LMFDB - Embedded newform 448.4.b.a.225.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(225,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("448.225");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.b (of order $2$, degree $1$, 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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			225.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.225
Dual form			448.4.b.a.225.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.66025i q^{3} +22.1962i q^{5} -7.00000 q^{7} -31.6795 q^{9} -40.1051i q^{11} -14.4833i q^{13} -170.028 q^{15} -111.569 q^{17} +117.804i q^{19} -53.6218i q^{21} +119.751 q^{23} -367.669 q^{25} -35.8461i q^{27} -92.8897i q^{29} +181.569 q^{31} +307.215 q^{33} -155.373i q^{35} -247.090i q^{37} +110.946 q^{39} +58.1333 q^{41} +249.885i q^{43} -703.163i q^{45} -101.913 q^{47} +49.0000 q^{49} -854.649i q^{51} +570.697i q^{53} +890.179 q^{55} -902.407 q^{57} +277.391i q^{59} +82.9526i q^{61} +221.756 q^{63} +321.474 q^{65} -397.969i q^{67} +917.325i q^{69} +6.22055 q^{71} -209.769 q^{73} -2816.44i q^{75} +280.736i q^{77} -271.902 q^{79} -580.756 q^{81} -1010.28i q^{83} -2476.41i q^{85} +711.559 q^{87} -730.420 q^{89} +101.383i q^{91} +1390.87i q^{93} -2614.79 q^{95} +707.451 q^{97} +1270.51i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 28 q^{7} - 196 q^{9} - 112 q^{15} - 280 q^{17} + 576 q^{23} - 764 q^{25} + 560 q^{31} + 1312 q^{33} - 928 q^{39} - 488 q^{41} + 784 q^{47} + 196 q^{49} + 928 q^{55} + 672 q^{57} + 1372 q^{63} + 2048 q^{65}+ \cdots - 856 q^{97}+O(q^{100}) $ 4 * q - 28 * q^7 - 196 * q^9 - 112 * q^15 - 280 * q^17 + 576 * q^23 - 764 * q^25 + 560 * q^31 + 1312 * q^33 - 928 * q^39 - 488 * q^41 + 784 * q^47 + 196 * q^49 + 928 * q^55 + 672 * q^57 + 1372 * q^63 + 2048 * q^65 + 496 * q^71 + 408 * q^73 + 880 * q^79 + 2596 * q^81 + 1904 * q^87 - 2312 * q^89 - 8256 * q^95 - 856 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	7.66025i	1.47422i	0.675775	+	0.737108i	$0.263809\pi$
$3$	7.66025i	1.47422i	−0.675775	+	0.737108i	$0.736191\pi$
$4$	0	0
$5$	22.1962i	1.98528i	0.121085	+	0.992642i	$0.461363\pi$
$5$	22.1962i	1.98528i	−0.121085	+	0.992642i	$0.538637\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−31.6795	−1.17331
$10$	0	0
$11$	− 40.1051i	− 1.09929i	−0.835400	−	0.549643i	$-0.814764\pi$
$11$	− 40.1051i	− 1.09929i	0.835400	−	0.549643i	$-0.185236\pi$
$12$	0	0
$13$	− 14.4833i	− 0.308997i	−0.987993	−	0.154498i	$-0.950624\pi$
$13$	− 14.4833i	− 0.308997i	0.987993	−	0.154498i	$-0.0493761\pi$
$14$	0	0
$15$	−170.028	−2.92674
$16$	0	0
$17$	−111.569	−1.59174	−0.795868	−	0.605470i	$-0.792985\pi$
$17$	−111.569	−1.59174	−0.795868	+	0.605470i	$0.792985\pi$
$18$	0	0
$19$	117.804i	1.42242i	0.702978	+	0.711212i	$0.251853\pi$
$19$	117.804i	1.42242i	−0.702978	+	0.711212i	$0.748147\pi$
$20$	0	0
$21$	− 53.6218i	− 0.557201i
$22$	0	0
$23$	119.751	1.08565	0.542823	−	0.839847i	$-0.317355\pi$
$23$	119.751	1.08565	0.542823	+	0.839847i	$0.317355\pi$
$24$	0	0
$25$	−367.669	−2.94135
$26$	0	0
$27$	− 35.8461i	− 0.255503i
$28$	0	0
$29$	− 92.8897i	− 0.594800i	−0.954753	−	0.297400i	$-0.903881\pi$
$29$	− 92.8897i	− 0.594800i	0.954753	−	0.297400i	$-0.0961194\pi$
$30$	0	0
$31$	181.569	1.05196	0.525981	−	0.850497i	$-0.323698\pi$
$31$	181.569	1.05196	0.525981	+	0.850497i	$0.323698\pi$
$32$	0	0
$33$	307.215	1.62059
$34$	0	0
$35$	− 155.373i	− 0.750367i
$36$	0	0
$37$	− 247.090i	− 1.09787i	−0.835864	−	0.548936i	$-0.815033\pi$
$37$	− 247.090i	− 1.09787i	0.835864	−	0.548936i	$-0.184967\pi$
$38$	0	0
$39$	110.946	0.455528
$40$	0	0
$41$	58.1333	0.221436	0.110718	−	0.993852i	$-0.464685\pi$
$41$	58.1333	0.221436	0.110718	+	0.993852i	$0.464685\pi$
$42$	0	0
$43$	249.885i	0.886210i	0.896470	+	0.443105i	$0.146123\pi$
$43$	249.885i	0.886210i	−0.896470	+	0.443105i	$0.853877\pi$
$44$	0	0
$45$	− 703.163i	− 2.32936i
$46$	0	0
$47$	−101.913	−0.316287	−0.158144	−	0.987416i	$-0.550551\pi$
$47$	−101.913	−0.316287	−0.158144	+	0.987416i	$0.550551\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	− 854.649i	− 2.34656i
$52$	0	0
$53$	570.697i	1.47908i	0.673112	+	0.739541i	$0.264957\pi$
$53$	570.697i	1.47908i	−0.673112	+	0.739541i	$0.735043\pi$
$54$	0	0
$55$	890.179	2.18240
$56$	0	0
$57$	−902.407	−2.09696
$58$	0	0
$59$	277.391i	0.612089i	0.952017	+	0.306044i	$0.0990056\pi$
$59$	277.391i	0.612089i	−0.952017	+	0.306044i	$0.900994\pi$
$60$	0	0
$61$	82.9526i	0.174115i	0.996203	+	0.0870573i	$0.0277463\pi$
$61$	82.9526i	0.174115i	−0.996203	+	0.0870573i	$0.972254\pi$
$62$	0	0
$63$	221.756	0.443471
$64$	0	0
$65$	321.474	0.613446
$66$	0	0
$67$	− 397.969i	− 0.725667i	−0.931854	−	0.362833i	$-0.881809\pi$
$67$	− 397.969i	− 0.725667i	0.931854	−	0.362833i	$-0.118191\pi$
$68$	0	0
$69$	917.325i	1.60048i
