LMFDB - Embedded newform 304.7.e.d.113.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,7,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	304.e (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:	$69.9364414204$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 483x^{6} + 75582x^{4} + 4242376x^{2} + 71047680 $ x^8 + 483x^6 + 75582x^4 + 4242376*x^2 + 71047680
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 29 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.4
Root			$-12.8592i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.7.e.d.113.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.6433i q^{3} -216.848 q^{5} +134.238 q^{7} +260.567 q^{9} +O(q^{10})$	$q-21.6433i q^{3} -216.848 q^{5} +134.238 q^{7} +260.567 q^{9} +610.546 q^{11} -3172.26i q^{13} +4693.32i q^{15} -4960.26 q^{17} +(-4433.12 - 5233.86i) q^{19} -2905.36i q^{21} +10967.2 q^{23} +31398.3 q^{25} -21417.5i q^{27} -43660.1i q^{29} +2147.30i q^{31} -13214.2i q^{33} -29109.3 q^{35} +32215.6i q^{37} -68658.3 q^{39} +1442.23i q^{41} -64084.1 q^{43} -56503.6 q^{45} +96956.3 q^{47} -99629.1 q^{49} +107356. i q^{51} -58504.5i q^{53} -132396. q^{55} +(-113278. + 95947.5i) q^{57} -78833.1i q^{59} -12464.6 q^{61} +34978.0 q^{63} +687901. i q^{65} +557643. i q^{67} -237368. i q^{69} +375099. i q^{71} -678557. q^{73} -679562. i q^{75} +81958.6 q^{77} -441287. i q^{79} -273592. q^{81} +286505. q^{83} +1.07562e6 q^{85} -944950. q^{87} +372535. i q^{89} -425839. i q^{91} +46474.6 q^{93} +(961316. + 1.13495e6i) q^{95} +392760. i q^{97} +159088. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	21.6433i		−	0.801604i	−0.916165	−	0.400802i	$-0.868732\pi$
$3$		−	21.6433i		−	0.801604i	0.916165	−	0.400802i	$-0.131268\pi$
$4$		0			0
$5$	−216.848			−1.73479			−0.867394	−	0.497622i	$-0.834207\pi$
$5$	−216.848			−1.73479			−0.867394	+	0.497622i	$0.834207\pi$
$6$		0			0
$7$	134.238			0.391365			0.195682	−	0.980667i	$-0.437308\pi$
$7$	134.238			0.391365			0.195682	+	0.980667i	$0.437308\pi$
$8$		0			0
$9$	260.567			0.357431
$10$		0			0
$11$	610.546			0.458712			0.229356	−	0.973343i	$-0.426338\pi$
$11$	610.546			0.458712			0.229356	+	0.973343i	$0.426338\pi$
$12$		0			0
$13$		−	3172.26i		−	1.44391i	−0.691941	−	0.721954i	$-0.743244\pi$
$13$		−	3172.26i		−	1.44391i	0.691941	−	0.721954i	$-0.256756\pi$
$14$		0			0
$15$			4693.32i			1.39061i
$16$		0			0
$17$	−4960.26			−1.00962			−0.504810	−	0.863231i	$-0.668437\pi$
$17$	−4960.26			−1.00962			−0.504810	+	0.863231i	$0.668437\pi$
$18$		0			0
$19$	−4433.12	−	5233.86i	−0.646322	−	0.763065i
$19$	−4433.12	−	5233.86i	−0.646322	−	0.763065i
$20$		0			0
$21$		−	2905.36i		−	0.313720i
$22$		0			0
$23$	10967.2			0.901393			0.450696	−	0.892677i	$-0.351176\pi$
$23$	10967.2			0.901393			0.450696	+	0.892677i	$0.351176\pi$
$24$		0			0
$25$	31398.3			2.00949
$26$		0			0
$27$		−	21417.5i		−	1.08812i
$28$		0			0
$29$		−	43660.1i		−	1.79016i	−0.445909	−	0.895078i	$-0.647119\pi$
$29$		−	43660.1i		−	1.79016i	0.445909	−	0.895078i	$-0.352881\pi$
$30$		0			0
$31$			2147.30i			0.0720787i	0.999350	+	0.0360393i	$0.0114742\pi$
$31$			2147.30i			0.0720787i	−0.999350	+	0.0360393i	$0.988526\pi$
$32$		0			0
$33$		−	13214.2i		−	0.367705i
$34$		0			0
$35$	−29109.3			−0.678935
$36$		0			0
$37$			32215.6i			0.636006i	0.948090	+	0.318003i	$0.103012\pi$
$37$			32215.6i			0.636006i	−0.948090	+	0.318003i	$0.896988\pi$
$38$		0			0
$39$	−68658.3			−1.15744
$40$		0			0
$41$			1442.23i			0.0209258i	0.999945	+	0.0104629i	$0.00333051\pi$
$41$			1442.23i			0.0209258i	−0.999945	+	0.0104629i	$0.996669\pi$
$42$		0			0
$43$	−64084.1			−0.806019			−0.403009	−	0.915196i	$-0.632036\pi$
$43$	−64084.1			−0.806019			−0.403009	+	0.915196i	$0.632036\pi$
$44$		0			0
$45$	−56503.6			−0.620067
$46$		0			0
$47$	96956.3			0.933862			0.466931	−	0.884294i	$-0.345360\pi$
$47$	96956.3			0.933862			0.466931	+	0.884294i	$0.345360\pi$
$48$		0			0
$49$	−99629.1			−0.846833
$50$		0			0
$51$			107356.i			0.809315i
$52$		0			0
$53$		−	58504.5i		−	0.392972i	−0.980507	−	0.196486i	$-0.937047\pi$
$53$		−	58504.5i		−	0.392972i	0.980507	−	0.196486i	$-0.0629531\pi$
$54$		0			0
$55$	−132396.			−0.795768
$56$		0			0
$57$	−113278.	+	95947.5i	−0.611676	+	0.518095i
$58$		0			0
$59$		−	78833.1i		−	0.383842i	−0.981410	−	0.191921i	$-0.938528\pi$
$59$		−	78833.1i		−	0.383842i	0.981410	−	0.191921i	$-0.0614718\pi$
$60$		0			0
$61$	−12464.6			−0.0549145			−0.0274573	−	0.999623i	$-0.508741\pi$
$61$	−12464.6			−0.0549145			−0.0274573	+	0.999623i	$0.508741\pi$
$62$		0			0
$63$	34978.0			0.139886
$64$		0			0
$65$			687901.i			2.50487i
$66$		0			0
$67$			557643.i			1.85410i	0.374943	+	0.927048i	$0.377662\pi$
$67$			557643.i			1.85410i	−0.374943	+	0.927048i	$0.622338\pi$
$68$		0			0
$69$		−	237368.i		−	0.722560i
$70$		0			0
$71$			375099.i			1.04802i	0.851712	+	0.524011i	$0.175565\pi$
$71$			375099.i			1.04802i	−0.851712	+	0.524011i	$0.824435\pi$
$72$		0			0
$73$	−678557.			−1.74429			−0.872143	−	0.489252i	$-0.837270\pi$
$73$	−678557.			−1.74429			−0.872143	+	0.489252i	$0.837270\pi$
$74$		0			0
$75$		−	679562.i		−	1.61081i
$76$		0			0
$77$	81958.6			0.179524
$78$		0			0
$79$		−	441287.i		−	0.895035i	−0.894275	−	0.447517i	$-0.852308\pi$
$79$		−	441287.i		−	0.895035i	0.894275	−	0.447517i	$-0.147692\pi$
$80$		0			0
$81$	−273592.			−0.514812
