LMFDB - Embedded newform 768.5.g.c.511.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(511,768)]

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, 0, 0]))

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

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

chi := DirichletCharacter("768.511");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.g (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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.1
Root			$1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.5.g.c.511.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -39.7490 q^{5} +46.0431i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} -39.7490 q^{5} +46.0431i q^{7} -27.0000 q^{9} -181.283i q^{11} -183.498 q^{13} +206.542i q^{15} +427.992 q^{17} +668.558i q^{19} +239.247 q^{21} +882.463i q^{23} +954.984 q^{25} +140.296i q^{27} -807.247 q^{29} +391.276i q^{31} -941.976 q^{33} -1830.17i q^{35} -466.980 q^{37} +953.484i q^{39} -2159.93 q^{41} +509.182i q^{43} +1073.22 q^{45} -2056.83i q^{47} +281.031 q^{49} -2223.91i q^{51} +753.725 q^{53} +7205.84i q^{55} +3473.93 q^{57} -1309.43i q^{59} -801.913 q^{61} -1243.16i q^{63} +7293.87 q^{65} -505.813i q^{67} +4585.41 q^{69} -2170.94i q^{71} +2297.97 q^{73} -4962.24i q^{75} +8346.85 q^{77} -10705.3i q^{79} +729.000 q^{81} +2977.81i q^{83} -17012.3 q^{85} +4194.58i q^{87} +6493.92 q^{89} -8448.82i q^{91} +2033.13 q^{93} -26574.5i q^{95} -8189.92 q^{97} +4894.65i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{5} - 108 q^{9}+O(q^{10}) $ 4 * q - 32 * q^5 - 108 * q^9	$ 4 q - 32 q^{5} - 108 q^{9} - 480 q^{13} + 696 q^{17} + 576 q^{21} + 1788 q^{25} - 2848 q^{29} - 720 q^{33} + 672 q^{37} + 504 q^{41} + 864 q^{45} + 5188 q^{49} - 160 q^{53} + 4752 q^{57} + 7968 q^{61} + 11904 q^{65} + 6912 q^{69} + 5128 q^{73} + 14592 q^{77} + 2916 q^{81} - 37824 q^{85} + 15816 q^{89} - 7488 q^{93} - 22600 q^{97}+O(q^{100}) $ 4 * q - 32 * q^5 - 108 * q^9 - 480 * q^13 + 696 * q^17 + 576 * q^21 + 1788 * q^25 - 2848 * q^29 - 720 * q^33 + 672 * q^37 + 504 * q^41 + 864 * q^45 + 5188 * q^49 - 160 * q^53 + 4752 * q^57 + 7968 * q^61 + 11904 * q^65 + 6912 * q^69 + 5128 * q^73 + 14592 * q^77 + 2916 * q^81 - 37824 * q^85 + 15816 * q^89 - 7488 * q^93 - 22600 * 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
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	−39.7490	−1.58996	−0.794980	−	0.606635i	$-0.792519\pi$
$5$	−39.7490	−1.58996	−0.794980	+	0.606635i	$0.792519\pi$
$6$	0	0
$7$	46.0431i	0.939655i	0.882758	+	0.469828i	$0.155684\pi$
$7$	46.0431i	0.939655i	−0.882758	+	0.469828i	$0.844316\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	− 181.283i	− 1.49821i	−0.662451	−	0.749105i	$-0.730484\pi$
$11$	− 181.283i	− 1.49821i	0.662451	−	0.749105i	$-0.269516\pi$
$12$	0	0
$13$	−183.498	−1.08579	−0.542894	−	0.839801i	$-0.682671\pi$
$13$	−183.498	−1.08579	−0.542894	+	0.839801i	$0.682671\pi$
$14$	0	0
$15$	206.542i	0.917964i
$16$	0	0
$17$	427.992	1.48094	0.740471	−	0.672089i	$-0.234603\pi$
$17$	427.992	1.48094	0.740471	+	0.672089i	$0.234603\pi$
$18$	0	0
$19$	668.558i	1.85196i	0.377571	+	0.925981i	$0.376759\pi$
$19$	668.558i	1.85196i	−0.377571	+	0.925981i	$0.623241\pi$
$20$	0	0
$21$	239.247	0.542510
$22$	0	0
$23$	882.463i	1.66817i	0.551635	+	0.834086i	$0.314004\pi$
$23$	882.463i	1.66817i	−0.551635	+	0.834086i	$0.685996\pi$
$24$	0	0
$25$	954.984	1.52797
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	−807.247	−0.959866	−0.479933	−	0.877305i	$-0.659339\pi$
$29$	−807.247	−0.959866	−0.479933	+	0.877305i	$0.659339\pi$
$30$	0	0
$31$	391.276i	0.407155i	0.979059	+	0.203577i	$0.0652569\pi$
$31$	391.276i	0.407155i	−0.979059	+	0.203577i	$0.934743\pi$
$32$	0	0
$33$	−941.976	−0.864992
$34$	0	0
$35$	− 1830.17i	− 1.49402i
$36$	0	0
$37$	−466.980	−0.341111	−0.170555	−	0.985348i	$-0.554556\pi$
$37$	−466.980	−0.341111	−0.170555	+	0.985348i	$0.554556\pi$
$38$	0	0
$39$	953.484i	0.626880i
$40$	0	0
$41$	−2159.93	−1.28491	−0.642454	−	0.766325i	$-0.722083\pi$
$41$	−2159.93	−1.28491	−0.642454	+	0.766325i	$0.722083\pi$
$42$	0	0
$43$	509.182i	0.275382i	0.990475	+	0.137691i	$0.0439682\pi$
$43$	509.182i	0.275382i	−0.990475	+	0.137691i	$0.956032\pi$
$44$	0	0
$45$	1073.22	0.529987
$46$	0	0
$47$	− 2056.83i	− 0.931116i	−0.885017	−	0.465558i	$-0.845854\pi$
$47$	− 2056.83i	− 0.931116i	0.885017	−	0.465558i	$-0.154146\pi$
$48$	0	0
$49$	281.031	0.117048
$50$	0	0
$51$	− 2223.91i	− 0.855022i
$52$	0	0
$53$	753.725	0.268325	0.134163	−	0.990959i	$-0.457166\pi$
$53$	753.725	0.268325	0.134163	+	0.990959i	$0.457166\pi$
$54$	0	0
$55$	7205.84i	2.38210i
$56$	0	0
$57$	3473.93	1.06923
$58$	0	0
$59$	− 1309.43i	− 0.376165i	−0.982153	−	0.188083i	$-0.939773\pi$
$59$	− 1309.43i	− 0.376165i	0.982153	−	0.188083i	$-0.0602272\pi$
$60$	0	0
$61$	−801.913	−0.215510	−0.107755	−	0.994177i	$-0.534366\pi$
$61$	−801.913	−0.215510	−0.107755	+	0.994177i	$0.534366\pi$
$62$	0	0
