LMFDB - Embedded newform 448.3.d.e.127.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(127,448)]

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([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("448.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.8
Root			$2.92812i$ of defining polynomial
Character	$\chi$	$=$	448.127
Dual form			448.3.d.e.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.85623i q^{3} +5.78167 q^{5} +2.64575i q^{7} -25.2955 q^{9} +O(q^{10})$	$q+5.85623i q^{3} +5.78167 q^{5} +2.64575i q^{7} -25.2955 q^{9} -3.01966i q^{11} -9.78167 q^{13} +33.8588i q^{15} -11.6027 q^{17} +25.5687i q^{19} -15.4941 q^{21} +26.1571i q^{23} +8.42771 q^{25} -95.4301i q^{27} -1.56334 q^{29} -12.0107i q^{31} +17.6839 q^{33} +15.2969i q^{35} +70.6721 q^{37} -57.2837i q^{39} -49.8775 q^{41} +73.2730i q^{43} -146.250 q^{45} -44.2248i q^{47} -7.00000 q^{49} -67.9479i q^{51} +54.2355 q^{53} -17.4587i q^{55} -149.736 q^{57} -12.4706i q^{59} -35.6770 q^{61} -66.9255i q^{63} -56.5544 q^{65} +24.4891i q^{67} -153.182 q^{69} +11.0480i q^{71} +74.3713 q^{73} +49.3547i q^{75} +7.98928 q^{77} -22.8912i q^{79} +331.202 q^{81} +48.1934i q^{83} -67.0828 q^{85} -9.15529i q^{87} +67.4801 q^{89} -25.8799i q^{91} +70.3376 q^{93} +147.830i q^{95} -7.75949 q^{97} +76.3838i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{9}+O(q^{10}) $ 8 * q - 40 * q^9	$ 8 q - 40 q^{9} - 32 q^{13} - 16 q^{17} + 104 q^{25} + 80 q^{29} + 176 q^{37} + 144 q^{41} - 256 q^{45} - 56 q^{49} - 48 q^{53} - 400 q^{57} + 192 q^{61} - 304 q^{65} - 576 q^{69} + 272 q^{73} + 112 q^{77} + 504 q^{81} + 160 q^{85} - 80 q^{89} + 608 q^{93} + 528 q^{97}+O(q^{100}) $ 8 * q - 40 * q^9 - 32 * q^13 - 16 * q^17 + 104 * q^25 + 80 * q^29 + 176 * q^37 + 144 * q^41 - 256 * q^45 - 56 * q^49 - 48 * q^53 - 400 * q^57 + 192 * q^61 - 304 * q^65 - 576 * q^69 + 272 * q^73 + 112 * q^77 + 504 * q^81 + 160 * q^85 - 80 * q^89 + 608 * q^93 + 528 * q^97

Display 100 coefficients 10 coefficients

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
$3$	5.85623i	1.95208i	0.217596	+	0.976039i	$0.430179\pi$
$3$	5.85623i	1.95208i	−0.217596	+	0.976039i	$0.569821\pi$
$4$	0	0
$5$	5.78167	1.15633	0.578167	−	0.815918i	$-0.303768\pi$
$5$	5.78167	1.15633	0.578167	+	0.815918i	$0.303768\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−25.2955	−2.81061
$10$	0	0
$11$	− 3.01966i	− 0.274515i	−0.990535	−	0.137257i	$-0.956171\pi$
$11$	− 3.01966i	− 0.274515i	0.990535	−	0.137257i	$-0.0438288\pi$
$12$	0	0
$13$	−9.78167	−0.752436	−0.376218	−	0.926531i	$-0.622776\pi$
$13$	−9.78167	−0.752436	−0.376218	+	0.926531i	$0.622776\pi$
$14$	0	0
$15$	33.8588i	2.25725i
$16$	0	0
$17$	−11.6027	−0.682510	−0.341255	−	0.939971i	$-0.610852\pi$
$17$	−11.6027	−0.682510	−0.341255	+	0.939971i	$0.610852\pi$
$18$	0	0
$19$	25.5687i	1.34572i	0.739769	+	0.672861i	$0.234935\pi$
$19$	25.5687i	1.34572i	−0.739769	+	0.672861i	$0.765065\pi$
$20$	0	0
$21$	−15.4941	−0.737816
$22$	0	0
$23$	26.1571i	1.13726i	0.822592	+	0.568632i	$0.192527\pi$
$23$	26.1571i	1.13726i	−0.822592	+	0.568632i	$0.807473\pi$
$24$	0	0
$25$	8.42771	0.337109
$26$	0	0
$27$	− 95.4301i	− 3.53445i
$28$	0	0
$29$	−1.56334	−0.0539083	−0.0269542	−	0.999637i	$-0.508581\pi$
$29$	−1.56334	−0.0539083	−0.0269542	+	0.999637i	$0.508581\pi$
$30$	0	0
$31$	− 12.0107i	− 0.387443i	−0.981057	−	0.193721i	$-0.937944\pi$
$31$	− 12.0107i	− 0.387443i	0.981057	−	0.193721i	$-0.0620557\pi$
$32$	0	0
$33$	17.6839	0.535875
$34$	0	0
$35$	15.2969i	0.437053i
$36$	0	0
$37$	70.6721	1.91006	0.955029	−	0.296513i	$-0.0958238\pi$
$37$	70.6721	1.91006	0.955029	+	0.296513i	$0.0958238\pi$
$38$	0	0
$39$	− 57.2837i	− 1.46881i
$40$	0	0
$41$	−49.8775	−1.21652	−0.608262	−	0.793736i	$-0.708133\pi$
$41$	−49.8775	−1.21652	−0.608262	+	0.793736i	$0.708133\pi$
$42$	0	0
$43$	73.2730i	1.70402i	0.523522	+	0.852012i	$0.324618\pi$
$43$	73.2730i	1.70402i	−0.523522	+	0.852012i	$0.675382\pi$
$44$	0	0
$45$	−146.250	−3.25000
$46$	0	0
$47$	− 44.2248i	− 0.940952i	−0.882413	−	0.470476i	$-0.844082\pi$
$47$	− 44.2248i	− 0.940952i	0.882413	−	0.470476i	$-0.155918\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 67.9479i	− 1.33231i
$52$	0	0
$53$	54.2355	1.02331	0.511655	−	0.859191i	$-0.329032\pi$
$53$	54.2355	1.02331	0.511655	+	0.859191i	$0.329032\pi$
$54$	0	0
$55$	− 17.4587i	− 0.317431i
$56$	0	0
$57$	−149.736	−2.62695
$58$	0	0
$59$	− 12.4706i	− 0.211367i	−0.994400	−	0.105683i	$-0.966297\pi$
$59$	− 12.4706i	− 0.211367i	0.994400	−	0.105683i	$-0.0337030\pi$
$60$	0	0
$61$	−35.6770	−0.584869	−0.292435	−	0.956285i	$-0.594465\pi$
$61$	−35.6770	−0.584869	−0.292435	+	0.956285i	$0.594465\pi$
$62$	0	0
$63$	− 66.9255i	− 1.06231i
$64$	0	0
$65$	−56.5544	−0.870068
$66$	0	0
$67$	24.4891i	0.365509i	0.983159	+	0.182754i	$0.0585013\pi$
$67$	24.4891i	0.365509i	−0.983159	+	0.182754i	$0.941499\pi$
$68$	0	0
