LMFDB - Embedded newform 768.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.31662$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +19.8997 q^{5} +19.8997 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +19.8997 q^{5} +19.8997 q^{7} +9.00000 q^{9} -48.0000 q^{11} +79.5990 q^{13} +59.6992 q^{15} -42.0000 q^{17} +92.0000 q^{19} +59.6992 q^{21} -39.7995 q^{23} +271.000 q^{25} +27.0000 q^{27} -19.8997 q^{29} -139.298 q^{31} -144.000 q^{33} +396.000 q^{35} +198.997 q^{37} +238.797 q^{39} +6.00000 q^{41} +92.0000 q^{43} +179.098 q^{45} +39.7995 q^{47} +53.0000 q^{49} -126.000 q^{51} -497.494 q^{53} -955.188 q^{55} +276.000 q^{57} -516.000 q^{59} -358.195 q^{61} +179.098 q^{63} +1584.00 q^{65} -524.000 q^{67} -119.398 q^{69} +994.987 q^{71} +430.000 q^{73} +813.000 q^{75} -955.188 q^{77} -1174.09 q^{79} +81.0000 q^{81} -432.000 q^{83} -835.789 q^{85} -59.6992 q^{87} -630.000 q^{89} +1584.00 q^{91} -417.895 q^{93} +1830.78 q^{95} +862.000 q^{97} -432.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 18 * q^9	$ 2 q + 6 q^{3} + 18 q^{9} - 96 q^{11} - 84 q^{17} + 184 q^{19} + 542 q^{25} + 54 q^{27} - 288 q^{33} + 792 q^{35} + 12 q^{41} + 184 q^{43} + 106 q^{49} - 252 q^{51} + 552 q^{57} - 1032 q^{59} + 3168 q^{65} - 1048 q^{67} + 860 q^{73} + 1626 q^{75} + 162 q^{81} - 864 q^{83} - 1260 q^{89} + 3168 q^{91} + 1724 q^{97} - 864 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 18 * q^9 - 96 * q^11 - 84 * q^17 + 184 * q^19 + 542 * q^25 + 54 * q^27 - 288 * q^33 + 792 * q^35 + 12 * q^41 + 184 * q^43 + 106 * q^49 - 252 * q^51 + 552 * q^57 - 1032 * q^59 + 3168 * q^65 - 1048 * q^67 + 860 * q^73 + 1626 * q^75 + 162 * q^81 - 864 * q^83 - 1260 * q^89 + 3168 * q^91 + 1724 * q^97 - 864 * 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	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	19.8997	1.77989	0.889944	−	0.456070i	$-0.150743\pi$
$5$	19.8997	1.77989	0.889944	+	0.456070i	$0.150743\pi$
$6$	0	0
$7$	19.8997	1.07449	0.537243	−	0.843428i	$-0.319466\pi$
$7$	19.8997	1.07449	0.537243	+	0.843428i	$0.319466\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	0	0
$13$	79.5990	1.69821	0.849107	−	0.528220i	$-0.177141\pi$
$13$	79.5990	1.69821	0.849107	+	0.528220i	$0.177141\pi$
$14$	0	0
$15$	59.6992	1.02762
$16$	0	0
$17$	−42.0000	−0.599206	−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000	−0.599206	−0.299603	+	0.954064i	$0.596854\pi$
$18$	0	0
$19$	92.0000	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$19$	92.0000	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	0	0
$21$	59.6992	0.620354
$22$	0	0
$23$	−39.7995	−0.360816	−0.180408	−	0.983592i	$-0.557742\pi$
$23$	−39.7995	−0.360816	−0.180408	+	0.983592i	$0.557742\pi$
$24$	0	0
$25$	271.000	2.16800
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−19.8997	−0.127424	−0.0637119	−	0.997968i	$-0.520294\pi$
$29$	−19.8997	−0.127424	−0.0637119	+	0.997968i	$0.520294\pi$
$30$	0	0
$31$	−139.298	−0.807055	−0.403527	−	0.914968i	$-0.632216\pi$
$31$	−139.298	−0.807055	−0.403527	+	0.914968i	$0.632216\pi$
$32$	0	0
$33$	−144.000	−0.759612
$34$	0	0
$35$	396.000	1.91246
$36$	0	0
$37$	198.997	0.884189	0.442094	−	0.896969i	$-0.354236\pi$
$37$	198.997	0.884189	0.442094	+	0.896969i	$0.354236\pi$
$38$	0	0
$39$	238.797	0.980465
$40$	0	0
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	0	0
$43$	92.0000	0.326276	0.163138	−	0.986603i	$-0.447838\pi$
$43$	92.0000	0.326276	0.163138	+	0.986603i	$0.447838\pi$
$44$	0	0
$45$	179.098	0.593296
$46$	0	0
$47$	39.7995	0.123518	0.0617591	−	0.998091i	$-0.480329\pi$
$47$	39.7995	0.123518	0.0617591	+	0.998091i	$0.480329\pi$
$48$	0	0
$49$	53.0000	0.154519
$50$	0	0
$51$	−126.000	−0.345952
$52$	0	0
$53$	−497.494	−1.28936	−0.644679	−	0.764453i	$-0.723009\pi$
$53$	−497.494	−1.28936	−0.644679	+	0.764453i	$0.723009\pi$
$54$	0	0
$55$	−955.188	−2.34177
$56$	0	0
$57$	276.000	0.641353
$58$	0	0
$59$	−516.000	−1.13860	−0.569301	−	0.822129i	$-0.692786\pi$
$59$	−516.000	−1.13860	−0.569301	+	0.822129i	$0.692786\pi$
$60$	0	0
$61$	−358.195	−0.751840	−0.375920	−	0.926652i	$-0.622673\pi$
$61$	−358.195	−0.751840	−0.375920	+	0.926652i	$0.622673\pi$
$62$	0	0
$63$	179.098	0.358162
$64$	0	0
$65$	1584.00	3.02263
$66$	0	0
$67$	−524.000	−0.955474	−0.477737	−	0.878503i	$-0.658543\pi$
$67$	−524.000	−0.955474	−0.477737	+	0.878503i	$0.658543\pi$
$68$	0	0
$69$	−119.398	−0.208317
$70$	0	0
$71$	994.987	1.66314	0.831572	−	0.555416i	$-0.187441\pi$
$71$	994.987	1.66314	0.831572	+	0.555416i	$0.187441\pi$
$72$	0	0
$73$	430.000	0.689420	0.344710	−	0.938709i	$-0.387977\pi$
$73$	430.000	0.689420	0.344710	+	0.938709i	$0.387977\pi$
$74$	0	0
$75$	813.000	1.25170
$76$	0	0
$77$	−955.188	−1.41369
$78$	0	0
