LMFDB - Embedded newform 768.4.a.m.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +2.82843 q^{5} -14.1421 q^{7} +9.00000 q^{9} -20.0000 q^{11} +39.5980 q^{13} +8.48528 q^{15} -34.0000 q^{17} -52.0000 q^{19} -42.4264 q^{21} -62.2254 q^{23} -117.000 q^{25} +27.0000 q^{27} +200.818 q^{29} +110.309 q^{31} -60.0000 q^{33} -40.0000 q^{35} -271.529 q^{37} +118.794 q^{39} -26.0000 q^{41} -252.000 q^{43} +25.4558 q^{45} -345.068 q^{47} -143.000 q^{49} -102.000 q^{51} -681.651 q^{53} -56.5685 q^{55} -156.000 q^{57} -364.000 q^{59} +735.391 q^{61} -127.279 q^{63} +112.000 q^{65} -628.000 q^{67} -186.676 q^{69} +333.754 q^{71} +338.000 q^{73} -351.000 q^{75} +282.843 q^{77} +789.131 q^{79} +81.0000 q^{81} -1036.00 q^{83} -96.1665 q^{85} +602.455 q^{87} +234.000 q^{89} -560.000 q^{91} +330.926 q^{93} -147.078 q^{95} -178.000 q^{97} -180.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 18 q^{9} - 40 q^{11} - 68 q^{17} - 104 q^{19} - 234 q^{25} + 54 q^{27} - 120 q^{33} - 80 q^{35} - 52 q^{41} - 504 q^{43} - 286 q^{49} - 204 q^{51} - 312 q^{57} - 728 q^{59} + 224 q^{65}+ \cdots - 360 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 18 * q^9 - 40 * q^11 - 68 * q^17 - 104 * q^19 - 234 * q^25 + 54 * q^27 - 120 * q^33 - 80 * q^35 - 52 * q^41 - 504 * q^43 - 286 * q^49 - 204 * q^51 - 312 * q^57 - 728 * q^59 + 224 * q^65 - 1256 * q^67 + 676 * q^73 - 702 * q^75 + 162 * q^81 - 2072 * q^83 + 468 * q^89 - 1120 * q^91 - 356 * q^97 - 360 * q^99

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
$5$	2.82843	0.252982	0.126491	−	0.991968i	$-0.459628\pi$
$5$	2.82843	0.252982	0.126491	+	0.991968i	$0.459628\pi$
$6$	0	0
$7$	−14.1421	−0.763604	−0.381802	−	0.924244i	$-0.624696\pi$
$7$	−14.1421	−0.763604	−0.381802	+	0.924244i	$0.624696\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	39.5980	0.844808	0.422404	−	0.906408i	$-0.361186\pi$
$13$	39.5980	0.844808	0.422404	+	0.906408i	$0.361186\pi$
$14$	0	0
$15$	8.48528	0.146059
$16$	0	0
$17$	−34.0000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−34.0000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−52.0000	−0.627875	−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000	−0.627875	−0.313937	+	0.949444i	$0.601648\pi$
$20$	0	0
$21$	−42.4264	−0.440867
$22$	0	0
$23$	−62.2254	−0.564126	−0.282063	−	0.959396i	$-0.591019\pi$
$23$	−62.2254	−0.564126	−0.282063	+	0.959396i	$0.591019\pi$
$24$	0	0
$25$	−117.000	−0.936000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	200.818	1.28590	0.642949	−	0.765909i	$-0.277711\pi$
$29$	200.818	1.28590	0.642949	+	0.765909i	$0.277711\pi$
$30$	0	0
$31$	110.309	0.639097	0.319549	−	0.947570i	$-0.396469\pi$
$31$	110.309	0.639097	0.319549	+	0.947570i	$0.396469\pi$
$32$	0	0
$33$	−60.0000	−0.316505
$34$	0	0
$35$	−40.0000	−0.193178
$36$	0	0
$37$	−271.529	−1.20646	−0.603231	−	0.797567i	$-0.706120\pi$
$37$	−271.529	−1.20646	−0.603231	+	0.797567i	$0.706120\pi$
$38$	0	0
$39$	118.794	0.487750
$40$	0	0
$41$	−26.0000	−0.0990370	−0.0495185	−	0.998773i	$-0.515769\pi$
$41$	−26.0000	−0.0990370	−0.0495185	+	0.998773i	$0.515769\pi$
$42$	0	0
$43$	−252.000	−0.893713	−0.446856	−	0.894606i	$-0.647456\pi$
$43$	−252.000	−0.893713	−0.446856	+	0.894606i	$0.647456\pi$
$44$	0	0
$45$	25.4558	0.0843274
$46$	0	0
$47$	−345.068	−1.07092	−0.535461	−	0.844560i	$-0.679862\pi$
$47$	−345.068	−1.07092	−0.535461	+	0.844560i	$0.679862\pi$
$48$	0	0
$49$	−143.000	−0.416910
$50$	0	0
$51$	−102.000	−0.280056
$52$	0	0
$53$	−681.651	−1.76664	−0.883320	−	0.468770i	$-0.844697\pi$
$53$	−681.651	−1.76664	−0.883320	+	0.468770i	$0.844697\pi$
$54$	0	0
$55$	−56.5685	−0.138685
$56$	0	0
$57$	−156.000	−0.362504
$58$	0	0
$59$	−364.000	−0.803199	−0.401600	−	0.915815i	$-0.631546\pi$
$59$	−364.000	−0.803199	−0.401600	+	0.915815i	$0.631546\pi$
$60$	0	0
$61$	735.391	1.54356	0.771780	−	0.635889i	$-0.219367\pi$
$61$	735.391	1.54356	0.771780	+	0.635889i	$0.219367\pi$
$62$	0	0
$63$	−127.279	−0.254535
$64$	0	0
$65$	112.000	0.213721
$66$	0	0
$67$	−628.000	−1.14511	−0.572555	−	0.819866i	$-0.694048\pi$
$67$	−628.000	−1.14511	−0.572555	+	0.819866i	$0.694048\pi$
$68$	0	0
$69$	−186.676	−0.325698
$70$	0	0
$71$	333.754	0.557878	0.278939	−	0.960309i	$-0.410017\pi$
$71$	333.754	0.557878	0.278939	+	0.960309i	$0.410017\pi$
$72$	0	0
$73$	338.000	0.541917	0.270958	−	0.962591i	$-0.412659\pi$
$73$	338.000	0.541917	0.270958	+	0.962591i	$0.412659\pi$
$74$	0	0
$75$	−351.000	−0.540400
$76$	0	0
$77$	282.843	0.418609
$78$	0	0
$79$	789.131	1.12385	0.561925	−	0.827188i	$-0.310061\pi$
$79$	789.131	1.12385	0.561925	+	0.827188i	$0.310061\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1036.00	−1.37007	−0.685035	−	0.728510i	$-0.740213\pi$