$63$	− 1243.16i	− 0.313218i
$64$	0	0
$65$	7293.87	1.72636
$66$	0	0
$67$	− 505.813i	− 0.112678i	−0.998412	−	0.0563392i	$-0.982057\pi$
$67$	− 505.813i	− 0.112678i	0.998412	−	0.0563392i	$-0.0179428\pi$
$68$	0	0
$69$	4585.41	0.963119
$70$	0	0
$71$	− 2170.94i	− 0.430658i	−0.976542	−	0.215329i	$-0.930918\pi$
$71$	− 2170.94i	− 0.430658i	0.976542	−	0.215329i	$-0.0690823\pi$
$72$	0	0
$73$	2297.97	0.431219	0.215610	−	0.976480i	$-0.430826\pi$
$73$	2297.97	0.431219	0.215610	+	0.976480i	$0.430826\pi$
$74$	0	0
$75$	− 4962.24i	− 0.882177i
$76$	0	0
$77$	8346.85	1.40780
$78$	0	0
$79$	− 10705.3i	− 1.71532i	−0.514219	−	0.857659i	$-0.671918\pi$
$79$	− 10705.3i	− 1.71532i	0.514219	−	0.857659i	$-0.328082\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	2977.81i	0.432256i	0.976365	+	0.216128i	$0.0693429\pi$
$83$	2977.81i	0.432256i	−0.976365	+	0.216128i	$0.930657\pi$
$84$	0	0
$85$	−17012.3	−2.35464
$86$	0	0
$87$	4194.58i	0.554179i
$88$	0	0
$89$	6493.92	0.819836	0.409918	−	0.912122i	$-0.365557\pi$
$89$	6493.92	0.819836	0.409918	+	0.912122i	$0.365557\pi$
$90$	0	0
$91$	− 8448.82i	− 1.02027i
$92$	0	0
$93$	2033.13	0.235071
$94$	0	0
$95$	− 26574.5i	− 2.94455i
$96$	0	0
$97$	−8189.92	−0.870435	−0.435217	−	0.900325i	$-0.643328\pi$
$97$	−8189.92	−0.870435	−0.435217	+	0.900325i	$0.643328\pi$
$98$	0	0
$99$	4894.65i	0.499403i
$100$	0	0
$101$	15576.5	1.52696	0.763479	−	0.645833i	$-0.223490\pi$
$101$	15576.5	1.52696	0.763479	+	0.645833i	$0.223490\pi$
$102$	0	0
$103$	− 6628.58i	− 0.624807i	−0.949950	−	0.312403i	$-0.898866\pi$
$103$	− 6628.58i	− 0.624807i	0.949950	−	0.312403i	$-0.101134\pi$
$104$	0	0
$105$	−9509.83	−0.862570
$106$	0	0
$107$	− 11796.4i	− 1.03035i	−0.857086	−	0.515173i	$-0.827728\pi$
$107$	− 11796.4i	− 1.03035i	0.857086	−	0.515173i	$-0.172272\pi$
$108$	0	0
$109$	−13286.1	−1.11826	−0.559130	−	0.829080i	$-0.688865\pi$
$109$	−13286.1	−1.11826	−0.559130	+	0.829080i	$0.688865\pi$
$110$	0	0
$111$	2426.50i	0.196940i
$112$	0	0
$113$	−11713.6	−0.917350	−0.458675	−	0.888604i	$-0.651676\pi$
$113$	−11713.6	−0.917350	−0.458675	+	0.888604i	$0.651676\pi$
$114$	0	0
$115$	− 35077.0i	− 2.65233i
$116$	0	0
$117$	4954.45	0.361929
$118$	0	0
$119$	19706.1i	1.39157i
$120$	0	0
$121$	−18222.7	−1.24463
$122$	0	0
$123$	11223.3i	0.741842i
$124$	0	0
$125$	−13116.5	−0.839459
$126$	0	0
$127$	− 5640.55i	− 0.349715i	−0.984594	−	0.174858i	$-0.944054\pi$
$127$	− 5640.55i	− 0.349715i	0.984594	−	0.174858i	$-0.0559465\pi$
$128$	0	0
$129$	2645.79	0.158992
$130$	0	0
$131$	− 31922.5i	− 1.86018i	−0.367335	−	0.930089i	$-0.619730\pi$
$131$	− 31922.5i	− 1.86018i	0.367335	−	0.930089i	$-0.380270\pi$
$132$	0	0
$133$	−30782.5	−1.74021
$134$	0	0
$135$	− 5576.63i	− 0.305988i
$136$	0	0
$137$	10515.6	0.560263	0.280132	−	0.959962i	$-0.409622\pi$
$137$	10515.6	0.560263	0.280132	+	0.959962i	$0.409622\pi$
$138$	0	0
$139$	− 14985.9i	− 0.775626i	−0.921738	−	0.387813i	$-0.873231\pi$
$139$	− 14985.9i	− 0.775626i	0.921738	−	0.387813i	$-0.126769\pi$
$140$	0	0
$141$	−10687.6	−0.537580
$142$	0	0
$143$	33265.2i	1.62674i
$144$	0	0
$145$	32087.3	1.52615
$146$	0	0
$147$	− 1460.28i	− 0.0675775i
$148$	0	0
$149$	11795.4	0.531298	0.265649	−	0.964070i	$-0.414414\pi$
$149$	11795.4	0.531298	0.265649	+	0.964070i	$0.414414\pi$
$150$	0	0
$151$	33792.9i	1.48208i	0.671460	+	0.741040i	$0.265667\pi$
$151$	33792.9i	1.48208i	−0.671460	+	0.741040i	$0.734333\pi$
$152$	0	0
$153$	−11555.8	−0.493647
$154$	0	0
$155$	− 15552.8i	− 0.647360i
$156$	0	0
$157$	28788.9	1.16796	0.583978	−	0.811770i	$-0.301496\pi$
$157$	28788.9	1.16796	0.583978	+	0.811770i	$0.301496\pi$
$158$	0	0
$159$	− 3916.47i	− 0.154918i
$160$	0	0
$161$	−40631.3	−1.56751
$162$	0	0
$163$	13409.0i	0.504684i	0.967638	+	0.252342i	$0.0812008\pi$
$163$	13409.0i	0.504684i	−0.967638	+	0.252342i	$0.918799\pi$
$164$	0	0
$165$	37442.6	1.37530
$166$	0	0
$167$	− 13206.7i	− 0.473546i	−0.971565	−	0.236773i	$-0.923910\pi$
$167$	− 13206.7i	− 0.473546i	0.971565	−	0.236773i	$-0.0760898\pi$
$168$	0	0
$169$	5110.53	0.178934
$170$	0	0
$171$	− 18051.1i	− 0.617320i
$172$	0	0
$173$	364.956	0.0121940	0.00609702	−	0.999981i	$-0.498059\pi$
$173$	364.956	0.0121940	0.00609702	+	0.999981i	$0.498059\pi$
$174$	0	0
$175$	43970.5i	1.43577i
$176$	0	0
$177$	−6804.00	−0.217179
$178$	0	0
$179$	− 14372.0i	− 0.448549i	−0.974526	−	0.224275i	$-0.927999\pi$
$179$	− 14372.0i	− 0.448549i	0.974526	−	0.224275i	$-0.0720013\pi$
$180$	0	0
$181$	4050.54	0.123639	0.0618195	−	0.998087i	$-0.480310\pi$
$181$	4050.54	0.123639	0.0618195	+	0.998087i	$0.480310\pi$
$182$	0	0
$183$	4166.86i	0.124425i
$184$	0	0
$185$	18562.0	0.542352
$186$	0	0
$187$	− 77587.9i	− 2.21876i
$188$	0	0
$189$	−6459.67	−0.180837
$190$	0	0
$191$	49880.9i	1.36731i	0.729805	+	0.683656i	$0.239611\pi$
$191$	49880.9i	1.36731i	−0.729805	+	0.683656i	$0.760389\pi$
$192$	0	0
$193$	−48425.7	−1.30005	−0.650026	−	0.759912i	$-0.725242\pi$
$193$	−48425.7	−1.30005	−0.650026	+	0.759912i	$0.725242\pi$
$194$	0	0
