LMFDB - Embedded newform 784.5.c.e.97.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,5,Mod(97,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	784.c (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:	$81.0420510577$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.11337408.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 18x^{4} + 81x^{2} + 12 $ x^6 + 18x^4 + 81x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{3}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.2
Root			$-3.17656i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.5.c.e.97.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.01997i q^{3} -31.0914i q^{5} -0.359775 q^{9} +145.721 q^{11} +209.930i q^{13} -280.444 q^{15} +187.147i q^{17} -177.283i q^{19} -333.685 q^{23} -341.677 q^{25} -727.372i q^{27} -1387.57 q^{29} -1553.93i q^{31} -1314.40i q^{33} -2277.98 q^{37} +1893.56 q^{39} -781.977i q^{41} -837.131 q^{43} +11.1859i q^{45} +1496.91i q^{47} +1688.06 q^{51} -4467.10 q^{53} -4530.67i q^{55} -1599.09 q^{57} -4405.75i q^{59} +2607.02i q^{61} +6527.01 q^{65} -5774.31 q^{67} +3009.82i q^{69} -1149.20 q^{71} +3688.65i q^{73} +3081.91i q^{75} -2698.32 q^{79} -6590.01 q^{81} +2689.70i q^{83} +5818.67 q^{85} +12515.9i q^{87} +5816.93i q^{89} -14016.4 q^{93} -5511.98 q^{95} -7890.42i q^{97} -52.4268 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 180 q^{9} + 270 q^{11} + 486 q^{15} + 486 q^{23} - 3756 q^{25} - 540 q^{29} - 4710 q^{37} + 13176 q^{39} + 948 q^{43} - 1782 q^{51} - 12582 q^{53} - 6894 q^{57} - 15336 q^{65} + 3318 q^{67} - 2268 q^{71}+ \cdots - 8100 q^{99}+O(q^{100}) $ 6 * q - 180 * q^9 + 270 * q^11 + 486 * q^15 + 486 * q^23 - 3756 * q^25 - 540 * q^29 - 4710 * q^37 + 13176 * q^39 + 948 * q^43 - 1782 * q^51 - 12582 * q^53 - 6894 * q^57 - 15336 * q^65 + 3318 * q^67 - 2268 * q^71 - 15546 * q^79 + 14958 * q^81 - 702 * q^85 - 2898 * q^93 - 41310 * q^95 - 8100 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 9.01997i	− 1.00222i	−0.865384	−	0.501109i	$-0.832925\pi$
$3$	− 9.01997i	− 1.00222i	0.865384	−	0.501109i	$-0.167075\pi$
$4$	0	0
$5$	− 31.0914i	− 1.24366i	−0.783153	−	0.621829i	$-0.786390\pi$
$5$	− 31.0914i	− 1.24366i	0.783153	−	0.621829i	$-0.213610\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.359775	−0.00444167
$10$	0	0
$11$	145.721	1.20431	0.602153	−	0.798381i	$-0.294310\pi$
$11$	145.721	1.20431	0.602153	+	0.798381i	$0.294310\pi$
$12$	0	0
$13$	209.930i	1.24219i	0.783736	+	0.621094i	$0.213311\pi$
$13$	209.930i	1.24219i	−0.783736	+	0.621094i	$0.786689\pi$
$14$	0	0
$15$	−280.444	−1.24642
$16$	0	0
$17$	187.147i	0.647567i	0.946131	+	0.323784i	$0.104955\pi$
$17$	187.147i	0.647567i	−0.946131	+	0.323784i	$0.895045\pi$
$18$	0	0
$19$	− 177.283i	− 0.491088i	−0.969385	−	0.245544i	$-0.921033\pi$
$19$	− 177.283i	− 0.491088i	0.969385	−	0.245544i	$-0.0789666\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−333.685	−0.630784	−0.315392	−	0.948962i	$-0.602136\pi$
$23$	−333.685	−0.630784	−0.315392	+	0.948962i	$0.602136\pi$
$24$	0	0
$25$	−341.677	−0.546683
$26$	0	0
$27$	− 727.372i	− 0.997767i
$28$	0	0
$29$	−1387.57	−1.64991	−0.824954	−	0.565199i	$-0.808799\pi$
$29$	−1387.57	−1.64991	−0.824954	+	0.565199i	$0.808799\pi$
$30$	0	0
$31$	− 1553.93i	− 1.61699i	−0.588500	−	0.808497i	$-0.700281\pi$
$31$	− 1553.93i	− 1.61699i	0.588500	−	0.808497i	$-0.299719\pi$
$32$	0	0
$33$	− 1314.40i	− 1.20698i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2277.98	−1.66397	−0.831985	−	0.554798i	$-0.812796\pi$
$37$	−2277.98	−1.66397	−0.831985	+	0.554798i	$0.812796\pi$
$38$	0	0
$39$	1893.56	1.24494
$40$	0	0
$41$	− 781.977i	− 0.465185i	−0.972574	−	0.232593i	$-0.925279\pi$
$41$	− 781.977i	− 0.465185i	0.972574	−	0.232593i	$-0.0747209\pi$
$42$	0	0
$43$	−837.131	−0.452748	−0.226374	−	0.974040i	$-0.572687\pi$
$43$	−837.131	−0.452748	−0.226374	+	0.974040i	$0.572687\pi$
$44$	0	0
$45$	11.1859i	0.00552391i
$46$	0	0
$47$	1496.91i	0.677641i	0.940851	+	0.338820i	$0.110028\pi$
$47$	1496.91i	0.677641i	−0.940851	+	0.338820i	$0.889972\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1688.06	0.649004
$52$	0	0
$53$	−4467.10	−1.59028	−0.795141	−	0.606424i	$-0.792603\pi$
$53$	−4467.10	−1.59028	−0.795141	+	0.606424i	$0.792603\pi$
$54$	0	0
$55$	− 4530.67i	− 1.49774i
$56$	0	0
$57$	−1599.09	−0.492178
$58$	0	0
$59$	− 4405.75i	− 1.26566i	−0.774292	−	0.632829i	$-0.781894\pi$
$59$	− 4405.75i	− 1.26566i	0.774292	−	0.632829i	$-0.218106\pi$
$60$	0	0
$61$	2607.02i	0.700624i	0.936633	+	0.350312i	$0.113924\pi$
$61$	2607.02i	0.700624i	−0.936633	+	0.350312i	$0.886076\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6527.01	1.54485
$66$	0	0
$67$	−5774.31	−1.28632	−0.643162	−	0.765730i	$-0.722378\pi$
$67$	−5774.31	−1.28632	−0.643162	+	0.765730i	$0.722378\pi$
$68$	0	0
$69$	3009.82i	0.632183i
$70$	0	0
$71$	−1149.20	−0.227971	−0.113986	−	0.993482i	$-0.536362\pi$
