LMFDB - Embedded newform 1344.3.m.e.127.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,3,Mod(127,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1344.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.m (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:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.489494783471841.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 $ x^12 - 3x^11 + 7x^10 - 11x^9 + 18x^8 - 22x^7 + 33x^6 - 44x^5 + 72x^4 - 88x^3 + 112x^2 - 96*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.5
Root			$1.10978 + 0.876576i$ of defining polynomial
Character	$\chi$	$=$	1344.127
Dual form			1344.3.m.e.127.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +6.29204 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +6.29204 q^{5} +2.64575i q^{7} -3.00000 q^{9} -16.5803i q^{11} -14.1288 q^{13} -10.8981i q^{15} -20.4687 q^{17} -6.39258i q^{19} +4.58258 q^{21} +29.0945i q^{23} +14.5898 q^{25} +5.19615i q^{27} -54.6984 q^{29} -14.7296i q^{31} -28.7179 q^{33} +16.6472i q^{35} -26.8040 q^{37} +24.4718i q^{39} -33.5802 q^{41} -3.83239i q^{43} -18.8761 q^{45} -27.4693i q^{47} -7.00000 q^{49} +35.4528i q^{51} +21.7113 q^{53} -104.324i q^{55} -11.0723 q^{57} +54.1344i q^{59} +35.7188 q^{61} -7.93725i q^{63} -88.8991 q^{65} -95.4255i q^{67} +50.3932 q^{69} -55.6524i q^{71} +130.476 q^{73} -25.2702i q^{75} +43.8673 q^{77} +15.6309i q^{79} +9.00000 q^{81} -49.4495i q^{83} -128.790 q^{85} +94.7403i q^{87} +39.2530 q^{89} -37.3813i q^{91} -25.5125 q^{93} -40.2224i q^{95} -171.943 q^{97} +49.7408i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{5} - 36 q^{9}+O(q^{10}) $ 12 * q - 8 * q^5 - 36 * q^9	$ 12 q - 8 q^{5} - 36 q^{9} + 24 q^{13} - 40 q^{17} + 180 q^{25} - 72 q^{29} + 88 q^{37} - 200 q^{41} + 24 q^{45} - 84 q^{49} - 104 q^{53} - 104 q^{61} + 176 q^{65} + 192 q^{69} + 312 q^{73} + 224 q^{77} + 108 q^{81} - 352 q^{85} - 552 q^{89} + 48 q^{93} - 264 q^{97}+O(q^{100}) $ 12 * q - 8 * q^5 - 36 * q^9 + 24 * q^13 - 40 * q^17 + 180 * q^25 - 72 * q^29 + 88 * q^37 - 200 * q^41 + 24 * q^45 - 84 * q^49 - 104 * q^53 - 104 * q^61 + 176 * q^65 + 192 * q^69 + 312 * q^73 + 224 * q^77 + 108 * q^81 - 352 * q^85 - 552 * q^89 + 48 * q^93 - 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	6.29204	1.25841	0.629204	−	0.777240i	$-0.283381\pi$
$5$	6.29204	1.25841	0.629204	+	0.777240i	$0.283381\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 16.5803i	− 1.50730i	−0.657278	−	0.753649i	$-0.728292\pi$
$11$	− 16.5803i	− 1.50730i	0.657278	−	0.753649i	$-0.271708\pi$
$12$	0	0
$13$	−14.1288	−1.08683	−0.543416	−	0.839463i	$-0.682869\pi$
$13$	−14.1288	−1.08683	−0.543416	+	0.839463i	$0.682869\pi$
$14$	0	0
$15$	− 10.8981i	− 0.726542i
$16$	0	0
$17$	−20.4687	−1.20404	−0.602020	−	0.798481i	$-0.705637\pi$
$17$	−20.4687	−1.20404	−0.602020	+	0.798481i	$0.705637\pi$
$18$	0	0
$19$	− 6.39258i	− 0.336452i	−0.985748	−	0.168226i	$-0.946196\pi$
$19$	− 6.39258i	− 0.336452i	0.985748	−	0.168226i	$-0.0538038\pi$
$20$	0	0
$21$	4.58258	0.218218
$22$	0	0
$23$	29.0945i	1.26498i	0.774568	+	0.632490i	$0.217967\pi$
$23$	29.0945i	1.26498i	−0.774568	+	0.632490i	$0.782033\pi$
$24$	0	0
$25$	14.5898	0.583591
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−54.6984	−1.88615	−0.943075	−	0.332579i	$-0.892081\pi$
$29$	−54.6984	−1.88615	−0.943075	+	0.332579i	$0.892081\pi$
$30$	0	0
$31$	− 14.7296i	− 0.475150i	−0.971369	−	0.237575i	$-0.923647\pi$
$31$	− 14.7296i	− 0.475150i	0.971369	−	0.237575i	$-0.0763525\pi$
$32$	0	0
$33$	−28.7179	−0.870238
$34$	0	0
$35$	16.6472i	0.475633i
$36$	0	0
$37$	−26.8040	−0.724431	−0.362216	−	0.932094i	$-0.617980\pi$
$37$	−26.8040	−0.724431	−0.362216	+	0.932094i	$0.617980\pi$
$38$	0	0
$39$	24.4718i	0.627483i
$40$	0	0
$41$	−33.5802	−0.819029	−0.409515	−	0.912304i	$-0.634302\pi$
$41$	−33.5802	−0.819029	−0.409515	+	0.912304i	$0.634302\pi$
$42$	0	0
$43$	− 3.83239i	− 0.0891253i	−0.999007	−	0.0445627i	$-0.985811\pi$
$43$	− 3.83239i	− 0.0891253i	0.999007	−	0.0445627i	$-0.0141894\pi$
$44$	0	0
$45$	−18.8761	−0.419469
$46$	0	0
$47$	− 27.4693i	− 0.584453i	−0.956349	−	0.292226i	$-0.905604\pi$
$47$	− 27.4693i	− 0.584453i	0.956349	−	0.292226i	$-0.0943961\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	35.4528i	0.695153i
$52$	0	0
$53$	21.7113	0.409648	0.204824	−	0.978799i	$-0.434338\pi$
$53$	21.7113	0.409648	0.204824	+	0.978799i	$0.434338\pi$
$54$	0	0
$55$	− 104.324i	− 1.89679i
$56$	0	0
$57$	−11.0723	−0.194250
$58$	0	0
$59$	54.1344i	0.917532i	0.888557	+	0.458766i	$0.151708\pi$
$59$	54.1344i	0.917532i	−0.888557	+	0.458766i	$0.848292\pi$
$60$	0	0
$61$	35.7188	0.585554	0.292777	−	0.956181i	$-0.405421\pi$
$61$	35.7188	0.585554	0.292777	+	0.956181i	$0.405421\pi$
$62$	0	0
$63$	− 7.93725i	− 0.125988i
$64$	0	0
$65$	−88.8991	−1.36768
$66$	0	0
$67$	− 95.4255i	− 1.42426i	−0.702047	−	0.712131i	$-0.747730\pi$