$70$	0	0
$71$	6.22055	0.0103978	0.00519889	−	0.999986i	$-0.498345\pi$
$71$	6.22055	0.0103978	0.00519889	+	0.999986i	$0.498345\pi$
$72$	0	0
$73$	−209.769	−0.336324	−0.168162	−	0.985759i	$-0.553783\pi$
$73$	−209.769	−0.336324	−0.168162	+	0.985759i	$0.553783\pi$
$74$	0	0
$75$	− 2816.44i	− 4.33619i
$76$	0	0
$77$	280.736i	0.415491i
$78$	0	0
$79$	−271.902	−0.387233	−0.193617	−	0.981077i	$-0.562022\pi$
$79$	−271.902	−0.387233	−0.193617	+	0.981077i	$0.562022\pi$
$80$	0	0
$81$	−580.756	−0.796648
$82$	0	0
$83$	− 1010.28i	− 1.33606i	−0.744136	−	0.668028i	$-0.767138\pi$
$83$	− 1010.28i	− 1.33606i	0.744136	−	0.668028i	$-0.232862\pi$
$84$	0	0
$85$	− 2476.41i	− 3.16005i
$86$	0	0
$87$	711.559	0.876863
$88$	0	0
$89$	−730.420	−0.869937	−0.434969	−	0.900446i	$-0.643241\pi$
$89$	−730.420	−0.869937	−0.434969	+	0.900446i	$0.643241\pi$
$90$	0	0
$91$	101.383i	0.116790i
$92$	0	0
$93$	1390.87i	1.55082i
$94$	0	0
$95$	−2614.79	−2.82392
$96$	0	0
$97$	707.451	0.740523	0.370262	−	0.928927i	$-0.379268\pi$
$97$	707.451	0.740523	0.370262	+	0.928927i	$0.379268\pi$
$98$	0	0
$99$	1270.51i	1.28981i
$100$	0	0
$101$	596.558i	0.587720i	0.955849	+	0.293860i	$0.0949399\pi$
$101$	596.558i	0.587720i	−0.955849	+	0.293860i	$0.905060\pi$
$102$	0	0
$103$	539.959	0.516541	0.258270	−	0.966073i	$-0.416847\pi$
$103$	539.959	0.516541	0.258270	+	0.966073i	$0.416847\pi$
$104$	0	0
$105$	1190.20	1.10620
$106$	0	0
$107$	660.497i	0.596754i	0.954448	+	0.298377i	$0.0964453\pi$
$107$	660.497i	0.596754i	−0.954448	+	0.298377i	$0.903555\pi$
$108$	0	0
$109$	234.157i	0.205763i	0.994694	+	0.102881i	$0.0328062\pi$
$109$	234.157i	0.205763i	−0.994694	+	0.102881i	$0.967194\pi$
$110$	0	0
$111$	1892.77	1.61850
$112$	0	0
$113$	−1740.74	−1.44916	−0.724578	−	0.689193i	$-0.757965\pi$
$113$	−1740.74	−1.44916	−0.724578	+	0.689193i	$0.757965\pi$
$114$	0	0
$115$	2658.02i	2.15532i
$116$	0	0
$117$	458.825i	0.362550i
$118$	0	0
$119$	780.985	0.601620
$120$	0	0
$121$	−277.420	−0.208430
$122$	0	0
$123$	445.316i	0.326445i
$124$	0	0
$125$	− 5386.32i	− 3.85414i
$126$	0	0
$127$	−341.377	−0.238522	−0.119261	−	0.992863i	$-0.538053\pi$
$127$	−341.377	−0.238522	−0.119261	+	0.992863i	$0.538053\pi$
$128$	0	0
$129$	−1914.18	−1.30647
$130$	0	0
$131$	− 405.014i	− 0.270124i	−0.990837	−	0.135062i	$-0.956877\pi$
$131$	− 405.014i	− 0.270124i	0.990837	−	0.135062i	$-0.0431233\pi$
$132$	0	0
$133$	− 824.627i	− 0.537626i
$134$	0	0
$135$	795.645	0.507246
$136$	0	0
$137$	−1590.40	−0.991803	−0.495902	−	0.868379i	$-0.665162\pi$
$137$	−1590.40	−0.991803	−0.495902	+	0.868379i	$0.665162\pi$
$138$	0	0
$139$	605.609i	0.369547i	0.982781	+	0.184774i	$0.0591552\pi$
$139$	605.609i	0.369547i	−0.982781	+	0.184774i	$0.940845\pi$
$140$	0	0
$141$	− 780.677i	− 0.466276i
$142$	0	0
$143$	−580.856	−0.339676
$144$	0	0
$145$	2061.79	1.18085
$146$	0	0
$147$	375.352i	0.210602i
$148$	0	0
$149$	1986.22i	1.09206i	0.837764	+	0.546032i	$0.183862\pi$
$149$	1986.22i	1.09206i	−0.837764	+	0.546032i	$0.816138\pi$
$150$	0	0
$151$	678.905	0.365884	0.182942	−	0.983124i	$-0.441438\pi$
$151$	678.905	0.365884	0.182942	+	0.983124i	$0.441438\pi$
$152$	0	0
$153$	3534.46	1.86761
$154$	0	0
$155$	4030.14i	2.08844i
$156$	0	0
$157$	− 3202.21i	− 1.62780i	−0.581006	−	0.813899i	$-0.697341\pi$
$157$	− 3202.21i	− 1.62780i	0.581006	−	0.813899i	$-0.302659\pi$
$158$	0	0
$159$	−4371.69	−2.18049
$160$	0	0
$161$	−838.259	−0.410336
$162$	0	0
$163$	− 340.941i	− 0.163831i	−0.996639	−	0.0819157i	$-0.973896\pi$
$163$	− 340.941i	− 0.163831i	0.996639	−	0.0819157i	$-0.0261038\pi$
$164$	0	0
$165$	6819.00i	3.21732i
$166$	0	0
$167$	−3038.90	−1.40812	−0.704062	−	0.710138i	$-0.748633\pi$
$167$	−3038.90	−1.40812	−0.704062	+	0.710138i	$0.748633\pi$
$168$	0	0
$169$	1987.23	0.904521
$170$	0	0
$171$	− 3731.97i	− 1.66895i
$172$	0	0
$173$	2611.06i	1.14749i	0.819035	+	0.573743i	$0.194509\pi$
$173$	2611.06i	1.14749i	−0.819035	+	0.573743i	$0.805491\pi$
$174$	0	0
$175$	2573.68	1.11173
$176$	0	0
$177$	−2124.88	−0.902351
$178$	0	0
$179$	− 2055.56i	− 0.858325i	−0.903227	−	0.429162i	$-0.858809\pi$
$179$	− 2055.56i	− 0.858325i	0.903227	−	0.429162i	$-0.141191\pi$
$180$	0	0
$181$	1986.95i	0.815961i	0.912991	+	0.407981i	$0.133767\pi$
$181$	1986.95i	0.815961i	−0.912991	+	0.407981i	$0.866233\pi$
$182$	0	0
$183$	−635.438	−0.256683
$184$	0	0
$185$	5484.44	2.17959
$186$	0	0
$187$	4474.50i	1.74977i
$188$	0	0
$189$	250.923i	0.0965711i
$190$	0	0
$191$	441.866	0.167394	0.0836971	−	0.996491i	$-0.473327\pi$
$191$	441.866	0.167394	0.0836971	+	0.996491i	$0.473327\pi$
$192$	0	0
$193$	−5059.63	−1.88705	−0.943524	−	0.331304i	$-0.892511\pi$
$193$	−5059.63	−1.88705	−0.943524	+	0.331304i	$0.892511\pi$
$194$	0	0
$195$	2462.58i	0.904352i
$196$	0	0
$197$	3150.30i	1.13934i	0.821875	+	0.569669i	$0.192928\pi$
$197$	3150.30i	1.13934i	−0.821875	+	0.569669i	$0.807072\pi$