$83$	−1036.00	−1.37007	−0.685035	+	0.728510i	$0.740213\pi$
$84$	0	0
$85$	−96.1665	−0.122714
$86$	0	0
$87$	602.455	0.742413
$88$	0	0
$89$	234.000	0.278696	0.139348	−	0.990243i	$-0.455499\pi$
$89$	234.000	0.278696	0.139348	+	0.990243i	$0.455499\pi$
$90$	0	0
$91$	−560.000	−0.645098
$92$	0	0
$93$	330.926	0.368983
$94$	0	0
$95$	−147.078	−0.158841
$96$	0	0
$97$	−178.000	−0.186321	−0.0931606	−	0.995651i	$-0.529697\pi$
$97$	−178.000	−0.186321	−0.0931606	+	0.995651i	$0.529697\pi$
$98$	0	0
$99$	−180.000	−0.182734
$100$	0	0
$101$	257.387	0.253574	0.126787	−	0.991930i	$-0.459534\pi$
$101$	257.387	0.253574	0.126787	+	0.991930i	$0.459534\pi$
$102$	0	0
$103$	−1886.56	−1.80474	−0.902371	−	0.430961i	$-0.858175\pi$
$103$	−1886.56	−1.80474	−0.902371	+	0.430961i	$0.858175\pi$
$104$	0	0
$105$	−120.000	−0.111531
$106$	0	0
$107$	−1404.00	−1.26850	−0.634251	−	0.773127i	$-0.718692\pi$
$107$	−1404.00	−1.26850	−0.634251	+	0.773127i	$0.718692\pi$
$108$	0	0
$109$	−39.5980	−0.0347963	−0.0173982	−	0.999849i	$-0.505538\pi$
$109$	−39.5980	−0.0347963	−0.0173982	+	0.999849i	$0.505538\pi$
$110$	0	0
$111$	−814.587	−0.696551
$112$	0	0
$113$	1378.00	1.14718	0.573590	−	0.819143i	$-0.305550\pi$
$113$	1378.00	1.14718	0.573590	+	0.819143i	$0.305550\pi$
$114$	0	0
$115$	−176.000	−0.142714
$116$	0	0
$117$	356.382	0.281603
$118$	0	0
$119$	480.833	0.370402
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−78.0000	−0.0571791
$124$	0	0
$125$	−684.479	−0.489774
$126$	0	0
$127$	1790.39	1.25096	0.625480	−	0.780241i	$-0.284903\pi$
$127$	1790.39	1.25096	0.625480	+	0.780241i	$0.284903\pi$
$128$	0	0
$129$	−756.000	−0.515985
$130$	0	0
$131$	−1572.00	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−1572.00	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	735.391	0.479447
$134$	0	0
$135$	76.3675	0.0486864
$136$	0	0
$137$	2854.00	1.77981	0.889904	−	0.456148i	$-0.150771\pi$
$137$	2854.00	1.77981	0.889904	+	0.456148i	$0.150771\pi$
$138$	0	0
$139$	−1964.00	−1.19845	−0.599224	−	0.800581i	$-0.704524\pi$
$139$	−1964.00	−1.19845	−0.599224	+	0.800581i	$0.704524\pi$
$140$	0	0
$141$	−1035.20	−0.618297
$142$	0	0
$143$	−791.960	−0.463126
$144$	0	0
$145$	568.000	0.325309
$146$	0	0
$147$	−429.000	−0.240703
$148$	0	0
$149$	1507.55	0.828882	0.414441	−	0.910076i	$-0.363977\pi$
$149$	1507.55	0.828882	0.414441	+	0.910076i	$0.363977\pi$
$150$	0	0
$151$	−2265.57	−1.22099	−0.610495	−	0.792020i	$-0.709029\pi$
$151$	−2265.57	−1.22099	−0.610495	+	0.792020i	$0.709029\pi$
$152$	0	0
$153$	−306.000	−0.161690
$154$	0	0
$155$	312.000	0.161680
$156$	0	0
$157$	−3529.88	−1.79436	−0.897181	−	0.441663i	$-0.854389\pi$
$157$	−3529.88	−1.79436	−0.897181	+	0.441663i	$0.854389\pi$
$158$	0	0
$159$	−2044.95	−1.01997
$160$	0	0
$161$	880.000	0.430768
$162$	0	0
$163$	−2932.00	−1.40891	−0.704454	−	0.709750i	$-0.748808\pi$
$163$	−2932.00	−1.40891	−0.704454	+	0.709750i	$0.748808\pi$
$164$	0	0
$165$	−169.706	−0.0800701
$166$	0	0
$167$	3676.96	1.70378	0.851890	−	0.523720i	$-0.175456\pi$
$167$	3676.96	1.70378	0.851890	+	0.523720i	$0.175456\pi$
$168$	0	0
$169$	−629.000	−0.286299
$170$	0	0
$171$	−468.000	−0.209292
$172$	0	0
$173$	−1445.33	−0.635180	−0.317590	−	0.948228i	$-0.602874\pi$
$173$	−1445.33	−0.635180	−0.317590	+	0.948228i	$0.602874\pi$
$174$	0	0
$175$	1654.63	0.714733
$176$	0	0
$177$	−1092.00	−0.463727
$178$	0	0
$179$	1308.00	0.546170	0.273085	−	0.961990i	$-0.411956\pi$
$179$	1308.00	0.546170	0.273085	+	0.961990i	$0.411956\pi$
$180$	0	0
$181$	1996.87	0.820034	0.410017	−	0.912078i	$-0.365523\pi$
$181$	1996.87	0.820034	0.410017	+	0.912078i	$0.365523\pi$
$182$	0	0
$183$	2206.17	0.891175
$184$	0	0
$185$	−768.000	−0.305213
$186$	0	0
$187$	680.000	0.265917
$188$	0	0
$189$	−381.838	−0.146956
$190$	0	0
$191$	−939.038	−0.355740	−0.177870	−	0.984054i	$-0.556921\pi$
$191$	−939.038	−0.355740	−0.177870	+	0.984054i	$0.556921\pi$
$192$	0	0
$193$	−2490.00	−0.928674	−0.464337	−	0.885659i	$-0.653707\pi$
$193$	−2490.00	−0.928674	−0.464337	+	0.885659i	$0.653707\pi$
$194$	0	0
$195$	336.000	0.123392
$196$	0	0
$197$	2723.78	0.985081	0.492540	−	0.870290i	$-0.336068\pi$
$197$	2723.78	0.985081	0.492540	+	0.870290i	$0.336068\pi$
$198$	0	0
$199$	−2158.09	−0.768758	−0.384379	−	0.923175i	$-0.625584\pi$
$199$	−2158.09	−0.768758	−0.384379	+	0.923175i	$0.625584\pi$
$200$	0	0
$201$	−1884.00	−0.661130
$202$	0	0
$203$	−2840.00	−0.981916
$204$	0	0
$205$	−73.5391	−0.0250546
$206$	0	0
$207$	−560.029	−0.188042
$208$	0	0
$209$	1040.00	0.344202
$210$	0	0
$211$	924.000	0.301473	0.150736	−	0.988574i	$-0.451836\pi$
$211$	924.000	0.301473	0.150736	+	0.988574i	$0.451836\pi$
$212$	0	0
$213$	1001.26	0.322091
