LMFDB - Embedded newform 768.5.b.g.127.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(127,768)]

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([1, 1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("768.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$79.3881316484$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.592240896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 $ x^8 - 7x^6 + 40x^4 - 63*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Root			$-1.99426 - 1.15139i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.5.b.g.127.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} -22.8444i q^{5} -56.8882i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} -22.8444i q^{5} -56.8882i q^{7} +27.0000 q^{9} +134.561 q^{11} -247.066i q^{13} -118.703i q^{15} -92.3112 q^{17} +29.5600 q^{19} -295.600i q^{21} -571.038i q^{23} +103.133 q^{25} +140.296 q^{27} -20.0891i q^{29} +474.736i q^{31} +699.199 q^{33} -1299.58 q^{35} -755.867i q^{37} -1283.79i q^{39} -541.822 q^{41} +3097.06 q^{43} -616.799i q^{45} +1050.16i q^{47} -835.266 q^{49} -479.663 q^{51} +1768.35i q^{53} -3073.97i q^{55} +153.598 q^{57} -2582.98 q^{59} +2403.33i q^{61} -1535.98i q^{63} -5644.09 q^{65} +379.816 q^{67} -2967.20i q^{69} +702.517i q^{71} -9824.92 q^{73} +535.894 q^{75} -7654.93i q^{77} -3756.40i q^{79} +729.000 q^{81} -10433.6 q^{83} +2108.79i q^{85} -104.386i q^{87} +11923.5 q^{89} -14055.2 q^{91} +2466.80i q^{93} -675.280i q^{95} -2199.06 q^{97} +3633.15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 216 q^{9}+O(q^{10}) $ 8 * q + 216 * q^9	$ 8 q + 216 q^{9} - 1200 q^{17} - 1944 q^{25} + 1440 q^{33} - 1104 q^{41} - 1144 q^{49} - 11232 q^{57} - 36384 q^{65} - 17680 q^{73} + 5832 q^{81} + 50160 q^{89} + 46096 q^{97}+O(q^{100}) $ 8 * q + 216 * q^9 - 1200 * q^17 - 1944 * q^25 + 1440 * q^33 - 1104 * q^41 - 1144 * q^49 - 11232 * q^57 - 36384 * q^65 - 17680 * q^73 + 5832 * q^81 + 50160 * q^89 + 46096 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	0	0
$5$	− 22.8444i	− 0.913776i	−0.889524	−	0.456888i	$-0.848964\pi$
$5$	− 22.8444i	− 0.913776i	0.889524	−	0.456888i	$-0.151036\pi$
$6$	0	0
$7$	− 56.8882i	− 1.16098i	−0.814266	−	0.580492i	$-0.802860\pi$
$7$	− 56.8882i	− 1.16098i	0.814266	−	0.580492i	$-0.197140\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	134.561	1.11207	0.556037	−	0.831157i	$-0.312321\pi$
$11$	134.561	1.11207	0.556037	+	0.831157i	$0.312321\pi$
$12$	0	0
$13$	− 247.066i	− 1.46193i	−0.682414	−	0.730966i	$-0.739070\pi$
$13$	− 247.066i	− 1.46193i	0.682414	−	0.730966i	$-0.260930\pi$
$14$	0	0
$15$	− 118.703i	− 0.527569i
$16$	0	0
$17$	−92.3112	−0.319416	−0.159708	−	0.987164i	$-0.551055\pi$
$17$	−92.3112	−0.319416	−0.159708	+	0.987164i	$0.551055\pi$
$18$	0	0
$19$	29.5600	0.0818836	0.0409418	−	0.999162i	$-0.486964\pi$
$19$	29.5600	0.0818836	0.0409418	+	0.999162i	$0.486964\pi$
$20$	0	0
$21$	− 295.600i	− 0.670294i
$22$	0	0
$23$	− 571.038i	− 1.07947i	−0.841836	−	0.539733i	$-0.818525\pi$
$23$	− 571.038i	− 1.07947i	0.841836	−	0.539733i	$-0.181475\pi$
$24$	0	0
$25$	103.133	0.165013
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	− 20.0891i	− 0.0238872i	−0.999929	−	0.0119436i	$-0.996198\pi$
$29$	− 20.0891i	− 0.0238872i	0.999929	−	0.0119436i	$-0.00380186\pi$
$30$	0	0
$31$	474.736i	0.494002i	0.969015	+	0.247001i	$0.0794452\pi$
$31$	474.736i	0.494002i	−0.969015	+	0.247001i	$0.920555\pi$
$32$	0	0
$33$	699.199	0.642056
$34$	0	0
$35$	−1299.58	−1.06088
$36$	0	0
$37$	− 755.867i	− 0.552131i	−0.961139	−	0.276065i	$-0.910969\pi$
$37$	− 755.867i	− 0.552131i	0.961139	−	0.276065i	$-0.0890306\pi$
$38$	0	0
$39$	− 1283.79i	− 0.844047i
$40$	0	0
$41$	−541.822	−0.322321	−0.161161	−	0.986928i	$-0.551524\pi$
$41$	−541.822	−0.322321	−0.161161	+	0.986928i	$0.551524\pi$
$42$	0	0
$43$	3097.06	1.67499	0.837496	−	0.546444i	$-0.184019\pi$
$43$	3097.06	1.67499	0.837496	+	0.546444i	$0.184019\pi$
$44$	0	0
$45$	− 616.799i	− 0.304592i
$46$	0	0
$47$	1050.16i	0.475401i	0.971338	+	0.237701i	$0.0763937\pi$
$47$	1050.16i	0.475401i	−0.971338	+	0.237701i	$0.923606\pi$
$48$	0	0
$49$	−835.266	−0.347882
$50$	0	0
$51$	−479.663	−0.184415
$52$	0	0
$53$	1768.35i	0.629532i	0.949169	+	0.314766i	$0.101926\pi$
$53$	1768.35i	0.629532i	−0.949169	+	0.314766i	$0.898074\pi$
$54$	0	0
$55$	− 3073.97i	− 1.01619i
$56$	0	0
$57$	153.598	0.0472755
$58$	0	0
$59$	−2582.98	−0.742024	−0.371012	−	0.928628i	$-0.620989\pi$
$59$	−2582.98	−0.742024	−0.371012	+	0.928628i	$0.620989\pi$
$60$	0	0
$61$	2403.33i	0.645883i	0.946419	+	0.322941i	$0.104672\pi$
$61$	2403.33i	0.645883i	−0.946419	+	0.322941i	$0.895328\pi$
$62$	0	0
$63$	− 1535.98i	− 0.386994i
$64$	0	0
$65$	−5644.09	−1.33588
$66$	0	0
$67$	379.816	0.0846104	0.0423052	−	0.999105i	$-0.486530\pi$
$67$	379.816	0.0846104	0.0423052	+	0.999105i	$0.486530\pi$
$68$	0	0
$69$	− 2967.20i	− 0.623230i
$70$	0	0
$71$	702.517i	0.139361i	0.997569	+	0.0696803i	$0.0221979\pi$
