LMFDB - Embedded newform 672.3.m.a.127.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,3,Mod(127,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, 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("672.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	672.m (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:	$18.3106737650$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.8
Root			$0.228425 + 1.39564i$ of defining polynomial
Character	$\chi$	$=$	672.127
Dual form			672.3.m.a.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +6.37780 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +6.37780 q^{5} +2.64575i q^{7} -3.00000 q^{9} -5.17303i q^{11} +8.09335 q^{13} +11.0467i q^{15} +25.1978 q^{17} -1.59045i q^{19} -4.58258 q^{21} +15.8485i q^{23} +15.6764 q^{25} -5.19615i q^{27} +23.0850 q^{29} +1.68295i q^{31} +8.95995 q^{33} +16.8741i q^{35} -17.2232 q^{37} +14.0181i q^{39} +12.0970 q^{41} +31.1494i q^{43} -19.1334 q^{45} -55.6971i q^{47} -7.00000 q^{49} +43.6438i q^{51} -56.8217 q^{53} -32.9926i q^{55} +2.75475 q^{57} +47.5731i q^{59} +101.351 q^{61} -7.93725i q^{63} +51.6178 q^{65} +58.9752i q^{67} -27.4505 q^{69} +13.3175i q^{71} +13.9050 q^{73} +27.1523i q^{75} +13.6865 q^{77} +45.8124i q^{79} +9.00000 q^{81} +145.317i q^{83} +160.706 q^{85} +39.9845i q^{87} +150.256 q^{89} +21.4130i q^{91} -2.91495 q^{93} -10.1436i q^{95} -90.5264 q^{97} +15.5191i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} - 24 q^{9}+O(q^{10}) $ 8 * q + 16 * q^5 - 24 * q^9	$ 8 q + 16 q^{5} - 24 q^{9} - 64 q^{13} + 64 q^{17} - 88 q^{25} + 64 q^{29} + 48 q^{33} + 128 q^{37} - 48 q^{45} - 56 q^{49} - 160 q^{53} + 48 q^{57} + 32 q^{61} - 32 q^{65} - 144 q^{69} - 112 q^{73} + 112 q^{77} + 72 q^{81} + 336 q^{85} + 352 q^{89} - 144 q^{93} - 240 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 - 24 * q^9 - 64 * q^13 + 64 * q^17 - 88 * q^25 + 64 * q^29 + 48 * q^33 + 128 * q^37 - 48 * q^45 - 56 * q^49 - 160 * q^53 + 48 * q^57 + 32 * q^61 - 32 * q^65 - 144 * q^69 - 112 * q^73 + 112 * q^77 + 72 * q^81 + 336 * q^85 + 352 * q^89 - 144 * q^93 - 240 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	6.37780	1.27556	0.637780	−	0.770218i	$-0.279853\pi$
$5$	6.37780	1.27556	0.637780	+	0.770218i	$0.279853\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 5.17303i	− 0.470275i	−0.971962	−	0.235138i	$-0.924446\pi$
$11$	− 5.17303i	− 0.470275i	0.971962	−	0.235138i	$-0.0755541\pi$
$12$	0	0
$13$	8.09335	0.622566	0.311283	−	0.950317i	$-0.399241\pi$
$13$	8.09335	0.622566	0.311283	+	0.950317i	$0.399241\pi$
$14$	0	0
$15$	11.0467i	0.736445i
$16$	0	0
$17$	25.1978	1.48222	0.741110	−	0.671383i	$-0.234299\pi$
$17$	25.1978	1.48222	0.741110	+	0.671383i	$0.234299\pi$
$18$	0	0
$19$	− 1.59045i	− 0.0837080i	−0.999124	−	0.0418540i	$-0.986674\pi$
$19$	− 1.59045i	− 0.0837080i	0.999124	−	0.0418540i	$-0.0133264\pi$
$20$	0	0
$21$	−4.58258	−0.218218
$22$	0	0
$23$	15.8485i	0.689066i	0.938774	+	0.344533i	$0.111963\pi$
$23$	15.8485i	0.689066i	−0.938774	+	0.344533i	$0.888037\pi$
$24$	0	0
$25$	15.6764	0.627054
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	23.0850	0.796036	0.398018	−	0.917378i	$-0.369698\pi$
$29$	23.0850	0.796036	0.398018	+	0.917378i	$0.369698\pi$
$30$	0	0
$31$	1.68295i	0.0542887i	0.999632	+	0.0271443i	$0.00864137\pi$
$31$	1.68295i	0.0542887i	−0.999632	+	0.0271443i	$0.991359\pi$
$32$	0	0
$33$	8.95995	0.271514
$34$	0	0
$35$	16.8741i	0.482117i
$36$	0	0
$37$	−17.2232	−0.465491	−0.232745	−	0.972538i	$-0.574771\pi$
$37$	−17.2232	−0.465491	−0.232745	+	0.972538i	$0.574771\pi$
$38$	0	0
$39$	14.0181i	0.359438i
$40$	0	0
$41$	12.0970	0.295048	0.147524	−	0.989059i	$-0.452870\pi$
$41$	12.0970	0.295048	0.147524	+	0.989059i	$0.452870\pi$
$42$	0	0
$43$	31.1494i	0.724405i	0.932100	+	0.362202i	$0.117975\pi$
$43$	31.1494i	0.724405i	−0.932100	+	0.362202i	$0.882025\pi$
$44$	0	0
$45$	−19.1334	−0.425187
$46$	0	0
$47$	− 55.6971i	− 1.18504i	−0.805554	−	0.592522i	$-0.798132\pi$
$47$	− 55.6971i	− 1.18504i	0.805554	−	0.592522i	$-0.201868\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	43.6438i	0.855761i
$52$	0	0
$53$	−56.8217	−1.07211	−0.536054	−	0.844184i	$-0.680085\pi$
$53$	−56.8217	−1.07211	−0.536054	+	0.844184i	$0.680085\pi$
$54$	0	0
$55$	− 32.9926i	− 0.599865i
$56$	0	0
$57$	2.75475	0.0483289
$58$	0	0
$59$	47.5731i	0.806323i	0.915129	+	0.403161i	$0.132089\pi$
$59$	47.5731i	0.806323i	−0.915129	+	0.403161i	$0.867911\pi$
$60$	0	0
$61$	101.351	1.66149	0.830747	−	0.556651i	$-0.187914\pi$
$61$	101.351	1.66149	0.830747	+	0.556651i	$0.187914\pi$
$62$	0	0
$63$	− 7.93725i	− 0.125988i
$64$	0	0
$65$	51.6178	0.794120
$66$	0	0
$67$	58.9752i	0.880226i	0.897942	+	0.440113i	$0.145062\pi$
$67$	58.9752i	0.880226i	−0.897942	+	0.440113i	$0.854938\pi$
$68$	0	0
$69$	−27.4505	−0.397833
$70$	0	0
