43496	−0.717482	+	0.696577i	$0.754706\pi$
$480$	0	0
$481$	−15.5861	−0.710666
$482$	42.0821	1.91679
$483$	0	0
$484$	−44.1312	−2.00596
$485$	−15.5861	−0.707729
$486$	0	0
$487$	16.9919	0.769975	0.384987	−	0.922922i	$-0.374206\pi$
$487$	16.9919	0.769975	0.384987	+	0.922922i	$0.374206\pi$
$488$	79.9306	3.61829
$489$	0	0
$490$	18.1312	0.819085
$491$	9.27058	0.418375	0.209188	−	0.977876i	$-0.432918\pi$
$491$	9.27058	0.418375	0.209188	+	0.977876i	$0.432918\pi$
$492$	0	0
$493$	−9.79306	−0.441057
$494$	18.9243	0.851444
$495$	0	0
$496$	−10.9243	−0.490515
$497$	−6.18038	−0.277228
$498$	0	0
$499$	−31.6151	−1.41529	−0.707643	−	0.706571i	$-0.750241\pi$
$499$	−31.6151	−1.41529	−0.707643	+	0.706571i	$0.750241\pi$
$500$	−5.12398	−0.229151
$501$	0	0
$502$	−23.7810	−1.06140
$503$	36.0821	1.60882	0.804410	−	0.594075i	$-0.202482\pi$
$503$	36.0821	1.60882	0.804410	+	0.594075i	$0.202482\pi$
$504$	0	0
$505$	−5.81962	−0.258970
$506$	0	0
$507$	0	0
$508$	80.7873	3.58436
$509$	−29.8196	−1.32173	−0.660866	−	0.750504i	$-0.729811\pi$
$509$	−29.8196	−1.32173	−0.660866	+	0.750504i	$0.729811\pi$
$510$	0	0
$511$	0.508012	0.0224731
$512$	−15.6618	−0.692162
$513$	0	0
$514$	56.5104	2.49257
$515$	5.58612	0.246154
$516$	0	0
$517$	18.6045	0.818227
$518$	−9.46214	−0.415743
$519$	0	0
$520$	−16.6763	−0.731305
$521$	20.4694	0.896780	0.448390	−	0.893838i	$-0.351998\pi$
$521$	20.4694	0.896780	0.448390	+	0.893838i	$0.351998\pi$
$522$	0	0
$523$	6.49593	0.284047	0.142024	−	0.989863i	$-0.454639\pi$
$523$	6.49593	0.284047	0.142024	+	0.989863i	$0.454639\pi$
$524$	39.7955	1.73847
$525$	0	0
$526$	−8.35748	−0.364403
$527$	−0.909808	−0.0396319
$528$	0	0
$529$	−23.0000	−1.00000
$530$	2.35748	0.102402
$531$	0	0
$532$	8.26334	0.358261
$533$	13.4057	0.580667
$534$	0	0
$535$	−8.90981	−0.385205
$536$	131.464	5.67836
$537$	0	0
$538$	10.6054	0.457233
$539$	10.4959	0.452092
$540$	0	0
$541$	5.53302	0.237883	0.118941	−	0.992901i	$-0.462050\pi$
$541$	5.53302	0.237883	0.118941	+	0.992901i	$0.462050\pi$
$542$	−67.6682	−2.90660
$543$	0	0
$544$	15.3719	0.659067
$545$	11.7665	0.504022
$546$	0	0
$547$	28.9508	1.23785	0.618924	−	0.785451i	$-0.287569\pi$
$547$	28.9508	1.23785	0.618924	+	0.785451i	$0.287569\pi$
$548$	−18.5113	−0.790764
$549$	0	0
$550$	−4.12398	−0.175847
$551$	−34.7173	−1.47901
$552$	0	0
$553$	3.63923	0.154756
$554$	−30.9243	−1.31385
$555$	0	0
$556$	25.1578	1.06693
$557$	17.6392	0.747398	0.373699	−	0.927550i	$-0.378089\pi$
$557$	17.6392	0.747398	0.373699	+	0.927550i	$0.378089\pi$