$79$	−1174.09	−1.67209	−0.836044	−	0.548663i	$-0.815137\pi$
$79$	−1174.09	−1.67209	−0.836044	+	0.548663i	$0.815137\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−432.000	−0.571303	−0.285652	−	0.958334i	$-0.592210\pi$
$83$	−432.000	−0.571303	−0.285652	+	0.958334i	$0.592210\pi$
$84$	0	0
$85$	−835.789	−1.06652
$86$	0	0
$87$	−59.6992	−0.0735682
$88$	0	0
$89$	−630.000	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$89$	−630.000	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$90$	0	0
$91$	1584.00	1.82471
$92$	0	0
$93$	−417.895	−0.465953
$94$	0	0
$95$	1830.78	1.97720
$96$	0	0
$97$	862.000	0.902297	0.451149	−	0.892449i	$-0.351014\pi$
$97$	862.000	0.902297	0.451149	+	0.892449i	$0.351014\pi$
$98$	0	0
$99$	−432.000	−0.438562
$100$	0	0
$101$	457.694	0.450914	0.225457	−	0.974253i	$-0.427613\pi$
$101$	457.694	0.450914	0.225457	+	0.974253i	$0.427613\pi$
$102$	0	0
$103$	537.293	0.513991	0.256996	−	0.966413i	$-0.417267\pi$
$103$	537.293	0.513991	0.256996	+	0.966413i	$0.417267\pi$
$104$	0	0
$105$	1188.00	1.10416
$106$	0	0
$107$	−1380.00	−1.24682	−0.623410	−	0.781896i	$-0.714253\pi$
$107$	−1380.00	−1.24682	−0.623410	+	0.781896i	$0.714253\pi$
$108$	0	0
$109$	1273.58	1.11915	0.559574	−	0.828780i	$-0.310965\pi$
$109$	1273.58	1.11915	0.559574	+	0.828780i	$0.310965\pi$
$110$	0	0
$111$	596.992	0.510487
$112$	0	0
$113$	2106.00	1.75324	0.876619	−	0.481186i	$-0.159794\pi$
$113$	2106.00	1.75324	0.876619	+	0.481186i	$0.159794\pi$
$114$	0	0
$115$	−792.000	−0.642212
$116$	0	0
$117$	716.391	0.566072
$118$	0	0
$119$	−835.789	−0.643838
$120$	0	0
$121$	973.000	0.731029
$122$	0	0
$123$	18.0000	0.0131952
$124$	0	0
$125$	2905.36	2.07891
$126$	0	0
$127$	139.298	0.0973285	0.0486643	−	0.998815i	$-0.484504\pi$
$127$	139.298	0.0973285	0.0486643	+	0.998815i	$0.484504\pi$
$128$	0	0
$129$	276.000	0.188376
$130$	0	0
$131$	540.000	0.360153	0.180076	−	0.983653i	$-0.442365\pi$
$131$	540.000	0.360153	0.180076	+	0.983653i	$0.442365\pi$
$132$	0	0
$133$	1830.78	1.19360
$134$	0	0
$135$	537.293	0.342540
$136$	0	0
$137$	1446.00	0.901753	0.450876	−	0.892586i	$-0.351112\pi$
$137$	1446.00	0.901753	0.450876	+	0.892586i	$0.351112\pi$
$138$	0	0
$139$	−1892.00	−1.15451	−0.577257	−	0.816563i	$-0.695877\pi$
$139$	−1892.00	−1.15451	−0.577257	+	0.816563i	$0.695877\pi$
$140$	0	0
$141$	119.398	0.0713132
$142$	0	0
$143$	−3820.75	−2.23432
$144$	0	0
$145$	−396.000	−0.226800
$146$	0	0
$147$	159.000	0.0892116
$148$	0	0
$149$	2805.86	1.54272	0.771360	−	0.636399i	$-0.219577\pi$
$149$	2805.86	1.54272	0.771360	+	0.636399i	$0.219577\pi$
$150$	0	0
$151$	−298.496	−0.160869	−0.0804347	−	0.996760i	$-0.525631\pi$
$151$	−298.496	−0.160869	−0.0804347	+	0.996760i	$0.525631\pi$
$152$	0	0
$153$	−378.000	−0.199735
$154$	0	0
$155$	−2772.00	−1.43647
$156$	0	0
$157$	−2348.17	−1.19366	−0.596829	−	0.802368i	$-0.703573\pi$
$157$	−2348.17	−1.19366	−0.596829	+	0.802368i	$0.703573\pi$
$158$	0	0
$159$	−1492.48	−0.744412
$160$	0	0
$161$	−792.000	−0.387692
$162$	0	0
$163$	3188.00	1.53192	0.765961	−	0.642887i	$-0.222263\pi$
$163$	3188.00	1.53192	0.765961	+	0.642887i	$0.222263\pi$
$164$	0	0
$165$	−2865.56	−1.35202
$166$	0	0
$167$	−1989.97	−0.922089	−0.461045	−	0.887377i	$-0.652525\pi$
$167$	−1989.97	−0.922089	−0.461045	+	0.887377i	$0.652525\pi$
$168$	0	0
$169$	4139.00	1.88393
$170$	0	0
$171$	828.000	0.370285
$172$	0	0
$173$	−378.095	−0.166162	−0.0830811	−	0.996543i	$-0.526476\pi$
$173$	−378.095	−0.166162	−0.0830811	+	0.996543i	$0.526476\pi$
$174$	0	0
$175$	5392.83	2.32948
$176$	0	0
$177$	−1548.00	−0.657372
$178$	0	0
$179$	−2844.00	−1.18754	−0.593772	−	0.804633i	$-0.702362\pi$
$179$	−2844.00	−1.18754	−0.593772	+	0.804633i	$0.702362\pi$
$180$	0	0
$181$	−2228.77	−0.915267	−0.457633	−	0.889141i	$-0.651303\pi$
$181$	−2228.77	−0.915267	−0.457633	+	0.889141i	$0.651303\pi$
$182$	0	0
$183$	−1074.59	−0.434075
$184$	0	0
$185$	3960.00	1.57376
$186$	0	0
$187$	2016.00	0.788366
$188$	0	0
$189$	537.293	0.206785
$190$	0	0
$191$	−875.589	−0.331704	−0.165852	−	0.986151i	$-0.553037\pi$
$191$	−875.589	−0.331704	−0.165852	+	0.986151i	$0.553037\pi$
$192$	0	0
$193$	2882.00	1.07488	0.537438	−	0.843304i	$-0.319392\pi$
$193$	2882.00	1.07488	0.537438	+	0.843304i	$0.319392\pi$
$194$	0	0
$195$	4752.00	1.74512
$196$	0	0
$197$	−4238.65	−1.53295	−0.766475	−	0.642274i	$-0.777991\pi$
$197$	−4238.65	−1.53295	−0.766475	+	0.642274i	$0.777991\pi$
$198$	0	0
$199$	3243.66	1.15546	0.577731	−	0.816227i	$-0.303938\pi$
$199$	3243.66	1.15546	0.577731	+	0.816227i	$0.303938\pi$
$200$	0	0
$201$	−1572.00	−0.551643
$202$	0	0
$203$	−396.000	−0.136915
$204$	0	0
$205$	119.398	0.0406788
$206$	0	0
$207$	−358.195	−0.120272
$208$	0	0
$209$	−4416.00	−1.46154
$210$	0	0
$211$	2356.00	0.768691	0.384345	−	0.923189i	$-0.374427\pi$