$195$	− 37900.0i	− 0.996714i
$196$	0	0
$197$	5556.74	0.143182	0.0715908	−	0.997434i	$-0.477192\pi$
$197$	5556.74	0.143182	0.0715908	+	0.997434i	$0.477192\pi$
$198$	0	0
$199$	− 60594.7i	− 1.53013i	−0.643952	−	0.765066i	$-0.722706\pi$
$199$	− 60594.7i	− 1.53013i	0.643952	−	0.765066i	$-0.277294\pi$
$200$	0	0
$201$	−2628.28	−0.0650549
$202$	0	0
$203$	− 37168.2i	− 0.901943i
$204$	0	0
$205$	85855.1	2.04295
$206$	0	0
$207$	− 23826.5i	− 0.556057i
$208$	0	0
$209$	121198.	2.77463
$210$	0	0
$211$	− 27539.9i	− 0.618583i	−0.950967	−	0.309292i	$-0.899908\pi$
$211$	− 27539.9i	− 0.618583i	0.950967	−	0.309292i	$-0.100092\pi$
$212$	0	0
$213$	−11280.6	−0.248640
$214$	0	0
$215$	− 20239.5i	− 0.437847i
$216$	0	0
$217$	−18015.6	−0.382585
$218$	0	0
$219$	− 11940.6i	− 0.248965i
$220$	0	0
$221$	−78535.7	−1.60799
$222$	0	0
$223$	3021.35i	0.0607564i	0.999538	+	0.0303782i	$0.00967116\pi$
$223$	3021.35i	0.0607564i	−0.999538	+	0.0303782i	$0.990329\pi$
$224$	0	0
$225$	−25784.6	−0.509325
$226$	0	0
$227$	7077.28i	0.137346i	0.997639	+	0.0686728i	$0.0218765\pi$
$227$	7077.28i	0.137346i	−0.997639	+	0.0686728i	$0.978124\pi$
$228$	0	0
$229$	−102761.	−1.95956	−0.979778	−	0.200088i	$-0.935877\pi$
$229$	−102761.	−1.95956	−0.979778	+	0.200088i	$0.935877\pi$
$230$	0	0
$231$	− 43371.5i	− 0.812795i
$232$	0	0
$233$	−22976.6	−0.423227	−0.211613	−	0.977353i	$-0.567872\pi$
$233$	−22976.6	−0.423227	−0.211613	+	0.977353i	$0.567872\pi$
$234$	0	0
$235$	81757.1i	1.48044i
$236$	0	0
$237$	−55626.4	−0.990339
$238$	0	0
$239$	35566.7i	0.622656i	0.950303	+	0.311328i	$0.100774\pi$
$239$	35566.7i	0.622656i	−0.950303	+	0.311328i	$0.899226\pi$
$240$	0	0
$241$	−92683.4	−1.59576	−0.797881	−	0.602816i	$-0.794045\pi$
$241$	−92683.4	−1.59576	−0.797881	+	0.602816i	$0.794045\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	−11170.7	−0.186101
$246$	0	0
$247$	− 122679.i	− 2.01084i
$248$	0	0
$249$	15473.2	0.249563
$250$	0	0
$251$	27590.5i	0.437937i	0.975732	+	0.218968i	$0.0702692\pi$
$251$	27590.5i	0.437937i	−0.975732	+	0.218968i	$0.929731\pi$
$252$	0	0
$253$	159976.	2.49927
$254$	0	0
$255$	88398.3i	1.35945i
$256$	0	0
$257$	92304.6	1.39752	0.698758	−	0.715358i	$-0.253736\pi$
$257$	92304.6	1.39752	0.698758	+	0.715358i	$0.253736\pi$
$258$	0	0
$259$	− 21501.2i	− 0.320526i
$260$	0	0
$261$	21795.7	0.319955
$262$	0	0
$263$	− 13884.4i	− 0.200732i	−0.994951	−	0.100366i	$-0.967999\pi$
$263$	− 13884.4i	− 0.200732i	0.994951	−	0.100366i	$-0.0320014\pi$
$264$	0	0
$265$	−29959.8	−0.426626
$266$	0	0
$267$	− 33743.4i	− 0.473333i
$268$	0	0
$269$	−52698.1	−0.728266	−0.364133	−	0.931347i	$-0.618635\pi$
$269$	−52698.1	−0.728266	−0.364133	+	0.931347i	$0.618635\pi$
$270$	0	0
$271$	− 47028.8i	− 0.640361i	−0.947356	−	0.320181i	$-0.896256\pi$
$271$	− 47028.8i	− 0.640361i	0.947356	−	0.320181i	$-0.103744\pi$
$272$	0	0
$273$	−43901.4	−0.589051
$274$	0	0
$275$	− 173123.i	− 2.28923i
$276$	0	0
$277$	108393.	1.41268	0.706340	−	0.707873i	$-0.250345\pi$
$277$	108393.	1.41268	0.706340	+	0.707873i	$0.250345\pi$
$278$	0	0
$279$	− 10564.4i	− 0.135718i
$280$	0	0
$281$	73090.7	0.925656	0.462828	−	0.886448i	$-0.346835\pi$
$281$	73090.7	0.925656	0.462828	+	0.886448i	$0.346835\pi$
$282$	0	0
$283$	44763.1i	0.558917i	0.960158	+	0.279458i	$0.0901549\pi$
$283$	44763.1i	0.558917i	−0.960158	+	0.279458i	$0.909845\pi$
$284$	0	0
$285$	−138085.	−1.70003
$286$	0	0
$287$	− 99449.9i	− 1.20737i
$288$	0	0
$289$	99656.3	1.19319
$290$	0	0
$291$	42556.1i	0.502546i
$292$	0	0
$293$	42030.2	0.489583	0.244792	−	0.969576i	$-0.421280\pi$
$293$	42030.2	0.489583	0.244792	+	0.969576i	$0.421280\pi$
$294$	0	0
$295$	52048.6i	0.598088i
$296$	0	0
$297$	25433.4	0.288331
$298$	0	0
$299$	− 161930.i	− 1.81128i
$300$	0	0
$301$	−23444.3	−0.258765
$302$	0	0
$303$	− 80937.9i	− 0.881590i
$304$	0	0
$305$	31875.3	0.342653
$306$	0	0
$307$	− 8820.31i	− 0.0935852i	−0.998905	−	0.0467926i	$-0.985100\pi$
$307$	− 8820.31i	− 0.0935852i	0.998905	−	0.0467926i	$-0.0149000\pi$
$308$	0	0
$309$	−34443.1	−0.360732
$310$	0	0
$311$	− 155059.i	− 1.60315i	−0.597893	−	0.801576i	$-0.703995\pi$
$311$	− 155059.i	− 1.60315i	0.597893	−	0.801576i	$-0.296005\pi$
$312$	0	0
$313$	179666.	1.83390	0.916951	−	0.398999i	$-0.130642\pi$
$313$	179666.	1.83390	0.916951	+	0.398999i	$0.130642\pi$
$314$	0	0
$315$	49414.6i	0.498005i
$316$	0	0
$317$	−6188.77	−0.0615865	−0.0307933	−	0.999526i	$-0.509803\pi$
$317$	−6188.77	−0.0615865	−0.0307933	+	0.999526i	$0.509803\pi$
$318$	0	0
$319$	146341.i	1.43808i
$320$	0	0
$321$	−61296.0	−0.594870
$322$	0	0
$323$	286138.i	2.74265i
$324$	0	0
$325$	−175238.	−1.65906
$326$	0	0
$327$	69036.4i	0.645628i
$328$	0	0
$329$	94703.1	0.874928
$330$	0	0
$331$	− 132159.i	− 1.20626i	−0.797643	−	0.603130i	$-0.793920\pi$
$331$	− 132159.i	− 1.20626i	0.797643	−	0.603130i	$-0.206080\pi$
$332$	0	0
$333$	12608.5	0.113704
$334$	0	0
$335$	20105.6i	0.179154i
$336$	0	0
$337$	70040.0	0.616718	0.308359	−	0.951270i	$-0.400220\pi$
$337$	70040.0	0.616718	0.308359	+	0.951270i	$0.400220\pi$