$558$	0	0
$559$	−17.3526	−0.733938
$560$	−5.46214	−0.230818
$561$	0	0
$562$	16.4959	0.695839
$563$	4.81567	0.202956	0.101478	−	0.994838i	$-0.467643\pi$
$563$	4.81567	0.202956	0.101478	+	0.994838i	$0.467643\pi$
$564$	0	0
$565$	13.5861	0.571573
$566$	4.74719	0.199539
$567$	0	0
$568$	−113.283	−4.75326
$569$	−19.5861	−0.821093	−0.410547	−	0.911840i	$-0.634662\pi$
$569$	−19.5861	−0.821093	−0.410547	+	0.911840i	$0.634662\pi$
$570$	0	0
$571$	37.9017	1.58614	0.793068	−	0.609133i	$-0.208483\pi$
$571$	37.9017	1.58614	0.793068	+	0.609133i	$0.208483\pi$
$572$	−15.8341	−0.662057
$573$	0	0
$574$	8.13846	0.339693
$575$	0	0
$576$	0	0
$577$	−31.3526	−1.30523	−0.652614	−	0.757691i	$-0.726327\pi$
$577$	−31.3526	−1.30523	−0.652614	+	0.757691i	$0.726327\pi$
$578$	2.66908	0.111019
$579$	0	0
$580$	50.1795	2.08359
$581$	2.64737	0.109831
$582$	0	0
$583$	1.36471	0.0565206
$584$	9.31160	0.385317
$585$	0	0
$586$	55.5973	2.29670
$587$	−7.62715	−0.314806	−0.157403	−	0.987534i	$-0.550312\pi$
$587$	−7.62715	−0.314806	−0.157403	+	0.987534i	$0.550312\pi$
$588$	0	0
$589$	−3.22536	−0.132899
$590$	−8.24797	−0.339563
$591$	0	0
$592$	−93.5731	−3.84583
$593$	−21.4323	−0.880119	−0.440059	−	0.897969i	$-0.645043\pi$
$593$	−21.4323	−0.880119	−0.440059	+	0.897969i	$0.645043\pi$
$594$	0	0
$595$	−0.454904	−0.0186492
$596$	70.5394	2.88940
$597$	0	0
$598$	0	0
$599$	−39.5330	−1.61528	−0.807638	−	0.589679i	$-0.799254\pi$
$599$	−39.5330	−1.61528	−0.807638	+	0.589679i	$0.799254\pi$
$600$	0	0
$601$	−43.7134	−1.78311	−0.891553	−	0.452916i	$-0.850384\pi$
$601$	−43.7134	−1.78311	−0.891553	+	0.452916i	$0.850384\pi$
$602$	−10.5346	−0.429357
$603$	0	0
$604$	41.2021	1.67649
$605$	8.61268	0.350155
$606$	0	0
$607$	−17.7255	−0.719455	−0.359728	−	0.933057i	$-0.617130\pi$
$607$	−17.7255	−0.719455	−0.359728	+	0.933057i	$0.617130\pi$
$608$	54.4950	2.21007
$609$	0	0
$610$	−25.5861	−1.03595
$611$	−24.0821	−0.974256
$612$	0	0
$613$	−28.1804	−1.13819	−0.569097	−	0.822270i	$-0.692707\pi$
$613$	−28.1804	−1.13819	−0.569097	+	0.822270i	$0.692707\pi$
$614$	0.623208	0.0251506
$615$	0	0
$616$	−5.86064	−0.236132
$617$	−10.4139	−0.419247	−0.209623	−	0.977782i	$-0.567224\pi$
$617$	−10.4139	−0.419247	−0.209623	+	0.977782i	$0.567224\pi$
$618$	0	0
$619$	28.9919	1.16528	0.582641	−	0.812730i	$-0.302019\pi$
$619$	28.9919	1.16528	0.582641	+	0.812730i	$0.302019\pi$
$620$	4.66184	0.187224
$621$	0	0
$622$	7.87602	0.315800
$623$	−6.26244	−0.250899
$624$	0	0