$211$	2356.00	0.768691	0.384345	+	0.923189i	$0.374427\pi$
$212$	0	0
$213$	2984.96	0.960217
$214$	0	0
$215$	1830.78	0.580735
$216$	0	0
$217$	−2772.00	−0.867169
$218$	0	0
$219$	1290.00	0.398037
$220$	0	0
$221$	−3343.16	−1.01758
$222$	0	0
$223$	−1771.08	−0.531839	−0.265920	−	0.963995i	$-0.585676\pi$
$223$	−1771.08	−0.531839	−0.265920	+	0.963995i	$0.585676\pi$
$224$	0	0
$225$	2439.00	0.722667
$226$	0	0
$227$	−3864.00	−1.12979	−0.564896	−	0.825162i	$-0.691084\pi$
$227$	−3864.00	−1.12979	−0.564896	+	0.825162i	$0.691084\pi$
$228$	0	0
$229$	557.193	0.160788	0.0803938	−	0.996763i	$-0.474382\pi$
$229$	557.193	0.160788	0.0803938	+	0.996763i	$0.474382\pi$
$230$	0	0
$231$	−2865.56	−0.816192
$232$	0	0
$233$	−2814.00	−0.791207	−0.395604	−	0.918421i	$-0.629465\pi$
$233$	−2814.00	−0.791207	−0.395604	+	0.918421i	$0.629465\pi$
$234$	0	0
$235$	792.000	0.219848
$236$	0	0
$237$	−3522.26	−0.965380
$238$	0	0
$239$	4616.74	1.24951	0.624754	−	0.780822i	$-0.285199\pi$
$239$	4616.74	1.24951	0.624754	+	0.780822i	$0.285199\pi$
$240$	0	0
$241$	362.000	0.0967571	0.0483786	−	0.998829i	$-0.484595\pi$
$241$	362.000	0.0967571	0.0483786	+	0.998829i	$0.484595\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1054.69	0.275026
$246$	0	0
$247$	7323.11	1.88647
$248$	0	0
$249$	−1296.00	−0.329842
$250$	0	0
$251$	5136.00	1.29156	0.645780	−	0.763524i	$-0.276532\pi$
$251$	5136.00	1.29156	0.645780	+	0.763524i	$0.276532\pi$
$252$	0	0
$253$	1910.38	0.474721
$254$	0	0
$255$	−2507.37	−0.615755
$256$	0	0
$257$	−174.000	−0.0422328	−0.0211164	−	0.999777i	$-0.506722\pi$
$257$	−174.000	−0.0422328	−0.0211164	+	0.999777i	$0.506722\pi$
$258$	0	0
$259$	3960.00	0.950048
$260$	0	0
$261$	−179.098	−0.0424746
$262$	0	0
$263$	−4775.94	−1.11976	−0.559880	−	0.828573i	$-0.689153\pi$
$263$	−4775.94	−1.11976	−0.559880	+	0.828573i	$0.689153\pi$
$264$	0	0
$265$	−9900.00	−2.29491
$266$	0	0
$267$	−1890.00	−0.433206
$268$	0	0
$269$	−1134.29	−0.257095	−0.128548	−	0.991703i	$-0.541032\pi$
$269$	−1134.29	−0.257095	−0.128548	+	0.991703i	$0.541032\pi$
$270$	0	0
$271$	−179.098	−0.0401454	−0.0200727	−	0.999799i	$-0.506390\pi$
$271$	−179.098	−0.0401454	−0.0200727	+	0.999799i	$0.506390\pi$
$272$	0	0
$273$	4752.00	1.05349
$274$	0	0
$275$	−13008.0	−2.85241
$276$	0	0
$277$	−6367.92	−1.38127	−0.690634	−	0.723205i	$-0.742668\pi$
$277$	−6367.92	−1.38127	−0.690634	+	0.723205i	$0.742668\pi$
$278$	0	0
$279$	−1253.68	−0.269018
$280$	0	0
$281$	−558.000	−0.118461	−0.0592304	−	0.998244i	$-0.518865\pi$
$281$	−558.000	−0.118461	−0.0592304	+	0.998244i	$0.518865\pi$
$282$	0	0
$283$	2068.00	0.434381	0.217191	−	0.976129i	$-0.430311\pi$
$283$	2068.00	0.434381	0.217191	+	0.976129i	$0.430311\pi$
$284$	0	0
$285$	5492.33	1.14154
$286$	0	0
$287$	119.398	0.0245570
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	2586.00	0.520942
$292$	0	0
$293$	−3363.06	−0.670553	−0.335276	−	0.942120i	$-0.608830\pi$
$293$	−3363.06	−0.670553	−0.335276	+	0.942120i	$0.608830\pi$
$294$	0	0
$295$	−10268.3	−2.02658
$296$	0	0
$297$	−1296.00	−0.253204
$298$	0	0
$299$	−3168.00	−0.612743
$300$	0	0
$301$	1830.78	0.350579
$302$	0	0
$303$	1373.08	0.260335
$304$	0	0
$305$	−7128.00	−1.33819
$306$	0	0
$307$	−6860.00	−1.27531	−0.637656	−	0.770321i	$-0.720096\pi$
$307$	−6860.00	−1.27531	−0.637656	+	0.770321i	$0.720096\pi$
$308$	0	0
$309$	1611.88	0.296753
$310$	0	0
$311$	10427.5	1.90125	0.950623	−	0.310348i	$-0.100446\pi$
$311$	10427.5	1.90125	0.950623	+	0.310348i	$0.100446\pi$
$312$	0	0
$313$	−6262.00	−1.13083	−0.565414	−	0.824807i	$-0.691284\pi$
$313$	−6262.00	−1.13083	−0.565414	+	0.824807i	$0.691284\pi$
$314$	0	0
$315$	3564.00	0.637488
$316$	0	0
$317$	776.090	0.137507	0.0687533	−	0.997634i	$-0.478098\pi$
$317$	776.090	0.137507	0.0687533	+	0.997634i	$0.478098\pi$
$318$	0	0
$319$	955.188	0.167650
$320$	0	0
$321$	−4140.00	−0.719851
$322$	0	0
$323$	−3864.00	−0.665631
$324$	0	0
$325$	21571.3	3.68173
$326$	0	0
$327$	3820.75	0.646141
$328$	0	0
$329$	792.000	0.132718
$330$	0	0
$331$	6460.00	1.07273	0.536365	−	0.843986i	$-0.319797\pi$
$331$	6460.00	1.07273	0.536365	+	0.843986i	$0.319797\pi$
$332$	0	0
$333$	1790.98	0.294730
$334$	0	0
$335$	−10427.5	−1.70064
$336$	0	0
$337$	−74.0000	−0.0119615	−0.00598077	−	0.999982i	$-0.501904\pi$
$337$	−74.0000	−0.0119615	−0.00598077	+	0.999982i	$0.501904\pi$
$338$	0	0
$339$	6318.00	1.01223
$340$	0	0
$341$	6686.32	1.06183
$342$	0	0
$343$	−5770.93	−0.908457
$344$	0	0
$345$	−2376.00	−0.370781
$346$	0	0
$347$	−6888.00	−1.06561	−0.532806	−	0.846238i	$-0.678862\pi$
$347$	−6888.00	−1.06561	−0.532806	+	0.846238i	$0.678862\pi$
$348$	0	0
$349$	−676.591	−0.103774	−0.0518870	−	0.998653i	$-0.516524\pi$
$349$	−676.591	−0.103774	−0.0518870	+	0.998653i	$0.516524\pi$