$71$	702.517i	0.139361i	−0.997569	+	0.0696803i	$0.977802\pi$
$72$	0	0
$73$	−9824.92	−1.84367	−0.921836	−	0.387581i	$-0.873311\pi$
$73$	−9824.92	−1.84367	−0.921836	+	0.387581i	$0.873311\pi$
$74$	0	0
$75$	535.894	0.0952701
$76$	0	0
$77$	− 7654.93i	− 1.29110i
$78$	0	0
$79$	− 3756.40i	− 0.601890i	−0.953641	−	0.300945i	$-0.902698\pi$
$79$	− 3756.40i	− 0.601890i	0.953641	−	0.300945i	$-0.0973021\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	−10433.6	−1.51452	−0.757262	−	0.653111i	$-0.773463\pi$
$83$	−10433.6	−1.51452	−0.757262	+	0.653111i	$0.773463\pi$
$84$	0	0
$85$	2108.79i	0.291875i
$86$	0	0
$87$	− 104.386i	− 0.0137913i
$88$	0	0
$89$	11923.5	1.50530	0.752651	−	0.658419i	$-0.228775\pi$
$89$	11923.5	1.50530	0.752651	+	0.658419i	$0.228775\pi$
$90$	0	0
$91$	−14055.2	−1.69728
$92$	0	0
$93$	2466.80i	0.285212i
$94$	0	0
$95$	− 675.280i	− 0.0748233i
$96$	0	0
$97$	−2199.06	−0.233718	−0.116859	−	0.993148i	$-0.537283\pi$
$97$	−2199.06	−0.233718	−0.116859	+	0.993148i	$0.537283\pi$
$98$	0	0
$99$	3633.15	0.370691
$100$	0	0
$101$	− 874.835i	− 0.0857597i	−0.999080	−	0.0428799i	$-0.986347\pi$
$101$	− 874.835i	− 0.0857597i	0.999080	−	0.0428799i	$-0.0136533\pi$
$102$	0	0
$103$	3036.94i	0.286260i	0.989704	+	0.143130i	$0.0457168\pi$
$103$	3036.94i	0.286260i	−0.989704	+	0.143130i	$0.954283\pi$
$104$	0	0
$105$	−6752.80	−0.612499
$106$	0	0
$107$	−19057.6	−1.66457	−0.832284	−	0.554350i	$-0.812967\pi$
$107$	−19057.6	−1.66457	−0.832284	+	0.554350i	$0.812967\pi$
$108$	0	0
$109$	13132.7i	1.10535i	0.833397	+	0.552675i	$0.186393\pi$
$109$	13132.7i	1.10535i	−0.833397	+	0.552675i	$0.813607\pi$
$110$	0	0
$111$	− 3927.60i	− 0.318773i
$112$	0	0
$113$	14541.5	1.13882	0.569408	−	0.822055i	$-0.307173\pi$
$113$	14541.5	1.13882	0.569408	+	0.822055i	$0.307173\pi$
$114$	0	0
$115$	−13045.0	−0.986391
$116$	0	0
$117$	− 6670.79i	− 0.487311i
$118$	0	0
$119$	5251.42i	0.370836i
$120$	0	0
$121$	3465.66	0.236709
$122$	0	0
$123$	−2815.39	−0.186092
$124$	0	0
$125$	− 16633.8i	− 1.06456i
$126$	0	0
$127$	− 25959.7i	− 1.60951i	−0.593609	−	0.804753i	$-0.702298\pi$
$127$	− 25959.7i	− 1.60951i	0.593609	−	0.804753i	$-0.297702\pi$
$128$	0	0
$129$	16092.8	0.967057
$130$	0	0
$131$	16440.3	0.958003	0.479002	−	0.877814i	$-0.340999\pi$
$131$	16440.3	0.958003	0.479002	+	0.877814i	$0.340999\pi$
$132$	0	0
$133$	− 1681.61i	− 0.0950655i
$134$	0	0
$135$	− 3204.98i	− 0.175856i
$136$	0	0
$137$	31618.7	1.68462	0.842312	−	0.538990i	$-0.181194\pi$
$137$	31618.7	1.68462	0.842312	+	0.538990i	$0.181194\pi$
$138$	0	0
$139$	−12895.9	−0.667453	−0.333726	−	0.942670i	$-0.608306\pi$
$139$	−12895.9	−0.667453	−0.333726	+	0.942670i	$0.608306\pi$
$140$	0	0
$141$	5456.80i	0.274473i
$142$	0	0
$143$	− 33245.5i	− 1.62578i
$144$	0	0
$145$	−458.924	−0.0218276
$146$	0	0
$147$	−4340.17	−0.200850
$148$	0	0
$149$	31474.8i	1.41772i	0.705350	+	0.708859i	$0.250790\pi$
$149$	31474.8i	1.41772i	−0.705350	+	0.708859i	$0.749210\pi$
$150$	0	0
$151$	− 39479.3i	− 1.73147i	−0.500502	−	0.865735i	$-0.666851\pi$
$151$	− 39479.3i	− 1.73147i	0.500502	−	0.865735i	$-0.333149\pi$
$152$	0	0
$153$	−2492.40	−0.106472
$154$	0	0
$155$	10845.1	0.451408
$156$	0	0
$157$	− 2619.07i	− 0.106255i	−0.998588	−	0.0531273i	$-0.983081\pi$
$157$	− 2619.07i	− 0.106255i	0.998588	−	0.0531273i	$-0.0169189\pi$
$158$	0	0
$159$	9188.64i	0.363460i
$160$	0	0
$161$	−32485.3	−1.25324
$162$	0	0
$163$	−7123.14	−0.268100	−0.134050	−	0.990975i	$-0.542798\pi$
$163$	−7123.14	−0.268100	−0.134050	+	0.990975i	$0.542798\pi$
$164$	0	0
$165$	− 15972.8i	− 0.586696i
$166$	0	0
$167$	− 48670.8i	− 1.74516i	−0.488472	−	0.872580i	$-0.662445\pi$
$167$	− 48670.8i	− 1.74516i	0.488472	−	0.872580i	$-0.337555\pi$
$168$	0	0
$169$	−32480.8	−1.13724
$170$	0	0
$171$	798.119	0.0272945
$172$	0	0
$173$	27575.9i	0.921377i	0.887562	+	0.460689i	$0.152397\pi$
$173$	27575.9i	0.921377i	−0.887562	+	0.460689i	$0.847603\pi$
$174$	0	0
$175$	− 5867.05i	− 0.191577i
$176$	0	0
$177$	−13421.6	−0.428408
$178$	0	0
$179$	−42280.0	−1.31956	−0.659780	−	0.751459i	$-0.729351\pi$
$179$	−42280.0	−1.31956	−0.659780	+	0.751459i	$0.729351\pi$
$180$	0	0
$181$	− 1006.79i	− 0.0307313i	−0.999882	−	0.0153657i	$-0.995109\pi$
$181$	− 1006.79i	− 0.0307313i	0.999882	−	0.0153657i	$-0.00489123\pi$
$182$	0	0
$183$	12488.1i	0.372901i
$184$	0	0
$185$	−17267.3	−0.504524
$186$	0	0
$187$	−12421.5	−0.355214
$188$	0	0
$189$	− 7981.19i	− 0.223431i
$190$	0	0
$191$	− 6906.62i	− 0.189321i	−0.995510	−	0.0946605i	$-0.969823\pi$
$191$	− 6906.62i	− 0.189321i	0.995510	−	0.0946605i	$-0.0301766\pi$
$192$	0	0
$193$	28207.8	0.757278	0.378639	−	0.925545i	$-0.376392\pi$
$193$	28207.8	0.757278	0.378639	+	0.925545i	$0.376392\pi$
$194$	0	0
$195$	−29327.5	−0.771270
$196$	0	0
$197$	38453.6i	0.990843i	0.868653	+	0.495422i	$0.164986\pi$
$197$	38453.6i	0.990843i	−0.868653	+	0.495422i	$0.835014\pi$
$198$	0	0
$199$	5490.10i	0.138635i	0.997595	+	0.0693176i	$0.0220822\pi$
