LMFDB - Embedded newform 768.6.d.n.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,6,Mod(385,768)]

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, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("768.385");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.d (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:	$123.174773616$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.6.d.n.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} -20.0000i q^{5} +122.000 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} -20.0000i q^{5} +122.000 q^{7} -81.0000 q^{9} +724.000i q^{11} +914.000i q^{13} +180.000 q^{15} +1006.00 q^{17} +2920.00i q^{19} +1098.00i q^{21} +3124.00 q^{23} +2725.00 q^{25} -729.000i q^{27} -6744.00i q^{29} -5010.00 q^{31} -6516.00 q^{33} -2440.00i q^{35} -5278.00i q^{37} -8226.00 q^{39} -5238.00 q^{41} +16752.0i q^{43} +1620.00i q^{45} -1108.00 q^{47} -1923.00 q^{49} +9054.00i q^{51} +22008.0i q^{53} +14480.0 q^{55} -26280.0 q^{57} -23716.0i q^{59} -45202.0i q^{61} -9882.00 q^{63} +18280.0 q^{65} -22756.0i q^{67} +28116.0i q^{69} +53436.0 q^{71} -4790.00 q^{73} +24525.0i q^{75} +88328.0i q^{77} +1886.00 q^{79} +6561.00 q^{81} -11268.0i q^{83} -20120.0i q^{85} +60696.0 q^{87} -73522.0 q^{89} +111508. i q^{91} -45090.0i q^{93} +58400.0 q^{95} +114154. q^{97} -58644.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 244 q^{7} - 162 q^{9}+O(q^{10}) $ 2 * q + 244 * q^7 - 162 * q^9	$ 2 q + 244 q^{7} - 162 q^{9} + 360 q^{15} + 2012 q^{17} + 6248 q^{23} + 5450 q^{25} - 10020 q^{31} - 13032 q^{33} - 16452 q^{39} - 10476 q^{41} - 2216 q^{47} - 3846 q^{49} + 28960 q^{55} - 52560 q^{57} - 19764 q^{63} + 36560 q^{65} + 106872 q^{71} - 9580 q^{73} + 3772 q^{79} + 13122 q^{81} + 121392 q^{87} - 147044 q^{89} + 116800 q^{95} + 228308 q^{97}+O(q^{100}) $ 2 * q + 244 * q^7 - 162 * q^9 + 360 * q^15 + 2012 * q^17 + 6248 * q^23 + 5450 * q^25 - 10020 * q^31 - 13032 * q^33 - 16452 * q^39 - 10476 * q^41 - 2216 * q^47 - 3846 * q^49 + 28960 * q^55 - 52560 * q^57 - 19764 * q^63 + 36560 * q^65 + 106872 * q^71 - 9580 * q^73 + 3772 * q^79 + 13122 * q^81 + 121392 * q^87 - 147044 * q^89 + 116800 * q^95 + 228308 * 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
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	0	0
$5$	− 20.0000i	− 0.357771i	−0.983870	−	0.178885i	$-0.942751\pi$
$5$	− 20.0000i	− 0.357771i	0.983870	−	0.178885i	$-0.0572491\pi$
$6$	0	0
$7$	122.000	0.941054	0.470527	−	0.882385i	$-0.344064\pi$
$7$	122.000	0.941054	0.470527	+	0.882385i	$0.344064\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	724.000i	1.80408i	0.431648	+	0.902042i	$0.357932\pi$
$11$	724.000i	1.80408i	−0.431648	+	0.902042i	$0.642068\pi$
$12$	0	0
$13$	914.000i	1.49999i	0.661445	+	0.749994i	$0.269944\pi$
$13$	914.000i	1.49999i	−0.661445	+	0.749994i	$0.730056\pi$
$14$	0	0
$15$	180.000	0.206559
$16$	0	0
$17$	1006.00	0.844259	0.422129	−	0.906536i	$-0.361283\pi$
$17$	1006.00	0.844259	0.422129	+	0.906536i	$0.361283\pi$
$18$	0	0
$19$	2920.00i	1.85566i	0.373001	+	0.927831i	$0.378329\pi$
$19$	2920.00i	1.85566i	−0.373001	+	0.927831i	$0.621671\pi$
$20$	0	0
$21$	1098.00i	0.543318i
$22$	0	0
$23$	3124.00	1.23138	0.615689	−	0.787989i	$-0.288878\pi$
$23$	3124.00	1.23138	0.615689	+	0.787989i	$0.288878\pi$
$24$	0	0
$25$	2725.00	0.872000
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	− 6744.00i	− 1.48910i	−0.667569	−	0.744548i	$-0.732665\pi$
$29$	− 6744.00i	− 1.48910i	0.667569	−	0.744548i	$-0.267335\pi$
$30$	0	0
$31$	−5010.00	−0.936340	−0.468170	−	0.883638i	$-0.655086\pi$
$31$	−5010.00	−0.936340	−0.468170	+	0.883638i	$0.655086\pi$
$32$	0	0
$33$	−6516.00	−1.04159
$34$	0	0
$35$	− 2440.00i	− 0.336682i
$36$	0	0
$37$	− 5278.00i	− 0.633819i	−0.948456	−	0.316909i	$-0.897355\pi$
$37$	− 5278.00i	− 0.633819i	0.948456	−	0.316909i	$-0.102645\pi$
$38$	0	0
$39$	−8226.00	−0.866019
$40$	0	0
$41$	−5238.00	−0.486638	−0.243319	−	0.969946i	$-0.578236\pi$
$41$	−5238.00	−0.486638	−0.243319	+	0.969946i	$0.578236\pi$
$42$	0	0
$43$	16752.0i	1.38164i	0.723026	+	0.690821i	$0.242751\pi$
$43$	16752.0i	1.38164i	−0.723026	+	0.690821i	$0.757249\pi$
$44$	0	0
$45$	1620.00i	0.119257i
$46$	0	0
$47$	−1108.00	−0.0731636	−0.0365818	−	0.999331i	$-0.511647\pi$
$47$	−1108.00	−0.0731636	−0.0365818	+	0.999331i	$0.511647\pi$
$48$	0	0
$49$	−1923.00	−0.114417
$50$	0	0
$51$	9054.00i	0.487433i
$52$	0	0
$53$	22008.0i	1.07619i	0.842883	+	0.538097i	$0.180857\pi$
$53$	22008.0i	1.07619i	−0.842883	+	0.538097i	$0.819143\pi$
$54$	0	0
$55$	14480.0	0.645449
$56$	0	0
$57$	−26280.0	−1.07137
$58$	0	0
$59$	− 23716.0i	− 0.886975i	−0.896281	−	0.443488i	$-0.853741\pi$
$59$	− 23716.0i	− 0.886975i	0.896281	−	0.443488i	$-0.146259\pi$
$60$	0	0
$61$	− 45202.0i	− 1.55537i	−0.628656	−	0.777684i	$-0.716394\pi$
$61$	− 45202.0i	− 1.55537i	0.628656	−	0.777684i	$-0.283606\pi$
$62$	0	0
$63$	−9882.00	−0.313685
$64$	0	0
$65$	18280.0	0.536652
$66$	0	0
$67$	− 22756.0i	− 0.619311i	−0.950849	−	0.309656i	$-0.899786\pi$