$67$	− 95.4255i	− 1.42426i	0.702047	−	0.712131i	$-0.252270\pi$
$68$	0	0
$69$	50.3932	0.730337
$70$	0	0
$71$	− 55.6524i	− 0.783837i	−0.920000	−	0.391919i	$-0.871811\pi$
$71$	− 55.6524i	− 0.783837i	0.920000	−	0.391919i	$-0.128189\pi$
$72$	0	0
$73$	130.476	1.78735	0.893673	−	0.448719i	$-0.148119\pi$
$73$	130.476	1.78735	0.893673	+	0.448719i	$0.148119\pi$
$74$	0	0
$75$	− 25.2702i	− 0.336936i
$76$	0	0
$77$	43.8673	0.569705
$78$	0	0
$79$	15.6309i	0.197859i	0.995094	+	0.0989295i	$0.0315418\pi$
$79$	15.6309i	0.197859i	−0.995094	+	0.0989295i	$0.968458\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 49.4495i	− 0.595777i	−0.954601	−	0.297889i	$-0.903718\pi$
$83$	− 49.4495i	− 0.595777i	0.954601	−	0.297889i	$-0.0962824\pi$
$84$	0	0
$85$	−128.790	−1.51517
$86$	0	0
$87$	94.7403i	1.08897i
$88$	0	0
$89$	39.2530	0.441045	0.220522	−	0.975382i	$-0.429224\pi$
$89$	39.2530	0.441045	0.220522	+	0.975382i	$0.429224\pi$
$90$	0	0
$91$	− 37.3813i	− 0.410784i
$92$	0	0
$93$	−25.5125	−0.274328
$94$	0	0
$95$	− 40.2224i	− 0.423394i
$96$	0	0
$97$	−171.943	−1.77261	−0.886303	−	0.463106i	$-0.846735\pi$
$97$	−171.943	−1.77261	−0.886303	+	0.463106i	$0.846735\pi$
$98$	0	0
$99$	49.7408i	0.502432i
$100$	0	0
$101$	−113.883	−1.12755	−0.563777	−	0.825927i	$-0.690652\pi$
$101$	−113.883	−1.12755	−0.563777	+	0.825927i	$0.690652\pi$
$102$	0	0
$103$	93.7629i	0.910319i	0.890410	+	0.455160i	$0.150418\pi$
$103$	93.7629i	0.910319i	−0.890410	+	0.455160i	$0.849582\pi$
$104$	0	0
$105$	28.8337	0.274607
$106$	0	0
$107$	− 48.2088i	− 0.450549i	−0.974295	−	0.225275i	$-0.927672\pi$
$107$	− 48.2088i	− 0.450549i	0.974295	−	0.225275i	$-0.0723279\pi$
$108$	0	0
$109$	−89.3790	−0.819991	−0.409996	−	0.912088i	$-0.634470\pi$
$109$	−89.3790	−0.819991	−0.409996	+	0.912088i	$0.634470\pi$
$110$	0	0
$111$	46.4258i	0.418251i
$112$	0	0
$113$	93.4667	0.827139	0.413569	−	0.910473i	$-0.364282\pi$
$113$	93.4667	0.827139	0.413569	+	0.910473i	$0.364282\pi$
$114$	0	0
$115$	183.064i	1.59186i
$116$	0	0
$117$	42.3865	0.362277
$118$	0	0
$119$	− 54.1551i	− 0.455085i
$120$	0	0
$121$	−153.905	−1.27194
$122$	0	0
$123$	58.1626i	0.472867i
$124$	0	0
$125$	−65.5016	−0.524013
$126$	0	0
$127$	208.300i	1.64016i	0.572248	+	0.820081i	$0.306071\pi$
$127$	208.300i	1.64016i	−0.572248	+	0.820081i	$0.693929\pi$
$128$	0	0
$129$	−6.63789	−0.0514565
$130$	0	0
$131$	− 67.7695i	− 0.517324i	−0.965968	−	0.258662i	$-0.916718\pi$
$131$	− 67.7695i	− 0.517324i	0.965968	−	0.258662i	$-0.0832816\pi$
$132$	0	0
$133$	16.9132	0.127167
$134$	0	0
$135$	32.6944i	0.242181i
$136$	0	0
$137$	−13.4127	−0.0979029	−0.0489514	−	0.998801i	$-0.515588\pi$
$137$	−13.4127	−0.0979029	−0.0489514	+	0.998801i	$0.515588\pi$
$138$	0	0
$139$	− 160.382i	− 1.15383i	−0.816804	−	0.576915i	$-0.804256\pi$
$139$	− 160.382i	− 1.15383i	0.816804	−	0.576915i	$-0.195744\pi$
$140$	0	0
$141$	−47.5782	−0.337434
$142$	0	0
$143$	234.260i	1.63818i
$144$	0	0
$145$	−344.164	−2.37355
$146$	0	0
$147$	12.1244i	0.0824786i
$148$	0	0
$149$	130.824	0.878015	0.439007	−	0.898483i	$-0.355330\pi$
$149$	130.824	0.878015	0.439007	+	0.898483i	$0.355330\pi$
$150$	0	0
$151$	− 175.855i	− 1.16460i	−0.812974	−	0.582300i	$-0.802153\pi$
$151$	− 175.855i	− 1.16460i	0.812974	−	0.582300i	$-0.197847\pi$
$152$	0	0
$153$	61.4061	0.401347
$154$	0	0
$155$	− 92.6795i	− 0.597932i
$156$	0	0
$157$	167.199	1.06496	0.532482	−	0.846441i	$-0.321259\pi$
$157$	167.199	1.06496	0.532482	+	0.846441i	$0.321259\pi$
$158$	0	0
$159$	− 37.6051i	− 0.236510i
$160$	0	0
$161$	−76.9769	−0.478118
$162$	0	0
$163$	220.591i	1.35332i	0.736295	+	0.676661i	$0.236574\pi$
$163$	220.591i	1.35332i	−0.736295	+	0.676661i	$0.763426\pi$
$164$	0	0
$165$	−180.694	−1.09511
$166$	0	0
$167$	− 64.6529i	− 0.387143i	−0.981086	−	0.193572i	$-0.937993\pi$
$167$	− 64.6529i	− 0.387143i	0.981086	−	0.193572i	$-0.0620072\pi$
$168$	0	0
$169$	30.6236	0.181205
$170$	0	0
$171$	19.1777i	0.112151i
$172$	0	0
$173$	−237.146	−1.37078	−0.685392	−	0.728174i	$-0.740369\pi$
$173$	−237.146	−1.37078	−0.685392	+	0.728174i	$0.740369\pi$
$174$	0	0
$175$	38.6009i	0.220577i
$176$	0	0
$177$	93.7636	0.529738
$178$	0	0
$179$	103.153i	0.576275i	0.957589	+	0.288137i	$0.0930360\pi$
$179$	103.153i	0.576275i	−0.957589	+	0.288137i	$0.906964\pi$
$180$	0	0
$181$	−43.5702	−0.240719	−0.120360	−	0.992730i	$-0.538405\pi$
$181$	−43.5702	−0.240719	−0.120360	+	0.992730i	$0.538405\pi$
$182$	0	0
$183$	− 61.8668i	− 0.338070i
$184$	0	0
$185$	−168.652	−0.911630
$186$	0	0
$187$	339.376i	1.81485i
$188$	0	0
$189$	−13.7477	−0.0727393
$190$	0	0
$191$	− 219.309i	− 1.14822i	−0.818779	−	0.574109i	$-0.805349\pi$
$191$	− 219.309i	− 1.14822i	0.818779	−	0.574109i	$-0.194651\pi$
$192$	0	0
$193$	9.40273	0.0487188	0.0243594	−	0.999703i	$-0.492245\pi$
$193$	9.40273	0.0487188	0.0243594	+	0.999703i	$0.492245\pi$
$194$	0	0
$195$	153.978i	0.789630i
$196$	0	0