$625$	1.00000	0.0400000
$626$	−43.6184	−1.74334
$627$	0	0
$628$	87.9903	3.51120
$629$	−7.79306	−0.310730
$630$	0	0
$631$	−3.59820	−0.143242	−0.0716211	−	0.997432i	$-0.522817\pi$
$631$	−3.59820	−0.143242	−0.0716211	+	0.997432i	$0.522817\pi$
$632$	66.7053	2.65339
$633$	0	0
$634$	−40.9774	−1.62742
$635$	−15.7665	−0.625675
$636$	0	0
$637$	−13.5861	−0.538302
$638$	40.3864	1.59891
$639$	0	0
$640$	−14.6691	−0.579846
$641$	35.1167	1.38703	0.693514	−	0.720443i	$-0.256062\pi$
$641$	35.1167	1.38703	0.693514	+	0.720443i	$0.256062\pi$
$642$	0	0
$643$	35.1312	1.38544	0.692720	−	0.721207i	$-0.256412\pi$
$643$	35.1312	1.38544	0.692720	+	0.721207i	$0.256412\pi$
$644$	0	0
$645$	0	0
$646$	9.46214	0.372283
$647$	5.45885	0.214610	0.107305	−	0.994226i	$-0.465778\pi$
$647$	5.45885	0.214610	0.107305	+	0.994226i	$0.465778\pi$
$648$	0	0
$649$	−4.77464	−0.187421
$650$	5.33816	0.209380
$651$	0	0
$652$	43.0507	1.68599
$653$	4.23349	0.165669	0.0828347	−	0.996563i	$-0.473603\pi$
$653$	4.23349	0.165669	0.0828347	+	0.996563i	$0.473603\pi$
$654$	0	0
$655$	−7.76651	−0.303463
$656$	80.4830	3.14233
$657$	0	0
$658$	−14.6199	−0.569944
$659$	18.2624	0.711404	0.355702	−	0.934599i	$-0.384242\pi$
$659$	18.2624	0.711404	0.355702	+	0.934599i	$0.384242\pi$
$660$	0	0
$661$	−30.2890	−1.17811	−0.589053	−	0.808095i	$-0.700499\pi$
$661$	−30.2890	−1.17811	−0.589053	+	0.808095i	$0.700499\pi$
$662$	8.49922	0.330332
$663$	0	0
$664$	48.5249	1.88313
$665$	−1.61268	−0.0625370
$666$	0	0
$667$	0	0
$668$	−61.4878	−2.37903
$669$	0	0
$670$	−42.0821	−1.62577
$671$	−14.8115	−0.571791
$672$	0	0
$673$	−20.4139	−0.786897	−0.393449	−	0.919347i	$-0.628718\pi$
$673$	−20.4139	−0.786897	−0.393449	+	0.919347i	$0.628718\pi$
$674$	12.4106	0.478038
$675$	0	0
$676$	−46.1158	−1.77369
$677$	18.5249	0.711969	0.355985	−	0.934492i	$-0.384146\pi$
$677$	18.5249	0.711969	0.355985	+	0.934492i	$0.384146\pi$
$678$	0	0
$679$	7.09019	0.272097
$680$	−8.33816	−0.319754
$681$	0	0
$682$	3.75203	0.143673
$683$	3.09019	0.118243	0.0591215	−	0.998251i	$-0.481170\pi$
$683$	3.09019	0.118243	0.0591215	+	0.998251i	$0.481170\pi$
$684$	0	0
$685$	3.61268	0.138033
$686$	−16.7472	−0.639411
$687$	0	0
$688$	−104.179	−3.97177
$689$	−1.76651	−0.0672986
$690$	0	0
$691$	20.8036	0.791406	0.395703	−	0.918379i	$-0.370501\pi$
$691$	20.8036	0.791406	0.395703	+	0.918379i	$0.370501\pi$
$692$	71.4636	2.71664
$693$	0	0
$694$	56.0289	2.12683
$695$	−4.90981	−0.186240
$696$	0	0
$697$	6.70287	0.253889
$698$	−49.7777	−1.88411