$67$	− 22756.0i	− 0.619311i	0.950849	−	0.309656i	$-0.100214\pi$
$68$	0	0
$69$	28116.0i	0.710936i
$70$	0	0
$71$	53436.0	1.25802	0.629011	−	0.777397i	$-0.283460\pi$
$71$	53436.0	1.25802	0.629011	+	0.777397i	$0.283460\pi$
$72$	0	0
$73$	−4790.00	−0.105203	−0.0526015	−	0.998616i	$-0.516751\pi$
$73$	−4790.00	−0.105203	−0.0526015	+	0.998616i	$0.516751\pi$
$74$	0	0
$75$	24525.0i	0.503449i
$76$	0	0
$77$	88328.0i	1.69774i
$78$	0	0
$79$	1886.00	0.0339996	0.0169998	−	0.999855i	$-0.494589\pi$
$79$	1886.00	0.0339996	0.0169998	+	0.999855i	$0.494589\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	− 11268.0i	− 0.179536i	−0.995963	−	0.0897680i	$-0.971387\pi$
$83$	− 11268.0i	− 0.179536i	0.995963	−	0.0897680i	$-0.0286126\pi$
$84$	0	0
$85$	− 20120.0i	− 0.302051i
$86$	0	0
$87$	60696.0	0.859730
$88$	0	0
$89$	−73522.0	−0.983880	−0.491940	−	0.870629i	$-0.663712\pi$
$89$	−73522.0	−0.983880	−0.491940	+	0.870629i	$0.663712\pi$
$90$	0	0
$91$	111508.i	1.41157i
$92$	0	0
$93$	− 45090.0i	− 0.540596i
$94$	0	0
$95$	58400.0	0.663902
$96$	0	0
$97$	114154.	1.23186	0.615931	−	0.787800i	$-0.288780\pi$
$97$	114154.	1.23186	0.615931	+	0.787800i	$0.288780\pi$
$98$	0	0
$99$	− 58644.0i	− 0.601361i
$100$	0	0
$101$	168024.i	1.63896i	0.573109	+	0.819479i	$0.305737\pi$
$101$	168024.i	1.63896i	−0.573109	+	0.819479i	$0.694263\pi$
$102$	0	0
$103$	38750.0	0.359897	0.179949	−	0.983676i	$-0.442407\pi$
$103$	38750.0	0.359897	0.179949	+	0.983676i	$0.442407\pi$
$104$	0	0
$105$	21960.0	0.194383
$106$	0	0
$107$	207172.i	1.74933i	0.484728	+	0.874665i	$0.338918\pi$
$107$	207172.i	1.74933i	−0.484728	+	0.874665i	$0.661082\pi$
$108$	0	0
$109$	− 39318.0i	− 0.316975i	−0.987361	−	0.158488i	$-0.949338\pi$
$109$	− 39318.0i	− 0.316975i	0.987361	−	0.158488i	$-0.0506618\pi$
$110$	0	0
$111$	47502.0	0.365935
$112$	0	0
$113$	5666.00	0.0417427	0.0208713	−	0.999782i	$-0.493356\pi$
$113$	5666.00	0.0417427	0.0208713	+	0.999782i	$0.493356\pi$
$114$	0	0
$115$	− 62480.0i	− 0.440551i
$116$	0	0
$117$	− 74034.0i	− 0.499996i
$118$	0	0
$119$	122732.	0.794494
$120$	0	0
$121$	−363125.	−2.25472
$122$	0	0
$123$	− 47142.0i	− 0.280960i
$124$	0	0
$125$	− 117000.i	− 0.669747i
$126$	0	0
$127$	−157894.	−0.868673	−0.434337	−	0.900751i	$-0.643017\pi$
$127$	−157894.	−0.868673	−0.434337	+	0.900751i	$0.643017\pi$
$128$	0	0
$129$	−150768.	−0.797691
$130$	0	0
$131$	− 94188.0i	− 0.479532i	−0.970831	−	0.239766i	$-0.922929\pi$
$131$	− 94188.0i	− 0.479532i	0.970831	−	0.239766i	$-0.0770707\pi$
$132$	0	0
$133$	356240.i	1.74628i
$134$	0	0
$135$	−14580.0	−0.0688530
$136$	0	0
$137$	29234.0	0.133072	0.0665360	−	0.997784i	$-0.478805\pi$
$137$	29234.0	0.133072	0.0665360	+	0.997784i	$0.478805\pi$
$138$	0	0
$139$	− 123556.i	− 0.542409i	−0.962522	−	0.271204i	$-0.912578\pi$
$139$	− 123556.i	− 0.542409i	0.962522	−	0.271204i	$-0.0874220\pi$
$140$	0	0
$141$	− 9972.00i	− 0.0422410i
$142$	0	0
$143$	−661736.	−2.70611
$144$	0	0
$145$	−134880.	−0.532755
$146$	0	0
$147$	− 17307.0i	− 0.0660585i
$148$	0	0
$149$	247836.i	0.914532i	0.889330	+	0.457266i	$0.151171\pi$
$149$	247836.i	0.914532i	−0.889330	+	0.457266i	$0.848829\pi$
$150$	0	0
$151$	−256670.	−0.916079	−0.458039	−	0.888932i	$-0.651448\pi$
$151$	−256670.	−0.916079	−0.458039	+	0.888932i	$0.651448\pi$
$152$	0	0
$153$	−81486.0	−0.281420
$154$	0	0
$155$	100200.i	0.334995i
$156$	0	0
$157$	591534.i	1.91527i	0.287981	+	0.957636i	$0.407016\pi$
$157$	591534.i	1.91527i	−0.287981	+	0.957636i	$0.592984\pi$
$158$	0	0
$159$	−198072.	−0.621341
$160$	0	0
$161$	381128.	1.15879
$162$	0	0
$163$	− 120056.i	− 0.353928i	−0.984217	−	0.176964i	$-0.943372\pi$
$163$	− 120056.i	− 0.353928i	0.984217	−	0.176964i	$-0.0566276\pi$
$164$	0	0
$165$	130320.i	0.372650i
$166$	0	0
$167$	−82544.0	−0.229031	−0.114516	−	0.993421i	$-0.536532\pi$
$167$	−82544.0	−0.229031	−0.114516	+	0.993421i	$0.536532\pi$
$168$	0	0
$169$	−464103.	−1.24996
$170$	0	0
$171$	− 236520.i	− 0.618554i
$172$	0	0
$173$	− 229404.i	− 0.582755i	−0.956608	−	0.291377i	$-0.905886\pi$
$173$	− 229404.i	− 0.582755i	0.956608	−	0.291377i	$-0.0941135\pi$
$174$	0	0
$175$	332450.	0.820599
$176$	0	0
$177$	213444.	0.512095
$178$	0	0
$179$	175788.i	0.410069i	0.978755	+	0.205034i	$0.0657306\pi$
$179$	175788.i	0.410069i	−0.978755	+	0.205034i	$0.934269\pi$
$180$	0	0
$181$	372934.i	0.846127i	0.906100	+	0.423063i	$0.139045\pi$
$181$	372934.i	0.846127i	−0.906100	+	0.423063i	$0.860955\pi$
$182$	0	0
$183$	406818.	0.897992
$184$	0	0
$185$	−105560.	−0.226762
$186$	0	0
$187$	728344.i	1.52311i
$188$	0	0
$189$	− 88938.0i	− 0.181106i
$190$	0	0
$191$	729016.	1.44595	0.722976	−	0.690874i	$-0.242774\pi$
$191$	729016.	1.44595	0.722976	+	0.690874i	$0.242774\pi$
$192$	0	0
$193$	−852962.	−1.64830	−0.824150	−	0.566371i	$-0.808347\pi$
$193$	−852962.	−1.64830	−0.824150	+	0.566371i	$0.808347\pi$
$194$	0	0
$195$	164520.i	0.309836i
$196$	0	0
$197$	214112.i	0.393075i	0.980496	+	0.196538i	$0.0629697\pi$
$197$	214112.i	0.393075i	−0.980496	+	0.196538i	$0.937030\pi$
$198$	0	0