$197$	−329.193	−1.67103	−0.835514	−	0.549469i	$-0.814830\pi$
$197$	−329.193	−1.67103	−0.835514	+	0.549469i	$0.814830\pi$
$198$	0	0
$199$	− 63.0050i	− 0.316608i	−0.987390	−	0.158304i	$-0.949397\pi$
$199$	− 63.0050i	− 0.316608i	0.987390	−	0.158304i	$-0.0506026\pi$
$200$	0	0
$201$	−165.282	−0.822298
$202$	0	0
$203$	− 144.718i	− 0.712898i
$204$	0	0
$205$	−211.288	−1.03067
$206$	0	0
$207$	− 87.2836i	− 0.421660i
$208$	0	0
$209$	−105.991	−0.507133
$210$	0	0
$211$	− 0.702405i	− 0.00332893i	−0.999999	−	0.00166447i	$-0.999470\pi$
$211$	− 0.702405i	− 0.00332893i	0.999999	−	0.00166447i	$-0.000529817\pi$
$212$	0	0
$213$	−96.3929	−0.452549
$214$	0	0
$215$	− 24.1135i	− 0.112156i
$216$	0	0
$217$	38.9710	0.179590
$218$	0	0
$219$	− 225.992i	− 1.03192i
$220$	0	0
$221$	289.199	1.30859
$222$	0	0
$223$	260.543i	1.16835i	0.811627	+	0.584177i	$0.198582\pi$
$223$	260.543i	1.16835i	−0.811627	+	0.584177i	$0.801418\pi$
$224$	0	0
$225$	−43.7693	−0.194530
$226$	0	0
$227$	− 222.146i	− 0.978617i	−0.872111	−	0.489308i	$-0.837249\pi$
$227$	− 222.146i	− 0.978617i	0.872111	−	0.489308i	$-0.162751\pi$
$228$	0	0
$229$	−124.781	−0.544893	−0.272447	−	0.962171i	$-0.587833\pi$
$229$	−124.781	−0.544893	−0.272447	+	0.962171i	$0.587833\pi$
$230$	0	0
$231$	− 75.9803i	− 0.328919i
$232$	0	0
$233$	332.321	1.42627	0.713135	−	0.701027i	$-0.247275\pi$
$233$	332.321	1.42627	0.713135	+	0.701027i	$0.247275\pi$
$234$	0	0
$235$	− 172.838i	− 0.735480i
$236$	0	0
$237$	27.0734	0.114234
$238$	0	0
$239$	− 285.085i	− 1.19282i	−0.802679	−	0.596412i	$-0.796593\pi$
$239$	− 285.085i	− 1.19282i	0.802679	−	0.596412i	$-0.203407\pi$
$240$	0	0
$241$	89.2332	0.370262	0.185131	−	0.982714i	$-0.440729\pi$
$241$	89.2332	0.370262	0.185131	+	0.982714i	$0.440729\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	−44.0443	−0.179773
$246$	0	0
$247$	90.3197i	0.365667i
$248$	0	0
$249$	−85.6490	−0.343972
$250$	0	0
$251$	83.4611i	0.332514i	0.986082	+	0.166257i	$0.0531682\pi$
$251$	83.4611i	0.332514i	−0.986082	+	0.166257i	$0.946832\pi$
$252$	0	0
$253$	482.395	1.90670
$254$	0	0
$255$	223.071i	0.874786i
$256$	0	0
$257$	−29.8619	−0.116194	−0.0580971	−	0.998311i	$-0.518503\pi$
$257$	−29.8619	−0.116194	−0.0580971	+	0.998311i	$0.518503\pi$
$258$	0	0
$259$	− 70.9166i	− 0.273809i
$260$	0	0
$261$	164.095	0.628717
$262$	0	0
$263$	− 245.304i	− 0.932717i	−0.884596	−	0.466358i	$-0.845566\pi$
$263$	− 245.304i	− 0.932717i	0.884596	−	0.466358i	$-0.154434\pi$
$264$	0	0
$265$	136.608	0.515504
$266$	0	0
$267$	− 67.9882i	− 0.254637i
$268$	0	0
$269$	18.9441	0.0704241	0.0352120	−	0.999380i	$-0.488789\pi$
$269$	18.9441	0.0704241	0.0352120	+	0.999380i	$0.488789\pi$
$270$	0	0
$271$	− 278.928i	− 1.02925i	−0.857414	−	0.514626i	$-0.827931\pi$
$271$	− 278.928i	− 1.02925i	0.857414	−	0.514626i	$-0.172069\pi$
$272$	0	0
$273$	−64.7464	−0.237166
$274$	0	0
$275$	− 241.902i	− 0.879644i
$276$	0	0
$277$	206.348	0.744938	0.372469	−	0.928045i	$-0.378511\pi$
$277$	206.348	0.744938	0.372469	+	0.928045i	$0.378511\pi$
$278$	0	0
$279$	44.1889i	0.158383i
$280$	0	0
$281$	−262.775	−0.935142	−0.467571	−	0.883956i	$-0.654871\pi$
$281$	−262.775	−0.935142	−0.467571	+	0.883956i	$0.654871\pi$
$282$	0	0
$283$	543.508i	1.92052i	0.279101	+	0.960262i	$0.409963\pi$
$283$	543.508i	1.92052i	−0.279101	+	0.960262i	$0.590037\pi$
$284$	0	0
$285$	−69.6672	−0.244446
$286$	0	0
$287$	− 88.8448i	− 0.309564i
$288$	0	0
$289$	129.967	0.449714
$290$	0	0
$291$	297.814i	1.02341i
$292$	0	0
$293$	−19.1934	−0.0655064	−0.0327532	−	0.999463i	$-0.510428\pi$
$293$	−19.1934	−0.0655064	−0.0327532	+	0.999463i	$0.510428\pi$
$294$	0	0
$295$	340.616i	1.15463i
$296$	0	0
$297$	86.1536	0.290079
$298$	0	0
$299$	− 411.072i	− 1.37482i
$300$	0	0
$301$	10.1395	0.0336862
$302$	0	0
$303$	197.251i	0.650994i
$304$	0	0
$305$	224.744	0.736866
$306$	0	0
$307$	− 543.121i	− 1.76912i	−0.466423	−	0.884562i	$-0.654458\pi$
$307$	− 543.121i	− 1.76912i	0.466423	−	0.884562i	$-0.345542\pi$
$308$	0	0
$309$	162.402	0.525573
$310$	0	0
$311$	183.154i	0.588920i	0.955664	+	0.294460i	$0.0951398\pi$
$311$	183.154i	0.588920i	−0.955664	+	0.294460i	$0.904860\pi$
$312$	0	0
$313$	513.858	1.64172	0.820859	−	0.571131i	$-0.193495\pi$
$313$	513.858	1.64172	0.820859	+	0.571131i	$0.193495\pi$
$314$	0	0
$315$	− 49.9415i	− 0.158544i
$316$	0	0
$317$	559.731	1.76571	0.882857	−	0.469642i	$-0.155617\pi$
$317$	559.731	1.76571	0.882857	+	0.469642i	$0.155617\pi$
$318$	0	0
$319$	906.914i	2.84299i
$320$	0	0
$321$	−83.5000	−0.260125
$322$	0	0
$323$	130.848i	0.405102i
$324$	0	0
$325$	−206.136	−0.634265
$326$	0	0
$327$	154.809i	0.473422i
$328$	0	0
$329$	72.6769	0.220902
$330$	0	0
$331$	379.794i	1.14741i	0.819061	+	0.573707i	$0.194495\pi$
$331$	379.794i	1.14741i	−0.819061	+	0.573707i	$0.805505\pi$
$332$	0	0
$333$	80.4119	0.241477
$334$	0	0
$335$	− 600.421i	− 1.79230i
$336$	0	0
$337$	−117.618	−0.349015	−0.174507	−	0.984656i	$-0.555833\pi$