$69$	−153.182	−2.22003
$70$	0	0
$71$	11.0480i	0.155606i	0.996969	+	0.0778030i	$0.0247905\pi$
$71$	11.0480i	0.155606i	−0.996969	+	0.0778030i	$0.975209\pi$
$72$	0	0
$73$	74.3713	1.01879	0.509393	−	0.860534i	$-0.329870\pi$
$73$	74.3713	1.01879	0.509393	+	0.860534i	$0.329870\pi$
$74$	0	0
$75$	49.3547i	0.658062i
$76$	0	0
$77$	7.98928	0.103757
$78$	0	0
$79$	− 22.8912i	− 0.289762i	−0.989449	−	0.144881i	$-0.953720\pi$
$79$	− 22.8912i	− 0.289762i	0.989449	−	0.144881i	$-0.0462800\pi$
$80$	0	0
$81$	331.202	4.08891
$82$	0	0
$83$	48.1934i	0.580644i	0.956929	+	0.290322i	$0.0937623\pi$
$83$	48.1934i	0.580644i	−0.956929	+	0.290322i	$0.906238\pi$
$84$	0	0
$85$	−67.0828	−0.789210
$86$	0	0
$87$	− 9.15529i	− 0.105233i
$88$	0	0
$89$	67.4801	0.758204	0.379102	−	0.925355i	$-0.376233\pi$
$89$	67.4801	0.758204	0.379102	+	0.925355i	$0.376233\pi$
$90$	0	0
$91$	− 25.8799i	− 0.284394i
$92$	0	0
$93$	70.3376	0.756318
$94$	0	0
$95$	147.830i	1.55610i
$96$	0	0
$97$	−7.75949	−0.0799947	−0.0399974	−	0.999200i	$-0.512735\pi$
$97$	−7.75949	−0.0799947	−0.0399974	+	0.999200i	$0.512735\pi$
$98$	0	0
$99$	76.3838i	0.771554i
$100$	0	0
$101$	102.524	1.01509	0.507546	−	0.861624i	$-0.330553\pi$
$101$	102.524	1.01509	0.507546	+	0.861624i	$0.330553\pi$
$102$	0	0
$103$	14.6874i	0.142596i	0.997455	+	0.0712980i	$0.0227141\pi$
$103$	14.6874i	0.142596i	−0.997455	+	0.0712980i	$0.977286\pi$
$104$	0	0
$105$	−89.5820	−0.853162
$106$	0	0
$107$	70.0212i	0.654404i	0.944954	+	0.327202i	$0.106106\pi$
$107$	70.0212i	0.654404i	−0.944954	+	0.327202i	$0.893894\pi$
$108$	0	0
$109$	65.9204	0.604774	0.302387	−	0.953185i	$-0.402217\pi$
$109$	65.9204	0.604774	0.302387	+	0.953185i	$0.402217\pi$
$110$	0	0
$111$	413.873i	3.72858i
$112$	0	0
$113$	83.2184	0.736446	0.368223	−	0.929737i	$-0.379966\pi$
$113$	83.2184	0.736446	0.368223	+	0.929737i	$0.379966\pi$
$114$	0	0
$115$	151.232i	1.31506i
$116$	0	0
$117$	247.432	2.11480
$118$	0	0
$119$	− 30.6978i	− 0.257965i
$120$	0	0
$121$	111.882	0.924642
$122$	0	0
$123$	− 292.094i	− 2.37475i
$124$	0	0
$125$	−95.8155	−0.766524
$126$	0	0
$127$	− 102.292i	− 0.805452i	−0.915321	−	0.402726i	$-0.868063\pi$
$127$	− 102.292i	− 0.805452i	0.915321	−	0.402726i	$-0.131937\pi$
$128$	0	0
$129$	−429.104	−3.32639
$130$	0	0
$131$	89.1759i	0.680732i	0.940293	+	0.340366i	$0.110551\pi$
$131$	89.1759i	0.680732i	−0.940293	+	0.340366i	$0.889449\pi$
$132$	0	0
$133$	−67.6484	−0.508635
$134$	0	0
$135$	− 551.745i	− 4.08700i
$136$	0	0
$137$	−11.3800	−0.0830660	−0.0415330	−	0.999137i	$-0.513224\pi$
$137$	−11.3800	−0.0830660	−0.0415330	+	0.999137i	$0.513224\pi$
$138$	0	0
$139$	121.256i	0.872343i	0.899863	+	0.436172i	$0.143666\pi$
$139$	121.256i	0.872343i	−0.899863	+	0.436172i	$0.856334\pi$
$140$	0	0
$141$	258.991	1.83681
$142$	0	0
$143$	29.5374i	0.206555i
$144$	0	0
$145$	−9.03872	−0.0623360
$146$	0	0
$147$	− 40.9936i	− 0.278868i
$148$	0	0
$149$	63.5731	0.426665	0.213333	−	0.976980i	$-0.431568\pi$
$149$	63.5731	0.426665	0.213333	+	0.976980i	$0.431568\pi$
$150$	0	0
$151$	122.414i	0.810689i	0.914164	+	0.405344i	$0.132848\pi$
$151$	122.414i	0.810689i	−0.914164	+	0.405344i	$0.867152\pi$
$152$	0	0
$153$	293.495	1.91827
$154$	0	0
$155$	− 69.4420i	− 0.448013i
$156$	0	0
$157$	−39.6070	−0.252274	−0.126137	−	0.992013i	$-0.540258\pi$
$157$	−39.6070	−0.252274	−0.126137	+	0.992013i	$0.540258\pi$
$158$	0	0
$159$	317.616i	1.99758i
$160$	0	0
$161$	−69.2051	−0.429845
$162$	0	0
$163$	− 32.9017i	− 0.201851i	−0.994894	−	0.100925i	$-0.967820\pi$
$163$	− 32.9017i	− 0.201851i	0.994894	−	0.100925i	$-0.0321803\pi$
$164$	0	0
$165$	102.242	0.619650
$166$	0	0
$167$	− 98.0405i	− 0.587069i	−0.955949	−	0.293534i	$-0.905168\pi$
$167$	− 98.0405i	− 0.587069i	0.955949	−	0.293534i	$-0.0948315\pi$
$168$	0	0
$169$	−73.3189	−0.433840
$170$	0	0
$171$	− 646.772i	− 3.78229i
$172$	0	0
$173$	336.891	1.94734	0.973672	−	0.227951i	$-0.0732028\pi$
$173$	336.891	1.94734	0.973672	+	0.227951i	$0.0732028\pi$
$174$	0	0
$175$	22.2976i	0.127415i
$176$	0	0
$177$	73.0309	0.412604
$178$	0	0
$179$	− 51.4537i	− 0.287451i	−0.989618	−	0.143725i	$-0.954092\pi$
$179$	− 51.4537i	− 0.287451i	0.989618	−	0.143725i	$-0.0459082\pi$
$180$	0	0
$181$	−281.807	−1.55695	−0.778473	−	0.627678i	$-0.784006\pi$
$181$	−281.807	−1.55695	−0.778473	+	0.627678i	$0.784006\pi$
$182$	0	0
$183$	− 208.933i	− 1.14171i
$184$	0	0
$185$	408.603	2.20866
$186$	0	0
$187$	35.0362i	0.187359i
$188$	0	0
$189$	252.484	1.33590
$190$	0	0
$191$	165.004i	0.863897i	0.901898	+	0.431949i	$0.142174\pi$
$191$	165.004i	0.863897i	−0.901898	+	0.431949i	$0.857826\pi$
$192$	0	0
$193$	−64.6933	−0.335199	−0.167599	−	0.985855i	$-0.553601\pi$
$193$	−64.6933	−0.335199	−0.167599	+	0.985855i	$0.553601\pi$
$194$	0	0
$195$	− 331.196i	− 1.69844i
$196$	0	0
$197$	−189.712	−0.963003	−0.481501	−	0.876445i	$-0.659908\pi$
$197$	−189.712	−0.963003	−0.481501	+	0.876445i	$0.659908\pi$