$71$	13.3175i	0.187570i	0.995592	+	0.0937851i	$0.0298967\pi$
$71$	13.3175i	0.187570i	−0.995592	+	0.0937851i	$0.970103\pi$
$72$	0	0
$73$	13.9050	0.190479	0.0952397	−	0.995454i	$-0.469638\pi$
$73$	13.9050	0.190479	0.0952397	+	0.995454i	$0.469638\pi$
$74$	0	0
$75$	27.1523i	0.362030i
$76$	0	0
$77$	13.6865	0.177747
$78$	0	0
$79$	45.8124i	0.579904i	0.957041	+	0.289952i	$0.0936394\pi$
$79$	45.8124i	0.579904i	−0.957041	+	0.289952i	$0.906361\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	145.317i	1.75080i	0.483396	+	0.875402i	$0.339403\pi$
$83$	145.317i	1.75080i	−0.483396	+	0.875402i	$0.660597\pi$
$84$	0	0
$85$	160.706	1.89066
$86$	0	0
$87$	39.9845i	0.459592i
$88$	0	0
$89$	150.256	1.68827	0.844135	−	0.536130i	$-0.180114\pi$
$89$	150.256	1.68827	0.844135	+	0.536130i	$0.180114\pi$
$90$	0	0
$91$	21.4130i	0.235308i
$92$	0	0
$93$	−2.91495	−0.0313436
$94$	0	0
$95$	− 10.1436i	− 0.106775i
$96$	0	0
$97$	−90.5264	−0.933262	−0.466631	−	0.884452i	$-0.654532\pi$
$97$	−90.5264	−0.933262	−0.466631	+	0.884452i	$0.654532\pi$
$98$	0	0
$99$	15.5191i	0.156758i
$100$	0	0
$101$	−157.083	−1.55528	−0.777640	−	0.628710i	$-0.783583\pi$
$101$	−157.083	−1.55528	−0.777640	+	0.628710i	$0.783583\pi$
$102$	0	0
$103$	− 142.058i	− 1.37920i	−0.724188	−	0.689602i	$-0.757785\pi$
$103$	− 142.058i	− 1.37920i	0.724188	−	0.689602i	$-0.242215\pi$
$104$	0	0
$105$	−29.2268	−0.278350
$106$	0	0
$107$	114.293i	1.06816i	0.845433	+	0.534082i	$0.179343\pi$
$107$	114.293i	1.06816i	−0.845433	+	0.534082i	$0.820657\pi$
$108$	0	0
$109$	171.256	1.57115	0.785577	−	0.618765i	$-0.212367\pi$
$109$	171.256	1.57115	0.785577	+	0.618765i	$0.212367\pi$
$110$	0	0
$111$	− 29.8314i	− 0.268751i
$112$	0	0
$113$	−113.011	−1.00010	−0.500051	−	0.865996i	$-0.666685\pi$
$113$	−113.011	−1.00010	−0.500051	+	0.865996i	$0.666685\pi$
$114$	0	0
$115$	101.079i	0.878946i
$116$	0	0
$117$	−24.2801	−0.207522
$118$	0	0
$119$	66.6670i	0.560227i
$120$	0	0
$121$	94.2398	0.778841
$122$	0	0
$123$	20.9525i	0.170346i
$124$	0	0
$125$	−59.4643	−0.475715
$126$	0	0
$127$	− 189.566i	− 1.49265i	−0.665583	−	0.746324i	$-0.731817\pi$
$127$	− 189.566i	− 1.49265i	0.665583	−	0.746324i	$-0.268183\pi$
$128$	0	0
$129$	−53.9523	−0.418235
$130$	0	0
$131$	119.818i	0.914641i	0.889302	+	0.457321i	$0.151191\pi$
$131$	119.818i	0.914641i	−0.889302	+	0.457321i	$0.848809\pi$
$132$	0	0
$133$	4.20794	0.0316387
$134$	0	0
$135$	− 33.1400i	− 0.245482i
$136$	0	0
$137$	−129.183	−0.942942	−0.471471	−	0.881881i	$-0.656277\pi$
$137$	−129.183	−0.942942	−0.471471	+	0.881881i	$0.656277\pi$
$138$	0	0
$139$	− 229.075i	− 1.64802i	−0.566572	−	0.824012i	$-0.691731\pi$
$139$	− 229.075i	− 1.64802i	0.566572	−	0.824012i	$-0.308269\pi$
$140$	0	0
$141$	96.4701	0.684185
$142$	0	0
$143$	− 41.8672i	− 0.292777i
$144$	0	0
$145$	147.232	1.01539
$146$	0	0
$147$	− 12.1244i	− 0.0824786i
$148$	0	0
$149$	−72.7587	−0.488313	−0.244157	−	0.969736i	$-0.578511\pi$
$149$	−72.7587	−0.488313	−0.244157	+	0.969736i	$0.578511\pi$
$150$	0	0
$151$	− 255.512i	− 1.69213i	−0.533078	−	0.846066i	$-0.678965\pi$
$151$	− 255.512i	− 1.69213i	0.533078	−	0.846066i	$-0.321035\pi$
$152$	0	0
$153$	−75.5933	−0.494074
$154$	0	0
$155$	10.7335i	0.0692485i
$156$	0	0
$157$	−37.2470	−0.237242	−0.118621	−	0.992940i	$-0.537847\pi$
$157$	−37.2470	−0.237242	−0.118621	+	0.992940i	$0.537847\pi$
$158$	0	0
$159$	− 98.4180i	− 0.618981i
$160$	0	0
$161$	−41.9313	−0.260443
$162$	0	0
$163$	− 213.932i	− 1.31247i	−0.754558	−	0.656233i	$-0.772149\pi$
$163$	− 213.932i	− 1.31247i	0.754558	−	0.656233i	$-0.227851\pi$
$164$	0	0
$165$	57.1448	0.346332
$166$	0	0
$167$	− 75.0723i	− 0.449535i	−0.974412	−	0.224767i	$-0.927838\pi$
$167$	− 75.0723i	− 0.449535i	0.974412	−	0.224767i	$-0.0721623\pi$
$168$	0	0
$169$	−103.498	−0.612412
$170$	0	0
$171$	4.77136i	0.0279027i
$172$	0	0
$173$	−33.5589	−0.193982	−0.0969909	−	0.995285i	$-0.530922\pi$
$173$	−33.5589	−0.193982	−0.0969909	+	0.995285i	$0.530922\pi$
$174$	0	0
$175$	41.4757i	0.237004i
$176$	0	0
$177$	−82.3989	−0.465531
$178$	0	0
$179$	− 284.970i	− 1.59201i	−0.605289	−	0.796006i	$-0.706942\pi$
$179$	− 284.970i	− 1.59201i	0.605289	−	0.796006i	$-0.293058\pi$
$180$	0	0
$181$	5.94109	0.0328237	0.0164119	−	0.999865i	$-0.494776\pi$
$181$	5.94109	0.0328237	0.0164119	+	0.999865i	$0.494776\pi$
$182$	0	0
$183$	175.545i	0.959263i
$184$	0	0
$185$	−109.846	−0.593762
$186$	0	0
$187$	− 130.349i	− 0.697052i
$188$	0	0
$189$	13.7477	0.0727393
$190$	0	0
$191$	− 22.3638i	− 0.117088i	−0.998285	−	0.0585440i	$-0.981354\pi$
$191$	− 22.3638i	− 0.117088i	0.998285	−	0.0585440i	$-0.0186458\pi$
$192$	0	0
$193$	29.0238	0.150382	0.0751911	−	0.997169i	$-0.476043\pi$
$193$	29.0238	0.150382	0.0751911	+	0.997169i	$0.476043\pi$
$194$	0	0
$195$	89.4047i	0.458486i
$196$	0	0
$197$	−252.577	−1.28212	−0.641059	−	0.767492i	$-0.721505\pi$
