LMFDB - Embedded newform 672.3.m.a.127.3

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			127.3
Root			$1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	672.127
Dual form			672.3.m.a.127.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +2.91370 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +2.91370 q^{5} +2.64575i q^{7} -3.00000 q^{9} -7.40998i q^{11} -24.0934 q^{13} -5.04668i q^{15} +17.2598 q^{17} -12.9926i q^{19} +4.58258 q^{21} -32.4315i q^{23} -16.5103 q^{25} +5.19615i q^{27} -38.8341 q^{29} -37.4320i q^{31} -12.8345 q^{33} +7.70893i q^{35} +70.3892 q^{37} +41.7309i q^{39} +24.9436 q^{41} -28.3154i q^{43} -8.74110 q^{45} +40.8631i q^{47} -7.00000 q^{49} -29.8948i q^{51} -89.0084 q^{53} -21.5905i q^{55} -22.5038 q^{57} -80.7391i q^{59} -61.6021 q^{61} -7.93725i q^{63} -70.2008 q^{65} -92.1412i q^{67} -56.1731 q^{69} +61.5975i q^{71} -52.4880 q^{73} +28.5968i q^{75} +19.6050 q^{77} +91.8555i q^{79} +9.00000 q^{81} -98.3206i q^{83} +50.2898 q^{85} +67.2626i q^{87} -67.5476 q^{89} -63.7450i q^{91} -64.8341 q^{93} -37.8564i q^{95} +41.1094 q^{97} +22.2299i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.91370	0.582740	0.291370	−	0.956610i	$-0.405889\pi$
$5$	2.91370	0.582740	0.291370	+	0.956610i	$0.405889\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 7.40998i	− 0.673634i	−0.941570	−	0.336817i	$-0.890650\pi$
$11$	− 7.40998i	− 0.673634i	0.941570	−	0.336817i	$-0.109350\pi$
$12$	0	0
$13$	−24.0934	−1.85333	−0.926667	−	0.375882i	$-0.877340\pi$
$13$	−24.0934	−1.85333	−0.926667	+	0.375882i	$0.877340\pi$
$14$	0	0
$15$	− 5.04668i	− 0.336445i
$16$	0	0
$17$	17.2598	1.01528	0.507640	−	0.861569i	$-0.330518\pi$
$17$	17.2598	1.01528	0.507640	+	0.861569i	$0.330518\pi$
$18$	0	0
$19$	− 12.9926i	− 0.683819i	−0.939733	−	0.341909i	$-0.888926\pi$
$19$	− 12.9926i	− 0.683819i	0.939733	−	0.341909i	$-0.111074\pi$
$20$	0	0
$21$	4.58258	0.218218
$22$	0	0
$23$	− 32.4315i	− 1.41007i	−0.709174	−	0.705033i	$-0.750932\pi$
$23$	− 32.4315i	− 1.41007i	0.709174	−	0.705033i	$-0.249068\pi$
$24$	0	0
$25$	−16.5103	−0.660414
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−38.8341	−1.33911	−0.669553	−	0.742764i	$-0.733514\pi$
$29$	−38.8341	−1.33911	−0.669553	+	0.742764i	$0.733514\pi$
$30$	0	0
$31$	− 37.4320i	− 1.20748i	−0.797180	−	0.603741i	$-0.793676\pi$
$31$	− 37.4320i	− 1.20748i	0.797180	−	0.603741i	$-0.206324\pi$
$32$	0	0
$33$	−12.8345	−0.388923
$34$	0	0
$35$	7.70893i	0.220255i
$36$	0	0
$37$	70.3892	1.90241	0.951205	−	0.308560i	$-0.0998469\pi$
$37$	70.3892	1.90241	0.951205	+	0.308560i	$0.0998469\pi$
$38$	0	0
$39$	41.7309i	1.07002i
$40$	0	0
$41$	24.9436	0.608380	0.304190	−	0.952611i	$-0.401614\pi$
$41$	24.9436	0.608380	0.304190	+	0.952611i	$0.401614\pi$
$42$	0	0
$43$	− 28.3154i	− 0.658498i	−0.944243	−	0.329249i	$-0.893204\pi$
$43$	− 28.3154i	− 0.658498i	0.944243	−	0.329249i	$-0.106796\pi$
$44$	0	0
$45$	−8.74110	−0.194247
$46$	0	0
$47$	40.8631i	0.869427i	0.900569	+	0.434714i	$0.143150\pi$
$47$	40.8631i	0.869427i	−0.900569	+	0.434714i	$0.856850\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 29.8948i	− 0.586172i
$52$	0	0
$53$	−89.0084	−1.67940	−0.839702	−	0.543048i	$-0.817270\pi$
$53$	−89.0084	−1.67940	−0.839702	+	0.543048i	$0.817270\pi$
$54$	0	0
$55$	− 21.5905i	− 0.392554i
$56$	0	0
$57$	−22.5038	−0.394803
$58$	0	0
$59$	− 80.7391i	− 1.36846i	−0.729267	−	0.684229i	$-0.760139\pi$
$59$	− 80.7391i	− 1.36846i	0.729267	−	0.684229i	$-0.239861\pi$
$60$	0	0
$61$	−61.6021	−1.00987	−0.504935	−	0.863157i	$-0.668484\pi$
$61$	−61.6021	−1.00987	−0.504935	+	0.863157i	$0.668484\pi$
$62$	0	0
$63$	− 7.93725i	− 0.125988i
$64$	0	0
$65$	−70.2008	−1.08001
$66$	0	0
$67$	− 92.1412i	− 1.37524i	−0.726070	−	0.687621i	$-0.758655\pi$
$67$	− 92.1412i	− 1.37524i	0.726070	−	0.687621i	$-0.241345\pi$
$68$	0	0
$69$	−56.1731	−0.814102
$70$	0	0
$71$	61.5975i	0.867571i	0.901016	+	0.433786i	$0.142822\pi$
$71$	61.5975i	0.867571i	−0.901016	+	0.433786i	$0.857178\pi$
$72$	0	0
$73$	−52.4880	−0.719014	−0.359507	−	0.933142i	$-0.617055\pi$
$73$	−52.4880	−0.719014	−0.359507	+	0.933142i	$0.617055\pi$
$74$	0	0
$75$	28.5968i	0.381290i
$76$	0	0
$77$	19.6050	0.254610
$78$	0	0
$79$	91.8555i	1.16273i	0.813643	+	0.581364i	$0.197481\pi$
$79$	91.8555i	1.16273i	−0.813643	+	0.581364i	$0.802519\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 98.3206i	− 1.18459i	−0.805723	−	0.592293i	$-0.798223\pi$
$83$	− 98.3206i	− 1.18459i	0.805723	−	0.592293i	$-0.201777\pi$
$84$	0	0
$85$	50.2898	0.591644
$86$	0	0
$87$	67.2626i	0.773133i
$88$	0	0
$89$	−67.5476	−0.758962	−0.379481	−	0.925200i	$-0.623897\pi$