$214$	0	0
$215$	−712.764	−0.226093
$216$	0	0
$217$	−1560.00	−0.488017
$218$	0	0
$219$	1014.00	0.312876
$220$	0	0
$221$	−1346.33	−0.409792
$222$	0	0
$223$	2276.88	0.683728	0.341864	−	0.939749i	$-0.388942\pi$
$223$	2276.88	0.683728	0.341864	+	0.939749i	$0.388942\pi$
$224$	0	0
$225$	−1053.00	−0.312000
$226$	0	0
$227$	−156.000	−0.0456127	−0.0228064	−	0.999740i	$-0.507260\pi$
$227$	−156.000	−0.0456127	−0.0228064	+	0.999740i	$0.507260\pi$
$228$	0	0
$229$	−639.225	−0.184459	−0.0922296	−	0.995738i	$-0.529399\pi$
$229$	−639.225	−0.184459	−0.0922296	+	0.995738i	$0.529399\pi$
$230$	0	0
$231$	848.528	0.241684
$232$	0	0
$233$	2826.00	0.794581	0.397291	−	0.917693i	$-0.369951\pi$
$233$	2826.00	0.794581	0.397291	+	0.917693i	$0.369951\pi$
$234$	0	0
$235$	−976.000	−0.270924
$236$	0	0
$237$	2367.39	0.648855
$238$	0	0
$239$	−2466.39	−0.667521	−0.333760	−	0.942658i	$-0.608318\pi$
$239$	−2466.39	−0.667521	−0.333760	+	0.942658i	$0.608318\pi$
$240$	0	0
$241$	−3354.00	−0.896474	−0.448237	−	0.893915i	$-0.647948\pi$
$241$	−3354.00	−0.896474	−0.448237	+	0.893915i	$0.647948\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−404.465	−0.105471
$246$	0	0
$247$	−2059.09	−0.530433
$248$	0	0
$249$	−3108.00	−0.791010
$250$	0	0
$251$	6396.00	1.60841	0.804207	−	0.594349i	$-0.202590\pi$
$251$	6396.00	1.60841	0.804207	+	0.594349i	$0.202590\pi$
$252$	0	0
$253$	1244.51	0.309255
$254$	0	0
$255$	−288.500	−0.0708492
$256$	0	0
$257$	6882.00	1.67038	0.835189	−	0.549962i	$-0.185358\pi$
$257$	6882.00	1.67038	0.835189	+	0.549962i	$0.185358\pi$
$258$	0	0
$259$	3840.00	0.921259
$260$	0	0
$261$	1807.36	0.428632
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−1928.00	−0.446929
$266$	0	0
$267$	702.000	0.160905
$268$	0	0
$269$	1434.01	0.325031	0.162515	−	0.986706i	$-0.448039\pi$
$269$	1434.01	0.325031	0.162515	+	0.986706i	$0.448039\pi$
$270$	0	0
$271$	5942.53	1.33204	0.666020	−	0.745934i	$-0.267997\pi$
$271$	5942.53	1.33204	0.666020	+	0.745934i	$0.267997\pi$
$272$	0	0
$273$	−1680.00	−0.372448
$274$	0	0
$275$	2340.00	0.513117
$276$	0	0
$277$	−1103.09	−0.239271	−0.119635	−	0.992818i	$-0.538173\pi$
$277$	−1103.09	−0.239271	−0.119635	+	0.992818i	$0.538173\pi$
$278$	0	0
$279$	992.778	0.213032
$280$	0	0
$281$	6266.00	1.33024	0.665121	−	0.746735i	$-0.268380\pi$
$281$	6266.00	1.33024	0.665121	+	0.746735i	$0.268380\pi$
$282$	0	0
$283$	8596.00	1.80558	0.902790	−	0.430082i	$-0.141515\pi$
$283$	8596.00	1.80558	0.902790	+	0.430082i	$0.141515\pi$
$284$	0	0
$285$	−441.235	−0.0917070
$286$	0	0
$287$	367.696	0.0756250
$288$	0	0
$289$	−3757.00	−0.764706
$290$	0	0
$291$	−534.000	−0.107573
$292$	0	0
$293$	−8397.60	−1.67438	−0.837189	−	0.546913i	$-0.815803\pi$
$293$	−8397.60	−1.67438	−0.837189	+	0.546913i	$0.815803\pi$
$294$	0	0
$295$	−1029.55	−0.203195
$296$	0	0
$297$	−540.000	−0.105502
$298$	0	0
$299$	−2464.00	−0.476578
$300$	0	0
$301$	3563.82	0.682442
$302$	0	0
$303$	772.161	0.146401
$304$	0	0
$305$	2080.00	0.390493
$306$	0	0
$307$	4940.00	0.918374	0.459187	−	0.888340i	$-0.348141\pi$
$307$	4940.00	0.918374	0.459187	+	0.888340i	$0.348141\pi$
$308$	0	0
$309$	−5659.68	−1.04197
$310$	0	0
$311$	−3382.80	−0.616788	−0.308394	−	0.951259i	$-0.599791\pi$
$311$	−3382.80	−0.616788	−0.308394	+	0.951259i	$0.599791\pi$
$312$	0	0
$313$	−3106.00	−0.560899	−0.280450	−	0.959869i	$-0.590484\pi$
$313$	−3106.00	−0.560899	−0.280450	+	0.959869i	$0.590484\pi$
$314$	0	0
$315$	−360.000	−0.0643927
$316$	0	0
$317$	6728.83	1.19220	0.596102	−	0.802909i	$-0.296715\pi$
$317$	6728.83	1.19220	0.596102	+	0.802909i	$0.296715\pi$
$318$	0	0
$319$	−4016.37	−0.704932
$320$	0	0
$321$	−4212.00	−0.732370
$322$	0	0
$323$	1768.00	0.304564
$324$	0	0
$325$	−4632.96	−0.790740
$326$	0	0
$327$	−118.794	−0.0200897
$328$	0	0
$329$	4880.00	0.817760
$330$	0	0
$331$	−2908.00	−0.482895	−0.241447	−	0.970414i	$-0.577622\pi$
$331$	−2908.00	−0.482895	−0.241447	+	0.970414i	$0.577622\pi$
$332$	0	0
$333$	−2443.76	−0.402154
$334$	0	0
$335$	−1776.25	−0.289693
$336$	0	0
$337$	4298.00	0.694739	0.347369	−	0.937728i	$-0.387075\pi$
$337$	4298.00	0.694739	0.347369	+	0.937728i	$0.387075\pi$
$338$	0	0
$339$	4134.00	0.662325
$340$	0	0
$341$	−2206.17	−0.350355
$342$	0	0
$343$	6873.08	1.08196
$344$	0	0
$345$	−528.000	−0.0823958
$346$	0	0
$347$	9996.00	1.54644	0.773218	−	0.634140i	$-0.218646\pi$
$347$	9996.00	1.54644	0.773218	+	0.634140i	$0.218646\pi$
$348$	0	0
$349$	3993.74	0.612550	0.306275	−	0.951943i	$-0.400917\pi$
$349$	3993.74	0.612550	0.306275	+	0.951943i	$0.400917\pi$
$350$	0	0
$351$	1069.15	0.162583
$352$	0	0
$353$	6738.00	1.01594	0.507971	−	0.861374i	$-0.330396\pi$
$353$	6738.00	1.01594	0.507971	+	0.861374i	$0.330396\pi$
$354$	0	0