$197$	−252.577	−1.28212	−0.641059	+	0.767492i	$0.721505\pi$
$198$	0	0
$199$	− 81.0611i	− 0.407342i	−0.979039	−	0.203671i	$-0.934713\pi$
$199$	− 81.0611i	− 0.407342i	0.979039	−	0.203671i	$-0.0652873\pi$
$200$	0	0
$201$	−102.148	−0.508199
$202$	0	0
$203$	61.0773i	0.300873i
$204$	0	0
$205$	77.1520	0.376351
$206$	0	0
$207$	− 47.5456i	− 0.229689i
$208$	0	0
$209$	−8.22746	−0.0393658
$210$	0	0
$211$	292.377i	1.38567i	0.721095	+	0.692836i	$0.243639\pi$
$211$	292.377i	1.38567i	−0.721095	+	0.692836i	$0.756361\pi$
$212$	0	0
$213$	−23.0666	−0.108294
$214$	0	0
$215$	198.665i	0.924022i
$216$	0	0
$217$	−4.45266	−0.0205192
$218$	0	0
$219$	24.0842i	0.109973i
$220$	0	0
$221$	203.934	0.922780
$222$	0	0
$223$	49.0529i	0.219968i	0.993933	+	0.109984i	$0.0350800\pi$
$223$	49.0529i	0.219968i	−0.993933	+	0.109984i	$0.964920\pi$
$224$	0	0
$225$	−47.0291	−0.209018
$226$	0	0
$227$	− 209.041i	− 0.920884i	−0.887690	−	0.460442i	$-0.847691\pi$
$227$	− 209.041i	− 0.920884i	0.887690	−	0.460442i	$-0.152309\pi$
$228$	0	0
$229$	−439.826	−1.92064	−0.960319	−	0.278904i	$-0.910029\pi$
$229$	−439.826	−1.92064	−0.960319	+	0.278904i	$0.910029\pi$
$230$	0	0
$231$	23.7058i	0.102622i
$232$	0	0
$233$	207.035	0.888563	0.444282	−	0.895887i	$-0.353459\pi$
$233$	207.035	0.888563	0.444282	+	0.895887i	$0.353459\pi$
$234$	0	0
$235$	− 355.225i	− 1.51159i
$236$	0	0
$237$	−79.3495	−0.334808
$238$	0	0
$239$	− 189.157i	− 0.791454i	−0.918368	−	0.395727i	$-0.870493\pi$
$239$	− 189.157i	− 0.791454i	0.918368	−	0.395727i	$-0.129507\pi$
$240$	0	0
$241$	179.181	0.743489	0.371745	−	0.928335i	$-0.378760\pi$
$241$	179.181	0.743489	0.371745	+	0.928335i	$0.378760\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−44.6446	−0.182223
$246$	0	0
$247$	− 12.8721i	− 0.0521138i
$248$	0	0
$249$	−251.696	−1.01083
$250$	0	0
$251$	191.872i	0.764428i	0.924074	+	0.382214i	$0.124838\pi$
$251$	191.872i	0.764428i	−0.924074	+	0.382214i	$0.875162\pi$
$252$	0	0
$253$	81.9849	0.324051
$254$	0	0
$255$	278.351i	1.09157i
$256$	0	0
$257$	319.778	1.24427	0.622135	−	0.782910i	$-0.286265\pi$
$257$	319.778	1.24427	0.622135	+	0.782910i	$0.286265\pi$
$258$	0	0
$259$	− 45.5682i	− 0.175939i
$260$	0	0
$261$	−69.2551	−0.265345
$262$	0	0
$263$	− 113.522i	− 0.431644i	−0.976433	−	0.215822i	$-0.930757\pi$
$263$	− 113.522i	− 0.431644i	0.976433	−	0.215822i	$-0.0692431\pi$
$264$	0	0
$265$	−362.397	−1.36754
$266$	0	0
$267$	260.251i	0.974724i
$268$	0	0
$269$	−406.524	−1.51124	−0.755620	−	0.655010i	$-0.772664\pi$
$269$	−406.524	−1.51124	−0.755620	+	0.655010i	$0.772664\pi$
$270$	0	0
$271$	− 508.180i	− 1.87520i	−0.347712	−	0.937601i	$-0.613041\pi$
$271$	− 508.180i	− 1.87520i	0.347712	−	0.937601i	$-0.386959\pi$
$272$	0	0
$273$	−37.0884	−0.135855
$274$	0	0
$275$	− 81.0943i	− 0.294888i
$276$	0	0
$277$	400.803	1.44694	0.723470	−	0.690355i	$-0.242546\pi$
$277$	400.803	1.44694	0.723470	+	0.690355i	$0.242546\pi$
$278$	0	0
$279$	− 5.04885i	− 0.0180962i
$280$	0	0
$281$	−544.569	−1.93797	−0.968984	−	0.247122i	$-0.920515\pi$
$281$	−544.569	−1.93797	−0.968984	+	0.247122i	$0.920515\pi$
$282$	0	0
$283$	185.268i	0.654656i	0.944911	+	0.327328i	$0.106148\pi$
$283$	185.268i	0.654656i	−0.944911	+	0.327328i	$0.893852\pi$
$284$	0	0
$285$	17.5692	0.0616464
$286$	0	0
$287$	32.0055i	0.111518i
$288$	0	0
$289$	345.927	1.19698
$290$	0	0
$291$	− 156.796i	− 0.538819i
$292$	0	0
$293$	−465.044	−1.58718	−0.793590	−	0.608452i	$-0.791791\pi$
$293$	−465.044	−1.58718	−0.793590	+	0.608452i	$0.791791\pi$
$294$	0	0
$295$	303.412i	1.02851i
$296$	0	0
$297$	−26.8798	−0.0905045
$298$	0	0
$299$	128.268i	0.428989i
$300$	0	0
$301$	−82.4136	−0.273799
$302$	0	0
$303$	− 272.076i	− 0.897941i
$304$	0	0
$305$	646.397	2.11933
$306$	0	0
$307$	274.356i	0.893669i	0.894617	+	0.446834i	$0.147449\pi$
$307$	274.356i	0.893669i	−0.894617	+	0.446834i	$0.852551\pi$
$308$	0	0
$309$	246.052	0.796284
$310$	0	0
$311$	− 14.2196i	− 0.0457223i	−0.999739	−	0.0228612i	$-0.992722\pi$
$311$	− 14.2196i	− 0.0457223i	0.999739	−	0.0228612i	$-0.00727757\pi$
$312$	0	0
$313$	−213.256	−0.681329	−0.340664	−	0.940185i	$-0.610652\pi$
$313$	−213.256	−0.681329	−0.340664	+	0.940185i	$0.610652\pi$
$314$	0	0
$315$	− 50.6222i	− 0.160706i
$316$	0	0
$317$	−3.37863	−0.0106582	−0.00532908	−	0.999986i	$-0.501696\pi$
$317$	−3.37863	−0.0106582	−0.00532908	+	0.999986i	$0.501696\pi$
$318$	0	0
$319$	− 119.420i	− 0.374356i
$320$	0	0
$321$	−197.962	−0.616704
$322$	0	0
$323$	− 40.0758i	− 0.124074i
$324$	0	0
$325$	126.874	0.390383
$326$	0	0
$327$	296.624i	0.907106i
$328$	0	0
$329$	147.361	0.447904
$330$	0	0
$331$	− 456.028i	− 1.37773i	−0.724891	−	0.688863i	$-0.758110\pi$
$331$	− 456.028i	− 1.37773i	0.724891	−	0.688863i	$-0.241890\pi$
$332$	0	0
$333$	51.6695	0.155164
$334$	0	0
$335$	376.132i	1.12278i
$336$	0	0
$337$	−149.399	−0.443321	−0.221661	−	0.975124i	$-0.571148\pi$