$350$	0	0
$351$	2149.17	0.326822
$352$	0	0
$353$	−1830.00	−0.275924	−0.137962	−	0.990438i	$-0.544055\pi$
$353$	−1830.00	−0.275924	−0.137962	+	0.990438i	$0.544055\pi$
$354$	0	0
$355$	19800.0	2.96021
$356$	0	0
$357$	−2507.37	−0.371720
$358$	0	0
$359$	−6646.52	−0.977130	−0.488565	−	0.872527i	$-0.662479\pi$
$359$	−6646.52	−0.977130	−0.488565	+	0.872527i	$0.662479\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	2919.00	0.422060
$364$	0	0
$365$	8556.89	1.22709
$366$	0	0
$367$	−3164.06	−0.450034	−0.225017	−	0.974355i	$-0.572244\pi$
$367$	−3164.06	−0.450034	−0.225017	+	0.974355i	$0.572244\pi$
$368$	0	0
$369$	54.0000	0.00761823
$370$	0	0
$371$	−9900.00	−1.38540
$372$	0	0
$373$	−9750.88	−1.35357	−0.676785	−	0.736181i	$-0.736627\pi$
$373$	−9750.88	−1.35357	−0.676785	+	0.736181i	$0.736627\pi$
$374$	0	0
$375$	8716.09	1.20026
$376$	0	0
$377$	−1584.00	−0.216393
$378$	0	0
$379$	−8332.00	−1.12925	−0.564625	−	0.825347i	$-0.690979\pi$
$379$	−8332.00	−1.12925	−0.564625	+	0.825347i	$0.690979\pi$
$380$	0	0
$381$	417.895	0.0561926
$382$	0	0
$383$	6765.91	0.902669	0.451334	−	0.892355i	$-0.350948\pi$
$383$	6765.91	0.902669	0.451334	+	0.892355i	$0.350948\pi$
$384$	0	0
$385$	−19008.0	−2.51620
$386$	0	0
$387$	828.000	0.108759
$388$	0	0
$389$	−5870.43	−0.765148	−0.382574	−	0.923925i	$-0.624962\pi$
$389$	−5870.43	−0.765148	−0.382574	+	0.923925i	$0.624962\pi$
$390$	0	0
$391$	1671.58	0.216203
$392$	0	0
$393$	1620.00	0.207934
$394$	0	0
$395$	−23364.0	−2.97613
$396$	0	0
$397$	2666.57	0.337106	0.168553	−	0.985693i	$-0.446091\pi$
$397$	2666.57	0.337106	0.168553	+	0.985693i	$0.446091\pi$
$398$	0	0
$399$	5492.33	0.689124
$400$	0	0
$401$	−2634.00	−0.328019	−0.164010	−	0.986459i	$-0.552443\pi$
$401$	−2634.00	−0.328019	−0.164010	+	0.986459i	$0.552443\pi$
$402$	0	0
$403$	−11088.0	−1.37055
$404$	0	0
$405$	1611.88	0.197765
$406$	0	0
$407$	−9551.88	−1.16331
$408$	0	0
$409$	−506.000	−0.0611738	−0.0305869	−	0.999532i	$-0.509738\pi$
$409$	−506.000	−0.0611738	−0.0305869	+	0.999532i	$0.509738\pi$
$410$	0	0
$411$	4338.00	0.520627
$412$	0	0
$413$	−10268.3	−1.22341
$414$	0	0
$415$	−8596.69	−1.01686
$416$	0	0
$417$	−5676.00	−0.666559
$418$	0	0
$419$	15480.0	1.80489	0.902443	−	0.430809i	$-0.141772\pi$
$419$	15480.0	1.80489	0.902443	+	0.430809i	$0.141772\pi$
$420$	0	0
$421$	−10666.3	−1.23478	−0.617390	−	0.786658i	$-0.711810\pi$
$421$	−10666.3	−1.23478	−0.617390	+	0.786658i	$0.711810\pi$
$422$	0	0
$423$	358.195	0.0411727
$424$	0	0
$425$	−11382.0	−1.29908
$426$	0	0
$427$	−7128.00	−0.807841
$428$	0	0
$429$	−11462.3	−1.28998
$430$	0	0
$431$	17153.6	1.91707	0.958537	−	0.284968i	$-0.0919828\pi$
$431$	17153.6	1.91707	0.958537	+	0.284968i	$0.0919828\pi$
$432$	0	0
$433$	2014.00	0.223526	0.111763	−	0.993735i	$-0.464350\pi$
$433$	2014.00	0.223526	0.111763	+	0.993735i	$0.464350\pi$
$434$	0	0
$435$	−1188.00	−0.130943
$436$	0	0
$437$	−3661.55	−0.400814
$438$	0	0
$439$	776.090	0.0843753	0.0421877	−	0.999110i	$-0.486567\pi$
$439$	776.090	0.0843753	0.0421877	+	0.999110i	$0.486567\pi$
$440$	0	0
$441$	477.000	0.0515063
$442$	0	0
$443$	408.000	0.0437577	0.0218789	−	0.999761i	$-0.493035\pi$
$443$	408.000	0.0437577	0.0218789	+	0.999761i	$0.493035\pi$
$444$	0	0
$445$	−12536.8	−1.33551
$446$	0	0
$447$	8417.59	0.890690
$448$	0	0
$449$	−11658.0	−1.22533	−0.612667	−	0.790341i	$-0.709903\pi$
$449$	−11658.0	−1.22533	−0.612667	+	0.790341i	$0.709903\pi$
$450$	0	0
$451$	−288.000	−0.0300696
$452$	0	0
$453$	−895.489	−0.0928780
$454$	0	0
$455$	31521.2	3.24777
$456$	0	0
$457$	16486.0	1.68749	0.843745	−	0.536745i	$-0.180346\pi$
$457$	16486.0	1.68749	0.843745	+	0.536745i	$0.180346\pi$
$458$	0	0
$459$	−1134.00	−0.115317
$460$	0	0
$461$	−4119.25	−0.416166	−0.208083	−	0.978111i	$-0.566722\pi$
$461$	−4119.25	−0.416166	−0.208083	+	0.978111i	$0.566722\pi$
$462$	0	0
$463$	−13472.1	−1.35227	−0.676137	−	0.736776i	$-0.736347\pi$
$463$	−13472.1	−1.35227	−0.676137	+	0.736776i	$0.736347\pi$
$464$	0	0
$465$	−8316.00	−0.829345
$466$	0	0
$467$	−4440.00	−0.439954	−0.219977	−	0.975505i	$-0.570598\pi$
$467$	−4440.00	−0.439954	−0.219977	+	0.975505i	$0.570598\pi$
$468$	0	0
$469$	−10427.5	−1.02664
$470$	0	0
$471$	−7044.51	−0.689159
$472$	0	0
$473$	−4416.00	−0.429277
$474$	0	0
$475$	24932.0	2.40833
$476$	0	0
$477$	−4477.44	−0.429786
$478$	0	0
$479$	−10626.5	−1.01364	−0.506822	−	0.862051i	$-0.669180\pi$
$479$	−10626.5	−1.01364	−0.506822	+	0.862051i	$0.669180\pi$
$480$	0	0
$481$	15840.0	1.50154
$482$	0	0
$483$	−2376.00	−0.223834
$484$	0	0
$485$	17153.6	1.60599
$486$	0	0
$487$	−1333.28	−0.124059	−0.0620296	−	0.998074i	$-0.519757\pi$
$487$	−1333.28	−0.124059	−0.0620296	+	0.998074i	$0.519757\pi$
$488$	0	0
$489$	9564.00	0.884456
$490$	0	0