$337$	−117.618	−0.349015	−0.174507	+	0.984656i	$0.555833\pi$
$338$	0	0
$339$	− 161.889i	− 0.477549i
$340$	0	0
$341$	−244.222	−0.716192
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	317.076	0.919061
$346$	0	0
$347$	− 302.706i	− 0.872351i	−0.899862	−	0.436175i	$-0.856333\pi$
$347$	− 302.706i	− 0.872351i	0.899862	−	0.436175i	$-0.143667\pi$
$348$	0	0
$349$	−83.1588	−0.238277	−0.119139	−	0.992878i	$-0.538013\pi$
$349$	−83.1588	−0.238277	−0.119139	+	0.992878i	$0.538013\pi$
$350$	0	0
$351$	− 73.4155i	− 0.209161i
$352$	0	0
$353$	−277.478	−0.786057	−0.393028	−	0.919526i	$-0.628573\pi$
$353$	−277.478	−0.786057	−0.393028	+	0.919526i	$0.628573\pi$
$354$	0	0
$355$	− 350.167i	− 0.986387i
$356$	0	0
$357$	−93.7993	−0.262743
$358$	0	0
$359$	− 76.3214i	− 0.212594i	−0.994334	−	0.106297i	$-0.966101\pi$
$359$	− 76.3214i	− 0.212594i	0.994334	−	0.106297i	$-0.0338995\pi$
$360$	0	0
$361$	320.135	0.886800
$362$	0	0
$363$	266.572i	0.734358i
$364$	0	0
$365$	820.962	2.24921
$366$	0	0
$367$	− 176.403i	− 0.480663i	−0.970691	−	0.240332i	$-0.922744\pi$
$367$	− 176.403i	− 0.480663i	0.970691	−	0.240332i	$-0.0772562\pi$
$368$	0	0
$369$	100.741	0.273010
$370$	0	0
$371$	57.4428i	0.154832i
$372$	0	0
$373$	−57.5272	−0.154229	−0.0771143	−	0.997022i	$-0.524571\pi$
$373$	−57.5272	−0.154229	−0.0771143	+	0.997022i	$0.524571\pi$
$374$	0	0
$375$	113.452i	0.302539i
$376$	0	0
$377$	772.823	2.04993
$378$	0	0
$379$	412.402i	1.08813i	0.839042	+	0.544066i	$0.183116\pi$
$379$	412.402i	1.08813i	−0.839042	+	0.544066i	$0.816884\pi$
$380$	0	0
$381$	360.787	0.946948
$382$	0	0
$383$	− 583.323i	− 1.52304i	−0.648143	−	0.761518i	$-0.724454\pi$
$383$	− 583.323i	− 1.52304i	0.648143	−	0.761518i	$-0.275546\pi$
$384$	0	0
$385$	276.015	0.716921
$386$	0	0
$387$	11.4972i	0.0297084i
$388$	0	0
$389$	−64.4937	−0.165794	−0.0828968	−	0.996558i	$-0.526417\pi$
$389$	−64.4937	−0.165794	−0.0828968	+	0.996558i	$0.526417\pi$
$390$	0	0
$391$	− 595.527i	− 1.52309i
$392$	0	0
$393$	−117.380	−0.298677
$394$	0	0
$395$	98.3500i	0.248987i
$396$	0	0
$397$	0.702051	0.00176839	0.000884195	−	1.00000i	$-0.499719\pi$
$397$	0.702051	0.00176839	0.000884195		1.00000i	$0.499719\pi$
$398$	0	0
$399$	− 29.2945i	− 0.0734198i
$400$	0	0
$401$	61.3178	0.152912	0.0764561	−	0.997073i	$-0.475640\pi$
$401$	61.3178	0.152912	0.0764561	+	0.997073i	$0.475640\pi$
$402$	0	0
$403$	208.113i	0.516408i
$404$	0	0
$405$	56.6284	0.139823
$406$	0	0
$407$	444.417i	1.09193i
$408$	0	0
$409$	−109.489	−0.267700	−0.133850	−	0.991002i	$-0.542734\pi$
$409$	−109.489	−0.267700	−0.133850	+	0.991002i	$0.542734\pi$
$410$	0	0
$411$	23.2315i	0.0565242i
$412$	0	0
$413$	−143.226	−0.346795
$414$	0	0
$415$	− 311.138i	− 0.749731i
$416$	0	0
$417$	−277.790	−0.666164
$418$	0	0
$419$	− 312.627i	− 0.746128i	−0.927806	−	0.373064i	$-0.878307\pi$
$419$	− 312.627i	− 0.746128i	0.927806	−	0.373064i	$-0.121693\pi$
$420$	0	0
$421$	−633.267	−1.50420	−0.752099	−	0.659050i	$-0.770958\pi$
$421$	−633.267	−1.50420	−0.752099	+	0.659050i	$0.770958\pi$
$422$	0	0
$423$	82.4078i	0.194818i
$424$	0	0
$425$	−298.633	−0.702667
$426$	0	0
$427$	94.5031i	0.221319i
$428$	0	0
$429$	405.750	0.945803
$430$	0	0
$431$	46.7851i	0.108550i	0.998526	+	0.0542750i	$0.0172848\pi$
$431$	46.7851i	0.108550i	−0.998526	+	0.0542750i	$0.982715\pi$
$432$	0	0
$433$	71.4224	0.164948	0.0824739	−	0.996593i	$-0.473718\pi$
$433$	71.4224	0.164948	0.0824739	+	0.996593i	$0.473718\pi$
$434$	0	0
$435$	596.110i	1.37037i
$436$	0	0
$437$	185.989	0.425605
$438$	0	0
$439$	− 0.829943i	− 0.00189053i	−1.00000		0.000945266i	$-0.999699\pi$
$439$	− 0.829943i	− 0.00189053i	1.00000		0.000945266i	$-0.000300887\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	− 770.065i	− 1.73830i	−0.494552	−	0.869148i	$-0.664668\pi$
$443$	− 770.065i	− 1.73830i	0.494552	−	0.869148i	$-0.335332\pi$
$444$	0	0
$445$	246.981	0.555014
$446$	0	0
$447$	− 226.594i	− 0.506922i
$448$	0	0
$449$	−80.0955	−0.178386	−0.0891932	−	0.996014i	$-0.528429\pi$
$449$	−80.0955	−0.178386	−0.0891932	+	0.996014i	$0.528429\pi$
$450$	0	0
$451$	556.769i	1.23452i
$452$	0	0
$453$	−304.589	−0.672382
$454$	0	0
$455$	− 235.205i	− 0.516934i
$456$	0	0
$457$	−178.912	−0.391493	−0.195747	−	0.980654i	$-0.562713\pi$
$457$	−178.912	−0.391493	−0.195747	+	0.980654i	$0.562713\pi$
$458$	0	0
$459$	− 106.358i	− 0.231718i
$460$	0	0
$461$	320.864	0.696017	0.348009	−	0.937491i	$-0.386858\pi$
$461$	320.864	0.696017	0.348009	+	0.937491i	$0.386858\pi$
$462$	0	0
$463$	597.651i	1.29082i	0.763835	+	0.645411i	$0.223314\pi$
$463$	597.651i	1.29082i	−0.763835	+	0.645411i	$0.776686\pi$
$464$	0	0
$465$	−160.526	−0.345216
$466$	0	0
$467$	− 578.307i	− 1.23834i	−0.785255	−	0.619172i	$-0.787468\pi$
$467$	− 578.307i	− 1.23834i	0.785255	−	0.619172i	$-0.212532\pi$
$468$	0	0
$469$	252.472	0.538320
$470$	0	0
$471$	− 289.598i	− 0.614858i
$472$	0	0
$473$	−63.5420	−0.134338
$474$	0	0