$71$	−1149.20	−0.227971	−0.113986	+	0.993482i	$0.536362\pi$
$72$	0	0
$73$	3688.65i	0.692185i	0.938200	+	0.346093i	$0.112492\pi$
$73$	3688.65i	0.692185i	−0.938200	+	0.346093i	$0.887508\pi$
$74$	0	0
$75$	3081.91i	0.547896i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2698.32	−0.432353	−0.216177	−	0.976354i	$-0.569359\pi$
$79$	−2698.32	−0.432353	−0.216177	+	0.976354i	$0.569359\pi$
$80$	0	0
$81$	−6590.01	−1.00442
$82$	0	0
$83$	2689.70i	0.390433i	0.980760	+	0.195217i	$0.0625410\pi$
$83$	2689.70i	0.390433i	−0.980760	+	0.195217i	$0.937459\pi$
$84$	0	0
$85$	5818.67	0.805352
$86$	0	0
$87$	12515.9i	1.65357i
$88$	0	0
$89$	5816.93i	0.734369i	0.930148	+	0.367184i	$0.119678\pi$
$89$	5816.93i	0.734369i	−0.930148	+	0.367184i	$0.880322\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14016.4	−1.62058
$94$	0	0
$95$	−5511.98	−0.610746
$96$	0	0
$97$	− 7890.42i	− 0.838604i	−0.907847	−	0.419302i	$-0.862275\pi$
$97$	− 7890.42i	− 0.838604i	0.907847	−	0.419302i	$-0.137725\pi$
$98$	0	0
$99$	−52.4268	−0.00534913
$100$	0	0
$101$	− 7921.56i	− 0.776548i	−0.921544	−	0.388274i	$-0.873071\pi$
$101$	− 7921.56i	− 0.776548i	0.921544	−	0.388274i	$-0.126929\pi$
$102$	0	0
$103$	1708.62i	0.161054i	0.996752	+	0.0805271i	$0.0256603\pi$
$103$	1708.62i	0.161054i	−0.996752	+	0.0805271i	$0.974340\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5349.92	−0.467283	−0.233641	−	0.972323i	$-0.575064\pi$
$107$	−5349.92	−0.467283	−0.233641	+	0.972323i	$0.575064\pi$
$108$	0	0
$109$	21025.8	1.76970	0.884851	−	0.465875i	$-0.154260\pi$
$109$	21025.8	1.76970	0.884851	+	0.465875i	$0.154260\pi$
$110$	0	0
$111$	20547.3i	1.66766i
$112$	0	0
$113$	−9886.76	−0.774278	−0.387139	−	0.922021i	$-0.626537\pi$
$113$	−9886.76	−0.774278	−0.387139	+	0.922021i	$0.626537\pi$
$114$	0	0
$115$	10374.7i	0.784478i
$116$	0	0
$117$	− 75.5275i	− 0.00551738i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6593.62	0.450353
$122$	0	0
$123$	−7053.40	−0.466217
$124$	0	0
$125$	− 8808.92i	− 0.563771i
$126$	0	0
$127$	12148.0	0.753178	0.376589	−	0.926380i	$-0.377097\pi$
$127$	12148.0	0.753178	0.376589	+	0.926380i	$0.377097\pi$
$128$	0	0
$129$	7550.90i	0.453753i
$130$	0	0
$131$	− 5623.28i	− 0.327678i	−0.986487	−	0.163839i	$-0.947612\pi$
$131$	− 5623.28i	− 0.327678i	0.986487	−	0.163839i	$-0.0523877\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−22615.0	−1.24088
$136$	0	0
$137$	29795.7	1.58750	0.793748	−	0.608247i	$-0.208127\pi$
$137$	29795.7	1.58750	0.793748	+	0.608247i	$0.208127\pi$
$138$	0	0
$139$	− 15694.4i	− 0.812298i	−0.913807	−	0.406149i	$-0.866871\pi$
$139$	− 15694.4i	− 0.812298i	0.913807	−	0.406149i	$-0.133129\pi$
$140$	0	0
$141$	13502.1	0.679144
$142$	0	0
$143$	30591.2i	1.49597i
$144$	0	0
$145$	43141.6i	2.05192i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−27456.8	−1.23674	−0.618369	−	0.785888i	$-0.712206\pi$
$149$	−27456.8	−1.23674	−0.618369	+	0.785888i	$0.712206\pi$
$150$	0	0
$151$	29942.1	1.31319	0.656596	−	0.754243i	$-0.271996\pi$
$151$	29942.1	1.31319	0.656596	+	0.754243i	$0.271996\pi$
$152$	0	0
$153$	− 67.3308i	− 0.00287628i
$154$	0	0
$155$	−48314.0	−2.01099
$156$	0	0
$157$	15490.8i	0.628455i	0.949348	+	0.314228i	$0.101746\pi$
$157$	15490.8i	0.628455i	−0.949348	+	0.314228i	$0.898254\pi$
$158$	0	0
$159$	40293.1i	1.59381i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−27338.5	−1.02896	−0.514482	−	0.857501i	$-0.672016\pi$
$163$	−27338.5	−1.02896	−0.514482	+	0.857501i	$0.672016\pi$
$164$	0	0
$165$	−40866.5	−1.50107
$166$	0	0
$167$	− 19753.9i	− 0.708306i	−0.935188	−	0.354153i	$-0.884769\pi$
$167$	− 19753.9i	− 0.708306i	0.935188	−	0.354153i	$-0.115231\pi$
$168$	0	0
$169$	−15509.5	−0.543029
$170$	0	0
$171$	63.7820i	0.00218125i
$172$	0	0
$173$	− 22946.9i	− 0.766711i	−0.923601	−	0.383355i	$-0.874768\pi$
$173$	− 22946.9i	− 0.766711i	0.923601	−	0.383355i	$-0.125232\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−39739.8	−1.26847
$178$	0	0
$179$	−14079.1	−0.439410	−0.219705	−	0.975566i	$-0.570509\pi$
$179$	−14079.1	−0.439410	−0.219705	+	0.975566i	$0.570509\pi$
$180$	0	0
$181$	− 12131.5i	− 0.370304i	−0.982710	−	0.185152i	$-0.940722\pi$
$181$	− 12131.5i	− 0.370304i	0.982710	−	0.185152i	$-0.0592778\pi$
$182$	0	0
$183$	23515.2	0.702178
$184$	0	0
$185$	70825.5i	2.06941i
$186$	0	0
$187$	27271.3i	0.779869i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−30005.3	−0.822490	−0.411245	−	0.911525i	$-0.634906\pi$
$191$	−30005.3	−0.822490	−0.411245	+	0.911525i	$0.634906\pi$
$192$	0	0
$193$	−18704.7	−0.502153	−0.251077	−	0.967967i	$-0.580785\pi$
$193$	−18704.7	−0.502153	−0.251077	+	0.967967i	$0.580785\pi$
$194$	0	0
$195$	− 58873.4i	− 1.54828i
$196$	0	0
$197$	−10271.9	−0.264678	−0.132339	−	0.991205i	$-0.542249\pi$
$197$	−10271.9	−0.264678	−0.132339	+	0.991205i	$0.542249\pi$
$198$	0	0
$199$	7188.50i	0.181523i	0.995873	+	0.0907616i	$0.0289301\pi$