$337$	−149.399	−0.443321	−0.221661	+	0.975124i	$0.571148\pi$
$338$	0	0
$339$	− 195.742i	− 0.577409i
$340$	0	0
$341$	8.70594	0.0255306
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−175.074	−0.507460
$346$	0	0
$347$	− 486.035i	− 1.40068i	−0.713811	−	0.700338i	$-0.753032\pi$
$347$	− 486.035i	− 1.40068i	0.713811	−	0.700338i	$-0.246968\pi$
$348$	0	0
$349$	−349.059	−1.00017	−0.500085	−	0.865977i	$-0.666698\pi$
$349$	−349.059	−1.00017	−0.500085	+	0.865977i	$0.666698\pi$
$350$	0	0
$351$	− 42.0543i	− 0.119813i
$352$	0	0
$353$	−47.2923	−0.133973	−0.0669863	−	0.997754i	$-0.521338\pi$
$353$	−47.2923	−0.133973	−0.0669863	+	0.997754i	$0.521338\pi$
$354$	0	0
$355$	84.9363i	0.239257i
$356$	0	0
$357$	−115.471	−0.323447
$358$	0	0
$359$	− 332.202i	− 0.925353i	−0.886527	−	0.462677i	$-0.846889\pi$
$359$	− 332.202i	− 0.925353i	0.886527	−	0.462677i	$-0.153111\pi$
$360$	0	0
$361$	358.470	0.992993
$362$	0	0
$363$	163.228i	0.449664i
$364$	0	0
$365$	88.6833	0.242968
$366$	0	0
$367$	191.517i	0.521843i	0.965360	+	0.260922i	$0.0840264\pi$
$367$	191.517i	0.521843i	−0.965360	+	0.260922i	$0.915974\pi$
$368$	0	0
$369$	−36.2909	−0.0983492
$370$	0	0
$371$	− 150.336i	− 0.405218i
$372$	0	0
$373$	−39.5852	−0.106126	−0.0530632	−	0.998591i	$-0.516898\pi$
$373$	−39.5852	−0.106126	−0.0530632	+	0.998591i	$0.516898\pi$
$374$	0	0
$375$	− 102.995i	− 0.274654i
$376$	0	0
$377$	186.835	0.495585
$378$	0	0
$379$	− 390.744i	− 1.03099i	−0.856894	−	0.515493i	$-0.827609\pi$
$379$	− 390.744i	− 1.03099i	0.856894	−	0.515493i	$-0.172391\pi$
$380$	0	0
$381$	328.338	0.861781
$382$	0	0
$383$	82.0354i	0.214192i	0.994249	+	0.107096i	$0.0341552\pi$
$383$	82.0354i	0.214192i	−0.994249	+	0.107096i	$0.965845\pi$
$384$	0	0
$385$	87.2901	0.226728
$386$	0	0
$387$	− 93.4482i	− 0.241468i
$388$	0	0
$389$	192.620	0.495166	0.247583	−	0.968867i	$-0.420364\pi$
$389$	192.620	0.495166	0.247583	+	0.968867i	$0.420364\pi$
$390$	0	0
$391$	399.347i	1.02135i
$392$	0	0
$393$	−207.531	−0.528068
$394$	0	0
$395$	292.183i	0.739703i
$396$	0	0
$397$	185.930	0.468337	0.234169	−	0.972196i	$-0.424763\pi$
$397$	185.930	0.468337	0.234169	+	0.972196i	$0.424763\pi$
$398$	0	0
$399$	7.28837i	0.0182666i
$400$	0	0
$401$	501.532	1.25070	0.625351	−	0.780343i	$-0.284956\pi$
$401$	501.532	1.25070	0.625351	+	0.780343i	$0.284956\pi$
$402$	0	0
$403$	13.6207i	0.0337983i
$404$	0	0
$405$	57.4002	0.141729
$406$	0	0
$407$	89.0959i	0.218909i
$408$	0	0
$409$	−604.348	−1.47762	−0.738812	−	0.673912i	$-0.764613\pi$
$409$	−604.348	−1.47762	−0.738812	+	0.673912i	$0.764613\pi$
$410$	0	0
$411$	− 223.752i	− 0.544408i
$412$	0	0
$413$	−125.866	−0.304761
$414$	0	0
$415$	926.801i	2.23326i
$416$	0	0
$417$	396.770	0.951487
$418$	0	0
$419$	129.024i	0.307932i	0.988076	+	0.153966i	$0.0492047\pi$
$419$	129.024i	0.307932i	−0.988076	+	0.153966i	$0.950795\pi$
$420$	0	0
$421$	97.4092	0.231376	0.115688	−	0.993286i	$-0.463093\pi$
$421$	97.4092	0.231376	0.115688	+	0.993286i	$0.463093\pi$
$422$	0	0
$423$	167.091i	0.395015i
$424$	0	0
$425$	395.009	0.929433
$426$	0	0
$427$	268.150i	0.627985i
$428$	0	0
$429$	72.5160	0.169035
$430$	0	0
$431$	189.028i	0.438580i	0.975660	+	0.219290i	$0.0703741\pi$
$431$	189.028i	0.438580i	−0.975660	+	0.219290i	$0.929626\pi$
$432$	0	0
$433$	679.795	1.56996	0.784982	−	0.619518i	$-0.212672\pi$
$433$	679.795	1.56996	0.784982	+	0.619518i	$0.212672\pi$
$434$	0	0
$435$	255.013i	0.586237i
$436$	0	0
$437$	25.2063	0.0576804
$438$	0	0
$439$	− 266.934i	− 0.608049i	−0.952664	−	0.304025i	$-0.901670\pi$
$439$	− 266.934i	− 0.608049i	0.952664	−	0.304025i	$-0.0983305\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	219.470i	0.495418i	0.968835	+	0.247709i	$0.0796776\pi$
$443$	219.470i	0.495418i	−0.968835	+	0.247709i	$0.920322\pi$
$444$	0	0
$445$	958.304	2.15349
$446$	0	0
$447$	− 126.022i	− 0.281928i
$448$	0	0
$449$	−93.1600	−0.207483	−0.103742	−	0.994604i	$-0.533082\pi$
$449$	−93.1600	−0.207483	−0.103742	+	0.994604i	$0.533082\pi$
$450$	0	0
$451$	− 62.5779i	− 0.138754i
$452$	0	0
$453$	442.560	0.976953
$454$	0	0
$455$	136.568i	0.300149i
$456$	0	0
$457$	32.7526	0.0716686	0.0358343	−	0.999358i	$-0.488591\pi$
$457$	32.7526	0.0716686	0.0358343	+	0.999358i	$0.488591\pi$
$458$	0	0
$459$	− 130.931i	− 0.285254i
$460$	0	0
$461$	4.03180	0.00874578	0.00437289	−	0.999990i	$-0.498608\pi$
$461$	4.03180	0.00874578	0.00437289	+	0.999990i	$0.498608\pi$
$462$	0	0
$463$	789.033i	1.70417i	0.523399	+	0.852087i	$0.324664\pi$
$463$	789.033i	1.70417i	−0.523399	+	0.852087i	$0.675336\pi$
$464$	0	0
$465$	−18.5910	−0.0399806
$466$	0	0
$467$	168.642i	0.361118i	0.983564	+	0.180559i	$0.0577907\pi$
$467$	168.642i	0.361118i	−0.983564	+	0.180559i	$0.942209\pi$
$468$	0	0
$469$	−156.034	−0.332694
$470$	0	0
$471$	− 64.5137i	− 0.136972i
$472$	0	0
$473$	161.137	0.340670
$474$	0	0
$475$	− 24.9325i	− 0.0524895i
$476$	0	0
$477$	170.465	0.357369