$198$	0	0
$199$	205.590i	1.03312i	0.856252	+	0.516559i	$0.172787\pi$
$199$	205.590i	1.03312i	−0.856252	+	0.516559i	$0.827213\pi$
$200$	0	0
$201$	−143.414	−0.713502
$202$	0	0
$203$	− 4.13621i	− 0.0203754i
$204$	0	0
$205$	−288.375	−1.40671
$206$	0	0
$207$	− 661.655i	− 3.19640i
$208$	0	0
$209$	77.2089	0.369421
$210$	0	0
$211$	220.725i	1.04609i	0.852305	+	0.523045i	$0.175204\pi$
$211$	220.725i	1.04609i	−0.852305	+	0.523045i	$0.824796\pi$
$212$	0	0
$213$	−64.6998	−0.303755
$214$	0	0
$215$	423.641i	1.97042i
$216$	0	0
$217$	31.7774	0.146440
$218$	0	0
$219$	435.536i	1.98875i
$220$	0	0
$221$	113.493	0.513545
$222$	0	0
$223$	− 163.489i	− 0.733134i	−0.930392	−	0.366567i	$-0.880533\pi$
$223$	− 163.489i	− 0.733134i	0.930392	−	0.366567i	$-0.119467\pi$
$224$	0	0
$225$	−213.183	−0.947480
$226$	0	0
$227$	− 221.227i	− 0.974569i	−0.873243	−	0.487285i	$-0.837987\pi$
$227$	− 221.227i	− 0.974569i	0.873243	−	0.487285i	$-0.162013\pi$
$228$	0	0
$229$	−233.563	−1.01993	−0.509964	−	0.860196i	$-0.670341\pi$
$229$	−233.563	−1.01993	−0.509964	+	0.860196i	$0.670341\pi$
$230$	0	0
$231$	46.7871i	0.202542i
$232$	0	0
$233$	−13.2573	−0.0568983	−0.0284491	−	0.999595i	$-0.509057\pi$
$233$	−13.2573	−0.0568983	−0.0284491	+	0.999595i	$0.509057\pi$
$234$	0	0
$235$	− 255.693i	− 1.08806i
$236$	0	0
$237$	134.056	0.565638
$238$	0	0
$239$	353.231i	1.47796i	0.673730	+	0.738978i	$0.264691\pi$
$239$	353.231i	1.47796i	−0.673730	+	0.738978i	$0.735309\pi$
$240$	0	0
$241$	181.261	0.752121	0.376060	−	0.926595i	$-0.377279\pi$
$241$	181.261	0.752121	0.376060	+	0.926595i	$0.377279\pi$
$242$	0	0
$243$	1080.72i	4.44742i
$244$	0	0
$245$	−40.4717	−0.165191
$246$	0	0
$247$	− 250.105i	− 1.01257i
$248$	0	0
$249$	−282.232	−1.13346
$250$	0	0
$251$	98.4522i	0.392240i	0.980580	+	0.196120i	$0.0628342\pi$
$251$	98.4522i	0.392240i	−0.980580	+	0.196120i	$0.937166\pi$
$252$	0	0
$253$	78.9856	0.312196
$254$	0	0
$255$	− 392.853i	− 1.54060i
$256$	0	0
$257$	−279.527	−1.08765	−0.543827	−	0.839197i	$-0.683025\pi$
$257$	−279.527	−1.08765	−0.543827	+	0.839197i	$0.683025\pi$
$258$	0	0
$259$	186.981i	0.721934i
$260$	0	0
$261$	39.5455	0.151515
$262$	0	0
$263$	45.3085i	0.172276i	0.996283	+	0.0861378i	$0.0274525\pi$
$263$	45.3085i	0.172276i	−0.996283	+	0.0861378i	$0.972547\pi$
$264$	0	0
$265$	313.572	1.18329
$266$	0	0
$267$	395.179i	1.48007i
$268$	0	0
$269$	220.227	0.818686	0.409343	−	0.912380i	$-0.365758\pi$
$269$	220.227	0.818686	0.409343	+	0.912380i	$0.365758\pi$
$270$	0	0
$271$	− 54.8463i	− 0.202385i	−0.994867	−	0.101193i	$-0.967734\pi$
$271$	− 54.8463i	− 0.202385i	0.994867	−	0.101193i	$-0.0322658\pi$
$272$	0	0
$273$	151.559	0.555160
$274$	0	0
$275$	− 25.4489i	− 0.0925413i
$276$	0	0
$277$	−320.078	−1.15552	−0.577758	−	0.816208i	$-0.696072\pi$
$277$	−320.078	−1.15552	−0.577758	+	0.816208i	$0.696072\pi$
$278$	0	0
$279$	303.817i	1.08895i
$280$	0	0
$281$	526.445	1.87347	0.936734	−	0.350041i	$-0.113832\pi$
$281$	526.445	1.87347	0.936734	+	0.350041i	$0.113832\pi$
$282$	0	0
$283$	331.932i	1.17290i	0.809984	+	0.586452i	$0.199476\pi$
$283$	331.932i	1.17290i	−0.809984	+	0.586452i	$0.800524\pi$
$284$	0	0
$285$	−865.726	−3.03763
$286$	0	0
$287$	− 131.963i	− 0.459803i
$288$	0	0
$289$	−154.378	−0.534180
$290$	0	0
$291$	− 45.4414i	− 0.156156i
$292$	0	0
$293$	240.715	0.821554	0.410777	−	0.911736i	$-0.365258\pi$
$293$	240.715	0.821554	0.410777	+	0.911736i	$0.365258\pi$
$294$	0	0
$295$	− 72.1011i	− 0.244410i
$296$	0	0
$297$	−288.167	−0.970259
$298$	0	0
$299$	− 255.860i	− 0.855718i
$300$	0	0
$301$	−193.862	−0.644061
$302$	0	0
$303$	600.407i	1.98154i
$304$	0	0
$305$	−206.273	−0.676304
$306$	0	0
$307$	− 221.862i	− 0.722678i	−0.932435	−	0.361339i	$-0.882320\pi$
$307$	− 221.862i	− 0.722678i	0.932435	−	0.361339i	$-0.117680\pi$
$308$	0	0
$309$	−86.0128	−0.278359
$310$	0	0
$311$	− 476.048i	− 1.53070i	−0.643614	−	0.765350i	$-0.722566\pi$
$311$	− 476.048i	− 1.53070i	0.643614	−	0.765350i	$-0.277434\pi$
$312$	0	0
$313$	−495.170	−1.58201	−0.791006	−	0.611808i	$-0.790442\pi$
$313$	−495.170	−1.58201	−0.791006	+	0.611808i	$0.790442\pi$
$314$	0	0
$315$	− 386.941i	− 1.22839i
$316$	0	0
$317$	493.029	1.55530	0.777649	−	0.628699i	$-0.216412\pi$
$317$	493.029	1.55530	0.777649	+	0.628699i	$0.216412\pi$
$318$	0	0
$319$	4.72077i	0.0147986i
$320$	0	0
$321$	−410.060	−1.27745
$322$	0	0
$323$	− 296.665i	− 0.918468i
$324$	0	0
$325$	−82.4371	−0.253653
$326$	0	0
$327$	386.045i	1.18057i
$328$	0	0
$329$	117.008	0.355647
$330$	0	0
$331$	− 586.929i	− 1.77320i	−0.462539	−	0.886599i	$-0.653061\pi$
$331$	− 586.929i	− 1.77320i	0.462539	−	0.886599i	$-0.346939\pi$
$332$	0	0
$333$	−1787.69	−5.36842
$334$	0	0
$335$	141.588i	0.422650i
$336$	0	0
$337$	22.2636	0.0660639	0.0330320	−	0.999454i	$-0.489484\pi$
$337$	22.2636	0.0660639	0.0330320	+	0.999454i	$0.489484\pi$
$338$	0	0
$339$	487.347i	1.43760i
$340$	0	0
$341$	−36.2683	−0.106359