$199$	7188.50i	0.181523i	−0.995873	+	0.0907616i	$0.971070\pi$
$200$	0	0
$201$	52084.1i	1.28918i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−24312.8	−0.578531
$206$	0	0
$207$	120.051	0.00280173
$208$	0	0
$209$	− 25833.8i	− 0.591421i
$210$	0	0
$211$	62160.8	1.39621	0.698107	−	0.715993i	$-0.254026\pi$
$211$	62160.8	1.39621	0.698107	+	0.715993i	$0.254026\pi$
$212$	0	0
$213$	10365.8i	0.228477i
$214$	0	0
$215$	26027.6i	0.563063i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	33271.5	0.693721
$220$	0	0
$221$	−39287.7	−0.804400
$222$	0	0
$223$	− 7060.74i	− 0.141984i	−0.997477	−	0.0709921i	$-0.977383\pi$
$223$	− 7060.74i	− 0.141984i	0.997477	−	0.0709921i	$-0.0226165\pi$
$224$	0	0
$225$	122.927	0.00242818
$226$	0	0
$227$	− 39109.4i	− 0.758978i	−0.925196	−	0.379489i	$-0.876100\pi$
$227$	− 39109.4i	− 0.758978i	0.925196	−	0.379489i	$-0.123900\pi$
$228$	0	0
$229$	362.549i	0.00691346i	0.999994	+	0.00345673i	$0.00110031\pi$
$229$	362.549i	0.00691346i	−0.999994	+	0.00345673i	$0.998900\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6876.32	0.126661	0.0633307	−	0.997993i	$-0.479828\pi$
$233$	6876.32	0.126661	0.0633307	+	0.997993i	$0.479828\pi$
$234$	0	0
$235$	46541.0	0.842752
$236$	0	0
$237$	24338.7i	0.433313i
$238$	0	0
$239$	−90119.7	−1.57770	−0.788849	−	0.614587i	$-0.789323\pi$
$239$	−90119.7	−1.57770	−0.788849	+	0.614587i	$0.789323\pi$
$240$	0	0
$241$	− 43376.1i	− 0.746822i	−0.927666	−	0.373411i	$-0.878188\pi$
$241$	− 43376.1i	− 0.746822i	0.927666	−	0.373411i	$-0.121812\pi$
$242$	0	0
$243$	524.548i	0.00888327i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	37216.9	0.610024
$248$	0	0
$249$	24261.0	0.391300
$250$	0	0
$251$	57720.4i	0.916183i	0.888905	+	0.458091i	$0.151467\pi$
$251$	57720.4i	0.916183i	−0.888905	+	0.458091i	$0.848533\pi$
$252$	0	0
$253$	−48624.9	−0.759657
$254$	0	0
$255$	− 52484.2i	− 0.807138i
$256$	0	0
$257$	− 67795.2i	− 1.02644i	−0.858258	−	0.513219i	$-0.828453\pi$
$257$	− 67795.2i	− 1.02644i	0.858258	−	0.513219i	$-0.171547\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	499.214	0.00732835
$262$	0	0
$263$	6767.47	0.0978397	0.0489198	−	0.998803i	$-0.484422\pi$
$263$	6767.47	0.0978397	0.0489198	+	0.998803i	$0.484422\pi$
$264$	0	0
$265$	138889.i	1.97777i
$266$	0	0
$267$	52468.5	0.735998
$268$	0	0
$269$	84044.2i	1.16146i	0.814097	+	0.580728i	$0.197232\pi$
$269$	84044.2i	1.16146i	−0.814097	+	0.580728i	$0.802768\pi$
$270$	0	0
$271$	38957.2i	0.530455i	0.964186	+	0.265228i	$0.0854471\pi$
$271$	38957.2i	0.530455i	−0.964186	+	0.265228i	$0.914553\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−49789.5	−0.658373
$276$	0	0
$277$	−71382.3	−0.930317	−0.465158	−	0.885227i	$-0.654003\pi$
$277$	−71382.3	−0.930317	−0.465158	+	0.885227i	$0.654003\pi$
$278$	0	0
$279$	559.066i	0.00718216i
$280$	0	0
$281$	57835.9	0.732462	0.366231	−	0.930524i	$-0.380648\pi$
$281$	57835.9	0.732462	0.366231	+	0.930524i	$0.380648\pi$
$282$	0	0
$283$	100996.i	1.26105i	0.776168	+	0.630526i	$0.217161\pi$
$283$	100996.i	1.26105i	−0.776168	+	0.630526i	$0.782839\pi$
$284$	0	0
$285$	49717.9i	0.612100i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	48497.0	0.580656
$290$	0	0
$291$	−71171.4	−0.840464
$292$	0	0
$293$	− 133132.i	− 1.55077i	−0.631487	−	0.775387i	$-0.717555\pi$
$293$	− 133132.i	− 1.55077i	0.631487	−	0.775387i	$-0.282445\pi$
$294$	0	0
$295$	−136981.	−1.57404
$296$	0	0
$297$	− 105993.i	− 1.20162i
$298$	0	0
$299$	− 70050.3i	− 0.783551i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−71452.2	−0.778271
$304$	0	0
$305$	81056.0	0.871335
$306$	0	0
$307$	− 7667.32i	− 0.0813517i	−0.999172	−	0.0406758i	$-0.987049\pi$
$307$	− 7667.32i	− 0.0813517i	0.999172	−	0.0406758i	$-0.0129511\pi$
$308$	0	0
$309$	15411.7	0.161411
$310$	0	0
$311$	− 1645.22i	− 0.0170099i	−0.999964	−	0.00850497i	$-0.997293\pi$
$311$	− 1645.22i	− 0.0170099i	0.999964	−	0.00850497i	$-0.00270725\pi$
$312$	0	0
$313$	112092.i	1.14416i	0.820198	+	0.572079i	$0.193863\pi$
$313$	112092.i	1.14416i	−0.820198	+	0.572079i	$0.806137\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3362.44	0.0334607	0.0167304	−	0.999860i	$-0.494674\pi$
$317$	3362.44	0.0334607	0.0167304	+	0.999860i	$0.494674\pi$
$318$	0	0
$319$	−202199.	−1.98700
$320$	0	0
$321$	48256.1i	0.468319i
$322$	0	0
$323$	33178.0	0.318013
$324$	0	0
$325$	− 71728.1i	− 0.679082i
$326$	0	0
$327$	− 189652.i	− 1.77363i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	86969.9	0.793803	0.396902	−	0.917861i	$-0.370085\pi$
$331$	86969.9	0.793803	0.396902	+	0.917861i	$0.370085\pi$
$332$	0	0
$333$	819.559	0.00739081
$334$	0	0
$335$	179531.i	1.59975i
$336$	0	0
$337$	−112228.	−0.988195	−0.494098	−	0.869406i	$-0.664502\pi$
$337$	−112228.	−0.988195	−0.494098	+	0.869406i	$0.664502\pi$
$338$	0	0
$339$	89178.2i	0.775996i
$340$	0	0
$341$	− 226441.i	− 1.94736i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	93579.7	0.786219
$346$	0	0