$199$	498902.	0.893064	0.446532	−	0.894768i	$-0.352659\pi$
$199$	498902.	0.893064	0.446532	+	0.894768i	$0.352659\pi$
$200$	0	0
$201$	204804.	0.357559
$202$	0	0
$203$	− 822768.i	− 1.40132i
$204$	0	0
$205$	104760.i	0.174105i
$206$	0	0
$207$	−253044.	−0.410459
$208$	0	0
$209$	−2.11408e6	−3.34777
$210$	0	0
$211$	− 474932.i	− 0.734388i	−0.930144	−	0.367194i	$-0.880319\pi$
$211$	− 474932.i	− 0.734388i	0.930144	−	0.367194i	$-0.119681\pi$
$212$	0	0
$213$	480924.i	0.726319i
$214$	0	0
$215$	335040.	0.494311
$216$	0	0
$217$	−611220.	−0.881147
$218$	0	0
$219$	− 43110.0i	− 0.0607390i
$220$	0	0
$221$	919484.i	1.26638i
$222$	0	0
$223$	−728118.	−0.980482	−0.490241	−	0.871587i	$-0.663091\pi$
$223$	−728118.	−0.980482	−0.490241	+	0.871587i	$0.663091\pi$
$224$	0	0
$225$	−220725.	−0.290667
$226$	0	0
$227$	− 355044.i	− 0.457317i	−0.973507	−	0.228659i	$-0.926566\pi$
$227$	− 355044.i	− 0.457317i	0.973507	−	0.228659i	$-0.0734339\pi$
$228$	0	0
$229$	− 987490.i	− 1.24435i	−0.782877	−	0.622177i	$-0.786248\pi$
$229$	− 987490.i	− 1.24435i	0.782877	−	0.622177i	$-0.213752\pi$
$230$	0	0
$231$	−794952.	−0.980192
$232$	0	0
$233$	−993210.	−1.19854	−0.599268	−	0.800548i	$-0.704542\pi$
$233$	−993210.	−1.19854	−0.599268	+	0.800548i	$0.704542\pi$
$234$	0	0
$235$	22160.0i	0.0261758i
$236$	0	0
$237$	16974.0i	0.0196297i
$238$	0	0
$239$	765984.	0.867411	0.433706	−	0.901055i	$-0.357206\pi$
$239$	765984.	0.867411	0.433706	+	0.901055i	$0.357206\pi$
$240$	0	0
$241$	1.66425e6	1.84577	0.922884	−	0.385079i	$-0.125826\pi$
$241$	1.66425e6	1.84577	0.922884	+	0.385079i	$0.125826\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	38460.0i	0.0409349i
$246$	0	0
$247$	−2.66888e6	−2.78347
$248$	0	0
$249$	101412.	0.103655
$250$	0	0
$251$	− 1.02534e6i	− 1.02727i	−0.858010	−	0.513634i	$-0.828299\pi$
$251$	− 1.02534e6i	− 1.02727i	0.858010	−	0.513634i	$-0.171701\pi$
$252$	0	0
$253$	2.26178e6i	2.22151i
$254$	0	0
$255$	181080.	0.174389
$256$	0	0
$257$	−862254.	−0.814334	−0.407167	−	0.913354i	$-0.633483\pi$
$257$	−862254.	−0.814334	−0.407167	+	0.913354i	$0.633483\pi$
$258$	0	0
$259$	− 643916.i	− 0.596458i
$260$	0	0
$261$	546264.i	0.496365i
$262$	0	0
$263$	233832.	0.208456	0.104228	−	0.994553i	$-0.466763\pi$
$263$	233832.	0.208456	0.104228	+	0.994553i	$0.466763\pi$
$264$	0	0
$265$	440160.	0.385031
$266$	0	0
$267$	− 661698.i	− 0.568044i
$268$	0	0
$269$	461848.i	0.389151i	0.980888	+	0.194576i	$0.0623330\pi$
$269$	461848.i	0.389151i	−0.980888	+	0.194576i	$0.937667\pi$
$270$	0	0
$271$	1.45220e6	1.20117	0.600584	−	0.799562i	$-0.294935\pi$
$271$	1.45220e6	1.20117	0.600584	+	0.799562i	$0.294935\pi$
$272$	0	0
$273$	−1.00357e6	−0.814971
$274$	0	0
$275$	1.97290e6i	1.57316i
$276$	0	0
$277$	− 1.53748e6i	− 1.20396i	−0.798513	−	0.601978i	$-0.794379\pi$
$277$	− 1.53748e6i	− 1.20396i	0.798513	−	0.601978i	$-0.205621\pi$
$278$	0	0
$279$	405810.	0.312113
$280$	0	0
$281$	−193266.	−0.146012	−0.0730062	−	0.997331i	$-0.523259\pi$
$281$	−193266.	−0.146012	−0.0730062	+	0.997331i	$0.523259\pi$
$282$	0	0
$283$	− 1.58718e6i	− 1.17804i	−0.808118	−	0.589020i	$-0.799514\pi$
$283$	− 1.58718e6i	− 1.17804i	0.808118	−	0.589020i	$-0.200486\pi$
$284$	0	0
$285$	525600.i	0.383304i
$286$	0	0
$287$	−639036.	−0.457953
$288$	0	0
$289$	−407821.	−0.287227
$290$	0	0
$291$	1.02739e6i	0.711215i
$292$	0	0
$293$	2.55274e6i	1.73715i	0.495555	+	0.868577i	$0.334965\pi$
$293$	2.55274e6i	1.73715i	−0.495555	+	0.868577i	$0.665035\pi$
$294$	0	0
$295$	−474320.	−0.317334
$296$	0	0
$297$	527796.	0.347196
$298$	0	0
$299$	2.85534e6i	1.84705i
$300$	0	0
$301$	2.04374e6i	1.30020i
$302$	0	0
$303$	−1.51222e6	−0.946253
$304$	0	0
$305$	−904040.	−0.556465
$306$	0	0
$307$	461868.i	0.279687i	0.990174	+	0.139843i	$0.0446599\pi$
$307$	461868.i	0.279687i	−0.990174	+	0.139843i	$0.955340\pi$
$308$	0	0
$309$	348750.i	0.207787i
$310$	0	0
$311$	1.13480e6	0.665301	0.332651	−	0.943050i	$-0.392057\pi$
$311$	1.13480e6	0.665301	0.332651	+	0.943050i	$0.392057\pi$
$312$	0	0
$313$	720986.	0.415974	0.207987	−	0.978132i	$-0.433309\pi$
$313$	720986.	0.415974	0.207987	+	0.978132i	$0.433309\pi$
$314$	0	0
$315$	197640.i	0.112227i
$316$	0	0
$317$	651792.i	0.364301i	0.983271	+	0.182151i	$0.0583059\pi$
$317$	651792.i	0.364301i	−0.983271	+	0.182151i	$0.941694\pi$
$318$	0	0
$319$	4.88266e6	2.68645
$320$	0	0
$321$	−1.86455e6	−1.00998
$322$	0	0
$323$	2.93752e6i	1.56666i
$324$	0	0
$325$	2.49065e6i	1.30799i
$326$	0	0
$327$	353862.	0.183006
$328$	0	0
$329$	−135176.	−0.0688509
$330$	0	0
$331$	2.10895e6i	1.05802i	0.848614	+	0.529012i	$0.177438\pi$
$331$	2.10895e6i	1.05802i	−0.848614	+	0.529012i	$0.822562\pi$
$332$	0	0
$333$	427518.i	0.211273i
$334$	0	0
$335$	−455120.	−0.221572
$336$	0	0
$337$	−1.45008e6	−0.695533	−0.347767	−	0.937581i	$-0.613060\pi$
$337$	−1.45008e6	−0.695533	−0.347767	+	0.937581i	$0.613060\pi$
$338$	0	0
$339$	50994.0i	0.0241002i
$340$	0	0
$341$	− 3.62724e6i	− 1.68924i
$342$	0	0
$343$	−2.28506e6	−1.04873
$344$	0	0
$345$	562320.	0.254352
$346$	0	0
