LMFDB - Embedded newform 368.8.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,8,Mod(1,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("368.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	368.a (trivial)

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:	yes
Analytic conductor:	$114.957689378$
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} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 $ x^8 - 832x^6 - 1059x^5 + 203052x^4 + 678328x^3 - 13424272x^2 - 73308944x - 37372224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 5 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-21.3077$ of defining polynomial
Character	$\chi$	$=$	368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-86.9475 q^{3} -188.239 q^{5} -639.155 q^{7} +5372.88 q^{9} +O(q^{10})$	$q-86.9475 q^{3} -188.239 q^{5} -639.155 q^{7} +5372.88 q^{9} +1629.29 q^{11} +1138.20 q^{13} +16366.9 q^{15} +28713.3 q^{17} -43387.0 q^{19} +55573.0 q^{21} +12167.0 q^{23} -42691.1 q^{25} -277004. q^{27} +62583.1 q^{29} -22533.6 q^{31} -141663. q^{33} +120314. q^{35} -277384. q^{37} -98963.5 q^{39} -130110. q^{41} -585763. q^{43} -1.01138e6 q^{45} +99374.3 q^{47} -415024. q^{49} -2.49655e6 q^{51} -636910. q^{53} -306696. q^{55} +3.77239e6 q^{57} +808062. q^{59} -2.40412e6 q^{61} -3.43410e6 q^{63} -214253. q^{65} +2.05060e6 q^{67} -1.05789e6 q^{69} +3.83810e6 q^{71} -2.78541e6 q^{73} +3.71188e6 q^{75} -1.04137e6 q^{77} -7.20890e6 q^{79} +1.23343e7 q^{81} +498585. q^{83} -5.40497e6 q^{85} -5.44145e6 q^{87} -1.37216e6 q^{89} -727485. q^{91} +1.95924e6 q^{93} +8.16712e6 q^{95} +6.57645e6 q^{97} +8.75398e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{3} + 444 q^{5} - 1446 q^{7} + 13878 q^{9}+O(q^{10}) $ 8 * q - 40 * q^3 + 444 * q^5 - 1446 * q^7 + 13878 * q^9	$ 8 q - 40 q^{3} + 444 q^{5} - 1446 q^{7} + 13878 q^{9} - 7588 q^{11} + 19862 q^{13} + 12770 q^{15} + 42070 q^{17} - 1050 q^{19} - 7698 q^{21} + 97336 q^{23} + 49496 q^{25} + 69500 q^{27} - 102578 q^{29} - 304172 q^{31} + 747242 q^{33} - 531048 q^{35} + 286472 q^{37} - 1032828 q^{39} + 1324414 q^{41} - 2052578 q^{43} + 2087442 q^{45} - 675556 q^{47} - 55404 q^{49} - 2775482 q^{51} + 203654 q^{53} + 1024444 q^{55} + 3908648 q^{57} + 748892 q^{59} + 61822 q^{61} - 1411632 q^{63} - 1571618 q^{65} - 3235604 q^{67} - 486680 q^{69} + 4951664 q^{71} + 11019370 q^{73} + 13607220 q^{75} - 5284888 q^{77} - 4202464 q^{79} + 10294096 q^{81} - 518568 q^{83} + 9854220 q^{85} - 4862532 q^{87} + 4203864 q^{89} - 2488406 q^{91} - 23367842 q^{93} + 44485300 q^{95} + 18621134 q^{97} + 64729930 q^{99}+O(q^{100}) $ 8 * q - 40 * q^3 + 444 * q^5 - 1446 * q^7 + 13878 * q^9 - 7588 * q^11 + 19862 * q^13 + 12770 * q^15 + 42070 * q^17 - 1050 * q^19 - 7698 * q^21 + 97336 * q^23 + 49496 * q^25 + 69500 * q^27 - 102578 * q^29 - 304172 * q^31 + 747242 * q^33 - 531048 * q^35 + 286472 * q^37 - 1032828 * q^39 + 1324414 * q^41 - 2052578 * q^43 + 2087442 * q^45 - 675556 * q^47 - 55404 * q^49 - 2775482 * q^51 + 203654 * q^53 + 1024444 * q^55 + 3908648 * q^57 + 748892 * q^59 + 61822 * q^61 - 1411632 * q^63 - 1571618 * q^65 - 3235604 * q^67 - 486680 * q^69 + 4951664 * q^71 + 11019370 * q^73 + 13607220 * q^75 - 5284888 * q^77 - 4202464 * q^79 + 10294096 * q^81 - 518568 * q^83 + 9854220 * q^85 - 4862532 * q^87 + 4203864 * q^89 - 2488406 * q^91 - 23367842 * q^93 + 44485300 * q^95 + 18621134 * q^97 + 64729930 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−86.9475	−1.85923	−0.929615	−	0.368533i	$-0.879860\pi$
$3$	−86.9475	−1.85923	−0.929615	+	0.368533i	$0.879860\pi$
$4$	0	0
$5$	−188.239	−0.673464	−0.336732	−	0.941600i	$-0.609322\pi$
$5$	−188.239	−0.673464	−0.336732	+	0.941600i	$0.609322\pi$
$6$	0	0
$7$	−639.155	−0.704309	−0.352154	−	0.935942i	$-0.614551\pi$
$7$	−639.155	−0.704309	−0.352154	+	0.935942i	$0.614551\pi$
$8$	0	0
$9$	5372.88	2.45673
$10$	0	0
$11$	1629.29	0.369083	0.184542	−	0.982825i	$-0.440920\pi$
$11$	1629.29	0.369083	0.184542	+	0.982825i	$0.440920\pi$
$12$	0	0
$13$	1138.20	0.143687	0.0718433	−	0.997416i	$-0.477112\pi$
$13$	1138.20	0.143687	0.0718433	+	0.997416i	$0.477112\pi$
$14$	0	0
$15$	16366.9	1.25212
$16$	0	0
$17$	28713.3	1.41746	0.708732	−	0.705477i	$-0.249267\pi$
$17$	28713.3	1.41746	0.708732	+	0.705477i	$0.249267\pi$
$18$	0	0
$19$	−43387.0	−1.45118	−0.725591	−	0.688127i	$-0.758433\pi$
$19$	−43387.0	−1.45118	−0.725591	+	0.688127i	$0.758433\pi$
$20$	0	0
$21$	55573.0	1.30947
$22$	0	0
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	0	0
$25$	−42691.1	−0.546446
$26$	0	0
$27$	−277004.	−2.70840
$28$	0	0
$29$	62583.1	0.476502	0.238251	−	0.971204i	$-0.423426\pi$
$29$	62583.1	0.476502	0.238251	+	0.971204i	$0.423426\pi$
$30$	0	0
$31$	−22533.6	−0.135851	−0.0679257	−	0.997690i	$-0.521638\pi$
$31$	−22533.6	−0.135851	−0.0679257	+	0.997690i	$0.521638\pi$
$32$	0	0
$33$	−141663.	−0.686211
$34$	0	0
$35$	120314.	0.474327
$36$	0	0
$37$	−277384.	−0.900276	−0.450138	−	0.892959i	$-0.648625\pi$
$37$	−277384.	−0.900276	−0.450138	+	0.892959i	$0.648625\pi$
$38$	0	0
$39$	−98963.5	−0.267146
$40$	0	0
$41$	−130110.	−0.294828	−0.147414	−	0.989075i	$-0.547095\pi$
$41$	−130110.	−0.294828	−0.147414	+	0.989075i	$0.547095\pi$