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−885.647	−2.56709
$346$	0	0
$347$	− 264.797i	− 0.763103i	−0.924348	−	0.381552i	$-0.875390\pi$
$347$	− 264.797i	− 0.763103i	0.924348	−	0.381552i	$-0.124610\pi$
$348$	0	0
$349$	−101.730	−0.291489	−0.145745	−	0.989322i	$-0.546558\pi$
$349$	−101.730	−0.291489	−0.145745	+	0.989322i	$0.546558\pi$
$350$	0	0
$351$	933.466i	2.65945i
$352$	0	0
$353$	472.399	1.33824	0.669121	−	0.743154i	$-0.266671\pi$
$353$	472.399	1.33824	0.669121	+	0.743154i	$0.266671\pi$
$354$	0	0
$355$	63.8760i	0.179933i
$356$	0	0
$357$	179.773	0.503567
$358$	0	0
$359$	− 473.453i	− 1.31881i	−0.751788	−	0.659405i	$-0.770808\pi$
$359$	− 473.453i	− 1.31881i	0.751788	−	0.659405i	$-0.229192\pi$
$360$	0	0
$361$	−292.758	−0.810965
$362$	0	0
$363$	655.205i	1.80497i
$364$	0	0
$365$	429.991	1.17806
$366$	0	0
$367$	− 467.459i	− 1.27373i	−0.770975	−	0.636866i	$-0.780231\pi$
$367$	− 467.459i	− 1.27373i	0.770975	−	0.636866i	$-0.219769\pi$
$368$	0	0
$369$	1261.67	3.41917
$370$	0	0
$371$	143.494i	0.386775i
$372$	0	0
$373$	−534.491	−1.43295	−0.716475	−	0.697613i	$-0.754246\pi$
$373$	−534.491	−1.43295	−0.716475	+	0.697613i	$0.754246\pi$
$374$	0	0
$375$	− 561.118i	− 1.49631i
$376$	0	0
$377$	15.2921	0.0405626
$378$	0	0
$379$	− 204.355i	− 0.539196i	−0.962973	−	0.269598i	$-0.913109\pi$
$379$	− 204.355i	− 0.539196i	0.962973	−	0.269598i	$-0.0868908\pi$
$380$	0	0
$381$	599.048	1.57231
$382$	0	0
$383$	− 622.590i	− 1.62556i	−0.582570	−	0.812781i	$-0.697953\pi$
$383$	− 622.590i	− 1.62556i	0.582570	−	0.812781i	$-0.302047\pi$
$384$	0	0
$385$	46.1914	0.119978
$386$	0	0
$387$	− 1853.48i	− 4.78934i
$388$	0	0
$389$	510.568	1.31251	0.656257	−	0.754537i	$-0.272139\pi$
$389$	510.568	1.31251	0.656257	+	0.754537i	$0.272139\pi$
$390$	0	0
$391$	− 303.492i	− 0.776194i
$392$	0	0
$393$	−522.235	−1.32884
$394$	0	0
$395$	− 132.349i	− 0.335062i
$396$	0	0
$397$	591.459	1.48982	0.744910	−	0.667165i	$-0.232492\pi$
$397$	591.459	1.48982	0.744910	+	0.667165i	$0.232492\pi$
$398$	0	0
$399$	− 396.165i	− 0.992895i
$400$	0	0
$401$	29.0538	0.0724534	0.0362267	−	0.999344i	$-0.488466\pi$
$401$	29.0538	0.0724534	0.0362267	+	0.999344i	$0.488466\pi$
$402$	0	0
$403$	117.485i	0.291526i
$404$	0	0
$405$	1914.90	4.72815
$406$	0	0
$407$	− 213.406i	− 0.524339i
$408$	0	0
$409$	120.749	0.295230	0.147615	−	0.989045i	$-0.452840\pi$
$409$	120.749	0.295230	0.147615	+	0.989045i	$0.452840\pi$
$410$	0	0
$411$	− 66.6442i	− 0.162151i
$412$	0	0
$413$	32.9942	0.0798891
$414$	0	0
$415$	278.638i	0.671418i
$416$	0	0
$417$	−710.102	−1.70288
$418$	0	0
$419$	379.985i	0.906885i	0.891286	+	0.453443i	$0.149804\pi$
$419$	379.985i	0.906885i	−0.891286	+	0.453443i	$0.850196\pi$
$420$	0	0
$421$	536.480	1.27430	0.637149	−	0.770740i	$-0.280113\pi$
$421$	536.480	1.27430	0.637149	+	0.770740i	$0.280113\pi$
$422$	0	0
$423$	1118.69i	2.64465i
$424$	0	0
$425$	−97.7840	−0.230080
$426$	0	0
$427$	− 94.3926i	− 0.221060i
$428$	0	0
$429$	−172.978	−0.403211
$430$	0	0
$431$	384.935i	0.893120i	0.894754	+	0.446560i	$0.147351\pi$
$431$	384.935i	0.893120i	−0.894754	+	0.446560i	$0.852649\pi$
$432$	0	0
$433$	−667.115	−1.54068	−0.770340	−	0.637633i	$-0.779914\pi$
$433$	−667.115	−1.54068	−0.770340	+	0.637633i	$0.779914\pi$
$434$	0	0
$435$	− 52.9329i	− 0.121685i
$436$	0	0
$437$	−668.802	−1.53044
$438$	0	0
$439$	521.733i	1.18846i	0.804296	+	0.594229i	$0.202543\pi$
$439$	521.733i	1.18846i	−0.804296	+	0.594229i	$0.797457\pi$
$440$	0	0
$441$	177.068	0.401515
$442$	0	0
$443$	204.156i	0.460848i	0.973090	+	0.230424i	$0.0740114\pi$
$443$	204.156i	0.460848i	−0.973090	+	0.230424i	$0.925989\pi$
$444$	0	0
$445$	390.148	0.876737
$446$	0	0
$447$	372.299i	0.832884i
$448$	0	0
$449$	513.424	1.14348	0.571742	−	0.820434i	$-0.306268\pi$
$449$	513.424	1.14348	0.571742	+	0.820434i	$0.306268\pi$
$450$	0	0
$451$	150.613i	0.333954i
$452$	0	0
$453$	−716.885	−1.58253
$454$	0	0
$455$	− 149.629i	− 0.328855i
$456$	0	0
$457$	−134.029	−0.293281	−0.146641	−	0.989190i	$-0.546846\pi$
$457$	−134.029	−0.293281	−0.146641	+	0.989190i	$0.546846\pi$
$458$	0	0
$459$	1107.24i	2.41230i
$460$	0	0
$461$	−8.06740	−0.0174998	−0.00874990	−	0.999962i	$-0.502785\pi$
$461$	−8.06740	−0.0174998	−0.00874990	+	0.999962i	$0.502785\pi$
$462$	0	0
$463$	− 837.707i	− 1.80930i	−0.426152	−	0.904652i	$-0.640131\pi$
$463$	− 837.707i	− 1.80930i	0.426152	−	0.904652i	$-0.359869\pi$
$464$	0	0
$465$	406.669	0.874556
$466$	0	0
$467$	− 495.780i	− 1.06163i	−0.847489	−	0.530814i	$-0.821886\pi$
$467$	− 495.780i	− 1.06163i	0.847489	−	0.530814i	$-0.178114\pi$
$468$	0	0
$469$	−64.7920	−0.138149
$470$	0	0
$471$	− 231.948i	− 0.492458i
$472$	0	0
$473$	221.260	0.467780
$474$	0	0
$475$	215.486i	0.453654i
$476$	0	0
$477$	−1371.91	−2.87613
$478$	0	0
$479$	− 225.993i	− 0.471802i	−0.971777	−	0.235901i	$-0.924196\pi$
$479$	− 225.993i	− 0.471802i	0.971777	−	0.235901i	$-0.0758042\pi$