$89$	−67.5476	−0.758962	−0.379481	+	0.925200i	$0.623897\pi$
$90$	0	0
$91$	− 63.7450i	− 0.700495i
$92$	0	0
$93$	−64.8341	−0.697140
$94$	0	0
$95$	− 37.8564i	− 0.398488i
$96$	0	0
$97$	41.1094	0.423809	0.211904	−	0.977290i	$-0.432033\pi$
$97$	41.1094	0.423809	0.211904	+	0.977290i	$0.432033\pi$
$98$	0	0
$99$	22.2299i	0.224545i
$100$	0	0
$101$	−24.8723	−0.246260	−0.123130	−	0.992391i	$-0.539293\pi$
$101$	−24.8723	−0.246260	−0.123130	+	0.992391i	$0.539293\pi$
$102$	0	0
$103$	− 10.8569i	− 0.105407i	−0.998610	−	0.0527036i	$-0.983216\pi$
$103$	− 10.8569i	− 0.105407i	0.998610	−	0.0527036i	$-0.0167839\pi$
$104$	0	0
$105$	13.3523	0.127164
$106$	0	0
$107$	29.7875i	0.278388i	0.990265	+	0.139194i	$0.0444512\pi$
$107$	29.7875i	0.278388i	−0.990265	+	0.139194i	$0.955549\pi$
$108$	0	0
$109$	60.4044	0.554169	0.277085	−	0.960846i	$-0.410632\pi$
$109$	60.4044	0.554169	0.277085	+	0.960846i	$0.410632\pi$
$110$	0	0
$111$	− 121.918i	− 1.09836i
$112$	0	0
$113$	21.5134	0.190384	0.0951922	−	0.995459i	$-0.469653\pi$
$113$	21.5134	0.190384	0.0951922	+	0.995459i	$0.469653\pi$
$114$	0	0
$115$	− 94.4958i	− 0.821702i
$116$	0	0
$117$	72.2801	0.617778
$118$	0	0
$119$	45.6650i	0.383740i
$120$	0	0
$121$	66.0922	0.546217
$122$	0	0
$123$	− 43.2035i	− 0.351248i
$124$	0	0
$125$	−120.949	−0.967590
$126$	0	0
$127$	− 84.9278i	− 0.668723i	−0.942445	−	0.334361i	$-0.891479\pi$
$127$	− 84.9278i	− 0.668723i	0.942445	−	0.334361i	$-0.108521\pi$
$128$	0	0
$129$	−49.0437	−0.380184
$130$	0	0
$131$	157.348i	1.20113i	0.799576	+	0.600565i	$0.205058\pi$
$131$	157.348i	1.20113i	−0.799576	+	0.600565i	$0.794942\pi$
$132$	0	0
$133$	34.3751	0.258459
$134$	0	0
$135$	15.1400i	0.112148i
$136$	0	0
$137$	−8.23390	−0.0601014	−0.0300507	−	0.999548i	$-0.509567\pi$
$137$	−8.23390	−0.0601014	−0.0300507	+	0.999548i	$0.509567\pi$
$138$	0	0
$139$	− 192.415i	− 1.38428i	−0.721764	−	0.692139i	$-0.756668\pi$
$139$	− 192.415i	− 1.38428i	0.721764	−	0.692139i	$-0.243332\pi$
$140$	0	0
$141$	70.7769	0.501964
$142$	0	0
$143$	178.531i	1.24847i
$144$	0	0
$145$	−113.151	−0.780351
$146$	0	0
$147$	12.1244i	0.0824786i
$148$	0	0
$149$	59.5926	0.399951	0.199975	−	0.979801i	$-0.435914\pi$
$149$	59.5926	0.399951	0.199975	+	0.979801i	$0.435914\pi$
$150$	0	0
$151$	163.350i	1.08179i	0.841091	+	0.540894i	$0.181914\pi$
$151$	163.350i	1.08179i	−0.841091	+	0.540894i	$0.818086\pi$
$152$	0	0
$153$	−51.7793	−0.338427
$154$	0	0
$155$	− 109.066i	− 0.703649i
$156$	0	0
$157$	194.988	1.24196	0.620981	−	0.783825i	$-0.286734\pi$
$157$	194.988	1.24196	0.620981	+	0.783825i	$0.286734\pi$
$158$	0	0
$159$	154.167i	0.969604i
$160$	0	0
$161$	85.8058	0.532955
$162$	0	0
$163$	− 70.8941i	− 0.434933i	−0.976068	−	0.217466i	$-0.930221\pi$
$163$	− 70.8941i	− 0.434933i	0.976068	−	0.217466i	$-0.0697793\pi$
$164$	0	0
$165$	−37.3958	−0.226641
$166$	0	0
$167$	153.404i	0.918589i	0.888284	+	0.459295i	$0.151898\pi$
$167$	153.404i	0.918589i	−0.888284	+	0.459295i	$0.848102\pi$
$168$	0	0
$169$	411.490	2.43485
$170$	0	0
$171$	38.9777i	0.227940i
$172$	0	0
$173$	70.9393	0.410054	0.205027	−	0.978756i	$-0.434272\pi$
$173$	70.9393	0.410054	0.205027	+	0.978756i	$0.434272\pi$
$174$	0	0
$175$	− 43.6823i	− 0.249613i
$176$	0	0
$177$	−139.844	−0.790080
$178$	0	0
$179$	288.549i	1.61201i	0.591911	+	0.806003i	$0.298374\pi$
$179$	288.549i	1.61201i	−0.591911	+	0.806003i	$0.701626\pi$
$180$	0	0
$181$	351.047	1.93949	0.969743	−	0.244127i	$-0.0785013\pi$
$181$	351.047	1.93949	0.969743	+	0.244127i	$0.0785013\pi$
$182$	0	0
$183$	106.698i	0.583049i
$184$	0	0
$185$	205.093	1.10861
$186$	0	0
$187$	− 127.894i	− 0.683927i
$188$	0	0
$189$	−13.7477	−0.0727393
$190$	0	0
$191$	163.611i	0.856601i	0.903636	+	0.428301i	$0.140888\pi$
$191$	163.611i	0.856601i	−0.903636	+	0.428301i	$0.859112\pi$
$192$	0	0
$193$	−137.688	−0.713408	−0.356704	−	0.934217i	$-0.616100\pi$
$193$	−137.688	−0.713408	−0.356704	+	0.934217i	$0.616100\pi$
$194$	0	0
$195$	121.591i	0.623546i
$196$	0	0
$197$	−170.743	−0.866716	−0.433358	−	0.901222i	$-0.642671\pi$
$197$	−170.743	−0.866716	−0.433358	+	0.901222i	$0.642671\pi$
$198$	0	0
$199$	− 202.599i	− 1.01809i	−0.860741	−	0.509043i	$-0.830000\pi$
$199$	− 202.599i	− 1.01809i	0.860741	−	0.509043i	$-0.170000\pi$
$200$	0	0
$201$	−159.593	−0.793996
$202$	0	0
$203$	− 102.745i	− 0.506134i
$204$	0	0
$205$	72.6781	0.354527
$206$	0	0
$207$	97.2946i	0.470022i
$208$	0	0
$209$	−96.2745	−0.460644
$210$	0	0
$211$	236.951i	1.12299i	0.827480	+	0.561496i	$0.189774\pi$
$211$	236.951i	1.12299i	−0.827480	+	0.561496i	$0.810226\pi$
$212$	0	0
$213$	106.690	0.500892
$214$	0	0
$215$	− 82.5026i	− 0.383733i
$216$	0	0
$217$	99.0357	0.456386
$218$	0	0