$699$	0	0
$700$	2.33092	0.0881005
$701$	−2.64737	−0.0999897	−0.0499948	−	0.998749i	$-0.515920\pi$
$701$	−2.64737	−0.0999897	−0.0499948	+	0.998749i	$0.515920\pi$
$702$	0	0
$703$	−27.6272	−1.04198
$704$	−26.2890	−0.990804
$705$	0	0
$706$	−51.1014	−1.92323
$707$	2.64737	0.0995645
$708$	0	0
$709$	37.4799	1.40759	0.703794	−	0.710404i	$-0.251488\pi$
$709$	37.4799	1.40759	0.703794	+	0.710404i	$0.251488\pi$
$710$	36.2624	1.36091
$711$	0	0
$712$	−114.787	−4.30184
$713$	0	0
$714$	0	0
$715$	3.09019	0.115567
$716$	−109.411	−4.08886
$717$	0	0
$718$	8.24797	0.307811
$719$	−17.5982	−0.656302	−0.328151	−	0.944625i	$-0.606426\pi$
$719$	−17.5982	−0.656302	−0.328151	+	0.944625i	$0.606426\pi$
$720$	0	0
$721$	−2.54115	−0.0946374
$722$	−17.1683	−0.638938
$723$	0	0
$724$	−60.2914	−2.24071
$725$	−9.79306	−0.363705
$726$	0	0
$727$	−17.5861	−0.652233	−0.326117	−	0.945330i	$-0.605740\pi$
$727$	−17.5861	−0.652233	−0.326117	+	0.945330i	$0.605740\pi$
$728$	7.58612	0.281160
$729$	0	0
$730$	−2.98068	−0.110320
$731$	−8.67632	−0.320905
$732$	0	0
$733$	34.9919	1.29245	0.646227	−	0.763145i	$-0.276346\pi$
$733$	34.9919	1.29245	0.646227	+	0.763145i	$0.276346\pi$
$734$	94.8404	3.50062
$735$	0	0
$736$	0	0
$737$	−24.3608	−0.897340
$738$	0	0
$739$	32.5249	1.19645	0.598224	−	0.801329i	$-0.295873\pi$
$739$	32.5249	1.19645	0.598224	+	0.801329i	$0.295873\pi$
$740$	39.9315	1.46791
$741$	0	0
$742$	−1.07243	−0.0393700
$743$	6.18038	0.226736	0.113368	−	0.993553i	$-0.463836\pi$
$743$	6.18038	0.226736	0.113368	+	0.993553i	$0.463836\pi$
$744$	0	0
$745$	−13.7665	−0.504366
$746$	−63.4347	−2.32251
$747$	0	0
$748$	−7.91705	−0.289476
$749$	4.05311	0.148097
$750$	0	0
$751$	−52.4428	−1.91367	−0.956833	−	0.290639i	$-0.906132\pi$
$751$	−52.4428	−1.91367	−0.956833	+	0.290639i	$0.906132\pi$
$752$	−144.579	−5.27227
$753$	0	0
$754$	−52.2769	−1.90381
$755$	−8.04103	−0.292643
$756$	0	0
$757$	−49.1722	−1.78720	−0.893598	−	0.448868i	$-0.851827\pi$
$757$	−49.1722	−1.78720	−0.893598	+	0.448868i	$0.851827\pi$
$758$	−27.3913	−0.994896
$759$	0	0
$760$	−29.5596	−1.07224
$761$	22.2335	0.805963	0.402982	−	0.915208i	$-0.367974\pi$
$761$	22.2335	0.805963	0.402982	+	0.915208i	$0.367974\pi$
$762$	0	0
$763$	−5.35263	−0.193778
$764$	77.7423	2.81262
$765$	0	0
$766$	96.1964	3.47572
$767$	6.18038	0.223161
$768$	0	0
$769$	−7.61268	−0.274520	−0.137260	−	0.990535i	$-0.543830\pi$
$769$	−7.61268	−0.274520	−0.137260	+	0.990535i	$0.543830\pi$
$770$	1.87602	0.0676070
$771$	0	0