$491$	−7404.00	−0.680525	−0.340263	−	0.940330i	$-0.610516\pi$
$491$	−7404.00	−0.680525	−0.340263	+	0.940330i	$0.610516\pi$
$492$	0	0
$493$	835.789	0.0763531
$494$	0	0
$495$	−8596.69	−0.780591
$496$	0	0
$497$	19800.0	1.78702
$498$	0	0
$499$	5740.00	0.514945	0.257473	−	0.966286i	$-0.417110\pi$
$499$	5740.00	0.514945	0.257473	+	0.966286i	$0.417110\pi$
$500$	0	0
$501$	−5969.92	−0.532368
$502$	0	0
$503$	−16039.2	−1.42177	−0.710887	−	0.703306i	$-0.751706\pi$
$503$	−16039.2	−1.42177	−0.710887	+	0.703306i	$0.751706\pi$
$504$	0	0
$505$	9108.00	0.802576
$506$	0	0
$507$	12417.0	1.08769
$508$	0	0
$509$	16377.5	1.42617	0.713084	−	0.701078i	$-0.247298\pi$
$509$	16377.5	1.42617	0.713084	+	0.701078i	$0.247298\pi$
$510$	0	0
$511$	8556.89	0.740772
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	10692.0	0.914846
$516$	0	0
$517$	−1910.38	−0.162511
$518$	0	0
$519$	−1134.29	−0.0959337
$520$	0	0
$521$	−22098.0	−1.85822	−0.929108	−	0.369807i	$-0.879424\pi$
$521$	−22098.0	−1.85822	−0.929108	+	0.369807i	$0.879424\pi$
$522$	0	0
$523$	16436.0	1.37418	0.687090	−	0.726572i	$-0.258888\pi$
$523$	16436.0	1.37418	0.687090	+	0.726572i	$0.258888\pi$
$524$	0	0
$525$	16178.5	1.34493
$526$	0	0
$527$	5850.53	0.483592
$528$	0	0
$529$	−10583.0	−0.869812
$530$	0	0
$531$	−4644.00	−0.379534
$532$	0	0
$533$	477.594	0.0388122
$534$	0	0
$535$	−27461.7	−2.21920
$536$	0	0
$537$	−8532.00	−0.685629
$538$	0	0
$539$	−2544.00	−0.203298
$540$	0	0
$541$	11143.9	0.885604	0.442802	−	0.896619i	$-0.353984\pi$
$541$	11143.9	0.885604	0.442802	+	0.896619i	$0.353984\pi$
$542$	0	0
$543$	−6686.32	−0.528430
$544$	0	0
$545$	25344.0	1.99196
$546$	0	0
$547$	1316.00	0.102867	0.0514334	−	0.998676i	$-0.483621\pi$
$547$	1316.00	0.102867	0.0514334	+	0.998676i	$0.483621\pi$
$548$	0	0
$549$	−3223.76	−0.250613
$550$	0	0
$551$	−1830.78	−0.141549
$552$	0	0
$553$	−23364.0	−1.79663
$554$	0	0
$555$	11880.0	0.908609
$556$	0	0
$557$	−3402.86	−0.258858	−0.129429	−	0.991589i	$-0.541314\pi$
$557$	−3402.86	−0.258858	−0.129429	+	0.991589i	$0.541314\pi$
$558$	0	0
$559$	7323.11	0.554087
$560$	0	0
$561$	6048.00	0.455164
$562$	0	0
$563$	−19560.0	−1.46422	−0.732110	−	0.681187i	$-0.761464\pi$
$563$	−19560.0	−1.46422	−0.732110	+	0.681187i	$0.761464\pi$
$564$	0	0
$565$	41908.9	3.12057
$566$	0	0
$567$	1611.88	0.119387
$568$	0	0
$569$	20838.0	1.53528	0.767640	−	0.640881i	$-0.221431\pi$
$569$	20838.0	1.53528	0.767640	+	0.640881i	$0.221431\pi$
$570$	0	0
$571$	−1100.00	−0.0806192	−0.0403096	−	0.999187i	$-0.512834\pi$
$571$	−1100.00	−0.0806192	−0.0403096	+	0.999187i	$0.512834\pi$
$572$	0	0
$573$	−2626.77	−0.191509
$574$	0	0
$575$	−10785.7	−0.782249
$576$	0	0
$577$	−1730.00	−0.124819	−0.0624097	−	0.998051i	$-0.519879\pi$
$577$	−1730.00	−0.124819	−0.0624097	+	0.998051i	$0.519879\pi$
$578$	0	0
$579$	8646.00	0.620579
$580$	0	0
$581$	−8596.69	−0.613857
$582$	0	0
$583$	23879.7	1.69639
$584$	0	0
$585$	14256.0	1.00754
$586$	0	0
$587$	−7644.00	−0.537482	−0.268741	−	0.963213i	$-0.586607\pi$
$587$	−7644.00	−0.537482	−0.268741	+	0.963213i	$0.586607\pi$
$588$	0	0
$589$	−12815.4	−0.896521
$590$	0	0
$591$	−12715.9	−0.885049
$592$	0	0
$593$	26874.0	1.86102	0.930508	−	0.366271i	$-0.119366\pi$
$593$	26874.0	1.86102	0.930508	+	0.366271i	$0.119366\pi$
$594$	0	0
$595$	−16632.0	−1.14596
$596$	0	0
$597$	9730.98	0.667106
$598$	0	0
$599$	9750.88	0.665125	0.332563	−	0.943081i	$-0.392087\pi$
$599$	9750.88	0.665125	0.332563	+	0.943081i	$0.392087\pi$
$600$	0	0
$601$	−650.000	−0.0441166	−0.0220583	−	0.999757i	$-0.507022\pi$
$601$	−650.000	−0.0441166	−0.0220583	+	0.999757i	$0.507022\pi$
$602$	0	0
$603$	−4716.00	−0.318491
$604$	0	0
$605$	19362.5	1.30115
$606$	0	0
$607$	16695.9	1.11642	0.558209	−	0.829701i	$-0.311489\pi$
$607$	16695.9	1.11642	0.558209	+	0.829701i	$0.311489\pi$
$608$	0	0
$609$	−1188.00	−0.0790479
$610$	0	0
$611$	3168.00	0.209760
$612$	0	0
$613$	−7522.11	−0.495620	−0.247810	−	0.968809i	$-0.579711\pi$
$613$	−7522.11	−0.495620	−0.247810	+	0.968809i	$0.579711\pi$
$614$	0	0
$615$	358.195	0.0234859
$616$	0	0
$617$	−2526.00	−0.164818	−0.0824092	−	0.996599i	$-0.526261\pi$
$617$	−2526.00	−0.164818	−0.0824092	+	0.996599i	$0.526261\pi$
$618$	0	0
$619$	4372.00	0.283886	0.141943	−	0.989875i	$-0.454665\pi$
$619$	4372.00	0.283886	0.141943	+	0.989875i	$0.454665\pi$
$620$	0	0
$621$	−1074.59	−0.0694391
$622$	0	0
$623$	−12536.8	−0.806225
$624$	0	0
$625$	23941.0	1.53222
$626$	0	0
$627$	−13248.0	−0.843818
$628$	0	0
$629$	−8357.89	−0.529811
$630$	0	0
$631$	23979.2	1.51283	0.756416	−	0.654091i	$-0.226949\pi$
$631$	23979.2	1.51283	0.756416	+	0.654091i	$0.226949\pi$
$632$	0	0
$633$	7068.00	0.443804
$634$	0	0
$635$	2772.00	0.173234
$636$	0	0
$637$	4218.75	0.262406
$638$	0	0
$639$	8954.89	0.554382