$347$	− 1.70980e6i	− 0.762291i	−0.924515	−	0.381145i	$-0.875530\pi$
$347$	− 1.70980e6i	− 0.762291i	0.924515	−	0.381145i	$-0.124470\pi$
$348$	0	0
$349$	− 1.75191e6i	− 0.769923i	−0.922933	−	0.384961i	$-0.874215\pi$
$349$	− 1.75191e6i	− 0.769923i	0.922933	−	0.384961i	$-0.125785\pi$
$350$	0	0
$351$	666306.	0.288673
$352$	0	0
$353$	−2.13401e6	−0.911508	−0.455754	−	0.890106i	$-0.650630\pi$
$353$	−2.13401e6	−0.911508	−0.455754	+	0.890106i	$0.650630\pi$
$354$	0	0
$355$	− 1.06872e6i	− 0.450083i
$356$	0	0
$357$	1.10459e6i	0.458701i
$358$	0	0
$359$	−387220.	−0.158570	−0.0792851	−	0.996852i	$-0.525264\pi$
$359$	−387220.	−0.158570	−0.0792851	+	0.996852i	$0.525264\pi$
$360$	0	0
$361$	−6.05030e6	−2.44348
$362$	0	0
$363$	− 3.26812e6i	− 1.30176i
$364$	0	0
$365$	95800.0i	0.0376386i
$366$	0	0
$367$	−1.30194e6	−0.504574	−0.252287	−	0.967652i	$-0.581183\pi$
$367$	−1.30194e6	−0.504574	−0.252287	+	0.967652i	$0.581183\pi$
$368$	0	0
$369$	424278.	0.162213
$370$	0	0
$371$	2.68498e6i	1.01276i
$372$	0	0
$373$	− 2.77593e6i	− 1.03309i	−0.856261	−	0.516544i	$-0.827218\pi$
$373$	− 2.77593e6i	− 1.03309i	0.856261	−	0.516544i	$-0.172782\pi$
$374$	0	0
$375$	1.05300e6	0.386679
$376$	0	0
$377$	6.16402e6	2.23363
$378$	0	0
$379$	2.08101e6i	0.744176i	0.928197	+	0.372088i	$0.121358\pi$
$379$	2.08101e6i	0.744176i	−0.928197	+	0.372088i	$0.878642\pi$
$380$	0	0
$381$	− 1.42105e6i	− 0.501529i
$382$	0	0
$383$	936408.	0.326188	0.163094	−	0.986611i	$-0.447853\pi$
$383$	936408.	0.326188	0.163094	+	0.986611i	$0.447853\pi$
$384$	0	0
$385$	1.76656e6	0.607402
$386$	0	0
$387$	− 1.35691e6i	− 0.460547i
$388$	0	0
$389$	923420.i	0.309404i	0.987961	+	0.154702i	$0.0494417\pi$
$389$	923420.i	0.309404i	−0.987961	+	0.154702i	$0.950558\pi$
$390$	0	0
$391$	3.14274e6	1.03960
$392$	0	0
$393$	847692.	0.276858
$394$	0	0
$395$	− 37720.0i	− 0.0121641i
$396$	0	0
$397$	4.76055e6i	1.51594i	0.652292	+	0.757968i	$0.273808\pi$
$397$	4.76055e6i	1.51594i	−0.652292	+	0.757968i	$0.726192\pi$
$398$	0	0
$399$	−3.20616e6	−1.00821
$400$	0	0
$401$	1.99791e6	0.620462	0.310231	−	0.950661i	$-0.399594\pi$
$401$	1.99791e6	0.620462	0.310231	+	0.950661i	$0.399594\pi$
$402$	0	0
$403$	− 4.57914e6i	− 1.40450i
$404$	0	0
$405$	− 131220.i	− 0.0397523i
$406$	0	0
$407$	3.82127e6	1.14346
$408$	0	0
$409$	−2.35352e6	−0.695681	−0.347840	−	0.937554i	$-0.613085\pi$
$409$	−2.35352e6	−0.695681	−0.347840	+	0.937554i	$0.613085\pi$
$410$	0	0
$411$	263106.i	0.0768292i
$412$	0	0
$413$	− 2.89335e6i	− 0.834692i
$414$	0	0
$415$	−225360.	−0.0642328
$416$	0	0
$417$	1.11200e6	0.313160
$418$	0	0
$419$	1.15232e6i	0.320654i	0.987064	+	0.160327i	$0.0512549\pi$
$419$	1.15232e6i	0.320654i	−0.987064	+	0.160327i	$0.948745\pi$
$420$	0	0
$421$	− 2.34116e6i	− 0.643763i	−0.946780	−	0.321882i	$-0.895685\pi$
$421$	− 2.34116e6i	− 0.643763i	0.946780	−	0.321882i	$-0.104315\pi$
$422$	0	0
$423$	89748.0	0.0243879
$424$	0	0
$425$	2.74135e6	0.736194
$426$	0	0
$427$	− 5.51464e6i	− 1.46369i
$428$	0	0
$429$	− 5.95562e6i	− 1.56237i
$430$	0	0
$431$	1.65631e6	0.429485	0.214742	−	0.976671i	$-0.431109\pi$
$431$	1.65631e6	0.429485	0.214742	+	0.976671i	$0.431109\pi$
$432$	0	0
$433$	1.80847e6	0.463544	0.231772	−	0.972770i	$-0.425548\pi$
$433$	1.80847e6	0.463544	0.231772	+	0.972770i	$0.425548\pi$
$434$	0	0
$435$	− 1.21392e6i	− 0.307586i
$436$	0	0
$437$	9.12208e6i	2.28502i
$438$	0	0
$439$	−3.34811e6	−0.829161	−0.414581	−	0.910013i	$-0.636072\pi$
$439$	−3.34811e6	−0.829161	−0.414581	+	0.910013i	$0.636072\pi$
$440$	0	0
$441$	155763.	0.0381389
$442$	0	0
$443$	1.59409e6i	0.385926i	0.981206	+	0.192963i	$0.0618097\pi$
$443$	1.59409e6i	0.385926i	−0.981206	+	0.192963i	$0.938190\pi$
$444$	0	0
$445$	1.47044e6i	0.352004i
$446$	0	0
$447$	−2.23052e6	−0.528005
$448$	0	0
$449$	2.55174e6	0.597339	0.298670	−	0.954357i	$-0.403457\pi$
$449$	2.55174e6	0.597339	0.298670	+	0.954357i	$0.403457\pi$
$450$	0	0
$451$	− 3.79231e6i	− 0.877936i
$452$	0	0
$453$	− 2.31003e6i	− 0.528898i
$454$	0	0
$455$	2.23016e6	0.505019
$456$	0	0
$457$	−996882.	−0.223282	−0.111641	−	0.993749i	$-0.535611\pi$
$457$	−996882.	−0.223282	−0.111641	+	0.993749i	$0.535611\pi$
$458$	0	0
$459$	− 733374.i	− 0.162478i
$460$	0	0
$461$	− 1.89230e6i	− 0.414703i	−0.978266	−	0.207352i	$-0.933516\pi$
$461$	− 1.89230e6i	− 0.414703i	0.978266	−	0.207352i	$-0.0664845\pi$
$462$	0	0
$463$	−3.30062e6	−0.715555	−0.357778	−	0.933807i	$-0.616465\pi$
$463$	−3.30062e6	−0.715555	−0.357778	+	0.933807i	$0.616465\pi$
$464$	0	0
$465$	−901800.	−0.193410
$466$	0	0
$467$	− 3.50482e6i	− 0.743658i	−0.928301	−	0.371829i	$-0.878731\pi$
$467$	− 3.50482e6i	− 0.743658i	0.928301	−	0.371829i	$-0.121269\pi$
$468$	0	0
$469$	− 2.77623e6i	− 0.582806i
$470$	0	0
$471$	−5.32381e6	−1.10578
$472$	0	0
$473$	−1.21284e7	−2.49260
$474$	0	0
$475$	7.95700e6i	1.61814i
$476$	0	0
$477$	− 1.78265e6i	− 0.358732i
$478$	0	0
$479$	3.32248e6	0.661642	0.330821	−	0.943694i	$-0.392674\pi$
$479$	3.32248e6	0.661642	0.330821	+	0.943694i	$0.392674\pi$
$480$	0	0
$481$	4.82409e6	0.950721
$482$	0	0
$483$	3.43015e6i	0.669030i