$198$	0	0
$199$	−539.846	−0.192305	−0.0961525	−	0.995367i	$-0.530654\pi$
$199$	−539.846	−0.192305	−0.0961525	+	0.995367i	$0.530654\pi$
$200$	0	0
$201$	3048.54	1.06979
$202$	0	0
$203$	650.228i	0.224813i
$204$	0	0
$205$	1290.34i	0.439614i
$206$	0	0
$207$	−3793.66	−1.27380
$208$	0	0
$209$	4724.54	1.56365
$210$	0	0
$211$	− 5413.14i	− 1.76614i	−0.469239	−	0.883071i	$-0.655472\pi$
$211$	− 5413.14i	− 1.76614i	0.469239	−	0.883071i	$-0.344528\pi$
$212$	0	0
$213$	47.6510i	0.0153286i
$214$	0	0
$215$	−5546.48	−1.75938
$216$	0	0
$217$	−1270.98	−0.397604
$218$	0	0
$219$	− 1606.88i	− 0.495814i
$220$	0	0
$221$	1615.89i	0.491841i
$222$	0	0
$223$	2289.18	0.687422	0.343711	−	0.939075i	$-0.388316\pi$
$223$	2289.18	0.687422	0.343711	+	0.939075i	$0.388316\pi$
$224$	0	0
$225$	11647.6	3.45113
$226$	0	0
$227$	6742.24i	1.97136i	0.168634	+	0.985679i	$0.446064\pi$
$227$	6742.24i	1.97136i	−0.168634	+	0.985679i	$0.553936\pi$
$228$	0	0
$229$	− 1536.97i	− 0.443520i	−0.975101	−	0.221760i	$-0.928820\pi$
$229$	− 1536.97i	− 0.443520i	0.975101	−	0.221760i	$-0.0711801\pi$
$230$	0	0
$231$	−2150.51	−0.612524
$232$	0	0
$233$	−1568.19	−0.440926	−0.220463	−	0.975395i	$-0.570757\pi$
$233$	−1568.19	−0.440926	−0.220463	+	0.975395i	$0.570757\pi$
$234$	0	0
$235$	− 2262.07i	− 0.627920i
$236$	0	0
$237$	− 2082.84i	− 0.570865i
$238$	0	0
$239$	4907.22	1.32812	0.664062	−	0.747677i	$-0.268831\pi$
$239$	4907.22	1.32812	0.664062	+	0.747677i	$0.268831\pi$
$240$	0	0
$241$	1403.57	0.375154	0.187577	−	0.982250i	$-0.439937\pi$
$241$	1403.57	0.375154	0.187577	+	0.982250i	$0.439937\pi$
$242$	0	0
$243$	− 5416.58i	− 1.42993i
$244$	0	0
$245$	1087.61i	0.283612i
$246$	0	0
$247$	1706.19	0.439524
$248$	0	0
$249$	7739.01	1.96964
$250$	0	0
$251$	3774.48i	0.949175i	0.880208	+	0.474588i	$0.157403\pi$
$251$	3774.48i	0.949175i	−0.880208	+	0.474588i	$0.842597\pi$
$252$	0	0
$253$	− 4802.64i	− 1.19344i
$254$	0	0
$255$	18969.9	4.65860
$256$	0	0
$257$	−3233.01	−0.784708	−0.392354	−	0.919814i	$-0.628339\pi$
$257$	−3233.01	−0.784708	−0.392354	+	0.919814i	$0.628339\pi$
$258$	0	0
$259$	1729.63i	0.414957i
$260$	0	0
$261$	2942.70i	0.697887i
$262$	0	0
$263$	3259.84	0.764298	0.382149	−	0.924101i	$-0.375184\pi$
$263$	3259.84	0.764298	0.382149	+	0.924101i	$0.375184\pi$
$264$	0	0
$265$	−12667.3	−2.93640
$266$	0	0
$267$	− 5595.21i	− 1.28248i
$268$	0	0
$269$	7328.77i	1.66113i	0.556925	+	0.830563i	$0.311981\pi$
$269$	7328.77i	1.66113i	−0.556925	+	0.830563i	$0.688019\pi$
$270$	0	0
$271$	−7009.29	−1.57116	−0.785579	−	0.618761i	$-0.787635\pi$
$271$	−7009.29	−1.57116	−0.785579	+	0.618761i	$0.787635\pi$
$272$	0	0
$273$	−776.622	−0.172173
$274$	0	0
$275$	14745.4i	3.23339i
$276$	0	0
$277$	3753.77i	0.814233i	0.913376	+	0.407116i	$0.133466\pi$
$277$	3753.77i	0.814233i	−0.913376	+	0.407116i	$0.866534\pi$
$278$	0	0
$279$	−5752.02	−1.23428
$280$	0	0
$281$	−36.9750	−0.00784961	−0.00392481	−	0.999992i	$-0.501249\pi$
$281$	−36.9750	−0.00784961	−0.00392481	+	0.999992i	$0.501249\pi$
$282$	0	0
$283$	3236.79i	0.679884i	0.940446	+	0.339942i	$0.110407\pi$
$283$	3236.79i	0.679884i	−0.940446	+	0.339942i	$0.889593\pi$
$284$	0	0
$285$	− 20030.0i	− 4.16306i
$286$	0	0
$287$	−406.933	−0.0836951
$288$	0	0
$289$	7534.69	1.53362
$290$	0	0
$291$	5419.25i	1.09169i
$292$	0	0
$293$	− 4412.89i	− 0.879876i	−0.898028	−	0.439938i	$-0.855000\pi$
$293$	− 4412.89i	− 0.879876i	0.898028	−	0.439938i	$-0.145000\pi$
$294$	0	0
$295$	−6157.01	−1.21517
$296$	0	0
$297$	−1437.61	−0.280871
$298$	0	0
$299$	− 1734.40i	− 0.335461i
$300$	0	0
$301$	− 1749.19i	− 0.334956i
$302$	0	0
$303$	−4569.78	−0.866426
$304$	0	0
$305$	−1841.23	−0.345667
$306$	0	0
$307$	− 588.428i	− 0.109392i	−0.998503	−	0.0546961i	$-0.982581\pi$
$307$	− 588.428i	− 0.109392i	0.998503	−	0.0546961i	$-0.0174190\pi$
$308$	0	0
$309$	4136.22i	0.761493i
$310$	0	0
$311$	−7495.29	−1.36662	−0.683310	−	0.730128i	$-0.739460\pi$
$311$	−7495.29	−1.36662	−0.683310	+	0.730128i	$0.739460\pi$
$312$	0	0
$313$	−6652.87	−1.20141	−0.600707	−	0.799469i	$-0.705114\pi$
$313$	−6652.87	−1.20141	−0.600707	+	0.799469i	$0.705114\pi$
$314$	0	0
$315$	4922.14i	0.880416i
$316$	0	0
$317$	− 7215.06i	− 1.27835i	−0.769060	−	0.639177i	$-0.779275\pi$
$317$	− 7215.06i	− 1.27835i	0.769060	−	0.639177i	$-0.220725\pi$
$318$	0	0
$319$	−3725.35	−0.653855
$320$	0	0
$321$	−5059.58	−0.879745
$322$	0	0
$323$	− 13143.3i	− 2.26412i
$324$	0	0
$325$	5325.08i	0.908868i
$326$	0	0
$327$	−1793.70	−0.303339
$328$	0	0
$329$	713.389	0.119545
$330$	0	0
$331$	− 1123.85i	− 0.186623i	−0.995637	−	0.0933116i	$-0.970255\pi$
$331$	− 1123.85i	− 0.186623i	0.995637	−	0.0933116i	$-0.0297453\pi$
$332$	0	0
$333$	7827.67i	1.28815i
$334$	0	0
$335$	8833.38	1.44065
$336$	0	0
$337$	−735.459	−0.118881	−0.0594407	−	0.998232i	$-0.518932\pi$