$347$	34291.4	0.284791	0.142396	−	0.989810i	$-0.454519\pi$
$347$	34291.4	0.284791	0.142396	+	0.989810i	$0.454519\pi$
$348$	0	0
$349$	− 133358.i	− 1.09488i	−0.836844	−	0.547441i	$-0.815602\pi$
$349$	− 133358.i	− 1.09488i	0.836844	−	0.547441i	$-0.184398\pi$
$350$	0	0
$351$	152697.	1.23941
$352$	0	0
$353$	− 152024.i	− 1.22001i	−0.792398	−	0.610005i	$-0.791167\pi$
$353$	− 152024.i	− 1.22001i	0.792398	−	0.610005i	$-0.208833\pi$
$354$	0	0
$355$	35730.4i	0.283518i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	81320.9	0.630977	0.315488	−	0.948929i	$-0.397832\pi$
$359$	81320.9	0.630977	0.315488	+	0.948929i	$0.397832\pi$
$360$	0	0
$361$	98891.8	0.758832
$362$	0	0
$363$	− 59474.2i	− 0.451352i
$364$	0	0
$365$	114686.	0.860841
$366$	0	0
$367$	− 202671.i	− 1.50474i	−0.658743	−	0.752368i	$-0.728912\pi$
$367$	− 202671.i	− 1.50474i	0.658743	−	0.752368i	$-0.271088\pi$
$368$	0	0
$369$	281.336i	0.00206620i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−76742.3	−0.551591	−0.275795	−	0.961216i	$-0.588941\pi$
$373$	−76742.3	−0.551591	−0.275795	+	0.961216i	$0.588941\pi$
$374$	0	0
$375$	−79456.2	−0.565022
$376$	0	0
$377$	− 291293.i	− 2.04950i
$378$	0	0
$379$	−13185.2	−0.0917926	−0.0458963	−	0.998946i	$-0.514614\pi$
$379$	−13185.2	−0.0917926	−0.0458963	+	0.998946i	$0.514614\pi$
$380$	0	0
$381$	− 109575.i	− 0.754849i
$382$	0	0
$383$	263893.i	1.79900i	0.436922	+	0.899499i	$0.356068\pi$
$383$	263893.i	1.79900i	−0.436922	+	0.899499i	$0.643932\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	301.179	0.00201096
$388$	0	0
$389$	101541.	0.671030	0.335515	−	0.942035i	$-0.391090\pi$
$389$	101541.	0.671030	0.335515	+	0.942035i	$0.391090\pi$
$390$	0	0
$391$	− 62448.1i	− 0.408475i
$392$	0	0
$393$	−50721.7	−0.328405
$394$	0	0
$395$	83894.5i	0.537699i
$396$	0	0
$397$	215443.i	1.36695i	0.729976	+	0.683473i	$0.239531\pi$
$397$	215443.i	1.36695i	−0.729976	+	0.683473i	$0.760469\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3251.28	0.0202193	0.0101096	−	0.999949i	$-0.496782\pi$
$401$	3251.28	0.0202193	0.0101096	+	0.999949i	$0.496782\pi$
$402$	0	0
$403$	326216.	2.00861
$404$	0	0
$405$	204893.i	1.24916i
$406$	0	0
$407$	−331949.	−2.00393
$408$	0	0
$409$	148944.i	0.890383i	0.895435	+	0.445191i	$0.146864\pi$
$409$	148944.i	0.890383i	−0.895435	+	0.445191i	$0.853136\pi$
$410$	0	0
$411$	− 268756.i	− 1.59102i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	83626.5	0.485565
$416$	0	0
$417$	−141563.	−0.814100
$418$	0	0
$419$	− 286696.i	− 1.63303i	−0.577324	−	0.816515i	$-0.695903\pi$
$419$	− 286696.i	− 1.63303i	0.577324	−	0.816515i	$-0.304097\pi$
$420$	0	0
$421$	−5357.63	−0.0302280	−0.0151140	−	0.999886i	$-0.504811\pi$
$421$	−5357.63	−0.0302280	−0.0151140	+	0.999886i	$0.504811\pi$
$422$	0	0
$423$	− 538.550i	− 0.00300985i
$424$	0	0
$425$	− 63943.8i	− 0.354014i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	275931.	1.49929
$430$	0	0
$431$	54289.1	0.292253	0.146126	−	0.989266i	$-0.453319\pi$
$431$	54289.1	0.292253	0.146126	+	0.989266i	$0.453319\pi$
$432$	0	0
$433$	206336.i	1.10052i	0.834992	+	0.550262i	$0.185472\pi$
$433$	206336.i	1.10052i	−0.834992	+	0.550262i	$0.814528\pi$
$434$	0	0
$435$	389136.	2.05647
$436$	0	0
$437$	59156.6i	0.309771i
$438$	0	0
$439$	− 73627.5i	− 0.382042i	−0.981586	−	0.191021i	$-0.938820\pi$
$439$	− 73627.5i	− 0.382042i	0.981586	−	0.191021i	$-0.0611799\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−144633.	−0.736987	−0.368493	−	0.929630i	$-0.620126\pi$
$443$	−144633.	−0.736987	−0.368493	+	0.929630i	$0.620126\pi$
$444$	0	0
$445$	180857.	0.913303
$446$	0	0
$447$	247660.i	1.23948i
$448$	0	0
$449$	305112.	1.51345	0.756723	−	0.653735i	$-0.226799\pi$
$449$	305112.	1.51345	0.756723	+	0.653735i	$0.226799\pi$
$450$	0	0
$451$	− 113950.i	− 0.560226i
$452$	0	0
$453$	− 270076.i	− 1.31610i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35774.6	0.171294	0.0856470	−	0.996326i	$-0.472704\pi$
$457$	35774.6	0.171294	0.0856470	+	0.996326i	$0.472704\pi$
$458$	0	0
$459$	136125.	0.646121
$460$	0	0
$461$	332913.i	1.56649i	0.621710	+	0.783247i	$0.286438\pi$
$461$	332913.i	1.56649i	−0.621710	+	0.783247i	$0.713562\pi$
$462$	0	0
$463$	45033.5	0.210075	0.105037	−	0.994468i	$-0.466504\pi$
$463$	45033.5	0.210075	0.105037	+	0.994468i	$0.466504\pi$
$464$	0	0
$465$	435790.i	2.01545i
$466$	0	0
$467$	− 108830.i	− 0.499015i	−0.968373	−	0.249508i	$-0.919731\pi$
$467$	− 108830.i	− 0.499015i	0.968373	−	0.249508i	$-0.0802688\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	139726.	0.629849
$472$	0	0
$473$	−121988.	−0.545247
$474$	0	0
$475$	60573.4i	0.268470i
$476$	0	0
$477$	1607.15	0.00706351
$478$	0	0
$479$	− 381929.i	− 1.66461i	−0.554320	−	0.832304i	$-0.687022\pi$
$479$	− 381929.i	− 1.66461i	0.554320	−	0.832304i	$-0.312978\pi$
$480$	0	0
$481$	− 478215.i	− 2.06696i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−245325.	−1.04294
$486$	0	0