$82$		0			0
$83$	286505.			0.501069			0.250535	−	0.968108i	$-0.419394\pi$
$83$	286505.			0.501069			0.250535	+	0.968108i	$0.419394\pi$
$84$		0			0
$85$	1.07562e6			1.75148
$86$		0			0
$87$	−944950.			−1.43500
$88$		0			0
$89$			372535.i			0.528442i	0.964462	+	0.264221i	$0.0851148\pi$
$89$			372535.i			0.528442i	−0.964462	+	0.264221i	$0.914885\pi$
$90$		0			0
$91$		−	425839.i		−	0.565095i
$92$		0			0
$93$	46474.6			0.0577786
$94$		0			0
$95$	961316.	+	1.13495e6i	1.12123	+	1.32376i
$96$		0			0
$97$			392760.i			0.430341i	0.976577	+	0.215170i	$0.0690307\pi$
$97$			392760.i			0.430341i	−0.976577	+	0.215170i	$0.930969\pi$
$98$		0			0
$99$	159088.			0.163958
$100$		0			0
$101$	825999.			0.801707			0.400853	−	0.916142i	$-0.368714\pi$
$101$	825999.			0.801707			0.400853	+	0.916142i	$0.368714\pi$
$102$		0			0
$103$			700092.i			0.640683i	0.947302	+	0.320342i	$0.103798\pi$
$103$			700092.i			0.640683i	−0.947302	+	0.320342i	$0.896202\pi$
$104$		0			0
$105$			630023.i			0.544237i
$106$		0			0
$107$			1.15315e6i			0.941315i	0.882316	+	0.470657i	$0.155983\pi$
$107$			1.15315e6i			0.941315i	−0.882316	+	0.470657i	$0.844017\pi$
$108$		0			0
$109$			1.22013e6i			0.942160i	0.882090	+	0.471080i	$0.156136\pi$
$109$			1.22013e6i			0.942160i	−0.882090	+	0.471080i	$0.843864\pi$
$110$		0			0
$111$	697252.			0.509825
$112$		0			0
$113$		−	1.42763e6i		−	0.989422i	−0.869058	−	0.494711i	$-0.835274\pi$
$113$		−	1.42763e6i		−	0.989422i	0.869058	−	0.494711i	$-0.164726\pi$
$114$		0			0
$115$	−2.37823e6			−1.56372
$116$		0			0
$117$		−	826588.i		−	0.516097i
$118$		0			0
$119$	−665856.			−0.395130
$120$		0			0
$121$	−1.39879e6			−0.789583
$122$		0			0
$123$	31214.6			0.0167742
$124$		0			0
$125$	−3.42041e6			−1.75125
$126$		0			0
$127$			1.69677e6i			0.828347i	0.910198	+	0.414174i	$0.135929\pi$
$127$			1.69677e6i			0.828347i	−0.910198	+	0.414174i	$0.864071\pi$
$128$		0			0
$129$			1.38699e6i			0.646108i
$130$		0			0
$131$	−976074.			−0.434179			−0.217090	−	0.976152i	$-0.569656\pi$
$131$	−976074.			−0.434179			−0.217090	+	0.976152i	$0.569656\pi$
$132$		0			0
$133$	−595094.	−	702584.i	−0.252948	−	0.298637i
$134$		0			0
$135$			4.64435e6i			1.88766i
$136$		0			0
$137$	−1.17898e6			−0.458505			−0.229253	−	0.973367i	$-0.573628\pi$
$137$	−1.17898e6			−0.458505			−0.229253	+	0.973367i	$0.573628\pi$
$138$		0			0
$139$	−4.38059e6			−1.63113			−0.815565	−	0.578665i	$-0.803574\pi$
$139$	−4.38059e6			−1.63113			−0.815565	+	0.578665i	$0.803574\pi$
$140$		0			0
$141$		−	2.09846e6i		−	0.748587i
$142$		0			0
$143$		−	1.93681e6i		−	0.662338i
$144$		0			0
$145$			9.46763e6i			3.10554i
$146$		0			0
$147$			2.15630e6i			0.678825i
$148$		0			0
$149$	−878444.			−0.265555			−0.132778	−	0.991146i	$-0.542390\pi$
$149$	−878444.			−0.265555			−0.132778	+	0.991146i	$0.542390\pi$
$150$		0			0
$151$			98177.8i			0.0285156i	0.999898	+	0.0142578i	$0.00453855\pi$
$151$			98177.8i			0.0285156i	−0.999898	+	0.0142578i	$0.995461\pi$
$152$		0			0
$153$	−1.29248e6			−0.360869
$154$		0			0
$155$		−	465638.i		−	0.125041i
$156$		0			0
$157$	2.56108e6			0.661797			0.330899	−	0.943666i	$-0.392648\pi$
$157$	2.56108e6			0.661797			0.330899	+	0.943666i	$0.392648\pi$
$158$		0			0
$159$	−1.26623e6			−0.315008
$160$		0			0
$161$	1.47222e6			0.352774
$162$		0			0
$163$	−2.58901e6			−0.597821			−0.298910	−	0.954281i	$-0.596623\pi$
$163$	−2.58901e6			−0.597821			−0.298910	+	0.954281i	$0.596623\pi$
$164$		0			0
$165$			2.86549e6i			0.637891i
$166$		0			0
$167$		−	5.42647e6i		−	1.16511i	−0.812790	−	0.582556i	$-0.802053\pi$
$167$		−	5.42647e6i		−	1.16511i	0.812790	−	0.582556i	$-0.197947\pi$
$168$		0			0
$169$	−5.23645e6			−1.08487
$170$		0			0
$171$	−1.15513e6	−	1.36377e6i	−0.231015	−	0.272743i
$172$		0			0
$173$			1.02784e6i			0.198512i	0.995062	+	0.0992562i	$0.0316463\pi$
$173$			1.02784e6i			0.198512i	−0.995062	+	0.0992562i	$0.968354\pi$
$174$		0			0
$175$	4.21484e6			0.786443
$176$		0			0
$177$	−1.70621e6			−0.307689
$178$		0			0
$179$		−	2.69303e6i		−	0.469550i	−0.972050	−	0.234775i	$-0.924565\pi$
$179$		−	2.69303e6i		−	0.469550i	0.972050	−	0.234775i	$-0.0754354\pi$
$180$		0			0
$181$			1.25185e6i			0.211113i	0.994413	+	0.105557i	$0.0336624\pi$
$181$			1.25185e6i			0.211113i	−0.994413	+	0.105557i	$0.966338\pi$
$182$		0			0
$183$			269774.i			0.0440197i
$184$		0			0
$185$		−	6.98590e6i		−	1.10333i
$186$		0			0
$187$	−3.02847e6			−0.463125
$188$		0			0
$189$		−	2.87505e6i		−	0.425853i
$190$		0			0
$191$	1.59280e6			0.228592			0.114296	−	0.993447i	$-0.463539\pi$
$191$	1.59280e6			0.228592			0.114296	+	0.993447i	$0.463539\pi$
$192$		0			0
$193$		−	8.49629e6i		−	1.18184i	−0.806731	−	0.590918i	$-0.798766\pi$
$193$		−	8.49629e6i		−	1.18184i	0.806731	−	0.590918i	$-0.201234\pi$
$194$		0			0
$195$	1.48884e7			2.00792
$196$		0			0
$197$	−8.44740e6			−1.10490			−0.552452	−	0.833545i	$-0.686308\pi$
$197$	−8.44740e6			−1.10490			−0.552452	+	0.833545i	$0.686308\pi$
$198$		0			0
$199$	−6.30988e6			−0.800685			−0.400343	−	0.916366i	$-0.631109\pi$
$199$	−6.30988e6			−0.800685			−0.400343	+	0.916366i	$0.631109\pi$
$200$		0			0
$201$	1.20693e7			1.48625
$202$		0			0
$203$		−	5.86086e6i		−	0.700605i
$204$		0			0
$205$		−	312745.i		−	0.0363019i
$206$		0			0
$207$	2.85770e6			0.322185
$208$		0			0
$209$	−2.70662e6	−	3.19551e6i	−0.296476	−	0.350027i
$210$		0			0