$772$	−59.3671	−2.13667
$773$	−53.8220	−1.93584	−0.967922	−	0.251252i	$-0.919158\pi$
$773$	−53.8220	−1.93584	−0.967922	+	0.251252i	$0.919158\pi$
$774$	0	0
$775$	−0.909808	−0.0326813
$776$	129.960	4.66528
$777$	0	0
$778$	−51.3140	−1.83970
$779$	23.7623	0.851373
$780$	0	0
$781$	20.9919	0.751148
$782$	0	0
$783$	0	0
$784$	−81.5659	−2.91307
$785$	−17.1722	−0.612904
$786$	0	0
$787$	−15.6803	−0.558941	−0.279470	−	0.960154i	$-0.590159\pi$
$787$	−15.6803	−0.558941	−0.279470	+	0.960154i	$0.590159\pi$
$788$	−0.924283	−0.0329262
$789$	0	0
$790$	−21.3526	−0.759693
$791$	−6.18038	−0.219749
$792$	0	0
$793$	19.1722	0.680827
$794$	−4.30437	−0.152756
$795$	0	0
$796$	−114.072	−4.04319
$797$	12.5225	0.443569	0.221785	−	0.975096i	$-0.428812\pi$
$797$	12.5225	0.443569	0.221785	+	0.975096i	$0.428812\pi$
$798$	0	0
$799$	−12.0410	−0.425981
$800$	15.3719	0.543480
$801$	0	0
$802$	54.6344	1.92921
$803$	−1.72548	−0.0608909
$804$	0	0
$805$	0	0
$806$	−4.85670	−0.171070
$807$	0	0
$808$	48.5249	1.70710
$809$	−36.7053	−1.29049	−0.645244	−	0.763976i	$-0.723244\pi$
$809$	−36.7053	−1.29049	−0.645244	+	0.763976i	$0.723244\pi$
$810$	0	0
$811$	50.3445	1.76783	0.883917	−	0.467643i	$-0.154897\pi$
$811$	50.3445	1.76783	0.883917	+	0.467643i	$0.154897\pi$
$812$	−22.8269	−0.801066
$813$	0	0
$814$	32.1385	1.12645
$815$	−8.40180	−0.294302
$816$	0	0
$817$	−30.7584	−1.07610
$818$	−98.5216	−3.44473
$819$	0	0
$820$	−34.3454	−1.19939
$821$	−32.8115	−1.14513	−0.572564	−	0.819860i	$-0.694051\pi$
$821$	−32.8115	−1.14513	−0.572564	+	0.819860i	$0.694051\pi$
$822$	0	0
$823$	−53.0860	−1.85046	−0.925231	−	0.379405i	$-0.876129\pi$
$823$	−53.0860	−1.85046	−0.925231	+	0.379405i	$0.876129\pi$
$824$	−46.5780	−1.62262
$825$	0	0
$826$	3.75203	0.130550
$827$	−27.5330	−0.957417	−0.478708	−	0.877974i	$-0.658895\pi$
$827$	−27.5330	−0.957417	−0.478708	+	0.877974i	$0.658895\pi$
$828$	0	0
$829$	−38.2359	−1.32799	−0.663994	−	0.747738i	$-0.731140\pi$
$829$	−38.2359	−1.32799	−0.663994	+	0.747738i	$0.731140\pi$
$830$	−15.5330	−0.539159
$831$	0	0
$832$	34.0289	1.17974
$833$	−6.79306	−0.235366
$834$	0	0
$835$	12.0000	0.415277
$836$	−28.0667	−0.970707
$837$	0	0
$838$	5.08690	0.175724
$839$	−30.4839	−1.05242	−0.526210	−	0.850355i	$-0.676387\pi$
$839$	−30.4839	−1.05242	−0.526210	+	0.850355i	$0.676387\pi$
$840$	0	0
$841$	66.9041	2.30704
$842$	−44.0998	−1.51978
$843$	0	0
$844$	102.480	3.52750
$845$	9.00000	0.309609
$846$	0	0
$847$	−3.91794	−0.134622
$848$	−10.6054	−0.364192
$849$	0	0