$42$	0	0
$43$	−585763.	−1.12352	−0.561762	−	0.827299i	$-0.689876\pi$
$43$	−585763.	−1.12352	−0.561762	+	0.827299i	$0.689876\pi$
$44$	0	0
$45$	−1.01138e6	−1.65452
$46$	0	0
$47$	99374.3	0.139615	0.0698074	−	0.997560i	$-0.477762\pi$
$47$	99374.3	0.139615	0.0698074	+	0.997560i	$0.477762\pi$
$48$	0	0
$49$	−415024.	−0.503949
$50$	0	0
$51$	−2.49655e6	−2.63539
$52$	0	0
$53$	−636910.	−0.587641	−0.293821	−	0.955861i	$-0.594927\pi$
$53$	−636910.	−0.587641	−0.293821	+	0.955861i	$0.594927\pi$
$54$	0	0
$55$	−306696.	−0.248564
$56$	0	0
$57$	3.77239e6	2.69808
$58$	0	0
$59$	808062.	0.512227	0.256114	−	0.966647i	$-0.417558\pi$
$59$	808062.	0.512227	0.256114	+	0.966647i	$0.417558\pi$
$60$	0	0
$61$	−2.40412e6	−1.35613	−0.678066	−	0.735001i	$-0.737182\pi$
$61$	−2.40412e6	−1.35613	−0.678066	+	0.735001i	$0.737182\pi$
$62$	0	0
$63$	−3.43410e6	−1.73030
$64$	0	0
$65$	−214253.	−0.0967678
$66$	0	0
$67$	2.05060e6	0.832949	0.416474	−	0.909147i	$-0.363266\pi$
$67$	2.05060e6	0.832949	0.416474	+	0.909147i	$0.363266\pi$
$68$	0	0
$69$	−1.05789e6	−0.387676
$70$	0	0
$71$	3.83810e6	1.27266	0.636329	−	0.771417i	$-0.280452\pi$
$71$	3.83810e6	1.27266	0.636329	+	0.771417i	$0.280452\pi$
$72$	0	0
$73$	−2.78541e6	−0.838029	−0.419015	−	0.907979i	$-0.637624\pi$
$73$	−2.78541e6	−0.838029	−0.419015	+	0.907979i	$0.637624\pi$
$74$	0	0
$75$	3.71188e6	1.01597
$76$	0	0
$77$	−1.04137e6	−0.259949
$78$	0	0
$79$	−7.20890e6	−1.64503	−0.822516	−	0.568742i	$-0.807430\pi$
$79$	−7.20890e6	−1.64503	−0.822516	+	0.568742i	$0.807430\pi$
$80$	0	0
$81$	1.23343e7	2.57881
$82$	0	0
$83$	498585.	0.0957120	0.0478560	−	0.998854i	$-0.484761\pi$
$83$	498585.	0.0957120	0.0478560	+	0.998854i	$0.484761\pi$
$84$	0	0
$85$	−5.40497e6	−0.954612
$86$	0	0
$87$	−5.44145e6	−0.885926
$88$	0	0
$89$	−1.37216e6	−0.206320	−0.103160	−	0.994665i	$-0.532895\pi$
$89$	−1.37216e6	−0.206320	−0.103160	+	0.994665i	$0.532895\pi$
$90$	0	0
$91$	−727485.	−0.101200
$92$	0	0
$93$	1.95924e6	0.252579
$94$	0	0
$95$	8.16712e6	0.977319
$96$	0	0
$97$	6.57645e6	0.731628	0.365814	−	0.930688i	$-0.380791\pi$
$97$	6.57645e6	0.731628	0.365814	+	0.930688i	$0.380791\pi$
$98$	0	0
$99$	8.75398e6	0.906739
$100$	0	0
$101$	−2.24529e6	−0.216844	−0.108422	−	0.994105i	$-0.534580\pi$
$101$	−2.24529e6	−0.216844	−0.108422	+	0.994105i	$0.534580\pi$
$102$	0	0
$103$	−2.14203e6	−0.193150	−0.0965751	−	0.995326i	$-0.530789\pi$
$103$	−2.14203e6	−0.193150	−0.0965751	+	0.995326i	$0.530789\pi$
$104$	0	0
$105$	−1.04610e7	−0.881883
$106$	0	0
$107$	−1.72640e7	−1.36238	−0.681189	−	0.732107i	$-0.738537\pi$
$107$	−1.72640e7	−1.36238	−0.681189	+	0.732107i	$0.738537\pi$
$108$	0	0
$109$	−2.52636e7	−1.86854	−0.934269	−	0.356569i	$-0.883947\pi$
$109$	−2.52636e7	−1.86854	−0.934269	+	0.356569i	$0.883947\pi$
$110$	0	0
$111$	2.41179e7	1.67382
$112$	0	0
$113$	3.18384e6	0.207576	0.103788	−	0.994599i	$-0.466904\pi$
$113$	3.18384e6	0.207576	0.103788	+	0.994599i	$0.466904\pi$
$114$	0	0
$115$	−2.29030e6	−0.140427
$116$	0	0
$117$	6.11540e6	0.353000
$118$	0	0
$119$	−1.83523e7	−0.998333
$120$	0	0
$121$	−1.68326e7	−0.863778
$122$	0	0
$123$	1.13128e7	0.548153
$124$	0	0
$125$	2.27423e7	1.04148
$126$	0	0
$127$	2.02393e7	0.876763	0.438381	−	0.898789i	$-0.355552\pi$
$127$	2.02393e7	0.876763	0.438381	+	0.898789i	$0.355552\pi$
$128$	0	0
$129$	5.09307e7	2.08889
$130$	0	0
$131$	4.46018e7	1.73342	0.866708	−	0.498816i	$-0.166232\pi$
$131$	4.46018e7	1.73342	0.866708	+	0.498816i	$0.166232\pi$
$132$	0	0
$133$	2.77310e7	1.02208
$134$	0	0
$135$	5.21430e7	1.82401
$136$	0	0
$137$	2.84803e7	0.946287	0.473144	−	0.880985i	$-0.343119\pi$
$137$	2.84803e7	0.946287	0.473144	+	0.880985i	$0.343119\pi$
$138$	0	0
$139$	4.10033e7	1.29499	0.647495	−	0.762070i	$-0.275817\pi$
$139$	4.10033e7	1.29499	0.647495	+	0.762070i	$0.275817\pi$
$140$	0	0
$141$	−8.64035e6	−0.259576
$142$	0	0
$143$	1.85446e6	0.0530323
$144$	0	0
$145$	−1.17806e7	−0.320907
$146$	0	0
$147$	3.60853e7	0.936957
$148$	0	0
$149$	−1.46291e6	−0.0362297	−0.0181148	−	0.999836i	$-0.505766\pi$
$149$	−1.46291e6	−0.0362297	−0.0181148	+	0.999836i	$0.505766\pi$
$150$	0	0
$151$	1.38217e7	0.326694	0.163347	−	0.986569i	$-0.447771\pi$
$151$	1.38217e7	0.326694	0.163347	+	0.986569i	$0.447771\pi$
$152$	0	0
$153$	1.54273e8	3.48233
$154$	0	0
$155$	4.24170e6	0.0914911
$156$	0	0
$157$	2.49008e7	0.513528	0.256764	−	0.966474i	$-0.417344\pi$
$157$	2.49008e7	0.513528	0.256764	+	0.966474i	$0.417344\pi$
$158$	0	0
$159$	5.53777e7	1.09256
$160$	0	0
$161$	−7.77660e6	−0.146859
$162$	0	0
$163$	−8.38251e7	−1.51606	−0.758032	−	0.652218i	$-0.773839\pi$
$163$	−8.38251e7	−1.51606	−0.758032	+	0.652218i	$0.773839\pi$
$164$	0	0
$165$	2.66665e7	0.462138
$166$	0	0
$167$	−3.89864e7	−0.647747	−0.323873	−	0.946100i	$-0.604985\pi$
$167$	−3.89864e7	−0.647747	−0.323873	+	0.946100i	$0.604985\pi$
$168$	0	0
$169$	−6.14530e7	−0.979354
$170$	0	0
$171$	−2.33113e8	−3.56517
$172$	0	0
$173$	9.35688e7	1.37395	0.686973	−	0.726683i	$-0.258939\pi$
$173$	9.35688e7	1.37395	0.686973	+	0.726683i	$0.258939\pi$
$174$	0	0
$175$	2.72862e7	0.384867
$176$	0	0
$177$	−7.02590e7	−0.952348
$178$	0	0
$179$	−3.65707e7	−0.476592	−0.238296	−	0.971193i	$-0.576589\pi$