$211$			279028.i			0.0297031i	0.999890	+	0.0148515i	$0.00472756\pi$
$211$			279028.i			0.0297031i	−0.999890	+	0.0148515i	$0.995272\pi$
$212$		0			0
$213$	8.11838e6			0.840099
$214$		0			0
$215$	1.38965e7			1.39827
$216$		0			0
$217$			288249.i			0.0282091i
$218$		0			0
$219$			1.46862e7i			1.39823i
$220$		0			0
$221$			1.57353e7i			1.45780i
$222$		0			0
$223$			1.67228e7i			1.50798i	0.656886	+	0.753990i	$0.271873\pi$
$223$			1.67228e7i			1.50798i	−0.656886	+	0.753990i	$0.728127\pi$
$224$		0			0
$225$	8.18135e6			0.718253
$226$		0			0
$227$		−	1.16656e7i		−	0.997312i	−0.866800	−	0.498656i	$-0.833827\pi$
$227$		−	1.16656e7i		−	0.997312i	0.866800	−	0.498656i	$-0.166173\pi$
$228$		0			0
$229$	221475.			0.0184424			0.00922120	−	0.999957i	$-0.497065\pi$
$229$	221475.			0.0184424			0.00922120	+	0.999957i	$0.497065\pi$
$230$		0			0
$231$		−	1.77385e6i		−	0.143907i
$232$		0			0
$233$	5.10799e6			0.403815			0.201907	−	0.979405i	$-0.435286\pi$
$233$	5.10799e6			0.403815			0.201907	+	0.979405i	$0.435286\pi$
$234$		0			0
$235$	−2.10248e7			−1.62005
$236$		0			0
$237$	−9.55091e6			−0.717463
$238$		0			0
$239$	1.49282e7			1.09349			0.546745	−	0.837299i	$-0.315867\pi$
$239$	1.49282e7			1.09349			0.546745	+	0.837299i	$0.315867\pi$
$240$		0			0
$241$			9.74453e6i			0.696161i	0.937465	+	0.348080i	$0.113166\pi$
$241$			9.74453e6i			0.696161i	−0.937465	+	0.348080i	$0.886834\pi$
$242$		0			0
$243$		−	9.69192e6i		−	0.675446i
$244$		0			0
$245$	2.16044e7			1.46908
$246$		0			0
$247$	−1.66032e7	+	1.40630e7i	−1.10179	+	0.933229i
$248$		0			0
$249$		−	6.20091e6i		−	0.401659i
$250$		0			0
$251$	−1.28110e7			−0.810141			−0.405071	−	0.914285i	$-0.632753\pi$
$251$	−1.28110e7			−0.810141			−0.405071	+	0.914285i	$0.632753\pi$
$252$		0			0
$253$	6.69600e6			0.413480
$254$		0			0
$255$		−	2.32801e7i		−	1.40399i
$256$		0			0
$257$		−	2.69491e7i		−	1.58761i	−0.608171	−	0.793806i	$-0.708097\pi$
$257$		−	2.69491e7i		−	1.58761i	0.608171	−	0.793806i	$-0.291903\pi$
$258$		0			0
$259$			4.32456e6i			0.248910i
$260$		0			0
$261$		−	1.13764e7i		−	0.639857i
$262$		0			0
$263$	−1.63453e7			−0.898514			−0.449257	−	0.893403i	$-0.648311\pi$
$263$	−1.63453e7			−0.898514			−0.449257	+	0.893403i	$0.648311\pi$
$264$		0			0
$265$			1.26866e7i			0.681723i
$266$		0			0
$267$	8.06290e6			0.423601
$268$		0			0
$269$		−	6.31414e6i		−	0.324383i	−0.986759	−	0.162191i	$-0.948144\pi$
$269$		−	6.31414e6i		−	0.324383i	0.986759	−	0.162191i	$-0.0518562\pi$
$270$		0			0
$271$	1.25433e7			0.630238			0.315119	−	0.949052i	$-0.397956\pi$
$271$	1.25433e7			0.630238			0.315119	+	0.949052i	$0.397956\pi$
$272$		0			0
$273$	−9.21657e6			−0.452982
$274$		0			0
$275$	1.91701e7			0.921777
$276$		0			0
$277$	−2.04081e7			−0.960202			−0.480101	−	0.877213i	$-0.659400\pi$
$277$	−2.04081e7			−0.960202			−0.480101	+	0.877213i	$0.659400\pi$
$278$		0			0
$279$			559515.i			0.0257631i
$280$		0			0
$281$			2.63413e7i			1.18719i	0.804765	+	0.593593i	$0.202291\pi$
$281$			2.63413e7i			1.18719i	−0.804765	+	0.593593i	$0.797709\pi$
$282$		0			0
$283$	3.93597e7			1.73657			0.868285	−	0.496066i	$-0.165223\pi$
$283$	3.93597e7			1.73657			0.868285	+	0.496066i	$0.165223\pi$
$284$		0			0
$285$	2.45642e7	−	2.08061e7i	1.06113	−	0.898784i
$286$		0			0
$287$			193602.i			0.00818963i
$288$		0			0
$289$	466619.			0.0193317
$290$		0			0
$291$	8.50063e6			0.344963
$292$		0			0
$293$		−	6.79927e6i		−	0.270308i	−0.990825	−	0.135154i	$-0.956847\pi$
$293$		−	6.79927e6i		−	0.270308i	0.990825	−	0.135154i	$-0.0431530\pi$
$294$		0			0
$295$			1.70948e7i			0.665885i
$296$		0			0
$297$		−	1.30764e7i		−	0.499135i
$298$		0			0
$299$		−	3.47910e7i		−	1.30153i
$300$		0			0
$301$	−8.60254e6			−0.315448
$302$		0			0
$303$		−	1.78774e7i		−	0.642651i
$304$		0			0
$305$	2.70292e6			0.0952651
$306$		0			0
$307$		−	3.78588e7i		−	1.30843i	−0.756308	−	0.654216i	$-0.772999\pi$
$307$		−	3.78588e7i		−	1.30843i	0.756308	−	0.654216i	$-0.227001\pi$
$308$		0			0
$309$	1.51523e7			0.513574
$310$		0			0
$311$	1.46077e7			0.485626			0.242813	−	0.970073i	$-0.421930\pi$
$311$	1.46077e7			0.485626			0.242813	+	0.970073i	$0.421930\pi$
$312$		0			0
$313$	−1.50273e7			−0.490059			−0.245030	−	0.969516i	$-0.578798\pi$
$313$	−1.50273e7			−0.490059			−0.245030	+	0.969516i	$0.578798\pi$
$314$		0			0
$315$	−7.58494e6			−0.242672
$316$		0			0
$317$		−	1.48184e7i		−	0.465183i	−0.972575	−	0.232591i	$-0.925279\pi$
$317$		−	1.48184e7i		−	0.465183i	0.972575	−	0.232591i	$-0.0747205\pi$
$318$		0			0
$319$		−	2.66565e7i		−	0.821166i
$320$		0			0
$321$	2.49580e7			0.754562
$322$		0			0
$323$	2.19895e7	+	2.59613e7i	0.652539	+	0.770405i
$324$		0			0
$325$		−	9.96036e7i		−	2.90152i
$326$		0			0
$327$	2.64075e7			0.755240
$328$		0			0
$329$	1.30152e7			0.365481
$330$		0			0
$331$			5.51403e7i			1.52050i	0.649633	+	0.760248i	$0.274923\pi$
$331$			5.51403e7i			1.52050i	−0.649633	+	0.760248i	$0.725077\pi$
$332$		0			0
$333$			8.39432e6i			0.227328i
$334$		0			0
$335$		−	1.20924e8i		−	3.21646i
$336$		0			0
$337$			4.52028e6i			0.118107i	0.998255	+	0.0590535i	$0.0188083\pi$
$337$			4.52028e6i			0.118107i	−0.998255	+	0.0590535i	$0.981192\pi$
$338$		0			0
$339$	−3.08987e7			−0.793125
$340$		0			0
$341$			1.31102e6i			0.0330634i
$342$		0			0
$343$	−2.91670e7			−0.722786
$344$		0			0
$345$			5.14728e7i			1.25349i
$346$		0			0