$484$	0	0
$485$	− 2.28308e6i	− 0.440724i
$486$	0	0
$487$	−6.61506e6	−1.26390	−0.631948	−	0.775011i	$-0.717744\pi$
$487$	−6.61506e6	−1.26390	−0.631948	+	0.775011i	$0.717744\pi$
$488$	0	0
$489$	1.08050e6	0.204340
$490$	0	0
$491$	1.03231e7i	1.93244i	0.257727	+	0.966218i	$0.417027\pi$
$491$	1.03231e7i	1.93244i	−0.257727	+	0.966218i	$0.582973\pi$
$492$	0	0
$493$	− 6.78446e6i	− 1.25718i
$494$	0	0
$495$	−1.17288e6	−0.215150
$496$	0	0
$497$	6.51919e6	1.18387
$498$	0	0
$499$	− 8.63184e6i	− 1.55186i	−0.630821	−	0.775929i	$-0.717282\pi$
$499$	− 8.63184e6i	− 1.55186i	0.630821	−	0.775929i	$-0.282718\pi$
$500$	0	0
$501$	− 742896.i	− 0.132231i
$502$	0	0
$503$	4.66876e6	0.822777	0.411388	−	0.911460i	$-0.365044\pi$
$503$	4.66876e6	0.822777	0.411388	+	0.911460i	$0.365044\pi$
$504$	0	0
$505$	3.36048e6	0.586372
$506$	0	0
$507$	− 4.17693e6i	− 0.721667i
$508$	0	0
$509$	− 4.27885e6i	− 0.732036i	−0.930608	−	0.366018i	$-0.880721\pi$
$509$	− 4.27885e6i	− 0.732036i	0.930608	−	0.366018i	$-0.119279\pi$
$510$	0	0
$511$	−584380.	−0.0990018
$512$	0	0
$513$	2.12868e6	0.357122
$514$	0	0
$515$	− 775000.i	− 0.128761i
$516$	0	0
$517$	− 802192.i	− 0.131993i
$518$	0	0
$519$	2.06464e6	0.336454
$520$	0	0
$521$	9.86240e6	1.59180	0.795900	−	0.605428i	$-0.206998\pi$
$521$	9.86240e6	1.59180	0.795900	+	0.605428i	$0.206998\pi$
$522$	0	0
$523$	3.30272e6i	0.527980i	0.964525	+	0.263990i	$0.0850386\pi$
$523$	3.30272e6i	0.527980i	−0.964525	+	0.263990i	$0.914961\pi$
$524$	0	0
$525$	2.99205e6i	0.473773i
$526$	0	0
$527$	−5.04006e6	−0.790513
$528$	0	0
$529$	3.32303e6	0.516292
$530$	0	0
$531$	1.92100e6i	0.295658i
$532$	0	0
$533$	− 4.78753e6i	− 0.729951i
$534$	0	0
$535$	4.14344e6	0.625859
$536$	0	0
$537$	−1.58209e6	−0.236753
$538$	0	0
$539$	− 1.39225e6i	− 0.206417i
$540$	0	0
$541$	− 1.91152e6i	− 0.280792i	−0.990095	−	0.140396i	$-0.955162\pi$
$541$	− 1.91152e6i	− 0.280792i	0.990095	−	0.140396i	$-0.0448376\pi$
$542$	0	0
$543$	−3.35641e6	−0.488512
$544$	0	0
$545$	−786360.	−0.113404
$546$	0	0
$547$	4.91005e6i	0.701645i	0.936442	+	0.350822i	$0.114098\pi$
$547$	4.91005e6i	0.701645i	−0.936442	+	0.350822i	$0.885902\pi$
$548$	0	0
$549$	3.66136e6i	0.518456i
$550$	0	0
$551$	1.96925e7	2.76326
$552$	0	0
$553$	230092.	0.0319955
$554$	0	0
$555$	− 950040.i	− 0.130921i
$556$	0	0
$557$	− 1.24364e7i	− 1.69847i	−0.528017	−	0.849234i	$-0.677064\pi$
$557$	− 1.24364e7i	− 1.69847i	0.528017	−	0.849234i	$-0.322936\pi$
$558$	0	0
$559$	−1.53113e7	−2.07245
$560$	0	0
$561$	−6.55510e6	−0.879371
$562$	0	0
$563$	8.30351e6i	1.10405i	0.833826	+	0.552027i	$0.186146\pi$
$563$	8.30351e6i	1.10405i	−0.833826	+	0.552027i	$0.813854\pi$
$564$	0	0
$565$	− 113320.i	− 0.0149343i
$566$	0	0
$567$	800442.	0.104562
$568$	0	0
$569$	2.46481e6	0.319156	0.159578	−	0.987185i	$-0.448987\pi$
$569$	2.46481e6	0.319156	0.159578	+	0.987185i	$0.448987\pi$
$570$	0	0
$571$	− 457540.i	− 0.0587271i	−0.999569	−	0.0293636i	$-0.990652\pi$
$571$	− 457540.i	− 0.0587271i	0.999569	−	0.0293636i	$-0.00934806\pi$
$572$	0	0
$573$	6.56114e6i	0.834820i
$574$	0	0
$575$	8.51290e6	1.07376
$576$	0	0
$577$	1.36331e7	1.70473	0.852364	−	0.522950i	$-0.175168\pi$
$577$	1.36331e7	1.70473	0.852364	+	0.522950i	$0.175168\pi$
$578$	0	0
$579$	− 7.67666e6i	− 0.951647i
$580$	0	0
$581$	− 1.37470e6i	− 0.168953i
$582$	0	0
$583$	−1.59338e7	−1.94155
$584$	0	0
$585$	−1.48068e6	−0.178884
$586$	0	0
$587$	− 570820.i	− 0.0683760i	−0.999415	−	0.0341880i	$-0.989115\pi$
$587$	− 570820.i	− 0.0683760i	0.999415	−	0.0341880i	$-0.0108845\pi$
$588$	0	0
$589$	− 1.46292e7i	− 1.73753i
$590$	0	0
$591$	−1.92701e6	−0.226942
$592$	0	0
$593$	9.51962e6	1.11169	0.555844	−	0.831287i	$-0.312395\pi$
$593$	9.51962e6	1.11169	0.555844	+	0.831287i	$0.312395\pi$
$594$	0	0
$595$	− 2.45464e6i	− 0.284247i
$596$	0	0
$597$	4.49012e6i	0.515611i
$598$	0	0
$599$	−1.58794e7	−1.80829	−0.904145	−	0.427225i	$-0.859491\pi$
$599$	−1.58794e7	−1.80829	−0.904145	+	0.427225i	$0.859491\pi$
$600$	0	0
$601$	6.15280e6	0.694843	0.347422	−	0.937709i	$-0.387057\pi$
$601$	6.15280e6	0.694843	0.347422	+	0.937709i	$0.387057\pi$
$602$	0	0
$603$	1.84324e6i	0.206437i
$604$	0	0
$605$	7.26250e6i	0.806673i
$606$	0	0
$607$	5.81252e6	0.640314	0.320157	−	0.947365i	$-0.396264\pi$
$607$	5.81252e6	0.640314	0.320157	+	0.947365i	$0.396264\pi$
$608$	0	0
$609$	7.40491e6	0.809052
$610$	0	0
$611$	− 1.01271e6i	− 0.109745i
$612$	0	0
$613$	− 8.28312e6i	− 0.890313i	−0.895453	−	0.445156i	$-0.853148\pi$
$613$	− 8.28312e6i	− 0.890313i	0.895453	−	0.445156i	$-0.146852\pi$
$614$	0	0
$615$	−942840.	−0.100519
$616$	0	0
$617$	3.87774e6	0.410078	0.205039	−	0.978754i	$-0.434268\pi$
$617$	3.87774e6	0.410078	0.205039	+	0.978754i	$0.434268\pi$
$618$	0	0
$619$	− 1.22670e7i	− 1.28680i	−0.765531	−	0.643400i	$-0.777523\pi$
$619$	− 1.22670e7i	− 1.28680i	0.765531	−	0.643400i	$-0.222477\pi$
$620$	0	0
$621$	− 2.27740e6i	− 0.236979i
$622$	0	0
$623$	−8.96968e6	−0.925885
$624$	0	0
$625$	6.17562e6	0.632384
$626$	0	0
$627$	− 1.90267e7i	− 1.93284i
$628$	0	0