$219$	90.9119i	0.415123i
$220$	0	0
$221$	−415.845	−1.88165
$222$	0	0
$223$	− 214.219i	− 0.960623i	−0.877098	−	0.480311i	$-0.840524\pi$
$223$	− 214.219i	− 0.960623i	0.877098	−	0.480311i	$-0.159476\pi$
$224$	0	0
$225$	49.5310	0.220138
$226$	0	0
$227$	− 316.287i	− 1.39334i	−0.717393	−	0.696668i	$-0.754665\pi$
$227$	− 316.287i	− 1.39334i	0.717393	−	0.696668i	$-0.245335\pi$
$228$	0	0
$229$	193.656	0.845660	0.422830	−	0.906209i	$-0.361037\pi$
$229$	193.656	0.845660	0.422830	+	0.906209i	$0.361037\pi$
$230$	0	0
$231$	− 33.9568i	− 0.146999i
$232$	0	0
$233$	222.042	0.952969	0.476485	−	0.879183i	$-0.341911\pi$
$233$	222.042	0.952969	0.476485	+	0.879183i	$0.341911\pi$
$234$	0	0
$235$	119.063i	0.506650i
$236$	0	0
$237$	159.098	0.671302
$238$	0	0
$239$	− 26.4216i	− 0.110551i	−0.998471	−	0.0552754i	$-0.982396\pi$
$239$	− 26.4216i	− 0.110551i	0.998471	−	0.0552754i	$-0.0176037\pi$
$240$	0	0
$241$	178.900	0.742324	0.371162	−	0.928568i	$-0.378959\pi$
$241$	178.900	0.742324	0.371162	+	0.928568i	$0.378959\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	−20.3959	−0.0832486
$246$	0	0
$247$	313.034i	1.26734i
$248$	0	0
$249$	−170.296	−0.683921
$250$	0	0
$251$	24.2905i	0.0967750i	0.998829	+	0.0483875i	$0.0154082\pi$
$251$	24.2905i	0.0967750i	−0.998829	+	0.0483875i	$0.984592\pi$
$252$	0	0
$253$	−240.317	−0.949869
$254$	0	0
$255$	− 87.1044i	− 0.341586i
$256$	0	0
$257$	308.081	1.19876	0.599380	−	0.800465i	$-0.295414\pi$
$257$	308.081	1.19876	0.599380	+	0.800465i	$0.295414\pi$
$258$	0	0
$259$	186.232i	0.719043i
$260$	0	0
$261$	116.502	0.446369
$262$	0	0
$263$	438.932i	1.66894i	0.551052	+	0.834471i	$0.314226\pi$
$263$	438.932i	1.66894i	−0.551052	+	0.834471i	$0.685774\pi$
$264$	0	0
$265$	−259.344	−0.978656
$266$	0	0
$267$	116.996i	0.438187i
$268$	0	0
$269$	−119.003	−0.442391	−0.221195	−	0.975229i	$-0.570996\pi$
$269$	−119.003	−0.442391	−0.221195	+	0.975229i	$0.570996\pi$
$270$	0	0
$271$	− 67.2292i	− 0.248078i	−0.992277	−	0.124039i	$-0.960415\pi$
$271$	− 67.2292i	− 0.248078i	0.992277	−	0.124039i	$-0.0395848\pi$
$272$	0	0
$273$	−110.410	−0.404431
$274$	0	0
$275$	122.341i	0.444877i
$276$	0	0
$277$	−182.471	−0.658739	−0.329369	−	0.944201i	$-0.606836\pi$
$277$	−182.471	−0.658739	−0.329369	+	0.944201i	$0.606836\pi$
$278$	0	0
$279$	112.296i	0.402494i
$280$	0	0
$281$	401.986	1.43056	0.715278	−	0.698840i	$-0.246300\pi$
$281$	401.986	1.43056	0.715278	+	0.698840i	$0.246300\pi$
$282$	0	0
$283$	222.644i	0.786726i	0.919383	+	0.393363i	$0.128688\pi$
$283$	222.644i	0.786726i	−0.919383	+	0.393363i	$0.871312\pi$
$284$	0	0
$285$	−65.5692	−0.230067
$286$	0	0
$287$	65.9945i	0.229946i
$288$	0	0
$289$	8.89923	0.0307932
$290$	0	0
$291$	− 71.2036i	− 0.244686i
$292$	0	0
$293$	505.760	1.72614	0.863072	−	0.505081i	$-0.168537\pi$
$293$	505.760	1.72614	0.863072	+	0.505081i	$0.168537\pi$
$294$	0	0
$295$	− 235.249i	− 0.797456i
$296$	0	0
$297$	38.5034	0.129641
$298$	0	0
$299$	781.384i	2.61333i
$300$	0	0
$301$	74.9155	0.248889
$302$	0	0
$303$	43.0801i	0.142179i
$304$	0	0
$305$	−179.490	−0.588492
$306$	0	0
$307$	− 156.777i	− 0.510675i	−0.966852	−	0.255337i	$-0.917813\pi$
$307$	− 156.777i	− 0.510675i	0.966852	−	0.255337i	$-0.0821866\pi$
$308$	0	0
$309$	−18.8048	−0.0608569
$310$	0	0
$311$	− 525.603i	− 1.69004i	−0.534734	−	0.845020i	$-0.679588\pi$
$311$	− 525.603i	− 1.69004i	0.534734	−	0.845020i	$-0.320412\pi$
$312$	0	0
$313$	−22.7440	−0.0726646	−0.0363323	−	0.999340i	$-0.511567\pi$
$313$	−22.7440	−0.0726646	−0.0363323	+	0.999340i	$0.511567\pi$
$314$	0	0
$315$	− 23.1268i	− 0.0734184i
$316$	0	0
$317$	−323.942	−1.02190	−0.510949	−	0.859611i	$-0.670706\pi$
$317$	−323.942	−1.02190	−0.510949	+	0.859611i	$0.670706\pi$
$318$	0	0
$319$	287.760i	0.902067i
$320$	0	0
$321$	51.5935	0.160728
$322$	0	0
$323$	− 224.248i	− 0.694267i
$324$	0	0
$325$	397.790	1.22397
$326$	0	0
$327$	− 104.624i	− 0.319950i
$328$	0	0
$329$	−108.114	−0.328613
$330$	0	0
$331$	− 454.289i	− 1.37247i	−0.727378	−	0.686237i	$-0.759261\pi$
$331$	− 454.289i	− 1.37247i	0.727378	−	0.686237i	$-0.240739\pi$
$332$	0	0
$333$	−211.168	−0.634137
$334$	0	0
$335$	− 268.472i	− 0.801408i
$336$	0	0
$337$	−112.431	−0.333622	−0.166811	−	0.985989i	$-0.553347\pi$
$337$	−112.431	−0.333622	−0.166811	+	0.985989i	$0.553347\pi$
$338$	0	0
$339$	− 37.2623i	− 0.109918i
$340$	0	0
$341$	−277.370	−0.813402
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−163.671	−0.474410
$346$	0	0
$347$	− 29.7063i	− 0.0856088i	−0.999083	−	0.0428044i	$-0.986371\pi$
$347$	− 29.7063i	− 0.0856088i	0.999083	−	0.0428044i	$-0.0136292\pi$
$348$	0	0
$349$	180.808	0.518075	0.259037	−	0.965867i	$-0.416595\pi$