$338$	0	0
$339$	60865.8i	0.529632i
$340$	0	0
$341$	70931.8	0.610004
$342$	0	0
$343$	123489.i	1.04964i
$344$	0	0
$345$	−182266.	−1.53132
$346$	0	0
$347$	157722.i	1.30988i	0.755680	+	0.654942i	$0.227307\pi$
$347$	157722.i	1.30988i	−0.755680	+	0.654942i	$0.772693\pi$
$348$	0	0
$349$	198747.	1.63174	0.815868	−	0.578238i	$-0.196259\pi$
$349$	198747.	1.63174	0.815868	+	0.578238i	$0.196259\pi$
$350$	0	0
$351$	− 25744.1i	− 0.208960i
$352$	0	0
$353$	13376.1	0.107345	0.0536724	−	0.998559i	$-0.482907\pi$
$353$	13376.1	0.107345	0.0536724	+	0.998559i	$0.482907\pi$
$354$	0	0
$355$	86292.9i	0.684729i
$356$	0	0
$357$	102396.	0.803426
$358$	0	0
$359$	190011.i	1.47432i	0.675721	+	0.737158i	$0.263832\pi$
$359$	190011.i	1.47432i	−0.675721	+	0.737158i	$0.736168\pi$
$360$	0	0
$361$	−316649.	−2.42976
$362$	0	0
$363$	94687.8i	0.718590i
$364$	0	0
$365$	−91342.0	−0.685622
$366$	0	0
$367$	− 94140.5i	− 0.698947i	−0.936946	−	0.349474i	$-0.886360\pi$
$367$	− 94140.5i	− 0.698947i	0.936946	−	0.349474i	$-0.113640\pi$
$368$	0	0
$369$	58318.1	0.428302
$370$	0	0
$371$	34703.9i	0.252133i
$372$	0	0
$373$	66152.4	0.475475	0.237738	−	0.971329i	$-0.423594\pi$
$373$	66152.4	0.475475	0.237738	+	0.971329i	$0.423594\pi$
$374$	0	0
$375$	68155.6i	0.484662i
$376$	0	0
$377$	148128.	1.04221
$378$	0	0
$379$	64395.2i	0.448306i	0.974554	+	0.224153i	$0.0719616\pi$
$379$	64395.2i	0.448306i	−0.974554	+	0.224153i	$0.928038\pi$
$380$	0	0
$381$	−29309.2	−0.201908
$382$	0	0
$383$	− 30714.5i	− 0.209385i	−0.994505	−	0.104693i	$-0.966614\pi$
$383$	− 30714.5i	− 0.209385i	0.994505	−	0.104693i	$-0.0333859\pi$
$384$	0	0
$385$	−331779.	−2.23835
$386$	0	0
$387$	− 13747.9i	− 0.0917941i
$388$	0	0
$389$	199122.	1.31589	0.657945	−	0.753066i	$-0.271426\pi$
$389$	199122.	1.31589	0.657945	+	0.753066i	$0.271426\pi$
$390$	0	0
$391$	377687.i	2.47046i
$392$	0	0
$393$	−165874.	−1.07397
$394$	0	0
$395$	425525.i	2.72729i
$396$	0	0
$397$	68565.8	0.435037	0.217519	−	0.976056i	$-0.430204\pi$
$397$	68565.8	0.435037	0.217519	+	0.976056i	$0.430204\pi$
$398$	0	0
$399$	159951.i	1.00471i
$400$	0	0
$401$	−21797.1	−0.135553	−0.0677765	−	0.997701i	$-0.521590\pi$
$401$	−21797.1	−0.135553	−0.0677765	+	0.997701i	$0.521590\pi$
$402$	0	0
$403$	− 71798.3i	− 0.442084i
$404$	0	0
$405$	−28977.0	−0.176662
$406$	0	0
$407$	84655.8i	0.511055i
$408$	0	0
$409$	−230211.	−1.37620	−0.688098	−	0.725618i	$-0.741554\pi$
$409$	−230211.	−1.37620	−0.688098	+	0.725618i	$0.741554\pi$
$410$	0	0
$411$	− 54640.6i	− 0.323468i
$412$	0	0
$413$	60290.3	0.353465
$414$	0	0
$415$	− 118365.i	− 0.687270i
$416$	0	0
$417$	−77868.9	−0.447808
$418$	0	0
$419$	− 17360.5i	− 0.0988859i	−0.998777	−	0.0494430i	$-0.984255\pi$
$419$	− 17360.5i	− 0.0988859i	0.998777	−	0.0494430i	$-0.0157446\pi$
$420$	0	0
$421$	−186829.	−1.05410	−0.527048	−	0.849836i	$-0.676701\pi$
$421$	−186829.	−1.05410	−0.527048	+	0.849836i	$0.676701\pi$
$422$	0	0
$423$	55534.5i	0.310372i
$424$	0	0
$425$	408726.	2.26284
$426$	0	0
$427$	− 36922.6i	− 0.202505i
$428$	0	0
$429$	172851.	0.939197
$430$	0	0
$431$	− 153753.i	− 0.827693i	−0.910347	−	0.413846i	$-0.864185\pi$
$431$	− 153753.i	− 0.827693i	0.910347	−	0.413846i	$-0.135815\pi$
$432$	0	0
$433$	9168.87	0.0489035	0.0244517	−	0.999701i	$-0.492216\pi$
$433$	9168.87	0.0489035	0.0244517	+	0.999701i	$0.492216\pi$
$434$	0	0
$435$	− 166730.i	− 0.881122i
$436$	0	0
$437$	−589978.	−3.08939
$438$	0	0
$439$	− 178760.i	− 0.927561i	−0.885950	−	0.463780i	$-0.846493\pi$
$439$	− 178760.i	− 0.927561i	0.885950	−	0.463780i	$-0.153507\pi$
$440$	0	0
$441$	−7587.85	−0.0390159
$442$	0	0
$443$	− 17798.2i	− 0.0906920i	−0.998971	−	0.0453460i	$-0.985561\pi$
$443$	− 17798.2i	− 0.0906920i	0.998971	−	0.0453460i	$-0.0144390\pi$
$444$	0	0
$445$	−258127.	−1.30351
$446$	0	0
$447$	− 61290.5i	− 0.306745i
$448$	0	0
$449$	242915.	1.20493	0.602464	−	0.798146i	$-0.294186\pi$
$449$	242915.	1.20493	0.602464	+	0.798146i	$0.294186\pi$
$450$	0	0
$451$	391559.i	1.92506i
$452$	0	0
$453$	175593.	0.855680
$454$	0	0
$455$	335832.i	1.62218i
$456$	0	0
$457$	190762.	0.913396	0.456698	−	0.889622i	$-0.349032\pi$
$457$	190762.	0.913396	0.456698	+	0.889622i	$0.349032\pi$
$458$	0	0
$459$	60045.6i	0.285007i
$460$	0	0
$461$	155121.	0.729909	0.364955	−	0.931025i	$-0.381084\pi$
$461$	155121.	0.729909	0.364955	+	0.931025i	$0.381084\pi$
$462$	0	0
$463$	− 230925.i	− 1.07723i	−0.842551	−	0.538616i	$-0.818947\pi$
$463$	− 230925.i	− 1.07723i	0.842551	−	0.538616i	$-0.181053\pi$
$464$	0	0
$465$	−80814.9	−0.373754
$466$	0	0
$467$	154112.i	0.706647i	0.935501	+	0.353323i	$0.114948\pi$
$467$	154112.i	0.706647i	−0.935501	+	0.353323i	$0.885052\pi$
$468$	0	0
$469$	23289.2	0.105879
$470$	0	0
$471$	− 149592.i	− 0.674319i
$472$	0	0
$473$	92306.3	0.412581
$474$	0	0
$475$	638462.i	2.82975i
$476$	0	0
$477$	−20350.6	−0.0894417
$478$	0	0
$479$	− 405668.i	− 1.76807i	−0.467420	−	0.884035i	$-0.654816\pi$
$479$	− 405668.i	− 1.76807i	0.467420	−	0.884035i	$-0.345184\pi$