$487$	270257.	1.13951	0.569756	−	0.821814i	$-0.307038\pi$
$487$	270257.	1.13951	0.569756	+	0.821814i	$0.307038\pi$
$488$	0	0
$489$	246593.i	1.03125i
$490$	0	0
$491$	240609.	0.998040	0.499020	−	0.866590i	$-0.333693\pi$
$491$	240609.	0.998040	0.499020	+	0.866590i	$0.333693\pi$
$492$	0	0
$493$	− 259680.i	− 1.06843i
$494$	0	0
$495$	1630.02i	0.00665248i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	186629.	0.749511	0.374756	−	0.927124i	$-0.377727\pi$
$499$	186629.	0.749511	0.374756	+	0.927124i	$0.377727\pi$
$500$	0	0
$501$	−178180.	−0.709877
$502$	0	0
$503$	349740.i	1.38232i	0.722700	+	0.691162i	$0.242901\pi$
$503$	349740.i	1.38232i	−0.722700	+	0.691162i	$0.757099\pi$
$504$	0	0
$505$	−246293.	−0.965759
$506$	0	0
$507$	139895.i	0.544234i
$508$	0	0
$509$	− 126445.i	− 0.488051i	−0.969769	−	0.244025i	$-0.921532\pi$
$509$	− 126445.i	− 0.488051i	0.969769	−	0.244025i	$-0.0784681\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−128951.	−0.489992
$514$	0	0
$515$	53123.5	0.200296
$516$	0	0
$517$	218131.i	0.816087i
$518$	0	0
$519$	−206980.	−0.768411
$520$	0	0
$521$	− 147502.i	− 0.543403i	−0.962382	−	0.271702i	$-0.912414\pi$
$521$	− 147502.i	− 0.543403i	0.962382	−	0.271702i	$-0.0875864\pi$
$522$	0	0
$523$	− 256600.i	− 0.938109i	−0.883169	−	0.469054i	$-0.844595\pi$
$523$	− 256600.i	− 0.938109i	0.883169	−	0.469054i	$-0.155405\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	290814.	1.04711
$528$	0	0
$529$	−168496.	−0.602112
$530$	0	0
$531$	1585.08i	0.00562163i
$532$	0	0
$533$	164160.	0.577847
$534$	0	0
$535$	166337.i	0.581139i
$536$	0	0
$537$	126993.i	0.440385i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	396687.	1.35536	0.677678	−	0.735359i	$-0.262986\pi$
$541$	396687.	1.35536	0.677678	+	0.735359i	$0.262986\pi$
$542$	0	0
$543$	−109426.	−0.371126
$544$	0	0
$545$	− 653723.i	− 2.20090i
$546$	0	0
$547$	266914.	0.892065	0.446032	−	0.895017i	$-0.352837\pi$
$547$	266914.	0.892065	0.446032	+	0.895017i	$0.352837\pi$
$548$	0	0
$549$	− 937.941i	− 0.00311194i
$550$	0	0
$551$	245993.i	0.810251i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	638844.	2.07400
$556$	0	0
$557$	411278.	1.32564	0.662819	−	0.748779i	$-0.269360\pi$
$557$	411278.	1.32564	0.662819	+	0.748779i	$0.269360\pi$
$558$	0	0
$559$	− 175739.i	− 0.562398i
$560$	0	0
$561$	245986.	0.781599
$562$	0	0
$563$	− 566949.i	− 1.78866i	−0.447410	−	0.894329i	$-0.647654\pi$
$563$	− 566949.i	− 1.78866i	0.447410	−	0.894329i	$-0.352346\pi$
$564$	0	0
$565$	307394.i	0.962937i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	597313.	1.84492	0.922459	−	0.386094i	$-0.126176\pi$
$569$	597313.	1.84492	0.922459	+	0.386094i	$0.126176\pi$
$570$	0	0
$571$	−248212.	−0.761290	−0.380645	−	0.924721i	$-0.624298\pi$
$571$	−248212.	−0.761290	−0.380645	+	0.924721i	$0.624298\pi$
$572$	0	0
$573$	270647.i	0.824315i
$574$	0	0
$575$	114012.	0.344839
$576$	0	0
$577$	− 61117.3i	− 0.183575i	−0.995779	−	0.0917873i	$-0.970742\pi$
$577$	− 61117.3i	− 0.183575i	0.995779	−	0.0917873i	$-0.0292580\pi$
$578$	0	0
$579$	168716.i	0.503267i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−650951.	−1.91519
$584$	0	0
$585$	−2348.26	−0.00686173
$586$	0	0
$587$	− 199552.i	− 0.579136i	−0.957157	−	0.289568i	$-0.906488\pi$
$587$	− 199552.i	− 0.579136i	0.957157	−	0.289568i	$-0.0935117\pi$
$588$	0	0
$589$	−275486.	−0.794087
$590$	0	0
$591$	92652.1i	0.265265i
$592$	0	0
$593$	156539.i	0.445156i	0.974915	+	0.222578i	$0.0714473\pi$
$593$	156539.i	0.445156i	−0.974915	+	0.222578i	$0.928553\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	64840.0	0.181926
$598$	0	0
$599$	−292624.	−0.815561	−0.407781	−	0.913080i	$-0.633697\pi$
$599$	−292624.	−0.815561	−0.407781	+	0.913080i	$0.633697\pi$
$600$	0	0
$601$	− 71047.1i	− 0.196697i	−0.995152	−	0.0983485i	$-0.968644\pi$
$601$	− 71047.1i	− 0.196697i	0.995152	−	0.0983485i	$-0.0313560\pi$
$602$	0	0
$603$	2077.45	0.00571342
$604$	0	0
$605$	− 205005.i	− 0.560085i
$606$	0	0
$607$	− 160393.i	− 0.435318i	−0.976025	−	0.217659i	$-0.930158\pi$
$607$	− 160393.i	− 0.435318i	0.976025	−	0.217659i	$-0.0698421\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−314245.	−0.841756
$612$	0	0
$613$	−67527.0	−0.179703	−0.0898517	−	0.995955i	$-0.528639\pi$
$613$	−67527.0	−0.179703	−0.0898517	+	0.995955i	$0.528639\pi$
$614$	0	0
$615$	219300.i	0.579815i
$616$	0	0
$617$	−18373.7	−0.0482644	−0.0241322	−	0.999709i	$-0.507682\pi$
$617$	−18373.7	−0.0482644	−0.0241322	+	0.999709i	$0.507682\pi$
$618$	0	0
$619$	591146.i	1.54281i	0.636342	+	0.771407i	$0.280447\pi$
$619$	591146.i	1.54281i	−0.636342	+	0.771407i	$0.719553\pi$
$620$	0	0
$621$	242713.i	0.629375i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−487430.	−1.24782
$626$	0	0
$627$	−233020.	−0.592733
$628$	0	0
$629$	− 426316.i	− 1.07753i
$630$	0	0
$631$	−729341.	−1.83177	−0.915887	−	0.401436i	$-0.868511\pi$
$631$	−729341.	−1.83177	−0.915887	+	0.401436i	$0.868511\pi$