$640$	0	0
$641$	1374.00	0.0846642	0.0423321	−	0.999104i	$-0.486521\pi$
$641$	1374.00	0.0846642	0.0423321	+	0.999104i	$0.486521\pi$
$642$	0	0
$643$	4196.00	0.257347	0.128673	−	0.991687i	$-0.458928\pi$
$643$	4196.00	0.257347	0.128673	+	0.991687i	$0.458928\pi$
$644$	0	0
$645$	5492.33	0.335287
$646$	0	0
$647$	−17392.4	−1.05682	−0.528412	−	0.848988i	$-0.677212\pi$
$647$	−17392.4	−1.05682	−0.528412	+	0.848988i	$0.677212\pi$
$648$	0	0
$649$	24768.0	1.49804
$650$	0	0
$651$	−8316.00	−0.500660
$652$	0	0
$653$	−18685.9	−1.11981	−0.559904	−	0.828558i	$-0.689162\pi$
$653$	−18685.9	−1.11981	−0.559904	+	0.828558i	$0.689162\pi$
$654$	0	0
$655$	10745.9	0.641032
$656$	0	0
$657$	3870.00	0.229807
$658$	0	0
$659$	16380.0	0.968246	0.484123	−	0.875000i	$-0.339139\pi$
$659$	16380.0	0.968246	0.484123	+	0.875000i	$0.339139\pi$
$660$	0	0
$661$	−5850.53	−0.344265	−0.172132	−	0.985074i	$-0.555066\pi$
$661$	−5850.53	−0.344265	−0.172132	+	0.985074i	$0.555066\pi$
$662$	0	0
$663$	−10029.5	−0.587500
$664$	0	0
$665$	36432.0	2.12447
$666$	0	0
$667$	792.000	0.0459766
$668$	0	0
$669$	−5313.23	−0.307057
$670$	0	0
$671$	17193.4	0.989185
$672$	0	0
$673$	−13250.0	−0.758915	−0.379458	−	0.925209i	$-0.623889\pi$
$673$	−13250.0	−0.758915	−0.379458	+	0.925209i	$0.623889\pi$
$674$	0	0
$675$	7317.00	0.417232
$676$	0	0
$677$	14347.7	0.814516	0.407258	−	0.913313i	$-0.366485\pi$
$677$	14347.7	0.814516	0.407258	+	0.913313i	$0.366485\pi$
$678$	0	0
$679$	17153.6	0.969505
$680$	0	0
$681$	−11592.0	−0.652285
$682$	0	0
$683$	28272.0	1.58389	0.791946	−	0.610591i	$-0.209068\pi$
$683$	28272.0	1.58389	0.791946	+	0.610591i	$0.209068\pi$
$684$	0	0
$685$	28775.0	1.60502
$686$	0	0
$687$	1671.58	0.0928307
$688$	0	0
$689$	−39600.0	−2.18961
$690$	0	0
$691$	−16828.0	−0.926436	−0.463218	−	0.886244i	$-0.653305\pi$
$691$	−16828.0	−0.926436	−0.463218	+	0.886244i	$0.653305\pi$
$692$	0	0
$693$	−8596.69	−0.471228
$694$	0	0
$695$	−37650.3	−2.05490
$696$	0	0
$697$	−252.000	−0.0136947
$698$	0	0
$699$	−8442.00	−0.456804
$700$	0	0
$701$	−18088.9	−0.974618	−0.487309	−	0.873230i	$-0.662021\pi$
$701$	−18088.9	−0.974618	−0.487309	+	0.873230i	$0.662021\pi$
$702$	0	0
$703$	18307.8	0.982206
$704$	0	0
$705$	2376.00	0.126930
$706$	0	0
$707$	9108.00	0.484500
$708$	0	0
$709$	−36854.3	−1.95218	−0.976089	−	0.217373i	$-0.930251\pi$
$709$	−36854.3	−1.95218	−0.976089	+	0.217373i	$0.930251\pi$
$710$	0	0
$711$	−10566.8	−0.557362
$712$	0	0
$713$	5544.00	0.291198
$714$	0	0
$715$	−76032.0	−3.97683
$716$	0	0
$717$	13850.2	0.721403
$718$	0	0
$719$	−21849.9	−1.13333	−0.566665	−	0.823948i	$-0.691767\pi$
$719$	−21849.9	−1.13333	−0.566665	+	0.823948i	$0.691767\pi$
$720$	0	0
$721$	10692.0	0.552276
$722$	0	0
$723$	1086.00	0.0558628
$724$	0	0
$725$	−5392.83	−0.276255
$726$	0	0
$727$	35282.3	1.79993	0.899963	−	0.435966i	$-0.143593\pi$
$727$	35282.3	1.79993	0.899963	+	0.435966i	$0.143593\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3864.00	−0.195506
$732$	0	0
$733$	8198.70	0.413132	0.206566	−	0.978433i	$-0.433771\pi$
$733$	8198.70	0.413132	0.206566	+	0.978433i	$0.433771\pi$
$734$	0	0
$735$	3164.06	0.158787
$736$	0	0
$737$	25152.0	1.25710
$738$	0	0
$739$	35404.0	1.76232	0.881162	−	0.472815i	$-0.156762\pi$
$739$	35404.0	1.76232	0.881162	+	0.472815i	$0.156762\pi$
$740$	0	0
$741$	21969.3	1.08915
$742$	0	0
$743$	9711.08	0.479495	0.239748	−	0.970835i	$-0.422935\pi$
$743$	9711.08	0.479495	0.239748	+	0.970835i	$0.422935\pi$
$744$	0	0
$745$	55836.0	2.74587
$746$	0	0
$747$	−3888.00	−0.190434
$748$	0	0
$749$	−27461.7	−1.33969
$750$	0	0
$751$	−21631.0	−1.05104	−0.525518	−	0.850783i	$-0.676128\pi$
$751$	−21631.0	−1.05104	−0.525518	+	0.850783i	$0.676128\pi$
$752$	0	0
$753$	15408.0	0.745682
$754$	0	0
$755$	−5940.00	−0.286329
$756$	0	0
$757$	26984.1	1.29558	0.647789	−	0.761820i	$-0.275694\pi$
$757$	26984.1	1.29558	0.647789	+	0.761820i	$0.275694\pi$
$758$	0	0
$759$	5731.13	0.274080
$760$	0	0
$761$	14382.0	0.685082	0.342541	−	0.939503i	$-0.388713\pi$
$761$	14382.0	0.685082	0.342541	+	0.939503i	$0.388713\pi$
$762$	0	0
$763$	25344.0	1.20251
$764$	0	0
$765$	−7522.11	−0.355506
$766$	0	0
$767$	−41073.1	−1.93359
$768$	0	0
$769$	32978.0	1.54645	0.773223	−	0.634134i	$-0.218643\pi$
$769$	32978.0	1.54645	0.773223	+	0.634134i	$0.218643\pi$
$770$	0	0
$771$	−522.000	−0.0243831
$772$	0	0
$773$	−28914.3	−1.34538	−0.672688	−	0.739926i	$-0.734861\pi$
$773$	−28914.3	−1.34538	−0.672688	+	0.739926i	$0.734861\pi$
$774$	0	0
$775$	−37749.8	−1.74970
$776$	0	0
$777$	11880.0	0.548510
$778$	0	0
$779$	552.000	0.0253883
$780$	0	0
$781$	−47759.4	−2.18818
$782$	0	0
$783$	−537.293	−0.0245227
$784$	0	0
$785$	−46728.0	−2.12458
$786$	0	0
$787$	24572.0	1.11296	0.556479	−	0.830862i	$-0.312152\pi$