$347$	2.13638e7			0.511317			0.255658	−	0.966767i	$-0.417708\pi$
$347$	2.13638e7			0.511317			0.255658	+	0.966767i	$0.417708\pi$
$348$		0			0
$349$	−2.39308e7			−0.562964			−0.281482	−	0.959567i	$-0.590826\pi$
$349$	−2.39308e7			−0.562964			−0.281482	+	0.959567i	$0.590826\pi$
$350$		0			0
$351$	−6.79420e7			−1.57115
$352$		0			0
$353$	−6.39808e7			−1.45454			−0.727270	−	0.686352i	$-0.759211\pi$
$353$	−6.39808e7			−1.45454			−0.727270	+	0.686352i	$0.759211\pi$
$354$		0			0
$355$		−	8.13396e7i		−	1.81810i
$356$		0			0
$357$			1.44113e7i			0.316738i
$358$		0			0
$359$	−2.48683e7			−0.537480			−0.268740	−	0.963213i	$-0.586607\pi$
$359$	−2.48683e7			−0.537480			−0.268740	+	0.963213i	$0.586607\pi$
$360$		0			0
$361$	−7.74071e6	+	4.64047e7i	−0.164535	+	0.986371i
$362$		0			0
$363$			3.02746e7i			0.632933i
$364$		0			0
$365$	1.47144e8			3.02596
$366$		0			0
$367$	8.86334e7			1.79308			0.896540	−	0.442964i	$-0.146073\pi$
$367$	8.86334e7			1.79308			0.896540	+	0.442964i	$0.146073\pi$
$368$		0			0
$369$			375797.i			0.00747953i
$370$		0			0
$371$		−	7.85354e6i		−	0.153796i
$372$		0			0
$373$		−	3.66672e7i		−	0.706563i	−0.935517	−	0.353282i	$-0.885066\pi$
$373$		−	3.66672e7i		−	0.706563i	0.935517	−	0.353282i	$-0.114934\pi$
$374$		0			0
$375$			7.40289e7i			1.40381i
$376$		0			0
$377$	−1.38501e8			−2.58482
$378$		0			0
$379$			7.73753e7i			1.42130i	0.703548	+	0.710648i	$0.251598\pi$
$379$			7.73753e7i			1.42130i	−0.703548	+	0.710648i	$0.748402\pi$
$380$		0			0
$381$	3.67238e7			0.664007
$382$		0			0
$383$			5.70315e7i			1.01512i	0.861616	+	0.507561i	$0.169453\pi$
$383$			5.70315e7i			1.01512i	−0.861616	+	0.507561i	$0.830547\pi$
$384$		0			0
$385$	−1.77726e7			−0.311436
$386$		0			0
$387$	−1.66982e7			−0.288096
$388$		0			0
$389$	3.18945e7			0.541836			0.270918	−	0.962602i	$-0.412673\pi$
$389$	3.18945e7			0.541836			0.270918	+	0.962602i	$0.412673\pi$
$390$		0			0
$391$	−5.44004e7			−0.910064
$392$		0			0
$393$			2.11255e7i			0.348040i
$394$		0			0
$395$			9.56924e7i			1.55269i
$396$		0			0
$397$	2.92281e7			0.467121			0.233560	−	0.972342i	$-0.424962\pi$
$397$	2.92281e7			0.467121			0.233560	+	0.972342i	$0.424962\pi$
$398$		0			0
$399$	−1.52062e7	+	1.28798e7i	−0.239388	+	0.202764i
$400$		0			0
$401$			9.77290e7i			1.51562i	0.652476	+	0.757810i	$0.273730\pi$
$401$			9.77290e7i			1.51562i	−0.652476	+	0.757810i	$0.726270\pi$
$402$		0			0
$403$	6.81179e6			0.104075
$404$		0			0
$405$	5.93281e7			0.893090
$406$		0			0
$407$			1.96691e7i			0.291743i
$408$		0			0
$409$		−	6.60177e7i		−	0.964919i	−0.875918	−	0.482459i	$-0.839744\pi$
$409$		−	6.60177e7i		−	0.964919i	0.875918	−	0.482459i	$-0.160256\pi$
$410$		0			0
$411$			2.55170e7i			0.367540i
$412$		0			0
$413$		−	1.05824e7i		−	0.150222i
$414$		0			0
$415$	−6.21281e7			−0.869249
$416$		0			0
$417$			9.48106e7i			1.30752i
$418$		0			0
$419$	−5.57280e7			−0.757586			−0.378793	−	0.925482i	$-0.623661\pi$
$419$	−5.57280e7			−0.757586			−0.378793	+	0.925482i	$0.623661\pi$
$420$		0			0
$421$		−	5.52129e7i		−	0.739937i	−0.929044	−	0.369968i	$-0.879369\pi$
$421$		−	5.52129e7i		−	0.739937i	0.929044	−	0.369968i	$-0.120631\pi$
$422$		0			0
$423$	2.52636e7			0.333791
$424$		0			0
$425$	−1.55744e8			−2.02882
$426$		0			0
$427$	−1.67322e6			−0.0214916
$428$		0			0
$429$	−4.19190e7			−0.530933
$430$		0			0
$431$		−	9.53718e7i		−	1.19121i	−0.803278	−	0.595605i	$-0.796913\pi$
$431$		−	9.53718e7i		−	1.19121i	0.803278	−	0.595605i	$-0.203087\pi$
$432$		0			0
$433$			4.42302e7i			0.544823i	0.962181	+	0.272412i	$0.0878212\pi$
$433$			4.42302e7i			0.544823i	−0.962181	+	0.272412i	$0.912179\pi$
$434$		0			0
$435$	2.04911e8			2.48942
$436$		0			0
$437$	−4.86192e7	−	5.74010e7i	−0.582590	−	0.687821i
$438$		0			0
$439$		−	1.47409e7i		−	0.174234i	−0.996198	−	0.0871168i	$-0.972235\pi$
$439$		−	1.47409e7i		−	0.174234i	0.996198	−	0.0871168i	$-0.0277653\pi$
$440$		0			0
$441$	−2.59601e7			−0.302684
$442$		0			0
$443$	−9.74634e7			−1.12106			−0.560532	−	0.828133i	$-0.689403\pi$
$443$	−9.74634e7			−1.12106			−0.560532	+	0.828133i	$0.689403\pi$
$444$		0			0
$445$		−	8.07837e7i		−	0.916735i
$446$		0			0
$447$			1.90124e7i			0.212870i
$448$		0			0
$449$			1.46013e8i			1.61307i	0.591189	+	0.806533i	$0.298659\pi$
$449$			1.46013e8i			1.61307i	−0.591189	+	0.806533i	$0.701341\pi$
$450$		0			0
$451$			880547.i			0.00959893i
$452$		0			0
$453$	2.12489e6			0.0228582
$454$		0			0
$455$			9.23425e7i			0.980319i
$456$		0			0
$457$	1.43785e8			1.50648			0.753241	−	0.657744i	$-0.228489\pi$
$457$	1.43785e8			1.50648			0.753241	+	0.657744i	$0.228489\pi$
$458$		0			0
$459$			1.06236e8i			1.09859i
$460$		0			0
$461$	5.83001e7			0.595068			0.297534	−	0.954711i	$-0.403836\pi$
$461$	5.83001e7			0.595068			0.297534	+	0.954711i	$0.403836\pi$
$462$		0			0
$463$	−2.61228e7			−0.263194			−0.131597	−	0.991303i	$-0.542011\pi$
$463$	−2.61228e7			−0.263194			−0.131597	+	0.991303i	$0.542011\pi$
$464$		0			0
$465$	−1.00779e7			−0.100234
$466$		0			0
$467$	−5.61432e7			−0.551248			−0.275624	−	0.961266i	$-0.588884\pi$
$467$	−5.61432e7			−0.551248			−0.275624	+	0.961266i	$0.588884\pi$
$468$		0			0
$469$			7.48570e7i			0.725628i
$470$		0			0
$471$		−	5.54303e7i		−	0.530499i
$472$		0			0
$473$	−3.91263e7			−0.369731
$474$		0			0
$475$	−1.39192e8	−	1.64334e8i	−1.29878	−	1.53337i
$476$		0			0