$632$	0	0
$633$	− 560689.i	− 1.39931i
$634$	0	0
$635$	− 377699.i	− 0.936695i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	413.455	0.00101257
$640$	0	0
$641$	−233516.	−0.568329	−0.284165	−	0.958775i	$-0.591716\pi$
$641$	−233516.	−0.568329	−0.284165	+	0.958775i	$0.591716\pi$
$642$	0	0
$643$	− 382516.i	− 0.925184i	−0.886571	−	0.462592i	$-0.846920\pi$
$643$	− 382516.i	− 0.925184i	0.886571	−	0.462592i	$-0.153080\pi$
$644$	0	0
$645$	234768.	0.564313
$646$	0	0
$647$	− 503696.i	− 1.20326i	−0.798775	−	0.601631i	$-0.794518\pi$
$647$	− 503696.i	− 1.20326i	0.798775	−	0.601631i	$-0.205482\pi$
$648$	0	0
$649$	− 642011.i	− 1.52424i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	241430.	0.566193	0.283096	−	0.959091i	$-0.408638\pi$
$653$	241430.	0.566193	0.283096	+	0.959091i	$0.408638\pi$
$654$	0	0
$655$	−174836.	−0.407519
$656$	0	0
$657$	− 1327.09i	− 0.00307446i
$658$	0	0
$659$	−652340.	−1.50211	−0.751057	−	0.660237i	$-0.770456\pi$
$659$	−652340.	−1.50211	−0.751057	+	0.660237i	$0.770456\pi$
$660$	0	0
$661$	15508.0i	0.0354938i	0.999843	+	0.0177469i	$0.00564931\pi$
$661$	15508.0i	0.0354938i	−0.999843	+	0.0177469i	$0.994351\pi$
$662$	0	0
$663$	354374.i	0.806184i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	463012.	1.04074
$668$	0	0
$669$	−63687.6	−0.142299
$670$	0	0
$671$	379898.i	0.843765i
$672$	0	0
$673$	62603.2	0.138219	0.0691093	−	0.997609i	$-0.477984\pi$
$673$	62603.2	0.138219	0.0691093	+	0.997609i	$0.477984\pi$
$674$	0	0
$675$	248526.i	0.545462i
$676$	0	0
$677$	393281.i	0.858075i	0.903287	+	0.429038i	$0.141147\pi$
$677$	393281.i	0.858075i	−0.903287	+	0.429038i	$0.858853\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−352765.	−0.760662
$682$	0	0
$683$	−440410.	−0.944096	−0.472048	−	0.881573i	$-0.656485\pi$
$683$	−440410.	−0.944096	−0.472048	+	0.881573i	$0.656485\pi$
$684$	0	0
$685$	− 926391.i	− 1.97430i
$686$	0	0
$687$	3270.17	0.00692879
$688$	0	0
$689$	− 937778.i	− 1.97543i
$690$	0	0
$691$	− 933893.i	− 1.95588i	−0.208897	−	0.977938i	$-0.566987\pi$
$691$	− 933893.i	− 1.95588i	0.208897	−	0.977938i	$-0.433013\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−487961.	−1.01022
$696$	0	0
$697$	146345.	0.301239
$698$	0	0
$699$	− 62024.1i	− 0.126942i
$700$	0	0
$701$	−903363.	−1.83834	−0.919171	−	0.393860i	$-0.871140\pi$
$701$	−903363.	−1.83834	−0.919171	+	0.393860i	$0.871140\pi$
$702$	0	0
$703$	403846.i	0.817157i
$704$	0	0
$705$	− 419798.i	− 0.844622i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	177182.	0.352474	0.176237	−	0.984348i	$-0.443607\pi$
$709$	177182.	0.352474	0.176237	+	0.984348i	$0.443607\pi$
$710$	0	0
$711$	970.788	0.00192037
$712$	0	0
$713$	518523.i	1.01997i
$714$	0	0
$715$	951123.	1.86048
$716$	0	0
$717$	812876.i	1.58120i
$718$	0	0
$719$	670072.i	1.29618i	0.761565	+	0.648088i	$0.224431\pi$
$719$	670072.i	1.29618i	−0.761565	+	0.648088i	$0.775569\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−391251.	−0.748478
$724$	0	0
$725$	474101.	0.901977
$726$	0	0
$727$	465279.i	0.880329i	0.897917	+	0.440164i	$0.145080\pi$
$727$	465279.i	0.880329i	−0.897917	+	0.440164i	$0.854920\pi$
$728$	0	0
$729$	−529060.	−0.995519
$730$	0	0
$731$	− 156667.i	− 0.293185i
$732$	0	0
$733$	187934.i	0.349782i	0.984588	+	0.174891i	$0.0559574\pi$
$733$	187934.i	0.349782i	−0.984588	+	0.174891i	$0.944043\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−841438.	−1.54913
$738$	0	0
$739$	−216461.	−0.396360	−0.198180	−	0.980166i	$-0.563503\pi$
$739$	−216461.	−0.396360	−0.198180	+	0.980166i	$0.563503\pi$
$740$	0	0
$741$	− 335696.i	− 0.611377i
$742$	0	0
$743$	−202338.	−0.366521	−0.183261	−	0.983064i	$-0.558665\pi$
$743$	−202338.	−0.366521	−0.183261	+	0.983064i	$0.558665\pi$
$744$	0	0
$745$	853672.i	1.53808i
$746$	0	0
$747$	− 967.686i	− 0.00173418i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−501585.	−0.889334	−0.444667	−	0.895696i	$-0.646678\pi$
$751$	−501585.	−0.889334	−0.444667	+	0.895696i	$0.646678\pi$
$752$	0	0
$753$	520636.	0.918215
$754$	0	0
$755$	− 930942.i	− 1.63316i
$756$	0	0
$757$	−21535.3	−0.0375802	−0.0187901	−	0.999823i	$-0.505981\pi$
$757$	−21535.3	−0.0375802	−0.0187901	+	0.999823i	$0.505981\pi$
$758$	0	0
$759$	438595.i	0.761342i
$760$	0	0
$761$	809136.i	1.39718i	0.715523	+	0.698589i	$0.246189\pi$
$761$	809136.i	1.39718i	−0.715523	+	0.698589i	$0.753811\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2093.41	−0.00357711
$766$	0	0
$767$	924899.	1.57218
$768$	0	0
$769$	356884.i	0.603496i	0.953388	+	0.301748i	$0.0975702\pi$
$769$	356884.i	0.603496i	−0.953388	+	0.301748i	$0.902430\pi$
$770$	0	0
$771$	−611510.	−1.02871
$772$	0	0
$773$	195745.i	0.327591i	0.986494	+	0.163795i	$0.0523737\pi$
$773$	195745.i	0.327591i	−0.986494	+	0.163795i	$0.947626\pi$
$774$	0	0
$775$	530942.i	0.883983i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−138631.	−0.228447
$780$	0	0