$478$	0	0
$479$	866.788i	1.80958i	0.425860	+	0.904789i	$0.359972\pi$
$479$	866.788i	1.80958i	−0.425860	+	0.904789i	$0.640028\pi$
$480$	0	0
$481$	−139.393	−0.289799
$482$	0	0
$483$	− 72.6271i	− 0.150367i
$484$	0	0
$485$	−577.360	−1.19043
$486$	0	0
$487$	16.5605i	0.0340051i	0.999855	+	0.0170026i	$0.00541234\pi$
$487$	16.5605i	0.0340051i	−0.999855	+	0.0170026i	$0.994588\pi$
$488$	0	0
$489$	370.541	0.757753
$490$	0	0
$491$	− 17.3669i	− 0.0353705i	−0.999844	−	0.0176852i	$-0.994370\pi$
$491$	− 17.3669i	− 0.0353705i	0.999844	−	0.0176852i	$-0.00562968\pi$
$492$	0	0
$493$	581.691	1.17990
$494$	0	0
$495$	98.9777i	0.199955i
$496$	0	0
$497$	−35.2347	−0.0708949
$498$	0	0
$499$	− 194.180i	− 0.389139i	−0.980889	−	0.194570i	$-0.937669\pi$
$499$	− 194.180i	− 0.389139i	0.980889	−	0.194570i	$-0.0623310\pi$
$500$	0	0
$501$	130.029	0.259539
$502$	0	0
$503$	839.393i	1.66877i	0.551179	+	0.834387i	$0.314178\pi$
$503$	839.393i	1.66877i	−0.551179	+	0.834387i	$0.685822\pi$
$504$	0	0
$505$	−1001.85	−1.98385
$506$	0	0
$507$	− 179.263i	− 0.353576i
$508$	0	0
$509$	217.634	0.427571	0.213785	−	0.976881i	$-0.431421\pi$
$509$	217.634	0.427571	0.213785	+	0.976881i	$0.431421\pi$
$510$	0	0
$511$	36.7892i	0.0719945i
$512$	0	0
$513$	−8.26424	−0.0161096
$514$	0	0
$515$	− 906.018i	− 1.75926i
$516$	0	0
$517$	−288.122	−0.557297
$518$	0	0
$519$	− 58.1256i	− 0.111995i
$520$	0	0
$521$	−511.156	−0.981106	−0.490553	−	0.871411i	$-0.663205\pi$
$521$	−511.156	−0.981106	−0.490553	+	0.871411i	$0.663205\pi$
$522$	0	0
$523$	− 414.920i	− 0.793346i	−0.917960	−	0.396673i	$-0.870165\pi$
$523$	− 414.920i	− 0.793346i	0.917960	−	0.396673i	$-0.129835\pi$
$524$	0	0
$525$	−71.8381	−0.136834
$526$	0	0
$527$	42.4065i	0.0804678i
$528$	0	0
$529$	277.824	0.525187
$530$	0	0
$531$	− 142.719i	− 0.268774i
$532$	0	0
$533$	97.9049	0.183687
$534$	0	0
$535$	728.941i	1.36251i
$536$	0	0
$537$	493.583	0.919149
$538$	0	0
$539$	36.2112i	0.0671822i
$540$	0	0
$541$	824.298	1.52366	0.761828	−	0.647780i	$-0.224302\pi$
$541$	824.298	1.52366	0.761828	+	0.647780i	$0.224302\pi$
$542$	0	0
$543$	10.2903i	0.0189508i
$544$	0	0
$545$	1092.23	2.00410
$546$	0	0
$547$	− 357.494i	− 0.653553i	−0.945102	−	0.326777i	$-0.894037\pi$
$547$	− 357.494i	− 0.653553i	0.945102	−	0.326777i	$-0.105963\pi$
$548$	0	0
$549$	−304.053	−0.553831
$550$	0	0
$551$	− 36.7157i	− 0.0666346i
$552$	0	0
$553$	−121.208	−0.219183
$554$	0	0
$555$	− 190.259i	− 0.342808i
$556$	0	0
$557$	−401.693	−0.721173	−0.360586	−	0.932726i	$-0.617423\pi$
$557$	−401.693	−0.721173	−0.360586	+	0.932726i	$0.617423\pi$
$558$	0	0
$559$	252.103i	0.450989i
$560$	0	0
$561$	225.771	0.402443
$562$	0	0
$563$	34.6358i	0.0615200i	0.999527	+	0.0307600i	$0.00979276\pi$
$563$	34.6358i	0.0615200i	−0.999527	+	0.0307600i	$0.990207\pi$
$564$	0	0
$565$	−720.765	−1.27569
$566$	0	0
$567$	23.8118i	0.0419961i
$568$	0	0
$569$	−650.576	−1.14337	−0.571684	−	0.820474i	$-0.693710\pi$
$569$	−650.576	−1.14337	−0.571684	+	0.820474i	$0.693710\pi$
$570$	0	0
$571$	624.560i	1.09380i	0.837198	+	0.546900i	$0.184192\pi$
$571$	624.560i	1.09380i	−0.837198	+	0.546900i	$0.815808\pi$
$572$	0	0
$573$	38.7353	0.0676008
$574$	0	0
$575$	248.447i	0.432082i
$576$	0	0
$577$	−885.685	−1.53498	−0.767491	−	0.641059i	$-0.778495\pi$
$577$	−885.685	−1.53498	−0.767491	+	0.641059i	$0.778495\pi$
$578$	0	0
$579$	50.2707i	0.0868232i
$580$	0	0
$581$	−384.472	−0.661741
$582$	0	0
$583$	293.940i	0.504185i
$584$	0	0
$585$	−154.853	−0.264707
$586$	0	0
$587$	593.145i	1.01047i	0.862982	+	0.505235i	$0.168594\pi$
$587$	593.145i	1.01047i	−0.862982	+	0.505235i	$0.831406\pi$
$588$	0	0
$589$	2.67665	0.00454440
$590$	0	0
$591$	− 437.477i	− 0.740231i
$592$	0	0
$593$	812.442	1.37005	0.685027	−	0.728518i	$-0.259790\pi$
$593$	812.442	1.37005	0.685027	+	0.728518i	$0.259790\pi$
$594$	0	0
$595$	425.189i	0.714603i
$596$	0	0
$597$	140.402	0.235179
$598$	0	0
$599$	− 987.235i	− 1.64814i	−0.566488	−	0.824070i	$-0.691698\pi$
$599$	− 987.235i	− 1.64814i	0.566488	−	0.824070i	$-0.308302\pi$
$600$	0	0
$601$	−875.190	−1.45622	−0.728112	−	0.685458i	$-0.759602\pi$
$601$	−875.190	−1.45622	−0.728112	+	0.685458i	$0.759602\pi$
$602$	0	0
$603$	− 176.925i	− 0.293409i
$604$	0	0
$605$	601.043	0.993459
$606$	0	0
$607$	− 1135.57i	− 1.87079i	−0.353601	−	0.935396i	$-0.615043\pi$
$607$	− 1135.57i	− 1.87079i	0.353601	−	0.935396i	$-0.384957\pi$
$608$	0	0
$609$	−105.789	−0.173709
$610$	0	0
$611$	− 450.776i	− 0.737768i
$612$	0	0
$613$	−458.347	−0.747710	−0.373855	−	0.927487i	$-0.621964\pi$
$613$	−458.347	−0.747710	−0.373855	+	0.927487i	$0.621964\pi$
$614$	0	0
$615$	133.631i	0.217286i
$616$	0	0
$617$	712.482	1.15475	0.577376	−	0.816479i	$-0.304077\pi$
$617$	712.482	1.15475	0.577376	+	0.816479i	$0.304077\pi$
$618$	0	0
$619$	1055.45i	1.70510i	0.522649	+	0.852548i	$0.324944\pi$