$355$	944.000	0.141133
$356$	0	0
$357$	1442.50	0.213852
$358$	0	0
$359$	−2132.63	−0.313527	−0.156763	−	0.987636i	$-0.550106\pi$
$359$	−2132.63	−0.313527	−0.156763	+	0.987636i	$0.550106\pi$
$360$	0	0
$361$	−4155.00	−0.605773
$362$	0	0
$363$	−2793.00	−0.403842
$364$	0	0
$365$	956.008	0.137095
$366$	0	0
$367$	−7628.27	−1.08499	−0.542496	−	0.840058i	$-0.682521\pi$
$367$	−7628.27	−1.08499	−0.542496	+	0.840058i	$0.682521\pi$
$368$	0	0
$369$	−234.000	−0.0330123
$370$	0	0
$371$	9640.00	1.34901
$372$	0	0
$373$	−8383.46	−1.16375	−0.581875	−	0.813278i	$-0.697681\pi$
$373$	−8383.46	−1.16375	−0.581875	+	0.813278i	$0.697681\pi$
$374$	0	0
$375$	−2053.44	−0.282771
$376$	0	0
$377$	7952.00	1.08634
$378$	0	0
$379$	12788.0	1.73318	0.866590	−	0.499020i	$-0.166307\pi$
$379$	12788.0	1.73318	0.866590	+	0.499020i	$0.166307\pi$
$380$	0	0
$381$	5371.18	0.722242
$382$	0	0
$383$	−2319.31	−0.309429	−0.154714	−	0.987959i	$-0.549446\pi$
$383$	−2319.31	−0.309429	−0.154714	+	0.987959i	$0.549446\pi$
$384$	0	0
$385$	800.000	0.105901
$386$	0	0
$387$	−2268.00	−0.297904
$388$	0	0
$389$	2684.18	0.349854	0.174927	−	0.984581i	$-0.444031\pi$
$389$	2684.18	0.349854	0.174927	+	0.984581i	$0.444031\pi$
$390$	0	0
$391$	2115.66	0.273641
$392$	0	0
$393$	−4716.00	−0.605320
$394$	0	0
$395$	2232.00	0.284314
$396$	0	0
$397$	2206.17	0.278903	0.139452	−	0.990229i	$-0.455466\pi$
$397$	2206.17	0.278903	0.139452	+	0.990229i	$0.455466\pi$
$398$	0	0
$399$	2206.17	0.276809
$400$	0	0
$401$	3582.00	0.446076	0.223038	−	0.974810i	$-0.428403\pi$
$401$	3582.00	0.446076	0.223038	+	0.974810i	$0.428403\pi$
$402$	0	0
$403$	4368.00	0.539915
$404$	0	0
$405$	229.103	0.0281091
$406$	0	0
$407$	5430.58	0.661385
$408$	0	0
$409$	5126.00	0.619717	0.309859	−	0.950783i	$-0.399718\pi$
$409$	5126.00	0.619717	0.309859	+	0.950783i	$0.399718\pi$
$410$	0	0
$411$	8562.00	1.02757
$412$	0	0
$413$	5147.74	0.613326
$414$	0	0
$415$	−2930.25	−0.346603
$416$	0	0
$417$	−5892.00	−0.691924
$418$	0	0
$419$	−2924.00	−0.340923	−0.170462	−	0.985364i	$-0.554526\pi$
$419$	−2924.00	−0.340923	−0.170462	+	0.985364i	$0.554526\pi$
$420$	0	0
$421$	7314.31	0.846741	0.423370	−	0.905957i	$-0.360847\pi$
$421$	7314.31	0.846741	0.423370	+	0.905957i	$0.360847\pi$
$422$	0	0
$423$	−3105.61	−0.356974
$424$	0	0
$425$	3978.00	0.454027
$426$	0	0
$427$	−10400.0	−1.17867
$428$	0	0
$429$	−2375.88	−0.267386
$430$	0	0
$431$	15844.8	1.77081	0.885405	−	0.464819i	$-0.153881\pi$
$431$	15844.8	1.77081	0.885405	+	0.464819i	$0.153881\pi$
$432$	0	0
$433$	−6274.00	−0.696326	−0.348163	−	0.937434i	$-0.613194\pi$
$433$	−6274.00	−0.696326	−0.348163	+	0.937434i	$0.613194\pi$
$434$	0	0
$435$	1704.00	0.187817
$436$	0	0
$437$	3235.72	0.354200
$438$	0	0
$439$	4596.19	0.499691	0.249846	−	0.968286i	$-0.419620\pi$
$439$	4596.19	0.499691	0.249846	+	0.968286i	$0.419620\pi$
$440$	0	0
$441$	−1287.00	−0.138970
$442$	0	0
$443$	5084.00	0.545255	0.272628	−	0.962120i	$-0.412107\pi$
$443$	5084.00	0.545255	0.272628	+	0.962120i	$0.412107\pi$
$444$	0	0
$445$	661.852	0.0705051
$446$	0	0
$447$	4522.65	0.478555
$448$	0	0
$449$	14190.0	1.49146	0.745732	−	0.666246i	$-0.232100\pi$
$449$	14190.0	1.49146	0.745732	+	0.666246i	$0.232100\pi$
$450$	0	0
$451$	520.000	0.0542923
$452$	0	0
$453$	−6796.71	−0.704939
$454$	0	0
$455$	−1583.92	−0.163198
$456$	0	0
$457$	−6474.00	−0.662672	−0.331336	−	0.943513i	$-0.607499\pi$
$457$	−6474.00	−0.662672	−0.331336	+	0.943513i	$0.607499\pi$
$458$	0	0
$459$	−918.000	−0.0933520
$460$	0	0
$461$	−6321.53	−0.638662	−0.319331	−	0.947643i	$-0.603458\pi$
$461$	−6321.53	−0.638662	−0.319331	+	0.947643i	$0.603458\pi$
$462$	0	0
$463$	−11435.3	−1.14783	−0.573915	−	0.818915i	$-0.694576\pi$
$463$	−11435.3	−1.14783	−0.573915	+	0.818915i	$0.694576\pi$
$464$	0	0
$465$	936.000	0.0933462
$466$	0	0
$467$	3796.00	0.376141	0.188071	−	0.982156i	$-0.439777\pi$
$467$	3796.00	0.376141	0.188071	+	0.982156i	$0.439777\pi$
$468$	0	0
$469$	8881.26	0.874411
$470$	0	0
$471$	−10589.6	−1.03598
$472$	0	0
$473$	5040.00	0.489935
$474$	0	0
$475$	6084.00	0.587691
$476$	0	0
$477$	−6134.86	−0.588880
$478$	0	0
$479$	10493.5	1.00096	0.500479	−	0.865749i	$-0.333157\pi$
$479$	10493.5	1.00096	0.500479	+	0.865749i	$0.333157\pi$
$480$	0	0
$481$	−10752.0	−1.01923
$482$	0	0
$483$	2640.00	0.248704
$484$	0	0
$485$	−503.460	−0.0471360
$486$	0	0
$487$	15406.4	1.43354	0.716769	−	0.697311i	$-0.245620\pi$
$487$	15406.4	1.43354	0.716769	+	0.697311i	$0.245620\pi$
$488$	0	0
$489$	−8796.00	−0.813433
$490$	0	0
$491$	−15452.0	−1.42024	−0.710121	−	0.704079i	$-0.751360\pi$
$491$	−15452.0	−1.42024	−0.710121	+	0.704079i	$0.751360\pi$
$492$	0	0
$493$	−6827.82	−0.623752
$494$	0	0
$495$	−509.117	−0.0462285