$349$	180.808	0.518075	0.259037	+	0.965867i	$0.416595\pi$
$350$	0	0
$351$	− 125.193i	− 0.356674i
$352$	0	0
$353$	−309.246	−0.876052	−0.438026	−	0.898962i	$-0.644322\pi$
$353$	−309.246	−0.876052	−0.438026	+	0.898962i	$0.644322\pi$
$354$	0	0
$355$	179.477i	0.505568i
$356$	0	0
$357$	79.0941	0.221552
$358$	0	0
$359$	21.7887i	0.0606927i	0.999539	+	0.0303464i	$0.00966103\pi$
$359$	21.7887i	0.0606927i	−0.999539	+	0.0303464i	$0.990339\pi$
$360$	0	0
$361$	192.194	0.532392
$362$	0	0
$363$	− 114.475i	− 0.315358i
$364$	0	0
$365$	−152.934	−0.418998
$366$	0	0
$367$	331.819i	0.904140i	0.891982	+	0.452070i	$0.149314\pi$
$367$	331.819i	0.904140i	−0.891982	+	0.452070i	$0.850686\pi$
$368$	0	0
$369$	−74.8307	−0.202793
$370$	0	0
$371$	− 235.494i	− 0.634755i
$372$	0	0
$373$	−297.079	−0.796458	−0.398229	−	0.917286i	$-0.630375\pi$
$373$	−297.079	−0.796458	−0.398229	+	0.917286i	$0.630375\pi$
$374$	0	0
$375$	209.489i	0.558638i
$376$	0	0
$377$	935.643	2.48181
$378$	0	0
$379$	− 568.577i	− 1.50020i	−0.661323	−	0.750101i	$-0.730005\pi$
$379$	− 568.577i	− 1.50020i	0.661323	−	0.750101i	$-0.269995\pi$
$380$	0	0
$381$	−147.099	−0.386087
$382$	0	0
$383$	34.1267i	0.0891036i	0.999007	+	0.0445518i	$0.0141860\pi$
$383$	34.1267i	0.0891036i	−0.999007	+	0.0445518i	$0.985814\pi$
$384$	0	0
$385$	57.1230	0.148371
$386$	0	0
$387$	84.9462i	0.219499i
$388$	0	0
$389$	−185.697	−0.477369	−0.238685	−	0.971097i	$-0.576716\pi$
$389$	−185.697	−0.477369	−0.238685	+	0.971097i	$0.576716\pi$
$390$	0	0
$391$	− 559.760i	− 1.43161i
$392$	0	0
$393$	272.535	0.693473
$394$	0	0
$395$	267.640i	0.677568i
$396$	0	0
$397$	−277.517	−0.699035	−0.349518	−	0.936930i	$-0.613654\pi$
$397$	−277.517	−0.699035	−0.349518	+	0.936930i	$0.613654\pi$
$398$	0	0
$399$	− 59.5394i	− 0.149221i
$400$	0	0
$401$	−530.463	−1.32285	−0.661425	−	0.750012i	$-0.730048\pi$
$401$	−530.463	−1.32285	−0.661425	+	0.750012i	$0.730048\pi$
$402$	0	0
$403$	901.862i	2.23787i
$404$	0	0
$405$	26.2233	0.0647489
$406$	0	0
$407$	− 521.582i	− 1.28153i
$408$	0	0
$409$	681.101	1.66528	0.832642	−	0.553812i	$-0.186827\pi$
$409$	681.101	1.66528	0.832642	+	0.553812i	$0.186827\pi$
$410$	0	0
$411$	14.2615i	0.0346996i
$412$	0	0
$413$	213.615	0.517229
$414$	0	0
$415$	− 286.477i	− 0.690306i
$416$	0	0
$417$	−333.272	−0.799214
$418$	0	0
$419$	− 208.004i	− 0.496430i	−0.968705	−	0.248215i	$-0.920156\pi$
$419$	− 208.004i	− 0.496430i	0.968705	−	0.248215i	$-0.0798439\pi$
$420$	0	0
$421$	58.5750	0.139133	0.0695665	−	0.997577i	$-0.477838\pi$
$421$	58.5750	0.139133	0.0695665	+	0.997577i	$0.477838\pi$
$422$	0	0
$423$	− 122.589i	− 0.289809i
$424$	0	0
$425$	−284.965	−0.670505
$426$	0	0
$427$	− 162.984i	− 0.381695i
$428$	0	0
$429$	309.225	0.720804
$430$	0	0
$431$	− 376.105i	− 0.872634i	−0.899793	−	0.436317i	$-0.856283\pi$
$431$	− 376.105i	− 0.872634i	0.899793	−	0.436317i	$-0.143717\pi$
$432$	0	0
$433$	−120.961	−0.279355	−0.139677	−	0.990197i	$-0.544607\pi$
$433$	−120.961	−0.279355	−0.139677	+	0.990197i	$0.544607\pi$
$434$	0	0
$435$	195.983i	0.450536i
$436$	0	0
$437$	−421.368	−0.964230
$438$	0	0
$439$	− 747.561i	− 1.70287i	−0.524459	−	0.851436i	$-0.675732\pi$
$439$	− 747.561i	− 1.70287i	0.524459	−	0.851436i	$-0.324268\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	654.425i	1.47726i	0.674112	+	0.738629i	$0.264526\pi$
$443$	654.425i	1.47726i	−0.674112	+	0.738629i	$0.735474\pi$
$444$	0	0
$445$	−196.813	−0.442277
$446$	0	0
$447$	− 103.217i	− 0.230912i
$448$	0	0
$449$	412.318	0.918303	0.459152	−	0.888358i	$-0.348153\pi$
$449$	412.318	0.918303	0.459152	+	0.888358i	$0.348153\pi$
$450$	0	0
$451$	− 184.831i	− 0.409825i
$452$	0	0
$453$	282.930	0.624570
$454$	0	0
$455$	− 185.734i	− 0.408206i
$456$	0	0
$457$	−860.397	−1.88271	−0.941353	−	0.337422i	$-0.890445\pi$
$457$	−860.397	−1.88271	−0.941353	+	0.337422i	$0.890445\pi$
$458$	0	0
$459$	89.6843i	0.195391i
$460$	0	0
$461$	−406.149	−0.881018	−0.440509	−	0.897748i	$-0.645202\pi$
$461$	−406.149	−0.881018	−0.440509	+	0.897748i	$0.645202\pi$
$462$	0	0
$463$	− 492.223i	− 1.06312i	−0.847022	−	0.531558i	$-0.821607\pi$
$463$	− 492.223i	− 1.06312i	0.847022	−	0.531558i	$-0.178393\pi$
$464$	0	0
$465$	−188.907	−0.406252
$466$	0	0
$467$	− 594.456i	− 1.27293i	−0.771307	−	0.636463i	$-0.780397\pi$
$467$	− 594.456i	− 1.27293i	0.771307	−	0.636463i	$-0.219603\pi$
$468$	0	0
$469$	243.783	0.519792
$470$	0	0
$471$	− 337.729i	− 0.717048i
$472$	0	0
$473$	−209.817	−0.443587
$474$	0	0
$475$	214.512i	0.451603i
$476$	0	0
$477$	267.025	0.559801
$478$	0	0
$479$	− 201.151i	− 0.419940i	−0.977708	−	0.209970i	$-0.932663\pi$
$479$	− 201.151i	− 0.419940i	0.977708	−	0.209970i	$-0.0673367\pi$
$480$	0	0