$199$	5490.10i	0.138635i	−0.997595	+	0.0693176i	$0.977918\pi$
$200$	0	0
$201$	1973.58	0.0488498
$202$	0	0
$203$	−1142.83	−0.0277326
$204$	0	0
$205$	12377.6i	0.294529i
$206$	0	0
$207$	− 15418.0i	− 0.359822i
$208$	0	0
$209$	3977.62	0.0910606
$210$	0	0
$211$	−46471.5	−1.04381	−0.521905	−	0.853004i	$-0.674778\pi$
$211$	−46471.5	−1.04381	−0.521905	+	0.853004i	$0.674778\pi$
$212$	0	0
$213$	3650.38i	0.0804599i
$214$	0	0
$215$	− 70750.5i	− 1.53057i
$216$	0	0
$217$	27006.9	0.573529
$218$	0	0
$219$	−51051.8	−1.06444
$220$	0	0
$221$	22807.0i	0.466964i
$222$	0	0
$223$	48307.2i	0.971409i	0.874123	+	0.485704i	$0.161437\pi$
$223$	48307.2i	0.971409i	−0.874123	+	0.485704i	$0.838563\pi$
$224$	0	0
$225$	2784.59	0.0550042
$226$	0	0
$227$	−19108.6	−0.370832	−0.185416	−	0.982660i	$-0.559363\pi$
$227$	−19108.6	−0.370832	−0.185416	+	0.982660i	$0.559363\pi$
$228$	0	0
$229$	− 73378.5i	− 1.39926i	−0.714506	−	0.699629i	$-0.753349\pi$
$229$	− 73378.5i	− 1.39926i	0.714506	−	0.699629i	$-0.246651\pi$
$230$	0	0
$231$	− 39776.2i	− 0.745417i
$232$	0	0
$233$	34041.9	0.627049	0.313524	−	0.949580i	$-0.398490\pi$
$233$	34041.9	0.627049	0.313524	+	0.949580i	$0.398490\pi$
$234$	0	0
$235$	23990.3	0.434411
$236$	0	0
$237$	− 19518.8i	− 0.347501i
$238$	0	0
$239$	− 35114.3i	− 0.614735i	−0.951591	−	0.307368i	$-0.900552\pi$
$239$	− 35114.3i	− 0.614735i	0.951591	−	0.307368i	$-0.0994482\pi$
$240$	0	0
$241$	−19621.0	−0.337821	−0.168910	−	0.985631i	$-0.554025\pi$
$241$	−19621.0	−0.337821	−0.168910	+	0.985631i	$0.554025\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	0	0
$245$	19081.2i	0.317887i
$246$	0	0
$247$	− 7303.28i	− 0.119708i
$248$	0	0
$249$	−54214.3	−0.874411
$250$	0	0
$251$	−560.729	−0.00890032	−0.00445016	−	0.999990i	$-0.501417\pi$
$251$	−560.729	−0.00890032	−0.00445016	+	0.999990i	$0.501417\pi$
$252$	0	0
$253$	− 76839.4i	− 1.20045i
$254$	0	0
$255$	10957.6i	0.168514i
$256$	0	0
$257$	80511.1	1.21896	0.609480	−	0.792801i	$-0.291378\pi$
$257$	80511.1	1.21896	0.609480	+	0.792801i	$0.291378\pi$
$258$	0	0
$259$	−42999.9	−0.641015
$260$	0	0
$261$	− 542.406i	− 0.00796240i
$262$	0	0
$263$	34821.7i	0.503430i	0.967801	+	0.251715i	$0.0809945\pi$
$263$	34821.7i	0.503430i	−0.967801	+	0.251715i	$0.919005\pi$
$264$	0	0
$265$	40397.0	0.575251
$266$	0	0
$267$	61956.3	0.869087
$268$	0	0
$269$	28469.5i	0.393438i	0.980460	+	0.196719i	$0.0630286\pi$
$269$	28469.5i	0.393438i	−0.980460	+	0.196719i	$0.936971\pi$
$270$	0	0
$271$	− 37653.7i	− 0.512706i	−0.966583	−	0.256353i	$-0.917479\pi$
$271$	− 37653.7i	− 0.512706i	0.966583	−	0.256353i	$-0.0825210\pi$
$272$	0	0
$273$	−73032.8	−0.979924
$274$	0	0
$275$	13877.7	0.183506
$276$	0	0
$277$	25920.1i	0.337814i	0.985632	+	0.168907i	$0.0540237\pi$
$277$	25920.1i	0.337814i	−0.985632	+	0.168907i	$0.945976\pi$
$278$	0	0
$279$	12817.9i	0.164667i
$280$	0	0
$281$	−49815.6	−0.630889	−0.315444	−	0.948944i	$-0.602154\pi$
$281$	−49815.6	−0.630889	−0.315444	+	0.948944i	$0.602154\pi$
$282$	0	0
$283$	−73403.9	−0.916530	−0.458265	−	0.888816i	$-0.651529\pi$
$283$	−73403.9	−0.916530	−0.458265	+	0.888816i	$0.651529\pi$
$284$	0	0
$285$	− 3508.86i	− 0.0431993i
$286$	0	0
$287$	30823.3i	0.374209i
$288$	0	0
$289$	−74999.6	−0.897974
$290$	0	0
$291$	−11426.6	−0.134937
$292$	0	0
$293$	− 130683.i	− 1.52224i	−0.648611	−	0.761120i	$-0.724650\pi$
$293$	− 130683.i	− 1.52224i	0.648611	−	0.761120i	$-0.275350\pi$
$294$	0	0
$295$	59006.8i	0.678044i
$296$	0	0
$297$	18878.4	0.214019
$298$	0	0
$299$	−141084.	−1.57811
$300$	0	0
$301$	− 176186.i	− 1.94464i
$302$	0	0
$303$	− 4545.78i	− 0.0495134i
$304$	0	0
$305$	54902.6	0.590192
$306$	0	0
$307$	139512.	1.48025	0.740126	−	0.672469i	$-0.234766\pi$
$307$	139512.	1.48025	0.740126	+	0.672469i	$0.234766\pi$
$308$	0	0
$309$	15780.4i	0.165273i
$310$	0	0
$311$	132034.i	1.36510i	0.730837	+	0.682552i	$0.239130\pi$
$311$	132034.i	1.36510i	−0.730837	+	0.682552i	$0.760870\pi$
$312$	0	0
$313$	−39526.3	−0.403458	−0.201729	−	0.979441i	$-0.564656\pi$
$313$	−39526.3	−0.403458	−0.201729	+	0.979441i	$0.564656\pi$
$314$	0	0
$315$	−35088.6	−0.353626
$316$	0	0
$317$	91959.1i	0.915116i	0.889180	+	0.457558i	$0.151276\pi$
$317$	91959.1i	0.915116i	−0.889180	+	0.457558i	$0.848724\pi$
$318$	0	0
$319$	− 2703.21i	− 0.0265643i
$320$	0	0
$321$	−99026.4	−0.961039
$322$	0	0
$323$	−2728.72	−0.0261549
$324$	0	0
$325$	− 25480.7i	− 0.241237i
$326$	0	0
$327$	68239.3i	0.638174i
$328$	0	0
$329$	59741.8	0.551933
$330$	0	0
$331$	−3244.08	−0.0296098	−0.0148049	−	0.999890i	$-0.504713\pi$
$331$	−3244.08	−0.0296098	−0.0148049	+	0.999890i	$0.504713\pi$
$332$	0	0
$333$	− 20408.4i	− 0.184044i
$334$	0	0
$335$	− 8676.68i	− 0.0773150i
$336$	0	0
$337$	5591.21	0.0492318	0.0246159	−	0.999697i	$-0.492164\pi$
$337$	5591.21	0.0492318	0.0246159	+	0.999697i	$0.492164\pi$
$338$	0	0
$339$	75560.0	0.657495
$340$	0	0
$341$	63881.0i	0.549367i
$342$	0	0
$343$	− 89071.8i	− 0.757098i
$344$	0	0