$619$	1055.45i	1.70510i	−0.522649	+	0.852548i	$0.675056\pi$
$620$	0	0
$621$	82.3514	0.132611
$622$	0	0
$623$	397.540i	0.638106i
$624$	0	0
$625$	−771.161	−1.23386
$626$	0	0
$627$	− 14.2504i	− 0.0227279i
$628$	0	0
$629$	−433.985	−0.689960
$630$	0	0
$631$	− 652.825i	− 1.03459i	−0.855808	−	0.517294i	$-0.826940\pi$
$631$	− 652.825i	− 1.03459i	0.855808	−	0.517294i	$-0.173060\pi$
$632$	0	0
$633$	−506.412	−0.800018
$634$	0	0
$635$	− 1209.02i	− 1.90396i
$636$	0	0
$637$	−56.6535	−0.0889380
$638$	0	0
$639$	− 39.9524i	− 0.0625234i
$640$	0	0
$641$	−193.690	−0.302168	−0.151084	−	0.988521i	$-0.548276\pi$
$641$	−193.690	−0.302168	−0.151084	+	0.988521i	$0.548276\pi$
$642$	0	0
$643$	227.685i	0.354098i	0.984202	+	0.177049i	$0.0566551\pi$
$643$	227.685i	0.354098i	−0.984202	+	0.177049i	$0.943345\pi$
$644$	0	0
$645$	−344.097	−0.533484
$646$	0	0
$647$	− 779.133i	− 1.20422i	−0.798412	−	0.602112i	$-0.794326\pi$
$647$	− 779.133i	− 1.20422i	0.798412	−	0.602112i	$-0.205674\pi$
$648$	0	0
$649$	246.097	0.379194
$650$	0	0
$651$	− 7.71224i	− 0.0118468i
$652$	0	0
$653$	267.074	0.408995	0.204498	−	0.978867i	$-0.434444\pi$
$653$	267.074	0.408995	0.204498	+	0.978867i	$0.434444\pi$
$654$	0	0
$655$	764.175i	1.16668i
$656$	0	0
$657$	−41.7150	−0.0634932
$658$	0	0
$659$	281.512i	0.427181i	0.976923	+	0.213590i	$0.0685158\pi$
$659$	281.512i	0.427181i	−0.976923	+	0.213590i	$0.931484\pi$
$660$	0	0
$661$	671.521	1.01592	0.507958	−	0.861382i	$-0.330400\pi$
$661$	671.521	1.01592	0.507958	+	0.861382i	$0.330400\pi$
$662$	0	0
$663$	353.225i	0.532767i
$664$	0	0
$665$	26.8374	0.0403570
$666$	0	0
$667$	365.864i	0.548522i
$668$	0	0
$669$	−84.9620	−0.126999
$670$	0	0
$671$	− 524.292i	− 0.781359i
$672$	0	0
$673$	−1104.65	−1.64138	−0.820690	−	0.571374i	$-0.806410\pi$
$673$	−1104.65	−1.64138	−0.820690	+	0.571374i	$0.806410\pi$
$674$	0	0
$675$	− 81.4568i	− 0.120677i
$676$	0	0
$677$	−245.606	−0.362785	−0.181393	−	0.983411i	$-0.558060\pi$
$677$	−245.606	−0.362785	−0.181393	+	0.983411i	$0.558060\pi$
$678$	0	0
$679$	− 239.510i	− 0.352740i
$680$	0	0
$681$	362.069	0.531673
$682$	0	0
$683$	502.169i	0.735240i	0.929976	+	0.367620i	$0.119827\pi$
$683$	502.169i	0.735240i	−0.929976	+	0.367620i	$0.880173\pi$
$684$	0	0
$685$	−823.904	−1.20278
$686$	0	0
$687$	− 761.801i	− 1.10888i
$688$	0	0
$689$	−459.878	−0.667457
$690$	0	0
$691$	324.045i	0.468951i	0.972122	+	0.234475i	$0.0753372\pi$
$691$	324.045i	0.468951i	−0.972122	+	0.234475i	$0.924663\pi$
$692$	0	0
$693$	−41.0596	−0.0592491
$694$	0	0
$695$	− 1461.00i	− 2.10215i
$696$	0	0
$697$	304.816	0.437326
$698$	0	0
$699$	358.596i	0.513012i
$700$	0	0
$701$	1218.75	1.73858	0.869291	−	0.494300i	$-0.164576\pi$
$701$	1218.75	1.73858	0.869291	+	0.494300i	$0.164576\pi$
$702$	0	0
$703$	27.3926i	0.0389653i
$704$	0	0
$705$	615.267	0.872720
$706$	0	0
$707$	− 415.603i	− 0.587840i
$708$	0	0
$709$	−617.631	−0.871130	−0.435565	−	0.900157i	$-0.643451\pi$
$709$	−617.631	−0.871130	−0.435565	+	0.900157i	$0.643451\pi$
$710$	0	0
$711$	− 137.437i	− 0.193301i
$712$	0	0
$713$	−26.6723	−0.0374085
$714$	0	0
$715$	− 267.020i	− 0.373455i
$716$	0	0
$717$	327.630	0.456946
$718$	0	0
$719$	− 693.411i	− 0.964410i	−0.876058	−	0.482205i	$-0.839836\pi$
$719$	− 693.411i	− 0.964410i	0.876058	−	0.482205i	$-0.160164\pi$
$720$	0	0
$721$	375.850	0.521290
$722$	0	0
$723$	310.350i	0.429254i
$724$	0	0
$725$	361.890	0.499158
$726$	0	0
$727$	407.809i	0.560947i	0.959862	+	0.280474i	$0.0904915\pi$
$727$	407.809i	0.560947i	−0.959862	+	0.280474i	$0.909508\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	784.895i	1.07373i
$732$	0	0
$733$	−251.068	−0.342521	−0.171260	−	0.985226i	$-0.554784\pi$
$733$	−251.068	−0.342521	−0.171260	+	0.985226i	$0.554784\pi$
$734$	0	0
$735$	− 77.3267i	− 0.105206i
$736$	0	0
$737$	305.080	0.413949
$738$	0	0
$739$	529.073i	0.715931i	0.933735	+	0.357965i	$0.116529\pi$
$739$	529.073i	0.715931i	−0.933735	+	0.357965i	$0.883471\pi$
$740$	0	0
$741$	22.2951	0.0300879
$742$	0	0
$743$	563.363i	0.758227i	0.925350	+	0.379114i	$0.123771\pi$
$743$	563.363i	0.758227i	−0.925350	+	0.379114i	$0.876229\pi$
$744$	0	0
$745$	−464.040	−0.622873
$746$	0	0
$747$	− 435.950i	− 0.583601i
$748$	0	0
$749$	−302.392	−0.403728
$750$	0	0
$751$	594.813i	0.792028i	0.918245	+	0.396014i	$0.129607\pi$
$751$	594.813i	0.792028i	−0.918245	+	0.396014i	$0.870393\pi$
$752$	0	0
$753$	−332.331	−0.441343
$754$	0	0
$755$	− 1629.61i	− 2.15842i
$756$	0	0
$757$	−239.379	−0.316221	−0.158111	−	0.987421i	$-0.550540\pi$
$757$	−239.379	−0.316221	−0.158111	+	0.987421i	$0.550540\pi$
$758$	0	0
$759$	142.002i	0.187091i
$760$	0	0
$761$	925.966	1.21678	0.608388	−	0.793640i	$-0.291817\pi$
$761$	925.966	1.21678	0.608388	+	0.793640i	$0.291817\pi$
$762$	0	0
$763$	453.100i	0.593840i
$764$	0	0
$765$	−482.119	−0.630221
$766$	0	0
$767$	385.026i	0.501989i
$768$	0	0
$769$	625.702	0.813657	0.406828	−	0.913505i	$-0.366635\pi$