$477$		−	1.52443e7i		−	0.140460i
$478$		0			0
$479$	1.94372e8			1.76859			0.884295	−	0.466929i	$-0.154640\pi$
$479$	1.94372e8			1.76859			0.884295	+	0.466929i	$0.154640\pi$
$480$		0			0
$481$	1.02196e8			0.918333
$482$		0			0
$483$		−	3.18638e7i		−	0.282785i
$484$		0			0
$485$		−	8.51695e7i		−	0.746550i
$486$		0			0
$487$			1.10939e8i			0.960497i	0.877133	+	0.480248i	$0.159453\pi$
$487$			1.10939e8i			0.960497i	−0.877133	+	0.480248i	$0.840547\pi$
$488$		0			0
$489$			5.60347e7i			0.479215i
$490$		0			0
$491$	−3.84300e7			−0.324658			−0.162329	−	0.986737i	$-0.551901\pi$
$491$	−3.84300e7			−0.324658			−0.162329	+	0.986737i	$0.551901\pi$
$492$		0			0
$493$			2.16566e8i			1.80738i
$494$		0			0
$495$	−3.44980e7			−0.284432
$496$		0			0
$497$			5.03526e7i			0.410159i
$498$		0			0
$499$	2.24551e8			1.80723			0.903616	−	0.428344i	$-0.140903\pi$
$499$	2.24551e8			1.80723			0.903616	+	0.428344i	$0.140903\pi$
$500$		0			0
$501$	−1.17447e8			−0.933959
$502$		0			0
$503$	−1.77754e8			−1.39674			−0.698370	−	0.715737i	$-0.746091\pi$
$503$	−1.77754e8			−1.39674			−0.698370	+	0.715737i	$0.746091\pi$
$504$		0			0
$505$	−1.79117e8			−1.39079
$506$		0			0
$507$			1.13334e8i			0.869635i
$508$		0			0
$509$		−	1.75276e8i		−	1.32914i	−0.747228	−	0.664568i	$-0.768616\pi$
$509$		−	1.75276e8i		−	1.32914i	0.747228	−	0.664568i	$-0.231384\pi$
$510$		0			0
$511$	−9.10882e7			−0.682652
$512$		0			0
$513$	−1.12096e8	+	9.49465e7i	−0.830308	+	0.703277i
$514$		0			0
$515$		−	1.51814e8i		−	1.11145i
$516$		0			0
$517$	5.91963e7			0.428374
$518$		0			0
$519$	2.22459e7			0.159128
$520$		0			0
$521$			1.55786e8i			1.10158i	0.834644	+	0.550789i	$0.185673\pi$
$521$			1.55786e8i			1.10158i	−0.834644	+	0.550789i	$0.814327\pi$
$522$		0			0
$523$		−	6.68971e7i		−	0.467630i	−0.972281	−	0.233815i	$-0.924879\pi$
$523$		−	6.68971e7i		−	0.467630i	0.972281	−	0.233815i	$-0.0751210\pi$
$524$		0			0
$525$		−	9.12232e7i		−	0.630416i
$526$		0			0
$527$		−	1.06511e7i		−	0.0727721i
$528$		0			0
$529$	−2.77554e7			−0.187491
$530$		0			0
$531$		−	2.05413e7i		−	0.137197i
$532$		0			0
$533$	4.57513e6			0.0302150
$534$		0			0
$535$		−	2.50059e8i		−	1.63298i
$536$		0			0
$537$	−5.82861e7			−0.376393
$538$		0			0
$539$	−6.08281e7			−0.388453
$540$		0			0
$541$	−1.85229e7			−0.116981			−0.0584907	−	0.998288i	$-0.518629\pi$
$541$	−1.85229e7			−0.116981			−0.0584907	+	0.998288i	$0.518629\pi$
$542$		0			0
$543$	2.70941e7			0.169229
$544$		0			0
$545$		−	2.64582e8i		−	1.63445i
$546$		0			0
$547$			9.09744e7i			0.555850i	0.960603	+	0.277925i	$0.0896466\pi$
$547$			9.09744e7i			0.555850i	−0.960603	+	0.277925i	$0.910353\pi$
$548$		0			0
$549$	−3.24785e6			−0.0196281
$550$		0			0
$551$	−2.28511e8	+	1.93551e8i	−1.36601	+	1.15702i
$552$		0			0
$553$		−	5.92376e7i		−	0.350285i
$554$		0			0
$555$	−1.51198e8			−0.884438
$556$		0			0
$557$	−2.06452e8			−1.19468			−0.597342	−	0.801987i	$-0.703777\pi$
$557$	−2.06452e8			−1.19468			−0.597342	+	0.801987i	$0.703777\pi$
$558$		0			0
$559$			2.03292e8i			1.16382i
$560$		0			0
$561$			6.55460e7i			0.371243i
$562$		0			0
$563$			2.53968e8i			1.42316i	0.702606	+	0.711579i	$0.252020\pi$
$563$			2.53968e8i			1.42316i	−0.702606	+	0.711579i	$0.747980\pi$
$564$		0			0
$565$			3.09580e8i			1.71644i
$566$		0			0
$567$	−3.67266e7			−0.201480
$568$		0			0
$569$		−	3.38715e8i		−	1.83865i	−0.393505	−	0.919323i	$-0.628738\pi$
$569$		−	3.38715e8i		−	1.83865i	0.393505	−	0.919323i	$-0.371262\pi$
$570$		0			0
$571$	−6.12557e7			−0.329032			−0.164516	−	0.986374i	$-0.552606\pi$
$571$	−6.12557e7			−0.329032			−0.164516	+	0.986374i	$0.552606\pi$
$572$		0			0
$573$		−	3.44735e7i		−	0.183241i
$574$		0			0
$575$	3.44352e8			1.81134
$576$		0			0
$577$	−1.60133e8			−0.833592			−0.416796	−	0.909000i	$-0.636847\pi$
$577$	−1.60133e8			−0.833592			−0.416796	+	0.909000i	$0.636847\pi$
$578$		0			0
$579$	−1.83888e8			−0.947365
$580$		0			0
$581$	3.84599e7			0.196101
$582$		0			0
$583$		−	3.57197e7i		−	0.180261i
$584$		0			0
$585$			1.79244e8i			0.895319i
$586$		0			0
$587$	2.31399e8			1.14405			0.572027	−	0.820235i	$-0.306157\pi$
$587$	2.31399e8			1.14405			0.572027	+	0.820235i	$0.306157\pi$
$588$		0			0
$589$	1.12386e7	−	9.51923e6i	0.0550007	−	0.0465861i
$590$		0			0
$591$			1.82830e8i			0.885695i
$592$		0			0
$593$	3.98181e8			1.90949			0.954743	−	0.297432i	$-0.0961301\pi$
$593$	3.98181e8			1.90949			0.954743	+	0.297432i	$0.0961301\pi$
$594$		0			0
$595$	1.44390e8			0.685466
$596$		0			0
$597$			1.36567e8i			0.641833i
$598$		0			0
$599$		−	5.91504e7i		−	0.275218i	−0.990487	−	0.137609i	$-0.956058\pi$
$599$		−	5.91504e7i		−	0.275218i	0.990487	−	0.137609i	$-0.0439418\pi$
$600$		0			0
$601$		−	8.49083e7i		−	0.391135i	−0.980690	−	0.195567i	$-0.937345\pi$
$601$		−	8.49083e7i		−	0.391135i	0.980690	−	0.195567i	$-0.0626549\pi$
$602$		0			0
$603$			1.45304e8i			0.662711i
$604$		0			0
$605$	3.03327e8			1.36976
$606$		0			0
$607$		−	1.86731e8i		−	0.834931i	−0.908693	−	0.417466i	$-0.862918\pi$
$607$		−	1.86731e8i		−	0.834931i	0.908693	−	0.417466i	$-0.137082\pi$
$608$		0			0
$609$	−1.26848e8			−0.561608
$610$		0			0
$611$		−	3.07571e8i		−	1.34841i
$612$		0			0
$613$	−3.22574e7			−0.140039			−0.0700193	−	0.997546i	$-0.522306\pi$
$613$	−3.22574e7			−0.140039			−0.0700193	+	0.997546i	$0.522306\pi$
$614$		0			0
$615$	−6.76884e6			−0.0290997
$616$		0			0