$345$	−67783.9	−0.569493
$346$	0	0
$347$	170630.	1.41708	0.708542	−	0.705668i	$-0.249353\pi$
$347$	170630.	1.41708	0.708542	+	0.705668i	$0.249353\pi$
$348$	0	0
$349$	96101.4i	0.789003i	0.918895	+	0.394502i	$0.129083\pi$
$349$	96101.4i	0.789003i	−0.918895	+	0.394502i	$0.870917\pi$
$350$	0	0
$351$	− 34662.5i	− 0.281349i
$352$	0	0
$353$	205914.	1.65248	0.826242	−	0.563316i	$-0.190474\pi$
$353$	205914.	1.65248	0.826242	+	0.563316i	$0.190474\pi$
$354$	0	0
$355$	16048.6	0.127344
$356$	0	0
$357$	27287.2i	0.214103i
$358$	0	0
$359$	78435.2i	0.608586i	0.952578	+	0.304293i	$0.0984202\pi$
$359$	78435.2i	0.608586i	−0.952578	+	0.304293i	$0.901580\pi$
$360$	0	0
$361$	−129447.	−0.993295
$362$	0	0
$363$	18008.1	0.136664
$364$	0	0
$365$	224445.i	1.68470i
$366$	0	0
$367$	− 221785.i	− 1.64664i	−0.567574	−	0.823322i	$-0.692118\pi$
$367$	− 221785.i	− 1.64664i	0.567574	−	0.823322i	$-0.307882\pi$
$368$	0	0
$369$	−14629.2	−0.107440
$370$	0	0
$371$	100599.	0.730876
$372$	0	0
$373$	− 25883.8i	− 0.186042i	−0.995664	−	0.0930208i	$-0.970348\pi$
$373$	− 25883.8i	− 0.186042i	0.995664	−	0.0930208i	$-0.0296523\pi$
$374$	0	0
$375$	− 86431.6i	− 0.614625i
$376$	0	0
$377$	−4963.35	−0.0349214
$378$	0	0
$379$	147801.	1.02896	0.514479	−	0.857503i	$-0.327985\pi$
$379$	147801.	1.02896	0.514479	+	0.857503i	$0.327985\pi$
$380$	0	0
$381$	− 134891.i	− 0.929249i
$382$	0	0
$383$	226231.i	1.54225i	0.636683	+	0.771126i	$0.280306\pi$
$383$	226231.i	1.54225i	−0.636683	+	0.771126i	$0.719694\pi$
$384$	0	0
$385$	−174872.	−1.17978
$386$	0	0
$387$	83620.6	0.558330
$388$	0	0
$389$	256419.i	1.69453i	0.531167	+	0.847267i	$0.321754\pi$
$389$	256419.i	1.69453i	−0.531167	+	0.847267i	$0.678246\pi$
$390$	0	0
$391$	52713.2i	0.344799i
$392$	0	0
$393$	85426.3	0.553103
$394$	0	0
$395$	−85812.7	−0.549993
$396$	0	0
$397$	56667.7i	0.359546i	0.983708	+	0.179773i	$0.0575364\pi$
$397$	56667.7i	0.359546i	−0.983708	+	0.179773i	$0.942464\pi$
$398$	0	0
$399$	− 8737.92i	− 0.0548861i
$400$	0	0
$401$	10912.0	0.0678605	0.0339302	−	0.999424i	$-0.489198\pi$
$401$	10912.0	0.0678605	0.0339302	+	0.999424i	$0.489198\pi$
$402$	0	0
$403$	117291.	0.722198
$404$	0	0
$405$	− 16653.6i	− 0.101531i
$406$	0	0
$407$	− 101710.i	− 0.614010i
$408$	0	0
$409$	289576.	1.73108	0.865538	−	0.500844i	$-0.166977\pi$
$409$	289576.	1.73108	0.865538	+	0.500844i	$0.166977\pi$
$410$	0	0
$411$	164296.	0.972618
$412$	0	0
$413$	146941.i	0.861477i
$414$	0	0
$415$	238348.i	1.38394i
$416$	0	0
$417$	−67008.8	−0.385354
$418$	0	0
$419$	69837.9	0.397798	0.198899	−	0.980020i	$-0.436263\pi$
$419$	69837.9	0.397798	0.198899	+	0.980020i	$0.436263\pi$
$420$	0	0
$421$	16207.0i	0.0914405i	0.998954	+	0.0457203i	$0.0145583\pi$
$421$	16207.0i	0.0914405i	−0.998954	+	0.0457203i	$0.985442\pi$
$422$	0	0
$423$	28354.4i	0.158467i
$424$	0	0
$425$	−9520.32	−0.0527077
$426$	0	0
$427$	136721.	0.749859
$428$	0	0
$429$	− 172749.i	− 0.938643i
$430$	0	0
$431$	− 105264.i	− 0.566662i	−0.959022	−	0.283331i	$-0.908560\pi$
$431$	− 105264.i	− 0.566662i	0.959022	−	0.283331i	$-0.0914395\pi$
$432$	0	0
$433$	131799.	0.702970	0.351485	−	0.936193i	$-0.385677\pi$
$433$	131799.	0.702970	0.351485	+	0.936193i	$0.385677\pi$
$434$	0	0
$435$	−2384.64	−0.0126021
$436$	0	0
$437$	− 16879.9i	− 0.0883906i
$438$	0	0
$439$	149053.i	0.773414i	0.922203	+	0.386707i	$0.126388\pi$
$439$	149053.i	0.773414i	−0.922203	+	0.386707i	$0.873612\pi$
$440$	0	0
$441$	−22552.2	−0.115961
$442$	0	0
$443$	248933.	1.26845	0.634227	−	0.773147i	$-0.281318\pi$
$443$	248933.	1.26845	0.634227	+	0.773147i	$0.281318\pi$
$444$	0	0
$445$	− 272385.i	− 1.37551i
$446$	0	0
$447$	163548.i	0.818520i
$448$	0	0
$449$	243749.	1.20906	0.604532	−	0.796581i	$-0.293360\pi$
$449$	243749.	1.20906	0.604532	+	0.796581i	$0.293360\pi$
$450$	0	0
$451$	−72908.1	−0.358445
$452$	0	0
$453$	− 205140.i	− 0.999665i
$454$	0	0
$455$	321082.i	1.55093i
$456$	0	0
$457$	−343278.	−1.64367	−0.821833	−	0.569729i	$-0.807048\pi$
$457$	−343278.	−1.64367	−0.821833	+	0.569729i	$0.807048\pi$
$458$	0	0
$459$	−12950.9	−0.0614716
$460$	0	0
$461$	155524.i	0.731804i	0.930653	+	0.365902i	$0.119239\pi$
$461$	155524.i	0.731804i	−0.930653	+	0.365902i	$0.880761\pi$
$462$	0	0
$463$	321457.i	1.49955i	0.661694	+	0.749774i	$0.269838\pi$
$463$	321457.i	1.49955i	−0.661694	+	0.749774i	$0.730162\pi$
$464$	0	0
$465$	56352.6	0.260620
$466$	0	0
$467$	−81219.7	−0.372415	−0.186208	−	0.982510i	$-0.559620\pi$
$467$	−81219.7	−0.372415	−0.186208	+	0.982510i	$0.559620\pi$
$468$	0	0
$469$	− 21607.1i	− 0.0982313i
$470$	0	0
$471$	− 13609.1i	− 0.0613461i
$472$	0	0
$473$	416743.	1.86271
$474$	0	0
$475$	3048.61	0.0135118
$476$	0	0
$477$	47745.6i	0.209844i
$478$	0	0
$479$	346051.i	1.50824i	0.656738	+	0.754119i	$0.271936\pi$
$479$	346051.i	1.50824i	−0.656738	+	0.754119i	$0.728064\pi$
$480$	0	0
$481$	−186749.	−0.807178
$482$	0	0
$483$	−168799.	−0.723560
$484$	0	0
$485$	50236.2i	0.213566i
$486$	0	0