$481$	−1695.91	−3.52580
$482$	0	0
$483$	− 148.620i	− 0.307702i
$484$	0	0
$485$	119.781	0.246970
$486$	0	0
$487$	376.266i	0.772619i	0.922369	+	0.386310i	$0.126250\pi$
$487$	376.266i	0.772619i	−0.922369	+	0.386310i	$0.873750\pi$
$488$	0	0
$489$	−122.792	−0.251109
$490$	0	0
$491$	186.938i	0.380729i	0.981713	+	0.190365i	$0.0609670\pi$
$491$	186.938i	0.380729i	−0.981713	+	0.190365i	$0.939033\pi$
$492$	0	0
$493$	−670.267	−1.35957
$494$	0	0
$495$	64.7714i	0.130851i
$496$	0	0
$497$	−162.972	−0.327911
$498$	0	0
$499$	298.003i	0.597200i	0.954378	+	0.298600i	$0.0965196\pi$
$499$	298.003i	0.597200i	−0.954378	+	0.298600i	$0.903480\pi$
$500$	0	0
$501$	265.704	0.530348
$502$	0	0
$503$	− 474.049i	− 0.942444i	−0.882015	−	0.471222i	$-0.843813\pi$
$503$	− 474.049i	− 0.942444i	0.882015	−	0.471222i	$-0.156187\pi$
$504$	0	0
$505$	−72.4704	−0.143506
$506$	0	0
$507$	− 712.721i	− 1.40576i
$508$	0	0
$509$	147.496	0.289776	0.144888	−	0.989448i	$-0.453718\pi$
$509$	147.496	0.289776	0.144888	+	0.989448i	$0.453718\pi$
$510$	0	0
$511$	− 138.870i	− 0.271762i
$512$	0	0
$513$	67.5113	0.131601
$514$	0	0
$515$	− 31.6339i	− 0.0614250i
$516$	0	0
$517$	302.794	0.585676
$518$	0	0
$519$	− 122.870i	− 0.236745i
$520$	0	0
$521$	700.270	1.34409	0.672044	−	0.740511i	$-0.265417\pi$
$521$	700.270	1.34409	0.672044	+	0.740511i	$0.265417\pi$
$522$	0	0
$523$	695.893i	1.33058i	0.746586	+	0.665289i	$0.231692\pi$
$523$	695.893i	1.33058i	−0.746586	+	0.665289i	$0.768308\pi$
$524$	0	0
$525$	−75.6599	−0.144114
$526$	0	0
$527$	− 646.067i	− 1.22593i
$528$	0	0
$529$	−522.804	−0.988288
$530$	0	0
$531$	242.217i	0.456153i
$532$	0	0
$533$	−600.974	−1.12753
$534$	0	0
$535$	86.7920i	0.162228i
$536$	0	0
$537$	499.782	0.930693
$538$	0	0
$539$	51.8698i	0.0962335i
$540$	0	0
$541$	596.690	1.10294	0.551470	−	0.834195i	$-0.314067\pi$
$541$	596.690	1.10294	0.551470	+	0.834195i	$0.314067\pi$
$542$	0	0
$543$	− 608.031i	− 1.11976i
$544$	0	0
$545$	176.000	0.322937
$546$	0	0
$547$	− 265.842i	− 0.486000i	−0.970026	−	0.243000i	$-0.921868\pi$
$547$	− 265.842i	− 0.486000i	0.970026	−	0.243000i	$-0.0781316\pi$
$548$	0	0
$549$	184.806	0.336623
$550$	0	0
$551$	504.554i	0.915705i
$552$	0	0
$553$	−243.027	−0.439470
$554$	0	0
$555$	− 355.231i	− 0.640057i
$556$	0	0
$557$	−1021.63	−1.83416	−0.917080	−	0.398703i	$-0.869460\pi$
$557$	−1021.63	−1.83416	−0.917080	+	0.398703i	$0.869460\pi$
$558$	0	0
$559$	682.213i	1.22042i
$560$	0	0
$561$	−221.520	−0.394866
$562$	0	0
$563$	402.546i	0.715002i	0.933913	+	0.357501i	$0.116371\pi$
$563$	402.546i	0.715002i	−0.933913	+	0.357501i	$0.883629\pi$
$564$	0	0
$565$	62.6837	0.110945
$566$	0	0
$567$	23.8118i	0.0419961i
$568$	0	0
$569$	275.167	0.483598	0.241799	−	0.970326i	$-0.422263\pi$
$569$	275.167	0.483598	0.241799	+	0.970326i	$0.422263\pi$
$570$	0	0
$571$	684.768i	1.19924i	0.800284	+	0.599622i	$0.204682\pi$
$571$	684.768i	1.19924i	−0.800284	+	0.599622i	$0.795318\pi$
$572$	0	0
$573$	283.382	0.494559
$574$	0	0
$575$	535.456i	0.931228i
$576$	0	0
$577$	226.867	0.393183	0.196592	−	0.980485i	$-0.437013\pi$
$577$	226.867	0.393183	0.196592	+	0.980485i	$0.437013\pi$
$578$	0	0
$579$	238.482i	0.411887i
$580$	0	0
$581$	260.132	0.447731
$582$	0	0
$583$	659.550i	1.13130i
$584$	0	0
$585$	210.602	0.360004
$586$	0	0
$587$	320.491i	0.545981i	0.962017	+	0.272991i	$0.0880128\pi$
$587$	320.491i	0.545981i	−0.962017	+	0.272991i	$0.911987\pi$
$588$	0	0
$589$	−486.337	−0.825699
$590$	0	0
$591$	295.735i	0.500398i
$592$	0	0
$593$	−814.649	−1.37378	−0.686888	−	0.726764i	$-0.741024\pi$
$593$	−814.649	−1.37378	−0.686888	+	0.726764i	$0.741024\pi$
$594$	0	0
$595$	133.054i	0.223621i
$596$	0	0
$597$	−350.912	−0.587792
$598$	0	0
$599$	505.842i	0.844478i	0.906485	+	0.422239i	$0.138756\pi$
$599$	505.842i	0.844478i	−0.906485	+	0.422239i	$0.861244\pi$
$600$	0	0
$601$	−506.284	−0.842403	−0.421201	−	0.906967i	$-0.638391\pi$
$601$	−506.284	−0.842403	−0.421201	+	0.906967i	$0.638391\pi$
$602$	0	0
$603$	276.423i	0.458414i
$604$	0	0
$605$	192.573	0.318302
$606$	0	0
$607$	− 286.599i	− 0.472156i	−0.971734	−	0.236078i	$-0.924138\pi$
$607$	− 286.599i	− 0.472156i	0.971734	−	0.236078i	$-0.0758622\pi$
$608$	0	0
$609$	−177.960	−0.292217
$610$	0	0
$611$	− 984.528i	− 1.61134i
$612$	0	0
$613$	−300.456	−0.490140	−0.245070	−	0.969505i	$-0.578811\pi$
$613$	−300.456	−0.490140	−0.245070	+	0.969505i	$0.578811\pi$
$614$	0	0
$615$	− 125.882i	− 0.204686i
$616$	0	0
$617$	930.734	1.50848	0.754241	−	0.656597i	$-0.228005\pi$
$617$	930.734	1.50848	0.754241	+	0.656597i	$0.228005\pi$
$618$	0	0
$619$	− 680.636i	− 1.09957i	−0.835305	−	0.549787i	$-0.814709\pi$