$480$	0	0
$481$	−691.292	−1.43720
$482$	0	0
$483$	− 405.281i	− 0.839091i
$484$	0	0
$485$	−44.8628	−0.0925006
$486$	0	0
$487$	728.173i	1.49522i	0.664137	+	0.747611i	$0.268799\pi$
$487$	728.173i	1.49522i	−0.664137	+	0.747611i	$0.731201\pi$
$488$	0	0
$489$	192.680	0.394028
$490$	0	0
$491$	724.908i	1.47639i	0.674587	+	0.738196i	$0.264322\pi$
$491$	724.908i	1.47639i	−0.674587	+	0.738196i	$0.735678\pi$
$492$	0	0
$493$	18.1389	0.0367930
$494$	0	0
$495$	441.626i	0.892174i
$496$	0	0
$497$	−29.2303	−0.0588135
$498$	0	0
$499$	− 691.520i	− 1.38581i	−0.721028	−	0.692906i	$-0.756330\pi$
$499$	− 691.520i	− 1.38581i	0.721028	−	0.692906i	$-0.243670\pi$
$500$	0	0
$501$	574.148	1.14600
$502$	0	0
$503$	− 518.951i	− 1.03171i	−0.856675	−	0.515856i	$-0.827474\pi$
$503$	− 518.951i	− 1.03171i	0.856675	−	0.515856i	$-0.172526\pi$
$504$	0	0
$505$	592.762	1.17379
$506$	0	0
$507$	− 429.373i	− 0.846889i
$508$	0	0
$509$	−823.772	−1.61841	−0.809206	−	0.587525i	$-0.800102\pi$
$509$	−823.772	−1.61841	−0.809206	+	0.587525i	$0.800102\pi$
$510$	0	0
$511$	196.768i	0.385065i
$512$	0	0
$513$	2440.02	4.75638
$514$	0	0
$515$	84.9177i	0.164889i
$516$	0	0
$517$	−133.544	−0.258305
$518$	0	0
$519$	1972.91i	3.80137i
$520$	0	0
$521$	−533.963	−1.02488	−0.512440	−	0.858723i	$-0.671258\pi$
$521$	−533.963	−1.02488	−0.512440	+	0.858723i	$0.671258\pi$
$522$	0	0
$523$	709.660i	1.35690i	0.734646	+	0.678451i	$0.237348\pi$
$523$	709.660i	1.35690i	−0.734646	+	0.678451i	$0.762652\pi$
$524$	0	0
$525$	−130.580	−0.248724
$526$	0	0
$527$	139.356i	0.264433i
$528$	0	0
$529$	−155.192	−0.293369
$530$	0	0
$531$	315.450i	0.594069i
$532$	0	0
$533$	487.885	0.915356
$534$	0	0
$535$	404.839i	0.756709i
$536$	0	0
$537$	301.325	0.561126
$538$	0	0
$539$	21.1376i	0.0392164i
$540$	0	0
$541$	666.282	1.23157	0.615787	−	0.787913i	$-0.288838\pi$
$541$	666.282	1.23157	0.615787	+	0.787913i	$0.288838\pi$
$542$	0	0
$543$	− 1650.33i	− 3.03928i
$544$	0	0
$545$	381.130	0.699321
$546$	0	0
$547$	− 65.3904i	− 0.119544i	−0.998212	−	0.0597718i	$-0.980963\pi$
$547$	− 65.3904i	− 0.119544i	0.998212	−	0.0597718i	$-0.0190373\pi$
$548$	0	0
$549$	902.467	1.64384
$550$	0	0
$551$	− 39.9726i	− 0.0725456i
$552$	0	0
$553$	60.5644	0.109520
$554$	0	0
$555$	2392.87i	4.31149i
$556$	0	0
$557$	671.474	1.20552	0.602759	−	0.797923i	$-0.294068\pi$
$557$	671.474	1.20552	0.602759	+	0.797923i	$0.294068\pi$
$558$	0	0
$559$	− 716.733i	− 1.28217i
$560$	0	0
$561$	−205.180	−0.365740
$562$	0	0
$563$	629.362i	1.11787i	0.829211	+	0.558936i	$0.188790\pi$
$563$	629.362i	1.11787i	−0.829211	+	0.558936i	$0.811210\pi$
$564$	0	0
$565$	481.142	0.851578
$566$	0	0
$567$	876.277i	1.54546i
$568$	0	0
$569$	−49.5333	−0.0870533	−0.0435267	−	0.999052i	$-0.513859\pi$
$569$	−49.5333	−0.0870533	−0.0435267	+	0.999052i	$0.513859\pi$
$570$	0	0
$571$	− 988.807i	− 1.73171i	−0.500294	−	0.865856i	$-0.666775\pi$
$571$	− 988.807i	− 1.73171i	0.500294	−	0.865856i	$-0.333225\pi$
$572$	0	0
$573$	−966.304	−1.68639
$574$	0	0
$575$	220.444i	0.383381i
$576$	0	0
$577$	−63.1355	−0.109420	−0.0547101	−	0.998502i	$-0.517423\pi$
$577$	−63.1355	−0.109420	−0.0547101	+	0.998502i	$0.517423\pi$
$578$	0	0
$579$	− 378.859i	− 0.654334i
$580$	0	0
$581$	−127.508	−0.219463
$582$	0	0
$583$	− 163.773i	− 0.280914i
$584$	0	0
$585$	1430.57	2.44542
$586$	0	0
$587$	9.88169i	0.0168342i	0.999965	+	0.00841711i	$0.00267928\pi$
$587$	9.88169i	0.0168342i	−0.999965	+	0.00841711i	$0.997321\pi$
$588$	0	0
$589$	307.098	0.521390
$590$	0	0
$591$	− 1111.00i	− 1.87986i
$592$	0	0
$593$	−147.865	−0.249351	−0.124676	−	0.992198i	$-0.539789\pi$
$593$	−147.865	−0.249351	−0.124676	+	0.992198i	$0.539789\pi$
$594$	0	0
$595$	− 177.484i	− 0.298293i
$596$	0	0
$597$	−1203.99	−2.01673
$598$	0	0
$599$	− 464.007i	− 0.774636i	−0.921946	−	0.387318i	$-0.873401\pi$
$599$	− 464.007i	− 0.774636i	0.921946	−	0.387318i	$-0.126599\pi$
$600$	0	0
$601$	215.359	0.358334	0.179167	−	0.983819i	$-0.442660\pi$
$601$	215.359	0.358334	0.179167	+	0.983819i	$0.442660\pi$
$602$	0	0
$603$	− 619.463i	− 1.02730i
$604$	0	0
$605$	646.863	1.06919
$606$	0	0
$607$	− 310.053i	− 0.510796i	−0.966836	−	0.255398i	$-0.917794\pi$
$607$	− 310.053i	− 0.510796i	0.966836	−	0.255398i	$-0.0822065\pi$
$608$	0	0
$609$	24.2226	0.0397744
$610$	0	0
$611$	432.592i	0.708007i
$612$	0	0
$613$	791.288	1.29084	0.645422	−	0.763826i	$-0.276681\pi$
$613$	791.288	1.29084	0.645422	+	0.763826i	$0.276681\pi$
$614$	0	0
$615$	− 1688.79i	− 2.74600i
$616$	0	0
$617$	1139.17	1.84631	0.923153	−	0.384433i	$-0.125603\pi$
$617$	1139.17	1.84631	0.923153	+	0.384433i	$0.125603\pi$
$618$	0	0
$619$	− 419.673i	− 0.677985i	−0.940789	−	0.338992i	$-0.889914\pi$
$619$	− 419.673i	− 0.677985i	0.940789	−	0.338992i	$-0.110086\pi$
$620$	0	0
$621$	2496.17	4.01960
$622$	0	0
$623$	178.536i	0.286574i
$624$	0	0
$625$	−764.666	−1.22347
$626$	0	0
$627$	452.153i	0.721138i
$628$	0	0