$487$	− 94439.7i	− 0.398196i	−0.979980	−	0.199098i	$-0.936199\pi$
$487$	− 94439.7i	− 0.398196i	0.979980	−	0.199098i	$-0.0638012\pi$
$488$	0	0
$489$	−37012.9	−0.154787
$490$	0	0
$491$	175728.	0.728915	0.364458	−	0.931220i	$-0.381254\pi$
$491$	175728.	0.728915	0.364458	+	0.931220i	$0.381254\pi$
$492$	0	0
$493$	1854.45i	0.00762995i
$494$	0	0
$495$	− 82997.1i	− 0.338729i
$496$	0	0
$497$	39964.9	0.161795
$498$	0	0
$499$	−37294.6	−0.149777	−0.0748885	−	0.997192i	$-0.523860\pi$
$499$	−37294.6	−0.149777	−0.0748885	+	0.997192i	$0.523860\pi$
$500$	0	0
$501$	− 252901.i	− 1.00757i
$502$	0	0
$503$	− 250727.i	− 0.990979i	−0.868614	−	0.495489i	$-0.834989\pi$
$503$	− 250727.i	− 0.990979i	0.868614	−	0.495489i	$-0.165011\pi$
$504$	0	0
$505$	−19985.1	−0.0783652
$506$	0	0
$507$	−168775.	−0.656588
$508$	0	0
$509$	− 151144.i	− 0.583383i	−0.956512	−	0.291692i	$-0.905782\pi$
$509$	− 151144.i	− 0.583383i	0.956512	−	0.291692i	$-0.0942181\pi$
$510$	0	0
$511$	558922.i	2.14047i
$512$	0	0
$513$	4147.15	0.0157585
$514$	0	0
$515$	69377.0	0.261578
$516$	0	0
$517$	141311.i	0.528682i
$518$	0	0
$519$	143289.i	0.531957i
$520$	0	0
$521$	179715.	0.662079	0.331039	−	0.943617i	$-0.392601\pi$
$521$	179715.	0.662079	0.331039	+	0.943617i	$0.392601\pi$
$522$	0	0
$523$	305226.	1.11588	0.557941	−	0.829881i	$-0.311592\pi$
$523$	305226.	1.11588	0.557941	+	0.829881i	$0.311592\pi$
$524$	0	0
$525$	− 30486.1i	− 0.110607i
$526$	0	0
$527$	− 43823.5i	− 0.157792i
$528$	0	0
$529$	−46243.2	−0.165248
$530$	0	0
$531$	−69740.6	−0.247341
$532$	0	0
$533$	133866.i	0.471211i
$534$	0	0
$535$	435360.i	1.52104i
$536$	0	0
$537$	−219694.	−0.761849
$538$	0	0
$539$	−112394.	−0.386871
$540$	0	0
$541$	282083.i	0.963789i	0.876229	+	0.481895i	$0.160051\pi$
$541$	282083.i	0.963789i	−0.876229	+	0.481895i	$0.839949\pi$
$542$	0	0
$543$	− 5231.43i	− 0.0177427i
$544$	0	0
$545$	300008.	1.01004
$546$	0	0
$547$	296270.	0.990176	0.495088	−	0.868843i	$-0.335136\pi$
$547$	296270.	0.990176	0.495088	+	0.868843i	$0.335136\pi$
$548$	0	0
$549$	64889.9i	0.215294i
$550$	0	0
$551$	− 593.834i	− 0.00195597i
$552$	0	0
$553$	−213695.	−0.698785
$554$	0	0
$555$	−89723.7	−0.291287
$556$	0	0
$557$	116573.i	0.375739i	0.982194	+	0.187869i	$0.0601582\pi$
$557$	116573.i	0.375739i	−0.982194	+	0.187869i	$0.939842\pi$
$558$	0	0
$559$	− 765179.i	− 2.44872i
$560$	0	0
$561$	−64543.9	−0.205083
$562$	0	0
$563$	117383.	0.370329	0.185164	−	0.982708i	$-0.440718\pi$
$563$	117383.	0.370329	0.185164	+	0.982708i	$0.440718\pi$
$564$	0	0
$565$	− 332193.i	− 1.04062i
$566$	0	0
$567$	− 41471.5i	− 0.128998i
$568$	0	0
$569$	222193.	0.686288	0.343144	−	0.939283i	$-0.388508\pi$
$569$	222193.	0.686288	0.343144	+	0.939283i	$0.388508\pi$
$570$	0	0
$571$	−117091.	−0.359131	−0.179565	−	0.983746i	$-0.557469\pi$
$571$	−117091.	−0.359131	−0.179565	+	0.983746i	$0.557469\pi$
$572$	0	0
$573$	− 35887.9i	− 0.109305i
$574$	0	0
$575$	− 58892.8i	− 0.178126i
$576$	0	0
$577$	−335932.	−1.00902	−0.504510	−	0.863406i	$-0.668327\pi$
$577$	−335932.	−1.00902	−0.504510	+	0.863406i	$0.668327\pi$
$578$	0	0
$579$	146572.	0.437214
$580$	0	0
$581$	593546.i	1.75834i
$582$	0	0
$583$	237952.i	0.700086i
$584$	0	0
$585$	−152390.	−0.445293
$586$	0	0
$587$	396842.	1.15170	0.575852	−	0.817554i	$-0.304670\pi$
$587$	396842.	1.15170	0.575852	+	0.817554i	$0.304670\pi$
$588$	0	0
$589$	14033.2i	0.0404507i
$590$	0	0
$591$	199811.i	0.572064i
$592$	0	0
$593$	13679.6	0.0389012	0.0194506	−	0.999811i	$-0.493808\pi$
$593$	13679.6	0.0389012	0.0194506	+	0.999811i	$0.493808\pi$
$594$	0	0
$595$	119965.	0.338862
$596$	0	0
$597$	28527.4i	0.0800411i
$598$	0	0
$599$	− 124021.i	− 0.345654i	−0.984952	−	0.172827i	$-0.944710\pi$
$599$	− 124021.i	− 0.345654i	0.984952	−	0.172827i	$-0.0552902\pi$
$600$	0	0
$601$	−484678.	−1.34185	−0.670926	−	0.741525i	$-0.734103\pi$
$601$	−484678.	−1.34185	−0.670926	+	0.741525i	$0.734103\pi$
$602$	0	0
$603$	10255.0	0.0282035
$604$	0	0
$605$	− 79170.9i	− 0.216299i
$606$	0	0
$607$	298924.i	0.811303i	0.914028	+	0.405651i	$0.132955\pi$
$607$	298924.i	0.811303i	−0.914028	+	0.405651i	$0.867045\pi$
$608$	0	0
$609$	−5938.34	−0.0160114
$610$	0	0
$611$	259460.	0.695004
$612$	0	0
$613$	− 155694.i	− 0.414334i	−0.978306	−	0.207167i	$-0.933576\pi$
$613$	− 155694.i	− 0.414334i	0.978306	−	0.207167i	$-0.0664243\pi$
$614$	0	0
$615$	64315.9i	0.170047i
$616$	0	0
$617$	625050.	1.64189	0.820946	−	0.571007i	$-0.193447\pi$
$617$	625050.	1.64189	0.820946	+	0.571007i	$0.193447\pi$
$618$	0	0
$619$	−368822.	−0.962576	−0.481288	−	0.876563i	$-0.659831\pi$
$619$	−368822.	−0.962576	−0.481288	+	0.876563i	$0.659831\pi$
$620$	0	0
$621$	− 80114.4i	− 0.207743i
$622$	0	0
$623$	− 678307.i	− 1.74763i
$624$	0	0
$625$	−315531.	−0.807758
$626$	0	0
$627$	20668.3	0.0525739
$628$	0	0
$629$	69775.0i	0.176359i
$630$	0	0
$631$	− 457172.i	− 1.14821i	−0.818782	−	0.574105i	$-0.805350\pi$
$631$	− 457172.i	− 1.14821i	0.818782	−	0.574105i	$-0.194650\pi$
$632$	0	0