$619$	− 680.636i	− 1.09957i	0.835305	−	0.549787i	$-0.185291\pi$
$620$	0	0
$621$	168.519	0.271367
$622$	0	0
$623$	− 178.714i	− 0.286861i
$624$	0	0
$625$	60.3504	0.0965606
$626$	0	0
$627$	166.752i	0.265953i
$628$	0	0
$629$	1214.90	1.93148
$630$	0	0
$631$	519.319i	0.823009i	0.911408	+	0.411504i	$0.134997\pi$
$631$	519.319i	0.823009i	−0.911408	+	0.411504i	$0.865003\pi$
$632$	0	0
$633$	410.412	0.648360
$634$	0	0
$635$	− 247.454i	− 0.389692i
$636$	0	0
$637$	168.653	0.264762
$638$	0	0
$639$	− 184.793i	− 0.289190i
$640$	0	0
$641$	540.111	0.842607	0.421303	−	0.906920i	$-0.361573\pi$
$641$	540.111	0.842607	0.421303	+	0.906920i	$0.361573\pi$
$642$	0	0
$643$	− 316.600i	− 0.492380i	−0.969222	−	0.246190i	$-0.920821\pi$
$643$	− 316.600i	− 0.492380i	0.969222	−	0.246190i	$-0.0791786\pi$
$644$	0	0
$645$	−142.899	−0.221548
$646$	0	0
$647$	− 745.515i	− 1.15226i	−0.817356	−	0.576132i	$-0.804561\pi$
$647$	− 745.515i	− 1.15226i	0.817356	−	0.576132i	$-0.195439\pi$
$648$	0	0
$649$	−598.275	−0.921841
$650$	0	0
$651$	− 171.535i	− 0.263494i
$652$	0	0
$653$	−794.499	−1.21669	−0.608345	−	0.793673i	$-0.708166\pi$
$653$	−794.499	−1.21669	−0.608345	+	0.793673i	$0.708166\pi$
$654$	0	0
$655$	458.465i	0.699947i
$656$	0	0
$657$	157.464	0.239671
$658$	0	0
$659$	− 26.4351i	− 0.0401140i	−0.999799	−	0.0200570i	$-0.993615\pi$
$659$	− 26.4351i	− 0.0401140i	0.999799	−	0.0200570i	$-0.00638477\pi$
$660$	0	0
$661$	497.881	0.753223	0.376612	−	0.926371i	$-0.377089\pi$
$661$	497.881	0.753223	0.376612	+	0.926371i	$0.377089\pi$
$662$	0	0
$663$	720.265i	1.08637i
$664$	0	0
$665$	100.159	0.150614
$666$	0	0
$667$	1259.45i	1.88823i
$668$	0	0
$669$	−371.038	−0.554616
$670$	0	0
$671$	456.470i	0.680283i
$672$	0	0
$673$	−94.9961	−0.141153	−0.0705766	−	0.997506i	$-0.522484\pi$
$673$	−94.9961	−0.141153	−0.0705766	+	0.997506i	$0.522484\pi$
$674$	0	0
$675$	− 85.7903i	− 0.127097i
$676$	0	0
$677$	757.820	1.11938	0.559690	−	0.828702i	$-0.310920\pi$
$677$	757.820	1.11938	0.559690	+	0.828702i	$0.310920\pi$
$678$	0	0
$679$	108.765i	0.160185i
$680$	0	0
$681$	−547.826	−0.804443
$682$	0	0
$683$	− 432.613i	− 0.633402i	−0.948525	−	0.316701i	$-0.897425\pi$
$683$	− 432.613i	− 0.633402i	0.948525	−	0.316701i	$-0.102575\pi$
$684$	0	0
$685$	−23.9911	−0.0350235
$686$	0	0
$687$	− 335.422i	− 0.488242i
$688$	0	0
$689$	2144.51	3.11250
$690$	0	0
$691$	− 407.736i	− 0.590067i	−0.955487	−	0.295034i	$-0.904669\pi$
$691$	− 407.736i	− 0.590067i	0.955487	−	0.295034i	$-0.0953308\pi$
$692$	0	0
$693$	−58.8149	−0.0848699
$694$	0	0
$695$	− 560.639i	− 0.806675i
$696$	0	0
$697$	430.520	0.617676
$698$	0	0
$699$	− 384.588i	− 0.550197i
$700$	0	0
$701$	−206.009	−0.293879	−0.146939	−	0.989145i	$-0.546942\pi$
$701$	−206.009	−0.293879	−0.146939	+	0.989145i	$0.546942\pi$
$702$	0	0
$703$	− 914.535i	− 1.30090i
$704$	0	0
$705$	206.223	0.292515
$706$	0	0
$707$	− 65.8059i	− 0.0930777i
$708$	0	0
$709$	511.639	0.721635	0.360817	−	0.932637i	$-0.382498\pi$
$709$	511.639	0.721635	0.360817	+	0.932637i	$0.382498\pi$
$710$	0	0
$711$	− 275.567i	− 0.387576i
$712$	0	0
$713$	−1213.98	−1.70263
$714$	0	0
$715$	520.186i	0.727533i
$716$	0	0
$717$	−45.7636	−0.0638265
$718$	0	0
$719$	1045.23i	1.45373i	0.686780	+	0.726866i	$0.259024\pi$
$719$	1045.23i	1.45373i	−0.686780	+	0.726866i	$0.740976\pi$
$720$	0	0
$721$	28.7248	0.0398402
$722$	0	0
$723$	− 309.864i	− 0.428581i
$724$	0	0
$725$	641.164	0.884364
$726$	0	0
$727$	− 193.079i	− 0.265584i	−0.991144	−	0.132792i	$-0.957606\pi$
$727$	− 193.079i	− 0.265584i	0.991144	−	0.132792i	$-0.0423942\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 488.717i	− 0.668560i
$732$	0	0
$733$	−851.928	−1.16225	−0.581124	−	0.813815i	$-0.697387\pi$
$733$	−851.928	−1.16225	−0.581124	+	0.813815i	$0.697387\pi$
$734$	0	0
$735$	35.3267i	0.0480636i
$736$	0	0
$737$	−682.764	−0.926410
$738$	0	0
$739$	189.437i	0.256342i	0.991752	+	0.128171i	$0.0409107\pi$
$739$	189.437i	0.256342i	−0.991752	+	0.128171i	$0.959089\pi$
$740$	0	0
$741$	542.191	0.731702
$742$	0	0
$743$	− 945.745i	− 1.27287i	−0.771329	−	0.636436i	$-0.780408\pi$
$743$	− 945.745i	− 1.27287i	0.771329	−	0.636436i	$-0.219592\pi$
$744$	0	0
$745$	173.635	0.233067
$746$	0	0
$747$	294.962i	0.394862i
$748$	0	0
$749$	−78.8104	−0.105221
$750$	0	0
$751$	902.515i	1.20175i	0.799342	+	0.600876i	$0.205181\pi$
$751$	902.515i	1.20175i	−0.799342	+	0.600876i	$0.794819\pi$
$752$	0	0
$753$	42.0724	0.0558731
$754$	0	0
$755$	475.953i	0.630401i
$756$	0	0
$757$	−1242.26	−1.64103	−0.820513	−	0.571627i	$-0.806312\pi$
$757$	−1242.26	−1.64103	−0.820513	+	0.571627i	$0.806312\pi$
$758$	0	0
$759$	416.241i	0.548407i
$760$	0	0
$761$	764.711	1.00488	0.502438	−	0.864613i	$-0.332436\pi$