$617$	7.13770e6			0.0303881			0.0151940	−	0.999885i	$-0.495163\pi$
$617$	7.13770e6			0.0303881			0.0151940	+	0.999885i	$0.495163\pi$
$618$		0			0
$619$	−2.27364e8			−0.958626			−0.479313	−	0.877644i	$-0.659114\pi$
$619$	−2.27364e8			−0.958626			−0.479313	+	0.877644i	$0.659114\pi$
$620$		0			0
$621$		−	2.34891e8i		−	0.980825i
$622$		0			0
$623$			5.00085e7i			0.206814i
$624$		0			0
$625$	2.51112e8			1.02856
$626$		0			0
$627$	−6.91614e7	+	5.85803e7i	−0.280583	+	0.237656i
$628$		0			0
$629$		−	1.59798e8i		−	0.642124i
$630$		0			0
$631$	1.66784e8			0.663845			0.331923	−	0.943307i	$-0.392303\pi$
$631$	1.66784e8			0.663845			0.331923	+	0.943307i	$0.392303\pi$
$632$		0			0
$633$	6.03910e6			0.0238101
$634$		0			0
$635$		−	3.67943e8i		−	1.43701i
$636$		0			0
$637$			3.16050e8i			1.22275i
$638$		0			0
$639$			9.77383e7i			0.374595i
$640$		0			0
$641$			2.98503e8i			1.13338i	0.823932	+	0.566689i	$0.191776\pi$
$641$			2.98503e8i			1.13338i	−0.823932	+	0.566689i	$0.808224\pi$
$642$		0			0
$643$	−2.52510e8			−0.949829			−0.474914	−	0.880032i	$-0.657521\pi$
$643$	−2.52510e8			−0.949829			−0.474914	+	0.880032i	$0.657521\pi$
$644$		0			0
$645$		−	3.00767e8i		−	1.12086i
$646$		0			0
$647$	−1.19544e8			−0.441381			−0.220690	−	0.975344i	$-0.570831\pi$
$647$	−1.19544e8			−0.441381			−0.220690	+	0.975344i	$0.570831\pi$
$648$		0			0
$649$		−	4.81312e7i		−	0.176073i
$650$		0			0
$651$	6.23867e6			0.0226125
$652$		0			0
$653$	2.99786e8			1.07664			0.538322	−	0.842739i	$-0.319059\pi$
$653$	2.99786e8			1.07664			0.538322	+	0.842739i	$0.319059\pi$
$654$		0			0
$655$	2.11660e8			0.753209
$656$		0			0
$657$	−1.76809e8			−0.623461
$658$		0			0
$659$			2.88850e8i			1.00929i	0.863327	+	0.504646i	$0.168377\pi$
$659$			2.88850e8i			1.00929i	−0.863327	+	0.504646i	$0.831623\pi$
$660$		0			0
$661$		−	4.67718e8i		−	1.61950i	−0.586777	−	0.809748i	$-0.699604\pi$
$661$		−	4.67718e8i		−	1.61950i	0.586777	−	0.809748i	$-0.300396\pi$
$662$		0			0
$663$	3.40563e8			1.16858
$664$		0			0
$665$	1.29045e8	+	1.52354e8i	0.438811	+	0.518071i
$666$		0			0
$667$		−	4.78831e8i		−	1.61363i
$668$		0			0
$669$	3.61938e8			1.20880
$670$		0			0
$671$	−7.61018e6			−0.0251900
$672$		0			0
$673$		−	3.36632e8i		−	1.10436i	−0.833725	−	0.552180i	$-0.813796\pi$
$673$		−	3.36632e8i		−	1.10436i	0.833725	−	0.552180i	$-0.186204\pi$
$674$		0			0
$675$		−	6.72472e8i		−	2.18657i
$676$		0			0
$677$			5.79462e7i			0.186749i	0.995631	+	0.0933747i	$0.0297655\pi$
$677$			5.79462e7i			0.186749i	−0.995631	+	0.0933747i	$0.970235\pi$
$678$		0			0
$679$			5.27234e7i			0.168420i
$680$		0			0
$681$	−2.52483e8			−0.799449
$682$		0			0
$683$			1.89792e8i			0.595683i	0.954615	+	0.297842i	$0.0962668\pi$
$683$			1.89792e8i			0.595683i	−0.954615	+	0.297842i	$0.903733\pi$
$684$		0			0
$685$	2.55660e8			0.795409
$686$		0			0
$687$		−	4.79344e6i		−	0.0147835i
$688$		0			0
$689$	−1.85592e8			−0.567415
$690$		0			0
$691$	6.07884e8			1.84241			0.921205	−	0.389077i	$-0.127206\pi$
$691$	6.07884e8			1.84241			0.921205	+	0.389077i	$0.127206\pi$
$692$		0			0
$693$	2.13557e7			0.0641673
$694$		0			0
$695$	9.49925e8			2.82966
$696$		0			0
$697$		−	7.15383e6i		−	0.0211271i
$698$		0			0
$699$		−	1.10554e8i		−	0.323700i
$700$		0			0
$701$	2.97048e8			0.862328			0.431164	−	0.902273i	$-0.358103\pi$
$701$	2.97048e8			0.862328			0.431164	+	0.902273i	$0.358103\pi$
$702$		0			0
$703$	1.68612e8	−	1.42816e8i	0.485313	−	0.411064i
$704$		0			0
$705$			4.55047e8i			1.29864i
$706$		0			0
$707$	1.10881e8			0.313760
$708$		0			0
$709$	2.19795e8			0.616708			0.308354	−	0.951272i	$-0.400222\pi$
$709$	2.19795e8			0.616708			0.308354	+	0.951272i	$0.400222\pi$
$710$		0			0
$711$		−	1.14985e8i		−	0.319913i
$712$		0			0
$713$			2.35499e7i			0.0649712i
$714$		0			0
$715$			4.19995e8i			1.14902i
$716$		0			0
$717$		−	3.23096e8i		−	0.876546i
$718$		0			0
$719$	−5.01794e8			−1.35002			−0.675008	−	0.737811i	$-0.735860\pi$
$719$	−5.01794e8			−1.35002			−0.675008	+	0.737811i	$0.735860\pi$
$720$		0			0
$721$			9.39791e7i			0.250741i
$722$		0			0
$723$	2.10904e8			0.558045
$724$		0			0
$725$		−	1.37085e9i		−	3.59730i
$726$		0			0
$727$	−5.67690e7			−0.147743			−0.0738717	−	0.997268i	$-0.523536\pi$
$727$	−5.67690e7			−0.147743			−0.0738717	+	0.997268i	$0.523536\pi$
$728$		0			0
$729$	−4.09214e8			−1.05625
$730$		0			0
$731$	3.17874e8			0.813772
$732$		0			0
$733$	−5.79284e8			−1.47089			−0.735444	−	0.677585i	$-0.763026\pi$
$733$	−5.79284e8			−1.47089			−0.735444	+	0.677585i	$0.763026\pi$
$734$		0			0
$735$		−	4.67591e8i		−	1.17762i
$736$		0			0
$737$			3.40467e8i			0.850496i
$738$		0			0
$739$	−3.13387e8			−0.776511			−0.388255	−	0.921552i	$-0.626922\pi$
$739$	−3.13387e8			−0.776511			−0.388255	+	0.921552i	$0.626922\pi$
$740$		0			0
$741$	3.04371e8	+	3.59348e8i	0.748081	+	0.883203i
$742$		0			0
$743$		−	4.31662e8i		−	1.05239i	−0.850363	−	0.526196i	$-0.823618\pi$
$743$		−	4.31662e8i		−	1.05239i	0.850363	−	0.526196i	$-0.176382\pi$
$744$		0			0
$745$	1.90489e8			0.460682
$746$		0			0
$747$	7.46537e7			0.179098
$748$		0			0
$749$			1.54797e8i			0.368398i
$750$		0			0
$751$			6.89806e8i			1.62857i	0.580463	+	0.814287i	$0.302872\pi$
$751$			6.89806e8i			1.62857i	−0.580463	+	0.814287i	$0.697128\pi$
$752$		0			0
$753$			2.77272e8i			0.649413i
$754$		0			0
$755$		−	2.12897e7i		−	0.0494685i