$337$	−735.459	−0.118881	−0.0594407	+	0.998232i	$0.518932\pi$
$338$	0	0
$339$	− 13334.5i	− 2.13637i
$340$	0	0
$341$	− 7281.85i	− 1.15641i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−20361.1	−3.17740
$346$	0	0
$347$	8412.83i	1.30151i	0.759287	+	0.650756i	$0.225548\pi$
$347$	8412.83i	1.30151i	−0.759287	+	0.650756i	$0.774452\pi$
$348$	0	0
$349$	8577.90i	1.31566i	0.753167	+	0.657829i	$0.228525\pi$
$349$	8577.90i	1.31566i	−0.753167	+	0.657829i	$0.771475\pi$
$350$	0	0
$351$	−519.171	−0.0789496
$352$	0	0
$353$	−34.2817	−0.00516892	−0.00258446	−	0.999997i	$-0.500823\pi$
$353$	−34.2817	−0.00516892	−0.00258446	+	0.999997i	$0.500823\pi$
$354$	0	0
$355$	138.072i	0.0206426i
$356$	0	0
$357$	5982.54i	0.886918i
$358$	0	0
$359$	1007.89	0.148174	0.0740872	−	0.997252i	$-0.476396\pi$
$359$	1007.89	0.148174	0.0740872	+	0.997252i	$0.476396\pi$
$360$	0	0
$361$	−7018.75	−1.02329
$362$	0	0
$363$	− 2125.11i	− 0.307271i
$364$	0	0
$365$	− 4656.07i	− 0.667698i
$366$	0	0
$367$	−13076.7	−1.85994	−0.929969	−	0.367638i	$-0.880166\pi$
$367$	−13076.7	−1.85994	−0.929969	+	0.367638i	$0.880166\pi$
$368$	0	0
$369$	−1841.63	−0.259815
$370$	0	0
$371$	− 3994.88i	− 0.559040i
$372$	0	0
$373$	8552.62i	1.18723i	0.804748	+	0.593616i	$0.202300\pi$
$373$	8552.62i	1.18723i	−0.804748	+	0.593616i	$0.797700\pi$
$374$	0	0
$375$	41260.6	5.68183
$376$	0	0
$377$	−1345.35	−0.183791
$378$	0	0
$379$	− 9753.81i	− 1.32195i	−0.750408	−	0.660975i	$-0.770143\pi$
$379$	− 9753.81i	− 1.32195i	0.750408	−	0.660975i	$-0.229857\pi$
$380$	0	0
$381$	− 2615.03i	− 0.351633i
$382$	0	0
$383$	−7141.71	−0.952805	−0.476402	−	0.879227i	$-0.658059\pi$
$383$	−7141.71	−0.952805	−0.476402	+	0.879227i	$0.658059\pi$
$384$	0	0
$385$	−6231.26	−0.824868
$386$	0	0
$387$	− 7916.22i	− 1.03980i
$388$	0	0
$389$	− 4854.81i	− 0.632773i	−0.948630	−	0.316386i	$-0.897530\pi$
$389$	− 4854.81i	− 0.632773i	0.948630	−	0.316386i	$-0.102470\pi$
$390$	0	0
$391$	−13360.6	−1.72806
$392$	0	0
$393$	3102.51	0.398221
$394$	0	0
$395$	− 6035.19i	− 0.768768i
$396$	0	0
$397$	− 10401.6i	− 1.31496i	−0.753471	−	0.657481i	$-0.771622\pi$
$397$	− 10401.6i	− 1.31496i	0.753471	−	0.657481i	$-0.228378\pi$
$398$	0	0
$399$	6316.85	0.792577
$400$	0	0
$401$	4386.56	0.546271	0.273135	−	0.961976i	$-0.411939\pi$
$401$	4386.56	0.546271	0.273135	+	0.961976i	$0.411939\pi$
$402$	0	0
$403$	− 2629.73i	− 0.325052i
$404$	0	0
$405$	− 12890.6i	− 1.58157i
$406$	0	0
$407$	−9909.56	−1.20688
$408$	0	0
$409$	−6474.49	−0.782746	−0.391373	−	0.920232i	$-0.628000\pi$
$409$	−6474.49	−0.782746	−0.391373	+	0.920232i	$0.628000\pi$
$410$	0	0
$411$	− 12182.9i	− 1.46213i
$412$	0	0
$413$	− 1941.74i	− 0.231348i
$414$	0	0
$415$	22424.3	2.65245
$416$	0	0
$417$	−4639.12	−0.544793
$418$	0	0
$419$	4781.90i	0.557544i	0.960357	+	0.278772i	$0.0899274\pi$
$419$	4781.90i	0.557544i	−0.960357	+	0.278772i	$0.910073\pi$
$420$	0	0
$421$	− 271.468i	− 0.0314265i	−0.999877	−	0.0157133i	$-0.994998\pi$
$421$	− 271.468i	− 0.0314265i	0.999877	−	0.0157133i	$-0.00500189\pi$
$422$	0	0
$423$	3228.54	0.371104
$424$	0	0
$425$	41020.6	4.68186
$426$	0	0
$427$	− 580.668i	− 0.0658091i
$428$	0	0
$429$	− 4449.50i	− 0.500756i
$430$	0	0
$431$	10024.3	1.12031	0.560153	−	0.828389i	$-0.310742\pi$
$431$	10024.3	1.12031	0.560153	+	0.828389i	$0.310742\pi$
$432$	0	0
$433$	14997.2	1.66448	0.832242	−	0.554413i	$-0.187057\pi$
$433$	14997.2	1.66448	0.832242	+	0.554413i	$0.187057\pi$
$434$	0	0
$435$	15793.9i	1.74082i
$436$	0	0
$437$	14107.2i	1.54425i
$438$	0	0
$439$	7619.62	0.828393	0.414196	−	0.910188i	$-0.364063\pi$
$439$	7619.62	0.828393	0.414196	+	0.910188i	$0.364063\pi$
$440$	0	0
$441$	−1552.30	−0.167616
$442$	0	0
$443$	3188.70i	0.341985i	0.985272	+	0.170993i	$0.0546975\pi$
$443$	3188.70i	0.341985i	−0.985272	+	0.170993i	$0.945303\pi$
$444$	0	0
$445$	− 16212.5i	− 1.72707i
$446$	0	0
$447$	−15215.0	−1.60994
$448$	0	0
$449$	3446.04	0.362202	0.181101	−	0.983464i	$-0.442034\pi$
$449$	3446.04	0.362202	0.181101	+	0.983464i	$0.442034\pi$
$450$	0	0
$451$	− 2331.44i	− 0.243422i
$452$	0	0
$453$	5200.58i	0.539392i
$454$	0	0
$455$	−2250.32	−0.231861
$456$	0	0
$457$	−7559.50	−0.773782	−0.386891	−	0.922126i	$-0.626451\pi$
$457$	−7559.50	−0.773782	−0.386891	+	0.922126i	$0.626451\pi$
$458$	0	0
$459$	3999.32i	0.406694i
$460$	0	0
$461$	10358.3i	1.04650i	0.852180	+	0.523249i	$0.175280\pi$
$461$	10358.3i	1.04650i	−0.852180	+	0.523249i	$0.824720\pi$
$462$	0	0
$463$	5546.46	0.556730	0.278365	−	0.960475i	$-0.410208\pi$
$463$	5546.46	0.556730	0.278365	+	0.960475i	$0.410208\pi$
$464$	0	0
$465$	−30871.9	−3.07882
$466$	0	0
$467$	7296.87i	0.723038i	0.932365	+	0.361519i	$0.117742\pi$
$467$	7296.87i	0.723038i	−0.932365	+	0.361519i	$0.882258\pi$
$468$	0	0
$469$	2785.78i	0.274276i
$470$	0	0
$471$	24529.8	2.39973
$472$	0	0
$473$	10021.7	0.974199
$474$	0	0
$475$	− 43312.8i	− 4.18385i