$761$	764.711	1.00488	0.502438	+	0.864613i	$0.332436\pi$
$762$	0	0
$763$	159.815i	0.209456i
$764$	0	0
$765$	−150.869	−0.197215
$766$	0	0
$767$	1945.27i	2.53621i
$768$	0	0
$769$	896.436	1.16572	0.582858	−	0.812574i	$-0.301934\pi$
$769$	896.436	1.16572	0.582858	+	0.812574i	$0.301934\pi$
$770$	0	0
$771$	− 533.612i	− 0.692104i
$772$	0	0
$773$	−483.067	−0.624925	−0.312463	−	0.949930i	$-0.601154\pi$
$773$	−483.067	−0.624925	−0.312463	+	0.949930i	$0.601154\pi$
$774$	0	0
$775$	618.015i	0.797438i
$776$	0	0
$777$	322.564	0.415140
$778$	0	0
$779$	− 324.081i	− 0.416021i
$780$	0	0
$781$	456.436	0.584426
$782$	0	0
$783$	− 201.788i	− 0.257711i
$784$	0	0
$785$	568.137	0.723742
$786$	0	0
$787$	1046.56i	1.32981i	0.746926	+	0.664907i	$0.231529\pi$
$787$	1046.56i	1.32981i	−0.746926	+	0.664907i	$0.768471\pi$
$788$	0	0
$789$	760.252	0.963564
$790$	0	0
$791$	56.9192i	0.0719585i
$792$	0	0
$793$	1484.20	1.87163
$794$	0	0
$795$	449.197i	0.565027i
$796$	0	0
$797$	820.370	1.02932	0.514661	−	0.857394i	$-0.327918\pi$
$797$	820.370	1.02932	0.514661	+	0.857394i	$0.327918\pi$
$798$	0	0
$799$	705.287i	0.882712i
$800$	0	0
$801$	202.643	0.252987
$802$	0	0
$803$	388.935i	0.484352i
$804$	0	0
$805$	250.012	0.310574
$806$	0	0
$807$	206.120i	0.255415i
$808$	0	0
$809$	−1254.40	−1.55056	−0.775279	−	0.631619i	$-0.782391\pi$
$809$	−1254.40	−1.55056	−0.775279	+	0.631619i	$0.782391\pi$
$810$	0	0
$811$	− 118.642i	− 0.146291i	−0.997321	−	0.0731454i	$-0.976696\pi$
$811$	− 118.642i	− 0.146291i	0.997321	−	0.0731454i	$-0.0233037\pi$
$812$	0	0
$813$	−116.444	−0.143228
$814$	0	0
$815$	− 206.564i	− 0.253453i
$816$	0	0
$817$	−367.889	−0.450293
$818$	0	0
$819$	191.235i	0.233498i
$820$	0	0
$821$	1103.74	1.34438	0.672190	−	0.740378i	$-0.265354\pi$
$821$	1103.74	1.34438	0.672190	+	0.740378i	$0.265354\pi$
$822$	0	0
$823$	− 178.011i	− 0.216295i	−0.994135	−	0.108147i	$-0.965508\pi$
$823$	− 178.011i	− 0.216295i	0.994135	−	0.108147i	$-0.0344919\pi$
$824$	0	0
$825$	211.901	0.256850
$826$	0	0
$827$	423.395i	0.511964i	0.966682	+	0.255982i	$0.0823988\pi$
$827$	423.395i	0.511964i	−0.966682	+	0.255982i	$0.917601\pi$
$828$	0	0
$829$	−1504.17	−1.81444	−0.907222	−	0.420651i	$-0.861802\pi$
$829$	−1504.17	−1.81444	−0.907222	+	0.420651i	$0.861802\pi$
$830$	0	0
$831$	316.048i	0.380323i
$832$	0	0
$833$	−120.818	−0.145040
$834$	0	0
$835$	446.974i	0.535299i
$836$	0	0
$837$	194.502	0.232380
$838$	0	0
$839$	− 448.309i	− 0.534337i	−0.963650	−	0.267168i	$-0.913912\pi$
$839$	− 448.309i	− 0.534337i	0.963650	−	0.267168i	$-0.0860880\pi$
$840$	0	0
$841$	667.084	0.793204
$842$	0	0
$843$	− 696.261i	− 0.825932i
$844$	0	0
$845$	1198.96	1.41889
$846$	0	0
$847$	174.864i	0.206451i
$848$	0	0
$849$	385.630	0.454217
$850$	0	0
$851$	− 2282.83i	− 2.68253i
$852$	0	0
$853$	120.642	0.141433	0.0707163	−	0.997496i	$-0.477471\pi$
$853$	120.642	0.141433	0.0707163	+	0.997496i	$0.477471\pi$
$854$	0	0
$855$	113.569i	0.132829i
$856$	0	0
$857$	−1440.05	−1.68034	−0.840169	−	0.542325i	$-0.817544\pi$
$857$	−1440.05	−1.68034	−0.840169	+	0.542325i	$0.817544\pi$
$858$	0	0
$859$	− 1531.82i	− 1.78326i	−0.452765	−	0.891630i	$-0.649562\pi$
$859$	− 1531.82i	− 1.78326i	0.452765	−	0.891630i	$-0.350438\pi$
$860$	0	0
$861$	114.306	0.132759
$862$	0	0
$863$	− 78.9293i	− 0.0914593i	−0.998954	−	0.0457296i	$-0.985439\pi$
$863$	− 78.9293i	− 0.0914593i	0.998954	−	0.0457296i	$-0.0145613\pi$
$864$	0	0
$865$	206.696	0.238955
$866$	0	0
$867$	− 15.4139i	− 0.0177785i
$868$	0	0
$869$	680.647	0.783254
$870$	0	0
$871$	2219.99i	2.54878i
$872$	0	0
$873$	−123.328	−0.141270
$874$	0	0
$875$	− 320.000i	− 0.365715i
$876$	0	0
$877$	1050.60	1.19795	0.598975	−	0.800768i	$-0.295575\pi$
$877$	1050.60	1.19795	0.598975	+	0.800768i	$0.295575\pi$
$878$	0	0
$879$	− 876.003i	− 0.996590i
$880$	0	0
$881$	−1135.28	−1.28862	−0.644312	−	0.764763i	$-0.722856\pi$
$881$	−1135.28	−1.28862	−0.644312	+	0.764763i	$0.722856\pi$
$882$	0	0
$883$	− 1107.98i	− 1.25479i	−0.778703	−	0.627393i	$-0.784122\pi$
$883$	− 1107.98i	− 1.25479i	0.778703	−	0.627393i	$-0.215878\pi$
$884$	0	0
$885$	−407.464	−0.460411
$886$	0	0
$887$	− 396.514i	− 0.447028i	−0.974701	−	0.223514i	$-0.928247\pi$
$887$	− 396.514i	− 0.447028i	0.974701	−	0.223514i	$-0.0717528\pi$
$888$	0	0
$889$	224.698	0.252753
$890$	0	0
$891$	− 66.6898i	− 0.0748482i
$892$	0	0
$893$	530.916	0.594530
$894$	0	0
$895$	840.746i	0.939381i
$896$	0	0
$897$	1353.40	1.50880
$898$	0	0
$899$	1453.64i	1.61695i
$900$	0	0
$901$	−1536.26	−1.70506
$902$	0	0
$903$	− 129.757i	− 0.143696i
$904$	0	0
$905$	1022.85	1.13022
$906$	0	0
$907$	815.997i	0.899666i	0.893113	+	0.449833i	$0.148516\pi$