$756$		0			0
$757$	−5.51771e8			−1.27195			−0.635977	−	0.771708i	$-0.719403\pi$
$757$	−5.51771e8			−1.27195			−0.635977	+	0.771708i	$0.719403\pi$
$758$		0			0
$759$		−	1.44924e8i		−	0.331447i
$760$		0			0
$761$	1.04177e8			0.236383			0.118192	−	0.992991i	$-0.462290\pi$
$761$	1.04177e8			0.236383			0.118192	+	0.992991i	$0.462290\pi$
$762$		0			0
$763$			1.63787e8i			0.368729i
$764$		0			0
$765$	2.80272e8			0.626031
$766$		0			0
$767$	−2.50079e8			−0.554232
$768$		0			0
$769$	−5.46341e7			−0.120139			−0.0600696	−	0.998194i	$-0.519132\pi$
$769$	−5.46341e7			−0.120139			−0.0600696	+	0.998194i	$0.519132\pi$
$770$		0			0
$771$	−5.83267e8			−1.27264
$772$		0			0
$773$		−	5.46712e8i		−	1.18364i	−0.806070	−	0.591821i	$-0.798409\pi$
$773$		−	5.46712e8i		−	1.18364i	0.806070	−	0.591821i	$-0.201591\pi$
$774$		0			0
$775$			6.74214e7i			0.144841i
$776$		0			0
$777$	9.35979e7			0.199528
$778$		0			0
$779$	7.54842e6	−	6.39358e6i	0.0159678	−	0.0135248i
$780$		0			0
$781$			2.29015e8i			0.480740i
$782$		0			0
$783$	−9.35091e8			−1.94791
$784$		0			0
$785$	−5.55367e8			−1.14808
$786$		0			0
$787$		−	3.32410e8i		−	0.681946i	−0.940073	−	0.340973i	$-0.889244\pi$
$787$		−	3.32410e8i		−	0.681946i	0.940073	−	0.340973i	$-0.110756\pi$
$788$		0			0
$789$			3.53766e8i			0.720253i
$790$		0			0
$791$		−	1.91643e8i		−	0.387225i
$792$		0			0
$793$			3.95409e7i			0.0792915i
$794$		0			0
$795$	2.74580e8			0.546472
$796$		0			0
$797$			4.45080e8i			0.879149i	0.898206	+	0.439575i	$0.144871\pi$
$797$			4.45080e8i			0.879149i	−0.898206	+	0.439575i	$0.855129\pi$
$798$		0			0
$799$	−4.80929e8			−0.942845
$800$		0			0
$801$			9.70704e7i			0.188881i
$802$		0			0
$803$	−4.14290e8			−0.800125
$804$		0			0
$805$	−3.19249e8			−0.611987
$806$		0			0
$807$	−1.36659e8			−0.260026
$808$		0			0
$809$	6.28653e8			1.18731			0.593657	−	0.804718i	$-0.297684\pi$
$809$	6.28653e8			1.18731			0.593657	+	0.804718i	$0.297684\pi$
$810$		0			0
$811$			5.18077e8i			0.971252i	0.874167	+	0.485626i	$0.161408\pi$
$811$			5.18077e8i			0.971252i	−0.874167	+	0.485626i	$0.838592\pi$
$812$		0			0
$813$		−	2.71479e8i		−	0.505202i
$814$		0			0
$815$	5.61423e8			1.03709
$816$		0			0
$817$	2.84093e8	+	3.35407e8i	0.520948	+	0.615044i
$818$		0			0
$819$		−	1.10960e8i		−	0.201982i
$820$		0			0
$821$	−1.15009e8			−0.207828			−0.103914	−	0.994586i	$-0.533137\pi$
$821$	−1.15009e8			−0.207828			−0.103914	+	0.994586i	$0.533137\pi$
$822$		0			0
$823$	−4.69442e8			−0.842137			−0.421068	−	0.907029i	$-0.638345\pi$
$823$	−4.69442e8			−0.842137			−0.421068	+	0.907029i	$0.638345\pi$
$824$		0			0
$825$		−	4.14904e8i		−	0.738900i
$826$		0			0
$827$		−	8.21593e8i		−	1.45258i	−0.687388	−	0.726290i	$-0.741243\pi$
$827$		−	8.21593e8i		−	1.45258i	0.687388	−	0.726290i	$-0.258757\pi$
$828$		0			0
$829$		−	3.49058e7i		−	0.0612681i	−0.999531	−	0.0306340i	$-0.990247\pi$
$829$		−	3.49058e7i		−	0.0612681i	0.999531	−	0.0306340i	$-0.00975264\pi$
$830$		0			0
$831$			4.41698e8i			0.769702i
$832$		0			0
$833$	4.94186e8			0.854980
$834$		0			0
$835$			1.17672e9i			2.02122i
$836$		0			0
$837$	4.59897e7			0.0784304
$838$		0			0
$839$		−	4.05151e8i		−	0.686010i	−0.939334	−	0.343005i	$-0.888555\pi$
$839$		−	4.05151e8i		−	0.686010i	0.939334	−	0.343005i	$-0.111445\pi$
$840$		0			0
$841$	−1.31138e9			−2.20466
$842$		0			0
$843$	5.70114e8			0.951654
$844$		0			0
$845$	1.13552e9			1.88202
$846$		0			0
$847$	−1.87772e8			−0.309015
$848$		0			0
$849$		−	8.51874e8i		−	1.39204i
$850$		0			0
$851$			3.53316e8i			0.573291i
$852$		0			0
$853$	−5.18312e8			−0.835111			−0.417555	−	0.908651i	$-0.637113\pi$
$853$	−5.18312e8			−0.835111			−0.417555	+	0.908651i	$0.637113\pi$
$854$		0			0
$855$	2.50487e8	+	2.95732e8i	0.400763	+	0.473151i
$856$		0			0
$857$		−	7.24971e8i		−	1.15180i	−0.817519	−	0.575902i	$-0.804651\pi$
$857$		−	7.24971e8i		−	1.15180i	0.817519	−	0.575902i	$-0.195349\pi$
$858$		0			0
$859$	1.77192e8			0.279554			0.139777	−	0.990183i	$-0.455362\pi$
$859$	1.77192e8			0.279554			0.139777	+	0.990183i	$0.455362\pi$
$860$		0			0
$861$	4.19019e6			0.00656484
$862$		0			0
$863$		−	3.44228e8i		−	0.535568i	−0.963479	−	0.267784i	$-0.913709\pi$
$863$		−	3.44228e8i		−	0.535568i	0.963479	−	0.267784i	$-0.0862913\pi$
$864$		0			0
$865$		−	2.22886e8i		−	0.344377i
$866$		0			0
$867$		−	1.00992e7i		−	0.0154963i
$868$		0			0
$869$		−	2.69426e8i		−	0.410563i
$870$		0			0
$871$	1.76899e9			2.67714
$872$		0			0
$873$			1.02340e8i			0.153817i
$874$		0			0
$875$	−4.59149e8			−0.685377
$876$		0			0
$877$			7.37805e8i			1.09381i	0.837194	+	0.546906i	$0.184194\pi$
$877$			7.37805e8i			1.09381i	−0.837194	+	0.546906i	$0.815806\pi$
$878$		0			0
$879$	−1.47159e8			−0.216680
$880$		0			0
$881$	7.86541e8			1.15025			0.575127	−	0.818064i	$-0.304953\pi$
$881$	7.86541e8			1.15025			0.575127	+	0.818064i	$0.304953\pi$
$882$		0			0
$883$	−7.61345e8			−1.10586			−0.552929	−	0.833229i	$-0.686490\pi$
$883$	−7.61345e8			−1.10586			−0.552929	+	0.833229i	$0.686490\pi$
$884$		0			0
$885$	3.69989e8			0.533776
$886$		0			0
$887$		−	1.05827e9i		−	1.51645i	−0.651995	−	0.758223i	$-0.726068\pi$
$887$		−	1.05827e9i		−	1.51645i	0.651995	−	0.758223i	$-0.273932\pi$
$888$		0			0
$889$			2.27772e8i			0.324186i
$890$		0			0
$891$	−1.67041e8			−0.236151
$892$		0			0
$893$	−4.29819e8	−	5.07456e8i	−0.603575	−	0.712597i
$894$		0			0