$629$	−819.985	−1.30363
$630$	0	0
$631$	− 506.397i	− 0.802530i	−0.915962	−	0.401265i	$-0.868571\pi$
$631$	− 506.397i	− 0.802530i	0.915962	−	0.401265i	$-0.131429\pi$
$632$	0	0
$633$	−1292.62	−2.04205
$634$	0	0
$635$	− 591.421i	− 0.931372i
$636$	0	0
$637$	68.4717	0.107491
$638$	0	0
$639$	− 279.465i	− 0.437347i
$640$	0	0
$641$	−501.692	−0.782671	−0.391335	−	0.920248i	$-0.627987\pi$
$641$	−501.692	−0.782671	−0.391335	+	0.920248i	$0.627987\pi$
$642$	0	0
$643$	− 691.217i	− 1.07499i	−0.843268	−	0.537493i	$-0.819371\pi$
$643$	− 691.217i	− 1.07499i	0.843268	−	0.537493i	$-0.180629\pi$
$644$	0	0
$645$	−2480.94	−3.84642
$646$	0	0
$647$	− 711.553i	− 1.09977i	−0.835239	−	0.549886i	$-0.814671\pi$
$647$	− 711.553i	− 1.09977i	0.835239	−	0.549886i	$-0.185329\pi$
$648$	0	0
$649$	−37.6571	−0.0580233
$650$	0	0
$651$	186.096i	0.285861i
$652$	0	0
$653$	−455.205	−0.697099	−0.348549	−	0.937290i	$-0.613326\pi$
$653$	−455.205	−0.697099	−0.348549	+	0.937290i	$0.613326\pi$
$654$	0	0
$655$	515.586i	0.787154i
$656$	0	0
$657$	−1881.26	−2.86341
$658$	0	0
$659$	− 156.616i	− 0.237656i	−0.992915	−	0.118828i	$-0.962086\pi$
$659$	− 156.616i	− 0.237656i	0.992915	−	0.118828i	$-0.0379138\pi$
$660$	0	0
$661$	−241.002	−0.364602	−0.182301	−	0.983243i	$-0.558354\pi$
$661$	−241.002	−0.364602	−0.182301	+	0.983243i	$0.558354\pi$
$662$	0	0
$663$	664.644i	1.00248i
$664$	0	0
$665$	−391.121	−0.588152
$666$	0	0
$667$	− 40.8924i	− 0.0613080i
$668$	0	0
$669$	957.429	1.43114
$670$	0	0
$671$	107.733i	0.160555i
$672$	0	0
$673$	623.074	0.925816	0.462908	−	0.886406i	$-0.346806\pi$
$673$	623.074	0.925816	0.462908	+	0.886406i	$0.346806\pi$
$674$	0	0
$675$	− 804.258i	− 1.19149i
$676$	0	0
$677$	−343.461	−0.507328	−0.253664	−	0.967292i	$-0.581636\pi$
$677$	−343.461	−0.507328	−0.253664	+	0.967292i	$0.581636\pi$
$678$	0	0
$679$	− 20.5297i	− 0.0302352i
$680$	0	0
$681$	1295.56	1.90243
$682$	0	0
$683$	− 909.578i	− 1.33174i	−0.746068	−	0.665870i	$-0.768061\pi$
$683$	− 909.578i	− 1.33174i	0.746068	−	0.665870i	$-0.231939\pi$
$684$	0	0
$685$	−65.7957	−0.0960521
$686$	0	0
$687$	− 1367.80i	− 1.99098i
$688$	0	0
$689$	−530.514	−0.769976
$690$	0	0
$691$	325.753i	0.471423i	0.971823	+	0.235711i	$0.0757420\pi$
$691$	325.753i	0.471423i	−0.971823	+	0.235711i	$0.924258\pi$
$692$	0	0
$693$	−202.093	−0.291620
$694$	0	0
$695$	701.061i	1.00872i
$696$	0	0
$697$	578.712	0.830290
$698$	0	0
$699$	− 77.6379i	− 0.111070i
$700$	0	0
$701$	414.315	0.591034	0.295517	−	0.955338i	$-0.404508\pi$
$701$	414.315	0.591034	0.295517	+	0.955338i	$0.404508\pi$
$702$	0	0
$703$	1806.99i	2.57040i
$704$	0	0
$705$	1497.40	2.12397
$706$	0	0
$707$	271.254i	0.383669i
$708$	0	0
$709$	−1146.48	−1.61703	−0.808517	−	0.588472i	$-0.799730\pi$
$709$	−1146.48	−1.61703	−0.808517	+	0.588472i	$0.799730\pi$
$710$	0	0
$711$	579.044i	0.814408i
$712$	0	0
$713$	314.165	0.440624
$714$	0	0
$715$	170.775i	0.238847i
$716$	0	0
$717$	−2068.60	−2.88508
$718$	0	0
$719$	387.316i	0.538687i	0.963044	+	0.269344i	$0.0868067\pi$
$719$	387.316i	0.538687i	−0.963044	+	0.269344i	$0.913193\pi$
$720$	0	0
$721$	−38.8592	−0.0538963
$722$	0	0
$723$	1061.51i	1.46820i
$724$	0	0
$725$	−13.1754	−0.0181730
$726$	0	0
$727$	620.563i	0.853594i	0.904347	+	0.426797i	$0.140358\pi$
$727$	620.563i	0.853594i	−0.904347	+	0.426797i	$0.859642\pi$
$728$	0	0
$729$	−3348.15	−4.59280
$730$	0	0
$731$	− 850.163i	− 1.16301i
$732$	0	0
$733$	−96.8796	−0.132169	−0.0660843	−	0.997814i	$-0.521051\pi$
$733$	−96.8796	−0.132169	−0.0660843	+	0.997814i	$0.521051\pi$
$734$	0	0
$735$	− 237.012i	− 0.322465i
$736$	0	0
$737$	73.9488	0.100338
$738$	0	0
$739$	− 782.913i	− 1.05942i	−0.848178	−	0.529711i	$-0.822300\pi$
$739$	− 782.913i	− 1.05942i	0.848178	−	0.529711i	$-0.177700\pi$
$740$	0	0
$741$	1464.67	1.97661
$742$	0	0
$743$	− 48.0641i	− 0.0646892i	−0.999477	−	0.0323446i	$-0.989703\pi$
$743$	− 48.0641i	− 0.0646892i	0.999477	−	0.0323446i	$-0.0102974\pi$
$744$	0	0
$745$	367.559	0.493368
$746$	0	0
$747$	− 1219.08i	− 1.63196i
$748$	0	0
$749$	−185.259	−0.247341
$750$	0	0
$751$	284.489i	0.378814i	0.981899	+	0.189407i	$0.0606565\pi$
$751$	284.489i	0.378814i	−0.981899	+	0.189407i	$0.939343\pi$
$752$	0	0
$753$	−576.559	−0.765683
$754$	0	0
$755$	707.757i	0.937427i
$756$	0	0
$757$	797.514	1.05352	0.526759	−	0.850014i	$-0.323407\pi$
$757$	797.514	1.05352	0.526759	+	0.850014i	$0.323407\pi$
$758$	0	0
$759$	462.558i	0.609431i
$760$	0	0
$761$	1004.15	1.31951	0.659756	−	0.751480i	$-0.270660\pi$
$761$	1004.15	1.31951	0.659756	+	0.751480i	$0.270660\pi$
$762$	0	0
$763$	174.409i	0.228583i
$764$	0	0
$765$	1696.89	2.21816
$766$	0	0
$767$	121.984i	0.159040i
$768$	0	0
$769$	−959.769	−1.24807	−0.624037	−	0.781395i	$-0.714509\pi$
$769$	−959.769	−1.24807	−0.624037	+	0.781395i	$0.714509\pi$
$770$	0	0
$771$	− 1636.98i	− 2.12319i
$772$	0	0
$773$	94.4740	0.122217	0.0611086	−	0.998131i	$-0.480536\pi$