$179$	−3.65707e7	−0.476592	−0.238296	+	0.971193i	$0.576589\pi$
$180$	0	0
$181$	8.44628e6	0.105874	0.0529372	−	0.998598i	$-0.483142\pi$
$181$	8.44628e6	0.105874	0.0529372	+	0.998598i	$0.483142\pi$
$182$	0	0
$183$	2.09033e8	2.52136
$184$	0	0
$185$	5.22145e7	0.606304
$186$	0	0
$187$	4.67824e7	0.523163
$188$	0	0
$189$	1.77049e8	1.90755
$190$	0	0
$191$	−1.13851e8	−1.18228	−0.591138	−	0.806570i	$-0.701321\pi$
$191$	−1.13851e8	−1.18228	−0.591138	+	0.806570i	$0.701321\pi$
$192$	0	0
$193$	1.22314e8	1.22469	0.612345	−	0.790591i	$-0.290226\pi$
$193$	1.22314e8	1.22469	0.612345	+	0.790591i	$0.290226\pi$
$194$	0	0
$195$	1.86288e7	0.179913
$196$	0	0
$197$	−1.16456e8	−1.08525	−0.542623	−	0.839976i	$-0.682569\pi$
$197$	−1.16456e8	−1.08525	−0.542623	+	0.839976i	$0.682569\pi$
$198$	0	0
$199$	1.76140e8	1.58443	0.792216	−	0.610241i	$-0.208928\pi$
$199$	1.76140e8	1.58443	0.792216	+	0.610241i	$0.208928\pi$
$200$	0	0
$201$	−1.78294e8	−1.54864
$202$	0	0
$203$	−4.00003e7	−0.335604
$204$	0	0
$205$	2.44919e7	0.198556
$206$	0	0
$207$	6.53718e7	0.512264
$208$	0	0
$209$	−7.06900e7	−0.535607
$210$	0	0
$211$	−1.86839e8	−1.36924	−0.684620	−	0.728901i	$-0.740032\pi$
$211$	−1.86839e8	−1.36924	−0.684620	+	0.728901i	$0.740032\pi$
$212$	0	0
$213$	−3.33713e8	−2.36616
$214$	0	0
$215$	1.10263e8	0.756654
$216$	0	0
$217$	1.44024e7	0.0956813
$218$	0	0
$219$	2.42185e8	1.55809
$220$	0	0
$221$	3.26814e7	0.203671
$222$	0	0
$223$	1.16234e8	0.701884	0.350942	−	0.936397i	$-0.385861\pi$
$223$	1.16234e8	0.701884	0.350942	+	0.936397i	$0.385861\pi$
$224$	0	0
$225$	−2.29374e8	−1.34247
$226$	0	0
$227$	1.60833e8	0.912609	0.456304	−	0.889824i	$-0.349173\pi$
$227$	1.60833e8	0.912609	0.456304	+	0.889824i	$0.349173\pi$
$228$	0	0
$229$	−1.58215e8	−0.870610	−0.435305	−	0.900283i	$-0.643359\pi$
$229$	−1.58215e8	−0.870610	−0.435305	+	0.900283i	$0.643359\pi$
$230$	0	0
$231$	9.05446e7	0.483304
$232$	0	0
$233$	6.74565e7	0.349364	0.174682	−	0.984625i	$-0.444110\pi$
$233$	6.74565e7	0.349364	0.174682	+	0.984625i	$0.444110\pi$
$234$	0	0
$235$	−1.87061e7	−0.0940256
$236$	0	0
$237$	6.26796e8	3.05849
$238$	0	0
$239$	−1.29369e8	−0.612968	−0.306484	−	0.951876i	$-0.599153\pi$
$239$	−1.29369e8	−0.612968	−0.306484	+	0.951876i	$0.599153\pi$
$240$	0	0
$241$	4.84417e7	0.222925	0.111463	−	0.993769i	$-0.464446\pi$
$241$	4.84417e7	0.222925	0.111463	+	0.993769i	$0.464446\pi$
$242$	0	0
$243$	−4.66633e8	−2.08619
$244$	0	0
$245$	7.81236e7	0.339392
$246$	0	0
$247$	−4.93830e7	−0.208515
$248$	0	0
$249$	−4.33508e7	−0.177951
$250$	0	0
$251$	−3.23668e8	−1.29194	−0.645968	−	0.763364i	$-0.723546\pi$
$251$	−3.23668e8	−1.29194	−0.645968	+	0.763364i	$0.723546\pi$
$252$	0	0
$253$	1.98236e7	0.0769592
$254$	0	0
$255$	4.69949e8	1.77484
$256$	0	0
$257$	3.44328e7	0.126534	0.0632670	−	0.997997i	$-0.479848\pi$
$257$	3.44328e7	0.126534	0.0632670	+	0.997997i	$0.479848\pi$
$258$	0	0
$259$	1.77292e8	0.634072
$260$	0	0
$261$	3.36251e8	1.17064
$262$	0	0
$263$	7.34996e7	0.249138	0.124569	−	0.992211i	$-0.460245\pi$
$263$	7.34996e7	0.249138	0.124569	+	0.992211i	$0.460245\pi$
$264$	0	0
$265$	1.19891e8	0.395756
$266$	0	0
$267$	1.19306e8	0.383595
$268$	0	0
$269$	−3.31972e8	−1.03984	−0.519922	−	0.854213i	$-0.674039\pi$
$269$	−3.31972e8	−1.03984	−0.519922	+	0.854213i	$0.674039\pi$
$270$	0	0
$271$	6.60070e7	0.201464	0.100732	−	0.994914i	$-0.467882\pi$
$271$	6.60070e7	0.201464	0.100732	+	0.994914i	$0.467882\pi$
$272$	0	0
$273$	6.32531e7	0.188153
$274$	0	0
$275$	−6.95562e7	−0.201684
$276$	0	0
$277$	−5.05242e8	−1.42830	−0.714151	−	0.699991i	$-0.753187\pi$
$277$	−5.05242e8	−1.42830	−0.714151	+	0.699991i	$0.753187\pi$
$278$	0	0
$279$	−1.21070e8	−0.333751
$280$	0	0
$281$	6.11069e8	1.64293	0.821463	−	0.570262i	$-0.193158\pi$
$281$	6.11069e8	1.64293	0.821463	+	0.570262i	$0.193158\pi$
$282$	0	0
$283$	−1.33286e8	−0.349567	−0.174784	−	0.984607i	$-0.555923\pi$
$283$	−1.33286e8	−0.349567	−0.174784	+	0.984607i	$0.555923\pi$
$284$	0	0
$285$	−7.10111e8	−1.81706
$286$	0	0
$287$	8.31608e7	0.207650
$288$	0	0
$289$	4.14116e8	1.00921
$290$	0	0
$291$	−5.71806e8	−1.36026
$292$	0	0
$293$	−4.33046e8	−1.00577	−0.502884	−	0.864354i	$-0.667728\pi$
$293$	−4.33046e8	−1.00577	−0.502884	+	0.864354i	$0.667728\pi$
$294$	0	0
$295$	−1.52109e8	−0.344967
$296$	0	0
$297$	−4.51321e8	−0.999626
$298$	0	0
$299$	1.38485e7	0.0299607
$300$	0	0
$301$	3.74394e8	0.791308
$302$	0	0
$303$	1.95222e8	0.403162
$304$	0	0
$305$	4.52550e8	0.913307
$306$	0	0
$307$	−9.27048e8	−1.82860	−0.914298	−	0.405042i	$-0.867257\pi$
$307$	−9.27048e8	−1.82860	−0.914298	+	0.405042i	$0.867257\pi$
$308$	0	0
$309$	1.86244e8	0.359110
$310$	0	0
$311$	9.34538e8	1.76172	0.880858	−	0.473381i	$-0.156967\pi$
$311$	9.34538e8	1.76172	0.880858	+	0.473381i	$0.156967\pi$
$312$	0	0
$313$	−7.84627e7	−0.144630	−0.0723149	−	0.997382i	$-0.523039\pi$
$313$	−7.84627e7	−0.144630	−0.0723149	+	0.997382i	$0.523039\pi$
$314$	0	0
$315$	6.46432e8	1.16530
$316$	0	0
$317$	1.46811e8	0.258851	0.129426	−	0.991589i	$-0.458687\pi$
$317$	1.46811e8	0.258851	0.129426	+	0.991589i	$0.458687\pi$
$318$	0	0
$319$	1.01966e8	0.175869
$320$	0	0
$321$	1.50106e9	2.53297
$322$	0	0
$323$	−1.24578e9	−2.05700
$324$	0	0
$325$	−4.85909e7	−0.0785169
$326$	0	0
$327$	2.19661e9	3.47404