$476$	0	0
$477$	− 18079.4i	− 1.73543i
$478$	0	0
$479$	19437.2	1.85409	0.927046	−	0.374946i	$-0.122339\pi$
$479$	19437.2	1.85409	0.927046	+	0.374946i	$0.122339\pi$
$480$	0	0
$481$	−3578.68	−0.339239
$482$	0	0
$483$	− 6421.28i	− 0.604924i
$484$	0	0
$485$	15702.7i	1.47015i
$486$	0	0
$487$	6540.19	0.608551	0.304276	−	0.952584i	$-0.401586\pi$
$487$	6540.19	0.608551	0.304276	+	0.952584i	$0.401586\pi$
$488$	0	0
$489$	2611.69	0.241523
$490$	0	0
$491$	8542.59i	0.785176i	0.919714	+	0.392588i	$0.128420\pi$
$491$	8542.59i	0.785176i	−0.919714	+	0.392588i	$0.871580\pi$
$492$	0	0
$493$	10363.6i	0.946764i
$494$	0	0
$495$	−28200.4	−2.56064
$496$	0	0
$497$	−43.5438	−0.00392999
$498$	0	0
$499$	− 17254.6i	− 1.54794i	−0.633224	−	0.773968i	$-0.718269\pi$
$499$	− 17254.6i	− 1.54794i	0.633224	−	0.773968i	$-0.281731\pi$
$500$	0	0
$501$	− 23278.7i	− 2.07588i
$502$	0	0
$503$	−5597.51	−0.496184	−0.248092	−	0.968736i	$-0.579804\pi$
$503$	−5597.51	−0.496184	−0.248092	+	0.968736i	$0.579804\pi$
$504$	0	0
$505$	−13241.3	−1.16679
$506$	0	0
$507$	15222.7i	1.33346i
$508$	0	0
$509$	− 5232.92i	− 0.455688i	−0.973698	−	0.227844i	$-0.926832\pi$
$509$	− 5232.92i	− 0.455688i	0.973698	−	0.227844i	$-0.0731676\pi$
$510$	0	0
$511$	1468.38	0.127118
$512$	0	0
$513$	4222.81	0.363434
$514$	0	0
$515$	11985.0i	1.02548i
$516$	0	0
$517$	4087.22i	0.347690i
$518$	0	0
$519$	−20001.4	−1.69164
$520$	0	0
$521$	7429.89	0.624778	0.312389	−	0.949954i	$-0.398871\pi$
$521$	7429.89	0.624778	0.312389	+	0.949954i	$0.398871\pi$
$522$	0	0
$523$	− 2963.01i	− 0.247732i	−0.992299	−	0.123866i	$-0.960471\pi$
$523$	− 2963.01i	− 0.247732i	0.992299	−	0.123866i	$-0.0395292\pi$
$524$	0	0
$525$	19715.1i	1.63893i
$526$	0	0
$527$	−20257.5	−1.67444
$528$	0	0
$529$	2173.37	0.178628
$530$	0	0
$531$	− 8787.60i	− 0.718172i
$532$	0	0
$533$	− 841.964i	− 0.0684231i
$534$	0	0
$535$	−14660.5	−1.18473
$536$	0	0
$537$	15746.1	1.26536
$538$	0	0
$539$	− 1965.15i	− 0.157041i
$540$	0	0
$541$	19355.9i	1.53822i	0.639118	+	0.769109i	$0.279299\pi$
$541$	19355.9i	1.53822i	−0.639118	+	0.769109i	$0.720701\pi$
$542$	0	0
$543$	−15220.6	−1.20290
$544$	0	0
$545$	−5197.38	−0.408498
$546$	0	0
$547$	7653.95i	0.598280i	0.954209	+	0.299140i	$0.0966997\pi$
$547$	7653.95i	0.598280i	−0.954209	+	0.299140i	$0.903300\pi$
$548$	0	0
$549$	− 2627.90i	− 0.204291i
$550$	0	0
$551$	10942.8	0.846057
$552$	0	0
$553$	1903.32	0.146360
$554$	0	0
$555$	42012.2i	3.21319i
$556$	0	0
$557$	10896.0i	0.828864i	0.910080	+	0.414432i	$0.136020\pi$
$557$	10896.0i	0.828864i	−0.910080	+	0.414432i	$0.863980\pi$
$558$	0	0
$559$	3619.16	0.273836
$560$	0	0
$561$	−34275.8	−2.57954
$562$	0	0
$563$	15441.8i	1.15594i	0.816057	+	0.577972i	$0.196156\pi$
$563$	15441.8i	1.15594i	−0.816057	+	0.577972i	$0.803844\pi$
$564$	0	0
$565$	− 38637.6i	− 2.87699i
$566$	0	0
$567$	4065.29	0.301104
$568$	0	0
$569$	−5615.59	−0.413739	−0.206870	−	0.978369i	$-0.566328\pi$
$569$	−5615.59	−0.413739	−0.206870	+	0.978369i	$0.566328\pi$
$570$	0	0
$571$	14371.0i	1.05325i	0.850097	+	0.526626i	$0.176543\pi$
$571$	14371.0i	1.05325i	−0.850097	+	0.526626i	$0.823457\pi$
$572$	0	0
$573$	3384.81i	0.246775i
$574$	0	0
$575$	−44028.9	−3.19327
$576$	0	0
$577$	−18933.5	−1.36605	−0.683027	−	0.730393i	$-0.739337\pi$
$577$	−18933.5	−1.36605	−0.683027	+	0.730393i	$0.739337\pi$
$578$	0	0
$579$	− 38758.1i	− 2.78192i
$580$	0	0
$581$	7071.96i	0.504982i
$582$	0	0
$583$	22887.9	1.62593
$584$	0	0
$585$	−10184.1	−0.719765
$586$	0	0
$587$	17871.6i	1.25663i	0.777959	+	0.628315i	$0.216255\pi$
$587$	17871.6i	1.25663i	−0.777959	+	0.628315i	$0.783745\pi$
$588$	0	0
$589$	21389.6i	1.49633i
$590$	0	0
$591$	−24132.1	−1.67963
$592$	0	0
$593$	45.4735	0.00314902	0.00157451	−	0.999999i	$-0.499499\pi$
$593$	45.4735	0.00314902	0.00157451	+	0.999999i	$0.499499\pi$
$594$	0	0
$595$	17334.9i	1.19439i
$596$	0	0
$597$	− 4135.36i	− 0.283499i
$598$	0	0
$599$	10336.6	0.705078	0.352539	−	0.935797i	$-0.385318\pi$
$599$	10336.6	0.705078	0.352539	+	0.935797i	$0.385318\pi$
$600$	0	0
$601$	4844.95	0.328835	0.164417	−	0.986391i	$-0.447426\pi$
$601$	4844.95	0.328835	0.164417	+	0.986391i	$0.447426\pi$
$602$	0	0
$603$	12607.5i	0.851435i
$604$	0	0
$605$	− 6157.67i	− 0.413793i
$606$	0	0
$607$	15710.1	1.05050	0.525249	−	0.850949i	$-0.323972\pi$
$607$	15710.1	1.05050	0.525249	+	0.850949i	$0.323972\pi$
$608$	0	0
$609$	−4980.91	−0.331423
$610$	0	0
$611$	1476.04i	0.0977317i
$612$	0	0
$613$	23320.2i	1.53653i	0.640130	+	0.768267i	$0.278881\pi$
$613$	23320.2i	1.53653i	−0.640130	+	0.768267i	$0.721119\pi$
$614$	0	0
$615$	−9884.30	−0.648087
$616$	0	0
$617$	−10540.0	−0.687720	−0.343860	−	0.939021i	$-0.611734\pi$
$617$	−10540.0	−0.687720	−0.343860	+	0.939021i	$0.611734\pi$
$618$	0	0
$619$	17788.6i	1.15507i	0.816367	+	0.577533i	$0.195984\pi$