$629$	− 5.30967e6i	− 0.535107i
$630$	0	0
$631$	9.29180e6	0.929023	0.464512	−	0.885567i	$-0.346230\pi$
$631$	9.29180e6	0.929023	0.464512	+	0.885567i	$0.346230\pi$
$632$	0	0
$633$	4.27439e6	0.423999
$634$	0	0
$635$	3.15788e6i	0.310786i
$636$	0	0
$637$	− 1.75762e6i	− 0.171624i
$638$	0	0
$639$	−4.32832e6	−0.419341
$640$	0	0
$641$	−5.39961e6	−0.519060	−0.259530	−	0.965735i	$-0.583568\pi$
$641$	−5.39961e6	−0.519060	−0.259530	+	0.965735i	$0.583568\pi$
$642$	0	0
$643$	− 5.49216e6i	− 0.523861i	−0.965087	−	0.261930i	$-0.915641\pi$
$643$	− 5.49216e6i	− 0.523861i	0.965087	−	0.261930i	$-0.0843591\pi$
$644$	0	0
$645$	3.01536e6i	0.285391i
$646$	0	0
$647$	1.15586e7	1.08554	0.542770	−	0.839882i	$-0.317376\pi$
$647$	1.15586e7	1.08554	0.542770	+	0.839882i	$0.317376\pi$
$648$	0	0
$649$	1.71704e7	1.60018
$650$	0	0
$651$	− 5.50098e6i	− 0.508730i
$652$	0	0
$653$	7.22411e6i	0.662981i	0.943459	+	0.331491i	$0.107552\pi$
$653$	7.22411e6i	0.662981i	−0.943459	+	0.331491i	$0.892448\pi$
$654$	0	0
$655$	−1.88376e6	−0.171562
$656$	0	0
$657$	387990.	0.0350677
$658$	0	0
$659$	1.57503e7i	1.41278i	0.707822	+	0.706391i	$0.249678\pi$
$659$	1.57503e7i	1.41278i	−0.707822	+	0.706391i	$0.750322\pi$
$660$	0	0
$661$	5.87009e6i	0.522566i	0.965262	+	0.261283i	$0.0841456\pi$
$661$	5.87009e6i	0.522566i	−0.965262	+	0.261283i	$0.915854\pi$
$662$	0	0
$663$	−8.27536e6	−0.731144
$664$	0	0
$665$	7.12480e6	0.624768
$666$	0	0
$667$	− 2.10683e7i	− 1.83364i
$668$	0	0
$669$	− 6.55306e6i	− 0.566082i
$670$	0	0
$671$	3.27262e7	2.80601
$672$	0	0
$673$	−1.38791e7	−1.18120	−0.590599	−	0.806965i	$-0.701108\pi$
$673$	−1.38791e7	−1.18120	−0.590599	+	0.806965i	$0.701108\pi$
$674$	0	0
$675$	− 1.98652e6i	− 0.167816i
$676$	0	0
$677$	− 1.13566e7i	− 0.952303i	−0.879363	−	0.476151i	$-0.842031\pi$
$677$	− 1.13566e7i	− 0.952303i	0.879363	−	0.476151i	$-0.157969\pi$
$678$	0	0
$679$	1.39268e7	1.15925
$680$	0	0
$681$	3.19540e6	0.264032
$682$	0	0
$683$	2.19391e7i	1.79957i	0.436338	+	0.899783i	$0.356275\pi$
$683$	2.19391e7i	1.79957i	−0.436338	+	0.899783i	$0.643725\pi$
$684$	0	0
$685$	− 584680.i	− 0.0476093i
$686$	0	0
$687$	8.88741e6	0.718428
$688$	0	0
$689$	−2.01153e7	−1.61428
$690$	0	0
$691$	1.91337e7i	1.52442i	0.647330	+	0.762210i	$0.275886\pi$
$691$	1.91337e7i	1.52442i	−0.647330	+	0.762210i	$0.724114\pi$
$692$	0	0
$693$	− 7.15457e6i	− 0.565914i
$694$	0	0
$695$	−2.47112e6	−0.194058
$696$	0	0
$697$	−5.26943e6	−0.410848
$698$	0	0
$699$	− 8.93889e6i	− 0.691975i
$700$	0	0
$701$	− 2.33384e7i	− 1.79381i	−0.442222	−	0.896905i	$-0.645810\pi$
$701$	− 2.33384e7i	− 1.79381i	0.442222	−	0.896905i	$-0.354190\pi$
$702$	0	0
$703$	1.54118e7	1.17615
$704$	0	0
$705$	−199440.	−0.0151126
$706$	0	0
$707$	2.04989e7i	1.54235i
$708$	0	0
$709$	− 3.44683e6i	− 0.257516i	−0.991676	−	0.128758i	$-0.958901\pi$
$709$	− 3.44683e6i	− 0.257516i	0.991676	−	0.128758i	$-0.0410991\pi$
$710$	0	0
$711$	−152766.	−0.0113332
$712$	0	0
$713$	−1.56512e7	−1.15299
$714$	0	0
$715$	1.32347e7i	0.968166i
$716$	0	0
$717$	6.89386e6i	0.500800i
$718$	0	0
$719$	5.42504e6	0.391364	0.195682	−	0.980667i	$-0.437308\pi$
$719$	5.42504e6	0.391364	0.195682	+	0.980667i	$0.437308\pi$
$720$	0	0
$721$	4.72750e6	0.338683
$722$	0	0
$723$	1.49783e7i	1.06565i
$724$	0	0
$725$	− 1.83774e7i	− 1.29849i
$726$	0	0
$727$	−9.03488e6	−0.633996	−0.316998	−	0.948426i	$-0.602675\pi$
$727$	−9.03488e6	−0.633996	−0.316998	+	0.948426i	$0.602675\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.68525e7i	1.16646i
$732$	0	0
$733$	− 5.58049e6i	− 0.383630i	−0.981431	−	0.191815i	$-0.938563\pi$
$733$	− 5.58049e6i	− 0.383630i	0.981431	−	0.191815i	$-0.0614373\pi$
$734$	0	0
$735$	−346140.	−0.0236338
$736$	0	0
$737$	1.64753e7	1.11729
$738$	0	0
$739$	10572.0i	0 0.000712108i	1.00000		0.000356054i	$0.000113336\pi$
$739$	10572.0i	0 0.000712108i	−1.00000		0.000356054i	$0.999887\pi$
$740$	0	0
$741$	− 2.40199e7i	− 1.60704i
$742$	0	0
$743$	1.51669e7	1.00792	0.503959	−	0.863728i	$-0.331876\pi$
$743$	1.51669e7	1.00792	0.503959	+	0.863728i	$0.331876\pi$
$744$	0	0
$745$	4.95672e6	0.327193
$746$	0	0
$747$	912708.i	0.0598453i
$748$	0	0
$749$	2.52750e7i	1.64621i
$750$	0	0
$751$	−2.15869e7	−1.39666	−0.698331	−	0.715775i	$-0.746074\pi$
$751$	−2.15869e7	−1.39666	−0.698331	+	0.715775i	$0.746074\pi$
$752$	0	0
$753$	9.22806e6	0.593093
$754$	0	0
$755$	5.13340e6i	0.327746i
$756$	0	0
$757$	1.07700e7i	0.683087i	0.939866	+	0.341543i	$0.110950\pi$
$757$	1.07700e7i	0.683087i	−0.939866	+	0.341543i	$0.889050\pi$
$758$	0	0
$759$	−2.03560e7	−1.28259
$760$	0	0
$761$	−1.75578e7	−1.09903	−0.549514	−	0.835485i	$-0.685187\pi$
$761$	−1.75578e7	−1.09903	−0.549514	+	0.835485i	$0.685187\pi$
$762$	0	0
$763$	− 4.79680e6i	− 0.298291i
$764$	0	0
$765$	1.62972e6i	0.100684i
$766$	0	0
$767$	2.16764e7	1.33045
$768$	0	0
$769$	−2.64544e7	−1.61318	−0.806589	−	0.591112i	$-0.798689\pi$
$769$	−2.64544e7	−1.61318	−0.806589	+	0.591112i	$0.798689\pi$
$770$	0	0
$771$	− 7.76029e6i	− 0.470156i
$772$	0	0
$773$	− 4.90282e6i	− 0.295119i	−0.989053	−	0.147559i	$-0.952858\pi$