$496$	0	0
$497$	−4720.00	−0.425998
$498$	0	0
$499$	−52.0000	−0.00466501	−0.00233250	−	0.999997i	$-0.500742\pi$
$499$	−52.0000	−0.00466501	−0.00233250	+	0.999997i	$0.500742\pi$
$500$	0	0
$501$	11030.9	0.983678
$502$	0	0
$503$	−12428.1	−1.10167	−0.550837	−	0.834613i	$-0.685691\pi$
$503$	−12428.1	−1.10167	−0.550837	+	0.834613i	$0.685691\pi$
$504$	0	0
$505$	728.000	0.0641497
$506$	0	0
$507$	−1887.00	−0.165295
$508$	0	0
$509$	−16362.5	−1.42486	−0.712429	−	0.701744i	$-0.752405\pi$
$509$	−16362.5	−1.42486	−0.712429	+	0.701744i	$0.752405\pi$
$510$	0	0
$511$	−4780.04	−0.413809
$512$	0	0
$513$	−1404.00	−0.120835
$514$	0	0
$515$	−5336.00	−0.456567
$516$	0	0
$517$	6901.36	0.587082
$518$	0	0
$519$	−4335.98	−0.366721
$520$	0	0
$521$	−714.000	−0.0600401	−0.0300201	−	0.999549i	$-0.509557\pi$
$521$	−714.000	−0.0600401	−0.0300201	+	0.999549i	$0.509557\pi$
$522$	0	0
$523$	−5980.00	−0.499975	−0.249988	−	0.968249i	$-0.580427\pi$
$523$	−5980.00	−0.499975	−0.249988	+	0.968249i	$0.580427\pi$
$524$	0	0
$525$	4963.89	0.412651
$526$	0	0
$527$	−3750.49	−0.310008
$528$	0	0
$529$	−8295.00	−0.681762
$530$	0	0
$531$	−3276.00	−0.267733
$532$	0	0
$533$	−1029.55	−0.0836673
$534$	0	0
$535$	−3971.11	−0.320909
$536$	0	0
$537$	3924.00	0.315332
$538$	0	0
$539$	2860.00	0.228551
$540$	0	0
$541$	−13729.2	−1.09106	−0.545530	−	0.838091i	$-0.683672\pi$
$541$	−13729.2	−1.09106	−0.545530	+	0.838091i	$0.683672\pi$
$542$	0	0
$543$	5990.61	0.473447
$544$	0	0
$545$	−112.000	−0.00880285
$546$	0	0
$547$	−18500.0	−1.44607	−0.723037	−	0.690809i	$-0.757255\pi$
$547$	−18500.0	−1.44607	−0.723037	+	0.690809i	$0.757255\pi$
$548$	0	0
$549$	6618.52	0.514520
$550$	0	0
$551$	−10442.6	−0.807382
$552$	0	0
$553$	−11160.0	−0.858176
$554$	0	0
$555$	−2304.00	−0.176215
$556$	0	0
$557$	−8765.30	−0.666782	−0.333391	−	0.942789i	$-0.608193\pi$
$557$	−8765.30	−0.666782	−0.333391	+	0.942789i	$0.608193\pi$
$558$	0	0
$559$	−9978.69	−0.755015
$560$	0	0
$561$	2040.00	0.153527
$562$	0	0
$563$	−268.000	−0.0200619	−0.0100310	−	0.999950i	$-0.503193\pi$
$563$	−268.000	−0.0200619	−0.0100310	+	0.999950i	$0.503193\pi$
$564$	0	0
$565$	3897.57	0.290216
$566$	0	0
$567$	−1145.51	−0.0848448
$568$	0	0
$569$	−13866.0	−1.02160	−0.510802	−	0.859698i	$-0.670652\pi$
$569$	−13866.0	−1.02160	−0.510802	+	0.859698i	$0.670652\pi$
$570$	0	0
$571$	5140.00	0.376712	0.188356	−	0.982101i	$-0.439684\pi$
$571$	5140.00	0.376712	0.188356	+	0.982101i	$0.439684\pi$
$572$	0	0
$573$	−2817.11	−0.205387
$574$	0	0
$575$	7280.37	0.528022
$576$	0	0
$577$	9386.00	0.677200	0.338600	−	0.940930i	$-0.390047\pi$
$577$	9386.00	0.677200	0.338600	+	0.940930i	$0.390047\pi$
$578$	0	0
$579$	−7470.00	−0.536170
$580$	0	0
$581$	14651.3	1.04619
$582$	0	0
$583$	13633.0	0.968477
$584$	0	0
$585$	1008.00	0.0712405
$586$	0	0
$587$	−8844.00	−0.621859	−0.310929	−	0.950433i	$-0.600640\pi$
$587$	−8844.00	−0.621859	−0.310929	+	0.950433i	$0.600640\pi$
$588$	0	0
$589$	−5736.05	−0.401273
$590$	0	0
$591$	8171.33	0.568737
$592$	0	0
$593$	−9406.00	−0.651363	−0.325681	−	0.945480i	$-0.605594\pi$
$593$	−9406.00	−0.651363	−0.325681	+	0.945480i	$0.605594\pi$
$594$	0	0
$595$	1360.00	0.0937051
$596$	0	0
$597$	−6474.27	−0.443843
$598$	0	0
$599$	23459.0	1.60018	0.800090	−	0.599880i	$-0.204785\pi$
$599$	23459.0	1.60018	0.800090	+	0.599880i	$0.204785\pi$
$600$	0	0
$601$	−1262.00	−0.0856540	−0.0428270	−	0.999083i	$-0.513636\pi$
$601$	−1262.00	−0.0856540	−0.0428270	+	0.999083i	$0.513636\pi$
$602$	0	0
$603$	−5652.00	−0.381704
$604$	0	0
$605$	−2633.27	−0.176955
$606$	0	0
$607$	−16288.9	−1.08920	−0.544602	−	0.838695i	$-0.683319\pi$
$607$	−16288.9	−1.08920	−0.544602	+	0.838695i	$0.683319\pi$
$608$	0	0
$609$	−8520.00	−0.566909
$610$	0	0
$611$	−13664.0	−0.904724
$612$	0	0
$613$	−7138.95	−0.470374	−0.235187	−	0.971950i	$-0.575570\pi$
$613$	−7138.95	−0.470374	−0.235187	+	0.971950i	$0.575570\pi$
$614$	0	0
$615$	−220.617	−0.0144653
$616$	0	0
$617$	16874.0	1.10101	0.550504	−	0.834833i	$-0.314436\pi$
$617$	16874.0	1.10101	0.550504	+	0.834833i	$0.314436\pi$
$618$	0	0
$619$	−20748.0	−1.34723	−0.673613	−	0.739085i	$-0.735258\pi$
$619$	−20748.0	−1.34723	−0.673613	+	0.739085i	$0.735258\pi$
$620$	0	0
$621$	−1680.09	−0.108566
$622$	0	0
$623$	−3309.26	−0.212813
$624$	0	0
$625$	12689.0	0.812096
$626$	0	0
$627$	3120.00	0.198725
$628$	0	0
$629$	9231.99	0.585220
$630$	0	0
$631$	14840.8	0.936294	0.468147	−	0.883651i	$-0.344922\pi$
$631$	14840.8	0.936294	0.468147	+	0.883651i	$0.344922\pi$
$632$	0	0
$633$	2772.00	0.174055
$634$	0	0
$635$	5064.00	0.316470
$636$	0	0
$637$	−5662.51	−0.352209
$638$	0	0
$639$	3003.79	0.185959
$640$	0	0
$641$	17758.0	1.09423	0.547113	−	0.837059i	$-0.315727\pi$