$475$	− 93.2663i	− 0.196350i
$476$	0	0
$477$	−65.1340	−0.136549
$478$	0	0
$479$	− 543.921i	− 1.13554i	−0.823189	−	0.567768i	$-0.807807\pi$
$479$	− 543.921i	− 1.13554i	0.823189	−	0.567768i	$-0.192193\pi$
$480$	0	0
$481$	378.708	0.787336
$482$	0	0
$483$	133.328i	0.276041i
$484$	0	0
$485$	−1081.87	−2.23066
$486$	0	0
$487$	25.1657i	0.0516750i	0.999666	+	0.0258375i	$0.00822524\pi$
$487$	25.1657i	0.0516750i	−0.999666	+	0.0258375i	$0.991775\pi$
$488$	0	0
$489$	382.076	0.781341
$490$	0	0
$491$	876.015i	1.78414i	0.451894	+	0.892072i	$0.350749\pi$
$491$	876.015i	1.78414i	−0.451894	+	0.892072i	$0.649251\pi$
$492$	0	0
$493$	1119.60	2.27100
$494$	0	0
$495$	312.971i	0.632265i
$496$	0	0
$497$	147.243	0.296263
$498$	0	0
$499$	9.06767i	0.0181717i	0.999959	+	0.00908584i	$0.00289215\pi$
$499$	9.06767i	0.0181717i	−0.999959	+	0.00908584i	$0.997108\pi$
$500$	0	0
$501$	−111.982	−0.223517
$502$	0	0
$503$	433.976i	0.862774i	0.902167	+	0.431387i	$0.141976\pi$
$503$	433.976i	0.862774i	−0.902167	+	0.431387i	$0.858024\pi$
$504$	0	0
$505$	−716.556	−1.41892
$506$	0	0
$507$	− 53.0416i	− 0.104619i
$508$	0	0
$509$	335.775	0.659675	0.329838	−	0.944038i	$-0.393006\pi$
$509$	335.775	0.659675	0.329838	+	0.944038i	$0.393006\pi$
$510$	0	0
$511$	345.208i	0.675553i
$512$	0	0
$513$	33.2168	0.0647502
$514$	0	0
$515$	589.960i	1.14555i
$516$	0	0
$517$	−455.448	−0.880944
$518$	0	0
$519$	410.748i	0.791423i
$520$	0	0
$521$	540.822	1.03805	0.519023	−	0.854760i	$-0.326296\pi$
$521$	540.822	1.03805	0.519023	+	0.854760i	$0.326296\pi$
$522$	0	0
$523$	− 337.109i	− 0.644568i	−0.946643	−	0.322284i	$-0.895549\pi$
$523$	− 337.109i	− 0.644568i	0.946643	−	0.322284i	$-0.104451\pi$
$524$	0	0
$525$	66.8587	0.127350
$526$	0	0
$527$	301.497i	0.572100i
$528$	0	0
$529$	−317.492	−0.600175
$530$	0	0
$531$	− 162.403i	− 0.305844i
$532$	0	0
$533$	474.449	0.890147
$534$	0	0
$535$	− 303.331i	− 0.566975i
$536$	0	0
$537$	178.667	0.332712
$538$	0	0
$539$	116.062i	0.215328i
$540$	0	0
$541$	−210.355	−0.388827	−0.194413	−	0.980920i	$-0.562280\pi$
$541$	−210.355	−0.388827	−0.194413	+	0.980920i	$0.562280\pi$
$542$	0	0
$543$	75.4657i	0.138979i
$544$	0	0
$545$	−562.376	−1.03188
$546$	0	0
$547$	− 228.588i	− 0.417894i	−0.977927	−	0.208947i	$-0.932996\pi$
$547$	− 228.588i	− 0.417894i	0.977927	−	0.208947i	$-0.0670036\pi$
$548$	0	0
$549$	−107.156	−0.195185
$550$	0	0
$551$	349.664i	0.634599i
$552$	0	0
$553$	−41.3554	−0.0747837
$554$	0	0
$555$	292.113i	0.526330i
$556$	0	0
$557$	−129.632	−0.232732	−0.116366	−	0.993206i	$-0.537125\pi$
$557$	−129.632	−0.232732	−0.116366	+	0.993206i	$0.537125\pi$
$558$	0	0
$559$	54.1471i	0.0968643i
$560$	0	0
$561$	587.817	1.04780
$562$	0	0
$563$	− 718.440i	− 1.27609i	−0.769998	−	0.638046i	$-0.779743\pi$
$563$	− 718.440i	− 1.27609i	0.769998	−	0.638046i	$-0.220257\pi$
$564$	0	0
$565$	588.096	1.04088
$566$	0	0
$567$	23.8118i	0.0419961i
$568$	0	0
$569$	736.365	1.29414	0.647069	−	0.762431i	$-0.275994\pi$
$569$	736.365	1.29414	0.647069	+	0.762431i	$0.275994\pi$
$570$	0	0
$571$	− 161.022i	− 0.282001i	−0.990010	−	0.141000i	$-0.954968\pi$
$571$	− 161.022i	− 0.282001i	0.990010	−	0.141000i	$-0.0450319\pi$
$572$	0	0
$573$	−379.855	−0.662923
$574$	0	0
$575$	424.483i	0.738230i
$576$	0	0
$577$	748.940	1.29799	0.648995	−	0.760793i	$-0.275190\pi$
$577$	748.940	1.29799	0.648995	+	0.760793i	$0.275190\pi$
$578$	0	0
$579$	− 16.2860i	− 0.0281278i
$580$	0	0
$581$	130.831	0.225183
$582$	0	0
$583$	− 359.980i	− 0.617461i
$584$	0	0
$585$	266.697	0.455893
$586$	0	0
$587$	− 261.696i	− 0.445819i	−0.974839	−	0.222910i	$-0.928444\pi$
$587$	− 261.696i	− 0.445819i	0.974839	−	0.222910i	$-0.0715555\pi$
$588$	0	0
$589$	−94.1605	−0.159865
$590$	0	0
$591$	570.178i	0.964769i
$592$	0	0
$593$	−284.492	−0.479750	−0.239875	−	0.970804i	$-0.577106\pi$
$593$	−284.492	−0.479750	−0.239875	+	0.970804i	$0.577106\pi$
$594$	0	0
$595$	− 340.746i	− 0.572682i
$596$	0	0
$597$	−109.128	−0.182794
$598$	0	0
$599$	358.608i	0.598678i	0.954147	+	0.299339i	$0.0967662\pi$
$599$	358.608i	0.598678i	−0.954147	+	0.299339i	$0.903234\pi$
$600$	0	0
$601$	−404.112	−0.672399	−0.336200	−	0.941791i	$-0.609142\pi$
$601$	−404.112	−0.672399	−0.336200	+	0.941791i	$0.609142\pi$
$602$	0	0
$603$	286.277i	0.474754i
$604$	0	0
$605$	−968.378	−1.60063
$606$	0	0
$607$	− 608.215i	− 1.00200i	−0.865447	−	0.501001i	$-0.832965\pi$
$607$	− 608.215i	− 1.00200i	0.865447	−	0.501001i	$-0.167035\pi$
$608$	0	0
$609$	−250.659	−0.411592
$610$	0	0
$611$	388.108i	0.635202i
$612$	0	0
$613$	741.906	1.21029	0.605143	−	0.796117i	$-0.293116\pi$
$613$	741.906	1.21029	0.605143	+	0.796117i	$0.293116\pi$
$614$	0	0
$615$	365.961i	0.595059i
$616$	0	0
$617$	788.372	1.27775	0.638875	−	0.769310i	$-0.279400\pi$
$617$	788.372	1.27775	0.638875	+	0.769310i	$0.279400\pi$
$618$	0	0
$619$	− 1053.85i	− 1.70251i	−0.524754	−	0.851254i	$-0.675843\pi$