$850$	2.66908	0.0915487
$851$	0	0
$852$	0	0
$853$	−50.7584	−1.73793	−0.868967	−	0.494870i	$-0.835216\pi$
$853$	−50.7584	−1.73793	−0.868967	+	0.494870i	$0.835216\pi$
$854$	11.6392	0.398286
$855$	0	0
$856$	74.2914	2.53923
$857$	−50.9919	−1.74185	−0.870924	−	0.491417i	$-0.836479\pi$
$857$	−50.9919	−1.74185	−0.870924	+	0.491417i	$0.836479\pi$
$858$	0	0
$859$	6.27452	0.214084	0.107042	−	0.994255i	$-0.465862\pi$
$859$	6.27452	0.214084	0.107042	+	0.994255i	$0.465862\pi$
$860$	44.4573	1.51598
$861$	0	0
$862$	49.1158	1.67289
$863$	3.04916	0.103795	0.0518974	−	0.998652i	$-0.483473\pi$
$863$	3.04916	0.103795	0.0518974	+	0.998652i	$0.483473\pi$
$864$	0	0
$865$	−13.9469	−0.474209
$866$	−5.47991	−0.186215
$867$	0	0
$868$	−2.12069	−0.0719809
$869$	−12.3608	−0.419310
$870$	0	0
$871$	31.5330	1.06846
$872$	−98.1110	−3.32246
$873$	0	0
$874$	0	0
$875$	−0.454904	−0.0153786
$876$	0	0
$877$	31.3792	1.05960	0.529800	−	0.848123i	$-0.322267\pi$
$877$	31.3792	1.05960	0.529800	+	0.848123i	$0.322267\pi$
$878$	−26.2093	−0.884522
$879$	0	0
$880$	18.5523	0.625399
$881$	−44.9943	−1.51590	−0.757948	−	0.652315i	$-0.773798\pi$
$881$	−44.9943	−1.51590	−0.757948	+	0.652315i	$0.773798\pi$
$882$	0	0
$883$	2.96292	0.0997101	0.0498550	−	0.998756i	$-0.484124\pi$
$883$	2.96292	0.0997101	0.0498550	+	0.998756i	$0.484124\pi$
$884$	10.2480	0.344676
$885$	0	0
$886$	−0.962916	−0.0323498
$887$	21.3526	0.716951	0.358476	−	0.933539i	$-0.383297\pi$
$887$	21.3526	0.716951	0.358476	+	0.933539i	$0.383297\pi$
$888$	0	0
$889$	7.17225	0.240550
$890$	36.7439	1.23166
$891$	0	0
$892$	108.758	3.64150
$893$	−42.6866	−1.42845
$894$	0	0
$895$	21.3526	0.713740
$896$	6.67302	0.222930
$897$	0	0
$898$	25.4444	0.849090
$899$	8.90981	0.297159
$900$	0	0
$901$	−0.883254	−0.0294255
$902$	−27.6425	−0.920395
$903$	0	0
$904$	−113.283	−3.76775
$905$	11.7665	0.391132
$906$	0	0
$907$	34.7705	1.15453	0.577267	−	0.816555i	$-0.304119\pi$
$907$	34.7705	1.15453	0.577267	+	0.816555i	$0.304119\pi$
$908$	1.84857	0.0613468
$909$	0	0
$910$	−2.42835	−0.0804990
$911$	−2.50801	−0.0830942	−0.0415471	−	0.999137i	$-0.513229\pi$
$911$	−2.50801	−0.0830942	−0.0415471	+	0.999137i	$0.513229\pi$
$912$	0	0
$913$	−8.99187	−0.297587
$914$	10.0531	0.332527
$915$	0	0
$916$	45.5176	1.50395
$917$	3.53302	0.116670
$918$	0	0
$919$	−15.5982	−0.514537	−0.257269	−	0.966340i	$-0.582823\pi$
$919$	−15.5982	−0.514537	−0.257269	+	0.966340i	$0.582823\pi$
$920$	0	0
$921$	0	0
$922$	−24.9629	−0.822110