$773$	− 4.90282e6i	− 0.295119i	0.989053	−	0.147559i	$-0.0471418\pi$
$774$	0	0
$775$	−1.36522e7	−0.816488
$776$	0	0
$777$	5.79524e6	0.344365
$778$	0	0
$779$	− 1.52950e7i	− 0.903035i
$780$	0	0
$781$	3.86877e7i	2.26958i
$782$	0	0
$783$	−4.91638e6	−0.286577
$784$	0	0
$785$	1.18307e7	0.685229
$786$	0	0
$787$	− 3.93416e6i	− 0.226420i	−0.993571	−	0.113210i	$-0.963887\pi$
$787$	− 3.93416e6i	− 0.226420i	0.993571	−	0.113210i	$-0.0361133\pi$
$788$	0	0
$789$	2.10449e6i	0.120352i
$790$	0	0
$791$	691252.	0.0392821
$792$	0	0
$793$	4.13146e7	2.33303
$794$	0	0
$795$	3.96144e6i	0.222298i
$796$	0	0
$797$	3.91796e6i	0.218481i	0.994015	+	0.109241i	$0.0348419\pi$
$797$	3.91796e6i	0.218481i	−0.994015	+	0.109241i	$0.965158\pi$
$798$	0	0
$799$	−1.11465e6	−0.0617690
$800$	0	0
$801$	5.95528e6	0.327960
$802$	0	0
$803$	− 3.46796e6i	− 0.189795i
$804$	0	0
$805$	− 7.62256e6i	− 0.414583i
$806$	0	0
$807$	−4.15663e6	−0.224677
$808$	0	0
$809$	8.08192e6	0.434154	0.217077	−	0.976155i	$-0.430348\pi$
$809$	8.08192e6	0.434154	0.217077	+	0.976155i	$0.430348\pi$
$810$	0	0
$811$	− 2.39763e7i	− 1.28006i	−0.768350	−	0.640030i	$-0.778922\pi$
$811$	− 2.39763e7i	− 1.28006i	0.768350	−	0.640030i	$-0.221078\pi$
$812$	0	0
$813$	1.30698e7i	0.693495i
$814$	0	0
$815$	−2.40112e6	−0.126625
$816$	0	0
$817$	−4.89158e7	−2.56386
$818$	0	0
$819$	− 9.03215e6i	− 0.470523i
$820$	0	0
$821$	2.04662e7i	1.05969i	0.848095	+	0.529845i	$0.177750\pi$
$821$	2.04662e7i	1.05969i	−0.848095	+	0.529845i	$0.822250\pi$
$822$	0	0
$823$	−2.21937e7	−1.14217	−0.571083	−	0.820892i	$-0.693477\pi$
$823$	−2.21937e7	−1.14217	−0.571083	+	0.820892i	$0.693477\pi$
$824$	0	0
$825$	−1.77561e7	−0.908265
$826$	0	0
$827$	1.56952e7i	0.798000i	0.916951	+	0.399000i	$0.130643\pi$
$827$	1.56952e7i	0.798000i	−0.916951	+	0.399000i	$0.869357\pi$
$828$	0	0
$829$	7.27719e6i	0.367771i	0.982948	+	0.183886i	$0.0588676\pi$
$829$	7.27719e6i	0.367771i	−0.982948	+	0.183886i	$0.941132\pi$
$830$	0	0
$831$	1.38373e7	0.695104
$832$	0	0
$833$	−1.93454e6	−0.0965972
$834$	0	0
$835$	1.65088e6i	0.0819406i
$836$	0	0
$837$	3.65229e6i	0.180199i
$838$	0	0
$839$	−2.14069e7	−1.04990	−0.524950	−	0.851133i	$-0.675916\pi$
$839$	−2.14069e7	−1.04990	−0.524950	+	0.851133i	$0.675916\pi$
$840$	0	0
$841$	−2.49704e7	−1.21741
$842$	0	0
$843$	− 1.73939e6i	− 0.0843003i
$844$	0	0
$845$	9.28206e6i	0.447201i
$846$	0	0
$847$	−4.43012e7	−2.12181
$848$	0	0
$849$	1.42846e7	0.680142
$850$	0	0
$851$	− 1.64885e7i	− 0.780471i
$852$	0	0
$853$	2.04603e7i	0.962810i	0.876498	+	0.481405i	$0.159873\pi$
$853$	2.04603e7i	0.962810i	−0.876498	+	0.481405i	$0.840127\pi$
$854$	0	0
$855$	−4.73040e6	−0.221301
$856$	0	0
$857$	3.91290e7	1.81990	0.909949	−	0.414721i	$-0.136121\pi$
$857$	3.91290e7	1.81990	0.909949	+	0.414721i	$0.136121\pi$
$858$	0	0
$859$	− 2.48593e7i	− 1.14949i	−0.818333	−	0.574745i	$-0.805101\pi$
$859$	− 2.48593e7i	− 1.14949i	0.818333	−	0.574745i	$-0.194899\pi$
$860$	0	0
$861$	− 5.75132e6i	− 0.264399i
$862$	0	0
$863$	2.90387e7	1.32724	0.663620	−	0.748070i	$-0.269019\pi$
$863$	2.90387e7	1.32724	0.663620	+	0.748070i	$0.269019\pi$
$864$	0	0
$865$	−4.58808e6	−0.208493
$866$	0	0
$867$	− 3.67039e6i	− 0.165830i
$868$	0	0
$869$	1.36546e6i	0.0613382i
$870$	0	0
$871$	2.07990e7	0.928959
$872$	0	0
$873$	−9.24647e6	−0.410620
$874$	0	0
$875$	− 1.42740e7i	− 0.630268i
$876$	0	0
$877$	− 3.91274e7i	− 1.71784i	−0.512111	−	0.858919i	$-0.671136\pi$
$877$	− 3.91274e7i	− 1.71784i	0.512111	−	0.858919i	$-0.328864\pi$
$878$	0	0
$879$	−2.29747e7	−1.00295
$880$	0	0
$881$	−8.65214e6	−0.375564	−0.187782	−	0.982211i	$-0.560130\pi$
$881$	−8.65214e6	−0.375564	−0.187782	+	0.982211i	$0.560130\pi$
$882$	0	0
$883$	3.32706e7i	1.43602i	0.696035	+	0.718008i	$0.254946\pi$
$883$	3.32706e7i	1.43602i	−0.696035	+	0.718008i	$0.745054\pi$
$884$	0	0
$885$	− 4.26888e6i	− 0.183213i
$886$	0	0
$887$	1.18364e7	0.505138	0.252569	−	0.967579i	$-0.418724\pi$
$887$	1.18364e7	0.505138	0.252569	+	0.967579i	$0.418724\pi$
$888$	0	0
$889$	−1.92631e7	−0.817469
$890$	0	0
$891$	4.75016e6i	0.200454i
$892$	0	0
$893$	− 3.23536e6i	− 0.135767i
$894$	0	0
$895$	3.51576e6	0.146711
$896$	0	0
$897$	−2.56980e7	−1.06640
$898$	0	0
$899$	3.37874e7i	1.39430i
$900$	0	0
$901$	2.21400e7i	0.908587i
$902$	0	0
$903$	−1.83937e7	−0.750671
$904$	0	0
$905$	7.45868e6	0.302720
$906$	0	0
$907$	− 1.41312e7i	− 0.570377i	−0.958471	−	0.285189i	$-0.907944\pi$
$907$	− 1.41312e7i	− 0.570377i	0.958471	−	0.285189i	$-0.0920562\pi$
$908$	0	0
$909$	− 1.36099e7i	− 0.546319i
$910$	0	0
$911$	3.78316e7	1.51028	0.755142	−	0.655561i	$-0.227568\pi$
$911$	3.78316e7	1.51028	0.755142	+	0.655561i	$0.227568\pi$
$912$	0	0
$913$	8.15803e6	0.323898
$914$	0	0
$915$	− 8.13636e6i	− 0.321275i
$916$	0	0
$917$	− 1.14909e7i	− 0.451265i
$918$	0	0
$919$	2.08170e7	0.813074	0.406537	−	0.913634i	$-0.366736\pi$
$919$	2.08170e7	0.813074	0.406537	+	0.913634i	$0.366736\pi$
$920$	0	0
$921$	−4.15681e6	−0.161477
$922$	0	0
$923$	4.88405e7i	1.88702i
$924$	0	0
$925$	− 1.43826e7i	− 0.552690i