$769$	625.702	0.813657	0.406828	+	0.913505i	$0.366635\pi$
$770$	0	0
$771$	553.871i	0.718380i
$772$	0	0
$773$	497.274	0.643304	0.321652	−	0.946858i	$-0.395762\pi$
$773$	497.274	0.643304	0.321652	+	0.946858i	$0.395762\pi$
$774$	0	0
$775$	26.3825i	0.0340419i
$776$	0	0
$777$	78.9264	0.101578
$778$	0	0
$779$	− 19.2396i	− 0.0246979i
$780$	0	0
$781$	68.8917	0.0882096
$782$	0	0
$783$	− 119.953i	− 0.153197i
$784$	0	0
$785$	−237.554	−0.302617
$786$	0	0
$787$	48.9026i	0.0621380i	0.999517	+	0.0310690i	$0.00989117\pi$
$787$	48.9026i	0.0621380i	−0.999517	+	0.0310690i	$0.990109\pi$
$788$	0	0
$789$	196.627	0.249210
$790$	0	0
$791$	− 299.000i	− 0.378003i
$792$	0	0
$793$	820.270	1.03439
$794$	0	0
$795$	− 627.691i	− 0.789548i
$796$	0	0
$797$	−547.361	−0.686777	−0.343388	−	0.939193i	$-0.611575\pi$
$797$	−547.361	−0.686777	−0.343388	+	0.939193i	$0.611575\pi$
$798$	0	0
$799$	− 1403.44i	− 1.75650i
$800$	0	0
$801$	−450.768	−0.562757
$802$	0	0
$803$	− 71.9310i	− 0.0895778i
$804$	0	0
$805$	−267.429	−0.332210
$806$	0	0
$807$	− 704.120i	− 0.872515i
$808$	0	0
$809$	−40.2545	−0.0497583	−0.0248792	−	0.999690i	$-0.507920\pi$
$809$	−40.2545	−0.0497583	−0.0248792	+	0.999690i	$0.507920\pi$
$810$	0	0
$811$	187.630i	0.231356i	0.993287	+	0.115678i	$0.0369041\pi$
$811$	187.630i	0.231356i	−0.993287	+	0.115678i	$0.963096\pi$
$812$	0	0
$813$	880.193	1.08265
$814$	0	0
$815$	− 1364.42i	− 1.67413i
$816$	0	0
$817$	49.5416	0.0606385
$818$	0	0
$819$	− 64.2390i	− 0.0784359i
$820$	0	0
$821$	−727.736	−0.886402	−0.443201	−	0.896422i	$-0.646157\pi$
$821$	−727.736	−0.886402	−0.443201	+	0.896422i	$0.646157\pi$
$822$	0	0
$823$	− 741.649i	− 0.901154i	−0.892738	−	0.450577i	$-0.851218\pi$
$823$	− 741.649i	− 0.901154i	0.892738	−	0.450577i	$-0.148782\pi$
$824$	0	0
$825$	140.459	0.170254
$826$	0	0
$827$	132.347i	0.160032i	0.996794	+	0.0800161i	$0.0254972\pi$
$827$	132.347i	0.160032i	−0.996794	+	0.0800161i	$0.974503\pi$
$828$	0	0
$829$	−1182.43	−1.42634	−0.713169	−	0.700992i	$-0.752741\pi$
$829$	−1182.43	−1.42634	−0.713169	+	0.700992i	$0.752741\pi$
$830$	0	0
$831$	694.211i	0.835392i
$832$	0	0
$833$	−176.384	−0.211746
$834$	0	0
$835$	− 478.797i	− 0.573409i
$836$	0	0
$837$	8.74486	0.0104479
$838$	0	0
$839$	− 499.514i	− 0.595368i	−0.954664	−	0.297684i	$-0.903786\pi$
$839$	− 499.514i	− 0.595368i	0.954664	−	0.297684i	$-0.0962142\pi$
$840$	0	0
$841$	−308.081	−0.366326
$842$	0	0
$843$	− 943.222i	− 1.11889i
$844$	0	0
$845$	−660.087	−0.781168
$846$	0	0
$847$	249.335i	0.294374i
$848$	0	0
$849$	−320.893	−0.377966
$850$	0	0
$851$	− 272.962i	− 0.320754i
$852$	0	0
$853$	−911.533	−1.06862	−0.534310	−	0.845288i	$-0.679429\pi$
$853$	−911.533	−1.06862	−0.534310	+	0.845288i	$0.679429\pi$
$854$	0	0
$855$	30.4308i	0.0355916i
$856$	0	0
$857$	−886.214	−1.03409	−0.517045	−	0.855958i	$-0.672968\pi$
$857$	−886.214	−1.03409	−0.517045	+	0.855958i	$0.672968\pi$
$858$	0	0
$859$	− 1636.74i	− 1.90540i	−0.303911	−	0.952700i	$-0.598293\pi$
$859$	− 1636.74i	− 1.90540i	0.303911	−	0.952700i	$-0.401707\pi$
$860$	0	0
$861$	−55.4352	−0.0643847
$862$	0	0
$863$	225.975i	0.261848i	0.991392	+	0.130924i	$0.0417944\pi$
$863$	225.975i	0.261848i	−0.991392	+	0.130924i	$0.958206\pi$
$864$	0	0
$865$	−214.032	−0.247436
$866$	0	0
$867$	599.163i	0.691076i
$868$	0	0
$869$	236.989	0.272715
$870$	0	0
$871$	477.307i	0.547999i
$872$	0	0
$873$	271.579	0.311087
$874$	0	0
$875$	− 157.328i	− 0.179803i
$876$	0	0
$877$	−388.417	−0.442893	−0.221446	−	0.975173i	$-0.571078\pi$
$877$	−388.417	−0.442893	−0.221446	+	0.975173i	$0.571078\pi$
$878$	0	0
$879$	− 805.480i	− 0.916359i
$880$	0	0
$881$	−653.052	−0.741262	−0.370631	−	0.928780i	$-0.620859\pi$
$881$	−653.052	−0.741262	−0.370631	+	0.928780i	$0.620859\pi$
$882$	0	0
$883$	386.470i	0.437678i	0.975761	+	0.218839i	$0.0702269\pi$
$883$	386.470i	0.437678i	−0.975761	+	0.218839i	$0.929773\pi$
$884$	0	0
$885$	−525.524	−0.593813
$886$	0	0
$887$	28.5610i	0.0321995i	0.999870	+	0.0160998i	$0.00512493\pi$
$887$	28.5610i	0.0321995i	−0.999870	+	0.0160998i	$0.994875\pi$
$888$	0	0
$889$	501.545	0.564168
$890$	0	0
$891$	− 46.5573i	− 0.0522528i
$892$	0	0
$893$	−88.5835	−0.0991977
$894$	0	0
$895$	− 1817.48i	− 2.03071i
$896$	0	0
$897$	−222.166	−0.247677
$898$	0	0
$899$	38.8509i	0.0432157i
$900$	0	0
$901$	−1431.78	−1.58910
$902$	0	0
$903$	− 142.744i	− 0.158078i
$904$	0	0
$905$	37.8911	0.0418686
$906$	0	0
$907$	541.169i	0.596658i	0.954463	+	0.298329i	$0.0964293\pi$
$907$	541.169i	0.596658i	−0.954463	+	0.298329i	$0.903571\pi$
$908$	0	0
$909$	471.250	0.518427
$910$	0	0
$911$	1175.01i	1.28981i	0.764265	+	0.644903i	$0.223102\pi$
$911$	1175.01i	1.28981i	−0.764265	+	0.644903i	$0.776898\pi$
$912$	0	0
$913$	751.727	0.823360
$914$	0	0
$915$	1119.59i	1.22360i
$916$	0	0
$917$	−317.009	−0.345702
$918$	0	0
$919$	959.135i	1.04367i	0.853046	+	0.521836i	$0.174753\pi$