$787$	24572.0	1.11296	0.556479	+	0.830862i	$0.312152\pi$
$788$	0	0
$789$	−14327.8	−0.646494
$790$	0	0
$791$	41908.9	1.88383
$792$	0	0
$793$	−28512.0	−1.27679
$794$	0	0
$795$	−29700.0	−1.32497
$796$	0	0
$797$	−19720.7	−0.876463	−0.438232	−	0.898862i	$-0.644395\pi$
$797$	−19720.7	−0.876463	−0.438232	+	0.898862i	$0.644395\pi$
$798$	0	0
$799$	−1671.58	−0.0740128
$800$	0	0
$801$	−5670.00	−0.250112
$802$	0	0
$803$	−20640.0	−0.907061
$804$	0	0
$805$	−15760.6	−0.690047
$806$	0	0
$807$	−3402.86	−0.148434
$808$	0	0
$809$	35286.0	1.53349	0.766743	−	0.641955i	$-0.221876\pi$
$809$	35286.0	1.53349	0.766743	+	0.641955i	$0.221876\pi$
$810$	0	0
$811$	−556.000	−0.0240737	−0.0120369	−	0.999928i	$-0.503832\pi$
$811$	−556.000	−0.0240737	−0.0120369	+	0.999928i	$0.503832\pi$
$812$	0	0
$813$	−537.293	−0.0231780
$814$	0	0
$815$	63440.4	2.72665
$816$	0	0
$817$	8464.00	0.362445
$818$	0	0
$819$	14256.0	0.608236
$820$	0	0
$821$	−33769.9	−1.43554	−0.717769	−	0.696281i	$-0.754837\pi$
$821$	−33769.9	−1.43554	−0.717769	+	0.696281i	$0.754837\pi$
$822$	0	0
$823$	−26247.8	−1.11171	−0.555856	−	0.831278i	$-0.687610\pi$
$823$	−26247.8	−1.11171	−0.555856	+	0.831278i	$0.687610\pi$
$824$	0	0
$825$	−39024.0	−1.64684
$826$	0	0
$827$	1116.00	0.0469252	0.0234626	−	0.999725i	$-0.492531\pi$
$827$	1116.00	0.0469252	0.0234626	+	0.999725i	$0.492531\pi$
$828$	0	0
$829$	−23242.9	−0.973775	−0.486888	−	0.873465i	$-0.661868\pi$
$829$	−23242.9	−0.973775	−0.486888	+	0.873465i	$0.661868\pi$
$830$	0	0
$831$	−19103.8	−0.797475
$832$	0	0
$833$	−2226.00	−0.0925886
$834$	0	0
$835$	−39600.0	−1.64121
$836$	0	0
$837$	−3761.05	−0.155318
$838$	0	0
$839$	−29650.6	−1.22009	−0.610044	−	0.792368i	$-0.708848\pi$
$839$	−29650.6	−1.22009	−0.610044	+	0.792368i	$0.708848\pi$
$840$	0	0
$841$	−23993.0	−0.983763
$842$	0	0
$843$	−1674.00	−0.0683934
$844$	0	0
$845$	82365.1	3.35319
$846$	0	0
$847$	19362.5	0.785480
$848$	0	0
$849$	6204.00	0.250790
$850$	0	0
$851$	−7920.00	−0.319029
$852$	0	0
$853$	17074.0	0.685348	0.342674	−	0.939454i	$-0.388667\pi$
$853$	17074.0	0.685348	0.342674	+	0.939454i	$0.388667\pi$
$854$	0	0
$855$	16477.0	0.659066
$856$	0	0
$857$	1206.00	0.0480702	0.0240351	−	0.999711i	$-0.492349\pi$
$857$	1206.00	0.0480702	0.0240351	+	0.999711i	$0.492349\pi$
$858$	0	0
$859$	−34756.0	−1.38051	−0.690256	−	0.723565i	$-0.742502\pi$
$859$	−34756.0	−1.38051	−0.690256	+	0.723565i	$0.742502\pi$
$860$	0	0
$861$	358.195	0.0141780
$862$	0	0
$863$	1034.79	0.0408164	0.0204082	−	0.999792i	$-0.493503\pi$
$863$	1034.79	0.0408164	0.0204082	+	0.999792i	$0.493503\pi$
$864$	0	0
$865$	−7524.00	−0.295750
$866$	0	0
$867$	−9447.00	−0.370054
$868$	0	0
$869$	56356.1	2.19994
$870$	0	0
$871$	−41709.9	−1.62260
$872$	0	0
$873$	7758.00	0.300766
$874$	0	0
$875$	57816.0	2.23376
$876$	0	0
$877$	19382.4	0.746289	0.373145	−	0.927773i	$-0.378280\pi$
$877$	19382.4	0.746289	0.373145	+	0.927773i	$0.378280\pi$
$878$	0	0
$879$	−10089.2	−0.387144
$880$	0	0
$881$	−5310.00	−0.203063	−0.101531	−	0.994832i	$-0.532374\pi$
$881$	−5310.00	−0.203063	−0.101531	+	0.994832i	$0.532374\pi$
$882$	0	0
$883$	−23308.0	−0.888309	−0.444154	−	0.895950i	$-0.646496\pi$
$883$	−23308.0	−0.888309	−0.444154	+	0.895950i	$0.646496\pi$
$884$	0	0
$885$	−30804.8	−1.17005
$886$	0	0
$887$	21969.3	0.831632	0.415816	−	0.909449i	$-0.363496\pi$
$887$	21969.3	0.831632	0.415816	+	0.909449i	$0.363496\pi$
$888$	0	0
$889$	2772.00	0.104578
$890$	0	0
$891$	−3888.00	−0.146187
$892$	0	0
$893$	3661.55	0.137211
$894$	0	0
$895$	−56594.9	−2.11370
$896$	0	0
$897$	−9504.00	−0.353767
$898$	0	0
$899$	2772.00	0.102838
$900$	0	0
$901$	20894.7	0.772591
$902$	0	0
$903$	5492.33	0.202407
$904$	0	0
$905$	−44352.0	−1.62907
$906$	0	0
$907$	27596.0	1.01026	0.505132	−	0.863042i	$-0.331444\pi$
$907$	27596.0	1.01026	0.505132	+	0.863042i	$0.331444\pi$
$908$	0	0
$909$	4119.25	0.150305
$910$	0	0
$911$	12099.0	0.440021	0.220011	−	0.975497i	$-0.429391\pi$
$911$	12099.0	0.440021	0.220011	+	0.975497i	$0.429391\pi$
$912$	0	0
$913$	20736.0	0.751655
$914$	0	0
$915$	−21384.0	−0.772605
$916$	0	0
$917$	10745.9	0.386979
$918$	0	0
$919$	14029.3	0.503574	0.251787	−	0.967783i	$-0.418982\pi$
$919$	14029.3	0.503574	0.251787	+	0.967783i	$0.418982\pi$
$920$	0	0
$921$	−20580.0	−0.736302
$922$	0	0
$923$	79200.0	2.82438
$924$	0	0
$925$	53928.3	1.91692
$926$	0	0
$927$	4835.64	0.171330
$928$	0	0
$929$	12438.0	0.439265	0.219633	−	0.975583i	$-0.429514\pi$
$929$	12438.0	0.439265	0.219633	+	0.975583i	$0.429514\pi$
$930$	0	0
$931$	4876.00	0.171648
$932$	0	0
$933$	31282.4	1.09768
$934$	0	0
$935$	40117.9	1.40320
$936$	0	0
$937$	−8278.00	−0.288613	−0.144307	−	0.989533i	$-0.546095\pi$
$937$	−8278.00	−0.288613	−0.144307	+	0.989533i	$0.546095\pi$
$938$	0	0
$939$	−18786.0	−0.652884