$781$	−167463.	−0.274547
$782$	0	0
$783$	1.00928e6i	1.64622i
$784$	0	0
$785$	481631.	0.781583
$786$	0	0
$787$	− 448355.i	− 0.723890i	−0.932199	−	0.361945i	$-0.882113\pi$
$787$	− 448355.i	− 0.723890i	0.932199	−	0.361945i	$-0.117887\pi$
$788$	0	0
$789$	− 61042.4i	− 0.0980567i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−547291.	−0.870306
$794$	0	0
$795$	1.25277e6	1.98215
$796$	0	0
$797$	536814.i	0.845099i	0.906340	+	0.422549i	$0.138865\pi$
$797$	536814.i	0.845099i	−0.906340	+	0.422549i	$0.861135\pi$
$798$	0	0
$799$	−280142.	−0.438818
$800$	0	0
$801$	− 2092.79i	− 0.00326182i
$802$	0	0
$803$	537515.i	0.833603i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	758075.	1.16403
$808$	0	0
$809$	−372528.	−0.569196	−0.284598	−	0.958647i	$-0.591860\pi$
$809$	−372528.	−0.569196	−0.284598	+	0.958647i	$0.591860\pi$
$810$	0	0
$811$	− 559118.i	− 0.850084i	−0.905174	−	0.425042i	$-0.860259\pi$
$811$	− 559118.i	− 0.850084i	0.905174	−	0.425042i	$-0.139741\pi$
$812$	0	0
$813$	351392.	0.531632
$814$	0	0
$815$	849994.i	1.27968i
$816$	0	0
$817$	148409.i	0.222339i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−343799.	−0.510057	−0.255029	−	0.966934i	$-0.582085\pi$
$821$	−343799.	−0.510057	−0.255029	+	0.966934i	$0.582085\pi$
$822$	0	0
$823$	−51722.5	−0.0763624	−0.0381812	−	0.999271i	$-0.512156\pi$
$823$	−51722.5	−0.0763624	−0.0381812	+	0.999271i	$0.512156\pi$
$824$	0	0
$825$	449099.i	0.659834i
$826$	0	0
$827$	1.27662e6	1.86659	0.933297	−	0.359105i	$-0.116918\pi$
$827$	1.27662e6	1.86659	0.933297	+	0.359105i	$0.116918\pi$
$828$	0	0
$829$	523547.i	0.761810i	0.924614	+	0.380905i	$0.124388\pi$
$829$	523547.i	0.761810i	−0.924614	+	0.380905i	$0.875612\pi$
$830$	0	0
$831$	643866.i	0.932381i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−614178.	−0.880889
$836$	0	0
$837$	−1.13029e6	−1.61338
$838$	0	0
$839$	− 1.14150e6i	− 1.62163i	−0.585303	−	0.810815i	$-0.699024\pi$
$839$	− 1.14150e6i	− 1.62163i	0.585303	−	0.810815i	$-0.300976\pi$
$840$	0	0
$841$	1.21808e6	1.72220
$842$	0	0
$843$	− 521678.i	− 0.734087i
$844$	0	0
$845$	482211.i	0.675342i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	910984.	1.26385
$850$	0	0
$851$	760125.	1.04961
$852$	0	0
$853$	587560.i	0.807521i	0.914865	+	0.403761i	$0.132297\pi$
$853$	587560.i	0.807521i	−0.914865	+	0.403761i	$0.867703\pi$
$854$	0	0
$855$	1983.07	0.00271273
$856$	0	0
$857$	1.21968e6i	1.66067i	0.557266	+	0.830334i	$0.311850\pi$
$857$	1.21968e6i	1.66067i	−0.557266	+	0.830334i	$0.688150\pi$
$858$	0	0
$859$	836087.i	1.13309i	0.824030	+	0.566546i	$0.191721\pi$
$859$	836087.i	1.13309i	−0.824030	+	0.566546i	$0.808279\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	767175.	1.03008	0.515042	−	0.857165i	$-0.327776\pi$
$863$	767175.	1.03008	0.515042	+	0.857165i	$0.327776\pi$
$864$	0	0
$865$	−713451.	−0.953525
$866$	0	0
$867$	− 437441.i	− 0.581945i
$868$	0	0
$869$	−393202.	−0.520686
$870$	0	0
$871$	− 1.21220e6i	− 1.59786i
$872$	0	0
$873$	2838.78i	0.00372480i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−493208.	−0.641255	−0.320628	−	0.947205i	$-0.603894\pi$
$877$	−493208.	−0.641255	−0.320628	+	0.947205i	$0.603894\pi$
$878$	0	0
$879$	−1.20085e6	−1.55421
$880$	0	0
$881$	− 549789.i	− 0.708344i	−0.935180	−	0.354172i	$-0.884763\pi$
$881$	− 549789.i	− 0.708344i	0.935180	−	0.354172i	$-0.115237\pi$
$882$	0	0
$883$	−177926.	−0.228202	−0.114101	−	0.993469i	$-0.536399\pi$
$883$	−177926.	−0.228202	−0.114101	+	0.993469i	$0.536399\pi$
$884$	0	0
$885$	1.23557e6i	1.57754i
$886$	0	0
$887$	26842.7i	0.0341176i	0.999854	+	0.0170588i	$0.00543025\pi$
$887$	26842.7i	0.0341176i	−0.999854	+	0.0170588i	$0.994570\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−960303.	−1.20963
$892$	0	0
$893$	265376.	0.332781
$894$	0	0
$895$	437740.i	0.546475i
$896$	0	0
$897$	−631851.	−0.785290
$898$	0	0
$899$	2.15619e6i	2.66789i
$900$	0	0
$901$	− 836005.i	− 1.02982i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−377187.	−0.460532
$906$	0	0
$907$	−1.22924e6	−1.49424	−0.747122	−	0.664687i	$-0.768565\pi$
$907$	−1.22924e6	−1.49424	−0.747122	+	0.664687i	$0.768565\pi$
$908$	0	0
$909$	2849.98i	0.00344917i
$910$	0	0
$911$	701772.	0.845589	0.422794	−	0.906226i	$-0.361049\pi$
$911$	701772.	0.845589	0.422794	+	0.906226i	$0.361049\pi$
$912$	0	0
$913$	391945.i	0.470201i
$914$	0	0
$915$	− 731122.i	− 0.873268i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.51023e6	−1.78819	−0.894094	−	0.447879i	$-0.852180\pi$
$919$	−1.51023e6	−1.78819	−0.894094	+	0.447879i	$0.852180\pi$
$920$	0	0
$921$	−69158.9	−0.0815322
$922$	0	0
$923$	− 241252.i	− 0.283183i
$924$	0	0
$925$	778331.	0.909664
$926$	0	0
$927$	− 614.720i	0 0.000715349i
$928$	0	0
$929$	927981.i	1.07525i	0.843186	+	0.537623i	$0.180677\pi$
$929$	927981.i	1.07525i	−0.843186	+	0.537623i	$0.819323\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14839.8	−0.0170477
$934$	0	0