$641$	17758.0	1.09423	0.547113	+	0.837059i	$0.315727\pi$
$642$	0	0
$643$	1148.00	0.0704086	0.0352043	−	0.999380i	$-0.488792\pi$
$643$	1148.00	0.0704086	0.0352043	+	0.999380i	$0.488792\pi$
$644$	0	0
$645$	−2138.29	−0.130535
$646$	0	0
$647$	−26988.9	−1.63994	−0.819970	−	0.572406i	$-0.806010\pi$
$647$	−26988.9	−1.63994	−0.819970	+	0.572406i	$0.806010\pi$
$648$	0	0
$649$	7280.00	0.440316
$650$	0	0
$651$	−4680.00	−0.281757
$652$	0	0
$653$	21069.0	1.26262	0.631311	−	0.775530i	$-0.282517\pi$
$653$	21069.0	1.26262	0.631311	+	0.775530i	$0.282517\pi$
$654$	0	0
$655$	−4446.29	−0.265238
$656$	0	0
$657$	3042.00	0.180639
$658$	0	0
$659$	−18356.0	−1.08505	−0.542525	−	0.840040i	$-0.682532\pi$
$659$	−18356.0	−1.08505	−0.542525	+	0.840040i	$0.682532\pi$
$660$	0	0
$661$	15250.9	0.897414	0.448707	−	0.893679i	$-0.351885\pi$
$661$	15250.9	0.897414	0.448707	+	0.893679i	$0.351885\pi$
$662$	0	0
$663$	−4038.99	−0.236594
$664$	0	0
$665$	2080.00	0.121292
$666$	0	0
$667$	−12496.0	−0.725408
$668$	0	0
$669$	6830.65	0.394751
$670$	0	0
$671$	−14707.8	−0.846184
$672$	0	0
$673$	−12082.0	−0.692016	−0.346008	−	0.938232i	$-0.612463\pi$
$673$	−12082.0	−0.692016	−0.346008	+	0.938232i	$0.612463\pi$
$674$	0	0
$675$	−3159.00	−0.180133
$676$	0	0
$677$	12742.1	0.723364	0.361682	−	0.932302i	$-0.382203\pi$
$677$	12742.1	0.723364	0.361682	+	0.932302i	$0.382203\pi$
$678$	0	0
$679$	2517.30	0.142276
$680$	0	0
$681$	−468.000	−0.0263345
$682$	0	0
$683$	−33508.0	−1.87723	−0.938615	−	0.344967i	$-0.887890\pi$
$683$	−33508.0	−1.87723	−0.938615	+	0.344967i	$0.887890\pi$
$684$	0	0
$685$	8072.33	0.450260
$686$	0	0
$687$	−1917.67	−0.106498
$688$	0	0
$689$	−26992.0	−1.49247
$690$	0	0
$691$	364.000	0.0200394	0.0100197	−	0.999950i	$-0.496811\pi$
$691$	364.000	0.0200394	0.0100197	+	0.999950i	$0.496811\pi$
$692$	0	0
$693$	2545.58	0.139536
$694$	0	0
$695$	−5555.03	−0.303186
$696$	0	0
$697$	884.000	0.0480400
$698$	0	0
$699$	8478.00	0.458752
$700$	0	0
$701$	−3849.49	−0.207408	−0.103704	−	0.994608i	$-0.533070\pi$
$701$	−3849.49	−0.207408	−0.103704	+	0.994608i	$0.533070\pi$
$702$	0	0
$703$	14119.5	0.757507
$704$	0	0
$705$	−2928.00	−0.156418
$706$	0	0
$707$	−3640.00	−0.193630
$708$	0	0
$709$	−23606.1	−1.25041	−0.625207	−	0.780459i	$-0.714986\pi$
$709$	−23606.1	−1.25041	−0.625207	+	0.780459i	$0.714986\pi$
$710$	0	0
$711$	7102.18	0.374617
$712$	0	0
$713$	−6864.00	−0.360531
$714$	0	0
$715$	−2240.00	−0.117163
$716$	0	0
$717$	−7399.17	−0.385393
$718$	0	0
$719$	15799.6	0.819507	0.409753	−	0.912196i	$-0.365615\pi$
$719$	15799.6	0.819507	0.409753	+	0.912196i	$0.365615\pi$
$720$	0	0
$721$	26680.0	1.37811
$722$	0	0
$723$	−10062.0	−0.517579
$724$	0	0
$725$	−23495.7	−1.20360
$726$	0	0
$727$	−4607.51	−0.235052	−0.117526	−	0.993070i	$-0.537496\pi$
$727$	−4607.51	−0.235052	−0.117526	+	0.993070i	$0.537496\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	8568.00	0.433514
$732$	0	0
$733$	26219.5	1.32120	0.660600	−	0.750738i	$-0.270302\pi$
$733$	26219.5	1.32120	0.660600	+	0.750738i	$0.270302\pi$
$734$	0	0
$735$	−1213.40	−0.0608935
$736$	0	0
$737$	12560.0	0.627752
$738$	0	0
$739$	−27924.0	−1.38999	−0.694994	−	0.719016i	$-0.744593\pi$
$739$	−27924.0	−1.38999	−0.694994	+	0.719016i	$0.744593\pi$
$740$	0	0
$741$	−6177.28	−0.306246
$742$	0	0
$743$	8937.83	0.441315	0.220658	−	0.975351i	$-0.429180\pi$
$743$	8937.83	0.441315	0.220658	+	0.975351i	$0.429180\pi$
$744$	0	0
$745$	4264.00	0.209692
$746$	0	0
$747$	−9324.00	−0.456690
$748$	0	0
$749$	19855.6	0.968633
$750$	0	0
$751$	14082.7	0.684270	0.342135	−	0.939651i	$-0.388850\pi$
$751$	14082.7	0.684270	0.342135	+	0.939651i	$0.388850\pi$
$752$	0	0
$753$	19188.0	0.928618
$754$	0	0
$755$	−6408.00	−0.308889
$756$	0	0
$757$	−14871.9	−0.714039	−0.357019	−	0.934097i	$-0.616207\pi$
$757$	−14871.9	−0.714039	−0.357019	+	0.934097i	$0.616207\pi$
$758$	0	0
$759$	3733.52	0.178549
$760$	0	0
$761$	−15834.0	−0.754247	−0.377124	−	0.926163i	$-0.623087\pi$
$761$	−15834.0	−0.754247	−0.377124	+	0.926163i	$0.623087\pi$
$762$	0	0
$763$	560.000	0.0265706
$764$	0	0
$765$	−865.499	−0.0409048
$766$	0	0
$767$	−14413.7	−0.678549
$768$	0	0
$769$	−16666.0	−0.781523	−0.390762	−	0.920492i	$-0.627788\pi$
$769$	−16666.0	−0.781523	−0.390762	+	0.920492i	$0.627788\pi$
$770$	0	0
$771$	20646.0	0.964394
$772$	0	0
$773$	−30957.1	−1.44043	−0.720214	−	0.693752i	$-0.755956\pi$
$773$	−30957.1	−1.44043	−0.720214	+	0.693752i	$0.755956\pi$
$774$	0	0
$775$	−12906.1	−0.598195
$776$	0	0
$777$	11520.0	0.531889
$778$	0	0
$779$	1352.00	0.0621828
$780$	0	0
$781$	−6675.09	−0.305830
$782$	0	0
$783$	5422.09	0.247471
$784$	0	0
$785$	−9984.00	−0.453942
$786$	0	0
$787$	−20228.0	−0.916201	−0.458101	−	0.888900i	$-0.651470\pi$