$619$	17788.6i	1.15507i	−0.816367	+	0.577533i	$0.804016\pi$
$620$	0	0
$621$	− 4292.62i	− 0.277386i
$622$	0	0
$623$	5112.94	0.328805
$624$	0	0
$625$	73597.0	4.71021
$626$	0	0
$627$	36191.2i	2.30516i
$628$	0	0
$629$	27567.6i	1.74752i
$630$	0	0
$631$	12278.6	0.774648	0.387324	−	0.921944i	$-0.373399\pi$
$631$	12278.6	0.774648	0.387324	+	0.921944i	$0.373399\pi$
$632$	0	0
$633$	41466.1	2.60368
$634$	0	0
$635$	− 7577.25i	− 0.473534i
$636$	0	0
$637$	− 709.684i	− 0.0441424i
$638$	0	0
$639$	−197.064	−0.0121999
$640$	0	0
$641$	15729.2	0.969216	0.484608	−	0.874732i	$-0.338962\pi$
$641$	15729.2	0.969216	0.484608	+	0.874732i	$0.338962\pi$
$642$	0	0
$643$	7527.17i	0.461652i	0.972995	+	0.230826i	$0.0741429\pi$
$643$	7527.17i	0.461652i	−0.972995	+	0.230826i	$0.925857\pi$
$644$	0	0
$645$	− 42487.4i	− 2.59371i
$646$	0	0
$647$	−4431.02	−0.269245	−0.134622	−	0.990897i	$-0.542982\pi$
$647$	−4431.02	−0.269245	−0.134622	+	0.990897i	$0.542982\pi$
$648$	0	0
$649$	11124.8	0.672860
$650$	0	0
$651$	− 9736.06i	− 0.586154i
$652$	0	0
$653$	− 4155.52i	− 0.249032i	−0.992218	−	0.124516i	$-0.960262\pi$
$653$	− 4155.52i	− 0.249032i	0.992218	−	0.124516i	$-0.0397378\pi$
$654$	0	0
$655$	8989.75	0.536272
$656$	0	0
$657$	6645.38	0.394613
$658$	0	0
$659$	− 10380.1i	− 0.613583i	−0.951777	−	0.306791i	$-0.900745\pi$
$659$	− 10380.1i	− 0.613583i	0.951777	−	0.306791i	$-0.0992554\pi$
$660$	0	0
$661$	− 17788.9i	− 1.04676i	−0.852099	−	0.523381i	$-0.824671\pi$
$661$	− 17788.9i	− 1.04676i	0.852099	−	0.523381i	$-0.175329\pi$
$662$	0	0
$663$	−12378.2	−0.725080
$664$	0	0
$665$	18303.5	1.06734
$666$	0	0
$667$	− 11123.7i	− 0.645742i
$668$	0	0
$669$	17535.7i	1.01341i
$670$	0	0
$671$	3326.82	0.191402
$672$	0	0
$673$	13619.4	0.780074	0.390037	−	0.920799i	$-0.372462\pi$
$673$	13619.4	0.780074	0.390037	+	0.920799i	$0.372462\pi$
$674$	0	0
$675$	13179.5i	0.751525i
$676$	0	0
$677$	− 10454.1i	− 0.593478i	−0.954959	−	0.296739i	$-0.904101\pi$
$677$	− 10454.1i	− 0.593478i	0.954959	−	0.296739i	$-0.0958992\pi$
$678$	0	0
$679$	−4952.16	−0.279892
$680$	0	0
$681$	−51647.3	−2.90621
$682$	0	0
$683$	− 11405.9i	− 0.638996i	−0.947587	−	0.319498i	$-0.896486\pi$
$683$	− 11405.9i	− 0.638996i	0.947587	−	0.319498i	$-0.103514\pi$
$684$	0	0
$685$	− 35300.8i	− 1.96901i
$686$	0	0
$687$	11773.6	0.653844
$688$	0	0
$689$	8265.60	0.457031
$690$	0	0
$691$	− 5921.64i	− 0.326006i	−0.986626	−	0.163003i	$-0.947882\pi$
$691$	− 5921.64i	− 0.326006i	0.986626	−	0.163003i	$-0.0521180\pi$
$692$	0	0
$693$	− 8893.57i	− 0.487502i
$694$	0	0
$695$	−13442.2	−0.733656
$696$	0	0
$697$	−6485.89	−0.352468
$698$	0	0
$699$	− 12012.8i	− 0.650021i
$700$	0	0
$701$	24236.4i	1.30584i	0.757426	+	0.652921i	$0.226457\pi$
$701$	24236.4i	1.30584i	−0.757426	+	0.652921i	$0.773543\pi$
$702$	0	0
$703$	29108.1	1.56164
$704$	0	0
$705$	17328.0	0.925690
$706$	0	0
$707$	− 4175.90i	− 0.222137i
$708$	0	0
$709$	− 30145.3i	− 1.59680i	−0.602128	−	0.798400i	$-0.705680\pi$
$709$	− 30145.3i	− 1.59680i	0.602128	−	0.798400i	$-0.294320\pi$
$710$	0	0
$711$	8613.73	0.454346
$712$	0	0
$713$	21743.1	1.14206
$714$	0	0
$715$	− 12892.8i	− 0.674353i
$716$	0	0
$717$	37590.6i	1.95794i
$718$	0	0
$719$	−5159.52	−0.267619	−0.133809	−	0.991007i	$-0.542721\pi$
$719$	−5159.52	−0.267619	−0.133809	+	0.991007i	$0.542721\pi$
$720$	0	0
$721$	−3779.71	−0.195234
$722$	0	0
$723$	10751.7i	0.553059i
$724$	0	0
$725$	34152.7i	1.74952i
$726$	0	0
$727$	22475.2	1.14657	0.573286	−	0.819355i	$-0.305668\pi$
$727$	22475.2	1.14657	0.573286	+	0.819355i	$0.305668\pi$
$728$	0	0
$729$	25812.0	1.31139
$730$	0	0
$731$	− 27879.4i	− 1.41061i
$732$	0	0
$733$	15258.4i	0.768871i	0.923152	+	0.384436i	$0.125604\pi$
$733$	15258.4i	0.768871i	−0.923152	+	0.384436i	$0.874396\pi$
$734$	0	0
$735$	−8331.38	−0.418106
$736$	0	0
$737$	−15960.6	−0.797715
$738$	0	0
$739$	27519.3i	1.36984i	0.728616	+	0.684922i	$0.240164\pi$
$739$	27519.3i	1.36984i	−0.728616	+	0.684922i	$0.759836\pi$
$740$	0	0
$741$	13069.9i	0.647954i
$742$	0	0
$743$	−38497.6	−1.90086	−0.950431	−	0.310937i	$-0.899357\pi$
$743$	−38497.6	−1.90086	−0.950431	+	0.310937i	$0.899357\pi$
$744$	0	0
$745$	−44086.5	−2.16806
$746$	0	0
$747$	32005.2i	1.56761i
$748$	0	0
$749$	− 4623.48i	− 0.225552i
$750$	0	0
$751$	25060.1	1.21765	0.608827	−	0.793303i	$-0.291641\pi$
$751$	25060.1	1.21765	0.608827	+	0.793303i	$0.291641\pi$
$752$	0	0
$753$	−28913.5	−1.39929
$754$	0	0
$755$	15069.1i	0.726384i
$756$	0	0
$757$	− 7440.21i	− 0.357225i	−0.983920	−	0.178612i	$-0.942839\pi$
$757$	− 7440.21i	− 0.357225i	0.983920	−	0.178612i	$-0.0571608\pi$
$758$	0	0
$759$	36789.4	1.75938
$760$	0	0
$761$	11058.6	0.526773	0.263386	−	0.964690i	$-0.415161\pi$
$761$	11058.6	0.526773	0.263386	+	0.964690i	$0.415161\pi$
$762$	0	0
$763$	− 1639.10i	− 0.0777710i
$764$	0	0
$765$	78451.3i	3.70773i
$766$	0	0
$767$	4017.55	0.189133
$768$	0	0