$619$	− 1053.85i	− 1.70251i	0.524754	−	0.851254i	$-0.324157\pi$
$620$	0	0
$621$	−151.180	−0.243446
$622$	0	0
$623$	103.854i	0.166699i
$624$	0	0
$625$	−776.883	−1.24301
$626$	0	0
$627$	183.581i	0.292793i
$628$	0	0
$629$	548.642	0.872245
$630$	0	0
$631$	729.494i	1.15609i	0.816004	+	0.578046i	$0.196185\pi$
$631$	729.494i	1.15609i	−0.816004	+	0.578046i	$0.803815\pi$
$632$	0	0
$633$	−1.21660	−0.00192196
$634$	0	0
$635$	1310.63i	2.06399i
$636$	0	0
$637$	98.9018	0.155262
$638$	0	0
$639$	166.957i	0.261279i
$640$	0	0
$641$	258.186	0.402787	0.201393	−	0.979510i	$-0.435453\pi$
$641$	258.186	0.402787	0.201393	+	0.979510i	$0.435453\pi$
$642$	0	0
$643$	844.440i	1.31328i	0.754203	+	0.656641i	$0.228023\pi$
$643$	844.440i	1.31328i	−0.754203	+	0.656641i	$0.771977\pi$
$644$	0	0
$645$	−41.7659	−0.0647533
$646$	0	0
$647$	− 94.5538i	− 0.146142i	−0.997327	−	0.0730709i	$-0.976720\pi$
$647$	− 94.5538i	− 0.146142i	0.997327	−	0.0730709i	$-0.0232799\pi$
$648$	0	0
$649$	897.563	1.38299
$650$	0	0
$651$	− 67.4997i	− 0.103686i
$652$	0	0
$653$	239.621	0.366954	0.183477	−	0.983024i	$-0.441265\pi$
$653$	239.621	0.366954	0.183477	+	0.983024i	$0.441265\pi$
$654$	0	0
$655$	− 426.408i	− 0.651005i
$656$	0	0
$657$	−391.429	−0.595782
$658$	0	0
$659$	150.618i	0.228555i	0.993449	+	0.114278i	$0.0364553\pi$
$659$	150.618i	0.228555i	−0.993449	+	0.114278i	$0.963545\pi$
$660$	0	0
$661$	575.893	0.871246	0.435623	−	0.900129i	$-0.356528\pi$
$661$	575.893	0.871246	0.435623	+	0.900129i	$0.356528\pi$
$662$	0	0
$663$	− 500.907i	− 0.755515i
$664$	0	0
$665$	106.418	0.160028
$666$	0	0
$667$	− 1591.42i	− 2.38594i
$668$	0	0
$669$	451.273	0.674549
$670$	0	0
$671$	− 592.227i	− 0.882604i
$672$	0	0
$673$	−614.833	−0.913571	−0.456785	−	0.889577i	$-0.650999\pi$
$673$	−614.833	−0.913571	−0.456785	+	0.889577i	$0.650999\pi$
$674$	0	0
$675$	75.8106i	0.112312i
$676$	0	0
$677$	308.666	0.455933	0.227966	−	0.973669i	$-0.426792\pi$
$677$	308.666	0.455933	0.227966	+	0.973669i	$0.426792\pi$
$678$	0	0
$679$	− 454.918i	− 0.669982i
$680$	0	0
$681$	−384.768	−0.565005
$682$	0	0
$683$	143.442i	0.210018i	0.994471	+	0.105009i	$0.0334871\pi$
$683$	143.442i	0.210018i	−0.994471	+	0.105009i	$0.966513\pi$
$684$	0	0
$685$	−84.3932	−0.123202
$686$	0	0
$687$	216.126i	0.314594i
$688$	0	0
$689$	−306.755	−0.445218
$690$	0	0
$691$	− 366.656i	− 0.530616i	−0.964164	−	0.265308i	$-0.914526\pi$
$691$	− 366.656i	− 0.530616i	0.964164	−	0.265308i	$-0.0854737\pi$
$692$	0	0
$693$	−131.602	−0.189902
$694$	0	0
$695$	− 1009.13i	− 1.45199i
$696$	0	0
$697$	687.343	0.986145
$698$	0	0
$699$	− 575.597i	− 0.823457i
$700$	0	0
$701$	−125.995	−0.179736	−0.0898681	−	0.995954i	$-0.528645\pi$
$701$	−125.995	−0.179736	−0.0898681	+	0.995954i	$0.528645\pi$
$702$	0	0
$703$	171.347i	0.243736i
$704$	0	0
$705$	−299.364	−0.424630
$706$	0	0
$707$	− 301.306i	− 0.426175i
$708$	0	0
$709$	159.688	0.225230	0.112615	−	0.993639i	$-0.464077\pi$
$709$	159.688	0.225230	0.112615	+	0.993639i	$0.464077\pi$
$710$	0	0
$711$	− 46.8926i	− 0.0659530i
$712$	0	0
$713$	428.552	0.601055
$714$	0	0
$715$	1473.97i	2.06150i
$716$	0	0
$717$	−493.781	−0.688677
$718$	0	0
$719$	− 1368.60i	− 1.90348i	−0.306899	−	0.951742i	$-0.599292\pi$
$719$	− 1368.60i	− 1.90348i	0.306899	−	0.951742i	$-0.400708\pi$
$720$	0	0
$721$	−248.073	−0.344068
$722$	0	0
$723$	− 154.556i	− 0.213771i
$724$	0	0
$725$	−798.036	−1.10074
$726$	0	0
$727$	510.747i	0.702540i	0.936274	+	0.351270i	$0.114250\pi$
$727$	510.747i	0.702540i	−0.936274	+	0.351270i	$0.885750\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	78.4440i	0.107311i
$732$	0	0
$733$	774.969	1.05726	0.528628	−	0.848853i	$-0.322707\pi$
$733$	774.969	1.05726	0.528628	+	0.848853i	$0.322707\pi$
$734$	0	0
$735$	76.2869i	0.103792i
$736$	0	0
$737$	−1582.18	−2.14679
$738$	0	0
$739$	1349.26i	1.82579i	0.408194	+	0.912895i	$0.366159\pi$
$739$	1349.26i	1.82579i	−0.408194	+	0.912895i	$0.633841\pi$
$740$	0	0
$741$	156.438	0.211118
$742$	0	0
$743$	853.994i	1.14939i	0.818369	+	0.574693i	$0.194878\pi$
$743$	853.994i	1.14939i	−0.818369	+	0.574693i	$0.805122\pi$
$744$	0	0
$745$	823.151	1.10490
$746$	0	0
$747$	148.348i	0.198592i
$748$	0	0
$749$	127.548	0.170292
$750$	0	0
$751$	− 509.524i	− 0.678460i	−0.940703	−	0.339230i	$-0.889833\pi$
$751$	− 509.524i	− 0.678460i	0.940703	−	0.339230i	$-0.110167\pi$
$752$	0	0
$753$	144.559	0.191977
$754$	0	0
$755$	− 1106.48i	− 1.46554i
$756$	0	0
$757$	−1373.83	−1.81484	−0.907418	−	0.420228i	$-0.861950\pi$
$757$	−1373.83	−1.81484	−0.907418	+	0.420228i	$0.861950\pi$
$758$	0	0
$759$	− 835.533i	− 1.10083i
$760$	0	0
$761$	−1165.32	−1.53130	−0.765652	−	0.643255i	$-0.777583\pi$
$761$	−1165.32	−1.53130	−0.765652	+	0.643255i	$0.777583\pi$
$762$	0	0
$763$	− 236.475i	− 0.309927i
$764$	0	0
$765$	386.370	0.505058
$766$	0	0
$767$	− 764.855i	− 0.997204i
$768$	0	0
$769$	−686.335	−0.892503	−0.446251	−	0.894908i	$-0.647241\pi$