$787$	−20228.0	−0.916201	−0.458101	+	0.888900i	$0.651470\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19487.9	−0.875991
$792$	0	0
$793$	29120.0	1.30401
$794$	0	0
$795$	−5784.00	−0.258034
$796$	0	0
$797$	9008.54	0.400375	0.200187	−	0.979758i	$-0.435845\pi$
$797$	9008.54	0.400375	0.200187	+	0.979758i	$0.435845\pi$
$798$	0	0
$799$	11732.3	0.519474
$800$	0	0
$801$	2106.00	0.0928987
$802$	0	0
$803$	−6760.00	−0.297080
$804$	0	0
$805$	2489.02	0.108977
$806$	0	0
$807$	4302.04	0.187657
$808$	0	0
$809$	−9242.00	−0.401646	−0.200823	−	0.979628i	$-0.564362\pi$
$809$	−9242.00	−0.401646	−0.200823	+	0.979628i	$0.564362\pi$
$810$	0	0
$811$	−10972.0	−0.475067	−0.237533	−	0.971379i	$-0.576339\pi$
$811$	−10972.0	−0.475067	−0.237533	+	0.971379i	$0.576339\pi$
$812$	0	0
$813$	17827.6	0.769053
$814$	0	0
$815$	−8292.95	−0.356429
$816$	0	0
$817$	13104.0	0.561139
$818$	0	0
$819$	−5040.00	−0.215033
$820$	0	0
$821$	−9336.64	−0.396895	−0.198448	−	0.980112i	$-0.563590\pi$
$821$	−9336.64	−0.396895	−0.198448	+	0.980112i	$0.563590\pi$
$822$	0	0
$823$	−3566.65	−0.151064	−0.0755319	−	0.997143i	$-0.524065\pi$
$823$	−3566.65	−0.151064	−0.0755319	+	0.997143i	$0.524065\pi$
$824$	0	0
$825$	7020.00	0.296249
$826$	0	0
$827$	−18876.0	−0.793691	−0.396846	−	0.917885i	$-0.629895\pi$
$827$	−18876.0	−0.793691	−0.396846	+	0.917885i	$0.629895\pi$
$828$	0	0
$829$	6974.90	0.292218	0.146109	−	0.989269i	$-0.453325\pi$
$829$	6974.90	0.292218	0.146109	+	0.989269i	$0.453325\pi$
$830$	0	0
$831$	−3309.26	−0.138143
$832$	0	0
$833$	4862.00	0.202231
$834$	0	0
$835$	10400.0	0.431026
$836$	0	0
$837$	2978.33	0.122994
$838$	0	0
$839$	30077.5	1.23765	0.618826	−	0.785528i	$-0.287608\pi$
$839$	30077.5	1.23765	0.618826	+	0.785528i	$0.287608\pi$
$840$	0	0
$841$	15939.0	0.653532
$842$	0	0
$843$	18798.0	0.768016
$844$	0	0
$845$	−1779.08	−0.0724287
$846$	0	0
$847$	13166.3	0.534121
$848$	0	0
$849$	25788.0	1.04245
$850$	0	0
$851$	16896.0	0.680596
$852$	0	0
$853$	−41159.3	−1.65213	−0.826065	−	0.563575i	$-0.809426\pi$
$853$	−41159.3	−1.65213	−0.826065	+	0.563575i	$0.809426\pi$
$854$	0	0
$855$	−1323.70	−0.0529470
$856$	0	0
$857$	−25194.0	−1.00421	−0.502107	−	0.864806i	$-0.667441\pi$
$857$	−25194.0	−1.00421	−0.502107	+	0.864806i	$0.667441\pi$
$858$	0	0
$859$	−9308.00	−0.369715	−0.184857	−	0.982765i	$-0.559182\pi$
$859$	−9308.00	−0.369715	−0.184857	+	0.982765i	$0.559182\pi$
$860$	0	0
$861$	1103.09	0.0436621
$862$	0	0
$863$	26802.2	1.05719	0.528596	−	0.848874i	$-0.322719\pi$
$863$	26802.2	1.05719	0.528596	+	0.848874i	$0.322719\pi$
$864$	0	0
$865$	−4088.00	−0.160689
$866$	0	0
$867$	−11271.0	−0.441503
$868$	0	0
$869$	−15782.6	−0.616098
$870$	0	0
$871$	−24867.5	−0.967399
$872$	0	0
$873$	−1602.00	−0.0621071
$874$	0	0
$875$	9680.00	0.373993
$876$	0	0
$877$	1436.84	0.0553235	0.0276617	−	0.999617i	$-0.491194\pi$
$877$	1436.84	0.0553235	0.0276617	+	0.999617i	$0.491194\pi$
$878$	0	0
$879$	−25192.8	−0.966703
$880$	0	0
$881$	−42830.0	−1.63789	−0.818944	−	0.573873i	$-0.805440\pi$
$881$	−42830.0	−1.63789	−0.818944	+	0.573873i	$0.805440\pi$
$882$	0	0
$883$	23964.0	0.913310	0.456655	−	0.889644i	$-0.349047\pi$
$883$	23964.0	0.913310	0.456655	+	0.889644i	$0.349047\pi$
$884$	0	0
$885$	−3088.64	−0.117315
$886$	0	0
$887$	28239.0	1.06897	0.534483	−	0.845179i	$-0.320506\pi$
$887$	28239.0	1.06897	0.534483	+	0.845179i	$0.320506\pi$
$888$	0	0
$889$	−25320.0	−0.955237
$890$	0	0
$891$	−1620.00	−0.0609114
$892$	0	0
$893$	17943.5	0.672405
$894$	0	0
$895$	3699.58	0.138171
$896$	0	0
$897$	−7392.00	−0.275152
$898$	0	0
$899$	22152.0	0.821814
$900$	0	0
$901$	23176.1	0.856947
$902$	0	0
$903$	10691.5	0.394008
$904$	0	0
$905$	5648.00	0.207454
$906$	0	0
$907$	31972.0	1.17047	0.585233	−	0.810865i	$-0.301003\pi$
$907$	31972.0	1.17047	0.585233	+	0.810865i	$0.301003\pi$
$908$	0	0
$909$	2316.48	0.0845246
$910$	0	0
$911$	−26858.7	−0.976806	−0.488403	−	0.872618i	$-0.662420\pi$
$911$	−26858.7	−0.976806	−0.488403	+	0.872618i	$0.662420\pi$
$912$	0	0
$913$	20720.0	0.751075
$914$	0	0
$915$	6240.00	0.225451
$916$	0	0
$917$	22231.4	0.800596
$918$	0	0
$919$	−40336.2	−1.44784	−0.723922	−	0.689882i	$-0.757662\pi$
$919$	−40336.2	−1.44784	−0.723922	+	0.689882i	$0.757662\pi$
$920$	0	0
$921$	14820.0	0.530223
$922$	0	0
$923$	13216.0	0.471300
$924$	0	0
$925$	31768.9	1.12925
$926$	0	0
$927$	−16979.0	−0.601580
$928$	0	0
$929$	−13650.0	−0.482069	−0.241034	−	0.970517i	$-0.577487\pi$
$929$	−13650.0	−0.482069	−0.241034	+	0.970517i	$0.577487\pi$
$930$	0	0
$931$	7436.00	0.261767
$932$	0	0
$933$	−10148.4	−0.356102
$934$	0	0
$935$	1923.33	0.0672723
$936$	0	0
$937$	7098.00	0.247472	0.123736	−	0.992315i	$-0.460512\pi$