$769$	39340.3	1.84479	0.922397	−	0.386243i	$-0.126227\pi$
$769$	39340.3	1.84479	0.922397	+	0.386243i	$0.126227\pi$
$770$	0	0
$771$	− 24765.7i	− 1.15683i
$772$	0	0
$773$	28953.3i	1.34719i	0.739100	+	0.673595i	$0.235251\pi$
$773$	28953.3i	1.34719i	−0.739100	+	0.673595i	$0.764749\pi$
$774$	0	0
$775$	−66757.4	−3.09419
$776$	0	0
$777$	−13249.4	−0.611736
$778$	0	0
$779$	6848.32i	0.314976i
$780$	0	0
$781$	− 249.476i	− 0.0114301i
$782$	0	0
$783$	−3329.73	−0.151973
$784$	0	0
$785$	71076.8	3.23164
$786$	0	0
$787$	− 15424.4i	− 0.698627i	−0.937006	−	0.349313i	$-0.886415\pi$
$787$	− 15424.4i	− 0.698627i	0.937006	−	0.349313i	$-0.113585\pi$
$788$	0	0
$789$	24971.2i	1.12674i
$790$	0	0
$791$	12185.2	0.547730
$792$	0	0
$793$	1201.43	0.0538008
$794$	0	0
$795$	− 97034.6i	− 4.32888i
$796$	0	0
$797$	− 13348.5i	− 0.593258i	−0.954993	−	0.296629i	$-0.904138\pi$
$797$	− 13348.5i	− 0.593258i	0.954993	−	0.296629i	$-0.0958625\pi$
$798$	0	0
$799$	11370.3	0.503446
$800$	0	0
$801$	23139.3	1.02071
$802$	0	0
$803$	8412.82i	0.369716i
$804$	0	0
$805$	− 18606.1i	− 0.814633i
$806$	0	0
$807$	−56140.2	−2.44886
$808$	0	0
$809$	18871.7	0.820139	0.410070	−	0.912054i	$-0.365504\pi$
$809$	18871.7	0.820139	0.410070	+	0.912054i	$0.365504\pi$
$810$	0	0
$811$	8113.79i	0.351312i	0.984452	+	0.175656i	$0.0562046\pi$
$811$	8113.79i	0.351312i	−0.984452	+	0.175656i	$0.943795\pi$
$812$	0	0
$813$	− 53692.9i	− 2.31623i
$814$	0	0
$815$	7567.57	0.325252
$816$	0	0
$817$	−29437.4	−1.26057
$818$	0	0
$819$	− 3211.77i	− 0.137031i
$820$	0	0
$821$	− 16368.2i	− 0.695804i	−0.937531	−	0.347902i	$-0.886894\pi$
$821$	− 16368.2i	− 0.695804i	0.937531	−	0.347902i	$-0.113106\pi$
$822$	0	0
$823$	−34575.7	−1.46444	−0.732220	−	0.681069i	$-0.761516\pi$
$823$	−34575.7	−1.46444	−0.732220	+	0.681069i	$0.761516\pi$
$824$	0	0
$825$	−112954.	−4.76672
$826$	0	0
$827$	43222.0i	1.81739i	0.417466	+	0.908693i	$0.362918\pi$
$827$	43222.0i	1.81739i	−0.417466	+	0.908693i	$0.637082\pi$
$828$	0	0
$829$	− 29288.6i	− 1.22706i	−0.789670	−	0.613532i	$-0.789748\pi$
$829$	− 29288.6i	− 1.22706i	0.789670	−	0.613532i	$-0.210252\pi$
$830$	0	0
$831$	−28754.9	−1.20036
$832$	0	0
$833$	−5466.89	−0.227391
$834$	0	0
$835$	− 67451.8i	− 2.79553i
$836$	0	0
$837$	− 6508.55i	− 0.268779i
$838$	0	0
$839$	−16518.3	−0.679708	−0.339854	−	0.940478i	$-0.610378\pi$
$839$	−16518.3	−0.679708	−0.339854	+	0.940478i	$0.610378\pi$
$840$	0	0
$841$	15760.5	0.646213
$842$	0	0
$843$	− 283.238i	− 0.0115720i
$844$	0	0
$845$	44108.9i	1.79573i
$846$	0	0
$847$	1941.94	0.0787792
$848$	0	0
$849$	−24794.6	−1.00230
$850$	0	0
$851$	− 29589.3i	− 1.19190i
$852$	0	0
$853$	− 8649.24i	− 0.347180i	−0.984818	−	0.173590i	$-0.944463\pi$
$853$	− 8649.24i	− 0.347180i	0.984818	−	0.173590i	$-0.0555367\pi$
$854$	0	0
$855$	82835.3	3.31334
$856$	0	0
$857$	16342.9	0.651413	0.325707	−	0.945471i	$-0.394398\pi$
$857$	16342.9	0.651413	0.325707	+	0.945471i	$0.394398\pi$
$858$	0	0
$859$	29613.4i	1.17625i	0.808772	+	0.588123i	$0.200133\pi$
$859$	29613.4i	1.17625i	−0.808772	+	0.588123i	$0.799867\pi$
$860$	0	0
$861$	− 3117.21i	− 0.123385i
$862$	0	0
$863$	20366.7	0.803348	0.401674	−	0.915783i	$-0.368429\pi$
$863$	20366.7	0.803348	0.401674	+	0.915783i	$0.368429\pi$
$864$	0	0
$865$	−57955.5	−2.27809
$866$	0	0
$867$	57717.6i	2.26089i
$868$	0	0
$869$	10904.7i	0.425680i
$870$	0	0
$871$	−5763.92	−0.224228
$872$	0	0
$873$	−22411.7	−0.868867
$874$	0	0
$875$	37704.3i	1.45673i
$876$	0	0
$877$	− 35210.6i	− 1.35573i	−0.735186	−	0.677866i	$-0.762905\pi$
$877$	− 35210.6i	− 1.35573i	0.735186	−	0.677866i	$-0.237095\pi$
$878$	0	0
$879$	33803.8	1.29713
$880$	0	0
$881$	−7354.03	−0.281230	−0.140615	−	0.990064i	$-0.544908\pi$
$881$	−7354.03	−0.281230	−0.140615	+	0.990064i	$0.544908\pi$
$882$	0	0
$883$	− 28837.9i	− 1.09906i	−0.835473	−	0.549531i	$-0.814806\pi$
$883$	− 28837.9i	− 1.09906i	0.835473	−	0.549531i	$-0.185194\pi$
$884$	0	0
$885$	− 47164.3i	− 1.79142i
$886$	0	0
$887$	10861.0	0.411136	0.205568	−	0.978643i	$-0.434096\pi$
$887$	10861.0	0.411136	0.205568	+	0.978643i	$0.434096\pi$
$888$	0	0
$889$	2389.64	0.0901528
$890$	0	0
$891$	23291.3i	0.875744i
$892$	0	0
$893$	− 12005.7i	− 0.449895i
$894$	0	0
$895$	45625.6	1.70402
$896$	0	0
$897$	13285.9	0.494542
$898$	0	0
$899$	− 16865.9i	− 0.625706i
$900$	0	0
$901$	− 63672.3i	− 2.35431i
$902$	0	0
$903$	13399.3	0.493798
$904$	0	0
$905$	−44102.7	−1.61992
$906$	0	0
$907$	18682.8i	0.683961i	0.939707	+	0.341980i	$0.111098\pi$
$907$	18682.8i	0.683961i	−0.939707	+	0.341980i	$0.888902\pi$
$908$	0	0
$909$	− 18898.6i	− 0.689580i
$910$	0	0
$911$	34392.2	1.25078	0.625392	−	0.780311i	$-0.284939\pi$
$911$	34392.2	1.25078	0.625392	+	0.780311i	$0.284939\pi$
$912$	0	0
$913$	−40517.4	−1.46871
$914$	0	0
$915$	− 14104.3i	− 0.509588i
$916$	0	0
$917$	2835.10i	0.102097i
$918$	0	0