$923$	−27.1722	−0.894385
$924$	0	0
$925$	−7.79306	−0.256234
$926$	41.1191	1.35126
$927$	0	0
$928$	−150.538	−4.94167
$929$	27.6947	0.908635	0.454317	−	0.890840i	$-0.349883\pi$
$929$	27.6947	0.908635	0.454317	+	0.890840i	$0.349883\pi$
$930$	0	0
$931$	−24.0821	−0.789258
$932$	101.283	3.31764
$933$	0	0
$934$	77.1303	2.52378
$935$	1.54510	0.0505300
$936$	0	0
$937$	42.9388	1.40275	0.701374	−	0.712793i	$-0.252570\pi$
$937$	42.9388	1.40275	0.701374	+	0.712793i	$0.252570\pi$
$938$	19.1433	0.625051
$939$	0	0
$940$	61.6980	2.01237
$941$	17.6392	0.575023	0.287511	−	0.957777i	$-0.407172\pi$
$941$	17.6392	0.575023	0.287511	+	0.957777i	$0.407172\pi$
$942$	0	0
$943$	0	0
$944$	37.1047	1.20765
$945$	0	0
$946$	35.7810	1.16334
$947$	17.4588	0.567336	0.283668	−	0.958923i	$-0.408449\pi$
$947$	17.4588	0.567336	0.283668	+	0.958923i	$0.408449\pi$
$948$	0	0
$949$	2.23349	0.0725022
$950$	9.46214	0.306992
$951$	0	0
$952$	3.79306	0.122934
$953$	56.9122	1.84357	0.921784	−	0.387705i	$-0.126732\pi$
$953$	56.9122	1.84357	0.921784	+	0.387705i	$0.126732\pi$
$954$	0	0
$955$	−15.1722	−0.490962
$956$	93.5764	3.02648
$957$	0	0
$958$	−83.8244	−2.70824
$959$	−1.64342	−0.0530689
$960$	0	0
$961$	−30.1722	−0.973298
$962$	−41.6006	−1.34126
$963$	0	0
$964$	80.7873	2.60198
$965$	11.5861	0.372971
$966$	0	0
$967$	−17.5041	−0.562893	−0.281446	−	0.959577i	$-0.590814\pi$
$967$	−17.5041	−0.562893	−0.281446	+	0.959577i	$0.590814\pi$
$968$	−71.8139	−2.30819
$969$	0	0
$970$	−41.6006	−1.33571
$971$	51.1722	1.64220	0.821098	−	0.570788i	$-0.193362\pi$
$971$	51.1722	1.64220	0.821098	+	0.570788i	$0.193362\pi$
$972$	0	0
$973$	2.23349	0.0716025
$974$	45.3526	1.45319
$975$	0	0
$976$	115.103	3.68435
$977$	−44.4507	−1.42210	−0.711052	−	0.703139i	$-0.751781\pi$
$977$	−44.4507	−1.42210	−0.711052	+	0.703139i	$0.751781\pi$
$978$	0	0
$979$	21.2706	0.679811
$980$	34.8075	1.11189
$981$	0	0
$982$	24.7439	0.789610
$983$	−3.45096	−0.110069	−0.0550343	−	0.998484i	$-0.517527\pi$
$983$	−3.45096	−0.110069	−0.0550343	+	0.998484i	$0.517527\pi$
$984$	0	0
$985$	0.180384	0.00574750
$986$	−26.1385	−0.832418
$987$	0	0
$988$	36.3300	1.15581
$989$	0	0
$990$	0	0
$991$	−48.9919	−1.55628	−0.778139	−	0.628092i	$-0.783836\pi$
$991$	−48.9919	−1.55628	−0.778139	+	0.628092i	$0.783836\pi$
$992$	−13.9855	−0.444041
$993$	0	0
$994$	−16.4959	−0.523219
$995$	22.2624	0.705767
$996$	0	0
$997$	36.6787	1.16163	0.580813	−	0.814037i	$-0.302735\pi$
$997$	36.6787	1.16163	0.580813	+	0.814037i	$0.302735\pi$