$328$	0	0
$329$	−6.35156e7	−0.0983320
$330$	0	0
$331$	5.09422e8	0.772111	0.386055	−	0.922476i	$-0.373837\pi$
$331$	5.09422e8	0.772111	0.386055	+	0.922476i	$0.373837\pi$
$332$	0	0
$333$	−1.49035e9	−2.21174
$334$	0	0
$335$	−3.86002e8	−0.560961
$336$	0	0
$337$	−1.03024e9	−1.46633	−0.733166	−	0.680050i	$-0.761958\pi$
$337$	−1.03024e9	−1.46633	−0.733166	+	0.680050i	$0.761958\pi$
$338$	0	0
$339$	−2.76827e8	−0.385931
$340$	0	0
$341$	−3.67138e7	−0.0501405
$342$	0	0
$343$	7.91636e8	1.05924
$344$	0	0
$345$	1.99136e8	0.261086
$346$	0	0
$347$	1.22094e9	1.56871	0.784354	−	0.620313i	$-0.212994\pi$
$347$	1.22094e9	1.56871	0.784354	+	0.620313i	$0.212994\pi$
$348$	0	0
$349$	1.05987e9	1.33464	0.667320	−	0.744771i	$-0.267441\pi$
$349$	1.05987e9	1.33464	0.667320	+	0.744771i	$0.267441\pi$
$350$	0	0
$351$	−3.15286e8	−0.389161
$352$	0	0
$353$	8.39230e8	1.01548	0.507738	−	0.861512i	$-0.330482\pi$
$353$	8.39230e8	1.01548	0.507738	+	0.861512i	$0.330482\pi$
$354$	0	0
$355$	−7.22480e8	−0.857090
$356$	0	0
$357$	1.59569e9	1.85613
$358$	0	0
$359$	−7.38651e8	−0.842576	−0.421288	−	0.906927i	$-0.638422\pi$
$359$	−7.38651e8	−0.842576	−0.421288	+	0.906927i	$0.638422\pi$
$360$	0	0
$361$	9.88556e8	1.10593
$362$	0	0
$363$	1.46355e9	1.60596
$364$	0	0
$365$	5.24323e8	0.564383
$366$	0	0
$367$	−1.62094e9	−1.71174	−0.855868	−	0.517194i	$-0.826976\pi$
$367$	−1.62094e9	−1.71174	−0.855868	+	0.517194i	$0.826976\pi$
$368$	0	0
$369$	−6.99067e8	−0.724314
$370$	0	0
$371$	4.07084e8	0.413881
$372$	0	0
$373$	1.09809e9	1.09562	0.547808	−	0.836604i	$-0.315463\pi$
$373$	1.09809e9	1.09562	0.547808	+	0.836604i	$0.315463\pi$
$374$	0	0
$375$	−1.97739e9	−1.93634
$376$	0	0
$377$	7.12320e7	0.0684669
$378$	0	0
$379$	−9.27372e7	−0.0875018	−0.0437509	−	0.999042i	$-0.513931\pi$
$379$	−9.27372e7	−0.0875018	−0.0437509	+	0.999042i	$0.513931\pi$
$380$	0	0
$381$	−1.75976e9	−1.63010
$382$	0	0
$383$	8.14572e8	0.740856	0.370428	−	0.928861i	$-0.379211\pi$
$383$	8.14572e8	0.740856	0.370428	+	0.928861i	$0.379211\pi$
$384$	0	0
$385$	1.96027e8	0.175066
$386$	0	0
$387$	−3.14723e9	−2.76020
$388$	0	0
$389$	−1.35871e9	−1.17032	−0.585159	−	0.810919i	$-0.698968\pi$
$389$	−1.35871e9	−1.17032	−0.585159	+	0.810919i	$0.698968\pi$
$390$	0	0
$391$	3.49355e8	0.295562
$392$	0	0
$393$	−3.87802e9	−3.22282
$394$	0	0
$395$	1.35700e9	1.10787
$396$	0	0
$397$	2.07777e9	1.66660	0.833300	−	0.552821i	$-0.186449\pi$
$397$	2.07777e9	1.66660	0.833300	+	0.552821i	$0.186449\pi$
$398$	0	0
$399$	−2.41114e9	−1.90028
$400$	0	0
$401$	−9.70183e8	−0.751361	−0.375680	−	0.926749i	$-0.622591\pi$
$401$	−9.70183e8	−0.751361	−0.375680	+	0.926749i	$0.622591\pi$
$402$	0	0
$403$	−2.56477e7	−0.0195200
$404$	0	0
$405$	−2.32181e9	−1.73673
$406$	0	0
$407$	−4.51940e8	−0.332277
$408$	0	0
$409$	−4.43084e7	−0.0320224	−0.0160112	−	0.999872i	$-0.505097\pi$
$409$	−4.43084e7	−0.0320224	−0.0160112	+	0.999872i	$0.505097\pi$
$410$	0	0
$411$	−2.47629e9	−1.75936
$412$	0	0
$413$	−5.16477e8	−0.360766
$414$	0	0
$415$	−9.38532e7	−0.0644586
$416$	0	0
$417$	−3.56513e9	−2.40768
$418$	0	0
$419$	−7.38219e8	−0.490271	−0.245135	−	0.969489i	$-0.578832\pi$
$419$	−7.38219e8	−0.490271	−0.245135	+	0.969489i	$0.578832\pi$
$420$	0	0
$421$	1.11608e8	0.0728967	0.0364483	−	0.999336i	$-0.488396\pi$
$421$	1.11608e8	0.0728967	0.0364483	+	0.999336i	$0.488396\pi$
$422$	0	0
$423$	5.33926e8	0.342996
$424$	0	0
$425$	−1.22580e9	−0.774567
$426$	0	0
$427$	1.53661e9	0.955136
$428$	0	0
$429$	−1.61240e8	−0.0985992
$430$	0	0
$431$	−1.89063e9	−1.13746	−0.568730	−	0.822524i	$-0.692565\pi$
$431$	−1.89063e9	−1.13746	−0.568730	+	0.822524i	$0.692565\pi$
$432$	0	0
$433$	−7.97013e8	−0.471800	−0.235900	−	0.971777i	$-0.575804\pi$
$433$	−7.97013e8	−0.471800	−0.235900	+	0.971777i	$0.575804\pi$
$434$	0	0
$435$	1.02429e9	0.596639
$436$	0	0
$437$	−5.27889e8	−0.302592
$438$	0	0
$439$	−3.82327e8	−0.215679	−0.107840	−	0.994168i	$-0.534393\pi$
$439$	−3.82327e8	−0.215679	−0.107840	+	0.994168i	$0.534393\pi$
$440$	0	0
$441$	−2.22987e9	−1.23807
$442$	0	0
$443$	2.31678e9	1.26611	0.633056	−	0.774106i	$-0.281800\pi$
$443$	2.31678e9	1.26611	0.633056	+	0.774106i	$0.281800\pi$
$444$	0	0
$445$	2.58294e8	0.138949
$446$	0	0
$447$	1.27196e8	0.0673593
$448$	0	0
$449$	4.47596e8	0.233358	0.116679	−	0.993170i	$-0.462775\pi$
$449$	4.47596e8	0.233358	0.116679	+	0.993170i	$0.462775\pi$
$450$	0	0
$451$	−2.11988e8	−0.108816
$452$	0	0
$453$	−1.20176e9	−0.607399
$454$	0	0
$455$	1.36941e8	0.0681544
$456$	0	0
$457$	2.31021e9	1.13226	0.566128	−	0.824317i	$-0.308441\pi$
$457$	2.31021e9	1.13226	0.566128	+	0.824317i	$0.308441\pi$
$458$	0	0
$459$	−7.95371e9	−3.83906
$460$	0	0
$461$	−1.90114e9	−0.903777	−0.451888	−	0.892075i	$-0.649249\pi$
$461$	−1.90114e9	−0.903777	−0.451888	+	0.892075i	$0.649249\pi$
$462$	0	0
$463$	−8.12980e8	−0.380668	−0.190334	−	0.981719i	$-0.560957\pi$
$463$	−8.12980e8	−0.380668	−0.190334	+	0.981719i	$0.560957\pi$
$464$	0	0
$465$	−3.68805e8	−0.170103
$466$	0	0
$467$	2.31278e9	1.05081	0.525406	−	0.850852i	$-0.323914\pi$
$467$	2.31278e9	1.05081	0.525406	+	0.850852i	$0.323914\pi$
$468$	0	0
$469$	−1.31065e9	−0.586653
$470$	0	0
$471$	−2.16506e9	−0.954767
$472$	0	0
$473$	−9.54379e8	−0.414674
$474$	0	0