$480$	0	0
$481$	85690.0	0.370373
$482$	0	0
$483$	211127.i	0.905000i
$484$	0	0
$485$	325541.	1.38396
$486$	0	0
$487$	− 426114.i	− 1.79667i	−0.439314	−	0.898334i	$-0.644778\pi$
$487$	− 426114.i	− 1.79667i	0.439314	−	0.898334i	$-0.355222\pi$
$488$	0	0
$489$	69675.0	0.291379
$490$	0	0
$491$	− 273125.i	− 1.13292i	−0.824090	−	0.566459i	$-0.808313\pi$
$491$	− 273125.i	− 1.13292i	0.824090	−	0.566459i	$-0.191687\pi$
$492$	0	0
$493$	−345495.	−1.42151
$494$	0	0
$495$	− 194558.i	− 0.794032i
$496$	0	0
$497$	99957.1	0.404670
$498$	0	0
$499$	103548.i	0.415855i	0.978144	+	0.207928i	$0.0666719\pi$
$499$	103548.i	0.415855i	−0.978144	+	0.207928i	$0.933328\pi$
$500$	0	0
$501$	−68624.2	−0.273402
$502$	0	0
$503$	− 217063.i	− 0.857925i	−0.903322	−	0.428963i	$-0.858879\pi$
$503$	− 217063.i	− 0.857925i	0.903322	−	0.428963i	$-0.141121\pi$
$504$	0	0
$505$	−619151.	−2.42780
$506$	0	0
$507$	− 26555.1i	− 0.103307i
$508$	0	0
$509$	−115098.	−0.444256	−0.222128	−	0.975018i	$-0.571300\pi$
$509$	−115098.	−0.444256	−0.222128	+	0.975018i	$0.571300\pi$
$510$	0	0
$511$	105806.i	0.405198i
$512$	0	0
$513$	−93796.1	−0.356410
$514$	0	0
$515$	263479.i	0.993418i
$516$	0	0
$517$	−372870.	−1.39501
$518$	0	0
$519$	− 1896.37i	− 0.00704024i
$520$	0	0
$521$	−288543.	−1.06300	−0.531502	−	0.847057i	$-0.678372\pi$
$521$	−288543.	−1.06300	−0.531502	+	0.847057i	$0.678372\pi$
$522$	0	0
$523$	− 176826.i	− 0.646461i	−0.946320	−	0.323231i	$-0.895231\pi$
$523$	− 176826.i	− 0.646461i	0.946320	−	0.323231i	$-0.104769\pi$
$524$	0	0
$525$	228477.	0.828942
$526$	0	0
$527$	167463.i	0.602973i
$528$	0	0
$529$	−498900.	−1.78280
$530$	0	0
$531$	35354.6i	0.125388i
$532$	0	0
$533$	396343.	1.39514
$534$	0	0
$535$	468896.i	1.63821i
$536$	0	0
$537$	−74678.9	−0.258970
$538$	0	0
$539$	− 50946.4i	− 0.175362i
$540$	0	0
$541$	−425220.	−1.45284	−0.726422	−	0.687249i	$-0.758818\pi$
$541$	−425220.	−1.45284	−0.726422	+	0.687249i	$0.758818\pi$
$542$	0	0
$543$	− 21047.2i	− 0.0713830i
$544$	0	0
$545$	528108.	1.77799
$546$	0	0
$547$	495505.i	1.65605i	0.560692	+	0.828024i	$0.310535\pi$
$547$	495505.i	1.65605i	−0.560692	+	0.828024i	$0.689465\pi$
$548$	0	0
$549$	21651.7	0.0718367
$550$	0	0
$551$	− 539691.i	− 1.77763i
$552$	0	0
$553$	492905.	1.61181
$554$	0	0
$555$	− 96451.0i	− 0.313127i
$556$	0	0
$557$	257895.	0.831251	0.415626	−	0.909536i	$-0.363563\pi$
$557$	257895.	0.831251	0.415626	+	0.909536i	$0.363563\pi$
$558$	0	0
$559$	− 93433.9i	− 0.299007i
$560$	0	0
$561$	−403158.	−1.28100
$562$	0	0
$563$	− 417799.i	− 1.31811i	−0.752096	−	0.659054i	$-0.770957\pi$
$563$	− 417799.i	− 1.31811i	0.752096	−	0.659054i	$-0.229043\pi$
$564$	0	0
$565$	465606.	1.45855
$566$	0	0
$567$	33565.4i	0.104406i
$568$	0	0
$569$	100935.	0.311756	0.155878	−	0.987776i	$-0.450179\pi$
$569$	100935.	0.311756	0.155878	+	0.987776i	$0.450179\pi$
$570$	0	0
$571$	453320.i	1.39038i	0.718828	+	0.695188i	$0.244679\pi$
$571$	453320.i	1.39038i	−0.718828	+	0.695188i	$0.755321\pi$
$572$	0	0
$573$	259189.	0.789418
$574$	0	0
$575$	842738.i	2.54892i
$576$	0	0
$577$	12119.6	0.0364031	0.0182015	−	0.999834i	$-0.494206\pi$
$577$	12119.6	0.0364031	0.0182015	+	0.999834i	$0.494206\pi$
$578$	0	0
$579$	251627.i	0.750586i
$580$	0	0
$581$	−137108.	−0.406172
$582$	0	0
$583$	− 136638.i	− 0.402008i
$584$	0	0
$585$	−196934.	−0.575453
$586$	0	0
$587$	327468.i	0.950370i	0.879886	+	0.475185i	$0.157619\pi$
$587$	327468.i	0.950370i	−0.879886	+	0.475185i	$0.842381\pi$
$588$	0	0
$589$	−261591.	−0.754035
$590$	0	0
$591$	− 28873.7i	− 0.0826660i
$592$	0	0
$593$	610182.	1.73520	0.867601	−	0.497260i	$-0.165661\pi$
$593$	610182.	1.73520	0.867601	+	0.497260i	$0.165661\pi$
$594$	0	0
$595$	− 783298.i	− 2.21255i
$596$	0	0
$597$	−314859.	−0.883422
$598$	0	0
$599$	499941.i	1.39337i	0.717379	+	0.696683i	$0.245342\pi$
$599$	499941.i	1.39337i	−0.717379	+	0.696683i	$0.754658\pi$
$600$	0	0
$601$	486965.	1.34818	0.674091	−	0.738649i	$-0.264536\pi$
$601$	486965.	1.34818	0.674091	+	0.738649i	$0.264536\pi$
$602$	0	0
$603$	13657.0i	0.0375595i
$604$	0	0
$605$	724334.	1.97892
$606$	0	0
$607$	− 41693.4i	− 0.113159i	−0.998398	−	0.0565796i	$-0.981981\pi$
$607$	− 41693.4i	− 0.113159i	0.998398	−	0.0565796i	$-0.0180195\pi$
$608$	0	0
$609$	−193131.	−0.520737
$610$	0	0
$611$	377425.i	1.01099i
$612$	0	0
$613$	542042.	1.44249	0.721244	−	0.692681i	$-0.243571\pi$
$613$	542042.	1.44249	0.721244	+	0.692681i	$0.243571\pi$
$614$	0	0
$615$	− 446116.i	− 1.17950i
$616$	0	0
$617$	146814.	0.385653	0.192827	−	0.981233i	$-0.438235\pi$
$617$	146814.	0.385653	0.192827	+	0.981233i	$0.438235\pi$
$618$	0	0
$619$	72720.6i	0.189791i	0.995487	+	0.0948956i	$0.0302517\pi$
$619$	72720.6i	0.189791i	−0.995487	+	0.0948956i	$0.969748\pi$
$620$	0	0
$621$	−123806.	−0.321040
$622$	0	0
$623$	299000.i	0.770363i
$624$	0	0
$625$	−75495.2	−0.193268
$626$	0	0
$627$	− 629766.i	− 1.60193i
$628$	0	0
$629$	−199864.	−0.505165
$630$	0	0
$631$	− 568570.i	− 1.42799i	−0.700151	−	0.713995i	$-0.746884\pi$