$919$	959.135i	1.04367i	−0.853046	+	0.521836i	$0.825247\pi$
$920$	0	0
$921$	−475.199	−0.515960
$922$	0	0
$923$	107.783i	0.116775i
$924$	0	0
$925$	−269.996	−0.291888
$926$	0	0
$927$	426.174i	0.459735i
$928$	0	0
$929$	−1042.64	−1.12232	−0.561161	−	0.827707i	$-0.689645\pi$
$929$	−1042.64	−1.12232	−0.561161	+	0.827707i	$0.689645\pi$
$930$	0	0
$931$	11.1332i	0.0119583i
$932$	0	0
$933$	24.6291	0.0263978
$934$	0	0
$935$	− 831.338i	− 0.889132i
$936$	0	0
$937$	354.563	0.378402	0.189201	−	0.981938i	$-0.439410\pi$
$937$	354.563	0.378402	0.189201	+	0.981938i	$0.439410\pi$
$938$	0	0
$939$	− 369.370i	− 0.393365i
$940$	0	0
$941$	1291.59	1.37257	0.686287	−	0.727331i	$-0.259240\pi$
$941$	1291.59	1.37257	0.686287	+	0.727331i	$0.259240\pi$
$942$	0	0
$943$	191.719i	0.203307i
$944$	0	0
$945$	87.6803	0.0927834
$946$	0	0
$947$	1678.61i	1.77256i	0.463151	+	0.886279i	$0.346719\pi$
$947$	1678.61i	1.77256i	−0.463151	+	0.886279i	$0.653281\pi$
$948$	0	0
$949$	112.538	0.118586
$950$	0	0
$951$	− 5.85197i	− 0.00615349i
$952$	0	0
$953$	−1129.47	−1.18517	−0.592586	−	0.805507i	$-0.701893\pi$
$953$	−1129.47	−1.18517	−0.592586	+	0.805507i	$0.701893\pi$
$954$	0	0
$955$	− 142.632i	− 0.149353i
$956$	0	0
$957$	206.841	0.216135
$958$	0	0
$959$	− 341.786i	− 0.356399i
$960$	0	0
$961$	958.168	0.997053
$962$	0	0
$963$	− 342.880i	− 0.356054i
$964$	0	0
$965$	185.108	0.191822
$966$	0	0
$967$	385.235i	0.398382i	0.979961	+	0.199191i	$0.0638314\pi$
$967$	385.235i	0.398382i	−0.979961	+	0.199191i	$0.936169\pi$
$968$	0	0
$969$	69.4134	0.0716341
$970$	0	0
$971$	1449.76i	1.49306i	0.665354	+	0.746528i	$0.268280\pi$
$971$	1449.76i	1.49306i	−0.665354	+	0.746528i	$0.731720\pi$
$972$	0	0
$973$	606.076	0.622895
$974$	0	0
$975$	219.753i	0.225387i
$976$	0	0
$977$	−7.93650	−0.00812333	−0.00406167	−	0.999992i	$-0.501293\pi$
$977$	−7.93650	−0.00812333	−0.00406167	+	0.999992i	$0.501293\pi$
$978$	0	0
$979$	− 777.279i	− 0.793952i
$980$	0	0
$981$	−513.767	−0.523718
$982$	0	0
$983$	− 149.757i	− 0.152347i	−0.997095	−	0.0761735i	$-0.975730\pi$
$983$	− 149.757i	− 0.152347i	0.997095	−	0.0761735i	$-0.0242703\pi$
$984$	0	0
$985$	−1610.89	−1.63542
$986$	0	0
$987$	255.236i	0.258598i
$988$	0	0
$989$	−493.672	−0.499163
$990$	0	0
$991$	− 463.743i	− 0.467955i	−0.972242	−	0.233977i	$-0.924826\pi$
$991$	− 463.743i	− 0.467955i	0.972242	−	0.233977i	$-0.0751741\pi$
$992$	0	0
$993$	789.863	0.795431
$994$	0	0
$995$	− 516.992i	− 0.519590i
$996$	0	0
$997$	669.502	0.671517	0.335758	−	0.941948i	$-0.391007\pi$
$997$	669.502	0.671517	0.335758	+	0.941948i	$0.391007\pi$
$998$	0	0
$999$	89.4942i	0.0895838i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.m.a.127.8	yes	8
3.2	odd	2		2016.3.m.b.127.2		8
4.3	odd	2	inner	672.3.m.a.127.4	✓	8
8.3	odd	2		1344.3.m.d.127.5		8
8.5	even	2		1344.3.m.d.127.1		8
12.11	even	2		2016.3.m.b.127.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.3.m.a.127.4	✓	8	4.3	odd	2	inner
672.3.m.a.127.8	yes	8	1.1	even	1	trivial
1344.3.m.d.127.1		8	8.5	even	2
1344.3.m.d.127.5		8	8.3	odd	2
2016.3.m.b.127.1		8	12.11	even	2
2016.3.m.b.127.2		8	3.2	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +6.37780 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +6.37780 q^{5} +2.64575i q^{7} -3.00000 q^{9} -5.17303i q^{11} +8.09335 q^{13} +11.0467i q^{15} +25.1978 q^{17} -1.59045i q^{19} -4.58258 q^{21} +15.8485i q^{23} +15.6764 q^{25} -5.19615i q^{27} +23.0850 q^{29} +1.68295i q^{31} +8.95995 q^{33} +16.8741i q^{35} -17.2232 q^{37} +14.0181i q^{39} +12.0970 q^{41} +31.1494i q^{43} -19.1334 q^{45} -55.6971i q^{47} -7.00000 q^{49} +43.6438i q^{51} -56.8217 q^{53} -32.9926i q^{55} +2.75475 q^{57} +47.5731i q^{59} +101.351 q^{61} -7.93725i q^{63} +51.6178 q^{65} +58.9752i q^{67} -27.4505 q^{69} +13.3175i q^{71} +13.9050 q^{73} +27.1523i q^{75} +13.6865 q^{77} +45.8124i q^{79} +9.00000 q^{81} +145.317i q^{83} +160.706 q^{85} +39.9845i q^{87} +150.256 q^{89} +21.4130i q^{91} -2.91495 q^{93} -10.1436i q^{95} -90.5264 q^{97} +15.5191i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 24 q^{9}+O(q^{10}) \) 8 * q + 16 * q^5 - 24 * q^9	\( 8 q + 16 q^{5} - 24 q^{9} - 64 q^{13} + 64 q^{17} - 88 q^{25} + 64 q^{29} + 48 q^{33} + 128 q^{37} - 48 q^{45} - 56 q^{49} - 160 q^{53} + 48 q^{57} + 32 q^{61} - 32 q^{65} - 144 q^{69} - 112 q^{73} + 112 q^{77} + 72 q^{81} + 336 q^{85} + 352 q^{89} - 144 q^{93} - 240 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 - 24 * q^9 - 64 * q^13 + 64 * q^17 - 88 * q^25 + 64 * q^29 + 48 * q^33 + 128 * q^37 - 48 * q^45 - 56 * q^49 - 160 * q^53 + 48 * q^57 + 32 * q^61 - 32 * q^65 - 144 * q^69 - 112 * q^73 + 112 * q^77 + 72 * q^81 + 336 * q^85 + 352 * q^89 - 144 * q^93 - 240 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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