$769$	−686.335	−0.892503	−0.446251	+	0.894908i	$0.647241\pi$
$770$	0	0
$771$	51.7223i	0.0670848i
$772$	0	0
$773$	−244.058	−0.315729	−0.157864	−	0.987461i	$-0.550461\pi$
$773$	−244.058	−0.315729	−0.157864	+	0.987461i	$0.550461\pi$
$774$	0	0
$775$	− 214.902i	− 0.277293i
$776$	0	0
$777$	−122.831	−0.158084
$778$	0	0
$779$	214.664i	0.275564i
$780$	0	0
$781$	−922.733	−1.18148
$782$	0	0
$783$	− 284.221i	− 0.362990i
$784$	0	0
$785$	1052.03	1.34016
$786$	0	0
$787$	959.027i	1.21859i	0.792945	+	0.609293i	$0.208547\pi$
$787$	959.027i	1.21859i	−0.792945	+	0.609293i	$0.791453\pi$
$788$	0	0
$789$	−424.880	−0.538504
$790$	0	0
$791$	247.290i	0.312629i
$792$	0	0
$793$	−504.665	−0.636399
$794$	0	0
$795$	− 236.613i	− 0.297626i
$796$	0	0
$797$	−24.5322	−0.0307807	−0.0153903	−	0.999882i	$-0.504899\pi$
$797$	−24.5322	−0.0307807	−0.0153903	+	0.999882i	$0.504899\pi$
$798$	0	0
$799$	562.260i	0.703705i
$800$	0	0
$801$	−117.759	−0.147015
$802$	0	0
$803$	− 2163.33i	− 2.69406i
$804$	0	0
$805$	−484.342	−0.601667
$806$	0	0
$807$	− 32.8121i	− 0.0406594i
$808$	0	0
$809$	673.148	0.832074	0.416037	−	0.909348i	$-0.363419\pi$
$809$	673.148	0.832074	0.416037	+	0.909348i	$0.363419\pi$
$810$	0	0
$811$	386.219i	0.476226i	0.971237	+	0.238113i	$0.0765288\pi$
$811$	386.219i	0.476226i	−0.971237	+	0.238113i	$0.923471\pi$
$812$	0	0
$813$	−483.117	−0.594239
$814$	0	0
$815$	1387.97i	1.70303i
$816$	0	0
$817$	−24.4989	−0.0299864
$818$	0	0
$819$	112.144i	0.136928i
$820$	0	0
$821$	−5.93033	−0.00722330	−0.00361165	−	0.999993i	$-0.501150\pi$
$821$	−5.93033	−0.00722330	−0.00361165	+	0.999993i	$0.501150\pi$
$822$	0	0
$823$	365.439i	0.444033i	0.975043	+	0.222016i	$0.0712638\pi$
$823$	365.439i	0.444033i	−0.975043	+	0.222016i	$0.928736\pi$
$824$	0	0
$825$	−418.987	−0.507863
$826$	0	0
$827$	1236.80i	1.49552i	0.663968	+	0.747761i	$0.268871\pi$
$827$	1236.80i	1.49552i	−0.663968	+	0.747761i	$0.731129\pi$
$828$	0	0
$829$	−899.957	−1.08559	−0.542797	−	0.839864i	$-0.682635\pi$
$829$	−899.957	−1.08559	−0.542797	+	0.839864i	$0.682635\pi$
$830$	0	0
$831$	− 357.405i	− 0.430090i
$832$	0	0
$833$	143.281	0.172006
$834$	0	0
$835$	− 406.799i	− 0.487184i
$836$	0	0
$837$	76.5375	0.0914427
$838$	0	0
$839$	− 473.095i	− 0.563880i	−0.959432	−	0.281940i	$-0.909022\pi$
$839$	− 473.095i	− 0.563880i	0.959432	−	0.281940i	$-0.0909779\pi$
$840$	0	0
$841$	2150.91	2.55756
$842$	0	0
$843$	455.140i	0.539905i
$844$	0	0
$845$	192.685	0.228029
$846$	0	0
$847$	− 407.195i	− 0.480750i
$848$	0	0
$849$	941.384	1.10881
$850$	0	0
$851$	− 779.849i	− 0.916391i
$852$	0	0
$853$	456.925	0.535668	0.267834	−	0.963465i	$-0.413692\pi$
$853$	456.925	0.535668	0.267834	+	0.963465i	$0.413692\pi$
$854$	0	0
$855$	120.667i	0.141131i
$856$	0	0
$857$	−1460.48	−1.70417	−0.852087	−	0.523401i	$-0.824663\pi$
$857$	−1460.48	−1.70417	−0.852087	+	0.523401i	$0.824663\pi$
$858$	0	0
$859$	− 329.072i	− 0.383087i	−0.981484	−	0.191544i	$-0.938651\pi$
$859$	− 329.072i	− 0.383087i	0.981484	−	0.191544i	$-0.0613493\pi$
$860$	0	0
$861$	−153.884	−0.178727
$862$	0	0
$863$	− 528.421i	− 0.612307i	−0.951982	−	0.306154i	$-0.900958\pi$
$863$	− 528.421i	− 0.612307i	0.951982	−	0.306154i	$-0.0990421\pi$
$864$	0	0
$865$	−1492.13	−1.72501
$866$	0	0
$867$	− 225.110i	− 0.259643i
$868$	0	0
$869$	259.164	0.298232
$870$	0	0
$871$	1348.25i	1.54793i
$872$	0	0
$873$	515.828	0.590869
$874$	0	0
$875$	− 173.301i	− 0.198058i
$876$	0	0
$877$	−532.559	−0.607250	−0.303625	−	0.952792i	$-0.598197\pi$
$877$	−532.559	−0.607250	−0.303625	+	0.952792i	$0.598197\pi$
$878$	0	0
$879$	33.2439i	0.0378201i
$880$	0	0
$881$	−326.517	−0.370621	−0.185310	−	0.982680i	$-0.559329\pi$
$881$	−326.517	−0.370621	−0.185310	+	0.982680i	$0.559329\pi$
$882$	0	0
$883$	31.0138i	0.0351232i	0.999846	+	0.0175616i	$0.00559032\pi$
$883$	31.0138i	0.0351232i	−0.999846	+	0.0175616i	$0.994410\pi$
$884$	0	0
$885$	589.964	0.666626
$886$	0	0
$887$	414.448i	0.467247i	0.972327	+	0.233623i	$0.0750583\pi$
$887$	414.448i	0.467247i	−0.972327	+	0.233623i	$0.924942\pi$
$888$	0	0
$889$	−551.111	−0.619923
$890$	0	0
$891$	− 149.222i	− 0.167477i
$892$	0	0
$893$	−175.600	−0.196640
$894$	0	0
$895$	649.044i	0.725189i
$896$	0	0
$897$	−711.997	−0.793753
$898$	0	0
$899$	805.688i	0.896204i
$900$	0	0
$901$	−444.402	−0.493232
$902$	0	0
$903$	− 17.5622i	− 0.0194487i
$904$	0	0
$905$	−274.145	−0.302923
$906$	0	0
$907$	388.080i	0.427872i	0.976848	+	0.213936i	$0.0686283\pi$
$907$	388.080i	0.427872i	−0.976848	+	0.213936i	$0.931372\pi$
$908$	0	0
$909$	341.649	0.375851
$910$	0	0
$911$	891.807i	0.978932i	0.872023	+	0.489466i	$0.162808\pi$
$911$	891.807i	0.978932i	−0.872023	+	0.489466i	$0.837192\pi$
$912$	0	0
$913$	−819.886	−0.898013
$914$	0	0
$915$	− 389.268i	− 0.425430i
$916$	0	0
$917$	179.301	0.195530
$918$	0	0
$919$	− 390.619i	− 0.425048i	−0.977156	−	0.212524i	$-0.931832\pi$
$919$	− 390.619i	− 0.425048i	0.977156	−	0.212524i	$-0.0681684\pi$