$475$	1.85224e9	0.792992
$476$	0	0
$477$	−3.42204e9	−1.44368
$478$	0	0
$479$	−2.05808e9	−0.855634	−0.427817	−	0.903865i	$-0.640717\pi$
$479$	−2.05808e9	−0.855634	−0.427817	+	0.903865i	$0.640717\pi$
$480$	0	0
$481$	−3.15718e8	−0.129358
$482$	0	0
$483$	6.76156e8	0.273044
$484$	0	0
$485$	−1.23794e9	−0.492725
$486$	0	0
$487$	2.99505e8	0.117504	0.0587520	−	0.998273i	$-0.481288\pi$
$487$	2.99505e8	0.117504	0.0587520	+	0.998273i	$0.481288\pi$
$488$	0	0
$489$	7.28838e9	2.81871
$490$	0	0
$491$	1.40636e9	0.536182	0.268091	−	0.963394i	$-0.413607\pi$
$491$	1.40636e9	0.536182	0.268091	+	0.963394i	$0.413607\pi$
$492$	0	0
$493$	1.79697e9	0.675424
$494$	0	0
$495$	−1.64784e9	−0.610657
$496$	0	0
$497$	−2.45314e9	−0.896345
$498$	0	0
$499$	6.52753e8	0.235178	0.117589	−	0.993062i	$-0.462483\pi$
$499$	6.52753e8	0.235178	0.117589	+	0.993062i	$0.462483\pi$
$500$	0	0
$501$	3.38977e9	1.20431
$502$	0	0
$503$	4.89782e9	1.71599	0.857995	−	0.513658i	$-0.171710\pi$
$503$	4.89782e9	1.71599	0.857995	+	0.513658i	$0.171710\pi$
$504$	0	0
$505$	4.22651e8	0.146037
$506$	0	0
$507$	5.34319e9	1.82084
$508$	0	0
$509$	−4.40431e9	−1.48035	−0.740177	−	0.672412i	$-0.765259\pi$
$509$	−4.40431e9	−1.48035	−0.740177	+	0.672412i	$0.765259\pi$
$510$	0	0
$511$	1.78031e9	0.590231
$512$	0	0
$513$	1.20184e10	3.93038
$514$	0	0
$515$	4.03213e8	0.130080
$516$	0	0
$517$	1.61910e8	0.0515295
$518$	0	0
$519$	−8.13557e9	−2.55448
$520$	0	0
$521$	4.10159e9	1.27063	0.635317	−	0.772251i	$-0.280869\pi$
$521$	4.10159e9	1.27063	0.635317	+	0.772251i	$0.280869\pi$
$522$	0	0
$523$	2.81959e8	0.0861846	0.0430923	−	0.999071i	$-0.486279\pi$
$523$	2.81959e8	0.0861846	0.0430923	+	0.999071i	$0.486279\pi$
$524$	0	0
$525$	−2.37247e9	−0.715555
$526$	0	0
$527$	−6.47014e8	−0.192565
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	0	0
$531$	4.34162e9	1.25841
$532$	0	0
$533$	−1.48091e8	−0.0423628
$534$	0	0
$535$	3.24976e9	0.917513
$536$	0	0
$537$	3.17973e9	0.886095
$538$	0	0
$539$	−6.76194e8	−0.185999
$540$	0	0
$541$	5.49037e9	1.49077	0.745386	−	0.666634i	$-0.232265\pi$
$541$	5.49037e9	1.49077	0.745386	+	0.666634i	$0.232265\pi$
$542$	0	0
$543$	−7.34384e8	−0.196845
$544$	0	0
$545$	4.75559e9	1.25839
$546$	0	0
$547$	1.46859e9	0.383659	0.191829	−	0.981428i	$-0.438558\pi$
$547$	1.46859e9	0.383659	0.191829	+	0.981428i	$0.438558\pi$
$548$	0	0
$549$	−1.29171e10	−3.33166
$550$	0	0
$551$	−2.71529e9	−0.691490
$552$	0	0
$553$	4.60761e9	1.15861
$554$	0	0
$555$	−4.53992e9	−1.12726
$556$	0	0
$557$	9.24360e8	0.226646	0.113323	−	0.993558i	$-0.463851\pi$
$557$	9.24360e8	0.226646	0.113323	+	0.993558i	$0.463851\pi$
$558$	0	0
$559$	−6.66715e8	−0.161435
$560$	0	0
$561$	−4.06761e9	−0.972679
$562$	0	0
$563$	6.05124e9	1.42911	0.714554	−	0.699581i	$-0.246630\pi$
$563$	6.05124e9	1.42911	0.714554	+	0.699581i	$0.246630\pi$
$564$	0	0
$565$	−5.99323e8	−0.139795
$566$	0	0
$567$	−7.88356e9	−1.81628
$568$	0	0
$569$	−2.41253e9	−0.549009	−0.274504	−	0.961586i	$-0.588514\pi$
$569$	−2.41253e9	−0.549009	−0.274504	+	0.961586i	$0.588514\pi$
$570$	0	0
$571$	8.05596e8	0.181089	0.0905443	−	0.995892i	$-0.471139\pi$
$571$	8.05596e8	0.181089	0.0905443	+	0.995892i	$0.471139\pi$
$572$	0	0
$573$	9.89905e9	2.19812
$574$	0	0
$575$	−5.19422e8	−0.113942
$576$	0	0
$577$	−5.18931e8	−0.112459	−0.0562295	−	0.998418i	$-0.517908\pi$
$577$	−5.18931e8	−0.112459	−0.0562295	+	0.998418i	$0.517908\pi$
$578$	0	0
$579$	−1.06349e10	−2.27698
$580$	0	0
$581$	−3.18673e8	−0.0674108
$582$	0	0
$583$	−1.03771e9	−0.216889
$584$	0	0
$585$	−1.15116e9	−0.237733
$586$	0	0
$587$	8.20772e9	1.67490	0.837450	−	0.546513i	$-0.184045\pi$
$587$	8.20772e9	1.67490	0.837450	+	0.546513i	$0.184045\pi$
$588$	0	0
$589$	9.77663e8	0.197145
$590$	0	0
$591$	1.01255e10	2.01772
$592$	0	0
$593$	−2.50638e9	−0.493577	−0.246789	−	0.969069i	$-0.579375\pi$
$593$	−2.50638e9	−0.493577	−0.246789	+	0.969069i	$0.579375\pi$
$594$	0	0
$595$	3.45461e9	0.672342
$596$	0	0
$597$	−1.53150e10	−2.94582
$598$	0	0
$599$	6.57651e9	1.25026	0.625131	−	0.780519i	$-0.285045\pi$
$599$	6.57651e9	1.25026	0.625131	+	0.780519i	$0.285045\pi$
$600$	0	0
$601$	7.94544e9	1.49299	0.746496	−	0.665390i	$-0.231735\pi$
$601$	7.94544e9	1.49299	0.746496	+	0.665390i	$0.231735\pi$
$602$	0	0
$603$	1.10176e10	2.04633
$604$	0	0
$605$	3.16855e9	0.581723
$606$	0	0
$607$	−2.68877e9	−0.487970	−0.243985	−	0.969779i	$-0.578455\pi$
$607$	−2.68877e9	−0.487970	−0.243985	+	0.969779i	$0.578455\pi$
$608$	0	0
$609$	3.47793e9	0.623965
$610$	0	0
$611$	1.13108e8	0.0200608
$612$	0	0
$613$	8.66814e9	1.51990	0.759949	−	0.649983i	$-0.225224\pi$
$613$	8.66814e9	1.51990	0.759949	+	0.649983i	$0.225224\pi$
$614$	0	0
$615$	−2.12951e9	−0.369161
$616$	0	0
$617$	1.12347e10	1.92559	0.962795	−	0.270232i	$-0.0871003\pi$
$617$	1.12347e10	1.92559	0.962795	+	0.270232i	$0.0871003\pi$
$618$	0	0
$619$	−2.76878e9	−0.469215	−0.234607	−	0.972090i	$-0.575380\pi$
$619$	−2.76878e9	−0.469215	−0.234607	+	0.972090i	$0.575380\pi$
$620$	0	0
$621$	−3.37031e9	−0.564741
$622$	0	0
$623$	8.77024e8	0.145313
$624$	0	0
$625$	−9.45748e8	−0.154951
$626$	0	0
$627$	6.14632e9	0.995816
$628$	0	0
$629$	−7.96462e9	−1.27611
$630$	0	0
$631$	2.81973e7	0.00446791	0.00223396	−	0.999998i	$-0.499289\pi$