$773$	94.4740	0.122217	0.0611086	+	0.998131i	$0.480536\pi$
$774$	0	0
$775$	− 101.223i	− 0.130610i
$776$	0	0
$777$	−1095.00	−1.40927
$778$	0	0
$779$	− 1275.30i	− 1.63710i
$780$	0	0
$781$	33.3613	0.0427162
$782$	0	0
$783$	149.190i	0.190536i
$784$	0	0
$785$	−228.994	−0.291713
$786$	0	0
$787$	− 108.864i	− 0.138328i	−0.997605	−	0.0691642i	$-0.977967\pi$
$787$	− 108.864i	− 0.138328i	0.997605	−	0.0691642i	$-0.0220332\pi$
$788$	0	0
$789$	−265.337	−0.336295
$790$	0	0
$791$	220.175i	0.278351i
$792$	0	0
$793$	348.981	0.440077
$794$	0	0
$795$	1836.35i	2.30987i
$796$	0	0
$797$	−887.200	−1.11317	−0.556587	−	0.830789i	$-0.687889\pi$
$797$	−887.200	−1.11317	−0.556587	+	0.830789i	$0.687889\pi$
$798$	0	0
$799$	513.125i	0.642209i
$800$	0	0
$801$	−1706.94	−2.13101
$802$	0	0
$803$	− 224.576i	− 0.279672i
$804$	0	0
$805$	−400.121	−0.497045
$806$	0	0
$807$	1289.70i	1.59814i
$808$	0	0
$809$	485.529	0.600160	0.300080	−	0.953914i	$-0.402987\pi$
$809$	485.529	0.600160	0.300080	+	0.953914i	$0.402987\pi$
$810$	0	0
$811$	1118.90i	1.37966i	0.723972	+	0.689830i	$0.242315\pi$
$811$	1118.90i	1.37966i	−0.723972	+	0.689830i	$0.757685\pi$
$812$	0	0
$813$	321.193	0.395071
$814$	0	0
$815$	− 190.227i	− 0.233407i
$816$	0	0
$817$	−1873.50	−2.29314
$818$	0	0
$819$	654.643i	0.799320i
$820$	0	0
$821$	486.539	0.592617	0.296308	−	0.955092i	$-0.404244\pi$
$821$	486.539	0.592617	0.296308	+	0.955092i	$0.404244\pi$
$822$	0	0
$823$	1049.45i	1.27515i	0.770389	+	0.637575i	$0.220062\pi$
$823$	1049.45i	1.27515i	−0.770389	+	0.637575i	$0.779938\pi$
$824$	0	0
$825$	149.035	0.180648
$826$	0	0
$827$	1472.00i	1.77993i	0.456031	+	0.889964i	$0.349271\pi$
$827$	1472.00i	1.77993i	−0.456031	+	0.889964i	$0.650729\pi$
$828$	0	0
$829$	−1049.47	−1.26595	−0.632973	−	0.774174i	$-0.718166\pi$
$829$	−1049.47	−1.26595	−0.632973	+	0.774174i	$0.718166\pi$
$830$	0	0
$831$	− 1874.45i	− 2.25566i
$832$	0	0
$833$	81.2187	0.0975014
$834$	0	0
$835$	− 566.838i	− 0.678848i
$836$	0	0
$837$	−1146.18	−1.36940
$838$	0	0
$839$	1456.31i	1.73576i	0.496771	+	0.867882i	$0.334519\pi$
$839$	1456.31i	1.73576i	−0.496771	+	0.867882i	$0.665481\pi$
$840$	0	0
$841$	−838.556	−0.997094
$842$	0	0
$843$	3082.98i	3.65716i
$844$	0	0
$845$	−423.906	−0.501664
$846$	0	0
$847$	296.011i	0.349482i
$848$	0	0
$849$	−1943.87	−2.28960
$850$	0	0
$851$	1848.58i	2.17224i
$852$	0	0
$853$	−121.003	−0.141855	−0.0709277	−	0.997481i	$-0.522596\pi$
$853$	−121.003	−0.141855	−0.0709277	+	0.997481i	$0.522596\pi$
$854$	0	0
$855$	− 3739.42i	− 4.37360i
$856$	0	0
$857$	1405.84	1.64041	0.820207	−	0.572066i	$-0.193858\pi$
$857$	1405.84	1.64041	0.820207	+	0.572066i	$0.193858\pi$
$858$	0	0
$859$	991.383i	1.15411i	0.816704	+	0.577057i	$0.195799\pi$
$859$	991.383i	1.15411i	−0.816704	+	0.577057i	$0.804201\pi$
$860$	0	0
$861$	772.808	0.897571
$862$	0	0
$863$	− 284.139i	− 0.329245i	−0.986357	−	0.164623i	$-0.947359\pi$
$863$	− 284.139i	− 0.329245i	0.986357	−	0.164623i	$-0.0526406\pi$
$864$	0	0
$865$	1947.79	2.25178
$866$	0	0
$867$	− 904.074i	− 1.04276i
$868$	0	0
$869$	−69.1238	−0.0795440
$870$	0	0
$871$	− 239.544i	− 0.275022i
$872$	0	0
$873$	196.280	0.224834
$874$	0	0
$875$	− 253.504i	− 0.289719i
$876$	0	0
$877$	255.174	0.290963	0.145481	−	0.989361i	$-0.453527\pi$
$877$	255.174	0.290963	0.145481	+	0.989361i	$0.453527\pi$
$878$	0	0
$879$	1409.68i	1.60374i
$880$	0	0
$881$	−421.803	−0.478778	−0.239389	−	0.970924i	$-0.576947\pi$
$881$	−421.803	−0.478778	−0.239389	+	0.970924i	$0.576947\pi$
$882$	0	0
$883$	− 1069.35i	− 1.21105i	−0.795827	−	0.605524i	$-0.792964\pi$
$883$	− 1069.35i	− 1.21105i	0.795827	−	0.605524i	$-0.207036\pi$
$884$	0	0
$885$	422.241	0.477108
$886$	0	0
$887$	619.581i	0.698513i	0.937027	+	0.349256i	$0.113566\pi$
$887$	619.581i	0.698513i	−0.937027	+	0.349256i	$0.886434\pi$
$888$	0	0
$889$	270.640	0.304432
$890$	0	0
$891$	− 1000.12i	− 1.12247i
$892$	0	0
$893$	1130.77	1.26626
$894$	0	0
$895$	− 297.488i	− 0.332389i
$896$	0	0
$897$	1498.37	1.67043
$898$	0	0
$899$	18.7769i	0.0208864i
$900$	0	0
$901$	−629.276	−0.698420
$902$	0	0
$903$	− 1135.30i	− 1.25726i
$904$	0	0
$905$	−1629.32	−1.80035
$906$	0	0
$907$	302.666i	0.333700i	0.985982	+	0.166850i	$0.0533595\pi$
$907$	302.666i	0.333700i	−0.985982	+	0.166850i	$0.946640\pi$
$908$	0	0
$909$	−2593.40	−2.85303
$910$	0	0
$911$	1004.97i	1.10315i	0.834127	+	0.551573i	$0.185972\pi$
$911$	1004.97i	1.10315i	−0.834127	+	0.551573i	$0.814028\pi$
$912$	0	0
$913$	145.528	0.159395
$914$	0	0
$915$	− 1207.98i	− 1.32020i
$916$	0	0
$917$	−235.937	−0.257293
$918$	0	0
$919$	73.2026i	0.0796546i	0.999207	+	0.0398273i	$0.0126808\pi$
$919$	73.2026i	0.0796546i	−0.999207	+	0.0398273i	$0.987319\pi$
$920$	0	0
$921$	1299.28	1.41072
$922$	0	0
$923$	− 108.068i	− 0.117084i
$924$	0	0
$925$	595.605	0.643897
$926$	0	0
$927$	− 371.525i	− 0.400782i
$928$	0	0
$929$	225.883	0.243146	0.121573	−	0.992582i	$-0.461206\pi$