$631$	− 568570.i	− 1.42799i	0.700151	−	0.713995i	$-0.253116\pi$
$632$	0	0
$633$	−143102.	−0.357139
$634$	0	0
$635$	224206.i	0.556033i
$636$	0	0
$637$	−51568.7	−0.127089
$638$	0	0
$639$	58615.5i	0.143553i
$640$	0	0
$641$	292494.	0.711870	0.355935	−	0.934511i	$-0.384162\pi$
$641$	292494.	0.711870	0.355935	+	0.934511i	$0.384162\pi$
$642$	0	0
$643$	− 103161.i	− 0.249514i	−0.992187	−	0.124757i	$-0.960185\pi$
$643$	− 103161.i	− 0.249514i	0.992187	−	0.124757i	$-0.0398151\pi$
$644$	0	0
$645$	−105167.	−0.252791
$646$	0	0
$647$	− 372569.i	− 0.890017i	−0.895526	−	0.445009i	$-0.853201\pi$
$647$	− 372569.i	− 0.890017i	0.895526	−	0.445009i	$-0.146799\pi$
$648$	0	0
$649$	−237378.	−0.563574
$650$	0	0
$651$	93611.6i	0.220886i
$652$	0	0
$653$	−85619.0	−0.200791	−0.100395	−	0.994948i	$-0.532011\pi$
$653$	−85619.0	−0.200791	−0.100395	+	0.994948i	$0.532011\pi$
$654$	0	0
$655$	1.26889e6i	2.95761i
$656$	0	0
$657$	−62045.1	−0.143740
$658$	0	0
$659$	− 258812.i	− 0.595954i	−0.954573	−	0.297977i	$-0.903688\pi$
$659$	− 258812.i	− 0.595954i	0.954573	−	0.297977i	$-0.0963119\pi$
$660$	0	0
$661$	327660.	0.749928	0.374964	−	0.927039i	$-0.377655\pi$
$661$	327660.	0.749928	0.374964	+	0.927039i	$0.377655\pi$
$662$	0	0
$663$	408084.i	0.928372i
$664$	0	0
$665$	1.22357e6	2.76686
$666$	0	0
$667$	− 712366.i	− 1.60122i
$668$	0	0
$669$	15699.4	0.0350777
$670$	0	0
$671$	145374.i	0.322880i
$672$	0	0
$673$	137121.	0.302742	0.151371	−	0.988477i	$-0.451631\pi$
$673$	137121.	0.302742	0.151371	+	0.988477i	$0.451631\pi$
$674$	0	0
$675$	133981.i	0.294059i
$676$	0	0
$677$	−573451.	−1.25118	−0.625589	−	0.780153i	$-0.715141\pi$
$677$	−573451.	−1.25118	−0.625589	+	0.780153i	$0.715141\pi$
$678$	0	0
$679$	− 377089.i	− 0.817909i
$680$	0	0
$681$	36774.6	0.0792965
$682$	0	0
$683$	− 175872.i	− 0.377012i	−0.982072	−	0.188506i	$-0.939636\pi$
$683$	− 175872.i	− 0.377012i	0.982072	−	0.188506i	$-0.0603644\pi$
$684$	0	0
$685$	−417984.	−0.890797
$686$	0	0
$687$	533962.i	1.13135i
$688$	0	0
$689$	−138307.	−0.291344
$690$	0	0
$691$	7216.66i	0.0151140i	0.999971	+	0.00755702i	$0.00240550\pi$
$691$	7216.66i	0.0151140i	−0.999971	+	0.00755702i	$0.997595\pi$
$692$	0	0
$693$	−225365.	−0.469267
$694$	0	0
$695$	595673.i	1.23321i
$696$	0	0
$697$	−924433.	−1.90287
$698$	0	0
$699$	119390.i	0.244350i
$700$	0	0
$701$	−502543.	−1.02267	−0.511337	−	0.859380i	$-0.670850\pi$
$701$	−502543.	−1.02267	−0.511337	+	0.859380i	$0.670850\pi$
$702$	0	0
$703$	− 312203.i	− 0.631723i
$704$	0	0
$705$	424823.	0.854731
$706$	0	0
$707$	717191.i	1.43481i
$708$	0	0
$709$	95591.1	0.190163	0.0950813	−	0.995470i	$-0.469689\pi$
$709$	95591.1	0.190163	0.0950813	+	0.995470i	$0.469689\pi$
$710$	0	0
$711$	289043.i	0.571772i
$712$	0	0
$713$	−345286.	−0.679204
$714$	0	0
$715$	− 1.32226e6i	− 2.58645i
$716$	0	0
$717$	184810.	0.359491
$718$	0	0
$719$	− 475789.i	− 0.920357i	−0.887826	−	0.460178i	$-0.847785\pi$
$719$	− 475789.i	− 0.920357i	0.887826	−	0.460178i	$-0.152215\pi$
$720$	0	0
$721$	305200.	0.587103
$722$	0	0
$723$	481597.i	0.921313i
$724$	0	0
$725$	−770908.	−1.46665
$726$	0	0
$727$	193262.i	0.365661i	0.983144	+	0.182830i	$0.0585259\pi$
$727$	193262.i	0.365661i	−0.983144	+	0.182830i	$0.941474\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	217926.i	0.407825i
$732$	0	0
$733$	650799.	1.21126	0.605632	−	0.795745i	$-0.292920\pi$
$733$	650799.	1.21126	0.605632	+	0.795745i	$0.292920\pi$
$734$	0	0
$735$	58044.8i	0.107446i
$736$	0	0
$737$	−91695.6	−0.168816
$738$	0	0
$739$	− 99472.6i	− 0.182144i	−0.995844	−	0.0910719i	$-0.970971\pi$
$739$	− 99472.6i	− 0.182144i	0.995844	−	0.0910719i	$-0.0290293\pi$
$740$	0	0
$741$	−637459.	−1.16096
$742$	0	0
$743$	− 715681.i	− 1.29641i	−0.761466	−	0.648205i	$-0.775520\pi$
$743$	− 715681.i	− 1.29641i	0.761466	−	0.648205i	$-0.224480\pi$
$744$	0	0
$745$	−468854.	−0.844744
$746$	0	0
$747$	− 80401.0i	− 0.144085i
$748$	0	0
$749$	543144.	0.968170
$750$	0	0
$751$	603532.i	1.07009i	0.844824	+	0.535045i	$0.179705\pi$
$751$	603532.i	1.07009i	−0.844824	+	0.535045i	$0.820295\pi$
$752$	0	0
$753$	143364.	0.252843
$754$	0	0
$755$	− 1.34324e6i	− 2.35645i
$756$	0	0
$757$	784192.	1.36846	0.684228	−	0.729268i	$-0.260139\pi$
$757$	784192.	1.36846	0.684228	+	0.729268i	$0.260139\pi$
$758$	0	0
$759$	− 831259.i	− 1.44296i
$760$	0	0
$761$	568076.	0.980927	0.490464	−	0.871462i	$-0.336827\pi$
$761$	568076.	0.980927	0.490464	+	0.871462i	$0.336827\pi$
$762$	0	0
$763$	− 611731.i	− 1.05078i
$764$	0	0
$765$	459331.	0.784880
$766$	0	0
$767$	240278.i	0.408435i
$768$	0	0
$769$	−531757.	−0.899209	−0.449605	−	0.893228i	$-0.648435\pi$
$769$	−531757.	−0.899209	−0.449605	+	0.893228i	$0.648435\pi$
$770$	0	0
$771$	− 479629.i	− 0.806856i
$772$	0	0
$773$	−260161.	−0.435394	−0.217697	−	0.976016i	$-0.569855\pi$
$773$	−260161.	−0.435394	−0.217697	+	0.976016i	$0.569855\pi$
$774$	0	0
$775$	373662.i	0.622122i
$776$	0	0
$777$	−111724.	−0.185056
$778$	0	0
$779$	− 1.44404e6i	− 2.37960i
$780$	0	0