$631$	2.81973e7	0.00446791	0.00223396	+	0.999998i	$0.499289\pi$
$632$	0	0
$633$	1.62452e10	2.54573
$634$	0	0
$635$	−3.80982e9	−0.590469
$636$	0	0
$637$	−4.72379e8	−0.0724107
$638$	0	0
$639$	2.06216e10	3.12658
$640$	0	0
$641$	6.22935e9	0.934199	0.467100	−	0.884205i	$-0.345299\pi$
$641$	6.22935e9	0.934199	0.467100	+	0.884205i	$0.345299\pi$
$642$	0	0
$643$	−9.36940e8	−0.138987	−0.0694934	−	0.997582i	$-0.522138\pi$
$643$	−9.36940e8	−0.138987	−0.0694934	+	0.997582i	$0.522138\pi$
$644$	0	0
$645$	−9.58714e9	−1.40679
$646$	0	0
$647$	1.71887e9	0.249504	0.124752	−	0.992188i	$-0.460186\pi$
$647$	1.71887e9	0.249504	0.124752	+	0.992188i	$0.460186\pi$
$648$	0	0
$649$	1.31657e9	0.189054
$650$	0	0
$651$	−1.25226e9	−0.177894
$652$	0	0
$653$	1.11139e10	1.56196	0.780981	−	0.624554i	$-0.214719\pi$
$653$	1.11139e10	1.56196	0.780981	+	0.624554i	$0.214719\pi$
$654$	0	0
$655$	−8.39579e9	−1.16739
$656$	0	0
$657$	−1.49657e10	−2.05881
$658$	0	0
$659$	−2.55105e9	−0.347232	−0.173616	−	0.984813i	$-0.555545\pi$
$659$	−2.55105e9	−0.347232	−0.173616	+	0.984813i	$0.555545\pi$
$660$	0	0
$661$	−1.34691e10	−1.81399	−0.906993	−	0.421146i	$-0.861628\pi$
$661$	−1.34691e10	−1.81399	−0.906993	+	0.421146i	$0.861628\pi$
$662$	0	0
$663$	−2.84157e9	−0.378670
$664$	0	0
$665$	−5.22006e9	−0.688334
$666$	0	0
$667$	7.61449e8	0.0993574
$668$	0	0
$669$	−1.01062e10	−1.30496
$670$	0	0
$671$	−3.91702e9	−0.500526
$672$	0	0
$673$	1.20855e9	0.152831	0.0764154	−	0.997076i	$-0.475652\pi$
$673$	1.20855e9	0.152831	0.0764154	+	0.997076i	$0.475652\pi$
$674$	0	0
$675$	1.18256e10	1.47999
$676$	0	0
$677$	1.03356e10	1.28019	0.640097	−	0.768294i	$-0.278894\pi$
$677$	1.03356e10	1.28019	0.640097	+	0.768294i	$0.278894\pi$
$678$	0	0
$679$	−4.20337e9	−0.515292
$680$	0	0
$681$	−1.39840e10	−1.69675
$682$	0	0
$683$	−4.58153e8	−0.0550223	−0.0275111	−	0.999621i	$-0.508758\pi$
$683$	−4.58153e8	−0.0550223	−0.0275111	+	0.999621i	$0.508758\pi$
$684$	0	0
$685$	−5.36111e9	−0.637291
$686$	0	0
$687$	1.37564e10	1.61866
$688$	0	0
$689$	−7.24929e8	−0.0844362
$690$	0	0
$691$	6.02963e9	0.695212	0.347606	−	0.937641i	$-0.386995\pi$
$691$	6.02963e9	0.695212	0.347606	+	0.937641i	$0.386995\pi$
$692$	0	0
$693$	−5.59515e9	−0.638625
$694$	0	0
$695$	−7.71841e9	−0.872130
$696$	0	0
$697$	−3.73590e9	−0.417908
$698$	0	0
$699$	−5.86518e9	−0.649548
$700$	0	0
$701$	9.96825e9	1.09296	0.546482	−	0.837471i	$-0.315967\pi$
$701$	9.96825e9	1.09296	0.546482	+	0.837471i	$0.315967\pi$
$702$	0	0
$703$	1.20349e10	1.30646
$704$	0	0
$705$	1.62645e9	0.174815
$706$	0	0
$707$	1.43509e9	0.152725
$708$	0	0
$709$	4.66397e9	0.491466	0.245733	−	0.969338i	$-0.420971\pi$
$709$	4.66397e9	0.491466	0.245733	+	0.969338i	$0.420971\pi$
$710$	0	0
$711$	−3.87325e10	−4.04140
$712$	0	0
$713$	−2.74166e8	−0.0283270
$714$	0	0
$715$	−3.49081e8	−0.0357154
$716$	0	0
$717$	1.12483e10	1.13965
$718$	0	0
$719$	3.16368e9	0.317426	0.158713	−	0.987325i	$-0.449266\pi$
$719$	3.16368e9	0.317426	0.158713	+	0.987325i	$0.449266\pi$
$720$	0	0
$721$	1.36909e9	0.136037
$722$	0	0
$723$	−4.21189e9	−0.414470
$724$	0	0
$725$	−2.67174e9	−0.260382
$726$	0	0
$727$	−5.00250e9	−0.482855	−0.241428	−	0.970419i	$-0.577616\pi$
$727$	−5.00250e9	−0.482855	−0.241428	+	0.970419i	$0.577616\pi$
$728$	0	0
$729$	1.35974e10	1.29990
$730$	0	0
$731$	−1.68192e10	−1.59256
$732$	0	0
$733$	−3.68820e9	−0.345900	−0.172950	−	0.984931i	$-0.555330\pi$
$733$	−3.68820e9	−0.345900	−0.172950	+	0.984931i	$0.555330\pi$
$734$	0	0
$735$	−6.79266e9	−0.631007
$736$	0	0
$737$	3.34102e9	0.307428
$738$	0	0
$739$	2.90694e9	0.264960	0.132480	−	0.991186i	$-0.457706\pi$
$739$	2.90694e9	0.264960	0.132480	+	0.991186i	$0.457706\pi$
$740$	0	0
$741$	4.29373e9	0.387678
$742$	0	0
$743$	1.70273e9	0.152295	0.0761473	−	0.997097i	$-0.475738\pi$
$743$	1.70273e9	0.152295	0.0761473	+	0.997097i	$0.475738\pi$
$744$	0	0
$745$	2.75376e8	0.0243994
$746$	0	0
$747$	2.67884e9	0.235139
$748$	0	0
$749$	1.10344e10	0.959535
$750$	0	0
$751$	6.10891e9	0.526289	0.263144	−	0.964756i	$-0.415240\pi$
$751$	6.10891e9	0.526289	0.263144	+	0.964756i	$0.415240\pi$
$752$	0	0
$753$	2.81421e10	2.40201
$754$	0	0
$755$	−2.60178e9	−0.220017
$756$	0	0
$757$	−4.90563e9	−0.411016	−0.205508	−	0.978655i	$-0.565885\pi$
$757$	−4.90563e9	−0.411016	−0.205508	+	0.978655i	$0.565885\pi$
$758$	0	0
$759$	−1.72361e9	−0.143085
$760$	0	0
$761$	1.56857e10	1.29020	0.645100	−	0.764098i	$-0.276816\pi$
$761$	1.56857e10	1.29020	0.645100	+	0.764098i	$0.276816\pi$
$762$	0	0
$763$	1.61473e10	1.31603
$764$	0	0
$765$	−2.90402e10	−2.34523
$766$	0	0
$767$	9.19734e8	0.0736001
$768$	0	0
$769$	−8.48511e9	−0.672845	−0.336423	−	0.941711i	$-0.609217\pi$
$769$	−8.48511e9	−0.672845	−0.336423	+	0.941711i	$0.609217\pi$
$770$	0	0
$771$	−2.99385e9	−0.235256
$772$	0	0
$773$	1.25408e10	0.976558	0.488279	−	0.872688i	$-0.337625\pi$
$773$	1.25408e10	0.976558	0.488279	+	0.872688i	$0.337625\pi$
$774$	0	0
$775$	9.61982e8	0.0742354
$776$	0	0
$777$	−1.54151e10	−1.17889
$778$	0	0
$779$	5.64510e9	0.427849
$780$	0	0
$781$	6.25338e9	0.469717
$782$	0	0
$783$	−1.73358e10	−1.29056
$784$	0	0
$785$	−4.68730e9	−0.345843
$786$	0	0
$787$	2.44401e10	1.78728	0.893639	−	0.448786i	$-0.148143\pi$