$926$	0	0
$927$	−3.13875e6	−0.119966
$928$	0	0
$929$	6.29002e6	0.239118	0.119559	−	0.992827i	$-0.461852\pi$
$929$	6.29002e6	0.239118	0.119559	+	0.992827i	$0.461852\pi$
$930$	0	0
$931$	− 5.61516e6i	− 0.212319i
$932$	0	0
$933$	1.02132e7i	0.384112i
$934$	0	0
$935$	1.45669e7	0.544926
$936$	0	0
$937$	−2.27077e7	−0.844937	−0.422468	−	0.906378i	$-0.638836\pi$
$937$	−2.27077e7	−0.844937	−0.422468	+	0.906378i	$0.638836\pi$
$938$	0	0
$939$	6.48887e6i	0.240163i
$940$	0	0
$941$	4.28149e7i	1.57623i	0.615525	+	0.788117i	$0.288944\pi$
$941$	4.28149e7i	1.57623i	−0.615525	+	0.788117i	$0.711056\pi$
$942$	0	0
$943$	−1.63635e7	−0.599235
$944$	0	0
$945$	−1.77876e6	−0.0647945
$946$	0	0
$947$	− 1.30949e7i	− 0.474491i	−0.971450	−	0.237246i	$-0.923755\pi$
$947$	− 1.30949e7i	− 0.474491i	0.971450	−	0.237246i	$-0.0762446\pi$
$948$	0	0
$949$	− 4.37806e6i	− 0.157803i
$950$	0	0
$951$	−5.86613e6	−0.210330
$952$	0	0
$953$	−2.06761e7	−0.737458	−0.368729	−	0.929537i	$-0.620207\pi$
$953$	−2.06761e7	−0.737458	−0.368729	+	0.929537i	$0.620207\pi$
$954$	0	0
$955$	− 1.45803e7i	− 0.517319i
$956$	0	0
$957$	4.39439e7i	1.55102i
$958$	0	0
$959$	3.56655e6	0.125228
$960$	0	0
$961$	−3.52905e6	−0.123268
$962$	0	0
$963$	− 1.67809e7i	− 0.583110i
$964$	0	0
$965$	1.70592e7i	0.589714i
$966$	0	0
$967$	5.52496e7	1.90004	0.950021	−	0.312187i	$-0.101062\pi$
$967$	5.52496e7	1.90004	0.950021	+	0.312187i	$0.101062\pi$
$968$	0	0
$969$	−2.64377e7	−0.904511
$970$	0	0
$971$	− 1.22186e7i	− 0.415887i	−0.978141	−	0.207943i	$-0.933323\pi$
$971$	− 1.22186e7i	− 0.415887i	0.978141	−	0.207943i	$-0.0666770\pi$
$972$	0	0
$973$	− 1.50738e7i	− 0.510436i
$974$	0	0
$975$	−2.24158e7	−0.755168
$976$	0	0
$977$	−6.95889e6	−0.233240	−0.116620	−	0.993177i	$-0.537206\pi$
$977$	−6.95889e6	−0.233240	−0.116620	+	0.993177i	$0.537206\pi$
$978$	0	0
$979$	− 5.32299e7i	− 1.77500i
$980$	0	0
$981$	3.18476e6i	0.105658i
$982$	0	0
$983$	−539272.	−0.0178002	−0.00890008	−	0.999960i	$-0.502833\pi$
$983$	−539272.	−0.0178002	−0.00890008	+	0.999960i	$0.502833\pi$
$984$	0	0
$985$	4.28224e6	0.140631
$986$	0	0
$987$	− 1.21658e6i	− 0.0397511i
$988$	0	0
$989$	5.23332e7i	1.70132i
$990$	0	0
$991$	2.14279e7	0.693100	0.346550	−	0.938031i	$-0.387353\pi$
$991$	2.14279e7	0.693100	0.346550	+	0.938031i	$0.387353\pi$
$992$	0	0
$993$	−1.89805e7	−0.610851
$994$	0	0
$995$	− 9.97804e6i	− 0.319512i
$996$	0	0
$997$	4.93966e7i	1.57383i	0.617059	+	0.786917i	$0.288324\pi$
$997$	4.93966e7i	1.57383i	−0.617059	+	0.786917i	$0.711676\pi$
$998$	0	0
$999$	−3.84766e6	−0.121978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.d.n.385.2		2
4.3	odd	2		768.6.d.e.385.1		2
8.3	odd	2		768.6.d.e.385.2		2
8.5	even	2	inner	768.6.d.n.385.1		2
16.3	odd	4		384.6.a.c.1.1	yes	1
16.5	even	4		384.6.a.d.1.1	yes	1
16.11	odd	4		384.6.a.b.1.1	yes	1
16.13	even	4		384.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.a.a.1.1	✓	1	16.13	even	4
384.6.a.b.1.1	yes	1	16.11	odd	4
384.6.a.c.1.1	yes	1	16.3	odd	4
384.6.a.d.1.1	yes	1	16.5	even	4
768.6.d.e.385.1		2	4.3	odd	2
768.6.d.e.385.2		2	8.3	odd	2
768.6.d.n.385.1		2	8.5	even	2	inner
768.6.d.n.385.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} -20.0000i q^{5} +122.000 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} -20.0000i q^{5} +122.000 q^{7} -81.0000 q^{9} +724.000i q^{11} +914.000i q^{13} +180.000 q^{15} +1006.00 q^{17} +2920.00i q^{19} +1098.00i q^{21} +3124.00 q^{23} +2725.00 q^{25} -729.000i q^{27} -6744.00i q^{29} -5010.00 q^{31} -6516.00 q^{33} -2440.00i q^{35} -5278.00i q^{37} -8226.00 q^{39} -5238.00 q^{41} +16752.0i q^{43} +1620.00i q^{45} -1108.00 q^{47} -1923.00 q^{49} +9054.00i q^{51} +22008.0i q^{53} +14480.0 q^{55} -26280.0 q^{57} -23716.0i q^{59} -45202.0i q^{61} -9882.00 q^{63} +18280.0 q^{65} -22756.0i q^{67} +28116.0i q^{69} +53436.0 q^{71} -4790.00 q^{73} +24525.0i q^{75} +88328.0i q^{77} +1886.00 q^{79} +6561.00 q^{81} -11268.0i q^{83} -20120.0i q^{85} +60696.0 q^{87} -73522.0 q^{89} +111508. i q^{91} -45090.0i q^{93} +58400.0 q^{95} +114154. q^{97} -58644.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 244 q^{7} - 162 q^{9}+O(q^{10}) \) 2 * q + 244 * q^7 - 162 * q^9	\( 2 q + 244 q^{7} - 162 q^{9} + 360 q^{15} + 2012 q^{17} + 6248 q^{23} + 5450 q^{25} - 10020 q^{31} - 13032 q^{33} - 16452 q^{39} - 10476 q^{41} - 2216 q^{47} - 3846 q^{49} + 28960 q^{55} - 52560 q^{57} - 19764 q^{63} + 36560 q^{65} + 106872 q^{71} - 9580 q^{73} + 3772 q^{79} + 13122 q^{81} + 121392 q^{87} - 147044 q^{89} + 116800 q^{95} + 228308 q^{97}+O(q^{100}) \) 2 * q + 244 * q^7 - 162 * q^9 + 360 * q^15 + 2012 * q^17 + 6248 * q^23 + 5450 * q^25 - 10020 * q^31 - 13032 * q^33 - 16452 * q^39 - 10476 * q^41 - 2216 * q^47 - 3846 * q^49 + 28960 * q^55 - 52560 * q^57 - 19764 * q^63 + 36560 * q^65 + 106872 * q^71 - 9580 * q^73 + 3772 * q^79 + 13122 * q^81 + 121392 * q^87 - 147044 * q^89 + 116800 * q^95 + 228308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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