$935$	847902.	0.969890
$936$	0	0
$937$	− 902133.i	− 1.02752i	−0.857933	−	0.513761i	$-0.828252\pi$
$937$	− 902133.i	− 1.02752i	0.857933	−	0.513761i	$-0.171748\pi$
$938$	0	0
$939$	1.01107e6	1.14670
$940$	0	0
$941$	− 878615.i	− 0.992246i	−0.868252	−	0.496123i	$-0.834756\pi$
$941$	− 878615.i	− 0.992246i	0.868252	−	0.496123i	$-0.165244\pi$
$942$	0	0
$943$	260934.i	0.293431i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.44236e6	1.60832	0.804162	−	0.594410i	$-0.202614\pi$
$947$	1.44236e6	1.60832	0.804162	+	0.594410i	$0.202614\pi$
$948$	0	0
$949$	−774358.	−0.859824
$950$	0	0
$951$	− 30329.1i	− 0.0335350i
$952$	0	0
$953$	−524371.	−0.577368	−0.288684	−	0.957424i	$-0.593218\pi$
$953$	−524371.	−0.577368	−0.288684	+	0.957424i	$0.593218\pi$
$954$	0	0
$955$	932907.i	1.02290i
$956$	0	0
$957$	1.82382e6i	1.99140i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.49118e6	−1.61467
$962$	0	0
$963$	1924.77	0.00207551
$964$	0	0
$965$	581556.i	0.624507i
$966$	0	0
$967$	−1.40754e6	−1.50525	−0.752624	−	0.658450i	$-0.771212\pi$
$967$	−1.40754e6	−1.50525	−0.752624	+	0.658450i	$0.771212\pi$
$968$	0	0
$969$	− 299264.i	− 0.318718i
$970$	0	0
$971$	375089.i	0.397828i	0.980017	+	0.198914i	$0.0637415\pi$
$971$	375089.i	0.397828i	−0.980017	+	0.198914i	$0.936259\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−646985.	−0.680589
$976$	0	0
$977$	1.29938e6	1.36128	0.680640	−	0.732618i	$-0.261702\pi$
$977$	1.29938e6	1.36128	0.680640	+	0.732618i	$0.261702\pi$
$978$	0	0
$979$	847650.i	0.884405i
$980$	0	0
$981$	−7564.57	−0.00786043
$982$	0	0
$983$	− 800216.i	− 0.828133i	−0.910247	−	0.414067i	$-0.864108\pi$
$983$	− 800216.i	− 0.828133i	0.910247	−	0.414067i	$-0.135892\pi$
$984$	0	0
$985$	319368.i	0.329169i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	279338.	0.285586
$990$	0	0
$991$	−923241.	−0.940086	−0.470043	−	0.882643i	$-0.655762\pi$
$991$	−923241.	−0.940086	−0.470043	+	0.882643i	$0.655762\pi$
$992$	0	0
$993$	− 784465.i	− 0.795564i
$994$	0	0
$995$	223501.	0.225752
$996$	0	0
$997$	− 704626.i	− 0.708872i	−0.935080	−	0.354436i	$-0.884673\pi$
$997$	− 704626.i	− 0.708872i	0.935080	−	0.354436i	$-0.115327\pi$
$998$	0	0
$999$	1.65694e6i	1.66025i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.5.c.e.97.2		6
4.3	odd	2		196.5.b.a.97.5		6
7.2	even	3		112.5.s.c.17.3		6
7.3	odd	6		112.5.s.c.33.3		6
7.6	odd	2	inner	784.5.c.e.97.5		6
28.3	even	6		28.5.h.a.5.1	✓	6
28.11	odd	6		196.5.h.c.117.3		6
28.19	even	6		196.5.h.c.129.3		6
28.23	odd	6		28.5.h.a.17.1	yes	6
28.27	even	2		196.5.b.a.97.2		6
84.23	even	6		252.5.z.f.73.3		6
84.59	odd	6		252.5.z.f.145.3		6
140.3	odd	12		700.5.o.a.649.2		12
140.23	even	12		700.5.o.a.549.5		12
140.59	even	6		700.5.s.a.201.3		6
140.79	odd	6		700.5.s.a.101.3		6
140.87	odd	12		700.5.o.a.649.5		12
140.107	even	12		700.5.o.a.549.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.h.a.5.1	✓	6	28.3	even	6
28.5.h.a.17.1	yes	6	28.23	odd	6
112.5.s.c.17.3		6	7.2	even	3
112.5.s.c.33.3		6	7.3	odd	6
196.5.b.a.97.2		6	28.27	even	2
196.5.b.a.97.5		6	4.3	odd	2
196.5.h.c.117.3		6	28.11	odd	6
196.5.h.c.129.3		6	28.19	even	6
252.5.z.f.73.3		6	84.23	even	6
252.5.z.f.145.3		6	84.59	odd	6
700.5.o.a.549.2		12	140.107	even	12
700.5.o.a.549.5		12	140.23	even	12
700.5.o.a.649.2		12	140.3	odd	12
700.5.o.a.649.5		12	140.87	odd	12
700.5.s.a.101.3		6	140.79	odd	6
700.5.s.a.201.3		6	140.59	even	6
784.5.c.e.97.2		6	1.1	even	1	trivial
784.5.c.e.97.5		6	7.6	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.01997i q^{3} -31.0914i q^{5} -0.359775 q^{9} +145.721 q^{11} +209.930i q^{13} -280.444 q^{15} +187.147i q^{17} -177.283i q^{19} -333.685 q^{23} -341.677 q^{25} -727.372i q^{27} -1387.57 q^{29} -1553.93i q^{31} -1314.40i q^{33} -2277.98 q^{37} +1893.56 q^{39} -781.977i q^{41} -837.131 q^{43} +11.1859i q^{45} +1496.91i q^{47} +1688.06 q^{51} -4467.10 q^{53} -4530.67i q^{55} -1599.09 q^{57} -4405.75i q^{59} +2607.02i q^{61} +6527.01 q^{65} -5774.31 q^{67} +3009.82i q^{69} -1149.20 q^{71} +3688.65i q^{73} +3081.91i q^{75} -2698.32 q^{79} -6590.01 q^{81} +2689.70i q^{83} +5818.67 q^{85} +12515.9i q^{87} +5816.93i q^{89} -14016.4 q^{93} -5511.98 q^{95} -7890.42i q^{97} -52.4268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 180 q^{9} + 270 q^{11} + 486 q^{15} + 486 q^{23} - 3756 q^{25} - 540 q^{29} - 4710 q^{37} + 13176 q^{39} + 948 q^{43} - 1782 q^{51} - 12582 q^{53} - 6894 q^{57} - 15336 q^{65} + 3318 q^{67} - 2268 q^{71}+ \cdots - 8100 q^{99}+O(q^{100}) \) 6 * q - 180 * q^9 + 270 * q^11 + 486 * q^15 + 486 * q^23 - 3756 * q^25 - 540 * q^29 - 4710 * q^37 + 13176 * q^39 + 948 * q^43 - 1782 * q^51 - 12582 * q^53 - 6894 * q^57 - 15336 * q^65 + 3318 * q^67 - 2268 * q^71 - 15546 * q^79 + 14958 * q^81 - 702 * q^85 - 2898 * q^93 - 41310 * q^95 - 8100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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