$787$	2.44401e10	1.78728	0.893639	+	0.448786i	$0.148143\pi$
$788$	0	0
$789$	−6.39061e9	−0.463204
$790$	0	0
$791$	−2.03497e9	−0.146198
$792$	0	0
$793$	−2.73637e9	−0.194858
$794$	0	0
$795$	−1.04243e10	−0.735800
$796$	0	0
$797$	9.51793e7	0.00665945	0.00332972	−	0.999994i	$-0.498940\pi$
$797$	9.51793e7	0.00665945	0.00332972	+	0.999994i	$0.498940\pi$
$798$	0	0
$799$	2.85337e9	0.197899
$800$	0	0
$801$	−7.37246e9	−0.506872
$802$	0	0
$803$	−4.53825e9	−0.309303
$804$	0	0
$805$	1.46386e9	0.0989040
$806$	0	0
$807$	2.88642e10	1.93331
$808$	0	0
$809$	−1.67314e10	−1.11100	−0.555498	−	0.831518i	$-0.687472\pi$
$809$	−1.67314e10	−1.11100	−0.555498	+	0.831518i	$0.687472\pi$
$810$	0	0
$811$	7.07377e9	0.465670	0.232835	−	0.972516i	$-0.425200\pi$
$811$	7.07377e9	0.465670	0.232835	+	0.972516i	$0.425200\pi$
$812$	0	0
$813$	−5.73915e9	−0.374568
$814$	0	0
$815$	1.57792e10	1.02101
$816$	0	0
$817$	2.54145e10	1.63044
$818$	0	0
$819$	−3.90869e9	−0.248621
$820$	0	0
$821$	−1.14579e9	−0.0722607	−0.0361304	−	0.999347i	$-0.511503\pi$
$821$	−1.14579e9	−0.0722607	−0.0361304	+	0.999347i	$0.511503\pi$
$822$	0	0
$823$	−1.74865e10	−1.09346	−0.546730	−	0.837309i	$-0.684128\pi$
$823$	−1.74865e10	−1.09346	−0.546730	+	0.837309i	$0.684128\pi$
$824$	0	0
$825$	6.04774e9	0.374977
$826$	0	0
$827$	−4.70708e9	−0.289389	−0.144694	−	0.989476i	$-0.546220\pi$
$827$	−4.70708e9	−0.289389	−0.144694	+	0.989476i	$0.546220\pi$
$828$	0	0
$829$	−3.27355e9	−0.199562	−0.0997811	−	0.995009i	$-0.531814\pi$
$829$	−3.27355e9	−0.199562	−0.0997811	+	0.995009i	$0.531814\pi$
$830$	0	0
$831$	4.39296e10	2.65554
$832$	0	0
$833$	−1.19167e10	−0.714330
$834$	0	0
$835$	7.33876e9	0.436234
$836$	0	0
$837$	6.24189e9	0.367940
$838$	0	0
$839$	7.60099e9	0.444328	0.222164	−	0.975009i	$-0.428688\pi$
$839$	7.60099e9	0.444328	0.222164	+	0.975009i	$0.428688\pi$
$840$	0	0
$841$	−1.33332e10	−0.772946
$842$	0	0
$843$	−5.31309e10	−3.05458
$844$	0	0
$845$	1.15679e10	0.659560
$846$	0	0
$847$	1.07586e10	0.608366
$848$	0	0
$849$	1.15889e10	0.649926
$850$	0	0
$851$	−3.37493e9	−0.187720
$852$	0	0
$853$	1.06027e10	0.584916	0.292458	−	0.956278i	$-0.405527\pi$
$853$	1.06027e10	0.584916	0.292458	+	0.956278i	$0.405527\pi$
$854$	0	0
$855$	4.38809e10	2.40101
$856$	0	0
$857$	3.25969e10	1.76906	0.884532	−	0.466479i	$-0.154478\pi$
$857$	3.25969e10	1.76906	0.884532	+	0.466479i	$0.154478\pi$
$858$	0	0
$859$	−3.28664e10	−1.76920	−0.884599	−	0.466353i	$-0.845568\pi$
$859$	−3.28664e10	−1.76920	−0.884599	+	0.466353i	$0.845568\pi$
$860$	0	0
$861$	−7.23063e9	−0.386069
$862$	0	0
$863$	−9.63622e9	−0.510351	−0.255176	−	0.966895i	$-0.582133\pi$
$863$	−9.63622e9	−0.510351	−0.255176	+	0.966895i	$0.582133\pi$
$864$	0	0
$865$	−1.76133e10	−0.925303
$866$	0	0
$867$	−3.60064e10	−1.87635
$868$	0	0
$869$	−1.17454e10	−0.607154
$870$	0	0
$871$	2.33398e9	0.119684
$872$	0	0
$873$	3.53344e10	1.79741
$874$	0	0
$875$	−1.45359e10	−0.733521
$876$	0	0
$877$	−9.13957e8	−0.0457538	−0.0228769	−	0.999738i	$-0.507283\pi$
$877$	−9.13957e8	−0.0457538	−0.0228769	+	0.999738i	$0.507283\pi$
$878$	0	0
$879$	3.76523e10	1.86995
$880$	0	0
$881$	−2.85437e10	−1.40635	−0.703176	−	0.711016i	$-0.748235\pi$
$881$	−2.85437e10	−1.40635	−0.703176	+	0.711016i	$0.748235\pi$
$882$	0	0
$883$	−6.16630e9	−0.301413	−0.150707	−	0.988579i	$-0.548155\pi$
$883$	−6.16630e9	−0.301413	−0.150707	+	0.988579i	$0.548155\pi$
$884$	0	0
$885$	1.32255e10	0.641372
$886$	0	0
$887$	4.16328e8	0.0200310	0.0100155	−	0.999950i	$-0.496812\pi$
$887$	4.16328e8	0.0200310	0.0100155	+	0.999950i	$0.496812\pi$
$888$	0	0
$889$	−1.29360e10	−0.617512
$890$	0	0
$891$	2.00963e10	0.951794
$892$	0	0
$893$	−4.31155e9	−0.202606
$894$	0	0
$895$	6.88402e9	0.320968
$896$	0	0
$897$	−1.20409e9	−0.0557038
$898$	0	0
$899$	−1.41022e9	−0.0647334
$900$	0	0
$901$	−1.82878e10	−0.832961
$902$	0	0
$903$	−3.25526e10	−1.47122
$904$	0	0
$905$	−1.58992e9	−0.0713026
$906$	0	0
$907$	8.01334e9	0.356605	0.178303	−	0.983976i	$-0.442939\pi$
$907$	8.01334e9	0.356605	0.178303	+	0.983976i	$0.442939\pi$
$908$	0	0
$909$	−1.20636e10	−0.532728
$910$	0	0
$911$	−1.58468e10	−0.694426	−0.347213	−	0.937786i	$-0.612872\pi$
$911$	−1.58468e10	−0.694426	−0.347213	+	0.937786i	$0.612872\pi$
$912$	0	0
$913$	8.12341e8	0.0353257
$914$	0	0
$915$	−3.93481e10	−1.69805
$916$	0	0
$917$	−2.85075e10	−1.22086
$918$	0	0
$919$	7.88167e9	0.334976	0.167488	−	0.985874i	$-0.446434\pi$
$919$	7.88167e9	0.334976	0.167488	+	0.985874i	$0.446434\pi$
$920$	0	0
$921$	8.06046e10	3.39978
$922$	0	0
$923$	4.36852e9	0.182864
$924$	0	0
$925$	1.18418e10	0.491952
$926$	0	0
$927$	−1.15089e10	−0.474518
$928$	0	0
$929$	1.82739e9	0.0747782	0.0373891	−	0.999301i	$-0.488096\pi$
$929$	1.82739e9	0.0747782	0.0373891	+	0.999301i	$0.488096\pi$
$930$	0	0
$931$	1.80066e10	0.731321
$932$	0	0
$933$	−8.12558e10	−3.27543
$934$	0	0
$935$	−8.80627e9	−0.352331
$936$	0	0
$937$	1.59441e10	0.633158	0.316579	−	0.948566i	$-0.397466\pi$
$937$	1.59441e10	0.633158	0.316579	+	0.948566i	$0.397466\pi$
$938$	0	0
$939$	6.82214e9	0.268900
$940$	0	0
$941$	1.62795e10	0.636909	0.318455	−	0.947938i	$-0.396836\pi$
$941$	1.62795e10	0.636909	0.318455	+	0.947938i	$0.396836\pi$
$942$	0	0
$943$	−1.58305e9	−0.0614759
$944$	0	0
$945$	−3.33275e10	−1.28467