$940$	0	0
$941$	5114.24	0.177172	0.0885862	−	0.996069i	$-0.471765\pi$
$941$	5114.24	0.177172	0.0885862	+	0.996069i	$0.471765\pi$
$942$	0	0
$943$	−238.797	−0.00824634
$944$	0	0
$945$	10692.0	0.368054
$946$	0	0
$947$	16980.0	0.582657	0.291328	−	0.956623i	$-0.405903\pi$
$947$	16980.0	0.582657	0.291328	+	0.956623i	$0.405903\pi$
$948$	0	0
$949$	34227.6	1.17078
$950$	0	0
$951$	2328.27	0.0793894
$952$	0	0
$953$	31998.0	1.08764	0.543818	−	0.839203i	$-0.316978\pi$
$953$	31998.0	1.08764	0.543818	+	0.839203i	$0.316978\pi$
$954$	0	0
$955$	−17424.0	−0.590395
$956$	0	0
$957$	2865.56	0.0967926
$958$	0	0
$959$	28775.0	0.968920
$960$	0	0
$961$	−10387.0	−0.348662
$962$	0	0
$963$	−12420.0	−0.415606
$964$	0	0
$965$	57351.1	1.91316
$966$	0	0
$967$	−8059.40	−0.268017	−0.134009	−	0.990980i	$-0.542785\pi$
$967$	−8059.40	−0.268017	−0.134009	+	0.990980i	$0.542785\pi$
$968$	0	0
$969$	−11592.0	−0.384302
$970$	0	0
$971$	29664.0	0.980395	0.490197	−	0.871612i	$-0.336925\pi$
$971$	29664.0	0.980395	0.490197	+	0.871612i	$0.336925\pi$
$972$	0	0
$973$	−37650.3	−1.24051
$974$	0	0
$975$	64714.0	2.12565
$976$	0	0
$977$	−3354.00	−0.109830	−0.0549150	−	0.998491i	$-0.517489\pi$
$977$	−3354.00	−0.109830	−0.0549150	+	0.998491i	$0.517489\pi$
$978$	0	0
$979$	30240.0	0.987206
$980$	0	0
$981$	11462.3	0.373050
$982$	0	0
$983$	29610.8	0.960772	0.480386	−	0.877057i	$-0.340497\pi$
$983$	29610.8	0.960772	0.480386	+	0.877057i	$0.340497\pi$
$984$	0	0
$985$	−84348.0	−2.72848
$986$	0	0
$987$	2376.00	0.0766250
$988$	0	0
$989$	−3661.55	−0.117726
$990$	0	0
$991$	−38506.0	−1.23429	−0.617146	−	0.786848i	$-0.711711\pi$
$991$	−38506.0	−1.23429	−0.617146	+	0.786848i	$0.711711\pi$
$992$	0	0
$993$	19380.0	0.619341
$994$	0	0
$995$	64548.0	2.05659
$996$	0	0
$997$	−54326.3	−1.72571	−0.862854	−	0.505453i	$-0.831326\pi$
$997$	−54326.3	−1.72571	−0.862854	+	0.505453i	$0.831326\pi$
$998$	0	0
$999$	5372.93	0.170162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.l.1.2		2
3.2	odd	2		2304.4.a.bl.1.1		2
4.3	odd	2		768.4.a.i.1.2		2
8.3	odd	2	inner	768.4.a.l.1.1		2
8.5	even	2		768.4.a.i.1.1		2
12.11	even	2		2304.4.a.y.1.1		2
16.3	odd	4		192.4.d.b.97.1	✓	4
16.5	even	4		192.4.d.b.97.2	yes	4
16.11	odd	4		192.4.d.b.97.4	yes	4
16.13	even	4		192.4.d.b.97.3	yes	4
24.5	odd	2		2304.4.a.y.1.2		2
24.11	even	2		2304.4.a.bl.1.2		2
48.5	odd	4		576.4.d.f.289.1		4
48.11	even	4		576.4.d.f.289.2		4
48.29	odd	4		576.4.d.f.289.3		4
48.35	even	4		576.4.d.f.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.d.b.97.1	✓	4	16.3	odd	4
192.4.d.b.97.2	yes	4	16.5	even	4
192.4.d.b.97.3	yes	4	16.13	even	4
192.4.d.b.97.4	yes	4	16.11	odd	4
576.4.d.f.289.1		4	48.5	odd	4
576.4.d.f.289.2		4	48.11	even	4
576.4.d.f.289.3		4	48.29	odd	4
576.4.d.f.289.4		4	48.35	even	4
768.4.a.i.1.1		2	8.5	even	2
768.4.a.i.1.2		2	4.3	odd	2
768.4.a.l.1.1		2	8.3	odd	2	inner
768.4.a.l.1.2		2	1.1	even	1	trivial
2304.4.a.y.1.1		2	12.11	even	2
2304.4.a.y.1.2		2	24.5	odd	2
2304.4.a.bl.1.1		2	3.2	odd	2
2304.4.a.bl.1.2		2	24.11	even	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +19.8997 q^{5} +19.8997 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +19.8997 q^{5} +19.8997 q^{7} +9.00000 q^{9} -48.0000 q^{11} +79.5990 q^{13} +59.6992 q^{15} -42.0000 q^{17} +92.0000 q^{19} +59.6992 q^{21} -39.7995 q^{23} +271.000 q^{25} +27.0000 q^{27} -19.8997 q^{29} -139.298 q^{31} -144.000 q^{33} +396.000 q^{35} +198.997 q^{37} +238.797 q^{39} +6.00000 q^{41} +92.0000 q^{43} +179.098 q^{45} +39.7995 q^{47} +53.0000 q^{49} -126.000 q^{51} -497.494 q^{53} -955.188 q^{55} +276.000 q^{57} -516.000 q^{59} -358.195 q^{61} +179.098 q^{63} +1584.00 q^{65} -524.000 q^{67} -119.398 q^{69} +994.987 q^{71} +430.000 q^{73} +813.000 q^{75} -955.188 q^{77} -1174.09 q^{79} +81.0000 q^{81} -432.000 q^{83} -835.789 q^{85} -59.6992 q^{87} -630.000 q^{89} +1584.00 q^{91} -417.895 q^{93} +1830.78 q^{95} +862.000 q^{97} -432.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 18 * q^9	\( 2 q + 6 q^{3} + 18 q^{9} - 96 q^{11} - 84 q^{17} + 184 q^{19} + 542 q^{25} + 54 q^{27} - 288 q^{33} + 792 q^{35} + 12 q^{41} + 184 q^{43} + 106 q^{49} - 252 q^{51} + 552 q^{57} - 1032 q^{59} + 3168 q^{65} - 1048 q^{67} + 860 q^{73} + 1626 q^{75} + 162 q^{81} - 864 q^{83} - 1260 q^{89} + 3168 q^{91} + 1724 q^{97} - 864 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 18 * q^9 - 96 * q^11 - 84 * q^17 + 184 * q^19 + 542 * q^25 + 54 * q^27 - 288 * q^33 + 792 * q^35 + 12 * q^41 + 184 * q^43 + 106 * q^49 - 252 * q^51 + 552 * q^57 - 1032 * q^59 + 3168 * q^65 - 1048 * q^67 + 860 * q^73 + 1626 * q^75 + 162 * q^81 - 864 * q^83 - 1260 * q^89 + 3168 * q^91 + 1724 * q^97 - 864 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more