$907$	815.997i	0.899666i	−0.893113	+	0.449833i	$0.851484\pi$
$908$	0	0
$909$	74.6169	0.0820868
$910$	0	0
$911$	1799.18i	1.97495i	0.157789	+	0.987473i	$0.449563\pi$
$911$	1799.18i	1.97495i	−0.157789	+	0.987473i	$0.550437\pi$
$912$	0	0
$913$	−728.553	−0.797977
$914$	0	0
$915$	310.886i	0.339766i
$916$	0	0
$917$	−416.304	−0.453984
$918$	0	0
$919$	1694.65i	1.84401i	0.387175	+	0.922006i	$0.373451\pi$
$919$	1694.65i	1.84401i	−0.387175	+	0.922006i	$0.626549\pi$
$920$	0	0
$921$	−271.546	−0.294838
$922$	0	0
$923$	− 1484.09i	− 1.60790i
$924$	0	0
$925$	−1162.15	−1.25638
$926$	0	0
$927$	32.5708i	0.0351357i
$928$	0	0
$929$	365.052	0.392952	0.196476	−	0.980509i	$-0.437050\pi$
$929$	365.052	0.392952	0.196476	+	0.980509i	$0.437050\pi$
$930$	0	0
$931$	90.9479i	0.0976884i
$932$	0	0
$933$	−910.370	−0.975745
$934$	0	0
$935$	− 372.646i	− 0.398552i
$936$	0	0
$937$	−622.594	−0.664455	−0.332228	−	0.943199i	$-0.607800\pi$
$937$	−622.594	−0.664455	−0.332228	+	0.943199i	$0.607800\pi$
$938$	0	0
$939$	39.3938i	0.0419529i
$940$	0	0
$941$	404.487	0.429848	0.214924	−	0.976631i	$-0.431050\pi$
$941$	404.487	0.429848	0.214924	+	0.976631i	$0.431050\pi$
$942$	0	0
$943$	− 808.958i	− 0.857856i
$944$	0	0
$945$	−40.0568	−0.0423881
$946$	0	0
$947$	1066.44i	1.12613i	0.826414	+	0.563063i	$0.190377\pi$
$947$	1066.44i	1.12613i	−0.826414	+	0.563063i	$0.809623\pi$
$948$	0	0
$949$	1264.61	1.33257
$950$	0	0
$951$	561.083i	0.589993i
$952$	0	0
$953$	1781.50	1.86936	0.934680	−	0.355491i	$-0.115686\pi$
$953$	1781.50	1.86936	0.934680	+	0.355491i	$0.115686\pi$
$954$	0	0
$955$	476.713i	0.499176i
$956$	0	0
$957$	498.414	0.520809
$958$	0	0
$959$	− 21.7848i	− 0.0227162i
$960$	0	0
$961$	−440.152	−0.458015
$962$	0	0
$963$	− 89.3626i	− 0.0927961i
$964$	0	0
$965$	−401.181	−0.415732
$966$	0	0
$967$	− 1048.75i	− 1.08454i	−0.840205	−	0.542269i	$-0.817565\pi$
$967$	− 1048.75i	− 1.08454i	0.840205	−	0.542269i	$-0.182435\pi$
$968$	0	0
$969$	−388.409	−0.400835
$970$	0	0
$971$	1439.69i	1.48268i	0.671127	+	0.741342i	$0.265810\pi$
$971$	1439.69i	1.48268i	−0.671127	+	0.741342i	$0.734190\pi$
$972$	0	0
$973$	509.082	0.523208
$974$	0	0
$975$	− 688.992i	− 0.706658i
$976$	0	0
$977$	562.504	0.575746	0.287873	−	0.957669i	$-0.407052\pi$
$977$	562.504	0.575746	0.287873	+	0.957669i	$0.407052\pi$
$978$	0	0
$979$	500.526i	0.511263i
$980$	0	0
$981$	−181.213	−0.184723
$982$	0	0
$983$	− 580.429i	− 0.590467i	−0.955425	−	0.295233i	$-0.904603\pi$
$983$	− 580.429i	− 0.590467i	0.955425	−	0.295233i	$-0.0953974\pi$
$984$	0	0
$985$	−497.494	−0.505070
$986$	0	0
$987$	187.258i	0.189725i
$988$	0	0
$989$	−918.312	−0.928526
$990$	0	0
$991$	− 1425.05i	− 1.43799i	−0.695013	−	0.718997i	$-0.744602\pi$
$991$	− 1425.05i	− 1.43799i	0.695013	−	0.718997i	$-0.255398\pi$
$992$	0	0
$993$	−786.851	−0.792398
$994$	0	0
$995$	− 590.313i	− 0.593279i
$996$	0	0
$997$	523.575	0.525151	0.262575	−	0.964912i	$-0.415428\pi$
$997$	523.575	0.525151	0.262575	+	0.964912i	$0.415428\pi$
$998$	0	0
$999$	365.753i	0.366119i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.m.a.127.3	✓	8
3.2	odd	2		2016.3.m.b.127.4		8
4.3	odd	2	inner	672.3.m.a.127.7	yes	8
8.3	odd	2		1344.3.m.d.127.2		8
8.5	even	2		1344.3.m.d.127.6		8
12.11	even	2		2016.3.m.b.127.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.3.m.a.127.3	✓	8	1.1	even	1	trivial
672.3.m.a.127.7	yes	8	4.3	odd	2	inner
1344.3.m.d.127.2		8	8.3	odd	2
1344.3.m.d.127.6		8	8.5	even	2
2016.3.m.b.127.3		8	12.11	even	2
2016.3.m.b.127.4		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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +2.91370 q^{5} +2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +2.91370 q^{5} +2.64575i q^{7} -3.00000 q^{9} -7.40998i q^{11} -24.0934 q^{13} -5.04668i q^{15} +17.2598 q^{17} -12.9926i q^{19} +4.58258 q^{21} -32.4315i q^{23} -16.5103 q^{25} +5.19615i q^{27} -38.8341 q^{29} -37.4320i q^{31} -12.8345 q^{33} +7.70893i q^{35} +70.3892 q^{37} +41.7309i q^{39} +24.9436 q^{41} -28.3154i q^{43} -8.74110 q^{45} +40.8631i q^{47} -7.00000 q^{49} -29.8948i q^{51} -89.0084 q^{53} -21.5905i q^{55} -22.5038 q^{57} -80.7391i q^{59} -61.6021 q^{61} -7.93725i q^{63} -70.2008 q^{65} -92.1412i q^{67} -56.1731 q^{69} +61.5975i q^{71} -52.4880 q^{73} +28.5968i q^{75} +19.6050 q^{77} +91.8555i q^{79} +9.00000 q^{81} -98.3206i q^{83} +50.2898 q^{85} +67.2626i q^{87} -67.5476 q^{89} -63.7450i q^{91} -64.8341 q^{93} -37.8564i q^{95} +41.1094 q^{97} +22.2299i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 24 q^{9}+O(q^{10}) \) 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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