$946$	0	0
$947$	1.58030e10	0.604665	0.302333	−	0.953203i	$-0.402235\pi$
$947$	1.58030e10	0.604665	0.302333	+	0.953203i	$0.402235\pi$
$948$	0	0
$949$	−3.17035e9	−0.120414
$950$	0	0
$951$	−1.27648e10	−0.481264
$952$	0	0
$953$	2.46781e9	0.0923607	0.0461804	−	0.998933i	$-0.485295\pi$
$953$	2.46781e9	0.0923607	0.0461804	+	0.998933i	$0.485295\pi$
$954$	0	0
$955$	2.14312e10	0.796221
$956$	0	0
$957$	−8.86571e9	−0.326980
$958$	0	0
$959$	−1.82033e10	−0.666478
$960$	0	0
$961$	−2.70049e10	−0.981544
$962$	0	0
$963$	−9.27572e10	−3.34700
$964$	0	0
$965$	−2.30243e10	−0.824785
$966$	0	0
$967$	2.14651e10	0.763379	0.381689	−	0.924291i	$-0.375342\pi$
$967$	2.14651e10	0.763379	0.381689	+	0.924291i	$0.375342\pi$
$968$	0	0
$969$	1.08318e11	3.82443
$970$	0	0
$971$	7.97270e9	0.279472	0.139736	−	0.990189i	$-0.455375\pi$
$971$	7.97270e9	0.279472	0.139736	+	0.990189i	$0.455375\pi$
$972$	0	0
$973$	−2.62074e10	−0.912073
$974$	0	0
$975$	4.22486e9	0.145981
$976$	0	0
$977$	9.53879e9	0.327237	0.163618	−	0.986524i	$-0.447683\pi$
$977$	9.53879e9	0.327237	0.163618	+	0.986524i	$0.447683\pi$
$978$	0	0
$979$	−2.23565e9	−0.0761491
$980$	0	0
$981$	−1.35738e11	−4.59050
$982$	0	0
$983$	9.66381e9	0.324497	0.162249	−	0.986750i	$-0.448125\pi$
$983$	9.66381e9	0.324497	0.162249	+	0.986750i	$0.448125\pi$
$984$	0	0
$985$	2.19215e10	0.730875
$986$	0	0
$987$	5.52253e9	0.182822
$988$	0	0
$989$	−7.12698e9	−0.234271
$990$	0	0
$991$	−2.82627e10	−0.922477	−0.461238	−	0.887276i	$-0.652595\pi$
$991$	−2.82627e10	−0.922477	−0.461238	+	0.887276i	$0.652595\pi$
$992$	0	0
$993$	−4.42930e10	−1.43553
$994$	0	0
$995$	−3.31565e10	−1.06706
$996$	0	0
$997$	2.32367e10	0.742576	0.371288	−	0.928518i	$-0.378916\pi$
$997$	2.32367e10	0.742576	0.371288	+	0.928518i	$0.378916\pi$
$998$	0	0
$999$	7.68365e10	2.43831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.8.a.h.1.1		8
4.3	odd	2		23.8.a.b.1.1	✓	8
12.11	even	2		207.8.a.f.1.8		8
20.19	odd	2		575.8.a.b.1.8		8
92.91	even	2		529.8.a.c.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.b.1.1	✓	8	4.3	odd	2
207.8.a.f.1.8		8	12.11	even	2
368.8.a.h.1.1		8	1.1	even	1	trivial
529.8.a.c.1.1		8	92.91	even	2
575.8.a.b.1.8		8	20.19	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:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	368.a (trivial)

\(f(q)\)	\(=\)	\(q-86.9475 q^{3} -188.239 q^{5} -639.155 q^{7} +5372.88 q^{9} +O(q^{10})\)	\(q-86.9475 q^{3} -188.239 q^{5} -639.155 q^{7} +5372.88 q^{9} +1629.29 q^{11} +1138.20 q^{13} +16366.9 q^{15} +28713.3 q^{17} -43387.0 q^{19} +55573.0 q^{21} +12167.0 q^{23} -42691.1 q^{25} -277004. q^{27} +62583.1 q^{29} -22533.6 q^{31} -141663. q^{33} +120314. q^{35} -277384. q^{37} -98963.5 q^{39} -130110. q^{41} -585763. q^{43} -1.01138e6 q^{45} +99374.3 q^{47} -415024. q^{49} -2.49655e6 q^{51} -636910. q^{53} -306696. q^{55} +3.77239e6 q^{57} +808062. q^{59} -2.40412e6 q^{61} -3.43410e6 q^{63} -214253. q^{65} +2.05060e6 q^{67} -1.05789e6 q^{69} +3.83810e6 q^{71} -2.78541e6 q^{73} +3.71188e6 q^{75} -1.04137e6 q^{77} -7.20890e6 q^{79} +1.23343e7 q^{81} +498585. q^{83} -5.40497e6 q^{85} -5.44145e6 q^{87} -1.37216e6 q^{89} -727485. q^{91} +1.95924e6 q^{93} +8.16712e6 q^{95} +6.57645e6 q^{97} +8.75398e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{3} + 444 q^{5} - 1446 q^{7} + 13878 q^{9}+O(q^{10}) \) 8 * q - 40 * q^3 + 444 * q^5 - 1446 * q^7 + 13878 * q^9	\( 8 q - 40 q^{3} + 444 q^{5} - 1446 q^{7} + 13878 q^{9} - 7588 q^{11} + 19862 q^{13} + 12770 q^{15} + 42070 q^{17} - 1050 q^{19} - 7698 q^{21} + 97336 q^{23} + 49496 q^{25} + 69500 q^{27} - 102578 q^{29} - 304172 q^{31} + 747242 q^{33} - 531048 q^{35} + 286472 q^{37} - 1032828 q^{39} + 1324414 q^{41} - 2052578 q^{43} + 2087442 q^{45} - 675556 q^{47} - 55404 q^{49} - 2775482 q^{51} + 203654 q^{53} + 1024444 q^{55} + 3908648 q^{57} + 748892 q^{59} + 61822 q^{61} - 1411632 q^{63} - 1571618 q^{65} - 3235604 q^{67} - 486680 q^{69} + 4951664 q^{71} + 11019370 q^{73} + 13607220 q^{75} - 5284888 q^{77} - 4202464 q^{79} + 10294096 q^{81} - 518568 q^{83} + 9854220 q^{85} - 4862532 q^{87} + 4203864 q^{89} - 2488406 q^{91} - 23367842 q^{93} + 44485300 q^{95} + 18621134 q^{97} + 64729930 q^{99}+O(q^{100}) \) 8 * q - 40 * q^3 + 444 * q^5 - 1446 * q^7 + 13878 * q^9 - 7588 * q^11 + 19862 * q^13 + 12770 * q^15 + 42070 * q^17 - 1050 * q^19 - 7698 * q^21 + 97336 * q^23 + 49496 * q^25 + 69500 * q^27 - 102578 * q^29 - 304172 * q^31 + 747242 * q^33 - 531048 * q^35 + 286472 * q^37 - 1032828 * q^39 + 1324414 * q^41 - 2052578 * q^43 + 2087442 * q^45 - 675556 * q^47 - 55404 * q^49 - 2775482 * q^51 + 203654 * q^53 + 1024444 * q^55 + 3908648 * q^57 + 748892 * q^59 + 61822 * q^61 - 1411632 * q^63 - 1571618 * q^65 - 3235604 * q^67 - 486680 * q^69 + 4951664 * q^71 + 11019370 * q^73 + 13607220 * q^75 - 5284888 * q^77 - 4202464 * q^79 + 10294096 * q^81 - 518568 * q^83 + 9854220 * q^85 - 4862532 * q^87 + 4203864 * q^89 - 2488406 * q^91 - 23367842 * q^93 + 44485300 * q^95 + 18621134 * q^97 + 64729930 * q^99

Introduction

L-functions

Modular forms

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Database

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