$895$			5.83979e8i			0.814570i
$896$		0			0
$897$	−7.52992e8			−1.04331
$898$		0			0
$899$	9.37512e7			0.129032
$900$		0			0
$901$			2.90198e8i			0.396752i
$902$		0			0
$903$			1.86187e8i			0.252864i
$904$		0			0
$905$		−	2.71461e8i		−	0.366236i
$906$		0			0
$907$		−	4.63722e8i		−	0.621493i	−0.950493	−	0.310747i	$-0.899421\pi$
$907$		−	4.63722e8i		−	0.621493i	0.950493	−	0.310747i	$-0.100579\pi$
$908$		0			0
$909$	2.15228e8			0.286555
$910$		0			0
$911$			8.15792e8i			1.07901i	0.841983	+	0.539503i	$0.181388\pi$
$911$			8.15792e8i			1.07901i	−0.841983	+	0.539503i	$0.818612\pi$
$912$		0			0
$913$	1.74924e8			0.229846
$914$		0			0
$915$		−	5.85001e7i		−	0.0763649i
$916$		0			0
$917$	−1.31026e8			−0.169922
$918$		0			0
$919$	−1.00276e9			−1.29197			−0.645984	−	0.763351i	$-0.723553\pi$
$919$	−1.00276e9			−1.29197			−0.645984	+	0.763351i	$0.723553\pi$
$920$		0			0
$921$	−8.19389e8			−1.04884
$922$		0			0
$923$	1.18991e9			1.51325
$924$		0			0
$925$			1.01151e9i			1.27805i
$926$		0			0
$927$			1.82421e8i			0.229000i
$928$		0			0
$929$	−3.77625e8			−0.470992			−0.235496	−	0.971875i	$-0.575672\pi$
$929$	−3.77625e8			−0.470992			−0.235496	+	0.971875i	$0.575672\pi$
$930$		0			0
$931$	4.41668e8	+	5.21445e8i	0.547327	+	0.646189i
$932$		0			0
$933$		−	3.16160e8i		−	0.389280i
$934$		0			0
$935$	6.56718e8			0.803423
$936$		0			0
$937$	9.00225e7			0.109429			0.0547145	−	0.998502i	$-0.482575\pi$
$937$	9.00225e7			0.109429			0.0547145	+	0.998502i	$0.482575\pi$
$938$		0			0
$939$			3.25241e8i			0.392833i
$940$		0			0
$941$		−	5.96862e8i		−	0.716316i	−0.933661	−	0.358158i	$-0.883405\pi$
$941$		−	5.96862e8i		−	0.716316i	0.933661	−	0.358158i	$-0.116595\pi$
$942$		0			0
$943$			1.58173e7i			0.0188624i
$944$		0			0
$945$			6.23450e8i			0.738764i
$946$		0			0
$947$	4.97294e7			0.0585549			0.0292775	−	0.999571i	$-0.490679\pi$
$947$	4.97294e7			0.0585549			0.0292775	+	0.999571i	$0.490679\pi$
$948$		0			0
$949$			2.15256e9i			2.51859i
$950$		0			0
$951$	−3.20719e8			−0.372893
$952$		0			0
$953$			9.85780e8i			1.13894i	0.822012	+	0.569470i	$0.192852\pi$
$953$			9.85780e8i			1.13894i	−0.822012	+	0.569470i	$0.807148\pi$
$954$		0			0
$955$	−3.45397e8			−0.396559
$956$		0			0
$957$	−5.76935e8			−0.658250
$958$		0			0
$959$	−1.58264e8			−0.179443
$960$		0			0
$961$	8.82893e8			0.994805
$962$		0			0
$963$			3.00473e8i			0.336455i
$964$		0			0
$965$			1.84241e9i			2.05024i
$966$		0			0
$967$	−1.19229e9			−1.31856			−0.659282	−	0.751896i	$-0.729140\pi$
$967$	−1.19229e9			−1.31856			−0.659282	+	0.751896i	$0.729140\pi$
$968$		0			0
$969$	5.61889e8	−	4.75925e8i	0.617560	−	0.523078i
$970$		0			0
$971$		−	3.55220e8i		−	0.388007i	−0.981001	−	0.194003i	$-0.937853\pi$
$971$		−	3.55220e8i		−	0.388007i	0.981001	−	0.194003i	$-0.0621473\pi$
$972$		0			0
$973$	−5.88043e8			−0.638367
$974$		0			0
$975$	−2.15575e9			−2.32587
$976$		0			0
$977$			2.18008e8i			0.233770i	0.993145	+	0.116885i	$0.0372910\pi$
$977$			2.18008e8i			0.233770i	−0.993145	+	0.116885i	$0.962709\pi$
$978$		0			0
$979$			2.27450e8i			0.242403i
$980$		0			0
$981$			3.17924e8i			0.336757i
$982$		0			0
$983$		−	1.82201e8i		−	0.191819i	−0.995390	−	0.0959093i	$-0.969424\pi$
$983$		−	1.82201e8i		−	0.191819i	0.995390	−	0.0959093i	$-0.0305759\pi$
$984$		0			0
$985$	1.83181e9			1.91677
$986$		0			0
$987$		−	2.81693e8i		−	0.292971i
$988$		0			0
$989$	−7.02826e8			−0.726539
$990$		0			0
$991$		−	1.11227e8i		−	0.114285i	−0.998366	−	0.0571426i	$-0.981801\pi$
$991$		−	1.11227e8i		−	0.114285i	0.998366	−	0.0571426i	$-0.0181990\pi$
$992$		0			0
$993$	1.19342e9			1.21884
$994$		0			0
$995$	1.36829e9			1.38902
$996$		0			0
$997$	−1.68213e9			−1.69736			−0.848679	−	0.528908i	$-0.822602\pi$
$997$	−1.68213e9			−1.69736			−0.848679	+	0.528908i	$0.822602\pi$
$998$		0			0
$999$	6.89978e8			0.692052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.d.113.4		8
4.3	odd	2		19.7.b.b.18.2	✓	8
12.11	even	2		171.7.c.d.37.7		8
19.18	odd	2	inner	304.7.e.d.113.5		8
76.75	even	2		19.7.b.b.18.7	yes	8
228.227	odd	2		171.7.c.d.37.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.b.b.18.2	✓	8	4.3	odd	2
19.7.b.b.18.7	yes	8	76.75	even	2
171.7.c.d.37.2		8	228.227	odd	2
171.7.c.d.37.7		8	12.11	even	2
304.7.e.d.113.4		8	1.1	even	1	trivial
304.7.e.d.113.5		8	19.18	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.6433i q^{3} -216.848 q^{5} +134.238 q^{7} +260.567 q^{9} +O(q^{10})\)	\(q-21.6433i q^{3} -216.848 q^{5} +134.238 q^{7} +260.567 q^{9} +610.546 q^{11} -3172.26i q^{13} +4693.32i q^{15} -4960.26 q^{17} +(-4433.12 - 5233.86i) q^{19} -2905.36i q^{21} +10967.2 q^{23} +31398.3 q^{25} -21417.5i q^{27} -43660.1i q^{29} +2147.30i q^{31} -13214.2i q^{33} -29109.3 q^{35} +32215.6i q^{37} -68658.3 q^{39} +1442.23i q^{41} -64084.1 q^{43} -56503.6 q^{45} +96956.3 q^{47} -99629.1 q^{49} +107356. i q^{51} -58504.5i q^{53} -132396. q^{55} +(-113278. + 95947.5i) q^{57} -78833.1i q^{59} -12464.6 q^{61} +34978.0 q^{63} +687901. i q^{65} +557643. i q^{67} -237368. i q^{69} +375099. i q^{71} -678557. q^{73} -679562. i q^{75} +81958.6 q^{77} -441287. i q^{79} -273592. q^{81} +286505. q^{83} +1.07562e6 q^{85} -944950. q^{87} +372535. i q^{89} -425839. i q^{91} +46474.6 q^{93} +(961316. + 1.13495e6i) q^{95} +392760. i q^{97} +159088. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more