$920$	0	0
$921$	−940.713	−1.02140
$922$	0	0
$923$	786.304i	0.851900i
$924$	0	0
$925$	−391.064	−0.422771
$926$	0	0
$927$	− 281.289i	− 0.303440i
$928$	0	0
$929$	−241.450	−0.259903	−0.129952	−	0.991520i	$-0.541482\pi$
$929$	−241.450	−0.259903	−0.129952	+	0.991520i	$0.541482\pi$
$930$	0	0
$931$	44.7481i	0.0480645i
$932$	0	0
$933$	317.232	0.340013
$934$	0	0
$935$	2135.37i	2.28382i
$936$	0	0
$937$	−753.075	−0.803709	−0.401855	−	0.915703i	$-0.631634\pi$
$937$	−753.075	−0.803709	−0.401855	+	0.915703i	$0.631634\pi$
$938$	0	0
$939$	− 890.028i	− 0.947847i
$940$	0	0
$941$	−1372.35	−1.45840	−0.729199	−	0.684302i	$-0.760107\pi$
$941$	−1372.35	−1.45840	−0.729199	+	0.684302i	$0.760107\pi$
$942$	0	0
$943$	− 977.000i	− 1.03606i
$944$	0	0
$945$	−86.5012	−0.0915357
$946$	0	0
$947$	− 498.081i	− 0.525957i	−0.964802	−	0.262979i	$-0.915295\pi$
$947$	− 498.081i	− 0.525957i	0.964802	−	0.262979i	$-0.0847048\pi$
$948$	0	0
$949$	−1843.48	−1.94255
$950$	0	0
$951$	− 969.483i	− 1.01944i
$952$	0	0
$953$	−117.526	−0.123322	−0.0616610	−	0.998097i	$-0.519640\pi$
$953$	−117.526	−0.123322	−0.0616610	+	0.998097i	$0.519640\pi$
$954$	0	0
$955$	− 1379.90i	− 1.44493i
$956$	0	0
$957$	1570.82	1.64140
$958$	0	0
$959$	− 35.4867i	− 0.0370038i
$960$	0	0
$961$	744.037	0.774233
$962$	0	0
$963$	144.626i	0.150183i
$964$	0	0
$965$	59.1623	0.0613081
$966$	0	0
$967$	− 1016.54i	− 1.05123i	−0.850723	−	0.525614i	$-0.823836\pi$
$967$	− 1016.54i	− 1.05123i	0.850723	−	0.525614i	$-0.176164\pi$
$968$	0	0
$969$	226.635	0.233886
$970$	0	0
$971$	− 760.606i	− 0.783323i	−0.920109	−	0.391661i	$-0.871900\pi$
$971$	− 760.606i	− 0.783323i	0.920109	−	0.391661i	$-0.128100\pi$
$972$	0	0
$973$	424.332	0.436107
$974$	0	0
$975$	357.038i	0.366193i
$976$	0	0
$977$	−1823.13	−1.86605	−0.933026	−	0.359809i	$-0.882842\pi$
$977$	−1823.13	−1.86605	−0.933026	+	0.359809i	$0.882842\pi$
$978$	0	0
$979$	− 650.825i	− 0.664786i
$980$	0	0
$981$	268.137	0.273330
$982$	0	0
$983$	− 1400.51i	− 1.42473i	−0.701809	−	0.712365i	$-0.747624\pi$
$983$	− 1400.51i	− 1.42473i	0.701809	−	0.712365i	$-0.252376\pi$
$984$	0	0
$985$	−2071.29	−2.10284
$986$	0	0
$987$	− 125.880i	− 0.127538i
$988$	0	0
$989$	111.502	0.112742
$990$	0	0
$991$	508.288i	0.512904i	0.966557	+	0.256452i	$0.0825536\pi$
$991$	508.288i	0.512904i	−0.966557	+	0.256452i	$0.917446\pi$
$992$	0	0
$993$	657.822	0.662459
$994$	0	0
$995$	− 396.430i	− 0.398422i
$996$	0	0
$997$	−1357.91	−1.36199	−0.680997	−	0.732286i	$-0.738453\pi$
$997$	−1357.91	−1.36199	−0.680997	+	0.732286i	$0.738453\pi$
$998$	0	0
$999$	− 139.277i	− 0.139417i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.m.e.127.5		12
4.3	odd	2	inner	1344.3.m.e.127.11		12
8.3	odd	2		84.3.g.a.43.5	✓	12
8.5	even	2		84.3.g.a.43.6	yes	12
24.5	odd	2		252.3.g.b.127.7		12
24.11	even	2		252.3.g.b.127.8		12
56.13	odd	2		588.3.g.d.295.6		12
56.27	even	2		588.3.g.d.295.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.g.a.43.5	✓	12	8.3	odd	2
84.3.g.a.43.6	yes	12	8.5	even	2
252.3.g.b.127.7		12	24.5	odd	2
252.3.g.b.127.8		12	24.11	even	2
588.3.g.d.295.5		12	56.27	even	2
588.3.g.d.295.6		12	56.13	odd	2
1344.3.m.e.127.5		12	1.1	even	1	trivial
1344.3.m.e.127.11		12	4.3	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +6.29204 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +6.29204 q^{5} +2.64575i q^{7} -3.00000 q^{9} -16.5803i q^{11} -14.1288 q^{13} -10.8981i q^{15} -20.4687 q^{17} -6.39258i q^{19} +4.58258 q^{21} +29.0945i q^{23} +14.5898 q^{25} +5.19615i q^{27} -54.6984 q^{29} -14.7296i q^{31} -28.7179 q^{33} +16.6472i q^{35} -26.8040 q^{37} +24.4718i q^{39} -33.5802 q^{41} -3.83239i q^{43} -18.8761 q^{45} -27.4693i q^{47} -7.00000 q^{49} +35.4528i q^{51} +21.7113 q^{53} -104.324i q^{55} -11.0723 q^{57} +54.1344i q^{59} +35.7188 q^{61} -7.93725i q^{63} -88.8991 q^{65} -95.4255i q^{67} +50.3932 q^{69} -55.6524i q^{71} +130.476 q^{73} -25.2702i q^{75} +43.8673 q^{77} +15.6309i q^{79} +9.00000 q^{81} -49.4495i q^{83} -128.790 q^{85} +94.7403i q^{87} +39.2530 q^{89} -37.3813i q^{91} -25.5125 q^{93} -40.2224i q^{95} -171.943 q^{97} +49.7408i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{5} - 36 q^{9}+O(q^{10}) \) 12 * q - 8 * q^5 - 36 * q^9	\( 12 q - 8 q^{5} - 36 q^{9} + 24 q^{13} - 40 q^{17} + 180 q^{25} - 72 q^{29} + 88 q^{37} - 200 q^{41} + 24 q^{45} - 84 q^{49} - 104 q^{53} - 104 q^{61} + 176 q^{65} + 192 q^{69} + 312 q^{73} + 224 q^{77} + 108 q^{81} - 352 q^{85} - 552 q^{89} + 48 q^{93} - 264 q^{97}+O(q^{100}) \) 12 * q - 8 * q^5 - 36 * q^9 + 24 * q^13 - 40 * q^17 + 180 * q^25 - 72 * q^29 + 88 * q^37 - 200 * q^41 + 24 * q^45 - 84 * q^49 - 104 * q^53 - 104 * q^61 + 176 * q^65 + 192 * q^69 + 312 * q^73 + 224 * q^77 + 108 * q^81 - 352 * q^85 - 552 * q^89 + 48 * q^93 - 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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