$919$	−14715.8	−0.528216	−0.264108	−	0.964493i	$-0.585078\pi$
$919$	−14715.8	−0.528216	−0.264108	+	0.964493i	$0.585078\pi$
$920$	0	0
$921$	4507.51	0.161268
$922$	0	0
$923$	− 90.0943i	− 0.00321288i
$924$	0	0
$925$	90847.3i	3.22923i
$926$	0	0
$927$	−17105.6	−0.606065
$928$	0	0
$929$	−36204.2	−1.27860	−0.639300	−	0.768957i	$-0.720776\pi$
$929$	−36204.2	−1.27860	−0.639300	+	0.768957i	$0.720776\pi$
$930$	0	0
$931$	5772.39i	0.203203i
$932$	0	0
$933$	− 57415.8i	− 2.01469i
$934$	0	0
$935$	−99316.6	−3.47380
$936$	0	0
$937$	43636.5	1.52139	0.760694	−	0.649110i	$-0.224859\pi$
$937$	43636.5	1.52139	0.760694	+	0.649110i	$0.224859\pi$
$938$	0	0
$939$	− 50962.7i	− 1.77114i
$940$	0	0
$941$	− 4751.54i	− 0.164608i	−0.996607	−	0.0823038i	$-0.973772\pi$
$941$	− 4751.54i	− 0.164608i	0.996607	−	0.0823038i	$-0.0262278\pi$
$942$	0	0
$943$	6961.54	0.240402
$944$	0	0
$945$	−5569.52	−0.191721
$946$	0	0
$947$	49387.3i	1.69469i	0.531044	+	0.847344i	$0.321800\pi$
$947$	49387.3i	1.69469i	−0.531044	+	0.847344i	$0.678200\pi$
$948$	0	0
$949$	3038.16i	0.103923i
$950$	0	0
$951$	55269.2	1.88457
$952$	0	0
$953$	−46336.1	−1.57500	−0.787499	−	0.616316i	$-0.788625\pi$
$953$	−46336.1	−1.57500	−0.787499	+	0.616316i	$0.788625\pi$
$954$	0	0
$955$	9807.72i	0.332325i
$956$	0	0
$957$	− 28537.2i	− 0.963924i
$958$	0	0
$959$	11132.8	0.374866
$960$	0	0
$961$	3176.38	0.106622
$962$	0	0
$963$	− 20924.2i	− 0.700180i
$964$	0	0
$965$	− 112304.i	− 3.74633i
$966$	0	0
$967$	21078.1	0.700957	0.350479	−	0.936571i	$-0.386019\pi$
$967$	21078.1	0.700957	0.350479	+	0.936571i	$0.386019\pi$
$968$	0	0
$969$	100681.	3.33781
$970$	0	0
$971$	− 27225.5i	− 0.899802i	−0.893079	−	0.449901i	$-0.851459\pi$
$971$	− 27225.5i	− 0.899802i	0.893079	−	0.449901i	$-0.148541\pi$
$972$	0	0
$973$	− 4239.26i	− 0.139676i
$974$	0	0
$975$	−40791.4	−1.33987
$976$	0	0
$977$	4699.41	0.153887	0.0769434	−	0.997035i	$-0.475484\pi$
$977$	4699.41	0.153887	0.0769434	+	0.997035i	$0.475484\pi$
$978$	0	0
$979$	29293.6i	0.956310i
$980$	0	0
$981$	− 7417.96i	− 0.241424i
$982$	0	0
$983$	48128.7	1.56161	0.780807	−	0.624773i	$-0.214808\pi$
$983$	48128.7	1.56161	0.780807	+	0.624773i	$0.214808\pi$
$984$	0	0
$985$	−69924.5	−2.26191
$986$	0	0
$987$	5464.74i	0.176236i
$988$	0	0
$989$	29924.0i	0.962111i
$990$	0	0
$991$	214.168	0.00686506	0.00343253	−	0.999994i	$-0.498907\pi$
$991$	214.168	0.00686506	0.00343253	+	0.999994i	$0.498907\pi$
$992$	0	0
$993$	8608.96	0.275123
$994$	0	0
$995$	− 11982.5i	− 0.381780i
$996$	0	0
$997$	− 12678.3i	− 0.402735i	−0.979516	−	0.201368i	$-0.935461\pi$
$997$	− 12678.3i	− 0.402735i	0.979516	−	0.201368i	$-0.0645385\pi$
$998$	0	0
$999$	−8857.20	−0.280510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.b.a.225.3	yes	4
4.3	odd	2		448.4.b.b.225.2	yes	4
8.3	odd	2		448.4.b.b.225.3	yes	4
8.5	even	2	inner	448.4.b.a.225.2	✓	4
16.3	odd	4		1792.4.a.b.1.2		2
16.5	even	4		1792.4.a.a.1.2		2
16.11	odd	4		1792.4.a.c.1.1		2
16.13	even	4		1792.4.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.4.b.a.225.2	✓	4	8.5	even	2	inner
448.4.b.a.225.3	yes	4	1.1	even	1	trivial
448.4.b.b.225.2	yes	4	4.3	odd	2
448.4.b.b.225.3	yes	4	8.3	odd	2
1792.4.a.a.1.2		2	16.5	even	4
1792.4.a.b.1.2		2	16.3	odd	4
1792.4.a.c.1.1		2	16.11	odd	4
1792.4.a.d.1.1		2	16.13	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.66025i q^{3} +22.1962i q^{5} -7.00000 q^{7} -31.6795 q^{9} -40.1051i q^{11} -14.4833i q^{13} -170.028 q^{15} -111.569 q^{17} +117.804i q^{19} -53.6218i q^{21} +119.751 q^{23} -367.669 q^{25} -35.8461i q^{27} -92.8897i q^{29} +181.569 q^{31} +307.215 q^{33} -155.373i q^{35} -247.090i q^{37} +110.946 q^{39} +58.1333 q^{41} +249.885i q^{43} -703.163i q^{45} -101.913 q^{47} +49.0000 q^{49} -854.649i q^{51} +570.697i q^{53} +890.179 q^{55} -902.407 q^{57} +277.391i q^{59} +82.9526i q^{61} +221.756 q^{63} +321.474 q^{65} -397.969i q^{67} +917.325i q^{69} +6.22055 q^{71} -209.769 q^{73} -2816.44i q^{75} +280.736i q^{77} -271.902 q^{79} -580.756 q^{81} -1010.28i q^{83} -2476.41i q^{85} +711.559 q^{87} -730.420 q^{89} +101.383i q^{91} +1390.87i q^{93} -2614.79 q^{95} +707.451 q^{97} +1270.51i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 28 q^{7} - 196 q^{9} - 112 q^{15} - 280 q^{17} + 576 q^{23} - 764 q^{25} + 560 q^{31} + 1312 q^{33} - 928 q^{39} - 488 q^{41} + 784 q^{47} + 196 q^{49} + 928 q^{55} + 672 q^{57} + 1372 q^{63} + 2048 q^{65}+ \cdots - 856 q^{97}+O(q^{100}) \) 4 * q - 28 * q^7 - 196 * q^9 - 112 * q^15 - 280 * q^17 + 576 * q^23 - 764 * q^25 + 560 * q^31 + 1312 * q^33 - 928 * q^39 - 488 * q^41 + 784 * q^47 + 196 * q^49 + 928 * q^55 + 672 * q^57 + 1372 * q^63 + 2048 * q^65 + 496 * q^71 + 408 * q^73 + 880 * q^79 + 2596 * q^81 + 1904 * q^87 - 2312 * q^89 - 8256 * q^95 - 856 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more