$937$	7098.00	0.247472	0.123736	+	0.992315i	$0.460512\pi$
$938$	0	0
$939$	−9318.00	−0.323835
$940$	0	0
$941$	−41326.1	−1.43166	−0.715831	−	0.698274i	$-0.753952\pi$
$941$	−41326.1	−1.43166	−0.715831	+	0.698274i	$0.753952\pi$
$942$	0	0
$943$	1617.86	0.0558693
$944$	0	0
$945$	−1080.00	−0.0371771
$946$	0	0
$947$	9900.00	0.339711	0.169856	−	0.985469i	$-0.445670\pi$
$947$	9900.00	0.339711	0.169856	+	0.985469i	$0.445670\pi$
$948$	0	0
$949$	13384.1	0.457815
$950$	0	0
$951$	20186.5	0.688319
$952$	0	0
$953$	−46938.0	−1.59546	−0.797729	−	0.603016i	$-0.793965\pi$
$953$	−46938.0	−1.59546	−0.797729	+	0.603016i	$0.793965\pi$
$954$	0	0
$955$	−2656.00	−0.0899960
$956$	0	0
$957$	−12049.1	−0.406993
$958$	0	0
$959$	−40361.7	−1.35907
$960$	0	0
$961$	−17623.0	−0.591554
$962$	0	0
$963$	−12636.0	−0.422834
$964$	0	0
$965$	−7042.78	−0.234938
$966$	0	0
$967$	−6989.04	−0.232422	−0.116211	−	0.993225i	$-0.537075\pi$
$967$	−6989.04	−0.232422	−0.116211	+	0.993225i	$0.537075\pi$
$968$	0	0
$969$	5304.00	0.175840
$970$	0	0
$971$	53052.0	1.75337	0.876684	−	0.481067i	$-0.159751\pi$
$971$	53052.0	1.75337	0.876684	+	0.481067i	$0.159751\pi$
$972$	0	0
$973$	27775.2	0.915139
$974$	0	0
$975$	−13898.9	−0.456534
$976$	0	0
$977$	−41890.0	−1.37173	−0.685865	−	0.727729i	$-0.740576\pi$
$977$	−41890.0	−1.37173	−0.685865	+	0.727729i	$0.740576\pi$
$978$	0	0
$979$	−4680.00	−0.152782
$980$	0	0
$981$	−356.382	−0.0115988
$982$	0	0
$983$	10861.2	0.352408	0.176204	−	0.984354i	$-0.443618\pi$
$983$	10861.2	0.352408	0.176204	+	0.984354i	$0.443618\pi$
$984$	0	0
$985$	7704.00	0.249208
$986$	0	0
$987$	14640.0	0.472134
$988$	0	0
$989$	15680.8	0.504166
$990$	0	0
$991$	−330.926	−0.0106077	−0.00530384	−	0.999986i	$-0.501688\pi$
$991$	−330.926	−0.0106077	−0.00530384	+	0.999986i	$0.501688\pi$
$992$	0	0
$993$	−8724.00	−0.278799
$994$	0	0
$995$	−6104.00	−0.194482
$996$	0	0
$997$	39948.7	1.26900	0.634498	−	0.772925i	$-0.281207\pi$
$997$	39948.7	1.26900	0.634498	+	0.772925i	$0.281207\pi$
$998$	0	0
$999$	−7331.28	−0.232184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.m.1.2		2
3.2	odd	2		2304.4.a.bh.1.1		2
4.3	odd	2		768.4.a.h.1.2		2
8.3	odd	2	inner	768.4.a.m.1.1		2
8.5	even	2		768.4.a.h.1.1		2
12.11	even	2		2304.4.a.bb.1.1		2
16.3	odd	4		384.4.d.d.193.1	✓	4
16.5	even	4		384.4.d.d.193.2	yes	4
16.11	odd	4		384.4.d.d.193.4	yes	4
16.13	even	4		384.4.d.d.193.3	yes	4
24.5	odd	2		2304.4.a.bb.1.2		2
24.11	even	2		2304.4.a.bh.1.2		2
48.5	odd	4		1152.4.d.n.577.2		4
48.11	even	4		1152.4.d.n.577.1		4
48.29	odd	4		1152.4.d.n.577.4		4
48.35	even	4		1152.4.d.n.577.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.d.193.1	✓	4	16.3	odd	4
384.4.d.d.193.2	yes	4	16.5	even	4
384.4.d.d.193.3	yes	4	16.13	even	4
384.4.d.d.193.4	yes	4	16.11	odd	4
768.4.a.h.1.1		2	8.5	even	2
768.4.a.h.1.2		2	4.3	odd	2
768.4.a.m.1.1		2	8.3	odd	2	inner
768.4.a.m.1.2		2	1.1	even	1	trivial
1152.4.d.n.577.1		4	48.11	even	4
1152.4.d.n.577.2		4	48.5	odd	4
1152.4.d.n.577.3		4	48.35	even	4
1152.4.d.n.577.4		4	48.29	odd	4
2304.4.a.bb.1.1		2	12.11	even	2
2304.4.a.bb.1.2		2	24.5	odd	2
2304.4.a.bh.1.1		2	3.2	odd	2
2304.4.a.bh.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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +2.82843 q^{5} -14.1421 q^{7} +9.00000 q^{9} -20.0000 q^{11} +39.5980 q^{13} +8.48528 q^{15} -34.0000 q^{17} -52.0000 q^{19} -42.4264 q^{21} -62.2254 q^{23} -117.000 q^{25} +27.0000 q^{27} +200.818 q^{29} +110.309 q^{31} -60.0000 q^{33} -40.0000 q^{35} -271.529 q^{37} +118.794 q^{39} -26.0000 q^{41} -252.000 q^{43} +25.4558 q^{45} -345.068 q^{47} -143.000 q^{49} -102.000 q^{51} -681.651 q^{53} -56.5685 q^{55} -156.000 q^{57} -364.000 q^{59} +735.391 q^{61} -127.279 q^{63} +112.000 q^{65} -628.000 q^{67} -186.676 q^{69} +333.754 q^{71} +338.000 q^{73} -351.000 q^{75} +282.843 q^{77} +789.131 q^{79} +81.0000 q^{81} -1036.00 q^{83} -96.1665 q^{85} +602.455 q^{87} +234.000 q^{89} -560.000 q^{91} +330.926 q^{93} -147.078 q^{95} -178.000 q^{97} -180.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 18 q^{9} - 40 q^{11} - 68 q^{17} - 104 q^{19} - 234 q^{25} + 54 q^{27} - 120 q^{33} - 80 q^{35} - 52 q^{41} - 504 q^{43} - 286 q^{49} - 204 q^{51} - 312 q^{57} - 728 q^{59} + 224 q^{65}+ \cdots - 360 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 18 * q^9 - 40 * q^11 - 68 * q^17 - 104 * q^19 - 234 * q^25 + 54 * q^27 - 120 * q^33 - 80 * q^35 - 52 * q^41 - 504 * q^43 - 286 * q^49 - 204 * q^51 - 312 * q^57 - 728 * q^59 + 224 * q^65 - 1256 * q^67 + 676 * q^73 - 702 * q^75 + 162 * q^81 - 2072 * q^83 + 468 * q^89 - 1120 * q^91 - 356 * q^97 - 360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more