$633$	−241473.	−0.602644
$634$	0	0
$635$	−593035.	−1.47073
$636$	0	0
$637$	206366.i	0.508580i
$638$	0	0
$639$	18967.9i	0.0464535i
$640$	0	0
$641$	−728866.	−1.77391	−0.886955	−	0.461855i	$-0.847184\pi$
$641$	−728866.	−1.77391	−0.886955	+	0.461855i	$0.847184\pi$
$642$	0	0
$643$	−201261.	−0.486786	−0.243393	−	0.969928i	$-0.578260\pi$
$643$	−201261.	−0.486786	−0.243393	+	0.969928i	$0.578260\pi$
$644$	0	0
$645$	− 367630.i	− 0.883673i
$646$	0	0
$647$	284857.i	0.680484i	0.940338	+	0.340242i	$0.110509\pi$
$647$	284857.i	0.680484i	−0.940338	+	0.340242i	$0.889491\pi$
$648$	0	0
$649$	−347569.	−0.825186
$650$	0	0
$651$	140332.	0.331127
$652$	0	0
$653$	− 639617.i	− 1.50001i	−0.661433	−	0.750005i	$-0.730051\pi$
$653$	− 639617.i	− 1.50001i	0.661433	−	0.750005i	$-0.269949\pi$
$654$	0	0
$655$	− 375569.i	− 0.875401i
$656$	0	0
$657$	−265273.	−0.614557
$658$	0	0
$659$	372398.	0.857505	0.428753	−	0.903422i	$-0.358953\pi$
$659$	372398.	0.857505	0.428753	+	0.903422i	$0.358953\pi$
$660$	0	0
$661$	− 137125.i	− 0.313843i	−0.987611	−	0.156922i	$-0.949843\pi$
$661$	− 137125.i	− 0.313843i	0.987611	−	0.156922i	$-0.0501570\pi$
$662$	0	0
$663$	118509.i	0.269602i
$664$	0	0
$665$	−38415.5	−0.0868686
$666$	0	0
$667$	−11471.7	−0.0257854
$668$	0	0
$669$	251011.i	0.560843i
$670$	0	0
$671$	323394.i	0.718269i
$672$	0	0
$673$	840931.	1.85665	0.928325	−	0.371770i	$-0.121249\pi$
$673$	840931.	1.85665	0.928325	+	0.371770i	$0.121249\pi$
$674$	0	0
$675$	14469.1	0.0317567
$676$	0	0
$677$	207006.i	0.451653i	0.974168	+	0.225827i	$0.0725083\pi$
$677$	207006.i	0.451653i	−0.974168	+	0.225827i	$0.927492\pi$
$678$	0	0
$679$	125100.i	0.271343i
$680$	0	0
$681$	−99291.3	−0.214100
$682$	0	0
$683$	241334.	0.517341	0.258670	−	0.965966i	$-0.416716\pi$
$683$	241334.	0.517341	0.258670	+	0.965966i	$0.416716\pi$
$684$	0	0
$685$	− 722311.i	− 1.53937i
$686$	0	0
$687$	− 381286.i	− 0.807862i
$688$	0	0
$689$	436901.	0.920333
$690$	0	0
$691$	−628208.	−1.31567	−0.657836	−	0.753161i	$-0.728528\pi$
$691$	−628208.	−1.31567	−0.657836	+	0.753161i	$0.728528\pi$
$692$	0	0
$693$	− 206683.i	− 0.430367i
$694$	0	0
$695$	294598.i	0.609903i
$696$	0	0
$697$	50016.2	0.102954
$698$	0	0
$699$	176887.	0.362027
$700$	0	0
$701$	470333.i	0.957127i	0.878053	+	0.478563i	$0.158842\pi$
$701$	470333.i	0.957127i	−0.878053	+	0.478563i	$0.841158\pi$
$702$	0	0
$703$	− 22343.4i	− 0.0452105i
$704$	0	0
$705$	124657.	0.250807
$706$	0	0
$707$	−49767.8	−0.0995656
$708$	0	0
$709$	− 666498.i	− 1.32589i	−0.748669	−	0.662944i	$-0.769307\pi$
$709$	− 666498.i	− 1.32589i	0.748669	−	0.662944i	$-0.230693\pi$
$710$	0	0
$711$	− 101423.i	− 0.200630i
$712$	0	0
$713$	271092.	0.533259
$714$	0	0
$715$	−759474.	−1.48560
$716$	0	0
$717$	− 182459.i	− 0.354918i
$718$	0	0
$719$	− 551482.i	− 1.06678i	−0.845870	−	0.533388i	$-0.820918\pi$
$719$	− 551482.i	− 1.06678i	0.845870	−	0.533388i	$-0.179082\pi$
$720$	0	0
$721$	172766.	0.332344
$722$	0	0
$723$	−101953.	−0.195041
$724$	0	0
$725$	− 2071.85i	− 0.00394169i
$726$	0	0
$727$	− 735388.i	− 1.39139i	−0.718339	−	0.695694i	$-0.755097\pi$
$727$	− 735388.i	− 1.39139i	0.718339	−	0.695694i	$-0.244903\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	−285893.	−0.535019
$732$	0	0
$733$	− 372734.i	− 0.693731i	−0.937915	−	0.346866i	$-0.887246\pi$
$733$	− 372734.i	− 0.693731i	0.937915	−	0.346866i	$-0.112754\pi$
$734$	0	0
$735$	99148.6i	0.183532i
$736$	0	0
$737$	51108.4	0.0940931
$738$	0	0
$739$	962125.	1.76174	0.880871	−	0.473356i	$-0.156957\pi$
$739$	962125.	1.76174	0.880871	+	0.473356i	$0.156957\pi$
$740$	0	0
$741$	− 37949.0i	− 0.0691136i
$742$	0	0
$743$	− 557423.i	− 1.00973i	−0.863197	−	0.504867i	$-0.831541\pi$
$743$	− 557423.i	− 1.00973i	0.863197	−	0.504867i	$-0.168459\pi$
$744$	0	0
$745$	719022.	1.29548
$746$	0	0
$747$	−281706.	−0.504841
$748$	0	0
$749$	1.08415e6i	1.93254i
$750$	0	0
$751$	− 619927.i	− 1.09916i	−0.835441	−	0.549580i	$-0.814788\pi$
$751$	− 619927.i	− 1.09916i	0.835441	−	0.549580i	$-0.185212\pi$
$752$	0	0
$753$	−2913.64	−0.00513860
$754$	0	0
$755$	−901881.	−1.58218
$756$	0	0
$757$	− 66502.8i	− 0.116051i	−0.998315	−	0.0580254i	$-0.981520\pi$
$757$	− 66502.8i	− 0.116051i	0.998315	−	0.0580254i	$-0.0184804\pi$
$758$	0	0
$759$	− 399269.i	− 0.693078i
$760$	0	0
$761$	−417057.	−0.720156	−0.360078	−	0.932922i	$-0.617250\pi$
$761$	−417057.	−0.720156	−0.360078	+	0.932922i	$0.617250\pi$
$762$	0	0
$763$	747093.	1.28329
$764$	0	0
$765$	56937.5i	0.0972916i
$766$	0	0
$767$	638169.i	1.08479i
$768$	0	0
$769$	−284602.	−0.481266	−0.240633	−	0.970616i	$-0.577355\pi$
$769$	−284602.	−0.481266	−0.240633	+	0.970616i	$0.577355\pi$
$770$	0	0
$771$	418348.	0.703767
$772$	0	0
$773$	690220.i	1.15512i	0.816347	+	0.577562i	$0.195996\pi$
$773$	690220.i	1.15512i	−0.816347	+	0.577562i	$0.804004\pi$
$774$	0	0
$775$	48960.9i	0.0815167i
$776$	0	0
$777$	−223434.	−0.370090
$778$	0	0
$779$	−16016.2	−0.0263928
$780$	0	0
$781$	94531.3i	0.154979i
$782$	0	0