$781$	−393556.	−0.645216
$782$	0	0
$783$	− 113254.i	− 0.184726i
$784$	0	0
$785$	−1.14433e6	−1.85700
$786$	0	0
$787$	− 724342.i	− 1.16948i	−0.811219	−	0.584742i	$-0.801196\pi$
$787$	− 724342.i	− 1.16948i	0.811219	−	0.584742i	$-0.198804\pi$
$788$	0	0
$789$	−72145.7	−0.115893
$790$	0	0
$791$	− 539332.i	− 0.861993i
$792$	0	0
$793$	147150.	0.233998
$794$	0	0
$795$	155676.i	0.246313i
$796$	0	0
$797$	−1.04249e6	−1.64117	−0.820586	−	0.571524i	$-0.806353\pi$
$797$	−1.04249e6	−1.64117	−0.820586	+	0.571524i	$0.806353\pi$
$798$	0	0
$799$	− 880309.i	− 1.37893i
$800$	0	0
$801$	−175336.	−0.273279
$802$	0	0
$803$	− 416584.i	− 0.646057i
$804$	0	0
$805$	1.61506e6	2.49227
$806$	0	0
$807$	273827.i	0.420465i
$808$	0	0
$809$	647222.	0.988909	0.494455	−	0.869204i	$-0.335368\pi$
$809$	647222.	0.988909	0.494455	+	0.869204i	$0.335368\pi$
$810$	0	0
$811$	1.24087e6i	1.88662i	0.331916	+	0.943309i	$0.392305\pi$
$811$	1.24087e6i	1.88662i	−0.331916	+	0.943309i	$0.607695\pi$
$812$	0	0
$813$	−244369.	−0.369713
$814$	0	0
$815$	− 532993.i	− 0.802428i
$816$	0	0
$817$	−340418.	−0.509997
$818$	0	0
$819$	228118.i	0.340089i
$820$	0	0
$821$	819276.	1.21547	0.607734	−	0.794141i	$-0.292079\pi$
$821$	819276.	1.21547	0.607734	+	0.794141i	$0.292079\pi$
$822$	0	0
$823$	515929.i	0.761710i	0.924635	+	0.380855i	$0.124370\pi$
$823$	515929.i	0.761710i	−0.924635	+	0.380855i	$0.875630\pi$
$824$	0	0
$825$	−899573.	−1.32169
$826$	0	0
$827$	− 140674.i	− 0.205685i	−0.994698	−	0.102842i	$-0.967206\pi$
$827$	− 140674.i	− 0.205685i	0.994698	−	0.102842i	$-0.0327938\pi$
$828$	0	0
$829$	−137443.	−0.199993	−0.0999964	−	0.994988i	$-0.531883\pi$
$829$	−137443.	−0.199993	−0.0999964	+	0.994988i	$0.531883\pi$
$830$	0	0
$831$	− 563229.i	− 0.815611i
$832$	0	0
$833$	120279.	0.173341
$834$	0	0
$835$	524955.i	0.752920i
$836$	0	0
$837$	−54894.5	−0.0783570
$838$	0	0
$839$	− 591422.i	− 0.840182i	−0.907482	−	0.420091i	$-0.861998\pi$
$839$	− 591422.i	− 0.840182i	0.907482	−	0.420091i	$-0.138002\pi$
$840$	0	0
$841$	−55633.2	−0.0786579
$842$	0	0
$843$	− 379790.i	− 0.534428i
$844$	0	0
$845$	−203138.	−0.284498
$846$	0	0
$847$	− 839029.i	− 1.16953i
$848$	0	0
$849$	232596.	0.322691
$850$	0	0
$851$	− 412093.i	− 0.569031i
$852$	0	0
$853$	−169773.	−0.233331	−0.116665	−	0.993171i	$-0.537220\pi$
$853$	−169773.	−0.233331	−0.116665	+	0.993171i	$0.537220\pi$
$854$	0	0
$855$	717512.i	0.981515i
$856$	0	0
$857$	−1.05083e6	−1.43077	−0.715384	−	0.698732i	$-0.753748\pi$
$857$	−1.05083e6	−1.43077	−0.715384	+	0.698732i	$0.753748\pi$
$858$	0	0
$859$	1.05896e6i	1.43514i	0.696487	+	0.717569i	$0.254745\pi$
$859$	1.05896e6i	1.43514i	−0.696487	+	0.717569i	$0.745255\pi$
$860$	0	0
$861$	−516757.	−0.697075
$862$	0	0
$863$	− 1.22011e6i	− 1.63824i	−0.573621	−	0.819121i	$-0.694462\pi$
$863$	− 1.22011e6i	− 1.63824i	0.573621	−	0.819121i	$-0.305538\pi$
$864$	0	0
$865$	−14506.6	−0.0193881
$866$	0	0
$867$	− 517829.i	− 0.688887i
$868$	0	0
$869$	−1.94069e6	−2.56991
$870$	0	0
$871$	92815.8i	0.122345i
$872$	0	0
$873$	221128.	0.290145
$874$	0	0
$875$	− 603927.i	− 0.788802i
$876$	0	0
$877$	707060.	0.919299	0.459650	−	0.888100i	$-0.347975\pi$
$877$	707060.	0.919299	0.459650	+	0.888100i	$0.347975\pi$
$878$	0	0
$879$	− 218395.i	− 0.282661i
$880$	0	0
$881$	430965.	0.555252	0.277626	−	0.960689i	$-0.410452\pi$
$881$	430965.	0.555252	0.277626	+	0.960689i	$0.410452\pi$
$882$	0	0
$883$	− 1.08382e6i	− 1.39007i	−0.718974	−	0.695037i	$-0.755388\pi$
$883$	− 1.08382e6i	− 1.39007i	0.718974	−	0.695037i	$-0.244612\pi$
$884$	0	0
$885$	270452.	0.345306
$886$	0	0
$887$	− 620229.i	− 0.788324i	−0.919041	−	0.394162i	$-0.871035\pi$
$887$	− 620229.i	− 0.788324i	0.919041	−	0.394162i	$-0.128965\pi$
$888$	0	0
$889$	259709.	0.328612
$890$	0	0
$891$	− 132156.i	− 0.166468i
$892$	0	0
$893$	1.37511e6	1.72439
$894$	0	0
$895$	571272.i	0.713176i
$896$	0	0
$897$	−841414.	−1.04574
$898$	0	0
$899$	− 315856.i	− 0.390814i
$900$	0	0
$901$	322589.	0.397374
$902$	0	0
$903$	121820.i	0.149398i
$904$	0	0
$905$	−161005.	−0.196581
$906$	0	0
$907$	− 155091.i	− 0.188526i	−0.995547	−	0.0942629i	$-0.969951\pi$
$907$	− 155091.i	− 0.188526i	0.995547	−	0.0942629i	$-0.0300494\pi$
$908$	0	0
$909$	−420566.	−0.508986
$910$	0	0
$911$	− 670134.i	− 0.807467i	−0.914877	−	0.403734i	$-0.867712\pi$
$911$	− 670134.i	− 0.807467i	0.914877	−	0.403734i	$-0.132288\pi$
$912$	0	0
$913$	539828.	0.647611
$914$	0	0
$915$	− 165629.i	− 0.197831i
$916$	0	0
$917$	1.46981e6	1.74793
$918$	0	0
$919$	− 631844.i	− 0.748134i	−0.927402	−	0.374067i	$-0.877963\pi$
$919$	− 631844.i	− 0.748134i	0.927402	−	0.374067i	$-0.122037\pi$
$920$	0	0
$921$	−45831.7	−0.0540314
$922$	0	0
$923$	398364.i	0.467602i
$924$	0	0
$925$	−445959.	−0.521208
$926$	0	0
$927$	178972.i	0.208269i
$928$	0	0
$929$	1.16204e6	1.34645	0.673225	−	0.739437i	$-0.264908\pi$
$929$	1.16204e6	1.34645	0.673225	+	0.739437i	$0.264908\pi$
$930$	0	0
$931$	187886.i	0.216768i
$932$	0	0
$933$	−805708.	−0.925580
$934$	0	0
$935$	3.08404e6i	3.52774i