$998$	−84.3831	−2.67110
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.a.k.1.3	✓	3
3.2	odd	2		765.2.a.l.1.1	yes	3
5.4	even	2		3825.2.a.be.1.1		3
15.14	odd	2		3825.2.a.bf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
765.2.a.k.1.3	✓	3	1.1	even	1	trivial
765.2.a.l.1.1	yes	3	3.2	odd	2
3825.2.a.be.1.1		3	5.4	even	2
3825.2.a.bf.1.3		3	15.14	odd	2

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:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66908 q^{2} +5.12398 q^{4} -1.00000 q^{5} +0.454904 q^{7} +8.33816 q^{8} -2.66908 q^{10} -1.54510 q^{11} +2.00000 q^{13} +1.21417 q^{14} +12.0072 q^{16} +1.00000 q^{17} +3.54510 q^{19} -5.12398 q^{20} -4.12398 q^{22} +1.00000 q^{25} +5.33816 q^{26} +2.33092 q^{28} -9.79306 q^{29} -0.909808 q^{31} +15.3719 q^{32} +2.66908 q^{34} -0.454904 q^{35} -7.79306 q^{37} +9.46214 q^{38} -8.33816 q^{40} +6.70287 q^{41} -8.67632 q^{43} -7.91705 q^{44} -12.0410 q^{47} -6.79306 q^{49} +2.66908 q^{50} +10.2480 q^{52} -0.883254 q^{53} +1.54510 q^{55} +3.79306 q^{56} -26.1385 q^{58} +3.09019 q^{59} +9.58612 q^{61} -2.42835 q^{62} +17.0145 q^{64} -2.00000 q^{65} +15.7665 q^{67} +5.12398 q^{68} -1.21417 q^{70} -13.5861 q^{71} +1.11675 q^{73} -20.8003 q^{74} +18.1650 q^{76} -0.702870 q^{77} +8.00000 q^{79} -12.0072 q^{80} +17.8905 q^{82} +5.81962 q^{83} -1.00000 q^{85} -23.1578 q^{86} -12.8833 q^{88} -13.7665 q^{89} +0.909808 q^{91} -32.1385 q^{94} -3.54510 q^{95} +15.5861 q^{97} -18.1312 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{5} + 9 q^{8} - 6 q^{11} + 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{17} + 12 q^{19} - 6 q^{20} - 3 q^{22} + 3 q^{25} + 15 q^{28} - 12 q^{29} + 18 q^{32} - 6 q^{37} + 3 q^{38} - 9 q^{40}+ \cdots - 21 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^5 + 9 * q^8 - 6 * q^11 + 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^17 + 12 * q^19 - 6 * q^20 - 3 * q^22 + 3 * q^25 + 15 * q^28 - 12 * q^29 + 18 * q^32 - 6 * q^37 + 3 * q^38 - 9 * q^40 + 6 * q^43 + 3 * q^44 - 3 * q^49 + 12 * q^52 + 12 * q^53 + 6 * q^55 - 6 * q^56 - 21 * q^58 + 12 * q^59 - 6 * q^61 + 6 * q^62 + 3 * q^64 - 6 * q^65 + 18 * q^67 + 6 * q^68 + 3 * q^70 - 6 * q^71 + 18 * q^73 - 21 * q^74 + 9 * q^76 + 18 * q^77 + 24 * q^79 - 12 * q^80 + 15 * q^82 + 12 * q^83 - 3 * q^85 - 48 * q^86 - 24 * q^88 - 12 * q^89 - 39 * q^94 - 12 * q^95 + 12 * q^97 - 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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