$783$	− 2818.43i	− 0.00459709i
$784$	0	0
$785$	−59831.1	−0.0970929
$786$	0	0
$787$	−855688.	−1.38155	−0.690774	−	0.723071i	$-0.742730\pi$
$787$	−855688.	−1.38155	−0.690774	+	0.723071i	$0.742730\pi$
$788$	0	0
$789$	180939.i	0.290655i
$790$	0	0
$791$	− 827242.i	− 1.32215i
$792$	0	0
$793$	593782.	0.944236
$794$	0	0
$795$	209909.	0.332122
$796$	0	0
$797$	− 579975.i	− 0.913046i	−0.889712	−	0.456523i	$-0.849095\pi$
$797$	− 579975.i	− 0.913046i	0.889712	−	0.456523i	$-0.150905\pi$
$798$	0	0
$799$	− 96941.7i	− 0.151851i
$800$	0	0
$801$	321935.	0.501768
$802$	0	0
$803$	−1.32205e6	−2.05030
$804$	0	0
$805$	742108.i	1.14518i
$806$	0	0
$807$	147932.i	0.227151i
$808$	0	0
$809$	60860.6	0.0929907	0.0464953	−	0.998919i	$-0.485195\pi$
$809$	60860.6	0.0929907	0.0464953	+	0.998919i	$0.485195\pi$
$810$	0	0
$811$	103330.	0.157103	0.0785515	−	0.996910i	$-0.474970\pi$
$811$	103330.	0.157103	0.0785515	+	0.996910i	$0.474970\pi$
$812$	0	0
$813$	− 195654.i	− 0.296011i
$814$	0	0
$815$	162724.i	0.244983i
$816$	0	0
$817$	91549.0	0.137154
$818$	0	0
$819$	−379489.	−0.565759
$820$	0	0
$821$	533333.i	0.791247i	0.918413	+	0.395623i	$0.129471\pi$
$821$	533333.i	0.791247i	−0.918413	+	0.395623i	$0.870529\pi$
$822$	0	0
$823$	− 728807.i	− 1.07600i	−0.842945	−	0.538000i	$-0.819180\pi$
$823$	− 728807.i	− 1.07600i	0.842945	−	0.538000i	$-0.180820\pi$
$824$	0	0
$825$	72110.5	0.105947
$826$	0	0
$827$	−768434.	−1.12356	−0.561779	−	0.827287i	$-0.689883\pi$
$827$	−768434.	−1.12356	−0.561779	+	0.827287i	$0.689883\pi$
$828$	0	0
$829$	401672.i	0.584471i	0.956346	+	0.292235i	$0.0943991\pi$
$829$	401672.i	0.584471i	−0.956346	+	0.292235i	$0.905601\pi$
$830$	0	0
$831$	134685.i	0.195037i
$832$	0	0
$833$	77104.4	0.111119
$834$	0	0
$835$	−1.11185e6	−1.59469
$836$	0	0
$837$	66603.7i	0.0950708i
$838$	0	0
$839$	429703.i	0.610443i	0.952281	+	0.305221i	$0.0987305\pi$
$839$	429703.i	0.610443i	−0.952281	+	0.305221i	$0.901270\pi$
$840$	0	0
$841$	706877.	0.999429
$842$	0	0
$843$	−258849.	−0.364244
$844$	0	0
$845$	742006.i	1.03919i
$846$	0	0
$847$	− 197155.i	− 0.274815i
$848$	0	0
$849$	−381418.	−0.529159
$850$	0	0
$851$	−431629.	−0.596007
$852$	0	0
$853$	− 732176.i	− 1.00628i	−0.864206	−	0.503138i	$-0.832179\pi$
$853$	− 732176.i	− 1.00628i	0.864206	−	0.503138i	$-0.167821\pi$
$854$	0	0
$855$	− 18232.6i	− 0.0249411i
$856$	0	0
$857$	−430159.	−0.585689	−0.292844	−	0.956160i	$-0.594602\pi$
$857$	−430159.	−0.585689	−0.292844	+	0.956160i	$0.594602\pi$
$858$	0	0
$859$	−489975.	−0.664030	−0.332015	−	0.943274i	$-0.607728\pi$
$859$	−489975.	−0.664030	−0.332015	+	0.943274i	$0.607728\pi$
$860$	0	0
$861$	160162.i	0.216050i
$862$	0	0
$863$	999093.i	1.34148i	0.741692	+	0.670740i	$0.234023\pi$
$863$	999093.i	1.34148i	−0.741692	+	0.670740i	$0.765977\pi$
$864$	0	0
$865$	629955.	0.841933
$866$	0	0
$867$	−389710.	−0.518445
$868$	0	0
$869$	− 505464.i	− 0.669347i
$870$	0	0
$871$	− 93839.8i	− 0.123695i
$872$	0	0
$873$	−59374.5	−0.0779062
$874$	0	0
$875$	−946265.	−1.23594
$876$	0	0
$877$	− 634599.i	− 0.825087i	−0.910938	−	0.412544i	$-0.864640\pi$
$877$	− 634599.i	− 0.825087i	0.910938	−	0.412544i	$-0.135360\pi$
$878$	0	0
$879$	− 679047.i	− 0.878865i
$880$	0	0
$881$	312649.	0.402815	0.201407	−	0.979508i	$-0.435448\pi$
$881$	312649.	0.402815	0.201407	+	0.979508i	$0.435448\pi$
$882$	0	0
$883$	209047.	0.268116	0.134058	−	0.990973i	$-0.457199\pi$
$883$	209047.	0.268116	0.134058	+	0.990973i	$0.457199\pi$
$884$	0	0
$885$	306608.i	0.391469i
$886$	0	0
$887$	178008.i	0.226252i	0.993581	+	0.113126i	$0.0360864\pi$
$887$	178008.i	0.226252i	−0.993581	+	0.113126i	$0.963914\pi$
$888$	0	0
$889$	−1.47680e6	−1.86861
$890$	0	0
$891$	98095.0	0.123564
$892$	0	0
$893$	31042.8i	0.0389276i
$894$	0	0
$895$	965863.i	1.20578i
$896$	0	0
$897$	−733095.	−0.911120
$898$	0	0
$899$	9537.04	0.0118003
$900$	0	0
$901$	− 163239.i	− 0.201082i
$902$	0	0
$903$	− 915490.i	− 1.12274i
$904$	0	0
$905$	−22999.5	−0.0280815
$906$	0	0
$907$	−62871.2	−0.0764253	−0.0382126	−	0.999270i	$-0.512166\pi$
$907$	−62871.2	−0.0764253	−0.0382126	+	0.999270i	$0.512166\pi$
$908$	0	0
$909$	− 23620.5i	− 0.0285866i
$910$	0	0
$911$	− 1.45791e6i	− 1.75669i	−0.478027	−	0.878345i	$-0.658648\pi$
$911$	− 1.45791e6i	− 1.75669i	0.478027	−	0.878345i	$-0.341352\pi$
$912$	0	0
$913$	−1.40395e6	−1.68426
$914$	0	0
$915$	285282.	0.340748
$916$	0	0
$917$	− 935258.i	− 1.11223i
$918$	0	0
$919$	− 315989.i	− 0.374146i	−0.982346	−	0.187073i	$-0.940100\pi$
$919$	− 315989.i	− 0.374146i	0.982346	−	0.187073i	$-0.0599001\pi$
$920$	0	0
$921$	724927.	0.854623
$922$	0	0
$923$	173568.	0.203736
$924$	0	0
$925$	− 77954.8i	− 0.0911086i
$926$	0	0
$927$	81997.3i	0.0954201i
$928$	0	0
$929$	122547.	0.141994	0.0709971	−	0.997477i	$-0.477382\pi$
$929$	122547.	0.141994	0.0709971	+	0.997477i	$0.477382\pi$
$930$	0	0
$931$	−24690.4	−0.0284859
$932$	0	0
$933$	686070.i	0.788143i
$934$	0	0
$935$	283761.i	0.324586i
$936$	0	0