$936$	0	0
$937$	1.18305e6	1.34748	0.673741	−	0.738967i	$-0.264686\pi$
$937$	1.18305e6	1.34748	0.673741	+	0.738967i	$0.264686\pi$
$938$	0	0
$939$	− 933570.i	− 1.05880i
$940$	0	0
$941$	159241.	0.179836	0.0899179	−	0.995949i	$-0.471340\pi$
$941$	159241.	0.179836	0.0899179	+	0.995949i	$0.471340\pi$
$942$	0	0
$943$	− 1.90606e6i	− 2.14345i
$944$	0	0
$945$	256766.	0.287523
$946$	0	0
$947$	− 940316.i	− 1.04851i	−0.851560	−	0.524257i	$-0.824343\pi$
$947$	− 940316.i	− 1.04851i	0.851560	−	0.524257i	$-0.175657\pi$
$948$	0	0
$949$	−421673.	−0.468213
$950$	0	0
$951$	32157.8i	0.0355570i
$952$	0	0
$953$	−577950.	−0.636362	−0.318181	−	0.948030i	$-0.603072\pi$
$953$	−577950.	−0.636362	−0.318181	+	0.948030i	$0.603072\pi$
$954$	0	0
$955$	− 1.98272e6i	− 2.17397i
$956$	0	0
$957$	760408.	0.830276
$958$	0	0
$959$	484170.i	0.526454i
$960$	0	0
$961$	770424.	0.834225
$962$	0	0
$963$	318504.i	0.343449i
$964$	0	0
$965$	1.92487e6	2.06703
$966$	0	0
$967$	785047.i	0.839542i	0.907630	+	0.419771i	$0.137890\pi$
$967$	785047.i	0.839542i	−0.907630	+	0.419771i	$0.862110\pi$
$968$	0	0
$969$	1.48681e6	1.58347
$970$	0	0
$971$	− 1.05236e6i	− 1.11616i	−0.829786	−	0.558082i	$-0.811538\pi$
$971$	− 1.05236e6i	− 1.11616i	0.829786	−	0.558082i	$-0.188462\pi$
$972$	0	0
$973$	689996.	0.728821
$974$	0	0
$975$	910562.i	0.957856i
$976$	0	0
$977$	20197.7	0.0211598	0.0105799	−	0.999944i	$-0.496632\pi$
$977$	20197.7	0.0211598	0.0105799	+	0.999944i	$0.496632\pi$
$978$	0	0
$979$	− 1.17724e6i	− 1.22829i
$980$	0	0
$981$	358723.	0.372754
$982$	0	0
$983$	1.43076e6i	1.48068i	0.672234	+	0.740338i	$0.265335\pi$
$983$	1.43076e6i	1.48068i	−0.672234	+	0.740338i	$0.734665\pi$
$984$	0	0
$985$	−220875.	−0.227653
$986$	0	0
$987$	− 492092.i	− 0.505140i
$988$	0	0
$989$	−449334.	−0.459385
$990$	0	0
$991$	787866.i	0.802241i	0.916025	+	0.401120i	$0.131379\pi$
$991$	787866.i	0.802241i	−0.916025	+	0.401120i	$0.868621\pi$
$992$	0	0
$993$	−686718.	−0.696434
$994$	0	0
$995$	2.40858e6i	2.43285i
$996$	0	0
$997$	−453709.	−0.456444	−0.228222	−	0.973609i	$-0.573291\pi$
$997$	−453709.	−0.456444	−0.228222	+	0.973609i	$0.573291\pi$
$998$	0	0
$999$	− 65515.5i	− 0.0656468i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.g.c.511.1		4
4.3	odd	2	inner	768.5.g.c.511.3		4
8.3	odd	2		768.5.g.g.511.2		4
8.5	even	2		768.5.g.g.511.4		4
16.3	odd	4		384.5.b.c.319.4	yes	8
16.5	even	4		384.5.b.c.319.1	✓	8
16.11	odd	4		384.5.b.c.319.5	yes	8
16.13	even	4		384.5.b.c.319.8	yes	8
48.5	odd	4		1152.5.b.k.703.7		8
48.11	even	4		1152.5.b.k.703.8		8
48.29	odd	4		1152.5.b.k.703.1		8
48.35	even	4		1152.5.b.k.703.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.c.319.1	✓	8	16.5	even	4
384.5.b.c.319.4	yes	8	16.3	odd	4
384.5.b.c.319.5	yes	8	16.11	odd	4
384.5.b.c.319.8	yes	8	16.13	even	4
768.5.g.c.511.1		4	1.1	even	1	trivial
768.5.g.c.511.3		4	4.3	odd	2	inner
768.5.g.g.511.2		4	8.3	odd	2
768.5.g.g.511.4		4	8.5	even	2
1152.5.b.k.703.1		8	48.29	odd	4
1152.5.b.k.703.2		8	48.35	even	4
1152.5.b.k.703.7		8	48.5	odd	4
1152.5.b.k.703.8		8	48.11	even	4

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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -39.7490 q^{5} +46.0431i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} -39.7490 q^{5} +46.0431i q^{7} -27.0000 q^{9} -181.283i q^{11} -183.498 q^{13} +206.542i q^{15} +427.992 q^{17} +668.558i q^{19} +239.247 q^{21} +882.463i q^{23} +954.984 q^{25} +140.296i q^{27} -807.247 q^{29} +391.276i q^{31} -941.976 q^{33} -1830.17i q^{35} -466.980 q^{37} +953.484i q^{39} -2159.93 q^{41} +509.182i q^{43} +1073.22 q^{45} -2056.83i q^{47} +281.031 q^{49} -2223.91i q^{51} +753.725 q^{53} +7205.84i q^{55} +3473.93 q^{57} -1309.43i q^{59} -801.913 q^{61} -1243.16i q^{63} +7293.87 q^{65} -505.813i q^{67} +4585.41 q^{69} -2170.94i q^{71} +2297.97 q^{73} -4962.24i q^{75} +8346.85 q^{77} -10705.3i q^{79} +729.000 q^{81} +2977.81i q^{83} -17012.3 q^{85} +4194.58i q^{87} +6493.92 q^{89} -8448.82i q^{91} +2033.13 q^{93} -26574.5i q^{95} -8189.92 q^{97} +4894.65i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{5} - 108 q^{9}+O(q^{10}) \) 4 * q - 32 * q^5 - 108 * q^9	\( 4 q - 32 q^{5} - 108 q^{9} - 480 q^{13} + 696 q^{17} + 576 q^{21} + 1788 q^{25} - 2848 q^{29} - 720 q^{33} + 672 q^{37} + 504 q^{41} + 864 q^{45} + 5188 q^{49} - 160 q^{53} + 4752 q^{57} + 7968 q^{61} + 11904 q^{65} + 6912 q^{69} + 5128 q^{73} + 14592 q^{77} + 2916 q^{81} - 37824 q^{85} + 15816 q^{89} - 7488 q^{93} - 22600 q^{97}+O(q^{100}) \) 4 * q - 32 * q^5 - 108 * q^9 - 480 * q^13 + 696 * q^17 + 576 * q^21 + 1788 * q^25 - 2848 * q^29 - 720 * q^33 + 672 * q^37 + 504 * q^41 + 864 * q^45 + 5188 * q^49 - 160 * q^53 + 4752 * q^57 + 7968 * q^61 + 11904 * q^65 + 6912 * q^69 + 5128 * q^73 + 14592 * q^77 + 2916 * q^81 - 37824 * q^85 + 15816 * q^89 - 7488 * q^93 - 22600 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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