$929$	225.883	0.243146	0.121573	+	0.992582i	$0.461206\pi$
$930$	0	0
$931$	− 178.981i	− 0.192246i
$932$	0	0
$933$	2787.85	2.98805
$934$	0	0
$935$	202.568i	0.216650i
$936$	0	0
$937$	−1168.51	−1.24708	−0.623540	−	0.781792i	$-0.714306\pi$
$937$	−1168.51	−1.24708	−0.623540	+	0.781792i	$0.714306\pi$
$938$	0	0
$939$	− 2899.83i	− 3.08821i
$940$	0	0
$941$	119.216	0.126690	0.0633452	−	0.997992i	$-0.479823\pi$
$941$	119.216	0.126690	0.0633452	+	0.997992i	$0.479823\pi$
$942$	0	0
$943$	− 1304.65i	− 1.38351i
$944$	0	0
$945$	1459.78	1.54474
$946$	0	0
$947$	875.062i	0.924036i	0.886871	+	0.462018i	$0.152874\pi$
$947$	875.062i	0.924036i	−0.886871	+	0.462018i	$0.847126\pi$
$948$	0	0
$949$	−727.476	−0.766571
$950$	0	0
$951$	2887.30i	3.03606i
$952$	0	0
$953$	−468.167	−0.491256	−0.245628	−	0.969364i	$-0.578994\pi$
$953$	−468.167	−0.491256	−0.245628	+	0.969364i	$0.578994\pi$
$954$	0	0
$955$	954.001i	0.998954i
$956$	0	0
$957$	−27.6459	−0.0288881
$958$	0	0
$959$	− 30.1088i	− 0.0313960i
$960$	0	0
$961$	816.743	0.849888
$962$	0	0
$963$	− 1771.22i	− 1.83927i
$964$	0	0
$965$	−374.036	−0.387602
$966$	0	0
$967$	1139.67i	1.17856i	0.807930	+	0.589279i	$0.200588\pi$
$967$	1139.67i	1.17856i	−0.807930	+	0.589279i	$0.799412\pi$
$968$	0	0
$969$	1737.34	1.79292
$970$	0	0
$971$	− 237.061i	− 0.244141i	−0.992521	−	0.122071i	$-0.961047\pi$
$971$	− 237.061i	− 0.244141i	0.992521	−	0.122071i	$-0.0389534\pi$
$972$	0	0
$973$	−320.812	−0.329715
$974$	0	0
$975$	− 482.771i	− 0.495150i
$976$	0	0
$977$	422.059	0.431994	0.215997	−	0.976394i	$-0.430700\pi$
$977$	422.059	0.431994	0.215997	+	0.976394i	$0.430700\pi$
$978$	0	0
$979$	− 203.767i	− 0.208138i
$980$	0	0
$981$	−1667.49	−1.69978
$982$	0	0
$983$	− 1614.06i	− 1.64197i	−0.570946	−	0.820987i	$-0.693424\pi$
$983$	− 1614.06i	− 1.64197i	0.570946	−	0.820987i	$-0.306576\pi$
$984$	0	0
$985$	−1096.85	−1.11355
$986$	0	0
$987$	685.224i	0.694250i
$988$	0	0
$989$	−1916.61	−1.93792
$990$	0	0
$991$	226.772i	0.228832i	0.993433	+	0.114416i	$0.0364996\pi$
$991$	226.772i	0.228832i	−0.993433	+	0.114416i	$0.963500\pi$
$992$	0	0
$993$	3437.19	3.46142
$994$	0	0
$995$	1188.66i	1.19463i
$996$	0	0
$997$	1555.56	1.56024	0.780120	−	0.625629i	$-0.215158\pi$
$997$	1555.56	1.56024	0.780120	+	0.625629i	$0.215158\pi$
$998$	0	0
$999$	− 6744.25i	− 6.75100i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.d.e.127.8		8
4.3	odd	2	inner	448.3.d.e.127.1		8
8.3	odd	2		224.3.d.b.127.8	yes	8
8.5	even	2		224.3.d.b.127.1	✓	8
16.3	odd	4		1792.3.g.f.127.7		8
16.5	even	4		1792.3.g.f.127.8		8
16.11	odd	4		1792.3.g.d.127.2		8
16.13	even	4		1792.3.g.d.127.1		8
24.5	odd	2		2016.3.m.c.127.8		8
24.11	even	2		2016.3.m.c.127.7		8
56.13	odd	2		1568.3.d.n.1471.8		8
56.27	even	2		1568.3.d.n.1471.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.1	✓	8	8.5	even	2
224.3.d.b.127.8	yes	8	8.3	odd	2
448.3.d.e.127.1		8	4.3	odd	2	inner
448.3.d.e.127.8		8	1.1	even	1	trivial
1568.3.d.n.1471.1		8	56.27	even	2
1568.3.d.n.1471.8		8	56.13	odd	2
1792.3.g.d.127.1		8	16.13	even	4
1792.3.g.d.127.2		8	16.11	odd	4
1792.3.g.f.127.7		8	16.3	odd	4
1792.3.g.f.127.8		8	16.5	even	4
2016.3.m.c.127.7		8	24.11	even	2
2016.3.m.c.127.8		8	24.5	odd	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.85623i q^{3} +5.78167 q^{5} +2.64575i q^{7} -25.2955 q^{9} +O(q^{10})\)	\(q+5.85623i q^{3} +5.78167 q^{5} +2.64575i q^{7} -25.2955 q^{9} -3.01966i q^{11} -9.78167 q^{13} +33.8588i q^{15} -11.6027 q^{17} +25.5687i q^{19} -15.4941 q^{21} +26.1571i q^{23} +8.42771 q^{25} -95.4301i q^{27} -1.56334 q^{29} -12.0107i q^{31} +17.6839 q^{33} +15.2969i q^{35} +70.6721 q^{37} -57.2837i q^{39} -49.8775 q^{41} +73.2730i q^{43} -146.250 q^{45} -44.2248i q^{47} -7.00000 q^{49} -67.9479i q^{51} +54.2355 q^{53} -17.4587i q^{55} -149.736 q^{57} -12.4706i q^{59} -35.6770 q^{61} -66.9255i q^{63} -56.5544 q^{65} +24.4891i q^{67} -153.182 q^{69} +11.0480i q^{71} +74.3713 q^{73} +49.3547i q^{75} +7.98928 q^{77} -22.8912i q^{79} +331.202 q^{81} +48.1934i q^{83} -67.0828 q^{85} -9.15529i q^{87} +67.4801 q^{89} -25.8799i q^{91} +70.3376 q^{93} +147.830i q^{95} -7.75949 q^{97} +76.3838i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{9}+O(q^{10}) \) 8 * q - 40 * q^9	\( 8 q - 40 q^{9} - 32 q^{13} - 16 q^{17} + 104 q^{25} + 80 q^{29} + 176 q^{37} + 144 q^{41} - 256 q^{45} - 56 q^{49} - 48 q^{53} - 400 q^{57} + 192 q^{61} - 304 q^{65} - 576 q^{69} + 272 q^{73} + 112 q^{77} + 504 q^{81} + 160 q^{85} - 80 q^{89} + 608 q^{93} + 528 q^{97}+O(q^{100}) \) 8 * q - 40 * q^9 - 32 * q^13 - 16 * q^17 + 104 * q^25 + 80 * q^29 + 176 * q^37 + 144 * q^41 - 256 * q^45 - 56 * q^49 - 48 * q^53 - 400 * q^57 + 192 * q^61 - 304 * q^65 - 576 * q^69 + 272 * q^73 + 112 * q^77 + 504 * q^81 + 160 * q^85 - 80 * q^89 + 608 * q^93 + 528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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