$937$	−495399.	−0.564256	−0.282128	−	0.959377i	$-0.591040\pi$
$937$	−495399.	−0.564256	−0.282128	+	0.959377i	$0.591040\pi$
$938$	0	0
$939$	−205385.	−0.232936
$940$	0	0
$941$	1.14226e6i	1.28999i	0.764188	+	0.644994i	$0.223140\pi$
$941$	1.14226e6i	1.28999i	−0.764188	+	0.644994i	$0.776860\pi$
$942$	0	0
$943$	309401.i	0.347935i
$944$	0	0
$945$	−182326.	−0.204166
$946$	0	0
$947$	527337.	0.588015	0.294007	−	0.955803i	$-0.405011\pi$
$947$	527337.	0.588015	0.294007	+	0.955803i	$0.405011\pi$
$948$	0	0
$949$	2.42741e6i	2.69532i
$950$	0	0
$951$	477833.i	0.528342i
$952$	0	0
$953$	259379.	0.285594	0.142797	−	0.989752i	$-0.454390\pi$
$953$	259379.	0.285594	0.142797	+	0.989752i	$0.454390\pi$
$954$	0	0
$955$	−157778.	−0.172997
$956$	0	0
$957$	− 14046.3i	− 0.0153369i
$958$	0	0
$959$	− 1.79873e6i	− 1.95582i
$960$	0	0
$961$	698146.	0.755962
$962$	0	0
$963$	−514556.	−0.554856
$964$	0	0
$965$	− 644391.i	− 0.691982i
$966$	0	0
$967$	1.03870e6i	1.11081i	0.831581	+	0.555403i	$0.187436\pi$
$967$	1.03870e6i	1.11081i	−0.831581	+	0.555403i	$0.812564\pi$
$968$	0	0
$969$	−14178.8	−0.0151005
$970$	0	0
$971$	1.15698e6	1.22713	0.613563	−	0.789646i	$-0.289736\pi$
$971$	1.15698e6	1.22713	0.613563	+	0.789646i	$0.289736\pi$
$972$	0	0
$973$	733622.i	0.774902i
$974$	0	0
$975$	− 132402.i	− 0.139278i
$976$	0	0
$977$	1.09732e6	1.14960	0.574799	−	0.818295i	$-0.305080\pi$
$977$	1.09732e6	1.14960	0.574799	+	0.818295i	$0.305080\pi$
$978$	0	0
$979$	1.60444e6	1.67401
$980$	0	0
$981$	354582.i	0.368450i
$982$	0	0
$983$	− 1.71299e6i	− 1.77276i	−0.462963	−	0.886378i	$-0.653214\pi$
$983$	− 1.71299e6i	− 1.77276i	0.462963	−	0.886378i	$-0.346786\pi$
$984$	0	0
$985$	878451.	0.905409
$986$	0	0
$987$	310427.	0.318659
$988$	0	0
$989$	− 1.76854e6i	− 1.80810i
$990$	0	0
$991$	− 1.03974e6i	− 1.05872i	−0.848399	−	0.529358i	$-0.822433\pi$
$991$	− 1.03974e6i	− 1.05872i	0.848399	−	0.529358i	$-0.177567\pi$
$992$	0	0
$993$	−16856.8	−0.0170953
$994$	0	0
$995$	125418.	0.126682
$996$	0	0
$997$	− 1.11868e6i	− 1.12542i	−0.826655	−	0.562709i	$-0.809759\pi$
$997$	− 1.11868e6i	− 1.12542i	0.826655	−	0.562709i	$-0.190241\pi$
$998$	0	0
$999$	− 106045.i	− 0.106258i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.b.g.127.6		8
4.3	odd	2	inner	768.5.b.g.127.2		8
8.3	odd	2	inner	768.5.b.g.127.7		8
8.5	even	2	inner	768.5.b.g.127.3		8
16.3	odd	4		12.5.d.a.7.4	yes	4
16.5	even	4		192.5.g.d.127.2		4
16.11	odd	4		192.5.g.d.127.4		4
16.13	even	4		12.5.d.a.7.3	✓	4
48.5	odd	4		576.5.g.m.127.2		4
48.11	even	4		576.5.g.m.127.1		4
48.29	odd	4		36.5.d.b.19.2		4
48.35	even	4		36.5.d.b.19.1		4
80.3	even	4		300.5.f.a.199.5		8
80.13	odd	4		300.5.f.a.199.3		8
80.19	odd	4		300.5.c.a.151.1		4
80.29	even	4		300.5.c.a.151.2		4
80.67	even	4		300.5.f.a.199.4		8
80.77	odd	4		300.5.f.a.199.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.5.d.a.7.3	✓	4	16.13	even	4
12.5.d.a.7.4	yes	4	16.3	odd	4
36.5.d.b.19.1		4	48.35	even	4
36.5.d.b.19.2		4	48.29	odd	4
192.5.g.d.127.2		4	16.5	even	4
192.5.g.d.127.4		4	16.11	odd	4
300.5.c.a.151.1		4	80.19	odd	4
300.5.c.a.151.2		4	80.29	even	4
300.5.f.a.199.3		8	80.13	odd	4
300.5.f.a.199.4		8	80.67	even	4
300.5.f.a.199.5		8	80.3	even	4
300.5.f.a.199.6		8	80.77	odd	4
576.5.g.m.127.1		4	48.11	even	4
576.5.g.m.127.2		4	48.5	odd	4
768.5.b.g.127.2		8	4.3	odd	2	inner
768.5.b.g.127.3		8	8.5	even	2	inner
768.5.b.g.127.6		8	1.1	even	1	trivial
768.5.b.g.127.7		8	8.3	odd	2	inner

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} -22.8444i q^{5} -56.8882i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} -22.8444i q^{5} -56.8882i q^{7} +27.0000 q^{9} +134.561 q^{11} -247.066i q^{13} -118.703i q^{15} -92.3112 q^{17} +29.5600 q^{19} -295.600i q^{21} -571.038i q^{23} +103.133 q^{25} +140.296 q^{27} -20.0891i q^{29} +474.736i q^{31} +699.199 q^{33} -1299.58 q^{35} -755.867i q^{37} -1283.79i q^{39} -541.822 q^{41} +3097.06 q^{43} -616.799i q^{45} +1050.16i q^{47} -835.266 q^{49} -479.663 q^{51} +1768.35i q^{53} -3073.97i q^{55} +153.598 q^{57} -2582.98 q^{59} +2403.33i q^{61} -1535.98i q^{63} -5644.09 q^{65} +379.816 q^{67} -2967.20i q^{69} +702.517i q^{71} -9824.92 q^{73} +535.894 q^{75} -7654.93i q^{77} -3756.40i q^{79} +729.000 q^{81} -10433.6 q^{83} +2108.79i q^{85} -104.386i q^{87} +11923.5 q^{89} -14055.2 q^{91} +2466.80i q^{93} -675.280i q^{95} -2199.06 q^{97} +3633.15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 216 q^{9}+O(q^{10}) \) 8 * q + 216 * q^9	\( 8 q + 216 q^{9} - 1200 q^{17} - 1944 q^{25} + 1440 q^{33} - 1104 q^{41} - 1144 q^{49} - 11232 q^{57} - 36384 q^{65} - 17680 q^{73} + 5832 q^{81} + 50160 q^{89} + 46096 q^{97}+O(q^{100}) \) 8 * q + 216 * q^9 - 1200 * q^17 - 1944 * q^25 + 1440 * q^33 - 1104 * q^41 - 1144 * q^49 - 11232 * q^57 - 36384 * q^65 - 17680 * q^73 + 5832 * q^81 + 50160 * q^89 + 46096 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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